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TRANSPORT PROPERTIES Introduction Transport properties determine a polymer’s ability to move through some medium or to have some penetrant medium move between its constituent segments. This definition encompasses processes with diverse driving forces such as concentration and pressure gradients, and even electrical or temperature gradients capable of motivating one component relative to another. This article emphasizes polymer transport properties under conditions of low to intermediate penetrant concentration, where extraordinary differences can exist between the diffusivities of penetrants having relatively small differences in molecular size or shape. This article focuses on transport that proceeds by the solution-diffusion mechanism. Transport by this mechanism requires that the penetrant sorb into the polymer at a high activity interface, diffuse through the polymer, and then desorb at a low activity interface. In contrast, the pore-flow mechanism transports penetrants by convective flow through porous polymers and will not be described in this article. Detailed models exist for the solution and diffusion processes of the solution-diffusion mechanism. The differences in the sorption and transport properties of rubbery and glassy polymers are reviewed and discussed in terms of the fundamental differences between the intrinsic characteristics of these two types of polymers. The transport properties of glassy and rubbery polymers are related to their microstructural morphology. For a penetrant to diffuse, a minimum characteristic packet of unoccupied volume is required. The penetrant diffuses by jumping through transient gaps between packets of unoccupied volume. The lifetime, size, and shape of these volume packets and the transient gaps that connect them are dependent 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.

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positron annihilation lifetime spectroscopy (PALS) and molecular modeling allow for elucidation of dynamic activities at the molecular scale. While a description at this length-scale offers fundamental information about the diffusion process, it is not yet possible to predict the macroscopic transport properties with sufficient accuracy for practical applications. In this regard, quantitative phenomenological models and more molecularly based treatments are complementary, and both will be considered.

Terminology Reference Frames and Fluxes. In speaking of a diffusional flux, it is necessary to specify a reference frame from which the diffusion process is to be observed. Generally, a reference velocity such as the mass, molar, or volume average bulk velocity (v,v∗ , or v , respectively) in the system is selected, and movement of the component of interest relative to this reference velocity is defined to be true diffusion. For practical reasons, a fixed reference frame is generally also considered to relate the mathematical treatment of this molecular scale transport process to an actual physical system with well-defined dimensions. In this case, the total flux relative to the fixed reference frame is partitioned into two parts: bulk flow and true molecular diffusion. This partitioning is necessary, because even in the absence of an externally imposed bulk flow, interdiffusion of molecules with respect to each other produces an effective bulk flow relative to fixed coordinates if the molecules have different masses (1,2). The definitions of the mass, molar, and volume average bulk velocities are given in Table 1 along with selected mass and molar flux expressions related to each 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 the mixture as defined below. The mutual diffusion coefficients appearing in Table 1 are identical in all of the expressions and DAB = DBA . Additional discussion of these definitions and relationships between the different reference frames and fluxes are discussed in detail elsewhere (3,4). In principle, any reference frame for analysis may be selected; however, a proper choice can reduce the mathematical difficulties. For example, in a one-dimensional diffusion process within fixed boundaries, where ideal mixing of components is a reasonable approximation, 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 + ωB vB ji = −ρDAB ∇ωi = ρ i (vi − v) Molar average v∗ = xA vA + xB vB ji ∗ = −M A CDAB ∇xA = Ci (vi − v∗ )  Volume average v = ρ A vA V A /M A + ρ B vB V B /M B ji  = −DAB ∇ ρ i = ρ i (vi − v ) aω

i and ρ i refer to the mass fraction and mass density of component i, respectively; ρ refers to the total solution mass density, and V i and M i are the molar volume and molecular weight of component i, respectively. The velocity of component i, vi , represents the average velocity of component i relative to fixed coordinates due to both bulk and true diffusive movement. b j and j  are the mass fluxes of the component i relative to the mass average and volume average i i velocities, respectively, and J i ∗ is the molar flux relative to the molar average velocity.

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the volume average bulk velocity as defined above is zero, and hence the fluxes viewed from both a fixed reference frame and a reference frame moving at the local volume average velocity are identical. In this case the fundamental differential equation of one-dimensional diffusion in an isotropic medium reduces to   ∂ρA ∂ ∂ρA = DAB ∂t ∂z ∂z

(1)

Documented solutions for equation 1 exist for a limited number of complex concentration-dependent diffusion coefficients and boundary conditions (5). Alternatively, for steady-state analysis involving a case with component B replaced by P, for polymer, a static reference frame and the mass average velocity as 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 + ωA (nA + nP ) ∂z

(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, and the last term is the flux of A due to bulk flow of fluid relative to the fixed coordinate system. However, since nP = 0 in this case (since vP = 0 for the membrane polymer relative to the fixed reference frame at steady state), the general expression for nA simplifies to nA =

− ρDAP ∂ωA (1 − 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, ωA is relatively small, and the 1/(1 − ωA ) factor can safely be neglected so that the flux 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 be of second-order importance compared to difficulties in accurately determining the mutual diffusion coefficient due to strong concentration dependencies. However, not accounting for the factor 1/(1 − ωA ) can lead to very significant errors in flux calculations in highly swollen systems (eg, 90–95% solvent), even if the mutual diffusion coefficient is accurately determined (6). Mutual Diffusion Coefficients. A better appreciation of the separation of bulk and true diffusive fluxes and the significance of the mutual diffusion coefficient DAP is useful prior to considering factors that determine this coefficient. Consider, for example, the transient problem of a block of polymer placed in contact with an external solvent (penetrant) phase. When a small penetrant moves into a polymer under transient conditions, most of the polymer movement, measured relative to a fixed coordinate, is thought to be due to bulk flow arising from the outward swelling of polymer segments into the region that was occupied previously by the external solvent phase (5). Because of its intrinsically high mobility, the penetrant (solvent) tends to interpose itself between the polymer segments

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and invade the domain initially occupied solely by the polymer, thereby acting as the 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 direction opposite to the invading solvent flux. This backward bulk motion thereby carries entrained solvent molecules against the direction of the simple concentrationdriven diffusive flux. Unlike the steady-state membrane case, it is clearly not acceptable to assume the mass flux of polymer is zero in this situation, and some relationship between nA and nP must be defined. This issue becomes important for cases involving the interaction of strong swelling solvents with glassy polymers in transient sorption processes. Also in this case, a very strong concentration dependence of DAP may exist, in some cases approaching a step function. In these situations, the “polymer fixed” frame of reference is often used for mathematical convenience. In this reference frame, DAP is defined relative to the penetrant concentration in the unswollen polymer. On the other hand, if the polymer was not assumed to be fixed, DAP would be defined relative to the concentration in the swollen 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 liquids, swelling of the polymer and non-Fickian complications are minimal. In these situations, solutions to equation 1, unconfused by bulk flow, provide an adequate description of the process and allow estimation of the mutual diffusion coefficient. In such cases with a constant diffusion coefficient, standard solutions of equation 1 for M t , the amount of material sorbed (or desorbed) at time t relative to M ∞ , the amount of material sorbed (or desorbed) at infinite time, can be used for this estimation of the mutual diffusion coefficient. If the initial and final concentrations in the sample of thickness  are uniform and the external penetrant activity is held constant, two mathematically equivalent solutions to equation 1 are given (5):      ∞  DAP t 0.5 1 n Mt n =4 + 2 ( − 1) ierfc M∞ 2 π 0.5 2(DAP t)0.5 n=1

(3a)

  ∞  Mt 8 2 2 DAP t =1− exp − (2n + 1) π 2 2 M∞ 2 n = 0 (2n + 1) π

(3b)

On the one hand, equation 3a is generally referred to as the short time solution because good accuracy is achieved for values of M t /M ∞ ≤ 0.6 even if the infinite summation term is neglected. On the other, equation 3b is referred to as the long time solution because good accuracy is achieved for values of M t /M ∞ ≥ 0.6 if only the first term in the summation is used. In cases where there are significant, but not extreme, concentration dependencies of the diffusion coefficient (changes in DAP of up to 50–100% over the course of an experiment), an average coefficient can be used in equations 1, 2, and ¯ is 3 with little loss in accuracy. For such cases, the average diffusion coefficient D

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defined as follows: (ω A )∞

¯= D

DAP dωA

(ωA )0

(ωA )∞ − (ωA )0

(4)

where (ωA )0 and (ωA )∞ refer to the uniform penetrant mass fractions in the polymer at the beginning and at the end of the sorption run, respectively. The experiment could also be run over a narrow concentration interval to ¯ In this latter case, the average determine the concentration dependence of D. diffusion coefficients determined from half-time analysis are good approximations ¯ is calculated simply by for DAP over that particular concentration interval. D rearrangement of equation 3a by neglecting the infinite summation terms with M t /M ∞ = 0.5: 2 ¯ = 0.0492  D t1/2

(5)

where t1/2 is the time required to achieve one-half of the mass uptake that ultimately occurs at equilibrium for the interval sorption experiment. Equation 5 applies for experiments in which the sample has two exposed faces. For a singlesided exposure the coefficient in equation 5 is replaced by 0.1968. Alternate halftime expressions for different geometries such as cylinders and spheres have been presented (5). To analyze the later stages of sorption using an average diffusion co¯ equation 3b indicates that a plot of ln(1 − M t /M ∞ ) vs t gives a straight efficient D, ¯ 2 /2 . line with a slope equal to π 2 D Self vs Mutual Diffusion Coefficients. The self-diffusion coefficient DA ∗ is measured by observing the rate of diffusion of a small amount of radioactivelytagged component A in a system composed of a uniform chemical composition of untagged components A and B. Because of the essentially identical physical natures of the tagged and untagged penetrant, observing the process of interdiffusional exchange of the tagged and untagged molecules allows measurement of the true mobility of the tagged molecules with respect to the stationary solution of known concentration. By varying the concentration of A and B in the presence of a small amount of tagged component A, the concentration dependence of the diffusion coefficient of the tagged molecule can be obtained, uncomplicated by bulk-flow considerations, namely D∗A = RTMA =

RT ζA

(6)

where MA is the mobility of A, ζ A is the molar friction coefficient, the product of the effective viscosity of the medium and the effective diameter of the penetrant (8). In addition to pure mobility considerations, thermodynamic factors enter in determining the concentration dependence of the mutual diffusion coefficient. A single mutual diffusion coefficient exists for a given binary pair under fixed local

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conditions (8). For molecules with similar sizes, shapes, and interaction potentials, the ratio of the self-diffusion coefficients of two components are related to the inverse ratio of the molar volumes, ie, DA ∗ /DB ∗ = V B /V A (8). Under these conditions the individual self-diffusion coefficients may be related to the corresponding mutual diffusion coefficient DAB in terms of the mole fractions of the two components and the partial derivative of the activity vs mole fraction relationship for the penetrant in the polymer at the local mole fraction of interest:  DAB =

∂ ln aA ∂ ln xA





∗

∗

DB xA + DA xB

(7)

T,p

The activity derivative can be readily evaluated from sorption vs activity measurements. The subscripts T and p, indicating which variables are held constant, will be dropped for simplicity in the subsequent equations. Equation 7 was given earlier (4) using classical diffusion analysis for systems in which there is negligible volume change of mixing. Although the inverse molar volume ratio relationship suggested above is probably adequate for small molecules with similar sizes, shapes, and interaction potentials, it is not clear that it applies in all cases where marked differences exist in the relative sizes of components A and B. A weight fraction based weighting of the respective Di ∗ values is probably more general. Except in the range of extremely high solvent fractions, the mobility, and hence the self-diffusion coefficient of a polymer in the presence of a low molecular weight solvent, is many orders of magnitude lower than that of the solvent. Therefore, with the reasonable approximation that DP ∗ = DB ∗ ≈ 0, equation 7 can be expressed in terms of the mass fraction of component A as follows: ∗

DAP = DA ωP



∂ ln aA ∂ ln ωA



  ∂ ln aA = DA (1 − ωA ) ∂ ln ωA ∗

(8)

Diffusion Coefficients in Multicomponent Systems. The value of the diffusion coefficient of a species in a binary system is often not the same as the value in a multicomponent system. The diffusion coefficients can be modified in multicomponent systems as a result of added frictional forces at the atomistic scale. The multiple diffusing species interact in various complex ways that can be described using equation 9, which is derived from the so-called Stefan–Maxwell relations (4): ni − x i Dim = −

n  j=1

n  j=1

nj

x i nj − x j ni Dij

(9)

where Dim is the diffusion coefficient of species “i” in the mixture, Dij are the binary diffusion 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

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form of equation 2a for multicomponent diffusion: ni = − cDim ∇xi + xi

n 

nj

(10)

j=1

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

Solution-Diffusion Mechanism Small molecule transport through nonporous polymers proceeds by the solutiondiffusion 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. While these 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 it are 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 solutiondiffusion mechanism states that the flux through the polymer is proportional to a chemical potential gradient. If a chemical potential gradient does not exist, in the absence of an imposed bulk flow, no net transport of penetrant occurs through the polymer. This proportionality is stated mathematically as JA = − L

∂µA ∂z

(11)

where J A is the flux of A through the polymer, µA is the chemical potential of penetrant A in the polymer phase, and L is the direct phenomenological transport coefficient in irreversible thermodynamics terminology. C is the concentration of A sorbed in the polymer, and MA , once again, is the mobility of A. In the absence of electromotive forces, the chemical potential of a dissolved penetrant in the polymer phase is given by

 µA = µA0 + RTln(γA CA ) + VA p − pA0

(12)

where µA 0 is the chemical potential of the pure penetrant at the reference pressure pA 0 . Moreover, γ A is the activity coefficient at concentration CA , and the partial molar volume of the penetrant is V A . The reference pressure is usually set as the pure component vapor pressure. With the knowledge that a chemical potential gradient must exist through the polymer, from inspection of equation 12 one can imagine three possible ways to affect such a gradient: (1) a concentration gradient 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 concentration gradient solely affects the gradient in chemical potential. The fact that no

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pressure gradient exists through the membrane interior can be proved using a simple mechanical argument. In the common laboratory setup for gas separations, the membrane is supported on the low pressure side by a porous metal support. If a pressure is exerted on the upstream side of the membrane by the gas, the metal support must exert an equal and opposite pressure on the downstream side of the membrane. For this reason, the pressure inside the membrane is equal to the upstream pressure (6). By using a constant pressure through the membrane, Paul was able to rationalize the observed dependence of flux on the external pressure difference for both hydraulic permeation and pervaporation systems (10). Furthermore, Paul conducted a series of experiments using a composite membrane consisting of a stack of three to four rubber membranes to measure the concentration gradient through the membrane. It was shown that the concentration gradient determined from these measurements alone accounted for the entire chemical potential gradient (11). This result excluded the possibility of a pressure gradient inside the membrane for solution-diffusion transport processes. With this knowledge of the physical situation, it is an easy exercise to derive Fick’s Law from the more general statement of equation 11.

J= −L

∂µA ∂µA ∂CA ∂CA = −L = − DAB ∂z ∂CA ∂z ∂z

(13)

Since DAB = (DA ∗ CA /RT)(∂µA /∂CA ) (5), the phenomenological coefficient L can be related back to the underlying penetrant mobility. ∗

L=

DA CA 1 = CA = MA CA RT ζA

(14)

Amorphous Rubbery Polymers Gases and Low Activity Vapors. At low concentration (ωA → 0) and in systems where Henry’s law applies (∂ ln aA /∂ ln ωA → 1), DAB approaches the infinite-dilution self-diffusion coefficient (DA ∗ )0 . Therefore, the mutual diffusion coefficient is a good estimator of the infinite-dilution self-diffusion coefficient for most gases in polymers at atmospheric pressure and below. Even at relatively high pressures with condensable gases and low activity vapors such as CO2 , SO2 and propane, deviations from Henry’s law can generally be described in terms of the Flory–Huggins isotherm or a similar simple expression. Therefore, self-diffusion coefficients, and thus the mobility of the penetrant, can be calculated using equations 5 and 6 or by using permeation data in conjunction with equilibrium sorption data to determine (∂ ln aA /∂ ln ωA ) at the appropriate ωA value. Rubbers are essentially high molecular weight liquids with the ability to adjust their segmental configurations rapidly over significant distances (>0.5–1 nm) and local volumes (12). Nevertheless, the rotational and translational motions of sorbed penetrants are rapid compared to the motions of the segments of the polymer (13,14). The limiting step in diffusion of such small molecules through the

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Fig. 1. Generation of a gap for the penetrant with subsequent collapse of the volume that previously housed the penetrant, emphasizing the mutual nature of the binary diffusion process.

rubber involves the generation of a sufficiently large gap for the penetrant to move into, with subsequent collapse of the sorbed cage that previously housed the penetrant (Fig. 1). This description emphasizes the mutual nature of the binary diffusion process, since both the penetrant and surrounding polymer segments tend 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 involved 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 crankshaft composed of four to five backbone carbon atoms (Fig. 2). For diene and other hydrocarbon polymers, similar cooperative motions involving several repeat units are also probably the most common types of motions observed over time scales of importance to diffusion. As expected, for penetrants such as gases that are clearly smaller than the size of the most common moving segment of the polymer, the infinite-dilution diffusion coefficient tends to show a steady drop with increasing penetrant size (Fig. 3). The tendency for diffusion coefficients of larger penetrants to approach an asymptotic plateau has been discussed (15). Branched penetrants tend 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 somewhat segmental fashion as does the polymer itself. In this case, the asymptotic limit of the linear penetrants in Figure 3 presumably reflects the mobility of the more or

Fig. 2. The crankshaft motion requiring the simultaneous rotation of several sequential CH2 moieties about bonds 1 and 7 or 1 and 5.

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Fig. 3. Diffusion coefficients for a variety of penetrants in natural rubber at 25◦ C. n-C3 ,nC4 , and n-C5 designate straight chain alkanes.

less freely orienting segments of the polymer that are themselves undergoing a self-diffusion process. Therefore, while the self-diffusion coefficient of the polymer center of mass may be orders of magnitude lower than that of intermediate-sized penetrants, segmental mobility in the polymer is still quite high. Not surprisingly, the asymptotic mobilities of penetrants in glassy polymers are orders of magnitude lower than those in the typical rubbery polymers. Concerted movements of several adjacent segments of chain comprising the sorbed cage of a simple gaseous penetrant are believed to provide the source of the penetrant-scale hole needed for diffusion. Lennard–Jones collision diameters (σ ), van der Waals volumes (b), and actual measured values of the partial molar volumes 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, very close to infinite-dilution partial molar volumes in several low molecular weight liquids (17), and are measures of the volume of the cage in which the sorbed penetrant resides at equilibrium. The occupied volume is calculated by viewing the molecule 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 actual equilibrium cage volume is occupied and the remainder is in a sense “free volume” available to be shared with the neighboring polymer segments to facilitate the mutual diffusion process. Estimates of the amounts of total free volume required for a diffusional jump of several gaseous penetrants in a copolymer of poly(vinyl chloride) and poly(vinyl acetate) have been made (18) (Table 2) (Fig. 4). Although the data in Table 2 are

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Table 2. Various Molecular Volumes for Gaseous Penetrants Volume measurement

He

van der Waals 0.0396 volume b, nm3 Lennard–Jones 0.258 diameter σ , nm 0.00899 Lennard–Jones occupied volume, nm3 Partial molar volumea in silicone rubber (PDMS), nm3 Free volume in sorbed stateb , nm3 Required free volume for 0.075 diffusional jump in PVC–PVA copolymerc , nm3

N2

O2

CH4

C2 H4

CO2

0.0642 0.0529 0.0709 0.0856 0.0711 0.368

0.343

0.382

0.422

0.400

0.0261 0.0211 0.0292 0.0393 0.0335 0.0549

0.0792 0.0940 0.0767

0.0288

0.0500 0.0547 0.0432 0.299

0.690

a Experimental b Partial c Ref.

(15). molar volume minus occupied volume.

17

for two different polymers, for an order of magnitude analysis they can be taken as representative of movement in typical rubbery polymers. The required free volumes 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 a momentary fluctuation in the rubbery polymer segmental position due to a local thermal fluctuation, rather than being locally present in the equilibrium sorbed cage. This fact can be treated in several ways, eg, via Activated State Theory or Free-Volume Theory. Activated State Theory. The diffusion process by which small molecules intermingle with a polymer can be considered a random walk of the penetrant among the segments of the polymer. Consistent with this qualitative description, the activated state theory assumes that holes covering a spectrum of different volumes and involving segments of several polymer molecules are continuously formed and destroyed because of thermal fluctuations. The rate of diffusion depends on the concentration of transient holes that are sufficiently large to accept diffusing molecules. Assuming a Boltzmann distribution, the concentration of a given size of holes decreases exponentially with the energy associated with its formation. The temperature dependence of the diffusion coefficient for the activated state theory can be expressed as 

− Ed D = D0 exp RT

 (15a)

where D0 is the preexponential factor and Ed is the energy of activation for diffusion. The indication by equation 15a that ln(D) is linearly dependent on 1/T has been verified for many systems well above the glass-transition temperature T g

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

(ie, T > T g > T g + 100 K). For penetrants smaller than the size of the primary moving segment of the polymer, the size of the molecule generally determines both 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 diffusion for the larger penetrants to approach an asymptotic value similar to that of the activation energy of viscous flow for the uncross-linked polymer had been noted (15). This asymptotic behavior is thought to represent a similarity between the mobilities of large penetrants and the primitive segmental mobility of the polymer that participates in viscous flow (19). The Eyring theory of rate processes (20) has been used as the basis of most theories for D0 :   S∗ D0 = eλ kT/ h exp R 2

(15b)

where S∗ is the entropy of activation and λ is the average jump length. For penetrants smaller than the average size of the jumping unit of the polymer, both the preexponential factor and activation energy increase with the size of the penetrant molecule. The actual diffusion coefficient of the penetrant decreases with penetrant size since the exponential weighting of Ed dominates the product in 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 cohesive energy density (CED) and the volume of 1 mol of cylindrical cavities having a length λ and a diameter equal to that of a diffusing molecule, dA , has been suggested (21). With this assumption, Ed = CED d2A π λ/4

(16)

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

An apparent diffusional jump length using such a cylindrical activation volume can be calculated. Assuming the total free volume for a diffusional jump to be equal to the activation volume, the value 0.69 nm3 per molecule for CO2 in Table 2 can 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 free volume associated with the diffusion jump for CO2 were converted to a spherical volume fluctuation. Even this length seems too high, since all of the activated volume may not be directly usable for a linear translation of position. Furthermore, this conceptualization of the activation energy is not in agreement with experimentally observed behaviors. Experimentally determined values of the activation energy extrapolate to zero at some value of the penetrant molecular diameter dA greater than zero, and correlations of Ed and dA obtained experimentally are not of 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 of Figure 1, it may be preferable to visualize the activated volume as a bulge in a tube

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Fig. 6. Activated volume as a bulge in a tube with the openings of the bulge smaller than the penetrant diameter. , Directly usable fraction of activation volume; , not directly usable 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 jump from the center of the bulge to one or the other side, even though the entire free volume must exist to accommodate the fact that polymer chains are not infinitely flexible, especially on the short time scale of a diffusional jump. In other words, the chains must have a considerable transition length over which the segments reapproach their original packing density; however, the volume associated with these regions is not directly usable by the penetrant in terms of ±λ translations of its center of mass. These ideas were formalized mathematically by Brandt (23), who assumed that the activation energy was composed of two contributions: (1) an energy Eα associated with bending two initially straight chain segments away from each other to accept the penetrant molecule, and (2). an energy Eβ needed to overcome repulsion of the bent chain segments by neighboring chains. The intermolecular energy Eα is calculated on the basis of the potential energy barrier to chain rotation, ψ 0 , a measure of chain stiffness. The repulsive component Eβ is calculated analogously to equation 16; however, the internal polymer pressure pI is used instead of the

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cohesive energy density. Using this framework, Brandt calculated the activation energy as follows: Ed = Eα + Eβ = 18ψ0 bd (dA − dS )2 ad3 + NA pI dp ad (dA − dS )/2

(17)

where N A is Avogadro’s number, bd is the length of a backbone bond projected onto the chain axis, ad is the length of the bent chain segments, and ds is the initial chain spacing prior to penetrant inclusion. This conceptualization of the activation energy accounts for many of the shortcomings of equation 16. The activation energy is equal to zero when dA ≤ ds , which is in accord with the nonzero extrapolations of Ed versus dA from experimental data. It is also apparent from inspection of equation 17 predicts that no simple relation should exist between the activation energy and the penetrant molecular diameter. Furthermore, equation 17 predicts that the dependence of the activation energy on the penetrant molecular diameter is weaker for flexible chain polymers, where ψ 0 is small. This model of the activation energy is the basis for more complex treatments by Pace and Daytner (24), and Dibenedetto and Paul (25). Free-Volume Theories. The basic idea of the free-volume theories is that the mobilities of the polymer segments and the penetrant molecules in a polymer– penetrant mixture are primarily determined by the amount of free volume in the system. As originally proposed (26),  MA = Ad exp

− Bd vf

 (18a)

where MA is the mobility of the penetrant, vf is the average locally available fractional free volume of the system, and Ad and Bd are empirical free-volume parameters that are assumed to be independent of penetrant concentration and temperature. The parameter Ad depends on the size and kinetic velocity of the penetrant. The parameter Bd is equivalent to the critical hole free volume necessary for a penetrant to make a diffusive jump (13). This expression was modified by arguing that crystalline material reduces the free volume in direct proportion to the amount of crystalline material present (27):  MA = Ad exp

− Bd  a vf

 (18b)

where a is the amorphous volume fraction of the penetrant free polymer at zero pressure and the temperature of the system. Alternate treatments of the effects of 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 vapors in polymers are strongly concentration-dependent, because mobilities are extremely sensitive to changes in the average free volume of the system. A small penetrant that is unconstrained on two sides by covalently bonded neighbors introduces much more free volume into the polymer–penetrant mixture than a polymer segment of equivalent size. Increased penetrant concentration thereby alters the effective viscosity of the medium, and a significant increase in the penetrant

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mobility results. The free-volume approach is simple and has evolved (28,29) into several readily useful forms. Its primary drawback lies in the difficulty of providing a precise physical definition for the parameter defined as the free volume. The native polymer, totally devoid of penetrant, still possesses a certain distribution of free-volume packets, which wander spontaneously and randomly through the polymer. Intuitively, proponents of free-volume theory argue that a penetrant can execute a diffusive jump when a free-volume element greater than or equal to a critical size presents itself to the penetrant. In fact, the polymer chains may also execute a self-diffusive jump when a packet of sufficient size presents itself to a segment. This is the mechanism that causes the slow interdiffusion of polymer chains. Increasing system temperature causes volume dilation, resulting in increased free volume. Thus the diffusion coefficient increases with temperature. The free-volume fraction vf may be represented as a linear addition of several variables (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 T s , usually the glass-transition temperature, and reference pressure ps , usually 101.3 kPa (1 atm). The penetrant volume fraction is A and the coefficients coefficients α, β, and γ are positive constants, the values of which are evaluated empirically. These coefficients characterize the effectiveness of temperature, pressure, and penetrant concentration, respectively, for changing the free volume in the amorphous phase:  α=

∂vf ∂T



 β= s

∂vf ∂p



 γ= s

∂vf ∂ϕ1

 (20) s

where s denotes some reference state. The free-volume thermal-expansion coefficient α can be estimated as the difference between the thermal expansion coefficient 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  =β = − 1 − v∗f 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 mobility leads to the following expression for the polymer self-diffusion coefficient, when

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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 molecule has been developed (29). In the absence of data for a given polymer, vfs and α can be 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 interprets the glass-transition point to be an isofree volume state. This approximation is valid in the temperature range from T g to 100◦ C above T g . In principle, each of the coefficients in equation 22 can be evaluated independently by observing the effects on DA ∗ over a sufficiently wide range of temperatures, 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 fixed partial pressure of the penetrant of interest. On the one hand, hydrostatic pressure is expected to have a rather small effect on DA ∗ since solid polymers are only slightly compressible. On the other, increases in temperature and sorbed penetrant concentration cause large increases in the free-volume fraction vf and in the self-diffusion coefficient DA ∗ . The permeability P is an important transport coefficient that represents the normalized molar flux N A across a polymer film of thickness  with a partial pressure driving force p at steady state: P=

NA p/

(24)

Under steady-state conditions, nP in equation 2a is zero, and for the low sorbed mass fractions typical of gases in polymers, an expression for N A = nA /M A can be substituted to yield P=

− ρDAP  ∂ωA MA p ∂z

(25a)

where M A is the molecular weight of component A. Integration between the upstream and downstream conditions gives −1 P= p2 − p1

ω A2

ωA1

− ρDAP (ω) dωA MA

(25b)

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

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to DA ∗ . Transformation of equation 25a from mass fractions to volume fractions gives −1 P= p2 − p1

ϕA2

ϕA1

− ρD∗A dωA dϕA MA dϕA

(25c)

where dωA /dϕ A can be evaluated using the specific volumes of the polymer and penetrant. Substitution for DA ∗ from equation 22 for the case of negligible downstream pressure yields the following expressions (32) when Henry’s law applies (ϕ A2 = kA p2 ) at the upstream membrane face: ln P = ln[RTAd kA ] −

   λkA Bd p 2 Bd 2βp2 − β + 1 + + a v∗f 2 v∗f a (v∗f )2

(26)

Therefore, the free-volume theory permits theoretical plots of permeability as a function of temperature, penetrant pressure, and amorphous volume fraction in 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 one presented above. In their approach, Vrentas and Duda include an energy factor, EJ , to the preexponential term Ad . This factor accounts for the energy required for a penetrant molecule to jump into an adjacent open hole. The exponential parameter Bd is modified to account for the molar volume of the moving polymer segment. Furthermore, Vrentas and Duda propose that only part of the total free  volume is available for diffusion and is denoted by Vfh . This concept of only a fraction of the free volume participating in diffusion is similar to the activation energy model of Brandt (see Fig. 6). The Vrentas and Duda model ultimately leads to a prediction of the thermodynamic diffusion coefficient: 

DT = Ad0

     − EJ wP MA Vˆ h∗ exp exp γov wA + RT MPJ Vˆ fh

(27)

where wA and wP are mass fractions of the penetrant and polymer, respectively, and M A and M PJ are the molecular weight of the penetrant and the jumping polymer segment respectively. V¯ h∗ is the specific volume of holes of the minimum size required for a diffusive jump. γ ov is a parameter, with a value between 0.5 and 1, originally introduced by Cohen and Turnbull that accounts for the ability of a free volume hole to be available to multiple jumping segments. Although the added sophistication of this model probably captures a more accurate picture of the diffusion process, the accompanying added mathematical complexity has hindered 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 large that 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

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do not operate strictly on the basis of size selection. Under a fixed partial pressure driving force for component i, the flux through a material of given thickness  is determined by the permeability, as can be seen by rearrangement of equation 24. Ni = pi Pi /

(28)

The permeability can be expressed as the product of a diffusivity and solubility coefficient: ¯ i S¯ i Pi = D

(29)

The diffusivity and solubility coefficients in the above expression are effective averages applying across a polymer film between its upstream and downstream faces. The presence of other copermeating components can often be neglected for gases and low activity vapors permeating through rubbers (37). The average mu¯ i is given below for penetrant A in the polymer compotual diffusion coefficient D nent P: ωA2

DAP dωA ωA1 ¯ Di = ωA2 − ωA1

(30)

When the presence of other components cannot be neglected, the system can still be treated approximately by considering the membrane in the presence of the additional component to be a new effective medium; however, this is at best a rough approximation since it neglects the bulk-flow contribution and thermodynamic influence of additional components on the flux of component A. Multicomponent thermodynamic issues are beyond the scope of this discussion, but have been treated (38). ¯ i , is found from combination of equations 25b, 29 The solubility coefficient, S and 30 as: ¯ i = Ci2 − Ci1 S pi2 − pi1

(31)

where Ci2 and Ci1 are the concentrations of “i” in the membrane at the upstream and downstream faces, respectively. pi1 and pi2 are the partial of “A” at the upstream and downstream faces, respectively. Under ideal conditions of fixed up¯ i parameter is equal stream pressure and negligible downstream pressure, the S to the secant slope of the sorption concentration vs pressure isotherm evaluated at the upstream partial pressure pi2 of the component ¯ i = Ci S pi2

(32)

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

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The temperature dependence of solubility coefficients are typically described in terms of van’t Hoff expressions shown below (39):    Si (T2 ) 1 Hs 1 ln − =− Si (T1 ) R T2 T1 

(33)

Equation 33 only applies when the solubility coefficient is independent of concentration or for solubility coefficients specified at the same concentration. The enthalpies of sorption H s are typically small and slightly negative (Fig. 7a). The solubilities and enthalpies of sorption of different penetrants are determined largely by their critical temperatures or other related measures of tendency to exist in a condensed phase such as normal boiling points or Lennard-Jones potential well depth /k¯ (Fig. 7b) (40). From the dependence of permeability on both solubility and diffusivity, it is clear that two opposing factors affect the permeability as temperature is increased, namely  ln

      D(T2 )S(T2 ) Ed + Hs 1 1 P(T2 ) − = ln =− P(T1 ) D(T1 )S(T1 ) R T2 T1

(34)

The positive diffusional activation energy is larger in absolute value than the negative H s , and so the overall permeability increases as temperature increases, but to a lower degree than the diffusion coefficient itself. It should be noted that the use of equation 34 is only strictly valid when the diffusion and solubility coefficients are independent of concentration. The separation factor or permselectivity between two penetrants A and B, α AB , is important in membrane separation systems. This factor is equal to the ratio of the downstream (permeate) mole fractions of component A relative to component B divided by the ratio of the upstream mole fractions of A relative to B. Under conditions of negligible downstream pressure, the permselectivity is simply equal to the ratio of the permeabilities of components A and B. In addition, since the presence of one component has a negligible effect on the permeability of the other at low pressures in rubbery materials, pure component permeabilities can often be used as good estimates of mixed-gas permeabilities:        xA DA xA SA PA αAB = = = PB xB 1 xB 2 DB SB

(35)

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

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Fig. 7. (a) Enthalpies of solution for various gases in elastomers: A, 䊊, natural rubber; B, , silicone rubber (30). To convert kJ/mol to kcal/mol, divide by 4.184. (b) Correlation of solubility of several gases in rubbery amorphous polymers: , silicone rubber; 䊊, natural rubber; ⵧ, styrene–acrylonitrile rubber; 䊉, 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

Polymer

312

Natural rubberd Polyethylenee Poly(methyl acrylate)f Poly(ethylene terephthalate)g Polychloropreneh Butyl rubberi

PHe , SHe , PO2 , SO2 , DHe , mmol/ cm3 (STP)/ DO2 , mmol/ cm3 (STP)/ (m · s · TPa)b PHe /PCH c (cm3 · GPa) SHe /SCH4 10 − 6 cm2 /s DHe /DCH4 (m · s · TPa)b PO2 /PO2 (cm3 · GPa)c SO2 /SN2 102 cm2 /s DO2 /DN2 10.4 1.64 3.55

1.05 1.7 45.1

105 54 101

0.042 0.048 0.069

22 6.8 7.88

25 36 650

8.04 0.97

3.0 3.0

1125 473

2.0 2.1

1.6 0.46

1.5 1.44

3.1

44

58

0.10

12

430

0.18

3.8

218

1.7

0.19

2.2

1.34 0.44

3.42 3.94

750 1200

2.13 2.19

0.4 0.081

1.6 1.8

25◦ C, unless otherwise specified. convert mmol/(m · s · TPa) to cm3 (STP)cm/(cm2 · s · cm Hg), multiply by 2.98 × 10 − 10 . c To convert cm3 (STP)/(cm3 · GPa) to cm3 (STP)/(cm3 · cm Hg), multiply by 1.333 × 10 − 6 . d Poly(cis-isoprene). e ρ = 0.914 g/mL; α = 0.57. f At 35◦ C. g At 100◦ C. h Neoprene. i Isobutylene/isoprene (98/2) copolymer. a At b To

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butadiene–acrylonitrile copolymers. The higher the nitrile content, the greater the attraction between neighboring chains. These interchain attractions are quantified by the cohesive energy density (CED). High CED tends to elevate the diffusion activation energy and hence depress the diffusivities according to equation 15b. The diffusivity selectivities of the various rubbers in Table 3 are generally low, consistent with the large scale of segmental motions in rubbery materials relative to the penetrant sizes that must be distinguished. Besides difficulties in achieving high selectivities, rubbery materials are generally unable to be selfsupporting under pressure in a high surface area hollow fiber form. This problem can be circumvented by placing a thin film of the rubber on a microporous support to make a composite membrane (42). Generally, however, the much higher selectivities of glassy polymers for separating gases make glasses the favored polymeric materials in separation applications. For cases involving a random copolymer or a miscible blend of two amorphous rubbery polymers, the behavior is generally a volume fraction weighted average of the permeabilities of the two homopolymers. On the other hand, the transport properties of immiscible blend systems depend significantly on the relative permeabilities and the morphology of the immiscible blend.

Nonideal Transport Effects in Rubbery Polymers Plasticization and Hydrostatic Effects. New complexity arises in the basic transport properties of a polymer at higher penetrant activities, since the local concentration of penetrant within the polymer can become quite high. At higher pressures (approaching the saturated vapor pressure), these high concentrations can occur in cases where favorable polymer–penetrant interactions promote solubility. This is common for organic vapors and even for small polar or quadrupolar molecules such as CO2 , H2 S and SO2 . Transport plasticization is defined as a significant increase in the diffusivity of a penetrant because of facilitation of local polymer segmental motion caused by another penetrant molecule in its neighborhood. This definition applies both to the increase in diffusivity of a penetrant caused by the presence of its own kind and increases caused by a different component. Even for pure component penetrants, a detailed fundamental analysis 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 applications because permselectivity is reduced as the diffusivity of a slower penetrant is increased. In the case of gases, plasticizing phenomena are suppressed because of hydrostatic compression in some polymers despite rather high sorption levels. Analysis in terms of the free-volume approach requires consideration of the compressibility of the rubbery matrix. In fact, exposure to the hydrostatic pressure of low solubility gases, such as helium or nitrogen, may actually compress out free volume and thereby reduce the ability of polymer segments to open gaps for movement of other segments or sorbed penetrants. Some data (44) indicate a marked reduction in diffusivity of n-butane as the external pressure of helium is increased, and it was concluded that helium gas applied to silicone rubber acts only as a pressuring medium within experimental

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limitations. The helium decreases the free volume of the silicone rubber and thus decreases its molecular motion. Plasticization and hydrostatic effects on the permeability of carbon dioxide, helium, nitrogen, methane, and ethylene in silicone rubber also have been studied (45). Over an extended pressure and concentration range, both compression of free volume and eventual plasticization have been observed for these different penetrants. As the amount of sorbed penetrant increases, the plasticization effect eventually overcomes the hydrostatic compression effect on the rubbery matrix for the more condensable penetrants such as carbon dioxide and ethylene. The local solubility and diffusivity must be considered to see when true transport plasticization occurs. For this it is more useful to consider the local mutual diffusion coefficient DAP (C) rather than the average value assessed from equation 30. The local mutual diffusion coefficient is a measure of the ability of a penetrant to move through the membrane at a point where the local conditions are well defined in terms of the local penetrant concentration CA , volume fraction ϕ A , weight fraction ωA , or penetrant fugacity, whichever is most convenient. Figure 8 (45) shows the permeation data for methane, carbon dioxide, ethylene, helium, and nitrogen in silicone rubber at 35◦ C over a broad range of pressures. The data agree reasonably well with the data reported previously over a smaller pressure range (46). For helium and nitrogen, which have low critical temperatures, the permeability decreases with increasing pressure, while it increases with increasing pressure for carbon dioxide and ethylene, which have relatively high critical temperatures. For methane, with an intermediate critical temperature, the permeability shows negligible pressure dependence. The sorption isotherms shown in Figure 9 (16) for each of the penetrants obey Henry’s law at low pressures, and the more strongly sorbing carbon dioxide and ethylene show Flory–Huggins swelling behavior at high pressures. The lines in these plots are, in fact, modified Flory–Huggins fits to the data according to equation 36, which includes the presence of the small amount of cross-linking in the sample (47):      p ve (1 − ϕA ) = ln(ϕA ) + (1 − ϕA ) + χ (1 − ϕA )2 + VA (1 − ϕA )1/3 − ln p0 V0 2 (36) where p is the penetrant pressure, p0 is the vapor pressure of the penetrant, V A is the partial molar volume of the penetrant, ve is the effective number of cross-links expressed in moles, V 0 is the volume of the penetrant-free polymer, and χ is the Flory–Huggins interaction parameter. The permeability and sorption isotherms can be used along with the following expression to calculate the local concentration-dependent mutual diffusion coefficient (48):  DAP (CA2 ) =

dP P(p) + p dp



dp dCA

   

(37) p2

The Flory–Huggins expression can be used to evaluate the dp/dCA term at the 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 diffusion coefficient for the various penetrants as a function of local fugacity in silicone

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Fig. 8. Comparison of the pure-gas permeabilities of various gases in silicone rubber at 35◦ C. (a) CO2 , C2 H4 , and CH4 ; (b) He and N2 · 1 Barrer = 10 − 10 [cm3 (STP)cm]/(cm2 · s · cm Hg) = 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 sorption isotherm data if desired. For carbon dioxide, the fugacity differs by as much as 22% from the pressure at the maximum pressure studied. The chemical potential difference is of course the true thermodynamic driving force for diffusion and should be used in mixed-gas calculations.

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Fig. 9. Sorption isotherms for Sorption and desorption of various gases in silicone rubber at 35◦ C. ⵧ, Sorption; 䊏, 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.

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The local diffusion coefficient of nitrogen decreases with increasing pressure until around 2.1 MPa (300 psia), where it levels off and slowly begins to rise with increasing pressure. These results indicate that the fractional free volume of the polymer is reduced because of nitrogen’s hydrostatic pressure. At higher nitrogen pressures the solubility level is sufficient to oppose the compressive effect and cause the diffusivity to begin to increase with pressure as free volume is added by the penetrant. The results for methane and carbon dioxide show a slight decrease in diffusivity with increasing pressure, but both gases reapproach their original diffusivity at higher pressures because of their higher solubilities. Ethylene exhibits an immediate increase in diffusivity with pressure that is not surprising, given its significant solubility and the large dilation in volume with increasing ethylene pressure, shown in Figure 11 (16). Even below 0.7 MPa (100 psia), more ethylene 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 nitrogen in Figure 8b and the positive permeability slope for the other gases in Figure 8a can be explained by these two competing effects caused by the penetrant pressure. The data in Figure 11b show an actual compression in volume, even though a 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 high compressibility of silicone rubber. Therefore, the two competing factors that determine the volume dilation of the polymer are (1) the sorption of gas into the polymer that results in an increase in free volume, and (2) the hydrostatic pressure on the film that reduces the free volume. The second effect is usually small and is overcome for gases that have significant sorption levels. The net result of these two opposing effects determines whether the permeability (and diffusivity) will increase or decrease with increasing pressure. These two competing factors can be described quantitatively by equations 23 and 26. The expression for the local mass diffusion flux accounting for the frame of reference term discussed in equation 2b is given below: nA =

− ρDAP ∂ωA 1 − ωA ∂z

(38)

Using equations 7 and 37, the effective mutual diffusion coefficient obtained from analysis of the permeability versus pressure data can be related to the selfdiffusion 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 such as 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 case where curved isotherms characterized by the Flory–Huggins equation apply. 

∂lnaA ∂lnωA

 =

ωP ρA2 (ωA ρP + ωP ρA )2



2χ ωA ρP 1− ωA ρP + ωP ρP

 (40)

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Fig. 11. Volume dilation isotherms for Sorption of various gases in silicone rubber at 35◦ C. ⵧ, Sorption; 䊏, desorption (46). To convert MPa to psia, multiply by 145.

where ρ A and ρ P are the mass densities of the pure penetrant and polymer, respectively. Penetrant mobilities (MA ) were determined as a function of local fugacity for silicone rubber (45). The mobility of nitrogen decreased with increasing pressure, but the mobility of carbon dioxide increased, suggesting that nitrogen acts predominantly as a pressurizing agent and CO2 acts primarily as a plasticizing agent. The permeability of helium and nitrogen decreased with increasing pressure because of 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 each component unless another component has sufficient solubility to overcome the

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hydrostatic effect. To test this hypothesis the silicone rubber films were exposed to both a 10/90 mol% mixture and a 50/50 mol% mixture of carbon dioxide and nitrogen. The presence of additional penetrants in the feed stream can be accounted for by the addition of a term in the free-volume equation to account for the presence of each penetrant and the applied hydrostatic pressure above ps = 101.3 kPa (1 atm) as follows: D∗A



− Bd = RTMA = Ad exp vfs + α[T − Tg ] − β[p − ps ] + γ1 ϕ1 + γ2 ϕ2 + · · ·

 (41)

The carbon dioxide permeability for both mixtures was depressed below the pure carbon dioxide values, but the carbon dioxide permeability for the 50/50 mixture did not display a decreasing tendency with pressure as was observed for the 10/90 CO2 /N2 mixture. Therefore, at the lower nitrogen partial pressures observed in the 50/50 mixture, the impact of the hydrostatic pressure effect is reduced by the swelling effect of CO2 . Although the swelling effect of CO2 cannot completely overcome the hydrostatic pressure effect of CO2 nitrogen, it greatly offsets its influence. Important extensions of the foregoing example, where hydrostatic pressure effects may be of considerable importance, involve supercritical extraction using high pressure CO2 or ethylene for processing of foods, pharmaceuticals, and purification of polymers by removal of trace monomers (49). This topic involves greater complexity than the binary diffusion case. Moreover, the factor involving the activity coefficient derivative may become very significant near the critical point. Transport behavior in these systems, however, can be anticipated on the basis of the results shown for the pure components. Little enhancement in the self-diffusion coefficient of the penetrant in these rubbery polymers is apparent at pressures below the critical pressure; however, both carbon dioxide and ethylene begin to overcome the effects of hydrostatic pressure and to show upturns in their mobilities as the critical conditions are approached. Most of the work in the area of mass transport under supercritical conditions in polymers has been rather qualitative, but more quantitative analysis should be possible using the free-volume model. Penetrant Clustering. Clustering may reduce the effective mobility of a penetrant such as water or methanol that can self-associate. The activated state representation of diffusion is typically used to discuss the suppressed diffusion coefficient for a clustered penetrant. If the diffusion process involves primarily jumps by monomeric, unassociated penetrants, energy must be available both to make a gap to allow the penetrant to jump into and to cause the dissociation of one of the penetrants in a local region adjacent to the gap. Therefore, higher activation energies for diffusion and, hence, lower diffusion coefficients at activities above which clustering occurs are expected if this visualization is correct. Work to document this point of view has been done for silicone rubbers (50), poly(alkyl methacrylates) (51), and hydrocarbon rubbers (52). Typical diffusion coefficients and activation energies for diffusion of water and methanol, both of which tend to cluster in hydrophobic environments, are shown for various rubbery materials

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Fig. 12. (a) Effects of water clustering on diffusion in rubbers. Concentration dependence of 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 the diffusion 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 over conditions prior to and after the onset of clustering. The sorption behavior in clustering systems is often discussed in terms of the cluster integral GAA given below in terms of the partial molar volume V A , the activity aA , and the volume fraction of penetrant ϕ A :   ∂(aA /ϕA ) GAA = − (1 − ϕA ) −1 VA ∂aA p,T

(42)

GAA may be determined from equilibrium sorption data as a function of penetrant partial pressure. Based on the theory of Zimm and Lundberg (53), the cluster function given by (ϕ A GAA /V A ) equals the excess number of A molecules

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Table 4. Activation Energies in Clustering Systems System Water cis-1,4-Polyisoprene

Polydimethylsiloxane

poly(3,3,3-trifluoropropylmethylsiloxane) Poly(butyl methacrylate)

Methanol Natural rubber

poly(3,3,3-trifluoropropylmethylsiloxane)

a To

C, cm3 [STP]/cm3 polymer ED , kJ/mola Ref. 0 0.3 0.5 0 0.45 0.6 0 0.7 0 2 4

42 58 58 14 46 75 17 50 28 44 50

0 5 8 0 4 8

42 54 71 25 29 38

50

48

48 49

50

49

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

above the number that would exist in the neighborhood of an arbitrarily chosen type A molecule if the mixing were totally random, ie, no clustering. As shown in Figure 12, the effective mutual diffusion coefficient is constant below the activity at which clustering occurs. The activity at which the cluster function deviates significantly from zero corresponds to the point at which the diffusion coefficient begins to drop significantly, and the activation energy begins to increase. Intermediate-Size Penetrants. These penetrants are defined as molecules with sizes approaching a polymer segment, but much smaller than an entire polymer chain. The infinite-dilution mutual diffusion coefficients for increasingly large, but not macromolecular, penetrants approach the diffusion coefficient for the segment of a polymer diffusing within itself. Movement of the entire polymer molecule itself is clearly a slower and more complex concerted process than the local movement of its segments; however, the segmental mobility and the self-diffusion coefficient of the entire molecule can be related as discussed later. For intermediate-size penetrants such as typical organic solvents, the selfdiffusion 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 intermediate penetrant concentration, equation 8 should still apply as it did in the case of gases and other small penetrants. The data in Figure 3 reflect the tendency to approach an infinite-dilution asymptotic mobility as the diameter of these intermediate-sized penetrants

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Fig. 13. Concentration dependence of the self-diffusion coefficient and the thermodynamic 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):  lim {DAP } = lim RTMA (1 − ωA )

ωA →0

ωA →0

∂lnaA ∂lnωA

 = RTMA =

RT ζA

(43)

Care must be used in applying such arguments when estimating practical diffusion coefficients, however, since increases in ω1 above the infinite-dilution limit can 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 as 10–20%, as was indicated in the case of carbon dioxide and ethylene earlier. In the case of o-xylene diffusion in polyethylene at 150◦ C, both the effects of plasticization and of the (∂ ln aA /∂ ln ωA ) term on the self-diffusion coefficient have been analyzed and are shown in Figure 13 (34). Except for supercritical extraction conditions, hydrostatic pressure effects are typically of negligible importance for simple solvent vapors diffusing in polymers, since the saturation vapor pressure is low, 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 of the square root of time. By contrast, Case II kinetics are characterized by linear mass uptake with time as shown in Figure 38, where uptake of n-pentane in polystyrene at high activity (penetrant partial pressure) shows a Fickian response in 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 that there is insufficient time to generate a Case II concentration profile. Apparently the diffusional equilibration in the small spheres is essentially complete before the complex step concentration profile associated with Case II sorption can be established. This behavior is similar to other observations (144). The general problem of two independent time scales for diffusive equilibration and molecular relaxation existing in a system has been considered (145). Fickian transport is obtained when the time scale of the relaxation is either effectively zero or infinite compared to the time required to establish a concentration profile in the sample. These two limits have been called the elastic and viscous Fickian diffusion limits, respectively.

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The elastic limit is characteristic of samples exposed to gases or low activities of vapors that experience little segmental disruption relative to the native polymer and are able to accommodate the sorption by elastic deformation of intrasegmental bonds or slight disruption of intersegmental bonds while maintaining the essential integrity of the entanglements of interpenetrating chains in the matrix. On removal of the perturbing penetrant, the matrix is able to recover to its original condition. The viscous limit applies when significant swelling occurs and reorganization of the entanglement network may occur sufficiently rapidly during the solvent invasion to relax any local stresses that may tend to build. In this situation, only a complex concentration-dependent diffusion coefficient is required to describe the locally changing environment through which a penetrant must move. An example of this behavior is the initial desorption of solvent from a highly swollen polymer. Between these two limiting Fickian cases lies an important domain in which significant swelling (greater than 10% by volume) occurs. Under such conditions, the polymer deformation in response to a local swelling stress caused by a local penetrant concentration is rapid (but not infinite) as compared to the time required to produce a change in local concentration. Typical uptake behavior in such Case II situations is linear with time as shown for the large spheres in Figure 38. The chief attributes (146) of these systems are a sharp advancing boundary existing between an inner glassy core and an outer swollen rubbery shell; a swollen gel behind the advancing penetrant front close to equilibrium with the external solvent supply; and a boundary between the swollen gel and glassy core, which advances at a constant velocity. An induction time prior to the onset of the linear uptake in mass with time is also a characteristic common in Case II systems (147). However, this behavior is somewhat difficult to discern using the gravimetric and optical methods usually employed. An induction time is consistent with the criterion for the relatively rapid polymer deformation in response to a local swelling stress compared to the time required to produce a change in concentration profile. The approach of the surface concentration to true equilibrium in such non-Fickian systems is protracted (148). A connection has been made between the effective deformation stress acting to dilate the matrix and a thermodynamically calculable osmotic swelling stress (149). Case II front movement is visualized to occur when local stresses build to a sufficient extent to allow crazing. Combining the two concepts of an induction time at the surface and an osmotically driven local dilational stress, the yielding problem has been formulated in terms of a concentration-dependent viscosity of the matrix (150). Thus, in essence, the Case II process is initiated when a surface element of sufficient thinness experiences a significant reduction in viscosity and is able to relax 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, further depressing the viscosity, and thereby autocatalytically accelerating the stress relaxation. This results in a surface concentration that exponentially approaches its equilibrium value. While this outermost element is undergoing its rapid, but not instantaneous, relaxation, the solvent that is now available to the next infinitesimally thin element invades it as has occurred in the first exposure of the surface element. The transport to this interior element is essentially instantaneous,

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because of the much lower resistance to diffusive supply through the swollen gel as compared to diffusive invasion of the virgin polymer in the second element. In this case, therefore, the second element experiences significant relaxation by the time the first is approaching its equilibrium conditions with respect to the external penetrant. A third, fourth, and progressive repetition of this process leads to a negligible concentration gradient in the swollen gel behind the front, and a small and very sharp concentration gradient in some characteristically small number of volume elements ahead of the front. Assuming the viscosity is dependent on the local penetrant volume fraction ϕ A 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 containing no penetrant. This expression is, again, related to the free-volume concepts discussed in relation to equations 18a and 19 with regard to diffusion coefficients. Further treatment (147) has focused on the magnitude of the induction time for front initiation, neglecting all diffusional resistance in the gel layer behind the ¯ of the penetrant in front and assuming an effective average diffusion coefficient D the partially plasticized thin element ahead of the front. The front velocity should be proportional to the square root of the ratio of the local effective diffusivity in each of the infinitesimal volume elements preceding the relaxation front and the unplasticized 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 concluded that the relative activation energies for the effective solvent diffusion coefficient and the intrinsic viscosity of the glass were 58.6 kJ/mol (14 kcal/mol) and 167 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 coefficient and viscosity of the glass are expected to be similar. The lower activation energy ¯ in the partially plasticized glass ahead of the front for the diffusion coefficient D as compared to the viscosity of the unplasticized glass, η0 , is not surprising. Recent theoretical and experimental work has significantly advanced the understanding of Case II transport; however, because of the general complexity of this work, a review of it is beyond the scope of this article. The interested reader is 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 the mass transfer through the polymer. This type of transport is very common in biological systems, and is capable of substantially increasing the flux and selectivity of synthetic polymers. Facilitated transport still proceeds by the solution-diffusion

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mechanism; however, the penetrant now has two diffusive pathways through the polymer. The penetrant can sorb into the membrane and diffuse down its chemical potential gradient, or the penetrant can complex with the carrier upon sorption and diffusion down a penetrant-carrier complex chemical potential gradient. The second pathway is only available to penetrants that complex with the carrier. The presence of a carrier usually increases the sorption of the complexing penetrant as well. The inclusion of the complexation reaction in facilitated transport greatly increases the mathematically complexity of the flux equations at steady state. The concentration through the polymer of the penetrant, cA , the carrier, cc , and the penetrant–carrier complex, cA–c , must satisfy the following continuity equations: DA

d2 cA − rA–c = 0 dx2

(72a)

Dc

d2 cc − rA–c = 0 dx2

(72b)

d2 cA–c + rA–c = 0 dx2

(72c)

DA–c

where DA , Dc , and DA–c are the diffusion coefficients of the penetrant, carrier, and penetrant-carrier complex, respectively. The net reaction rate of complexation is rA–c . The boundary conditions for the above equations are determined from an appropriate sorption model. Cussler solved the above set of equations analytically for the two limiting cases of fast reaction and fast diffusion (154). Except for these limiting cases, numerical methods are required to determine the flux of each species. Facilitated transport has been reported in a number of systems. One of the most common systems uses silver as a carrier for olefins (155–157). Olefins are able to complex with silver, while paraffins are not. Transport of carbon dioxide has been facilitated through the use of amine complexes (158–160). Porphyrins have been used to facilitate the transport of oxygen and increase oxygen/nitrogen selectivity (161–163). Several methods have been used for carrier inclusion in the polymer: (1). carriers are blended as ionic salts with the polymer, (2). carriers are bonded to the polymer backbone through the use of crown ethers, and (3). carriers are used as the exchange sites on ionic exchange membranes. Regardless of the carrier and the inclusion technique, all facilitated transport membranes still currently suffer from unstable performance. The carrier is often poisoned by contaminants in the feed stream, and in some cases the carrier simply leaches out of the membrane. These issues must be resolved before there is widespread industrial adoption of facilitated transport membranes. Contradictory to facilitated transport, the use of scavengers attempts to inhibit transport of particular penetrants. Scavengers are adsorbents that are blended with the polymer. This technology is often used in food packaging.

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

Proton Exchange Membranes In recent years, a great deal of research has been focused on proton exchange membranes (PEMs) for use in fuel cells. Fuel cell based vehicles are touted as a more efficient, environmentally-friendly replacement for internal combustion engines. The fuel cell harnesses energy from the chemical reaction of hydrogen and oxygen to form water. In this reaction, the cathode and anode are separated by the PEM which allows passage of protons and theoretically blocks passage of other species. The electrons produced from the reaction are routed through an external circuit to provide power. Nafion produced by Dupont is the most commonly used PEM material. Nafion is a hydrated perfluorosulfonic acid (PFSA) polymer. The Nafion backbone is hydrophobic, while side-chain sulfonic acid groups are hydrophilic. The repeat unit of Nafion is given as structure (2).

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

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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, nonporous polymers, the rest of this discussion is limited to transport where a solutiondiffusion mechanism is assumed to occur. In this regime, the flux of “A” is given by (175)  jA = − DA

dcA F dψ + zA cA dx RT dx

 (73)

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

DH + cH + F ψ RT L

(74)

where L is the membrane thickness. In the literature, the flux is often reported as a current density i, which equals i = FjA =

σ ψ L

(75)

where the conductivity σ = DA cA F 2 /RT. In the case of noncharged penetrants, the charge equals zero, and equation 73 reduces to Fick’s law. The above equations are exact only when the diffusion coefficient is constant, the membrane is uniformly charged, and electroosmosis effects are neglected.

Positron Annihilation Lifetime Spectroscopy From the models of transport through both rubbery and glassy polymers given previously, it becomes apparent that knowing the free-volume distribution of the material is crucial for a microscale description of gas transport. Recently, positron annihilation lifetime spectroscopy (PALS) has been developed to probe the free volume of glassy polymers. PALS injects positrons, antielectrons, into the material where 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 is sensitive to the electron density of the environment. This makes PALS a modeldependent technique which is capable of measuring free-volume holes that exist for 10◦− 9 s or longer. PALS is able to resolve free-volume hole sizes in the range of 1–10A;. Reviews of PALS are available (176–178). The positron source is usually 22 Na, which releases a positron every 1.5 ms as it decays into 22 Ne. Approximately 3 ps after the positron is emitted from the source, gamma radiation of energy 1.28 MeV is released. This energy release is detected and marks the positron “birth.” Once inside the sample, the positron annihilates by one of the three possible modes. The first and shortest lived mode results from the para-positronium, p-Ps, species which is formed when the positron

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extracts an electron from the sample with opposite spin. The p-Ps species annihilates with its own electron after a lifetime of 125 ps. The lifetime of p-Ps is independent of the environment. The second mode of annihilation results from free positrons which have a lifetime of 100–500 ps. The third and longest lived mode results from the ortho-positronium, o-Ps, species which is formed from a positron and an extracted electron with opposing spins. The o-Ps decays after approximately 103 ps when it extracts or picks off another electron from the sample. After each annihilation event, regardless of the mode, gamma radiation of energy 0.511 MeV is released. This energy is detected and marks the “death” of the positron. The spectrum produced from these annihilation events is most commonly fit using three exponential decays. The resulting parameters are I1 , I2 , and I3 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 volume, the parameters associated with it, I3 and τ 3 , can be used to characterize the glassy structure. The probability of o-Ps formation and the concentration of free-volume holes are related to the intensity I3 . The time constant τ 3 is sensitive to the size of the free-volume holes. Simplistically, the time constant is short in small free-volume holes because the close proximity of the o-Ps to neighboring mass within the sample makes it easier for the o-Ps to pick off an electron. The semiempirical relationship between the time constant τ 3 and the free-volume hole radius R is given below (179).

τ3 =

   2π R − 1 1 1 R sin + 1− 2 R0 2π R0

(76)

The value R0 is equal to R + R where R is the electron layer thickness. ◦ of R was found empirically and is usually assumed to equal 1.66A;. Equation 76 assumes that the o-Ps resides in the center of a spherical free-volume hole, and so it follows that the hole volume V h is equal to 4/3π R3 . The fractional free volume of the material is given by (180) vf = CVh I3

(77)

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

 Vh p dvf = − 3.32 cos

  2πR α(λ) −1 R + 1.66 (R + 1.66)2 K(R)4πR2

(78)

where the fraction of holes between V h and dV h is V h pdvf dV h . In equation 78, α(λ) is the annihilation probability-density function. This function is obtained from a continuous fit of the positron lifetime spectra. K(R) is a correction for the capture probability of o-Ps in different size holes. A detailed description of equation 78 is given elsewhere (182). Because PALS is still a very new technique for characterizing glassy polymers, only a limited number of materials have been investigated which are also of interest from a gas transport standpoint (181,183–187). The most thorough

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Fig. 39. Free-volume hole distributions of various polycarbonates. Dashed lines are the molecular volumes of common penetrants. Reprinted from Ref. 181. This material is used by 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). Jean and 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 sizes in these polymers is given in Figure 39 along with the molecular volume of common penetrants. The materials with larger average free-volume hole size also had a wider distribution of hole sizes. The diffusion coefficient of O2 and CO2 through these materials correlated well with the experimentally determined fractional free volume from equation 77. The ability of PALS to determine both the hole size distribution and the fractional free volume opens up a number of very interesting questions. For example, how will the transport differ in materials of the same fractional free volume but with different free-volume distributions? Another interesting set of articles (188,189) uses PALS to investigate the behavior 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-volume hole radius obtained from the positron lifetime spectra increases. Continuing to increase the N2 pressure from 4.14 MPa (600 psi) to 8.28 MPa (1200 psi) results in a decrease in the free-volume hole radius. No hysteretic effects were observed upon depressurization. This behavior was explained as plasticization and hydrostatic effects similar to those already discussed in the context of rubbery polymers. The N2 is claimed to initially plasticize or soften the polymer resulting in larger freevolume holes. These results appear rather contrary to concepts noted with regard to the effect of pressurization on free volume in Fig. 8. As the pressure is increased more, the polymer compresses resulting in smaller free-volume holes, which is consistent with Fig. 8. A very different effect was observed when PC was exposed to high pressure CO2 . The free-volume hole size was observed to monotonically increase with CO2 . Upon depressurization, a large hysteresis was seen. After CO2 exposure, the fractional free volume and average hole size were larger. This

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behavior results from the strong conditioning effect of CO2 that was described in a previous section. Although PALS has been applied successfully to determine the glassy polymer microstructure, the technique has several limitations. In many low polar liquids, the positronium species forms a “bubble.” This bubble results from repulsions between the electron in the positronium and electrons in the surrounding. Obviously, formation of such a bubble would affect the free-volume characterization. This bubble formation process may be possible in rubbery polymers, but is generally not thought to occur in glassy polymers (176). Inhibition or quenching effects can also affect the lifetime of the positronium species. Halogen-containing polymers can lead to inhibition effects when the positronium interacts with the halogen atom. Nitroaromatics, such as polyimides, may react with the positron causing quenching (190). Perhaps, the largest disadvantage of PALS is that it is a model-dependent technique. In this respect, characterization of free-volume distributions in polymers using PALS is analogous to obtaining the pore-size distribution 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 significant body of research, which seeks to describe penetrant sorption and diffusion in amorphous polymers through the use of computer simulations. Two main goals exist for computer simulations: first, to gain a better fundamental understanding of the transport mechanism through amorphous polymers; and second, to accurately predict the transport properties (ie, the diffusion and solubility coefficients) of novel polymers. The first goal is motivated by the fact that small-molecule transport is determined by the structure of the penetrant–polymer environment on an atomistic scale. Excluding the recent use of PALS structural information on this length-scale is not accessible from traditional macroscopic transport experiments but is accessible from molecular dynamic (MD) simulations. The second goal, accurate prediction of transport properties, seeks to eliminate expensive and laborintensive experiments. Several reviews are available on computer simulations of gas diffusion through amorphous polymers (191,192). The starting point for all computer simulation methods is the choice of a potential energy function or force field, (R). This function gives the potential energy of a system of N atoms at positions ri . The positions of all atoms in the system are represented by the three dimensional vector R = (r1 , r2 , . . . , rN ). The force-field accounts for all bonded and nonbonded interactions between atoms in the system. Bonded interactions include bondlength and bond-angle deformations and conformational deformations. These interactions are usually described as mechanical springs with the spring constant related to experimentally known bonding energies. The nonbonded interactions include van der Waals forces and other molecular forces and are accounted for empirically by a Lennard–Jones potential. Since the nonbonded interactions include contributions from each atom pair in the system, current computational power limits the number of atoms in the system to less than 10,000 (191). Many force-field models exist and proper selection is crucial for accurate results.

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Once the force-field is chosen the motion of atoms in the system is simulated using classical equations of motions: 1 dui (t) = Fi [R(t)] dt mi

(79a)

dri (t) = ui (t) dt

(79b)

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

∂(R) ∂ri

(79c)

Equations 79a, 79b, and 79c are a coupled set of second-order differential equations. 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 H bond oscillation time. Because such extremely small timesteps are required, an upper limit of several nanoseconds is imposed on current MD simulations (191). The limit of MD models to at most 10,000 atoms on current workstations results in model systems that are cubic volume elements with a side length of just a few nanometers. Many approaches have been attempted to initially populate the system with atoms. The approach described by Theodorou and Suter seems to have gained the most use in recent literature (193,194). In this method the cubic volume element is filled by growing the chain one backbone bond at a time. Rotational isomeric states (RIS) theory is used to calculate the conditional probability of each conformation. As the chain is grown, the conformation angle is selected statistically to agree with the RIS information. After the chain has grown a few repeat units, it is likely to cross the boundary of the volume element. When this happens, the chain growth is continued by reentering the volume element on the opposite side. Chains grown in this manner typically achieve the overall density target, but the chain growth often results in regions of severe atomic overlap. To combat this effect, a series of static energy minimizations and dynamic simulations are carried out to equilibrate the structure. After equilibration, it is possible to get the density of the volume element to within a few percent of the experimentally determined density; however, such a high level of agreement is not currently possible for all materials because of deficiencies in the force field. The particle trajectories from the MD simulations can be used to determine the self-diffusion coefficient using the Einstein equation: D∗A = |r(t) − r(0)|2 /6t =

λ2 f 6

(80)

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

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penetrant. f simple indicates the average frequency of a diffusional jump of length λ. The diffusion coefficient in equation 80 ideally allows one to estimate the selfdiffusion coefficient in equation 6. The Einstein equation gives accurate results for rubbery polymers; however, in glassy polymers, anomalous diffusion effects act to degrade the accuracy (195). In anomalous diffusion the penetrant is not proceeding by a truly random walk from one free-volume hole to the next which is assumed by the Einstein equation. This effect results from penetrant oscillations inside the free-volume holes that occur between diffusive jumps from one hole to the next. Since the simulation time is only a few nanoseconds, these oscillations have a nonnegligible effect on the mean-squared displacement. It is important to note that this definition of anomalous diffusion is different than non-Fickian diffusion, discussed previously, which is also commonly referred to as anomalous diffusion in classic texts (5). Gusev and Suter have developed a transition-state theory that is based on a Monte Carlo scheme which requires less computational power than the MD simulations and therefore is able to calculate diffusion coefficients where anomalous diffusion effects are negligible (192,196–198). In this method a three-dimensional grid is overlaid on a fully equilibrated MD volume element. A typical grid spacing is 0.03 nm (191). The insertion energy of a penetrant molecule is calculated at each grid location. In this calculation the insertion energy comes from the nonbonded contribution of the force field. The volume element is then mapped as regions of high and low packing on the basis of whether the insertion energy was high or low. This mapping allows for the identification of pathways between adjacent low energy regions. Each of these pathways is assigned a jump probability which is based on the Boltzmann distribution. This jump probability is affected by the thermal motion of the polymer atoms which is characterized by the meansquared displacement of polymer atoms, 2 . This parameter is often gained from a short MD simulation. It should be noted that the Gusev and Suter method does not account for polymer relaxation as a result of penetrant insertion, and so this theory is only reasonable for small penetrants up to methane. With the appropriate jump probabilities, the penetrant diffusion is simulated using a Monte Carlo scheme. The insertion energy Eins calculated during this process can also be used to calculate the solubility coefficient S: S = exp( − µex /RT)

(81)

where the excess chemical potential µex = RT ln exp(−Eins /kB T) (199). Diffusion and solubility coefficients calculated using the above techniques are usually within a factor of 3 to 5 of experimentally determined values. This degree of agreement is not accurate enough for many applications. Molecular modeling of the type described above has been conducted on a number of polymer systems. Of these systems the great majority have been based on rubbery polymers. It is only recently that molecular simulations based on glassy polymer systems are appearing in the literature (200–206). From these studies, differences in the diffusion mechanism of rubbery and glassy polymers are starting to emerge. In both cases, the penetrant can spend relatively long periods of time (several hundred picoseconds) bouncing around inside the free-volume hole. At some point in time, a transient gap opens up because of thermally induced polymer

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◦

Fig. 40. H2 diffusion through PEEK shows diffusion jumps of 5–10A; separated by several hundred picoseconds of oscillations in free volume. Reprinted from Ref. 200, with permission from Elsevier.

motion between free-volume holes. While the gap is open, the penetrant may make ◦ a 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 is only open for a few picoseconds, and the penetrant does not have the opportunity to make a backwards jump into the original free-volume hole. This is not the case in glassy polymers, where the channel between voids can be open for an extended period of time (several nanoseconds). This allows the penetrant molecule to jump back and forth numerous times between the two connected free-volume holes. The back-and-forth motion experienced by the penetrant in this behavior does not add 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 been observed in many glassy polymers. Reprinted from Ref. 200, with permission from Elsevier.

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as a contributing factor to explain why rubbery polymers exhibit much higher diffusion coefficients than glassy polymers (191), since it reduces the frequency f of a successful jump in equation 80. However, there has been recent evidence that even amorphous rubbery polymers may be significantly affected by anomalous diffusion because of alignment of the polymer chains at the nanoscale (207). The frontier in the field of transport properties of polymers involves bridging the gap between the molecular scale penetrant–polymer environment and the macroscale solubility and diffusion coefficients. Impressive efforts to achieve this goal have already been made through the use of PALS and molecular modeling. The continued increase in computing power will allow for molecular dynamic simulations of larger volume elements for longer time periods. These improvements should cure many of the current inaccuracies inherent in molecular modeling, and allow for the first time a truly predictive method of determining polymer transport properties. The topics discussed in this article only dealt with small-molecule diffusion through polymers. Another interesting and complex topic involves the diffusion of large polymeric penetrants through a polymeric medium. de Gennes provided a theoretical treatment of a polymer chain diffusing through a cross-linked rubber (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 polymer chain is confined to a tube defined by the surrounding cross-linked matrix which restricts transverse motions. More complex treatments of reptation allow for the self-diffusion of the surrounding polymer matrix (209). The diffusion of macromolecular penetrants is of importance in a number of industrial applications including crack healing and adhesion between polymer surfaces, and in the use of polymer composites and blends.

BIBLIOGRAPHY “Transport Properties” in EPSE 2nd ed., Suppl. Vol., pp. 724–802, by William J. Koros and Mark W. Hellums, University of Texas at Austin. 1. L. S. Darken, Trans. AIME. 174, 184 (1948). 2. A. D. Smigelskas and E. O. Kirkendall, Technical Publication No. 2071, NorthHolland, Amsterdam, The Netherlands, 1963. 3. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., New York, 1960. 4. A. L. Hines and R. N. Maddox, Mass Transfer Fundamentals and Applications, Prentice-Hall, Englewood Cliffs, N.J., 1985. 5. J. Crank, The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975. 6. D. R. Paul, J. Appl. Polym. Sci. 16, 771 (1972). 7. J. H. Petropoulos, in D. R. Paul and Y. P. Yampolskii, eds., Polymeric Gas Separation Membranes, CRC Press, London, 1994. 8. R. J. Bearman, J. Phys. Chem. 65, 1961 (1961). 9. R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley & Sons, Inc., New York, 1993. 10. D. R. Paul and O. M. Ebra-Lima, J. Appl. Polym. Sci. 14, 2201 (1970). 11. D. R. Paul and O. M. Ebra-Lima, J. Appl. Polym. Sci. 15, 2199 (1971). 12. R. H. Boyd and S. M. Breitling, Macromolecules 7(6), 855 (1974).

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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.