The entropy theory of glass formation after 40 years

COMPUTATIONAL MATERIALS SCIENCE ELSEVIER

Computational Materials Science 4 (1995) 317-324

The entropy theory of glass formation after 40 years

1

Edmund A. Di Marzio Polymers Division, National Institute of Standards

and Technology, Gaithersburg

MD 20899, USA

Received 3 July 1995

Abstract It is argued that an equilibrium theory of the glass transition transition, as observed kinetically, can be obtained. The lattice

is required

before

a full understanding

of the glass

model provides a basis for such a theory and predicts

that vitrification occurs as the configurational entropy S, approaches zero. Comparison with nine different classes of polymer experiments shows semiquantitative agreement in each case. An important aspect of glass formation in polymers is that the crystal phase is not ubiquitous so that amorphous polymer material exists necessarily at low temperatures. A low-temperature equilibrium theory for these materials is required. Using equilibrium theory as a base, a kinetic theory is developed and an equation for the relaxation function is derived which predicts the proper temperature dependence of the viscosity and shows stretched exponential-like behavior for the relaxation.

1. Introduction We define a glass to be a material which is an ordinary liquid at high temperatures and whose thermodynamic extensive quantities, volume V, and entropy S, fall out of equilibrium as we lower the temperature past some temperature Tg which depends on the rate of cooling. Above Tg the relaxation times associated with viscosity are less than the time scale of the experiment, while below Tg they are greater. The above definition describes the formation of a crystal as well as a 1

Contribution to proceedings from a workshop on Glasses and the Glass Transition: Challenges in Materials Theory and Simulation, organized by the Center for Theoretical and Computational Materials Science at the National Institute of Standards and Technology. The workshop was held in Kent Island, Maryland, 16-18 February, 1995. 0927-0256/95/$09.50

glass so we augment our definition by requiring that the extensive thermodynamic quantities be continuous at Tg and that there be no change of spatial symmetry as we cross Tg. This operational definition immediately suggests a number of questions which must be answered if we are to understand glasses. (1) Why does the glass transition occur at one temperature, Tg, rather than some other temperature? (2) What are the V(T, P) and NT, P) equations of state in the high-temperature side of Tg? (3) What are the properties well below Tg where the relaxation times for diffusion of molecules are so long that only oscillatory motions occur and experimentally the glass is known to be an elastic solid? (4) What is the viscosity q(T, P, o), where w is frequency?

0 1995 Elsevier Science B.V. All rights reserved

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318

EA. Dl Marzio /Computational

The first three questions are concerned with the equilibrium properties of glass-forming materials. In Section 2 the rationale for formulating an equilibrium theory of glasses is presented. Section 3 outlines the arguments supporting a vanishing of the configurational entropy S,, as an important feature of glass formation. Section 4 describes briefly experimental findings that support the S, = 0 hypothesis, while Section 5 offers a set of requirements for equilibrium theories of glass formation. In Section 6 some insights into glass kinetics obtained from equilibrium considerations are offered. The final section contains suggestions for future work.

2. Necessity for an equilibrium theory of those materials that form glasses There are four reasons to formulate an equilibrium theory of glasses [1,2]. They are: (1) Glasses have equilibrium properties above 7”; it is sensible to ask what they are. (2) An equilibrium theory is needed to resolve Kauzmann’s paradox. Such a theory allows us to extrapolate equilibrium quantities through the glass transition to see how the “negative entropy” and “volume less than crystal volume” catastrophes are avoided. For polymer glasses the sharp leveling off of the experimental thermodynamic quantities must also occur in a correct equilibrium theory. This either is a second-order transition or it approximates one. Either case allows us to calculate a T2 to which the Tg tends in very long time-scale experiments. (3) It was proved in Ref. [2] that the crystal phase is not ubiquitous. Therefore an equilibrium theory is needed for those materials which are necessarily amorphous at low temperatures. (4) An equilibrium theory is a necessary prerequisite for an understanding of the kinetics. 3. Is configurational entropy the Rosetta stone of glass formation? Once one is convinced that the equilibrium properties of glassy materials exist there are no

Materials Science 4 (1995) 317-324

options. One simply evaluates the partition function for the material and then the two equations of state V(T, I’) and S(T, PI. It is required, of course, that the important characteristics of the molecules are taken into account, at least within a minimal model. This minimal model must have both intermolecular energy to allow for volume changes and intramolecular energy to allow for temperature dependent shape changes of the molecules. The lattice model of Gibbs and Di Marzio [3-51 is a minimal model for polymers which incorporates an intermolecular bond energy, E,, which regulates the number of empty lattice sites (volume) and intramolecular stiffness energy, AE, which controls the temperature-dependent shape changes. When this was done within the framework of the Flory-Huggins (F-H) approximation it was discovered that a secondorder transition (in the Ehrenfest sense) was obtained and that the T,(P) line separating the liquid state and the glassy state was given by the vanishing of the configurational entropy S,( T,, P) = 0.

(1)

The configurational entropy is easily evaluated in the F-H approximation. (More generally, for non-polymer systems S, might be defined as the total entropy minus the vibrational part of the entropy). The volume on the T,(P) line is not constant; neither is the number of holes in the lattice model. In fact, the configurational entropy can be expressed [6] as a function S&f, +J, of the fraction of flexed bonds, f, and the number of holes, n,. The condition S,(f, n,) = 0 forces no to vary along the T,(P) transition line which separates the liquid from the glass phase. Therefore, the existence of a critical volume cannot be a criterion for glass formation. Since the two independent equations of state completely characterize the thermodynamics, within the accuracy of the lattice model calculation there can be no thermodynamic criterion of glass formation other than S, = 0. This important conclusion is reinforced by the connection between S, and the viscosity q(T, P> noted in the original Gibbs-Di Marzio paper [4]. If the number of configurations becomes smaller and smaller as we approach the glass temperature from above, flow - which is a

E.A. Di Marzio /Computational Materials Science 4 (1995) 317-324

moving or jumping from one allowed configuration to another - becomes more and more difficult and consequently the viscosity becomes larger and larger. This suggests that the S, + 0 criterion is the universal criterion for glass formation. We will now quantify the implications of the above statements.

4. Evaluation of T, from the S, = 0 condition If we identify the transition temperature, T2, as the point at which the configurational entropy equals zero then Eq. (1) can be used to determine T,. However, the glass temperature, Tg, lies above T2 by an amount which is a function of the time scale of the experiment. If we assume that the time scales are such that the ratio TJT, is a constant then it is easy to see that the only effect is to rescale the energies (Ae/kT* + (AeT,/ Thus Eq. GMT,, E&T2 + (E,T,/T,)/kT,). (1) with resealed energies can be used to determine Tg. We have S,(T,,

P) = 0.

(2)

We have done this for nine separate classes of experiments on polymers: (1) Tg versus molecular weight for linear polymers [ 1,4]. (2) Tg versus molecular weight for ring polymers

[7,81. (3) Tg versus copolymer composition [9]. (4) Tg versus blend composition [lO,ll]. (5) Tg versus pressure [6,121. (6) Tg versus cross-links in rubber [131. (7) Tg versus strain in rubber [13]. (8) Tg versus ACP at Tg for large molecular weight polymers [14]. (9) Tg versus plasticizer (diluent) content 115,161. In all cases the experimental data are well described by the theory. There are several interesting aspects to these calculations. First, they are essentially no parameter fits to experiment. In item (1) of the above list we fit to the glass temperature at infinite molecular weight in order to determine the stiffness energy Ae (one parameter). The intermolecular energy parameter, E,, is determined from the thermal expansion above

319

and below Tg. Item (5) requires that the size of a lattice site be known as a function of pressure. All the other experiments agree with the theoretical predictions without any adjustable parameters. Each class of experiments illustrates a feature of polymer glasses. Item (9) illustrates the colligative-like properties of glasses. The initial glass temperature suppression by low molecular weight diluent is predicted to obey the equation [16] y dT,/dm

= -3Tp,

(3)

where m is the total mole fraction of diluent expressed in terms of mole fraction of the monomers, and y is the number of flexible bonds per monomer. One notices the universal character of the prediction. (5) predicts in accordance with experiment that Tg versus pressure has a horizontal asymptote at high pressure (in contrast, the free volume theory which assumes that the glass transition occurs when the hole fraction reaches a critically small value (usually 0.025) predicts a vertical asymptote). In (8) the specific heat for large molecular weight polymer is given to within 10% by [14] AC,=Rf(l

-f)(Ae/kT,)’

+ RT,Aa(4

- T,Aa/0.06)

+ 0.5T,AcKP( Tg-),

(4)

where R is the universal gas constant, f is the fraction of flexed bonds at Tg, Aa is the change in the thermal-expansion coefficient as we pass through the glass transition, and Cp(TgF) is the specific heat just below Tg. Notice that this is a no-parameter prediction since Tg, Aa, and C,(T;) are known from experiment, Ae/kT, is determined from the condition that S,(T,) = 0, and f is a function of only Ac/kT,. In (2) the glass temperature is predicted to rise as we lower the molecular weight in accordance with experiment [7,8]. This is purely an entropy effect arising from the observation that a ring of molecular weight x has more entropy than two rings each of molecular weight x/2. Thus a bulk system of the larger rings, since it has the larger entropy, must be cooled further to reach the S, = 0 condition

320

E.A. Di Murzio / Computational

which defines T,. Finally, it should be noted that the fits of theory to experiment have all been made with the original Gibbs-Di Marzio theory [4]. We have not needed to adjust the theory to account for new experimental data. It is worth noting that a perfect fit to experiment would require: (1) that the kinetics have no sensible effect on the comparison with experiment. We would argue that since the experimentally observed T, is affected by kinetics, perfect accord with experiments is not expected. The theory predicts an underlying transition temperature T,, and the relation between T2 and the experimental Tg requires further elucidation: (2) that the F-H calculation be perfect. It is not, because the statistics are approximate and because the molecules are modeled imperfectly; (3) that the experimental data are excellent, including the use of well characterized polymer material. Mention should also be made of the attempts to predict the glass temperature of an infinite molecular weight polymer by simply knowing its chemical structure. Fig. 10 of Ref. [ll] and Fig. 6 of Ref. [17] correlate T, to chemical structure (unfortunately the sample scatter is larger than desired). Further progress in this area would be valuable. In both of these predictions the entropy criterion is used.

5. Requirements of an equilibrium theory of glasses An equilibrium theory of glasses should satisfy the following: (1) Give accurate predictions of thermodynamic quantities without multiplication of parameters. (2) Explain the ubiquitous nature of glass formation. (3) Should provide a foundation for kinetic theory. (4) Predictions for other phenomena should also be correct. We believe that the entropy theory of glasses satisfies criterion (1). It accurately predicts glass temperatures as a function of a range of parame-

Materials Science 4 (1995) 317-324

ters from pressure to molecular architecture. However, two questions that naturally arise are “is there really a second-order transition?“, and “does S, really becomes zero at T2?” To settle these questions an improved theory is required. Such a theory should derive the P-V-T and S-P-T equations of state to equal accuracy. A theory that gives a poor S-P-T equation of state is sure to give undue stress to imagined implications of the P-V-T equation of state. The theory must allow the molecules to have shape-dependent energies, since these are undoubtedly very important to vitrification in polymers. We stress that an improved theory may not show an actual underlying second-order transition as the GibbsDi Marzio theory does (there may be a rounding), but it should approximate one. A reason for the configurational entropy to be somewhat greater than zero at the glass transition has to do with the concept of “percolation of frustration” as a criterion for glass formation [18]. As an entre to this problem we express the configurational entropy as a function S,(f, n,) of the two order parameters f and n,. The equation S,(f,

no) = 0

(5)

divides f, IE~ parameter space into two regions, the small f, n, region being the glass region. Because there are spatio-temporal fluctuations in f andno,if we are in the liquid region just above the glass region there will be clusters of polymer for which the f, n, values are appropriate to the glassy state, and clusters for which the f, n, values are appropriate to the liquid state. As we lower the temperature these glass-like clusters grow until they span the space. However, as is characteristic for percolation there will be pockets of liquid-like clusters 1191(material for which the f, no values are appropriate to the liquid phase). The glass temperature would be defined as the highest temperature for which there is percolation of the glass-like structures. Because of the existence of the liquid-like pockets this T, would correspond to a configurational entropy somewhat greater than zero. Criterion (2) may be met by first defining S, for all materials as the total entropy minus the extrapolation of the vibrational entropy. Any

E.A. Di Mark /Computational Materials Science 4 (1995) 317-324

method of evaluating the partition function from first principles which gives the proper equilibrium behavior above Tg is viable. One would then identify the glass transition as the place where S, becomes smaller than some critical value as we cool the system. The following systems need to be examined for their glassy behavior: (1) Polymer glasses, (2) low molecular weight glasses, (3) the classic inorganic glasses, (4) metallic glasses, (5) liquid crystals and plastic crystals, (6) systems composed of plate-like molecules, (7) spin glasses. Under criterion (4) above we have the happy circumstance that the F-H lattice model predicts the formation of liquid crystals. As Onsager originally observed [20], the nematic phase of liquid crystals occurs because of the increased difficulty of packing rigid rod molecules together in space as we increase their concentration. Thus, the isotropic to nematic transition in liquid crystals occurs because it is entropy driven - configurational entropy given. The nematic liquid crystal phase occurs for the same reason as glasses and the correctness of the F-H calculations for liquid crystals argues for their correctness for glasses, and conversely. The transition from random order to parallel alignment for a system of plate-like molecules is also entropy driven [21].

321

V. More generally the F-D theorem relates x (defined as the response of a material at (r, t> arising from an impulsive force at (0, 0)) to the correlation in fluctuation at these two space-time points 1221. Since the fluctuations of a system at equilibrium show a discontinuity of the same character as the thermodynamic extensive variables, so do also the dissipative quantities. Thus, for a system undergoing a first-order liquid to crystal transition the viscosity q(T, P, w) will show a discontinuity as a function of T, P since the volume and entropy do. Similarly, for a system undergoing a second-order transition the viscosity will show discontinuities in slope since the volume and entropy do. There are many examples in the literature of dissipative quantities such as viscosity, diffusion coefficient, electrical conductivity, particle conductivity and thermal conductivity which show breaks as a function of temperature as we pass through the glass transition. However, it is also true that a genuine falling out of equilibrium will also cause the same kind of behavior. It is not clear how one distinguishes between the two effects. Movement of the transition point as a function of the time scale of the experiment seems not to be a distinguishing characteristic since this happens also for systems known to have genuine first-order transitions. 6.2. A remark on the topology of phase space

6. Insights into glass kinetics obtained from equilibrium considerations 6.1. Fluctuation -diwipation theorem

Q=lexp(-(K((..Qj,~j..))

Whenever there is a thermodynamic phase transition the fluctuation-dissipation (F-D) theorem shows that dissipative quantities have the same discontinuities as the underlying thermodynamic phase transition: A simple example of a F-D theorem is the Kubo relation [221 D= (1/3)/m(v(O) 0

*u(t)>

dt,

The potential energy surface of a liquid E’(..qj..) appears in the partition function Q

(6)

which relates the diffusion coefficient D to the autocorrelation function of the particle velocity

+E’( ..s,..))/kT)T

dqjv

dpj

(7) where qj, pi are the position and momentum coordinates, K is the kinetic energy and E’ is the potential energy. In polymers, even if E’ is pairwise additive, E is not [22]. The simplification of Eq. (7) allows us to work exclusively in configuration soace. This is aenerallvI reDresented as in Fig. L

E.A. Di Marzio /Computational

322

/-/

,

i’ \

,

Fig. 1. Two insights into the topology of configuration space. Fig. l(a) is meant to show that the potential energy surface is highly corrugated with the very deep wells representing the glassy state. The abscissa is really 3N coordinates. Flow is conceived as the process of jumping from one deep well to another. Fig. l(b) shows that the part of configuration space visited by a phase point is an infinitesimal fraction (equal to exp( - N) of the total. Flow is then thought of as the wandering along gossamer threads and the occasional jumping from one thread to another.

l(a) as a multi-well potential energy surface, where the abscissa represents the 3N position coordinates. As one approaches Tg from above the wells effectively become very deep because of the l/kT term. One then talks about flow as a motion from one deep well to another deep well via the higher energy continuum. We wish to emphasize a different aspect of the topology. Consider the configuration space of N identical particles on a line of length L. The partition function is given by Q=LN/N!.

(8)

The volume of phase space for this system is given by LN. Now consider the case where the particles each have a diameter d. The partition function is Qd=(L-Nd)N,‘N!.

(9)

The ratio of the two phase space volumes is given by QJQ

= (1

-

Nd/L)

= exp( -4N),

N G exp( -N 2d/L) (10)

where 4, being the volume fraction occupied by the particles, is on the order of 1. Since N is on the order of Avogadro’s number we see that the

Materials Science 4 (1995) 317-324

fraction of the volume of phase space occupied by the extended particles is infinitesimally small. According to this picture a point in configuration space wanders on the finest of gossamer threads which pervade the N-dimensional hypercube of phase space as a fine network whose total volume is an infinitesimal fraction of LN (see Fig. l(b)). The application to glasses is in the observation that as we lower T the effective value of d increases, resulting in even fewer and finer gossamer threads for the phase point to travel on. Thus, not only are the number of paths (threads) between two phase points fewer as we decrease T but also as one traverses a given thread the potential energy minima are deeper and the barriers higher. Another aspect of configuration space is that it is not really a space; it is a manifold. In order to have a space one needs to define a metric on the manifold. The statement “distance between wells” has meaning only if a metric is defined. For now, lacking any insight on this question, we will simply assume a Euclidean metric. 6.3. Detailed balance makes a significant statement concerning the kinetics of glasses [24]

Boltzmann’s law gives exp(-E/kT) as the fraction of time that a system spends in state i but it does not say how often the system jumps from state i to j. To determine this we use the principle of detailed balance in the form N,a,; = Njaji, aij/cyji = h$/Ni =exp(-[E,--E,]/kT),

(11)

where ajj is the rate of jumping from state i to j and N, is the fraction of time a system spends in state i. In using Eq. (11) one must first decide how the energy is apportioned into forward and backward transitions. For deep wells it is sensible to assume that all of the barrier is in preventing the phase point from jumping out of the well. It does this at a rate given by l/7 where 7 is the average time to exit the well. If we also recognize that the probability of jumping out of the well is exponential in time [25] we have p( t, T) = 7-l exp( -t/T) and 7-l --bij exp(-[I?,-E,]/kT),

(12)

E.A. Di Marzio /Computational

Materials Science 4 (1995) 317-324

323

where p is the normalized probability of exiting the well at time f. It is imagined that once the phase point has escaped the well it wanders around in the configurational sea of the high-energy region of phase space until it falls into a low-lying well, starting the flow process all over again. This configurational sea consists of many shallow energy wells, so it is expected that jumping out of the deep wells are the rate-determining steps.

experiment is left for future work but it is clear that Eq. (14) qualitatively gives both proper longtime behavior and temperature dependence. True progress would consist in relating the three parameters B, E,, and 0 to the topology of phase space.

6.4. The relaxation function for flow can be obtained by summing over the distribution of well depths

(1) We should settle on several model experimental systems that are also amenable to computer modeling. A suggestion for polymer glasses is a polymer diluent system with the diluent being monomer. Polystyrene is a good choice since the monomer is easily supercooled and the glass temperature of polystyrene is convenient. The fluctuating bond Monte Carlo method [28] should be able to probe long times in such a system. There should also be model systems for inorganic glasses, plastic crystal glasses [29], and spin glasses [30]. We might go very far with plastic crystal glasses because, if we allow only rotations about fixed centers of the molecules and induce frustration with nearest-neighbor angular pair potentials that do not have the symmetry of any lattice, then the relatively few variables (one per particle in 2D) should allow speedy computations. For such a system we should be able to choose an angular potential for which there is no crystalline phase. That is to say, there will be no spatially periodic repeat of rotation angle. (2) It is important to seek equilibrium properties on a computer to see what thermodynamic quantities correlate with glass formation, We need better equations of state (both P-V-T and S-PT or E-V-T) for various glass-forming systems. The excellent recent work of Poole et al. [31] on supercooled water should be continued. The equilibrium values of relevant order parameters should be calculated in order to effect a tie-in with future molecular dynamic calculations. Perhaps such computer calculations can provide insight into the thermodynamic phase behavior of glasses. (3) It is important to continue the exploration of the topology of the potential energy surface in

Let Q(E) be the distribution of wells. An estimate of the relaxation function P(t, T) describing the exiting from a well is made by weighting each of the wells by the distribution p(t, T), P(t,

T) =/Q(E) xexp(

exp( -E/kT-bt -E/kT))

dE//Q(

E) dE, (13)

We can argue that Q(E) is given by [24,26] Q(E) = exp( - 0( E - E,)*). With this substitution P(t, T) is given by the “aftereffect function” tabulated by Janke and Emde [27]. It has a time dependence which looks very much like the stretched exponential [24]. Does the temperature dependence look anything like the Arrhenius or Vogel expressions? If we assume that the long-time viscosity is proportional to (t) then we obtain in(v)

=ln B+ln +E/kT)

i

/exp(-B(E-E,)2

dE

, 1

(14)

which behaves much like the Vogel-Fulcher (VF) and Arrhenius expressions in that the viscosity is rising towards 00 as T decreases and dq/dT is negative and rising steeply as temperature decreases. This equation has the same number of parameters as the VF equation and is expressible as an error integral. Quantitative comparison to

7. Suggestions arising from workshop discussions

324

E.A. Di Marzio / Computational Materials Science 4 (1995) 317-324

configuration space [26]. Toy models for determining jump rates in a highly corrugated one-dimensional potential V(x) which contains deep wells could be illustrative. The fractal aspects of the motion of the phase point among trapping wells need to be accommodated [32]. Can the energy metric of Thirumalai and Mountain [33] be used as a model for defining a useful metric for configuration space? (4) One should track different order parameters simultaneously to see how they relax with different time dependences [34]. Volume and P&an, - O,+j) for chemical bonds (0, -Oi+j is the angle between bonds i and j in real space and P2 the second-order Legendre function) should behave differently and relate to the order parameters n, and f of the entropy theory. A related phenomenon is the strong history dependence and the non-linearities displayed by glasses [35]. Computer modeling may have a pivotal role to play in elucidating these phenomena because the analogue of linear response theory does not exist for non-linear systems.

Acknowledgement

The author thanks G.B. McKenna for a critical reading of the manuscript.

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