Polymer Glass Formation: Role of Activation Free Energy, Configurational Entropy, and Collective Motion
aa r X i v : . [ c ond - m a t . s o f t ] F e b Polymer Glass Formation: Role of ActivationFree Energy, Configurational Entropy, andCollective Motion
Wen-Sheng Xu, ∗ , † Jack F. Douglas, ∗ , ‡ and Zhao-Yan Sun ∗ , † † State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of AppliedChemistry, Chinese Academy of Sciences, Changchun 130022, P. R. China ‡ Materials Science and Engineering Division, National Institute of Standards andTechnology, Gaithersburg, Maryland 20899, United States
E-mail: [email protected]; [email protected]; [email protected] bstract We provide a perspective on polymer glass formation, with an emphasis on mod-els in which the fluid entropy and collective particle motion dominate the theoreticaldescription and data analysis. The entropy theory of glass formation has its originsin experimental observations relating to correlations between the fluid entropy andliquid dynamics going back nearly a century ago, and it has entered a new phase inrecent years. We first discuss the dynamics of liquids in the high temperature Arrhe-nius regime, where transition state theory is formally applicable. We then summarizethe evolution of the entropy theory from a qualitative framework for organizing andinterpreting temperature-dependent viscosity data by Kauzmann to the formulationof a hypothetical ‘ideal thermodynamic glass transition’ by Gibbs and DiMarzio, fol-lowed by seminal measurements linking entropy and relaxation by Bestul and Changand the Adam-Gibbs (AG) model of glass formation rationalizing the observations ofBestul and Chang. These developments laid the groundwork for the generalized entropytheory (GET), which merges an improved lattice model of polymer thermodynamicsaccounting for molecular structural details and enabling the analytic calculation of theconfigurational entropy with the AG model, giving rise to a highly predictive model ofthe segmental structural relaxation time of polymeric glass-forming liquids. The devel-opment of the GET has occurred in parallel with the string model of glass formation inwhich concrete realizations of the cooperatively rearranging regions are identified andquantified for a wide range of polymeric and other glass-forming materials. The stringmodel has shown that many of the assumptions of AG are well supported by simula-tions, while others are certainly not, giving rise to an entropy theory of glass formationthat is largely in accord with the GET. As the GET and string models continue tobe refined, these models progressively grow into a more unified framework, and thisPerspective reviews the present status of development of this promising approach tothe dynamics of polymeric glass-forming liquids. Introduction
Glasses have been central to fabrication technologies since the dawn of civilization, andthese materials are becoming increasingly important in modern technologies. Glasses arenot only encountered as structural materials in our environment, such as window panes, ves-sels, and plastic polymeric materials, but they are also used in high technology applicationsas materials for non-volatile electronic memory, organic light emitting diodes and organicelectronics, commercial aircraft, molecular separations in manufacturing and water de-salination and purification, etc. The physics of glass formation is also highly relevant tobiological substances whose properties and functions are dictated by their complex molecularstructure and intermolecular interactions. Polymeric materials have a general propensity to form glasses at low temperatures ( T )and exhibit glassy dynamics over a large T range, even if some of them ultimately formsemicrystalline materials composed of a mixture of crystalline and noncrystalline regions intheir solid state. Since polymers exhibit many of the same features in glass formation asother glass-forming (GF) materials while at the same time the many complications associatedwith crystallization that often arise in GF liquids composed of atomic mixtures can beavoided, these materials have been widely used to investigate the fundamental nature ofglass formation for decades. On the practical side, polymer glasses have the advantage ofbeing light, ductile, easy to process, and, of course, relatively inexpensive, giving rise tonumerous applications in daily life and emerging technologies. We are truly living in the‘Polymer Age’. From our own standpoint, the rich chemistry of polymers allows for anexploration of the large molecular parameter space of possible types of glass formation, thesame attribute that makes these materials so attractive for applications in materials science.As a complement to our work emphasizing the entropy theory of glass formation, andsimulations for purposes of validation and refinement of the present theoretical perspectiveon the dynamics of GF liquids, we mention some excellent summaries of the experimentalobservations on polymer glass formation. T g from above where the fluid is still in equilibrium and where strong changes in thefluid dynamics still occur. The most basic phenomenology is that the dynamic propertiesof GF liquids, such as shear viscosity η and structural relaxation time τ α , display a dramatic T dependence upon cooling toward T g . For example, the shear viscosity of a GF materialcan alter by over orders of magnitude over a relatively narrow T range. Angell hasintroduced the concept of ‘fragility’ to quantify the strength of the T dependence of thedynamics of GF liquids. The dynamics of ‘strong’ liquids, such as SiO , has a nearly Arrhe-nius T dependence, and in contrast, ‘fragile’ liquids, such as o -terphenyl, exhibit dynamicproperties whose T dependence is highly non-Arrhenius. For polymers, T g can be tunedroughly from K to
K, and fragility varies by nearly an order of magnitude fromapproximately to over . While models and theories are being constantly introduced for understanding the originand nature of the slowing down of the dynamics of GF liquids, some ideas remain invariantfor qualitatively understanding the basic phenomenology of glass formation. It has beengenerally appreciated since the works of Simon and Kauzmann that the rapid increasein η and τ α upon cooling is accompanied by a drop in the fluid entropy. Much of the earlyinteresting literature attempting to define the glass transition and understand the role ofthe fluid entropy in the glass transition is difficult to access, but Schmelzer and Tropin have provided an informative review of these early studies of glass formation, along with adiscussion of how these early studies relate to more recent works. Kauzmann discussed thefate of equilibrium cooled liquids based on the excess entropy, S exc ≡ S liq − S xtal , where S liq and S xtal are the entropies of a material in the liquid and crystalline states, respectively. Thedrops in S exc upon lowering T are so rapid that, if S exc in the T regime above T g is formallyextrapolated to much lower T where the material can no longer remain in equilibrium, S exc would vanish at a finite characteristic temperature called the ‘Kauzmann temperature’ T K ,4elow which S exc would formally become negative. The inferred vanishing of S exc is termedthe ‘Kauzmann paradox’ or the ‘entropy crisis’, but this situation may instead represent theresult of an unwarranted extrapolation, and thus have no physical meaning. Nonetheless, thereality of the drop in the entropy upon lowering T is an unequivocal feature of GF liquids,and the general idea that there is some sort of ‘entropy crisis’ associated with intrinsic glassformation remains a prevalent model of the physical nature of glass formation, an approachthat we develop in our own work. In more recent works, Martinez and Angell have reviewedthe parallelism between the T dependences of the viscosity and fluid entropy in diverse GFliquids, and Angell has emphasized the role of the fluid entropy in understanding thefragility of GF liquids.Another more recently identified feature that is often invoked to explain the dynamics ofGF liquids is the occurrence of dynamic heterogeneity or collective motion of particles. A significant body of evidence has indicated the presence of transient clusters of particleswith excessively high or low mobilities relative to simple Brownian particles. In many cases,the specific nature of dynamic heterogeneity has often been rather vague, and there arecurrently many uncertainties about how the various types of dynamic heterogeneity that onecan imagine or identify might be related to the dynamics of GF liquids. This list seems tobe essentially endless and ever growing, and we believe that the time has come to cull thisproliferation by investigating which of these heterogeneity measures have a direct relationto the overall dynamics of the fluids. Interestingly, both mobile and immobile particles arefound to exhibit the property of forming ‘fractal’ dynamic polymeric clusters, regardless ofthe molecular bond connectivity of individual molecules, and the temperature and structuralcharacteristics of these types of heterogeneity conform to well-known phenomenologies of self-assembling systems.
This finding gives us hope for an overall generalized theoreticaltreatment of GF materials. It must be acknowledged at the outset that the theory of glassformation is still in a model building stage, however. The entropy theory is just one of anumber of promising theoretical frameworks for glass formation. For instance, the nonlinear5angevin equation (NLE) theory and the more recent elastically collective nonlinearLangevin (ECNLE) equation theory of Schweizer and coworkers emphasize the centralrole of the long wavelength limit of the static structure factor, S (0) , in determining thedynamics of GF liquids. The free volume model of the dynamics of GF liquids also remainspopular in the polymer science community. Schweizer and coworkers have previouslydiscussed many of these models, so we do not reproduce such a discussion in this Perspective.Here, we focus on models of glass formation where the entropy and collective particleexchange motion are taken to be of primary importance in the description of the dynamicsof GF liquids, as postulated by Adam and Gibbs (AG). The present perspective on thenature of polymer glass formation thus emphasizes the consequence of changes in the fluidentropy on the dynamics of cooled liquids in the low T regime where the T dependencesof relaxation and diffusion are observed to be non-Arrhenius. We strictly avoid the term‘supercooled’ since it is clear that not all GF liquids, and particularly not all polymericliquids, can crystallize, even in principle. It is then possible that the glass state ofsome materials at low T is the true thermodynamic state rather than just a metastablethermodynamic condition.We first discuss transition state theory (TST) describing the dynamics of liquids inthe high T Arrhenius regime, which, in our view, should be a natural starting point forany theory of glass formation based on the idea of thermally activated transport. We thensummarize the historical development of the entropy theory of glass formation which has been‘under construction’ for almost a century, a development that started from observations ofcorrelations between the fluid entropy and heats of vaporization and dynamics in the 1930s,to the organization and interpretation of these data by Kauzmann in 1948, and thenthe formulation of the Gibbs-DiMarzio (GD) theory in 1958 based on the then newlydeveloped theory of polymer thermodynamics formulated by Flory in his classical workaimed originally at understanding the crystallization and equation of state properties ofpolymer materials. A precise fundamental link between dynamics and thermodynamics was6rst discovered by Bestul and Chang in 1964 by comparing data for relaxation and excessentropy obtained from specific heat measurements of molecular and polymer GF liquids,which, in turn, led to the introduction of the AG model in 1965 to rationalize theseobservations. The AG model later gave rise to the current prevailing conceptual frameworkand language in which the dynamics of GF liquids is characterized by fragility andother metrics related to the T variation of the configurational entropy S c associated with acomplex energy landscape that gives rise to the complex dynamics of cooled liquids. Thesedevelopments led to the generalized entropy theory (GET) and the string model, whichare built on concepts and methods developed in a vast number of previous experimentalstudies, so these newer works are just part of a long historical development for a descriptionof liquid dynamics to increased precision and validation. For completeness, we also providea brief overview of other models of liquid dynamics emphasizing the fluid entropy, includingthe random first-order transition (RFOT) and the Rosenfeld and excess-entropy scalingapproaches to relaxation and diffusion in fluids. When cooling upon the approach to T g , the T dependence of the rapidly growing relaxationtime of GF liquids is described in most experimental studies by the Vogel-Fulcher-Tammann(VFT) equation, τ α = τ exp (cid:18) DT VFT T − T VFT (cid:19) , (1)where τ is a prefactor, D is a ‘fragility parameter’ quantifying the strength of the T depen-dence of τ α , and T VFT is the temperature where τ α extrapolates to infinity. Although theVFT equation is perhaps the best-known phenomenological relationship in GF liquids, therehave been attempts to challenge the supremacy of this equation. Nonetheless, most re-searchers persist in fitting their relaxation time, viscosity, and diffusion data to this simpleequation because of its relative simplicity and apparent effectiveness in fitting data for a7ast range of materials. It should be mentioned, however, that even some of the earliestquantitative works on relaxation in polymer materials recognized that the VFT equation,and the mathematically equivalent and heavily utilized expression of Williams, Landel andFerry (WLF), are limited to a T range between T g and a temperature about K higherthan T g , which corresponds to a temperature comparable to the crossover temperature T c in our discussion of the GET model in Section 4. It is notable that the VFT relation maybe derived from the GET for a prescribed T range between T c and T g , along with an analogof this equation when glass formation is achieved by increasing pressure ( P ) at fixed T .Moreover, the VFT parameters D and T VFT , or their WLF equivalent, may be calculatedfrom the GET so these are not just adjustable empirical parameters in the GET. The GETalso specifies an alternative functional form for the T regime above T c but below the on-set temperature T A , where relaxation and diffusion become non-Arrhenius, again with allmodel parameters fixed in terms of molecular parameters describing the thermodynamics ofpolymer fluids.At sufficiently high T , a large body of evidence indicates that the structural relaxationtime τ α of GF liquids generally exhibits an Arrhenius T dependence, τ α = A exp (cid:18) ∆ H k B T (cid:19) , (2)where A is a prefactor, k B is Boltzmann’s constant, and ∆ H is the activation enthalpy.The prefactor A in eq 2 involves a vibrational attempt frequency τ multiplied by a factorinvolving the activation entropy ∆ S , exp( − ∆ S /k B ) . We append an ‘ o ’ subscript to theactivation energetic parameters to denote that these quantities are determined in the high T Arrhenius regime where relaxation times are relatively short and the fluid is relatively‘dynamically homogeneous’.To highlight the presence of Arrhenius dynamics at high T , we show the segmentalstructural relaxation time τ α of poly(dimethylsiloxane) versus /T at ambient pressure8 /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) ExperimentVFT equationArrhenius equation
Figure 1: Illustration of the temperature ( T ) dependence of the dynamics of glass-formingliquids. The figure displays the logarithm of the segmental structural relaxation time, log τ α ,versus inverse temperature /T for poly(dimethylsiloxane) with the molar mass of g/mol at ambient pressure ( P ). Experimental data were provided to us by the authors of ref88. The solid line is a fit to the Vogel-Fulcher-Tammann (VFT) equation (eq 1). The dashedline is a fit to the Arrhenius equation (eq 2) with the fitted parameters, A = 1 . × − s and ∆ H /k B = 2181 K. Despite the dramatic increase of τ α with lowering T upon approachingthe glass transition temperature T g , an Arrhenius dynamics applies instead at sufficientlyhigh T .measured by the Rössler group in Figure 1. Measurements of dynamics in the Arrheniusregime are generally challenging experimentally due to the thermal degradation of poly-mers, but the field-cycling H nuclear magnetic resonance relaxometry has recently changedthis situation, at least for some polymers.
It is thus well established that the fluid dy-namics is observed to be homogeneous in the high T regime where the Arrhenius equationformally holds, and the dynamic heterogeneity sets in below the onset temperature T A fornon-Arrhenius relaxation where the dynamics deviates from the Arrhenius equation.While the physical mechanisms for explaining the dynamics of fluids must be differentin the high T Arrhenius and low T non-Arrhenius regimes, it is essential to build an ex-perimentally validated theoretical foundation for understanding and ultimately controllingthe dynamics of cooled liquids, where collective many-body effects are prevalent, based on athorough understanding of the dynamics in the high T Arrhenius regime. In our view, thisis a natural starting point for any theory of glass formation based on the idea of thermallyactivated transport. Molecular dynamics (MD) modeling becomes a particularly attractive9ool at present because of the relative small magnitude of structural relaxation and relativelyshort equilibration times in this regime. It should then be possible to make relatively rapidprogress in understanding the relation between molecular structure and other molecular pa-rameters and factors controlling structural relaxation and diffusion in the high T Arrheniusregime. We expect that this information should be useful for understanding the dynamicsof polymers, even though many polymer materials would physically degrade before reach-ing such high T at which the Arrhenius equation describes the dynamics of the materials.These observations again serve to underscore the fact that we need to be concerned aboutmore than just the glass transition temperature T g , since T A is often more than twice T g so that glass formation encompasses the entire T regime in which polymer materials arenormally utilized and processed and thermally stable to chemical decomposition. This large T range above T g thus has fundamental importance for processing and field use applicationsof polymers, and for understanding the nature of glass formation, regardless of whether thepolymer materials crystallize or not at temperatures lower than T A .Historically, TST of Eyring and others has provided significant advances for under-standing the dynamics of liquids in the Arrhenius regime. If viscous flow is viewed as achemical reaction in which the elementary process is the passing of a single molecule fromone equilibrium position to anther over a potential barrier, viscosity, plasticity, and diffu-sion of fluids can all be treated as examples of absolute reaction rates within TST. Thecentral quantity in TST is the activation free energy ∆ G , ∆ G = ∆ H − T ∆ S , (3)which contains both enthalpic and entropic contributions. The TST-based approach hasproven to be a reliable general theoretical framework for describing the dynamic propertiesof condensed fluids at elevated T , even though the actual development of Eyring’s absoluterate theory for the dynamics of real liquids is based on the rather idealized view of10he dynamics of real fluids that enables transforming the problem of explaining gas phasedynamics into a model of the dynamics of condensed liquids in terms of assumptions moreacceptable for gases than for pure fluids. A rigorous TST has been developed for idealizedcrystalline materials, in which the particles interact with purely harmonic interactions.Viewing a liquid as a highly defective crystal has provided an alternative model of the Ar-rhenius dynamics of liquids. This alternative view of structural relaxation and diffusion incondensed materials emphasizes the collective nature of transport in the condensed state,but there is still no satisfying analytic theory of dynamics, even in the Arrhenius regimewhere TST is expected to apply. Numerical implementations of TST in ordered condensedmaterials have shown that the TST framework still applies even though it currently does notallow any direct analytic treatment.
We thus adopt TST as a working framework forunderstanding the dynamics of cooled liquids. Notably, the AG theory implicitly adoptsthe TST framework in its formulation, and thus, this assumption extends to the GET, which combines the AG relation with a thermodynamic theory that enables calculating theconfigurational entropy of polymer fluids. While the TST-based approach to glass formationdoes not require the many simplifying assumptions of Eyring, its practical implementa-tion requires MD simulations to estimate the activation free energy parameters. Thus, thetreatment of the dynamics of condensed fluids remains largely semi-empirical and requiressimulation studies or experimental measurements to determine parameters characterizingthe free energy of activation for fluids with very different chemical natures. Over the years,correlations between the energetic parameters of activation and thermodynamic propertieshave been observed, such as a strong correlation between ∆ H and the heat of vaporiza-tion, and changes in ∆ H and ∆ S have been found to occur in a parallel fashion whenchanging molecular mass, film thickness, or additive concentration, a phenomenon termed the ‘entropy-enthalpy compensation’ (EEC) effect. Thesecorrelations provide some sign posts indicating what properties control the basic energeticparameters, ∆ H and ∆ S , which are attractive targets for future simulation and theoretical11odeling efforts.We have observed the EEC effect to be a significant factor in simulations of polymersystems and experiments of polymer nanocomposites and thin films. Notably, the magnitude of the EEC effect on relaxation and diffusion can be rather signif-icant, even meriting the term ‘astounding’. For example, the prefactor of eq 2 can changeby a factor of about as the activation energy grows upon approaching the melting tem-perature from below. Extremely large changes in the prefactor are also encountered throughthis mechanism in the diffusion coefficient of metallic glasses; e.g., see Figure 13 in ref 125.Calculations based on TST have provided insight into these astronomical changes of theprefactor in the context of surface dislocation nucleation of metallic materials. NumericalTST calculations have also given some insight into the EEC phenomenon in the interfacialdynamics of crystalline materials.
An implication of these results is that the neglect ofthe activation entropy ∆ S , which is a basic assumption in the classical AG theory andthe more recent GET, can lead to serious errors in the TST description of the dynamics ofGF liquids. This is perhaps the most serious assumption in the GET that we must addressin the future to improve this model. This point is further discussed in Section 4. In this section, we introduce the classical entropy theories of glass formation, namely, thetheories of GD and AG. We present main ideas behind these theories and discuss theirstrengths and weaknesses. The current section augments a previous discussion given in ref19, so our discussion is relatively brief.
In 1958, GD developed a systematic statistical mechanical theory of polymer glass forma-tion based on the lattice model of semiflexible polymers that had been developed earlier by12lory for the description of polymer crystallization. Since the entropy and other thermo-dynamic properties are purely configurational in the lattice model, the term ‘configurationalentropy’ has been widely adopted in the literature. The GD theory provided the first theoret-ical underpinning for how estimates of the fluid entropy might be utilized to make predictionsof relevance for understanding the dynamics of GF liquids by identifying an ‘ideal glass tran-sition’ with a thermodynamic event, the vanishing of the configurational entropy S c . Thisresult was a direct theoretical ‘echo’ of the Kauzmann’s concept of an entropy catastrophepossibly underlying glass formation. In addition to providing a conceptually clear pictureof the origin of glass formation upon cooling at some point where the system has ‘no placeto go’ dynamically and thus becomes arrested, the GD theory allowed for quantitative pre-dictions for the glass transition temperature T g , which was heuristically identified by GD asbeing a non-equilibrium analog of the true ideal glass transition temperature T that couldnot be reached because of the extreme slowing of the dynamics upon approaching this tem-perature, but which ‘tracks’ T . DiMarizo and Yang have reviewed the many successes ofthe GD theory, which is commonly invoked in experimental studies to the present time.In a more recent work, DiMarzio revised his interpretation of the model prediction ofa vanishing of the configuraional entropy in light of information that later became available.In particular, DiMarzio suggested that S c became ‘critically small’ at the glass transition,rather than actually vanishing, so that the spirit of the GD theory is preserved. Corsi andGudrati have reassessed the initial negative view of the GD theory by formulating a theoryof semiflexible polymers on Husimi lattices to allow calculations without approximation andtheir work has provided an expanded view of the original work by GD that confirms theessential idea of an entropy crisis in this type of thermodynamic model. We further discussthe technical revision of the GD picture and its ramifications for glass formation based onthe GET below in Section 4.The central quantity in the GD model is the configurational entropy S c , defined as thetotal entropy with vibrational contributions being excluded. In physical terms, the configu-13ational entropy is related to the number of distinct configurational states of the fluid. Unfor-tunately, the conceptual clarity of the notion of configurational entropy remains a questioneven in current studies of glass formation, posing difficulties and challenges for determin-ing this quantity both experimentally and computationally. Berthier and coworkers haverecently reviewed the methods of estimating S c from a computational perspective. In par-ticular, a new Monte Carlo swap algorithm has been developed to estimate S c at extremelylow T , and these simulation results have been interpreted as suggesting the existenceof a Kauzmann temperature in the particular model fluids studied. Unfortunately, thisnovel simulation method seems to be only applicable to fluids of spherical particles havinga high size polydispersity, a property that is not encountered in real atomic or molecularmaterials. It seems unlikely to us that the swap algorithm can be extended to polymeric andmolecular GF liquids generally. There is still no proof that S c actually vanishes at a finite T . Personally, we are skeptical about the existence of a finite Kauzmann temperature in realmolecular fluids, even though the GET itself predicts such a temperature rather generallyin semiflexible polymer melts, as we discuss further in Section 4.In addition, strong criticism has been often raised against fundamental tenets of theGD theory concerning the identification of a vanishing of S c with a second-order phasetransition. Even more seriously, the vanishing of S c has been suggested to be an artifactof the inaccuracy of the mean-field calculations of S c for dense polymer fluids based on thelattice model. Gujrati and Goldstein were the first to demonstrate that the GD modelviolated rigorous bounds on the entropy of polymer melts, clearly bringing the validity ofthe GD model into question, despite its impressive empirical success in identifying essentialtrends between the experimental T g and the theoretical ‘ideal glass transition temperature’ T calculated from the lattice model. Binder and coworkers clarified the situation somewhatbased on simulations of a bond fluctuation lattice model of flexible polymers, which suggestthat S c approaches a constant positive value at low T , but the validity of these resultshas remained a question because of the inherently difficulty of estimating thermodynamic14roperties reliably from simulations at such low T .The original GD theory of glass formation in polymer liquids involves other assump-tions that significantly limit the predictive capacity of this pioneering model of polymerglass formation. First, the theory is preoccupied with the general philosophical problem oflocating and explaining the existence of an ‘ideal glass transition temperature’, T , which,as we have discussed above, probably does not really exist. Unfortunately, a fluid cannotremain in equilibrium near T because the structural relaxation time becomes astronomicalin magnitude near this temperature so this issue is a bit academic. A more recent worksuggests that the vanishing of S c at a finite temperature T in the lattice model is justan artifact of the high T expansion involved in the lattice model calculations for the freeenergy. Explicit calculations of S c for flexible polymer melts show that S c does not van-ish at any finite positive T when the high T expansion associated with the treatment ofchain stiffness is avoided. When combined with the AG model, this finding also impliesthat no divergence generally exists in the AG theory and its extensions such as the GET. Second, we also emphasize that the GD model is based on a highly simplified descriptionof polymer chains as semiflexible self-avoiding walks interacting with uniform short-rangeinteractions and composed of structureless monomer units. This extremely coarse-grainedmodel of polymer fluids does not allow one to consider how monomer structure and chaintopology affect polymer glass formation. Moreover, there is more to glass formation thanknowing T g . Large complex changes in the dynamics of GF liquids initiate at T well above T g , but below T A for non-Arrhenius dynamics. Thus, we really need a theoretical frameworkthat can predict the breadth of the transition temperature range by estimating temperaturescharacterizing the beginning, middle, and end of this broad transition phenomenon havingboth well-defined dynamic and thermodynamic signatures. Any discussion of this kind re-quires something like the AG model to interrelate the structural relaxation time to S c or some other thermodynamic property. The original GD theory describes glass formationfrom a purely thermodynamic perspective. It is the ‘marriage’ of the powerful lattice cluster15heory (LCT) extension of the lattice model of type studied by GD to treat polymermelts having general monomer structure and the AG model formally linking the melt ther-modynamics to the segmental structural relaxation time. The resulting GET is a highlypredictive model of the segmental dynamics of polymeric liquids over the entire temperatureand pressure regimes of glass formation. Of course, the question remains of whether thisframework is true to the physics of real and simulated materials. This is why validation bysimulation and measurement is so important in further developments of the entropy theoryof glass formation. Although a large body of evidence indicates that the Arrhenius equation describes the T dependence of relaxation and diffusion in liquids at elevated T , this description ceasesto apply as liquids are cooled to low T (e.g., see Figure 1). The term ‘fragility’ has beenintroduced to quantify the degree to which the dynamics deviates from the Arrhenius be-havior. The structural relaxation time provides an important quantity whose change inmagnitude can be as large as between the high T Arrhenius regime and the low T non-Arrhenius regime where the material progressively acquires the rheological properties ofa solid as T g is approached. Of course, the apparently universal and practical nature of thisphenomenon has naturally prompted many speculations as to its cause.Early experimental studies attempting to rationalize the non-Arrhenius T dependenceof the shear viscosity and other fluid transport properties emphasized that large changesin the dynamics correlate with corresponding changes in the fluid entropy, and thiscorrelation led some influential scientists to infer that the thermodynamic changes originatefrom the emergence of collective motion within cooled liquids. Following up on an earliersuggestion of Goldstein that some other thermodynamic property than density, such asentropy and enthalpy, might be a better ‘control parameter’ for glass formation, Bestul andChang took a first step forward to transform this loose connection of qualitative ideas and16bservations into a quantitative relationship between the entropy and the rate of relaxationby noticing a direct relationship between the excess entropy S exc and the segmental structuralrelaxation time τ α . Note that Bestul and Chang defined an excess entropy by taking thesolid reference state to be the glass state, while some authors later chose the crystallinestate as the reference state, in line with the definition of Kauzmann. This choice iscontingent on what form of the solid exists for a given material, since some materials haveno available, or even perhaps existing, crystalline form. Regardless of how S exc is defined,this quantity only provides a rather rough estimate of S c , which is unfortunately not adirectly observable property. It should also be noted that there is another excess entropydefined as the difference of entropies between the liquid and gas states, which we designateas b S ex to avoid confusion with other definitions of the excess entropy discussed above. b S ex has attracted significant interest in relation to the dynamics of GF liquids, and Dyre hasrecently provided a comprehensive review on this topic. We discuss this alternative theoryof the dynamics of liquids built around b S ex in Section 6.2.Bestul and Chang also introduced a practical method of estimating S c from S exc fromspecific heat measurements, in addition to performing such measurements at high resolution.They clearly made many notable contributions to the emergence of a quantitative entropytheory of glass formation, and, accordingly, we believe that their pioneering contribution tounderstanding of the dynamics of GF liquids deserves greater recognition.The seminal observations of Bestul and Chang were later rationalized by AG withformal arguments that remain highly influential to the present. In the AG model, therelaxation of the fluids is assumed to be described by an extension of TST with a T -dependent barrier height ∆ G , τ α = τ exp (cid:18) ∆ Gk B T (cid:19) , (4)where τ is the high T limit of τ α . This is the first postulate of the AG model. Thecentral idea expressed by this extension of TST to describe GF liquids, which was widelyaccepted as a general formulation of liquid dynamics at the time, is that the dynamic slowing17own of cooled liquids has its origin in the presence of dynamic clusters, which AG termedthe ‘cooperatively rearranging regions’ (CRR) to embody the prevailing idea of why simpleTST ‘breaks down’ at low T . AG were rather vague about the form of these clusters andabout how they might be identified in practice, but they postulated that these hypotheticalclusters should have properties consistent with experimental observations. In particular,AG made a second postulate that the average number of particles in these CRR should beproportional to the activation free energy ∆ G . The same hypothesis was introduced earlierby Mott; see the paper of Nachtreib and Handler for a discussion of this model ofthe T dependence of the activation energy and its application to understand the stronglynon-Arrhenius T dependence of the measured diffusion coefficient of white phosphorous. Itis notable that Nachtreib and Handler also invoked EEC with the melting point beingthe compensation temperature in a heated crystalline material. Finally, AG assumed thatthe average number of particles in these hypothetical CRR should be inversely proportionalto the fluid configurational entropy S c , thereby tying the concepts of growing collectivemotion in cooled liquids to the observed decreasing fluid entropy, whereupon they adoptedan estimate of S c introduced by Bestul and Chang to finish their model of the dynamicsof glass formation.We may translate the AG arguments into a more formal symbolic representation. AGfirst posited that the activation free energy ∆ G of TST at high T should be multipliedby a factor z , corresponding to the number of molecules or segments participating in theabstract CRR. AG then further assumed that the size of the CRR should scale inverselywith S c , z = S ∗ c /S c , where S ∗ c is the high T limit of S c , a quantity that is also assumed toexist by AG. In other words, AG basically postulated that the activation free energy ∆ G is a T -dependent quantity, ∆ G ( T ) = z ( T )∆ G = [ S ∗ c /S c ( T )]∆ G , (5)18hich reduces to standard TST at high T where ∆ G ( T ) reduces to ∆ G . AG were silentabout the characteristic temperature or the T range at which the change to non-cooperativeliquid dynamics should occur, however. This physically attractive picture of glass formationthen indicates that collective motion simply renormalizes the activation energy by a factorequal to the number of particles involved in the collective particle exchange motion, z , whichbasically rationalized the empirical relation of Bestul and Chang. It is evident that the AG model is clearly more a series of hypotheses than a real theory,but this model has provided a highly influential general conceptual framework for under-standing the dynamics of GF liquids, despite its apparent theoretical weaknesses. The AGmodel has entered a new phase in recent years as MD simulations enable estimates of col-lective motion and calculations of the thermodynamic properties of polymeric and other GFliquids, so we can now test this model quantitatively and extend the model as necessary.It is truly remarkable how much of the AG theory has survived in the more modern andvalidated description of GF liquids, which is a testament to the deep physical understandingthat AG had of the essential nature of GF liquids. In particular, the AG theory has held upremarkably well over the last 50 years in comparison with numerous experiments and simulations of diverse GF liquids. Efforts have also been made to place the AGmodel on a sounder theoretical foundation by extending TST to account for barrier crossingprocesses that involve many particles, basically formalizing the heuristic ideas intro-duced long before Mott and AG. The development of TST by Bonzel in the context ofunderstanding the non-Arrhenius dynamics of crystalline materials is particularly a detailedreformulation of TST to treat this type of problem in an analytic framework, and the theo-retical framework developed by Freed shares many mathematical features, but this theoryis not formulated in a chemically specific way, so we view this theory as a promising workin progress rather than a complete theory. The RFOT theory also offers great potentialfor developing a sounder foundation for the AG model, as we discuss in Section 6.1.While the AG theory represents an important step in our understanding of the dynamics19f GF liquids, in the sense that it provides a rationale for a quantitative link between τ α and S c , there are several drawbacks of the AG model that limit its predictive capacity. First,no effort was made by AG to calculate S c from theoretical considerations. Originally, AG argued for approximating S c by the excess entropy S exc to make contact with the VFTequation and to rationalize the correlation of Bestul and Chang, while Martinez andAngell emphasized the use of S exc as a thermodynamic predictor of the dynamics of GFliquids in the spirit of the AG model. Second, the combination of the AG relation witha thermodynamic model, such as the GD model, was never made, so there have beenno previous predictions regarding the influence of molecular structure and interactions onpolymer glass formation, a problem of both fundamental and practical importance. Thus, theAG theory has been essentially used as a framework for understanding qualitative featuresof glass formation, such as the origin of the VFT equation and the origin of fragility, a measure of the degree of non-Arrhenius relaxation. The AG model in its original form haslimited value in relation to the design of new materials and prior predictions of the propertiesof materials in terms of given molecular parameters. Moreover, AG offered no quantitativepicture of the structure and polydispersity of the hypothetical CRR so that the originalAG theory has some serious shortcomings. Finally, as we have mentioned in Section 2,AG made the completely unwarranted assumption of neglecting the activation entropy ∆ S ,presumably for reasons of mathematical expediency. It is known from even standard TSTthat ∆ G contains an entropic contribution that can sometimes be quite large. This isjust one of the things that must be fixed in any extended model of TST that incorporatesthe ideas of AG for the purpose of making quantitative comparisons with relaxation dataobtained from real GF materials.We again emphasize that despite the above problems in the classical theories of GD and AG, these models of glass formation have been quite influential in understanding thenature of glass formation and general trends of T g with molecular structure. In particular,these classical theories have set the stage for the development of new models probing the20rigin of the slowing down of the dynamics of GF liquids from a molecular perspective. In thefollowing, we discuss the development of the entropy theory along two lines of research. Thefirst line of research merges the AG relation with explicit computations of the configurationalentropy based on a statistical mechanical model of polymers, yielding the GET of segmentalrelaxation in polymeric GF liquids, which we discuss in Section 4. The second thrustof our work is built on the platform of MD simulations and tools drawn from liquid statetheory to quantify collective motion and calculate structural relaxation time, along with otherdynamical and thermodynamic properties. The identification of well-defined cooperativemotion in cooled liquids that appears to conform exactly to the main hypotheses of AG and the finding that the geometrical properties of these clusters accord with equilibriumpolymerization led to the development of the highly quantitative string model of glassformation, as discussed in Section 5. The development of these more recent works hascontinuously addressed the shortcomings of the classical GD and AG models. Againwe underscore that like all extant theories of glass formation, the entropy theory of glassformation remains in an exploratory stage so that some ‘loose ends’ are to be expected. As acomplement to our works on the GET and the string model, we also discuss the RFOT and the Rosenfeld and excess-entropy scaling approaches to liquid dynamics since thesemodels are likewise based on the fluid entropy, and may thus be characterized as alternativeentropy theories of glass formation. The GET takes advantage of the LCT for the thermodynamics of polymer systems,which significantly extends the models of Flory and GD in the sense that it includes21xplicitly the description of monomer structure. The LCT represents individual monomersin terms of a set of united-atom groups with each occupying a single lattice site of volume V cell and enables thermodynamic descriptions of the basic molecular characteristics of polymers,including excluded volume interactions, chain connectivity, cohesive interaction strength,chain stiffness, and monomer molecular structure. For simplicity, we discuss the LCT forsingle-component polymer melts with the reminder that this theory enables the descriptionof multi-component polymer systems, such as polymer blends. Extensions of the GEThas been made for the glass formation of polymers with additives and polymer blends. For a compressible polymer melt, the LCT yields an analytic expression for the Helmholtzfree energy f per lattice site in the following general form, βf = βf mf − X i =1 C i φ i , (6)where β = ( k B T ) − and φ is the lattice ‘filling fraction’ defined by the ratio of the totalnumber of united-atom groups to the total number N l of lattice sites. The term βf mf corresponds to a contribution from the zeroth-order mean-field approximation, βf mf = φM ln (cid:18) φz L M (cid:19) + φ (cid:18) − M (cid:19) + (1 − φ ) ln(1 − φ ) − φ N i M ln( z b ) , (7)where M is the molecular mass corresponding to the total number of united-atom groups ina single chain, z represents the lattice coordination number, which is related to the spatialdimension via z = 2 d for a d -dimensional hypercubic lattice, L is the number of subchains, N i is the number of sequential bond pairs lying along the identical subchains, z b = ( z/ −
1) exp( − βE b ) + 1 with E b being the bending energy parameter. The second term on theright-hand side of eq 6 represents corrections to the zeroth-order mean-field contribution,where the coefficients C i ( i = 1 , ..., are obtained by collecting terms corresponding to agiven power of φ , C i = C i, + C i,ǫ ( βǫ ) + C i,ǫ ( βǫ ) . (8)22ere, ǫ is the microscopic cohesive energy parameter describing the net attractive van derWaals interactions between nearest-neighbor united-atom groups. C i, , C i,ǫ , and C i,ǫ aretabulated in the Appendix A of ref 138 and are generally functions of z , T , E b , M , as well asa set of geometrical indices that reflect the size, shape, and bonding patterns of monomers.The free energy enables computations of all thermodynamic properties.
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T [K] c o n f i g u r a t i o n a l e n t r o p y / k B per lattice siteper occupied lattice site T A T c T g T Figure 2: Illustration of the T dependence of the two different definitions of configurationalentropy predicted by the generalized entropy theory (GET). The figure displays the config-urational entropy per lattice site versus per occupied lattice site as a function of T at zeropressure. The symbols cross, triangle, square, and diamond indicate the positions of the onset T A , crossover T c , glass transition T g , and ideal glass transition temperatures T , respectively.The vanishing of s c for semiflexible polymer melts is probably spurious due to the high T expansion associated with the treatment of chain stiffness in the lattice cluster theory. Thecalculations are performed for a melt of chains with the structure of polypropylene (PP),where the cell volume parameter is V cell = 2 . Å , the chain length is N c = 8000 , the cohesiveenergy parameter is ǫ/k B = 200 K, and the bending energy parameter is E b /k B = 600 K.The above parameter set is utilized for the calculations considered in Figures 3, 8, 9, and 10.Before the development of the GET, the LCT has been primarily used to explain enig-matic observations in polymer blends, including the existence of an entropic contributionto the effective Flory interaction parameter χ . The generalization of the LCT to considerpolymer glass formation was made possible first by arriving at an analytical expression forthe configurational entropy in terms of the natural logarithm of the density of states Ω( U ) in the microcanonical ensemble, S c ( T ) = k B ln Ω( U ) | U = U ( T ) , (9)23here U ( T ) is the internal energy at the temperature T . The explicit form for S c derived fromthe LCT turns out to be quite complicated mathematically, but a more recent work indicates that S c approximates very closely the ordinary entropy S calculated from the freeenergy F = N l f , i.e., S c ≈ S = − ∂F/∂T | φ , greatly simplifying the evaluation of S c fromthe LCT. While the ability to calculate S c within the LCT already enables investigatingthe ideal glass transition temperature T , which is the primary focus in the GD theory, the GET combines the LCT with the AG relation to explore other key aspects of polymerglass formation. This is achieved by realizing that one has to utilize the physically consistentconfigurational entropy when invoking the AG relation. We discuss below what we mean by‘physically consistent’. For simplicity, here we perform illustrative calculations for a meltof chains with the structure of polypropylene (PP) at zero pressure, where a single bendingenergy E b is required for the backbone. As shown in Figure 2, the configurational entropyper occupied lattice site increases progressively with increasing T , showing no tendency tolevel off at high T , as AG suggested. Note that AG did not consider fluids at constantpressure. On the other hand, the T dependence of the configurational entropy per latticesite or the configurational entropy density ( s c = S c /N l ) does plateau at high T , making it asensible candidate for use in the AG relation. This point has been elucidated in detail in ref19, and we briefly discuss some of the ramifications of the T dependence of the configurationalentropy density s c that serves to define the characteristic temperatures within this model ofglass formation.The GET predicts that polymer glass formation is a broad thermodynamic transitionwith four characteristic temperatures. Figure 3 summarizes the determinations of thesecharacteristic temperatures, along with the fragility index. The first one is called the Ar-rhenius or onset temperature, T A , which signals the onset of non-Arrhenius behavior of τ α .In particular, s c exhibits a maximum s ∗ c at a temperature identified with T A (Figure 3a),and the maximum s ∗ c is chosen to be the high T limit of s c appearing in the AG relation. Following GD, the hypothetical ‘ideal glass transition temperature’ T is identified from24
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T [K] s c / k B (a) ∂s c /∂(k B T)|
P, T = T A = 0s c (T )/k B → 0 250 300 350 400 450 T [K] ∂ ( T s c ) / ∂ ( k B T ) (b) ∂ (Ts c )/∂(k B T) | P, T = T c = 0200 300 400 500 600 T [K] -12-8-404 l og ( τ α / [ s ] ) (c) τ α (T g ) = 100 s 0.6 0.7 0.8 0.9 1.0 1.1 T g /T -12-8-404 l og ( τ α / [ s ] ) (d) m = ∂logτ α /∂(T g /T) | P, T = T g Figure 3: Determination of the characteristic temperatures and fragility of glass formationin the GET. (a) Configurational entropy density s c /k B versus T . T A and T are determinedfrom the temperatures at which s c displays a maximum and is extrapolated to zero, respec-tively. (b) ∂ ( T s c ) /∂ ( k B T ) versus T . T c is determined from the temperature at which the T dependence of T s c displays an inflection point. (c) log τ α versus T . T g is determined fromthe temperature at which τ α = 100 s. (d) Angell plot of log τ α . The fragility index m isdetermined from its standard definition. The calculations are performed at zero pressureand utilize the parameter set specified in the caption of Figure 2.the condition of s c ( T ) → (Figure 3a). The crossover temperature T c separates two regimesof T with qualitatively different dependences of τ α on T and is estimated from an inflectionpoint in the T dependence of T s c (Figure 3b). To determine τ α and thus T g , the GETcombines information on s c with the AG relation, τ α = τ exp (cid:18) ∆ G k B T s ∗ c s c (cid:19) , (10)where s ∗ c and s c are inputs directly determined from the LCT, and τ and ∆ G are the high T limit of τ α and the high T activation free energy of TST, respectively.25ecause the LCT does not provide a method for evaluating τ and ∆ G , these fun-damental parameters have been obtained empirically so that assumptions are necessarilyinvoked. The GET takes τ = 10 − s, following typical experimentally reported values forpolymers and arguments based on TST suggest that this time scale should be on theorder of an inverse molecular vibrational frequency. This time scale can be refined by calcu-lating the decay time of the velocity autocorrelation function. In practice, the variations of τ with molecular parameters such as chain stiffness have been found to be rather limited.As anticipated from TST, there should be both enthalpic and entropic contributionsto the activation free energy (eq 3). However, as ∆ S is notoriously difficult to estimatefrom a theoretical viewpoint, the GET follows the original treatment by AG and assumesthat the entropic contribution to the activation free energy is negligible, i.e., ∆ S = 0 .The GET also assumes based on experimental evidence that ∆ G = ∆ H ≈ k B T c . Wehave discussed this relation at length in previous works, so the reader is referred tothese necessarily technical discussions for details. We emphasize here that these simplifyingassumptions are not required by the GET, but they allow for the prediction of polymerproperties without any further information than the determination of molecular parameters(e.g., monomer structure, cohesive interaction strength, and chain stiffness) and thermody-namic parameters (e.g., T and P ) required to specify the thermodynamics of polymer melts,properties that the LCT has been established to describe with good reliability. The enthalpy ∆ H and entropy ∆ S of activation can be independently determined from either simulationor measurement, and in Section 4.5 we explore an extension of the GET where these pa-rameters are determined from experiment to gain some insight into general trends in theseenergetic parameters. Jeong and Douglas have studied the dependence of these energeticparameters on molecular mass in unentangled linear alkane chains by MD simulations, forwhich there are substantial experimental data to compare to and validate the simulation re-sults. We expect this activity of studying the activation free energy parameters of polymersand GF liquids to expand in the future, given the fundamental importance of these energetic26arameters revealed by recent simulation and experimental studies. Once τ α is computed as a function of T , then T g can be determined using the commonempirical definition, τ α ( T g ) = 100 s (Figure 3c). The fragility index proposed by Angell, m = ∂ log τ α ∂ ( T g /T ) (cid:12)(cid:12)(cid:12)(cid:12) P,T = T g , (11)is also readily obtained. Alternatively, we can determine m by fitting τ α in the T rangebetween T c and T g to the VFT equation (eq 1). Since the VFT equation turns out to bea rather good description for the T variation of τ α calculated from the GET in the T rangebetween T c and T g , T VFT is basically equivalent to T so that we do not differentiate them. m is related to D simply via m = DT g T VFT / [( T g − T VFT ) ln 10] .Because the LCT is a powerful theoretical framework for addressing the thermo-dynamics of a vast array of polymer systems, a number of important problems relevant topolymer glass formation can be investigated by the GET. For instance, the GET has beendemonstrated to quantitatively describe the characteristic properties of poly( α -olefins), toclarify the meaning of activation volume of polymer liquids, to elucidate the influence ofcohesive energy and chain stiffness on polymer glass formation, and to better under-stand a variety of phenomena related to polymer glass formation, such as plasticization andantiplasticization of polymer melts by molecular additives, density-temperature scalingof the segmental dynamics of polymer melts, two glass transitions in miscible polymerblends, and the interpretation of the ‘universal’ WLF parameters of polymer liquids and the related observation that the structural relaxation time depends nearly universallyon T − T g in related families of GF liquids. The LCT has also been extended to modelpolymers with specific interactions where cohesive interaction strengths are differentbetween united-atom groups and telechelic polymers where associative groups at thechain ends have strong interactions, enabling predictions of glass formation in these polymersystems when combined with the AG model. It is also worth mentioning that the GET27as been shown to be useful for coarse-grained modeling of polymeric GF materials.
Thus, the GET has the unique capacity to investigate the influence of structural detailsand interactions on polymer glass formation and better understand a variety of importantproblems in polymer glass formation.As we have shown above, a particular advantage of the GET is that the characteristictemperatures ( T A , T c , T g , and T ) of glass formation can be calculated precisely in thetheory because of the analytic definitions of these fundamental characteristic temperaturesof glass formation. This stands in contrast to experimental and computational studies ofglass formation, where large uncertainties often arise in the estimates of these temperatures.It is also interesting to note that the ECNLE theory of Schweizer and coworkers allowsfor predictions of the characteristic temperatures of glass formation. It would be quiteinteresting to compare predictions of this model with the GET to better understand thephysical meaning of the characteristic temperatures of glass formation. From the AG relation (eq 10), it is evident that the T dependence of s c , along with ∆ G ,determines the dynamics of GF liquids within the GET. It is thus crucial to know how s c varies with T and then better comprehend the implications of this information on glassformation. While GD focused primarily on a low T regime where the configurationalentropy extrapolates to zero and AG essentially made some guesses about the T dependenceof the configurational entropy, one advantage of the GET is that it allows us to analyze s c over the entire T regime of glass formation in great detail. In particular, the GET has madedetailed predictions for the variation of s c with T in both the high and low T regimes of glassformation. This has been discussed at length in the original review on the GET. However,our more recent calculations based on the GET have revealed certain features in the T dependence of s c that are not anticipated from the original GET. Here, we summarize thesefeatures, along with those basic aspects identified in the original GET.
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T [K] s c / k B (a) withoutbending constraints withbending constraints 0 50 100 150 200 250 300 350 T [K] S c [ J K − m o l − ] (b) propenebutenepentene3MP MTHFETBtoluenePC Figure 4: T dependence of the configurational entropy in the entire T regime of glass for-mation. (a) s c /k B as a function of T predicted by the GET for polymer melts with andwithout bending constraints at zero pressure, where E b /k B = 600 K and K, respectively.Dashed lines are the descriptions based on eq 12. The symbols cross, triangle, and squareindicate the positions of T A , T c , and T g , respectively. The calculations are performed for amelt of chains with the structure of PP at zero pressure, where V cell = 2 . Å , N c = 8000 ,and ǫ/k B = 200 K. (b) Configurational entropy S c determined from the heat capacity mea-surements as a function of T for various molecular liquids. 3MP, MTHF, ETB, and PC areshort for 3-methylpentane, 2-methyl-tetrahedrofuran, ethylbenzene, and propylene carbon-ate, respectively. Panel (b) was adapted with permission from ref 186.We begin by noting the behavior of s c at T well below T g . While the vanishing of s c at low T shown in Figures 2 and 3 was found to be a typical behavior in the originalGET for polymer melts with bending constraints, as in the GD model, our more recentcalculations have demonstrated that this potentially spurious behavior can be avoidedin certain models. Specifically, s c was found to level off rather than vanish at low T inpolymer models with variable spatial dimension d when d is above a critical value in thevicinity of d = 8 and in fully flexible polymer models in the physical dimension of d =3 . Here, we take the fully flexible polymer model to illustrate our point, as shown inFigure 4a, where we compare the T dependence of s c of a polymer melt with bendingconstraints to that without bending constraints. As can be seen, s c for fully flexible polymermelts no longer vanishes, but instead evidently approaches a positive constant s r at low T , providing a different perspective for the nature of glass formation than the original GDtheory. This finding is consistent with the suggestion of DiMarzio in his later work that29 c should become ‘critically small’ rather than vanishing at the glass transition. The explicitexpression for s r from the GET has been derived in ref 136 as a function of molecularparameters, such as chain length and monomer structure. Interestingly, S c has also beenestimated to be effectively constant at low T in the heat capacity measurements of a numberof molecular liquids by Yamamuro and coworkers. Although this type of observationis often interpreted to be an inherently nonequilibrium phenomenon, this result is highlyattractive as a possibly equilibrium phenomenon from a viewpoint of the entropy theory, sowe reproduce the excess entropy data of ref 186 for an array of materials in Figure 4b.Philosophically, the presence of a positive residual configurational entropy has importantramifications for glass formation within the entropy theory. First, the Kauzmann’s ‘entropycrisis’ is naturally avoided in the GET for fully flexible polymer melts. Moreover, whencombined with the AG relation in eq 10, a finite residual configurational entropy evidentlyindicates that structural relaxation becomes of an Arrhenius form. In this case, τ α does notdiverge at any finite T , and thus, the material is a ‘liquid’ at low T from a mathematicalstandpoint. However, since the activation energy ∆ G ( T ) in the low T regime below T g ishigher than that which the fluid has at high T above T A and since liquids are cooled to suchlow T , the relaxation times at low T are much larger than those in the high T regime, andby all practical measures, the polymer melt in the low T regime can be considered to bea ‘solid’ in a rheological sense. We are thus tempted to term the low T state as being atype of a ‘glass’. O’Connell and McKenna and Novikov and Sokolov have tentativelysuggested that relaxation in GF materials generally approaches an Arrhenius behavior at low T , consistent with the prediction of the GET for fully flexible polymers. While our findingof a finite residual configurational entropy and a corresponding highly viscous ‘glass state’ atlow T in flexible polymer melts provides a clear mechanism of how the ‘entropy crisis’ mightbe avoided in real polymer melts and a counterexample to the claim that the AG modelpredicts a diverging structural relaxation time at a finite low T , we must admit that it doesnot imply that s c always exhibits a plateau at low T . There is also no doubt that it would30e extremely difficult for materials to reach equilibrium over ‘reasonable’ timescales at low T due to the very long relaxation times, making it quite challenging to separate equilibriumfrom non-equilibrium contributions to the ‘effective’ configurational entropy. This situationmakes the utility of the GET of uncertain value below T g .We can analyze the variation of s c with T more quantitatively. In particular, we haveshown in ref 136 that T dependence of s c can be reasonably described by the followingempirical equation describing the generally sigmoidal variation of s c , s c = s r + ( s ∗ c − s r )[1 + exp( a + b/T )] c , (12)where the constants a , b , and c are obtained by numerically fitting ( s c − s r ) / ( s ∗ c − s r ) as afunction of T since this normalized quantity varies from zero to unity as T varies, therebymotivating the fitting functional form, [1 + exp( a + b/T )] c . The description based on eq 12is shown as a dashed line in Figure 4a, where we see that this functional form providesan excellent description of s c ( T ) for polymer melts without bending constraints over theentire T regime of glass formation. The origin of eq 12, and a similar expression utilized inour recent study of polymer coarse-graining, can be traced back to the string model ofglass formation, which establishes a relation between emergent elasticity and cooperativemotion in polymeric GF liquids and a quantitative inverse relation between the extentof cooperative motion and the configurational entropy, as suggested by AG. A relationbetween the activation free energy and material stiffness is further found to qualitativelyaccord with the shoving model of Dyre and coworkers.
Motivated by these observationsand arguments, eq 12 is obtained by first invoking a simple two-state model of the shearmodulus G of amorphous materials, G ( T ) /G (0) = 1 / [1 + exp( a + b/T )] , and then byutilizing the observation that the activation free energy for relaxation scales approximatelyas a positive power of G in a model GF material. While these arguments are admittedlyintuitive, eq 12 turns out to provide a good numerical approximation for s c ( T ) calculated31rom the full GET, whose complexity does not lend itself readily to the simple analyticforms that seem to be highly desired by experimentalists and engineers. Since the vanishingof s c in polymer melts with bending constraints is likely an artifact of the high T expansioninvolved in the lattice model calculations for the free energy when the bending constraints areintroduced, we may also invoke eq 12 to describe the variation of s c with T in polymer meltswith bending constraints, for which eq 12 is also a reasonable approximation, as evidencedin Figure 4a. Note that we choose a somewhat arbitrary value of s r /k B = 0 . for thepolymer melt with bending constraints shown in Figure 4a.Given the good description of eq 12 for s c ( T ) predicted by the GET, we can suggest thefollowing functional form for τ α , τ α = τ exp (cid:26) ∆ G /k B Tδs + (1 − δs )[1 + exp( a + b/T )] c (cid:27) , (13)where δs ≡ s r /s ∗ c is the ratio of the activation energy in the high T regime to that in the low T regime. This approximation is expected to work in the entire T regime of glass formationbelow T A . (T A − T)/T A s ∗ c / s c − (a) E b /k B = 0 KE b /k B = 600 K 0 100 200 300 400 T [K] T s c / k B [ K ] (b) E b /k B = 0 KE b /k B = 600 K Figure 5: T dependence of the configurational entropy in the high and low T regimes of glassformation predicted by the GET. (a) s ∗ c /s c − versus ( T A − T ) /T A and (b) T s c /k B versus T for polymer melts with E b /k B = 0 K and
K, respectively. Dashed lines in panels (a)and (b) correspond to eqs 14 and 15, respectively. The symbols cross, triangle, and squareindicate the positions of T A , T c , and T g , respectively. The calculations are performed for amelt of chains with the structure of PP at zero pressure, where V cell = 2 . Å , N c = 8000 ,and ǫ/k B = 200 K. 32hile the mathematical forms given by eqs 12 and 13 are a bit complicated in comparisonto the VFT relation (eq 1) and may not be utilized readily for describing experimental andsimulation results, the GET predicts that the functional form for describing the T variationof s c can be simplified greatly when we focus on certain restricted regimes of glass formation.For instance, in the high T regime of glass formation between T A and T c , the GET predictsa simple functional form for s c ( T ) as, s ∗ c /s c − C s [( T A − T ) /T A ] , T c < T A −
100 K < T < T A , (14)where the quantity C s measures the steepness of the T dependence of s c . C s appears to pro-vide a measure of fragility in the high T regime of glass formation where the VFT equation is not valid. Figure 5a indicates that eq 14 provides a good description for both polymer meltswith and without bending constraints. Equation 14 has been confirmed by simulations ofa coarse-grained polymer melt with and without antiplasticizer additives, and this func-tional form has been applied in studies of metallic GF materials as a method for estimating T A . The quantity C s should be of great interest from a simulation viewpoint sincesimulations are mostly restricted to a T regime above T c in GF liquids because of the verylong relaxation times at lower T , so we think that eq 14 deserves greater consideration inthe future studies.Kivelson and coworkers have suggested a similar form as eq 14 based on experimen-tal estimates of the reduced activation energy for diverse fluids, and the particular exponentwas reported to be / for their analog of eq 14, a value motivated by a ‘frustrated-limitedcluster model’ of glass formation. Chandler and coworkers have also argued for the same T dependence of the activation energy, as the GET predicts for eq 14, which is termedthe ‘parabolic law’ involving a characteristic temperature comparable to T A , along with anadjustable parameter corresponding to C s , and they have made comparisons with experi-mental data for structural relaxation times for numerous GF liquids, where a good ‘fitting’33as claimed. In comparison of the parabolic law proposed by Chandler and coworkers tomeasurements, a high T Arrhenius term is added to the parabolic law to improve thecomparison of the model to experiments at high T where relaxation is Arrhenius, and thismodification leads to an expression for ∆ G ( T ) exactly in accord with eq 14. We also mentionthat the parabolic functional form for the activation energy can be derived from the NLEand ECNLE theories of Schweizer and coworkers, and the details have been given in aprevious review on the GET. The above comparisons seem to indicate that the GET hasmuch in common with other models of glass formation. Note that C s and T A are derived interms of molecular structure and interactions in the GET. We again emphasize that eq 14is only applicable in the high T regime above T c but below T A . The GET predicts thatthe VFT relation applies in the low T regime of glass formation below T c where the VFTparameters are likewise calculated from the theory as a function of molecular parameters, aswe discuss below based on the T dependence of s c .Figure 5b shows the prediction from the GET for the product T s c /k B as a function of T for both polymer melts with and without bending constraints. Over a limited T range inthe low T regime of glass formation between T c and T , a linear relation between T s c /k B and T approximately holds with the slope K T , T s c /k B = K T ( T /T K − , (15)where T K is the Kauzmann temperature at which s c is extrapolated to zero. Since K T bearsno direct relation to the strength of the T dependence of the relaxation time, we prefer tocall this quantity the low T fragility parameter. We see from Figure 5b that eq 15 holdsfor T down to T for the polymer melt with bending constraints. In the absence of bendingconstraints, however, the T range where eq 15 is applicable seems to be limited, which clearlyarises from the sigmoidal variation of s c with T , as discussed earlier.34 .3 Packing Frustration and Glass Formation Despite the fact that the GET is essentially a thermodynamic theory of glass formationthat relies on predictions of fluid dynamics based on the configurational entropy, calculationsbased on this model have provided molecular insight into the physical nature of polymer glassformation and the most relevant factors governing this phenomenon. Here, we discuss howthe GET is utilized to understand the fragility and T g of GF polymers, two of the mostimportant parameters of polymer materials from a practical viewpoint. A previous work based on the GET investigated the influence of molecular parameters, such as cohesive inter-action strength, chain stiffness, and molecular mass, on fragility quantified by the index m or /D in the VFT relation (eq 1), and this work indicated a remarkable regularity. Specif-ically, the changes in fragility arising from varying molecular parameters can be understoodat fixed measures of cohesive interaction strength from how these parameters influence the packing frustration of the molecules, as measured quantitatively by the ‘free volume’ at T g ,i.e., φ v ( T g ) , where φ v characterized by the extent to which the lattice is not occupied by chainsegments, i.e., φ v = 1 − φ . The concept of free volume has a long history in modeling the dy-namics of cooled liquids going back to Batschinski, Hildebrand,
Doolittle,
Ferry, and many others, and this concept remains popular in the polymer science community.
The most widely used definitions of free volume in the literature and the differences betweendifferent definitions have been recently reviewed by White and Lipson, who have developeda ‘cooperative free volume’ model to explain the dynamics of GF liquids. While the general concept of packing frustration is rather vague, the GET allows us toquantify this property of fluids in terms of how molecular parameters, such as the relativerigidities of the polymer backbone and side chains, chain length, monomer structure, andcohesive interaction strength, influence macroscopic properties such as the thermal expan-sion coefficient and isothermal compressibility that are highly dependent on the efficiencyof molecular packing and the complex intermolecular interactions of molecular fluids. Earlystudies with the GET indicated that these molecular parameters can greatly influence35igure 6: Schematic illustration of chain configurations for general classes of polymers.Polymers are categorized on the basis of the relative rigidities of the backbone and sidechains, leading to flexible-flexible (F-F), flexible-stiff (F-S), stiff-flexible (S-F), and stiff-stiff(S-S) polymers.packing efficiency, and, accordingly, polymers were classified into general classes based onthe results of the model and measurements that also suggested such a classification. Specif-ically, this early work led to a classification of polymers into classes of polymers sharingsimilar packing characteristics. In particular, the flexible-flexible (F-F), flexible-stiff (F-S),stiff-flexible (S-F), and stiff-stiff (S-S) classes of polymers were defined to be chains with aflexible backbone and flexible side groups, chains with a flexible backbone and relatively rigidside branches, chains with a relatively stiff backbone and flexible side groups, and chains withboth a stiff backbone and stiff side groups, respectively. The S-S class of polymers has notbeen encountered frequently in the past, but such polymers have recently become of interestin connection with the applications of molecular filtration in which the large free volumeand T g of such ‘polymers of intrinsic microporsity’ (PIMs) are highly physical attributes, motivating the theoretical study of S-S polymers in more detail. For illustrative purposes,Figure 6 displays representative chain configurations for these general classes of polymers.Intuitively, F-F polymers can pack efficiently in space, and accordingly, these polymersystems exhibit a relatively weak T dependence of s c and are relatively strong glass-formers.In comparison, polymers with either a stiff backbone or stiff side groups exhibit more packingfrustration, and hence, these polymers have a relatively strong variation of s c with T , pro-vided that the cohesive interaction strength is fixed, as in the case of molecules having vander Waals or other fixed types of intermolecular interactions. The GET then predicts that36ariable fragility is a direct outgrowth of the extent of packing frustration of the moleculescomposing the material, a property that can be engineered through exertion of control ofmolecular structure by chemical synthesis. We note that the experimental measurements ofSokolov and coworkers have confirmed the general trends predicted by the GET of fragilitywith varying molecular structure. φ v (T g ) m (a) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) ˜ǫ P (T g ) m (b) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) ˜κ T (T g ) m (c) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) φ v (T g ) T g [ K ] (d) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) ˜ǫ P (T g ) T g [ K ] (e) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) ˜κ T (T g ) T g [ K ] (f) variableE b (PP)(cid:3) (PP)N c (PP)n (F-F)n (F-S)n (S-F) Figure 7: Correlations between packing frustration and the characteristic properties of glassformation predicted by the GET. (a–c) m and (d–f) T g versus thermodynamic metrics forpacking frustration for varying individual molecular parameters. The proposed metrics, φ v , ˜ α P , and ˜ κ T , are the ‘free volume’ defined in terms of lattice occupancy, reduced thermalexpansion coefficient, and reduced isothermal compressibility, respectively, which are allgiven at T g . In these calculations, the variables E b , ǫ , and N c are systematically varied forpolymer melts with the structure of PP, and the variable side group length n is systematicallyvaried for F-F, F-S, and S-F polymers. Adapted with permission from ref 175.The above discussion naturally leads us to expect that m increases with the extent ofpacking frustration, as quantified by the thermodynamic measures discussed above. Thisgeneral trend is verified in Figure 7a, where φ v ( T g ) is utilized as a candidate for a quantitativemeasure of packing frustration. In Figure 7, we have systematically varied E b , ǫ , and N c forpolymer melts with the structure of PP and the side group length n for F-F, F-S, and S-Fpolymers. Since S-S polymers exhibit the same trends in m and T g as F-F polymers, this37lass of polymers is not included here for our analysis. Reference 175 provides more detailsregarding the calculations. We next discuss these measures of packing frustration in detailand their determination in the GET as well as their correlation with the fragility of glassformation as the structural properties of polymers are modified.From an experimental point of view, φ v ( T g ) is difficult to measure, and it is thus naturalto compare m to properties that are related to φ v ( T g ) , but more readily measured, such asthe thermal expansion coefficient α P and isothermal compressibility κ T , α P = 1 V ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12) P , κ T = − V ∂V∂P (cid:12)(cid:12)(cid:12)(cid:12) T . (16)These properties are of both fundamental and practical interest and can be measured experi-mentally for polymers and other materials. For equilibrium fluids, κ T is related to the longwavelength limit of the static structure factor, S (0) , via κ T = ρk B T /S (0) with ρ being thenumber density. The density is given by ρ = φ/V cell in the GET. Instead of focusing directlyon α P and κ T , the GET suggests that the reduced properties be considered as measures ofpacking frustration, ˜ α P = T α P , ˜ κ T = ρk B T κ T . (17)The experimental studies of Simha and Boyer have demonstrated that the reduced ther-mal expansion coefficient ˜ α P can be used to estimate the dependence of T g on both cohesiveinteraction strength and chain stiffness for many polymers, thereby emphasizing the signif-icance of ˜ α P rather than α P itself. The NLE and ECNLE theories of glass formation ofSchweizer and coworkers also emphasize the central role of S (0) in predicting the dy-namics of GF liquids. Sanchez has utilized the dimensionless thermodynamics to discussthe liquid state properties, and this interesting work is recommended to the reader for fur-ther details on reduced thermodynamic properties. We have investigated the dimensionlessthermodynamic properties in our simulation studies of polymer glass formation.
38e show a comparison m versus ˜ α P and ˜ κ T in Figures 7b and 7c, respectively, whereboth quantities are given at T g . We again see the expected trend that m varies with packingfrustration, as quantified by ˜ α P and ˜ κ T . We emphasize that our discussion here is based onthe reduced thermal expansion coefficient and isothermal compressibility.We may also utilize the concept of packing frustration to understand T g , a quantity thatis naturally appreciated from the famous WLF relation and its universal parameters C and C , which are often invoked in the description of the dynamics of polymer materials.The WLF relation can be derived from the VFT expression for τ α if the fragility parameteris taken to be simply proportional to T g , an assumption that accounts for why fragility isnot an explicit parameter in the WLF equation and for why the dynamics depends on thetemperature difference T − T g for materials for which the WLF equation applies. A detaileddiscussion of the WLF relation along with references relevant to C and C has been given inref 92. Figures 7d–f show that T g strongly correlates with the metrics for packing frustration,provided that materials are compared at fixed cohesive interaction strength, which informsus that materials with inherently high packing frustration exhibit an inherently higher T g and fragility. This finding accords with the use of high free volume polymer materials orPIMs for applications in gas separation processes, heterogeneous catalysis, and hydrogenstorage as well as the socially important problem of large-scale water desalination, to namea few emerging applications of these new materials. T g in some commercial PIMs is sohigh that its estimates have previously been difficult, but recent techniques have overcomethis difficulty, at least in some PIMs. Finally, it is worth noting in Figures 7d–f that aninversion of the trend of T g with the measures of packing frustration occurs when varying ǫ ,a result first identified in ref 174. This more complicated behavior should be borne in mindwhen discussing the glass formation of polar and charged polymer fluids, which generallyexhibit high cohesive interaction strength. 39 .4 Glass Formation under Pressure It has been well established that glass formation can often be induced by increasing P atconstant T . The large alterations in the dynamics of GF liquids upon the applicationof pressure are evidently of significance in numerous manufacturing applications. In partic-ular, there has been intense interest in separating out aspects of glass formation relating tothe effects of attractive intermolecular interactions, temperature, and density,
Theseefforts have naturally led to systematic studies of the dynamics of GF liquids over a widerange of P to complement the traditional studies where glass formation occurs uponcooling. The GET is naturally suitable for addressing polymer glass formation under appliedpressures since calculations can be performed under different constant P conditions. In thissection, we utilize the GET to discuss several interesting aspects of polymer glass formationwhen P is a variable. φ /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) P [MPa]01020 304050 /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) Figure 8: Density-temperature scaling of τ α predicted by the GET. The main plot shows log τ α versus φ γ /T with γ = 7 . for a range of fixed P . The inset displays log τ α versus /T for the same P . The calculations utilize the parameter set specified in the caption ofFigure 2.We begin by noting the extremely interesting density-temperature scaling phenomenon, which indicates that dynamic properties such as structural relaxation time and diffusion co-efficient in suitably reduced units become universal functions of T times density to a power γ characteristic of the material. Dyre and coworkers have suggested that the existence ofthis type of scaling should play the role of a ‘filter’ for acceptable models of GF liquids. In40 previous study based on the GET, we indeed found density-temperature scaling to holdfor τ α over the entire T regime of glass formation in polymer melts. The GET thus passesthis test. For illustrative purposes, we show the density-temperature scaling of τ α for a rangeof fixed P predicted by the GET in Figure 8, the inset of which displays log τ α as a functionof /T . Experimental and computational studies indicate that the scaling exponent γ obtained from estimates of viscosity and relaxation time lies in the broad range of . to . . Our previous study indicates that the scaling exponent γ varies in a large rangewhen varying molecular parameters, such as chain length, chain stiffness, cohesive interac-tion strength, and side group length, factors that evidently influence the fragility of glassformation. Our calculations based on the GET also confirmed the approximate inverserelation between γ and the constant volume fragility parameter m V found experimentallyby Casalini and Roland. Recently, we have made additional progress on the mysterious density-temperature scal-ing phenomenon based on the GET and simulations. We have also noticed that some otherpopular models of glass formation do not pass the test associated with density-temperaturescaling. We will report on these findings in a separate paper, where we discuss the origin ofdensity-temperature scaling and its many implications. It is notable that the GET is basedon a lattice model where the polymer intersegment interactions are described by the roughequivalent of a square potential in off-lattice liquids. The scaling exponent γ depends onmolecular parameters discussed above, but it has nothing to do with the shape of the pairpotential in our calculations. Density-temperature scaling in polymer and other molecularfluids then seems to arise from the variation of the anharmonic intermolecular interactionsthat arise from packing frustration, and the GET provides a powerful computational frame-work for studying this phenomenon that fundamentally links the thermodynamics and dy-namics of fluids and the relative balance of repulsive and attractive interparticle interactionsgoverning the dynamics of real molecular fluids.The GET also allows us to quantitatively analyze the variation of τ α with P at constant T .41
20 40 60 80 100
P [MPa] l n [ τ α ( P ) / τ α ( ) ] (a) T [K]350310290280 275270267264 0 2 4 6 8 10
P [MPa] l n [ τ α ( P ) / τ α ( ) ] (b) T [K]350310290280 2752702672640 50 100 150 200 D P P/(P − P) l n [ τ α ( P ) / τ α ( ) ] (c) T [K]350310290280 275270267264 0 2 4 6 8
P∆V /k B T l n [ τ α ( P ) / τ α ( ) ] (d) T [K]350310290280 275270267264
Figure 9: P dependence of τ α at constant T predicted by the GET. Panels (a) and (b) show ln[ τ α ( P ) /τ α (0)] as a function of P for a range of T in a wide P range and in the low P limit, respectively. Lines in panels (a) and (b) are the descriptions based on eqs 18 and 19,respectively. Panels (c) and (d) show the universal reduction of the P dependence of τ α . Linesindicate ln[ τ α ( P ) /τ α (0)] = D P P/ ( P − P ) in panel (c) and ln[ τ α ( P ) /τ α (0)] = P ∆ V /k B T in panel (d), respectively. The calculations utilize the parameter set specified in the captionof Figure 2.Experimental and computational studies have indicated that the P dependenceof τ α generally displays a pressure analog of the VFT equation (PVFT) in non-associatingGF liquids, in which P and the critical pressure P , respectively, replace T and the criticaltemperature T VFT in the conventional VFT equation given in eq 1, τ α ( P ) = τ α (0) exp (cid:18) D P PP − P (cid:19) , (18)where τ α (0) is the structural relaxation time at zero pressure. The GET predicts the PVFTrelation and further provides a rationale based on the variation of the configurational entropyof the fluid with P . We show an illustrative result for the P dependence of τ α calculated42rom the GET for a range of T in a wide P range in Figure 9a, along with a universalreduction of the data based on the PVFT relation in Figure 9c.While the PVFT relation is evidently required to describe the P variation of τ α in a large P range, there is a simple linear variation of ln τ α with P in the low P limit, τ α ( P ) = τ α (0) exp (cid:18) P ∆ V k B T (cid:19) , (19)where ∆ V defines the activation volume. Figure 9b displays ln[ τ α ( P ) /τ α (0)] as a functionof P in a much narrower regime of P than that in Figure 9a. We see that the GET predictsthe linear variation of ln τ α with P when P is restricted to small values. Again, we show auniversal reduction of the data from the GET based on eq 19 in Figure 9d. This analysisevidently indicates an onset pressure P A above which the variation of ln τ α with P deviatesfrom the linear relation given by eq 19, the analog of the Arrhenius regime of GF liquids when T is instead varied at fixed P . We have also found that the P dependence of τ α calculatedfrom the GET seems to follow a power law in a restricted P range, τ α ( P ) ∼ ( P c − P ) − γ c ,which P c may be defined by a crossover pressure and γ c is an effective T -dependent ‘crossoverexponent’. We do not think that this exponent estimated from the GET has anything to dowith the mode-coupling theory, but it is sometimes attributed to this theory because itlikewise predicts a power-law relation of this kind, albeit in a T range closer to T A than T c . Moreover, a ‘glass transition pressure’ P g can be identified for a given T by the pressure atwhich τ α = 100 s, and as discussed above, we may also define a pressure analog P of theVFT temperature by a limiting pressure at which τ α extrapolates to infinity. Therefore, theGET allows us to estimate the different characteristic pressures ( P A , P c , P g , and P ) of glassformation when P is used as a control variable for glass formation at fixed T . There is thenan interesting ‘duality’ between glass formation with variable T at fixed P and with variable P at fixed T . This is another topic that we will investigate in the future.Parenthetically, the activation volume ∆ V is often encountered in an industrial and43aterials science setting when materials are subjected to large changes in P or appliedsteady stresses of various kinds, but the physical meaning of ∆ V is often rather unclear.In our recent work, we have systematically investigated ∆ V based on the GET andsimulations. Our study indicates that ∆ V is related to the differential change of theactivation free energy as a function of T , and thus bears a close relationship to the fragilityof glass formation. The GET is just one of a number of promising models of glass formation which has theparticular advantage of predicting the segmental structural relaxation time of polymeric GFliquids in terms of essential molecular parameters over the entire T regime of glass forma-tion. The model is also advantageous because it describes the equation of state and otherbasic thermodynamic properties of polymer melts within a mature and validated theoreticalmodel in a consistent framework. Moreover, the GET naturally allows for the validation ofthermodynamic and dynamic properties based on the string model of glass formation inconjunction with MD simulations of coarse-grained polymer melts that cover a wide rangeof molecular parameters of interest in understanding the thermodynamic and dynamic prop-erties of real polymer materials. As discussed earlier, there are assumptions in the GETrelating to the precise relation between thermodynamic and dynamic properties whose va-lidity needs to be further assessed and modified if necessary, but this type of limitation existsfor all current models of GF liquids. Here, we discuss how the GET might be improved.We first note that, while ∆ H can be estimated from an Arrhenius fit to the dynamicsof fluids at high T if such information is available, it remains a difficult matter to determineand understand ∆ S . The theoretical estimation of ∆ S is generally appreciated to be theprimary source of uncertainty in TST, and inevitably, ∆ S must be estimated from eitherexperiment or simulation. Theoretical attempts to calculate ∆ S has a long historyin condensed materials, which we briefly mention here. Vineyard proposed a rigorous44ST for idealized crystalline materials with ideal harmonic interactions, which has beenvalidated for simple ordered molecular clusters. Unfortunately, anharmonic effects, asfound characteristically in cooled liquids, heated crystals, and the interfacial dynamics ofcrystals for
T > T m / with T m being the melting temperature, cause the Vineyard theoryto break down. A general discussion of TST for crystalline materials has been providedby Rice, who emphasized the relation between the Arrhenius prefactor and the stableand saddle point frequencies. This work is particularly interesting from a philosophicalviewpoint in the sense that it explains why one must consider a free energy of activation, ∆ G , in condensed materials rather than just an activation energy, ∆ H . Collective motionassociated with many-body effects is thus essential for barrier crossing events associated withrelaxation and diffusion in condensed materials.While the first principles analytic estimation of ∆ S for fluids from TST has remainedchallenging, numerical implementations of TST have recently enabled precise estimationsof ∆ S for model condensed materials. A large body of semi-empirical works havecombined careful measurements with physical arguments to explain ∆ S in terms of a pre-sumed connection with local elastic distortions required for a particle or defect to movein the condensed state. This line of argument can be traced back to the pioneering workof Zener in the context of metallic crystalline materials and was taken further by Law-son by directly relating the activation volume ∆ V to ∆ S . Keyes argued for acorresponding relation between ∆ V and ∆ H . These relations have provided a rationaliza-tion of the EEC effect between ∆ H and ∆ S that has been validated empirically in manypolymeric and other condensed materials. Exactly solvable toy models of dynamics onhierarchical potential energy landscapes, as naturally found in disordered condensedmaterials, also give rise to the EEC. The large entropy of activation in condensed materialsis found in these models to arise from the rapid proliferation of transition state paths overhigh barriers in such hierarchical spaces, an intuitive physical picture supporting the formaltheoretical results of Yelon and coworkers.
A more physically tangible real-space ap-45roach to understanding relaxation in condensed fluids starts from a consideration of thenature of collective motion in the fluids required to enable atomic displacement. Based onthis point of view, Barrer has offered an interesting heuristic ‘zone theory’ of the EECphenomenon in the high T Arrhenius regime of liquids, thus having features in common withthe heuristic arguments of AG in cooled liquids, as discussed in Section 3.2. Therefore, aphysical approach of this kind focusing on the precise nature of collective motion in densefluids might offer a deep understanding of the link between ∆ H and ∆ S , as found in therelaxation, diffusion, and chemical reaction processes of so many condensed materials. -8 -7 -6 -5 -4 -3 -2 -1 0 ∆S /k B T g [ K ] (a) ∆H /k B T c
345 678 -8 -7 -6 -5 -4 -3 -2 -1 0 ∆S /k B m (b) ∆H /k B T c
345 678
Figure 10: Influence of the high T activation free energy parameters on polymer glass forma-tion predicted by the GET. (a) T g and (b) m as a function of the activation entropy ∆ S /k B for a range of ∆ H . The calculations utilize the parameter set specified in the caption ofFigure 2.Table 1: Basic properties of the polymers considered in Figure 11. The listed propertiesinclude the molar mass M , glass transition temperature T g , and fragility index m , alongwith the model parameters, ǫ , E b , ∆ H , and ∆ S , used for the fits to the experimentalresults based on the GET.polymer M [g / mol] T g [K] m ǫ/k B [K] E b /k B [K] ∆ H /k B T c ∆ S /k B PI 15700 207 84 . . . − . PPG 18000 202 105 . . . − . PB 87000 174 90 . . . − . PPS 44000 229 116 . . . − . Although the GET does not provide a method for estimating ∆ H and ∆ S , we mayexamine the influence of these basic parameters on polymer glass formation within this46 .5 3.0 3.5 4.0 4.5 5.0 /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) (a) poly(isoprene) ExperimentGET with ∆S GET without ∆S /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) (b) poly(propylene glycol) ExperimentGET with ∆S GET without ∆S /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) (c) poly(butadiene) ExperimentGET with ∆S GET without ∆S /T [K −1 ] -12-8-404 l og ( τ α / [ s ] ) (d) poly(propylene sulfide) ExperimentGET with ∆S GET without ∆S Figure 11: Description of experimental results for the T dependence of τ α for several polymersbased on the GET. (a) Poly(isoprene). (b) Poly(propylene glycol). (c) Poly(butadiene). (d)Poly(propylene sulfide). Experimental data were provided to us by the authors of ref 88.Solid and dashed lines represent our best fits based on the GET with and without ∆ S ,respectively. The calculations based on the GET are performed for a melt of chains withthe structure of PP at zero pressure, where V cell = 2 . Å and N c = 8000 . The molecularparameters ǫ and E b and energetic parameters ∆ H and ∆ S for the polymers are given inTable 1, along with the basic properties of polymers.TST-based model. As noted above, we have utilized simplifying approximations in the GETto avoid estimating these energetic parameters, i.e., ∆ H = 6 k B T c and ∆ S /k B = 0 , withthe second simplifying assumption being ‘inherited’ from AG. Here, we tentatively treatboth ∆ H and ∆ S as adjustable parameters to better understand their potential relevanceto polymer glass formation. Figure 10 shows both T g and m as a function of ∆ S for arange of ∆ H . Our calculations also include the predictions based on the assumptions, ∆ H /k B T c = 6 and ∆ S /k B = 0 , of the original GET. We see that T g increases linearlywith | ∆ S | , while m drops considerably as | ∆ S | elevates for the range of ∆ S considered47ere. This result clearly demonstrates that precise estimations of both ∆ H and ∆ S arecrucial for transforming the GET into a quantitative predictive molecular theory. Next,we present our preliminary results on comparisons between the GET and experiment, inparticular when the activation entropy is present.Figure 11 shows our GET description of experimental results for the T dependenceof τ α for several polymers, including poly(isoprene) (PI), poly(propylene glycol) (PPG),poly(butadiene) (PB), and poly(propylene sulfide) (PPS). These representative polymersare selected to demonstrate the significance of including ∆ S in the GET. All the exper-imental data were provided to us by the authors of ref 88, and the molar mass for eachpolymer corresponds to the highest value reported there. For simplicity, our calculationsbased on the GET are performed for a melt of chains with the structure of PP at zero pres-sure, where V cell = 2 . Å and N c = 8000 , for all the polymers considered. In an effort todescribe the experimental data, we first adopt the original GET, where ∆ H = 6 k B T c and ∆ S = 0 , to fit the experimental values of T g and m for each polymer by adjusting ǫ and E b simultaneously. The fitted results are shown as dashed lines in Figure 11, and the best valuesfor ǫ and E b are summarized in Table 1. We see that the GET, even without ∆ S , leads toa reasonable quantitative description of the experimental results in the T regime near T g ,but deviations are noticeable at high T . Based on the obtained molecular parameters ǫ and E b , we then allowed both ∆ H and ∆ S to be adjusted. Utilizing the values of ∆ H and ∆ S in Table 1, Figure 11 exhibits our calculations from the GET with ∆ S as solid lines,indicating that the inclusion of ∆ S enables a quantitative and accurate description of theexperimental data for the T dependence of log τ α over the entire T regime of glass formationaccessible to experiment.In the future work, we need to compare the GET predictions to measurements for morepolymers to better understand both ∆ H and ∆ S , as well as the interrelation of these twoenergetic parameters. Jeong and Douglas have explored the determination of ∆ H and ∆ S for alkane melts based on MD simulations, and we have also investigated these param-48ters in coarse-grained polymer melts with variable pressure, cohesive interactionstrength, and chain stiffness. The computational machinery certainly exists to makeprogress in understanding the activation parameters based on simulations, and we hope thatsuch analyses will lead to corresponding advances in analytic theories of these fundamentalenergetic parameters. We think that a comprehensive understanding of these energetic pa-rameters will provide the foundation on which a real theory of GF liquids will ultimatelybe based. For the present, we must admit that TST of liquid dynamics has some seriousshortcomings that are an impediment to the GET and other models of glass formation basedon the foundation of thermally activated transport.
As in the case of the relation between the fluid entropy and the dynamics in GF liquids, theidea that the growth of the structural relaxation time should be associated with increasedcollective motion that accompanies the reduction of configurational entropy long predates theAG model. Most of the arguments along this line in the older literature of GF liquids arerather qualitative, but we mention a historically interesting suggestion by Mott at a Solvayconference on condensed materials in 1951, in which the activation energy of materialsin a relatively disordered state was suggested to grow extensively with the mass of somedynamic cluster in the material. Later, this suggestion was subjected to experimental tests byNachtreib and Handler to interpret their self-diffusion measurements of white phosphorousover a large T range, and the measurements seemed to confirm the heuristic picture of the T -dependent activation energy of Mott for diffusion in this material. Although these historicalcontributions are not often discussed now, we also acknowledge the historical contributionsby Orowan to the understanding of the nature of molecular motions underlying flowprocesses in viscous fluids and solid crystalline materials. See the paper of Goldstein for49 recounting of this now largely inaccessible work. Moreover, the influential ‘zone theory’of Barrer likewise emphasized the necessity of collective atomic motion involvingmany particles in connection with molecular diffusion. These are the shoulders upon whichlater models of the dynamics of GF liquids rest.With this kind of historical background in view, it seems natural that AG would pro-pose that the activation energy in cooled liquids should grow in proportion to the numberof particles in some sort of hypothetical CRR, thereby creating an enduring conceptualconstruct in modeling the dynamics of liquids. As discussed in Section 3.2, AG furtherproposed that the number of particles in the vaguely defined CRR should scale inverselywith the configurational entropy S c . The greatest strength of this assumption is apparentlythat the VFT relation can be rationalized when this assumption is combined with roughestimates of S c based on an approximation suggested earlier by Bestul and Chang. TheAG model is not so much a theory, but rather a set of hypotheses introduced to conform toprevailing theoretical ideas and correlations between properties seen experimentally. This isprobably a quite good description of more recent theories of glass formation, so we do notthink that the AG model should be judged harshly given its rather empirical origin. This‘semi-empirical’ model has been immensely successful for creating a qualitative picture ofthe origin of the strong variation of the relaxation and diffusion properties with T and poly-mer concentration, and it has largely determined the language used to describe GF liquids,regardless of whether or not one believes in the assumptions of the AG model. The RFOTtheory is based upon similar ideas to rationalize the slowing down of the dynamics ofGF liquids in terms of an ‘entropic droplet model’.It is worth noting that DiMarzio never accepted the AG model, explaining why he nevercombined the AG model with the lattice model of GD developed for calculating S c . This situation is apparent in his later work, where DiMarzio introduced an alternativeto the AG model by asserting a rather different relation between the thermodynamics andthe structural relaxation time in cooled liquids. We also mention a formal critique of the50ssumptions underlying the AG model by Dyre and coworkers.
As a counterweight tothese criticisms, we mention some theoretical discussions that formalize the idea of a growingactivation energy as T is varied in the interfacial dynamics of crystalline materials andin GF liquids, showing that there are significant merits to the assumptions of the AGmodel. Dudowicz et al. have shown that the fluid configurational entropy scales inversely withthe average length of dynamic polymers calculated from a model of living polymerization,a finding that has taken great significance that dynamic polymeric structures involving string-like structures have been observed in many simulations of GF liquids to date, in which the collective motion has been examined in great detail and the thermo-dynamic, geometric, and dynamic properties of the dynamic stringlike structures closelyconform to equilibrium polymerization. Historically, the findings of Dudowicz et al. servedas a great impetus for the development of the string model of glass formation, in which thedynamic clusters in cooled liquids were identified as being equilibrium polymers and realiza-tions of the hypothetical CRR of AG. Douglas et al. have comparatively discussed thevariation of the configurational entropy of fluids exhibiting equilibrium polymerization andthe configurational entropy calculated from the GET, and strong similarities in the T varia-tion of the configurational entropies from the two models are indicated, such as analogs of theonset temperature T A , the crossover temperature T c , and the glass transition temperature T g , below which the configurational entropy saturates to a finite value in the polymerizationmodel. Douglas et al. have also shown that the ‘cooperativity’ of the equilibrium poly-merization transition, a description of the extent to which the transition resembles a phasetransition, plays the role of fragility in the correspondence between the polymerization modeland the GET. 51 .2 String Model Essentials As just discussed above and elaborated on in ref 19, some of the assumptions made by AG now have some support from simulations and the analytic theory of fluids exhibiting self-assembly, providing the starting point for developing the string model of glass formation. Asubsequent work has found that the stringlike collective motion conforms quantitatively tothe equilibrium polymerization model over the entire T range below T A for which simulationscan currently be performed for GF liquids. As we shall see below, the only unspecifiedparameters in these comparisons between the simulation data and the string model are theactivation enthalpy ∆ H and activation entropy ∆ S of TST, which are determined in thehigh T Arrhenius regime, i.e., ∆ H from the slope of the variation of ln τ α with /T and ∆ S from the prefactor in the Arrhenius relation (eq 2). Our lack of a theoretical understandingof the dynamics of fluids in the Arrhenius regime is the greatest hurdle in predicting thedynamics of GF liquids from the string model.As noted briefly above, the development of the string model was motivated by extensivesimulations of both polymeric and other GF liquids indicating thatthe most mobile particles in GF liquids form stringlike structures defined in terms of theircooperative exchange motion over the lifetime in which the strings exist. The technicaldetails for quantifying the stringlike cooperative motion have been repeatedly describedin previous works and we refer the reader to ref 241 for a discussion of this procedure.Such cooperative motion is broadly consistent with the philosophical assumptions of the AGmodel. Figure 12a shows a snapshot of stringlike clusters of polymer segments undergoingcooperative exchange motion over the lifetime over which these clusters persist in a simulatedpolymer melt.
We note that there is little correlation between these stringlike motions andchain connectivity, so these stringlike motions are not ‘reptative’, i.e., the segments in thesemotions are not correlated with the chain backbone. Such collective motion occurs in a rathersimilar fashion even in metallic GF liquids where there are no chemical bonds. Starr et al. have investigated what measure of dynamic heterogeneity may appropriately quantify the52 a) S c (b) z/T l n τ α (c) c l n τ α (d) Figure 12: Test of the average string length L as a quantitative realization of the hypotheticalcooperatively rearranging regions (CRR) of AG based on simulation results of a coarse-grained polymer melt. (a) Snapshot of stringlike clusters of polymer segments undergoingcooperative exchange motion over the lifetime over which these clusters persist. Each stringis shown by large spheres in a different color. Only strings of length larger than are shownto aid the visualization, and the other segments are displayed as translucent think cylinders.(b) Configurational entropy S c versus /L . (c) ln τ α versus z/T , where the size of the CRR is z = L/L A with L A being the value of L at T A . (d) ln τ α versus /T S c . Lines in panels (b–d)are a guide to the eye. Here and in Figures 13 and 14, all quantities from the coarse-grainedsimulations are expressed in standard reduced Lennard-Jones units. See ref 217 for detailsabout reduced units and their mapping to laboratory units. Panels (b–d) were adapted withpermission from ref 161.size scales of the CRR envisioned by AG based on simulations of a coarse-grained polymermelt, where S c has been calculated. It has been shown that S c is inversely proportional tothe average string length L to a good approximation, as illustrated in Figure 12b. Moreover,taking the size of the CRR as z = L/L A with L A being the value of L at T A , the linearrelation between ln τ α and z/T is confirmed by the analysis in Figure 12b. Figure 12dalso verifies the linear relation between ln τ α and /T S c . Further, L/L A has been shown to53rack the normalized activation free energy, ∆ G/ ∆ G . Therefore, these results convincinglydemonstrate that the dynamic strings provide a quantitative realization of the hypotheticalCRR of AG. In the string model of glass formation, applicable in the T range below T A where relax-ation is non-Arrhenius, the activation free energy ∆ G for structural relaxation is proportionalto the average string length L normalized by its value at the onset temperature T A . As aformal extension of TST, the string model of glass formation defines the T -dependentactivation free energy as ∆ G ( T ) = ∆ G ( L/L A ) , leading to the following expression for τ α , τ α = τ exp (cid:18) ∆ G k B T LL A (cid:19) . (20)The parameter τ can be eliminated from a knowledge of τ α at T A , and ∆ H may bedetermined from the Arrhenius equation (eq 2) in the high T regime where standard TSTis assumed to be applicable as a descriptive framework for liquid dynamics, resulting in thefollowing equation with ∆ S being the only fitting parameter, τ α = τ α ( T A ) exp (cid:18) ∆ H − T ∆ S k B T LL A − ∆ H − T A ∆ S k B T A (cid:19) . (21)Notably, simulation studies indicate that collective motion does not completely vanish inthe Arrhenius regime, i.e., L A is not equal to unity. Moreover, the string model assertsthat ∆ S cannot be neglected, but the spirit of the AG model is certainly preserved in thismodel, despite this and other technical differences. Therefore, the essence of the stringmodel of glass formation is that the activation free energy in the high T Arrhenius regimeis ‘renormalized’ by the T -dependent factor L/L A quantifying the change in the extent ofcollective motion in the non-Arrhenius regime below T A .Equation 20 of the string model has been quantitatively confirmed in simulations ofa number of polymeric and other GF liquids, including knotted ring and star polymermelts, thin films on solid substrates, polymer nanocomposites having a range54 ∆G (L/L A )/k B T l n ( τ α / τ ) (a) P: pressureP0.01.02.0 3.04.05.0 5 10 15 20 25 ∆G (L/L A )/k B T l n ( τ α / τ ) (b) a θ : chain stiffness parametera θ ∆G (L/L A )/k B T l n ( τ α / τ ) (c) α: cohesive energy parameterα11.522.53 3.544.55 Figure 13: String model description of the relationship between τ α and L . (a–c) Results fora simulated coarse-grained polymer melt with variable pressure, chain stiffness, and cohesiveinteraction strength, respectively. Lines indicate ln( τ α /τ ) = ∆ G ( L/L A ) /k B T . Adaptedwith permission from refs 217, 121, and 122.of concentrations and polymer-surface interaction strengths, polymer melts with vari-able pressure, chain stiffness, and cohesive interaction strength, polymernanofibers, metallic liquids, and superionic UO . For the purposes of illustration, weshow in Figure 13 that the string model describes well the simulation results in a coarse-grained polymer melt, where the pressure, chain stiffness, or cohesive interaction strengthis systematically varied. The string model is further justified by a theoretical analysis byFreed, where TST is extended to account for stringlike cooperative barrier crossing eventsin GF liquids. However, much work needs to be done to extend the string model to a widerange of other materials. We again emphasize that even standard TST in the Arrheniusregime remains in a relatively rudimentary theoretical state.In passing, we mention that relaxation in GF liquids occurs as a multi-stage hierarchicalprocess.
While we evidently focus on the long-time structural relaxation process involv-ing both large-scale diffusive molecular motion and momentum diffusion, there is a ‘fast’relaxation process dominated by the inertial motion of the molecules whose amplitude growsupon heating. It is worth pointing out that the fast dynamics can be considered in a unifiedway with the long-time structural relaxation within the string model, as indicated in a recentwork. .3 Temperature Dependence of String Length The string model not only identifies the CRR with the stringlike cooperative motion, butit also provides an explanation for the origin and geometrical nature of these strings based onthe framework of treating them as ‘initiated equilibrium polymers’. This has been discussedin detail in ref 20. Here, we briefly review the most interesting predictions from the livingpolymerization model of strings.In the model of initiated equilibrium polymerization, the strings are dynamic or ‘equi-librium’ polymers that form and disintegrate in equilibrium. These dynamic polymers havea T -dependent average length, or polymerization index, L ≡ h L i . The fraction of linkedmobile particles Φ serves as the order parameter for the self-assembly process of strings,which is related to L via the relation, L = 11 − Φ + δr/ , (22)where δr is the ratio of the initiator to the monomer volume fraction φ . Φ is limited to therange between δr and . Taking the onset temperature T A as the reference, along with theapproximation of δr ≈ Φ A with Φ A being the value of Φ at T A , eq 22 can be written as L = L A (1 − Φ A / − Φ + Φ A / . (23)In the living polymerization model, Φ is the extent of polymerization defined by thefraction of monomers forming polymeric structures, and, in simulations, this quantity can beinterpreted to be the fraction of the highly mobile particles participating in the strings. Thevariation of Φ with T is well described by the prediction from the polymerization model, Φ = 1 − φ ( T ) /φ , (24)where the explicit expression for φ ( T ) is given in refs 20 and 50. Φ is directly related to S c in56he theory of equilibrium polymerization, a relation that has been validated in simulationsof a coarse-grained GF polymer melt, where Φ and S c are independently calculated. Φ is notably much easier to calculate than S c in simulations. As we have discussed earlierin Figure 12, a simulation study based on the same polymer melt has also confirmed aninverse scaling between the average string length L and S c to a very good approximation, a basic tenet of the AG theory and the thermodynamic relationship in the equilibriumpolymerization model. While the full expression for L predicted by the string model is complicated mathe-matically, this expression can be simplified by using the high T expansion of Φ , L = L A (cid:18) − Φ A (cid:19) (cid:26) A (cid:20) φ exp (cid:18) − ∆ G p k B T (cid:19)(cid:21)(cid:27) , (25)where the free energy ∆ G p = ∆ H p − T ∆ S p describes the thermodynamics of string poly-merization rather than the activation free energy of the fluid. Equation 25 is expected tobe valid in the T range above T c but below T A , which we have confirmed in simulationsof coarse-grained polymer melts with variable cohesive energy and pressure. It is animportant feature of equilibrium polymerization that the excess degree of polymerizationin the high T regime, L A − , is directly related to the plateau of L in the low T regime,where both the degree of polymerization and the fluid configurational entropy saturate tofinite values associated with the fully polymerized state. This feature of the polymerizationmodel has highly non-trivial implications for the string model of glass formation, since itlinks collective motion in the high T Arrhenius regime to the scale of collective motion in theequilibrium glass state, where relaxation again becomes Arrhenius in the string model of thedynamics of GF liquids under equilibrium conditions. Of course, equilibrium conditions atlow T are normally difficult to achieve in practice, so this prediction is somewhat of academicinterest, just as the question of whether or not S c actually vanishes at a finite T .Figure 14a examines the validity of the polymerization model in describing the variation57 .0 0.2 0.4 0.6 0.8 1.0 T L (a) T g SimulationHigh T expansionFull string model T S c (b) T g SimulationHigh T expansionFull string model T g /T l og τ α (c) T g SimulationHigh T expansionFull string model
Figure 14: T dependence of the thermodynamic and dynamic properties of glass formationpredicted by the string model, along with comparisons with simulation results of a coarse-grained polymer melt. (a) L and (b) S c as a function of T . (c) Angell plot of log τ α . Solidand dashed lines correspond to the full form of the string model and its approximant basedon the high T expansion, respectively. The filled regions indicate the T regime below T g estimated from the full string model, which is determined by the condition, τ α = 100 s,corresponding to in standard reduced Lennard-Jones units. Adapted with permissionfrom ref 20.of L with T determined from simulations of a coarse-grained polymer melt. As can be seen,the string model, either in its full form (given in ref 20) or its simplified form based on thehigh T expansion (eq 25), provides an excellent description of the available simulation data,which are restricted to a T range between T A and T c . However, large differences arise at low T between the full and simplified string models, which has important implications for glassformation. While L from the high T expansion increases rapidly upon cooling at low T , thefull form for L instead exhibits a plateau at low T . Since L inversely scales with S c in thestring model, we see from Figure 14b that the string model makes similar predictions for S c .Note that the plateau of S c at low T in the full string model is reminiscent of the predictionfrom the GET for fully flexible polymer melts, as discussed in Section 4.2. We note thata recent study on a metallic GF liquid, where the estimation of S c is less difficult than forpolymers because the large vibrational contributions associated with molecular bonding tothe fluid entropy are absent, has revealed the presence of an extended low T plateau in S c . ? Combing the functional form for L provided by the living polymerization model witheq 20, the string model goes beyond the relatively high T regime of glass formation accessibleto present simulations and make predictions for τ α over the entire T regime of glass formation.58his is illustrated in Figure 14c, where we show the Angell plot of log τ α determined fromsimulations, along with the descriptions based on the string model. It is evident that thefull string model indicate a return to Arrhenius relaxation in the glass state, albeit witha higher effective activation free energy. We emphasize that the string model implies nodivergence in τ α at any finite T . This is in accord with the GET predictions for flexiblepolymer melts and the observations on a number of liquids, such as water, silica, andmetallic GF liquids, which show a ‘fragile-to-strong’ transition upon sufficient cooling,corresponding to a return to Arrhenius relaxation. The interesting thing about GF liquidsexhibiting a fragile-to-strong transition is that the Arrhenius relaxation seems to occur ina T regime where the liquid is apparently fully in equilibrium. Note that the position ofthe transition to a low T Arrhenius behavior as well as the fragility of glass formation andthe low T plateau in the activation energy all depend on Φ A in the string model, a quantitydetermined in the Arrhenius regime. While the string model aims to describe the dynamics of GF liquids in terms of parametersand quantities with well-defined meanings, there are popular models of glass formation thatare highly phenomenological in nature. Here, we utilize the string model to understandsome of these phenomenological models, which has been discussed in refs 20 and 122. Inparticular, we show that the string model enables the rationalization of the empirical fittingfunction for τ α introduced by Rössler and coworkers and the Mauro-Yue-Ellison-Gupta-Allan (MYEGA) model. As discussed in Section 5.3, the order parameter Φ for the assembly of mobile particlesinto strings can be derived from the equilibrium polymerization theory. However, the exactmathematical description of Φ is complicated. Douglas and coworkers found that a59imple two-state model can be used to describe Φ in the cooperative polymerization model, Φ = 11 + exp[(∆ H p − T ∆ S p ) /k B T ] . (26)This two-state expression arises in many physical contexts and has often been further ap-proximated by expanding around the free energy of association about the polymerizationtransition point T p corresponding to the condition of Φ = 1 / , which gives rise to the fol-lowing approximation, Φ ≈
11 + exp[ − ∆ H p ( T − T p ) /k B T p ] ≡
11 + exp[( T − T p ) / D ] , (27)where D ≡ k B T p / | ∆ H p | reflects the width of the clustering transition. Reference 51 providesmore details regarding the above argument. The two-state model predicts that L/L A shouldequal at T p at which the T dependences of both Φ and s c exhibit an inflection point, inaccord with the observation of Ngai and coworkers that the apparent activation energynear T c is normally about twice the high T activation energy. In the Stockmayer fluid ofdipolar particles and tabletop measurements of driven magnetic particles thermalized byvertical shaking, the average string length L has been observed to be generally near . Inthe GET, the crossover temperature T c of glass formation is identified by the temperature atwhich the T dependence of T s c has an inflection point, so we may expect a close connectionbetween T p and T c .If the above approximations for Φ are combined with the string model, an approximationfor the activation free energy emerges as, ∆ G ( L/L A ) ≈ ∆ G { − ( T − T p ) / D ] } , (28)which is the same form as the expression for the activation free energy proposed by Rösslerand coworkers. In this two-state model expression, the activation free energy is somewhat60rtificially broken into a ‘local’ contribution ∆ G and a ‘cooperative’ contribution represent-ing the stringlike collective motion. This type of two-state model has a long history in themodeling of GF liquids. The ECNLE model of Schweizer and coworkers assumesa similar decomposition of the activation free energy in local and collective parts where thecollective motions are conceived to arise from the energetic costs arising from the emergenceof caging in liquids at low T or high densities. Tanaka and coworkers have recently invokeda mean-field two-state model of liquid dynamics that seems similar in spirit for the activationenergy for structural relaxation in water. Note that eq 28 is a rough mean-field model thatneglects the distribution of barrier heights associated with the structural organization of thedynamic heterogeneity, an aspect that is emphasized in the string model. We may also formally recover the recent fashionable MYEGA expression for the T de-pendence of τ α by approximating L by its high T approximation in its well-known equilibriumpolymerization model estimate, L ≈ L exp[∆ H p / (2 k B T )] , (29)where L is a constant determined by molecular parameters of the polymerization model.The Arrhenius variation of L holds exactly for purely uncooperative or ‘isodesmic’ equilib-rium polymerization and provides a generic approximation for L for equilibrium polymeriza-tion processes near and above the thermodynamic transition temperature T p . However, thissimple relation for L no longer holds generally at low T in the case in which the polymer-ization process becomes cooperative. Correspondingly, we may thus expect the MYEGAequation, as well as the popular VFT relation, to only hold over a limited T range.Therefore, we can also understand the success of these phenomenological fitting functionsfor τ α based on the string model. 61 .5 Opportunities and Challenges Although stringlike collective motion is a conspicuous feature of the dynamics of simulatedGF liquids broadly in the regime where the dynamics becomes non-Arrhenius and has alsobeen observed in GF colloidal fluids, the grain boundaries of colloidal and granularsystems, and colloidal fluids approaching their melting, there is currently noobvious experimental method for identifying the strings in molecular fluids. There is clearlya need for getting information about the strings in materials of practical interest.In our recent works, we hypothesized that the activation volume ∆ V might berelated to the average string length L , based on a number of studies showingthat the growth of ∆ V upon cooling occurs in parallel with the apparent activation energyof GF liquids below T A . If such a relation could be established between ∆ V and L , thenit would provide an accessible method for determining L , the central quantity in the stringmodel of the dynamics of GF liquids. Since measurements of the length scales characterizingthe extent of collective motion are often quite challenging to make and involve advancedinstrumentation that is often not available in an industrial research setting, while ∆ V can be straightforwardly measured experimentally, a direct relation between ∆ V and L isparticularly appealing from an experimental viewpoint. Unfortunately, our study based onsimulations and the GET indicates that there is no direct relation between ∆ V and theextent of collective motion quantified by L . Although the strings can be directly observed in measurements of colloidal suspensions,we continue the search for an accessible experimental metric for L based on correlation stud-ies. For instance, previous works on energy and mobility fluctuations in the glassyinterfacial dynamics of nanoparticles indicate that the color of these fluctuations correlatesstrongly with L . The basic idea here is that mobility fluctuations associated with mobileparticles, such as strings, should lead to observable effects on properties that measure fluc-tuations in the mobility, such as electrical conductivity. In the future work, we needto check these relationships between noise color associated with mobility fluctuations and L
62n cooled liquids below T A and see if we can develop an accessible method for estimating L in molecular and polymeric liquids. There is also the prospect of observing stringlike collec-tive motion on the surface of crystalline and GF materials in real space based on ultrahighresolution imaging. The imaging studies, even in the case of the strings in the col-loidal systems, needs improvement in order to compare quantitatively with the stringmodel. Another promising possibility seems to be offered by single molecule fluorescencemicroscopy.
This technique allows for investigations of the rotational motion of singlefluorescence probes with different sizes doped in a GF polymer, leading to the identificationof a characteristic length scale that might be related to L , but much work remains to bedone to establish such a relationship quantitatively. We look forward to participating inthese developments in the future. The strong correlation between the fluid entropy and the dynamics of GF liquids has led toalternative models to the AG model and its antecedents, the GET and the string modeldiscussed above. For completeness, we provide a brief overview of these alternative models,with an emphasis on the hints provided by these model into how the GET might be developedin the future. The RFOT theory is close in spirit to the AG model in the sense that it is predicatedon the idea of dynamic domains in cooled liquids which grow upon cooling, accompaniedby a corresponding drop in the configurational entropy, S c . This mean-field model, initiallyformulated in the context of spin glass materials, has also elements similar to the GDmodel of polymer glass formation in the sense that it predicts S c to vanish at a Kauzmanntemperature, T K . 63he RFOT theory envisions that the liquid is divided into metastable regions witha characteristic correlation length ξ , describing the average ‘entropic droplet’ or ‘mosaic’size. A scaling relation between ξ and S c is suggested by a consideration of the balancebetween the surface and bulk free energies of these regions. This theory also assumes thatthe activation energy for relaxation scales with ξ with a power. It should thus be appreciatedthat the general qualitative findings of this interesting mean-field model have many pointsof contact with the AG and GET models. A particularly interesting aspect of the RFOTtheory is that it focuses on the geometrical size of what may be identified with the CRR ofthe AG model, rather than the number of particles within these hypothetical clusters.As in the AG model, a basic problem about the implementation of the RFOT model for polymeric fluids is that S c can only be roughly estimated based on similar approximationsintroduced by Bestul and Chang and AG, where the uncertainties are especially large inpolymer fluids because of the relatively large vibrational entropy associated with molecularbonding in such fluids. We note that an interesting methodology for estimating S c for non-molecular GF liquids based on a RFOT framework has been introduced, ? so this situationcould conceivably change in the future if this methodology can be extended to treat molecularGF liquids. Steven and Wolynes have performed a test of the RFOT model based onestimates of S c obtained from specific heat data for a range of small-molecule liquids wherethis type of S c estimate is expected to be a reasonable rough approximation. We view theestimation of S c from specific heat measurements to be inherently unreliable for quantitativeanalysis, however, and we strictly avoid any discussion of this method for estimating S c .Although S c can be estimated for molecular fluids from simulations, including polymers,the method is limited in the T range in which equilibrium simulations can be performed.The RFOT theory offers no method for calculating S c and other parameters of themodel. The GET is founded on a statistical mechanical model of polymer fluids, whichenables calculations of all the thermodynamic properties in terms of molecular parameters,including S c . The LCT for polymer thermodynamics, which is the thermodynamic64omponent of the GET, has been validated by many studies. By combining the LCTwith the AG model, the GET is a highly predictive theory with no free parametersbeyond the molecular and thermodynamic parameters governing the thermodynamic stateof polymer fluids. Of course, the extension of the GET to include the entropy of activation,a quantity neglected by AG, makes the GET no longer fully predictive. We consider belowhow this limitation of the GET might be overcome. An interesting difference between theAG and RFOT models is that the dimensionless activation free energy is deduced to scale inthe RFOT theory as ∆ G/k B T ∼ /S c , which is consistent with the phenomenologicalexpression first noted by Bestul and Chang rather than the AG expression, ∆ G/k B T ∼ / ( T S c ) . This RFOT expression is then consistent with the GET when the enthalpy ∆ H of activation is set to zero, in which case the free energy of activation is taken to be dominatedby the entropy of activation, ∆ S > , an assumption suitable for the description of systemswhere particle interactions are dominated by repulsive interactions, as in the case of hard-sphere fluids at constant density. Note that the RFOT model emphasizes the configurationalentropy per unit mass rather than the entropy density, which has ramifications for consideringthe glass formation of materials at constant pressure.It might be possible to combine the LCT with the activation free energy scaling argumentsunderlying the RFOT approach to glass formation, but the model seems to involveunspecified parameters that we currently do not know how to specify and various assumptionsthat require careful assessment, as we have tried to perform in the GET. We plan to explorethis alternative approach to the dynamics of GF liquids in the future.At this stage, we mention an interesting exploration of the compatibility of the stringmodel of glass formation and the RFOT scaling arguments for the dynamics of GF liq-uids by Starr et al. This work adopts the hypothesis that the ‘string’ clusters exhibitingcooperative exchange motion, identified successfully as having the main attributes of thehypothetical CRR of AG, can also be identified with the hypothetical ‘entropic droplets’of the RFOT model as the logical compatibility of the AG and RFOT models would formally65uggest. Verification of such a relation would be important because the RFOT theory offersno explicit algorithm for what might constitute an ‘entropic droplet’ and is equally vagueas the AG model regarding the physical nature of the CRR. It has often been assumed intheoretical discussions of the AG and RFOT models that the CRR are more or less spheri-cal, in line with the term droplet, but the basis of this assumption is rather unclear.Klein and coworkers have considered the role of fluctuations on the formation of dy-namic clusters in model GF liquids, indicating that while compact clump-like structures arefound in mean-field theory descriptions of GF materials, the clusters become progressivelymore ramified, i.e., fractal, when fluctuation effects associated with the short-range attrac-tive interactions of real liquids are incorporated into the theory, a finding having profoundimplications for both glass formation and crystal nucleation at high undercooling. Re-solving the physical nature of the ‘entropic droplets’ is then a central question in developingthe RFOT theory into a quantitative model of the dynamics of GF liquids and we discussthis matter below.The RFOT scaling expression for the activation free energy reads ∆ G ∼ ξ ψ , where ψ isa surface tension exponent whose value depends on the assumed geometry of the entropicdroplets. See ref 161 for an extended discussion of ψ . As noted above, identifying ξ with theradius of gyration of the strings R g , string indicates that ∆ G = AR ψ g , string with A and ψ beingadjustable constants. This preliminary test of the RFOT model indicates compatibilitybetween the AG and RFOT models in the qualitative sense that the free energy barriershould grow with the size of regions exhibiting cooperative motion. However, the exponent ψ is not compatible with compact liquid droplet estimates of this scaling exponent, suggestingthat these field excitations should be modeled as being fractal structures, as in the case ofthe string model. We may also view this scaling of the activation free energy with as beingcompatible with the string model based on the observed scaling relation, R g , string ∼ L ν ,where ψ = 1 /ν in this interpretation of the ∆ G scaling with /ν being the fractal dimensionof the self-avoiding polymeric strings. This interpretation of the free energy scaling with66he average string size would lead to ψ = 5 / if we assume the Flory estimate of ν . This estimate of ψ roughly accords with the value of ψ ≈ . estimated from simulations ofpolymer melts by Starr et al. Reference 161 has discussed the theoretical RFOT estimates of ψ in comparison to com-putational estimates, along with the implications of the observed scaling of ∆ G with R g , string ,so we do not elaborate on this matter further here. Our main point here is that the RFOTapproach appears to be promising, provided that a more realistic physical description of the‘entropic droplets’ is considered. Despite the apparent success of the RFOT scaling, thereare still issues that must be faced. In particular, this scaling approach still involves a num-ber of practical issues regarding predictions of relaxation times for particular materials. Wecurrently do not understand the meaning of the parameters A and ψ in the RFOT model sothat quantitative predictions of relaxation times remain elusive. In our view, there seems tobe no demonstrated advantage of the RFOT model over the AG model. This exerciseis valuable, however, in the sense that it provides a new perspective on the AG model andindicates that further attention should be given to the geometrical structure of the strings indeveloping the entropy theory of glass formation into a more general and validated theoreticalframework. This effort remains a work in progress.
There is another interesting approach to the dynamics of liquids based on an attempt tointerrelate the rate of diffusion and relaxation to the fluid entropy. Based on a conceptiongoing back to Boltzmann that both the thermodynamics and dynamics of liquids is domi-nated by hard-core intermolecular interactions,
Rosenfeld developed a scaling modelof the dynamics of liquids emphasizing a formal mapping of liquids onto hard spheres basedon a consideration of the entropy in the liquid state, S liq , relative to the entropy in the idealgas state, S id , rather than the crystal or glass state as in the case of AG. See ref 77 fora detailed discussion of Rosendeld’s works. This ‘excess entropy’, relative to the ideal gas ,67enoted as b S ex = S liq − S id , should not be confused with other definitions of the ‘excessentropy’ S exc in the context of the AG model. In particular, Rosenfeld argued that transportproperties of liquids such as the diffusion coefficient D should scale with b S ex as, D ∼ exp( B b S ex ) , (30)where B is an adjustable positive constant. This particular exponential variation of dy-namic properties is termed ‘Rosenfeld scaling’ and the description of D and τ α by moregeneral functional forms is termed ‘excess-entropy scaling’. These two distinct terms in-dicate that Rosenfeld scaling has its limitations, as we shall discuss below. There is also apopular extension of the Rosenfeld scaling model introduced by Dzugutov based on anapproximation of b S ex by its two-body estimate from the radial distribution function g ( r ) , S = − πρ Z ∞ { g ( r ) ln g ( r ) − [ g ( r ) − } r dr, (31)where r is the spatial distance. For readers familiar with the AG model, Rosenfeld scalingappears to be rather odd, but it must be remembered that the sign of b S ex is opposite to S c .Note also that b S ex is in the numerator of eq 30, while S c is in the denominator in the corre-sponding AG relation for D and τ α . It is then easy to appreciate why confusion is sometimesencountered regarding how the fluid entropy should be related to liquid dynamics. Interest-ingly, simulation results have indicated that b S ex actually scales inversely with the product ofthe numerically estimated S c and T in model GF liquids, as required by self-consistencybetween Rosenfeld scaling and the AG model, a relation that seems rather mysterious to us.Particularly clear expositions of the excess-entropy scaling approach to liquid dynamics havebeen made by Chakraborty and coworkers and Sastry and coworkers. An extensive andexcellent review on Rosenfeld scaling has recently been given by Dyre. Speedy et al. have discussed the limitations of the hard-core approximation in describing the dynamics ofliquids. 68e can calculate b S ex from the LCT to implement this type of entropy theory.However, we did not do so because numerous simulations have shown that while this modelseems to hold rather well in the high T regime, large deviations from Rosenfeld scaling arenormally observed in GF liquids at low T . In many fluids, the simple exponential form of D and τ α in terms of b S ex is superseded by other functional forms that can be applied withimpressive accuracy over a wide range of temperatures and densities, accounting for theintense interest in this approach to understanding and correlating the dynamic propertiesof liquids in terms of thermodynamic properties of liquids. Unfortunately, the particularfunctional form relating transport properties to b S ex has been found to be highly dependenton molecular structure in molecular fluids so that this approach to liquid dynamics offersno direct general relationship between transport properties and b S ex . Thus, the applicabilityof this approach is largely limited to atomic and small-molecule liquids. Chopra et al. have discussed the origin of the failure of density-temperature scaling in molecular fluidsand related this change of scaling to changes in the density-temperature scaling of transportproperties, a phenomenon that is deeply related to the cooperative dynamics of GF liquids, aswe have recently found elsewhere. We thus do not see how one could develop a quantitativetheory of the dynamics of real polymeric GF liquids based on this theoretical frameworksince many polymer materials are only thermally stable against degradation below T A , i.e.,non-Arrhenius collective dynamics is inherent to this class of materials. Nonetheless, recentworks suggest that the excess-entropy scaling approach has some important conceptual ideasto offer for the future development of the GET, which we briefly discuss below.Given the apparently general observation that the excess entropy scaling is restricted toelevated temperatures and the emphasis of this model on repulsive interactions, it is naturalto compare this model to the standard theory of activated transport where D formally scalesas, D ∼ exp(∆ S /k B ) , for molecular diffusion in systems in which attractive interactionsare entirely neglected. This might then provide an approach to calculating ∆ S , one of themost elusive quantities in TST. We hope to pursue this approach to understanding69nd numerically estimating ∆ S in the future.Finally, recent studies appear to indicate that b S ex might enable the estimation of thecharacteristic temperatures of glass formation. For instance, refs 304 and 305 suggest thatthe onset temperature T A may be calculated as the temperature at which ∆ b S ex , definedas b S ex minus it two-body estimate S , changes sign. It has also been suggested that thecrossover temperature T c might be inferred from the T variation of b S ex . It would bean invaluable advance to the GET to have an alternative thermodynamic-based approachfor calculating T A , and we plan to test this proposal against independent estimates of T A for our polymer model in the future. Over time, we hope to see the various ‘pieces’ of theentropy theory of glass formation to grow together to form a highly predictive and validatedframework from theoretical, computational, and experimental perspectives. The entropy perspective on the dynamics of glass-forming liquids and the general conceptionthat collective motion accompanies the drop of configurational entropy has been developingsteadily over the last century, and over this period, empirical correlations between thermo-dynamic and dynamic properties have gradually been superseded by quantitative relationsorganized around increasingly sophisticated theoretical frameworks for quantifying the dy-namics and thermodynamics of condensed materials and for quantifying collective motionand its physical consequences on relaxation and diffusion. The generalized entropy theoryand the string model are just the most recent manifestations of this conceptual development,along with the necessary experimental and computational works that give credence to thesemodels through validation studies.Although much progress has been made, there are still many ‘gaps’ in the entropy ap-proach to the dynamics of glass-forming liquids that will require significant effort in thefuture to obtain a more predictive model of the dynamics of glass-forming liquids. For ex-70mple, our understanding of the activation free energy parameters has not advanced verymuch beyond the descriptions introduced by Eyring and coworkers.
A significant effortalso needs to be made to develop a fundamental understanding of the activation parametersin the Arrhenius regime where the complicated effects of dynamic heterogeneity are not animportant issue. It is very helpful that relaxation times are generally relatively short in thehigh temperature Arrhenius regime, which is a favorable aspect for simulation studies thatcould provide much information on which a sound theoretical treatment of the activationenergy parameters could be made. There are also many experimental data now available forthis purpose, so we are hopeful that much progress in this direction will be made in the nearfuture. Simulations have shown that molecular topology can greatly influence the dynamicsof polymeric glass-forming materials, so computational and experimental investigationsof how molecular topology influence the high temperature activation energetic parametersand the overall dynamics of glass-forming liquids promise to be a future of development.We also anticipate that studies of nanocomposites, thin supported films, polymers with sol-vent additives, and polymer blends all offer promising areas for examining the activationfree energy parameters and their large influence on the dynamics of glass-forming liquids. Inshort, we envision this type of study to form the foundation for a fundamental and predictivetheory of glass-forming liquids.There are also many issues to resolve in the entropy theory relating to the question ofwhy dynamic heterogeneity in glass-forming liquids, such as the strings, generally adoptsstructural forms and obeys the thermodynamics consistent with existing models of self-assembly.
Answering these questions will require a theoretical understanding of adifferent order. Equilibrium polymerization has long been understood to arise in connec-tion with the description of fluctuation effects in materials undergoing second-order phasetransitions, and we expect that a deeper understanding of glass formation withinappropriate field theoretical frameworks will naturally evolve in time. We also expect thatthe theoretical concepts and methods derived from the alternative entropy theories of glass71ormation, namely, the random first-order transition theory and the excess-entropy scal-ing approach, will play a large role in developing the generalized entropy theory further.We await these theoretical developments and quantitative computational investigations tovalidate these new concepts. For the present, the generalized entropy theory and the stringmodel emphasized in the present work are promising working models. It seems inevitablethat many of the currently disparate models of glass formation will come together to forma unified theory of glass formation. Hopefully, we will not have to wait another century forthese developments to occur. 72 iographies
Wen-Sheng Xu is a professor in the State Key Laboratory of Polymer Physics and Chem-istry at Changchun Institute of Applied Chemistry, Chinese Academy of Sciences in China,where he has been a faculty member since 2019. He received a B.E. degree in MaterialsScience and Engineering at Tianjin University in 2007 and a Ph.D. degree in Chemistryand Physics of Polymers at Changchun Institute of Applied Chemistry, Chinese Academy ofSciences in 2012 under the guidance of Professors Li-Jia An and Zhao-Yan Sun. From 2013to 2018, he was a postdoc first working with Professor Karl F. Freed at the University ofChicago and then with Dr. Yangyang Wang at Oak Ridge National Laboratory. His researchis focused on the glass formation and rheology of polymeric materials.Jack F. Douglas is a NIST Fellow in the Materials Science and Engineering Division of73he National Institute of Standards and Technology (NIST) at the facility in Gaithersburg,Maryland. He obtained a B.S. degree in Chemistry and an M.S. degree in Mathematics fromVirginia Commonwealth University and then a Ph.D. degree in Chemistry at the Universityof Chicago. After receiving his Ph.D. degree, he was a NATO Fellow at the CavendishLaboratory, Cambridge and a NRC postdoctoral fellow at NIST before becoming a researchscientist at NIST. His field of research includes the equilibrium and dynamic properties ofpolymer solutions and melts, fractional calculus and path integration, phase separation andcritical phenomena, relaxation processes in glass-forming liquids, polycrystalline materialsand nanoparticles, the elastic properties of gels, and the thermodynamics and dynamics ofself-assembly processes.Zhao-Yan Sun is a professor at Changchun Institute of Applied Chemistry, ChineseAcademy of Sciences in China and a deputy director of the State Key Laboratory of PolymerPhysics and Chemistry. She received a Ph.D. degree in Polymer Chemistry and Physics atJilin University in 2001. She was a postdoctoral fellow in the Department of Chemistry atUniversity of Dortmund from 2001 to 2002. Since 2003, she has been a faculty member at thepresent institution, initially as an assistant professor and then an associate professor, beforebecoming a full professor in 2010. She was awarded the Young Chemistry Award of the Chi-nese Chemical Society in 2005. Her research group focuses on the structure and dynamics74f polymers and nanocomposites and the development of computer simulation methods.
Acknowledgement
W.-S.X. acknowledges the support from the National Natural Science Foundation of China(No. 21973089). Z.-Y.S acknowledges the support from the National Natural Science Foun-dation of China (Nos. 21833008 and 21790344), the National Key R&D Program of China(No. 2018YFB0703701), the Jilin Provincial science and technology development program(No. 20190101021JH), and the Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-SLH027). J.F.D. thanks his long-term collaborators, Karl F. Freed, Jacek Dudowicz,Francis W. Starr, Hao Zhang, and Wen-Sheng Xu, and many postdocs associated with theresearch groups at the National Institute of Standards and Technology, the University ofChicago, the University of Alberta, Wesleyan University, and Changchun Institute of Ap-plied Chemistry, Chinese Academy of Sciences for the many collaborative efforts over manyyears that made his contributions to the entropy theory of glass formation possible. J.F.Dalso acknowledges the National Institute of Standards and Technology for its long-term sup-port of this research effort. Z.-Y.S thanks Professor Li-Jia An for helpful discussions on glassformation over the years. W.-S.X. thanks Professor Karl F. Freed and Dr. Jack F. Dou-glas for numerous discussions concerning polymer glass formation over the years. W.-S.X.also thanks Professors Li-Jia An and Zhao-Yan Sun for their support and encouragementat every stage of his academic career. We are grateful to the authors of ref 88 for sharingtheir experimental data with us and to Dr. Beatriz A. Pazmiño Betancourt for help withthe visualization of strings in simulations.
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