application of free-volume theory to self diffusion of solvents in polymers below the glass...

Application of Free-Volume Theory to Self Diffusion of Solvents inPolymers Below the Glass Transition Temperature: A Review

N. Ramesh,1 P. K. Davis,2 J. M. Zielinski,3 R. P. Danner,4 J. L. Duda4y1Dow Chemical Company, Building1381, Midland, Michigan 48667

2GE Global Research, One Research Circle, Niskayuna, New York 2309

3Intertek Analytical Services America, 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195

4Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802

Correspondence to: N. Ramesh (E-mail: [email protected])

Received 3 May 2011; revised 25 August 2011; accepted 26 August 2011; published online 28 September 2011

DOI: 10.1002/polb.22366

ABSTRACT: Theories based on free-volume concepts have

been developed to characterize the self and mutual-diffusion

coefficients of low molecular weight penetrants in rubbery

and glassy polymer-solvent systems. These theories are ap-

plicable over wide ranges of temperature and concentration.

The capability of free-volume theory to describe solvent dif-

fusion in glassy polymers is reviewed in this article. Two al-

ternative free-volume based approaches used to evaluate

solvent self-diffusion coefficients in glassy polymer-solvent

systems are compared in terms of their differences and

applicability. The models can correlate/predict temperature

and concentration dependencies of the solvent diffusion

coefficient. With the appropriate accompanying thermody-

namic factors they can be used to model concentration pro-

files in mutual diffusion processes that are Fickian such as

drying of coatings. The free-volume methodology has been

found to be consistent with molecular dynamics simulations.

VC 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym

Phys 49: 1629–1644, 2011

KEYWORDS: diffusion; free volume; glass transition; polymers

INTRODUCTION The diffusion of small molecules in poly-mers is of considerable practical importance and has beenstudied extensively. Diffusion theories based on the free-vol-ume concept have been used extensively to correlate andpredict solvent self-diffusion coefficients in rubbery polymer-solvent systems.1–6 These methods provide accurate predic-tions over a wide range of temperatures and concentrationsabove the glass transition temperature of the pure polymer.As polymer solutions are cooled over practical time scales,the rate of cooling exceeds the rate of relaxation of the poly-mer, and a nonequilibrium state referred to as the glassystate results. This phenomenon causes volume to be trappedin the polymer in excess of that expected at equilibrium.Free-volume theory presumes that this extra volume is avail-able to facilitate mass transport in the glassy state. Based onthis concept, the original free-volume theory was adapted todescribe the diffusion of a trace amount of a solvent in aglassy polymer,7,8 and it was further extended to describeself-diffusion below the mixture glass transition temperatureat finite concentrations of the solvent.9,10

Free-volume approaches to diffusion analyses are based onapproximations that relate the free volume of the polymer

above and below the glass transition temperature to thethermal expansion coefficients of the polymer. In this article,the applicability of two different free-volume approachesused to correlate and predict the self-diffusion of the solventbelow the glass is discussed. Examples include the diffusionof infinitely dilute solvent in a polymer matrix where the sol-vent-self and binary mutual diffusion coefficients are identi-cal, as well as solvent self diffusion in finite concentrationsolvent solutions that are homogeneous and glassy. If onecan describe well the diffusion at these conditions, than onecan predict mutual diffusion by introducing the appropriatethermodynamic chemical potential gradient.11

Thermal expansion coefficients are temperature dependent,but data typically do not exist to characterize this tempera-ture dependence. Consequently, most free-volume analysesrely on the implementation of constant thermal expansioncoefficients which describe both the rubbery and glassyregimes. The literature has demonstrated that some prob-lems exist in the original extension of the free-volume theoryat temperatures below the glass transition of the pure poly-mer.12,13 These problems are associated with the limitationsof assuming an average value of the expansion coefficient for

†Deceased

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the temperature range of interest which can lead to the pre-diction of negative free volume for the polymer. Methodsthat have been proposed to counter these difficulties are dis-cussed by Vrentas and Vrentas.4,10

Two distinct points of view are presented in the literatureregarding the nature of the free volume and correspondingmechanisms for sorption and diffusion in the glassy state. Bothschools of thought agree that when a polymer enters the glassystate, excess free volume is incorporated, or at least temporar-ily frozen, into the matrix. Meares14 is the first investigator tosuggest that this excess free volume consists of micro voids orholes which are fixed in the polymer and that this componentof free volume is not continuously being redistributed by thenatural thermal fluctuations. Based on this micro void conceptand the unusual shape of organic vapor sorption isotherms,Barrer et al.15 suggested that sorption in the glassy state iscaused by two concurrent mechanisms. There is the conven-tional solution of low molecular weight species utilizing theequilibrium free volume and additional ‘‘hole filling’’ adsorp-tion associated with the excess micro void free volume. Thisdual mode sorption has been the basis for numerous modelsto describe the thermodynamic adsorption and molecular mo-bility in glassy polymers.16

This review focuses not on this dual mode sorption, butrather on the concept that all the additional free-volumetrapped below the glass transition temperature (i.e., hole-freevolume) is continuously redistributed by the natural thermalfluctuations, and that slow relaxation processes eventuallylead to the free volume which is characteristic of a system atthermodynamic equilibrium. Because of the very slow volu-metric relaxation in the glassy state the enhanced free volumepersists in the system to influence observed thermodynamicsorption and molecular diffusion behavior. In addition, com-plications associated with Case II diffusion, plasticization, anda moving glassy-rubbery front encountered during the disso-lution of glassy polymer systems are beyond the scope of thisreview. We note, however, that proper application of free-vol-ume theory does describe well the diffusion characteristics atthese conditions for dissolution processes that are Fickian.17

Other non-free-volume theories such as the mode-couplingtheory18 present relationships for viscosity and matrix self-diffusion in super-cooled liquids. The relationship betweenthe results of these theories and the self-diffusion of a pene-trant in the super-cooled matrix has not been formulated.

FREE-VOLUME THEORY

According to the free-volume theory, the rate of moleculardiffusion is based on the amount of free space or free-vol-ume between the molecules and was first used to describethe diffusion in liquids. Cohen and Turnbull19 envisionedtranslational motion in a liquid consisting of spherical mole-cules to occur when a hole sufficiently large to accommodatethat molecule is formed adjacent to the molecule throughrandom thermal fluctuations. A molecule did not need toattain a specific energy to overcome an activation energybarrier as previous researchers had suggested,20–22 butcould undergo translational motion simply by jumping into a

free-volume hole arising from the continuous redistributionof free volume within the material. Cohen and Turnbulldeveloped the following relationship for the probability of amolecule finding a hole large enough to accommodate it.

D1 ¼ AO exp �c �V��

Vf

� �(1)

D1 is the self-diffusion coefficient of the molecule, V * is theminimum volume of a hole into which a molecule can jump,and c is a numerical factor introduced to account for freevolume being shared by neighboring molecules. Vf is the av-erage free volume per spherical molecule in the liquid andA0 is a proportionality constant which Cohen and Turnbullconsider to be related to the gas kinetic velocities. The free-volume concepts developed by Cohen and Turnbull todescribe the self-diffusion in a liquid have been extended todescribe self and mutual binary diffusion consisting of apolymeric species and a relatively low molecular weightsolvent.2,3

For the case of the self-diffusion coefficient, D1, of a solvent(component 1) within a polymer matrix (component 2), thefree-volume formalism may be written as:

�Vf ¼ VFH

x1=M1j þ x2=M2j(2)

D1 ¼ D01 expð�E=RTÞ exp½�ðx1V�1 þ x2nV

�2Þ=ðVFH=cÞ� (3)

VFH

c¼ x1

K11

c1ðK21 � Tg1 þ TÞ þ x2

K12

c2ðK22 � Tg2 þ TÞ (4)

Mij (i ¼ 1,2) is the molecular weight of a jumping unit, xi isthe weight fraction of species i, V�

i is the specific hole freevolume required for a diffusional step of component i, Tgi isthe glass transition temperature of component i, K1i and K2i

are the free-volume parameters of component i, ci representsthe overlap factor for the free volume of component i, andVFH is the specific hole free volume of the system. D01, isassumed to be a constant pre-exponential factor and is inde-pendent of the temperature of the system. The parameter n,is the ratio of the molar volume of the jumping unit of thesolvent to the molar volume of the jumping unit of the poly-mer. In this formulation, it is often assumed that for the dif-fusion of small molecules, the complete solvent molecule isthe solvent jumping unit, while the polymer is made up ofseveral small units that jump independently.23 E, is theenergy per mole that a molecule requires to overcomeattractive forces between it and its neighbors, T is the tem-perature of the system, and R is the gas constant. E in eq 3is different from the overall activation energy for diffusion.The overall activation energy for diffusion can be obtainedfrom an Arrhenius equation correlation of the diffusivity as afunction of temperature and can be calculated from free-vol-ume parameters.24 Techniques have been developed so thatall the parameters, (K11/c1,K12/c2,K21 � Tg1 and K22 � Tg2)can be estimated from properties of the two pure compo-nents comprising the mixture.5,6,25–28 The free-volume

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parameters are defined in terms of the thermal expansioncoefficients of the two components.2,3 This formulation isextremely useful to correlate and predict self-diffusion coeffi-cients above the glass transition temperature.

The total volume of a liquid can be considered to be com-posed of two parts, the volume actually occupied by the mol-ecules and the space or free volume between the molecules.This total free volume is the free volume that appears inmodels to describe the thermodynamic behavior of liquids.Investigators have shown that transport properties such asviscosity and molecular diffusion coefficients cannot be cor-related with this total free volume since this volume cannotbe continuously redistributed without overcoming activationenergy barriers. Vrentas and Duda followed the concepts ofKaelble29 and divided the total free volume into two types.One portion, denoted as interstitial free volume, requires alarge redistribution energy and is not implicated in facilitat-ing transport through the liquid. The remaining free-volumeallotment, which is presumed to dictate molecular transport,is termed the hole free volume and is redistributed effort-lessly. These volumes are depicted in Figure 1. The hole freevolume is defined by eq 4. The free-volume parameters inthat equation can be obtained by analysis of the temperaturebehavior of the polymer and solvent. The key to the applic-ability of the Vrentas-Duda theory is that the hole free vol-ume responsible for molecular diffusion can be obtainedfrom the correlation of viscosity measurements in the limitof zero shear rate with the Williams–Landel–Ferry (WLF)model.30 Wang et al.28 have presented an alternate methodof determining free-volume parmeters from the Sanchez-Lacombe equation of state. Lv et al.27 have invoked theSimha-Somcynsky equation of state in conjunction with posi-tron annihilation lifetime spectroscopy to predict the freevolume available for transport.

Practical application of the free volume theory via solutionof continuum mass transport equations for polymer-solvent

mixtures requires the mutual binary diffusion coefficient, D.Following the Bearman formalism31 of relating friction coeffi-cients to diffusion coefficients and further assuming that thecross-friction coefficient is well represented from the purecoefficients by a geometric mixing rule, this mutual diffusiv-ity can be calculated as follows:

D ¼ ðD1x2 þ D2x1ÞRT

@l1@ ln x1

� ��T;P

(5)

Here, D1 is the solvent self-diffusion coefficient, D2 is thepolymer self-diffusion coefficient, l1 is the chemical potentialof the solvent, and xi represents the mole fraction of thepolymer and solvent. Using free-volume theory to predict thesolvent and polymer self-diffusion coefficients, Vrentas andDuda2,3 were able to show that in certain cases for most ofthe concentration regime (0-~80% solvent by weight), the mu-tual diffusivity is well approximated by the following expres-sion:

D ¼ ðD1x1x2ÞRT

@l1@ lnx1

� ��T;P

(6)

xi is the weight fraction of either the polymer or solvent.This simplification allows calculation of the mutual diffusiv-ity from only the solvent self-diffusivity and an appropriatethermodynamic model for the binary solution. Assuming thatthe Flory-Huggins model describes the system thermody-namics, this expression can written as:

D ¼ D1ð1� /1Þ2ð1� 2v/1Þ (7)

u1 is the volume fraction of the solvent and v is the Flory-Huggins binary interaction parameter. Finally, in the concen-tration limit where the polymer-solvent mixture is infinitelydilute in solvent (f ! 0), the mutual diffusivity becomesequivalent to the solvent self-diffusivity:

D ¼ D1 (8)

In this review, we will cover the various free-volume meth-ods available for determining the solvent self-diffusion coeffi-cient (D1) below and at the mixture glass transitiontemperature. Such solvent self-diffusivities are equivalent tothe mutual diffusivity only in the limit of infinite solventdilution. However, if an appropriate nonequilibrium thermo-dynamic model is available for the glassy system, then eq 6may be used to calculate the system’s mutual diffusivity.

Diffusion Below Tg—The k ApproachAs polymer solutions are cooled, the motion of individualpolymer chains becomes so constrained that the cooling ratebecomes faster than the rate at which the polymer samplecan relax. The resulting nonequilibrium condition is referredto as the glassy state. At temperatures above the glass transi-tion temperature, the polymer chains are capable of achiev-ing equilibrium configurations, while they are incapable ofattaining equilibrium configurations below the glass

FIGURE 1 Extra hole free-volume—k approach.



transition temperature in conventional time scales.7,32 Fromthe free volume point of view, the extra free volume trappedin the polymer as it is cooled through the glass transitiontemperature is available for molecular transport and is con-sidered extra hole free volume as shown in Figure 1.

Although the rate of molecular motion prevents volumerelaxation from reaching an equilibrium state in glassypolymers, molecular motion is not eliminated in the glassystate. Using these ideas, the free-volume framework wasfirst extended to predict the self-diffusion coefficient forpolymer-solvent mixtures below the system glass transitiontemperature by Vrentas and Duda.8 In the limit of a traceamount of solvent in a polymer, the glass transition temper-ature of the mixture can be assumed to be the same as theglass transition temperature of the pure polymer. This idealeads to an extension of the free-volume formulation todescribe diffusion of a trace amount of solvent below theglass transition temperature of the pure polymer. In thelimit as x1 ! 0 the solvent self-diffusion coefficient, D1, isequal to the mutual binary diffusion coefficient D, and thefree-volume formulation for the solvent self-diffusion canbe expressed as:

D ¼ D1 ¼ D01 exp�cnV�

2

VFH2g

!(9)

VFH2g

c¼ K12

c2K22 þ k½T � Tg2��

(10)

Equation 9 assumes that the activation energy term in eq 3equals zero. This is a reasonable approximation for most sys-tems since the temperature dependence of the free-volumeterm is often orders of magnitude greater than the energyrequired for a migrating solvent to overcome the attractiveforces of its neighbors and jump into a free-volume holewhich appears due to natural thermal fluctuations. This isparticularly true near the glass transition where the temper-ature dependence of the free-volume term can often beequivalent to an activation energy of 30 kcal/mol or greater.Consequently, the approximation, E ¼ 0, is employed in allthe calculations presented in this review.

The parameters K12 and K22 are the same as defined before.The key parameter in this formulation is k, which is relatedto the change in the expansion coefficient of the polymerabove the glass transition temperature, a2, and the expansioncoefficient of the glassy polymer, a2g.

8

k ¼ 1� V02 ðTg2Þ½a2 � a2g�

K12(11)

This formulation has been extended to describe diffusion atfinite concentrations of solvent and temperatures betweenthe glass transition temperature of the mixture and the glasstransition temperature of the pure polymer.9 The polymer isassumed to possess an equilibrium structure betweenTg2,the glass of the pure polymer, and Tgm, the glass of the mix-ture. Under these conditions, the thermal expansion coeffi-

cient characterizing the change in free volume withtemperature is assumed to be the same as that above theglass transition temperature of the pure polymer. Theexpression for the free volume of a polymer-solvent mixturetherefore is the same as that for a rubbery polymer-solventmixture. At finite concentrations of the solvent and tempera-tures below the glass transition temperature of the mixture,however, the expansion coefficient describing the reductionin polymer free volume with temperature undergoes achange. Therefore, the free volume of the polymer decreasesat a different rate compared to that above the glass transi-tion temperature of the mixture. It is possible to describethe self-diffusion coefficient if appropriate expressions areavailable to estimate the free volume of the glassy polymer-solvent mixture. The free volume of a glassy polymer-solventmixture will depend on the sample preparation method,because the properties of a glassy polymer-solvent mixtureare history dependent. For a simple sample preparation his-tory, expressions have been derived for the free volume of aglassy polymer-solvent mixture.9,33

VFH ¼ x1VFH1 þ x2VFH2 þ x2V02 ðTg2Þ½1� Ax1a2�½T � Tg2 þ Ax1�½a2g � a2� ð12Þ

VFH1 ¼ K11ðK21 � Tg1 þ TÞ (13)

VFH2 ¼ K12ðK22 � Tg2 þ TÞ (14)

A simple sample preparation history is essentially one over aconventional experimental time scale, that is, not rapidquenching or extremely long times (centuries). In the devel-opment of the above equations, a linear approximation hasbeen assumed to describe the concentration dependence ofthe glass transition temperature of the mixture.

Tgm ¼ Tg2 � Ax1 (15)

A is a factor, that is, a measure of the depression in glasstransition temperature as a function of solvent weightfraction.

Diffusion Below Tg—The a ApproachOne of the problems arising from the k approach is thataccording to eq 14 the polymer contribution to the hole freevolume in an equilibrium polymer-solvent mixture becomesnegative at T < Tg2–K22. This is a consequence of theassumption that the thermal expansion coefficient of thepolymer at temperatures below Tg2 or Tgm, is a constant. Inreality, the thermal expansion coefficient of the polymer istemperature dependent. To counter this problem, a modifiedapproach was developed by Vrentas and Vrentas.4,10 Threedistinct regions are envisaged to exist as shown in Figure 2.Region 1 and region 2 are rubbery polymer-solvent mixtures.Region 1 exists above the glass transition temperature of thepure polymer, while region 2 exists between the glass transi-tion temperature of the pure polymer and the glass of themixture. Region 3 is a glassy polymer solvent mixture, andexists below Tgm, the glass transition temperature of the mix-ture. For rubbery polymer-solvent systems, expressions for



the hole free volume above and below the glass transitiontemperature of the pure polymer are given as follows:4

VFH

c¼ x1

K11


VFH2

c2(16)

VFH2 ¼ V02 ðTg2Þ½f GH2 �

ZTg2

T

ða2 � ac2ÞdT 0� ðT < Tg2Þ (17)

VFH2 is the specific hole free volume of the equilibrium liquidpolymer at any temperature, V0

2(Tg2), is the specific volume ofthe pure equilibrium liquid polymer at Tg2, fGH2 is the frac-tional hole free volume of the polymer at its glass transitiontemperature Tg2, a2 is the thermal expansion coefficient of theequilibrium liquid polymer, and ac2 is the thermal expansioncoefficient for the sum of the specific occupied volume andthe specific interstitial volume of the equilibrium liquid poly-mer. It is assumed that a2 is a constant above Tg2, but may betemperature dependent below Tg2. It is also assumed that ac2is a constant below Tg2, but drops rapidly to zero above Tg2,and that this drop is idealized as a step change at Tg2. Thequantities fGH2, ac2, and c2 can be computed as follows:

f GH2 ¼ a2K22 (18)

ac2 ¼ln

V02 ðTg2Þð1�f GH2Þ

V02 ð0Þ

h iTg2

(19)

c2 ¼ V02 ðTg2Þa2K12=c2

(20)

V�1 ¼ V0

1 ð0Þ (21)

V�2 ¼ V0

2 ð0Þ (22)

V0i (0) is the specific volume of component i at 0 K and K12

and K22 are the free-volume parameters for the polymer.This idea has also been extended for glassy polymer solventsystems, and the specific hole free volume of the system canbe calculated using the following expression.10

VFH

c¼ x1

K11


VFH2g

c2(23)

VFH2g is the specific hole free volume of the glassy polymerat any temperature below Tgm, the glass transition tempera-ture of the mixture. An expression for the specific hole freevolume of the glassy polymer can be derived by integratingthe defining equations for a2g, the thermal expansion coeffi-cient of the glassy polymer and ac2g, the thermal expansioncoefficient for the sum of the specific occupied volume andspecific interstitial free volume of the polymer. a2g and ac2gare defined in eqs. 24 and 25.

@ lnðV02gÞ

@T¼ a2g (24)

@ lnðVFI2g þ V2g0Þ@T

¼ ac2g (25)

VFH2g ¼ V2g � ½VFI2g þ V2g0� (26)

V02g is the specific volume of the glassy polymer, VFI2g is the

specific interstitial volume of the glassy polymer, and V2g0 isthe specific occupied volume of the glassy polymer. An expres-sion of the form exp (aT) ¼ (1 þ aT) is assumed to be a suffi-ciently accurate representation in the temperature range ofinterest. Using the above definitions and assumptions, theexpression for the free volume of the polymer below the glasstransition temperature of the mixture is written as,

VFH2g ¼ V02 ðTg2Þ½f GH2 �

ZTg2

Tgm

ða2 � ac2ÞdT 0

�ZTgmT

ða2g � ac2gÞdT 0� T < Tgm ð27Þ

To estimate ac2g, the assumption that (ac2g/ac2) ¼ (a2g/a2)T ¼ Tg2

is invoked.

This equation reduces to a simple form if it is assumed thata trace amount of solvent does not alter the glass transitiontemperature of the mixture significantly from the glass tran-sition temperature of the pure polymer.

VFH2g ¼ V02 ðTg2Þ½f GH2 � ða2g � ac2gÞðTg2 � TÞ� (28)

Although eq 28 extends the temperature range over whichthe polymer hole free volume is positive, it does not fullyeliminate this problem. The equivalent definition for k usingthis approach is:

k ¼ ða2g � ac2gÞ=a2� �

(29)

FIGURE 2 Extra hole free-volume—a approach. Bold solid line

represents free volume in an equilibrium structure.



Comparison of k and a Approaches withExperimental ResultsThe differences between the k approach and the a approachare compared using the polystyrene(PS)-toluene system. Thevalues of the free-volume parameters for this system areshown in Table 1. Figure 3 shows the hole free volume ofthe pure PS at various temperatures above and below theglass transition temperature of the pure polymer. Above theglass transition temperature, identical values of free volumeare obtained using both approaches. A break is observed atthe glass transition temperature because of a change in theexpansion coefficient at this point. The free volume calcu-lated using the a approach is greater than that calculatedusing the k approach below the glass transition temperature.This is a consequence of the assumption that a change in theexpansion coefficient for the sum of the occupied and inter-stitial free volume occurs at the glass transition temperature.This change is idealized as a step change at the glass transi-tion temperature.

Figures 4 and 5 compare the hole free volume of a PS-tolu-ene mixture at different weight fractions of the solvent, 5and 15 wt. % respectively. In both of these figures, the free-volume behavior is terminated at the low temperature forwhich the free volume of the polymer is calculated tobecome negative according to the k approach. As indicatedearlier, this physically impossible behavior is due to theassumption that the expansion coefficient of the polymer is aconstant and independent of temperature.

Since the addition of a solvent to a polymer depresses theglass transition temperature of the mixture, the three regionspredicted by the a approach (Fig. 2) are observed in Figure5 for the calculated free volume. Region 1 lies above theglass transition temperature of the pure polymer. Region 2exists between the glass transition temperature of the purepolymer and the glass transition temperature of the mixture,while Region 3 is located below the glass transition tempera-ture of the mixture. In this example, the depression of theglass transition temperature was accounted for by using the

approximate linear relationship given in eq 15 with A ¼ 500.This value results from plotting the approximate theory ofChow36 in the concentration range of 0–16 wt % solvent. Abreak in the hole free volume is observed at the glass transi-tion temperature of the mixture. This occurs because thereis a change in the thermal expansion coefficient of the equi-librium liquid polymer and the sum of the occupied and in-terstitial free volumes at this point. The break in free volumebecomes more difficult to detect at high solvent concentra-tions, since the contribution of the polymer free volume issmall compared to the solvent. Using the k approach the freevolume of the polymer becomes zero at T ¼ Tg2 � K22 andassumes negative values at temperatures below this value.Since negative values of free volumes are not consistent withthe free-volume theory, the free volume in this region wasset to zero. This leads to an artificial break in the graph offree volume as a function of temperature using the kapproach. The calculated free volume increases after the freevolume of the polymer is set to zero. This is an artificialeffect resulting from the approximation that the expansioncoefficients are assumed to be constant in the different tem-perature regions. If the thermal expansion coefficients areavailable as a function of temperature, they should be usedand, thus, eliminate the weakness of assuming constantexpansion coefficients. Similarly, if data are available for thedepression of the glass transition temperature with the addi-tion of solvent, they should be used. In the absence of anydata, the approximate theory developed by Chow36 may beused.

TABLE 1 Free-Volume Parameters for the Polystyrene/Toluene

System (Figs. 3–5)5,35

K11/c1 (cm3/g K) 1.57 � 10�3

K12/c2 (cm3/g K) 5.39 � 10�4

K21-Tg1 (K) �90.5

K22 (K) 50

fGH2 0.0265

V02 (Tg2) (cm

3/g) 0.972

a2 (K�1) 5.3 � 10�4

ac2 (K�1) 2.88 � 10�4

a2g (K�1) 1.6 � 10�4

ac2g (K�1) 8.7 � 10�5

A (K) 500

k 0.3

Tg2 (K) 373

FIGURE 3 Hole free-volume of the pure polymer above and

below Tg2. Bold solid line represents free volume in an equilib-

rium structure: dotted line free volume using k approach:

dashed line free volume using the a approach. Reproduced

from Ref. 34 with permission from Springer Science.



Diffusion of a tracer, tetra-hydrothiophene indigo (TTI) hasbeen studied in PS by Ehlich and Sillescu.37 Table 2 lists thevalues of the parameters used in this correlation. The valuesof a2, a2g, and Tg2 were used from the data provided byEhlich and Sillescu, while the values of D0 and n wereregressed from the data above the glass transition tempera-ture of the pure polymer. In the analysis, the glass transitiontemperature was taken as 369 K, the temperature on the Dversus T curve where a break in diffusivity is observed. Thetwo parameters D0 and n were correlated independently forthe two approaches. Their values, however, were identical.

Figure 6 compares the diffusion coefficient of this tracer,TTI in PS calculated using the a approach and the kapproach. The diffusion coefficient calculated using the twoapproaches is the same above the glass transition tempera-ture of pure polystyrene. This is consistent with the resultsof Figure 6, where no difference in the hole free volumeabove the glass transition temperature of the pure polymeris observed. A difference is observed, however, in the esti-mated diffusion coefficient below the glass transition tem-perature. The predicted value of diffusivity using the kapproach is smaller than that using the a approach, espe-cially at lower temperatures. This is a consequence of thehigher free volume predicted by the a approach when com-pared to the k approach. For diffusion of TTI in PS, the kapproach does a better job of correlating the data of Ehlichand Sillescu. At the lowest temperature, however, deviationsare seen between the correlation and the experimentalpoint for both approaches.

The diffusion of styrene in polystyrene has been measuredby Murphy et al.38 at different residual levels of styrene andat different temperatures below the glass transition tempera-ture. These experimental results have been compared withthe predictions of the two free-volume approaches. Thesedata were obtained over a small range of concentration anda relatively large range of temperature. In this analysis, thethermodynamic term relating the change in activity withchange in solvent concentration has been neglected, since in

FIGURE 4 Hole free-volume of a 5 wt % polystyrene/toluene

mixture. Bold solid line represents free volume in an equilib-




FIGURE 5 Hole free-volume of a 15 wt % polystyrene/toluene

mixture. Bold solid line represents free volume in an equilib-




TABLE 2 Free-Volume Parameters for the

Tetra-Hydrothiophene Indigo System (Fig. 6)37

D01 (cm2/s) 1.7 � 10�3 (a), 1.56 � 10�3 (k)a

V02 (Tg2) (cm

3/g) 0.972

fGH2 0.028

a2 (K�1) 5.6 � 10�4

a2g (K�1) 1.9 � 10�4

ac2g (K�1) 9.7 � 10�5

K22 (K) 50

A (K) 500

k 0.34

Tg2 (K) 369

c2 1.0

n 0.9

K12/c2 (cm3/g K) 5.44 � 10�4

E (kcal/mole) 0

a The a and k approaches result in different values for D01.



the limit of pure polymer, the thermodynamic termapproaches one. Hence, the predicted values of the self-diffu-sion coefficient are equivalent to the mutual binary diffusioncoefficient. Since free-volume parameters were not available forstyrene, free-volume parameters for ethylbenzene, a molecule ofsimilar size and structure were used. As a first approximationthe activation energy for diffusion as defined in eq 3 wasassumed to be zero. The parameters used to predict the self dif-fusion-coefficient of styrene in polystyrene were obtained fromseveral sources6,10,25 and are listed in Table 3. The free-volume

predictions for the diffusion of styrene in polystyrene using thea approach and the k approach are shown in Figures 7 and 8,respectively. The self-diffusion coefficient of styrene is a weakfunction of the concentration of styrene in the concentrationrange studied by Murphy et al.38

At low concentration levels, the agreement between pre-dicted values using the a approach and experimentally meas-ured values are good at all temperatures except 82.2�C. Atlow temperatures the difference between the predicted val-ues of diffusivity using the two approaches becomes clearlyevident, and values predicted using the k approach aresmaller than the experimentally measured values at all tem-peratures. This clearly illustrates the effect of the smallerfree volume predicted by the k approach.

The diffusion coefficient of a small molecule at high polymerconcentrations is controlled by the hole free volume of thesystem. An increase in temperature as well as an increase inthe concentration of the low molecular weight penetrant cancause an increase in the free volume of the system andhence an increase in the diffusion coefficient.

Diffusion at the Polymer-Solvent Mixture GlassTransition Temperature, TgmNumerous studies have evaluated the free-volume theoryabove and below the glass transition temperature of thepolymer. The theory has not been evaluated, however, at theglass transition temperature of a mixture, that is, as a func-tion of the solvent concentration.

The relationship between bulk viscosity, g, and probe self-diffusion, D1, most commonly cited in polymeric systems isthe WLF equation.30

FIGURE 6 Comparison of correlation for tetra hydrothiophene

indigo, TTI, in polystyrene using the a and k approaches. Bold

solid line represents free-volume in an equilibrium structure:

dotted line free volume using k approach: dashed line free vol-

ume using the a approach. Data from Ref. 37. Reproduced


TABLE 3 Free-Volume Parameters for the Polystyrene/Styrene

System (Figs. 7 and 8)6,11,25

D01 (cm2/s) 4.61 � 10�4

V02 (Tg2) (cm

3/g) 0.972

fGH2 0.0265

a2 (K�1) 5.3 � 10�4

a2g (K�1) 1.6 � 10�4

ac2g (K�1) 8.7 � 10�5

K22 (K) 50

k 0.30

Tg2 (K) 373

c2 0.956

n 0.9

K12/c2 (cm3/g K) 5.39 � 10�4

E (kcal/mole) 0

FIGURE 7 Prediction of styrene diffusion in polystyrene using

the a approach. Data from Ref. 38. Reproduced from Ref. 39

with permission from Elsevier.



log aT ¼ loggðTÞgðTgÞ¼

�C1mðT � TgÞC2m þ T � Tg

¼ � 1nlog

D1ðTÞD1ðTgÞ (30)

C1m and C2m are the WLF constants for the matrix materialand aT is a temperature shift factor. In the context of free-volume theory, the parameter n is the ratio of jumping unitof the solvent to the jumping unit of the polymer. For scien-tists examining relaxation processes, however, n is generallyidentified more closely as a decoupling parameter which dis-tinguishes between differences in the fundamental mecha-nisms of diffusion and rheology.40–43

According to free-volume theory,2,3 eq 30 is strictly applica-ble to describe solvent self-diffusion in either the pure poly-mer or pure solvent limit. For the case of trace amounts ofsolvent in an essentially pure polymer matrix, a value of ncan be determined by comparing the temperature depend-ence of the polymer viscosity to that of the self-diffusioncoefficient of the solvent molecule. In the case of the puresolvent limit, the parameter n must be equal to one, sincethe matrix and the diffusing species are identical. At concen-trations between these two extremes, free-volume theorysuggests a somewhat different relationship. We will returnto this point shortly. Equation 30 indeed holds for the major-ity of materials investigated. There does appear, however, tobe a decoupling between viscosity and diffusion for super-cooled liquids when viscosity and diffusion data are acquiredover a broad temperature range. A cross-over from n ¼ 1 ton < 1 is commonly observed in pure-component liquids.40–43

Consequently, the applicability of free-volume theory to uni-versally describe transport in liquids has been questioned.

For polymer-solvent systems, however, it appears that if thetheory is applied properly, a reasonable description of mo-lecular transport can be obtained.13,44 Further, it is arguedthat molecular interactions are captured in the WLF modelthrough changes in viscosity behavior and are translated tofree volume parameter estimations.

Although free volume has been applied to describe the tem-perature and concentration dependence of small diffusionmolecules in polymers both above4,6 and below10,33,45 thepure polymer glass transition temperature, there appearsnever to have been a direct examination of the theory’s abil-ity to describe diffusion behavior at the mixture glass-transi-tion temperature (Tgm), that is, while solvent concentrationincreases and the system temperature decreases. There havebeen numerous excellent experimental investigations of dif-fusion of tracer components in polymers by Sillescu and co-workers both by Forced Rayleigh Scattering (FRS) and field-gradient NMR. Among them are two that are of significancewith regards to diffusion at Tgm.

In a study by Lohfink and Sillescu46 FRS was applied toexamine the self-diffusion behavior of trace quantities of aphotochromic dye (TTI) in binary mixtures of polymethyl-phenyl-siloxane (PMPS) and bis (3-methoxyphenyl)cyclohex-ane (BMC) of varied composition. The principal components,PMPS and BMC, were selected to maintain a constant mix-ture glass transition temperature at all compositions, sincePMPS and BMC have the same Tg, �30 �C. The glass transi-tion temperature of the pure components and mixtures wereverified by differential scanning calorimetry. Experimentalself-diffusion data were measured as a function of both tem-perature and composition.

In their analysis, Lohfink and Sillescu invoked eq 30 for thetemperature shift of D1 by treating the PMPS/BMC/TTI mix-ture as a pseudo-binary system comprised of trace amountsof TTI in a host matrix of PMPS/BMC. Generally systemssuch as these have been treated as ternary mixtures, inwhich case the free-volume expression of Vrentas et al.33 isused to describe diffusion behavior at Tgm.

D1 ¼ Do1 exp�cðx2

n13n23V�2 þ x3V

�3n13Þ

VFH

!

¼ Do1 exp�cðx2

n13n23V�2 þ x3V

�3n13Þ

x2K12K22þx3K13K23

!ð31Þ

The subscripts 1, 2, and 3 refer to TTI, BMC, and PMPS,respectively. From analysis of diffusion data for the two bi-nary systems TTI/PMPS and TTI/BMC Lohfink and Sillescureported values of 0.87 for n12 and 0.70 for n13. Based ontheir WLF constants the free-volume parameters K12/c andK22 for BMC were calculated to be 3.69 � 10�4 cm3/g K and83.5 K and for PMPS K13/c and K23 were 6.25 � 10-4 cm3/gK and 49.3 K. Since the system was treated as pseudo-binary,no value of n23 was determined.

By assigning typical values to the specific occupied volumes(V�

2 ¼ V�3 � 1) and assuming that the jump size units of

FIGURE 8 Prediction of styrene diffusion in polystyrene using

the k approach. Data from Ref. 38. Reproduced from Ref. 34

with permission from Springer Science.



PMPS and BMC to be comparable, that is, n23�1, one can gen-erate a free-volume curve of the predicted dependence of TTIself-diffusion in mixtures of PMPS and BMC. A n23 value of0.9 provided a better representation of the data and was usedin the calculations. A comparison of the free-volume (dotted)line and the experimental data is provided in Figure 9.

The trend line that would be obtained from free-volumetheory (eq 31) for this system yields an accurate representa-tion for the majority of the experimental data. The experi-mental diffusivity in the pure PMPS limit, however, is signifi-cantly higher than the experimental data for mixturescontaining relatively small amounts of BMC. Lohfink and Sil-lescu suggest that this result indicates that a small fractionof the sorbed molecules fill special sites within the polymerthus blocking diffusion over large distances.

This dramatic reduction in D(Tgm) upon addition of a littlesolvent (BMC) is not predicted from the assumption of theadditivity of pure-component free volumes commonlyemployed in free-volume models. This result, however, doesnot contradict free-volume theory itself, but rather revealsinteresting rheological and PVT behavior for this mixture.

The concentration dependence of the mixture hole-free vol-ume can be ascertained from the reported WLF parameters.

VFH

c¼ K1mK2m

c¼ V�

1m

2:303C1m(32)

V�1m is the average value of the specific occupied volume of

the mixture. (In the present case, V�1m � 1). As defined by eq

30, Cim is a matrix WLF parameter which is commonlydeduced from rheological studies. In this case, the matrix isa binary mixture of PMPS and BMC which reflects the free-volume characteristics of the mixture at a particular blendcomposition. Using the WLF parameters reported by Lohfinkand Sillescu,46 the revised prediction of the self-diffusionbehavior of TTI in BMC/PMPS mixtures at Tgm is obtained asshown in Figure 9 as a solid line. The free-volume predictionfrom eq 31 is now in good agreement with the experimentaldata. The remarkable diffusion behavior witnessed in thepure PMPS is directly attributable to the significant rheologi-cal impact BMC has on the blended matrix as perceivedthrough the C1m values.

Another system for which diffusion data exist at the mixtureglass transition temperature is toluene/polystyrene. Diffusiondata at Tgm, acquired at different concentrations and temper-atures have been inferred from capillary column inverse gaschromatography (CCIGC),45 static gradient NMR (SG-NMR),47

and FRS48 measurements. These data are illustrated in Fig-ure 10. Just as in the case of the TTI/BMC/PMPS system, theaddition of a small amount of solvent radically alters the sys-tem molecular dynamics at the mixture glass transitiontemperature.

FIGURE 9 Experimental data46 and free-volume predictions for

TTI self-diffusion coefficients in mixtures of BMC and PMPS at

the mixture glass-transition temperature, D(Tgm). The solid line

is the free-volume prediction based on the assumption of addi-

tivity of the pure-component free volumes: the dotted line rep-

resents the free-volume prediction based on WLF fits of

rheological data taken on the mixtures.

FIGURE 10 Experimental data45,47,48 and free-volume theory

predictions (solid line) of toluene self-diffusion coefficients in

toluene/polystyrene solutions at the mixture glass-transition

temperature, D(x1,Tgm). The free-volume prediction is shown

up to the point where the polystyrene contribution to the mix-

ture free-volume becomes negative.



At one time the glass transition temperature was consideredan iso-free-volume state, that is, all materials would begin toexhibit glassy characteristics when cooled to a temperaturewhere their free volumes were the same.49 The datadepicted in Figure 10 reveal that this clearly is not the case,or at least that the molecular mobility is strongly influencedby the nature (perhaps fragility) of the glass transitionpoint.50

The modified free-volume theory referred to in the previoussection as the a approach results in eq 28 for a trace amountof solvent in a polymer. When this approach is expanded toconsider the influence of a finite concentration of solvent onthe free volume, a more general equation is obtained.

VFH

c¼ x1

K11


Vo2 Tg2

� �c2

fGH2 � a2 � a2cð Þ Tg2 � T� �� ð33Þ

Combining eqs 33 and 3 gives a description of a modifiedfree-volume theory below the glass transition temperature ofthe polymer when there is a finite concentration of solvent.The parameters for eqs 3 and 33 have been reported byVrentas and Vrentas4,35 for the polystyrene/toluene system(Table 1). Correlation of the data for the polystyrene/tolueneglass-transition temperature measured by Adachi et al.51

shows that they are well-represented by the following corre-lation.

Tgm ¼ 372:79� 491:68 x1 þ 250:92x12 (34)

Consequently, the self-diffusion behavior of toluene at T ¼Tgm can be predicted from eqs 3, 33, and 34 as a function oftoluene weight fraction, x1. These predictions are comparedto the experimental data in Figure 10 using parameterslisted in Table 4. The prediction highlights the weakness dis-cussed previously of applying free-volume theory at temper-atures well-below the pure polymer glass transitiontemperature; namely, the free-volume contribution of the

polymer becomes negative at sufficiently low temperatures.As a result, the free-volume curve depicted in Figure 10 ispredicted only up to the point at which the polymer free vol-ume becomes negative, that is, at approximately 250 K. Touse the free-volume formulation to predict diffusive behaviorat temperatures far below the glass transition temperatureof the pure polymer, one would have to incorporate informa-tion describing the temperature behavior of the expansioncoefficients of the solvent and the polymer.

Despite this general weakness, the dramatic initial rise ofD(Tgm), that is, four orders of magnitude, is predicted quitewell by free-volume theory. No conclusions may be drawn,however, regarding the ability of the theory to predict therelative concentration independence of D(Tgm) observedexperimentally at higher toluene concentrations. Both sys-tems examined here behave similarly in that a dramaticchange in D(Tgm) occurs upon addition of a few weight per-cent of solvent while D(Tgm) remains relatively concentrationindependent upon subsequent solvent addition.

MOLECULAR DYNAMICS SIMULATIONS OF DIFFUSION NEAR

THE GLASS TRANSITION TEMPERATURE

Molecular dynamics (MD) simulations are normally con-ducted in a particle box of approximately 50 angstromsinside of which are placed polymer chains and penetrant mol-ecules. An approximation of a bulk system is achieved throughthe application of periodic boundary conditions, so that whena molecule (or portion of a molecule) exits a face of the box,it reenters on the opposite side. The bulk system is repre-sented as a very large collection of the same simulation boxes,so computation needs to be performed only on a single box.Although the polymer is confined to periodic cells, theunfolded polymer configurations span many cells and needsto be selected from ensembles that properly reflect the poly-mers Gaussian backbone configurations. Thus, for MD simula-tion of polymers, great effort is made to ensure that the poly-mer micro structure is representative of the bulk material,despite the relatively small simulation cell size.

MD simulates the molecules motions over a very short time(order nanoseconds). Thus, to obtain realistic diffusion coef-ficients, one must ensure that the system is starting off in alikely or realistic configuration. The initial starting configura-tion can be obtained in a number of ways. For example, thechain can be grown atom by atom according to a specificmodel.52 The starting configuration can also be obtainedthough a molecular mechanics (MM) calculation which findsconfigurations that produce local minima in the systemspotential energy.53 The potential energy is a highly convo-luted function of the coordinates of all polymer and solventmolecules in the system, rp and rs. Theodorou

54 presentedthe polymer’s potential energy plotted against rp, resultingin a curve.

In rubbery polymers, all of the different energy minima canbe visited by the chain in experimental time scales since thechain is free to move around and has enough thermal energyto pass over the high energy maxima. In glassy polymers,

TABLE 4 Toluene/Polystyrene Parameters for Binary Free-

Volume Model (eqs 3 and 33)4

Do1 (cm2/s) 4.17 � 10�4

E 0

V�1 (cm3/g) 0.917

V�2 (cm3/g) 0.850

n 0.575

K11/c1 (cm3/g K) 1.57 � 10�3

K21-Tg1 (K) �90.54

V02 (Tg2) 0.972

c2 0.956

fGH2 0.0265

Tg2 (K) 273.15

a2 (K�1) 5.30 � 10�4

a2c (K�1) 3.88 � 10�4



however, the chain motion is hindered and the polymer findsitself somewhat trapped in one or a few of the energy min-ima. It is very difficult for the chain to visit the many differ-ent minima in a reasonable amount of time since it does notpossess the thermal energy required to overcome the highenergy barriers. Of course the particular configuration that aglassy polymer is in depends upon the history of the vitrifi-cation. The problem can usually be overcome by performingmultiple simulations with different starting minimum config-urations and then averaging the results.55

The key to any MD simulation is proper parameterization ofthe force field. There are numerous force fields available, butall involve the summation of different energetic contributionssuch as bond stretching, bond angle energies, torsional ener-gies, and of course nonbonded energies like van der Waalsforces and electronic interactions. Different force field pa-rameters define the energetic effects. Some of the parame-ters like bond angles and lengths can be obtained fromquantum chemistry while others (the nonbonded interac-tions) are fit to experimental spectrographic and thermody-namic data.54

Once the initial system configuration is determined, a MDsimulation proceeds by integrating the equations of motionfor each atom in the system. From statistical mechanics, thepenetrant self-diffusion coefficient can be obtained from thetime evolution of its position vector:

D1;i ¼ limt!1

1

6tr1;i tð Þ � r1;i t ¼ 0ð Þ�� 2� �

(35)

r1,i is the position vector of penetrant molecule i. The trueself-diffusion coefficient is obtained by averaging eq 35 forall penetrant molecules in the system. The mutual diffusivitycan be obtained from MD simulation if the simulation is alsoused to obtain the sorption isotherm. For a sufficiently lowsolvent concentration, the mutual diffusivity can be obtainedfrom a thermodynamic term and the self diffusivity by:

D ¼ D1x1 1� x1ð ÞRT

@l1@x1

� �T;P

(36)

x1 is the weight fraction of solvent and l1 is the chemicalpotential of the solvent. In most MD diffusion simulations,the penetrant is infinitely dilute and the mutual diffusivityapproaches the solvent self diffusivity.

D ¼ D1 as x1 ! 0 (37)

MD simulation where each atom is parameterized in theforce field is often referred to as completely atomistic MD.Simulations of this type have been quite successful in pre-dicting self-diffusion coefficients of small penetrants inglassy polymers (on the order of 10�7 cm2/s or larger). Oneimpressive work comes from Muller-Plathe56 where water’sself diffusivity in poly(vinyl alcohol) was predicted by atom-istic MD to match experimental NMR data over the entireconcentration regime. Fried et al.57,58 have also used atomis-

tic MD to accurately predict diffusion coefficients for variousgases in glassy poly(dibutoxyphosphazenes) and poly(1-(tri-methylsilyl-1-propyne)). Hofman et al.59 have also used thistechnique to predict diffusion coefficients for different pene-trant gases in glassy poly(amide imide).

Often times, each atom itself is not parameterized in theforce field. Rather small groups of atoms are lumped to-gether into a ‘‘united atom’’. This united atom is then para-meterized in the force filed. An example of this is the unitedatom polyethylene model where each methylene is a unitedatom.60 The polyethylene chain is then just a large numberof attached united atoms. This avoids having to solve thegoverning equations for each of the atoms. Most often, MDsimulation systems are a combination of united atoms andreal atoms. A good example is model polystyrene, where thearomatic C-H can be described as a united atom without lossof calculation accuracy.61 It is naturally desired to performcomplete atomized MD without united atoms, but often,available computation power does not allow it. United atomapproaches have been used by several investigators to studydiffusion in glassy polypropylene.62–64 The technique hasalso been used in other studies for gas diffusion in glassypoly(dimethyl siloxane),65 poly(isobutylene),66 polyimide,67

polyesters,68 and polybutadiene.69 All of these studies pro-duced diffusion coefficients as well as PVT thermodynamicdata that were in good agreement with experiment. Becausepolymer diffusion is so closely related to free volume, MDsimulations used to obtain diffusion coefficients must beable to accurately represent the system’s actual PVT behav-ior. It has become common practice for those working in thisfield to present both PVT and diffusion data. Thus, the forcefield used for the simulations must be capable of predictingthe system density. Early united atom models were unableto accurately reproduce experimental density data, but thesemore recent hybrid models (partial united atom models)have been able to meet this strict-but-necessary test.

Diffusion of gases in rubbery polymers has been studiedextensively by atomistic and united atom MD. Many investi-gators have been able to accurately match experimental sol-vent self-diffusion coefficients with these types of simula-tions. A better way to put this is to say that atomistic MDcan accurately predict diffusivities for systems that have highdiffusion coefficients (greater than 10�7 cm2/s) irrespectiveof the glass transition. The problem with performing MDnear and below the glass transition is that the solvent self-diffusion coefficient gets quite small. Because of this, thecharacteristic time needed to obtain an accurate diffusioncoefficient gets into the realm of hundreds of nanosecondsor greater.54 These large times are currently not computa-tionally accessible. Fully atomistic MD below the glass usu-ally results in watching the solvent molecule rattle aboutinside of a cavity of free volume.70 A molecule is seldomseen to jump into a nearby cavity and, thus, not enoughjumps are made to obtain good diffusion statistics. Thisproblem has led to the development of a transition statetheory (TST) approach to MD diffusion modeling of polymerglasses.71–75



Multiple MD diffusion simulations have shown that diffusionof penetrants in low temperature melts and polymer glassesoccurs as a series of jumps between neighboring voids infree volume. As the temperature becomes progressivelylower, these jumps become quite infrequent, making the pro-cess a good candidate for a transition state approach tomodeling the behavior. In the TST framework, the polymermatrix is described by a series of macrostates. Each macro-state corresponds to the system configuration at each poten-tial energy minima. In certain cases, the energy barrier sepa-rating different macrostates is small, meaning that a smallcollection of macrostates can be visited by the penetrantwith ease. The diffusive jump of a penetrant can be modeledas a first order reaction (rate constant defined as ki!j)where the penetrant jumps from macrostate i to macrostatej. The jump is accomplished by the penetrant passingthrough the transition state, which is defined as the highestenergy barrier on the lowest energy pathway to the neigh-boring macrostate. If pi is the probability of finding the sys-tem in macrostate i, this probability will be governed by:54

@pi@t

¼ �Xj

ki!jpi þXj

kj!ipj (38)

The pi can then be related to the position vectors of the pen-etrant, and ultimately to the self-diffusion coefficient via eq35. Thus, the problem of calculating the diffusivity nowreduces to finding the probabilities and determining the rate

constants. This is accomplished by multidimensional transi-tion state theory, the details of which are beyond the scopeof this review and can be found in the works of Theodorouand coworkers.54,73 The major point is that if the rate con-stants are sufficiently small as they should be if jumps areinfrequent; then much larger time steps can be taken than inatomistic MD. This makes slow diffusion process simulationaccessible with current available computing power. TSTapproaches have accurately predicted gas self-diffusion coef-ficients as low as 10�9 cm2/s.74,75 Gas diffusion and sorptionin glassy polyimide has been studied by TST, and predictionshave matched experimental values reasonably well.76,77 Onevery recent investigation has looked at gas diffusion in dif-ferent styrene copolymers.78 The results, however, match ex-perimental diffusivities only to within an order ofmagnitude. Figure 11 shows the comparison of various MDsimulated and measured diffusion coefficients. Also in thisfigure are predicted values of the diffusivities in differentgas-polystyrene systems using the k free-volume approachdescribed in the preceeding sections of this review. For thesepredictions, the polymer free-volume parameters were thesame as those given for polystyrene in Table 4. The D0 and nwere estimated using a linear correlation with the penetrantsize following the approach described in a previous study byCai et al.24

Interpretation of MD ResultsOne common thread that all of the previously mentioned MDresults have is the nature of the low temperature diffusionmechanism. In a glass or a low temperature melt, diffusion ofa penetrant consists of a series of jumps between voids infree volume. The major distinguishing feature of glassy

FIGURE 11 Comparison of simulated-MD and measured

solvent self-diffusion coefficients. Numbered circles indicate

MD-simulated values and lettered boxes indicate free-volume

predictions. The key is given in Table 5.

TABLE 5 Summary of Data and Key for Figure 11

Figure 11 Key Polymer Solvent Ref.

1, A Poly(styrene) CH4 78

2, B Poly(styrene) N2 78

3, C Poly(styrene) O2 78

6 Styrene butadiene rubber N2 78

8 Poly(dibutoxyphosphazene) CO2 58

4 Poly(imide) N2 76

5 Poly(ethylene) CH4 65

7 Styrene butadiene rubber O2 78

12 Poly(dimethyl siloxane) CH3CH2OH 65

9, D Poly(styrene) Ne 78

10 Poly(imide) O2 76

13 Poly(dibutoxyphosphazene) CO2 58

11 Poly(amide imide) H2 76

14, E Poly(styrene) H2 78

15 Poly(dibutoxyphosphazene) O2 58

16 Styrene butadiene rubber H2 78

17, F Poly(styrene) He 78

18 Poly(dimethyl siloxane) H2O 65

19 Poly(dimethyl siloxane) CH4 76



diffusion is the inability of the polymer to make large coordi-nated motions (a-relaxation) in a reasonable amount of timebecause of its frozen nature. Rather, the polymer makessmaller, faster, less coordinated motions (b-relaxation) due toside chain and small backbone section relaxations. These tworelaxation mechanisms have been extensively studied by a va-riety of scattering, NMR, and spectroscopic techniques.79–83 Infact, MD simulations have also been used to predict the differ-ent relaxation phenomena.84,85 MD has proven successful inpredicting a and b-relaxation times that were measured fromscattering and dielectric experiments.86–89

These MD relaxation and diffusion simulation results can beinterpreted in terms of the macroscopic Vrentas-Duda free-volume theory of diffusion. Actually, the MD diffusion studiessupport the concepts underlying the free-volume theory.That is that solvent self diffusion occurs as series of jumpsbetween voids in free volume. MD simulations show thatdynamically connected cavities and the fluctuations in theseconnections, which are due to beta-relaxations and atomisticmotion of the polymer, control the rate of self diffusion forsmall molecules in glassy polymers. This provides a molecu-lar interpretation of the meaning of free volume in glassypolymers and highlights the importance of coordinated poly-mer thermal motion in regulating solvent diffusion. Further-more, there is little or no evidence to suggest that a dualmode type model is the responsible diffusion mechanismbelow the glass transition temperature. All MD simulationshave shown that diffusion in a glass is quite similar to thatin a low temperature melt. In fact, the whole basis of theoverwhelmingly successful transition state theory is thatpenetrant molecules hop between cages of empty space.

The relaxation MD results also fit well into the free volumepicture of diffusion. Above and slightly below the glass tran-sition, free volume is redistributed by the large coordinatedmotion of the polymer (the a-relaxation process). Far belowthe glass transition, such large coordinated motion is notpossible, and free-volume voids are opened and closedbecause of secondary, faster (b-relaxation) processes.

The only piece of the puzzle that remains to be investigatedby MD is the so-called break in the diffusivity at the glasstransition. So far, this characteristic change in the apparentdiffusion activation energy has not been simulated by MD.The most probable reason for this is that only very small sol-vent molecules have been simulated so far. Experimental evi-dence reveals that small penetrants (like gases) require sucha small amount of free volume to diffuse that the break is ei-ther minor or even negligible.90 Because of this, the extrafrozen-in free volume created in a glassy polymer does notsignificantly raise the diffusion coefficient as it does for alarge penetrant that requires more free volume fortransport.

CONCLUSIONS

Two approaches to correlate and predict self-diffusion coeffi-cients below the glass transition temperature of the purepolymer and mixture are the a approach and the k approach.

These approaches have been compared in terms of the val-ues that they yield for free volume of the polymer and freevolume of the mixture. The free volume calculated using thek approach decreases quickly, and this leads to the nonphysi-cal result of a zero or even negative free volume of the poly-mer in equilibrium mixtures. The a approach however, obvi-ates this difficulty to some extent. Both of these approachesprovide approximations to the actual behavior and the kapproach is applicable for diffusion of trace amounts of apenetrant, in the vicinity of Tg2. If methods become availableto accurately determine the occupied and interstitial volumeat all temperatures, and especially at temperatures belowthe glass transition temperature, the basic equations given inthe preceding sections can be used to obtain more precisepredictions.

Diffusion of the tracer, TTI in PS has been used to illustratethe differences in results that can be obtained using the twoapproaches. A higher diffusion coefficient has been predictedbelow the glass transition temperature using the a approach,while a smaller value was predicted using the k approach atthe lowest temperature. This result is consistent with thefree-volume behavior calculated using the two approaches.This result also indicates that the actual behavior of free vol-ume may be something between the values predicted by thetwo approaches.

The predictions have been compared to the experimentalresults of Murphy et al.38 and the results indicate that thepredictions are consistent with experimental results. Thesedata clearly elucidate the differences in the predictionsbetween the two approaches, which become apparent at lowtemperatures.

The a approach has been used successfully to predict the dif-fusion coefficient of styrene in general-purpose polystyreneat low concentrations of styrene and temperatures below theglass transition temperature of the pure polymer. One of thepractical problems, however, with the a approach is that allthe parameters are usually not available. Much of the dataavailable below the glass are at an infinite dilution of thepenetrant. Therefore, in many cases, the k approach is usedto correlate diffusion coefficients with k as an adjustable pa-rameter. While this weakens the predictive capability of thefree-volume model by adding another fitting parameter, itcan be estimated rather easily from limited diffusivity databelow Tg. Since the glassy state is a nonequilibrium state,the thermal expansion coefficients of the polymer are historydependent and are likely to be different for the same poly-mer depending on the history of the sample. At high solventconcentrations and low temperatures, diffusion is typicallycoupled with relaxation. Analysis of data obtained underthese conditions is not trivial. This is one of the reasons thatthe preponderance of data available are at infinite dilution.

An evaluation of free-volume theories to predict diffusioncoefficients at the glass transition temperature of a polymer-solvent mixture indicates the experimental results are con-sistent with the free-volume approach. This evaluationclearly demonstrates, however, the limitations of the



available free-volume models at temperatures far below Tg.These models which are based on constant expansion coeffi-cients in the different temperature regimes will eventuallypredict negative free volumes at low temperatures.

NOMENCLATURE

A0 Pre-exponential factor in eq 1A Constant characterizing the depression in Tg

(K)C1m, C2m WLF constants for matrix materialD Mutual binary diffusion coefficient (cm2/s)D1 Self-diffusion coefficient (cm2/s)D01 Constant pre-exponential factor (cm2/s)E Energy/mole required to over-come

attractive forces of neighboringmolecules

fGH2 Fractional hole free volume of polymer atthe glass

K11c1

Free-volume parameter of component 1(cm3/g K)

K12c2

Free-volume parameter of component 2(cm3/g K)

K21 Free-volume parameter of component 1 (K)K21 Free-volume parameter of component 2 (K)R Gas constantT Temperature (K)Tg1 Glass transition temperature of component

1 (K)Tg2 Glass transition temperature of component

2 (K)Tgm Glass transition temperature of the mixture

(K)VFHc Hole free volume of system (cm3/g)

V02 (Tg2) Specific volume of polymer at Tg2 (cm3/g)

V�1 Specific hole free volume for component 1

to jump (¼V01(0)) (cm

3/g)V�2 Specific hole free volume for component 2

to jump (¼V02(0))(cm

3/g)VFH2g Hole free volume of a glassy polymer

(cm3/g)V * Minimum size of hole required for

molecular jump (cm3/g)VF Average free volume per molecule (cm3/g)VFH1 Specific hole free volume of component 1

(cm3/g)VFH2 Specific hole free volume of component 2

(cm3/g)VFH2g Specific hole free volume of glassy polymer

(cm3/g)V02g Specific volume of glassy polymer (cm3/g)

VF12g Specific interstitial volume of glassypolymer (cm3/g)

V2g0 Specific occupied volume of glassy polymer(cm3/g)

a2 Thermal expansion coefficient of polymerabove Tg (K

�1)

a2g Thermal expansion coefficient of polymerbelow Tg (K

�1)aC2 Thermal expansion coefficient of occupied

and interstitial free volume below Tg2(K�1)

ac2g Thermal expansion coefficient of occupiedand interstitial free volume below Tgm(K�1)

c Parameter describing sharing of free volumek Parameter characterizing nonequilibrium

nature of glassy stateg Viscosity at zero shear ratex1 Weight fraction of component 1x2 Weight fraction of component 2n Ratio of molar volume of jumping unit of

the solvent to that of the polymer

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application of free-volume theory to self diffusion of solvents in polymers below the glass...

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