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Vol. 12 TRANSPORT PROPERTIES 291

TRANSPORT PROPERTIES

Introduction

Transport properties determine a polymer’s ability to move through some mediumor to have some penetrant medium move between its constituent segments. Thisdefinition encompasses processes with diverse driving forces such as concentra-tion and pressure gradients, and even electrical or temperature gradients capableof motivating one component relative to another. This article emphasizes poly-mer transport properties under conditions of low to intermediate penetrant con-centration, where extraordinary differences can exist between the diffusivities ofpenetrants having relatively small differences in molecular size or shape.

This article focuses on transport that proceeds by the solution-diffusionmechanism. Transport by this mechanism requires that the penetrant sorb intothe polymer at a high activity interface, diffuse through the polymer, and then des-orb at a low activity interface. In contrast, the pore-flow mechanism transportspenetrants by convective flow through porous polymers and will not be describedin this article. Detailed models exist for the solution and diffusion processes ofthe solution-diffusion mechanism. The differences in the sorption and transportproperties of rubbery and glassy polymers are reviewed and discussed in termsof the fundamental differences between the intrinsic characteristics of these twotypes of polymers.

The transport properties of glassy and rubbery polymers are related to theirmicrostructural morphology. For a penetrant to diffuse, a minimum characteris-tic packet of unoccupied volume is required. The penetrant diffuses by jumpingthrough transient gaps between packets of unoccupied volume. The lifetime, size,and shape of these volume packets and the transient gaps that connect them aredependent upon the micromotions of the polymeric media. New techniques such as

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

292 TRANSPORT PROPERTIES Vol. 12

positron annihilation lifetime spectroscopy (PALS) and molecular modeling allowfor elucidation of dynamic activities at the molecular scale. While a descriptionat this length-scale offers fundamental information about the diffusion process, itis not yet possible to predict the macroscopic transport properties with sufficientaccuracy for practical applications. In this regard, quantitative phenomenologicalmodels and more molecularly based treatments are complementary, and both willbe considered.

Terminology

Reference Frames and Fluxes. In speaking of a diffusional flux, it isnecessary to specify a reference frame from which the diffusion process is to beobserved. Generally, a reference velocity such as the mass, molar, or volume aver-age bulk velocity (v,v∗, or v�, respectively) in the system is selected, and movementof the component of interest relative to this reference velocity is defined to be truediffusion. For practical reasons, a fixed reference frame is generally also consideredto relate the mathematical treatment of this molecular scale transport process toan actual physical system with well-defined dimensions. In this case, the total fluxrelative to the fixed reference frame is partitioned into two parts: bulk flow andtrue molecular diffusion. This partitioning is necessary, because even in the ab-sence of an externally imposed bulk flow, interdiffusion of molecules with respectto each other produces an effective bulk flow relative to fixed coordinates if themolecules have different masses (1,2).

The definitions of the mass, molar, and volume average bulk velocities aregiven in Table 1 along with selected mass and molar flux expressions related toeach of the specified reference velocities for a binary system of components A and B.The mutual diffusion coefficient DAB is the diffusivity of component A and B in themixture as defined below. The mutual diffusion coefficients appearing in Table 1are identical in all of the expressions and DAB = DBA.

Additional discussion of these definitions and relationships between the dif-ferent reference frames and fluxes are discussed in detail elsewhere (3,4). In princi-ple, any reference frame for analysis may be selected; however, a proper choice canreduce the mathematical difficulties. For example, in a one-dimensional diffusionprocess within fixed boundaries, where ideal mixing of components is a reasonableapproximation, selecting the volume average frame of reference is wise because

Table 1. Average Velocities and Forms of Fick’s First Law for Binary Diffusiona

Average velocity Definition of average velocity Form of Fick’s first lawb

Mass average v = ωA vA + ωBvB ji = −ρDAB ∇ωi = ρi(vi − v)Molar average v∗ = xA vA + xBvB ji

∗ = −MA CDAB ∇xA = Ci(vi − v∗)Volume average v� = ρAvAVA/MA + ρBvBVB/MB ji

� = −DAB ∇ ρi = ρi(vi − v�)aωi and ρi refer to the mass fraction and mass density of component i, respectively; ρ refersto the total solution mass density, and Vi and Mi are the molar volume and molecular weightof component i, respectively. The velocity of component i, vi, represents the average velocity ofcomponent i relative to fixed coordinates due to both bulk and true diffusive movement.bji and ji� are the mass fluxes of the component i relative to the mass average and volume averagevelocities, respectively, and Ji

∗ is the molar flux relative to the molar average velocity.


the volume average bulk velocity as defined above is zero, and hence the fluxesviewed from both a fixed reference frame and a reference frame moving at the localvolume average velocity are identical. In this case the fundamental differentialequation of one-dimensional diffusion in an isotropic medium reduces to

∂ρA

∂t= ∂

∂z

(DAB

∂ρA

∂z

)(1)

Documented solutions for equation 1 exist for a limited number of complexconcentration-dependent diffusion coefficients and boundary conditions (5).

Alternatively, for steady-state analysis involving a case with component Breplaced by P, for polymer, a static reference frame and the mass average velocityas a reference velocity may be selected. The mass flux of component A (penetrant)relative to the fixed reference frame is given in general by

nA = − ρDAP∂ωA

∂z+ ωA(nA + nP) (2a)

where nA is the total mass flux of A relative to a fixed reference coordinate system,the first term on the right is the flux of A relative to the mass average velocity, andthe last term is the flux of A due to bulk flow of fluid relative to the fixed coordinatesystem. However, since nP = 0 in this case (since vP = 0 for the membrane polymerrelative to the fixed reference frame at steady state), the general expression fornA simplifies to

nA = − ρDAP

(1 − ωA)∂ωA

∂z(2b)

The 1/(1 − ωA) term is commonly referred to as the frame of reference term.For many cases of importance in polymeric systems such as in gas permeation, ωAis relatively small, and the 1/(1 − ωA) factor can safely be neglected so that theflux relative to fixed coordinates is equal to the flux relative to moving coordinates.Even for intermediate concentrations (0.1 < ωA < 0.5), this factor may often beof second-order importance compared to difficulties in accurately determining themutual diffusion coefficient due to strong concentration dependencies. However,not accounting for the factor 1/(1 − ωA) can lead to very significant errors in fluxcalculations in highly swollen systems (eg, 90–95% solvent), even if the mutualdiffusion coefficient is accurately determined (6).

Mutual Diffusion Coefficients. A better appreciation of the separationof bulk and true diffusive fluxes and the significance of the mutual diffusion co-efficient DAP is useful prior to considering factors that determine this coefficient.Consider, for example, the transient problem of a block of polymer placed in con-tact with an external solvent (penetrant) phase. When a small penetrant movesinto a polymer under transient conditions, most of the polymer movement, mea-sured relative to a fixed coordinate, is thought to be due to bulk flow arising fromthe outward swelling of polymer segments into the region that was occupied previ-ously by the external solvent phase (5). Because of its intrinsically high mobility,the penetrant (solvent) tends to interpose itself between the polymer segments


and invade the domain initially occupied solely by the polymer, thereby acting asthe prime mover for the mixing process.

The initial invasion of the solvent effectively induces a local swelling stress.The polymer sample responds to this swelling stress and moves in a directionopposite to the invading solvent flux. This backward bulk motion thereby carriesentrained solvent molecules against the direction of the simple concentration-driven diffusive flux. Unlike the steady-state membrane case, it is clearly notacceptable to assume the mass flux of polymer is zero in this situation, and somerelationship between nA and nP must be defined. This issue becomes important forcases involving the interaction of strong swelling solvents with glassy polymersin transient sorption processes. Also in this case, a very strong concentrationdependence of DAP may exist, in some cases approaching a step function. In thesesituations, the “polymer fixed” frame of reference is often used for mathematicalconvenience. In this reference frame, DAP is defined relative to the penetrantconcentration in the unswollen polymer. On the other hand, if the polymer wasnot assumed to be fixed, DAP would be defined relative to the concentration in theswollen polymer. DAP for these two reference frames are not necessarily equal;however, diffusivities in different reference frames can be readily interconverted.A more thorough treatment of this subject is given in References 5 and 7.

However, for most cases involving gases and even low sorbing vapors or liq-uids, swelling of the polymer and non-Fickian complications are minimal. In thesesituations, solutions to equation 1, unconfused by bulk flow, provide an adequatedescription of the process and allow estimation of the mutual diffusion coefficient.In such cases with a constant diffusion coefficient, standard solutions of equation 1for Mt, the amount of material sorbed (or desorbed) at time t relative to M∞, theamount of material sorbed (or desorbed) at infinite time, can be used for this esti-mation of the mutual diffusion coefficient. If the initial and final concentrations inthe sample of thickness � are uniform and the external penetrant activity is heldconstant, two mathematically equivalent solutions to equation 1 are given (5):

Mt

M∞= 4

[DAPt�2

]0.5{ 1π0.5

+ 2∞∑

n = 1

( − 1)n ierfc[

n�

2(DAPt)0.5

]}(3a)

Mt

M∞= 1 −

∞∑n = 0

8

(2n + 1)2π2

exp[

− (2n + 1)2π2 DAPt

�2

](3b)

On the one hand, equation 3a is generally referred to as the short time so-lution because good accuracy is achieved for values of Mt/M∞ ≤ 0.6 even if theinfinite summation term is neglected. On the other, equation 3b is referred to asthe long time solution because good accuracy is achieved for values of Mt/M∞ ≥0.6 if only the first term in the summation is used.

In cases where there are significant, but not extreme, concentration depen-dencies of the diffusion coefficient (changes in DAP of up to 50–100% over thecourse of an experiment), an average coefficient can be used in equations 1, 2, and3 with little loss in accuracy. For such cases, the average diffusion coefficient D̄ is


defined as follows:

D̄ =

(ωA)∞∫(ωA)0

DAPdωA

(ωA)∞ − (ωA)0(4)

where (ωA)0 and (ωA)∞ refer to the uniform penetrant mass fractions in the poly-mer at the beginning and at the end of the sorption run, respectively.

The experiment could also be run over a narrow concentration interval todetermine the concentration dependence of D̄. In this latter case, the averagediffusion coefficients determined from half-time analysis are good approximationsfor DAP over that particular concentration interval. D̄ is calculated simply byrearrangement of equation 3a by neglecting the infinite summation terms withMt/M∞ = 0.5:

D̄ = 0.0492�2

t1/2(5)

where t1/2 is the time required to achieve one-half of the mass uptake that ul-timately occurs at equilibrium for the interval sorption experiment. Equation 5applies for experiments in which the sample has two exposed faces. For a single-sided exposure the coefficient in equation 5 is replaced by 0.1968. Alternate half-time expressions for different geometries such as cylinders and spheres have beenpresented (5). To analyze the later stages of sorption using an average diffusion co-efficient D̄, equation 3b indicates that a plot of ln(1 − Mt/M∞) vs t gives a straightline with a slope equal to π2D̄2/�2.

Self vs Mutual Diffusion Coefficients. The self-diffusion coefficient DA∗

is measured by observing the rate of diffusion of a small amount of radioactively-tagged component A in a system composed of a uniform chemical compositionof untagged components A and B. Because of the essentially identical physicalnatures of the tagged and untagged penetrant, observing the process of interdif-fusional exchange of the tagged and untagged molecules allows measurement ofthe true mobility of the tagged molecules with respect to the stationary solutionof known concentration. By varying the concentration of A and B in the pres-ence of a small amount of tagged component A, the concentration dependence ofthe diffusion coefficient of the tagged molecule can be obtained, uncomplicated bybulk-flow considerations, namely

D∗A = RTMA = RT

ζA(6)

where MA is the mobility of A, ζA is the molar friction coefficient, the product ofthe effective viscosity of the medium and the effective diameter of the penetrant(8).

In addition to pure mobility considerations, thermodynamic factors enter indetermining the concentration dependence of the mutual diffusion coefficient. Asingle mutual diffusion coefficient exists for a given binary pair under fixed local


conditions (8). For molecules with similar sizes, shapes, and interaction poten-tials, the ratio of the self-diffusion coefficients of two components are related tothe inverse ratio of the molar volumes, ie, DA

∗/DB∗ = VB/VA (8). Under these condi-

tions the individual self-diffusion coefficients may be related to the correspondingmutual diffusion coefficient DAB in terms of the mole fractions of the two compo-nents and the partial derivative of the activity vs mole fraction relationship forthe penetrant in the polymer at the local mole fraction of interest:

DAB =(

∂ ln aA

∂ ln xA

)T,p

[D

∗BxA + D

∗AxB

](7)

The activity derivative can be readily evaluated from sorption vs activitymeasurements. The subscripts T and p, indicating which variables are held con-stant, will be dropped for simplicity in the subsequent equations. Equation 7 wasgiven earlier (4) using classical diffusion analysis for systems in which there isnegligible volume change of mixing.

Although the inverse molar volume ratio relationship suggested above isprobably adequate for small molecules with similar sizes, shapes, and interactionpotentials, it is not clear that it applies in all cases where marked differences existin the relative sizes of components A and B. A weight fraction based weightingof the respective Di

∗ values is probably more general. Except in the range of ex-tremely high solvent fractions, the mobility, and hence the self-diffusion coefficientof a polymer in the presence of a low molecular weight solvent, is many orders ofmagnitude lower than that of the solvent. Therefore, with the reasonable approx-imation that DP

∗ = DB∗ ≈ 0, equation 7 can be expressed in terms of the mass

fraction of component A as follows:

DAP = D∗AωP

(∂ ln aA

∂ ln ωA

)= D

∗A(1 − ωA)

(∂ ln aA

∂ ln ωA

)(8)

Diffusion Coefficients in Multicomponent Systems. The value of thediffusion coefficient of a species in a binary system is often not the same as thevalue in a multicomponent system. The diffusion coefficients can be modified inmulticomponent systems as a result of added frictional forces at the atomisticscale. The multiple diffusing species interact in various complex ways that can bedescribed using equation 9, which is derived from the so-called Stefan–Maxwellrelations (4):

Dim = −ni − xi

n∑j = 1

nj

n∑j = 1

xinj − xjni

Dij

(9)

where Dim is the diffusion coefficient of species “i” in the mixture, Dij are the binarydiffusion coefficients, xi is the mole fraction of species “i”, and ni is the molar flux of“i”. The diffusion coefficients calculated from equation 9 are used in a generalized


form of equation 2a for multicomponent diffusion:

ni = − cDim∇xi + xi

n∑j = 1

nj (10)

More detailed treatments of multicomponent diffusion are available (4,9).

Solution-Diffusion Mechanism

Small molecule transport through nonporous polymers proceeds by the solution-diffusion mechanism. This a three-step mechanism where penetrant molecules(1). sorb into the polymer phase from a high activity external gas or liquid phase,(2). diffuse across the polymer driven by a chemical potential gradient, and (3).desorb from the polymer phase to a low activity gas or liquid external phase. Whilethese three general steps of the solution-diffusion mechanism are agreed upon,the specifics of sorption into and out of the polymer phase and diffusion across itare still active areas of research. As will be seen in later sections of this article,there are many ways to conceptualize the sorption and diffusion processes.

Regardless of how one conceptualizes the diffusion process, the solution-diffusion mechanism states that the flux through the polymer is proportional toa chemical potential gradient. If a chemical potential gradient does not exist, inthe absence of an imposed bulk flow, no net transport of penetrant occurs throughthe polymer. This proportionality is stated mathematically as

JA = − L∂µA

∂z(11)

where JA is the flux of A through the polymer, µA is the chemical potential ofpenetrant A in the polymer phase, and L is the direct phenomenological transportcoefficient in irreversible thermodynamics terminology. C is the concentration ofA sorbed in the polymer, and MA, once again, is the mobility of A.

In the absence of electromotive forces, the chemical potential of a dissolvedpenetrant in the polymer phase is given by

µA = µ 0A + RTln(γACA) + VA

(p − p 0

A

)(12)

where µA0 is the chemical potential of the pure penetrant at the reference pres-

sure pA0. Moreover, γ A is the activity coefficient at concentration CA, and the

partial molar volume of the penetrant is VA. The reference pressure is usuallyset as the pure component vapor pressure. With the knowledge that a chemicalpotential gradient must exist through the polymer, from inspection of equation 12one can imagine three possible ways to affect such a gradient: (1) a concentrationgradient across the polymer, (2) a pressure gradient across the polymer, or (3)the coexistence of both pressure and concentration gradients across the polymer.In an elegant series of articles, Paul and co-workers showed that a concentra-tion gradient solely affects the gradient in chemical potential. The fact that no


pressure gradient exists through the membrane interior can be proved using asimple mechanical argument. In the common laboratory setup for gas separa-tions, the membrane is supported on the low pressure side by a porous metalsupport. If a pressure is exerted on the upstream side of the membrane by thegas, the metal support must exert an equal and opposite pressure on the down-stream side of the membrane. For this reason, the pressure inside the membraneis equal to the upstream pressure (6). By using a constant pressure through themembrane, Paul was able to rationalize the observed dependence of flux on theexternal pressure difference for both hydraulic permeation and pervaporation sys-tems (10). Furthermore, Paul conducted a series of experiments using a compositemembrane consisting of a stack of three to four rubber membranes to measure theconcentration gradient through the membrane. It was shown that the concentra-tion gradient determined from these measurements alone accounted for the entirechemical potential gradient (11). This result excluded the possibility of a pressuregradient inside the membrane for solution-diffusion transport processes.

With this knowledge of the physical situation, it is an easy exercise to deriveFick’s Law from the more general statement of equation 11.

J = − L∂µA

∂z= − L

∂µA

∂CA

∂CA

∂z= − DAB

∂CA

∂z(13)

Since DAB = (DA∗CA/RT)(∂µA/∂CA) (5), the phenomenological coefficient L

can be related back to the underlying penetrant mobility.

L = D∗ACA

RT= 1

ζACA =MACA (14)

Amorphous Rubbery Polymers

Gases and Low Activity Vapors. At low concentration (ωA → 0) and insystems where Henry’s law applies (∂ ln aA/∂ ln ωA → 1), DAB approaches theinfinite-dilution self-diffusion coefficient (DA

∗)0. Therefore, the mutual diffusioncoefficient is a good estimator of the infinite-dilution self-diffusion coefficient formost gases in polymers at atmospheric pressure and below. Even at relatively highpressures with condensable gases and low activity vapors such as CO2, SO2 andpropane, deviations from Henry’s law can generally be described in terms of theFlory–Huggins isotherm or a similar simple expression. Therefore, self-diffusioncoefficients, and thus the mobility of the penetrant, can be calculated using equa-tions 5 and 6 or by using permeation data in conjunction with equilibrium sorptiondata to determine (∂ ln aA/∂ ln ωA) at the appropriate ωA value.

Rubbers are essentially high molecular weight liquids with the ability to ad-just their segmental configurations rapidly over significant distances (>0.5–1 nm)and local volumes (12). Nevertheless, the rotational and translational motions ofsorbed penetrants are rapid compared to the motions of the segments of the poly-mer (13,14). The limiting step in diffusion of such small molecules through the


Fig. 1. Generation of a gap for the penetrant with subsequent collapse of the volume thatpreviously housed the penetrant, emphasizing the mutual nature of the binary diffusionprocess.

rubber involves the generation of a sufficiently large gap for the penetrant tomove into, with subsequent collapse of the sorbed cage that previously housed thepenetrant (Fig. 1). This description emphasizes the mutual nature of the binarydiffusion process, since both the penetrant and surrounding polymer segmentstend to undergo a minute translation in their positions as a result of the event.Given the overall mass of the polymer and the small fraction of the total chain in-volved in a primitive diffusion jump by a small penetrant, this change is miniscule,but nonzero, even for the polymer.

In typical polyolefins, the most common moving segment is a crankshaftcomposed of four to five backbone carbon atoms (Fig. 2). For diene and otherhydrocarbon polymers, similar cooperative motions involving several repeat unitsare also probably the most common types of motions observed over time scales ofimportance to diffusion. As expected, for penetrants such as gases that are clearlysmaller than the size of the most common moving segment of the polymer, theinfinite-dilution diffusion coefficient tends to show a steady drop with increasingpenetrant size (Fig. 3). The tendency for diffusion coefficients of larger penetrantsto approach an asymptotic plateau has been discussed (15). Branched penetrantstend to approach a lower, but still similar, asymptotic limit in a given medium.This effect suggests that larger penetrants are capable of moving in a somewhatsegmental fashion as does the polymer itself. In this case, the asymptotic limit ofthe linear penetrants in Figure 3 presumably reflects the mobility of the more or

Fig. 2. The crankshaft motion requiring the simultaneous rotation of several sequentialCH2 moieties about bonds 1 and 7 or 1 and 5.


Fig. 3. Diffusion coefficients for a variety of penetrants in natural rubber at 25◦C. n-C3,n-C4, and n-C5 designate straight chain alkanes.

less freely orienting segments of the polymer that are themselves undergoing aself-diffusion process. Therefore, while the self-diffusion coefficient of the polymercenter of mass may be orders of magnitude lower than that of intermediate-sizedpenetrants, segmental mobility in the polymer is still quite high. Not surprisingly,the asymptotic mobilities of penetrants in glassy polymers are orders of magnitudelower than those in the typical rubbery polymers.

Concerted movements of several adjacent segments of chain comprising thesorbed cage of a simple gaseous penetrant are believed to provide the source of thepenetrant-scale hole needed for diffusion. Lennard–Jones collision diameters (σ ),van der Waals volumes (b), and actual measured values of the partial molar vol-umes of various gases in polydimethylsiloxane (PDMS) (16) are shown in Table 2.These partial molar volumes in PDMS are close to the van der Waals volumes, veryclose to infinite-dilution partial molar volumes in several low molecular weightliquids (17), and are measures of the volume of the cage in which the sorbed pen-etrant resides at equilibrium. The occupied volume is calculated by viewing themolecule acting as a freely rotating sphere of diameter equivalent to the Lennard–Jones diameter σ . Based on these data, it is clear that only a fraction of the actualequilibrium cage volume is occupied and the remainder is in a sense “free vol-ume” available to be shared with the neighboring polymer segments to facilitatethe mutual diffusion process.

Estimates of the amounts of total free volume required for a diffusional jumpof several gaseous penetrants in a copolymer of poly(vinyl chloride) and poly(vinylacetate) have been made (18) (Table 2) (Fig. 4). Although the data in Table 2 are


Table 2. Various Molecular Volumes for Gaseous Penetrants

Volume measurement He N2 O2 CH4 C2H4 CO2

van der Waalsvolume b, nm3

0.0396 0.0642 0.0529 0.0709 0.0856 0.0711

Lennard–Jonesdiameter σ , nm

0.258 0.368 0.343 0.382 0.422 0.400

Lennard–Jonesoccupied volume, nm3

0.00899 0.0261 0.0211 0.0292 0.0393 0.0335

Partial molar volumea

in silicone rubber(PDMS), nm3

0.0549 0.0792 0.0940 0.0767

Free volume in sorbedstateb, nm3

0.0288 0.0500 0.0547 0.0432

Required free volumefor 0.075 diffusionaljump in PVC–PVAcopolymerc, nm3

0.299 0.690

aExperimental (15).bPartial molar volume minus occupied volume.cRef. 17

for two different polymers, for an order of magnitude analysis they can be takenas representative of movement in typical rubbery polymers. The required free vol-umes for diffusional jumps are significantly greater than the penetrant volumes.Therefore, most of the free volume involved in a given jump must be supplied by amomentary fluctuation in the rubbery polymer segmental position due to a localthermal fluctuation, rather than being locally present in the equilibrium sorbedcage. This fact can be treated in several ways, eg, via Activated State Theory orFree-Volume Theory.

Activated State Theory. The diffusion process by which small moleculesintermingle with a polymer can be considered a random walk of the penetrantamong the segments of the polymer. Consistent with this qualitative description,the activated state theory assumes that holes covering a spectrum of differentvolumes and involving segments of several polymer molecules are continuouslyformed and destroyed because of thermal fluctuations. The rate of diffusion de-pends on the concentration of transient holes that are sufficiently large to acceptdiffusing molecules. Assuming a Boltzmann distribution, the concentration of agiven size of holes decreases exponentially with the energy associated with its for-mation. The temperature dependence of the diffusion coefficient for the activatedstate theory can be expressed as

D = D0 exp( − �Ed

RT

)(15a)

where D0 is the preexponential factor and �Ed is the energy of activation fordiffusion.

The indication by equation 15a that ln(D) is linearly dependent on 1/T hasbeen verified for many systems well above the glass-transition temperature Tg


Fig. 4. Estimated required free volume for a diffusional jump for various penetrants inPVC–PVA copolymer (17).

(ie, T > Tg > Tg + 100 K). For penetrants smaller than the size of the primarymoving segment of the polymer, the size of the molecule generally determinesboth the size of the required hole and the activation energy required for diffusion,as shown in Figure 5 (15). This tendency for the activation energy of diffusionfor the larger penetrants to approach an asymptotic value similar to that of theactivation energy of viscous flow for the uncross-linked polymer had been noted(15). This asymptotic behavior is thought to represent a similarity between themobilities of large penetrants and the primitive segmental mobility of the polymerthat participates in viscous flow (19).

The Eyring theory of rate processes (20) has been used as the basis of mosttheories for D0:

D0 = eλ2kT/h exp(

�S∗

R

)(15b)

where �S∗ is the entropy of activation and λ is the average jump length. Forpenetrants smaller than the average size of the jumping unit of the polymer,both the preexponential factor and activation energy increase with the size of thepenetrant molecule. The actual diffusion coefficient of the penetrant decreaseswith penetrant size since the exponential weighting of Ed dominates the productin equation 15a.

Molecular models of the diffusion process help clarify the meaning of �Ed.A molecular rationalization of the value of �Ed in terms of the product of cohe-sive energy density (CED) and the volume of 1 mol of cylindrical cavities havinga length λ and a diameter equal to that of a diffusing molecule, dA, has beensuggested (21). With this assumption,

�Ed = CED d2Aπλ/4 (16)


Fig. 5. Activation energy required for diffusion for various penetrants. To convert kJ/molto kcal/mol, divide by 4.184.

An apparent diffusional jump length using such a cylindrical activation vol-ume can be calculated. Assuming the total free volume for a diffusional jump to beequal to the activation volume, the value 0.69 nm3 per molecule for CO2 in Table 2can be used to calculate λ = 8.4 nm. This seems physically unrealistic. Moreover,a large jump length (1.1 nm) would still be calculated if this same amount of freevolume associated with the diffusion jump for CO2 were converted to a sphericalvolume fluctuation. Even this length seems too high, since all of the activated vol-ume may not be directly usable for a linear translation of position. Furthermore,this conceptualization of the activation energy is not in agreement with experi-mentally observed behaviors. Experimentally determined values of the activationenergy extrapolate to zero at some value of the penetrant molecular diameter dAgreater than zero, and correlations of Ed and dA obtained experimentally are notof the simple proportionality expected from equation 16.

To account for the experimentally observed behavior of the activation energy,a more refined picture of the diffusion jump process must be introduced. Instead ofFigure 1, it may be preferable to visualize the activated volume as a bulge in a tube


Fig. 6. Activated volume as a bulge in a tube with the openings of the bulge smaller thanthe penetrant diameter. , Directly usable fraction of activation volume; , not directlyusable fraction of activation volume.

with the openings of the bulge smaller than the penetrant diameter (Fig. 6) (22).In this case, on the average, the penetrant can only execute its ±λ diffusional jumpfrom the center of the bulge to one or the other side, even though the entire freevolume must exist to accommodate the fact that polymer chains are not infinitelyflexible, especially on the short time scale of a diffusional jump. In other words,the chains must have a considerable transition length over which the segmentsreapproach their original packing density; however, the volume associated withthese regions is not directly usable by the penetrant in terms of ±λ translationsof its center of mass.

These ideas were formalized mathematically by Brandt (23), who assumedthat the activation energy was composed of two contributions: (1) an energy Eα as-sociated with bending two initially straight chain segments away from each otherto accept the penetrant molecule, and (2). an energy Eβ needed to overcome repul-sion of the bent chain segments by neighboring chains. The intermolecular energyEα is calculated on the basis of the potential energy barrier to chain rotation, ψ0, ameasure of chain stiffness. The repulsive component Eβ is calculated analogouslyto equation 16; however, the internal polymer pressure pI is used instead of the


cohesive energy density. Using this framework, Brandt calculated the activationenergy as follows:

�Ed = Eα + Eβ = 18ψ0bd(dA − dS)2/a 3d + NApIdpad(dA − dS)/2 (17)

where NA is Avogadro’s number, bd is the length of a backbone bond projectedonto the chain axis, ad is the length of the bent chain segments, and ds is theinitial chain spacing prior to penetrant inclusion. This conceptualization of theactivation energy accounts for many of the shortcomings of equation 16. The acti-vation energy is equal to zero when dA ≤ ds, which is in accord with the nonzeroextrapolations of �Ed versus dA from experimental data. It is also apparent frominspection of equation 17 predicts that no simple relation should exist betweenthe activation energy and the penetrant molecular diameter. Furthermore, equa-tion 17 predicts that the dependence of the activation energy on the penetrantmolecular diameter is weaker for flexible chain polymers, where ψ0 is small. Thismodel of the activation energy is the basis for more complex treatments by Paceand Daytner (24), and Dibenedetto and Paul (25).

Free-Volume Theories. The basic idea of the free-volume theories is thatthe mobilities of the polymer segments and the penetrant molecules in a polymer–penetrant mixture are primarily determined by the amount of free volume in thesystem. As originally proposed (26),

MA = Ad exp[ − Bd

vf

](18a)

where MA is the mobility of the penetrant, vf is the average locally availablefractional free volume of the system, and Ad and Bd are empirical free-volumeparameters that are assumed to be independent of penetrant concentration andtemperature. The parameter Ad depends on the size and kinetic velocity of thepenetrant. The parameter Bd is equivalent to the critical hole free volume neces-sary for a penetrant to make a diffusive jump (13). This expression was modifiedby arguing that crystalline material reduces the free volume in direct proportionto the amount of crystalline material present (27):

MA = Ad exp( − Bd

�avf

)(18b)

where �a is the amorphous volume fraction of the penetrant free polymer at zeropressure and the temperature of the system. Alternate treatments of the effectsof crystallinity will be considered later in terms of a chain immobilization factor(see eq. 45).

According to free-volume theories, the diffusion coefficients of organic va-pors in polymers are strongly concentration-dependent, because mobilities areextremely sensitive to changes in the average free volume of the system. A smallpenetrant that is unconstrained on two sides by covalently bonded neighbors in-troduces much more free volume into the polymer–penetrant mixture than a poly-mer segment of equivalent size. Increased penetrant concentration thereby altersthe effective viscosity of the medium, and a significant increase in the penetrant


mobility results. The free-volume approach is simple and has evolved (28,29) intoseveral readily useful forms. Its primary drawback lies in the difficulty of provid-ing a precise physical definition for the parameter defined as the free volume.

The native polymer, totally devoid of penetrant, still possesses a certaindistribution of free-volume packets, which wander spontaneously and randomlythrough the polymer. Intuitively, proponents of free-volume theory argue that apenetrant can execute a diffusive jump when a free-volume element greater thanor equal to a critical size presents itself to the penetrant. In fact, the polymer chainsmay also execute a self-diffusive jump when a packet of sufficient size presentsitself to a segment. This is the mechanism that causes the slow interdiffusion ofpolymer chains.

Increasing system temperature causes volume dilation, resulting in in-creased free volume. Thus the diffusion coefficient increases with temperature.The free-volume fraction vf may be represented as a linear addition of severalvariables (28,29):

vf (T,p,ϕ1) = vfs(Ts,ps,0) + α(T − Ts) −β( p − ps) + γ ϕA (19)

where vfs is the fractional free volume of the pure, penetrant-free, amorphous,rubbery polymer at some reference temperature Ts, usually the glass-transitiontemperature, and reference pressure ps, usually 101.3 kPa (1 atm). The penetrantvolume fraction is �A and the coefficients coefficients α, β, and γ are positiveconstants, the values of which are evaluated empirically. These coefficients char-acterize the effectiveness of temperature, pressure, and penetrant concentration,respectively, for changing the free volume in the amorphous phase:

α =(

∂vf

∂T

)sβ =

(∂vf

∂p

)sγ =

(∂vf

∂ϕ1

)s

(20)

where s denotes some reference state. The free-volume thermal-expansion coeffi-cient α can be estimated as the difference between the thermal expansion coeffi-cient for an amorphous material above and below its glass-transition temperature.The parameter β can be related to the conventional compressibility coefficient β ′

by (30)

β

1 − v∗f

= β′ = − 1

V

(∂V∂p

)T

(21)

where v∗f is the fractional free volume of the pure polymer at zero pressure. If the

glass-transition temperature is used as the reference temperature, v∗f is given by

v∗f = vfs + α(T − Tg) + βps (22)

Combination of equations 18 and 19 and use of the definition of mobilityleads to the following expression for the polymer self-diffusion coefficient, when


the reference temperature is taken to be the glass-transition temperature:

D∗A = Ad exp

( − Bd

�a{vfs + α[T − Tg] −β[ p − ps] + γ ϕA})

(23)

Thus the self-diffusion coefficient D∗A and its variation with concentration

and temperature can be estimated if Ad, Bd, and γ are known. A correlation for γ

as a function of the fractional increase in volume caused by a penetrant moleculehas been developed (29). In the absence of data for a given polymer, vfs and α canbe approximated as the universal values of 0.025 and 4.8 × 10− 4/◦C, respectively,in accordance with the theory of Williams, Landel, and Ferry (31) that interpretsthe glass-transition point to be an isofree volume state. This approximation isvalid in the temperature range from Tg to 100◦C above Tg.

In principle, each of the coefficients in equation 22 can be evaluated inde-pendently by observing the effects on DA

∗ over a sufficiently wide range of tem-peratures, external hydrostatic pressures, and sorbed penetrant concentrations.Hydrostatic effects on β can be decoupled from penetrant sorptive effects on γ

by using a very low sorbing penetrant, such as helium in the presence of a fixedpartial pressure of the penetrant of interest. On the one hand, hydrostatic pres-sure is expected to have a rather small effect on DA

∗ since solid polymers are onlyslightly compressible. On the other, increases in temperature and sorbed pene-trant concentration cause large increases in the free-volume fraction vf and in theself-diffusion coefficient DA

∗.The permeability P is an important transport coefficient that represents the

normalized molar flux NA across a polymer film of thickness � with a partialpressure driving force �p at steady state:

P = NA

�p/�(24)

Under steady-state conditions, nP in equation 2a is zero, and for the low sorbedmass fractions typical of gases in polymers, an expression for NA = nA/MA can besubstituted to yield

P = − ρDAP�

MA�p∂ωA

∂z(25a)

where MA is the molecular weight of component A. Integration between the up-stream and downstream conditions gives

P = − 1p2 − p1

ωA2∫ωA1

− ρDAP(ω) dωA

MA(25b)

where ωA2 and ωA1 are the equilibrium penetrant weight fractions in the polymerat upstream (p2) and downstream (p2) faces. At low concentrations, DAP reduces


to DA∗. Transformation of equation 25a from mass fractions to volume fractions

gives

P = − 1p2 − p1

ϕA2∫ϕA1

− ρD∗A

MA

dωA

dϕAdϕA (25c)

where dωA/dϕA can be evaluated using the specific volumes of the polymer andpenetrant. Substitution for DA

∗ from equation 22 for the case of negligible down-stream pressure yields the following expressions (32) when Henry’s law applies(ϕA2 = kAp2) at the upstream membrane face:

ln P = ln[RTAdkA] − Bd

�av∗f

+ p2Bd

�a(v∗f )2

[− β + λkA

2

(1 + 2βp2

v∗f

)](26)

Therefore, the free-volume theory permits theoretical plots of permeabilityas a function of temperature, penetrant pressure, and amorphous volume fractionin the rubbery polymer.

It is worth noting that in a series of articles, Vrentas and Duda (33–36)have introduced and applied a more rigorous free volume theory than the onepresented above. In their approach, Vrentas and Duda include an energy factor,EJ, to the preexponential term Ad. This factor accounts for the energy requiredfor a penetrant molecule to jump into an adjacent open hole. The exponentialparameter Bd is modified to account for the molar volume of the moving polymersegment. Furthermore, Vrentas and Duda propose that only part of the total freevolume is available for diffusion and is denoted by

�

Vfh. This concept of only afraction of the free volume participating in diffusion is similar to the activationenergy model of Brandt (see Fig. 6). The Vrentas and Duda model ultimately leadsto a prediction of the thermodynamic diffusion coefficient:

DT = Ad0 exp( − EJ

RT

)exp

[γov

(wA + wPMA

MPJ

)V̂∗

h

V̂fh

](27)

where wA and wP are mass fractions of the penetrant and polymer, respectively,and MA and MPJ are the molecular weight of the penetrant and the jumpingpolymer segment respectively. V̄∗

h is the specific volume of holes of the minimumsize required for a diffusive jump. γ ov is a parameter, with a value between 0.5and 1, originally introduced by Cohen and Turnbull that accounts for the abilityof a free volume hole to be available to multiple jumping segments. Althoughthe added sophistication of this model probably captures a more accurate pictureof the diffusion process, the accompanying added mathematical complexity hashindered widespread adoption of the model.

Membrane and Barrier Implications. In the case of typical rubbers,crankshaft and other related rotational motions of the repeat units are so largethat little difference in diffusion coefficients exists for simple gaseous penetrants,the sizes of which differ by less than a few hundredths of a nanometer. Of course,since permeation is a solution-diffusion process, membranes or selective barriers


do not operate strictly on the basis of size selection. Under a fixed partial pressuredriving force for component i, the flux through a material of given thickness � isdetermined by the permeability, as can be seen by rearrangement of equation 24.

Ni = �piPi/� (28)

The permeability can be expressed as the product of a diffusivity and solu-bility coefficient:

Pi = D̄iS̄i (29)

The diffusivity and solubility coefficients in the above expression are effectiveaverages applying across a polymer film between its upstream and downstreamfaces. The presence of other copermeating components can often be neglected forgases and low activity vapors permeating through rubbers (37). The average mu-tual diffusion coefficient D̄i is given below for penetrant A in the polymer compo-nent P:

D̄i =

ωA2∫ωA1

DAP dωA

ωA2 − ωA1(30)

When the presence of other components cannot be neglected, the system canstill be treated approximately by considering the membrane in the presence of theadditional component to be a new effective medium; however, this is at best a roughapproximation since it neglects the bulk-flow contribution and thermodynamicinfluence of additional components on the flux of component A. Multicomponentthermodynamic issues are beyond the scope of this discussion, but have beentreated (38).

The solubility coefficient, S̄i, is found from combination of equations 25b, 29and 30 as:

S̄i = Ci2 − Ci1

pi2 − pi1(31)

where Ci2 and Ci1 are the concentrations of “i” in the membrane at the upstreamand downstream faces, respectively. pi1 and pi2 are the partial of “A” at the up-stream and downstream faces, respectively. Under ideal conditions of fixed up-stream pressure and negligible downstream pressure, the S̄i parameter is equalto the secant slope of the sorption concentration vs pressure isotherm evaluatedat the upstream partial pressure pi2 of the component

S̄i = Ci

pi2(32)

where Ci typically has units of cm3(STP)/cm3 polymer, and Si can be related to theHenry’s Law constant ki in equation 26 for the gas in the polymer using partialmolar volumes such as those in Table 2 in a subsequent expression (see eq. 32).


The temperature dependence of solubility coefficients are typically described interms of van’t Hoff expressions shown below (39):

ln(

Si(T2)Si(T1)

)= − �Hs

R

[1

T2− 1

T1

](33)

Equation 33 only applies when the solubility coefficient is independent ofconcentration or for solubility coefficients specified at the same concentration.The enthalpies of sorption �Hs are typically small and slightly negative (Fig. 7a).The solubilities and enthalpies of sorption of different penetrants are determinedlargely by their critical temperatures or other related measures of tendency to ex-ist in a condensed phase such as normal boiling points or Lennard-Jones potentialwell depth ε/k̄ (Fig. 7b) (40).

From the dependence of permeability on both solubility and diffusivity, it isclear that two opposing factors affect the permeability as temperature is increased,namely

ln(

P(T2)P(T1)

)= ln

(D(T2)S(T2)D(T1)S(T1)

)= −

[Ed + �Hs

R

][1

T2− 1

T1

](34)

The positive diffusional activation energy is larger in absolute value than thenegative �Hs, and so the overall permeability increases as temperature increases,but to a lower degree than the diffusion coefficient itself. It should be noted thatthe use of equation 34 is only strictly valid when the diffusion and solubilitycoefficients are independent of concentration.

The separation factor or permselectivity between two penetrants A andB, αAB, is important in membrane separation systems. This factor is equal tothe ratio of the downstream (permeate) mole fractions of component A rela-tive to component B divided by the ratio of the upstream mole fractions of Arelative to B. Under conditions of negligible downstream pressure, the perms-electivity is simply equal to the ratio of the permeabilities of components Aand B. In addition, since the presence of one component has a negligible ef-fect on the permeability of the other at low pressures in rubbery materials,pure component permeabilities can often be used as good estimates of mixed-gaspermeabilities:

αAB = PA

PB=

(xA

xB

)1

/(xA

xB

)2=

(DA

DB

)(SA

SB

)(35)

The subscripts 1 and 2 refer to downstream and upstream conditions, respec-tively. The ratio of the two diffusion coefficients is commonly called the diffusivityselectivity of a membrane and the ratio of the two solubilities, the solubility selec-tivity. A typical range of permeabilities and selectivities achievable with rubberymaterials is indicated in Table 3 for several gas pairs of commercial interest.More extensive tabulations are available (41). The presence of polar groups inthe polymer molecule generally leads to low diffusivity for penetrants (15); eg,the diffusivity decreases strongly on increasing the nitrile content in a series of


Fig. 7. (a) Enthalpies of solution for various gases in elastomers: A, �, natural rubber;B, , silicone rubber (30). To convert kJ/mol to kcal/mol, divide by 4.184. (b) Correlation ofsolubility of several gases in rubbery amorphous polymers: , silicone rubber; �, naturalrubber; �, styrene–acrylonitrile rubber; �, amorphous polyethylene. Temperature, 25◦C(30). To convert [cm3(STP)]/(cm3 · MPa) to [cm3(STP)]/(cm3 · cm Hg), multiply by 1.333 ×10− 3.

Table 3. Permeabilities, Solubilities, and Diffusivities of Various Gas Pairs in Rubbery Polymersa

PHe, SHe, PO2 , SO2 ,mmol/ cm3(STP)/ DHe, mmol/ cm3(STP)/ DO2 ,

Polymer (m · s · TPa)b PHe/PCHc (cm3 · GPa) SHe/SCH4 10− 6cm2/s DHe/DCH4 (m · s · TPa)b PO2 /PO2 (cm3 · GPa)c SO2 /SN2 102 cm2/s DO2 /DN2

Natural rubberd 10.4 1.05 105 0.042 22 25 8.04 3.0 1125 2.0 1.6 1.5Polyethylenee 1.64 1.7 54 0.048 6.8 36 0.97 3.0 473 2.1 0.46 1.44Poly(methylacrylate)f

3.55 45.1 101 0.069 7.88 650

Poly(ethyleneterephthalate)g

3.1 44 58 0.10 12 430 0.18 3.8 218 1.7 0.19 2.2

Polychloropreneh 1.34 3.42 750 2.13 0.4 1.6Butyl rubberi 0.44 3.94 1200 2.19 0.081 1.8aAt 25◦C, unless otherwise specified.bTo convert mmol/(m · s · TPa) to cm3(STP)cm/(cm2 · s · cm Hg), multiply by 2.98 × 10− 10.cTo convert cm3(STP)/(cm3 · GPa) to cm3(STP)/(cm3 · cm Hg), multiply by 1.333 × 10− 6.dPoly(cis-isoprene).eρ = 0.914 g/mL; α = 0.57.f At 35◦C.gAt 100◦C.hNeoprene.iIsobutylene/isoprene (98/2) copolymer.

312


butadiene–acrylonitrile copolymers. The higher the nitrile content, the greater theattraction between neighboring chains. These interchain attractions are quanti-fied by the cohesive energy density (CED). High CED tends to elevate the diffusionactivation energy and hence depress the diffusivities according to equation 15b.

The diffusivity selectivities of the various rubbers in Table 3 are generallylow, consistent with the large scale of segmental motions in rubbery materialsrelative to the penetrant sizes that must be distinguished. Besides difficulties inachieving high selectivities, rubbery materials are generally unable to be self-supporting under pressure in a high surface area hollow fiber form. This problemcan be circumvented by placing a thin film of the rubber on a microporous supportto make a composite membrane (42). Generally, however, the much higher selec-tivities of glassy polymers for separating gases make glasses the favored polymericmaterials in separation applications.

For cases involving a random copolymer or a miscible blend of two amorphousrubbery polymers, the behavior is generally a volume fraction weighted averageof the permeabilities of the two homopolymers. On the other hand, the trans-port properties of immiscible blend systems depend significantly on the relativepermeabilities and the morphology of the immiscible blend.

Nonideal Transport Effects in Rubbery Polymers

Plasticization and Hydrostatic Effects. New complexity arises in thebasic transport properties of a polymer at higher penetrant activities, since thelocal concentration of penetrant within the polymer can become quite high. Athigher pressures (approaching the saturated vapor pressure), these high con-centrations can occur in cases where favorable polymer–penetrant interactionspromote solubility. This is common for organic vapors and even for small polaror quadrupolar molecules such as CO2, H2S and SO2. Transport plasticization isdefined as a significant increase in the diffusivity of a penetrant because of facil-itation of local polymer segmental motion caused by another penetrant moleculein its neighborhood. This definition applies both to the increase in diffusivity of apenetrant caused by the presence of its own kind and increases caused by a dif-ferent component. Even for pure component penetrants, a detailed fundamentalanalysis of this phenomenon on a molecular basis has not been achieved; however,various free-volume analyses are available (43).

Plasticization is typically an unfavorable phenomenon in separation appli-cations because permselectivity is reduced as the diffusivity of a slower penetrantis increased. In the case of gases, plasticizing phenomena are suppressed becauseof hydrostatic compression in some polymers despite rather high sorption levels.Analysis in terms of the free-volume approach requires consideration of the com-pressibility of the rubbery matrix. In fact, exposure to the hydrostatic pressureof low solubility gases, such as helium or nitrogen, may actually compress outfree volume and thereby reduce the ability of polymer segments to open gaps formovement of other segments or sorbed penetrants.

Some data (44) indicate a marked reduction in diffusivity of n-butane as theexternal pressure of helium is increased, and it was concluded that helium gasapplied to silicone rubber acts only as a pressuring medium within experimental


limitations. The helium decreases the free volume of the silicone rubber and thusdecreases its molecular motion. Plasticization and hydrostatic effects on the per-meability of carbon dioxide, helium, nitrogen, methane, and ethylene in siliconerubber also have been studied (45). Over an extended pressure and concentrationrange, both compression of free volume and eventual plasticization have been ob-served for these different penetrants. As the amount of sorbed penetrant increases,the plasticization effect eventually overcomes the hydrostatic compression effecton the rubbery matrix for the more condensable penetrants such as carbon dioxideand ethylene.

The local solubility and diffusivity must be considered to see when truetransport plasticization occurs. For this it is more useful to consider the localmutual diffusion coefficient DAP(C) rather than the average value assessed fromequation 30. The local mutual diffusion coefficient is a measure of the ability ofa penetrant to move through the membrane at a point where the local conditionsare well defined in terms of the local penetrant concentration CA, volume fractionϕA, weight fraction ωA, or penetrant fugacity, whichever is most convenient.

Figure 8 (45) shows the permeation data for methane, carbon dioxide, ethy-lene, helium, and nitrogen in silicone rubber at 35◦C over a broad range of pres-sures. The data agree reasonably well with the data reported previously over asmaller pressure range (46). For helium and nitrogen, which have low critical tem-peratures, the permeability decreases with increasing pressure, while it increaseswith increasing pressure for carbon dioxide and ethylene, which have relativelyhigh critical temperatures. For methane, with an intermediate critical tempera-ture, the permeability shows negligible pressure dependence.

The sorption isotherms shown in Figure 9 (16) for each of the penetrantsobey Henry’s law at low pressures, and the more strongly sorbing carbon dioxideand ethylene show Flory–Huggins swelling behavior at high pressures. The linesin these plots are, in fact, modified Flory–Huggins fits to the data according toequation 36, which includes the presence of the small amount of cross-linking inthe sample (47):

ln(

pp0

)= ln(ϕA) + (1 − ϕA) + χ (1 − ϕA)2 + VA

(ve

V0

)((1 − ϕA)1/3 − (1 − ϕA)

2

)(36)

where p is the penetrant pressure, p0 is the vapor pressure of the penetrant,VA is the partial molar volume of the penetrant, ve is the effective number ofcross-links expressed in moles, V0 is the volume of the penetrant-free polymer,and χ is the Flory–Huggins interaction parameter. The permeability and sorptionisotherms can be used along with the following expression to calculate the localconcentration-dependent mutual diffusion coefficient (48):

DAP(CA2) =[{

P(p) + pdPdp

}(dp

dCA

))∣∣∣∣p2

(37)

The Flory–Huggins expression can be used to evaluate the dp/dCA term atthe upstream pressure for conditions where Henry’s law does not apply; however,only the region of Henry’s law is discussed here. Figure 10 shows the local diffusioncoefficient for the various penetrants as a function of local fugacity in silicone


Fig. 8. Comparison of the pure-gas permeabilities of various gases in silicone rubber at35◦C. (a) CO2, C2H4, and CH4; (b) He and N2 · 1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s · cmHg) = 0.335 mmol/(m · s · TPa). To convert MPa to psi, multiply by 145.

rubber (45). This can be easily converted to a plot of DAP(C) vs C using the sorptionisotherm data if desired. For carbon dioxide, the fugacity differs by as much as22% from the pressure at the maximum pressure studied. The chemical potentialdifference is of course the true thermodynamic driving force for diffusion andshould be used in mixed-gas calculations.


Fig. 9. Sorption isotherms for Sorption and desorption of various gases in silicone rubberat 35◦C. �, Sorption; �, desorption. To convert MPa to psia, multiply by 145.

Fig. 10. The local effective diffusion coefficient of various penetrants in silicone rubber.To convert MPa to psia, multiply by 145.


The local diffusion coefficient of nitrogen decreases with increasing pressureuntil around 2.1 MPa (300 psia), where it levels off and slowly begins to rise withincreasing pressure. These results indicate that the fractional free volume of thepolymer is reduced because of nitrogen’s hydrostatic pressure. At higher nitrogenpressures the solubility level is sufficient to oppose the compressive effect andcause the diffusivity to begin to increase with pressure as free volume is added bythe penetrant. The results for methane and carbon dioxide show a slight decreasein diffusivity with increasing pressure, but both gases reapproach their originaldiffusivity at higher pressures because of their higher solubilities. Ethylene ex-hibits an immediate increase in diffusivity with pressure that is not surprising,given its significant solubility and the large dilation in volume with increasingethylene pressure, shown in Figure 11 (16). Even below 0.7 MPa (100 psia), moreethylene is sorbed than nitrogen in the silicone rubber at 7 MPa (1000 psia).

The negative slope of the permeability vs pressure plot for helium and nitro-gen in Figure 8b and the positive permeability slope for the other gases in Figure8a can be explained by these two competing effects caused by the penetrant pres-sure. The data in Figure 11b show an actual compression in volume, even thougha small but measurable amount of sorption was observed under these conditions.The results are believed to occur because of the low solubility of helium and highcompressibility of silicone rubber. Therefore, the two competing factors that de-termine the volume dilation of the polymer are (1) the sorption of gas into thepolymer that results in an increase in free volume, and (2) the hydrostatic pres-sure on the film that reduces the free volume. The second effect is usually smalland is overcome for gases that have significant sorption levels. The net result ofthese two opposing effects determines whether the permeability (and diffusivity)will increase or decrease with increasing pressure. These two competing factorscan be described quantitatively by equations 23 and 26.

The expression for the local mass diffusion flux accounting for the frame ofreference term discussed in equation 2b is given below:

nA = − ρDAP

1 − ωA

∂ωA

∂z(38)

Using equations 7 and 37, the effective mutual diffusion coefficient obtainedfrom analysis of the permeability versus pressure data can be related to the self-diffusion coefficient of the penetrant, namely

DAP = D∗A(1 −ωA)

(∂lnaA

∂lnωA

)(39)

The factor (∂ ln aA/∂ ln ωA) is available from the sorption measurements suchas those in Figure 9. For the cases considered here, where Henry’s law applies,this factor is simply equal to unity, whereas it is given by equation 40 for the casewhere curved isotherms characterized by the Flory–Huggins equation apply.

(∂lnaA

∂lnωA

)= ωPρ2

A

(ωAρP + ωPρA)2

(1 − 2χωAρP

ωAρP + ωPρP

)(40)


Fig. 11. Volume dilation isotherms for Sorption of various gases in silicone rubber at35◦C. �, Sorption; �, desorption (46). To convert MPa to psia, multiply by 145.

where ρA and ρP are the mass densities of the pure penetrant and polymer, re-spectively.

Penetrant mobilities (MA) were determined as a function of local fugacity forsilicone rubber (45). The mobility of nitrogen decreased with increasing pressure,but the mobility of carbon dioxide increased, suggesting that nitrogen acts predom-inantly as a pressurizing agent and CO2 acts primarily as a plasticizing agent. Thepermeability of helium and nitrogen decreased with increasing pressure becauseof the reduction in free volume caused by the hydrostatic pressure of the gas.Therefore, at high pressure, these gases tend to lower the permeability of eachcomponent unless another component has sufficient solubility to overcome the


hydrostatic effect. To test this hypothesis the silicone rubber films were exposedto both a 10/90 mol% mixture and a 50/50 mol% mixture of carbon dioxide andnitrogen.

The presence of additional penetrants in the feed stream can be accounted forby the addition of a term in the free-volume equation to account for the presenceof each penetrant and the applied hydrostatic pressure above ps = 101.3 kPa(1 atm) as follows:

D∗A = RTMA = Adexp

( − Bd

vfs + α[T − Tg] −β[p − ps] + γ1ϕ1 + γ2ϕ2 + · · ·)

(41)

The carbon dioxide permeability for both mixtures was depressed below thepure carbon dioxide values, but the carbon dioxide permeability for the 50/50mixture did not display a decreasing tendency with pressure as was observedfor the 10/90 CO2/N2 mixture. Therefore, at the lower nitrogen partial pressuresobserved in the 50/50 mixture, the impact of the hydrostatic pressure effect isreduced by the swelling effect of CO2. Although the swelling effect of CO2 cannotcompletely overcome the hydrostatic pressure effect of CO2 nitrogen, it greatlyoffsets its influence.

Important extensions of the foregoing example, where hydrostatic pressureeffects may be of considerable importance, involve supercritical extraction us-ing high pressure CO2 or ethylene for processing of foods, pharmaceuticals, andpurification of polymers by removal of trace monomers (49). This topic involvesgreater complexity than the binary diffusion case. Moreover, the factor involvingthe activity coefficient derivative may become very significant near the criticalpoint. Transport behavior in these systems, however, can be anticipated on thebasis of the results shown for the pure components. Little enhancement in theself-diffusion coefficient of the penetrant in these rubbery polymers is apparentat pressures below the critical pressure; however, both carbon dioxide and ethy-lene begin to overcome the effects of hydrostatic pressure and to show upturnsin their mobilities as the critical conditions are approached. Most of the work inthe area of mass transport under supercritical conditions in polymers has beenrather qualitative, but more quantitative analysis should be possible using thefree-volume model.

Penetrant Clustering. Clustering may reduce the effective mobility of apenetrant such as water or methanol that can self-associate. The activated staterepresentation of diffusion is typically used to discuss the suppressed diffusioncoefficient for a clustered penetrant. If the diffusion process involves primarilyjumps by monomeric, unassociated penetrants, energy must be available both tomake a gap to allow the penetrant to jump into and to cause the dissociation ofone of the penetrants in a local region adjacent to the gap. Therefore, higher ac-tivation energies for diffusion and, hence, lower diffusion coefficients at activitiesabove which clustering occurs are expected if this visualization is correct. Workto document this point of view has been done for silicone rubbers (50), poly(alkylmethacrylates) (51), and hydrocarbon rubbers (52). Typical diffusion coefficientsand activation energies for diffusion of water and methanol, both of which tendto cluster in hydrophobic environments, are shown for various rubbery materials


Fig. 12. (a) Effects of water clustering on diffusion in rubbers. Concentration dependenceof the diffusion coefficient for water in Cariflex IR305: A, 37◦C; B, 42◦C; C, 50◦C; D, 61◦C.(b) Effects of methanol clustering on diffusion in rubbers. Concentration dependence of thediffusion coefficient for methanol in Cariflex IR305: A, 26◦C; B, 31.1◦C; C, 50◦C; D, 60◦C.

in Figures 12a and 12b (52) and in Table 4 as a function of concentration overconditions prior to and after the onset of clustering.

The sorption behavior in clustering systems is often discussed in terms ofthe cluster integral GAA given below in terms of the partial molar volume VA, theactivity aA, and the volume fraction of penetrant ϕA:

GAA

VA= − (1 − ϕA)

(∂(aA/ϕA)

∂aA

)p,T

− 1 (42)

GAA may be determined from equilibrium sorption data as a function of pen-etrant partial pressure. Based on the theory of Zimm and Lundberg (53), thecluster function given by (ϕA GAA/VA) equals the excess number of A molecules


Table 4. Activation Energies in Clustering Systems

C, cm3[STP]/cm3

System polymer �ED, kJ/mola Ref.

Watercis-1,4-Polyisoprene 0 42 50

0.3 580.5 58

Polydimethylsiloxane 0 14 480.45 460.6 75

poly(3,3,3-trifluoropropylmethylsiloxane) 0 170.7 50 48

Poly(butyl methacrylate) 0 28 492 444 50

MethanolNatural rubber 0 42 50

5 548 71

poly(3,3,3-trifluoropropylmethylsiloxane) 0 254 29 498 38

aTo convert kJ/mol to kcal/mol, divide by 4.184.

above the number that would exist in the neighborhood of an arbitrarily chosentype A molecule if the mixing were totally random, ie, no clustering. As shown inFigure 12, the effective mutual diffusion coefficient is constant below the activityat which clustering occurs. The activity at which the cluster function deviatessignificantly from zero corresponds to the point at which the diffusion coefficientbegins to drop significantly, and the activation energy begins to increase.

Intermediate-Size Penetrants. These penetrants are defined asmolecules with sizes approaching a polymer segment, but much smaller thanan entire polymer chain. The infinite-dilution mutual diffusion coefficients forincreasingly large, but not macromolecular, penetrants approach the diffusion co-efficient for the segment of a polymer diffusing within itself. Movement of theentire polymer molecule itself is clearly a slower and more complex concerted pro-cess than the local movement of its segments; however, the segmental mobilityand the self-diffusion coefficient of the entire molecule can be related as discussedlater.

For intermediate-size penetrants such as typical organic solvents, the self-diffusion coefficient of the polymer is much smaller than that for the penetrant.Consistent with this expectation, equation 8 has been found to be appropriate (36)for mass fractions of toluene as high as 85%. Therefore, for low and intermediatepenetrant concentration, equation 8 should still apply as it did in the case of gasesand other small penetrants.

The data in Figure 3 reflect the tendency to approach an infinite-dilutionasymptotic mobility as the diameter of these intermediate-sized penetrants


Fig. 13. Concentration dependence of the self-diffusion coefficient and the thermo-dynamic factor Q for the toluene–polystyrene system at 110◦C and for the o-xylene–polyethylene system at 150◦C. Q = (1 − ωA)(∂ ln aA/∂ ln ωA. From Ref. 45.

increases (see eq. 6):

limωA→0

{DAP} = limωA→0

RTMA(1 − ωA)(

∂lnaA

∂lnωA

)= RTMA = RT

ζA(43)

Care must be used in applying such arguments when estimating practical dif-fusion coefficients, however, since increases in ω1 above the infinite-dilution limitcan cause significant changes in the effective mobility of the penetrant. Moreover,the (∂ ln aA/∂ ln ωA) term can deviate from unity at weight fractions as small as10–20%, as was indicated in the case of carbon dioxide and ethylene earlier. Inthe case of o-xylene diffusion in polyethylene at 150◦C, both the effects of plasti-cization and of the (∂ ln aA/∂ ln ωA) term on the self-diffusion coefficient have beenanalyzed and are shown in Figure 13 (34).

Except for supercritical extraction conditions, hydrostatic pressure effectsare typically of negligible importance for simple solvent vapors diffusing inpolymers, since the saturation vapor pressure is low, <101.3 kPa (1 atm), in mostapplications. The predictive power of the approach is indicated by the resultsfor the mutual diffusion coefficient of toluene in a toluene–polystyrene system(Fig. 14) (35).

The rather complex behavior represented in Figure 14 was analyzed in termsof the respective contributions to the self-diffusion coefficient and the significantreduction in the thermodynamic (1 − ωA)(∂ ln aA/∂ ln ωA) terms in equation 39.These two constituent contributions indicate that the dramatic increases in the


Fig. 14. Dependence of diffusivity on mass fraction for the toluene–polystyrene system.

mutual diffusion coefficient at low solvent mass fractions in the polystyrenesystem at 100◦C is primarily due to dramatic transport plasticization-inducedchanges in the penetrant self-diffusion coefficient, rather than due to the thermo-dynamic term.

The strong temperature dependence of the infinite-dilution diffusion coeffi-cient in the vicinity of the glass-transition temperature of the polymer (Tg =105◦Cfor polystyrene) cannot be predicted by a simple exponential expression such asthe activated state model (equation 15a) unless the energy of activation is allowedto vary. Variation in the activation energy in this region, where backbone segmen-tal motions are beginning a transition from a primarily oscillatory nature aroundrelatively fixed positions to include large rotary crankshaft motions, is not unrea-sonable, but it cannot be predicted directly from the activated state theory. Onthe other hand, such near- Tg behavior is a natural expectation of the free-volumetheory based on the Williams–Landel–Ferry (WLF) description of the effects offree volume on polymer properties.

Recent attempts to apply the free-volume approach below the glass transi-tion have also been made (33). This sub- Tg problem often includes additionalfactors not related simply to Fickian transport processes, since the movementsof intermediate-sized penetrants in contact with glassy polymers often inducecomplex time-dependent relaxations.


Semicrystalline and Cross-Linked Rubbery Polymers

Gases and Low Activity Vapors. Regions of crystallinity or points ofchemical cross-linking have somewhat similar effects on the transport propertiesof a rubber causing restriction of swelling and suppression of long-range chainsegmental motion. Semicrystalline polymers, however, are more complex thancross-linked materials because of tortuosity caused by the presence of the typi-cally impermeable crystalline regions. Similarly, adding fillers such as glass, talc,or mica particles to an amorphous matrix prior to cross-linking can make the trans-port properties of the resultant material take on a semicrystalline-like characterif the interfacial adhesion between the filler and matrix phase is good. Promotionof adhesion can be achieved by using interfacial wetting agents such as silanesand titanates that bond to the inorganic surface and can either chemically bondor physically interact with the elastomer matrix. In the absence of such adhesionpromoters, the barrier properties may be degraded because of interfacial defectsnot found in semicrystalline materials under most conditions.

The effects of fillers or crystallites can be influenced considerably not onlyby the volume fraction of this phase, which affects the solubility of a componentdirectly, but also by the area-to-thickness aspect ratio and orientation of the fillerwith respect to the direction of permeation. The orientation factor affects the ap-parent diffusivity by determining how the crystallites or fillers behave as tortuousimpediments to penetrants. This effect is not considered in the simple free-volumeanalysis (eq. 18a). Since the effect of orientation can be significant, approacheswhich account for tortuosity and chain mobility are more desirable. Examplesof materials where orientation effects are significant include phase-separatedurethane–elastomer block copolymers and random copolymers of ethylene andvinyl alcohol since the two components are able to cocrystallize in the same crys-tal lattice. This latter copolymer is used as a barrier in which the permeability ofthe material can be tailored by varying the amount of the highly polar, diffusion-inhibiting (vinyl alcohol) component without strongly affecting the crystallinity.

Studies of gas sorption and transport strongly support the notion of theimpenetrability of crystalline domains by even tiny gas molecules. As shown inFigure 15, the solubility coefficient (S̄i = Ci/pi) of both gases and low activityvapors are essentially proportional to the volume fraction of amorphous material�a and the intrinsic solubility coefficient for the totally amorphous materialS̄a (54), ie

S̄

S̄a= �a (44)

The effects of crystallinity on the diffusion coefficient can be more complexthan this simple volumetric exclusion, since the crystallites may act not only astortuous barriers, but also as effective restrictors of chain motion analogous tochemical cross-linking. This twofold effect has been treated (54) in terms of atortuosity factor τ , and a chain immobilization factor β∗, both of which increasewith increasing crystalline fraction. The parameters D and Da are the diffusion


Fig. 15. Linear relationship between the infinite dilution solubility coefficient and theamorphous fraction in polyethylene at 25◦C (30). To convert to [cm3(STP)cm]/(cm3 · MPa)to [cm3(STP)]/(cm3 · atm), multiply by 0.1013.

coefficients in the actual sample and in a totally amorphous sample, respectively:

D = Da

τβ∗ (45)

Chain restriction effects due to the effective cross-linking of crystallites maybe negligible for helium compared to other penetrants due to the much smallerfree volume required for an activated jump by helium. For example, 0.075 nm3

is the required free volume for a jump of helium in poly(vinyl acetate) as op-posed to 0.3 nm3 for O2 and 0.69 nm3 for CO2. This premise allowed estimationof the two factors in equation 45 for polyethylene (40). A 65% reduction in he-lium diffusivity for a 50% amorphous fraction material relative to a hypotheti-cal branched polyethylene with no crystallinity was attributed to tortuosity. Thediffusivities of the amorphous polyethylene were approximated as equivalent to


Fig. 16. Relative mobilities in semicrystalline and amorphous polyethylene for severalpenetrants as a function of penetrant van der Waals volume.

those of unvulcanized natural rubber. The reduction in diffusivity from simpletortuosity in polyethylene was estimated to be uniformly equal to 65% for all pen-etrants, and additional depressions in diffusivity relative to the amorphous refer-ence were attributed to chain restriction effects due to the crystallites. The rela-tive mobilities in the semicrystalline and amorphous samples are summarized inFigure 16.

In a treatment of the effects of crystallinity on transport in rubbery polymersusing a modification of free-volume expressions (26), it was suggested that theintroduction of crystallinity reduces the freedom of motion of amorphous chainsegments between crystals. Although these ideas are physically consistent withthose described above in terms of the chain immobilization factor β∗, a ratherdifferent analytical form of the diffusion coefficient is obtained.

The complementary effects of orientation and crystallinity can be treated interms of the τ factor, since the aspect ratio and degree of orientation may alterthe degree to which tortuosity influences the suppression in D. However, τ mustcurrently be used for characterization rather than being an a priori predictablefactor. A thorough treatment of the effect of uniaxial elongation at room tempera-ture has been given for CO2 diffusion in low density polyethylene (55). The effectsof the orientation are complex. Prior to the onset of plastic deformation at 6% elon-gation, the solubility increases about 3%, and the diffusivity increases about 10%,and so a net increase of about 13% is observed in the permeability, presumablybecause of the introduction of free volume in the sample that dilates during theelongation. Such dilation is characteristic of materials with a Poisson ratio lessthan 0.5.

On the basis of increase in the sorption coefficient, it is suggested that thedeformation causes dilation of the amorphous regions. At larger deformations, Dfalls below its original value for the unoriented crystalline material, since smallincreases in free volume are unable to offset the very large increase in tortuosity.


This increase in tortuosity can be anticipated as crystalline regions become moredeformed by slippage and crystal lamellae rotate to adopt orientations more par-allel to the film surface. Such organization makes efficient use of the lamellaeas overlapping tortuous barriers capable of producing larger reductions in theeffective diffusion coefficient than would be possible with spherical crystallitesor even randomly oriented lamellae of equivalent volume fraction. Moreover, thealignment of chains in the amorphous regions between the plastically deformedspherulites may also act to increase the chain immobilization factor β∗.

Larger permanent deformations (up to 300% elongation), induced by eitheruniaxial drawing or roll processing at room temperature, reportedly produce re-ductions in diffusivity, solubility, and permeability (56). Larger reductions wereseen for roll processing as compared to the simple drawing process, presumablydue to the elimination of interfacial defects between the crystalline and amor-phous phases in the rolling process.

The effects of crystalline regions on glassy materials have been found to beless extreme than for rubbery materials. For example, the chain immobilizationfactor affects the diffusion of different sizes of penetrants only in rubbery mate-rials (57). This effect has been explained in terms of the low mobility of chainsegments in glassy environments compared to the rubbery state. The concept ofchain immobilization by crystallites loses significance in glasses, since the rigid-ity of the polymer backbone apparently outweighs any additional restriction onmobility imposed by crystallinity.

Although it appears in general that the assumption of impenetrability ofcrystalline domains is valid, poly(4-methyl-1-pentene) (PMP) may have crystalsthat do not fit this pattern (58). Gaps of almost 0.4 nm appear to exist between thechain segments in this material, which is consistent with the low crystal densityof <0.85 g/cm3 (Fig. 17). Contrary to the polyethylene case, these results provideprovocative support for the possibility that measurable sorption does occur in thecrystalline regions of this polymer. The two independent measures of crystallinityoffered by X-ray and thermal analysis were found to be in good agreement.

Although the crystalline regions of PMP were found to be penetrable, the CO2and CH4 permeabilities are still dependent on �a, since the solubilities and diffu-sivities in the more structured phase were lower than in the amorphous phase. Itwas concluded that gases such as CO2 and CH4 dissolve in the crystalline regionsof PMP at about 1/3 to 1/4 the extent in the amorphous phase (58). Moreover, thediffusion coefficients of CO2 and CH4 are about 37% and 64% lower, respectively,in the crystalline phase than the amorphous phase.

The nonzero intercept in the solubility coefficient vs amorphous fraction plotin Figure 18 (59) supports the notion that sorption occurs in the crystalline regions.Although the data provide rather strong support for the ability of small gases topenetrate the crystals of PMP, it is possible that defects at the interface betweenthe crystalline and amorphous regions are the source of the nonzero intercept. Thismechanism could also explain the tendency for the solubility associated with thecrystalline phase to increase proportionally to the crystalline fraction since moreinterfacial defects would accompany the increasing crystalline fraction. However,it is unclear why permeation and diffusion through such regions would have lowerdiffusion coefficients as opposed to the amorphous phase, unless severe restrictionof chain segmental mobility occurs in this region. For PMP, close to its Tg, the


Fig. 17. Crystalline domains of poly(4-methyl-1-pentene).

concept of chain immobilization by crystallites is expected to be less significantthan in the case of rubbery polyethylene because the rigidity of the glassy PMPbackbone is expected to outweigh immobility imposed by neighboring crystallinity.Nevertheless, complementary proof of the important conclusions is desirable, andmay be possible by consideration of the data in Table 5.

Fig. 18. Solubility coefficient vs amorphous fraction for poly(4-methyl-1-pentene). To con-vert [cm3(STP)]/(cm3 · MPa) to [cm3(STP)]/(cm3 · atm), multiply by 0.1013.


Table 5. Kinetic Diametersa of VariousPenetrantsb, nm

Molecule Diameter Molecule Diameter

He 0.26 C2H4 0.39H2 0.289 Xe 0.396NO 0.317 C3H8 0.43CO2 0.33 n-C4H10 0.43Ar 0.34 CF2Cl2 0.44O2 0.346 C3H6 0.45N2 0.364 CF4 0.47CO 0.376 i-C4H10 0.50CH4 0.38aZeolite sieving diameters.bRef. 53

The zeolite sieving diameters in this table represent effective minimum di-mensions as compared with the Lennard–Jones diameters for various penetrants.For more or less spherical penetrants such as CH4, the two diameters agree ratherwell. In the case of a nonspherical molecule such as CO2, however, restricted envi-ronments such as a zeolite can detect a minimum projected dimension that differsfrom the Lennard–Jones diameter characteristic of tumbling in a relatively un-restricted liquid or gaseous environment. The crystalline cage of the PMP maybe acting similarly to a zeolite, and if this is the case it is anticipated that largerpenetrants such as propane may, indeed, be effectively excluded from entering thecrystalline region. In this case, the same type of behavior is expected for the solu-bility coefficient for propane in PMP as is illustrated for propane in polyethylene,ie, S̄/S̄a directly proportional to �a and passing through the origin.

High Activity Vapor and Liquid Transport in SemicrystallineRubbery Polymers

In considering the transport behavior of high activity vapors and liquids insemicrystalline polymers at low solubility (eg, water in polyethylene), small de-grees of transport plasticization or clustering may need to be considered to gen-eralize the D̄ and S̄ to include dependence on activity. Combination of these moregeneral relations for D̄ and S̄ into the permeability relationship of equation 29should approximately describe the steady-state permeability of the amorphousfraction of the material. Generally, the permeability of the crystalline fraction isassumed to be zero. Tortuosity and chain-restriction effects should be of decreas-ing importance as the extent of swelling increases. At high degrees of swelling(above 70% solvent fraction), frame of reference corrections must be incorporatedin formulating steady-state permeability expressions.

For high solubility penetrants, great complexity is often observed in sorption–desorption kinetics or transient permeation due to time-dependent rearrange-ments of crystalline regions in response to swelling-induced stresses (59,60). These


are non-Fickian diffusional behaviors. Protracted approaches to equilibrium in thesurface concentration may make the approach to sorption equilibrium or steady-state permeation behavior much slower than expected on the basis of the effectivediffusion coefficient determined from equation 37 (5).

Using sorption–desorption kinetics, the extent of non-Fickian behavior de-creases as the thickness of the film increases. Presumably, as the ratio of the timescale for diffusion to the time scale for stress-induced relaxation (�2/D) becomeslarge, the time-dependent relaxation occurs so rapidly (relatively) that it has asmall effect on the diffusion process. In other words, in the limit where molecular-scale responses to swelling stresses are rapid compared to the rate of diffusion,the process appears to be Fickian. This is the case for totally amorphous rubberypolymers that behave as high molecular weight liquids.

Film formation protocols may cause significant nonuniformity in structurebetween a thin surface layer and the bulk of the film, especially when casting ona chilled roll. This effect can cause apparent thickness-dependent behavior of per-meability and diffusivity parameters. Effectively, the film behaves as a laminatewith different transport properties that can be treated according to a simple lam-inate model involving series resistances. Of course, assessment of the thicknessof the surface layer must be possible for this approach to have utility. The skinlayer reportedly increases in thickness as the rate of crystallization increases (61)because of contact with surfaces that promote surface nucleation. The diffusioncoefficient of ethane in the 25-µm-thick surface layer of a quenched polyethylenefilm has been reported to be about twice as high as that in the bulk, becauseof favorable orientation of crystallites in this skin region of the semicrystallinefilm (62). In some cases, true laminate structures composed of chemically differ-ent materials are needed to obtain optimum performance in barrier applications.An example is the commercial food packaging made from ethylene–vinyl alcohol(EVOH) copolymers coextruded with polypropylene to form a laminate (63). Inthe dry state, the EVOH copolymers are among the best, easily processible oxy-gen barriers known. However, at elevated relative humidities (>80–90%), waterdisturbs the strong hydrogen bonds between the vinyl alcohol repeat units in theamorphous regions and reduces its barrier properties significantly. At still higherrelative humidities and temperatures typical of retort conditions needed to killmicroorganisms, actual disruption of crystalline domains is likely. To mitigatesuch problems, polypropylene layers are used to sandwich the barrier layer usingmultilayer coextrusion technology. This minimizes the quantity of the more ex-pensive oxygen barrier (EVOH) but provides a package with sufficient mechanicalstrength.

Modeling of the transient-state transport process of water and oxygenthrough such a laminated barrier material also containing intermediate tie layersto promote adhesion is complex. Reported partial solutions use different physicalapproximations (64). Typically, the approach is to model the total amount of wa-ter, which is the major plasticizing agent transported into the barrier film throughthe outer polyolefin guard layer during retort, and relate this weight fraction to anew effective oxygen permeability of the partially humidified barrier layer. Dur-ing subsequent cooling and drying of the container, the water is removed by backpermeation out of the package into the dry storage environment over an extendedperiod of time determined by the relative thicknesses of the outer guard layer and


Fig. 19. Permeability of toluene in nylon–polyethylene barriers. The data (�) for a laminarblend is compared to theoretical permeabilities in a true laminate (upper curve) and ina homogeneous blend (lower curve). The permeability factor is the permeability in purepolyethylene divided by the permeability in the mixture.

the inside oxygen barrier layer. During this drying period a significant oxygeninvasion can occur. Consequently, more complex structures are sometimes usedto provide a desiccant material in the tie layer between the outer guard layer andthe barrier layer to prevent the penetration of water into the barrier layer duringthe retort operations (65).

A laminar-blend technique can be used to achieve properties intermediate be-tween those of the relatively expensive laminates formed by coextrusion and thoseof an inexpensive, but less effective, dispersion of a barrier component in a matrixof another material. For the laminate, the total barrier resistance is composed ofthe additive resistances contributed by each of the layers in the structure. For asimple dispersion of component A in a matrix of B, a volume fraction weightingof the permeabilities of the two components applies unless large aspect ratios areachieved (66). Laminar-blend technology achieves large aspect ratios by smear-ing out the dispersed barrier layer into thin lamellae with large area to thicknessaspect ratio. Figure 19 shows the data and results of model calculations based onthe two limiting cases of a true laminate or a homogeneous dispersion in com-parison to the barrier properties of a laminar blend of nylon-6,6 and polyethylenefor use in hydrocarbon barrier applications. The larger the aspect ratio of the pri-mary barrier component, the closer the approach to the true multilayer laminateproperties due to overlapping of the barrier lamellae (67). Such barrier-containerissues involve not only a consideration of transport properties of materials, butalso an economic weighing of the costs of processing and materials to achieve anoptimum structure.


Glassy Polymers

Glassy polymers, polymers existing at a temperature below their Tg, are differen-tiated from rubbery polymers in discussions of physical properties because theirresponses in most experiments are quite different. Essentially all of the differ-ences arise from the characteristic scales of the micromotions that occur at thesegmental level in rubbery and glassy polymers. As noted earlier, the high molec-ular weight liquid-like nature of rubbers provides them with the ability to adjusttheir segmental configurations rapidly over significant distances by large-scalemotions, such as the crankshaft example (Fig. 2); however, in the glassy state,only severely hindered torsional motions are possible.

The frequency and amplitude of torsional segmental motions increase asthe temperature is raised and the glass transition is approached. Close to Tg,the frequency of torsional motions increases dramatically and gives rise to truerotational motions typical of rubbery materials. However, in glasses well belowTg, the intramolecular backbone motions occurring over time scales of secondsor microseconds are much less extensive than in rubbers and are believed to beprimarily torsional oscillations (68).

A distribution of these torsional oscillations results from subtle differences inchain packing throughout the glass. The average of this distribution is sensed bymost probes of the material, including spectroscopic and diffusion experiments. Acorresponding cumulative distribution of average gap sizes forms spontaneouslyand randomly throughout the glass in response to the concerted torsional motionof repeat units. For penetrants smaller than the limiting moving segment size,the tendency of the penetrant mobility to decrease with increasing penetrant sizeshould be clear. This visualization is in effect the basis for applying free-volumetheories, discussed previously in the context of rubbery materials, to glassy poly-mers.

If the available gap is larger than the critical dimension of a penetrant, adiffusion jump can be taken. If the available gap is too small, a jump is precluded.The fraction of diffusion jumps available to penetrant A vs B that are actuallyselective is small in rubbers, unless the size difference between penetrants is verylarge. For glassy materials, on the other hand, the range of motions, and hence thesize of transiently opening gaps, is more narrowly distributed, thereby providinga much more selective environment for penetrant diffusion. Indeed, the infinite-dilution diffusion coefficients of glassy PVC and natural rubber in Figure 20(69) illustrate the tendency of the glass to maintain size and shape selectivityover a larger range of penetrant sizes than the rubber. The diffusivities are morestrongly dependent on penetrant size in the glass. At the same time, the morehighly restricted range of molecular motion possible in the glass leads to lowerdiffusion coefficients.

The higher diffusivity selectivities often make glasses more favorable thanrubbers for small molecule permselective membranes. The lower diffusivities alsofavor glasses over rubbers for barrier applications. On the other hand, rubbers areused in some membranes when a very large solubility selectivity is available, suchas in separating organic vapors from air. Complex non-Fickian and plasticizationphenomena can occur in applications where a significant amount of penetrant is


Fig. 20. Diffusion coefficients for a variety of penetrants in natural rubber at 25◦C andrigid poly(vinyl chloride) at 30◦C.

present in a glass. The diffusion coefficients of penetrants in glasses may increaseby several orders of magnitude if the weight fraction of the penetrant reduces theTg to the use temperature.

This plasticization phenomena can be observed in some glassy polymers ex-posed to high pressures of condensable gases, such as CO2. Moreover, a pressuredependence of permeability occurs for glassy polymers prior to plasticization,whereas rubbery materials tend to have essentially pressure-independent per-meabilities. The decrease in permeability of component A with increasing partialpressure of either component A or B prior to the onset of plasticization (70) shownin Figure 21 (71) is characteristic of glassy polymers.

The pressure at which plasticizing effects begin to overcome permeabilitydepression depends primarily on the ratio of the partial pressure of the penetrantin the feed stream to the corresponding vapor pressure of the penetrant at the sys-tem temperature. As the pressure approaches the vapor pressure, solubility tendsto increase rapidly, thereby inducing plasticization or clustering responses suchas those already discussed for rubbery polymers. Even below the plasticizationpoint, additional complexities not encountered with rubbery polymers are oftenobserved in glassy materials. For example, low vapor pressure components suchas water can cause depression of the permeability of other gases when water is


Fig. 21. Permeability of a glassy polymer to penetrant A (PA) in the presence of varyingpartial pressures of penetrant B.

present at even a few hundred pascals (70). Components such as CO2 that areonly slightly supercritical can also produce a reduction in the permeability of an-other component if plasticization is not occurring. Although the vapor pressureof a supercritical component is not defined, a useful effective value can be ob-tained by extrapolation of a semi-log plot of vapor pressure versus the inverseabsolute temperature to the system temperature. For instance, an effective vaporpressure of 7.3 MPa (72 atm) is obtained for CO2 at 35◦C by this method. There-fore 2 MPa (20 atm) of CO2 should have flux depressing effects similar to only1.6 kPa (12 torr) of water vapor having a vapor pressure of only 5.6 kPa (42 torr)at 35◦C. This approach cannot be applied unambiguously to highly supercriticalgases.

Sorption Models for Glassy Polymers. The sorption isotherms of glassypolymers are typically concave to the pressure axis at low pressures and tend toapproach a straight line as the pressure is increased (see Fig. 22). This sorptionbehavior contrasts with that of a rubbery polymer, where the isotherm is typi-cally linear at low pressures and then becomes convex to the pressure axis athigh pressures, indicating the onset of plasticization or clustering. While there isgeneral agreement in the literature on the mechanism of sorption into a rubberypolymer, there are many competing models of sorption in glassy polymers. Thedual-mode model was the first successful attempt to describe sorption of smallmolecules in glassy polymers. The dual-mode model is still very popular todaybecause of its simplicity and wide applicability. However, the physical basis of thedual-mode model, which rests on the concept of two dissolved penetrant popula-tions, has never been conclusively proven. Furthermore, the parameters of thedual-mode must be measured for each system independently, and the parametersfor sorption differ from those for desorption. In light of these shortcomings, multi-ple models have been proposed as alternatives to the dual-mode model, althoughnone as yet enjoys the same widespread acceptance. A fairly recent review ofthese models is available (72). In this section, the dual-mode model, along withtwo alternatives, the site-distribution model and the nonequilibrium lattice fluid


Fig. 22. Sorption isotherms for various gases in bisphenol A polycarbonate at 35◦C. Toconvert MPa to atm, multiply by 10.

model, is presented. The two alternate models were chosen for review becausethey illustrate two rather different approaches. The site-distribution model isa phenomenological model based on the free-volume distribution of the glassystate, while the nonequilibrium lattice fluid model is based on nonequilibriumthermodynamics.

Dual-Mode Model. The physical basis of the dual-mode model rests on theconcept of unrelaxed free volume in the glassy state. The concept of unrelaxed freevolume Vg − V�, where Vg is the specific volume of the glass and V� is the extrapo-lated specific volume of a hypothetical rubber, is illustrated in Figure 23 (71). Theextremely long relaxation times for segmental motion in the glassy state lead totrapping of nonequilibrium chain conformations in glasses, thereby presumablypermitting miniscule gaps to exist between chain segments. This results in theadditional unrelaxed free volume, which causes the specific volume of a glass tobe higher than the extrapolation for a hypothetical rubber.

The dual-mode sorption model considers the glassy solid to consist primar-ily of an equilibrium-densified matrix with a small volume fraction of uniformly


Fig. 23. Schematic representation of the unrelaxed volume, Vg − V�, in a glassy polymer.Vg = glass specific volume; V� = equilibrium volume of densified glass. The unrelaxedvolume disappears at the glass-transition temperature.

distributed molecular-scale gaps, or holes, throughout the matrix. Gaps that aresmaller in size than a penetrant molecule may be locally redistributed duringthe sorption process in the initial exposure of the polymer to a penetrant (71).This model has proved to be very useful for interpreting a wide spectrum ofphenomena including pure component and multicomponent permeation and sorp-tion and penetrant-induced dilation in gas–glassy polymer systems. The develop-ment of the dual-mode model has been reviewed (73,74).

The dual-mode model has also been applied with some success to the sorp-tion and transport behavior of small- and intermediate-size organic molecules inglasses (75). Above some size, however, the penetrant may be too large to be ac-commodated solely within individual holes, and two truly distinct environmentsmay not actually exist for these penetrants. Even in this case, however, the conceptof unrelaxed free volume remains valid, and observations of sorption and dilationbehavior that apply to gases also apply qualitatively to larger penetrants up tothe point of extreme plasticization where an actual glass transition occurs.

If a hypothetical equilibrium-densified glass, V� in Figure 23, is exposed toa given pressure of a penetrant, a sorption concentration CD, characteristic oftrue molecular dissolution, occurs, as it does in rubbery polymers or liquids. Ina nonequilibrium glass Vg, the additional volume present in the form of inter-segmental gaps, or holes, provides for additional low energy berths for penetrantsthat do not require the energy expense of dilating the matrix as in the case of thedissolved mode (CD) population. In this case, an additional sorption populationwith concentration CH can occupy these sites in local equilibrium with the dis-solved mode population. This hole-filling population is often described by a Lang-muir isotherm. The dual-mode model can be written analytically up to reasonably


high pressures for a penetrant indicated by subscript A, in terms of the sum of aHenry’s law expression for CD and a Langmuir expression for CH, namely

CA = CDA + CHA (46)

CA = kDApA + C′HA

bApA

1 + bApA(47)

where kDA is the coefficient that characterizes sorption of the penetrant in thedensified regions of the matrix and pA is the penetrant partial pressure. The valueof kDA has been correlated with the Lennard–Jones potential of the gas (76) andthe free volume of the polymer (77). The bA and C

′HA

parameters are the Langmuiraffinity and capacity constants, respectively. Several researchers have shown thatC

′HA

approaches zero at the glass transition in generally the same way that Vg −V� approaches zero (78,79). In fact, the value of C

′HA

has been predicted well as theproduct of the unrelaxed volume fraction [(Vg − V�)/V�] and the inverse equivalentsorbed volume of penetrant molecules. The volume per molecule can be estimatedwith reasonable accuracy using data from gas adsorption on zeolites or activatedcarbons. The value of [(Vg − V�)/V�] can be estimated using dilatometry in mostcases (80).

The dual-mode model is easily extended to include gas mixtures (81). In termsof partial pressures, the expressions for mixed-gas sorption are usually written as

CA = kDApA + C′HA

bApA

1 + bApA + bBpB(48)

CB = kDBpB + C′HE

bBpB

1 + bApA + bBpB(49)

The dual-mode model thus predicts a reduction in sorption of any gas, at afixed partial pressure, as the partial pressure of a second gas is increased. Sorp-tion of binary mixtures of CO2/N2O and CO2/CH4 in poly(methyl methacrylate)(PMMA) is predicted to within 2% by this model. Moreover, Figure 24 shows pureand mixed-gas data (82) for the sorption of CO2 and CH4 in poly(phenylene oxide)(PPO). The solid lines represent dual-mode model predictions for a partial pres-sure of 0.5 MPa (5 atm) for each component using parameters from the pure-gasisotherm. The dual-mode sorption model provides an accurate description of thesolubility of both pure and mixed-gases for these polymers over the range of partialpressures studied. The small differences between mixed-gas permeability modelpredictions and experimental data may be due to differences in conditioning ofthe polymer in single- and mixed-gas environments. These differences might alsobe attributable to changes in the diffusivity of mixture components due to thepresence of another component; diffusivity is not as easily predicted as solubilitybecause of potential plasticization effects for strongly sorbing penetrants. Theseeffects appear to be less significant for equilibrium sorption coefficients than fordiffusion coefficients.


Fig. 24. Mixed-gas sorption for the PPO/CO2/CH4 system at 35◦C. Dotted lines show pure-gas sorption and solid lines show model predictions for pCO2 = pCH4 = 0.5 MPa (5 atm) (76).To convert MPa to atm, multiply by 10.

Site-Distribution Model. In the dual-mode model, penetrants are picturedas being sorbed into either a low energy microvoid site or a high energy den-sified site within the glassy polymer matrix. These two idealized sorbed pop-ulations are probably an oversimplification of the actual penetrant–polymerenvironment. Experimental evidence to validate the assumption of two dis-solved penetrant populations has been conflicting and somewhat inconclusive(44,83–85). In the site-distribution model, the glassy matrix is pictured as con-taining free volume distributed throughout as spherical holes. The distribution ofhole volumes in the glass is assumed to be Gaussian. This Gaussian distributionof free volume throughout the polymer results in a corresponding Gaussian dis-tribution of sorption site energies. As penetrant is dissolved into the polymer, thelow energy sorption sites are preferentially occupied. As the penetrant activity isincreased, higher energy sorption sites are filled. The site-distribution model usesFermi–Dirac statistics to determine the occupancy at each energy level, wherethe integral of the site density times the occupancy level is the total penetrantconcentration (86).

The distribution of free volume pictured in the site-distribution model is theresult of volume fluctuations in the liquid polymer being frozen into the glass asthe polymer is quenched below Tg. The shape of the free-volume distribution isassumed to be Gaussian and is expressed by

n(Vh) = 1σv

√π

exp(

− (Vh − Vh0)2

σ 2v

)(50)


where, σv = √2kBTgVh0/B is the width of the distribution. Vh0 is the average hole

volume and B is the bulk modulus at the Tg. When a gas molecule of volume, Vg,is sorbed into a hole an elastic contribution, Gel, is made to the total Gibbs energy:

Gel = 23

µs(Vg − Vh)

Vh(51)

where, µs is the shear modulus. The total Gibbs energy, G, consists of this elasticcontribution plus an enthalpic contribution from penetrant–polymer interactionsand an entropic contribution from penetrant confinement in the sorption site. Onlythe elastic contribution to the Gibbs energy is assumed to vary with hole volume.Through the use of Fermi–Dirac statistics the occupancy of the sorption sites iscalculated, with the total penetrant concentration equaling

CA = VmolN0

NA

∞∫− ∞

1σG

√π

exp(

− (G − G0)2

σ 2G

)1

1 + exp[(G −µ)/RT]dG (52)

The Vmol/NA prefactor is necessary to put the concentration into the commonunits of cc STP/cc polymer. Vmol equals 22,400 cc STP/mole, and NA is Avogadro’snumber. G0 is the Gibbs energy of a hole of average size Vh0. The chemical potentialof the penetrant gas phase is given by µ. The width of the Gibbs energy distributionis given by σG and is related to the width of the free volume hole size distribution,σ v. N0 is the number of sorption sites.

The site-distribution model results in four fitting parameters: H0, S0, N0,and σG. The parameters H0 and S0 are used to calculate G0 = H0 − TS0. S0 is thechange in entropy from the gas phase to solution in a polymer. The value of S0 inmost cases is roughly equal to the change in entropy from the gas phase to the solid.The number of sorption sites, N0, can be treated as a purely adjustable parameterbut is often estimated as 6.7 × 1021/cm3 (86). The distribution width σG can becalculated from measurable quantities or treated as an adjustable parameter.Correlations do exist between the distribution width and Tg and the volume ofthe gas molecule, Vg (87). For high Tg polymers, more volume fluctuations exist inthe liquid polymer state. When these high Tg polymers are quenched to the glassystate, the volume fluctuations are frozen and a wide distribution of sorption sitesresults. When large gas molecules are dissolved, there is a greater elastic strainwhich results in a large energy distribution.

The site-distribution model has recently been extended to mixed-gas sorp-tion, and is able to account for competitive sorption effects (88). As with the dual-mode model, the site-distribution model allows for the prediction of mixed-gassorption by using parameters obtained from experimentally less complicated pure-gas sorption measurements. Like the dual-mode model, the site-distribution modelis phenomenological, while the dual-mode model uses the bulk specific volume Vgand extrapolated equilibrium volume V� as key parameters; the site-distributionmodel relies upon the bulk shear modulus µs as a key parameter. The use of bulkparameters to characterize sorption which occurs at the atomistic scale is ques-tionable from a fundamental point of view; nevertheless, such models provide use-ful tools to describe and even predict the effect of conditioning and physical aging


on the free-volume distribution in glassy polymers. Conditioning of the polymerwith a highly soluble gas such as CO2 produces packing disruptions throughoutthe glass which remain after conditioning as low energy sorption sites. Physicalaging, on the other hand, reduces the number of low energy sites and resultsin a narrower energy distribution (89). The physical insight provided by the site-distribution model and the possibility of future development into a truly predictivemodel are advantages which must be weighed against the added mathematicalcomplexity.

Nonequilibrium Lattice Fluid Model. The nonequilibrium lattice fluid(NELF) model is a nonequilibrium thermodynamics approach to modeling sorptionin a glassy polymer. The model pictures penetrants as a single dissolved popula-tion in a nonequilibrium glassy matrix (90). The penetrant concentration in theglassy phase is calculated by requiring that the chemical potential of the pene-trant in the gaseous phase equal the chemical potential of the penetrant in theglassy phase. NELF is a modified form of the Sanchez–Lacombe (SL) lattice fluidmodel for a rubbery polymer. The SL lattice fluid model places polymer segmentsand vacancies onto a lattice (91,92). The placements of vacancies on the latticesimulate free volume in the polymer. The number of vacancies is dependent on thepressure and temperature of the system. Within this conceptual framework, theSL lattice fluid model expression for the Gibbs free energy of a pure component is(92)

G = rnRT∗{

− ρ̃ + P̃ρ̃

+ T̃ρ̃

[(1 − ρ̃)ln(1 − ρ̃) + ρ̃

rln(ρ̃)

]}(53)

In the above equation ρ̃, T̃, and P̃ are the reduced density, temperature, andpressure, respectively. These variables are defined as: ρ̃ = ρ/ρ∗, T̃ = T/T ∗, and P̃= p/P ∗. The parameters ρ∗, T ∗, and P ∗ are obtained from the P-V-T properties ofthe pure component. The parameter r in equation 53 is the number of lattice sitesoccupied by a single molecule.

In order to extend the SL lattice fluid model to a penetrant-polymer mixture,a set of mixing rules is used to calculate ρ∗, T ∗, and P ∗ for the mixture from thepenetrant and polymer parameters. The proper set of mixing rules is given byDoghieri and Sarti (90). Using ρ∗, T ∗, and P ∗ for a mixture of n1 moles penetrantand n2 moles polymer, the Gibbs free energy of the mixture is obtained from (90)

G = RT ∗(r1n1 + r2n2)

{− ρ̃ + P̃

ρ̃+ T̃

ρ̃

[(1 − ρ̃)ln(1 − ρ̃) + ϕ1

r1ln(ϕ1ρ̃)

]}(54)

where r1 and r2 are the number of lattice sites occupied by a molecule of “1” and “2,”respectively, in the mixture. ϕ1 is the volume fraction of penetrant in the mixture.

In the SL lattice fluid model approach, the equilibrium density is found byminimizing the Gibbs free energy. The following equilibrium conditions are en-forced for the pure component and the mixture (90):

(∂G∂ρ

)T,p,n

= 0 at equilibrium for the pure component (55a)


(∂G∂ρ

)T,p,n1,n2

= 0 at equilibrium for the mixture (55b)

The chemical potential of the system is then calculated at values of the den-sity which satisfy the above conditions at each temperature and pressure.

In a nonequilibrium system, the conditions for equilibrium (eqs. 55a and 55b),no longer hold. NELF bypasses the need for an equilibrium condition through theuse of an internal order parameter. The order parameter characterizes the degreethe system is out of equilibrium. One can imagine multiple possible choices oforder parameters in the glassy state, such as the fractional free volume at theTg (93) or simply the mixture density (94). NELF specifically makes use of thepolymer partial density, ρ2, as its order parameter. A discussion of the rationaleand implications of picking an order parameter is given by Doghieri and Sarti(90). Nevertheless, with NELF the order parameter ρ2 is related to the reduceddensity of the mixture by

ρ̃ = ρ2

ω2

1ρ∗ (56)

The chemical potential of the dissolved penetrant is then given as (95)

µ1

RT= ln

(ω1ρ2

ω2ρ∗1

)− M1P ∗

1

ρ∗1RT ∗

1

[1 +

(T ∗

1 P ∗

T ∗P ∗1

− 1)

ω2ρ∗

ρ2

]ln

(1 − ρ2

ω2ρ∗

)

+ T ∗1 P ∗

T ∗P ∗1+ ρ2

ω2ρ∗

[T ∗

1

T

(1 + P ∗

P ∗1

−(

ω2ρ∗

ρ∗2

)2 �P ∗12

P ∗1

)]

+ 1

(57)where ω1 and ω2 are the weight fractions of penetrant and polymer. �P ∗

12 is abinary interaction term and is assumed equal to (

√P ∗

1 − √P ∗

2 )2 (91). The weightfraction of penetrant in the glassy phase is solved for by requiring that the chem-ical potential of dissolved penetrant, given by equation 57, equal the chemicalpotential of the penetrant in the gas phase. The chemical potential in the gasphase may be calculated using the SL lattice fluid model or an equation of statesuch as Peng–Robinson.

The truly impressive feature of NELF is that it is a completely predic-tive model of small molecule sorption in glassy polymers, if information aboutthe polymer partial density, ρ2, is known. Unfortunately, ρ2 is obtained fromdilatometry data, and these data exist for only a very small number of penetrant–polymer systems, which limits the general practical applicability of the model. Inlight of this shortcoming, two approaches have been used to make NELF morewidely applicable. The first approach assumes that the polymer partial density isequal to the density of the pure polymer, ρ2

0 (96). This approach limits NELF tolow pressures, where no polymer swelling occurs; however, in this pressure range,NELF is still completely predictive. The second approach to increase the applica-bility of NELF introduces one or two adjustable parameters (95). In this approach,the polymer partial density is assumed to vary linearly with pressure such that

ρ2 = ρ02 (1 − kp) (58)


where k is the swelling coefficient. If the pure polymer density is known,equation 58 is substituted into equation 57 and k is treated as an adjustable pa-rameter. If the pure polymer density is unknown, then both ρ2

0 and k are treatedas adjustable parameters. This approach allows for the prediction of the entireisotherm from just one or two sorption measurements. Another drawback of NELFis that values of ρ∗, T ∗, and P ∗ are required for both components. These valuesare available for most common penetrants (92,97). However, fewer of these val-ues exist for polymers because they are obtained from the P-V-T behavior of thepolymer above Tg. Many high performance glassy polymers of interest have ex-tremely high glass transitions, which makes obtaining P-V-T data in the rubberystate experimentally challenging.

Transport Models for Glassy Polymers

The following three multicomponent transport models have been used to explainthe depression of the permeability of a component in a mixture relative to itspure component value (Fig. 21): the Petropoulos model and the competitive sorp-tion model, both of which assume that direct competition for diffusive pathwayswithin the glass is negligible, and a more general permeability model in whichdirect competition can occur between penetrant molecules for both sorption sitesand diffusion pathways. All three of the models presented here are based uponthe framework of the dual-mode model. It is worth mentioning that the site-distribution model has recently been extended to account for diffusion (98) andthat free volume models exist for transport in glassy polymers (99).

Simple Dual-Mode Transport Models. As a direct extension of the dual-mode sorption model, it is convenient to treat pure-gas permeability as the sumof two terms, which are characteristic of the two sorbed penetrant populations.Petropoulos first developed such an expression in terms of the chemical potentialgradient of the sorbed gas (100). For the case where there is a vacuum downstreamand a pressure pA2 of component A upstream, the following expression results:

PA = kDADTDA + C′HA

DTHA

pA2

ln(1 + bApA2) (59)

where DTDA and DTHA are diffusion coefficients characteristic of the Henry’s law(dissolved), and the Langmuir (hole-filling) populations of sorbed gas molecules.Extension of the Petropoulos model to include gas mixtures involves numericalintegration of the flux equations for components A and B (101).

Taking an approach similar to that of Petropoulos, a model was developed(102) representing the flux for gas A in terms of concentration gradients ratherthan chemical potential gradients:

JA = − DDA

dCDA

dx− DHA

dCHA

dx(60)

where DDA and DHA are diffusion coefficients characteristic of Henry’s law and theLangmuir populations of sorbed gas molecules. After substitution of equation 47


into equation 60, the flux expression can be integrated analytically to give theequation for pure-gas permeability, again for the condition of zero downstreampressure and an upstream pressure of pA2:

PA = kDADDA

{1 + FAKA

1 + bApA2

}(61)

where FA = DHA/DDA ,KA = C′HA

bA/kDA, and pA2 is the upstream pressure of purecomponent A, with the downstream pressure assumed to be negligible. This equa-tion represents the partial immobilization dual-mode transport model of Korosand Paul. By plotting pure-gas permeability data for component A, values of FA,DDA, and DHA can be determined. The sorption parameters kDA, C

′HA

, and bA can bedetermined independently in equilibrium sorption measurements, and so the pa-rameters in either equation 59 or equation 61 can be easily determined. Therefore,by making only pure-gas sorption and permeation measurements, all of the modelparameters can be determined in either of these simple dual-mode formulationsof the transport expressions.

Assuming that the diffusion coefficients DDi and DHi for components in a gasmixture are unchanged from their pure-gas values, the simple expression repre-sented above can be extended to gas mixtures. After substitution of equation 48into equation 60, the flux equation can be analytically integrated to give the fol-lowing expression for the permeability of component A in a binary mixture of gasesA and B (103):

PA = kDADDA

{1 + FAKA

1 + bApA2 + bBpB2

}(62)

Similarly, for component B,

PB = kDBDDB

{1 + FBKB

1 + bApA2 + bBpB2

}(63)

Assuming that DTDi and DTHi are constant in mixed feed situations, thePetropoulos expressions can be generalized for mixed-gas feeds; however, numer-ical solutions are required in this case. Equations 62 and 63 and the analogouspredictions in terms of the chemical potential driving force model of Petropoulosindicate that the presence of a second gas B lowers the permeability of gas A onlyby reducing the sorption of gas A.

Advanced Dual-Mode Transport Models. To account for the effect ofnearby gas molecules on the ability of a given gas molecule to find sites intowhich it might jump, Barrer proposed and developed a model that adopts asomewhat more complex description of the diffusion process in glassy materi-als (104). A gas molecule is capable of executing four distinct types of diffusionaljumps, depending on which of the two distinct sorption environments it is diffus-ing from and diffusing into (104,105). Denoting the Henry’s law and Langmuir


environments as D and H, respectively, the four possible jumps can be depicted asfollows:

D → D D → H(dissolved to dissolved) (dissolved to hole)

H → D H → D(hole to dissolved) (hole to hole)

Each jump is described by a distinct primitive diffusion coefficient DDD, DDH,DHD, or DHH that is related to the average mobility of a penetrant in each of thetwo environments DDA and DHA as shown below:

DDA = DDDA + DDHA(1 − θA) (64)

DHA = DHDA + DHHA (65)

where θA = CHA/C′HA

represents the fraction of Langmuir sorption sites occupiedby sorbed gas A molecules. Consequently, the model predicts that DDA decreases asthe concentration of gas in the polymer increases. In effect, this model predicts areduction in the number of diffusional pathways available to a penetrant molecule.When combined with the dual-mode sorption model for describing the fraction offilled holes, the flux equations can be integrated to give the permeability of apure-gas A:

PA = kDADDDA + C′HA

bA(DHHA + DHDA) − kDADDHA

1 + bApA2

+ 2kDADDHA

ln(1 + bApA2)bApA2

(66)Using nonlinear regression techniques to fit the pure-gas permeability vspressure data for gas A, values for DDDA + DDHA, and (DHHA + DHDA) can bedetermined.

The extension of this model to mixed-gases is considerably more difficultthan the extension of the simple dual-mode transport models. To include gasmixtures, the Barrer model must be modified to accurately describe the frac-tion of filled Langmuir sorption sites when more than one gas species is presentin the polymer. To account for the presence of a second gas, the flux expres-sions can be rewritten using the following definition for the fraction of filledholes:

θtotal = CHA

C′HA

+ CHB

C′HB

(67)

where θ total represents the fraction of the total unrelaxed volume occupied byeither penetrant A or B, and therefore is not available for allowing a penetrant tojump into it.

This model results in a system of nonlinear differential equations that mustbe solved numerically. Figure 25 (82) shows pure-gas permeability data for CO2


Fig. 25. Permeability of pure CO2 in poly(phenylene oxide). The solid lines represent(a) the Koros and Paul, (b) the Petropoulos model, and (c) the Barrer model. To con-vert MPa to atm, multiply by 10.1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s · cm Hg) = 0.335mmol/(m · s · TPa).

in poly(phenylene oxide) along with the theoretical fits for the Petropoulos, Korosand Paul, and Barrer models, respectively. The Barrer model provides a slightlymore accurate fit of the permeability data than either of the simpler models, espe-cially in the low pressure region. This is expected because of the additional modelparameter.


Figures 26 and 27 (82) show mixed-gas permeability data for an equimolarmixture of CO2 and CH4 in PPO along with predictions made using the Petropou-los, Koros and Paul, and Barrer models. Mixed-gas permeability is depressed rela-tive to the pure-gas values for both components. As expected, the mixed-gas Barrermodel predicts lower permeability for both components than either of the simplermodels.

Below about 1 MPa (10 atm) total pressure, all three mixed-gas models givefairly accurate predictions for CO2 permeability in the mixture. Above this pres-sure, however, CO2 data lie below all model predictions. By contrast, permeabilitydata for CH4 in the equimolar mixture lie above all three predictions over theentire range of upstream partial pressures.

The Barrer model provides a better prediction for mixed-gas carbon dioxidepermeability data, while methane permeability is predicted more accurately bythe simpler model of Koros and Paul. Thus none of the models emerge as havinga clear advantage over the other two in predicting PPO/CO2/CH4 permeability.However, several factors can be suggested as possible sources of error inherent inall of the models.

One possible discrepancy involves fugacity effects at high pressures, but fu-gacity effects do not fully account for the differences between permeability dataand model predictions. The relative deviations of fugacity-based predictions fromexperimental data are somewhat smaller than those with pressure-based calcu-lations for all three models (82).

It is unlikely that deviations from the mixed-gas sorption model are to blamefor the discrepancies between mixed-gas permeability data and model predic-tions because the mixed-gas sorption model provides an accurate description ofPPO/CO2/CH4 sorption. However, different conditioning effects between the equi-librium sorption and permeation conditions may introduce a significant sourceof error. In equilibrium sorption the polymer experiences uniform conditioning,while during permeation a conditioning profile exists through the polymer. Thesedifferent conditioning histories ultimately may lead to nonnegligible differencesin sorption.

From a purely practical point of view, the Koros and Paul model for mixed-gas permeability is significantly more straightforward than either the Petropoulosor Barrer model in terms of parameter calculation and comparisons between dif-ferent gas–polymer systems. Although the Barrer model provides a tighter fit ofpure-gas permeability data, it does not provide the more accurate prediction ofmixed-gas permeability for which it was initially proposed. The inability of themore complex model to correct for the small differences between mixed-gas per-meability data and the two simpler model predictions suggests that these subtledeviations may be the result of the fundamental assumptions of all three mod-els. In terms of transition-state theory, all three of these models assume thatthe activated state is the same regardless of whether the penetrant was origi-nally sorbed in a Henry’s law or a Langmuir site, which leads to independent,additive fluxes through the Henry’s law and Langmuir modes. This assumptionis not valid if the local density around a Langmuir site is less than the den-sity of the dissolved polymer matrix. In this case, the activated state of a pene-trant sorbed in a Langmuir site will be lower in energy than that of a penetrantsorbed in a Henry’s law site. Petropoulos has recently developed a more general


Fig. 26. CO2 permeability for the PPO/CO2/CH4 system at 35◦C using an equimolar gasmixture, compared with the three permeability model predictions: A, the Koros and Paulmodel; B, the Petropoulos model; and C the Barrer model. To convert MPa to atm, multiplyby 10.1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s · cm Hg) = 0.335 mmol/(m · s · TPa).

dual-mode transport model that incorporates this more complex picture of thediffusion process (106).

Another likely source of error in all three models is related to the veryhigh solubility of CO2 in PPO. In both the Koros and Paul and Barrer mod-els, the characteristic diffusivity parameters are assumed constant on the

Fig. 27. CH4 permeability for the PPO/CO2/CH4 system at 35◦C using an equimolar gasmixture, compared with the three permeability model predictions: A, the Koros and Paulmodel; B, the Petropoulos model; and C, the Barrer model. To convert MPa to atm, multiplyby 10.1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s · cm Hg) = 0.335 mmol/(m · s · TPa).


basis of the assumption that polymer segmental mobility does not significantlychange as a result of penetrant-induced dilation. This approximation ultimatelybreaks down and plasticization begins to dominate the behavior of the system(Fig. 21).

Polymers highly dilated with CO2 have significantly decreased glass-transition temperatures, implying that polymer segmental motion is facilitatedby high CO2 concentration (107,108). Such increased polymer motions make iteasier for gas molecules to execute diffusional jumps from one point to another inthe polymer matrix. Thus, dilation may not only manifest itself in higher sorbedgas concentrations, but in increased gas diffusivity as well. As noted earlier, suchchanges in transport characteristics of a polymer due to high concentrations ofsorbed penetrants are termed transport plasticization. Failure to account for thismolecular-scale phenomenon is probably a significant source of the small but mea-surable inadequacies of the mixed-gas permeability models. The diffusivity pa-rameters of CO2 as determined by pure-gas sorption and permeability may, infact, represent apparent values, which reflect the effects of CO2 dilation. On theother hand, pure-gas diffusivity parameters for CH4 cannot supply any informa-tion as to how much the methane diffusivity would be influenced by CO2-inducedplasticization. Thus, the mixed-gas permeability data for methane may be higherthan model predictions because of increased methane diffusivity in the plasticizedpolymer.

Conversely, the presence of CH4 may suppress CO2-induced plasticizationof the polymer, relative to its pure CO2-dilated state, by compression of the freevolume introduced by CO2 sorption. It may thereby suppress CO2 diffusivity be-low model predictions. Studies have also sought to refine the original idea of aconstant C′

H parameter by considering plasticization-induced reductions in C′H

due to depression in the effective Tg (109).It is most likely that the prediction of the permeability of gas mixtures con-

taining CO2 could be improved by accounting for the effects of CO2-induced plas-ticization on transport parameters. This could be tested best at higher CO2 pres-sures where such effects should be significant and, in fact, would correspond toconditions of practical importance in enhanced oil recovery operations where CO2and CH4 must be separated at pressures as high as 6.9 MPa (1000 psia) and withCO2 contents above 50%. The dual-mode transport model of Petropoulos has re-cently been extended to account for the effects of plasticization in both pure- andmixed-gas environments (110).

Although extensive data for mixed-gas permeation in the plasticizationregime are not available, good pure-gas values are available and illustrate someimportant points. The permeability of CO2 in various substituted polycarbonatesshown in Figure 28 (111) increases with pressure above a certain characteristicvalue for each of the polymers. The shapes of the permeability curves are likethe curve shown schematically in Figure 21. Plasticization occurs despite the factthat the downstream pressure is maintained at a vacuum of less than 1330 Pa(10 mm Hg). To determine when true transport plasticization occurs, the localdiffusion coefficient DAP(CA), which measures the ability of a penetrant to movethrough the membrane at a point where the local concentration of penetrantis equal to CA, must be considered. The sorption isotherms for CO2 in thevarious materials are shown in Figure 29 (111), and application of the standard


Fig. 28. Carbon dioxide permeability for (a) poly-carbonate (PC), (b) silicone rubber, (c) tetramethylpolycarbonate (TMPC), and the fluori-nated species wherein the methyls on the central carbon are fluoromethyl (CF3) groups,ie, hexafluoropolycarbonate (HFPC) and tetramethylhexafluorocarbonate (TMHFPC). Toconvert MPa to psia, multiply by 145.1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s · cm Hg) =0.335 mmol/(m · s · TPa).


Fig. 29. Sorption isotherms of carbon dioxide in (a) polycarbonate (PC), (b) silicone rubber,(c) tetramethylpolycarbonate (TMPC), and the fluorinated species wherein the methyls onthe central carbon are fluoromethyl (CF3) groups, ie, hexafluoropolycarbonate (HFPC) andtetramethylhexafluorocarbonate (TMHFPC). To convert MPa to psia, multiply by 145.

formula (eq. 37) for determination of DAP(CA) leads to the results shown inFigure 30 (111).

The characteristic tendency of the DAP(CA) curves to rise to a plateau atintermediate concentrations does not reflect true plasticization, but instead re-flects the saturation of preexisting intersegmental packing defects as consideredin the dual-mode theory. The deviation upward from the plateau at higher concen-trations indicates a breakdown in the constancy of the local diffusion coefficientfor the normally-packed matrix and thereby signals the onset of extreme trans-port plasticization. Despite the markedly different thermal-transition points ofthe various glassy materials, it is interesting that all of them show plasticizinginflections at around 35 ± 2 cm3 polymer. On the other hand, the local partialpressure (or fugacity) at which plasticization is reached varies greatly for thedifferent materials, suggesting that the useful operating range for the variousmaterials will also vary considerably. For bisphenol A polycarbonate (PC), theupturn point in local diffusion coefficient is not reached until roughly 2.76 MPa(400 psia); for tetramethylhexafluoro-bisphenol A polycarbonate (TMHFPC), thepoint occurs at ∼0.93 MPa (135 psia) because of the much higher solubility in thisopen, packing-inhibited material.


Fig. 30. Effective diffusion coefficient for carbon dioxide at 35◦C. The arrows show theapproximate concentration at which plasticization occurs as indicated by the upward in-flection in the curve.

Measurements on PC and tetramethylbisphenol A polycarbonate (TMPC) in-dicate that the overall volume dilations due to sorption of 35 cm3 (STP) CO2/cm3

polymer are 3.8% and 2.9%, respectively (111), suggesting that less volume di-lation is required to induce plasticization in the more rigid TMPC (Tg = 198◦C)than the PC (Tg = 150◦C). X-ray diffraction measurements indicate higher in-tersegmental spacings in TMPC, than in PC (112). It may be that more dilationis needed to free a neighboring segment of the more packable PC. In addition,because of the larger sorption coefficient in TMPC, the plasticization condition isreached at a lower CO2 pressure.

History Effects in Glassy Polymers

Penetrant-Induced Effects. History-dependent properties of glassy poly-mers are well documented in terms of thermal (113,114) and mechanical (115)properties. Studies have related excess enthalpy changes measured by differentialscanning calorimetry to changes in small-molecule sorption (116,117). Many stud-ies have reported penetrant-induced history-dependent (or conditioning) effectsfor gas and vapor sorption and transport properties in glassy polymers. The abil-ity to display long-lived changes in sorption, diffusion, and permeation properties


Table 6. Partial Molar Volumes of CO2 in OrganicLiquids

Medium Partial molar volume, cm3/mol

Carbon tetrachloride 48.2Chlorobenzene 44.6Benzene 47.9Acetone 44.7Methyl acetate 44.5Methanol 43.0Average 45.5

distinguishes glassy materials from rubbery ones and is believed to arise fromthe long relaxation times in glasses that reflect their inhibited segmentalmotions (68).

If a sufficient amount of some penetrant is absorbed into a glassy polymerand then totally removed, the amount of excess volume and enthalpy of the sam-ple will have been changed. In a real sense, the material is no longer the same,since it is now in a different nonequilibrium state. If the upstream face of a glassyfilm is exposed to a conditioning agent while maintaining the downstream faceat vacuum, it is possible to induce a conditioning profile of excess volume andenthalpy. A complete treatment of this phenomenon is exceedingly complex, andsince the excess enthalpy and excess volume of a nonequilibrium glass tend tochange correspondingly, it is common to use excess volume as the primary vari-able for discussion. This is convenient since it allows direct monitoring of changesin sorption-induced nonequilibrium properties by dilatometry. Experiments illus-trate that dilation and sorption for the glass have marked hystereses (Fig. 31)(16), whereas rubbers behave simply as equilibrium materials with no measur-able hysteresis.

The essentially linear nature of the dilation response of the glass duringsorption may seem surprising at first, given the marked nonlinearity of the sorp-tion isotherm. This response, however, is consistent with the dual-mode sorptionmodel and, in fact, is rather an important independent verification that the modelhas a realistic physical basis. The dual-mode viewpoint suggests that only thefraction of the total sorption associated with the actual separation of chain seg-ments to accommodate penetrant will cause volume dilation. Since the gap-fillingpopulation in a glass contributes substantially less to dilation than the Henry’slaw population, the infinite-dilution partial specific volume of a penetrant is lowerin the glassy polymer than in low molecular weight solvents or rubbery polymersin which no gap filling occurs.

Infinite-dilution partial molar volumes for CO2 in various low molecularweight liquids (Table 6) are insensitive to the type of medium and decrease onlyslightly as interactions between the medium and CO2 increase. This approxima-tion may be somewhat inappropriate for highly supercritical gases such as ni-trogen or methane, but for CO2 near ambient temperature it appears valid. Theaverage of the CO2 partial molar volumes in the organic liquids is approximately46 cm3/mol. The value for CO2 in silicone rubber determined from analysis of


Fig. 31. (a) Sorption and (b) dilation isotherms for CO2 in bisphenol A polycarbonate at35◦C. �, Sorption; �, desorption. To convert MPa to psia, multiply by 145.

sorption and dilation data is also approximately 46 cm3/mol. In measurements(118) of the partial molar volume for CO2 in low density polyethylene over atemperature range of 25–55◦C, no strong temperature dependence is observed.The average value reported over this temperature range, 44.5 cm3/mol, is also


in good agreement with the value in low molecular weight liquids and siliconerubber.

An impressive quantitative test of the dual-mode model is possible usingpartial molar volume results. Dilation of the glassy matrix is predictable usingthe average value of the partial molar volume from Table 6 (119) for CO2. If theLangmuir sorption term in the dual-mode sorption model corresponds to a truehole-filling process, the volume dilation relative to V0, the unswollen polymervolume, can be predicted using the fraction of CO2 sorbed according to linearHenry’s law sorption terms, kDp. Volume dilation predictions assuming that theentire population causes dilation correspond to the dashed line in Figure 32,whereas the solid line is the prediction based on the assumption that only theHenry’s law population causes the dilation. The solid line is remarkably effectivefor predicting dilation (120). Similar success has been demonstrated using theHenry’s law prediction for other glassy polymers (118).

The effects of conditioning on subsequent sorption behavior of glasses, as wellas the hystereses seen in both the dilation and mass uptake curves in Figure 31,have been studied. On total removal of the CO2, the nonzero intercept on thevolume dilation axis represents the additional excess volume introduced duringsorption–desorption cycles. For this same system of CO2 and bisphenol A polycar-bonate, the experiments measuring the permeability while steadily increasing theupstream pressure, then monotonically decreasing the upstream pressure back tozero, add insight. During the experiment, the downstream face of the film is alwaysmaintained under vacuum. The CO2 permeability in PC shown in Figure 33 (121)displays the shape typical of a dual-mode response at low pressures, followed by

Fig. 32. Volume dilation by sorption of CO2 in bisphenol A polycarbonate at 35◦C. Thedashed line is the prediction assuming that the entire population causes dilation, whereasthe solid line is the prediction based on the assumption that only the Henry’s law populationcauses the dilation. To convert MPa to psia, multiply by 145.


Fig. 33. Permeability of CO2 through bisphenol A polycarbonate at 35◦C illustrating thehysteretic behavior due to conditioning. �, pressure increasing; �, pressure decreasing. Toconvert MPa to psia, multiply by 145.

plasticization at high pressures during the pressurization leg of the cycle. Duringdepressurization, the permeability increases steadily as the pressure is reduced.The nominal time to reach steady state for the 0.127-mm sample of polycarbonateis much less than one day if simple Fickian kinetics applies; longer times wereused to guarantee true equilibration because polymer relaxation effects mightprotract the process. The permeability at 3.4 MPa (500 psia) following the normalequilibration time (5 days) was shown to be negligibly different from the valueafter a 63-day equilibration time (121).

At low pressures, the permeabilities for the depressurization legs are about100% larger than the permeability for the pressurization leg of the cycle. Theincrease in the permeability for the depressurization can be explained in terms ofthe enhanced solubility and diffusivity in the disturbed matrix. The disturbancesto the matrix appear to be more subtle than those responsible for the true packingdefects characterized by the C′

H parameter. In the dilated state, the disturbancescan be thought of as causing a generalized reduction in the cohesive energy densityof the polymer, thereby making it energetically easier to make a cavity to insert agas molecule into a dissolved state. On total removal of the disturbing penetrant,some of these subtle disturbances may coalesce as a result of slow segmentalmovement to produce new packing defects of sufficient size to accommodate apenetrant.

Typically, even at high pressure the lower sorbing gases, such as nitrogen,helium, and even methane, are not able to induce the hysteretic responses (condi-tioning) observed with CO2. The lower condensability, and hence lower solubilityof these supercritical gases, is presumably responsible for their lack of condition-ing capability. Although these low sorbing gases are not able to induce a hysteretic


response, they can maintain the disturbed matrix in an open state if the solubilityis sufficiently high under the conditions used after the exposure to the primaryconditioning gas (122).

Similar effects have been noted for gases using intermediate-size penetrantsas the disturbing and probing agents (123–126). The use of PMMA, PVC, orpolystyrene monodisperse microspheres allows study of these larger penetrantswith exceedingly small diffusion coefficients. Effects consistent with the generalideas noted above are seen in these cases, but additional complexity often entersbecause of protracted glassy state relaxation.

Sample Dimension Dependent Effects. Recently, it has been noted thatthe permeability of glassy polymers decreases over time and the selectivity in-creases over time. Interestingly, the decreases in permeability and increases inselectivity are accelerated for thin films, on the order of a micrometer in thickness.The study of the gas transport properties in thin films is not only academic becausefilm thicknesses of this order are often used in semiconductors, in adhesives, andin packaging. Furthermore, the skin thicknesses of asymmetric hollow fibers, themost common industrial membrane geometry, are usually 1 µm or less.

The work of Pinnau and co-workers with asymmetric films was the first toshow dramatic time-dependent gas transport properties (127,128). Later, Pfrommand co-workers showed similar behavior in free-standing polyimide and polysul-fone films (129). Figure 34 shows that for a 0.5-µm thick polyimide film, the N2 per-meability decreases by approximately 50% over a 6000-h period, while the He/N2selectivity increases by approximately 25% in the same time period. The decreasein permeability is not nearly as dramatic for the thick films. Rezac and co-workers(130) and McCaig and co-workers (131) saw similar effects in polyimide/ceramicand polycarbonate/ceramic composite membranes, respectively. Chung and Teohobserved the same effects in asymmetric hollow fiber gas separation membranes(132).

The time-dependent gas transport properties illustrated above are usuallydescribed in terms of physical aging. Physical aging is a well-known phenomenonin glassy polymers that results from their nonequilibrium nature. As alreadymentioned, the segmental motions of the polymer chains are severely hindered inthe glassy state, but these motions are not completely excluded. The hinderedsegmental motions allow for the recovery of unrelaxed free volume, Vg − V�

(see Fig. 22). The gradual approach to the equilibrium volume, V�, is called phys-ical aging (68).

Two mechanisms are often used to describe physical aging: lattice contractionand diffusion of free volume. In the lattice-contraction mechanism, the polymerchains are pictured as uniformly moving closer together, as if the lattice spacingthroughout the polymer has decreased. The kinetics of the lattice contraction arenot dependent on the sample dimensions. The diffusion of free volume or holeswas first proposed by Alfrey (134). In this mechanism, packets of free-volume arepictured as diffusing from the interior to the surface of the sample where they areeliminated. Since the diffusion of free-volume process is assumed to be Fickian,the loss of free volume becomes dependent on the sample dimensions.

Curro and co-workers (135) proposed that contributions from both the latticecontraction and diffusion of free volume mechanisms represented the overall vol-ume recovery in the glass. McCaig and co-workers extended this idea to capture


Fig. 34. (a) Decrease in N2 permeability and (b) increase in He/N2 selectivity for polyimidefilms of varying film thickness: �, 28.45 µm; ×, 2.54 µm; and �, 0.5 µm. Arrows indicatestorage in dry air at 35◦C. From Ref. 133.

the effect the free volume recovery had on the permeability of the glass (136). Inthis approach, the free volume is related to the permeability by the commonlyused correlation:

P(t) = Aexp[ − B/vf(t)] (68)

where A and B are parameters dependent on the penetrant–polymer pair andvf(t) is the time-dependent fractional free volume. The time dependence of thefree volume loss from the lattice contract mechanism was modeled as a first-orderexponential with time constant τ . The time dependence of the free-volume lossfrom the free-volume diffusion mechanism was calculated by solving the diffusionequation with a concentration-dependent diffusion coefficient. The concentration


Fig. 35. Oxygen permeability decay through BPA–BnzDCA films illustrating physicalaging effects for the following film thicknesses: �, 33 µm; �, 28 µm; �, 9.7 µm; �, 4.4 µm;�, 1.85 µm;, �, 0.99 µm; �, 0.74 µm; �, 0.58 µm; , 0.25 µm. Reprinted from Ref. 131 withpermission from Elsevier..

dependence of the free volume diffusion coefficient was assumed to be of the form:

D = Dg exp[

− Z(

1vf(t)

− 1vg

f

)](69)

where vgf is the fractional free volume of the slowly cooled glassy polymer, Dg is the

diffusion coefficient when vf = vgf, and Z is a material parameter. The concentra-

tion dependence of the diffusion coefficient captures the experimentally observedself-retarding nature of physical aging. The diffusion coefficient of free volumedecreases as free volume is eliminated from the sample. This is the same concen-tration dependence seen for the diffusion coefficient with many small penetrants.

The dual-mechanism approach of McCaig and co-workers described abovewas able to describe the oxygen permeability decay in bisphenol. A benzophenone-dicarboxylic acid (BPA–BnzDCA) films of varying thickness remarkably well. Thisdata is illustrated in Figure 35. All films, regardless of thickness, experience aninitial sharp decline in permeability. This drop is due to the lattice-contractionmechanism, which is independent of sample dimension. Only in the thin filmswhere the half-thickness � was less than 2.5 µm was a contribution from thefree-volume diffusion mechanism observed.

Orientation-Induced Effects. Orientation and combined heat and ori-entation processing affect the transport properties of glassy polymers. Especiallywhen crystallites are present, the effects can become surprisingly large. As notedfor rubbery semicrystalline materials, the obvious improvements in barrier prop-erties associated with organization of lamellar crystalline domains with theirplatelets perpendicular to the direction of penetrant flow can produce significant


Table 7. Transport Properties of Kapton H and PMDA–ODAPolyimide Films at 2 MPa

CO2 CO2 solubility, CO2

permeability, cm3(STP)/ diffusivity, CO2/CH4

Sample Barrersa (cm3 · MPa)b 10− 10 cm2/s selectivity

Kapton H 0.3 22 11 55.0PMDA–ODA 2.6 37 56 44.6

a1 Barrer = 10− 10 [cm3(STP)cm]/(cm2 · s· cm Hg) = 0.335, mmol/(m · s · TPa).bTo convert cm3(STP)/(cm3 · MPa) to cm3(STP)/(cm3 · atm), multiply by 0.1.

reductions in permeability. Simple crystallinity appears to act principally as tor-tuous blocking elements to increase the path length of penetrants in their randomwalks across the film (eqs. 18b and 45).

Depending on the aspect ratio, orientation of the crystallites can have mini-mal or profound effects. The results in Table 7 show roughly an order of magnitudereduction in the permeability of CO2 in a sample of biaxially oriented Kapton poly-imide as compared to a chemically identical, but essentially unoriented, sample ofthe same polyimide of pyromellitic dianhydride and oxydianiline (PMDA–ODA).The reductions in permeability caused by the biaxial orientation are greater forthe larger CH4 molecule as indicated by the higher permselectivity in the Kapton.X-ray scattering measurements indicate the presence of small platelet-like aggre-gates in the Kapton (137). The orientation of the platelets perpendicular to thedirection of permeation adds significantly to the tortuosity of the Kapton samples.This is reflected in the solubility and diffusivity data in Table 7. The primary effecton permeability is due to the diffusivity component. It has been suggested thatthe platelet dimensions are on the order of 20 nm × 20 nm × 5 nm.

In studies of oxygen and water vapor permeability in PVC having low crys-tallinity (ca 3–5%) (138), the effects of biaxial orientation, draw rates, temperatureof draw, and subsequent annealing temperatures have been reported. As expected,the effects of orientation are much less significant in PVC than in more highly crys-talline materials in which the effectiveness of flow impediments can be influencedgreatly by orientation. The oxygen and water permeability typically decrease byless than 50% with the various processing parameters, and in fact show increasesin some cases.

In another study of the effects of uniaxial orientation on PVC and other es-sentially amorphous glasses (139), a wide range of penetrants has been consideredand permeability, solubility, and diffusivity effects caused by various orientationvariables have been explicitly determined (Fig. 36) (139). The orientation, mea-sured by birefringence, leads to improved barrier properties, and appears to havean effect on both the apparent solubility coefficient ka = C/p and the apparentdiffusivity determined by time-lag measurements.

One of the few studies of transport in liquid crystalline glassy polymers hasyielded some interesting results (140). The liquid and more ordered nematic mor-phologies characteristic of these highly rigid rod-like materials is compared inFigure 37 to more flexible chained materials such as glassy poly(ethylene tereph-thalate) (PET), PVC, or rubbery polyethylene discussed previously.


Fig. 36. Effects of uniaxial orientation on permeability at 35◦C. Drawing temperature =100◦C. (a) Permeability, relative to “as received” value P0 at 35◦C for �, He; �, Ar; and �,N2 in PVC as a function of draw ratio. (b) Relative permeability at 35◦C for �, He; �, Ar;and �, N2 as a function of birefringence. (c) Apparent solubility for Ar and N2 in PVC at35◦C vs film birefringence. (d) Apparent diffusion coefficients for Ar and N2 in PVC at 35◦Cas a function of film birefringence.

The liquid crystalline material with the approximate repeat unit shown asstructure (1) above has properties that approach those of the excellent barrier


Table 8. Gas Transport Properties at 35◦C of Liquid Crystalline Polymer andPolyacrylonitrilea

P, 10− 5 Barrersb Da, 10− 10 cm2/s Sa, cm3(STP)/(cm3 · MPa)c

Gas LCP PAN LCP PAN LCP PAN

Helium 17,700 71,000 6,600 270 0.002 0.20Nitrogen 3.0 2.9 1.4 0.042 0.0016 0.0520Oxygen 47 54 7.1 0.14 0.0050 0.29Argon 10 18 1.5 0.042 0.0054 0.33Carbon dioxide 70 280 0.96 0.023 0.0510 9.2aRef. 116.b1 Barrer = 10− 10 [cm3(STP)/cm]/(cm2 · s · cm Hg) = 0.335 mmol/(m · s · TPa).cTo convert cm3(STP)/(cm3 · MPa) to cm3(STP)/(cm3 · atm), multiply by 0.1.

polyacrylonitrile (PAN) and are much superior to those of the structurally simi-lar, but nonliquid crystalline PET (Table 8). PAN provides a permeation barrierlargely on the basis of its tremendous resistance to penetrant diffusion. The lowdiffusion coefficients are associated with the intersegmental polar attractions ofthe acrylonitrile pendent groups. The strong polar attractions produce a highcohesive energy density that in turn increases activation energy for diffusion andthereby suppresses the diffusion coefficient. Resistances to intrasegmental mo-tion caused by the size of the acrylonitrile group are believed to be of secondaryimportance in suppressing diffusion compared to the strong intersegmental forces.

The rigid liquid crystal material does not have particularly strong interseg-mental attractions, and although it has significant intrasegmental resistance torotation, it is not even as rigid as the polyimide materials noted earlier. Becauseof the highly ordered structure of the liquid crystalline material, the extremely

Fig. 37. Comparison of liquid crystal and conventional random coil polymer properties.


low gas solubilities reported are not surprising; however, it is surprising that thediffusivities in this highly crystalline medium are not significantly different fromthose of PET, which has a much lower crystalline fraction.

Non-Fickian Transport Behavior

The topic of non-Fickian transport encompasses all cases in which the simpleform of the flux equation given by equation 1, with a concentration-dependentdiffusion coefficient and constant-system boundary conditions, is insufficient todescribe the observed time-dependent behavior of the system. This definitionincludes cases with position-dependent diffusion and solubility coefficients. Forexample, a film conditioned with a swelling agent (eg, CO2) used to induce dif-ferent properties at the upstream and downstream faces falls in this category.Another variation of this problem includes cases in which quenching or orienta-tion produces transverse swelling stresses during the sorption kinetics process.Such position dependencies, superimposed on concentration-dependent transportcoefficients, considerably complicate the response characteristics of a system. Nev-ertheless, mathematical analysis of the problem is possible by generalizing pro-cedures used for the case of concentration-dependent, time-invariant coefficientsif the position-dependent profile of the transport properties can be defined.

Complex concentration dependence of the sorption and diffusion coefficientscan be accommodated within the framework of Fickian transport if nonisotropicvariations in properties are not too serious. The use of an average diffusion co-efficient and an average solubility coefficient defined over a particular range ofexternal penetrant boundary conditions are acceptable for many practical appli-cations.

Unlike the situation with Fickian systems, when dealing with non-Fickiansystems data derived from the kinetic responses of a system cannot be readilyextended to other configurations with significantly different boundary conditionsor sample dimensions. By far the most difficult problems encountered in the do-main of non-Fickian processes involve time-dependent changes in transport andsorption properties, ie, additional time dependence, beyond that caused by simpletime variation in the local concentration. The additional time-dependent problemis the most difficult to treat because it generally involves introducing additionalphysical expressions. Transverse swelling stresses typify such cases (141). In thisexample, the additional physics are accommodated by a linear viscoelastic approx-imation to account for the time-dependent relaxation of stresses induced in themedium by swelling.

Another example of this class is solvent-induced crystallization (142). Theeffects of crystallization kinetics and rate of movement of the boundary betweenpenetrated and unpenetrated regions have been treated. Moreover, effects of in-creasing impediments to diffusion caused by the increasing crystalline fractionduring the same time scale as that of the solvent transport are included in themodel of the system. Such models become exceedingly complex and fall outside thedomain of the present discussion; however, they clearly require the input of infor-mation related to transport properties and their dependence on local conditions.


Fig. 38. Comparison of theoretical curves and experimental data for n-hexane sorptionin the large- and small-diameter spheres at P/P0 = 0.75 and 30◦C. A, Fickian model,diameter = 184 µm; B, Case II model, diameter = 0.534 µm.

Testing samples of different macroscopic dimensions of a given material undercertain conditions can bring unusual factors into play.

Indeed, entirely different responses can be observed as the time requiredfor relaxation of stresses varies relative to the time required to change the localconcentration at a point in the material. The dependence of mass absorbed Mton time at early time in a sorption run is often used to categorize the type oftransport process occurring. A simple dependence on the square root of time (t0.5)is an indication of Fickian transport (recall eq. 3a), whereas dependence ta, a >

0.5 on is indicative of non-Fickian transport and is termed anomalous.A limiting case of non-Fickian response in which a = 1.0 is typically referred

to as Case II diffusion to differentiate it from normal Fickian (Case I) diffusion.Case I (Fickian) diffusion shows a linear increase in sorption as a function ofthe square root of time. By contrast, Case II kinetics are characterized by linearmass uptake with time as shown in Figure 38, where uptake of n-pentane inpolystyrene at high activity (penetrant partial pressure) shows a Fickian responsein small spheres and a non-Fickian response in larger diameter spheres (143).The data strongly suggest that diffusion into the small spheres is so rapid thatthere is insufficient time to generate a Case II concentration profile. Apparentlythe diffusional equilibration in the small spheres is essentially complete beforethe complex step concentration profile associated with Case II sorption can beestablished. This behavior is similar to other observations (144).

The general problem of two independent time scales for diffusive equilibra-tion and molecular relaxation existing in a system has been considered (145).Fickian transport is obtained when the time scale of the relaxation is either effec-tively zero or infinite compared to the time required to establish a concentrationprofile in the sample. These two limits have been called the elastic and viscousFickian diffusion limits, respectively.


The elastic limit is characteristic of samples exposed to gases or low activitiesof vapors that experience little segmental disruption relative to the native polymerand are able to accommodate the sorption by elastic deformation of intrasegmentalbonds or slight disruption of intersegmental bonds while maintaining the essen-tial integrity of the entanglements of interpenetrating chains in the matrix. Onremoval of the perturbing penetrant, the matrix is able to recover to its originalcondition.

The viscous limit applies when significant swelling occurs and reorganizationof the entanglement network may occur sufficiently rapidly during the solventinvasion to relax any local stresses that may tend to build. In this situation, onlya complex concentration-dependent diffusion coefficient is required to describe thelocally changing environment through which a penetrant must move. An exampleof this behavior is the initial desorption of solvent from a highly swollen polymer.

Between these two limiting Fickian cases lies an important domain in whichsignificant swelling (greater than 10% by volume) occurs. Under such conditions,the polymer deformation in response to a local swelling stress caused by a localpenetrant concentration is rapid (but not infinite) as compared to the time requiredto produce a change in local concentration. Typical uptake behavior in such Case IIsituations is linear with time as shown for the large spheres in Figure 38. The chiefattributes (146) of these systems are a sharp advancing boundary existing betweenan inner glassy core and an outer swollen rubbery shell; a swollen gel behind theadvancing penetrant front close to equilibrium with the external solvent supply;and a boundary between the swollen gel and glassy core, which advances at aconstant velocity.

An induction time prior to the onset of the linear uptake in mass with time isalso a characteristic common in Case II systems (147). However, this behavior issomewhat difficult to discern using the gravimetric and optical methods usuallyemployed. An induction time is consistent with the criterion for the relatively rapidpolymer deformation in response to a local swelling stress compared to the timerequired to produce a change in concentration profile. The approach of the surfaceconcentration to true equilibrium in such non-Fickian systems is protracted (148).A connection has been made between the effective deformation stress acting todilate the matrix and a thermodynamically calculable osmotic swelling stress(149). Case II front movement is visualized to occur when local stresses build toa sufficient extent to allow crazing. Combining the two concepts of an inductiontime at the surface and an osmotically driven local dilational stress, the yieldingproblem has been formulated in terms of a concentration-dependent viscosity ofthe matrix (150).

Thus, in essence, the Case II process is initiated when a surface element ofsufficient thinness experiences a significant reduction in viscosity and is able torelax some of the stresses due to the elastically deformed chains of the polymer.This relaxation of stresses allows more solvent to invade the thin element, furtherdepressing the viscosity, and thereby autocatalytically accelerating the stress re-laxation. This results in a surface concentration that exponentially approaches itsequilibrium value. While this outermost element is undergoing its rapid, but notinstantaneous, relaxation, the solvent that is now available to the next infinites-imally thin element invades it as has occurred in the first exposure of the sur-face element. The transport to this interior element is essentially instantaneous,


because of the much lower resistance to diffusive supply through the swollen gelas compared to diffusive invasion of the virgin polymer in the second element. Inthis case, therefore, the second element experiences significant relaxation by thetime the first is approaching its equilibrium conditions with respect to the exter-nal penetrant. A third, fourth, and progressive repetition of this process leads to anegligible concentration gradient in the swollen gel behind the front, and a smalland very sharp concentration gradient in some characteristically small numberof volume elements ahead of the front.

Assuming the viscosity is dependent on the local penetrant volume fractionϕA in the volume element under consideration (147),

η = η0 exp( − mϕA) (70)

where m is an empirical constant and η0 is the viscosity of the virgin polymer con-taining no penetrant. This expression is, again, related to the free-volume conceptsdiscussed in relation to equations 18a and 19 with regard to diffusion coefficients.Further treatment (147) has focused on the magnitude of the induction time forfront initiation, neglecting all diffusional resistance in the gel layer behind thefront and assuming an effective average diffusion coefficient D̄ of the penetrant inthe partially plasticized thin element ahead of the front. The front velocity shouldbe proportional to the square root of the ratio of the local effective diffusivity ineach of the infinitesimal volume elements preceding the relaxation front and theunplasticized viscosity of the glassy material prior to penetration, namely

V(T) ∝(

D(T)η0(T)

)0.5

(71)

Evidence is given to support this suggestion (147). In the study, it was con-cluded that the relative activation energies for the effective solvent diffusion coef-ficient and the intrinsic viscosity of the glass were 58.6 kJ/mol (14 kcal/mol) and167 kJ/mol (40 kcal/mol), respectively. Based on the earlier discussion of Figure 5,the infinite-dilution values of the activation energy for the diffusion coefficientand viscosity of the glass are expected to be similar. The lower activation energyfor the diffusion coefficient D̄ in the partially plasticized glass ahead of the frontas compared to the viscosity of the unplasticized glass, η0, is not surprising.

Recent theoretical and experimental work has significantly advanced theunderstanding of Case II transport; however, because of the general complexity ofthis work, a review of it is beyond the scope of this article. The interested readeris directed to the work of Petropoulos and co-workers (151), Wu and Peppas (152),Huang and Durning (153), and the references therein.

Facilitated Transport

Facilitated transport involves the coupling of a reversible chemical reaction to themass transfer through the polymer. This type of transport is very common in bio-logical systems, and is capable of substantially increasing the flux and selectivityof synthetic polymers. Facilitated transport still proceeds by the solution-diffusion


mechanism; however, the penetrant now has two diffusive pathways through thepolymer. The penetrant can sorb into the membrane and diffuse down its chemicalpotential gradient, or the penetrant can complex with the carrier upon sorptionand diffusion down a penetrant-carrier complex chemical potential gradient. Thesecond pathway is only available to penetrants that complex with the carrier. Thepresence of a carrier usually increases the sorption of the complexing penetrantas well.

The inclusion of the complexation reaction in facilitated transport greatlyincreases the mathematically complexity of the flux equations at steady state. Theconcentration through the polymer of the penetrant, cA, the carrier, cc, and thepenetrant–carrier complex, cA–c, must satisfy the following continuity equations:

DAd2cA

dx2− rA–c = 0 (72a)

Dcd2cc

dx2− rA–c = 0 (72b)

DA–cd2cA–c

dx2+ rA–c = 0 (72c)

where DA, Dc, and DA–c are the diffusion coefficients of the penetrant, carrier, andpenetrant-carrier complex, respectively. The net reaction rate of complexation isrA–c. The boundary conditions for the above equations are determined from anappropriate sorption model. Cussler solved the above set of equations analyticallyfor the two limiting cases of fast reaction and fast diffusion (154). Except forthese limiting cases, numerical methods are required to determine the flux ofeach species.

Facilitated transport has been reported in a number of systems. One of themost common systems uses silver as a carrier for olefins (155–157). Olefins areable to complex with silver, while paraffins are not. Transport of carbon dioxidehas been facilitated through the use of amine complexes (158–160). Porphyrinshave been used to facilitate the transport of oxygen and increase oxygen/nitrogenselectivity (161–163). Several methods have been used for carrier inclusion in thepolymer: (1). carriers are blended as ionic salts with the polymer, (2). carriersare bonded to the polymer backbone through the use of crown ethers, and (3).carriers are used as the exchange sites on ionic exchange membranes. Regardlessof the carrier and the inclusion technique, all facilitated transport membranesstill currently suffer from unstable performance. The carrier is often poisoned bycontaminants in the feed stream, and in some cases the carrier simply leachesout of the membrane. These issues must be resolved before there is widespreadindustrial adoption of facilitated transport membranes.

Contradictory to facilitated transport, the use of scavengers attempts toinhibit transport of particular penetrants. Scavengers are adsorbents that areblended with the polymer. This technology is often used in food packaging.


Unfortunately, most of the literature on scavenging techniques is only availablein patents (164–166). Scavengers typically attempt to limit transport of oxygen,water, and organoleptics. Anthraquinone and iron have been used as oxygen scav-engers. Clays, molecular sieves, and other common desiccants have been used aswater scavengers. Antioxidants, such as butylated hydroxytoluene (167), are oftenused as organoleptic scavengers.

Proton Exchange Membranes

In recent years, a great deal of research has been focused on proton exchangemembranes (PEMs) for use in fuel cells. Fuel cell based vehicles are touted asa more efficient, environmentally-friendly replacement for internal combustionengines. The fuel cell harnesses energy from the chemical reaction of hydrogenand oxygen to form water. In this reaction, the cathode and anode are separatedby the PEM which allows passage of protons and theoretically blocks passage ofother species. The electrons produced from the reaction are routed through anexternal circuit to provide power.

Nafion produced by Dupont is the most commonly used PEM material. Nafionis a hydrated perfluorosulfonic acid (PFSA) polymer. The Nafion backbone is hy-drophobic, while side-chain sulfonic acid groups are hydrophilic. The repeat unitof Nafion is given as structure (2).

Hydrated Nafion is thought to have a morphology composed of a hydrophobicmatrix with dispersed hydrophilic sulfonic acid clusters connected by pores (168).At low hydration levels, the pores between sulfonic acid clusters are thought tobe collapsed. At high hydration, the pores become open and the sulfonic acid clus-ters are truly interconnected. As a result of this rather complex morphology, agreat detail of debate exists in the literature about the mechanism of transportin Nafion. Models of transport based on the solution-diffusion mechanism havebeen proposed (169,170). These models picture Nafion as a one-phase homogenousmaterial where transport is driven by chemical potential gradients. Alternativemodels, based on the pore-flow mechanism, have also been proposed for transportin Nafion (171,172). In these models, Nafion is pictured as a porous material, wheretransport is achieved by convective flow through pores. Neither type of model isable to describe all of the observed transport phenomena in Nafion. Recently, ithas been argued that a combination of the solution-diffusion and pore-flow mod-els may be appropriate for Nafion (173,174). At low hydration levels, solution-diffusion based transport is the dominant mode because no interconnected poresexist. At high hydration levels, pore flow based transport is dominant because now


an interconnected pore network does exist. At intermediate levels of hydration,the two mechanisms of transport are both present.

Since the scope of this article is limited to transport in dense, nonporouspolymers, the rest of this discussion is limited to transport where a solution-diffusion mechanism is assumed to occur. In this regime, the flux of “A” is givenby (175)

jA = − DA

(dcA

dx+ zAcA

FRT

dψ

dx

)(73)

where zA is the charge of “A”, F is Faraday’s constant, and ψ is the electrostaticpotential. For protons, which have a charge of zA = 1, the gradient in electrostaticpotential is the dominant driving force, and the flux equation reduces to

jH+ = − DH+ cH+ FRT

�ψ

L(74)

where L is the membrane thickness. In the literature, the flux is often reportedas a current density i, which equals

i = FjA = σ�ψ

L(75)

where the conductivity σ = DAcA F2/RT. In the case of noncharged penetrants, thecharge equals zero, and equation 73 reduces to Fick’s law. The above equations areexact only when the diffusion coefficient is constant, the membrane is uniformlycharged, and electroosmosis effects are neglected.

Positron Annihilation Lifetime Spectroscopy

From the models of transport through both rubbery and glassy polymers givenpreviously, it becomes apparent that knowing the free-volume distribution of thematerial is crucial for a microscale description of gas transport. Recently, positronannihilation lifetime spectroscopy (PALS) has been developed to probe the freevolume of glassy polymers. PALS injects positrons, antielectrons, into the materialwhere they have an extended lifetime because of interactions with the sample.The lifetime of the positronium species is on the order of a few nanoseconds and issensitive to the electron density of the environment. This makes PALS a model-dependent technique which is capable of measuring free-volume holes that existfor 10− 9 s or longer. PALS is able to resolve free-volume hole sizes in the range of1–10

◦A;. Reviews of PALS are available (176–178).The positron source is usually 22Na, which releases a positron every 1.5 ms

as it decays into 22Ne. Approximately 3 ps after the positron is emitted from thesource, gamma radiation of energy 1.28 MeV is released. This energy release isdetected and marks the positron “birth.” Once inside the sample, the positronannihilates by one of the three possible modes. The first and shortest lived moderesults from the para-positronium, p-Ps, species which is formed when the positron


extracts an electron from the sample with opposite spin. The p-Ps species anni-hilates with its own electron after a lifetime of 125 ps. The lifetime of p-Ps isindependent of the environment. The second mode of annihilation results fromfree positrons which have a lifetime of 100–500 ps. The third and longest livedmode results from the ortho-positronium, o-Ps, species which is formed from apositron and an extracted electron with opposing spins. The o-Ps decays afterapproximately 103 ps when it extracts or picks off another electron from the sam-ple. After each annihilation event, regardless of the mode, gamma radiation ofenergy 0.511 MeV is released. This energy is detected and marks the “death” ofthe positron. The spectrum produced from these annihilation events is most com-monly fit using three exponential decays. The resulting parameters are I1, I2, andI3 for the intensity of each mode and the time constants τ 1, τ 2, and τ 3.

Since the behavior of the o-Ps species is dependent on the polymer free vol-ume, the parameters associated with it, I3 and τ 3, can be used to characterizethe glassy structure. The probability of o-Ps formation and the concentration offree-volume holes are related to the intensity I3. The time constant τ 3 is sensitiveto the size of the free-volume holes. Simplistically, the time constant is short insmall free-volume holes because the close proximity of the o-Ps to neighboringmass within the sample makes it easier for the o-Ps to pick off an electron. Thesemiempirical relationship between the time constant τ 3 and the free-volume holeradius R is given below (179).

τ3 = 12

[1 − R

R0+ 1

2πsin

(2πRR0

)]− 1

(76)

R0 is equal to R + �R where �R is the electron layer thickness. The valueof �R was found empirically and is usually assumed to equal 1.66

◦A;. Equation 76

assumes that the o-Ps resides in the center of a spherical free-volume hole, and soit follows that the hole volume Vh is equal to 4/3πR3. The fractional free volumeof the material is given by (180)

vf = CVhI3 (77)

where C is a material parameter determined from the P-V-T properties of thepolymer. The free volume probability-density function, Vh pdvf, is given as (181)

Vhp dvf = − 3.32{

cos[

2πRR + 1.66

]− 1

}α(λ)

(R + 1.66)2K(R)4πR2(78)

where the fraction of holes between Vh and dVh is Vh pdvf dVh. In equation 78, α(λ)is the annihilation probability-density function. This function is obtained from acontinuous fit of the positron lifetime spectra. K(R) is a correction for the captureprobability of o-Ps in different size holes. A detailed description of equation 78 isgiven elsewhere (182).

Because PALS is still a very new technique for characterizing glassy poly-mers, only a limited number of materials have been investigated which are alsoof interest from a gas transport standpoint (181,183–187). The most thorough


Fig. 39. Free-volume hole distributions of various polycarbonates. Dashed lines are themolecular volumes of common penetrants. Reprinted from Ref. 181. This material is usedby permission of John Wiley & Sons, Inc.

study of this kind yet conducted was carried out by Shantarovich and co-workers,where positron lifetime spectra were collected for 10 glassy polymers (187). Jeanand co-workers used PALS to study the same polycarbonates shown in Figure 28:PC, TMPC, HFPC, and TMHFPC (181). The distribution of free-volume hole sizesin these polymers is given in Figure 39 along with the molecular volume of com-mon penetrants. The materials with larger average free-volume hole size also hada wider distribution of hole sizes. The diffusion coefficient of O2 and CO2 throughthese materials correlated well with the experimentally determined fractionalfree volume from equation 77. The ability of PALS to determine both the hole sizedistribution and the fractional free volume opens up a number of very interestingquestions. For example, how will the transport differ in materials of the samefractional free volume but with different free-volume distributions?

Another interesting set of articles (188,189) uses PALS to investigate thebehavior of the free-volume as the polymer is subjected to high pressure gases.As PC is pressurized to approximately 4.14 MPa (600 psi) by N2, the free-volumehole radius obtained from the positron lifetime spectra increases. Continuing toincrease the N2 pressure from 4.14 MPa (600 psi) to 8.28 MPa (1200 psi) results ina decrease in the free-volume hole radius. No hysteretic effects were observed upondepressurization. This behavior was explained as plasticization and hydrostaticeffects similar to those already discussed in the context of rubbery polymers. TheN2 is claimed to initially plasticize or soften the polymer resulting in larger free-volume holes. These results appear rather contrary to concepts noted with regardto the effect of pressurization on free volume in Fig. 8. As the pressure is increasedmore, the polymer compresses resulting in smaller free-volume holes, which isconsistent with Fig. 8. A very different effect was observed when PC was exposedto high pressure CO2. The free-volume hole size was observed to monotonicallyincrease with CO2. Upon depressurization, a large hysteresis was seen. AfterCO2 exposure, the fractional free volume and average hole size were larger. This


behavior results from the strong conditioning effect of CO2 that was described ina previous section.

Although PALS has been applied successfully to determine the glassy poly-mer microstructure, the technique has several limitations. In many low polarliquids, the positronium species forms a “bubble.” This bubble results from repul-sions between the electron in the positronium and electrons in the surrounding.Obviously, formation of such a bubble would affect the free-volume characteriza-tion. This bubble formation process may be possible in rubbery polymers, but isgenerally not thought to occur in glassy polymers (176). Inhibition or quenchingeffects can also affect the lifetime of the positronium species. Halogen-containingpolymers can lead to inhibition effects when the positronium interacts with thehalogen atom. Nitroaromatics, such as polyimides, may react with the positroncausing quenching (190). Perhaps, the largest disadvantage of PALS is that itis a model-dependent technique. In this respect, characterization of free-volumedistributions in polymers using PALS is analogous to obtaining the pore-size dis-tribution of a porous material from vapor sorption–desorption experiments.

Molecular Modeling of Transport in Amorphous Polymers

The rapid increase in computing power over recent years has led to a signifi-cant body of research, which seeks to describe penetrant sorption and diffusion inamorphous polymers through the use of computer simulations. Two main goalsexist for computer simulations: first, to gain a better fundamental understandingof the transport mechanism through amorphous polymers; and second, to accu-rately predict the transport properties (ie, the diffusion and solubility coefficients)of novel polymers. The first goal is motivated by the fact that small-molecule trans-port is determined by the structure of the penetrant–polymer environment on anatomistic scale. Excluding the recent use of PALS structural information on thislength-scale is not accessible from traditional macroscopic transport experimentsbut is accessible from molecular dynamic (MD) simulations. The second goal, ac-curate prediction of transport properties, seeks to eliminate expensive and labor-intensive experiments. Several reviews are available on computer simulations ofgas diffusion through amorphous polymers (191,192).

The starting point for all computer simulation methods is the choice of apotential energy function or force field, �(R). This function gives the potentialenergy of a system of N atoms at positions ri. The positions of all atoms in thesystem are represented by the three dimensional vector R = (r1, r2, . . . , rN). Theforce-field accounts for all bonded and nonbonded interactions between atomsin the system. Bonded interactions include bondlength and bond-angle deforma-tions and conformational deformations. These interactions are usually describedas mechanical springs with the spring constant related to experimentally knownbonding energies. The nonbonded interactions include van der Waals forces andother molecular forces and are accounted for empirically by a Lennard–Jones po-tential. Since the nonbonded interactions include contributions from each atompair in the system, current computational power limits the number of atoms inthe system to less than 10,000 (191). Many force-field models exist and properselection is crucial for accurate results.


Once the force-field is chosen the motion of atoms in the system is simulatedusing classical equations of motions:

dui(t)dt

= 1mi

Fi[R(t)] (79a)

dri(t)dt

= ui(t) (79b)

where ui(t) is the velocity of atom i with mass mi, at time t. The force on i, Fi, isgiven by

Fi = − ∂�(R)∂ri

(79c)

Equations 79a, 79b, and 79c are a coupled set of second-order differential equa-tions. Methods exist to solve these equations numerically only for small timesteps.The timestep can not be greater than about 1 fs, which is about a tenth of the C Hbond oscillation time. Because such extremely small timesteps are required, anupper limit of several nanoseconds is imposed on current MD simulations (191).

The limit of MD models to at most 10,000 atoms on current workstationsresults in model systems that are cubic volume elements with a side length of justa few nanometers. Many approaches have been attempted to initially populatethe system with atoms. The approach described by Theodorou and Suter seems tohave gained the most use in recent literature (193,194). In this method the cubicvolume element is filled by growing the chain one backbone bond at a time. Rota-tional isomeric states (RIS) theory is used to calculate the conditional probabilityof each conformation. As the chain is grown, the conformation angle is selectedstatistically to agree with the RIS information. After the chain has grown a fewrepeat units, it is likely to cross the boundary of the volume element. When thishappens, the chain growth is continued by reentering the volume element on theopposite side. Chains grown in this manner typically achieve the overall densitytarget, but the chain growth often results in regions of severe atomic overlap. Tocombat this effect, a series of static energy minimizations and dynamic simula-tions are carried out to equilibrate the structure. After equilibration, it is possibleto get the density of the volume element to within a few percent of the experimen-tally determined density; however, such a high level of agreement is not currentlypossible for all materials because of deficiencies in the force field.

The particle trajectories from the MD simulations can be used to determinethe self-diffusion coefficient using the Einstein equation:

D∗A = 〈|r(t) − r(0)|2〉/6t = λ2f

6(80)

where r(t) is the position of a penetrant atom at time t, r(0) is the initial position ofthe penetrant, and the angular brackets indicate an average over all penetrantsin the system. The term 〈|r(t) − r(0)|2〉 is the mean-squared displacement of the


penetrant. f simple indicates the average frequency of a diffusional jump of lengthλ. The diffusion coefficient in equation 80 ideally allows one to estimate the self-diffusion coefficient in equation 6. The Einstein equation gives accurate resultsfor rubbery polymers; however, in glassy polymers, anomalous diffusion effectsact to degrade the accuracy (195). In anomalous diffusion the penetrant is notproceeding by a truly random walk from one free-volume hole to the next which isassumed by the Einstein equation. This effect results from penetrant oscillationsinside the free-volume holes that occur between diffusive jumps from one hole tothe next. Since the simulation time is only a few nanoseconds, these oscillationshave a nonnegligible effect on the mean-squared displacement. It is importantto note that this definition of anomalous diffusion is different than non-Fickiandiffusion, discussed previously, which is also commonly referred to as anomalousdiffusion in classic texts (5).

Gusev and Suter have developed a transition-state theory that is based on aMonte Carlo scheme which requires less computational power than the MD sim-ulations and therefore is able to calculate diffusion coefficients where anomalousdiffusion effects are negligible (192,196–198). In this method a three-dimensionalgrid is overlaid on a fully equilibrated MD volume element. A typical grid spacingis 0.03 nm (191). The insertion energy of a penetrant molecule is calculated ateach grid location. In this calculation the insertion energy comes from the non-bonded contribution of the force field. The volume element is then mapped asregions of high and low packing on the basis of whether the insertion energy washigh or low. This mapping allows for the identification of pathways between ad-jacent low energy regions. Each of these pathways is assigned a jump probabilitywhich is based on the Boltzmann distribution. This jump probability is affectedby the thermal motion of the polymer atoms which is characterized by the mean-squared displacement of polymer atoms, 〈�〉2. This parameter is often gained froma short MD simulation. It should be noted that the Gusev and Suter method doesnot account for polymer relaxation as a result of penetrant insertion, and so thistheory is only reasonable for small penetrants up to methane. With the appropri-ate jump probabilities, the penetrant diffusion is simulated using a Monte Carloscheme. The insertion energy Eins calculated during this process can also be usedto calculate the solubility coefficient S:

S = exp( − µex/RT) (81)

where the excess chemical potential µex = RT ln 〈exp(−Eins/kBT)〉 (199). Diffusionand solubility coefficients calculated using the above techniques are usually withina factor of 3 to 5 of experimentally determined values. This degree of agreementis not accurate enough for many applications.

Molecular modeling of the type described above has been conducted on anumber of polymer systems. Of these systems the great majority have been basedon rubbery polymers. It is only recently that molecular simulations based on glassypolymer systems are appearing in the literature (200–206). From these studies,differences in the diffusion mechanism of rubbery and glassy polymers are startingto emerge. In both cases, the penetrant can spend relatively long periods of time(several hundred picoseconds) bouncing around inside the free-volume hole. Atsome point in time, a transient gap opens up because of thermally induced polymer


Fig. 40. H2 diffusion through PEEK shows diffusion jumps of 5–10◦A; separated by several

hundred picoseconds of oscillations in free volume. Reprinted from Ref. 200, with permis-sion from Elsevier.

motion between free-volume holes. While the gap is open, the penetrant may makea diffusive jump between holes. These jumps are usually estimated to be 5–10

◦A;

in length (see Figs. 40 and 41). In rubbers the gaps between free-volume holes isonly open for a few picoseconds, and the penetrant does not have the opportunityto make a backwards jump into the original free-volume hole. This is not the casein glassy polymers, where the channel between voids can be open for an extendedperiod of time (several nanoseconds). This allows the penetrant molecule to jumpback and forth numerous times between the two connected free-volume holes.The back-and-forth motion experienced by the penetrant in this behavior does notadd to the overall diffusion coefficient (see eq. 80). This effect has been suggested

Fig. 41. N2 diffusion through PEEK shows back-and-forth diffusion jumps that have beenobserved in many glassy polymers. Reprinted from Ref. 200, with permission from Elsevier.


as a contributing factor to explain why rubbery polymers exhibit much higherdiffusion coefficients than glassy polymers (191), since it reduces the frequency fof a successful jump in equation 80. However, there has been recent evidence thateven amorphous rubbery polymers may be significantly affected by anomalousdiffusion because of alignment of the polymer chains at the nanoscale (207).

The frontier in the field of transport properties of polymers involves bridg-ing the gap between the molecular scale penetrant–polymer environment and themacroscale solubility and diffusion coefficients. Impressive efforts to achieve thisgoal have already been made through the use of PALS and molecular modeling.The continued increase in computing power will allow for molecular dynamic sim-ulations of larger volume elements for longer time periods. These improvementsshould cure many of the current inaccuracies inherent in molecular modeling, andallow for the first time a truly predictive method of determining polymer transportproperties.

The topics discussed in this article only dealt with small-molecule diffusionthrough polymers. Another interesting and complex topic involves the diffusionof large polymeric penetrants through a polymeric medium. de Gennes provideda theoretical treatment of a polymer chain diffusing through a cross-linked rub-ber (208). In this treatment the polymer chain motion is referred to as reptation,since the motion of the chain is similar to that of a snake. The diffusing poly-mer chain is confined to a tube defined by the surrounding cross-linked matrixwhich restricts transverse motions. More complex treatments of reptation allowfor the self-diffusion of the surrounding polymer matrix (209). The diffusion ofmacromolecular penetrants is of importance in a number of industrial applica-tions including crack healing and adhesion between polymer surfaces, and in theuse of polymer composites and blends.

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WILLIAM J. KOROS

WILLIAM C. MADDEN

Georgia Institute of Technology

TRIBOLOGICAL PROPERTIES OF POLYMERS.See SCRATCH BEHAVIOR OF POLYMERS; SURFACE MECHANICAL

DAMAGE AND WEAR.

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