Long range interactions in nanoscale science
Roger H. French
*
DuPont Co. Central Research, E400-5207 Experimental Station, Wilmington, Delaware
19880, USA
and Department of Materials Science and Engineering, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA
V. Adrian Parsegian
?
Laboratory of Physical and Structural Biology, NICHD, National Institutes of Health,
Bethesda, Maryland 20892-0924, USA
Rudolf Podgornik
Laboratory of Physical and Structural Biology, NICHD, National Institutes of Health,
Bethesda, Maryland 20892-0924, USA;
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana SI-1000, Slovenia;
and Department of Theoretical Physics, J. Stefan Institute, Ljubljana 1000, Slovenia
Rick F. Rajter
Department of Materials Science and Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139-4307, USA
Anand Jagota
Department of Chemical Engineering and Bioengineering Program, Lehigh University,
Bethlehem, Pennsylvania 18015, USA
Jian Luo
School of Materials Science and Engineering, Clemson University, Clemson, South
Carolina 29634, USA
Dilip Asthagiri
Department of Chemical and Biomolecular Engineering, Johns Hopkins University,
Baltimore, Maryland 21218, USA
Manoj K. Chaudhury
Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015,
USA
Yet-ming Chiang
Department of Materials Science and Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
Steve Granick
Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA
Sergei Kalinin
Materials Science and Technology Division and The Center for Nanophase Materials
Science, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Mehran Kardar
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA
Roland Kjellander
Department of Chemistry, University of Gothenburg, SE-412 96 Gothenburg, Sweden
David C. Langreth
Center for Materials Theory, Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854-8019, USA
REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL?JUNE 2010
0034-6861/2010/82H208492H20850/1887H2084958H20850 ?2010 The American Physical Society1887
Jennifer Lewis
Frederick Seitz Materials Research Laboratory, Materials Science and Engineering
Department, University of Illinois, Urbana, Illinois 61801, USA
Steve Lustig
DuPont Co. Central Research, E400-5472 Experimental Station, Wilmington, Delaware
19880, USA
David Wesolowski
Chemical Sciences Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge,
Tennessee 37831-6110, USA
John S. Wettlaufer
Department of Geology and Geophysics, Department of Physics, Program in Applied
Mathematics, Yale University, New Haven, Connecticut 06520-8109, USA
Wai-Yim Ching
Department of Physics, University of Missouri?Kansas City, Kansas City, Missouri 64110,
USA
Mike Finnis
Department of Materials and Department of Physics, Imperial College London, Exhibition
Road, London SW7 2AZ, United Kingdom
Frank Houlihan
AZ Electronic Materials Corporation USA, 70 Meister Avenue, Somerville, New Jersey
08876, USA
O. Anatole von Lilienfeld
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories,
Albuquerque, New Mexico 87185, USA
Carel Jan van Oss
Department of Microbiology and Immunology, Department of Chemical and Biological
Engineering, and Department of Geology, State University of New York at Buffalo, Buffalo,
New York 14260, USA
Thomas Zemb
Institut de Chimie S?parative de Marcoule, UMR 5257, 30207 Bagnols sur C?ze,
France
H20849Published 11 June 2010H20850
Our understanding of the ?long range? electrodynamic, electrostatic, and polar interactions that
dominate the organization of small objects at separations beyond an interatomic bond length is
reviewed. From this basic-forces perspective, a large number of systems are described from which one
can learn about these organizing forces and how to modulate them. The many practical systems that
harness these nanoscale forces are then surveyed. The survey reveals not only the promise of new
devices and materials, but also the possibility of designing them more effectively.
DOI: 10.1103/RevModPhys.82.1887 PACS numberH20849sH20850: 73.22.H11002f, 78.40.H11002q, 78.67.Sc, 79.60.Jv
*
Corresponding author.
rogerhfrench@longrangeinteractions.com
?
Present address: Department of Physics, University of Mas-
sachusetts, Amherst, MA 01003, USA.
1888
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
CONTENTS
I. Introduction 1889
II. Fundamental Interactions 1890
A. Electrodynamic interactions 1891
1. Lifshitz theory from optical properties 1892
a. Hamaker coefficients 1892
b. Full spectral optical properties 1893
c. Model systems: PS-water-PS and
SiO
2
-water-SiO
2
1893
d. Advanced effects and phenomenology 1894
2. Ab initio optical properties of complex
materials 1895
3. Electrodynamic interactions for arbitrary
shapes 1895
a. Cylinders and plates 1896
b. Spheres and compact shapes 1897
4. Noncovalent interactions from electronic
structure calculations 1897
a. Recently evolving density functional theory
methods 1898
i. Many-body methods using Kohn-Sham
orbitals 1898
ii. Explicit density functional methods 1898
b. C
6
coefficients of polarizabilities, and
damping functions 1899
5. Challenges and opportunities 1900
B. Electrostatic interactions 1900
1. Electrostatics in equilibrium statistical
mechanics: Electrostatic double layers 1901
a. Model system: DNA-DNA interactions in
charged electrolytes 1901
b. Electrostatics with surfaces and interfaces 1902
2. Electrostatics in density functional theory 1902
3. Challenges and opportunities 1903
C. Polar interactions 1904
1. Motivation and recent advances 1904
2. Challenges and opportunities 1905
III. Instructive Systems 1906
A. Atoms and molecules 1906
1. Optical spectra and Lifshitz theory for
complex biomolecular systems 1906
a. Optical properties of SWCNTs 1907
b. Optical properties of B-DNA 1907
2. Hydration interaction and ionic specificity 1908
3. Extraction, separation, and phase transfer
reactions 1908
4. DFT results on DNA base-pair vdW
interactions 1909
B. Interfaces, surfaces, and defects in solids 1909
1. Impurity-based quasiliquid surficial and
interfacial films 1909
2. Charged defects in solids 1910
C. Solid/liquid interfaces and suspensions 1911
1. Water and ice 1911
a. The phase architecture of ice 1911
b. Optical properties of ice and water 1912
2. Hydration 1912
3. Structure and dynamics at oxide/electrolyte
interfaces 1913
4. Colloidal suspensions 1916
5. Solution-based manipulation of SWCNT 1917
a. Dispersion and structure 1918
b. Sorting and placement 1918
IV. Harnessing LRIs 1919
A. Surfaces and interfaces 1919
1. Proton exchange membranes for hydrogen
fuel cells 1920
2. Intergranular and surficial films 1922
3. Premelting dynamics and its manipulation 1923
B. Colloids and self-assembly 1925
1. Tailored building blocks: From hard spheres
to patchy colloids 1925
2. Synthesis and assembly of designer colloidal
building blocks 1925
3. Challenges and opportunities 1926
C. Self-assembly and emerging device applications 1927
1. Electrochemical devices: Li-ion batteries
from heterogeneous colloids 1927
2. Active electronic devices: Single-walled
carbon nanotubes 1927
D. Nanoscale probes of long range interactions 1929
1. Scanning probes 1929
a. The SPM approach 1929
i. Direct force measurements 1930
ii. Voltage modulation approaches 1930
iii. Functional probes 1931
iv. Probing dissipative dynamics 1931
b. Future developments 1931
2. Scattering probes 1932
a. X-ray scattering 1932
b. Neutron scattering 1932
V. Findings and Recommendations: LRI in NS
Workshop 1934
A. Recent scientific advances in LRI in NS 1935
B. Challenges and needs in LRI in NS 1935
C. Transformative opportunities from LRI in NS 1936
VI. Conclusions 1936
Acknowledgments 1936
References 1937
I. INTRODUCTION
In his famous musing, Plenty of Room at the Bottom,
Feynman H208491960H20850 noted ??as we go down in size, there
are a number of interesting problems that arise. All
things do not simply scale down in proportion. There is
the problem that materials stick together by the molecu-
lar H20849van der WaalsH20850 attractions. It would be like this:
After you have made a part and you unscrew the nut
from a bolt, it isn?t going to fall down because the grav-
ity isn?t appreciable; it would even be hard to get it off
the bolt. It would be like those old movies of a man with
his hands full of molasses, trying to get rid of a glass of
water. There will be several problems of this nature that
we will have to be ready to design for.?
The importance of long range interactions H20849LRIsH20850 in
the synthesis, design, and manipulation of materials at
the nanometer scale was thus recognized from the very
1889
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
beginning of nanoscience. Yet it is only recently that the
intricacies of not only van der Waals forces, referred to
by Feynman, but all LRIs have emerged in unexpectedly
many research areas. These areas include quantum field
theory, quantum and classical density functional theo-
ries, various mean-field and strong-coupling statistical
mechanical formulations, liquid state integral equations,
and computer simulations. These theoretical repercus-
sions lead to novel experimental designs and methods
with concomitant novelty and prospects in technology.
The role of LRIs in self-assembling active devices
constructed of heterogeneous components is fundamen-
tal. They govern the stability of component clusters, es-
sential for the design of nanodevices and nanoactuators.
The new technological paradigms that might be devel-
oped as a consequence of these fundamental studies
promise new ways of thinking that bring old problems
closer to solution.
The present review is the outcome of a workshop,
Long Range Interactions in Nanoscale Science, con-
vened under the auspices of the United States Depart-
ment of Energy, Council for the Division of Materials
Sciences and Engineering, Basic Energy Sciences. The
panel was charged to survey, identify, report, and assess
basic research challenges, needs, and opportunities from
working with long range interactions. The group exam-
ined recent advances in the theory, computation, and
measurement of the primary LRIs?electrodynamic,
electrostatic, and polar?as well as secondary LRIs, in-
cluding hydrogen bonding, hydrophobic/hydrophilic/
hydration, steric, structural, and entropic interactions.
The aim was to create a comprehensive framework and
language of these forces in nanoscience as well as to
identify strategies to harness them for the design of new
materials and devices.
This endeavor requires spanning a vast range of sci-
ence from field theory to colloid science, from physical
sciences to chemistry and biology, and from theory to
experiment and computation. The aim is to couple the
fundamentals of LRIs to experimentally accessible sys-
tems that can be manipulated on the nanoscale and that
can in the foreseeable future enable technological appli-
cations. Our review focuses first on the fundamentals of
electrodynamic, electrostatic, and polar H20849acid and/or
baseH20850 interactions. This focus includes instructive sys-
tems that reveal different aspects of LRIs, such as at-
oms, molecules, nanoscopic, mesoscopic, and macro-
scopic interfaces, surfaces and defects, as well as
chemical equilibria in liquids, suspensions, and colloidal
aggregates. We then assess this understanding in order
to harness the properties of LRIs in nanoscale systems
and to guide the design and chemical construction of
electronic, optical, and sensing devices.
The scope of this review is broader than found in a
traditional setting. We chose the format to represent the
kind of thinking that inspired this conference. It relies
on commonality and shared potential of different fields
of science so as to weave them into a unifed whole.
II. FUNDAMENTAL INTERACTIONS
At the most fundamental level, all atomistic interac-
tions are electromagnetic. In spite of this unifying and
underlying fundamental principle, various types of
atomic and molecular interactions show sufficient speci-
ficity either in the underlying theories or in their relative
strength within different regimes of interatomic or inter-
molecular separations. Consequently, they have occu-
pied disjoint scientific communities that study, character-
ize, classify, and use distinct kinds of interactions in
different ways.
The clearest distinction is between electrodynamic
van der Waals interactions and electrostatic or Coulomb
interactions. Electrodynamic interactions are formulated
within the rules of quantum electrodynamics; electro-
static interactions are formulated within the framework
of classical electrostatics and statistical mechanics. This
fundamental division is subdivided further on an ever-
increasing ladder of energy and length scales; see Table
I.
The limits of this classification are soon encountered,
especially when one approaches the nanoscale. The hard
edges that used to separate these interactions begin to
blur and merge in such a way that their conceptual de-
scription and formal models cease to be independent of
each other. This leads one to wonder what exactly con-
stitutes a ?primary interaction.? To some extent this dis-
tinction is arbitrary. We selected the van der Waals, the
electrostatic, and the acid-base interactions as the pri-
mary interactions. All secondary interactions either are
subsets of these primary interactions, specific to a class
of materials, or are a more macroscopic phenomenon
that does not persist down to the atomic interaction
level.
The distinction between these long and short ranged
basic primary interactions and effective interactions gen-
erated by collective phenomena can be properly quanti-
fied in terms of the appropriate correlation functions.
These correlation functions can be in their turn again
TABLE I. Different levels of long range interactions.
Primary Secondary
Hydrogen bonding
Hydrophobic
Electrodynamic H20849van der WaalsH20850: Hydration
London, dispersion, Debye, Osmotic
Induction, Keesom, orientation Disjoining
Structural
Electrostatic H20849CoulombicH20850 Steric
Depletion
Entropy-driven
Enthalpy-driven
Cross-binding
Polar H20849acid-baseH20850 Specific
Magnetic
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
either short or long ranged. If the underlying primary
interactions are long ranged, the correlation functions
are often long ranged, but short ranged interactions can
nevertheless generate long ranged correlation functions
as can be clearly seen close to phase transitions or in the
polymer scaling limit. A clear example of this type is the
critical Casimir interaction that can be observed close to
second-order phase transition H20849Hertlein et al., 2008H20850.Itis
important to note here that what constitutes a basic pri-
mary long range interaction or an effective long range
interaction of course depends crucially on the specific
problem under investigation, i.e., on relevant degrees of
freedom in the Hamiltonian of the system H20849electrons,
atoms, molecules, etc.H20850.
One of the outcomes of this review was a redefinition
of what should be considered ?long range? on the nano-
scale. On the one hand, gravity can reach out to the
cosmos and is certainly a long range interaction, but it is
not of the type that would be relevant at the nanoscale.
On the other hand, the hydrogen bond that mostly
reaches across only a single bond to the next neighbor
can qualify as an effective long range interaction since
collective phenomena and ensuing effects can extend its
range into the nanoscale. Because these effects can
propagate well beyond the spatial scale set by nearest
neighbors and can affect even the macroscopic proper-
ties of materials, one can propose the definition of long
range on the nanoscale starting with ?extending beyond
a single bond.?
A viable distinction between the long versus short
range interactions would be the algebraically decaying
interaction potentials versus exponentially decaying
ones. This choice is useful because these classes of inter-
action potentials lead to qualitatively different proper-
ties of resulting correlation functions. However, this
definition nevertheless creates a problem since an inter-
action reveals itself as being long ranged only at large
distances where it asymptotically dominates any expo-
nentially decaying short range potential. It thus appears
that the distinction between long and short ranged nano-
scale interactions is blurred and to some extent idiosyn-
cratic, manifesting itself clearly only at the upper end of
the nanoscale or even after entering the mesoscale, with
a consequence that on the nanoscale long and short
ranged interactions appear to be equally important and
difficult to distinguish. Thus here too what constitutes a
long range as opposed to short range interaction de-
pends primarily on the specific problem under investiga-
tion.
A. Electrodynamic interactions
Everyday condensed matter is mostly bound by elec-
tromagnetic H20849EMH20850 forces between neutral objects, forces
that are animated by electromagnetic fields from the
?coordinated dance of fluctuating charges? H20849Parsegian,
2005H20850. From the atomistic perspective, dilute gases expe-
rience these attractive interactions H20849i.e., the Keesom,
Debye, and London dispersion contributionsH20850 asa1/R
6
function of atomic separation R. However, the collective
behavior of atoms in condensed matter is better formu-
lated in terms of its macroscopic continuum-dielectric
properties. In 1948, by focusing on the quantum fluctua-
tions of the EM field, Casimir computed the force be-
tween two parallel ideally metallic plates in vacuum H20849Ca-
simir, 1948H20850. This approach was later generalized to
realistic dielectric materials by Lifshitz, who took into
account the fluctuating charge sources in the media H20849Lif-
shitz, 1955, 1956; Dzyaloshinskii et al., 1961H20850. It follows
from this formulation, later corroborated by experi-
ments H20849Munday et al., 2009H20850, that the electrodynamic in-
teractions can be attractive as well as repulsive.
The following decades were marked by theoretical ad-
vances as well as precise measurements H20851see, e.g.,
Sabisky and Anderson H208491973H20850 and Derjaguin et al.
H208491978H20850H20852. As the most relevant interaction between neu-
tral bodies at short distance, electrodynamic long range
interactions play an important role in, e.g., microelectro-
mechanical systems H20849Capasso et al., 2007H20850, where they
cause metals to attract and to stick at short distances,
also known as ?stiction? H20849Serry, 1998H20850.
Given that the theoretical foundation and early ex-
periments were developed in the 1940s and 1950s, why is
there now such a burst of activity in electrodynamic?
?London,? ?dispersion,? ?van der Waals,? ?Casimir,? or
?Lifshitz??forces? Was it the advent of high-precision
measurements, the ability to manipulate materials, or
the technical ability to measure spectra for computa-
tion? Unfortunately, it is clear that the different lan-
guages and training of those working on different facets
of the same problem still impede progress and inhibit
constructive collaboration.
Even words pose barriers. The diverse nomenclature
of interactions that collectively fall under ?electrody-
namic? can be separated into two distinct categories:
continuum methods using bulk or macroscopic materials
properties H20849Lifshitz, Casimir, etc.H20850 and atomistic H20851classi-
cal force field, density functional theory H20849DFTH20850, etc.H20852.
These different limits coexist as separate entities and
comfortably give meaningful results and insights within
their particular regimes. However, the nanotech, bio-
tech, and other popular fields are placing ever-increasing
demands upon these formulations so as to be applicable
to systems of all sizes and separations, requiring a way to
blend them in a straightforward and fundamentally
sound manner. This is not easy, but it provides opportu-
nities to develop what is typically seen by outsiders as a
?mature? field. The following overview of the founda-
tions of electrodynamic forces includes specific examples
of where challenging areas remain.
Though formally not of electrodynamic origin, ?criti-
cal Casimir interactions? H20849Krech and Dietrich, 1992;
Krech, 1994H20850 might be nevertheless classified with the
electrodynamic interactions because they are similar to
the original Casimir effect. They arise close to a second-
order phase transition as a consequence of a broken
continuous symmetry in the bulk, where the correlation
functions acquire an algebraic decay, thus giving rise to
bona fide long range effective interactions generated by
collective phenomena H20849Fisher and de Gennes, 1978H20850 on
1891
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
top of and in addition to the electrodynamic dispersion
force H20849Dantchev et al., 2007H20850. The formal point of corre-
spondence between the two Casimir effects is that the
effective Hamiltonian of a critical fluid corresponds to
massless fields and is thus formally equivalent to the
electrodynamic Hamiltonian. Though there is formal
similarity between standard and critical Casimir interac-
tions, the latter profit from a different type of theoretical
approach based on the theory of finite size scaling
H20849Krech and Dietrich, 1991; Macio?ek et al., 2007H20850. Critical
Casimir interactions present a new pathway to generate
long ranged interactions in systems that are nominally
governed by short range microscopic interactions.
Critical Casimir interactions are particularly impor-
tant and have been observed experimentally in the case
of thinning of
4
He films near the superfluid transition
H20849Garcia and Chan, 1999; Ganshin et al., 2006H20850, near the
3
He-
4
He tricritical point H20849Garcia and Chan, 2002; Maci-
olek and Dietrich, 2006H20850, in binary wetting films H20849Fukuto
et al., 2005H20850, and in colloidal suspensions in the vicinity
of chemically patterned surfaces H20849Soyka et al., 2008H20850.
However, in the case of
4
He films above and below the
superfluid transition, the experimentally observed differ-
ence in thickness is larger than can be accounted for by
the critical Casimir force. It appears that surface fluctua-
tions of the film surface give rise to an additional force,
similar in form but larger in magnitude than the critical
Casimir force, which is needed to account quantitatively
for the observations H20849Zandi, Rudnick, and Kardar, 2004H20850.
1. Lifshitz theory from optical properties
The van der Waals?London dispersion H20849vdW-LdH20850 in-
teractions influence properties ranging from colloidal
forces in solution to the fracture of bulk materials. They
can significantly affect a given system even when ?stron-
ger? forces, such as electrostatic or polar interactions,
are acting. One example is the single-wall carbon nano-
tube separation experiments by Zheng and Semke
H208492007H20850. Although single-stranded DNA coatings wrap
the different chiralities with an equivalent surface
charge density, one is able to separate them reliably and
repeatably during a salt elution experiment. How could
this be if only electrostatics were involved? One theory
notes the chirality-dependent optical properties of the
single-wall carbon nanotube core. In fact, calculations
via the Lifshitz formulation have shown that these dif-
ferences do exist and can be potentially used in experi-
mental design H20849Rajter and French, 2008H20850.
At least in principle, proper understanding and a con-
sistent theoretical formulation of the vdW-Ld interac-
tion has been fully achieved within the Lifshitz theory of
dispersion interactions H20849Parsegian, 2005H20850. It provides the
link between optical properties or ?London dispersion
spectra? and the magnitude of these interactions for ge-
ometries that are either analytically tractable or easily
approximated with simpler geometries. For this reason
we review the vdW-Ld energy calculations from the
bulk-material perspective.
a. Hamaker coefficients
In the framework of the Lifshitz theory H20849Parsegian,
2005H20850, the nonretarded dispersion interaction free en-
ergy per unit area is
GH20849lH20850 =?
A
123
12H9266l
2
, H208491H20850
where l is the surface to surface separation thickness
between two semi-infinite half-spaces H20849Fig. 1H20850 and A
123
is
the effective Hamaker coefficient, which is defined in
this case as
a)
b)
FIG. 1. H20849Color onlineH20850 A comparison of Lifshitz geometries. H20849aH20850
Schematic view of the isotropic, plane-plane configuration for
the Hamaker coefficient computation between two optically
isotropic, semi-infinite half-spaces. H20849bH20850 Schematic view of the
anisotropic, cylinder-cylinder configuration for the Hamaker
coefficient computation between two identical single-walled
carbon nanotubes H20849H2085110,0,sH20852 in the case depictedH20850, which is al-
ways attractive irrespective of the medium in between, or two
different single-walled carbon nanotubes H20849H2085110,0,sH20852 and
H208516,6,mH20852 in the case depictedH20850, which can be repulsive in an
appropriately chosen intervening medium.
1892
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
A
123
=?
3
2
k
B
T
H20858
n=0
H11009
H20885
0
H11009
udu lnH208511?H9004
32
H20849iH9264
n
H20850H9004
12
H20849iH9264
n
H20850e
?u
H20852
H11015
3
2
kT
H20858
n=0
H11009
H9004
32
H20849iH9264
n
H20850H9004
12
H20849iH9264
n
H20850, H208492H20850
Here 1 and 3 H20849of subscripts 123H20850 represent the left and
right infinite half-spaces separated by medium 2 of
thickness l; u is the magnitude of the wave vector in the
plane of the two opposing interfaces. The summation is
over a discrete set of Matsubara, or boson, frequencies
H9264
n
=2H9266H20849k
B
TH20850n/H6036, where k
B
is the Boltzmann constant
and 2H9266H6036 is the Planck constant. At room temperature,
the interval between successive photon energies H6036H9264
n
is
H110150.16 eV. The prime in the summation signifies that the
first, n=0, term is taken with weight 1/2.
The Hamaker coefficient is largely determined by ma-
terials properties. Its magnitude and sign depend on the
values of H9004?s that describe the relative optical spectral
mismatches or contrast between neighboring materials
at the frequency H9264
n
involved in the interaction,
H9004
ij
H20849iH9264
n
H20850 =
H9255
i
H20849iH9264
n
H20850 ? H9255
j
H20849iH9264
n
H20850
H9255
i
H20849iH9264
n
H20850 + H9255
j
H20849iH9264
n
H20850
. H208493H20850
The dielectric function at imaginary values of the fre-
quency argument H9255H20849iH9264H20850, the fundamental ingredient of
the Lifshitz theory of vdW-Ld interactions, can be ob-
tained via the Kramers-Kronig H20849KKH20850 transform in the
form
H9255H20849iH9264H20850 =1+
2
H9266
H20885
0
H11009
H9255H11033H20849H9275H20850dH9275
H9275
2
+ H9264
2
, H208494H20850
where H9255H11033H20849H9275H20850 is the imaginary part of the dielectric re-
sponse function in real frequency space, i.e., H9255H20849H9275H20850
=H9255H11032H20849H9275H20850+iH9255H11033H20849H9275H20850. H9255H20849iH9264H20850 is referred to as the van der Waals?
London dispersion spectrum. The magnitude of H9255H20849iH9264H20850 de-
scribes how well the material responds and is polarized
by fluctuations up to a given frequency. While the inte-
gration in Eq. H208494H20850 requires spectra to infinite frequencies,
in practice this is unnecessary as long as all interband
transition energies are either known or properly ap-
proximated. For simple systems, it may be acceptable to
use this simple formulation for planar geometries to es-
timate the magnitude and sign of the Hamaker coeffi-
cient. However, there are situations where the geometry
can also influence the Hamaker coefficient by altering
the form of H9004
ij
H20849Rajter et al., 2007H20850.
b. Full spectral optical properties
Traditionally, optical spectra were obtained either ex-
perimentally or approximated using damped-oscillator
models. Recently it has been possible to use ab initio
calculations. The particular method used is relevant only
if caveats are considered, e.g., particular frequency or
energy ranges where data are not trustworthy. Some-
times it is possible to adequately approximate gaps in
spectral properties, such as using analytical wings and
oscillator extensions H20849French, 2000H20850. Once sufficiently
obtained and/or approximated, the various experimental
or ab initio optical properties are interchangeable and
can be directly transformed into the required vdW-Ld
spectra using the appropriate Kramers-Kronig H20849KKH20850
transforms. For example, valence electron energy loss
spectrum measurement might give the frequency-
dependent results in J
cv
H20849eVH20850H20849interband transition
strength form in eV frequency unitsH20850 while the ab initio
codes give the imaginary part of the dielectric spectrum
over real frequencies H9255H11033H20849H9275H20850. One can convert between
the two via
J
cv
H20849H9275H20850 =
m
0
2
H9275
2
e
2
H6036
2
8H9266
2
H20851H9255H11033H20849H9275H20850 + iH9255H11032H20849H9275H20850H20852. H208495H20850
From this, one can use H9255H11033H20849H9275H20850 in the KK transform H20851Eq.
H208494H20850H20852 to determine the vdW-Ld spectra.
c. Model systems: PS-water-PS and SiO
2
-water-SiO
2
The many layers of abstraction within the Lifshitz for-
mulation can hide the linkage between optical proper-
ties and underlying material properties. Knowing how
optical properties depend on the underlying material
composition and crystal structure can benefit materials
design and engineering.
For example, the interband optical properties of poly-
styrene in the vacuum ultraviolet H20849VUVH20850 can be investi-
gated using combined spectroscopic ellipsometry and
VUV spectroscopy H20849French et al., 2007H20850. Over the range
1.5?32 eV, the optical properties exhibit electronic tran-
sitions that can be assigned to three groupings, corre-
sponding to a hierarchy of interband transitions of aro-
matic H20849H9266?H9266
*
H20850, nonbonding H20849n?H9266
*
, n?H9268
*
H20850, and
saturated H20849H9268?H9268
*
H20850 orbitals. In polystyrene there are
strong features in the interband transitions arising from
the side-chain H9266 bonding of the aromatic ring consisting
of a shoulder at 5.8 eV and a peak at 6.3 eV, and from
the H9268 bonding of the C?C backbone at 12 and 17.1 eV.
These transitions have characteristic critical point line
shapes associated with one-dimensionally delocalized
electron states in the polymer backbone. A small shoul-
der at 9.9 eV is associated with excitations possibly from
residual monomer or impurities.
Having obtained polystyrene?s optical spectra from
the above experiments, it is then trivial to calculate Ha-
maker coefficients and vdW-Ld interaction energies in
the plane-parallel geometry H20849Fig. 1H20850 at all separation dis-
tances. One can also vary the intervening medium H20849by
changing its full spectral optical propertiesH20850 in order to
get the Hamaker coefficients and the vdW-Ld compo-
nent of the surface free energy for polystyrene im-
mersed in these other materials; see Table II.
Similar and related calculations and analysis can also
be done for two semi-infinite slabs of SiO
2
interacting
across various media H20849Tan et al., 2005H20850. The interband
optical properties of crystalline quartz and amorphous
SiO
2
in the VUV have been investigated using com-
bined spectroscopic ellipsometry and VUV spectros-
copy. Over the range of 1.5?42 eV, the optical proper-
ties exhibit similar exciton and interband transitions,
1893
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
with crystalline SiO
2
exhibiting larger transition
strengths and indices of refraction. Crystalline SiO
2
has
more sharp features in the interband transition strength
spectrum than amorphous SiO
2
, the energy of the ab-
sorption edge for crystalline SiO
2
is about 1 eV higher
than that for amorphous SiO
2
, and the direct band-gap
energies for X- and Z-cut quartz are 8.30 and 8.29 eV
within the absorption coefficient range 2?20 cm
?1
.
By calculating and analyzing the Hamaker coeffi-
cients, we can determine the degree to which changes in
crystal structure or composition influence the vdW-Ld
interaction. Comparing c-SiO
2
or a-SiO
2
, there is an ap-
preciable increase in the overall vdW-Ld strength for the
c-SiO
2
, which is a result of its increased physical density,
index of refraction, transition strengths, and oscillator
strengths compared with a-SiO
2
H20849Table IH20850.
d. Advanced effects and phenomenology
There are many other effects that occur as we make
the system more complex. The first is that the Lifshitz
formulation allows for attractive as well as repulsive Ha-
maker coefficients, based on the magnitude and spatial
arrangement of the optical contrast at each interface in
the system. One example of this common phenomenon
is liquid helium films where the substrate material pre-
fers as much contact as possible with helium rather than
with air H20849Sabisky and Anderson, 1973H20850. There is ?com-
plete wetting? that occurs because the system prefers to
make the medium as thick as possible.
Wetting phenomena are relevant in the context of
LRIs not only in the framework of liquid helium, but
from a general point of view. On the nanoscale they are
very sensitive to the range of the underlying interactions
in two ways. First, there is a qualitative difference be-
tween short and long range forces in that for the former
capillary wavelike fluctuations are relevant, whereas for
the latter these fluctuations are not. Second, the depen-
dence of wetting film thickness on thermodynamic vari-
ables, e.g., pressure, directly reveals the algebraic decay
behavior of the underlying forces H20849Dietrich, 1988H20850.Ina
similar vein, the LRIs also give rise to algebraically de-
caying H20849van der WaalsH20850 tails in the density profiles of thin
fluid films H20849Dietrich and Napi?rkowski, 1991H20850.
In the context of wetting one should also note that
LRIs translate nanosculptured substrate morphologies
into distinct wetting properties H20849Gang et al., 2005; Ta-
sinkevych and Dietrich, 2006, 2007H20850.
The next effect is that of retardation and/or sign re-
versal. Because of the finite speed of light, the high-
energy frequency contributions to the summation in Eq.
H208492H20850 dampen quickly as the separation distance increases.
Thus the Hamaker coefficient itself has a nonlinear de-
pendence on separation. For example, consider a system
where the higher-energy part of the spectrum gives rise
to primarily repulsive terms and the low-energy part
contributed attractive terms. At large separations, where
retardation eliminates all but the low-energy terms, the
overall interaction is positive. However, as the particles
attract and get closer, the Hamaker coefficient would
continue to add repulsive terms to the overall summa-
tion until it hits zero and actually changes sign. This
combination of attractive and repulsive contributions
can lead to interesting effects, such as an equilibrium
1?2 nm vdW-Ld separation energy well and the exis-
tence of premelting in surface layers of ice H20849see Sec.
II.C.1H20850.
Multilayers H20849such as coatingsH20850 experience a set of com-
peting interactions because the effective optical proper-
ties of a layered object can change drastically as a func-
tion of separation distance H20849Podgornik et al., 2006H20850.
Optical anisotropy will create a configurational or direc-
tional component such as to cause alignment forces and
torques H20849Parsegian and Weiss, 1972H20850. Many such effects,
important on the nanoscale, are only beginning to be
recognized, probably because the necessary optical
properties and extensions to the Lifshitz formulations
are still under development.
We also note that although this review is devoted to
static structural LRI dependent properties, there are
well-known specific influences of LRIs on the dynamics
of nanoscale objects. As an example, note that under the
action of the long ranged tails of long range dispersion
TABLE II. Full spectral Hamaker coefficients for vdW-Ld interactions of different physical configurations with polystyrene,
amorphous SiO
2
, or water, using interband transition strength spectra.
Physical geometry
Hamaker
coeff. H20851zJH20852
a
Physical geometry
Hamaker
coeff. H20851zJH20852 Physical geometry
Hamaker
coeff. H20851zJH20852
H20851PS|vacuum|PSH20852 70.9 H20851SiO
2
H20841vacuumH20841SiO
2
H20852 71.6 H20851water|vacuum|waterH20852 34.2
H20851PS|water|PSH20852 7.71 H20851SiO
2
H20841waterH20841SiO
2
H20852 8.06
H20851PSH20841H9251-SiO
2
H20841PSH20852 1.53 H20851SiO
2
H20841SiO
2
H20841SiO
2
H20852 0 H20851waterH20841SiO
2
H20841waterH20852 8.04
Literature results H20851PS|water|PSH20852 reported by Bos et al. H208491999H20850
H20851PS|water|PSH20852 5 H20851PS|water|PSH20852 from 3.5?5.3
from contact angles macroscopic measurements
H20851PS|vacuum|PSH20852 8.90 H20851PS|vacuum|PSH20852 from 9.63
from Dagastine et al. H208492004H20850 Dagastine et al. H208492004H20850 using
Parsegian and Weiss H208491972H20850 PS
a
1zJ=10
?21
J.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
forces nanodroplets move in lateral directions which are
opposite to the ones expected for micron-sized drops
H20849Moosavi et al., 2006, 2008a, 2008bH20850.
2. Ab initio optical properties of complex materials
The previous sections outlined the important effects
that result from even small changes in the full spectral
optical properties. Given the ubiquity of vdW-Ld inter-
actions in condensed matter, one might naively assume
that there is a large catalog of these spectra available.
This information could then be data-mined and used in
all sorts of ways, particularly in experimental design.
Unfortunately, obtaining full optical spectra is a more
difficult endeavor than most realize. While vacuum ul-
traviolet spectroscopy and valence electron energy loss
spectroscopy H20849V-EELSH20850 are well established and have
proven critical to reliable force computation H20849French,
2000; French et al., 2007H20850, stringent sample preparation
specifications H20849among other factorsH20850 make a generalized
cataloging of hundreds of materials cost and time pro-
hibitive. Liquids are particularly difficult to characterize
under the required vacuum conditions with consequent
problems in containment. To experimentally character-
ize spatially varying, deep UV optical properties of a
complex material in a liquid medium would be difficult
or impossible.
More recently, ab initio codes have proven to be a
viable alternative to analyze those materials whose spec-
tra cannot be cleanly obtained experimentally H20849e.g., bio-
logical molecules immersed in a liquid solutionH20850.Inor-
der to capture all possible electron transitions between
the valence and conduction bands of the total electronic
structure, these orthogonalized linear combination of
atomic orbitals?DFT codes use very large basis sets. Test
calculations on several ceramic crystals showed that the
calculated Hamaker coefficients using theoretical spec-
tra do not differ much from those obtained using experi-
mental spectra H20849Ahuja et al., 2004H20850. Recently this ap-
proach has been applied to obtain Hamaker constants
for both metallic and semiconducting single-wall carbon
nanotubes H20849SWCNTH20850 and multiwall carbon nanotubes
H20849MWCNTH20850 of different chiralities with considerable suc-
cess H20849Rajter, French, et al., 2008; Rajter et al., 2008H20850.
In Figs. 2 and 3 we show H9255H11033H20849H9275H20850 and the corresponding
vdW-Ld spectra properties of H208516,5,sH20852 and H208519,3,mH20852
SWCNTs. It is important to capture all of these inter-
band transitions, out to at least 30 eV, because all areas
are adding to the overall summations and can shift the
magnitude of the resulting Hamaker coefficients.
vdW-Ld interactions require accurate information for
electronic transitions well beyond the band gap, while
device performance studies have typically been focused
near the band gap energy. But beyond a 30?40 eV cut-
off, the transitions become less important because they
are considerably dampened by the KK transform and
therefore need to be much larger in order to be signifi-
cant.
While the ascertaining and usage of ab initio optical
properties for vdW-Ld interactions is still a new and
small field, its speed, low cost, and broad utility makes it
an appealing solution to the long-standing problem of a
dearth of optical property catalogs. It even provides ex-
citing new possibilities, such as spatially resolving the
optical properties for, e.g., biological materials, which
will be described in Sec. III.A.1.
3. Electrodynamic interactions for arbitrary shapes
The impact of shape or geometry is a more complex
but equally important component in determining the
multibodied behavior of electrodynamic LRIs. Recall
that the sources of these LRIs arise from the quantum
fluctuations of the electromagnetic H20849EMH20850 field as they
are modified by the presence, positions, and shapes of
metallic H20849CasimirH20850 or dielectric H20849LifshitzH20850 objects H20849Ca-
simir, 1948; Lifshitz, 1956H20850. The advent of high-precision
measurements H20849Lamoreaux, 1997; Mohideen and Roy,
1998; Roy et al., 1999H20850, and the possibility that they can
be applied to nanoscale electromechanical devices
H20849Serry et al., 1998; Chan et al., 2001; Decca et al., 2003H20850,
has stimulated interest in developing a practical way to
calculate the dependence of electrodynamic energies on
the shapes of the objects.
The simplest and most commonly used methods for
dealing with complex shapes rely on pairwise summa-
tions. In the proximity force approximation H20849PFAH20850, also
referred to as the Derjaguin approximation, the energy
is obtained as an integral over infinitesimal parallel sur-
FIG. 2. H20849Color onlineH20850 Ab initio dielectric function H20849H9255H11033H20850 spectra
for the H20849aH20850 semiconducting H208516,5,sH20852 and H20849bH20850 metallic H208519,3,mH20852
SWCNTs in the radial and axial directions from 0 to 30 eV.
Each spectra depends on direction and chirality. From Rajter
et al., 2007.
FIG. 3. H20849Color onlineH20850 The van der Waals?London dispersion
spectra for the H20849aH20850 semiconducting H208516,5,sH20852 and H20849bH20850 metallic
H208519,3,mH20852 SWCNTs in the radial and axial directions from
0 to 30 eV. The differences are large enough to create chirality
and orientation-dependent vdW-Ld interactions. From Rajter
et al., 2007.
1895
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
face elements at their local separation measured perpen-
dicular to a surface H20849Rempel et al., 2001; Parsegian, 2005H20850
and works well when two objects are very close together
as, for example, in the case of SWCNTs at small separa-
tions H20849Rajter et al., 2007H20850. A second method described by
Sedmik et al. H208492007H20850 is based on addition of Casimir-
Polder ?atomic? interactions H20849CPIH20850. Unfortunately, both
methods are heuristic and not easily amenable to sys-
tematic improvements.
Balian and Duplantier H208491978H20850 proposed a method,
based on a multiple scattering expansion, for calculating
Casimir energies for arbitrary shape. However, this
method has not proved workable in practice. An ap-
proach based on path integral methods has also been
used to compute corrections to the parallel-plate result
by perturbation for small deformations H20849Emig et al.,
2003H20850. The limitations of perturbation theory weaken
the usefulness of this approach, but in some cases it can
be overcome by specialized numerical methods H20849B?scher
and Emig, 2005H20850. There is also a numerical implementa-
tion of the path integral method H20849Gies and Klingmuller,
2006H20850, which has so far been applied only to scalar fields.
Recent numerical approaches have been based either on
an explicit discretization of the EM fields in space and
computation of the mean stress tensor H20849Rodriguez, Iba-
nescu, Iannuzzi, Capasso, et al., 2007H20850 or on the bound-
ary element method by rewriting the van der Waals in-
teraction energy exclusively in terms of surface integrals
of surface operators H20849Veble and Podgornik, 2007aH20850.In
principle, both schemes can deal with arbitrary geom-
etries and a spatially varying dielectric constant.
a. Cylinders and plates
The geometry of the cylinder, as intermediate be-
tween sphere and plate, is ideally suited to testing the
limitations of PFA and CPI approximations. It is also
relevant to important experimental systems such as car-
bon nanotubes H20851see Fig. 1H20849bH20850H20852 and stiff polymers such as
DNA. The translational symmetry along the cylinder
axis considerably simplifies the problem, as the electro-
magnetic H20849EMH20850 field H20849with metallic objectsH20850 can be de-
composed into transverse magnetic H20849TMH20850 and transverse
electric H20849TEH20850 components, with Dirichlet and Neumann
boundary conditions, respectively. Emig et al. H208492006H20850 ex-
ploited this to find the exact Casimir force between a
plate and a cylinder, and there have been further elabo-
rations H20849Brown-Hayes et al., 2005; Bordag, 2006H20850. The
force has an unexpectedly weak decay
FH20849HH20850 H11008
L
H
3
lnH20849H/RH20850
H208496H20850
at large plate-cylinder separations H H20849L and R are the
cylinder length and radiusH20850, due to transverse magnetic
modes. Path integral quantization with a partial wave
expansion additionally provides the density of states,
and corrections at finite temperatures. An example of
the failure of pairwise additivity was obtained by Rod-
riguez, Jesse, et al. H208492007H20850, in which the Casimir force
between two squares exhibits a nonmonotonic depen-
dence on the distance from enclosing sidewalls. Figure 4,
left panel, shows the same effect for two cylinders, with
one or two nearby sidewalls H20849all metalsH20850. The right panel
of Fig. 4 shows results for the force between two cylin-
ders, normalized by the PFA result. The solid and
dashed lines correspond to one and two sidewalls, re-
spectively; in each case the contributions of TM and TE
modes to the total force are also depicted. For the plot-
ted separation of a/R=2, PFA overestimates the force
by roughly a factor of 2. More significantly, in pairwise
approximations H20849PFA or CPIH20850 the sidewallH20849sH20850 have no
effect on the force between the cylinders. The nonmono-
tonic variation with the separation H to the plates is thus
a direct illustration of the importance of three-body ef-
fects.
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h/R
F/F
PFA
TE (Neumann)
TM (Dirichlet)
total
h
R
a
h
R
a
x
y
z
FIG. 4. H20849Color onlineH20850 Casimir force per unit length between two cylinders H20849blackH20850 vs the ratio of sidewall separation to cylinder
radius h/R, at fixed a/R=2, normalized by the total ?proximity force approximation? force per unit length between two isolated
cylinders. From Rahi et al., 2008.
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b. Spheres and compact shapes
A potentially quite powerful approach for computing
electrodynamic forces for materials of arbitrary shape
and composition was developed recently H20849Emig et al.,
2007H20850. There are dual equivalent perspectives on the Ca-
simir interaction: in terms of the fluctuating EM field or
the fluctuating sources H20849charges and currentsH20850 in the ma-
terial bodies H20849Schwinger, 2004H20850; Emig et al. employ the
latter. The fluctuating sources on the different objects
H20849labeled by greek lettersH20850 are indicated by H20853Q
H9251
H20854. Each Q
H9251
carries multiple indices that designate the source H20849charge
or currentH20850, partial wave H20849l,mH20850 in a multipole represen-
tation, and frequency H20849after Fourier transformation in
timeH20850, which will be suppressed. In path integral quanti-
zation, each configuration is weighted using an action
SH20851H20853Q
H9251
H20854H20852, which is quadratic and comprised of several
parts. The off-diagonal elements of the action,
SH20851H20853Q
H9251
H20854H20852 = Q
H9251
V
H9251H9252
Q
H9252
, H208497H20850
represent the interaction between charges. As is familiar
from electrostatics, we expect that the lowest multipoles
dominate the interaction at large separations. The ma-
trix elements V
H9251H9252
are thus a function of the separation
D
H9251H9252
and the implicit multipoles. The ?diagonal? compo-
nents,
SH20851H20853Q
H9251
H20854H20852 =
1
2
Q
H9251
T
H9251
?1
Q
H9251
, H208498H20850
are more interesting and represent the self-energy H20849ac-
tionH20850 of the source. The crucial observation is that the
matrices T
H9251
, which encode all relevant shape and mate-
rial properties of the objects, are directly related to scat-
tering from the object H20849Newton, 1966H20850. This connection
was also noted by Kenneth and Klich H208492006H20850, and pro-
vides a link to the mature and well-developed field of
scattering of EM waves from different objects.
The T matrix can be obtained for dielectric objects of
arbitrary shape by integrating the standard vector solu-
tions of the Helmholtz equation in dielectric media over
the object?s surface H20849Waterman, 1971H20850, and both analyti-
cal and numerical results are available for many shapes
H20849Mishchenko et al., 2004H20850. For the specific case of two
dielectric spheres, for which explicit formulas for the T
matrix are available, the Casimir force can be obtained
at all separations. Focusing on low-order multipoles
gives an expansion in powers of the ratio of sphere ra-
dius to separation H20849R/DH20850. Due to alternating signs, the
convergence of this series is problematic, but a conver-
gent approach can be obtained by including all terms
coming from a given order in the multipole order l H20849ir-
respective the power in R/DH20850, and extrapolating to l
?H11009. This procedure yields a curve for the force that
interpolates all the way from the Casimir-Polder limit to
the PFA result at short separations.
This concludes our basic overview of the fundamen-
tals of continuum model, electrodynamic LRIs. We now
turn our attention to the atomistic methods, specifically
the wide variety available using ab initio quantum codes.
4. Noncovalent interactions from electronic structure
calculations
Approximate numerical solutions to the electronic
Schr?dinger equation have become a standard tool for
ab initio prediction of materials properties in the fields
of computational physics, chemistry, and biology. Quan-
tum Monte Carlo and post Hartree-Fock methods, such
as coupled cluster or configuration interaction, are able
to reach an accuracy which is more than sufficient for
comparison to spectroscopic experiments. They suffer,
however, from a prohibitive computational cost with an
increasing number of electrons. Alternatively, the solu-
tion of the Schr?dinger equation within the Kohn-Sham
density functional theory H20849KS-DFTH20850 framework H20849Hohen-
berg and Kohn, 1964; Kohn and Sham, 1965; Parr and
Yang, 1989H20850 frequently proves to represent not only a
reasonable trade-off between accuracy and computa-
tional cost H20849Koch and Holthausen, 2001H20850, but also a con-
ceptually more appealing view on the electronic many-
body problem in terms of the single-particle electron
density, nH20849rH20850. Furthermore, the modest computational
cost of DFT led to the development of ab initio molecu-
lar dynamics methods such as Born-Oppenheimer or
Car-Parrinello molecular dynamics H20849Iftimie et al., 2005H20850.
In principle, KS-DFT is an exact theory that yields the
exact electronic ground state and interatomic potential.
In practice, however, the exchange-correlation energy
E
xc
H20851nH20849rH20850H20852, the unknown term in the KS-Hamiltonian,
must be approximated, thereby rendering the accuracy
less than perfect.
DFT has been widely successful in describing the
properties in dense materials and isolated molecules,
where local and semilocal approximations and their gen-
eralizations typically give satisfactory results H20849Staroverov
et al., 2003, 2004H20850; hybrid functionals H20849Sousa et al., 2007H20850
play a special role in the molecular case. However,
sparse systems, soft matter, molecular van der Waals
complexes, biomolecules, and the like cannot be ad-
equately described by the previously standard DFT ap-
proximations. These ?weak? LRIs are among the phe-
nomena for which the first generations of
approximations to the exchange-correlation interaction
v
xc
, the local density and generalized gradient approxi-
mations, LDA and GGA, yielded qualitatively errone-
ous predictions. Kristy?n and Pulay H208491994H20850 and P?rez-
Jord? et al. H208491994H20850 showed this more than a decade ago
for rare gas, as did Meijer and Sprik H208491996H20850 for the ben-
zene dimer and benzene crystal. For a recent assessment
of the performance of 44 approximations to v
xc
for de-
scribing nonbonded dimers see Zhao and Truhlar
H208492005aH20850. These difficulties have now become widely rec-
ognized, and a number of different techniques are being
developed and exploited to deal with the situation
where the LRIs are important. We start by reviewing
these new techniques. We then discuss additional ap-
proaches of incorporating vdW-Ld alongside DFT.
These approaches represent only a beginning to the so-
lution of a critical problem.
1897
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
It is also necessary to point out that there exists not
only an electronic density functional theory, but a so-
phisticated and properly formulated classical density
functional theory describing the structural properties of
inhomogeneous classical liquids on the molecular and
nanoscale H20849see Evans, 1979, 1992H20850.
a. Recently evolving density functional theory methods
We characterize emerging DFT methods as H20849iH20850 many-
body, using Kohn-Sham orbitals; H20849iiH20850 empirical and non-
empirical explicit density functional; and H20849iiiH20850 perturba-
tive, which leave the electronic structure uncorrected
and aim solely at predicting the correct interatomic po-
tentials, usually requiring the input of C
6
coefficients,
the polarizabilities, or damping functions which are con-
sistent with the employed exchange-correlation interac-
tion v
xc
. This characterization scheme is to some extent
arbitrary. Another scheme might be according to
whether or not it would be necessary to identify distinct
fragments of matter in order to define and apply the
method; this would split the methods in each of the cat-
egories above.
i. Many-body methods using Kohn-Sham orbitals. We begin
by discussing an important type of method whose use
has recently expanded and is based on the random phase
approximation H20849RPAH20850, often enhanced in various ways,
such as using corrections in the style of time-dependent
density functional theory to the RPA density-density
correlation function. The RPA approximates long range
correlations including van der Waals, but when applied
to uniform systems it is less accurate than the modern
form of LDA based on fitting of exact limits and quan-
tum Monte Carlo simulations, so it was abandoned de-
cades ago as a DFT technique. A suggestion by Kurth et
al. H208491999H20850 to simply apply the full RPA and to correct the
above error with an extra local or semilocal correction
seems to have resurrected its use. RPA methods have
recently been applied not only to model systems H20849Dob-
son and Wang, 1999; Pitarke and Perdew, 2003; Jung et
al., 2004H20850, but also to molecules H20849Aryasetiawan et al.,
2002; Fuchs and Gonze, 2002; Furche and Van Voorhis,
2005H20850 and solids H20849Miyake et al., 2002H20850; many of these are
not obviously vdW systems, but of recent interest is the
application by Marini et al. H208492006H20850 to a vdW-bonded lay-
ered solid H20849BNH20850. By separating the layers, the system can
be brought into the region where vdW-Ld is predomi-
nant, where both the strength and weakness of the
method is shown. Its strength is that there is no empiri-
cal input, distinct fragments do not need to be defined,
and it can be applied to both finite and extended sys-
tems. Its weakness is shown by the error bars: it is com-
putationally intensive; one must calculate excited Kohn-
Sham states accurately, which implies either a fine grid
or a large basis set, depending on the method used.
Symmetry-adapted perturbation theory H20849SAPTH20850 is a
quantum chemical method which treats the interaction
between monomers via perturbation theory. Years ago a
method was introduced that mimics the SAPT proce-
dure using KS orbitals H20849Williams and Chabalowski,
2001H20850, which we term SAPT-DFT. The initial version was
rather inaccurate, but various improvements, some of
which are semiempirical, have yielded a method capable
of giving good results H20849He?elmann and Jansen, 2003;
Misquitta et al., 2003H20850. However, there are also weak-
nesses. First, the method has not been developed for
application to extended systems. For finite systems, al-
though the scaling with system size is superior to the
state-of-the-art coupled-cluster wave function methods,
it is nevertheless significantly worse than for either stan-
dard DFT methods or the recent DFT methods that in-
clude vdW-Ld, as discussed below. Its application neces-
sitates the identification of individual fragments, so it
cannot be seamless as fragments merge together to be-
come single entities.
ii. Explicit density functional methods. The exchange and
correlation energies are given as explicit functionals; this
means specifically that once the occupied Kohn-Sham
orbitals and hence the density are obtained, the ex-
change and correlation energy components are simply
evaluated, without the need to calculate unoccupied KS
orbitals. This requirement guarantees that the scaling of
the computational requirements with system size will
not destroy the cubic scaling enjoyed by ordinary DFT.
The modern version of the functional nonempirical
vdW-DF was introduced several years ago H20849Dion et al.,
2004H20850, with the fully self-consistent version H20849Thonhauser
et al., 2007H20850 coming more recently. It is completely con-
sistent with the result stemming from a properly refor-
mulated Lifshitz theory H20849Veble and Podgornik, 2007bH20850.
The new version supplants the obsolete functional for
planar systems H20849Rydberg et al., 2000H20850. Like the RPA
methods, the correlation functional of vdW-DF is non-
empirical. Unlike them, however, it does not automati-
cally provide its own exchange functional. For this the
revised Perdew-Burke-Ernzerhof H20849revPBEH20850 functional
H20849Zhang and Yang, 1998H20850 version of the generalized gra-
dient approximation H20849GGAH20850 was used, because it ap-
peared to give the best agreement with Hartree-Fock
calculations when the correlation functional was omit-
ted. The vdW-DF method has shown promise for a va-
riety of system types where vdW-Ld interactions are im-
portant H20849Kleis et al., 2007; Thonhauser et al., 2007;
Chakarova-K?ck et al., 2008; Cooper et al., 2008H20850. The
method appears to be most accurate for larger systems,
where the multiplicity of probable particle-hole excita-
tions more closely matches the assumptions under which
it was derived. Even for small systems like rare-gas
dimers, it qualitatively captures the vdW-Ld interaction,
which is missed by standard density functionals. It can
handle extended systems with large unit cells, as well as
large finite systems, as its scaling with system size is the
same as ordinary DFT. It is fully self-consistent, which
means that it can produce the Hellmann-Feynman inter-
nuclear forces that are crucial for relaxation and mo-
lecular dynamics. It does not require the identification of
fragments, and if they physically exist, the theory is
seamless as they merge. Prototypical results, using this
approach for large systems, include physisorption of
benzene and naphthalene on graphite, the structure and
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binding of a polyethylene crystal, and a DNA base-pair
dimer, whose sequence-dependent twist matched trends
from high-resolution data H20849Olson et al., 1998H20850.
One can aim to improve the electron-electron interac-
tion such that the corrected electron density yields cor-
rect atomic forces. This is tantamount to improving v
xc
empirically, as it has already been done for functionals
which yield reasonable atomic and intramolecular ener-
gies. Usually, a small set of parameters is fitted to many
reliable reference results, in this case nonbonded com-
plexes. The recently introduced X3LYP functional H20849Xu
and Goddard, 2004H20850 was quickly shown to fail com-
pletely for stacked nucleic acid bases and amino acid
pairs H20849Cerny and Hobza, 2005H20850, although hydrogen
bonding was found to be well described. It thereby fol-
lows the typical pattern of conventional density func-
tionals that fail to properly describe the dispersion inter-
action. Zhao and Truhlar H208492005a, 2005bH20850 presented
various empirical functionals for weak interactions.
Alternatively, inspired by the idea that molecular
properties can be influenced through the parametriza-
tion of effective core potentials H20849Hellmann, 1935H20850,an
electron-nucleus correction, ?London dispersion-
corrected atom-centered potentials? H20849DCACPH20850, has
been introduced H20849von Lilienfeld et al., 2004H20850. In that ap-
proach a set of parameters, H20853H9268
i
H20854, in an atom-centered
potential is calibrated for every atom I in the periodic
table and added to a GGA exchange-correlation poten-
tial,
v
xc
= v
xc
GGA
+
H20858
I
v
I
DCACP
H20849H20853H9268
iI
H20854H20850. H208499H20850
Conventionally, the DCACP has the functional form of
the analytical pseudopotentials proposed by Goedecker
et al. H208491996H20850. Current versions of this correction use two
H9268 parameters per atom which are calibrated to experi-
mental or highly accurate theoretical results for proto-
typical van der Waals complexes. Thereafter, the same
atomic potential is employed for the different chemical
environments. The latest generation of calibrated poten-
tials can be found by Lin, Coutinho, et al. H208492007H20850. This
scheme has already successfully been applied to a range
of systems and situations, such as small rare-gas clusters,
the hydrogen bromide dimer, and conformational
changes in cyclooctane H20849von Lilienfeld, Tavernelli, et al.,
2005H20850 to the adsorption of Ar on graphite H20849Tkatchenko
and von Lilienfeld, 2006H20850, to the coarse graining of inter-
molecular potentials of discotic aromatic materials H20849von
Lilienfeld and Andrienko, 2006H20850, to the dimers of small
organic molecules, and to cohesive energies and the lat-
tice constant of the benzene crystal H20849Tapavicza et al.,
2007H20850, to the intermolecular binding of DNA base-pairs
and base-pair or intercalator drug candidate H20849Lin, von
Lilienfeld, et al., 2007H20850, and liquid water. However, for all
these empirical approaches to the exact form of the ex-
change interaction v
xc
, the fundamental problem re-
mains that a priori not only the parameters in v
xc
are
unknown but even its functional form.
b. C
6
coefficients of polarizabilities, and damping functions
We use the term DFT-dampedC
6
to describe any of
the methods that treat short range interactions by DFT,
but treat vdW-Ld via a damped interaction directly be-
tween atomic nuclei. This is based on London?s original
perturbational work yielding the C
6
/R
6
asymptotic dis-
sociative scaling between two atoms at distance R
H20849Heitler and London, 1927H20850. The total energy is then
extended by
H20858
IH11021J
C
6J
C
6I
/R
IJ
6
, H2084910H20850
and must be damped to avoid any spurious effects on
the repulsive potential which originates in the nuclear
and electronic Coulomb, and the Pauli repulsion. The
vdW-Ld interaction is thus not treated by DFT at all, but
rather by a force-field method with a dissociative
Lennard-Jones potential. There exists a multiplicity of
such methods H20849Elstner et al., 2001; Wu et al., 2001; Wu
and Yang, 2002H20850 that generally require a substantial
amount of empirical input. Recently, Grimme and co-
workers systematically tabulated C
6
coefficients for us-
age in various GGA functionals. They employed them
for studies of supramolecular host guest systems H20849Parac
et al., 2005H20850, for organic reactions involving anthracene,
for supramolecular aggregates of bio-organic com-
pounds H20849Grimme et al., 2007H20850, and even fullerenes and
graphene sheets. Ortmann et al. H208492005H20850 used the same
C
6
/r
6
approach to study the adsorption of a DNA base
on graphite, as well as solid state properties H20849Ortmann et
al., 2006H20850. While they do find significant improvement
for isolated dimers or surface adsorption, condensed
phase properties such as the lattice constants and the
bulk moduli of Ne and Ar crystals show no improve-
ment at all. This is discouraging because it indicates an
intrinsic limit to the applicability of the C
6
correction
when dealing with all condensed phase systems. One of
the reasons for the failure to improve upon crystal prop-
erties is that many-body contributions to the vdW-Ld
forces themselves in the condensed phase are not ac-
counted for correctly here H20849Tkatchenko and von Lilien-
feld, 2008H20850. Yet, because of their low computational re-
quirements, methods of this type are used by many
groups.
The C
6
coefficients can be determined from atomic
static polarizabilities H20849Parsegian, 2005H20850, on-the-fly for at-
oms in molecules using scaled atomic volumes and po-
larizabilities H20849Tkatchenko and Scheffler, 2009H20850, from
electronic excitations H20849Marques et al., 2007H20850, or from the
exchange-correlation hole as proposed by Becke and
Johnson H208492005H20850. The latter method has been applied as
the dispersion energy part of a density functional H20849Becke
and Johnson, 2005; Johnson and Becke, 2006H20850.Ithas
some similarities to DFT-dampedC
6
, but with less semi-
empirical input, and more importantly with a carefully
reasoned, although heuristic argument for its validity.
This method requires the monomer static polarizabilities
as input, much like the early asymptotic functionals that
either required these polarizabilities H20849Hult et al., 1998H20850 or
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a cutoff that determined them H20849Andersson et al., 1996H20850,
the C
6
coefficient, and eventually higher-order C?s. How-
ever, instead of the multiparameter damping functions
used in DFT-dampedC
6
type theories, it was found that
a single universal parameter could be used.
5. Challenges and opportunities
The encouraging progress of the past several years
only makes it clearer where more work must be done.
We are still learning the importance of shape and topol-
ogy. We are learning the consequences of material prop-
erties, especially as they are modified by temperature,
pressure, defects, and impurities. On the longest scales,
materials can be approximated as continua with empha-
sis on the consequences of shape and orientation. On
the shortest scales, we must recognize atoms and mol-
ecules. Here the need is for accurate ab initio ap-
proaches and density functional methods. On intermedi-
ate scales, we need effective potentials that capture the
crossover between the atomic and the continuum and
that can enable efficient and accurate numerical simula-
tions.
Although the foundations of the modern day con-
tinuum models were laid over 50 years ago, many still
prefer to employ and to speak in terms of the even ear-
lier H208491930sH20850 picture that made little conceptual distinc-
tion between gasses and condensed materials. Much of
this resistance to embracing the Lifshitz or Casimir for-
mulations comes from the perception that they are too
unwieldy or difficult to use. Paradoxically modern for-
mulations are in many ways simpler than the approxi-
mate earlier methods. Given recent attempts to make
the modern theory accessible, it might be time for all to
enjoy modern thinking and the possibility of linking
force computation to practical spectroscopy.
Here there is another tension between tradition and
modern practice. Biologists and chemists typically think
about optical properties in the visible and infrared re-
gions that are useful for material characterization. As it
happens, dispersion forces are often dominated by opti-
cal properties at deep UV frequencies. When difficulties
in obtaining full spectral optical properties are encoun-
tered, the temptation has often been to give up and sim-
ply approximate or ignore what are evasively imagined
to be ?weak? interactions. Eliminating this barrier be-
tween solid electrodynamics and practice in other fields
would liberate creative thinking.
Because of the speed, ease, and low cost by which
information can be generated, ab initio codes may pro-
vide provisional spectra. But at the atomic level, ab ini-
tio calculations are still limited to relatively small sys-
tems. The long range of the vdW-Ld interaction requires
that many atoms be considered together. Large biologi-
cal systems, such as interacting proteins and DNA and
systems of like size found in many fields of physics and
chemistry, still remain out of reach.
We need ways to handle metals and semimetals,
graphene, and nanotubes. An important issue is that the
nonretarded asymptotic forms differ from the integral
inverse power laws in certain geometries of reduced di-
mensionality H20851see, e.g., Barash and Notysh H208491988H20850; Bos-
tr?m and Sernelius H208492000H20850; Dobson et al. H208492006H20850H20852. The
long range limits of the vdW interaction between infinite
metallic fragments of reduced dimensionality have long
range tails that decay more slowly than expected from
r
?6
summation. Polarizabilities of fragments may be-
come singular for a small frequency and wave vector.
The possibilities are particularly intriguing with
graphene layers, which can be weak metals in some of
their forms.
A summary list of needs and opportunities working
with electrodynamic forces includes the following: a
common language between the continuum and atomistic
communities; procedures to interpolate between the
continuum and atomistic regimes, or at least to identify
the distance limits when each formulation and its as-
sumptions break down; efficient and reliable analytical
and numerical methods for computing forces H20849and
torquesH20850 between objects of arbitrary shape and mate-
rial; accounting for anisotropy in material properties of
nanotubes and other objects; alternative methods for
electronic structure calculations incorporating disper-
sion forces; interaction potentials for complex biomol-
ecules such as DNA; interactions of biopolymers with
substrates; temperature or pressure dependence of force
and associated Hamaker coefficients; wetting on pat-
terned substrates and more complex geometries; ther-
modynamic ?buoyancy? to account for the properties of
complex media; dynamic Casimir phenomena such as
dissipation between mutually moving plates; multibody
and/or repulsive vdW-Ld effects for ab initio codes; and
vdW-Ld functionals for DFT codes that can coexist with
methods for solving cavitation and solvation energies.
B. Electrostatic interactions
Not only for their inverse-square-power reach but also
for their dielectric-breakdown strengths, the fields
around the simplest ions frustrate us. Looking back to
the triumphs of the 1920s that gave some idea of ion
?activity? H20849Debye, HuckelH20850 and self-energy H20849BornH20850,we
realize how little we have advanced beyond that pio-
neering decade. Work in the decade that followed H20849On-
sager, Kirkwood, and LangmuirH20850 taught us when and
when not to use the simple ideas, while the 1940s and
1950s were a time when these ideas entered common
practice with their limitations progressively forgotten.
Now, the founding ideas of ionic self-energies and
electrostatic double layers permeate most current think-
ing so strongly as to trap us from learning precisely what
we need to learn on the nanoscale where electric fields
are strongest and where the structure of media dominate
their dielectric response. The most facile common prac-
tice treats liquid water on the molecular scale as a con-
tinuum, little different from the dielectric material ex-
amined between the plates of a macroscopic capacitor;
frequently imagines ions as featureless charges; and
sometimes assumes that spatial averages are the same as
time averages H20849Ben-Yaakov et al., 2009H20850. There are clear
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exceptions of which we consider, but these exceptions
too often tend to improve one feature of traditional pic-
tures without concomitant correction of others. There
do exist cases where continuum electrostatics can be a
good approximation, for example, phenomena that are
dominated by averages over many molecules.
As is usual, our best guide to properly formulate these
additional features of electrostatic interactions on the
nanolevel is the experiment itself. The direct measure-
ments of interactions between charged macroions in
ionic solutions, be it with the surface force apparatus
H20849Israelachvili and Adams, 1978H20850, the osmotic stress tech-
nique H20849Parsegian et al., 1986H20850, or the atomic force micros-
copy H20849Munday and Capasso, 2007H20850, provide us with the
details of the measured electrostatic interactions that
cannot be captured by the traditional point of view
found in the classical literature H20849Verwey and Overbeek,
1948; Derjaguin, Churaev, and Miller, 1987; Evans and
Wennerstr?m, 1999; Israelachvili, 2006H20850.
1. Electrostatics in equilibrium statistical mechanics:
Electrostatic double layers
Much of the current understanding of electrostatics is
based on mean-field Poisson-Boltzmann H20849PBH20850 thinking.
When the PB approximation does not agree with experi-
ments it is still common to patch up the PB theory by
introducing various devices rather than questioning its
basis. There are cases where the PB approximation
works, e.g., for dilute univalent aqueous electrolytes and
weak fields far from charged surfaces H20849Hunter, 2001H20850, but
for other systems patched up PB approaches more often
hinder rather than advance understanding. Where to
look for next steps?
The most convenient place is theory itself. For ex-
ample, effects of many-body correlations were surprising
when they were initially found, but suggested possibili-
ties of like-charge attractions H20849Oosawa, 1971; Guldbrand
et al., 1984; Kjellander and Marcelja, 1984H20850 and effective
charge reversals H20849Lozada-Cassou et al., 1982; Valleau
and Torrie, 1982; Outhwaite and Bhuiyan, 1983; Ennis et
al., 1996H20850. The thought is that ion-ion correlations in di-
valent and multivalent solutions contribute to these
kinds of effects in real systems and sometimes cause
them H20849even in the absence of specific ion adsorptionH20850
H20849Netz, 2001; Moreira and Netz, 2002H20850. Nonzero effective
charges for electroneutral particles, e.g., a hard sphere
or surface in asymmetric electrolytes, is another ex-
ample. These are cases where solution composition de-
termines even qualitative features of the interactions.
The importance of ion-ion correlation effects between
multivalent counterions for the appearance of attrac-
tions between equally charged particles has been recog-
nized in many kinds of systems, although it is still taking
a long time for this knowledge to spread H20849Boroudjerdi et
al., 2005; Naji et al., 2005H20850. Early examples include DNA
H20849Guldbrand et al., 1986H20850, lamellar surfactant phases
H20849Wennerstr?m et al., 1991H20850, clay minerals H20849Kjellander et
al., 1988H20850 and mica surfaces H20849K?kicheff et al., 1993H20850.An
example of practical value is the cohesion of cement
paste H20849J?nsson et al., 2005; Labbez et al., 2007H20850.
Better than theory is the recognition of measured
forces between charged materials. Thus we keep clearest
in mind lessons from nature not from computers. To il-
lustrate we focus on the properties of DNA in electro-
lyte solution as a model system.
a. Model system: DNA-DNA interactions in charged electrolytes
In the simplest limits, predictions of the PB theory for
interactions between charged macroions in electrolyte
solutions of univalent salts conform to osmotic stress
measurements on ordered DNA arrays H20849Strey et al.,
1998; Podgornik et al., 1998H20850. With some adjustment for
assumed charge density on the molecule, but partly com-
pensated by bound ions, there is near quantitative agree-
ment between theory and experiment H20849Strey et al., 1998H20850.
This agreement holds at distances greater than a Debye
length between molecules. At closer separations, there
are qualitative deviations from the simplest electrostatic
double-layer theories of charged-rod repulsions. The
sources of these deviations are likely due in part to the
nonuniform charge of the molecule and to powerful sol-
vation forces H20849Leikin et al., 1993H20850. The latter can be de-
scribed as structural forces or as structure-dependent di-
electric response. The major feature is a failure of
classical PB theory, a failure that occurs precisely in the
important nanometer range that is the focus of our re-
view and that is relevant to macromolecular assembly.
In salt solutions containing at least higher valence
counterions, such as Mn
2+
,CoH20849NH
3
H20850
6
3+
, or various
polyamines, PB predictions lose all agreement with ex-
periment. Not only does the PB theory give the wrong
numerical values for the strength of the electrostatic in-
teractions but also, and more importantly, misses their
sign. Measurements point to the existence of attractions
that do not follow mean-field electrostatic double-layer
theory H20849Rau and Parsegian, 1992H20850. This attraction is de-
duced from the shape of the osmotic pressure as a func-
tion of density of DNA as well as from direct magnetic-
tweezer measurements of the attraction that condenses
the double helices H20849Todd and Rau, 2008; Todd et al.,
2008H20850. The relevant measurements resemble pressure
versus volume isotherms for gasses and liquids. Rather
than attractive van der Waals interactions between gas
molecules, for DNA the drive comes from the polyva-
lent counterion. For sufficiently large concentrations of
CoH20849NH
3
H20850
6
3+
, the DNA array spontaneously precipitates
or condenses into an ordered high-density phase. One
thus concludes that the polyvalent counterion confers
attractions on nominally equally charged DNA mol-
ecules H20849Podgornik et al., 2008H20850.
van der Waals interactions are much too small to ac-
count for the strong attractions seen with the addition of
polyvalent counterions. Electrostatics on a mean-field
level in cases of univalent counterions cannot give at-
tractions H20849Andelman, 1995H20850. In the presence of polyva-
lent counterions, one must go beyond the PB approxi-
mation and include the effects of ion-ion correlations as
in simulations H20849Guldbrand et al., 1984H20850, integral equation
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theory H20849Kjellander and Marcelja, 1984H20850, or strong-
coupling and strong-correlation approximations H20849Gros-
berg et al., 2002; Boroudjerdi et al., 2005; Naji et al.,
2005H20850. However, this may not be sufficient unless hydra-
tion effects and the details of the DNA structure are
included. One way is to describe DNA attraction or re-
pulsion at nanometer separations with an ad hoc order-
parameter formalism built on perturbation of hydration
forces H20849Leikin et al., 1993H20850. There is clear deviation from
the simplest electrostatic double-layer models routinely
used in current theorizing. Inclusion of helical structure
brings out some features that might explain certain fac-
ets of measurement as indicated by Kornyshev et al.
H208492007H20850.
b. Electrostatics with surfaces and interfaces
Stabilization of colloidal dispersions has been ob-
served to be due to the appearance of substantial effec-
tive charges for weakly charged macroparticles from cor-
relation effects among smaller charged particles H20849Liu
and Luijten, 2005; Martinez et al., 2005H20850. Clustering and
phase separation in systems with highly and equally
charged colloidal particles can be created by counterion-
counterion correlations H20849Hribar and Vlachy, 2000; Re?~i~
and Linse, 2001; Linse, 2005; Hynninen and Panagioto-
poulos, 2007H20850. In some cases, various forms of depletion
interactions can play a decisive role; they can for in-
stance be caused by electrostatic ion-ion correlation or
excluded-volume effects H20849Kanduc, Dobnikar, and
Podgornik, 2009H20850.
A common kind of depletion interaction H20849Verma et al.,
1998; Roth et al., 2000H20850 is an attraction between large
colloidal particles that are in a suspension containing
small colloidal particles, e.g., polymers H20849Schlesener et al.,
2001H20850 or disklike platelets H20849Harnau and Dietrich, 2004H20850.
The origin of this attraction is in the fact that small col-
loidal particles and large colloidal particles cannot over-
lap. Thus small colloidal particles are excluded from the
depletion region in the vicinity of large colloidal par-
ticles. Furthermore, if the depletion regions of two large
colloidal particles overlap, there is an osmotic pressure
pushing the particles together, creating an effective at-
tractive interaction potential H20849Triantafillou and Kamien,
1999H20850. The range of the depletion interaction is given by
the hard core radius of the small colloidal particles or
equivalently by the polymer radius of gyration. One is
thus in a position to vary the range H20849polymer molecular
weightH20850 or the strength H20849the polymer concentrationH20850 of
an attractive depletion potential between large colloidal
particles.
The study of polar liquids and electric double layers
near one surface and interactions between two surfaces
H20849e.g., double-layer interactionsH20850 are closely related to the
study of properties of electrolytes and polar liquids in
pores and other confined geometries. All are special
cases of inhomogeneous fluids. In the presence of sur-
faces, electrostatic correlations, and thereby the electric
screening, are profoundly changed compared to bulk
electrolytes H20851power-law screening along a surface H20849Jan-
covici, 1982H20850, exponential screening perpendicular to itH20852.
The treatment of nonlocal dielectric response, dielectric
saturation, and other effects on solvent due to electro-
static fields from surfaces, molecules, and other particles
is a long-standing issue H20849Bopp et al., 1998; Yeh and
Berkowitz, 1999; Ballenegger and Hansen, 2005; Balle-
negger et al., 2006H20850.
The complexity of electrostatic interactions and their
coupling to other modes of interactions is illustrated by
the interdependence of dispersion and electrostatic in-
teractions H20849Ninham and Yaminsky, 1997; Kunz et al.,
2004; Tavares et al., 2004; Wernersson and Kjellander,
2007H20850. Ion-macromolecule dispersion interactions can
explain some of the ion specificity seen in effects of elec-
trolytes on macromolecular interactions. vdW-Ld inter-
actions between the constituent molecules of electro-
lytes affect the screening of electrostatic interactions in
nontrivial ways, for instance, by changing the exponen-
tial screening in bulk to a power-law screening in both
quantum H20849Brydges and Martin, 1999H20850 and classical H20849Kjel-
lander and Forsberg, 2005H20850 statistical mechanics theory.
Especially with the added consideration of related inter-
actions, such as ion-solvent induced-dipole interactions,
more attention should be spent on the correct treatment
of the screening by electrolytes of the static part of the
van der Waals interaction H20849Mahanty and Ninham, 1976;
Parsegian, 2005H20850, which is relevant in some systems like
aqueous electrolytes common in biology. Likewise, other
static polarization effects like the ion-solvent induced-
dipole interactions are also screened in electrolytes,
while the high-frequency part of the dispersion interac-
tion is not.
It was recently realized that in electrolytes the anisot-
ropy of the electrostatic potential from a molecule ex-
tends to the far field region H20849Rowan et al., 2000; Hoff-
mann et al., 2004H20850. The full directional dependence of
the electrostatic potential from a charged or uncharged
molecule in electrolytes remains in the longest range tail
H20849i.e., from all multipole momentsH20850H20849Ramirez and Kjel-
lander, 2006H20850. In particular, the range of the potential
from an ion and that from an electroneutral polar par-
ticle is the same. This is contrary to the case in vacuum
or pure polar liquids, where the potential from a single
charge is longer ranged than that from a dipole, which in
turn is more long ranged than from a quadrupole, etc.
The orientational dependence of the electrostatic inter-
action between two molecules in electrolytes is therefore
complex even at large distances, and the consequences
of this complexity must be further explored H20849Trizac et al.,
2002; Agra et al., 2004H20850.
2. Electrostatics in density functional theory
Following the Born-Oppenheimer approximation, we
refer here to all properties of matter for which the elec-
trons may be treated as in their ground state. Nuclei or
ions may be treated as classical particles in whose poten-
tial the electrons move. In this approximation then ions
are in no sense in their ground state and can redistribute
in response to gradients in temperature, stress, electric
fields, chemical potential, etc.
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The range of validity of the Born-Oppenheimer ap-
proximation includes most of the chemistry as well as
equilibrium and nonequilibrium thermodynamics of ma-
terials and much of biology. It includes all interatomic
forces between ions, thermally activated atoms, in elec-
trolytes, metals, insulators and semiconductors, van der
Waals solids, liquids and gases, while excluding fast-
moving ions in radiation damage events, radiative pro-
cesses, such as luminescence, and electron transport in
which quantum electrodynamics or electromagnetism
start to play a role. It matches the range of validity of
density functional theory H20849Hohenberg and Kohn, 1964;
Kohn and Sham, 1965H20850, which is a unifying principle, the
basis of many practical schemes of calculation, and a
source of insights into the long range electrostatic inter-
actions H20849Finnis, 2003H20850.
The LDA or GGA paradigms have led to codes that
are fast compared to any tractable quantum chemical
techniques that might be more accurate, such as those
involving multiconfiguration wave functions, or quan-
tum Monte Carlo. Nevertheless, it should be recognized
that the inherent absolute errors in standard DFT or
LDA may be of order 0.1 eV H20849H110111200 KH20850 per bond or
more in some cases, which is too large for many ques-
tions in chemistry and particularly biology.
3. Challenges and opportunities
As with well-established vdW-Ld formulations, there
is sometimes a perception that there is little else funda-
mental to discover in electrostatics and that at least by
computer simulation we can solve every system of inter-
est. Although the present practice of electrostatics rests
on some solid long-standing foundations, both the quan-
tum mechanical and classical statistical mechanical as-
pects require enormous improvement.
Within its framework, density functional theory al-
ready provides a systematic way to obtain all kinds of all
LRIs in condensed materials. Besides the difficulty of
applying the procedure, it requires that the Born-
Oppenheimer approximation be valid and that electrons
be close to their ground state. The recent introduction of
nonlocal functionals by Langreth and co-workers H20849see
Sec. II.A.4.aH20850 promised to extend the practical scope of
DFT to situations in which dispersion forces are impor-
tant, by including nonlocal contributions to correlation
energy. At least for some systems this might give a useful
approximation to the dispersion interaction and super-
sede the addition of two-body potentials. However,
there are significant impediments to progress. In metals,
long ranged electrostatic forces manifest themselves as
anomalies in phonon spectra, and even in elastic moduli.
Charged point defects in insulators require long length
and time scales to simulate the equilibrium distributions,
together with the detailed electronic structure calcula-
tions needed to obtain their formation and segregation
energies.
In the domain of electrostatic double layers and
charge interactions in general, with mobile ions whose
distributions depend on the very potentials they them-
selves create, the situation is daunting. Particularly on
the nanometer scale, when continuum electrostatics is
inappropriate, the need to incorporate solvent structure
as well as ionic personality immediately removes us
from the domain of traditional theories. There is a need
to look at real experimental data that probe situations of
ignorance. Elaborate simulations alone do not suffice.
Simulations need to be validated against measured
forces before they can be trusted to teach us where di-
rect experimental data do not exist.
A significant body of data already exists on directly
measured forces over the few nanometers approaching
molecular contact. The extent to which the simplest
theories must be recast to recognize molecular structure
and structural forces is still completely open. Without
accurate representation of measured forces, simulations
are in danger of being hypothetical exercises. One can
obtain exact statistical mechanical results in certain lim-
its H20849e.g., low densities, high or low temperatures, or large
distancesH20850. Such limits do provide estimates from which
important aspects of the systems might be recognized
H20849Boroudjerdi et al., 2005; Naji et al., 2005H20850.
Theoretical problems for nanoscale systems often in-
volve several simultaneously important length scales.
One strategy might be to integrate simulations with re-
sults from formal theory in ways that enhance applica-
bility of simulations. For example, there exist formal re-
sults for the long range tails of various distribution
functions that rely on the behavior at short range
H20849Gonz?lez-Mozuelos and Bagatella-Flores, 2000; Kjel-
lander and Ramirez, 2005H20850. There have been attempts to
construct effective interaction potentials where some
molecular degrees of freedom are included implicitly
H20849Lyubartsev and Laaksonen, 1995, 2004; Ayton et al.,
2007; T?th, 2007H20850. The idea is to treat several length
scales at the same time; for example, the small size of
the solvent molecules and dissolved salt ions, the appre-
ciably larger size of the macroparticles or aggregates,
and the average separation between particles H20849Lobaskin
et al., 2001H20850. Even when the main interest concerns phe-
nomena on the large scale, the effects of the molecular
details somehow have to be taken into account without
losing their important features. ?First principles? simu-
lations are still far too expensive and idealized to treat
most problems in soft matter science, even for simple
aqueous systems H20851see Schwegler H208492007H20850H20852.
Ideally, one should first eliminate the possibility that
the deviations, too often dismissed, occur because the
theory used is too approximate. One example of this is
the PB treatment of primitive model electrolyte systems.
When the true properties of the primitive model were
evaluated with accurate statistical mechanical methods,
many new features were found beyond PB approxima-
tion H20849Ben-Yaakov et al., 2009; Kanduc, Naji, Jho, Pincus,
and Podgornik, 2009H20850. Several of the observed devia-
tions between experiments and the PB prediction were
due to the mathematical approximations implicit in the
PB approach and not due to the underlying model. Even
worse, comparisons of any model with measurements
continue to be avoided. Until measurements are given
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
precedence over models, our idealized pictures of elec-
trostatics forces will continue to cripple productive
thinking.
C. Polar interactions
1. Motivation and recent advances
Although not as long ranged and clearly defined as
electrostatic and vdW-Ld interactions, polar H20851or some-
times referred to as acid-base H20849ABH20850H20852 interactions play
important roles in, for example, chemical reaction, ad-
hesion, triboelectrification, and colloidal interactions.
There are several ways to speak of them: donor-acceptor
interactions H20849Gutmann, 1978H20850, hard-soft AB H20849Parr and
Pearson, 1983H20850, and through the equations of Drago and
Wayland H208491965H20850. What can be learned from these various
theories is that a polar, or AB, interaction is composed
of Coulombic, covalent, and charge-transfer interac-
tions.
The first two H20849electrostatic and covalentH20850 are often ap-
proximately expressed as H20849Hudson and Klopman, 1967H20850
H9004E =?
q
1
q
2
R
12
H9255
+2
H20858
m
occ
H20858
n
unocc
H20875
H20849c
1
m
H20850
2
H20849c
2
n
H20850
2
H9252
2
E
m
*
? E
n
*
H20876
. H2084911H20850
The first term is an idealized Coulombic interaction; and
the second term refers to outer orbital interactions. In
Eq. H2084911H20850, m and n denote the donor and acceptor orbit-
als; c
1
m
and c
2
n
are the coefficients of atomic orbitals par-
ticipating in interaction; and H9252 is a resonance integral.
E
m
*
and E
n
*
are the energy of the donor and acceptor
orbitals, equivalent to the energies of their highest occu-
pied molecular orbital H20849HOMOH20850 and the lowest unoccu-
pied molecular orbital H20849LUMOH20850. Drago?s empirical ver-
sion of Eq. H2084911H20850 describes the heat of formation H20849?H9004HH20850
of AB complex:
? H9004H = E
A
E
B
+ C
A
C
B
, H2084912H20850
E and C are the electrostatic and covalent interaction
constants, respectively. Drago?s equation was used by
Fowkes H208491963H20850 to study the effects of AB interactions on
wetting, adsorption, and adhesion.
Among their many idealizations, these ancient rela-
tions, Eqs. H2084911H20850 and H2084912H20850, do not recognize the important
charge-transfer interaction for which Parr and Pear-
sons?s HSAB H20849hard-soft ABH20850 principle has been used
with two important parameters: absolute electronegativ-
ity H20849H9273H20850 and absolute hardness H20849H9257H20850. Using density func-
tional theory, Parr and Pearson showed that the absolute
electronegativity is a chemical potential of electrons,
and hardness is the derivative of this chemical potential
with respect to the number of electrons. Formally,
H9273 =?
H20875
H11509E
H11509N
H20876
Z
=
1
2
H20849I + AH20850H2084913H20850
and
H9257 =
1
2
H20875
H11509
2
E
H11509N
2
H20876
Z
=
1
2
H20849I ? AH20850. H2084914H20850
I is the ionization potential and A is the electron affinity
of a species. Parr and Pearson also estimated the change
H9004E of the electronic energy associated with charge
transfer from a donor to an acceptor as
H9004E =?
H20849H9273
A
0
? H9273
B
0
H20850
2
4H20849H9257
A
+ H9257
B
H20850.
H2084915H20850
Because the softness of a species is a chemical poten-
tial of electrons, H9273
0
corresponds to the standard chemical
state. According to the HSAB principle, strong AB in-
teractions result when the gap between the HOMO of
the donor and the LUMO of the acceptor is very low,
leading to a ?soft? AB interaction. If the above gap is
large, there will be little AB interaction via charge-
transfer complexation so that the primary AB interac-
tion is due to electrostatics.
According to Derjaguin et al. H208491973H20850, as electrons are
transferred across an interface from the donor to the
acceptor sites, an electric potential difference H20851H9004VH20849nH20850H20852
develops depending upon the numbers of electrons
transferred, or the AB pairs H20849nH20850 formed, across the inter-
face. By minimizing the total free energy of the system
with respect to n,
n
N ? n
= exp
H20875
? H9004E ? eH9004VH20849nH20850 ? enH11509VH20849nH20850/H11509n
kT
H20876
. H2084916H20850
Treated as an experimentally determined parameter,
H9004E should be exactly the same as in Eq. H2084915H20850. Another
equation relating H9004VH20849nH20850 and n is, however, necessary to
obtain the optimum number of AB pair and the result-
ing electrical potential across an interface. Equation H2084915H20850
in conjunction with Eq. H2084916H20850 form the basis for a well-
known phenomenon?triboelectrification?on the basis
of AB interaction across an interface. Fowkes H208491953H20850
first proposed that the interfacial interaction can be de-
composed into two terms, one arising from the disper-
sion forces and the other arising from the AB interac-
tion. The treatment of Van Oss et al. H208491987, 1988H20850 is
similar to Fowkes in that the interfacial energy is ex-
pressed as H9253
TOT
=H9253
LW
+H9253
AB
. While the vdW-Ld compo-
nent of the adhesive interaction follows the geometric
combining rule H20849i.e., W
12
LW
=2
H20881
H9253
1
LW
H9253
2
LW
H20850, the AB compo-
nent does not. The AB component of the adhesive inter-
action is expressed as
W
12
AB
=2
H20881
H9253
1
+
H9253
2
?
+2
H20881
H9253
1
?
H9253
2
+
. H2084917H20850
The geometric-mean combining rule used is a good ap-
proximation for describing surface properties of media
where nonpolar interactions dominate and is based on
the assumption that the principal dielectric absorption
frequencies of the media are similar H20849Israelachvili, 2006H20850.
Thus, the AB component of interfacial energy H9253
AB
comprises two nonadditive parameters, an electron-
acceptor surface tension parameter H20849H9253
+
H20850 and an electron-
donor surface tension parameter H20849H9253
?
H20850. The total AB con-
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
tribution to the surface tension is given by H9253
AB
=2
H20881
H9253
+
H9253
?
. The total interfacial tension between con-
densed phases i and j is described by
H9253
ij
AB
=2H20849
H20881
H9253
i
+
H9253
i
?
+
H20881
H9253
j
+
H9253
j
?
?
H20881
H9253
i
+
H9253
j
?
?
H20881
H9253
i
?
H9253
j
+
H20850H2084918H20850
which is not a geometric combining rule, but rather ex-
presses the doubly asymmetric interaction between two
different materials resulting from the fact that a material
can be a good electron donor, electron acceptor, neither
H20849apolarH20850, or both H20849bipolarH20850. This theory predicts that mo-
nopolar H20849predominantly acidic or basicH20850 materials will
strongly interact with bipolar materials or with monopo-
lar materials of the opposite type, and if this adhesive
interaction is strong enough, the interfacial interaction
between two condensed phases can become negative.
Based on the Dupr? equation, which can be applied for
both polar and nonpolar materials, the polar interac-
tions between two solid materials 1 and 2 in a liquid
medium 3 using the interfacial energy given by Eq. H208494H20850,is
H9004G
132
AB
= H9253
12
? H9253
13
? H9253
23
=2H20851H20849
H20881
H9253
1
+
?
H20881
H9253
2
+
H20850H20849
H20881
H9253
1
?
?
H20881
H9253
2
?
H20850
? H20849
H20881
H9253
1
+
?
H20881
H9253
3
+
H20850H20849
H20881
H9253
1
?
?
H20881
H9253
3
?
H20850
? H20849
H20881
H9253
2
+
?
H20881
H9253
3
+
H20850H20849
H20881
H9253
2
?
?
H20881
H9253
3
?
H20850H20852. H2084919H20850
For two identical or different polar materials separated
by a solvent, it is then possible for the interaction to be
repulsive or attractive.
These concepts have been applied to a variety of phe-
nomena in condensed phases. In numerous liquids and
polymers, the quantitative interpretation of surface ten-
sions is incomplete without inclusion of AB interaction
energies as shown in Fig. 5. The same is true of the
solubility of various polymers in solvents. AB interac-
tions can also generate osmotic pressure more than a
hundredfold greater than that due to van?t Hoff. The AB
approach can give clear indications about the nature of
complex surfaces, for example, the preferential segrega-
tion of certain groups on the surface of solid copolymers
H20849Ad?o et al., 1999H20850. Parameters have also been deter-
mined for biological surfaces such as skin H20849Mavon et al.,
1998H20850 and bacterial cells H20849Ong et al., 1999H20850. Empirical
methods for estimating the strength of AB interactions
are summarized by Chaudhury H208491996H20850. There nonethe-
less remains a lack of consensus regarding the most ap-
propriate approach to quantification of the strength of
the AB interaction H20849Correia et al., 1997; Douillard, 1997;
Lee, 1998H20850.
AB interactions are therefore important as part of the
LRIs ?tool kit? for understanding physical behavior,
and, in the future, for the manipulation and design of
new materials and devices. One recent example con-
cerns the use of AB interactions to produce self-
organized devices such as lithium-ion batteries H20849Fig. 6H20850.
Cho et al. H208492007H20850 proposed a general approach to the
direct formation of bipolar devices from heterogeneous
colloids in which attractive and repulsive interactions
could be combined to produce a network of one mate-
rial H20849e.g., an anodeH20850 that is everywhere separated from a
network of a second H20849e.g., a cathodeH20850 An ensuing search
for suitable combinations of conductive device materials
and solvents using atomic force microscopy H20849AFMH20850 mea-
surements showed first that inclusion of AB interactions
was essential to understanding experimental data for the
inorganic compounds studied, and second led to the suc-
cessful identification of several electronically conductive
materials H20851carbon, indium tin oxide H20849ITOH20850, LiCoO
2
H20852 be-
tween which repulsive AB interactions are obtained in
an appropriately chosen liquid medium and in the ab-
sence of electrostatics. As a result, a colloidal-scale self-
organized lithium rechargeable battery based on
graphite-LiCoO
2
was demonstrated H20849Fig. 6H20850.
2. Challenges and opportunities
Given the vintage of many ideas still being used as
well as limited experimental information to allow prac-
tical application, it is clear that this subject needs refor-
mulation and quantitative measurement. There have
been few studies attempting to generalize the methods
for estimating AB interactions based on molecular con-
FIG. 5. Role of acid-base interaction in polymer adsorption.
The adsorption of a basic polymer, poly-methymethacrylate
H20849PMMAH20850, onto an acidic silica is maximal when the adsorption
occurs in a neutral solvent, carbon tetrachloride. However,
when the adsorption occurs in either acidic or basic solvent,
adsorption is reduced as the solvent interacts either with
PMMA or silica. From Fowkes, 1983.
Percolating Network of
Material 1
Material 1: anode
storage compound
Material 3: cathode
storage compound
A
121
>0
(attraction)
A
123
<0
BipolarJunction
Material 2:
electrolyte/
binder
Percolating Network
of Material 3
A
323
>0
(attraction)
(repulsion)
Load
/
so
ur
ce
FIG. 6. H20849Color onlineH20850 Colloidal-scale self-organizing lithium-
ion battery concept, demonstrated in graphite?LiCoO
2
system,
making use of acid or base forces for junction formation and
particle assembly.
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
cepts of hardness and softness in acids and bases to
other classes of materials such as metals and semicon-
ductors H20849Cain et al., 1969; Ho et al., 1991H20850. While the
current theories of AB interactions can reasonably pre-
dict whether a specific AB interaction will prevail, the
ability to predict quantitatively the strength of AB inter-
action is poor. Our belief is that they remain conceptu-
ally important.
Experimentally, few data exist for classes of materials
besides the most simple liquids and polymers. A major
difference from Hamaker coefficients is that a system-
atic cataloging of more coefficients is straightforward
compared with similar systematic cataloging of AB in-
teractions. This is an area ripe for scientific advance.
While AB interactions have the potential to be a key
tool in material and device design, improved fundamen-
tal understanding is essential for their use in the synthe-
sis of materials as well as in the production of self-
organizing junctions, subassemblies, devices, and other
applications. Metals, semiconductors, insulators, and
biological materials are all likely components of such
?engineered? devices.
So far, only indirect methods such as wetting and solu-
bility have been used to characterize AB interactions. It
is time that more modern methods be used to measure
interactions directly and at nanoscale spatial resolution,
methods such as scanning probes and force microscopy
with chemically modified tips.
Many more measurements are needed to connect po-
lar behavior with fundamental properties. There are few
monopolar materials of the electron-acceptor type H20849Lee
and Sigmund, 2002H20850. One important class of such com-
pounds may be fluoroalcohols and fluoroalcohol-bearing
polymers H20849see Fig. 7H20850. These materials have been shown
experimentally to be hydrogen bond donors only, with
no capability to act as hydrogen acceptors. It may be
that ab initio codes can enhance or extend needed mea-
surement. If first-principles calculations were able to
agree with currently available experimental results, it
might be possible to enlarge the list of well-
characterized materials. This enlargement would be use-
ful for chemists, biologists, and others for material selec-
tion and experimental design purposes.
III. INSTRUCTIVE SYSTEMS
Having identified fundamental forces, how do we now
learn about them? In Sec. III.A, we discuss computation
of optical spectra and Hamaker coefficients of complex
biomolecular systems including single-wall carbon nano-
tubes H20849SWCNTH20850 and B-DNA. We then consider several
examples of macromolecules and polyelectrolytes:
anionic-cationic polyelectrolyte complexes, hydration in-
teractions with ionic specificity, extraction or separation
in phase transfer reactions, and surfactant-decorated in-
terfaces. Impurity-based quasiliquid films and space
charges at solid interfaces are discussed in Sec. III.B;
these two phenomena often intermingle at the grain
boundaries and surfaces of many structural and func-
tional ceramics. Section III.C presents instructive sys-
tems related to aqueous solutions and suspensions: pre-
melting and related phenomena in ice and water, ion
hydration, oxide/electrolyte interfaces, colloidal suspen-
sions, and SWCNT hybrids. Further aspects are deferred
to Sec. IV for practical implications and technological
importance.
A. Atoms and molecules
1. Optical spectra and Lifshitz theory for complex
biomolecular systems
For systems of high complexity and fragility, such as
biomolecular membranes or proteins, experimental
measurement of optical spectra using vacuum ultraviolet
spectroscopy is not yet possible. Although theoretical
calculations of optical spectra for such systems are
daunting, new computational methodologies and theory
finally make such calculations feasible. Calculation of
optical properties via ab initio theories has been dis-
cussed in Sec. II.A.3. Here we present the example of
SWCNTs.
As demonstrated for ceramic crystals H20849Ahuja et al.,
2004H20850 and SWCNTs H20849Rajter et al., 2008H20850, calculation of
optical properties at the level of local density approxi-
mation H20849LDAH20850 of the density functional theory H20849DFTH20850 in
the random phase approximation H20849RPAH20850 seems to be ad-
equate. Higher-level theory, such as time-dependent
density functional theory H20849TDDFTH20850, or theories that in-
clude some aspects of many-body corrections or self-
interaction correction H20849SICH20850 might be sometimes neces-
sary.
In addition to LDA, DFT, and RPA, it is necessary to
make further approximations. First, for geometrical
shapes of real objects, in order to use available analytic
formulas one might describe a small-radius SWCNT as a
cylinder, a bucky ball as a sphere, graphene as an infinite
planar sheet, or biomembranes as plane-parallel blocks.
Second, in real situations where several media are in-
volved, there are additional approximations for averag-
ing or mixing the dielectric functions H20849Brosseau, 2006H20850.
Third, practical calculation of optical dielectric functions
for large complex biological systems is still computation-
ally prohibitive. Further developments of computational
CF
3
F
3
C
OH
F
F
FF
F
F
OH
F
F
FF
F
F
OH
F
F
F
F
F
F
OH
FIG. 7. Examples of fluoropolymers which may serve as a
H9253+-reference solid to be used for the analysis of monopolar
surfaces and liquids interactions.
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
codes, identified criteria of accuracy, and new computa-
tional resources are necessary.
a. Optical properties of SWCNTs
Computed Hamaker coefficients for two types of
CNTs with gold in a water medium are shown in Fig. 8.
First, look at the changes in the Hamaker coefficient
versus separation along the radial direction of the
SWCNT-water-Au substrate system H20849Rajter et al., 2007H20850.
This calculation demonstrates the relative magnitude
differences that occur because small changes in the
chirality of the SWCNTs create large changes in optical
spectra. It also highlights the importance of access to the
full spectrum. Glancing at the vdW-Ld spectra curves in
Sec. II.A.3, it might seem that the large low-energy wing
in the H208519,3,mH20852 spectrum would make the interaction
much larger than for the semiconducting H208516,5,sH20852. How-
ever, the H208516,5,sH20852 has stronger interactions in the remain-
der of the Matsubara summation in both the near- and
far-separation formulations, to create a stronger overall
vdW-Ld interaction H20849Fig. 8H20850.
The orientation dependence of the Hamaker coeffi-
cient itself is seen with anisotropic materials. In the far
limit of the anisotropic cylinder-cylinder interaction, we
see an increase of the Hamaker coefficient by a factor of
nearly 30%. Similar results were reported for the Al
2
O
3
substrate-substrate system by Knowles H208492005H20850, although
to a lesser extent because of a smaller degree of anisot-
ropy. The potential implications for design and manipu-
lation of pieces during construction of nanodevices are
real.
As we incorporate the effects of a changing internal
medium of the SWCNT core, and adding the effects of
surfactant layers, etc., additional effects will likely be-
come clear.
b. Optical properties of B-DNA
Ab initio optical properties of B-DNA and collagen
have been obtained using the DFT-based orthogonalized
linear combination of atomic orbitals H20849OLCAOH20850 method
H20849Ching, 1990H20850. In these calculations, Na ions were added
to the bare DNA model to neutralize the negatively
charged PO
4
groups from the DNA backbone. Without
the compensating counter ions, the self-consistent po-
tential in the electronic structure calculation will not
converge. One such calculation is a periodic model H20849in
the z directionH20850 with 10 CG base pairs and 20 Na coun-
terions for b-DNA. This model contains a total of 650
atoms and 2220 valence electrons. The ab initio calcula-
tion shows it to be an insulator with a band gap of about
2.5 eV. Figures 9 and 10 show the calculated total den-
sity of states H20849TDOSH20850 and the imaginary part of the
frequency-dependent dielectric function H9255H11033H20849H9275H20850 of the
model H20849Rulis, Liang, and Ching, 2009H20850.
These calculations use several advantages of the
OLCAO method. First, the local orbital basis expansion
keeps the total dimension of the Kohn-Sham equation at
a manageable level for more than thousands of atoms.
Second, the effective Gaussian representation in both
the basis function and atom-centered potential functions
facilitates the evaluation of multicenter integrals. Third,
the inclusion of optical matrix elements in the calcula-
tion for transitions up to high-unoccupied states pro-
vides the needed accuracy at both the low and high-
frequency limits. Finally, the ability to explore the
interatomic, intermolecular, and intramolecular bonding
using the concept of partial density of states H20849PDOSH20850
from different groups of atoms facilitates the interpreta-
FIG. 8. H20849Color onlineH20850 The Hamaker coefficients H20849AH20850 of
SWCNTs. For H208519,3,mH20852 and H208516,5,sH20852 SWCNTs in water as a
function of l for gold substrate. From Rajter et al., 2007.
FIG. 9. The calculated total density of states H20849TDOSH20850 of a
periodic B-DNA model with 10 CG base pairs and with 20 Na
ions as counter ions added. The calculation shows this model
has an insulating gap of about 2.0 eV.
FIG. 10. The calculated imaginary part of the dielectric func-
tion of the b-DNA model of Fig. 9.
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
tion of the calculated results. How such features affect
force computation is as yet undetermined.
It is hoped that the same approach can be used to
calculate optical spectra in other systems such as colloi-
dal suspensions, intergranular films H20849IGFsH20850 in polycrys-
talline ceramics H20849Secs. III.B.1 and IV.A.2H20850, and interfa-
cial models of different materials, which then can be
used for quantitative estimation of the long range dis-
persion forces. We still have no reliable strategy how to
include fluids and counter ions in the calculation of op-
tical properties for force computation. On a larger scale,
there is a critical need for realistic atomic-scale struc-
tural models for large biological molecules or complex
microstructures in ceramics that can be used for ab initio
electronic and optical properties calculations.
2. Hydration interaction and ionic specificity
We focus here on aspects of phase stability of surfac-
tant solutions or colloidal microcrystals when stability,
coexistence, or swelling is due to a hydration force, and
is not of immediate electrostatic origin. In such situa-
tions, the absence of an identified ?electrostatic effect?
such as a link between Debye lengths and phase limits is
due either to the absence of charge or to an effect inde-
pendent of the presence of added salt H20849Hao and Zemb,
2007H20850. It may seem paradoxical to attribute long range to
the hydration force, which can only persist for a length
where the drive for structural alignment of the solvent
around the solute can overcome the effects of Brownian
motion H20849Todd et al., 2007H20850. This force is ?long? versus
hydrogen bonding, complexation, and other nearest-
neighbor interactions considered in the chemistry of col-
loids. For good model systems, in the absence of salt, the
hydration force can be detected by applied osmotic pres-
sure as low as a few hundred Pa with typical distances
between surfactant aggregates of up to H110113nmH20849Carriere
et al., 2007H20850.AtH110111 nm, the hydration pressure can grow
to hundreds of atmospheres between planar surfaces.
Compared with the ultralong range of electrostatics in
the absence of screening, the hydration force is short
range. A combination of hydration force and electrostat-
ics is the source of several behaviors of surfactants sys-
tems that can be explained only if hydration is consid-
ered as a fundamental repulsive mechanism which can
dominate even in the absence of structural net charge:
measurements mixing anionic and cationic components,
producing colloidal aggregates of known charge, play an
instructive role here H20849Ricoul et al., 1998H20850.
The distance dependence characterizing exclusion of
small solutes from macromolecular surfaces follows the
same exponential behavior as the hydration force be-
tween macromolecules at close spacings. Similar repul-
sive forces are seen for the exclusion of nonpolar alco-
hols from highly charged DNA and of salts and small
polar solutes from hydrophobically modified cellulose
H20849Bonnet-Gonnet et al., 2001H20850. Exclusion magnitudes for
different salts follow the Hofmeister series that has long
been thought connected with water structuring H20849Todd et
al., 2008H20850.
One feature is the intriguing connection with the dis-
tributions of salts in thin liquid films on ice. The connec-
tion between hydration effects in water and the Bjerrum
defect distribution in ice has been noted before H20849Gruen
and Marcelja, 1983H20850 and is due to the structuring of wa-
ter molecules close to macroscopic surfaces. In ice this is
described by a redistribution of orientational Bjerrum
defects, whereas in water it is usually discussed within
water solvation or hydration models. In both cases, how-
ever, ion redistribution couples with hydration patterns.
Solvation of interacting macromolecular surfaces and
modulation of this solvation by cosolutes such as salts
exquisitely regulates equilibria of specific association in
chemistry and biology. Depending on whether the coso-
lute is preferentially excluded from, or attracted to, the
surfaces of the macromolecules, a cosolute can either
increase or decrease complex stability H20849Harries and Par-
segian, 2004H20850. However, the dynamic action of cosolute
on complexation is not yet understood, and there is no
way to predict which kinetic constant, the ?on rate? or
the ?off rate,? has the greater impact.
A decade ago ?molecular Coulter counting? demon-
strated that a single protein nanopore could be used to
detect polymer exchange between pore and bulk, a dem-
onstration that stimulated modern development of
nanosensors H20849Bezrukov et al., 1994H20850. This same method
also allows one to address the dynamic side of preferen-
tial solvation. Using an alpha-hemolysin nanopore as a
sensor, it is possible to follow the effect of solutes on a
simple complexation reaction at the single-molecule
level H20849Gu et al., 1999H20850. Monitoring transient obstruction
of current through a nanopore complex reveals the ki-
netics underlying the reaction equilibrium in the pres-
ence of various cosolute salts. Measurements with alpha-
hemolysin progressively blocked by cyclodextrin and
adamantane reveal changes in on rates as well as off
rates, depending on the type of salt used H20849Gurnev et al.,
2009H20850. Chloride and bromide salts mainly impact the off
rates; sulfate changes the on rate, revealing qualitatively
different dynamic action of different cosolutes H20849Harries
and Rosgen, 2008H20850.
3. Extraction, separation, and phase transfer reactions
Liquid-liquid phase transfer of metals in the form of
ions is a crucial step in re-processing nuclear fuel. Cur-
rently available technologies rely on liquid-liquid extrac-
tion.
The existence and localization of the critical point po-
sition in the phase diagram of the extracting media have
demonstrated the predominance of vdW forces, a long
range interaction in phase separation of a dispersion of
water nanodroplets covered by extracting molecules
H20849Nave et al., 2004H20850.
The extracting process involves the cations, associated
anions, and co-extracted water. All these species are
confined in a dehydrated form in the polar core of
formed micelles. Using a series of homologous acids, it
has been proven that the Hofmeister series H20849Collins,
2004H20850 of the anion involved in the co-extraction process
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
profoundly influences the efficiency and selectivity of
the extraction as well as the overall stability of the dis-
persion of reverse micelles involved in the process H20849Tes-
tard et al., 2007H20850.
4. DFT results on DNA base-pair vdW interactions
In Sec. II.A.4, we reviewed a number of recently de-
veloped methods for including the vdW-Ld interaction
in electronic DFT. Prior to these developments, DFT
had been generally successful in the description of dense
matter and isolated molecules. The newer methods are
beginning to extend this success to sparse matter, bio-
logical matter, and vdW molecular complexes, and
thereby to systems where DFT with conventional func-
tionals generally have failed. Some of these methods are
highly empirical, while others are not. Here we give an
example of using nonempirical vdW density functional
H20849vdW-DFH20850H20849Dion et al., 2004; Thonhauser et al., 2007H20850 as
applied to nucleic acid-base-pair steps H20849Cooper et al.,
2008H20850.
The latter work considers Watson-Crick base pairs in
a stacking geometry as shown in Fig. 11. The work uses
vdW-DF to calculate the interaction energy between the
two components of each of the ten possible DNA base-
pair duplexes as a function of the so-called twist angle
H20849see Fig. 11H20850 with the separation H20849riseH20850 between each
component being optimized. In this way one could de-
termine whether the twist and each step of a DNA poly-
mer has its precursor within the properties of the iso-
lated duplex step. By comparing the results with
analyses H20849Olson et al., 2001H20850 of high-resolution crystalline
data from the nucleic acid database, they imply that on a
broad scale the answer to this question is yes. They find
a mean twist of 34? ?10?, where the ? sign indicates the
standard deviation arising from sequence dependence.
The Olson experimental database indicates a corre-
sponding twist of 36? ?7?.
More detailed results from Cooper et al. H208492008H20850 are
shown in Fig. 12 for three steps which show revealing
behavior. Simply stated base pairs gain on the order of
10 or more kcal/mol by stacking in an untwisted configu-
ration, and typically gain several more kcal/mol by twist-
ing by an amount of the same order of magnitude as
found in high-resolution studies of crystalline DNA. As
for the sequence dependence of the twist, Cooper et al.
pointed out that their results generally follow the trend
variations of the databases, but with larger fluctuations.
From the above one can conclude that vdW-DF is suf-
ficiently accurate to obtain meaningful results for this
type of problem. Combining this information with the
fact that the scaling with system size for vdW-DF is no
worse than for standard DFT suggests that vdW-DF can
be applied to systems that are significantly larger than
those considered above, with reasonable confidence of
obtaining valid predictions.
B. Interfaces, surfaces, and defects in solids
1. Impurity-based quasiliquid surficial and interfacial films
Nanoscale, impurity-based, quasiliquid interfacial
films of similar character have been found in an increas-
ing number of different material systems and configura-
tions H20851Fig. 13; see Luo H208492007H20850, and references thereinH20852.
These include silicate-based IGFs in Si
3
N
4
, SiC, and sev-
eral oxides H20849where SiO
2
additive was once considered
essential to stabilize such nanoscale IGFsH20850; IGFs in
ZnO-Bi
2
O
3
and H20849Sr,BaH20850TiO
3
, where SiO
2
is not in-
volved; IGFs at oxide-oxide heterointerfaces;
SiO
2
-enriched IGFs at metal-oxide interfaces; analogous
IGFs in metal systems, e.g., Ni-doped W; and surficial
amorphous films H20849SAFsH20850. Despite the partial structural
order within them, these intergranular or surficial films
are often referred to as ?glassy? or ?amorphous.?
Systematic data have been collected for several SAF
systems, namely, Bi
2
O
3
on ZnO H20849Luo, Chiang, and Can-
non, 2005; Luo and Chiang, 2008H20850,VO
x
on TiO
2
H20849Qian
and Luo, 2007H20850 and SiO
x
on Si H20849Tang et al., 2008H20850, where
film stability and thickness have been measured as func-
tions of temperature and composition H20849or dopant activi-
tiesH20850. Thus, they can be considered as ?instructive sys-
tems? to illustrate the thermodynamic stability of the
FIG. 11. H20849Color onlineH20850 A typical stacking configuration used
in this study. The two base pairs are attracted by the vdW-Ld
interaction. In the lowest energy configuration they undergo a
helical twist as shown, which reduces the Pauli repulsion. This
is the so-called AT:AT step, where the first AT labels the
nucleobases going up one strand while the second labels those
going down the other.
FIG. 12. H20849Color onlineH20850 Stacking energy vs twist angle for the
indicated base-pair steps, labeled as described in Fig. 11. The
TA:TA, CG:CG, and CA:TG curves are typical. The AT:AT
one is not; its kink at about 20? was attributed to the hydrogen-
nitrogen interaction indicated by the double-arrowed lines in
Fig. 11.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
aforementioned broader class of interfacial films H20849Luo,
2007H20850.
Experimental observations and thermodynamic mod-
els for SAFs in two analogous binary oxide systems
H20849Bi
2
O
3
on ZnO and VO
x
on TiO
2
H20850 have recently been
reviewed H20849Luo and Chiang, 2008H20850 and are discussed here
as illustrative examples. In both systems, the equilibrium
film thickness was found to decrease monotonically with
decreasing temperature in the subeutectic regimes. Fur-
thermore, dewetting transitions H20849from a nanoscale SAF
to Langmuir submonolayer adsorptionH20850 were observed
at lower temperatures in both systems. A hysteresis loop
in film thickness versus temperature curve was also ob-
served for VO
x
on TiO
2
. This indicates the existence of
a first-order monolayer-to-multilayer adsorption transi-
tion. Premeltinglike force-balance models H20849with the
volumetric free energy penalty for forming undercooled
liquids being the dominating attractive forceH20850 predicted
subeutectic SAF stability and thickness that agree with
experiments for both systems H20849Luo et al., 2006; Qian and
Luo, 2007H20850. This suggests an analogy between the stabi-
lization of subeutectic quasiliquid SAFs in these binary
systems and premelting in unary systems H20849the latter is
discussed in Sec. III.C.1H20850. For Bi
2
O
3
on ZnO, SAFs of
similar character were also observed in single-phase
ZnO samples containing Bi
2
O
3
concentrations below the
bulk solid-solubility limits, where the films are thinner.
For Bi
2
O
3
on ZnO, nanometer-thick SAFs persist into
the solid-liquid coexistence regime, in equilibrium with
partial-wetting drops H20849Luo, Chiang, and Cannon, 2005;
Luo and Chiang, 2008; Qian et al., 2008H20850, where an anal-
ogy to the phenomena of frustrated-complete wetting
H20849Bertrand et al., 2000H20850 and pseudopartial wetting H20849Bro-
chard and de Gennes, 1991H20850 can be made. The average
SAF composition is markedly different from the associ-
ated bulk liquid phase even when these quasiliquid
SAFs are in thermodynamic equilibration with the bulk
liquid phase H20849Luo and Chiang, 2008H20850.
In a diffuse-interface theory H20849Luo et al., 2006H20850, these
SAFs are alternatively considered as multilayer adsor-
bates formed from coupled prewetting and premelting
transitions. For Bi
2
O
3
on ZnO H20853112
?
0H20854 surfaces where
nanometer-thick SAFs are present in equilibrium with
partially wetting drops, the measured contact angle de-
creases with increasing temperature in the solid-liquid
coexistence regime H20849Qian, Luo, and Chiang, 2008H20850.In
contrast, with increasing temperatures the contact angle
is virtually a constant on the H2085311
?
00H20854 surfaces where SAFs
are not present. This observation suggests that wetting
in the presence of nanoscale SAFs follows a generalized
Cahn wetting model H20849Cahn, 1977H20850. However, an ex-
pected complete wetting transition is inhibited by the
presence of an attractive vdW-Ld force of significant
strength H20849Qian, Luo, and Chiang, 2008H20850.
The technological importance of the nanoscale inter-
granular and surficial films in ceramics and metals is dis-
cussed in Sec. IV.A.2.
2. Charged defects in solids
As a second example related to solid surfaces and in-
terfaces, space charges related to interfacial segregation
of charged defects are discussed here. This is an issue of
practical importance for many technological ceramics,
where the space charges can often coexist and interact
with IGF or SAF formation. Analogous space charges
effects are present at both internal interfaces H20849such as
grain boundariesH20850 and free surfaces.
While their individual energies can be calculated by
DFT and the LDA, charged defects present several
technical challenges because of their long range electro-
static interactions. One in particular is worth describing
here. It is frequently observed by electron microscopy
that interfaces in ceramics are charged, and these
charges are compensated by space charges, in the form
of screening distributions of electrons or other mobile
FIG. 13. Representative impurity-based quasiliquid interfacial films. From Luo, 2007.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
charged species. Such space charges are expected to
have a profound effect on the electronic conductivity
and capacitance of boundaries, but their occurrence and
extent is unpredictable. The difficulty in understanding
and therefore predicting the occurrence of such space
charges lies in the length scales involved. The segrega-
tion of charged species to the boundary can be under-
stood only at the atomic level. It depends on the details
of the atomic structure of the boundary, and atomic-
scale calculations are necessary to predict the segrega-
tion energy. On the other hand, space charges may ex-
tend over a length scale up to microns. This is far too
large for an atomistic description, and the appropriate
physics in this regime is described by a continuum ap-
proach, embodied in the Poisson-Boltzmann equation.
An example of setting up and solving the Poisson-
Bolzmann equation was given in a study of the energy of
a system of mobile charges within an IGF H20849Johnston and
Finnis, 2002H20850.
Consider a planar boundary in the y-z plane. We sup-
pose it somehow lowers its energy by picking up a
charge H9268 per unit area. By how much is the energy low-
ered? We can imagine this as a two-stage process. In the
absence of space charge, and omitting electrostatic self-
energy of the segregated charge H20849which would be infinite
in the absence of compensationH20850 it would cost a chemical
energy of segregation E
seg
to move the charge carriers to
the boundary, which must be a negative energy. Then a
space charge, consisting in general of positive and/or
negative mobile carriers of density n
+
and n
?
, develops
to compensate the boundary charge. These carriers have
charges q
+
and q
?
. The distribution of the carriers as a
function of x in the continuum approximation can be
obtained by solving the Poisson-Boltzmann equation.
Within the simplest classical density functional theory,
assuming the carrier densities are low and linearizing the
Poisson-Boltzmann equation, the boundary energy
would be lowered by
H9004H9253H20849equilH20850 = E
seg
+
1
2
V
H
H208490H20850H9268
?
1
2
H20885
0
H11009
H20851n
+
H20849xH20850q
+
+ n
?
H20849xH20850q
?
H20852V
H
H20849xH20850dx. H2084920H20850
This simple example illustrates how the chemical segre-
gation term is modified by the entropy and separation
energy of charged defects. The approach is readily gen-
eralized to include the classical correlation energy of the
carriers. However, a computational strategy for welding
the continuum Poisson-Boltzmann equation onto the
discrete atomic sites at a boundary that are available for
segregation has yet to be established.
C. Solid/liquid interfaces and suspensions
1. Water and ice
Water and ice nicely illustrate the effects of LRIs. One
phenomenon of water and ice, with significant geophys-
ical and ecological implications, is premelting or surface
melting, which can occur at ice surfaces, grain bound-
aries, and interfaces with inert walls. Furthermore, ice
and water can also be used as an instructive system to
understand analogous, but usually more complex, inter-
facial behaviors in multicomponent ceramics and metals,
e.g., the impurity-based quasiliquid surficial and interfa-
cial films discussed in Secs. III.B.1 and IV.A.2. Premelt-
ing dynamics and its implications are further discussed
in Sec. IV.A.3.
Lifshitz theory has had remarkable success in predict-
ing the nature of the surface melting or premelting of
many materials H20849Dash et al., 2006H20850. Of special interest
here is ice because of the novel influence of the retarda-
tion effects responsible for the change from complete to
incomplete surface melting. This is because retardation
effects attenuate the LRIs that are driving the film
growth H20849Elbaum and Schick, 1991; Wilen et al., 1995H20850.
The basic idea is this. When the polarizability of the
substrate is greater than that of the film, wetting occurs.
Therefore, when dispersion forces dominate, the wetting
of the ice by water at temperatures below T
m
will be
driven when the polarizability of the water lies between
that of the ice and the other material H20849vapor phase or
chemically inert solidH20850. However, the net wetting forces
depend on the entire frequency spectrum that underlies
the polarizability of the system. The novelty of the ice-
water system, first pointed out by Stranski H208491942H20850, is that
the polarizability of ice is greater than that of water at
frequencies higher than the ultraviolet whereas it is
smaller at lower frequencies. Therefore, while the sur-
face melted layer of water is thin, the polarizabilities at
all frequencies contribute to drive surface melting. How-
ever, when the film thickens, the finite speed of light
attenuates the wetting forces by favoring those in which
the polarizability of water dominates over that of the
ice. Hence, the self-attraction of the water begins to
dominate and the film of water on ice stops growing.
We also note here the more familiar mechanisms as-
sociated with the extension of the equilibrium domain of
the liquid phase into the solid region of the bulk phase
diagram: Gibbs-Thomson and colligative effects. Cumu-
latively we refer to these effects as premelting. Premelt-
ing phenomena characterize the equilibrium structure of
the material. As noted above, the premelting phenom-
ena observed in ice and other materials have recently
been reviewed H20849Dash et al., 2006H20850, so here we describe
only the basic tenets of this aqueous system as they may
apply to materials of interest across a range of disci-
plines.
a. The phase architecture of ice
Naturally occurring polycrystalline ice holds much in
common with all polycrystalline materials but with some
distinct advantages for study. It is transparent, optically
birefringent, can be easily doped, and can be held near
its local melting point without sophisticated cryogenic
systems. Like all materials held sufficiently close to their
bulk melting points, ice is not entirely solid. Ice has a
phase geometry that is characterized by a closely packed
hexagonal crystal structure, interlaced with liquid films
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
threading through its volume. Where three ice grains
come together a thin liquid vein exists, and where four
grains join a node of liquid water forms H20849Fig. 14H20850.
The mechanisms responsible for this liquid water are
the Gibbs-Thomson and colligative effects. While seen
on the scale accessible with an optical microscope there
is a finite dihedral angle, but a probe that can penetrate
into the grain boundary on a finer scale can assess the
conditions under which a film between two grain bound-
aries, grain boundary premelting, may exist. Namely,
moving away from the veins and nodes into the planar
interface between two grains, the interfacial curvature
effects disappear and the existence of liquid, if present,
depends on long ranged intermolecular and electrostatic
forces. Here specific models of electrostatic interactions,
modeled, for example, using Poisson-Boltzmann theory,
can compete admirably with vdW attraction; they are
both required for a complete study of grain boundary
melting H20849Benatov and Wettlaufer, 2004H20850. It was shown
there that when the film contains an electrolyte, the
solid grains may be held apart by repulsive screened
Coulomb interactions against the attractive dispersion
interactions, but the latter must be taken out to their full
range using Lifshitz theory to ensure that the longest
range behavior is captured.
In a thought experiment Benatov and Wettlaufer
doped the grain boundary with a 1-1 electrolyte, NaCl.
Because missing bonds at any ice surface give rise to an
increase in the Bjerrum defect density, they treated the
surface as having a finite charge density screened in a
manner that depends on the number density of impurity
ions. Hence, the detailed consequences rely on a correct
treatment of the frequency-dependent dispersion forces
and the peculiar functional dependence of the range and
amplitude of the repulsive Coulomb interaction on the
dopant concentration. In such circumstances, research-
ers in colloid and interface science usually treat the De-
bye length as a constant; in these systems this is well
justified and it is experimentally realizable. In grain
boundary premelting H20849and indeed surface meltingH20850 it is
not necessarily realizable. Due to the nearly perfect re-
jection of salt from the ice lattice the electrolyte remains
in the film and thus an increase H20849decreaseH20850 in film thick-
ness is accommodated by melting H20849freezingH20850 of the solid
so that, up to the solubility limit, the dopant level is
simply inversely proportional to the film volume. Hence,
when the film thins at low temperatures the impurity
concentration in the film increases and the Debye length
decreases through the impurity effect. The novelty then
is that the Debye length, and the amplitude and range of
the Coulomb interaction, are themselves a function of
temperature through the dilution or concentration of the
film. This influences the abruptness of the transition
from premelted to dry grain boundaries. In the surface
melting of ice this same confluence of effects is at work
H20849Wettlaufer, 1999H20850 and is the most likely explanation of
the wide variation of experimental data across laborato-
ries that use the same methodology H20849Dash et al., 2006H20850.
b. Optical properties of ice and water
As described, the finite travel time of photons is mani-
fested in the incomplete surface melting of ice against a
pure vapor phase. The origin is in the frequency depen-
dence of the polarizabilities of water and ice. Two enor-
mously important tasks lie ahead. First, we need to de-
velop novel and quantitatively accurate methodologies
to refine and expand our experimental understanding of
these data. At present, we must use limited spectra from
a variety of sources to fit the data to a damped-oscillator
model H20849Dash et al., 2006H20850. While it is possible to use
ultrahigh vacuum H20849UHVH20850 to obtain the full VUV optical
properties for solid materials, we need modern methods
that can obtain such properties for high vapor pressure
materials, such as most liquids. This will be impeded by
the possibility that the spectra themselves differ in an
interfacial environment in which they are ultimately of
interest, whereas they may be more easily obtained in
bulk samples. It is essential to overcome such impedi-
ments if we are to understand the phase behavior not
only of ice and water but of mixtures across all colloidal
and engineering materials.
2. Hydration
The ability to predict the properties of solutes in liq-
uid water, a prerequisite for rationally guiding nanoscale
processes in aqueous media, continues to be a challenge.
Since the earliest realistic simulation of liquid water
H20849Rahman and Stillinger, 1971H20850, computer simulations
have proven to be vital complements to experiments in
understanding the phenomenon of hydration. For simu-
lation studies of hydration, the most common target is
the excess free energy of hydration H9262
ex
. The excess free
energy is the reversible work done to transfer a solute
from some phase H20849typically the vaporH20850 into liquid water.
This quantity directly informs us about the solubility of
the solute and, with realistically applied extensions,
about the interaction of the solute with other solutes and
interfaces.
Though the original perturbative ideas were devel-
oped for weak coupling H20849beyond the reference stateH20850,
this technique can be used to study hydration of solutes
that interact strongly with water. For example, the dif-
FIG. 14. Liquid film vein structure of polycrystalline ice. Left:
Photograph of four veins intersecting at a node between four
grains in polycrystalline ice near the bulk melting temperature.
Right: Schematic of the vein-node network. From Dash et al.,
2006.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
ference in hydration free energy of Na
+
H20849aqH20850 and K
+
H20849aqH20850
is of much interest in the study of ion channels H20849Astha-
giri et al., 2006H20850. In perturbative or coupling-parameter
approaches, this free energy change is calculated by in-
troducing many artificial states X
i
intermediate in prop-
erties between Na
+
H20849aqH20850 and K
+
H20849aqH20850H20849aqH20850:
Na
+
?X
1
? ? ?X
i
? ? ?K
+
. H2084921H20850
Thus X
1
H20849aqH20850 is only slightly different from Na
+
H20849aqH20850, such
that Na
+
H20849aqH20850 is a useful reference system. Then the free
energy changes between X
i
and X
i?1
are estimated and
the net change between Na
+
and K
+
assessed. H20849Note that
in the limit of a continuum of X we recover the coupling
parameter approach.H20850 Such techniques have greatly ad-
vanced; these developments are summarized by Chipot
and Pohorille H208492007H20850.
Though the above techniques are by now standard in
computer simulation studies of hydration, in an era of
increasing sophistication of simulations including ab ini-
tio approaches H20849cf. Sec. II.A.5H20850, the above approaches
are neither physically revealing nor readily applicable.
For example, if intermolecular potentials from an ab ini-
tio calculation are desired, then no real chemical object
corresponds to X
i
. Quantum chemistry can provide in-
formation only about how Na
+
and K
+
interact with wa-
ter.
In the past decade or so, a new theoretical approach
has been advanced based on the insight that the distri-
bution of potential a solute feels in a solvent can be
parsed on the basis of local chemical structures H20851see, e.g.,
Chipot and Pohorille H208492007H20850H20852. These quasichemical gen-
eralizations of the potential distribution provide a theo-
retical framework within which to investigate such prob-
lems. These techniques have revealed interesting
insights about hydrophobic and hydrophilic hydration.
Several features of the above development are note-
worthy. First, H9262
ex
is transparently linked to the hydration
structures that the solute forms. In fact, the theory al-
lows prediction of the most optimal hydration struc-
tures. In the cases of hydration of ions with high charge
density, these predictions have been supported by results
based on ab initio molecular dynamics simulations H20849Ast-
hagiri et al., 2005H20850. Second, we avoid all considerations of
intermediate states that are not physically realizable.
Last, it is possible to study directly configurations result-
ing from an ab initio simulation, i.e., avoiding the mixing
of ab initio and classical types of treatment above. So far
for reasons of computational cost this has been done
only for a classical model of liquid water H20849Paliwal et al.,
2006H20850. The hydration structures predicted by the primi-
tive quasichemical approach are in good accord with hy-
dration structures observed in ab initio simulations.
A clear understanding of the hydration of various ions
and their role in biological structures and long range
interactions in biological self-assembly is still elusive, de-
spite the fact that over a century has transpired since the
first identification of these effects H20849Kunz et al., 2004H20850.
Whereas ab initio simulations of ions in water can pro-
vide chemical insights, they are still much too expensive
and approximate for large-scale simulations. An inter-
esting development in the area of simulations has been
the use of classical potentials that attempt to include
polarization H20851see, e.g., Grossfield et al. H208492003H20850H20852. Such po-
larization effects have been shown to be important in
how ions partition near an air-water interface H20849Jungwirth
and Tobias, 2006H20850. These simulation efforts can be
complemented by the theoretical directions laid out
above and these can begin to better illuminate specific
ion effects and thus also hydration in nanoscale science.
3. Structure and dynamics at oxide/electrolyte interfaces
The interface between oxides and aqueous solutions
controls ionic and molecular adsorption H20849and thus con-
taminant transportH20850, mineral dissolution or precipitation
kinetics, corrosion rates, heterogeneous catalysis, nutri-
ent and energy supply to bacterial communities, charge-
transfer processes, and in deep subsurface settings, frac-
ture propagation and hydrous melt formation.
Crystalline phases with oxygen as the dominant anion
H20849oxides, silicates, carbonates, phosphates, etc.H20850 are ubiq-
uitous in natural and industrial environments H20849Brown et
al., 1999H20850. In the Earth?s crust and in many industrial
settings such as nuclear and fossil power plants and in
chemical and materials industries, interactions between
liquid aqueous electrolytes and oxide surfaces, over
wide ranges of temperature, pressure, and chemical
and/or mineralogical composition, are the dominant pro-
cesses controlling mass transport, solution chemistry,
and mineral transformations. In many natural and indus-
trial systems, nothing interesting happens until aqueous
solutions encounter solid surfaces, and the rates of fluid-
fluxed reactions are so much greater than anhydrous
processes that they completely dominate such subjects
as geochemistry and corrosion science.
There is no more fundamental process at oxide/water
interfaces than the charging of the surface and the struc-
turing of the adjacent fluid phase due to the undercoor-
dination of atoms at the crystal termination. Typically,
when oxide surfaces come into contact with water,
monovalent cations are rapidly leached out and multiva-
lent cations immediately react with water to produce a
surface that is completely covered with variably proto-
nated oxygens bonded to underlying metal ions of the
bulk crystal. Commonly this process is modeled using
hypothetical reactions such as the ?two-pK? model
H20849Stumm, 1992H20850,
H11022SO
?
+H
+
? SiOH
0
,
H2084922H20850
K
H1
= H20853 H11022 SiOH
0
H20854/H20849H20853H
+
H20854H20853H11022 SiO
?
H20854H20852,
H11022SiOH
0
+H
+
? SiOH
2
+
,
H2084923H20850
K
H2
= H20853 H11022 SiOH
2
+
H20854/H20849H20853H
+
H20854H20853H11022 SiOH
0
H20854H20852,
where H11022S is a generic surface site, H20851iH20852 are site fractions,
and H20853H
+
H20854 is the bulk solution H
+
activity adjusted for the
work of bringing the ion to the charged surface. Here
the critical role of water dissociation to supply H
+
is
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
apparent. The solution pH can often be considered the
master variable in aqueous processes and the pHat
which oxide surfaces have equal concentrations of nega-
tively and positively charged surface sites, referred to as
the pH
pzc
or point of zero charge H20849PZCH20850H20849Sposito, 1998H20850,
is a fundamental parameter, obtainable directly from pH
titrations with careful mass and charge balance. For the
generic two-pK surface protonation scheme, pH
pzc
H11013PZC=
1
2
H20849log
10
K
H1
+log
10
K
H2
H20850, which perhaps ex-
plains its popularity. Another fundamental variable is
the zeta potential H20849ZPH20850 measured by electrokinetic meth-
ods such as electrophoresis or streaming potential. The
pH at which ZP=0 is referred to as the isoelectric point
H20849pH
iep
or simply IEPH20850. For H20849?indifferent?H20850 electrolytes
whose cations and anions interact nearly equally with
the surface, IEPH11015PZC. The exact meaning of ZP is,
however, more ambiguous and highly model dependent
H20849Hunter, 1989; Knecht et al., 2008H20850. In recent years, new
experimental approaches have been developed to deter-
mine zeta potential, surface charge density, and PZC?s
for a number of oxides at temperatures above 100 ?C
H20849Wesolowski et al., 2000; Machesky et al., 2001; Fedkin et
al., 2003; Zhou et al., 2003H20850.
The surface charge density at pH?s other than the
PZC is governed by the site densities of the surface spe-
cies and the screening of surface charge buildup by wa-
ter dipoles and charged, polar, and/or polarizable species
in the solution. Electrostatic screening effects are taken
into consideration by any of a number of parallel-plate
capacitor-type models of the electrostatic double layer
H20849EDLH20850H20849Holm et al., 2001; Poon and Andelman, 2006H20850.
However, very few real oxides H20849with the notable excep-
tion of quartz and other crystalline and amorphous
forms of silicaH20850 exhibit surfaces characterized by the
simple stoichiometry of reactions H2084922H20850 and H2084923H20850. Rather
most oxide and silicate surfaces are characterized by
oxygens bonded to as many as three or four underlying
metal cations, and these can be dissimilar cations with
formal valencies ranging from one to five or more H20849Ko-
retsky et al., 1998H20850. Furthermore, the activity coefficients
of surface sites are typically assumed to be unity and
equated to their volume or mass concentration or mole
fraction, ignoring steric and electrostatic attractive or re-
pulsive forces that might alter their thermodynamic con-
centrations H20849Sverjensky, 2003H20850.
A major advance in predicting the surface charging
process has been the development of the multisite-
complexation model H20849Hiemstra et al., 1996H20850, which ap-
plies the Pauling bond-valence principle to calculate the
unsatisfied valence of oxygen atoms in specific bonding
configurations on oxide surfaces, and incorporates hy-
drogen bonding with sorbed water molecules to provide
a truly predictive capability for estimating surface site
densities, PZC?s, and the protonation states of individual
surface sites. The protonation constant for an individual
surface oxygen in this approach depends on its local
bonding environment and is defined as
log
10
K
H
=?AH20851V + H9018
S
MeO
+ mH20849s
H
H20850 + nH208491?s
H
H20850H20852, H2084924H20850
where V is the formal valence of oxygen H20849?2H20850, the sum-
mation totals the bond-valence contribution to the oxy-
gen from all metal ions of the substrate bonded to the
surface oxygen H20849a function of bond length and charge of
the cationH20850, m is the number of donating H bonds from
adsorbed water molecules, n is the number of H bonds
contributed by any hydrogen directly bonded to the sur-
face oxygen to adsorbed water molecules, and s
H
is the
assumed bond-valence contribution of H
+
. The A pa-
rameter in Eq. H2084924H20850 is a regression constant derived from
a large number of hydrolysis reactions of hydrated metal
ions in aqueous solution. Machesky et al. H208492001H20850 ex-
tended this model to 300 ?C and demonstrated its valid-
ity for the very few oxides H20849magnetite, rutile, zirconia,
and nickel ferriteH20850 for which PZC data are available at
temperatures above 100 ?C. This and other surface pro-
tonation models and temperature extrapolation ap-
proaches are reviewed by L?tzenkirchen H208492006H20850, who
also provided a review of current ?surface-
complexation-models? describing ion adsorption and de-
scriptions of the ion, charge density, and electrical po-
tential distributions in the EDL based on these
simplified concepts. These generally involve Gouy-
Chapman approximations for the effect of surface
charge density on ion distributions in the diffuse layer,
together with various Stern or Helmholtz planes of spe-
cific ion binding. Sverjensky and co-workers H20849Criscenti
and Sverjensky, 1999; Sverjensky, 2006; Fukushi and
Sverjensky, 2007H20850 analyzed a large database of ion ad-
sorption studies on a large number of oxide and silicate
surfaces, mainly conducted at room temperature, using
the two-pK surface protonation model and the ?triple
layer? description of the EDL, providing a semiempir-
ical predictive capability for modeling electrolyte oxide
interactions based on Born solvation principles and tak-
ing into consideration the dielectric properties of the
substrate as well as the solution.
All such electrostatic and structural models of the
EDL are characterized by numerous adjustable param-
eters H20849capacitance terms, specific ion binding constants,
estimations of the solvent dielectric properties in the
double layer, etc.H20850. They generally lack a predictive ca-
pability for Stern layer capacitances and ion binding en-
ergies, and the defining parameters are highly covariant
and difficult to render physically meaningful, even at
room temperature where abundant experimental data
are available. Most importantly, they are largely based
on hypothetical interfacial structures that ignore the dis-
crete atomic nature of the interface at the angstrom-
nanometer scale. In recent years, there has been a con-
certed effort to elucidate the actual structure and
dynamics of the oxide/water interface using a variety of
analytic and computational approaches. Surface forces
have been directly measured to determine the dynamics
of the electrolyte layer between mica sheets brought
into nanometer-scale contact H20849Zhu and Granick, 2001;
Raviv and Klein, 2002H20850. Synchrotron-based extended
x-ray-absorption fine structure, x-ray standing wave, and
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reflectivity measurements are being used to map out the
three-dimensional H208493DH20850 distributions of atoms of the
crystal surface, the solvent, and the ionic species in the
EDL with sub-angstrom resolution H20849Fenter et al., 2002;
Fenter and Sturchio, 2004H20850. Second harmonic generation
H20849SHGH20850 studies have been applied to determine the point
of zero charge of individual crystal faces H20849Stack et al.,
2001; Eisenthal, 2006H20850. To illustrate these approaches,
we review recent integrated studies of the interaction of
water and aqueous electrolytes with the H20855110H20856 crystal sur-
face of rutile H20849H9251-TiO
2
H20850, perhaps the most intensely stud-
ied of all metal-oxide surfaces H20849Diebold, 2003H20850.
Bandura and Kubicki H208492003H20850 used ab initio DFT to
calculate the minimum energy configuration of the rutile
H20855110H20856 surface in contact with a significant number of wa-
ter molecules. Fitts et al. H208492005H20850 used the DFT relaxed
bond lengths and partial charges of surface oxygen at-
oms as input into the multisite-complexation model H20851Eq.
H2084929H20850H20852 to calculate the protonation constants for the reac-
tive surface oxygen atoms, obtaining a calculated PZC
H20849H110115.0 at 25 ?CH20850 in quantitative agreement with SHG
measurements of real rutile H20855110H20856 single-crystal surfaces
in contact with dilute aqueous sodium nitrate solutions.
Figure 15 shows the protonation scheme for the reactive
oxygen atoms on this surface, namely, ?bridging? oxygen
atoms each bonded to two underlying Ti atoms, and
?terminal? oxygen atoms, which result from the chemi-
sorption of water molecules onto bare five-coordinated
Ti atoms exposed on the H20855110H20856 surface. The ab initio op-
timized surface, and interaction potentials of water and
ions with the surface oxygen atoms determined from the
DFT calculations were also used by P1edota, Bandura, et
al. H208492004H20850 and P1edota and Vlcek H208492007H20850 as input into
large-scale classical molecular dynamics H20849MDH20850 simula-
tions of the interface between the rutile H20855110H20856 surface
and 40 ? layers of SPC/E model water H20849Berendsen et al.,
1987H20850 at the density H208491.0 g/cm
3
H20850 of real liquid water.
Figure 16 shows typical MD results for SPC/E con-
taining about 2 mol kg
?1
dissolved SrCl
2
in contact with
uncharged and negatively charged surfaces at 1 atm and
298 K. On the negatively charged surface, the MD simu-
lations predict sorption of solution cations at ?inner
sphere? sites in direct contact with the surface oxygens.
Using synchrotron x-ray standing wave H20849XSWH20850 and crys-
tal truncation rod H20849CTRH20850 techniques, Zhang and Glotzer
H208492004H20850 and Zhang et al. H208492007H20850 were able to image the
H20855110H20856 surfaces of real rutile single crystals in contact with
real bulk water containing a variety of dissolved ions, at
sub-angstrom resolution. As shown in Fig. 17, many ions
were found to sorb at a ?tetradentate? site in contact
with two bridging and two terminal oxygen atoms, while
smaller, transition metal cations sorb at ?monodentate?
and ?bidentate? sites that are approximately the same as
Ti lattice-equivalent sites in the bulk crystal structure.
Figure 18 shows the remarkable agreement obtained
from the synchrotron XSW, CTR, and x-ray absorption
FIG. 15. H20849Color onlineH20850 Protonation constants of bridging
H20849Ti
2
O
?0.516
H20850 and terminal H20849TiOH
?0.483
H20850 oxygen atoms on the
rutile H20855110H20856 surface calculated using ab initio?optimized bond
lengths and surface atom partial charges as input into Eq. H2084930H20850.
An O
II
ion resides at the corner of each coordination polyhe-
dron, which also contains a central Ti
IV
ion.
FIG. 16. H20849Color onlineH20850H110111 m SrCl
2
solution H20849in SPC/E waterH20850
at 25 ?C at liquid density sandwiched between rutile H20855110H20856 sur-
face slabs. Oxygen and titanium atoms are minimized for clar-
ity. Also shown are Sr
2+
ions, Cl
?
ions and hydrogen atoms.
Left: Uncharged, ?nonhydroxylated? surfaces H20849P1edota,
Zhang, et al., 2004H20850 with full 40 ? water layer shown. Note
strong layering of near-surface water, an inner layer of chemi-
sorbed water molecules atop bare fivefold titanium ions and a
distinct second layer associated with the bridging oxygens.
Right: Negatively charged H20849?0.2 C/m
2
H20850, ?hydroxylated? sur-
face attracts Sr
2+
ions into ?inner sphere? sorption sites, dis-
placing second layer water molecules and interacting directly
with bridging and terminal oxygen atoms.
FIG. 17. H20849Color onlineH20850 ?Inner sphere? sorption site geom-
etries on the rutile H20855110H20856 surface as identified by synchrotron
x-ray standing wave and crystal truncation rod studies. Also
shown are oxygen atoms, titanium atoms, and ions sorbed at
the ?tetradentate? site H20849Rb
+
,Sr
2+
,Y
3+
H20850: ?monodentate? and
?bidentate? sites preferred by transition-metal cations
H20849Zn
2+
,Co
2+
H20850.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
fine structure H20849EXAFSH20850 measurements of real electro-
lyte solutions in contact with real rutile H20855110H20856 surfaces,
compared with results of the DFT calculations and clas-
sical MD simulations. The latter agreement is surprising,
given the simplicity of the nondissociating, nonpolariz-
able SPC/E water model.
These integrated computational, chemical imaging,
and macroscopic experimental studies reveal features of
the rutile/aqueous electrolyte interface that may be rep-
resentative of other oxide/water interfaces. The true re-
laxed state of the crystal surface in contact with bulk
water H20849both real water and SPC/E model waterH20850 is shown
to be more similar to the undistorted bulk termination
than previously indicated from studies of the dry surface
under ultrahigh vacuum conditions. The ordering of wa-
ter molecules adjacent to neutral and charged surfaces is
shown to extend only a few monolayers H20849H110111.5 nmH20850 be-
fore bulk solvent properties are observed. The first few
water layers are highly ordered, in registry with the bulk
and surface crystal structures including strong dipole re-
orientations and H bonding within these layers H20849with no
resemblance to liquid water or iceH20850. Cations that sorb as
inner sphere complexes in direct contact with surface
oxygen atoms are absolutely ordered with respect to the
crystal surface structure, and solvent-separated ion pairs
further out in the EDL show lateral and axial ordering
related to both the crystal structure and the distribution
of sorbed water dipoles. These studies provide the first
direct evidence that cations of nominally ?indifferent?
background electrolyte media H20849Na
+
,K
+
, and Rb
+
H20850 also
bind at inner sphere sites, competing for sorption at such
sites with multivalent trace cations that are much more
strongly attracted H20849much higher binding constants in the
thermodynamic senseH20850. These studies also provide fairly
direct confirmation of the general features of the
multisite-complexation model and the Guoy-Chapman-
Stern models of surface protonation and binding of
counterions in discrete and well-defined layers that
screen most of the surface charge within 1 nm of the
surface.
The following needs emerge as necessary at this stage
in our learning: more accurate water models; calibration
of interactions of ions and surfaces with these model
waters for possible use in large-scale molecular dynam-
ics simulations; measurements of ion adsorption and sur-
face charging at high temperatures and pressures as well
as in low-density supercritical water; development of
parallel codes for making high-level static and dynamic
quantum mechanical calculations of interfaces involving
meaningful numbers of solid and solution species, that
will capture the collective H20849nonpairwiseH20850 interactions of
complex systems; and chemical imaging of interfacial
structures and dynamics at the sub-angstrom to micron
scales.
Similar ideas, for metal/electrolyte interfaces, are be-
ginning to appear in integrated experimental and com-
putational studies of such interfaces H20851see, e.g.,
Ogasawara et al. H208492002H20850; Denzler et al. H208492003H20850; Schiros et
al. H208492007H20850H20852.
4. Colloidal suspensions
What can we learn from traditional colloidal building
blocks? Before increasing system complexity, it is in-
structive to examine the behavior of model colloidal
building blocks composed of either hard or attractive
H20849i.e., stickyH20850 spheres. The phase behavior, structure, and
dynamics of colloidal suspensions composed of tradi-
tional building blocks depend both on the colloid vol-
ume fraction H9278 and on their interparticle interactions. In
hard-sphere suspensions, the critical parameter for de-
termining phase behavior is H9278 H20849Prasad et al., 2007H20850; see
Fig. 19. For H9278 just below 0.49, the suspension forms a
dense liquid with particle positions that are disordered.
The radial distribution function gH20849rH20850 is a measure of the
probability of a particle center being located at a dis-
tance r from a given particle, relative to a uniform dis-
FIG. 18. H20849Color onlineH20850 Sorption heights H20849?H20850 of cations above
the Ti-O surface plane of rutile H20855110H20856 vs bare cation radius for
ions in the tetradentate, monodentate, and bidentate sites de-
termined by x-ray and computational approaches. From Weso-
lowski et al., 2008.
FIG. 19. H20849Color onlineH20850 Phase diagram of hard-sphere colloids.
Micrographs and schematic plots of gH20849rH20850 for representative col-
loidal dense liquid, crystal, and glass suspensions.
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Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
tribution. For a dense colloidal fluid approaching strong
correlation in spacings, gH20849rH20850 contains local maxima and
minima at integral multiples of the interparticle spacing.
As H9278 is increased toward 0.49, the particle positions be-
come increasingly correlated, as shown in the micro-
graph of a colloidal fluid suspension H20849Fig. 19H20850. When H9278 is
increased above 0.49 and the system is allowed to equili-
brate, entropy drives the formation of crystalline do-
mains. From H9278=0.49 to 0.54, the liquid and crystal
phases coexist. Above H9278=0.54, the state of the suspen-
sion is a crystalline solid with a high degree of positional
order, as shown in Fig. 19. The sharp maxima in the
crystal gH20849rH20850 correspond to spacing of particles within this
ordered structure. By contrast, when H9278 is increased
above 0.49 in systems that are not allowed to equilibrate,
the suspension resides in a supercooled fluid state, which
is a metastable solid that eventually relaxes. From H9278
=0.58 to 0.63, the suspension retains its glassy nature
and is unable to relax over the experimental time frame.
The particle arrangements in colloidal glasses closely re-
semble those of liquids H20849Fig. 19H20850. Similarly, gH20849rH20850 for a
glass is like that of a dense fluid, with correlations ex-
tending over multiple coordination shells. What distin-
guishes a glass from a liquid are the particle dynamics;
local caging of particles leads to dynamic arrest in the
solid glass, whereas in the liquid the particles freely re-
arrange. The repulsive potential in soft-sphere systems
increases the particle?s effective size and therefore phase
transitions occur at lower H9278, while the liquid-crystal co-
existence phase is relatively smaller than in hard spheres
H20849Liu et al., 2002H20850.
Colloidal mixtures exhibit rich phase behavior. The
simple act of mixing two particle populations together
can lead to unexpectedly rich phase behavior. For ex-
ample, van Blaaderen and co-workers H20849Leunissen et al.,
2005H20850 demonstrated that oppositely charged micro-
spheres can self-assemble into ionic crystals under ap-
propriate solution conditions. Figure 20 shows a CsCl
lattice assembled from 1 H9262m spheres that are weakly
positive and negative. In another important example,
Lewis and co-workers H20849Mohraz et al., 2008H20850 recently de-
veloped biphasic mixtures of attractive and repulsive mi-
crospheres, in which both the structure of colloidal gels
formed by the attractive species and the dynamics of the
repulsive species can be tuned solely by varying the ratio
of the two constituents.
As the above examples illustrate, there is still much to
learn even in colloidal systems based on simple mix-
tures. Nevertheless, there is a strong drive to introduce
greater complexity by tailoring colloidal building blocks
at the subparticle level. Manipulating colloids and self-
assembly via harnessing LRIs are discussed in Sec. IV.B.
5. Solution-based manipulation of SWCNT
Carbon nanotubes have been studied intensively over
the last 15 years. Their physical and electronic structure
is well known as are many other characteristics such as
elasticity, strength, transport, and optical properties
H20849Saito et al., 1999H20850. There are several optical techniques
including absorbance, Raman, circular dichroism, and
fluorescence spectroscopy that have been developed to
help study and identify CNTs. Scanning probe tech-
niques such as STM and AFM have been used widely to
study their structure and properties.
How can manipulation of LRIs be used in solution-
based processing of single-walled carbon nanotubes
H20849single layers of carbon atoms rolled into a seamless
tube with diameter of about 1 nmH20850? To first order, if one
regards SWCNT band structure as that of graphite with
some k vectors disallowed due to circumferential sym-
metry, one finds that two-thirds of all allowed SWCNT
have a band gap and the remaining are metallic H20849Saito et
al., 1999H20850. This fact is simultaneously the bane and the
promise of these materials. While each form has highly
desirable properties, all known syntheses result in some
mixture. The ability either to synthesize a known type,
or to sort individual types in a mixture, is therefore a
critical problem. For electronic applications, there are
two broad approaches. One may either grow the
SWCNT on a substrate H20849Javey et al., 2003H20850 or process it
in a liquid and attempt to place it on the substrate H20849Ar-
nold et al., 2006; McLean et al., 2006H20850. In the latter case,
the process is almost invariably governed by long range
interactions such as between the SWCNT and a sub-
strate or another interface. Here we focus our attention
on solution-based processing which is applicable more
generally.
Noncovalent modification using surfactants and bio-
logical molecules H20849Bachilo et al., 2002; Zheng, Jagota,
Semke, et al., 2003; Zheng, Jagota, Strano, et al., 2003H20850
works with long range interactions to create solution-
based manipulation techniques. Several solution-based
techniques for sorting by metallic versus semiconducting
character, or by diameter, have also been demonstrated
H20849Weisman, 2003H20850 based on selective adsorption H20849Chatto-
padhyay et al., 2003H20850, dielectrophoresis H20849Krupke et al.,
2003H20850, density differentiation H20849Arnold et al., 2006H20850, and
using a DNA-CNT hybrid H20849Zheng, Jagota, Semke, et al.,
2003; Zheng, Jagota, Strano, et al., 2003; Tu et al., 2009H20850.
We discuss how control of long range interactions in
solution has contributed to solving problems associated
with dispersion, sorting, and placement of SWCNT. We
consider approaches where dispersion is achieved by
(100m)
FIG. 20. H20849Color onlineH20850 Ionic crystal H20849CsClH20850 assembled from
oppositely charged colloidal microspheres. From Leunissen et
al., 2005.
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
forming a hybrid of the SWCNT with small-molecule or
polymeric surfactants and focus on a few successful ex-
amples.
a. Dispersion and structure
Surfactants have been used to disperse SWCNTs in
water. These include molecules with charged head
groups and flexible alkyl tails such as sodium dodecyl
sulfate H20849SDSH20850H20849Bachilo et al., 2002H20850 and, more rigid, pla-
nar surfactants such as sodium cholate H20849SCH20850, sodium
deoxycholate, and sodium taurodeoxycholate H20849Arnold et
al., 2006H20850. A signature of successful dispersion is the ap-
pearance of distinct band-gap absorbance peaks in the
near infrared, and of corresponding band-gap fluores-
cence. Resulting dispersions are stable as long as suffi-
cient excess surfactant is maintained.
Qualitatively, the nature of the hydrophobic interac-
tion that controls surfactant-SWCNT hybrid formation
can be readily understood by analogy with micelle for-
mation. However, there are two basic issues about which
there is currently little quantitative understanding: the
structure of the resulting hybrid and the binding
strength. Knowledge of the structure determines the ef-
fective density of the hybrid material, and this is a deli-
cate matter. Small differences in density have been in-
voked to explain the basic dispersion mechanism, i.e.,
why individual SWCNT hybrids remain suspended while
small clusters are dispatched to the bottom of the cen-
trifuge tube. More importantly, as discussed next, modu-
lation of the effective density of the SWCNT-surfactant
hybrid both by the diameter of the SWCNT core and by
its electronic properties H20849e.g., metallic versus semicon-
ducting characterH20850 allows their separation by centrifuga-
tion in a density gradient. Direct observation of the
structure is difficult and has rarely been reported. Ex-
perimental and theoretical studies of the structure and
binding thermodynamics and kinetics are much needed.
The structure seems to depend on the electronic charac-
ter of the core SWCNT and suggests that electronic re-
sponse to charges on the surfactant plays an important
role.
Dispersion of SWCNT has also been demonstrated by
coating with water-soluble polymers H20849O?Connell et al.,
2001; Zheng, Jagota, Semke, et al., 2003H20850. As for small-
molecule surfactants, pertinent issues relate to the bind-
ing free energy and the structure of the resulting hybrid.
DNA-dispersed SWCNT H20849DNA-CNTH20850, because the two
constituents themselves have been studied extensively,
may prove to be a good model system.
Single and double-stranded DNA have been studied
even more extensively than CNTs. It is believed that the
interaction between the DNA and SWCNTs is mediated
by stacking of bases onto the CNT side-wall and that the
entire hybrid is rendered water soluble because of the
charged sugar phosphate backbone H20849Fig. 21H20850. By subject-
ing it to a combination of ion-exchange and size exclu-
sion chromatography, one can obtain excellent model
systems, for instance, consisting primarily of one type of
CNT with controlled length wrapped by ssDNA with
known sequence and length H20849Zheng, Jagota, Semke, et
al., 2003; Zheng, Jagota, Strano, et al., 2003; Tu et al.,
2009; see Fig. 21H20850.
Various contributions to the free energy of binding of
DNA homopolymers to a SWCNT have been discussed
in Manohar et al. H208492007H20850. Dispersion efficiency appears
to be optimal for sequences with about 30 bases al-
though it has also been achieved with small dsDNA mol-
ecules H208496-mersH20850H20849Vogel et al., 2007H20850. Dispersion by indi-
vidual nucleotides, which is not as effective as with
longer strands, shows that purines are more effective
than pyrimidines and that charge on the phosphate plays
an important role. Binding strength decreases in the se-
quence guanineH11022adenineH11022H20849cytosine/thymine/urasilH20850
H20849Ikeda et al., 2006H20850. Similar findings have been reported
for binding of bases onto graphite; the sequence is G
H11022AH11022TH11022C H20849Sowerby, Cohn, et al., 2001; Sowerby,
Morth, and Holm, 2001H20850 with binding enthalpy of ad-
enine reported to be about 20 kJ/mol. Since the nominal
Kuhn length of ssDNA H20851H110111.6 nm H20849Bustamante et al.,
2003H20850H20852 includes several bases, this suggests that binding
is strong enough to overcome entropic increase in free
energy, and that all bases should be bound. However,
experiments find that the binding strength of homopoly-
mers follows a different sequence H20849TH11022AH11022CH20850 and simu-
lations suggest that the difference is because steric hin-
drance between adjacent bases results in only partial
base stacking with effective binding strength in the se-
quence TH11022GH11022AH11022C. That the DNA binds in helical
fashion has been suggested by molecular models
H20849Johnson, Johnson, and Klein, 2008H20850 and confirmed by
AFM H20849Zheng, Jagota, Semke, et al., 2003; Zheng, Jagota,
Strano, et al., 2003H20850. It has been suggested that certain
structures are further stabilized by unconventional base
pairing H20849Saenger, 1984; Zheng, Jagota, Semke, et al.,
2003; Zheng, Jagota, Strano, et al., 2003; Tu et al., 2009H20850.
For example, adenine is known to form hydrogen-
bonded monolayers on graphite H20849Sowerby, Cohn, et al.,
2001H20850. There is currently a lack of measurements on
binding free energies and structure.
Single-stranded DNA in water carries a charge on its
backbone about every 0.6 nm. On wrapping around the
CNT, it therefore renders the hybrid a highly charged
rod. Therefore, electrostatics is expected to play an im-
portant role in determining the structure, in sorting, and
during placement. Handling electrostatic interactions for
DNA-CNT, even within the Poisson-Boltzmann ap-
proximation, raises new questions because of the high
linear charge density and the variable electronic proper-
ties of the SWCNT core. This issue will be discussed
further in the following subsection.
b. Sorting and placement
The ability to create hybrids of surfactant and poly-
meric molecules with individual SWCNT has made it
possible to sort dispersions. While techniques based on
covalent modification of the SWCNT have been demon-
strated, we consider only those that rely on noncovalent
H20849long rangeH20850 interactions. Separation by metallic versus
semiconducting character and by diameter has been
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French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
demonstrated by differential adsorption H20849Chattopadhyay
et al., 2003H20850; by dielectrophoresis H20849Krupke et al., 2003H20850
that differentiates based on difference in polarizability
with respect to that of water, by density differentiation
H20849Arnold et al., 2006H20850; and by ion-exchange chromatogra-
phy H20849Zheng, Jagota, Semke, et al., 2003; Zhang, Jagota,
Strano, et al., 2003; Tu et al., 2009H20850.
Figure 22 shows that hybrids of sodium cholate with
SWCNTs can be separated by ultracentrifugation in a
medium with a density gradient H20849Arnold et al., 2006H20850.
SC/SWCNT hybrids travel to the location in the medium
that has the same effective density; separation is there-
fore an indication that hybrid effective density depends
systematically on SWCNT diameter and electronic prop-
erties. As discussed previously, little is known about how
interactions between the SC molecules and SWCNT
control structure. For DNA-CNT, ion-exchange chroma-
tography is understood to differentiate between
SWCNT cores because the latter modulates the electro-
static interactions of the hybrid with external substrates
or fields H20849Lustig et al., 2005H20850. In both cases, it is likely
that hydrophobic, van der Waals, and electrostatic inter-
actions will all play an important role. Much more work
is needed to understand how the charged SWCNT rod
interacts with an external field, with other rods, and with
a substrate H20849either for deposition or for sortingH20850.
The examples cited here work by converting the
SWCNT into a hybrid with an amphiphilic molecule
H20849surfactants or polyelectrolytesH20850. Systematic variations in
the structure and effective electrostatics of the hybrid
rely on delicate control of long range interactions be-
tween the constituents of the hybrid, and between the
hybrid and external fields or substrates. Both because of
the interest in this application and because SWCNT hy-
brids provide model systems, there is scope for consid-
erable further work on these materials to understand
how they work and to improve dispersion, sorting, and
placement techniques.
IV. HARNESSING LRIs
A. Surfaces and interfaces
The interface between solids and liquids H20849or quasiliq-
uidsH20850 is a complex region where electrostatic, vdW-Ld,
400 600 800 1000 1200
Abs
Wavelength (nm)
HiPco
(starting material)
(9,1)
(8,3)
(6,5)
(7,5)
(10,2)
(8,4)
(9,4)
(7,6)
(8,6)
(9,5)
(10,5)
(8,7)
!"# !$#
!%# !&#
FIG. 21. H20849Color onlineH20850 DNA forms a hybrid with carbon nanotubes, enabling their dispersion and separation. H20849aH20850 Atomic force
microscopy image of DNA-CNT deposited on a mica surface following separation and length fractionation. H20849bH20850 Absorption spectra
of all 12 semiconducting species separated by ion-exchange chromatography of a starting mixture H20849top plotH20850 using different DNA
sequences. H20849cH20850 A starting dispersion H20849blackH20850 and three sorted fractions rich in metallic SWCNT and semiconducting SWCNT with
different diameters. H20849dH20850 Proposed H9252-barrel structure of the DNA-CNT hybrid. H20849aH20850 Adapted from Zheng and Semke, 2007. H20849bH20850 and
H20849dH20850 Adapted from Tu et al., 2009. H20849cH20850 Adapted from Zheng, Jagota, Semke, et al., 2003.
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polar H20849ABH20850, solvation, steric, and related interactions
create distinct properties dictated by the nature of the
bounding bulk phases and by temperature and pressure.
This region is often poorly characterized at the atomic
level. In this section, three examples are given of the
potential to manipulate such interfaces for desired prop-
erties.
We first discuss proton-conducting polymer mem-
branes, a system whose function is entirely dictated by
an interplay of long range forces and interactions. The
hope is that these can be manipulated and tailored to
optimize performance. However, such ?tailoring? has
been achieved largely by trial and error. Many variables,
such as polymer backbone, side chain and functional
group properties and distributions, and the type, size,
and distribution of inorganic additives and processing
methods are simply altered by many hundreds of re-
searchers in search of ideal characteristics.
Second, we discuss the formation of amorphous inter-
granular and surface films during ceramic and metals
processing. The thickness, composition, and structural or
dynamic properties of such films are clearly driven by
the bulk compositions and structures of the substrates
on which they form, by trace levels of contaminants that
often migrate to these films, and by environmental pa-
rameters such as temperature, pressure, and their gradi-
ents. Intergranular films strongly influence the tough-
ness, strength, permeability, and many other
physicochemical properties of the processed material,
and surface films can alter catalytic activity, resistance to
corrosion, or sequestration of contaminants. As for ion-
conducting membranes, an atomic-level predictive un-
derstanding of the origins and controls on intergranular
and surface film structures and properties would enable
efficient manipulation of input parameters for desired
functionality.
Finally, we describe the more general phenomenon of
premelting at grain boundaries and surfaces. Premelting
is driven by temperature and compositional gradients,
by juxtaposition of particles with different crystallo-
graphic orientations or bulk properties, and by the inter-
play of forces at their interface so as to induce the for-
mation of a thin liquid film at a temperature below the
T
m
of the bulk phase. It is shown that manipulation of
temperature and compositional gradients in such sys-
tems might be used to redistribute nanoparticles embed-
ded within a polycrystalline material during warming, or
from a particle suspension in a liquid during cooling.
One intriguing consequence of premelting in polycrys-
talline ice is the interpretation of paleoclimate from the
record of trapped water, trace elements, and gases in
continental glaciers. If neglected, the consequences of
premelting phenomena can lead to erroneous interpre-
tations of the evolution of the Earth?s atmosphere, hy-
drosphere, and climate spanning geologic time scales.
1. Proton exchange membranes for hydrogen fuel cells
In a hydrogen fuel cell, H
2
is catalytically split into
protons and electrons at an anode H20849usually Pt nanocata-
lyst on a carbon supportH20850. Protons traverse a proton-
permeable membrane, where they are catalytically re-
combined at a cathode with O
2
and electrons to form
water. Electrons perform work in an external circuit, for
example, to power a hybrid vehicle or a static power
generator. Particularly for transportation applications,
novel proton exchange membranes H20849PEMsH20850, to serve as
proton conductors as well as electrical insulators and
barriers to fuel-oxidant mixing, are needed for fuel cells
that operate efficiently and reliably from
?40 ?C to 130 ?C and at low relative humidities H20849Kreuer
et al., 2004; Eikerling et al., 2006H20850. The rational design of
advanced membranes requires control and understand-
FIG. 22. H20849Color onlineH20850 Centrifugation of surfactant H20849sodium cholateH20850 dispersed SWCNT in a density gradient has been demon-
strated to result in their separation. It has been proposed that the effective density of the surfactant-SWCNT hybrid increases with
increase in diameter and on the electronic character of the SWCNT. From Arnold et al., 2006.
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ing of membrane structure and dynamics at the nano-
scale. Exploiting unique phenomena resulting from re-
active interactions in such a complex, multicomponent,
multiscale system is a scientific and technical challenge.
Nafion?, the industry standard PEM H20849Mauritz and
Moore, 2004H20850, performs poorly at H1102280 ?C and H11021100%
relative humidity. Other ionomers are being considered
for PEM fuel cells H20849Hickner et al., 2004; Harrison et al.,
2005; Hickner and Pivovar, 2005H20850, but they all fail to
meet required performance characteristics. Nafion is a
statistical copolymer of tetrafluoroethylene and tet-
rafluoroethylene substituted with pendant perfluoroet-
her chains terminated with sulfonic acid groups. This
material is highly polydisperse, molecular weight is not
well known, and placement of the sulfonic acid groups
along the backbone is not readily controlled. Although
statistical copolymers usually do not form multiphase
morphologies, in the case of Nafion the very high incom-
patibility between the two types of units does result in a
nanometer-sized phase-separated morphology. Figure 23
shows the H20849hypotheticalH20850 complexity of Nafion-like ran-
dom copolymer membranes at atomic to macroscopic
scales. Attempts to model structure and dynamics of
such membranes at these scales have thus far met with
limited success H20849Jang et al., 2004; Kreuer et al., 2004;
Blake et al., 2005H20850.
The morphology assumed by uncharged diblock co-
polymers reflects, to a good approximation, only the vol-
ume fraction of the two components H20849Khandpur et al.,
1995; Bates and Fredrickson, 1999H20850. The excluded-
volume and component-specific affinity effects of inor-
ganic nanoparticles on diblock copolymer morphology,
chemical and mechanical stability have been clearly
demonstrated for the simpler case of nanoparticle-
guided self-assembly of uncharged block copolymers
H20849Balazs et al., 2006H20850. Such approaches have not been ex-
tensively applied to pure ionomer or inorganic compos-
ite membrane systems.
The addition of inorganic particles H20849oxides and other
compounds of Al, Mo, P, Si, Ti, W, Zr, etc.H20850 to Nafion
H20849and to a lesser extent other PEMsH20850 can dramatically
improve PEM fuel cell power output, membrane water
uptake, and retention and membrane tensile strength
while reducing polymer decomposition and fuel, oxygen,
and water transport H20849Alberti and Casciola, 2003; Alberti
et al., 2005; Chalkova et al., 2006; Licoccia and Traversa,
2006H20850. Inorganic additives generally increase the glass
transition temperature of the polymer H20849Adjemian et al.,
2006H20850, suggesting a correlation with improved high-
temperature performance. Surface functionalization of
inorganic additives provides the opportunity to tailor the
surface to achieve desired characteristics H20849Gomes et al.,
2006; Kim et al., 2006H20850. Since the polymer functional
groups are highly acidic, releasing H
+
to form sites of
fixed negative charge, one could reasonably postulate
that inorganic phases that become positively or nega-
tively charged at the low-pH operating conditions might
alter PEM performance through electrostatic interac-
tions with the functional groups. Inorganic additives also
exhibit a wide range of bulk dielectric constants H20849H9255H20850, such
as 120 for rutile H20849H9251-TiO
2
H20850 to 4.6 for quartz H20849H9251-SiO
2
H20850
H20849Sverjensky, 2001H20850. Both electrostatic and vdW-Ld inter-
actions of inorganic nanoparticle and macroparticle ad-
ditives with both charged and uncharged regions of
ionomer membranes may dictate the structure, dynam-
ics, and performance characteristics of composite mem-
branes for fuel cell and similar applications.
FIG. 23. H20849Color onlineH20850 Hypothetical Nafion membrane self-assembly. From Eikerling et al., 2006.
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2. Intergranular and surficial films
Impurity-based, nanometer-thick intergranular films
H20849IGFsH20850 have been widely observed in ceramics and met-
als. Clarke originally proposed that these IGFs exhibit
an equilibrium thickness H20849Clarke, 1987H20850. A recent critical
review H20849Luo, 2007H20850 pointed out the following general
character of IGFs: First, IGFs exhibit a self-selecting
H20849equilibriumH20850 thickness for a given set of equilibration
conditions, and the equilibrium thickness can be altered
by changing the temperature, dopant activities, and
other thermodynamic potentials. Second, the average
composition of an IGF differs markedly from that of the
bulk liquid phase, and it can lie within a bulk miscibility
gap. Third, quasiliquid IGFs can be stabilized at subsoli-
dus temperatures. Finally, IGFs generally exhibit partial
structural order and through-thickness compositional or
structural gradients.
Surficial amorphous films H20849SAFsH20850 of similar character
have been observed H20849Luo and Chiang, 2008H20850. Systematic
data have been collected for two SAF systems, providing
insights into understanding analogous IGFs where sys-
tematic experiments of a similar level have not been
conducted. These observations are reviewed in Section
III.B.1.
In a phenomenological model, the excess film free en-
ergy is written as
G
x
H20849hH20850 = H20849H9253
1
+ H9253
2
H20850 + H20849H9004G
vol
hH20850
+
? A
12H9266h
2
+ H9268
elec
H20849hH20850 + H9268
short range
H20849hH20850 + ? .
H2084925H20850
The term H9253
1
+H9253
2
represents the sum of the excess free
energies of two independent H20849well separatedH20850 interfaces
H20851H20849H9253
1
+H9253
2
H20850=2H9253
cl
for an IGF or H20849H9253
1
+H9253
2
H20850=H9253
cl
+H9253
lv
for an
SAFH20852. H9004G
vol
is a volumetric free energy for forming a
hypothesized uniform liquid film from a mixture of equi-
librium bulk phases. When the film is thin, additional
interfacial interactions may arise, e.g., a vdW-Ld force,
an electrostatic double-layer interaction, and short range
interactions of structural and chemical origin. Capillary
and applied pressures, if they are present, also affect film
thickness. An ?equilibrium? thickness corresponds to a
balance among these attractive and repulsive interfacial
interactions, i.e., a local or global minimum in the excess
free energy. This model, initially carried over from the
wetting community and proposed for IGFs by Clarke
H208491987H20850, can be viewed as a high-temperature DLVO
theory, in which H9004G
vol
h is an additional significant at-
tractive interaction for quasiliquid films formed under
subsolidus conditions H20849being analogous to the volumetric
term in premelting theoriesH20850. Recently significant
progress has been made in quantifying vdW-Ld forces
H20849French, 2000H20850. There is a critical need to quantify other
interfacial interactions. Further discussions of interfacial
interactions can be found in Luo H208492007H20850.
Alternatively, these nanometer-thick IGFs and SAFs
can be described as multilayer adsorbates with composi-
tions set by bulk chemical potentials H20849Cannon and Es-
posito, 1999; Cannon et al., 2000H20850. These IGFs and SAFs
can be understood as graded multilayer adsorbates
formed from coupled premelting and prewetting transi-
tions in diffuse-interface models H20849Luo et al., 2006; Tang
et al., 2006H20850 that were extended from the critical point
wetting model H20849Cahn, 1977H20850.
A challenge is to develop quantitative and realistic
models to predict the stability of IGFs. In general, sys-
tematic measurements of the film thickness and other
characteristics as a function of temperature and chemi-
cal potentials are required to validate interfacial thermo-
dynamic models. Despite wide observations of IGFs,
such measurements have not been conducted because of
difficulties in controlling impurities in ceramics and effi-
ciently preparing TEM specimens.
SAFs do offer a good platform to conduct critical and
systematic measurements that are difficult to conduct on
IGFs. Moreover, vdW-Ld forces, which are always at-
tractive for IGFs of symmetrical configurations, can be
repulsive for SAFs. Stable subsolidus SAFs have been
found in systems with repulsive vdW-Ld forces. Experi-
ments of SAF film stability and related wetting transi-
tions in these systems provided insights into understand-
ing the thermodynamic stability of IGFs.
The technological importance of SAFs and IGFs has
recently been reviewed H20849Luo, 2007H20850. IGFs play impor-
tant roles in fracture toughness, strength, fatigue, creep
resistance, and oxidation of Si
3
N
4
- and SiC-based struc-
tural ceramics, mechanical properties and erosive wear
behaviors of Al
2
O
3
, superplasticity of ZrO
2
, tunable
conductivities for ruthenate-based thick-film resistors,
nonlinear I-V characteristics for ZnO-Bi
2
O
3
?based
varistors, and functions of H20849Sr,BaH20850TiO
3
?based perov-
skite sensors and actuators. IGFs have also been found
in Synroc, AlN substrates, and high-T
c
superconductors,
with attendant detrimental implications on corrosion re-
sistance and thermal or electrical conductivity. Under-
standing and control of SAFs are also technologically
important for tailoring supported oxide catalysts and ul-
trathin dielectric films as well as for manipulating the
shape and growth kinetics of nanoparticles and nano-
wires.
Furthermore, IGFs and SAFs have important techno-
logical roles in materials fabrication. For example, Dil-
lon et al. recently revealed the existence of six distinct
grain boundary ?complexions? H20849IGFs and derivative
structuresH20850 with increasing structural disorder coupled
with increasing mobility in doped Al
2
O
3
H20849Dillon et al.,
2007H20850. This revelation explains the abnormal grain
growth mechanism in this system.
Finally, a related long range scientific goal is to de-
velop quantitative interface complexion H20849phaseH20850 dia-
grams to represent the stability of IGFs, SAFs, and
other interfacial structures H20849Dillon et al., 2007; Luo,
2008H20850. Such diagrams are useful for designing not only
fabrication recipes to use the most appropriate interface
structures during processing, but also heat treatment
recipes to tune the final interface structures for desired
performance properties.
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The necessity of developing grain boundary complex-
ion diagrams is demonstrated by recent research on ac-
tivated sintering. Studies using ZnO-Bi
2
O
3
H20849Luo et al.,
1999H20850 and W-Ni H20849Luo, et al., 2005; Gupta et al., 2007H20850 as
model ceramic and metallic materials showed that nano-
scale quasiliquid IGFs form well below the bulk
eutectic-solidus temperature, where the bulk liquid is
not yet stable. Nonetheless, short-circuit diffusion in
these subsolidus quasiliquid IGFs leads to activated sin-
tering in these systems, phenomenologically similar to
liquid-phase sintering but able to initiate at H1102160% of
T
solidus
. Hence bulk phase diagrams are not adequate for
designing optimal activated sintering protocols. A re-
cently developed quantitative model can predict grain
boundary disordering and related onset activated sinter-
ing for five tungsten-based binary alloys H20849Luo and Shi,
2008H20850. Furthermore, a variety of discrete liquidlike inter-
facial ?phases? can form from the interplay of multiple
interfacial interactions and a finite atomic size effect.
Related concepts and thermodynamic theories are fur-
ther elaborated in Luo H208492008H20850H20850.
3. Premelting dynamics and its manipulation
As discussed in Secs. III.A.3 and II.C.1, Lifshitz
theory has successfully predicted the nature of the sur-
face melting of many materials. Premelting of a material
refers to the persistence of a film of its liquid phase at
temperatures below the normal melting temperature.
Most commonly discussed are the Gibbs-Thomson and
colligative or impurity effects. The former is a conse-
quence of a solid phase being convex to its melt phase
and thus having a lower freezing point than the bulk,
and the latter originates in the lowering of the chemical
potential of a solvent in the presence of a solute. All
crystals, whether in contact with their vapor phase or
with another material, have surfaces where the process
of melting is initiated: if there is a layer of liquid at the
surface, at temperatures below the bulk melting point
T
m
, then there is little need to activate the melting pro-
cess. The ease with which liquids can be supercooled
tells us that the melting process is inherently asymmetri-
cal about the transition point.
Interfacial premelting occurs at the surfaces of solid
rare gases, quantum solids, metals, semiconductors, and
molecular solids and is characterized by the appearance
of an interfacial thin film of liquid that grows in thick-
ness as the bulk melting temperature T
m
is approached
from below H20849Dash et al., 2006H20850. When interfacial pre-
melting occurs at vapor surfaces, it is referred to as ?sur-
face melting;? when it occurs at the interface between a
solid and a chemically inert substrate, ?interfacial melt-
ing;? and when at the interface between two grains of
the same material, ?grain boundary melting.? When
films at such solid surfaces diverge at the bulk transition
the melting is complete, but where retarded potential
effects intervene and attentuate the intermolecular wet-
ting forces the film growth may be blocked and thereby
be finite at the bulk transition. This latter circumstance,
in which the behavior is discontinuous, is referred to as
incomplete melting. Because near the bulk transition we
can view these phenomena as wetting transitions H20851sur-
face melting is a special case of triple point wetting; see
Dietrich H208491988H20850H20852, the role of LRIs, and particularly long
range van der Waals effects, has a broad context and
setting which is particularly relevant to the role of geom-
etry in these volume-volume interactions H20849Lamoreaux,
2005; Dantchev et al., 2007H20850.
We now consider the dynamics of premelted liquid
driven by variations in temperature and/or composition.
Although the physics of the surface and that of the bulk
are most often taught and studied in isolation, it is the
confluence of dimensionality and phase behavior found
at the surface that provides a wide-ranging area of ex-
ploration in science and engineering. Here we summa-
rize aspects of a recent review H20849Wettlaufer and Worster,
2006H20850.
Two aspects of surfaces familiar to those working in
materials are wetting phenomena and thermophoretic or
Marangoni flows. Important and present in many set-
tings, premelting dynamics makes some contact with
these phenomena, but it is distinct in its origin and con-
sequences. The fluid dynamics of interfacially premelted
fluid H20849the focus hereH20850 would not exist in the absence of
interfaces, but it requires neither contact lines nor gra-
dients in the coefficients of surface energy.
During complete interfacial premelting under long
range forces that are of a power-law form, the film thick-
ness d is related to the temperature T through d=H9261H20849T
m
?TH20850
?1/H9263
, in which H9261 and H9263 are positive constants, the lat-
ter is an integer that depends on the nature of the inter-
actions driving melting of the system. The dynamical
consequences emerge from the fact that the LRIs pro-
vide the field energy per molecule that shifts the equi-
librium domain of the liquid phase into the solid region
of the bulk phase diagram H20849Dash et al., 2006; Wettlaufer
and Worster, 2006H20850. For an interfacial film of thickness d,
the shift is then written as
H9262
d
H20849T,p,dH20850 = H9262
l
H20849T,pH20850 + UH11032H20849dH20850 = H9262
S
H20849T,pH20850, H2084926H20850
where H9262
l
H20849T,pH20850 and H9262
S
H20849T,pH20850 are the bulk chemical poten-
tials of the liquid and solid, respectively, and UH11032H20849dH20850 is the
derivative, with respect to d, of the underlying effective
interfacial free energy UH20849dH20850 which itself depends on the
nature of the intermolecular interactions in the system.
For example, in the case of nonretarded vdW-Ld forces,
a phenomenological description is given by
UH11032H20849dH20850 =
2H9004H9253H9268
2
H9267
l
d
3
, H2084927H20850
where H9268 is of order a molecular length, H9267
l
is the liquid
density, and H9004H9253 is the difference between the interfacial
energy of the dry interface between the solid phase and
the third phase, be it the vapor, a wall, or a different
orientation of the solid phase. The coefficient in Eq. H2084927H20850
is related to the Hamaker coefficient A
?
by A
?
/12H9266
=H9004H9253H9268
2
. Thus we see that interfacial melting occurs when
A
?
H110210, so that there is a force of repulsion?disjoining
pressure or thermomolecular pressure?between the
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media bounding the liquid film, written here for nonre-
tarded vdW-Ld forces. When the external pressure ap-
plied to the third phase, such as a wall, p
w
equal to that
applied to the solid p
S
balances the thermomolecular
pressure p
T
=?A
?
/6H9266d
3
, then p
w
=p
S
=p
T
+p
l
, where the
hydrodynamic pressure is p
l
. Combining this with the
Gibbs-Duhem relationship, which can be applied on
each side of a solid-melt interface, allows one to show in
general that when the system is in equilibrium at tem-
perature T then
p
S
? p
l
= p
T
=
H9267
S
q
m
T
m
H20849T
m
? TH20850, H2084928H20850
where H9267
S
is the density of the solid and q
m
is the latent
heat of fusion H20849Dash et al., 2006; Wettlaufer and Worster,
2006H20850. We imagine fixing the pressure in the bulk solid
phase and increasing the temperature toward T
m
from
below; the film thickens, the repulsive thermomolecular
pressure decreases, and hence the hydrodynamic pres-
sure increases, driving unfrozen liquid from high to low
temperatures. It is important to emphasize that the
transport of unfrozen liquid is driven by thermomolecu-
lar pressure gradients brought about by temperature
gradients and not capillarity.
We discuss a number of applications below but also
refer the interested reader to Dash et al. H208492006H20850 and Wet-
tlaufer and Worster H208492006H20850 for a more thorough treat-
ment.
To enhance the ready sintering of any material it has
long been known empirically that to bring the system
close to its melting point is advantageous. We now un-
derstand that this is due to the enhanced liquidity of the
system due to effects of premelting.
A temperature gradient parallel to the surface is re-
sponsible for a gradient in the film thickness and thus a
thermomolecular pressure gradient. If the film is suffi-
ciently thick, it can be treated as a thin viscous fluid and
its flow described by lubrication theory. The effect can
be responsible for trapping particles at a bulk solidifica-
tion front H20849Rempel and Worster, 2001H20850 and, if the tem-
perature gradient is maintained, the particle within the
solid will continue to move. Indeed, a simple analysis
reveals that the force on the particle can be written in a
manner directly analogous to Archimedean buoyancy
F
H6023
B
=?m
S
G
H6023
. Here m
S
is the mass of solid displaced by the
particle against which it premelts; G
H6023
=H11612
H6023
H20849H9004H9262H20850 is the gra-
dient in the generalized departure of the system from
bulk coexistence H9004H9262H11008T
m
?T, which is determined here
by a departure of temperature from the bulk melting
temperature. This is referred to as thermodynamic
buoyancy H20849Rempel et al., 2001H20850. As an example, a par-
ticle of a micron in radius moving under the influence of
a very weak temperature gradient H208490.025 K/mH20850 has G
H6023
H110153gH6023 it moves at about 1 H9262m/year for a nanometer-scale
film thickness. From the materials standpoint, we can
control the temperature and temperature gradient of
many systems accurately and over a wide range. Hence,
there is a potential to redistribute particles included in a
host solid using premelting dynamics. The idea is to en-
sure that they enter the host solid through quenching
and then redistribute them by manipulation of the tem-
perature field of the system.
The transport properties of unfrozen water in ice
polycrystals bears strongly on the redistribution of pa-
leoclimate records, retrieved from the ice cores, that are
redistributed by the premelted water H20849Dash et al., 2006H20850.
In fact, as mentioned, any physical or biological process
relying on the presence of the liquid phase will be
strongly influenced by grain boundary melting because
the surface area of a polycrystal is dominated by grain
boundary interfaces. Soluble species reside principally in
the liquid phase, and their diffusivity is orders of magni-
tude greater in the liquid than in the solid, hence the
fate and evolution of such species is controlled by the
volume fraction of liquid in the material. Indeed, a
proper homogenization of a two-phase, multicomponent
polycrystalline ice system shows a wide variety of
coupled processes. For example, a major species con-
trols the H20849subfreezingH20850 liquid fraction H20849through the liqui-
dusH20850 and thereby influences the minor species in striking
ways H20849Rempel et al., 2002H20850. In Fig. 24 we show an ex-
ample of how a major species can be deposited in the
sample out of phase with a minor species and after some
time the minor species is drawn into phase with the ma-
jor species. Such a homogenization of multicomponent
polycrystalline materials has not been harnessed for the
purposes of tailoring materials properties, and thus it
seems a unique area ripe for exploration.
Manipulation by freezing rate is also a possible mo-
dality of using long ranged interactions to control the
morphology and properties of a system. For example, it
is possible to treat a colloidal suspension as a ?binary
alloy? and yet, as shown in Fig. 25, at a solidification
FIG. 24. H20849Color onlineH20850 The time evolution of two periodic
profiles of bulk H20849salt concentration in the solid plus liquid sys-
temH20850 solute fields deposited on a polycrystalline ice sheet asyn-
chronously H20849Rempel et al., 2002H20850. The initial profiles are la-
beled 0. The units are dimensionless, with the horizontal axis
being the distance along the sample, but the essential point is
that the major species c
B2
H20849solidH20850 controls the liquid fraction
and it drives the minor species c
B1
H20849dashedH20850 into phase with it
on a time scale sufficiently rapidly that its original profile is
unchanged.
1924
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
front premelting can conspire to reject particles from the
advancing front to create a range of microscopic pat-
terns depending on concentration H20849Peppin et al., 2007,
2008H20850.
B. Colloids and self-assembly
1. Tailored building blocks: From hard spheres to patchy
colloids
The programmable assembly of designer colloidal
building blocks with engineered size, shape, and chemi-
cal anisotropy is the next frontier in the scientific and
technological development of soft materials H20849van Blaad-
eren, 2006H20850. Establishing this mastery will enable the as-
sembly of new photonic, electronic, plasmonic, and pho-
tovoltaic devices with unparalleled complexity and
precision, akin to that achieved in biological systems. As
an important first step toward realizing this goal, several
synthetic pathways have been recently reported that of-
fer precise control over colloid geometry at the nano-
scale H20849e.g., ellipsoids, triangular plates, cubes, and tetra-
podsH20850 and microscale H20849e.g., unary and binary colloidal
?molecules?H20850. Despite this impressive geometric control,
all of the building blocks shown in Fig. 26 are chemically
homogeneous. To date, the ability to spatially encode
their surfaces with the desired chemical heterogeneity
H20849or ?patchy? structureH20850 remains an elusive goal. Recent
simulations demonstrate the transformational impact
that this would have on their ability to self-organize into
unique structures H20849Zhang and Glotzer, 2004; van Blaad-
eren, 2006H20850; see Fig. 27.
To fully harness the potential that spatially and chemi-
cally anisotropic colloids H20849?patchy particles?H20850 provide,
the following questions must be addressed. First, how do
various particle motifs, e.g., janus spheres H20849particles that
consist of oppositely charged hemispheresH20850, ringlike H20849or
stripedH20850, or distinct patches influence colloidal assembly?
What role do vdW-Ld, electrostatic, and hydrophobic
forces play in controlling their assembly? Finally, how do
critical parameters such as patch size and charge density,
colloid size, density, volume fraction, and solution con-
ditions affect their ability to assemble into the desired
equilibrium H20849crystallineH20850 and nonequilibrium H20849gel or
glassyH20850 phases?
2. Synthesis and assembly of designer colloidal building blocks
Current research efforts focus on moving beyond the
traditional systems by enabling the creation of designer
colloidal building blocks such as colloidal molecules, ja-
nus spheres, and other patchy particle motifs. Myriad
particle motifs are envisioned, in which surface chemis-
try, shape anisotropy, faceting, pattern quantization, and
branching are controlled H20849Glotzer and Solomon, 2007H20850.
To date, significant progress has been made toward the
realization of many of these motifs. Below, we highlight
those assembly routes that yield chemically heteroge-
neous H20849or patchyH20850 colloids.
Granick and colleagues H20849Hong et al., 2006H20850 recently
reported a highly scalable synthetic pathway for creating
bipolar janus spheres. Such species, probably the sim-
plest example of patchy colloids, exhibit orientation-
dependent interactions that go with presentation of like
or unlike charges. This heterogeneous interaction land-
scape promotes the formation of colloidal molecules
that are themselves patchy in nature, somewhat akin to
globular proteins.
Recently lithographic patterning has been employed
to create more complex, patchy colloidal spheres and
FIG. 25. Colloidal suspensions frozen upward in a cell that is
free at its upper end The structure of the ice H20849dark regionsH20850
depends on the conditions of freezing and the particle concen-
trations. While H20849aH20850 and H20849dH20850 exhibit ice dendrites that align col-
loids in H20849cH20850 and H20849fH20850 a polygonal structure forms and there are
mixed states between the two geometries initiated by side
branching as shown in H20849eH20850. From Peppin et al., 2006, 2007.
(a) (b) (c) (d) (e) (f)
FIG. 26. Images of synthetic nanoscale and microscale building blocks with controlled architectures: H20849aH20850 Fe
2
O
3
ellipsoids. Adapted
from Wang, Brandl, et al., 2006. H20849bH20850 Ag triangular plates. Adapted from Sun, Mayers, et al., 2003. H20849cH20850 Ag cubes. From Sun and Xia,
2002. H20849dH20850 CdSe tetrapods. Adapted from Manna et al., 2000. H20849eH20850 Tetrahedral cluster of polystyrene microspheres H20849844 nm in
diameterH20850H20849e.g., unaryH20850. From Manoharan et al., 2003. H20849fH20850 Binary cluster of silica microspheres H208492.3 and 0.23 H9262m in diameter, where
the number of larger particles equals 8. Adapted from Cho, Yi, et al., 2005.
1925
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
wires. M?hwald and co-workers H20849Edwards et al., 2007H20850
created polystyrene microspheres decorated with gold
dots with sp valence H20849Fig. 28H20850. By rendering these
patches attractive, one may be able to assemble the
spheres into linear particle chains H20849or stringsH20850. Using on-
wire lithography, Mirkin and co-workers H20849Qin et al.,
2005H20850 created silver-gold metallic nanowires whose spa-
tial composition can be exquisitely tuned H20849Fig. 28H20850.
While neither approach is readily scalable to bulk quan-
tities, they represent the current state of the art in de-
signing chemically heterogeneous building blocks for
colloidal assembly.
3. Challenges and opportunities
Novel synthetic pathways must be developed that en-
able the creation of bulk quantities of designer colloidal
building blocks of controlled size, shape, and chemical
functionality. These pathways must be extended beyond
polymeric and silica-based colloids to functional build-
ing blocks, such as metals, semiconductors, and complex
oxides. A theoretical and computational framework
must be developed that is capable of predicting the
phase behavior, structure, and assembly of a diverse ar-
ray of particle motifs. The fundamental understanding
of the behavior of designer colloids that would emerge
from such predictive tools would provide the synthetic
guidelines for creating building blocks that self-assemble
into desired equilibrium H20849crystallineH20850 and nonequilib-
rium H20849gel or glassyH20850 phases, while avoiding undesired H20849or
jammedH20850 states.
If the above obstacles can be successfully overcome,
?colloids by design? offers enormous potential. Ad-
vances in our current synthetic capabilities and funda-
mental understanding of long range interactions at the
nanoscale would open new avenues to engineer crystal-
line and amorphous phases. For example, the elusive
goal of producing a diamond crystal with a lattice con-
stant suitable for photonic band-gap applications could
finally be realized. Additionally, colloidal gels could be
created with controlled connectivity and elasticity, which
may find potential application in the self-assembly of
novel Li-ion batteries H20849Cho et al., 2007H20850 or as inks for
direct-write assembly H20849Smay et al., 2002H20850 of highly effi-
cient solar arrays.
FIG. 27. H20849Color onlineH20850 Sche-
matic representations of patchy
particles and their predicted
equilibrium structures H20849side and
face viewsH20850 formed via self-
assembly: H20849aH20850 particles with four
circular patches assembled at a
dimensionless temperature of
kT/H9255=1.0, and H20849bH20850 particles
with a ringlike patch arranged
on an equatorial plane as-
sembled at kT/H9255=0.5. Adapted
from Zhang and Glotzer, 2004.
FIG. 28. H20849Color onlineH20850 Scan-
ning electron micrographs of
lithographically patterned col-
loids H20849leftH20850 and nanowires
H20849rightH20850. Left image: From Ed-
wards et al., 2007. Right image:
From Qin, Park et al., 2005.
1926
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
C. Self-assembly and emerging device applications
1. Electrochemical devices: Li-ion batteries from heterogeneous
colloids
Self-assembly, using intrinsic or applied forces, is a
widely accepted concept for the design of novel materi-
als and composites over a wide range of length scales.
By contrast, the self-assembly of subassemblies and
complete devices has been more challenging, although
good examples exist H20849Colvin et al., 1994; Rueckes et al.,
2000; Huang et al., 2001; Whitesides and Boncheva,
2002; Whitesides and Grzybowski, 2002; Gur et al.,
2005H20850. Recently it has been demonstrated H20849Cho et al.,
2007H20850 that electrochemical junctions can be formed be-
tween conductive device materials using combined
vdW-Ld and AB interactions, and in an additional step
that the simultaneous implementation of repulsive and
attractive interactions can be used to fabricate complete
colloidal-scale self-organizing and self-wiring devices.
These concepts have been demonstrated in prototype
self-organizing lithium rechargeable batteries.
With the advent of nanotechnology, numerous devices
have been created that are based on contact junctions
between components, including bistable carbon nano-
tube memory, diodes, light-emitting diodes H20849LEDsH20850, logic
gates, and solar cells H20849Colvin et al., 1994; Rueckes et al.,
2000; Huang et al., 2001; Gur et al., 2005H20850. It is likely that
attractive vdW-Ld forces provide the intimate contact
between materials that allows electronic transport across
these nanoscale Ohmic and pn junctions. In contrast,
bipolar electrochemical devices such as batteries, fuel
cells, electrochromic displays, and certain types of sen-
sors are based on the separation of electronically con-
ducting electrodes by ionically conducting but electroni-
cally insulating electrolytes. Electrode materials of the
same type H20849e.g., cathode or anodeH20850 simultaneously need
to be continuously connected to their respective current
collectors. In principle, the simultaneous control of re-
pulsive and attractive forces can enable the ?bottom-up?
self-organization of dissimilar colloids into complete bi-
polar devices as conceptualized in Fig. 29.
Although current theories provide general guidelines
for materials selection, the paucity of physical properties
data for lithium-active compounds require direct experi-
mentation. Using liquid-cell atomic force microscopy
H20849AFMH20850 and graphite tips H20851in the form of mesocarbon
microbeads H20849MCMBH20850H20852, numerous solid compounds and
solvents were screened to identify cathode-solvent-
anode combinations between which repulsive interac-
tions exist. Figure 30 shows typical results in which four
solvents and five different solids were characterized, and
shows a range of interactions from strongly attractive to
strongly repulsive. Detailed analysis showed that these
interactions cannot be explained on the basis of vdW-Ld
forces alone, but rather they indicate a strong and occa-
sionally dominant role of AB interactions. Using
LiCoO
2
as the positive and graphite as the negative elec-
trode material, and solvents providing a repulsive inter-
action H20849modified to obtain a lithium conducting electro-
lyteH20850, self-assembling batteries exhibiting reversible
Faradaic storage were demonstrated H20849Fig. 31H20850. This gen-
eral approach to colloidal-scale self-assembly of hetero-
geneous colloids could be extended to a range of bipolar
device types, and would benefit from improved funda-
mental understanding of the long range interactions be-
tween device materials.
2. Active electronic devices: Single-walled carbon nanotubes
Many novel applications of SWCNT have been pro-
posed and demonstrated H20849Baughman et al., 2002H20850.We
mention here only some representative examples in
electronics where the control of LRI is likely to play an
enabling role.
Semiconducting SWCNT channels have been used to
construct field effect transistors. They have some of the
highest mobilities and can act as ballistic conductors
H20849Tans et al., 1998; Appenzeller et al., 2002; Javey et al.,
A C
B
Substrate
Cathode Anode
A
123
<0
A
123
>0A
123
>0
(+) (-)
(+)
(-)
(-)
(+)
Graphite
FIG. 29. H20849Color onlineH20850 Schemes for self-organization of bipolar electrochemical devices, exemplified using repulsive short range
van der Waals?London dispersion H20849vdW-LdH20850 interaction H20849negative Hamaker constant, A
123
H110210H20850 to form the electrochemical junc-
tion while simultaneously using attractive vdW-Ld H20849A
121
H110220H20850 to form percolating networks of a single active material and/or to
selectively adhere to current collectors H20849A
123
H110220H20850. H20849aH20850 Battery formed from a single-particle pair. H20849bH20850 Nanorod-based batteries. H20849cH20850
Layered lithium-ion battery using repulsive short range forces to separate LiCoO
2
and graphite, while using vdW-Ld attraction to
form continuous percolating network of graphite anode.
1927
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
2003H20850. While individual devices with impressive perfor-
mance and some simple circuits have been demon-
strated, considerable progress is required in fabrication,
materials, and device design so that these devices can
find application as general-purpose circuit elements. Im-
portant issues include the development of effective gate
dielectric materials, fabrication of good contacts be-
tween SWCNT and electrodes, large-scale fabrication,
demonstration of high-frequency operability, and ability
to dope controllably with n- and p-type charge carriers.
Heinze et al. H208492002H20850 showed that CNT-FET operation
can be governed by modulation of the Schottky barrier
with the metal contact. Wind et al. H208492003H20850 measured per-
formance of CNT-FETs with multiple, individually ad-
dressable gate segments, suggesting a transition of
switching from Schottky barrier to the nanotube chan-
nel. Javey et al. H208492003H20850, using Pd contacts in CNT-FET,
reduced the Schottky barrier resistance and achieved
ballistic transport at room temperature. Guo and co-
workers H20849Guo, Goasguen, et al., 2002; Guo, Lundstrom,
and Datta, 2002H20850 provided a theoretical study on the
electrostatics of ballistic CNT-FET in one and two di-
mensions by solving the Poisson equation self-
consistently with carrier statistics. Because of their ro-
bustness and small dimensions, carbon nanotubes have
proven to be effective in field emission devices, serving
as field concentration points on an electrode from which
electrons are easily emitted H20849de Heer et al., 1995H20850. They
are being developed for use in field emission displays
H20849Lee et al., 2001H20850.
-5 0 5 10 15 20 25 30
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
F
o
r
ce
/
R
ad
iu
s
(
m
N
/m
)
Separation Distance (nm)
Teflon
LiCoO2
ITO
Si3N4
HOPG
-5 0 5 10 15 20 25 30
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
F
o
r
ce
/
R
ad
iu
s
(
m
N
/m
)
Separation Distance (nm)
Teflon
LiCoO2
ITO
Si3N4
HOPG
A
-5 0 5 10 15 20 25 30
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
F
o
r
ce
/
R
ad
i
u
s
(
m
N
/m
)
Separation Distance (nm)
Teflon
LiCoO2
ITO
Si3N4
HOPG
-5 0 5 10 15 20 25 30
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
F
o
r
ce
/
R
ad
iu
s
(
m
N
/m
)
Separation Distance (nm)
Teflon
LiCoO2
ITO
Si3N$
HOPG
B
C D
FIG. 30. H20849Color onlineH20850 Force curves measured between MCMB probes and five substrates H20849PTFE, LiCoO
2
, ITO, Si
3
N
4
, and
HOPGH20850 in H20849aH20850 acetonitrile, H20849bH20850 m-xylene, H20849cH20850 ethanol, and H20849dH20850 methylethyl ketone H20849MEK; 2-butanoneH20850 using liquid-cell atomic force
microscope H20849AFMH20850 show behavior ranging from strong attraction to strong repulsion.
FIG. 31. H20849Color onlineH20850 Three-electrode cells using lithium titanate reference electrodes H20849labeled RH20850 allowed working voltage as
well as the potentials at the working H20849LiCoO
2
labeled WH20850 and counter H20849MCMB carbon labeled CH20850 electrodes to be independently
measured. Right: Galvanostatic cycling H2084911th cycleH20850 of a self-organized LiCoO
2
-graphite rechargeable cell, charging at 100 H9262A and
discharging at ?20 H9262A conducted in MEK+0.1 M LiClO
4
+1 wt % PEG 1500.
1928
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
SWCNT-based channels expose all their atoms to the
environment. This exposure allows use as very sensitive
sensors H20849Qi et al., 2003; Staii et al., 2005; Gruner, 2006H20850.
Such devices can be gated efficiently in solutions with
possible use as sensors of dissolved species or of the
electrochemical environment H20849Rosenblatt et al., 2002H20850.
Rosenblatt et al. fabricated devices that, using electro-
lyte gating, achieved low contact resistance, excellent
gate coupling, and considerably higher mobility than
back-gated CNT-FETs. Larrimore et al. H208492006H20850 showed
that such devices can be used to measure the change in
solution electrostatic potential H20849for given gate voltageH20850
due to the presence of redox-active species in solution.
D. Nanoscale probes of long range interactions
1. Scanning probes
Scanning probe microscopy H20849SPMH20850 offers a way to
probe LRIs on the nanoscale directly H20849Hofer et al., 2003;
Butt et al., 2005; Foster and Hofer, 2006H20850. In the last
decade, SPM-based force spectroscopy and optical twee-
zers technology have led to new areas of learning, in-
cluding thermodynamics and kinetics of single-molecule
reactions H20849Ritort, 2006H20850. Given the large number of re-
views, we describe here only the general principles of
SPM as applied to LRIs, identify some of the challenges,
and outline some of the possible pathways for develop-
ment.
a. The SPM approach
The essence of an SPM approach is a combination of
a local probe bearing a specific aspect of studied func-
tionality H20851e.g., charged H20849Nonnemmacher et al., 1991H20850 or
chemically functionalized H20849Noy et al., 1995H20850H20852 combined
with a positioning system that locates it with respect to
the studied system, and a detection system that mea-
sures forces or currents on or other interactions through
the probes, thus linking nanoscale probe-surface interac-
tions to the macroscopic world.
From the perspective of LRI measurement function-
ality, there are two parameters that can be controlled in
an SPM experiment, namely, tip-surface separation and
probe bias. Detected are the probe current and force
components acting on the probe. Hence, the SPM imag-
ing mechanism can be represented as force-distance-bias
surface F
c
=F
c
H20849h,V
tip
,H9262H20850, where h is the tip-surface sepa-
ration H20849for noncontact methodsH20850 or indentation depth
H20849for contactH20850, V
tip
is the probe bias, and H9262 are parameters
describing chemical functionality of the probe H20849Fig. 32H20850.
One of the primary challenges in SPM probing of the
LRI is the decoupling of multiple interactions simulta-
neously present in the tip-surface junction that all con-
tribute to the forces, F
c
. One approach to measurement
of LRI is based on measurement force at several sepa-
rations from the surface. The alternative is offered by
modulated methods in which a specific functionality of
the probe is modulated and the oscillatory response is
detected. This allows both probing a subset of tip-
surface interactions H20849e.g., oscillating electrostatic poten-
tial does not affect VdW forcesH20850, minimizing noise level
utilizing resonance enhancement, and probing linear re-
sponse dynamics though measurement of both ampli-
tude and phase of response. Practically, only some as-
pects of probe functionality can be modulated at rates of
H110221 kHz, required for imaging. These include position h
H20849e.g., acoustic drivingH20850 or force F
c
H20849e.g., magnetic drivingH20850
and electrical bias V
tip
. Chemical functionality, hydro-
phobicity, and other chemical functionalities H9262 do not
offer obvious universal strategies for modulation, even
though optically and bias-induced transformations pro-
vide some possibilities. Below we discuss SPM tech-
niques based on whether the separation between differ-
ent interactions is achieved through force-based
Tip-surfaceseparation
Repulsive
Attractive
IC
C NC
Interleave
(a)
(b)
CID11CID12
CID11CID12
2
'
surftip
znc
VV
zCF
CID16
CID32
CID11 CID12
tipcc
VhFF ,CID32
EFM
SSPM
PFM
AFAM
UFM
HUEFM
Distance
Bias
Force
FIG. 32. The SPM imaging mechanism can be represented as a force distance bias surface. H20849aH20850 Force-based scanning probe
microscopy H20849SPMH20850 can be conveniently described using a force-distance curve, showing the regimes in which contact H20849CH20850, noncon-
tact H20849NCH20850, intermittent contact H20849ICH20850, and interleave imaging are performed. Also shown are domains of repulsive and attractive
tip-surface interactions. H20849bH20850 Voltage modulation SPMs can be described using force-distance bias surface. In the small-signal limit,
signal in techniques such as piezoresponse force microscopy, atomic force acoustic microscopy, electrostatic force microscopy, and
Kelvin probe force microscopy is directly related to the derivative in bias or distance direction. Adapted from Kalinin et al., 2007.
1929
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
measurements or voltage modulated measurements.
i. Direct force measurements. The most straightforward
application of SFM to probe long range interaction
forces is direct measurements of static H20849dcH20850 force be-
tween functionalized probe and the surface using the
cantilever position detection. The experimentally
achievable force sensitivity level is of the order of 1?10
pN, sufficient to probe a single hydrogen bond H20849Rief et
al., 1997H20850.
One of the most exciting examples of this approach is
molecular unfolding spectroscopy, in which force-
distance curves obtained at different rates contain infor-
mation on thermodynamics and kinetics of force-
induced reaction on a single molecule level.
Alternatively, the force-distance curves can be used to
determine local adhesion, indentation modulus, or long
range electrostatic interactions. Chemical functionaliza-
tion of the probe allows controlling the nature of tip-
surface interaction so as to probe hydrogen bonding and
polar interactions. Butt et al. H208492005H20850 provided an in-
depth account of force-based studies of materials.
The direct force measurements at each point bring a
dual challenge of large data acquisition times and rela-
tively low pixel density, and the necessity to analyze the
large 3D arrays of data to extract relevant parameters
that can be plotted as 2D maps. Existing approaches for
measuring LRIs are based on the decoupling topogra-
phy from measured force components either through H20849aH20850
height variation, H20849bH20850 use of force-modulation ap-
proaches, H20849cH20850 methods using fast spectroscopy, or H20849dH20850 de-
tection through complementary mechanical degrees of
freedom.
Examples of the height variation approach are well-
known double-pass or interleave modes. In these, the
first AFM scan is used to determine the position of the
surface, i.e., the condition at which the measured signal
RH20849h,V
0
H20850=R
0
, where R
0
is a set-point value. The feed-
back signal R can be static deflection for contact AFM,
oscillation amplitude for amplitude-based detection sig-
nal, or frequency shift for frequency-tracking methods.
The second scan is performed to determine interactions
at different distance and bias conditions to measure R
=RH20849h+H9254,V
1
H20850. As a typical example, magnetic and elec-
trostatic force microscopies utilize the fact that these
forces are relatively long range and weak compared to
the vdW interactions. Hence, once the position of the
surface has been determined, force measurement at sig-
nificant H20849H9254=10?500 nmH20850 separations yield magnetic H20849if
the probe is magnetizedH20850 or electrostatic H20849if the probe is
biasedH20850 force components.
The alternative approach to force detection is based
on ac modulation approaches, e.g., atomic force acoustic
microscopy, force modulation, and similar methods. In
these, the condition of RH20849h,V
0
H20850=R
0
is used to determine
the position of the surface, and modulation of the probe
height or bias is used to obtain additional information
on the distance or bias derivative of the force-distance-
bias surface. For example, in the small-signal approxi-
mation, the AFAM signal is related to H20849H11509h/H11509FH20850
V=const
.
This decoupling can be performed dynamically by re-
sponse phase and amplitude in AFM phase imaging
H20849amplitude yield information on topography and phase
yield information on elasticity and adhesionH20850.
A number of approaches for LRI detection are based
on data processing beyond simple lock-in or PLL detec-
tion. For example, the use of the functional agent
coupled to the probe with a flexible linker, combined
with separate detection of the signal from top and bot-
tom of the trajectory, forms the basis of molecular rec-
ognition imaging H20849Raab et al., 1999H20850. A number of meth-
ods have been developed to sample a larger region of
the force-distance phase space of the system beyond
small-signal approximation, e.g., pulsed force mode
H20849Miyatani et al., 1997H20850.
Finally, different interactions can be decoupled using
flexural and torsional degrees of freedom of the probe,
with normal force mapping the topography, while fric-
tion forces are sensitive to adhesion and chemical inter-
actions. In lateral force microscopy H20849Mate et al., 1987H20850,
the topography is detected from the normal force signal,
while the friction force detected from the lateral signal
provides information on the short range tip-surface in-
teractions that are sensitive to local chemical composi-
tion, molecular orientation, etc. Recent development of
the harmonic detection method H20849Sahin et al., 2007H20850 al-
lows reconstruction of tip-surface interactions based on
decoupling between torsional and lateral oscillations.
ii. Voltage modulation approaches. An efficient approach
to decoupling vdW-Ld and electrostatic components is
based on the use of voltage modulation. Electrically
modulated SPMs include Kelvin probe force microscopy
H20849KPFMH20850 with amplitude H20849measured forceH20850 and frequency
H20849measures force gradientH20850 feedback, and piezoresponse
force microscopy. Applications of these modes are well
known in areas as diversified as local work function im-
aging H20849Henning et al., 1995H20850, mapping electrostatic po-
tential in operational devices H20849Shikler et al., 1999H20850, and
sub-10-nm structural imaging of calcified tissues based
on piezoelectricity H20849Kalinin et al., 2005H20850. The spatial res-
olution and sensitivity of these methods are dictated by
environmental limitations and the nature of tip-surface
interactions to H110111 mV and H1101130 nm for KPFM and
1 pm/V and H110115 nm for PFM.
Progress in areas such as high-resolution imaging,
probing macromolecular transformations, or cellular
and subcellular electrophysiology necessitates imple-
mentation of electrically modulated SPM in liquid con-
ductive environments. The key task here is the capabil-
ity to control dc and ac electric potential on small length
scales. Experimentally, it has been shown that the use of
sufficiently high ac frequencies accessed through direct
imaging or frequency mixing down conversion allows
probing ac behavior H20849ac field is localizedH20850H20849Lynch et al.,
2006; Rodriguez et al., 2006H20850. At the same time, dc fields
are not localized in most solvents H20849Fig. 33H20850H20849Rodriguez,
Callahan, et al., 2007H20850. The development of insulated and
shielded probes H20849Rosner and van der Weide, 2000; Fre-
derix et al., 2005H20850 offers a pathway to future progress.
1930
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
iii. Functional probes. In the last several years, a number
of approaches for imaging have emerged based on
probes carrying more complex local functionalities, in-
cluding field effect transistors H20849Park et al., 2004H20850 and
electrochemical probes, among others. While the field is
still nascent, there is tremendous potential for progress.
iv. Probing dissipative dynamics. Understanding long
range interactions ideally requires probing not only the
conservative, but also the dissipative part of the force.
SFM offers a natural approach to probing local dissipa-
tion by comparing the quality factor of the cantilever
interacting with the surface and away from the surface.
Assuming that the internal dissipation of a cantilever
does not change and that environmental dissipation
changes only insignificantly, changes in the Q factor will
be due only to the tip-surface dissipation. Simple esti-
mates suggest that detection limits set by the thermome-
chanical noise are of the order of 3H1100310
?17
W H20849the ex-
perimentally achieved limit is H110110.03?0.1 pWH20850H20849Proksch
et al., 1999H20850. However, until recently dissipation probing
in SPMs has been limited. Indeed, cantilever dynamics
in the vicinity of resonance in the simplest harmonic os-
cillator approximation is described by three parameters:
resonance amplitude, resonance frequency, and Q factor.
Experimentally, SFM based on sinusoidal modulations,
which are now the vast majority of experimental plat-
forms, measure only two independent parameters, am-
plitude and phase in lock-in detection, and resonance
frequency and amplitude for phase-locked loop-based
frequency-tracking methods. Hence, dissipation is not
addressed directly. Practically, with a single-frequency
modulation, dissipation can be determined if the driving
force acting on the system is known. This provides an
additional constraint on the signal, an assumption im-
plicitly used by Cleveland et al. H208491998H20850 and Garcia
H20849Tamayo and Garcia, 1998H20850. In frequency-tracking meth-
ods H20849Albrecht et al., 1991H20850 the presence of an additional
constraint implies that the response amplitude is in-
versely proportional to the Q factor. These approaches
have been implemented by several groups to study mag-
netic dissipation H20849Gr?tter et al., 1997; Proksch et al.,
1999H20850, electrical dissipation H20849Denk and Pohl, 1991H20850, and
mechanical dissipation on atomic H20849Kantorovich and Tre-
vethan, 2004H20850 and molecular levels H20849Farrell et al., 2005H20850.
This approximation is applicable in a limited number
of cases, such as magnetic driving H20849if the sample is para-
magneticH20850 or acoustic driving with a flat H20849no dispersionH20850
transfer function. Even with acoustic driving by the pi-
ezo element, adopted in most SPMs, the nonidealities in
the transfer function of the piezodriver lead to the quali-
tatively incorrect results in, for example, MFM dissipa-
tion studies. For methods based on electric excitation,
the relationship between the driving voltage and the lo-
cal force is position dependent, and this dependence is
convoluted with cantilever response function, and the
two cannot be distinguished by single-frequency mea-
surements. Simple frequency sweeps or ring-down mea-
surements are usually time consuming, and thus have
only limited applicability to imaging applications. Re-
cently the advent of multiple-modulation methods such
as dual ac H20849Rodriguez, Callahan, et al., 2007H20850 and band
excitation H20849Jesse et al., 2007H20850 that allow simultaneous
probing of finite region of Fourier space offer a potential
path forward.
b. Future developments
One of the least understood and least used aspects of
nanoscale is the conversion between electrical and me-
chanical phenomena beyond simple capacitive forces.
Electromechanical coupling occurs everywhere in bio-
logical systems, in processes from hearing to motion to
cardiac activity; it also occurs in soft condensed matter
systems such as polyelectrolytes, redox-active molecules,
ferroelectric polymers, etc. However, electromechanical
properties are traditionally difficult to access even in
macroscopic systems due to the smallness of correspond-
ing coupling coefficients. On the nanoscale, even thin-
film measurements have become possible only with the
introduction of double-beam interferometer systems in
the 1990s. This is in stark contrast to the mechanical and
transport measurements that evolved continuously from
macroscale to nanoscale. SPM methods combining high
field localization and sensitivity to extremely small me-
chanical response offer a unique capability to study elec-
tromechanical coupling on the nanoscale. While intro-
duced only in 1996 for ferroelectrics, these methods
have become the mainstay for characterization of ferro-
electric materials. Their applicability to piezoelectric
biopolymers and III-V nitrides and imaging in liquid has
recently been demonstrated. These materials will open
the pathway for probing and controlling electromechani-
cal conversion on a single-molecule level and harnessing
these for device applications H20849Fig. 34H20850.
(e)
(f) 2CID80m
(c)
(d)
2CID80m
(a)
(b)
400nm
FIG. 33. H20849Color onlineH20850 Three cases of electrically modulated
SPM in liquid conductive environments. Schematics H20851H20849aH20850,H20849cH20850,H20849eH20850H20852
and PFM phase images H20851H20849bH20850,H20849dH20850,H20849fH20850H20852 of switching in H20849aH20850,H20849bH20850 local,
H20849cH20850,H20849dH20850 fractal, and H20849eH20850,H20849fH20850 nonlocal cases. The size of switched
region illustrates the spatial extent of the electric field. The
localized switching was observed only in ambient environment,
while fractal-like clusters are observed in methanol and isopro-
panol. In aqueous solutions and DI water, only nonlocal
switching is observed, indicative of nonlocalized dc electric
field. From Rodriquez et al., 2007.
1931
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
2. Scattering probes
a. X-ray scattering
The worldwide availability of synchrotron x-ray and
neutron scattering user facilities H20849Brown et al., 2006H20850 has
made it routinely possible to determine the structure
and dynamics of both free and buried interfaces at sub-
angstrom to millimeter length scales H20849Fenter and Stur-
chio, 2004H20850 and femtosecond to millisecond time scales
H20849Rheinst?dter et al., 2006H20850. Advances in computational
theory, algorithms, and processing power are beginning
to permit quantitative comparison of atomic and mo-
lecular interactions in condensed phases over similar
length and time scales H20849Cygan and Kubicki, 2001;
Schoen and Klapp, 2007H20850. This convergence of capabili-
ties offers unprecedented opportunities for exploring
LRIs at the nanoscale for materials science and engi-
neering.
Electromagnetic radiation is elastically or inelastically
scattered by the electrons surrounding atomic nuclei,
and the penetrating power of x rays has long been ap-
plied in imaging biological and materials structures. The
physics of interaction of electromagnetic radiation with
atoms is precisely known, permitting crystalline struc-
tures to be determined with extreme precision. Synchro-
tron x-ray sources provide highly intense beams, the en-
ergies and polarizations of which are highly tunable.
These enhancements over conventional x-ray sources
permit a wide range of unique studies of condensed mat-
ter, as recently reviewed for earth science applications
by Fenter et al. H208492002H20850, and references therein. The high
intensities and wide range of wavelengths available from
synchrotron sources permit determination of local or-
dering, even in glasses and other amorphous or weakly
crystalline materials H20849Egami, 2004, 2007H20850. X-ray absorp-
tion and fluorescence provide direct identification of the
specific elements interacting with the beam, and meth-
ods such as absorption fine structure, near-edge, and
photoelectron spectroscopy can be used to determine
the valence state of the target element and its nearest-
neighbor distributions and elemental compositions
H20849Evans et al., 2003; Glover and Chantler, 2007; Wada et
al., 2007H20850. 3D tomographic imaging of structures and el-
emental distributions within condensed matter is now
routinely performed at synchrotron light sources H20849Sutton
et al., 2002H20850, even at high temperatures and pressures
H20849Wang et al., 2005H20850.
Synchrotron x-ray methods for probing elemental dis-
tributions at interfaces have been highly developed in
recent years. Grazing-incidence small-angle scattering
and absorption fine structure can give precise informa-
tion on the bonding configuration of sorbed ions and the
distribution and orientation of nanoparticles on solid
substrates H20849Waychunas, 2002; Waychunas et al., 2005;
Saint-Lager et al., 2007H20850. Crystal truncation rod studies
can provide three-dimensional information, at sub-
angstrom resolution, on the relaxation of crystalline sur-
faces and the distributions of water and ions in liquid
electrolytes at the crystal-water interface H20849Catalano et
al., 2007; Lee et al., 2007; Zhang et al., 2007H20850. Some ex-
amples for cations on the rutile H20849110H20850 surface are de-
scribed in Sec. III.C.3. Resonant anomalous reflectivity
can probe a wide length scale on either side of a crystal/
fluid interface and can give information on the distribu-
tion of elements not specifically oriented relative to the
underlying crystal structure H20849Fenter et al., 2007; Park and
Fenter, 2007H20850. Reflectivity studies have also revealed the
interaction of water with hydrophobic surfaces H20849Poynor
et al., 2006H20850. X-ray standing wave studies can probe the
distribution of fluorescent trace elements at interfaces
with crystalline phases, also at sub-angstrom resolution,
and distinguish the ordered fraction of the total distribu-
tion of the trace element, relative to the underlying crys-
tal structure H20849Zhang et al., 2004H20850. Buried interfaces be-
tween dense solids have also been probed by
nonspecular scattering and absorption spectroscopy
H20849Lutzenkirchen-Hecht et al., 2007H20850.
b. Neutron scattering
The dependence of x-ray scattering on atomic number
presents problems for H20849aH20850 probing deeply into dense
matter, especially if the material is composed of heavy
elements; and H20849bH20850 detection of elements lighter than car-
bon. Neutrons, on the other hand, are scattered by
atomic nuclei, which occupy an exceedingly small vol-
ume of even the densest phases, such that neutrons can
penetrate many centimeters of dense matter. This makes
high pressure?temperature studies readily possible, and
important advances in deep Earth petrology and mate-
rials are being made at neutron diffraction facilities H20849Su-
zuki et al., 2001; Matthies et al., 2005H20850. The coherent scat-
Force
Electric
Field
Force or Bias
pH
Ionic strength
Solvent
Redox
potential
Binding site
Binding site
H-bonded
regions
Coordinate
V
GCID39
Reduced
Oxidized
(a)
(b)
FIG. 34. H20849Color onlineH20850 Probing and controlling electrome-
chanical conversion of a single molecule level. H20849aH20850 Schematic of
molecular transformation change in electric field in the tip-
surface junction. H20849bH20850 Corresponding force-bias-distance sur-
face. The redox potential of the molecule is expected to de-
pend on the fore acting on the molecule.
1932
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
tering length can be positive or negative, depending on
whether the interaction with the nucleus is attractive or
repulsive. Furthermore, the scattering lengths are a com-
plex function of the atomic weight H20849rather than atomic
numberH20850, and thus different isotopes of the same ele-
ment interact differently with neutron beams, and the
light elements interact strongly with neutrons, making
detection independent of atomic number H20849Bee, 1988;
Kreitmeir et al., 2007H20850. Hydrogen, which is nearly impos-
sible to detect by x-ray scattering, has a coherent neu-
tron scattering length comparable to heavier elements,
and a very large incoherent scattering length, useful for
probing the dynamics of hydrogen-bearing compounds.
The coherent scattering lengths of hydrogen and deute-
rium are opposite in sign, making many unique experi-
ments possible. Wenk H208492006H20850 provided a recent and thor-
ough review of neutron scattering applications in the
Earth sciences, highlighting techniques readily appli-
cable to materials science.
Neutron diffraction is complementary to x-ray diffrac-
tion. Light elements as well as heavy elements are
readily detected H20849Goncharenko, 2005; Kim et al., 2007H20850,
and isotope substitution can be used to isolate specific
substructures within complex crystals and weakly crys-
talline materials. A unique aspect of neutron scattering
is its sensitivity to magnetic substructures within solid
phases H20849Hayward et al., 2005; Kimber et al., 2006; Ape-
trei et al., 2007H20850. Isotope substitution can even be used to
determine the hydration and complexation structures of
ions in homogeneous liquids H20849Enderby, 1995; Neilson et
al., 2001; Muenter et al., 2007H20850. Neutron wide-angle,
small-angle, and ultrasmall-angle scattering has been ex-
tensively used to identify structures H20849scattering density
contrastsH20850 in complex fluids, polymer blends, and solid
phases angstrom to millimeter length scales. This tech-
nique is also widely employed in determining surface
fractals, particle size distributions, and pore-filling char-
acteristics of ceramics, metals, and mesoporous materi-
als H20849Ficker et al., 2007; Kaewsaiha et al., 2007; Rother et
al., 2007H20850.
Another unique feature of neutron scattering is the
strong incoherent scattering of hydrogen which results
from the gain or loss of energy of incident neutrons in-
teracting with the same hydrogen nucleus within a
sample at different times. Neutron inelastic H20849INSH20850, quasi-
elastic H20849QENSH20850, and spin-echo H20849NSEH20850 spectroscopies can
be used to probe dynamics ranging from vibrational
densities of states to translational and diffusional dy-
namics of water molecules and other hydrogen-bearing
molecules in bulk systems and as surface coatings at
open or buried interfaces H20849Cole et al., 2006H20850. QENS has
been extensively employed in studies of the dynamics of
bulk water H20849Teixeira et al., 1985H20850 and water confined
within nanoscale pores in a variety of inorganic sub-
strates, predominantly various silica matrices, such as
Vycor and Gelsil glass and MCM zeolites H20849Bellissent-
Funel et al., 1993, 1995; Takamuku et al., 1997; Takahara
et al., 1999, 2005; Zanotti et al., 1999; Crupi et al., 2002a,
2002b; Mansour et al., 2002; Faraone, Liu, Mou, Shih,
Brown, et al., 2003; Faraone, Liu, Mou, Shih, Copley,
and Chen, 2003; Liu et al., 2004H20850 and thin films on the
surfaces of metal-oxide nanoparticles H20849Mamontov, 2004,
2005a, 2005bH20850. These studies demonstrated that water in
such nanoscale environments is very different from bulk
water, exhibiting liquidlike dynamic motions at tempera-
tures far below the freezing temperature of bulk water.
Mamontov et al. H208492007H20850 showed that water sorbed on
rutile-structured TiO
2
and SnO
2
nanoparticles is struc-
turally similar to bulk water in contact with the macro-
scopic H20855110H20856 surfaces of these phases. As shown in Fig.
35 the first structural layer H20849L
1
H20850 is composed of water
molecules chemisorbed strongly to undercoordinated
metal atoms at the crystal surface, and the second struc-
tural layer H20849L
2
H20850 consists of water molecules strongly
hydrogen-bonded to L
1
and bridging surface oxygen at-
oms. Coupling QENS and MD, Mamontov et al. H208492007,
2008H20850 determined the rotational and translational dy-
namics of these and more loosely bound L
3
water mol-
ecules and uniquely assign the dynamic features to H20849aH20850
hindered rotations of water molecules within their
hydrogen-bonded cages in all layers, with characteristic
relaxation times H20849H9270H20850 in the 1?10 ps range; H20849bH20850 coupled
rotational-translational motions of water molecules with
undersaturated hydrogen-bonding environments in L
3
H2084910?100 ps rangeH20850; and H20849cH20850 translational jumps of L
2
wa-
ter molecules into L
3
H20849100?1000 ps rangeH20850. QENS is sen-
sitive only to hydrogen dynamics, but the MD simula-
tions capture both hydrogen and oxygen dynamics in
water molecules. As shown in Fig. 36, this enabled
unique identification of the origin of the slow transla-
tional component, and demonstration that L
1
water
molecules do not undergo translations on the time scale
detectable by QENS. The excellent agreement between
QENS and MD determinations of the rotational and
coupled rotation-translation components is shown in
Fig. 37.
The series of studies of water at metal-oxide surfaces
summarized in the Secs. III.C.3 and IV.D.2 demonstrates
the power of combining synchrotron x-ray reflectometry
with inelastic neutron scattering and ab initio optimized
classical MD simulations to uniquely determine the
structure and dynamics of interfacial water at the ang-
FIG. 35. H20849Color onlineH20850 Hydrogen-bonding configuration of
chemisorbed L
1
water molecules which sit atop bare five-
coordinated metal atoms at the H20855110H20856 surface of rutile H20849H9251-TiO
2
H20850
and cassiterite H20849H9251-SnO
2
H20850 as either intact water molecules H20849leftH20850
or dissociated hydroxyl groups H20849rightH20850 at ?terminal? H20849TH20850 and
?bridging? H20849BH20850 surface oxygen sites, and physisorbed L
2
water
molecules hydrogen bonded to the surface groups.
1933
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
strom and picosecond length and time scales. The com-
putational approaches treat LRIs H20849all of which arise
from interatomic and intra-atomic electron interactionsH20850
at varying degrees of rigor. It has been demonstrated in
this and many other systems that high-level treatment of
LRIs at the electronic structure level is only practical for
simple systems composed of very few atoms. However,
new experimental probes are available to directly assess
the level of rigor needed to capture the essential fea-
tures of interfacial systems at the nanoscale. Further-
more, advances in computational capabilities can in turn
be used to extract far more detail from scattering experi-
ments than is apparent from the scattering signal alone,
because the scattering signal can be simulated, and fea-
tures of that signal can be assigned.
V. FINDINGS AND RECOMMENDATIONS: LRI IN NS
WORKSHOP
From the workings of natural systems, by trial-and-
error synthesis of new materials, by playing on the
probabilistic aspects of the very small, by applying the
mechanics of the very large, by mimicking the design of
biological models, and by learning to work between the
atomic microscopic and the materials macroscopic, we
are creating a new science and engineering of ?nano-
scale? forces.
In this undertaking, there is healthy tension between
the need to learn basic interaction fundamentals and the
drive to realize practical powers of nanoscale forces. The
particularities of solvation, fluctuation, structure, and
electromagnetic fields acting at nanometer distances im-
mediately compel us to learn new physics and chemistry
as well as motivate us to design materials to new degrees
of detail.
The concept of harnessing nanoscale forces evokes
images of small low-energy control systems, colloids by
design of controlled size, shape, and chemical function-
ality for catalysts; electronic devices, micro and nanoma-
chines, and materials of ?superdiamond? strength. Pro-
totypical batteries and capacitors of high-energy density,
solar arrays, photocatalysts, and high efficiency fuel cells
are already being synthesized and tested. Most tantaliz-
ing is the potential mimicry of biological systems for en-
ergy conversion, efficient direct imaging, self-
reproduction, and self-organization.
Systematic progress and leaps of invention both re-
quire knowledge of fundamental interactions beyond
what we now possess. Intelligent recognition of this gap
in determining research support can remove heavy im-
pediments to practical invention. The greatest need in
building a systematic science of nanometer-scale materi-
als is to achieve reliable understanding and manipula-
tion of their organizing forces.
Finally, because a line of inquiry can stop at a local
minimum, it is beneficial to have external guidance and
new perspectives to find global answers. For example,
the discovery of the Casimir force was reportedly
prompted by a simple remark by Niels Bohr during a
conversation with Hendrik Casimir. Gathering together
scientists of different fields facilitates these conversa-
tions. Fundamental insights created in this way can lead
to transformative opportunities. Having prepared this
review from such a gathering, we believe our work
would not be complete without giving our recommenda-
tions for new lines of inquiry.
FIG. 36. H20849Color onlineH20850 MD simulation results of oxygen atom
trajectories above the rutile H20855110H20856 surface for water molecules
that originated in L
1
and L
2
, as a function of time. Adapted
from Mamontov et al., 2008.
FIG. 37. H20849Color onlineH20850 Characteristic residence times H20849H9270H20850 for
fast H20849rotationH20850 and intermediate H20849rotation-translationH20850 diffu-
sional components H20849open symbolsH20850 extracted from the QENS
scattering from rutile and cassiterite nanoparticles surfaces
H20849open symbols and dashed linesH20850, compared with H9270?s extracted
form MD H20849filled symbolsH20850 over the same energy transfer range
as the NIST DCS. Adapted from Mamontov et al., 2007.
1934
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
A. Recent scientific advances in LRI in NS
Perhaps the most pleasant result from the reports on
basic science had to do with electrodynamic H20849London
dispersion, van der Waals, Hamaker, Casimir, and Lif-
shitzH20850 forces. There has been rapid progress in formula-
tions of geometries relevant to nanoscale systems. There
is encouraging work on quantum mechanical computa-
tion of the material polarizabilities that animate these
forces. There is also good progress in techniques to col-
lect spectra needed for quantitative formulations.
Paradoxically, on the nanometer scale, the classic
Coulomb and double-layer electrostatic forces that are
widely encountered in most assembly processes are
dauntingly intractable, this despite the statistical me-
chanical analyses that are being carried out on various
idealized systems. Long range electric fields were, until
the Debye-Huckel theory, the bane of early studies of
dilute salt solutions. The correspondingly strong electric
fields near ions still pose qualitative problems. Their po-
larizing forces are so strong as to reorganize water sol-
vent so that continuum models break down. Ionic fields
are so powerful as to approach those known to break
down dielectrics. The chemical details of the ion orbitals
enforce strict geometric conditions on surrounding wa-
ters. Even the polarizability of different ions creates
charge-fluctuation forces as well as additional dielectric
forces that are too often neglected. For these reasons,
energies of individual ion solvation as well as ion inter-
action and ion approach to interfaces all occur under the
dominance of high electric fields.
In the area of polar H20849ABH20850 and H-bonded systems there
has been good progress on specific problems but still
little understanding of the powerful ?hydration? forces
that dominate interactions at approximately nanometer
separations or even of the solvation of ions. As with
electrostatics, there is an intrinsic connection between
field-based and structural forces; and again as with elec-
trostatics, there are few rigorously formulated and com-
puted cases.
Over all these considerations is the obvious fact that
real materials are organized by combinations of LRIs.
They vary differently with solution conditions. The bal-
ance between them varies even more. The different de-
gree of rigor and accuracy with which they can be com-
puted frustrated attempts to achieve reliable
combination and balance.
B. Challenges and needs in LRI in NS
Given the examples presented, it is at first difficult to
see the defects and deficiencies impeding vigorous
progress. The first point is the unevenness of that
progress and the need to locate areas requiring new
ideas and practices not currently the focus of active ef-
fort.
There is a striking disparity in the level of accuracy
with which we can speak of the different kinds of inter-
actions reviewed here. The general impression is that all
kinds of interactions must be simultaneously connected.
When it comes to combining component forces in order
to work with real systems, the chain of reasoning is as
strong as its weakest link. That is to say, interactions are
treated with such different degrees of approximation
that their combination limits the strength of the enter-
prise to its most poorly understood component. We need
reliable intermolecular force fields for computation. We
need sufficient understanding to model many-body in-
teractions. Particularly when working in the ?nano?
range, we must learn how to formulate effective interac-
tions for various length scales.
Electrostatic and solvation interactions involve polar-
ization, hydration, dielectric saturation, and structural
components that carry us far beyond the continuum-
dielectric models that are still most popular. Reversing
the scene of a half-century ago, it is the far more sophis-
ticated theory of electrodynamic forces H20849van der Waals,
Lifshitz, and CasimirH20850 that earns reliability. Electrostatic
and hydrogen-bonding polar interactions still frustrate
quantitative evaluation. There are not enough test sys-
tems where measured forces are examined in sufficient
detail that it is possible to test and validate quantitative
theories and experimental procedures. Even within the
world of computation and simulation there are not
enough criteria procedures where results of the same
problem can be compared so as to test different algo-
rithms. There are even impediments in language: physi-
cists think in terms of electrostatics and electrodynamics
that sometimes do not easily relate to the chemist?s
structural approach.
The need for a merger of approaches is being system-
atically met on many fronts. Casimir forces are being
formulated for arbitrary geometries. Simple analytical
approximations are emerging that establish limits on be-
havior for different governing physics. There is an
emerging, if overdue, recognition that it is necessary to
join molecular dynamics with more accurate dispersion
forces, generalized density functional theory, and quan-
tum chemistry. More rigorous theory and computational
algorithms, development of scattering, spectroscopic,
and local probes of structure, forces, and dynamics at
relevant time, space, and energy scales will combine to
create practical theoretical tools. In this way one can
expect new theoretical principles and practical imple-
mentations for real systems. Pushed to a natural limit,
this effort will provide in-depth understanding of self-
assembly for device manufacture.
How to proceed in this merger while keeping close to
real systems? Many kinds of phenomena can prove in-
structive: collective dynamics, colloidal crystal self-
assembly, and effective properties of zeolites and porous
solids; properties of inhomogeneous liquids; processes
driven by the defect interactions; ordered and amor-
phous interfaces of finite thickness; dimensionality and
charge effects; charged defects and electrostatic fields on
local phase stability and microscopic mechanisms of
phase transitions and front motion; phase transitions in
confined geometries?surface films and liquids; and con-
centrated solutions.
1935
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
From these we may expect new theoretical principles
and implementations for real systems. The possibility
exists to construct colloids by design?scalable synthetic
methods that control size, shape, and chemical function-
ality of colloidal building blocks; the development of
predictive tools for phase behavior, structure, and as-
sembly for diverse motifs; the capability to control two-
and three-dimensional electric and magnetic fields at
nanoscale; and so on.
In this way, we expect to be able to merge not only
physics, chemistry, and materials science and engineer-
ing approaches but also computational, experimental,
and chemical methods for realistic interpretation and
prediction.
C. Transformative opportunities from LRI in NS
The magic of the nanoscale H20849mixing the macromolecu-
lar with the macroscopicH20850 creates a learning path to cre-
ate new materials organized by long range forces. Even
with the current rough ideas of colloid organizing forces,
nanometer-size colloids or ?tailored building blocks? are
being synthesized with reliable uniformity so as to be
suspended or manipulated to create instructive assem-
blies. But the range of such materials is still limited;
theory and computation do not yet serve to describe or
to design desired assemblies; and synthetic pathways are
not yet ready for large supply.
Given needed support, the first two impediments will
likely yield to current effort. The third, synthesis, needs
bolder study but is not beyond achievement given the
rapid progress in the theory and measurement of colloi-
dal forces. Design of materials with temperature sensi-
tive vdW-Ld forces and ion-conducting membranes
show similar promise. Among the many tantalizing ap-
plications of carbon nanotubes as substrates for assem-
bly and for elements in construction is the qualitative
change to be expected from minor changes in carbon-
carbon networks. Given the dozens of windings in car-
bon atoms along the cylindrical shell, it becomes clear
that some CNTs are simply dielectrics while others are
conductors and still others are semiconductors whose
electrical and charge-fluctuation properties can be
switched by small changes in temperature. Obstacles to
progress are the difficulties in synthesis of precisely de-
fined CNTs as well as means to manipulate and watch
them. None of these presents a major obstacle in prin-
ciple, but practical procedures are nowhere within sight.
Batteries and high storage capacitors are perennial fa-
vorites on the must-make list for energy transforming
materials. New materials such as the microbatteries
from assembled AB colloids and capacitors from highly
polar nonreactive materials might begin to satisfy desid-
erata such as high-energy density, nontoxicity, durability,
and inexpensive synthesis.
Manipulation of material polarizabilities creates the
possibility of switchable friction between bodies whose
polarizabilities can be changed by applied fields. The ex-
istence of such repulsion between surfaces has recently
been reported. There is good reason to expect rapid
progress in materials design.
VI. CONCLUSIONS
Evidence gathered here reveals not only abundant
creativity in the design of devices but also inspiring re-
search on physical forces governing organization on the
nanometer scale. Careful consideration of this research
also shows an unevenness in our grasp of the basic orga-
nizing forces. Perhaps the greatest surprise is the inad-
equacy of theories of polar and electrostatic interactions
compared with the present-day sophistication in formu-
lating and computing charge-fluctuation forces. We can-
not avoid these areas of ignorance. Electrostatics and
polar interactions need conceptual advances. There is
still no good algorithm to handle the strong electric
fields near ions nor any language to include the powerful
solvation forces surrounding even the simplest mol-
ecules.
Realizing that few systems operate by only one kind
of interaction, we are faced with the paradox that the
most sophisticated and accurate theory about one kind
of force is vitiated when combined with less reliably for-
mulated interactions. We were unanimous in our ardent
plea that attention and significant research support be
devoted to fundamental science. With better force mea-
surement, theoretical formulation, and potentials for
computer simulation, design of materials will accelerate
and likely move faster than has been possible with trial-
and-error approaches.
A second fundamental need is education. There can
be a healthy change in emphasis on learning to learn:
better modes of teaching about forces as the need to
learn them is recognized; better preparation in the basics
of several sciences so as to remove the daunting fear of
new learning. The possibilities can be realized in many
ways, through supplementary coursework, improved
computer facilities, and specialized texts that are written
at a friendly level.
Then, third and greatest, there is the heavy work of
designing and making materials, testing them, and creat-
ing synthetic pathways to provide needed supply. While
theory and computation still fall short, the creation of
materials is simultaneously a source of testing design
ideas and of providing samples with systematically var-
ied properties for systematic construction. It would
qualitatively improve this iteration if material synthesis
were made more accessible. With the magnificent facili-
ties now being developed in national centers, including
Department of Energy nanoscience centers, people will
have new possibilities for design and application of de-
sign ideas. Training and linking programs that facilitate
use of existing facilities is an economically practical
strategy.
ACKNOWLEDGMENTS
We thank Harriet Kung, Director of Science, Office of
Basic Energy Sciences H20849OBESH20850 of the U.S. Department
1936
French et al.: Long range interactions in nanoscale science
Rev. Mod. Phys., Vol. 82, No. 2, April?June 2010
of Energy H20849DOEH20850, and Frances Hellman, chairperson of
the Council on Materials Science and Engineering of the
Division of Materials Science and Engineering, in the
OBES of the U.S. DOE and Arvind Kini of the U.S.
DOE, OBES, Division of Materials Science and Engi-
neering for sponsoring the workshop; Christie Ashton
and Sophia Kitts for organizational assistance; and Bar-
bara Brown French and Valerie Parsegian for assistance
editing the manuscript. J.L. acknowledges support from
an NSF CAREER award H20849Grant No. DMR 0448879, for
studying SAFsH20850, an AFOSR Young Investigator award
H20849Grant No. FA9550-07-1-0125, for studying sintering and
IGFs in WH20850, and a DOE-BES grant H20849Grant No. DE-
FG02-08ER46511, for studying GB transitions in SiH20850.
O.A.v.L. acknowledges support from SNL Truman Pro-
gram LDRD under Project No. 120209. Sandia is a mul-
tiprogram laboratory operated by Sandia Corporation, a
Lockheed Martin Co., for the United States Department
of Energy?s National Nuclear Security Administration
under Contract No. DE-AC04-94AL85000. R.P. ac-
knowledges support from the European Commission
under Contract No. NMP3-CT-2005-013862
H20849INCEMSH20850.
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