Polymer Physics (From Suspensions to Nanocomposites and Beyond) || Volume Relaxation and the Lattice-Hole Model

4VOLUME RELAXATION ANDTHE LATTICE–HOLE MODEL

Richard E. RobertsonDepartments of Materials Science and Engineering, and Macromolecular Science andEngineering, The University of Michigan, Ann Arbor, MI, USA

Robert SimhaDepartment of Macromolecular Science and Engineering, Case Western Reserve University,Cleveland, OH, USA

4.1 Introduction

4.2 Lattice–hole model

4.3 Free volume and molecular mobility

4.4 Stochastic functions and master equations

4.5 Comparisons with experiments4.5.1 Single temperature steps4.5.2 Multiple temperature steps4.5.3 Stepwise cooling4.5.4 Pressure steps

4.6 Continuum limit of free-volume states and the rate equations4.6.1 Derivation4.6.2 Example application of the continuum model

4.7 Summary, conclusions, and outlook

Polymer Physics: From Suspensions to Nanocomposites and Beyond, Edited by Leszek A. Utracki andAlexander M. JamiesonCopyright © 2010 John Wiley & Sons, Inc.

161

162 VOLUME RELAXATION AND THE LATTICE–HOLE MODEL

4.1 INTRODUCTION

It has been known for some time that the physical properties of polymer glasseschange with time, especially when the glass is kept at temperatures within 25◦C or sobelow the glass transition temperature (Tg ). The properties undergoing change includeelastic modulus, yield strength, and thermodynamic properties such as volume andenthalpy. These changes can be reversed in the sense that were the glass to be heatedabove the glass transition and cooled again at the same rate, the initial properties wouldbe restored. Because of the reversibility, this change of properties with time is oftencalled physical aging, as opposed to chemical aging, which is largely irreversible.To understand the nature of physical aging, attempts have been made to replicate itskinetics theoretically. That is the principal theme of this chapter.

For many years (for many, many years, going back to Greek science of the fourthcentury b.c. [Lucretius, 1982; West, 1986; Robertson, 1992]) changes in structure,such as occurs during physical aging, have been linked to the presence of free spaceor free volume. A successful measure of free volume has been derived from the stateequations of Simha and Somcynsky, based on lattice–hole theory. The kinetics of phys-ical aging pursued in this chapter use this Simha–Somcynsky (S-S) measure of freevolume. Of the various physical properties changing during physical aging, we focuson volume, allowing us to use the celebrated experimental measurements of Kovacsand Rehage for comparison with theory. Another advantage of using volume (orenthalpy, for that matter) is that its value at equilibrium is knowable. So there are twoessential motivations for pursuing the subject. First, one expects other useful physicalproperties, which are likely to be related to volume, to exhibit similar physical agingbehavior and kinetics. Second, considering the agent driving this behavior (i.e., localmolecular motions involving polymer structure), one may hope for insights into theseelements by appropriate molecular theories and experiments of the aging kinetics.

The basic process to be explored is illustrated by the simple experiment depicted inFigure 4.1. A melt is cooled at a given rate, passing through the glass transition region

FIGURE 4.1 The basic process to be explored, involving the volume, V. When the liquidis cooled from the melt, the volume deviates at Tg from equilibrium, indicated by the dashedline. The downward arrow indicates that over time, the volume approaches equilibrium.

LATTICE–HOLE MODEL 163

and terminating at some sub-glass temperature. At this fixed temperature, the volumeis observed to decrease in the course of time toward the extrapolated equilibriumvolume. Thus, we have to deal with a process that starts out from a nonequilibriumstate and ends when the system reaches an equilibrium state.

We want to pursue the subject by starting with a brief review for the present pur-poses of the essentials of lattice–hole theory, then follow with a consideration offree-volume mobility connections, and continue with some comparisons of exper-iment versus theory. Finally, we propose and sketch modifications of the theory.These may open the way to generalizations and more insightful relations to empiri-cal formulations, such as the KAHR model [Kovacs et al., 1977, 1979] for volumerelaxation.

4.2 LATTICE–HOLE MODEL

The lattice–hole model is discussed at length in Chapter 6, so a brief recapitulationof assumptions and results will suffice. As implied by the name, a polymer melt isrepresented by a lattice with sites occupied by chain segments or, as the case may be,by solvent-type small molecules. The melt is distinguished from the perfect crystalby the existence of a fraction h = 1 − y of holes or vacancies. These holes provideadditional disorder and entropy. Their number fraction is a measure of free volume.In what follows it is viewed as a free-volume quantity and referred to simply assuch. What is required, then, is the volume-dependent part of the Helmholtz energyF[V,T,y(V,T)], which also contains a free volume–related quantity, y. F is derived froma partition function evaluated by the methods of statistical thermodynamics containingthree contributions [Simha and Somcynsky, 1969]. First is the combinatory entropy,arising from the mixing of chains and holes. Second, there is a free-volume quantityvf internal to the occupied sites and obtained by averaging linearly over modes ofmotion under solid and gaslike conditions. These involve the number 3c of external,volume-dependent degrees of freedom. For flexible chains, these degrees of freedomare essentially those associated with internal segmental motions. Third, there is thecontribution from the cell potential, that is, the potential energy arising from theLennard-Jones interactions, assumed to be 6–12, between nonbonded segments. Abasic simplification is a mean-field approximation. It permits a given segment toexecute its thermal motions in a field defined by placing surrounding segments intomean positions (rather than letting them fluctuate) defined by the lattice.

Obtaining F in this manner, the first matter is computation of the free-volumefraction h = 1 − y. At equilibrium this is done by the minimization of F at the V andT specified: (

∂F

∂y

)V,T

= 0 (4.1)

The second matter is the PVT relation, the equation of state:

−P =(

∂F

∂V

)T

(4.2)


Combining Eqs. (4.1) and (4.2) yields the final equation of state. It is useful to scalethe variables of state and the resulting thermodynamic functions, such as the internalenergy. With the explicit expressions at hand, for sufficiently long chains this lendsitself to an effective principle of corresponding states, with the scaling parametersdefined as follows for an s-mer:

P∗ = qz�∗

sv∗ T ∗ = qz�∗

ckV ∗ = Nsv∗ (4.3)

Here �* is the maximum intersegmental attraction energy, v* is the intersegmentalrepulsion volume, and qz = s(z − 2) + 2 represents the number of intersegmentalnonbonded sites per chain of the s-mer. These parameters are obtained by fits of thetheoretical equation of state to the experimental PVT surface of a given system. Theyare available for more than 50 polymer melts [Rodgers, 1993].

These considerations apply to a polymer (or oligomer) equilibrium melt.Analogous information is required for the steady-state glass, that is, for rapid experi-mentation in which recovery is not measurable. In view of the quantitative success ofthe lattice–hole theory for the melt, we wish to retain it for the glass, with appropriatedistinctions between the two states. Clearly, Eq. (4.1) must be abandoned. Making thisby assumption the single modification and retaining the scaling parameters, we canwrite the reduced-pressure equation, indicated by tildes, as [McKinney and Simha,1976]

−P =(

∂F

∂V

)T ,y

+(

∂F

∂y

)V ,T

(∂y

∂V

)T

(4.2a)

For a given experimental pressure (or corresponding reduced pressure) and introduc-ing the explicit expressions for the two free-energy derivatives, a partial differentialequation for y ensues. Its solution provides the V, T dependence of the free volume.It will be noted that y does not freeze at Tg but only exhibits a reduced temperaturecoefficient [McKinney and Simha, 1976]. In subsequent discussion, as in most appli-cations, an approximation has been used that avoids the complications attendant onto Eq. (4.2a). That is, the first term on the right-hand side has been used togetherwith an experimental pressure to solve for y as an adjustable quantity. A compari-son with the result obtained from the full Eq. (4.2a) for the case of two poly(vinylacetate) (PVAc) glasses shows this to be an acceptable approximation not too farbelow Tg .

4.3 FREE VOLUME AND MOLECULAR MOBILITY

Volume relaxation is considered to be determined by free-volume relaxation. Theagents of this process, segmental conformational changes, are therefore to be coupledto local free-volume states. These states are therefore to be viewed as mobility mea-sures. Thus, we have the connections of mobility → rate → free volume f = h. (The

FREE VOLUME AND MOLECULAR MOBILITY 165

theoretical hole fraction, h, is only a measure of the free volume, not identical to it[Utracki and Simha, 2001], although as mentioned above, h is treated here as the freevolume, with a correcting scaling factor below.) The local molecular rearrangementsto be considered involve a few chain backbone units in a dense environment of similarunits, altogether a number Ns of about 20 to 40 units, forming a rearrangement cell.In such a cell, there are assumed to be i = 0, 1, 2, . . ., n possible states of free-volumesize, where n is a sufficiently large number so that states of equal or larger free-volumesize are very improbable. We note at this point that we assumed discrete states as amatter of convenience. We revert to this point again in what follows.

The literature offers empirical expressions that relate free volume to relaxationtimes. In particular, we refer to the Vogel and Williams–Landel–Ferry (WLF) rela-tions derived from fluidity measurements. These macroscopically defined equationsprovide relaxation rates (i.e., reciprocal relaxation times, τ) as functions of temper-ature. We can convert these to functions of free volume, f, or lattice–hole fraction,h. Due to the essentially linear dependence of h on T, the mathematical form of theoriginal equation is preserved, and thus one has [Robertson, 1992]

τ−1 = τ−1g exp

(2.303c1

f − fg

c2f ∗/T ∗ + f−fg

)(4.4)

with two parameters, c1 and c2. The subscript g refers to the glass transition tem-perature. The decisive step taken is to apply Eq. (4.4) on the microscopic scale, butincorporating a scaling factor R. Microscopic scale implies a rearrangement cell, asdescribed above, formed by Ns units. At this scale, thermal fluctuations can be signifi-cant, and Eq. (4.4) involves average values of τ and f. The mean-squared free-volumefluctuations are given by

⟨δf 2

⟩= kT

(∂2F

∂y2

)V,T

= F (y, V , T )

Ns

(4.5)

with F a function derived from the free-energy expression, both above and below Tg

[Robertson et al., 1984; Robertson, 1992]. With the mean 〈f 〉 obtained from f = 1 − y,or as shown below, a two-parameter binominal size distribution, or a fitted Gaussiandistribution, can be defined, leading to a corresponding relaxation time distribution.

We proceed by introducing a free-volume unit β and discrete free-volume parcelsof size fj = jβ for 0 ≤ j ≤ n. For a binomial distribution with two parameters, unitprobability pr , and number of states n + 1, we have

〈f 〉 = nprβ (4.6)⟨δf 2

⟩= npr(1−pr)β

2 (4.7)


Combining these two equations gives for β,

β =⟨δf 2

⟩〈f 〉 + 〈f 〉

n(4.8)

(pr is also obtained, but we make no further use of this quantity.)

4.4 STOCHASTIC FUNCTIONS AND MASTER EQUATIONS

For the development of time dependencies, two probability functions are introduced.One, wi (t), is the probability of free-volume state i at time t. Thus, the mean freevolume at time t is given by

〈f (t)〉 =n∑

i=0

fiwi(t) (4.9)

The second is a transition probability, Pij (t − t0), which as written here is the prob-ability of a transition to state jβ at t from state iβ at t0. The connection between thetwo functions is established by the following equation:

wj(t) =n∑

i=0

wi(t0)Pij(t − t0) (4.10)

Further, a relation between transitions at times t and t + tδ can be defined for any timeinterval tδ ≥ 0:

Pij(t + tδ) =n∑

k=0

Pik(t)Pkj(tδ) (4.11)

which is the Chapman–Kolmogorov relation. Differentiation of Eq. (4.11) withrespect to tδ, in the limit tδ → 0, yields the set of differential equations

dPij

dt=

n∑k=0

Pik(t)Akj(tδ) Akj = limtδ→0

dPkj(tδ)

dtδ(4.12)

For sufficiently small time intervals, we consider only transitions between adjacentstates:

Pk,k±1 = tδλ±k + O(t2

δ ) Pk,k = 1−tδ(λ+k + λ−

k ) (4.13)

with λ±k being the up and down rates from state k.

STOCHASTIC FUNCTIONS AND MASTER EQUATIONS 167

The free-volume dependence of the λ±k ’s is derived from Eq. (4.4). The explicit

dependencies are [Robertson, 1992]

λ−k = Rτ−1

g

(ξk−1

ξk

)1/2

β−2 exp

(2.303c1

f k−fg

c2f ∗/T ∗ + f k−fg

)(4.4a)

and

λ+k = λ−

k+1ξk+1

ξk

(4.4b)

The latter relationship between the two rate parameters and the equilibrium state occu-pancies ξk arises from the prescription of detailed balancing (i.e., the numbers of unitsentering and leaving free volume state k need to be equal at equilibrium). In Eq. (4.4a),R is a constant parameter that adjusts the global kinetics to that locally and compen-sates numerically for the other terms in the front factor. The factor (ξk−1/ξk)1/2 isplaced in Eq. (4.4a) so that it enters both the upward and downward rates similarly.The factor β−2, where β is the step between adjacent free-volume states, arises fromthe discrete random walk representation of a continuous diffusion process. For relax-ation and associated molecular rearrangements, an enlarged region has been assumedto be controlling, and the free-volume function for the kth state of this is written asf k and is given by

ζf k = kβ + (ζ−1) 〈f 〉 (4.14)

f k includes the region of interest itself, of Ns monomers, with free volume fk = kβas well as ζ neighbors of similar size. The ζ neighbors, with ζ often set equal to 12,are assumed together to have an average free volume equal to the global average 〈f 〉.This mean-field treatment is assumed because of the relatively large size of the region.

In the following, λ−k [Eq. (4.4a)] will be written as

λ−k = H(ξk−1/ξk)1/2β−2g(kβ) (4.15)

where

H = Rτ−1g exp(2.303c1) (4.16a)

and

g(kβ) = exp

(− 2.303c1c2f

∗/T ∗

c2f ∗/T ∗ + f k−fg

)(4.16b)


With all this said and done, we arrive at the final rate equations for the transitionprobabilities:

dPi0/dt = λ−1 Pi1 − λ+

0 Pi0

dPij/dt = λ+j−1Pi,j−1 − (λ+

j + λ−j )Pij + λ−

j+1Pi,j+1, 2 ≤ j ≤ n − 1

dPin/dt = λ+n−1Pi,n−1 − λ−

n Pin (4.17)

4.5 COMPARISONS WITH EXPERIMENTS

4.5.1 Single Temperature Steps

The comparison with volume recovery data requires the input of two sets of parametervalues. These are given in Table 4.1. First, there are the scaling quantities P*, V*,and T*, retrieved from equation-of-state measurements. As stated previously, theseare amply available. Second, there are the time–temperature shift parameters c1 andc2, obtainable from viscosity or other dynamic relaxation data. The use of these twoconstants resulted in satisfactory agreement for PVAc, considered below, at shorttimes. This is when only the larger free-volume states of the distribution are engagedprimarily in the recovery process. At longer times and the involvement of smallerfree-volume states, the recovery predicted was too slow. Associating this distinction

TABLE 4.1 Parameters for Volume Recovery Kinetics of PVAc

Simha–Somcynsky characteristic parametersP* = 9380 bara

V* = 814.1 mm3/ga

T* = 9419. Ka

Time–temperature shift parametersc1 = 12.81b

c2 = 28.74 Kb

C1 = 11.24c

C2 = 45.96 Kc

Additional parametersTg = 308 K Glass transition temperatureτg = 1 h (3600 s) Nominal relaxation time at Tg

τb = 36,000 sc Relaxation time of sub-Tg motion at Tg

Ns = 26 Number of monomer segments in free-volume transition regionz = 13 Size ratio for region controlling free-volume changesR = 0.0022 Translation factor between macroscopic and microscopic processes

a McKinney and Simha [1974].b Plazek [1980].c Robertson [1985].

COMPARISONS WITH EXPERIMENTS 169

with higher and lower temperature dynamic data, a second set of shift parameters C1and C2 was introduced. It was suggested that they were needed to account for sub-Tg motions. This results in the viscosity or relaxation time rising less rapidly withdecreasing temperature at low temperatures than would be suggested by extrapolationof the WLF curve obtained from above Tg . C1 and C2 were obtained by fitting to oneset of Kovacs aging data [Robertson, 1985].

In addition, there is the input of Tg , τg , and τb , corresponding to the sub-Tg

motions. Also needed are numbers for Ns , the number of units in the rearrangementcell, and for R, the scaling factor from macro to micro behavior.

Now consider the pioneering experiments of Kovacs [1963] on PVAc. In theseexperiments the temperature was stepped rapidly from an initial value T0 to a finalvalue T1. Either of these two temperatures may be in either the glassy or melt state.The Tg of the samples is taken as 35◦C. In Figure 4.2, specimens at different initialtemperatures are stepped downward to Tg − 5◦C (30◦C). Some are stepped from amelt state (T > Tg) and others from a glassy state (T ≤ Tg) to a different glassystate. In Figure 4.3, specimens at different initial temperatures are stepped upwardto Tg + 5◦C (40◦C); and in Figure 4.4, specimens at initial temperatures above andbelow Tg are stepped to Tg.

To obtain theoretical curves for comparison with Kovacs data, the rate equationsgiven in Eqs. (4.17) must be solved using the parameters for PVAc given in Table4.1. Because Eqs. (4.17) are coupled to one another, they need to be solved together.This was done by expressing all of the equations in Eqs. (4.17) as a single matrixequation. After some manipulation, a square, symmetric matrix was obtained onthe right-hand side, and this was diagonalized to give a set of independent decayequations with a spectrum of decay rates. Further details of this procedure have beengiven by Robertson [1992]. As can be seen in Figures 4.2, 4.3 and 4.4, satisfactory,

FIGURE 4.2 Relative deviation of volume V(t) from equilibrium V1 versus time for PVAcspecimens after being stepped downward to Tg − 5◦C (30◦C) from different initial temperatures,where specimens had been in equilibrium. (Adapted from Robertson et al. [1984]; data fromKovacs [1963].)


FIGURE 4.3 Relative deviation of volume V(t) from equilibrium V1 versus time for PVAcspecimens after being stepped upward to Tg + 5◦C (40◦C) from different initial temperatures,where specimens had been in equilibrium. (Adapted from Robertson et al. [1984]; data fromKovacs [1963].)

FIGURE 4.4 Relative deviation of volume V(t) from equilibrium V1 versus time for PVAcspecimens after being stepped to Tg from initial temperatures above and below Tg, where speci-mens had been in equilibrium. (Adapted from Robertson et al. [1984]; data from Kovacs [1963].)

although not perfect agreement is obtained between experiment (symbols) and theory(lines).

4.5.2 Multiple Temperature Steps

We continue by turning to multiple temperature steps [Kovacs, 1963]. In such situ-ations, extrema in volume are observed as the system evolves toward equilibrium.Table 4.2 lists the histories to be explored [Robertson et al., 1988]. Experiment 1 isa direct down-step from the initial (T0) to the final temperature (T1), with no detour.


TABLE 4.2 Double-Step Experiments

Experiment T0 (K) Ti (K) ti (105 s) T1 (K)

1 313 — — 3032 313 283 5.75 3033 313 288 5.0 3034 313 298 3.25 303

The next three experiments involve detours: down-steps from the same temperatureT0 to different subglass temperatures Ti , which after different resident times, ti , aretaken to the final temperature T1.

The comparison between theory and experiment can be seen in Figure 4.5. Theparameter values used for the theory were those already employed in the figures forthe single-step analyses and are given in Table 4.1. The abscissa in the graphs is thetime elapsed since the final step was made to T1, whether from T0 or from Ti . Thequality of the fit between theory and experiment is rather modest. At least, maximaare seen for the multistep experiments, their sequential ordering is in accord withobservation, and all of the theoretical curves converge at large times to equilibrium,the horizontal line. Also, the fit for experiment 1, the direct step from T0 to T1, issatisfactory.

The discrepancy between prediction and observation in Figure 4.5 starts with thevolume predicted right at the beginning of recovery at T1, after stepping from Ti . Thisis particularly noticeable with the two lower Ti values, of experiments 2 and 3. Thevolumes predicted are too low and too far from the eventual equilibrium volume atT1. This then seems to cause the maxima calculated to deviate from experiment in

FIGURE 4.5 Relative deviation of volume V(t) from equilibrium V1 versus time for PVAcfollowing temperature step sequences listed in Table 4.2. (Experiment 1 was a direct downstepfrom the initial, T0, to the final temperature, T1.) The abscissa is the time elapsed since the finalstep was made to T1, whether from T0 or from Ti . (Adapted from Robertson et al. [1988]; datafrom Kovacs [1963].)


FIGURE 4.6 Same experiment as in Figure 4.5 with the theory adjusted for retarded motionin the glass at low temperatures. (Adapted from Robertson et al. 1988; data from Kovacs[1963].)

two ways. The maxima appear too late, and their heights are reduced. The fact that theearly volumes calculated for experiments 2 and 3 are too low in Figure 4.5 suggeststhat the response calculated from the residence at Ti was too great, with the volumesfalling too far. A possible explanation is that the effect of thermal agitation needs tobe expressed explicitly, and distinguished from the effect of structure. In the theorypresented so far, only the effect of structure (free volume) is assumed to control therelaxation kinetics. But as the temperature falls, thermal agitation and its effect onthe kinetics decrease. To unbundle the thermal agitation effect from the structure (freevolume), an activation enthalpy was introduced. Further details of how this was donehave been given by Robertson et al. [1988]. The result of these changes producedthe comparison of theory and experiment shown in Figure 4.6. By slowing the fallin volume at low temperatures, the match between theory and experiment was muchmore satisfactory.

4.5.3 Stepwise Cooling

Vleehouwers and Nies have continued these theoretical lines by employing two mod-ifications [Vleeshouwers and Nies, 1992; Vleeshouwers, 1993]. One concerns theequation of state in the lattice–hole theory. Instead of the site fractions used by Floryfor the entropy of mixing and then in the other terms of the free energy, they employedthe expressions resulting from Huggins’ use of contact fractions. We recall that thetwo equations become formally identical for a monomer fluid or for the coordinationnumber ζ tending to infinity. (In the calculations described above, ζ was assumed tobe 12.)

Their second and main concern was the effect of the glass formation history onphysical properties. We illustrate this by examining the effect of cooling rate on Tg

and the glassy volume. The basic idea is sketched in Figure 4.7, where the free volumeis plotted as a function of time. The solid line represents the equilibrium volumes, as


Time(a)

Free

vol

ume

(b) Time

Free

vol

ume

FIGURE 4.7 Effect of cooling rate on Tg and the glassy volume. Shown is free volume versustemperature for (a) slow cooling and (b) faster cooling. Solid lines represent equilibrium, from astepwise reduction of temperature. Dashed lines depict schematically the trend of experimentalbehavior. (Adapted from Vleeshouwers [1993].)

computed for a reduction in temperature for a given cooling rate in multiple steps. Thedashed lines depict schematically the trend in experimental behavior. At sufficientlyelevated temperatures, there is adequate time for the system to achieve equilibriumat each step. At lower temperatures, experiment continues increasingly to lag behindprediction as cooling proceeds (Figure 4.7a versus b). Figure 4.8 shows the volumeof PVAc as the cooling rate was increased in steps over a range of 104. The slopechange (Tg) occurs at increasingly higher temperature as the rate is increased.

4.5.4 Pressure Steps

In addition to temperature, pressure can also be introduced as a variable. Three typesof volume-recovery experiments can be considered in which the pressure is dif-ferent from, or is not maintained at, 1 bar. In the first, the temperature is steppedup or down at constant pressure but for pressures exceeding 1 bar; in the second,the pressure is stepped at constant temperature; and in the third, volume recoveryis examined for densified glasses formed by compressing the liquid, cooling it tobelow Tg, and then releasing the pressure [Robertson et al., 1985]. For the first two


FIGURE 4.8 Increase in Tg and glassy volume of PVAc as the cooling rate is increased overthe range of 104. (Adapted from from Vleeshouwers [1993] and Vleeshouwers and Nies [1994].)

experiments, polystyrene (PS) was used as the example material, allowing comparisonwith the experimental results [Rehage and Goldbach, 1966; Goldbach and Rehage,1967; Rehage, 1970; Oels and Rehage, 1977].

To compute the recovery kinetics of PS under different histories of pressure andtemperature, various material parameters are needed, those given in Table 4.3 andothers described below. The first three quantities in Table 4.3 are the S-S characteristicparameters for the pressure–volume–temperature properties of liquid PS. From these,all of the desired PVT properties and the free-volume functions for the liquid can bederived. For the glass, however, further PVT properties are needed. These include Tg

as a function of pressure, Tg(P), the thermal expansivity of the glass as a functionof pressure, αglass(P), and the bulk modulus of the glass as a function of pressure,Bglass(P). For use in the calculations, these data can be used in the form of empiricalequations. (Further details can be found in the Appendix of Robertson et al. 1985).Other quantities in Table 4.3 are the nominal glass transition temperature, Tg, the time-temperature shift parameter of the liquid as a function of temperature and pressure,aT,p , and the transition region size parameters Ns and ζ.

TABLE 4.3 Parameters for Recovery Kinetics of PS

P* = 7453 bara S-S characteristic parametersV* = 959.8 mm3/ga

T* = 12,680 Ka

cl = 13.3b time–temperature shift parametersc2 = 47.5 Kb

Tg = 373 Kτg = 1 h (3600 s) nominal relaxation time at glass transitionNs = 40 number of segments in free-volume regionζ = 12 size ratio for region controlling free volumeR = 5.3 translation factor between macroscopic and microscopic processes

a Quach and Simha [1977].b Plazek [1965].


We first show the comparison of theory with Rehage and Goldbach’s volumerecovery data from temperature steps for PS at 1 bar. These data are a companionto volume-recovery data from pressure steps, described below. The computation ofvolume recovery for PS is the same as that described above for PVAc. The polymer isassumed to be in equilibrium at the initial temperature T0, and then at time t = 0, thetemperature is suddenly stepped to the final temperature T1. The data of Rehage andGoldbach [Rehage and Goldbach, 1966; Goldbach and Rehage, 1967] are shown inFigure 4.9 for steps from various initial temperatures, T0, to the final temperature, T1,of 90.70◦C. The thermally polymerized PS examined had a number-average molec-ular weight of 500 kg/mol. The results basically agree with those of Kovacs [1958],Hozumi et al. [1970], Uchidoi et al. [1978], and Adachi and Kotaka [1982]. Theordinate in Figure 4.9 is the relative deviation of the volume from equilibrium; thevolume difference in the denominator, (V0 − V1), is that existing immediately afterthe step in temperature. The curves computed, given by the solid lines, were movedalong the time axis until a reasonable fit with the data was obtained. The fit to thedata in Figure 4.9 is fairly good and yields the parameter of R in Table 4.3. [The largedifference between the value of R for PS (5.3) and that used previously for PVAc(0.0022) is believed to arise largely from different relaxation times at the assumedglass transition temperatures.]

The computation of volume recovery following pressure steps at constant tem-perature is analogous to that described above for temperature steps under constant

FIGURE 4.9 Relative change in volume versus time of PS specimens after temperature stepsof various magnitude at 1-bar pressure to 90.70◦C. (V0, initial volume; V1, volume at 90.70◦C.(Adapted from Robertson et al. [1985]; data from Rehage and Goldbach [1966] and Goldbachand Rehage [1967].)


pressure. The polymer is assumed to be in equilibrium at the initial pressure p0and temperature T. Then at time t = 0, the pressure is suddenly stepped to the finalvalue p1 without change in temperature. Rehage and Goldbach have measured thevolume recovery following pressure steps from several elevated pressures down toatmospheric pressure at the temperature of 91.84◦C [Rehage and Goldbach, 1966;Goldbach and Rehage, 1967]. Their data are shown in Figure 4.10 along with thepredictions computed. The parameters in the computation were the same as thoseused for the volume recovery following temperature steps discussed above, includingthe value of R obtained from fitting those data.

The fit between experiment and computation in Figure 4.10 is only approximate.One feature reproduced in the calculation is the degree of spread between the datain Figures 4.9 and 4.10. Rehage and Goldbach drew particular attention to the muchlarger spread in the temperature-step than in the pressure-step data. This differenceis also predicted by the calculation. However, it is unclear why there is not betteragreement between theory and experiment. In contrast to the assumptions of thecomputation, the data seem to suggest that recovery from pressure steps is basicallydifferent from recovery from temperature steps. Although recovery from the pressuresteps occurred at a higher temperature (91.84◦C) than recovery from the tempera-ture steps (90.7◦C), the recovery from the pressure steps is slightly slower. Lackingthe usual S-shape slowdown as equilibrium is reached, however, the recovery fromthe pressure steps appears to be complete at nearly the same time as that from thetemperature steps.

FIGURE 4.10 Relative change in volume versus time of PS specimens after pressure steps ofvarious magnitude at 91.84◦C to 1 bar. V0, initial volume; V1, volume at 1 bar. (Adapted fromRobertson et al. [1985]; data from Rehage and Goldbach [1966] and Goldbach and Rehage[1967].)


FIGURE 4.11 Paths for attaining equilibrium volume below the glass transition temperatureat atmospheric pressure with densified glass. (Adapted from Robertson et al. [1985].)

Densified glasses allow further exhibitions of structural recovery. For example,one can consider the following question: Is it possible to reach rapidly an equilibriumliquid state below the glass transition temperature at atmospheric pressure by pres-surizing the liquid, cooling it below the glass transition, and then depressurizing it?This sequence of steps is shown schematically in Figure 4.11. The liquid at A is pres-surized to B, cooled through the glass transition at C to D, and then depressurized toE, the equilibrium volume at that temperature at atmospheric pressure. An alternativepath, indicated by the dashed line, would be to cool the pressurized glass to roomtemperature at D′ before releasing the pressure and then heating the depressurizedglass from D′′ to E. It would seem that for either path, the equilibrium volume atE would be more quickly reached in this way than if the glass had been cooled atatmospheric pressure, along the upper, glass line to the temperature of D and E, andallowing it to recover to E.

Although the densified glass at E in Figure 4.11 has the equilibrium volume, wouldit be in equilibrium? Various indirect experiments suggest that it would not. Forexample, Oels and Rehage [1977] found that all of their densified glasses, producedunder pressures up to 5000 bar, tended at 22◦C to expand with time. Yet one ofthese densified glasses had a volume very close to equilibrium at 22◦C immediatelyfollowing depressurization.

Since we are generally assuming that the total volume can be divided into filled andunfilled space, specimens of the same material having the same volume at the sametemperature and pressure are assumed to have the same free volume. However, there


FIGURE 4.12 Development of the volume in time at 86, 90, and 94◦C for a glass having theequilibrium volume and free volume at 90◦C and a mean-squared fluctuation in free volumeassumed to be 15% smaller than equilibrium. (Adapted from Robertson et al. [1985].)

is a parameter other than volume and average free volume that is used by the kinetictheory to describe the structure of the glass, and that is the free-volume distribution.Therefore, in the following, only the free-volume distribution will be assumed not tobe in equilibrium on arrival at E. There is still the question of whether the distributionhas the symmetry of the equilibrium distribution. We assume that it does and consideronly two distortions of the equilibrium free-volume distribution. We suppose that thetemperature at which the PS is depressurized, corresponding to point E, is 90◦C andthe breadth of the free-volume distributions for 40 monomer-size regions is roughly8% narrower or 8% broader than the equilibrium distribution. These distributionshave mean-square fluctuations smaller and larger by 15% than at equilibrium. Thetime developments of the volume for these distributions are shown in Figures 4.12and 4.13.

If maintained at 90◦C, the densified glass with the narrower free-volume distribu-tion is seen in Figure 4.12 to go through a maximum before returning to the initial(and equilibrium) volume. The reason for the maximum is that the higher free-volumehalf of the distribution moves up toward equilibrium before the lower free-volumehalf moves down. This causes the average free volume to go through a maximum,and hence so does the total volume. Later, the lower half of the initial distributionwill move downward to bring the entire distribution into equilibrium. In contrast, thedensified glass with the broader free volume is seen in Figure 4.13 to go through aminimum. This occurs in like manner because the upper half of the initial distributionmoves downward toward the equilibrium distribution before the lower half of the


FIGURE 4.13 Development of the volume in time at 86, 90, and 94◦C for a glass having theequilibrium volume and free volume at 90◦C and a mean-squared fluctuation in free volumeassumed to be 15% larger than equilibrium. (Adapted from Robertson et al. [1985].)

initial distribution moves upward. An equilibrium distribution, of course, would nothave changed.

Two other curves are shown in Figures 4.12 and 4.13. These assume that afterdepressurization at 90◦C, the temperature is moved up or down by 4◦C before recoverybegins. (The initial displacements of these curves from the curves for 90◦C arise fromthe thermal expansion of the glass.) If the free-volume distribution of the densifiedglass is narrower than at equilibrium, a maximum is superimposed on the volumecurve at the beginning or the end of the transition, whichever is at the higher volume.But the height of the maximum decreases as the temperature differs from that atdepressurization. In contrast, if the free-volume distribution of the densified glassis broader than at equilibrium, a minimum is superimposed on the volume curve atthe beginning or end of the transition, whichever has the smaller volume, althoughthe minimum has essentially disappeared for the downstep to 86◦C in Figure 4.13.Although either a narrower or a broader distribution of free volume than at equilibriumseems possible for densified glasses, the results of Oels and Rehage [1977], and ofKogowski and Filisko [1986], suggest that densified PS has a narrowed free-volumedistribution.

In the computation of the volume–recovery curves in Figures 4.12 and 4.13, onlythermal agitation, or Brownian motion, was assumed for the driving force for recovery.Stress field effects were not taken into account. However, the release of pressure fromthe densified glass is equivalent to an application of a negative pressure (i.e., an


expansion) to the stable mechanical system that had existed under the pressure, andthis stress application could be a further driving force for the expansion of the glass.

Vleeshouwers and Nies have extended this work somewhat to include both temper-ature and pressure in the formation of glasses [Vleeshouwers and Nies, 1992, 1994,1996]. Their work follows along lines similar to the above. Vleeshouwers and Niesfound that at higher pressures, the free volume alone, as in Eq. (4.4), is no longersufficient to describe mobility. However, by adding temperature as a second indepen-dent variable, in addition to free volume, they were able to describe the results ofexperiment satisfactorily.

4.6 CONTINUUM LIMIT OF FREE-VOLUME STATES AND THERATE EQUATIONS

4.6.1 Derivation

The discussion and results presented so far were based on discrete free-volume statesand thus on discrete sets of state and transition probabilities, and corresponding dis-crete coupled differential equations for their time dependencies. We now wish toproceed to the continuum limit, leading to partial differential equations [Simha andRobertson, 2006]. There are several motivations for attempting this modification.First, there was the earlier computational necessity for short time intervals, whichmight be avoided with the continuum model. Second, the approach may serve asa starting point for generalizations of the relaxation condition. Consider, for exam-ple, physical aging under mechanical stress. A linear theory of glassy elastic modulias functions of temperature based on the lattice hole model exists [Papazoglou andSimha, 1988]. It would be of interest to combine it with the kinetic theory. Finally,a comparison of results from the stochastic theory with empirical rate equations isdesirable.

The discrete quantities that need to be taken to the continuum limit are wk (t),Pik (t), λ±

k , ξk , and g(kβ). The corresponding continuous quantities are written asw(x,t), P(x|x′,t), λ±(x), W∞(x), and g(x). [W∞(x) is written in this form to showthat the equilibrium distribution is assumed to be attained when t → ∞.] Here, themagnitude of the free volume, f, is written as x or x′. Thus, for the two probabilityfunctions for regions of Ns monomer units, w(x, t) and P(x|x′, t), represent respectively,a free-volume state x at time t and the probability of passing from a free-volume statey at t = 0 to a state x at t. The relationship between these two, corresponding toEq. (4.9), is

w(x, t) =∫

P(x∣∣x′, t)w(x′, 0) dx′ (4.18)

The continuous variables are chosen to equal the discrete quantities when x = f = kβfor k = 0, 1, . . ., n. Although the time dependence is not shown explicitly, λ±(x) andg(x) can vary slightly with time because of the dependence of the global average freevolume 〈f 〉 on time.

CONTINUUM LIMIT OF FREE-VOLUME STATES AND THE RATE EQUATIONS 181

Being continuous, the new quantities can be expanded in a Taylor’s series forneighboring points; for example:

w[x = (k ± 1)β, t] = w(x = kβ, t) ± β∂w

∂x

∣∣∣∣x=kβ

+ β2 ∂2w

∂x2

∣∣∣∣x=kβ

+ O(β3)

(4.19a)

P[x = (k ± 1)β |iβ, t ] = P(x = kβ |iβ, t ) ± β∂P

∂x

∣∣∣∣x=kβ

+ β2 ∂2P

∂x2

∣∣∣∣x=kβ

+ O(β3)

(4.19b)

Equation (4.19b) and the corresponding expansions for λ±k±1 can be used in Eq. (4.17)

to give

∂P

∂t= −β

∂

∂x[P(λ+ − λ−)] + 1

2β2 ∂2

∂x2 [P(λ+ + λ−)] (4.20)

This is a Fokker–Planck–Kolmogorov equation. With the substitutions

a(x) = β(λ+−λ−)

2D(x) = β2(λ+ + λ−)

the Fokker–Planck–Kolmogorov equation becomes

∂P

∂t= ∂2

∂x2 [D(x)P]− ∂

∂x[a(x)P] (4.21)

From Eqs. (4.4b) and (4.15), we have for the discrete jump rate variables,

(λ+k ± λ−

k ) = Hβ−2

[(ξk+1

ξk

)1/2

g[(k + 1)β] ±(

ξk−1

ξk

)1/2

g(kβ)

](4.22)

Now using continuous variables, λ±, W∞, and g(x,t), again expanding them in aTaylor’s series, and keeping only lowest powers of β, we have

a(x) = β(λ+ − λ−) = H

(∂g

∂x+ dlnW∞

dxg

)(4.23a)

and

2D(x) = β2(λ+ + λ−) = 2Hg (4.23b)

where D(x,t) = Hg(x,t) is a “diffusion” function.


The (discrete) free-volume state occupancy, wj (t), can be put into the followingform from Eqs. (4.10) and (4.11):

wj(t + tδ) =n∑

i=0

wi(t)Pij(tδ) (4.24)

and taking the derivative of this with respect to tδ and letting tδ → 0 gives, with Eqs.(4.12) and (4.13),

dwj

dt= wj−1λ

+j−1 − wj

(λ+

j + λ−j

)+ wj+1λ

−j+1 (4.25)

and at equilibrium

dξj

dt= ξj−1λ

+j−1 − ξj(λ+

j + λ−j ) + ξj+1λ

−j+1 = 0 (4.26)

which can be subtracted from Eq. (4.25) to give

d(wj − ξj)

dt= (wj−1−ξj−1)λ+

j−1 − (wj−ξj)(λ+j + λ−

j ) + (wj+1 − ξj+1)λ−j+1

(4.27)

Using Taylor’s expansions for the corresponding continuous functions w(x,t), λ±(x),and W∞(x), and keeping only powers of β up to β 2, the analogous equation to Eq.(4.25) is

∂W(x, t)

∂t= −β

∂

∂x[W(λ+−λ−)] + 1

2β2 ∂2

∂x2 [W(λ+ + λ−)] (4.28)

where W(x,t) = w(x,t) − W∞. [The resemblance between Eqs. (4.28) and (4.20), withW taking the place of P, was of course expected.] Equation (4.28) can be written, withEqs. (4.23a) and (4.23b) and a little math as

∂W(x, t)

∂t= ∂

∂x

[D(x)

∂W

∂x−D

dlnW∞(x)

dxW

](4.29)

Equation (4.29) is a continuity relationship in x-space. The bracketed term representsa current. The first term is a diffusion flux. The second term is the product of aconcentration W and a velocity as the ratio of a drag force and a frictional factorkT/D. Thus, the force is kT(d ln W∞/dx) and arises from the gradient of entropy. Anequation similar to that in Eq. (4.29) has also been derived by Drozdov [1999], usinga similar approach.


One may attempt direct solutions of Eq. (4.29) or try to simplify it by transforma-tions of dependent and independent variables. We tentatively adopt the second route[Simha and Robertson, 2006] by putting

W(x, t) = W1/2∞ D−1/4U(y, t) (4.30)

and

dy = D−1/2 dx (4.31)

This yields, from Eq. (4.29),

∂U(y, t)

∂t= ∂2U

∂y2 − W−1/2∞ D−1/4 d2

dy2 (W1/2∞ D1/4)U (4.32)

with

y =∫ x

0D(x′)−1/2

dx′ (4.33)

We arrive at time-independent solutions by setting

U = e−ωtψ(y) (4.34)

and obtain

d2ψ

dy2 −[V (y) − ω]ψ = 0 (4.35)

Equation (4.35) has the form of a Schrodinger equation with the potential energyV(y), where

V (y) = W−1/2∞ D−1/4 d2

dy2 (W1/2∞ D1/4) (4.36)

A possibly convenient form for Eq. (4.36) is

V (y) = d2ln(W1/2∞ D1/4)

dy2 +[

dln(W1/2∞ D1/4)

dy

]2

(4.37)

It is of interest to note that nearly 80 years ago Furth [1933] discussed the relationshipbetween classical statistics with quantum mechanics in the Schrodinger formulation.


FIGURE 4.14 Distribution of free volume in PVAc immediately after the temperature wasstepped to 30◦C after equilibrium had been established at 40◦C (w0) and after equilibrium isattained at 30◦C (W∞). Shown is the relative fraction of states having the free volume indicatedby the abscissa (x).

4.6.2 Example Application of the Continuum Model

To apply the Schrodinger form of the continuum model, the steps involved are:(1) calculate the potential, V(y); (2) solve the Schrodinger equation for ψ and ω;and (3) transform back to the stochastic function, W(x,t). To show these steps, weconsider the example of PVAc undergoing a downward step from 40◦C to 30◦C. Thisis one of Kovac’s experiments shown in Figure 4.2.

The material is assumed to be in equilibrium at 40◦C when its temperature issuddenly (“instantaneously”) changed to 30◦C. At equilibrium at 40◦C and 1 barpressure, the average free volume is calculated to be 0.08361 according to the S-Stheory. The average fluctuation in the free-volume quantity Ns

⟨δf 2

⟩is 0.01401. When

the temperature is stepped to 30◦C, the average free volume becomes 0.08253 andthe average fluctuation in the free-volume quantity becomes 0.01358. At equilibriumat 30◦C, these will become 0.07751 and 0.01344, respectively.

The distribution in free volume immediately after the downstep from 40◦C to 30◦C,as well as the eventual equilibrium distribution at 30◦C, is shown in Figure 4.14. Thedifference between the two is shown in Figure 4.15.

Using Eq. (4.37) to compute V(y) for the initial state, which occurs just after thedownstep from 40◦C to 30◦C, we find the potential shown in four views in Figure 4.16a

FIGURE 4.15 Difference between the two curves in Figure 4.14.


to d. The potential is seen in Figure 4.16a to rise rapidly toward infinity at y = 1893.81.For y in the range between 0 and 1893.81, the potential is quite small (although notprecisely zero, as can be seen in Figure 4.16b to d). The potential also becomessingular as y approaches zero, as suggested in Figure 4.16d. This behavior of V(y), of

FIGURE 4.16 The “potential” V(y) in Eqs. (4.36) and (4.37) versus y: (a) V multiplied by10−9; (b) V; (c) V multiplied by 102; (d) V multiplied by 104.


being small between y = 0 and y = 1893.81 and effectively infinite at the end pointssuggests that the potential be treated as a square well. The solutions of Eq. (4.35)for a one-dimensional square well are well known. The solutions are of the formψ = A sin (by) [Atkins, 1986], where A and b are constants. b is determined by thecondition that by = mπ when y = ymax = 1893.81, where m is an integer. ψ is thenzero at both y = 0 and y = ymax, where the potential is essentially infinite. Puttingthese together gives the set of m eigenfunctions:

ψm(y) = Amsin

(mπ

ymaxy

)(4.38)

When these functions are used in Eq. (4.35), and assuming V(y) to be zero betweeny = 0 and ymax, the eigenvalues are found to be

ωm = m2π2

y2max

(4.39)

This allows Eq. (4.34) to be written as

U =∑m

Amsin

(mπ

ymaxy

)exp

(−m2π2

y2max

t

)(4.40)

with the Am values to be determined by the initial conditions.Although a square well is suggested for the potential V, especially in Figure 4.16a,

V is not really zero between y = 0 and y = 1893.8. Perhaps the easiest way to treatthe nonzero potential between y = 0 and y = 1893.8 is to treat it as a perturbation ofthe square well solution [Morse and Feshbach, 1953]. But even with the perturbationit can be seen that the solutions must still go to zero at the endpoints (0 and 1893.8).Since the sine functions form a basis set, even the perturbed solution can be expressedin terms of sine functions such as Eq. (4.40). Hence, the effect of the perturbationwould be absorbed into the Am values.

Continuing with the square-well potential for the initial conditions just after thetemperature step (t = 0), we can use Eq. (4.30) to obtain U from W(x,0) = w0 − W∞:

U(y, 0) =∑m

Amsin

(mπ

ymaxy

)= W−1/2

∞ D1/4W(x, 0) (4.41)

with y = ∫ x

0 D(x′)−1/2dx′. This gives the plot of U0 = U(y,0) versus y shown in

Figure 4.17. Figure 4.17a shows U for high y; the rest of the behavior is shown inFigure 4.17b and c. The inclusion of the perturbation from nonzero V(y) between theendpoints would cause the curves in Figures 4.17 to change slightly. The negative Vat high y will cause U0 in this region to increase. The positive V at midvalues of ywill cause U0 in this region to decrease. But the general shape of these curves shouldremain relatively unchanged.


FIGURE 4.17 The function U from Eq. (4.30), for t = 0, immediately after the step from 40to 30◦C, versus y: (a) 1780 < y ≤ 1893.81; (b) 1100 < y < 1893.81; (c) 0 ≤ y < 1800.

The unknown coefficients Am can then be determined by applying Fourier analysisor a Fourier transform to U. The shape of U suggests that many terms in Eq. (4.41)will be nonzero, meaning that W(x,t) decays toward zero at a number of relaxationrates:

W(x, t) = W1/2∞ D−1/4

∑m

Amsin

(mπ

ymaxy

)exp

(−m2π2

y2max

t

)(4.42)

A multiplicity of relaxation times also arose in our previous, discrete analysis[Robertson et al., 1984].

The example above is for the single downward temperature step from 40◦C to 30◦C.The general behavior described also occurs with other temperature jump experimentswith PVAc and probably all other glassy polymers slightly below their Tg values.The potential V(y) can again be treated approximately as a square well. The main


difference is that the upper singular value of y (ymax) no longer occurs at 1893.81,which is the ymax only for the step 30 to 40◦C.

We stop our calculations at this point for two reasons. First, the calculation seemsto be going in the same direction as the discrete calculation, and the final results areexpected to be qualitatively similar. Second, the calculation remains computationallyintensive, unfortunately. (None of the equations in the formulation of the Schrodingerapproach above were optimized for numerical computation, however.) Moreover, anexplicit equation of U in terms of y was not obtained. Rather, the parts in Figure 4.17are parametric plots, with both U and y being functions of x.

What seems remarkable to us is that we have been able to obtain the curves inFigures 4.14 to 4.17 using Mathematica running on a several-year-old desktop iMacwith no special computational pedigree. [For interested computer historians: TheiMac used has a PowerPC G4 (3.3) CPU, running at 1.25 GHz, with 768 MB of RAMmemory.] In contrast, the previous discrete calculations had to be solved with a largemainframe computer, preferably one having a vector processor, for diagonalizing themultielement matrices involved.

4.7 SUMMARY, CONCLUSIONS, AND OUTLOOK

The intent of this chapter has been to demonstrate the utility of the S-S definition offree volume for predicting volume relaxation in the glass. The S-S definition is notthe only one or even the most common. More common is the WLF definition, wherethe free volume is independent of temperature below Tg, and its magnitude is smallerthan that given by S-S. Using the S-S free volume, agreement between theory andexperiment for volume changes of PVAc and PS following temperature steps is quitesatisfactory. Also, agreement between theory and experiment for volume changes ofPS with pressure steps is fair, although as pointed out by Vleeshouwers and Nies[1994, 1996], for higher pressures, temperature may have to be added as anotherindependent variable besides free volume. A form of temperature and free volumetogether as independent variables was also suggested above when the glass is taken tolower temperatures even at atmospheric pressure, as in the multiple-temperature-stepexperiments.

It seems to us that further research in this area of volume relaxation can go in threedirections. The first is to make use of the mathematical tools developed for solvingthe Schrodinger equation. The example given was intended to show the nature of thesolutions, not to exhibit these tools. The continuum solutions are expected to be moreaccurate than the discrete model at both long and short times, when the relaxationbehavior is controlled by only a few states in the discrete model.

The second direction is the behavior of the elastic moduli as volume relaxationoccurs. This was foreseen by Papazoglou and Simha [1988] and was mentionedabove. It has long been noticed, at least with polycarbonate (see, e.g., [Robertson andJoynson, 1972]) that measurable changes in Young’s modulus occur with annealing(physical aging).

REFERENCES 189

A third direction, which is perhaps related to the elastic moduli, is the change inductility with physical aging. This is again strongly exhibited by polycarbonate, aglassy polymer noted for its ductility. A sign of a decrease in ductility with physicalaging is exemplified by the increase in yield stress [Robertson and Joynson, 1972]that occurs with heat aging.

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Polymer Physics (From Suspensions to Nanocomposites and Beyond) || Volume Relaxation and the Lattice-Hole Model