polymers Article Raising Two More Fundamental Questions Regarding the Classical Views on the Rheology of Polymer Melts Jean Pierre Ibar † Rheology Department, Polymat Institute, University of the Basque Country (UPV/EHU), 20018 Donostia (St Sebastian), Gipuskoa, Spain; jpibar@alum.mit.edu † Current address: Aurrera Center of Dissipative Interactions (ACDI), 20305 Irun, Gipuskoa, Spain. Abstract: The current paradigm of polymer flow assumes that (i) the effect of the molecular weight of the macromolecules, M, and of the temperature, T, on the expression of the viscosity of polymer melts separate; (ii) the molecular weight for entanglement, Mc, is independent of T; and (iii) the determination of Mc by the break in the log viscosity curve against log M unequivocally differentiates un-entangled melts from entangled melts. We use reliable rheological data on monodispersed polystyrene samples from very low molecular weight (M/Mc = 0.015) to relatively high molecular weight (M/Mc = 34) to test the separation of M and T in the expression of the viscosity; we reveal that an overall illusion of the validity of the separation of T and M is mathematically comprehensible, especially at high temperature and for M > 2Mc, but that, strictly speaking, the separation of M and T is not valid, except for certain periodic values of M equal to Mc, 2Mc, 4Mc, 8Mc, 16Mc, etc. (period doubling) organized around a “pole reference” value MR = 4Mc. We also reveal, for M < Mc, the existence of a lower molecular weight limit, M’c = Mc/8 for the onset of the macromolecular behavior (macro-coil). The discrete and periodic values of M that validate the separation of the effect of M and T on the viscosity generate the fragmentation of the molecular range into three rheological ranges. Likewise, we show that the effect of temperature is also fragmented into three rheological ranges for T > Tg: Tg < T< (Tg + 23 ◦), (Tg + 23 ◦) < T < TLL and T > TLL’ where TLL is the liquid-liquid temperature. Our conclusion is that the classical formulation of the viscosity of polymer melts is so overly simplified that it is missing important experimental facts, such as period doubling for the separation of T and M, TLL, M’c, and Mc, resulting in its inability to understand the true nature of Citation: Ibar, J.P. Raising Two More entanglements. We present in the discussion of the paper the alternative approach to the viscoelastic Fundamental Questions Regarding behavior, “the duality and cross-duality” of the Dual-conformers, showing how this model formalism the Classical Views on the Rheology was used to test mathematically and invalidate the separation of T and M in the classical formulation of Polymer Melts. Polymers 2024, 16, 2042. https://doi.org/10.3390/ of viscosity. polym16142042 Keywords: rheology; dual-phase rheology; newtonian viscosity; entanglements; entanglement Academic Editors: Alexander Malkin molecular weight; TLL transition; separation of variables; T and M dependency of viscosity; grain- and Martin Kröger field statistics; cross-dual-phase Received: 12 June 2024 Revised: 10 July 2024 Accepted: 11 July 2024 Published: 17 July 2024 1. Introduction Polymers are made up of long-chain molecules, which have been studied for more than 70 years, leading to our current understanding of how they flow under stress at high molding temperatures [1]. Viscosity, the stress divided by the strain rate, is the physical Copyright: © 2024 by the author. variable to follow to describe the effect of the molecular weight of the macromolecules, M, Licensee MDPI, Basel, Switzerland. This article is an open access article and of the temperature, T, on the ability of the polymer melts to flow in molds, extruders distributed under the terms and and injection molding machines. Fundamentally, the current paradigm of polymer flow [1] conditions of the Creative Commons relies on assumptions made at the beginning of the investigations in polymer science that Attribution (CC BY) license (https:// have rarely been questioned, if they ever were: creativecommons.org/licenses/by/ • Is the separation of the M and T variables in the expression of the viscosity truly validated? 4.0/). Polymers 2024, 16, 2042. https://doi.org/10.3390/polym16142042 https://www.mdpi.com/journal/polymers Polymers 2024, 16, 2042 2 of 34 • Is the molecular weight for entanglement, Mc, truly independent of T, and is the deter- mination of Mc by the break in the log viscosity curve against log M, unequivocally assigned to characterize “entanglements”? The objective of this communication is to provide qualified experimental and math- ematical evidence to be able to answer both questions. We analyze reliable rheological data [2,3] on a series of monodispersed polystyrene samples from very low molecular weight (M/Mc = 0.015) to relatively high molecular weight (M/Mc = 34). The qualification of these data is critically examined. Our tentative conclusions lead to answer no to both questions, yet their generalization will require that the experiments be repeated and ex- tended to include more molecular weight samples. This new tentative invalidation of the currently accepted views in the rheology of polymer melts comes after the publication [4] of inconsistencies found in the two most popular dynamic models of the rheology of melts, the Rouse model [5] and the reptation model [6]. In the Development section, we show that our viscosity equation explains the ap- pearance of validity of the separation of T and M, especially at high temperature and for M > 2Mc, but invalidates, strictly speaking, the separation of M and T except for certain periodic values of M equal to Mc, 2Mc, 4Mc, 8Mc, etc. (period doubling) organized around a “pole reference” value MR = 4Mc. This paper also suggests, for M < Mc, the existence of a lower molecular weight limit, M’c = Mc/8 that we assign to the onset of the macromolecular behavior (the formation of stable macro-coils). The discrete and periodic values of M that validate the separation of the effect of M and T on the viscosity generate the fragmentation of the molecular range into three rheological ranges for M > M’c and/or M > Mc as well as the fragmentation of the temperature range above Tg into three rheological ranges: Tg < T < (T + 23 ◦ g ), (Tg + 23 ◦) < T < TLL, and T > TLL, where TLL is the liquid-liquid tem- perature [4,7–12]. This fragmentation of the rheological data range challenges the classical claims that molecular motions involving the macromolecules occur identically over the full temperature and that the molecular range is only fragmented into two ranges by Mc. We introduce in the Discussion section the reasons that led to the formulation of the empirical viscosity equation used to test the separation of M and T and explain how their conception was inspired by the dual-split general concepts and the simulation work specifically done to simulate viscosity [7–9]. Figure 1 schematically summarizes the classical description of the rheology of polymer melts. Our disruptive new perspective of the effect of M and T to characterize the changes of viscosity presented in this paper reinforces what we already suggested somewhere else [4,10] that the current rheology paradigm has overly simplified the presentation of the experimental facts, perhaps to be able to understand them within the current paradigm framework, and has defensively declared artifacts any experimental deviation from their expectations when there was no possible explanation for those facts by the current paradigm. In our discussion, we briefly present “the duality and cross-duality” of the Dual-conformers, an alternative approach to the viscoelastic behavior of polymer macromolecules. In this new physics of polymer interactions, the Dual-conformers belong to the same macromolecule or different ones and assemble as dissipative open statistical systems of interactions [4,7–9]; we conclude that this model explains, at least qualitatively, all the new and provocative results regarding the effect of M and T on the viscosity introduced and analyzed in this paper. Polymers 2024, 16, x FOR PEER REVIEW 3 of 37 Polymers 20 24, 16, 2042 3 of 34 Figure 1. ClaFssigicuarlev i1e.w CloafstshiceaLl ovgieowf Nofe wthteo nLioang ovifs Ncoeswitytoonfiaanp ovliysmcoesritmy eoltf ap lpoottleydmaegra imnsetltt hpelotted against the Log Log of its moloefc uitlsa rmwoeliegchut laatrT wcoenigsthatn ta.tS Tu cchoanpstloatnits. sShuocwhn a, f oprloint sitsa nscheo, ownnp, .5fo0r( Finigsutraen5c.e4,) oonf p.50 (Figure 5.4) of Graessley’s boGorkae[1s]s.lTeyh’es sbloopoek o[f1]1. fTohr eM sl Mc. The coGnLceEpDt”o fr“eegniotann gwleimthe nat ”slionppeo loyfm 1e.r sTwhea stdweori vreedgifornoms afirgeu srienpgaorauttewd hbyya Mc, the molecular sharp breakwineitghehtv fisocro esintytaoncgcluermreednatts.M Tchien lFairgguer ein1c, rseeapsaera itnin tghteh eeffshecotr toefr tmhoel emcuollaercular weight on the weights fromvitshceolsointgye irsm reomlecaurlkaar bwleyi gshtatst.eWd hbeyn tthhee itnemcrpeearsaetu orfe tishceh salnogpeed ftrooamd i1ff etore 3n.t4, which occurs for value, Figure 1 remains basically unchanged, except for a vertical shift of the lines, their M > M . The concept of “entanglement” in polymers was derived from figuring out why slopes remaining cconstant to their respective values across Mc, and their intersection’s x-coordinateaa slshoarrepm barineainkg icno tnhseta vntisactoMsict’y. Aosccaucrornesde qaut eMncce ,inth FeiNguewret o1n, isaenpvairsacotisnitgy tµhoe shorter molecular is the produwcteoigf htwtso firnodmep tehned leonntgteerrm ms,oolenceuvlaarry winegigwhittsh. tWemhpeenr athtuer teeomnplye, rTa,taunrde tihs echanged to a differ- other one wietnhtm voalleuceu,l aFriwgueirgeh 1t ornelmy,aMin. sT hbiassiiscwalhlyat uisndcehsaignngaetded, ebxyc“etphte fsoerp aar avtieornticoaf l shift of the lines, M and T in the expression of viscosity”. Generally speaking, the experimental evidence their slopes remaining constant to their respective values across M , and their intersec- backing up the veracity of the features in Figure 1 is overwhelming and has been verifiecd and agreed ttoiobny’sa xm-cuoltoitruddineaotfeq aulasloifi reedmpaeienrisntgh rcooungshtoauntt tahte Mlitce,r.a Atusr ea [c1o].nTsheqeuauenthcoer,s the Newtonian vis- of the data tchoastiwtye µre- aisn athlyez eprinodthuisctp oafp tewr, oDo i rn. Pdieeprseonndaenndt tDerr.mSus,s uoknie[ 2v,a3r]yvienrgifi weditahn tdemperature only, T, explicitly claimed that their data validated the classical formula, thus the separation of M and T; theayndco tmhpe aortehdetrh oeinred watiathto motohleercurelsaera wrcehiegrhs’td oantaly, ,a gMai. nTchlaisi misi nwghaact oims dpelestiegnated by “the sep- compatibilitya.rUatsiionng aofd Miffe arenndt pTe risnp ethcteiv eextphraenscsliaosnsi coafl ,vthisicsoasuitthyo”r. aGlseoneexraamlliyn esdpcelaokseinlyg, the experimental the experimenvtiadlernescuel tbsainck[i2n,3g] uanpd threep voretreadciinty[ 1o0f] tahned feCahtauprteesr i3no Ff i[g7u],rteh e1 iims poavcetrowf htheelming and has been measuremenvteurinficedrt aaindtie asgorneethde tsolo bpye sa, imntuelrtcietputds,ea onfd qtuhaeldifieteedrm pieneartsio tnhroofuMgchaotutht ethire literature [1]. The intersectionaiuntFhiogrusr eof1 tfhoer tdhaetad atthaaot fwPeie rreso-annaanlydzSeu isnu kthi.isIt pwaapserre, pDorr.t ePdieirnso[7n,1 a0n] dth Datr. Susuki [2,3] veri- what is showfinedh earneda seFxipgluicreitl1yw calas,imineddee tdh,a“tv tihsueairl ldyaotbas veravleidda”t,epdro tvhied icnlgastshiecaalp fpoarrmenutla, thus the separa- validation otfiothne ocfla Mssi caanl dap Tp;r othacehy. cFoumrthpearrmedo rteh, esierv edraatlar etsoe aortchheerr sroesfetharecUhenrivs’e rdsaittya, again claiming a of Pau and Pays de l’Adour (UPPA) [13–16] worked on the rheology of monodispersed complete compatibility. Using a different perspective than classical, this author also ex- Polystyrene polymers to study their blends [13,15,16] or validate rheological models based on tube dynaammiincsed[1 5c]l,oasnedlyc tohme peaxrpedertihmeeirnrteaslu rletssuwltitsh inth [o2s,e3]d aonnde prerepvoiortuesdly inb y[1o0th] earnd Chapter 3 of [7], scientists, extphleic iimtlypvaaclti doaft itnhge tmheeiar sruesruelmtsewnitt hunthcoesretaoifnPtiieerss oonn athnde sSluospueksi., Iinntseurmcempatsr,y ,and the determina- tion of Mc at their intersection in Figure 1 for the data of Pierson and Susuki. It was re- ported in [7,10] that what is shown here as Figure 1 was, indeed, “visually observed”, providing the apparent validation of the classical approach. Furthermore, several re- searchers of the University of Pau and Pays de l’Adour (UPPA) [13–16] worked on the rheology of monodispersed Polystyrene polymers to study their blends [13,15,16] or Polymers 2024, 16, 2042 4 of 34 the data of Susuki and Pierson, reanalyzed in the present paper from another angle than the classical rheological approach, are reliable results: they have been verified by us and by several authors who have all claimed the validation of the classical view. The only difference between our own conclusion and these authors’ conclusions was the presence of the word “apparent” before validation in our conclusion. This paper is divided into three sections: in the first section (A), illustrated by Figures 2–8, we analyze Newtonian viscosity data plotted like in Figure 1 at various temper- atures, showing that the classical picture of the effect of T and M on the viscosity and on the definition of Mc depicted by Figure 1 may be incomplete and too simplistic to be practically useful and leads to the wrong conclusions regarding the nature of entanglements. In the second section (B), illustrated in Figures 9–21, we introduce a new empirical formulation of the viscosity (inspired by the Cross-Duality concept) to address the testing of the sepa- ration of M and T. This mathematical formula quantitatively describes the viscosity data “perfectly” (r2 = 1.0) when M and T vary and can be plotted like in Figure 1 to determine the slopes projected by the classical formulation of viscosity (Figure 1). This mathematical fitting exercise reveals the invalidation of the separation of M and T except at values of M corresponding to period doubling multiples of Mc. We also conclude in section (B) on the futility of finding a theoretical explanation for the value of the slopes, 1 and 3.4, respectively, for M < Mc and M > Mc in Figure 1. Finally, in the last section (Discussion), we point to the main differences between the admitted polymer dynamic models’ approach to mechanical flow and our disruptive model of polymer interactions that is manifested by a new interpretation of the viscoelastic effects and entanglements. A. The Effect of Molecular weight on the Newtonian Viscosity at various constant T. Figures 2–7 plot Log (µo) vs. Log (M/Me) data obtained at various temperatures for a series of monodispersed Polystyrene (PS) samples. µo is the Newtonian viscosity (in Poises), and Me is the molecular weight between entanglements, known to correspond to ~Mc/2 where Mc demarks the break between the un-entangled and entangled data in Figure 1. The data were first generated and analyzed by Pierson in his PhD thesis at the University of Strasbourg [2]. The same monodispersed PS samples were part of an extended study and retested by Susuki [3], also at the University of Strasbourg, revealing an excellent repeatability of the results of Pierson and a concurring analysis using the classical formula of rheology. As already mentioned, Pierson and Susuki’s viscosity results on monodispersed PS were tested and validated over the years by several scientists at the university of Pau and Pays de l’Adour (UPPA), for instance, by Majeste [16], Montfort [13], Marin [14], and Cassagnau [15]. The Susuki’s data will be used in section B of the paper because it involves more samples with M > Mc. Theses authors adhered to the established practice of the separation of the effect of T and M during their analysis. This provides an excellent example of rheological data that can be analyzed with an “apparent great satisfaction” according to the classical views, yet with great concerns when our approach analyzes the same data from a different angle. For instance, Pierson and Susuki use the Vogel-Fulcher equation and the WLF equations (Equations (1) and (2)) to express the temperature dependence of the friction factor of the Newtonian viscosity at constant M. Logµo = A + B/(T − T2) (1) where A, B, and T2 are fitting parameters that vary with M [1,17]. This equation can be rewritten with respect to a temperature reference, for instance, Tg at which the viscosity is µog: log (µo/(µog) = −C1g(T − Tg)/(C2g + (T − Tg) (2) with C1g = B/(Tg- T2) and C2g = (Tg − T2) (3) where the fitting constants, C1g and C2g are often called the WLF constants [1,17]. The viscosity is now expressed as a function of the free volume, proportional to (T − Tg). The Polymers 2024, 16, x FOR PEER REVIEW 5 of 37 log (µo/(µog) = −C1g(T − Tg)/(C2g + (T − Tg) (2) with C1g = B/(Tg– T2) and C2g = (Tg − T2) (3) where the fitting constants, C1g and C2g are often called the WLF constants [1,17]. The Polymers 2024, 16, 2042 viscosity is now expressed as a function of the free volume, proportional to (T − Tg). The 5 of 34 validity of the Vogel-Fulcher and the WLF equations over the temperature range is re- stricted by the classical models to hold between Tg to (Tg + 100°); beyond that, Equation validity of the Vogel-Fulcher and the WLF equations over the temperature range is restricted (1) is often replaced by a thermally activated expression providing an Arrhenius descrip- by the classical models to hold between Tg to (T + 100 ◦); beyond that, Equation (1) is often tion of the temperature dependence of the Newtoniagn viscosity Chapter 11 of [17]. The replaced by a thermally activated expression providing an Arrhenius description of the fragmentation of the rheological data into two ranges has been critically reviewed by this temperature dependence of the Newtonian viscosity Chapter 11 of [17]. The fragmentation author in several publications [10–12], Chapter 3 of [6]; it is also an important issue raised of the rheological data into two ranges has been critically reviewed by this author in in the present communication. several publications [10–12], Chapter 3 of [6]; it is also an important issue raised in the Fipgruersee 2n tcocorrmesmpuonndicsa ttoio tnh.e data of Pierson [2] at T = 210 °C plotted pursuant to Figure 1. The unit fFoirg uvirseco2scitoyr rise sPpooinsed (si.teo., t0h.1e Pdaa-tsa). oTfhPe ixe rasxoins i[s2 ]loagt (TM=/M2e1)0 w◦hCerpel oMtte d~ Mpucr/s2u. ant to The vaFluige uorfe M1.e iTs heequanli ttof oρrRvTi/sGcoNs pituyrissuPanoti steo (tih.ee. ,th0.e1oPrya -osf) .ruTbhbeerx ealxaisstiicsitlyo gw(hMer/eM ρ ei)s where the meMlt de e~nMsitcy/; 2R. Tish tehve aglause coofnMsteanist;e Tq uisa tlhtoe aρbRsTo/luGtNe tpeumrspuearnattutroet;h aentdh GeoNr yiso tfhreu pblbaeteraeula sticity elastic wmhoedruelρusis otfh eanm eenlttadnegnlseidty m; Relits [t1h,4e].g aWsec ounsest arendt; sTqiusatrhee saybmsobloultse ftoerm Mp e>r aMture; acand nd GN is the plateau elastic modulus of an entangled melt [1,4]. We use red square symbols for dark dots for M < Mc. Above Mc, the data conform with what is classically expected: the M > Mc and dark dots for M < Mc. Above Mc, the data conform with what is classically log-loge pxploetc itse da :stthraeiglohgt- llionge pwloitthi sslaosptera =i g3h.4t7l5in aenwd iitnhteslrocpepet= 13.1.46745. and intercept 1.164. Figure 2F.i glougrµeo2(.Mlo)g vµeor(sMus) Mve/rMsuespMlott/Mede pulortstueadnpt utors Fuiagnutrteo 1F. iDguatrae o1f. PDiaetrasonf P[2ie] rasto Tn =[2 2]1a0t °TC=. 210 ◦C. Below M , the log-log plot of the data should be a straight line with slope 1 according Below Mc, the lo cg-log plot of the data should be a straight line with slope 1 according to classical predictions; instead, the dark dots align according to a curve more or less to classsmicaolo pthrleyd. iTchtiiosncsu; rivnasttueraed, ,a pthpea rdeanrtkly dcontst raaldigicnti nagcctohredpinregd ticot iao ncuofrvthe emRorues eomr loedsesl of the smoothNlye.w Tthoinsi acunrviastcuorsei,t yapproapreonrttliyo ncaolnittyratdoiMctinfogr tuhne -penretadnicgtlieodn poof ltyhme eRrosu(Mse m< Modce)l, hoaf s been the Nerweptoonrtieadn avnisdcoesxiptyla ipnreodpboyrtisoenvaelriatlya tuot hMor fso[r1 ,u1n0-,1e6n,t1a7n]g, clehda pptoerly3moefr[s6 ](.MIn

Mc, continuesM3.5, an exponent thought to be characteristic, precisely, to the presence of entanglements? to increase slightly as T decreases: 3.61 for T = 125 ◦C, and 3.63 for T = 115 ◦C. As already mentioned, this slope is not independent of T, as it should be to validate the separation of M and T in the formulation of the Newtonian viscosity. It is remarkable that Figures 5 and 6 show no entanglement break from M = 0.25 Me to 100 Me, raising legitimate concerns regarding the concept of entanglement and its origin. Polymers 2024, 16, x FOR PEER REVIEW 7 of 37 Polymers 2024, 16, 2042 7 of 34 FiguFreig 4u. rLeo4g.µLo g(Mµo) d(Mat)ad oaft aPioefrsPoiner [s2o]n a[t2 T] a=t T ◦ 145= °1C4 5ploCttpo ed lo pttuedrsupuanrst utoan FtigtourFeig 1u. re 1. Additionally, the meaning of the slopes of Logµo vs. Log M themselves raise questions, inFcigluudrien 4g ainppthlieese ntota Tn g=l e1m45e °nCt ,s eshctoiowninogf Mth.eT thhereter ardeigtiioonnsa laanpdp trhoeaicrh recospnesicdtievres sthloepcehsa, nge fromo fletfht etov arilguheto: 4f .t3h3e5,e 2x.p3o73n,e annt,dt y3p.5i7c6a,l lryesfproemctiv1etloy.3 T.4h,et obrbeeakims opcocrutar natt, Mev’ce n= e0s.2se5n Mtiael , to Polymers 2024, 16, x FOR PEER REVIaEnWd cMhar a=c t2e rMize. tFhiegumreosle 5c ualnard w6,e ifgohr tTd e=p 1e2n5d e°Cnc aenodf v1i1s5c o°sCit, yr.eWspheactiavbeloyu, tairfet hdiisffe8e xropef no3tn7 ent c e turns out to be the wrong variable to tell what happens to the statistical properties of a set because they only reveal two log-log straight lines, not 3. The originality is the absence of of macromolecules when their molecular weight increases? the break at M = Mc! The slope in region 1, for M < 0.25 Me, continues to sharply increase as T decreases, 6.0 for 125 °C, 6.38 for 115 °C. The slope in region 3, for M > Mc, continues to increase slightly as T decreases: 3.61 for T = 125 °C, and 3.63 for T = 115 °C. As already mentioned, this slope is not independent of T, as it should be to validate the separation of M and T in the formulation of the Newtonian viscosity. It is remarkable that Figures 5 and 6 show no entanglement break from M = 0.25 Me to 100 Me, raising legitimate concerns regarding the concept of entanglement and its origin. Additionally, the meaning of the slopes of Logµo vs. Log M themselves raise ques- tions, including in the entanglement section of M. The traditional approach considers the change of the value of the exponent, typically from 1 to 3.4, to be important, even essential, to characterize the molecular weight dependence of viscosity. What about if this exponent turns out to be the wrong variable to tell what happens to the statistical properties of a set of macromolecules when their molecular weight increases? FiguFrieg u5r. eL5o.gLµoog (µMo)( dMa)tad aotfa PoiefrPsioenrs [o2n] a[2t ]Ta =t T12=5 1°2C5 p ◦lCottpeldot pteudrspuuarnstu taon tFitgouFrieg u1.r e 1. Figure 6. Logµo (M) data of Pierson [2] at T = 115 °C plotted pursuant to Figure 1. The regressions consider the 2 ranges across Mc separate. If we fit one single range starting from M’c the slope is 3.604 (r2 = 0.998). Polymers 2024, 16, 2042 8 of 34 ◦ Polymers 2024, 16, x FOR PEER REVIEWF igure 6. Logµo (M) data of Pierson [2] at T = 115 C plotted pursuant to Figure 1. The reg9r eosfs i3o7n s consider the 2 ranges across Mc separate. If we fit one single range starting from M’c the slope is 3.604 (r2 = 0.998). FiFgiugruer 7e. 7L.oLgo µgoµ o(M(M) d) adtat aofo Pf iPeiresrosno n[2[]2 a]ta Tt Texetxrtarpaoploaltaetde dtot o∞∞ anadn dplpolttoetdte dpuprusursaunatn ttot FoiFgiugruer 1e. 1. Take Figure 7 for instance: it is a plot of Log µ vs. Log M for T = ∞ in the Vogel- Take Figure 7 for instance: it is a plot of Log µ ovs. Log M for T = ∞ in the Vogel- Fulcher equation. This figure could be used as a operfect illustration of the teaching of Futhlcehcelra sesqiucaaltivoine.w Tshriesg fiagrduirneg coeuntldan bgel eumseedn tass: taw poerrfeegcito inllsuosftrMatiaorne ovfi stihbele t,ewacehllinsgep oafr tahteed , claanssdicwale vlliedwessc rreigbaerddbinygt henetcalnagssleicmaletnotos:l stw, wo irtehgaiosnlos poef Mof a3r.3e7 v, iaslimbleo,s wt peellr sfeecptalyra3te.4d,,a abnodv e wMellc ,daenscdriabesdlo bpye tohfe1 c.5labsseilcoawl toMocls. , Awlittho au gslhopthee osf l3o.p3e7, baelmlowostM pcerifsenctolyt 13,.4w, aitbhopveo sMsicb,l e anrdea as osnlospgei voefn 1.b5e bfoerloe,wth Me b.r Aealkthboeutgwhe ethnet hsleohpieg bhealonwd lMow ims nooletc 1u,l awriwthe pigohstssibisles rheaarsponanc c s d well defined. The problem, though, is that the Mc value at this sharp break, Mc = 47,663, giivsesnig bneiffiocraen, ttlhyeg brreeaatekr b(e+t3w6e%e)nt hthaen hthigehM an~d 3l5o,w00 m0 collaescsuiclaarl lywaeisgshigtns eisd sthoatrhpe amnodl ewcuellla c r dewfieniegdh.t Tfohre epnrtoabnlgelmem, tehnotus gfohr, PisS t,hpa.t5 5th[e1 ]M. Fci gvualruee2 acto trhreisc tslhyasrhpo bwrseaMk, Matc3 =5 ,0407,066fo3r, itsh c e significantly greater (+36%) than the Mc~ 35,000 classically assigned to the molecular 1 weight for entanglements for PS, p.55 [1]. Figure 2 correctly shows Mc at 35,000 for the same data. Would this be a sign to indicate that Mc varies with T, which it should not do according to the classical paradigm? Figures 5 and 6 show the break in the viscosity curve occurring at 0.25 Me, not 2Me as projected by the classical explanation of entanglements. One sees that using the break in the viscosity curve to define Mc, like it is illustrated in Figure 1, is not coherent with the claim that Mc does not vary with temperature: Figures 5, 6, and 7 provide clear evidence that the value of Mc varies with T when it is defined by the break of the log viscosity-logM curve (Figure 1). An alternative explanation is based on the instability of the entanglement state [18], resulting in the possibility of manipulat- ing the value of Mc, refer to the chapter 8 of [7]. Finally, there is the question of the meaning of M’c~0.25 Me = Mc/8 = 4375 g/mole. This molecular weight stands out in all the figures, separating the very low M and the region below Mc. Both Pierson [2] and Susuki [3] admit the existence of a deviation from the standard analysis in this region without raising the possibility of a molecular transi- tion. To the best of our knowledge, the recognition of such a deviation as a true “molecular transition” below Mc has never been reported in the literature, except recently while ana- lyzing TLL (M) in Ref. [12]. The molecular weight of a monomeric repeat unit for PS is 104.15 g/mole, and M’c corresponds to 42 such units. As T decreases from Figures 2–7 Polymers 2024, 16, 2042 9 of 34 same data. Would this be a sign to indicate that Mc varies with T, which it should not do according to the classical paradigm? Figures 5 and 6 show the break in the viscosity curve occurring at 0.25 Me, not 2Me as projected by the classical explanation of entanglements. One sees that using the break in the viscosity curve to define Mc, like it is illustrated in Figure 1, is not coherent with the claim that Mc does not vary with temperature: Figure 5, Figure 6, and Figure 7 provide clear evidence that the value of Mc varies with T when it is defined by the break of the log viscosity-logM curve (Figure 1). An alternative explanation is based on the instability of the entanglement state [18], resulting in the possibility of manipulating the value of Mc, refer to the chapter 8 of [7]. Finally, there is the question of the meaning of M’c~0.25 Me = Mc/8 = 4375 g/mole. This molecular weight stands out in all the figures, separating the very low M and the region below Mc. Both Pierson [2] and Susuki [3] admit the existence of a deviation from the standard analysis in this region without raising the possibility of a molecular transition. To the best of our knowledge, the recognition of such a deviation as a true “molecular transition” below Mc has never been reported in the literature, except recently while analyzing TLL (M) in Ref. [12]. The molecular weight of a monomeric repeat unit for PS is 104.15 g/mole, and M’c corresponds to 42 such units. As T decreases from Figures 2–7 from T infinity to T = 115 ◦C, respectively, the slope of log µo vs. M in the region M < M’c monotonically increases from 1.5 to 6, a four-fold increase of the exponent of the effect of M on viscosity. It is clear that the admitted validation of the separation of the effect of M and T on viscosity badly fails in this region. Figure 8 is a plot of the slope of log µo vs. M in the entanglement region (i.e., for M > Mc) in Figure 1. The classical value of the slope is 3.4, as said earlier, yet Figure 8 shows that, although the value of the slope is not far from 3.4, it is not constant when T increases and, even more problematically, it decreases monotonously, not randomly, from 3.635 at T = 115 ◦C to 3.475 at 210 ◦C, a clear indication that the exponent does not strictly remain constant when T varies. As we reported earlier, such a small change in the slope is not easily visually perceptible, giving the illusion of a constant slope for M > Mc when T varies. A regression analysis performed in the M > Mc range at various T unequivocally produces a small but systematic non-randomness of the deviation from 3.4 (or from whatever average value is chosen). Figure 8, already presented in a previous paper [10], was our first exposure to the possible failure of the separation of T and M in the expression of the viscosity, an indication that motivated the research presented in this communication. The second section of this paper, section B, will quantitatively explain the conditions rendering the effect of M and T on the viscosity separable. Figure 8. Slope of Log µo vs. M in the entanglement region (i.e., for M > Mc) in Figure 1. 1 Polymers 2024, 16, 2042 10 of 34 We conclude from Figures 2–8 that the basic foundation of the classical understanding of the effect of molecular weight and temperature on the Newtonian viscosity of polymer melts (Figure 1) is possibly contradicted by the above observations. To pursue this idea that the classical views might have oversimplified the effect of M and T in the formulation of the melt viscosity of polymers, we present in section B a mathematical demonstration that allows us to determine whether the effect of M and T on the viscosity do separate or not. B TESTING MATHEMATICALLY THE SEPARATION OF M AND T IN THE EXPRES- SION OF VISCOSITY. B1 Summary of the classical approach. For polymers of low molecular weight (M < Mc) in Figure 1, the Newtonian viscosity takes the form: µo(M,T) = K(T) M (M < Mc) (4) where M is the molecular weight of the chains, µo is the Newtonian viscosity at temperature T, and K is a constant that varies with (T − Tg) to make the free volume constant in Equation (3). Note that the slope of 1 in Figure 1 for M < Mc requires in Equation (4) that K(T) be actually K(T − Tg) to work at free volume constant, where Tg is a function of M below Mc (i.e., Tg(M)). This means that the classical viscosity equation below Mc admits that M and T do not separate. This should be pointed out. For longer chains (M > Mc), the classical and quasi-universal value of 3.4 reflects the power law dependence of the entanglements on the viscosity: µo (M,T) = K’(T) M 3.4 (M > Mc) (5) The critical molecular weight, Mc’ is obtained from intersecting the straight lines Log µo vs. Log M drawn in the two regions M < Mc and M > Mc. Formula (5) simply states that molecular weight and temperature effects separate in the expression of the viscosity of polymer melts. Since Tg is constant for M > Mc, K’(T − Tg) does not make this term function of M, unlike for M < Mc. The temperature dependence of K(T) or K’(T) in Equations (4) and (5) is often written, as we already said, with the WLF expression in Equation (3). The WLF equation explains the curvature observed in Arrhenius plots of log(µo) vs. 1/T and is claimed to work well between Tg and Tg + 100, with the reservations made previously regarding the veracity of that statement. The constants C1g and C2g in Equation (3) are often admitted to have the universal value of 17.44 and 51.6, respectively, a statement discussed and refuted in a previous communication [10]; in fact, it was shown that C1g and C2g are not constant when M is varied and that their molecular dependence revealed the fragmentation of the rheological range, an observation that will also be forthcoming from the Dual-Phase approach presented next. B2 The Dual-Phase Inspired Approach. Let us consider Equations (6) and (7), inspired by the classical formulation of rhe- ology, yet modified pursuant to the two-phase nature of the Dual-conformers. This for- mulation of the effect of M and T on the Newtonian viscosity is empirical, as much as Formulas (4) and (5) are also empirical. The main difference between the classical empirical formula, (4) and (5) and the Dual-Phase inspired empirical formula, (6) and (7), is in the interpretation of the physical parameters entering the equations. This will be explained in the Discussion. In the following, we refer to “phase 1” and “phase 2” or their respective Newtonian viscosity, µo1 and µo2 pursuant to our assumption of a possible “split” of Logµo Polymers 2024, 16, 2042 11 of 34 into a structure: Logµo1 and Logµo2, corresponding to (µo = µo1*µo2). The split is expressed in the first line of Equation (6): Logµo(T, M) = a1(M, T) + LogµoR(T) = Logµo1(T, M) + Logµo2(T, M) a1(M, T) = a11(M, T) + a12(M, T) Logµo1(T, M) = a11(M, T) + Logµo1R(T) (6) Logµo2(T, M) = a12(M, T) + Logµo2R(T) LogµoR(T) = Logµo(T, M = MR) where the sub-index “11” refers to phase 1 and “12” to phase 2. When a1(M,T) = a1(M), Log µo(T,M) expresses the separation of M and T: a12 = b1 + b2 exp(∆H2R/T) + b4 exp(∆H2/T) ′ ′ (7)a11 = b 1 + b 2 exp(∆H1R/T) + b ′ 4 exp(∆H1/T) where ∆H2R and ∆H1R are constant (the value of ∆H2 and ∆H1 at M = MR, respectively), ∆H1 and ∆H2 are a function of M. The values of b1, b2, and b4 and b’1, b’2, and b’4 can be set to 0 or split into two terms (e.g., b1 = b1(a11) + b1(a12), b’1 = b’1(a11) + b’1(a12) etc.), which may both vary with M or T, M and T, or remain constant, for instance to 0. The testing of the different assumptions generates a matrix of testing possibilities, called “configurations”, that are exhaustively studied but not presented in this first presentation on the subject. For each set of assumptions, ∆H2R, b1 b2 b4 and ∆H2 or ∆H1R, b’1 b’2 b’4 and ∆H1 are found by non-linear regression of the data. When b2 and b4 simultaneously equal 0 in Equation (7) or when (b2exp(∆H2R/T) +b4exp(∆H2/T)) = 0 and then a1(M,T) = a1(M) = b1 and there is true separation of the vari- ables M and T. Same thing for the b’2 and b’4 possibility. For the sake of this mathematical demonstration, there is no need to attribute any physical meaning to the split of the viscos- ity into two separate components. One can consider the split, at this stage, as a possible way to improve the fit of the experimental values in a way similar to the improvement of the fit of the melt modulus E(t) by using two relaxation terms or even a spectrum of relaxation terms instead of a single relaxation term (Maxwell or Voigt models [10]). As we already said, the physical interpretation of the “Dual-Phases”, phase 1 and phase 2 will be explicit in the Discussion. Equations (6) and (7) were used to fit the PS monodispersed Newtonian viscosity data of Susuki [3] for M > Mc. We generated a set of “synthetic data” from the data in [3] in order to increase the number of T and M pairs, and the following analysis in the paper relates to the synthetic data. The procedure used to obtain the synthetic data was as follows: the original Susuki data were first curve-fitted with Equations (3) or (4), depending on the value of M. The viscosity values were re-tabulated every 5◦ from 110 to 210 ◦C for each M, providing a matrix of 420 values. The matrix was then transposed to obtain the same data but with a different line-up to permit plots of Log(µo) vs. Log(M) at the same temperature T. The synthesized data were re-analyzed by non-linear regression for each T, using Equations (6) and (7). The regressions results were statistically “perfect” (r2 = 1 with random residuals) for all combinations of T and M, validating its use for our purpose of testing the separation of M and T. Note the absence of bias in the starting equation, in the sense that the possibility to find the classical assumption of a full separation for all M and T is included in our validation test matrix. The results of the tests are described in Figures 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18b corresponding to test configuration (0, 0), the simplest formulation for M > Mc. Another example of configuration, (0, 1), will be given in the Discussion (Figure 19), among the four configurations possible in the matrix of tests. The complete analysis will be published separately [8]. Polymers 2024, 16, x FOR PEER REVIEW 13 of 37 Polymers 2024, 16, 2042 12 of 34 Figure 9. PlFoitg oufr ae 9.(PTl)o, thoef bal1a1c(Tk )s,qthuearbelsa,c ak sq(Tu)a, rtehse, rae1d2( Td)o,ttsh, eanredd ado=t (sa, an+daa1 =), (tah1e1 b+luae1 2t)r,iathneglbelsu, e triangles,11 12 1 11 12 against 1/Ta gfoari nMst =1 /1T34f,o0r00M p=ur1s3u4a,0n0t 0top uErqsuuaatniotntos (E6q) uaantdio n(7s).( 6T)haen rded(7 l)i.nTehs epraesdsinlign etshrpoausgsihn gthteh rough the points are tphoei nctusravreefitths eobcutarivneefidt sboyb rteaginreesdsiboyn roefg Ereqsusiaotnioonf (E7)q. uation (7). Figure 9 iFsi gau prelo9t ios fa tphleo tvoafluthees voafl uaes (oTf11 ) a 1a1n(dT) aand12(T a)1 2f(oTu)nfodu bnyd breygrreegsrseiossni ofnorf oMr M = = 134,000 according to Equations (6) and (7). The black squares represent a11, and the red dots 134,000 acrceoprrdeisnegn ttoa Eq. uTahteiobnlsu e(6t)r iaanndg l(e7s).r Tefheer btolatchke isrqsuuamresa re(Tp)re=seant (a , and the red 12 1 11 T1)1+ a12(T) in Equation (6). dots repreTshenetv aalid. aTt12 h ioen btleuset torfiatnhgelseesp raerfearti toon thofeiMr saunmd aT cons1(T) = iasts11( inT)d+e atermi12(T) n iinn gEqwuhaetitohne r the sum a1 = a11+ a12 of Equation (6) remains constant when 1/T varies (and the blue triangles (6). The varelimdaatiinonh oteriszt oonf ttahlley saelpigarnaetdio)nf oorf tMhi sapndar Tti ccuolnasrisvtasl uine doeftMer.mFiingiunrge w9hsehtohwers tthhea t the hor- sum a1 = aiz11o+n taa1l2it yofo Efqauaitsionne (v6e)r reremaaliiznesd cofonrstMant= w1h34en,0 010/Ts vinacreieas ((aMnd) sthlieg hbltulye vtraiarinegslwes 1 1 ith T. The remain hosrliozpoentiasllcylo asleigr ntoed0) ifnort htheish ipgahretircuTlarre gviaolnue( tohfe Mlo. wFiegru1r/e T9 vshalouwes) t.hTath tehsee hpoaria-tion of M zontality aonf da T1 i iss ninevaerli dreaatelidzefdo rftohri sMm =o l1e3c4u,l0a0r0w sienigceh ta(1(3M4,)0 s0l0iggh/tlmy ovlae). W1 ries ewriethp eTa.t Tt he same test in Figure 9 for all the other M values available. For certain discrete values of M, e.g., slope is closer to 0 in the higher T region (the lower 1/T values). The separation of M and 2Mc, 4Mc, 8Mc, etc. the line of blue triangles remains strictly horizontal when T varies: T is invalidated for this molecular weight (134,000 g/mole). We repeat the same test in a1(M,T) = a1(M) for those discrete values of M. For the other values of M, a “structure of Figure 9 for all the other M values available. For certain discrete values of M, e.g., 2M , a1(M) appears visible, fragmenting a11(M,T) and a12(M,T) into ranges of sequecntial M, for 4Mc, 8Mcw, ehtcic. hthceo lminpee onfs abtliuoen tsroiacncgulre. sI treamppaeinarss sthriacttltyh ehocorimzopnetnasla wtiohnenp Toi vnatsriaelsw: a1y(sMco,Tin) cide with the values of M for which the blue lines are horizontal. Depending on the configuration = a1(M)for those discrete values of M. For the other values of M, a “structure of a1(M) status of our assumptions in Equation (7), some period doubling compensation may be appears vmisiisbslien, gfr(ea.ggm., e1n6tMincgf oar1c1o(Mnfi,Tgu) raantido na1(12,(0M) [,8T])) .into ranges of sequential M, for which compenWsaetiiollnuss otrcactuert.h Iet saepopbesaerrsv tahtaiot nthsei ncoFmigpuerenssa1t0i–o1n4 pboyinptlso attliwngayas1 2c(oMin)cvids.e lwogitMh for three the valuesf roafg mMe fnotre dwrhainchg etshoe fbMlu,eT lvinaleuse asr.eF ihgourriezo1n0tashl.o Dwesptehnedsienpga orant itohne ocfotnhfiegmuroalteicounl ar weight range in three sub-ranges, all a ◦status of our assumptions in Equation (b7o),v seoMmec, pfoerriao1d2( Mdo)uabtlTin=g 4co23mKpe(n15sa0tioCn) .may be missing (e.g., 1I6nMFig fuorre c1o1n,fithgeurvac at liuoens (o1f,0a)1 2[8(M])., T). for Mc < M < 2Mc) (i.e., in “range-1”), at various T form a collection of straight lines compensating at M = 188,000 ~5Mc when M is plotted on We ilalulostgrastcea lteh.eTshe eoyb-sceorovradtiionnaste inof Fthigeucroems 1p0e–n1s4a tbioyn pploottinitnigs ap1o2s(iMtiv)e v. sT. hleogli nMes fcoorm pensate three fragamseTnitnedcr eransegsesfr ofm M1,2T0 v◦aClutoes2. 1F0ig◦uCr.eT 1h0e sshlopwes otfhae1 s2e(pTa)rvasti.olno goMf thre amchoelescaunlaars ymptotic weight ranvagleu ieni tnhtrhee shuigbh-rearnTgerse,g aiolln a,bwovheic Mh li,n feosr uap t(hMis rangc 12 ) at T = e -412o3f Kd a(1ta50w °iCth). th e data located in the other ranges, range 2 and range-3, at higher M (Figures 12 and 13). Polymers 2024, 16, x FOR PEER REVIEW 14 of 37 Polymers 2024, 16, 2042 13 of 34 Figure 12 explores “range-2” of the values of the molecular weight, showing a com- pensation point at M = MR = 144,000 ~ 4Mc. The value of a12 at the compensation point is zero, which is also the case for a11 (not shown), indicating that a1 = a11 + a12 = 0 is a Polymers 2024, 16, x FOR PEER REVIEW 14 of 37 special solution for the separation of M and T corresponding to a11 = a12 = 0. In Figure 12 we see that in range-2 (M < MR), the compensation lines converge to the compensation point, whereas they emerge from the compensation point in range 2R (M > MR). Figure 10. Plot of a12(M) at constant T (T = 423 °K is represented by the blue triangles) delimitating 3 ranges: 1, 2, 3. When T varies, the three ranges still exist, but the respective disposition of the three segments is different. The straight line corresponds to T = 483 °K for which the three ranges almost fuse into a single one (range-2 points) is slightly above the straight line yet parallel to it. The T = 423 and T = 483 lines, and the lines for the other T, all cross at a universal “pole R” corresponding to M = 4M and a12 = a11 = 0 whatever T. c In Figure 11, the values of a12(M,T). for Mc < M < 2Mc) (i.e., in “range-1”), at various T form a collection of straight lines compensating at M = 188,000 ~5M when M is plotted FFigiguurere1 01.0P. lPoltootf a ◦ c of1 2a(M) at constant T (T = 412(M) at constant T (T 23 = 4K2i3s °reKp rise sreenpterdesbeynttehde bblyu ethtreia bnlgulee st)rdiaenligmleitsa)t idneglimitating o3n a log scale.3r raannggese:s1: ,12, ,23,. 3W Thheen Ty-vcaorioers,dtihneathtere oe fr atnhgee scsotmill pexeinsts, abutitotnhe preosipnetc tiisv epdoisspitive. The lines compen-. When T varies, the three ranges still exist, but the respectiovseit idoinspofotshiteiothnr eoef the three sae te as T seggmmenetnstiss indcifrferaeses from 120 °C is diffenrte.nTth. eTshtera sigtrhatiglihne toco 2r1re0s p°oC. The st line corrnedsspotonTds= lo4pe ◦of a (T) vs. log M reaches an asymp- to8 3T =K 4f8o13r2 w°Kh ifcohrt whehtihcrhe ethraen tgherseael mraonsgt es almost ffuusseei nintotoa asi nsignlegloen oen(rea n(rgaen-2gpe-o2in ptso)inistssl)i gish tsllyigahbtolvye atbhoevster athigeh sttlrine yettotic value in the higher T regio , which lines up this ranagigeh-1t l pinareallelof d yaetat tpoairt. T witahll h e e tlh t To=e 423 dita. tTah leo Tca =t e4d23 aannddT T= =4 48833l ilniense,sa, nadndth tehlein leins efos rfothre thoteh oerthTe,ra Tll,c arolls csraotsas uant iav eurnsaivl e“rpsoalle “Rp”ocleo rRre”s pcoonding toin the other ranges, range 2 and range-3, at higher M (Figures 12 and 13). rresponding to M M= 4=M4M ac nandd aa1212 = = aa11 = 011 = 0 wwhhaatetvevererT .T. c In Figure 11, the values of a12(M,T). for Mc < M < 2Mc) (i.e., in “range-1”), at various T form a collection of straight lines compensating at M = 188,000 ~5Mc when M is plotted on a log scale. The y-coordinate of the compensation point is positive. The lines compen- sate as T increases from 120 °C to 210 °C. The slope of a12 (T) vs. log M reaches an asymp- totic value in the higher T region, which lines up this range-1 of data with the data located in the other ranges, range 2 and range-3, at higher M (Figures 12 and 13). FFiigguurree1 11.1S. aSmaemfieg ufiregausreF igausr eF1i0gurerset ri1c0te dretosttrhicet“erda ntgoe -1th”eM “vraalnuegsea-1n”d vMar iovuasluTe. sT haenidso tvhaerrmioauls T. The isseogtmheernmtsaolf ase12g(mMe) noftsr aonfg ea-1 a(lMl c)o mofp ernansagtee-a1s aTliln ccoremaspees.nTsahtees laosp eTr eiancchrees12 as aensa. sTymhep tsoltoicpleim rietaches an for T > 453 ◦K (approximately). asymptotic limit for T > 453 °K (approximately). Figure 11. Same figure as Figure 10 restricted to the “range-1” M values and various T. The isothermal segments of a12(M) of range-1 all compensate as T increases. The slope reaches an asymptotic limit for T > 453 °K (approximately). Polymers 2024, 16, x FOR PEER REVIEW 15 of 37 Polymers 2024, 16, x FOR PEER REVIEW 15 of 37 Polymers 2024, 16, 2042 14 of 34 Figure 12. This figure explores range-2 of Figure 10 at various T indicated by the different symbols. There is a double-compensation point at the pole M ~4Mc delimitating two ranges 2L and 2R The R value of a at the compensation point is zero, which also occurs for a (M) vs. log M (not shown). 12 11 This figure should be compared to Figures 11 and 13. Figure 12 explores “range-2” of the values of the molecular weight, showing a com- pensation point at M = MR = 144,000 ~ 4Mc. The value of a12 at the compensation point is zero, which is also the case for a11 (not shown), indicating that a1 = a11 + a12 = 0 is a special solution for the separation of M and T c orresponding to a11 = a12 = 0. In Figure 12 FFiwgiugerue rs1e2 .e1T 2th.h iTsahfitgi siu nrfie greuaxpnrelgo erex-sp2r lao(nMrgees -2< r oaMfnFgRigeu)-,2r e toh1f0e Fa itcgvouamrrieop u1es0nT asitan vdtaiocraintoe udlisbn yTe tsihn ecddoiicnffaevtrednrtg bsey m ttobh oetl shd.eiff ceoremnpt seynmsabtoilosn. TThehreeries aisd aou dboleu-cbolme-pcensation point at the pole MR~4Mc delimitating two ranges 2L and 2R Thepoint, whereas thoemyp eemnseartgioen f proomint tahte t hceo mpoplee nMsat~i4oMn cp doeilnimt iinta trianngg tew 2oR r a(nMge >s M2L a)n. d 2R The value of a12 at the compensation point is zero, which also occur Rs for a11(M) vs. log M (not shown). R Tvhiaslfiugeu oref sahou ladt btehceo cmopmarpedentosaFtigiounre sp1o1inant dis1 z3.ero, which also occurs for a (M) vs. log M (not shown). 12 11 This figure should be compared to Figures 11 and 13. Figure 12 explores “range-2” of the values of the molecular weight, showing a com- pensation point at M = MR = 144,000 ~ 4Mc. The value of a12 at the compensation point is zero, which is also the case for a11 (not shown), indicating that a1 = a11 + a12 = 0 is a special solution for the separation of M and T corresponding to a11 = a12 = 0. In Figure 12 we see that in range-2 (M < MR), the compensation lines converge to the compensation point, whereas they emerge from the compensation point in range 2R (M > MR). Figure 13. Same as in Figure 10, with a12(M,T) now focusing on range 3 and showing a negative Figure 13. Same as in Figure 10, with a (M,T) now focusing on range 3 and showing a negative compensation point at M~2Mc. Note the y-coordina1te2value of the compensation point to be compared wicthomthpateonfstahteicoonm ppeonisnatt ioant pMoin~t2iMn ra.n gNe o1t(eF igtuhree 11)c y-c : othoerirdsiunmateeq uvaalslu−e1 .of the compensation point to be comInpFairgeudr ew1it3h, tthheata of( tTh,Me )covmalpueesnsoaftiMona preoilonct aitne drainngrea n1 g(eF-i3g,uarned 11w):h tehneiTr vsuarmie se,quals −1. 12 the isotherms compensate at M~2Mc with a negative value of the y-coordinate of the compenIsna tFioinguporein 1t.3I,n ttehree sating(lTy,,Mif o)n12 v eacloumeps aoref sMth earther eloecraantegdes iinn rFaignugrees-31,1 a–1n3d, t whehen T varies, the higher M range (range-3 in Figure 13) corresponds to a “negative” compensation; that is,isthoethiseortmhesr mcosmempeerngesafrtoem ata Mlo~w2eMr Mc wvailtuhe ,aw nheigleatthieveco vmapluenes oatfi othnei sy“-pcosoitridvein”ainte of the compen- range-1 (Figure 11), i.e., the isotherms converge to a higher M value. The situation in sation point. Interestingly, if one compares the three ranges in Figures 11–13, the higher range-2 is intermediate with the combination of a positive and a negative compensation usMin grtahnegsea m(reacnogmep-3e nisna tiFoinguporien t1a3t) Mco=rrMesp. oTnhdesd R etosc raip “tinonegoaftaive(”M ,cTo)mcapnetnhsuastion; that is, the 12 be described as a succession of positive and negative compensations, a very interesting network structure threaded by the interdependence of the effect of M and T on viscosity. Figure 13. Same as in Figure 10, with a (M,T) now focusing on range 3 and showing a negative 12 compensation point at M~2Mc. Note the y-coordinate value of the compensation point to be compared with that of the compensation point in range 1 (Figure 11): their sum equals −1. In Figure 13, the a12(T,M) values of M are located in range-3, and when T varies, the isotherms compensate at M~2Mc with a negative value of the y-coordinate of the compen- sation point. Interestingly, if one compares the three ranges in Figures 11–13, the higher M range (range-3 in Figure 13) corresponds to a “negative” compensation; that is, the Polymers 2024, 16, x FOR PEER REVIEW 16 of 37 isotherms emerge from a lower M value, while the compensation is “positive” in range-1 (Figure 11), i.e., the isotherms converge to a higher M value. The situation in range-2 is intermediate with the combination of a positive and a negative compensation using the same compensation point at M = MR. The description of a12(M,T) can thus be described as a succession of positive and negative compensations, a very interesting network struc- Polymers 2024, 16, 2042 ture threaded by the interdependence of the effect of M and T on viscosity. Also1 5noft3ic4eable are the complementarities of the y-coordinate values of the compensations for Figures 11 and 13, which add up to −1. Also noticeTahbel evarreiatthioenc omf aple11( mMe,nTt)a irsi tsieims oilfatrh teoy a-coordinate values of the compensations12(M,T) for the effect of M at T constant, yet for Figures 11 and 13, which add up to −1. almost symmetrically “opposite” to a (M,T) for the effect of T at M constant. In Figures The variation of a11(M,T) is similar t1o2 a12(M,T) for the effect of M at T constant, yet al1m4 otost 1s6y, ma1m1 eatnridc al1ly2 a“roep plosttiteed” atgoaain12s(tM lo,gT )Mfo art tTh ecoenffsetacnt to, fshToawtinMg ctwonos taemntp. eIrnatures Figuroens l1y4: –a1 l6o,wa1 1T a=n 4d03a 1°2Ka r(1e3p0l o°Ctt)e danadg aai nhsigthlo Tg =M 47a8t T°Kc o(2n0s5ta °nCt,))s. howing two tempera- tures only: a low T = 403 ◦K (130 ◦C) and a high T = 478 ◦K (205 ◦C). FigureFi1g4u. rCe o1m4.p Caorimngpaar1i1n(gM a)1a1n(Md a) 1a2n(Md a)1fo2(rMT)= fo4r0 T3 ◦=K 4.0T3 h°eKa. 1T2h(Me a)1v2a(Mlue) sv(arleudesd (ortesd) hdaovtse) ahlarevaed aylready been pbreesne nptreedseinteFdig iunr eFi1g0u,rfer a1g0m, fernatgimngenthtiengM thraen Mge rianntoget hinreteo stuhrbe-era snugbe-sr.anTghees.t wThoes trwuoct ustrraulctural compoconmenptsonoefnat1s corfo sas1 actrothsse apto tlhee( Mpo=leM (MR) =, wMhe)r, ewtherye btehceoym bec0o.me 0. R We seWe ein sFeeig iunr Fei1g4u,rfeo r14th, fisorlo twhisT lovwal ◦ uTe v(aTlu=e4 (0T3 = K40),3t °hKe)a, tphpee aaprapnecaeraonf csey mofm syemtrymfeotrry for a11(Ma) an(Md )a a12n(dM a) ar(oMu)n adrothuenidr mtheeairn mvalue, but in real11 12 ean value, but in it yre, afolirtys,u fcohr lsouwcht elomwp teermatpuererastuupres up to T = 448 ◦K, the axis of symmetry is different for the values of M superior or inferior to MR =to4M T =c( a4t4M8 °K, b, oththe aR axisa 11 nodf saymamreeterqyu ias ldtoiff0e,raens tw foers athwe ivnaFluigesu roef 1M2) s. uDpaetraiopr or in12 oints aferreior to Polymers 2024, 16, x FOR PEERm RiEssViIMnEgWRa =t b4Mothc(aext tMreRm, ibtioetsh oaf1t1haenpdl ao1t2to arceo neqfiurmal ttoh e0,p aoss swibei sliatwy f ionr Faigu(Mre) 1a2n).d Daata(M po) itno1t7s oafre 11 12 37 crossmonistshinegs yamt bmoethtr yexlitnreems aittiMes o~fM thcea pnldotM to~ 5c5o.n5fiMrmc, cthoen fipromssinibgiltihtye fporre saenc(Me o) f11 a pnedr iaodi12( cM) to anchors for discrete M values. cross on the symmetry lines at M ~ Mc and M~55.5 Mc, confirming the presence of periodic anchors for discrete M values. FigureF1i5g.uSrep l1i5t .o Sfpal1iti noft oa1a 1i1n(tMo )aa1n1(dMa)1 2a(nMd) aa1t2h(Mig)h aetr hTi,g4h7e8r KT,, i4n78th Kis, ignr atphihs. gTrhapeha.v Tehraeg aevvearaluge, value, a1/2, reapr1/2 e,s reenpterdesbenyttehde bdyo tttheed dlointtee,da plipneea, rasptpoebaerst htoe bsye mthme estyrmicamleatxriiscaolf atxhies doaf ttah,et hdeatsaa,m theea scarmoses across the pole MR. The green triangles display the behavior of a1(M) vs. log M, which can be fitted by athe pole M . The green triangles display the behavior of a1(M) vs. log M, which can be fitted by a straight line with Rslope ~ 3.4, the classical result. straight line with slope ~ 3.4, the classical result. Figure 15 displays the equivalent of Figure 14 at a higher temperature: T = 478 K (205 °C). Like at lower T, a11(M) and a12(M) cross at MR = 4Mc, but the disposition of the red dots (a12) and of the black squares (a11) on the graph looks different than at lower tem- peratures. This implies that the influence of T on the interdependence of M and T is not the same in the three sub-ranges of M, range-1, 2, or 3. Likewise, the influence of M on the interdependency of T and M is not the same in the three temperature sub-ranges 1, 2, or 3, delimited by Tg, Tg + 23 °C and TLL (see the discussion). The green triangles represent a1(M) = a11(M) + a12(M) vs. log M (i.e., a plot of log µo vs. log M pursuant to Equation (6)). These points can be fitted by a straight line across MR (r 2 = 0.9983) with slope = 3.372, which is the classical exponent value characteristic of PS melts. This is expected since we used data generated by the classical formula (Equations (3–(5)). One sees that a two-phase formulation of the viscosity, Equations (6) and (7), that assumes viscosity to be the product of two terms for which the effect of M and T are not separable, provides the same value of the slope for log µo vs. log M for M > Mc than in the classical view, that assumes that the effect of M and T on the viscosity separates (Figure 1). This apparent paradox proves, at least, that the veracity of the features in Figure 1 does not uniquely imply the separation of M and T in the viscosity. On the contrary, Figures 10–14 demonstrate the particularities of the dependence of the effect of T and M on the viscosity in ways that cannot be deducted if M and T were separable. In particular, among the special features of the dependence between M and T is the finding of a special molecular weight MR = 4 Mc. Figure 15 shows the average value of a11(M) and a12(M), a1/2, represented by the dotted line, which ap- pears to be the symmetrical axis across this M value. This “pole” molecular weight, a R clear marker of the dual-phase viscosity formulation in range M > Mc, is not Mc itself but 4Mc, a value of M that does not resonate to be significant in the classical paradigm. Like- wise for the T scale in the T > Tg range, as we further explain in the Discussion section, the TLL transition [11,12] has no physical reason to exist in the classical theories of melt viscoelasticity and is vehemently described as an artifact by the adepts of the classical views [12], yet erroneously so, in our opinion, since this temperature determines the Polymers 2024, 16, 2042 16 of 34 Figure 15 displays the equivalent of Figure 14 at a higher temperature: T = 478 K (205 ◦C). Like at lower T, a11(M) and a12(M) cross at MR = 4Mc, but the disposition of the red dots (a12) and of the black squares (a11) on the graph looks different than at lower temperatures. This implies that the influence of T on the interdependence of M and T is not the same in the three sub-ranges of M, range-1, 2, or 3. Likewise, the influence of M on the interdependency of T and M is not the same in the three temperature sub-ranges 1, 2, or 3, delimited by Tg, Tg + 23 ◦C and TLL (see the discussion). The green triangles represent a1(M) = a11(M) + a12(M) vs. log M (i.e., a plot of log µo vs. log M pursuant to Equation (6)). These points can be fitted by a straight line across M 2R (r = 0.9983) with slope = 3.372, which is the classical exponent value characteristic of PS melts. This is expected since we used data generated by the classical formula (Equations (3–(5)). One sees that a two-phase formulation of the viscosity, Equations (6) and (7), that assumes viscosity to be the product of two terms for which the effect of M and T are not separable, provides the same value of the slope for log µo vs. log M for M > Mc than in the classical view, that assumes that the effect of M and T on the viscosity separates (Figure 1). This apparent paradox proves, at least, that the veracity of the features in Figure 1 does not uniquely imply the separation of M and T in the viscosity. On the contrary, Figures 10–14 demonstrate the particularities of the dependence of the effect of T and M on the viscosity in ways that cannot be deducted if M and T were separable. In particular, among the special features of the dependence between M and T is the finding of a special molecular weight MR = 4 Mc. Figure 15 shows the average value of a11(M) and a12(M), a1/2, represented by the dotted line, which appears to be the symmetrical axis across this MR value. This “pole” molecular weight, a clear marker of the dual-phase viscosity formulation in range M > Mc, is not Mc itself but 4Mc, a value of M that does not resonate to be significant in the Polymers 2024, 16, x FOR PEEcRla RsEsVicIaElWp aradigm. Likewise for the T scale in the T > Tg range, as we further explain18 of 37 in the Discussion section, the TLL transition [11,12] has no physical reason to exist in the classical theories of melt viscoelasticity and is vehemently described as an artifact by the adepts soufctcheescsl aosrs itchael fvaiielwurse[ 1o2f ]a, yneetwer rpornoecoeusssilnygs ob,einneofiut rfoorp itnhieo np,osliynmceerth riesstienms pinedrautsutrrey: “sus- determtianiensedth oersieunctcaetsiosno”r [t4h,e7]f.a ilure of a new processing benefit for the polymer resins industry: “sustained orientation” [4,7]. Figure 1F6i.gTuhries 1fi6g. uTrheiiss fithgeusrae mise tahse Fsiagmuree a1s5 ,Ffiogcuursei n1g5,o fnocduifsfienrge notna sdpiffecetrse.nFti rasstp, tehcetsl.i nFeirasrti,t ythoef ltihnearity of variationthoef va1a1r(iMati)oann dofa 1a21(1M(M) v) sa. nlodg aM12f(oMr )M v>s. Mlocga tMh ifgohre rMT >a nMdcs eacto nhdig,hthere pTo assnidb lesescigonidfi,c tahnec epossible of the slsoipgeniofifctahnectew oof t“hpeh salosepse” ogfe tnher tawteod “bpyhaEsqeus”a tgioenser(a6t)eadn bdy( 7E)qwuahteionnpsl (o6t)t eadnda g(7a)i nwsthleong pMlo.tted against log M: Figure 16 may possibly be the simplest explanation of the classically acclaimed formulation Figouf vreis1co6smitya ybyp Eoqsusiabtlioynb (e5)t hfoer sMim >p Mlecs.t Ae xDpulaaln-Pathiaosne oinfttehrperceltaastisoicna ollfy thaec ccllaasimsiceadllyfo arcmclaui-med 3.4 lation oefxvpiosncoensitt yis bthyaEt q3u.4a =ti o(3n +( 50).4f)o ranMd >thMat co.nAe sDhuoaull-dP sheapsaeraintet etrhper eutnadtieornstaonfdtihnegc olaf stshiec aelxlyponent 3 acclaimferodm3 .4theex epxopnoennetnits 0t.h4a, tin3s.4te=ad( 3o+f t0r.y4i)nagn tdo tmhaotdoifnye thshe oeuxlpdlasneaptaiorna toeft htheeu enxdpeornsetnatn d3 i(nsgay by de of the eGxepnonnees n[t5]3) fbroym intchoerpeoxrpatoionne notf 0a.4d,diintisotneald moefcthrayninisgmtso smucohd iafsy cthonetoeuxpr llaenagtiho nfluocftuation, the expconestnrat i3nt( rsealyeabsey adned Gcheanin estsre[t5c]h)ibngy (iCnhcaoprpteor r7a otifo [n7]o) ftoa rdedacihti othnea vl amlueec ohfa 3n.4is fmors tshue cehxponent. as contAou Dr uleanl-gPthhasfleu ecxtpulaatniaotnio,nc oisn tshtrea sienptarrealteioanse oaf nthdec phhaainsess tirne ttchhei nvigsc(oCshitayp etqeura7tioonf [i7n]s)tetaod of the separation of the effect of M and T. Figure 16 is the same as Figure 15 plotted differently, showing different aspects of this graph: the variation of a11(M), the black squares with slope 0.414, and of a12(M), the red dots with slope ~3 at high T. The difference between the lower temperature range of the melt (e.g., in Figure 10) and the higher temperature range (Figure 16) is now clearer: the lines in Figure 16 represent the log µo1(M), and the log µo2(M) vs. log M viscosity plots of the two-phases 1 and 2 at T = 478 °K, and these lines are straight all across and not fragmented segments as they appear at lower T (403 °K in Figure 15). This, again, clearly reveals not only the separate and distinct role of M and T on the viscosity but also their compensating interactive effects. Speaking in terms of viscosity, Figure 16 represents the molecular weight depend- ence of the log of viscosity of “phase 1 and phase 2” pursuant to the “Dual-Phase equa- tions”, Equations (6) and (7): it provides the dual-structure of log µ into two terms, log o µo1, represented by the black squares for a11(M) with slope 0.414 (phase 1) and log µo2, the red dots for a12(M), with slope 2.93 (phase 2). Note that if we change the temperature from T = 478 °K T = 483 °K, the respective slopes of the lines in Figure 16 become 0.328 and 3.0 for the respective phases, giving a total exponent of 3.328 for the variation of a1(M) = log µo(M) in Equation (7). It is, indeed, remarkable to find for the slope of phase 2, the “core-phase”, the value of 3, which is the value found by de Gennes in the original reptation calculation of the molecular weight dependence of the Newtonian viscosity [6] and to attribute to phase 1 “the entanglement phase” the “corrective” additional 0.4 exponent value classically assigned to sophisticated perfections of the original de Gennes’ tube calculations (see Figure 16 caption). Polymers 2024, 16, 2042 17 of 34 reach the value of 3.4 for the exponent. A Dual-Phase explanation is the separation of the phases in the viscosity equation instead of the separation of the effect of M and T. Figure 16 is the same as Figure 15 plotted differently, showing different aspects of this graph: the variation of a11(M), the black squares with slope 0.414, and of a12(M), the red dots with slope ~3 at high T. The difference between the lower temperature range of the melt (e.g., in Figure 10) and the higher temperature range (Figure 16) is now clearer: the lines in Figure 16 represent the log µo1(M), and the log µo2(M) vs. log M viscosity plots of the two-phases 1 and 2 at T = 478 ◦K, and these lines are straight all across and not fragmented segments as they appear at lower T (403 ◦K in Figure 15). This, again, clearly reveals not only the separate and distinct role of M and T on the viscosity but also their compensating interactive effects. Speaking in terms of viscosity, Figure 16 represents the molecular weight dependence of the log of viscosity of “phase 1 and phase 2” pursuant to the “Dual-Phase equations”, Equations (6) and (7): it provides the dual-structure of log µo into two terms, log µo1, represented by the black squares for a11(M) with slope 0.414 (phase 1) and log µo2, the red dots for a12(M), with slope 2.93 (phase 2). Note that if we change the temperature from T = 478 ◦K to T = 483 ◦K, the respective slopes of the lines in Figure 16 become 0.328 and 3.0 for the respective phases, giving a total exponent of 3.328 for the variation of a1(M) = log µo(M) in Equation (7). It is, indeed, remarkable to find for the slope of phase 2, the “core-phase”, the value of 3, which is the value found by de Gennes in the original reptation calculation of the molecular weight dependence of the Newtonian viscosity [6] Polymers 2024, 16, x FOR PEER REVIEW 19 of 37 and to attribute to phase 1 “the entanglement phase” the “corrective” additional 0.4 expo- nent value classically assigned to sophisticated perfections of the original de Gennes’ tube calculations (see Figure 16 caption). FigurFeig1u7.reT h17is. Tphloits (pfolortM (fo>r Mc >) sMhco)w sshothwesa tvheer agvervaagleu evableutew beeetnwae1e1n(T a)1a1n(Td) a1n2d( Ta)1,2w (Th)ic, hwihsichha lifs half the vathluee voaflua1e (oTf) a=1l o(Tg)µ =o (lTog), pµloo(tTte)d, palgoattinedst aloggaiMns.tF lorgM M>. F2oMr cM(~ 7>7 2,0M00c) (,~a7s7i,n0g0l0e),s tar asiignhgtleli nsteragiogehst line througgohetsh tehdroautag,hw thhiech draetnad, ewrhs iµcoh( Tre)ninddeersp eµnd(eTn) to in ofdTep: TenadnednMt osfe Tp:a Tra taen.dB eMlo wsep2Marca,tew. eBoebloswer v2eMac, we compoebnsseartvioen a ocfolminpesenassaTtiovna roief sliinnesste aasd To fvtahreieus niniqsuteeasdt roafi gthhet luinnei:qTuea nstdraMighnto llionne:g Ter asnedp aMra tneo. longer separate. Figure 17 illustrates how the separation of M and T for M > 2Mc is simply the re- sult of an illusion created by the use of log axes that compresses the resolution of a sig- Figure 17 illustrates how the separation of M and T for M > 2M is simply the result nificant residual periodically fluctuating around 0. This figure plots hcalf the value of log µoof a=na i1lluasgiaoin sctrelaotgedM byfo trhea lul steh eofT lovga aluxeess tahvaati claobmlepr(eTss=es3 t9h3e– 4re8s3o ◦luKt)i.onW oef oab ssiegrnvieficant that rae1s(iMdu)/a2l p=e(raio11d+ical1l2y) /fl2uvctauraietisnwg iatrhouTnadn 0d. TMhiisn fitghuerleo pwleortsm hoallfe ctuhela vrawlueei gohf tloragn µgoe = a1 (Mc (T2 =M 3c9)3th–4er8e3 i°sKa).s iWngel eobstsrearivgeh tthliante aw(iMth)/2 = all the points superposing; that is, it apparently looks temperature independent, vary1ing only(wa1it1h +M a1; t2h)/e2e vffaerciteos fwMitha nTd aTndap Mpe ianr sthteo lsoewpaerra mteoilnecthuelaerx wpreeisgshiot nraonfgteh e(Mtoct aat2 iMn oc.thIenrt rhainsgresg ioofn Mo f(M >a 2nMd cT), ttheeres liosp ae soinfgal1e vstsr.aliogghtM linise 2w*it1h. 6a8ll6 th=e3 p.3o7i2n,tsth seuper- expepctoesdinvga;l utheaftr oism, itth aeppclaarsesnitclayl lmoookdse lteomf vpiesrcaotsuitrye. iBnedloepweMnd=en2t,M vac,rhyionwge ovnelry, w weitohb sMer; vthee ef- fect of M and T appear to separate in the expression of the total viscosity for M > 2Mc. In this region of M and T, the slope of a1 vs. log M is 2 * 1.686 = 3.372, the expected value from the classical model of viscosity. Below M = 2 Mc, however, we observe a spectrum of compensating lines as T varies, the visible proof using these axes that T and M at least do not separate in this range. The details for M < 2Mcis blown up in Figure 18a, and the com- pensation search is shown in Figure 18b. We can clearly see that the slope of each isotherm increases with T up to T ~ 428–448 °K (155–175 °C) and that for T beyond that temperature, the slope asymptotically converges to a value close to (but not equal to) the 3.372 slope of the T independent region (Chapter 4 of [8]). In range-2 and 3 for M, the a11(T) and a12(T) are more symmetrically disposed with respect to their average value when M varies, especially at high temperatures (e.g., Figure 15), and therefore their half sum gives the illusion to lie more closely on the average value line, giving the appearance of being independent of T. However, if we zoom on the points of the M > 2Mc region in Figure 17 or if we analyze the residuals of their linear regression, the illusion of the separation of T and M reveals itself unequivocally (see Figures 12 and 13). Polymers 2024, 16, 2042 18 of 34 a spectrum of compensating lines as T varies, the visible proof using these axes that T and M at least do not separate in this range. The details for M < 2Mcis blown up in Figure 18a, and the compensation search is shown in Figure 18b. We can clearly see that the slope of each isotherm increases with T up to T ~ 428–448 ◦K (155–175 ◦C) and that for T beyond that temperature, the slope asymptotically converges to a value close to (but not equal to) the 3.372 slope of the T independent region (Chapter 4 of [8]). In range-2 and 3 for M, the a11(T) and a12(T) are more symmetrically disposed with re- spect to their average value when M varies, especially at high temperatures (e.g., Figure 15), and therefore their half sum gives the illusion to lie more closely on the average value line, giving the appearance of being independent of T. However, if we zoom on the points of the M > 2Mc region in Figure 17 or if we analyze the residuals of their linear regression, the illusion of the separation of T and M reveals itself unequivocally (see Figures 12 and 13). Another interesting observation consists of comparing the compensation coordinates for the a1(M,T) data in range-1 (Figure 18a) and for the a12(M,T) data in range-3 (Figure 13): they have the same value of M at the compensation but their y-coordinates are shifted by approximately 1, a clear indication that the fragmentation of the T range and the M range for T > Tg and M > Mc, respectively, are dependent and coupled in the expression of the viscosity of polymer melts, an important conclusion totally missed by the current paradigm of rheology. In summary, the regression of the Newtonian viscosity data as a function of T and M by Equations (6) and (7) show that b2 and b4 become simultaneously zero for periodic values of M, which are multiples of Mc (33,000 to 36,000 g/mole), pursuant to period doubling: 2Mc, 4Mc, 8Mc, 16Mc. For all other values of M, the values of b2 and b4 are quite small but not zero, yet small enough to make the sum (b2exp(∆H2R/T) + b4exp(∆H2/T)) give the appearance that the separation of the variables is justified, although it is not. Furthermore, we observe that the sum (b2exp(∆H2R/T) + b4exp(∆H2/T)), although small, structures with M in a way that makes the periodic molecular weights for the zeros the compensation points for the other M values when M is expressed as log M (see the Discussion at Figure 19 for a visual summary). As we mentioned at the onset of section B, our current presentation of the periodic dependence of T and M in the expression of the viscosity has been simplified in this paper. A more complete and thorough mathematical proof will be provided separately [8]. The key result of this section B is the understanding that all current models of melt viscoelasticity of polymers based on the separation of M and T are approximations that overly simplify the effect of M and T, miss out some fundamental aspects of the interactive coupling between T and M, and are, as proven mathematically, incorrectly describing the interactions in polymers, theoretically speaking. Of course, it should also be added that the separation of T and M becomes a better approximation at higher melt temperature and for M > 2Mc (Figure 17), which are the usual ranges of use of polymer resins, a choice that may explain why the previous models [4,8] remained falsely convinced of the true separation of T and M, despite of the contrary in reality. Polymers 2024, 16, x FOR PEER REVIEW 20 of 37 Polymer s 2024, 16, 2042 19 of 34 (a) (b) FigureF1ig8u. r(ea )1C8.o (ma)p Cenosmaptieonnsaotfiothne oifs oththee irsmotahlesremgaml esnegtsmfeonrttsh feoar vtehrea gaveeorfagae o(Mf a)1a1n(Md )a an11 12(M d a) 1in2(M) in range-r1an(Mge-<1 2(M c)<. 2TMhce).c oTmhep ecnosmatpioensSateiaornc hSeisardcihs pilsa ydeisdpliany(ebd) .in( b()b)C. o(mb)p Cenosmatpioenssaetiaornch setoarch to determdienteerthmeincoe mthpee ncosamtipoennpsoatiniotnc opooridnitn caoteosrdininFaitgeusr ein1 F8aig; uthre c1o8oar; dtihnea tceosoarrdeincaltceus larted cafrlocumlattheed from slope athned silnotpeerc aenpdt oinfttehrececpotm opf ethnes actoiomnpleinesaitniotnh ilsinpel oint: tihnitse prcloept: ti:n−t0e.r4c6e7p9t:;−s0l.o4p6e7:9;− s4lo.8p9e0: 2−24..8T9h0e22. The value of M at the compensation point is, there4f.o89r0e2, 2104.89022value of M at the compensation point is, therefore, 10 = 77,6 6=4 7, 7a,n6d64th, aenyd-c tohoer dyi-ncoatoerdisin−a0te.4 i6s7 −90. .4679. 3. DiscussiAonother interesting observation consists of comparing the compensation coordinates Tfors uthmem a1a(rMiz,eTt)h dearteas iunl trsapnrgees-e1n (tFeidguinres e1c8tai)o nansdA faonr dthBe, aw12e(cMo,nTc)l uddateat ihna traFnigue-r3e (1F,igure repres1e3n)t:i tnhgeyth heacvoen tcheep stsamofet hvealcuuer orefn Mt p aatr tahdei gcommopfernhseaotlioogny b, uist itnhceoimr yp-lceotoeradnidnamteiss laeraed sshifted the cobmyp arpephreonxsiimonatoeflyth 1e, fla ocwleaorf imndeilctsatbioenca tuhsaet itthies ftroaogsmimenptlaitsitoicn; oitf dtohees Tn oratncgoer raenctdly the M descrirbaentghee foefrf eTc >t oTf taenmdp Mer a>t Mure,, rTe,soprecthtieveelfyfe, catreo fdtehpeemndoelnect ualnadr wcoeuigphletd, M in, tahned emxpisses out the consequencesgof their couc ression of pling on the fragmentation of rheological ranges affecting the Nethwet voinsicaonsivtyis coof spitoyly; imnemr maneyltws, aayns imitspsohrotarntcto cmonincglussbioenco tmotealilnyc monisssiestde nbcyi ethsele caudrirnegnt par- to falsaediignmter opfr rehtaetoiolongsy. . For instance, the classical interpretation of the viscosity break at Mc as tIhne spurmoomfafroyr, tehnet arenggrleesmsieonnt sof( Fthigeu Nreew1)toanpiapne avrissctoosibtye ddairtaec atsly a cfuhnalclteinonge odf Tb yand M the wboyr kEqpureasteionntes d(6i)n atnhdis (7c)o mshmowun tihcaatt ibo2n a(nFdig ubr4e bse2c–o7m).e Tsihmeuflatailnueroeuoslfyt hzeercol afossri cpaelriodic interpvraeltuateiso nofo Mf e, nwtahnicghl eamree mntus lhtiapslebse eonf Mrepc ( o3r3te,0d00a ntod 3d6o,0c0u0m ge/nmteodleb),y puusrsbueafnotr eto[ 4p,7e,r1io8]d, dou- showing its negative impact on the polymer processing industry [4,7]. This paper makes use ofbmlinatgh:e 2mMacti,c 4aMl toc,o 8lsMdce,v 1e6loMpce.d Fforor malli doethaserin vsapliureeds obfy Mth,e tdhue avlaalnudesc roofs bs-2d aunadl a bs4p eacrtes quite Polymers 2024, 16, 2042 20 of 34 of the rheology of polymer melts [7–9] to reveal fundamental inconsistencies in the current assumptions leading to the classical understanding of the rheology of polymer melts [10]. In section A, the graphs in Figures 1–7 encompass the M < Mc and M > Mc regions to evaluate the classical view assumption that the entanglement molecular weight is defined at the crossing of the distinct behaviors below and above Mc. Figure 1 schematizes the break at Mc that is supposed to be independent of T. In the other graphs, in Figures 2–6, the change of color for the squares, from red at high M to black at lower M, starts at the value of Mc projected for PS: 33,000 to 36,000 g/mole [1]. The highest T is in Figure 7 (T = infinity), followed by, in descending order, Figures 2–6. Figure 7 is the only graph resembling Figure 1: 2 straight lines and a sharp break at Mc, accepting the reservations we made for the 1.51 slope for M < Mc (instead of 1) and the higher value of Mc at the break than the quoted values in the literature (48,000 instead of 36,000). Only Figure 2 (at T = 210 ◦C) could validate with a certain certitude the classical view for Mc and above Mc, yet needed the reserves we made for the description of the M < Mc distinct curvature from the expected straight line one in Figure 1. Mc is at its correct literature value in Figure 2. As T decreases, from Figure 3 to 6, the overall aspect of the graph starts to systematically diverge from the classical expected view of Figure 1, displaying three straight lines (not two) in Figures 3 and 4, along with two molecular breaks, Mc and M’c, not just Mc; and then, only two straight lines with a break at M’c and no break at Mc for the two lowest T (Figures 5 and 6). For instance, in Figure 5 (T = 125 ◦C), there is no break at the change of color from red to black at Mc, and the straight line passing through the “entangled melt” continues unbroken down to M’c = Mc/8. Is the entanglement molecular weight now defined by M’c = Mc/8? Do we need to define another criterion than the break of the molecular weight in a graph like Figure 1 to characterize Mc? This is the first fundamental challenge raised in this paper to the current paradigm of rheology. Along with it, what is the physical significance of the Mc/8 molecular weight? The second fundamental question raised concerns the separation of the effect of M and T in the formulation of the viscosity. This question already surfaces in Figure 8, at the end of section A, by showing that the molecular weight exponent for the Newtonian viscosity, the famous 3.4 slope in Figure 1 for M > Mc, explained with great details by the reptation school [6], seemingly varies systematically with temperature, which would contradict the separation of T and M if it were confirmed. Section B presents a simplified version of a mathematical proof that we have developed [8] to evaluate the classical assumption of the separation of T and M in the formulation of viscosity. The overall conclusion is that the separation of T and M in the expression of viscosity is invalid except for a series of specific molecular weight values that are multiples of Mc: 2Mc, 4Mc, 8Mc, 16Mc, etc. We found that the value of MR = 4Mc holds a special role among all the solutions, the role of a “pole reference” for the melt (a11 = a12 = a1 = 0), symbolized by the presence of the subscript “R” in some of the terms in Equation (7). In other words, we discovered in section A the existence of a certain special molecular weight M’c = Mc/8, and in section B the existence of another special molecular weight, MR = 4 Mc, possibly indicating that the period doubling observed above Mc could actually start at M’c? Note that playing along with this period doubling picture, the value of Me of the rubber elasticity theory falls right between M’c and Mc with Me = Mc/2 = 4 M’c. Figure 19 summarizes the period doubling fragmentation of the M range at various values of T pursuant to “configuration (0,1)”. Figures 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18a,b above referred to configuration (0,0), the simplest to describe. The difference between the configurations bears on the complexity and the status of certain constants to be found by regression, which results in certain solutions being skipped for period doubling: for instance, 16Mc is not a solution for configuration (1,0). These complications are not essential for this introductory presentation. Figure 19 shows the sum of the b’2 and the b’4 terms in Equation (7), configuration (0,1)), plotted against log M. This graph summarizes the various facets of the sophisti- cation of the coupling between the T and M variables introduced in the previous fig- ures (Figures 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18b): it shows the true values of the zeros Polymers 2024, 16, 2042 21 of 34 (b’2 = b’4 = 0) corresponding to the compensation of the non-zero regions, it shows the fragmentation of M in three regions and the period doubling of Mc. The low value of the sum of the b’2 and b’4 terms in Figure 19 explains the visual illusion of the separation of M Polymers 2024, 16, x FOR PEER REVIEW and T perceived in Figure 17. Also, the sum of the b’ and b’ terms be2c3o omf e37s 2 4 smaller for M > 2Mc and as T increases, making the separation of T and M appear more real. Figure 19. PrFoipgousreed1 s9t.ruPcrtouproastieodns btryu tchtue rcaotiuopnlibnyg tohfe Tc oaunpdl iMng towf Taradnsd vMaritaobwlea sredpsavrartiioabnl,e wshepicahr ation, which occurs at M =o c2cMurcs, a4tMMc, =8M2Mc, c1,64MMc cf,o8rM coc,n1fi6gMurcaftoiornc o(0n,fi1)g uinr aEtiqouna(ti0o,1n) (i7n).E Tqhuiast ifiognu(r7e) .shTohwissfi tghuer e shows the alignment of athlieg ndmatean ptooifntths eodn asttarapiogihntt lsinoens,s ttrhaeiigrh ctolminpees,ntshaetiironcos mocpceunrsraintigo nosn othcceu zrerrinog hoonrizthoentzaelr o horizontal line and the slyinseteamnadtitch deespyastretumraet fircodmep tahret ucrrietefrrioam fotrh tehcer istepriaarafotirotnh oefs Mep arnadti oTn wohfoMsea anmdpTliwtuhdoes e amplitude gets smaller agse tMs s imncarlelearseass (M i>n c2rMeacs)e, sex(Mpla>in2iMngc )t,heex applapienainragntchee oafp tphe aMra nacnedo Tf tsheepaMraatinodn Tmsoerpea ration more real at higherr eMal vaatlhuiegsh. eSreeM [8v]a floure ds.eStaeiels[.8 ] for details. We mentionWede in mseevnertiaol ninedstancines ofs ethviesr palreseinntsattainocne asnd oinf prtehviisousp pruesbelinctaattiioonns and in [7,10] the neperde vtoio furasgpmuebnlitc tahtieo mnse[l7t ,t1e0m] ptheerantuereed rtaongfrea ignm 3e rnhtetohloegmiceallt rtaenmgepse rfoartu Tr e> rTagn, ge in 3 rhe- in a way simoilloagr itcoa lthraen fgraegsmfoernTta>tioTng, iinn tahrweea yrhseimoliolagrictaol trhaenfgreasg omf etnhtea Mtio nmionletchurelaerr rhaenogloeg ical ranges in Figures 1o0f–t1h3e aMndm 1o9l:e cMu’la r’c M< M. T MR. The fragmentaticon of the tcempcerature deRlimitates T Rg < T < Tg + 23 ◦C, Tg + 23 < T < TLL, the temperaatnudreT de>liTmitawtehse Treg o cTiLaLt ewdhwerieth TtLhL eisD ual-Phase the liquid-limquoide lt’esmdpisesriaptautirvee atsrsaoncsiiattioedn twemithp etrhaet uDrue a[4l-,P7–h1a2s]e. mTLoLdveal’rsi edsiwssiitphaMtiv[e1 1tr]abnusti-takes values tion temperbaetutwree e[4n,71–5152t]o. 1T7L5L v ◦Carfieosr PwSi.thT gMis [a1l1s]o bvuatr itaabklesw viathluMes bbeeltoweMenc 1r5ea5c thoi n1g75a n°Ca symptote in ◦ ◦ for PS. Tg ist haelsMo v>aMriacbrlaen wgeithso Mth bateltohwe vMalcu ereoafchTign+g 2a3n aCsyfmorpMtot>e Minc tshtea bMili z>e Ms acr oraunngde1 25 C. [10]. Figures 20 and 21 give a possible explanation for the temperature fragmentation of the melt so that the value of Tg + 23 °C for M > M stabilizes around 125 °C. [10]. Figures 20 and 21 above Tg using the mathema ctical framework of Equations (6) and (7). give a possible exFpiglaunreasti2o0n afnodr t2h1es theomwpethraetsuarme efrpaglomt e(wnthaitcihonc oouf ltdheh amveelbt eaebnovcoe mTgb iunseidngin one graph, the mathemyaetitctahle fyraamreeswpolirtkf orf cElqaruiatytiotons h(6o)w anthde i(r7)d. ifferent poles for selected different values of M). Figure 20 only has the selected group of M values shown in the insert that belong to either range-1 or range-3, but none of range-2 of the molecular weight fragmentation. The other values of M giving the alternative pole are shown in Figure 21. All the curves in Figure 20 compensate at T ~ T ◦g + 23 C., and all the curves in Figure 21 compensate at T ~ TLL. Note: in the insert, 38K refers to M = 38,000, etc. Polymers 2024, 16, x FOR PEER REVIEW 24 of 37 Polymers 2024, 16, x FOR PEER REVIEW 24 of 37 Polymers 2024, 16, 2042 22 of 34 Figure 20. The data a (M,T) and a (M,T) are now plotted against T at constant M (the reverse of 11 12 Figures 11–13). Figures 20 and 21 show the same plot (which could have been combined in one graph, yet they are split for clarity to show their different poles for selected different values of M). Figure 20 only has the selected group of M values shown in the insert that belong to either range-1 or range-3, but none of range-2 of the molecular weight fragmentation. The other values of M giving the alternative pole are shown in Figure 21. All the curves in Figure 20 compensate at T ~ Tg + 23 °C., and all the curves in Figure 21 compensate at T ~ TLL. Note: in the insert, 38K refers to M = 38,000, Figureet c2.F 0i. gTuhree d20a.taT ahe(dMa,tTa)a a1n1(dM a,T()Man,Td) aa1r2e( Mno,Tw) aprleottneodw apglaointtsetd Ta gata icnosntsTtaantt cMon (stthaen treMve(rtshee oref verse of 11 12 Figures F11ig–u1r3e).s F1i1g–u1r3e).s 20 and 21 show the same plot (which could have been combined in one graph, yet they are split for clarity to show their different poles for selected different values of M). Figure 20 only has the selected group of M values shown in the insert that belong to either range-1 or range-3, but none of range-2 of the molecular weight fragmentation. The other values of M giving the alternative pole are shown in Figure 21. All the curves in Figure 20 compensate at T ~ Tg + 23 °C., and all the curves in Figure 21 compensate at T ~ TLL. Note: in the insert, 38K refers to M = 38,000, etc. FigFuirgeu 2r1e. 2T1h. iTs hfiigs ufirgeu irse ciosmcopmlepmleemnetanrtya rtyo tFoigFuigrue r2e02 a0nadn dshsohwows sthteh esasmame epplolot tbbuut tfofor rddiffifeferreenntt vvaall-ues ueso offM M, ,a allllb beleolonnggininggt otor arnangge-e2-.2.A Allllt htheec ucurvrvesesi ninddiciactaetdedc ocommppenensastaetea tatT T~ ~T TLL.. Allssoo nnoottiiccee tthhaattt heLL they y--ccoooorrddiinnaattee ooff tthhee ccoomppeennssaattiioonn ppoollee iiss 00, ,aa sistiutuaatitoionn alaslos oococcucrurrinrign gfofro Mr M iRn FinigFuirgeu 1r1e, 1m1,amkiankgi ng R thesthe etsweotw “tora“ntrsaitniosintiso”n rse”larteeldat e[8d].[ 8N].oNte:o itne: tihne tihneseinrts,e 1r2t,51K2 5reKferresf etros Mto =M 12=51,02050,0, 0e0tc,.e tc. Both Figures 20 and 21 are plots of (a 211 − a 2 12 ) vs. T for various values of M remaining constant when T varies. In a certain sense, these figures ar e similar to Figures 11–13, which Figure 2d1.i sTphliasy fiegdutrhe eisf rcaogmmpleenmtaetion ntary o tfot hFeigMurer a2n0 gaendin sthhorwees sthueb s-raamneg peslo. tF biguut froers d2i0ffaenrednt2 1v,alh-owever, ues of Mu, saell abeslloignhgitnlyg dtoi frfaenregne-t2y. A-clol othrde icnuarvtee,s ainnddictahteedr ocloemopfeMnsaatne datT Ti s~ rTev.e Arslseod n: oTtivcea rtiheast, and M LL is constant in Figures 20 and 21, whereas M varies and T is constant in Figures 11–13. Yet the y-coordinate of the compensation pole is 0, a situation also occurring for M in Figure 11, making the presence of poles (compensations) is visible when T is scaRnned from T ◦g to 210 C. In these two “transitions” related [8]. Note: in the insert, 125K refers to M = 125,000, etc. Figure 20, the curves compensate at T = 401 ◦K (127 ◦C), but only for a discrete set of M Polymers 2024, 16, 2042 23 of 34 values listed, not for all the values of M (the other values of M are in Figure 21); additionally, the compensation y-coordinate in Figure 20 is not 0, but 0.4623. Figure 21 applies to all the M values not listed in Figure 20 (i.e., for M = 125,000 to 190,000), and it is shown that the curves compensate at a different temperature, T = 428 ◦K (155 ◦C), and on the y-coordinate axis (a11 = a12 = 0). The values found for the two compensations, 127 ◦C and 155 ◦C in Figures 20 and 21, respectively, appear to match the value given for Tg + 23 ◦C and TLL from other estimation sources [9–12]. In other words, the fragmentation of the temperature and of the molecular weight ranges into rheological sub-ranges is due to the interactive coupling between the effect of M and T in the definition of the viscosity (shown as an example in Equations (6) and (7)). Figures 20 and 21 reveal an added complexity, which is absent in Figures 11–13: whether the curves, for any given value of M, belong to either the pole group in Figure 20 or Figure 21. This complexity in the inter-coupling between M and T, and of its incidence on the fragmentation, has been eluded to earlier when we noted that the existence of pole 16Mc was part of the period-doubling solutions for configuration (0, 1) in Figure 19, but not for configuration (1,0). This shows that the rheology of the coupling µo(M,T) is, indeed, more sophisticated than what has been introduced in this communication and needs more elaboration [8]. Let us now provide a short presentation of the differences between our challenging model of polymer physics to explain the rheology of polymer melts and the classical molecular dynamic models of rheology currently admitted as the current paradigm. The reader should refer to the Preamble of [7,18] as well as [4] for a brief presentation of the theoretical background of our new model of interactive dissipative coupling applied to polymers [7,18] and of its practical application to “disentangle” polymer melts during processing to temporarily decrease their viscosity during processing [4]. These three refer- ences are fully available from the ResearchGate archive platform (“Preamble: Introduction to the Dual-Phase Model of Polymer Interactions and to the Cross-Dual-Phase Model of Entanglements”). The present understanding of the physics of macromolecules is based on an analysis of the properties of a single chain. The presence of the other chains is perceived as a mean field influence on the properties of that chain. The reptation model of de Gennes [6] considers that “entanglements” arise at Mc when the homogeneous field, creating the environment of the chains, is modified due to the higher molecular weight of the macromolecules to include obstacles restricting the motion of the single chain. In other words, topological constraints occurring for higher molecular weights chain lengths create the obstacles that explain the rise of the viscosity at Mc compared to the viscosity in the absence of such obstacles, which has been modelized by Rouse [5]. For the classical view of rheology, the relaxation time of the molecular motions occurring within the chain, in the presence or not of the topological constraints, generates the stresses and strains when the fluid is submitted to a strain rate tensor to put it in motion. The challenge is to correlate the relaxation times obtained from the analysis of the rheological parameters, for example, from the Newtonian viscosity or the cross-over relaxation time to the relaxation times calculated from the conformation and configuration description of the polymer chain. In conclusion, to briefly summarize the foundation of the classical dynamic models of rheology [5,6], their statistical system is the macromolecule, and the chain dynamic is described by the chain relaxation time (Rouse or reptation time). For the Dual-conformers approach, the statistical system studied to define the stresses and the strains is not a single chain, and the molecular motions are not apprehended by their relaxation times. We refer to a basic unit of deformation within the chain, the Dual-conformer, and consider the properties of a statistical system consisting of several Dual-conformers belonging to the same macromolecule but also to other macromolecules, all submitted to the constraints of a statistical model, the Grain-Field Statistics Theory, that defines their evolving interactive coupling as an open dissipative system [4,7,18]. The assumptions of the model define mathematically what “evolving interactive coupling” Polymers 2024, 16, 2042 24 of 34 means when the full collection of Dual-conformers is brought out of equilibrium by an outside “deformation” (thermal or mechanical) to determine how many Dual-conformers of a certain type dynamically cooperate in an active system at any instant, how many systems are active and how many relax, and where the cooperative Dual-conformers are located: on a single chain or on several chains. The physics of dealing with all the chains at once in the statistics, redefining the coupling between the covalent and the inter-molecular interactions, is the model that we have adopted to describe the deformation of polymer melts and solids, above Tg and below Tg [7–9,18]. The theory not only addresses the spatial conformation of the Dual-conformers (cis, gauche, trans) but also their type (either b or F), defining at minimum six possible Dual-conformer “thermokinetics states”. The deformation of a set of macromolecules is quantified by the determination of the distribution of the total number of Dual-conformers among the thermokinetics states and by the determination of the strain and the stress generated by the evolution of the population in those states due to external constraints imposed on the collective system. The single dissipative system that assembles all the Dual-conformers from all the macromolecules is called “a single- dual-phase” system, the duality coming from the b or F character of the Dual-conformers (which is also the reason for the “Dual” connotation). In the Dual-Phase model of rheology the chains are there but are not the statistical systems (they can become the statistical system but only above TLL), and the phenomenon of entanglement, although initiated by an increase of the length of the chain, is not due to a variation in the sequence of various molecular events, say reptation vs. Rouse, happening to the chain deformation, but to the instability of the single dual-phase solution representing the statistical system of Dual-conformers interacting as a single system. At the critical molecular weight of the chain, Mc, corresponding to the triggering of the instability of the single dual-phase, the system of single dual-phase conformers splits into two sub-systems of dual-phases, phase 1 and phase 2, to release from saturation the dissipative energy of the collective system of interactions. We call phase 1 and phase 2 the “Cross-Dual phases”, the word “cross” indicating that the two phases are themselves inter-coupled and dissipative. The split at Mc can be considered “a statistical fold” occurring at the scale of the collective level of the single dual-phase system; it could be imagined similar to the division of a biological cell into two biological cells, although in the case of the split of a single dual-phase the remaining “phases-cells” are not independent clones, they are interactively coupled. Note that at the local scale of the Dual-conformer dynamics, the definition of the b or F status for its Dual-conformers is equivalent to splitting the population of the Dual-conformers in type F and type b conformational states to form a single dual-phase dissipative system, In a sense, this split is “an internal fold” of the system, another form of duality, explaining the terminology of “Dual-Phase” for the dissipative systems just described, which are stable for M < Mc [4,7–9]. We have symbolized the “internal split” leading to a single Dual-Phase system of Dual-conformers by: [(b/F) ↔ (c,g,t)] and the “the external split” of a single Dual-Phase leading to two cross-dual-phases is symbolized in Equation (8): [(b/F) ↔ (c, g, t)] ↔ [(b/F) ↔ (c, g, t)]1 + [(b/F) ↔ (c, g, t)] ↔ [(b/F) ↔ (c, g, t)]2 (8) SINGLE DUAL-PHASE CROSS-DUAL-PHASE where b and F refer to an intermolecular bonding “conformation”, and (c,g,t) (i.e., cis, gauche, trans) refers to an “intra-chain conformation bonding” [7–9]. Figure 22 is a cartoon sketch of a single Dual-Phase system of Dual-conformers forming b-grains locally inter- facing with F-conformers around them (left) and a Cross-Dual-Phase system where two different rheological states of b-grains surrounded by F-conformers coexist and are made stable (below TLL) by compensation of their enthalpic and entropic thermokinetic terms to generate an elastic dissipative shear wave (right) [7,18]. The dissipative statistics equations of a system of Dual-conformers can be simulated (the “EKTOR” simulator) to characterize the interactions between the Dual-conformers of a single chain, and it is shown that the chain can assume the shape of a macro-coil, see (a) in Figure 22, when the number of Dual-conformers in the macromolecule reaches a certain value, estimated around 42 for PS (our value for M’c in Figures 2–8). The interpenetration Polymers 2024, 16, x FOR PEER REVIEW 27 of 37 [(b/F) ↔(c,g,t)] ↔ [(b/F) ↔ (c,g,t)]1 + [(b/F) ↔ (c,g,t)]2 (8) SINGLE DUAL-PHASE CROSS-DUAL-PHASE where b and F refer to an intermolecular bonding “conformation”, and (c,g,t) (i.e., cis, gauche, trans) refers to an “intra-chain conformation bonding” [7–9]. Figure 22 is a cartoon sketch of a single Dual-Phase system of Dual-conformers forming b-grains locally inter- facing with F-conformers around them (left) and a Cross-Dual-Phase system where two different rheological states of b-grains surrounded by F-conformers coexist and are made stable (below TLL) by compensation of their enthalpic and entropic thermokinetic terms to generate an elastic dissipative shear wave (right) [7,18]. The dissipative statistics equations of a system of Dual-conformers can be simulated (the “EKTOR” simulator) to characterize the interactions between the Dual-conformers of a single chain, and it is shown that the chain can assume the shape of a macro-coil, see (a) Polymers 2024, 16, 2042 in Figure 22, when the number of Dual-conformers in the macromolecule reaches a ce2r5toafin34 value, estimated around 42 for PS (our value for M’c in Figures 2–8). The interpenetration of 3 of these macro-coils can also be simulated to see if the resulting state after full or of 3 of these macro-coils can also be simulated to see if the resulting state after full or partial partial penetration differs from the state of a “grown macro-coil” (i.e., a similar macro-coil penetration differs from the state of a “grown macro-coil” (i.e., a similar macro-coil but bwuitt hwbitihg gbeigrgbe-rg rba-ignrsa,ifnosr, ifnosr tiannsctea)n.ce). FFiigguurree 2222. .LLeeftf:t :ccaartrotooonn rreepprreesseenntatatitoionn oof fththee ssiningglele DDuuaal-lP-Phhaassee ssyysstetemm wwhheerree tthhee dduuaal-l-pphhaasseess aarree rreepprreesseenntteedd bbyy tthhee bb-g-grraaininss (b(blalacckk bbaalllsls) )ssuurrrorouunnddeedd bbyy wwhhitiete spspaacece rereppreresesenntitningg ththee FF-c-oconnfoformrmeresr.s . Assuming that single macromolecules can take the shape of “macro-coils”, represented by the larger Assuming that single macromolecules can take the shape of “macro-coils”, represented by the larger spheres, which they can do for M > M’ , (3 macro-coils are represented in (a), their interpenetration spheres, which they can do for M > Mc’c, (3 macro-coils are represented in (a), their interpenetration rreemmaaininss aa ssiningglele DDuuala-lP-Phhasaes easa slolnogn gasa Ms M rermemainaisn bsebloewlo wMMc, w, hwichhic ihs sishoshwonw inn i(nb)(. Rc b) i.gRhitg: hwth: ewnh Men >M M>c,M th,e tihneteinrpteernpeetnraettiroanti oonf tohe macro-cc f the macroo-iclos ialss aas sainsignleg lDe Duaula-Pl-hPahsaes eisi sinisntsatbalbel eanandd reresusultlst sinin tthhee ssppliltit ininttoo ttwwoo inintteerrlalacceedd ((““eennttaannggleledd””)) ssiningglele DDuuaal-lP-Phhaasseess (c(c),) ,wwhhicichh aarree ccoonnsstatanntltyly rreeddeefifinniningg ththeeirir inintteerrffaaccee ttoo rreemmaaiinn ssttaabbllee, ,ggeenneerraattiinngg tthhee pprreesseennccee ooff aann eellaassttiicc ddiissssiippaattiivvee sshheeaarr waavvee inin ththee mmooltlteenn ssttaattee [[77,1,188]]. . TThhee ssttaabiilliitty off a diissiipattiive ssysstteemo offD Duuaal-lc-coonnfoformrmeresrsis isc ocnotnrtorlolelldedby btyh tehDe iDssiispsaiptiav-e ttievrem terinmt ihne thEeK ETKOTROsRe tseotf oefq euqautiaotniosn, sw, whihcihchis isd deefifinneeddb byy tthhee value of IIdd = LLnn(N(Nbb/N/NF)f,) , whheerree N b aanndd N f aarree tthhee ppooppuullaattiioonn ooff tthhee bb aanndd Ffb F Du uaall--ccoonnffoorrmeerrss iirrrreessppeeccttiivvee ooff tthheeiirr (c,t,g) spatial conformation, Chapter 3 of [9] and Appendix A. When the length of the (c,t,g) spatial conformation, Chapter 3 of [9] and Appendix A. When the length of the mac- macromolecules interpenetrating as macro-coils is below a certain value, which we have romolecules interpenetrating as macro-coils is below a certain value, which we have as- assumed corresponds to the manifestation of M , the simulation shows [8,18] that the one sumed corresponds to the manifestation of M cc, the simulation shows [8,18] that the one Dual-Phase solution is stable (Figure 22 left box (b) state), i.e., its state does not differ Dfruoaml-Pthheasreh seoolluotgioicna liss statatebloef (Faiggruorwe 2n2s lyesftt ebmoxw (bit)h staatger)e, ai.tee.r, intsu smtabtee rdoofeus nniotts dinifftehre frtoomtal tshyes rtehmeolpoogpicualla stitoante. oHfo aw gervoewr,nt hsyesstiemu wlaittiho na sghroeawtesrt hnautmI begrr oowf usneixtsp oinn ethnet itaod lltyalt soywstaermds paonpauslaytmiopnt.o Hteo. wAetveMr, tahne dsibmeuyloantido,na sthwowo-ss c itnhgalte Isdy gstreomwsi sexcrpeoanteednti(aFlilgyu troew2a2rdrisg ahnt baos-x y(mc))ptotere. dAut cMe cth aenvda blueeyoonfdI,d ai ntwthoe-stiwngoled suyasltepmha isse scrceraetaetde d(F. igTuhries s2p2 lirtigohftt hboexo r(cig))i ntoal single Dual-Phase system into a more stable Cross-Dual-Phase system is what we suggest corresponds to the manifestation of the entanglements in polymer melt rheology. In this Dual-Phase comprehension of entanglements, the thermodynamic aspect of the stability of the interactions is responsible for its occurrence. The value of Mc can be changed by manipulating externally-mechanically and thermally- the constraints on the Dual-Phase system, and this new conception of the entanglements gave birth to the technology of Rheo-Fluidification of melts to control their viscosity by “disentanglement” manipulation [4,7]. The Rheo-extrusion of hundreds of kilos of “sustained-oriented” pellets demonstrated that the “disentangled melts” produced by reheating the treated pellets exhibiting “Sustained-Orientation” behaved in a way incomprehensible to an interpretation of melt deformation based on the classical reptation relaxation time explanation. We call the “thermokinetics parameters” of the Dual-Phase dissipative statistics the value of the population of the b and F states and the (cis, gauche, trans) states of the Dual- conformers statistics. These values can be obtained by simulation [8], assuming certain values for the statistical constants [9], or by Depolarization spectroscopy [9], but not directly Polymers 2024, 16, 2042 26 of 34 by rheological experimental evidence. The simulation of the basic equations by the EKTOR simulator [8,9] permitted to define the strain produced by the system and quantify the corresponding stress generated by the system when submitted to strain rate deformations at various T for various M. This simulation work was capable of predicting all the rheological results found for a typical amorphous melt with M < Mc: the proportionality of the viscosity with M (Equation (4), the Vogel-Fulcher dependence of the viscosity with T (Equation (3)), the effect of pressure, and the shear-thinning decrease of the viscosity at higher shear rates [7]. This success [8] proved that it was indeed possible to correlate the viscosity of the polymer melts to the population thermokinetic parameters. The remaining important issue is to find an empirical formulation of the viscosity measured experimentally so that we have access, directly or indirectly, to the thermokinetic parameters of the model. This research is ongoing and currently presents different possible solutions, one of which is presented here for the first time. In the criteria considered to achieve a satisfactory empirical formulation of the experimental viscosity dependence on T, M, the strain-rate and pressure, we have assumed that we could rely on our cartoon representations of the dissipative interactions in Figure 22 to define the b-grains “conformational enthalpy” and the “dynamic free volume” surrounding the b-grains [7,18]. In that regard, the formulation of the viscosity presented by Equations (6) and (7) offers the advantage of being able to extract from the fitting parameters pseudo thermodynamic quantities, ∆H1(M,T), ∆S1(M,T) for phase 1, and ∆H2(M,T), ∆S2(M,T) for phase 2, which qualitatively look like they could be related to the Dual-Phase thermokinetic parameters extracted from Figure 22. The enthalpies are clearly visible in Equation (7), and the entropies can be calculated by “entering” inside the exponent the values of the pre-exponential factors: b2 and b4 or b’2 and b’4. The other advantage of this empirical formula is that it can be applied to both the case of one single Dual-Phase (for M < Mc) and to the case of two Dual-Phases (for M > Mc), since the presence of the second phase, which is the mathematical clone of the first phase with different fitting parameters, can be eliminated by forcing the regressions to determine the first phase constants only. The veracity of our statement that a single exponential term fits better the M < Mc data range and that the M > Mc requires two exponential terms (to achieve the same level of fitting accuracy) can easily be proven by following the variation of the r2 and Syx of the regressions in both the M < Mc range and the M > Mc range. Finally, to explain the choice of an exponential function, e.g., (a1exp(a2/T) + a3), instead of a hyperbolic function such as the Vogel-Fulcher Fulcher equation (Equation (3)), to describe the temperature dependence of Log µo(T) at M given, we have already introduced before [10,11] the notion of “isomorphic functions”, which provide identical curves passing through the same data set when they are both used to fit those data. For usual rheological data, from, say, Tg to Tg + 150 ◦, the Log µo(T) vs. 1/T fitted by the exponential growth function provides the isomorph of Log µo(T) vs. A + B/(T − T ) with an r 2 2 difference smaller than 10 −5 in most cases (i.e., with an uncertainty less than the experimental error to determine the value of log µo). In this paper, our choice of the empirical function to describe the effect of T is the exponential growth function (or its Dual version above Mc), but in other instances [11,12], our choice was to modify the Vogel-Fulcher equation to make its potential thermodynamic attributes ∆H and ∆S [11,12]. The classical interpretations of the viscosity results are “molecular dynamic models”, based on the description of the properties of the macromolecular chain during flow and in particular of the molecular mechanisms occurring to the chain that produce the strain and require the stress needed for the chain motion. In other words, the classical models of rheology try to correlate the viscosity to the chain molecular description and to its evolution [1,5,6]. For instance, for the entangled melt of Figure 23, the attention is on understand how the molecular weight of the chain, actually M to the power 3.4, can be calculated and thus understood from molecular parameters arising from the chain’s description: from its length, from its location among its neighboring chains, from its displacement, its stretching, its relaxation time, etc. Let us critically examine if this power equation description of the viscosity could be understood from a Dual-Phase viewpoint. Polymers 2024, 16, x FOR PEER REVIEW 29 of 37 provide identical curves passing through the same data set when they are both used to fit those data. For usual rheological data, from, say, Tg to Tg + 150°, the Log µo(T) vs. 1/T fitted by the exponential growth function provides the isomorph of Log µo(T) vs. A + B/(T 2 −5 − T2) with an r difference smaller than 10 in most cases (i.e., with an uncertainty less than the experimental error to determine the value of log µo). In this paper, our choice of the empirical function to describe the effect of T is the exponential growth function (or its Dual version above Mc), but in other instances [11,12], our choice was to modify the Vogel- Fulcher equation to make its potential thermodynamic attributes ΔH and ΔS [11,12]. The classical interpretations of the viscosity results are “molecular dynamic models”, based on the description of the properties of the macromolecular chain during flow and in particular of the molecular mechanisms occurring to the chain that produce the strain and require the stress needed for the chain motion. In other words, the classical models of rheology try to correlate the viscosity to the chain molecular description and to its evolu- tion [1,5,6]. For instance, for the entangled melt of Figure 23, the attention is on understand how the molecular weight of the chain, actually M to the power 3.4, can be calculated and thus understood from molecular parameters arising from the chain’s description: from its length, from its location among its neighboring chains, from its displacement, its stretch- ing, its relaxation time, etc. Let us critically examine if this power equation description of Polymers 2024, 16, 2042 27 of 34 the viscosity could be understood from a Dual-Phase viewpoint. FFiigguurree 2233.. TThhee M >> Mc ddaattaa aarree fifitttteedd wiitthh EEqquuaattiioonnss ((44)) aanndd ((55)),, tthhee ccllaassssiiccaall ffoorrmuullaa.. TThhee bbeesstt regressiionall fifitt off aa11v vss.. LLoogg Mh haasss sloloppee3 3.3.36699a annddi nintetercreceppt t− −17..44444 ((rr 22 = 0..9982,, Syxx//DoF = 0..0883),, whiich sshoouulldd bbeec coommppaarreeddt otot htheefi fit to of ft hthees asmame ed adtaatau suisnigngM Min isntesatedadof olof gloMgMan adnfidt tfiettdeadc caocrcdoirndgintog tthoe thCer oCsrso-sdsu-dalu-palh-pasheasmeo mdoeldoefl othf etheen etanntagnlegmleemnet nmt emlteults uinsginagd ao duobulebleex epxopnoenetinatliatel rtmersmfsu fnucnticotnio.n. The classical models are interested in determining in a plot like in Figure 1 whether the sllope ooff llooggµ µoov vs.s.l ologgM Mis i1s 1o ro3r. 43.t4o tkon konwowif tihf ethpeo lpyomlyemr iesru ins -uent-aengtalendgloerde onrta enngtlaend-, agnleddt, haenDd uthael- PDhuaasle-Pmhoasdee lmnoededels ntoeeddest etrom dienteerwmhienteh werhoenthlyera osninlyg lae osirntgwleo oDr utwal-oP Dhausaels- aPrheapseress aernet ipnrethseenmt einlt .thFeig mureelt2. 3Fdigeufirnei t2e3ly daeffifinrmiteslyth aaffit trhme sm tehltati sthene tmanegllte ids iennvtaienwgleodf iitns pvoiewwe roef xitpso pnoewnteerq euxaplotone3n.3t7 e.qTuhael dtoa t3a.3in7.F Tighuer dea2t3a iisnr eF-ipgluorttee 2d3i nisF rieg-uprleott24edto inu sFeigaulirnee 2a4r stoca ulesefo ar Mlinienasrt esacdaloe ffaorl oMga rinitshtmeaidc socfa lae ,lologgarMith. Tmhiec rsecdalceu, rlvoge pMas. sTinhge trherdo ucguhrvteh epapsosiinntgs is obtained by fitting the linear scale data with a pair of exponential terms of M, i.e., exp through the points is obtained by fitting the linear scale data with a pair of exponential (-M/M1) and exp(-M/M2), which could be attributed to the presence of two Dual-Phasesterms of M, i.e., exp (-M/M ) and exp(-M/M ), which could be attributed to the presence and thus to an entangled me1lt (M > Mc) pursu 2ant to the Cross-Dual-Phase criterion. We find Mof tw= o7 5D,3u8a0l-~P2haMsesa annddM thu=s t1 c 2 84 o4 a,4n6 1e.nTtahnegrleegdr emsseilot n(Mby >t hMecp) apirurosfueaxnpto tnoe tnhtiea Cl treorsms-sDoufaMl- in Figure 23 is at least as good, perhaps even slightly better, than the classical power law of Figure 23 (see the captions of Figures 23 and 24). Now, if we can fit the M < Mcviscosity data with a single dual-phase instead of 2, i.e., with a single exponential of M, this could be considered a simple validation of the single vs. cross dual-phase criterion for entanglements. Figures 2–8 of section A show that the “un-entangled” state of the melt, below Mc, is not as simple as a set of parallel lines with slope 1 when the temperature varies, which is what would be expected from the classical view. In fact, we saw that the location of the entanglement molecular weight, Mc, could not be ascertained from the break of the viscosity extrapolated from the entanglement and the un-entanglement sides. However, if we select the highest temperature of the melt, T = 210 ◦C, corresponding to Figure 2, we see that the points below Mc make up a continuous curve without a break, which can be re-plotted (not shown) against M instead of logM, like it was done in Figure 23, and fitted by regression with either a single exponential of M or with two exponential terms of M: we verify that the fit is better with a single exponential term (r2 = 0.99873) than with 2 exponential terms (r2 = 0.9931), although the opposite is true for melts above Mc, for which the 2 exponential terms solution is better. One could conclude from these simple validation tests, involving the difference between a regression using one exponential or two exponential terms of M, that the Dual-Phase explanation of entanglements is satisfied. The reality is unfortunately more complex, and a convincing proof requires a much more sophisticated analysis [8]. Nevertheless, we want to emphasize that the use, in Figure 23, of a different analytical formula to describe the molecular weight dependence of the viscosity of a polymer melt in Polymers 2024, 16, x FOR PEER REVIEW 30 of 37 Phase criterion. We find M1 = 75,380 ~ 2 Mc and M2 = 844,461. The regression by the pair of exponential terms of M in Figure 23 is at least as good, perhaps even slightly better, than the classical power law of Figure 23 (see the captions of Figures 23 and 24). Now, if we can fit the M < Mcviscosity data with a single dual-phase instead of 2, i.e., with a single exponential of M, this could be considered a simple validation of the single vs. cross dual- phase criterion for entanglements. Figures 2–8 of section A show that the “un-entangled” state of the melt, below Mc, is not as simple as a set of parallel lines with slope 1 when the temperature varies, which is what would be expected from the classical view. In fact, we saw that the location of the entanglement molecular weight, Mc, could not be ascertained from the break of the viscosity extrapolated from the entanglement and the un-entangle- ment sides. However, if we select the highest temperature of the melt, T = 210 °C, corre- sponding to Figure 2, we see that the points below Mc make up a continuous curve with- out a break, which can be re-plotted (not shown) against M instead of logM, like it was done in Figure 23, and fitted by regression with either a single exponential of M or with two exponential terms of M: we verify that the fit is better with a single exponential term (r2 = 0.99873) than with 2 exponential terms (r2 = 0.9931), although the opposite is true for melts above Mc, for which the 2 exponential terms solution is better. One could conclude from these simple validation tests, involving the difference between a regression using one exponential or two exponential terms of M, that the Dual-Phase explanation of entan- glements is satisfied. The reality is unfortunately more complex, and a convincing proof Polymers 2024, 16, 2042 requires a much more sophisticated analysis [8]. Nevertheless, we want to empha2s8izoef 3t4hat the use, in Figure 23, of a different analytical formula to describe the molecular weight dependence of the viscosity of a polymer melt in lieu of the sacrosanct formula that uses lietuheo fptohwe esar cerxopsoancetnfto irnm Fuiglautrhe a2t3u rseepsrtehseenptosw a efirresxt pstoenpe ntot dinemFigyustriefy2 t3hriesp mreissleenatdsiangfi rcsltas- stespictaol dfoermmyusltaif yoft hmiselmt irshleeaodloinggy c(Elaqsusiactailofnosr m(4u) laanodf (m5)e.l tInr hoeuorl oogpyin(Eioqnu, atthioenses f(o4)rmanudla(s5 )a.re Ineomupr ioripcainl ieoxnp,rethsseisoenfso wrmithu laa struaree veamlupe ilriimcaitleedx tpor eesnsgiionneserwinigth paurtpruoesevs aolnuley.l iTmhiet evdartioous enagttineemepritnsg top auttrpriobsuetse oan plyh.ysTihcael vseanrisoeu tso atthteesme pmtsatthoeamttartiibcualt eexapprehsyssiiocnasl, seesnpseecitaolltyh feosre M ma>t hMem (aet.gic.a, lbeyx tphree rsesipotnasti,oens pscehcioaollly afc d oerpMts [>6]M), ca(ree. go.n, lbyy ththeeorreetpictaatliloyn jusscthifioeodl aifd tehpet sp[r6e]d),ic- are only theoretically justified if the predictions of the theory succeed in explaining all tions of the theory succeed in explaining all the other experimental facts, such as the frag- the other experimental facts, such as the fragmentation of the rheological ranges for M mentation of the rheological ranges for M and T, as an example, or the “sustained-Orien- and T, as an example, or the “sustained-Orientation” of Rheo-fluidified melts, as a second tation” of Rheo-fluidified melts, as a second example [10]. example [10]. FigFuigreur2e4 .2S4a. mSaemdea dtaataas ains iFni gFuigreur2e3 ,2r3e, prleoptltoettdeads asf ua nfuctnioctnioonf oMf Man danfidt tfiedttebdy bryeg rreegsrseiossnioann alnyasliyssis usiunsginag dao duobuleb-leex-pexopnoenetinatliadle dcaeycafyu nfucnticotnioonf oMf :Mw: iwthitthh ethfeo lflolwloiwngintgw towtoe rtmersm(sr e(dreldin lei)n:ee):x epx(-p(- M/M84/844,446,416)1a)n adnedx pex(-pM(-/M7/57,358,308).0)r.2 r=2 =0 .09.9989181S ySxy/xD/DoFoF= =0 .00.006046.4T. hTihsigs rgarpahphd edmemysytisfitiefisetsh ethues uesoef othf ethe power exponent 3.4 to describe the dependence of viscosity on the molecular weight and contributes to the validation of the existence of two Cross-Dual-Phases for entangled melts [8]. The formulation of the viscous flow behavior of M > Mc melts from the Cross-Dual- Phase perspective offers new arguments regarding the reptation model historical dilemma: while the original de Gennes’s theory predicted a power exponent of 3 for the molecular weight dependence of Newtonian melt viscosity [6], several authors successfully tweaked the mathematics of the initial reptation model to explain the “reality”, i.e., that viscosity appeared to follow a 3.4 exponent behavior instead of 3. This is discussed in [4,7,10]. Are the dual-phase results of Figures 15 and 16 actually suggesting that the original reptation exponent of 3, calculated by de Gennes, adequately describes the viscosity in phase 2, the core phase of the Cross-Dual-phase model, and that the coupling with the “entangled phase”, phase 1, is responsible for the correction to the power exponent, not the tube fluctuations nor any other possible tweaking due to molecular reasons?. In the (0,0) configuration Cross-Dual-Phase treatment of viscosity, we find that viscosity is the product of two terms, each M and T dependent. At high T, one of the terms varies with M with a power exponent 3, while the other term varies with M with a power exponent 0.373, for PS. The slope 3 for phase 2 is what the Dual-Phase model predicts at a high T, just like what de Gennes projected starting from molecular dynamics considerations. We show in Refs. [4,9,11,12] that for T > TLL, which is the case in Figures 15 and 16, the Dual-Phase dissipative term, Id, relaxes out, making the statistics converge with the non-dissipative Boltzmann’s statistics, rendering legit both the computations of de Gennes and the cross- dual phase explanations of the results. This is the reason why, under such conditions of equivalence, i.e., for T > TLL only, we expect to find for phase 2 the de Gennes’s slope of 3 Polymers 2024, 16, 2042 29 of 34 when the log µo2, calculated from b1, b2, b4 in Equation (7) is plotted like in Figure 16. If the equivalence between de Gennes’s calculations and the Dual-Phase approach is justified at higher T, one may question the utility of the subsequent improvements by many authors to correct the original calculations of de Gennes: were they actually necessary? As far as the Dual-Phase model of the entanglement is concerned, the requirement to have two Cross-dual-phases for M > Mc, and simply one dual-phase for M < Mc is a theoretical statement from a theory, hypothetically predicting what changes at Mc. But the parameters entering the theory are not directly available, and we cannot validate it directly. But the consequence of having a split of the system at Mc guides us in the choice of the empirical formula that may describe the experimental facts: the single exponential growth function of 1/T has a r2 close to 1 when fitting all the data for M < Mc, yet starts to deteriorate in fitting quality around Me and becomes questionable, even poor, at Mc; the addition of another exponential growth term for the M > Mc data makes r 2 raise again to almost perfection. These empirical formulas are “inspired” by the theoretical model of a split at Mc and describe the facts well, although they do not directly address the issues of the stability of a single phase or a cross-dual-phase dissipative system, which are the true theoretical issues that explain the cause of the split at Mc. The “Dissipative function” of the EKTOR simulations cannot be measured directly from rheological data, but its assumed existence and the ability to simulate its relaxation behavior continue to inspire new perspectives about the origin of Mc with, as a consequence, the so-called Rheo-Fluidification (aka“disentanglement”) and Sustained-Orientation technologies [4,7,8]. Theoretically, in our view, the possibility to create disentangled states in polymer melts to reduce favorably their processing viscosity is due to the inducement, by the effect of shear and vibration, of the instability of the entanglement state of the melt. This instability results from the consequence of the effect of the Grain-Field Statistics on the interplay between the size of the system of interactions (related to M), its incidence on the b-grain population (vs. the dynamic free volume), and the value of the thermokinetic constants of the dissipative open system [7–9]. An explication of the Rheo-Fluidification of melts to produce Sustained-Orientation does not seem possible using the current understanding of entanglements by the current paradigm of rheology [4,7]. 4. Conclusions There are sophisticated theories that have been advanced for the last 70 years that derive from the assumptions expressed by Figure 1 (Rouse [5] and reptation [6], for instance). Actually, all the theories from the past regarding viscous flow have assumed the separation of the effect of M and T in the expression of viscosity. It is classically admitted [1] that the (Newtonian) viscosity depends on the product of two parameters: a friction factor which is controlled solely by local features such as the free volume, itself a function of T only, and a molecular weight dependent factor which is a function of the configuration of the chains, itself controlled by M. The difference between the various theories arose from the different statistical treatments of the molecular weight factor of the macromolecules. This was the case for the molecular dynamic theories of Rouse [5] applicable to melts with M < Mc and of de Gennes [6] for the reptation model applicable to entangled melts with M > Mc. The failure of both the Rouse and the reptation models to adequately describe the experimental data has been suggested in another publication [4]. In this paper, we summarize the results of a thorough mathematical analysis of viscosity data on a series of monodispersed PS samples [2,3,7,8,10] to test the separation of M and T in the expression of the viscosity equations. The different approach in this paper is the use of a mathematical formula in which M and T do not separate to express the dependence of M and T on the melt viscosity; it succeeds to fit the data points “perfectly”, at least as well as the classical formula. However, it concludes the opposite of what the classical formula claims regarding the separation of M and T, which is summarized in Figure 1. Our paper analyzes the relative impact of the fitting parameters magnitude on the residual curves of the regressions when M Polymers 2024, 16, 2042 30 of 34 or T varies and concludes that the features in Figure 1 are only valid for certain values of M and T, otherwise are only approximations. The classical formulation of viscosity, Equations (3)–(5) of the Development section, may be good “isomorphs” of our new for- mulation, Equations (6) and (7), in the data range chosen, leading to wrong conclusions when trying to understand them theoretically, or it could be the other way around, our contradictors could rightfully say. For instance, our new empirical equation of viscosity generates data that can be plotted like in Figure 1, validating the classical slopes of 1 and 3.4 when those data are fitted pursuant to the classical formula. However, the understanding of the new viscosity equation by the Dual-Phase and Cross-Dual-Phase concepts is possible and induces a very different understanding of entanglements at Mc. This ambiguity of having two distinct mathematical formulas provide similarly satisfactory descriptions of the experimental results, making the two formulas “isomorphs”, can be resolved by considering the consequences of the two different theoretical approaches and the possibility to test experimentally their projections. In that sense, the new rheological formula proposed projects the fragmentation of the rheological ranges defined by the effect of M and the effect of T, thus predicting the existence of “new” transitions in the melt, Tg + 23 and TLL for temperature, Mc, 4Mc and M’c = Mc/8 for the molecular weight. The classical approach does not predict the existence of such fragmentation of the rheological ranges, nor does it understand its origin. The other projection of the new viscosity equation is the “period-doubling” of the values of the molecular weights that lead to the true separation of the effect of T and M: Our contradictors may ask why so many experienced scientists could have missed the possibility that T and M did not separate in the viscosity expression. The use of the log-log plot in Figure 1 was possibly one of the reasons for the visual success impression that may have misled the scientific community in the belief that the separation of M and T was validated [10]. Another possible reason was the coherence with previous work and the simplifications the separation of T and M implied theoretically: the influence of the molecular weight of the chains could be treated like in the rubber theory, by consider- ations on the statistical configuration of the chains [1], and, separately, the influence of temperature was explained by the free volume, successfully addressing the non-Arrhenius behavior [1,10]. In reality, the main reason why these results were not apparent to previ- ous investigators is explained in this paper: for T above approximately 448 ◦K (175 ◦C), the Newtonian viscosity of the PS melts demonstrates an apparent satisfactory classical behavior for all M > 2Mc, which is the usual range tested. The mathematical testing of the separation of M and T explains that the apparent validation becomes systematically worse as T and M are decreased. By re-plotting the viscous data according to the classical paradigm of rheology (Figure 1) at various temperatures (Figures 2–8), we reveal that the current definition of Mc by the break of the viscosity curves across Mc, does not validate that Mc is independent of T, again defaulting the assumption of the separation of M and T. The problem is that, in general, rheologists test and report high-temperature data only and rarely (if not ever) show the evolution of their Figure 1 at several lower temperatures, as we did (Figures 2–8). In summary, the separation of T and M in the expression of viscosity was an illusion that was hard to reveal. The realization that it was an illusion and is actually not true has consequences on the understanding of viscoelasticity: for instance, it implies that the time-temperature superposition principle (TTS) is itself invalid, a conclusion also validated by the fragmentation of the rheological ranges for M and T. A second consequence of the illusion of the separation of M and T is that it precisely hides the existence of the fragmentation of the molecular range with its period doubling structure (Figure 19), one of the most interesting differences between the classical and the dual-phase views of the melt rheology. Overall, the current understanding of entanglements is challenged by the results presented in this communication. However, as is concluded below, a word of caution must be added. In previous communications [4,7], we pointed to convincing Rheo-SANS experimental evidence challenging the predictions of the popular reptation model of polymer melt defor- Polymers 2024, 16, 2042 31 of 34 mation and concluded that the mechanism of reptation of the macromolecules through a set of topological obstacles was probably inadequate to describe the concept of entanglements and of shear-thinning. In this paper, we show that a new way to analyze classical data raises intriguing new questions regarding the effect of molecular weight and temperature on viscosity, taking us far, very far, from a molecular dynamic explanation of the rheology of melts. We conclude once more [4] on the inadequacy of the current modelization of the physical properties of polymers [5,6,19] and on the need to change the paradigm. We briefly presented in the paper the basics of the “Dual-conformers” Dissipative Statistics, which assumes that Dual-conformers are the statistical units of the macro- molecules that need to be characterized, not the single macromolecules themselves, to understand the properties of a set of macromolecules responding collectively to a deforma- tion (thermal or mechanical). This new approach to the interactions in polymers, which is described in more detail in previous publications [7,9,18], is summarized in Figure 22 to explain the possible physical reason behind the appearance of the M’c transition in Figures 2–8, as the formation of stable macro-coils, and to visualize the split of a single Dual-Phase system into a Cross-Dual Phase system at Mc, providing for Mc, the entangle- ment molecular weight, a very different explanation from the current admitted view. It will be shown in future presentations [8] that the new Dual-Phase and Cross-Dual-Phase model not only proposes a new understanding of “entanglements” but also of the dynamic melt properties G′(ω,T) and G′′(ω,T) and the normal stresses. It predicts shear-thinning and strain softening in shear mode and strain-hardening in extensional mode; it success- fully describes the transitional behavior at Tg, from a solid-like to a liquid-like behavior, predicts [11,12] the existence and the characteristics of the TLL upper melt transition tem- perature as the end of dissipative modulation in a Dissipative system [9]. Finally, the theory addresses the stability (or the strain induced lack of stability) of the Cross-Dual Phase entanglement network, leading to the Rheo-Fluidification technologies [7]. The theoretical assumptions of the new model and the quantitative descriptions it generates constitute a whole new understanding of the viscoelastic properties of polymers that could be considered the premises of a new paradigm in that field of physics. However, our last word must address a word of caution necessary before we can generalize these conclusions to the status of a new reality. Our model is in its infancy and in constant improvement. What we found in this paper is, indeed, quite unexpected and intriguing: the effect of T and M combine to create a fragmentation of the rheological ranges resulting in a periodical validation of the criteria for variable separation, but only at discrete specific values of M that verify period doubling at Mc/2: Mc, 2Mc, 4Mc, etc.). We believe that such an important conclusion (Figure 19) is based on experimental results that should be duplicated and even extended using more modern automated instruments, permitting the use of a greater number of samples. We suggest tripling the current set of 22 samples to cover the same range of molecular weight (from M = 550 to M = 1.2 Million g/mole). We also suggest repeating the rheological data using a dynamic rheometer instead of a viscometer operating in the linear visco-elastic range and covering the temperature span from (Tg + 10) to (Tg + 110), spacing the temperature range every 5 ◦C to obtain the maximum number of points on the temperature scale. In conclusion, the results of a new analysis of the Newtonian data for a series of monodispersed PS samples raise more fundamental questions regarding the validity of the current established views in polymer physics. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Data Availability Statement: Data are contained within the article. Conflicts of Interest: The author declares no conflict of interest. Polymers 2024, 16, 2042 32 of 34 Appendix A (Notions of Grain-Field-Statistics) Dual-Conformers 3 to 5 successive covalent bonds of the macromolecular chain define a Dual-conformer. Their spatial conformational state can be cis, gauche, trans, which can be reduced to 2: cis-gauche (“cg”) and trans (“t”) to simplify. Dual-conformers also have a dissipative confor- mational state. Their dissipative state can be “f” or “b” (free or bonded inter-molecularly). The duality of the Dual-conformers implies that we split each state cg and t into an F and b category, giving 4 states cgf, cgb, tf, tb; the result is a dual partition, between cg and t units on one hand, and between the F and b units on the other hand. The two partition functions are coupled by kinetic equations and energy equations. The energy equation includes a Dissipative term Id = Ln (Nb/Nf) where Nb is the total number of b dual-conformers and Nf is the total number of f dual-conformers. Id collapses to zero at equilibrium. The Dual-Split Dissipative equations (the EKTOR model) present the most basic assumptions controlling the dynamics of a dissipative system of Dual-conformers in inter- actions. The new equations converge to traditional kinetic equations at long times or under “true” equilibrium conditions. Under non-isothermal conditions, triggered by heating or cooling, the system of interactions becomes self-dissipative, and the conformational duality is responsible for a structure of the Free Energy. Mechanical “straining” of the dynamic systems of interactive dual-conformers, i.e., favoring an increase of the population of the ‘t” conformers in the population statistics, requires to apply a stress to the system. The stress can be quantified from the modification of (∆e − ∆eo) by the non-equilibrium states, where ∆eo is the conformational energy difference between a flexed (cg) and a trans (t) spatial conformation at equilibrium. Activated and Relaxed Open Dissipated Systems. The dynamics of a single dissipative system of Dual-conformers is essential but not sufficient to describe the plurality of mechanical modes of deformation available to a dissipative strained system. In the Grain Field Statistics approach of mechanical deformation, one is concerned with all the possible organizational changes occurring to the Dual-conformers statistics by application of a stress field. This includes the possibility to generate a network of open dissipative systems where some systems are actively stretched (S) while other are Relaxing (R). The deformation of a polymeric material consists of a change in the partition function of the conformers via a dynamic structuration of the interactions into Stretched (∆e ̸= ∆eo) and Relaxing (∆e = ∆eo) open dissipative systems. The process of activation (S)—deactivation (R) can be viewed as a fluctuating grain field in the temporal and space dimensions. Open Dissipative Network Dynamics of Dual-Conformers. -A Dual-Split Dissipative Equations (∆e = ∆eo): (Bor type) Nb = ntb + ncgb Nf = ntf + ncgf (A1) Bo = (Nb + Nf) dntb/dt = 0.5dNb/dt − k1*ntb + k2*(Nb − ntb) dntf/dt = 0.5dNf/dt − k1*ntf + k2*(Nf − ntf) k1 = νm exp(−∆1/T) (A2) k2 = νm exp(−∆2/T) ∆m = (∆1 + ∆2)/2 ∆eo = (∆1 − ∆2)/2 Ln (ntb/ncgb) + Ln (ntf/ncgf) + Ln (Nb/Nf) = 2 Ln (k2/k1) = 4∆eo/T (A3) (Note: the gas constant is equated to 1) Id = Ln(Nb/Nf) is the dissipative term. Polymers 2024, 16, 2042 33 of 34 Note: Equations (A1)–(A3) apply to “Bor” systems (∆e = ∆eo) and correspond to an internal split of Bo into Nb, Nf) to generate the inter mode of dissipation. In addition to the Dual-Split Kinetics constants (∆m, νm and ∆e), the study of the mechanical properties of a polymer requires the knowledge of: (1) The amount of strain induced by the rotation of a conformer from one spatial confor- mation to the other. (2) The relationship for a stretched system between the kinetic stretch rate, Rs, and (∆e − ∆eo). (3) The mechanical behavior of open “stretched” and “relaxed” systems, i.e., during and after it is submitted to a given deformation. -B Mechanically Disturbed “Stretched” Systems (Bos type). l = “kinetic length” l = ntb + ntF (total population of t-conformers) Additional constraint: R = [dntb/dt + dntf/dt] is imposed Stress is determined from (∆e − ∆eo) but is not proportional to it. 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