Local Upstream Concentration, C2FIGURE 20.3-11 Demonstration of the marked difference in concentration dependence of the true localdiffusion coefficient (solid line) and the apparent diffusion coefficient (dashed line) for a hypotheticalexponentially varying case such as shown in Fig. 20.3-6a.

(20.3-16) is clearly not very serious. Several rather extensive examples of comparisons of measured andcalculated time lags for CO2 and other gases in glassy polymers have been reported recently. In most casesup to the 20 atm limit which have been investigated, Fick's Law appeared to be obeyed well.

20.4 FUNDAMENTALS OF SORPTION AND TRANSPORT PROCESSES IN POLYMERS

20.4-1 General Discussion

The following two subsections provide physical interpretations of the forms of sorption isotherms andconcentration-dependent diffusion coefficients observed for rubbery and glassy polymers, respectively.These sections are not required for simulation of module operations if a complete set of empirical pressure,temperature, and composition-dependent permeation data are available. A simple polynomial fit of perme-ability data as a function of all operating variables would suffice for design and simulation.

Generally, such detailed data are not available, and a theoretical framework is valuable to aid in thephysical interpretation of new data. Moreover, in some cases an appreciation of fundamentals can explainobservations that otherwise might be termed "anomalous" and cast doubt on the validity of a perfectlygood particular set of data. Examples of this sort of situation can be cited for the case of glassy polymersin which the following consistent, but surprising, observations were reported by two separate investigatorsregarding H2 permeation in the presence of a slower permeating gas (either CO or CH4):

There is some interaction between components of the binary mixture during permeation whicheffectively decreases the permeability of H2 relative to CO.1

The fast gas was slowed down by the slow gas and the slow gas was speeded up by the fast gas.2

Such behavior is not consistent with plasticization, since the slightly sorbing H2 would not be expectedto plasticize the polymer while the more condensable (and hence more strongly sorbing) CO or CH4 actsas an antiplasticizer. As is discussed in the section on glassy polymers, such behavior as described here isnot anomalous and can be shown to be consistent with inherent properties of glassy polymers under certainconditions.

The major emphasis of the following discussion is on glassy polymer membranes since the more rigidsize- and shape-selecting glassy materials tend to have substantially higher selectivities than do flexiblerubbers, which are essentially high-molecular-weight liquids. This effect can be seen clearly from com-parison of the ideal separation factor data in Tables 20.4-1, 20.4-2, and 20.4-3 for rubbery and glassypolymers.3 However, consistent with the preceding discussion of the two-part nature (solubility and mo-bility) of both permeability and selectivity, one might envision a rubbery polymer with sufficiently favorablesolubility for one of the components in the mixture to make it useful as a thin film coating on a compositemembrane. In such a case, plasticization of the separating membrane could be a less serious problem thanfor the materials whose permselection action relies on molecular sieving action based on penetrant size andshape. Therefore, treatment of transport in rubbers is reviewed prior to consideration of the more importantglassy polymer case.

Actual Local

Previous Page

TABLE 20.4-1 Permeabilities and Separation Factors for Typical Rubbery Polymers at Low Pressures

PHJPCH*PoJPu2PHJPQH*P^

(Barrer)"(Barrer)"(Barrer)"(Barrer)"(Barrer)"P»e

(BarrerfT(0C)Rubber

3.38

4.45.9

6.68.54.34.4

0.8

9.3

0.69

1.631.91

9.177.69

10.1-0.6*

6.36

2.02.29

2.58

2.962.713.63

4.72

3.963.642.992.932.68

2.28

4.25

0.38

1.031.09

10.675.01.692.90

44-2.7*

1.00

4.23

25070

33

8.16.31 .1

0.18

0.331.10.970.142.20.79

0.92

0.008

500160

85

24173.84

0.85

1.304.02.90.415.9

2.1

0.034

800

3021

0.782.62.90.39

1.4-23*

30

1.7

2,700970

670

13112430.9

4.3

5.182212.6

1.7

2.625

15.8

0.17

550210

123

4940.115.9

7.5

7.220

14.112C

10.8

1.0

300

86

3123.112.2

7.8

8.42134.91.14

621130

7.2

2525

25

252525

25

25232525303025

23

25

Silicone rubberSilicone-polycarbonate

block copolymerSilicone-nitrite

copolymerNatural rubberSBRAcrylonitrile-butadiene

copolymer (27% AN)Acrylonitrile-isoprene

(26% AN)Butyl rubberPolychloroprenePolyethylene p = 0.914Polyethylene p = 0.964FEP copolymer''Teflonrf 75% crystallinePolyethylene-vinyl

acetate copolymerChlorosulfonated

polyethyleneNylon 66

"1 Barrer « 10"10 [cm3 (STP)-cm]/[cm2-sec-cm Hg].^Highly pressure dependent at low temperature. Above 600C, pressure dependence diminishes greatly.Temperature = 33°C.rfThese fluorocarbon polymers are difficult to describe unambiguously as rubber or glassy because of mutliple thermal transitions.

Source: Stannett et al.3

TABLE 2.4-2 Helium and CO2 Permeabilities and Separation Factors Relative to CH4 in a Varietyof Glassy Polymers

P P^Polymer Structure (Barrer)0 (Barrer)" PWJ^CHA PCO^JPCH4

Poly(tetramethyl bis L 148 83.0 16 34terephthalate)

Poly(tetraisopropyl bis A- 110 98.0 25 15sulfone)

Poly(2-6-dimethyl-l,4- 60.0 50.0 27 18phenylene oxide)

Poly(tetramethylcyclo- 75.0 32.0 38 14butane diol carbonate)

Poly(tetramethyl bis A 43.0 29.0 49 33carbonate)

Poly(tetramethyl bis L 48.3 65.0 28 24sulfone)

Poly(tetramethyl bis A 31.0 — 60 —isophthalate)

Poly(tetramethyl bis A 38.4 21.0 67 68sulfone)

Poly(bis A carbonate) 15.0 10.0 35 20Lexan®

Poly(bis A-4,4'-N-methyl 6.05 - 93 -phenyl sulfone urethane)

Poly(ethylene terephthalate) 1.30 — 57 —

"1 Barrer = HT10 [cm3 (STP)-cm]/[cm2-sec cm Hg].hLikely to be somewhat pressure dependent.

Source: Stannett et al.3 No temperature was reported in the reference; presumably, these data apply at~30°C.

20.4-2 Rubbery Polymers

**Rubbery polymers" refers to amorphous polymeric materials that are above their softening or glasstransition temperatures under the conditions of use.4 Regions of crystallinity in rubbery matrices act asfillers, restricting segmental motion, and also as tortuous blocks to facile movement of penetrants. Reductionin the sorption level of a desirable permeating component also obviously results from reduction in thevolume fraction of material available for accommodating permeant.

As shown in Fig. 20.4-1, the solubility coefficient [CIp in Eq. (20.3-8)] of a large number of penetrantsin polyethylene is essentially directly proportional to the volume fraction of amorphous material in thepolymer.5 Therefore, except to improve mechanical properties and resistance to dissolution by highlysorbing vapors, crystallinity generally is not too desirable since it can reduce productivity greatly. However,Michaels and Bixler6 and Bixler and Sweeting5 have discussed cases in which, besides reducing productivity,the introduction of crystallinity can impart an ability to discriminate among molecules of increasing size.The diffiisivity of a penetrant in a semicrystalline polymer decreases relative to a totally amorphous sample.This reduction occurs because of the combined effects of added tortuosity and the restriction of amorphouschain mobility necessary for the opening of a transient molecular-scale passageway into which the penetrantcan jump.

In the work of Bixler and Sweeting5 it was assumed that chain restriction effects are negligible for thecompact helium molecule, so that only tortuosity (added path length) contributes to the reduction of thediffusion coefficient for this penetrant. Furthermore, for the 50% crystalline polyethylene sample studied,these authors suggest that the tortuosity effect is essentially nondiscriminating and causes about a 65%reduction in mobility compared to the hypothetical totally amorphous case. The transport properties of ahypothetical totally amorphous branched polyethylene were assumed by these authors to be equivalent tothose of unvulcanized natural rubber.

The ratios of diffusivities for a series of penetrants in the 50% crystalline polyethylene (D05) and inthe hypothetical amorphous sample (D, 0) are shown in Fig. 20.4-2. In addition to the substantial nondis-criminating reductions in mobility, there is an additional size-dependent reduction in mobility due to chainrestriction effects. Note that since the solubilities of all the components are proportional to the amorphousfraction (Fig. 20.4-1), selectivity enhancements resulting from the introduction of crystallinity rely solelyon the chain restriction effect.

TABLE 2.4-3 Hydrogen Permeabilities and Separation Factors Relative to CH4 and CO for aVariety of Glassy Polymers

p

Material (Barrer)* P\iJPc\u fV^co*

3,5DBA-6F 33.6 340.6 -1,5NMF* 78.1 108.5 -PPD-6F' 48.2 152.5 -Polystyrene 11.0 — 12PVC 8.0 - 12.9Parylene®^ 2.8 — 25.4Cellulose acetate* 13.0 — 37.2Sulfone* 14.0 — 37.8Polyvinyl fluoride' 0.6 — 66.7Mylar® type Sj 1.4 — 74.0Polyimide* 2.0 — 74.0Parylene®C' 1.4 — 110.0Polycaprolactamm 1.5 — 115.0

M Barrer = 10" l0 [cm3(STP)-cm]/[cm2-s-cm Hg].bPossibly dependent on driving pressure.cSoluble polyimide prepared by reaction of 4,4'-hexafluorisopropylidene diphthalic anhydride (6F) with3,5-diaminobenzoic acid.''Soluble polyimide prepared by reaction of (6F) with 1,5-diaminonaphthalene.e Soluble polyimide prepared by reaction of (6F) with 1,4-diaminobenzene.•^Poly(para-xylylene) Union Carbide registered trademark.*du Pont 100 CA-43.h Union Carbide Sulfone 47.'du Pont Tedlar®.Jdu Pont semicrystalline polyethylene terephthalate) film.*du Pont Kapton® polyimide.'Poly^monochloro-paraxylylene) Union Carbide registered trademark.mAllied Chemical, amorphous nylon film.

Source: Stannett et al.3

Polyethylene

Solu

bilit

y C

onst

ont,

(C/p

)

Amorphous Volume FractionFIGURE 20.4-1 Linear relationship between the infinite dilution solubility coefficient and the amorphousfraction in the polymer. The data are for polyethylene at 250C.5

As shown in Fig. 20.4-3, this added effect can improve helium selectivity relative to methane by afactor of 1.75. At the same time, a factor of 2.0 loss in helium solubility and a factor of 2.8 reduction inhelium diffiisivity due to the crystallinity produces a 5.6-fold reduction in helium productivity relative tothe hypothetical amorphous film case. Interestingly, the effects of chain restriction on different sizes ofpenetrants were found by Michaels to be effective only for rubbery materials, not glassy ones. A logicalexplanation for this fact was advanced in terms of the already low mobility of chain segments in the glasscompared to the rubbery state:

In glassy amorphous or crystalline polyethylene terephthalate, a diffusing molecule must pass be-tween chain segments which have little rotational mobility. Physically, the concept of chain im-mobilization by crystallites loses its significance, since the rigidity of the polymer backbone itselfapparently outweighs any additional restriction on mobility imposed by crystallinity. A consequenceof the absence of this effect in polyethylene terephthalate is the virtual equality, within precisionlimits, of apparent activation energies in the glassy amorphous and crystalline polymers.7

Kreituss and Frisch9 also have treated the effect of crystallinity on transport in rubbery polymers usingthe so-called free-volume approach, which is discussed below. In essence, their ideas suggest that intro-duction of crystallinity reduces the freedom of motion of amorphous chain segments between crystals. By

Frac

tiona

l M

obili

ty L

oss

Non Selective MobilityLoss Due to Tortuosity

Selective Chain'Restriction

EffectsFIGURE 20.4-2 Ratio of diffusivities in a 50% crys-talline and a 0% crystalline polyethylene as a functionof the van der Waals volume of the penetrant.8 Thediffusion coefficients for amorphous polyethylene weretaken to be equivalent to those of unvulcanized nat-ural rubber.5

van der Waals1 Volume

Hel

ium

Sel

ectiv

ity I

mpr

ovem

ent

Rat

io

van der Waals1 Volume (cmVmole)FIGURE 20.4-3 Ratio of selectivities of helium relative to component i (N2, CH4, or C3H8) for 50%crystalline and amorphous polyethylene as a function of the van der Waals volume of penetrant /8.

expressing these concepts in the terminology of the free-volume theory, a different analytical form of thediffusion coefficient results from that suggested by Michaels and coworkers. The physical ideas, however,are consistent with those discussed by Bixler and Sweeting5 and Michaels and Bixler.6

Several versions of the free-volume theory of diffusion have been promulgated following its introductionfor polymeric systems in the work of Fujita.10 Other more fundamentally based molecular approaches havebeen attempted at times based on actual molecular properties of the polymer and penetrant; however, theirgreater complexity has discouraged widespread application. The free-volume theory is simple and hasevolved through the work of Vrentas and Duda1 '*l2 and Stern, Frisch, and coworkers13*l4 to a readily usefulform. Its primary drawback lies in the difficulty in providing a precise physical definition for the parameterdefined as the free volume. Excellent discussions of the theory have been offered, and the reader is referredto them for greater detail than that provided here.1214

In general, D can be a function of local concentration as discussed in the context of Fig. 20.3-6. Asnoted by Stern, a convenient form of Fick's Law can be written as shown in Eq. (20-4.1) in terms of thelocal volume fraction of the penetrant in the polymer, v9 and the mutual diffusion coefficient D.13

(20.4-1)

The volume fraction of penetrant is related easily to penetrant sorption concentration it tne partial moiaivolume of the penetrant can be approximated by its molar volume as a hypothetical liquid. Values of theinfinite dilution partial molar volume of various gases in a series of low-molecular-weight liquids are shownin Table 20.4-4.l5 Since rubbery polymers are essentially high-molecular-weight liquids, the average valuesshown in the table for the various penetrants would be reasonable estimates of the partial molar volume inpolymeric media also.

The so-called thermodynamic diffusion coefficient D7 is related to D by Eq. (20.4-2):

(20.4-2)

For gases, because of their relatively low solubilities, [(d In a)/(d In v)]T may be taken as a constant closeto 1.0,13 so that Eq. (20.4-2) becomes

(20.4-3)

The diffusion coefficient appearing in Eq. (20.4-3) is a true measure of the molecular mobility of thepenetrant in question. Intuitively, the free-volume theory proponents argue that a penetrant can execute adiffusive jump when a free-volume element greater than or equal to a critical size presents itself to apenetrant. The native polymer, totally devoid of penetrant, still possesses a certain amount of free-volumepackets of distributed size which wander spontaneously and randomly through the rubbery matrix. In fact,when a packet of sufficient size presents itself to a polymer segment, the polymer may execute a self-dimasive motion, and this is the action that causes slow interdiffusion of polymer chains.

Kreituss and Frisch have modified Fujita's ideas by arguing that introduction of crystalline materialreduces the free volume in the composite semicrystalline structure in direct proportion to the amount ofcrystalline material present.9 Thus,

(20.4-4)

TABLE 20.4-4 Partial MoIaI Volumes P 0 0 of Gases in Liquid Solution at 25 0C

H2 N2 CO O2 CH4 C2H2 C2H4 C2H6 CO2 SO2

Ethyl ether 50 66 62 56 58 - - - - —Acetone 38 55 53 48 55 49 58 64 - 68Methyl acetate 38 54 53 48 53 49 62 69 - 47Carbon 38 53 53 45 52 54 61 67 - 54

tetrachlorideBenzene 36 53 52 46 52 51 61 67 - 48Methanol 35 52 51 45 52 — — — 43 —Chlorobenzene 34 50 46 43 49 50 58 64 - 48

Source: Reid et al.15

where Ad and Bd are free-volume parameters that must be evaluated empirically. The parameter <f>a is theamorphous volume fraction, and Vf is the so-called free-volume fraction of a totally amorphous sample inthe presence of penetrant at the system hydrostatic pressure and temperature.9

Increases in system temperature give rise to volume dilation, resulting in increased amounts of freevolume. This explains the strong tendency for the diffusion coefficient to increase with temperature. Thefree-volume fraction in Eq. (20.4-4) may be represented as a linear addition of the several variables thataffect its value:

(20.4-5)

where Vfx is the fractional free volume of the pure, penetrant-free, amorphous, rubbery polymer at somereference temperature 7; (usually the glass transition) and reference pressure p, (usually 1 atm). Thecoefficients a, /3, and 7 are positive constants whose values are evaluated empirically. These coefficientscharacterize the effectiveness of temperature, pressure, and penetrant concentration for increasing (or inthe case of pressure, decreasing) the fractional free volume in the amorphous phase that is accessible topenetrant movement.

In principle, one can evaluate each of these coefficients somewhat independently by observing theeffects on D7 over a sufficiently wide range of temperatures, external hydrostatic pressures, and sorbedconcentrations. The values of D7 for analysis in this manner can be determined by the techniques describedearlier using Eq. (20.3-6) under different temperature and hydrostatic pressure conditions. The temperature,pressure, and penetrant volume fraction effects on the observed values of D7 then can be fit by nonlinearleast squares to the expression for D7 given by substitution of Eq. (20.4-5) into Eq. (20.4-4):

(20.4-6)

One could decouple hydrostatic effects from penetrant sorptive effects by using an increasingly highpressure of an effectively nonsorbing pressurizing medium such as helium in the presence of a fixed partialpressure of the penetrant of interest. Hydrostatic pressure would be expected to have a rather small negativeeffect on D7 (and VJ) since solid polymers are slightly compressible. On the other hand, increases intemperature and sorbed penetrant concentrations may cause large increases in D7 (and Vf). Because of thelow solubility of supercritical gases, Henry's Law typically is obeyed up to high pressures for the solubilityrelationship, and dCldp is a constant in Eq. (20.3-3). Because of the relatively low volume fractions ofsorbed material present in such cases, strong plasticization generally is not observed for supercriticalcomponents.

For more strongly sorbing penetrants, such as CO2 and C2 and C1 hydrocarbons, BET III isothermsindicative of Flory-Huggins swelling behavior are common. The Flory-Huggins model, shown below,relates the penetrant activity (a) to the sorbed volume fraction v of the component. Typically, the activityis replaced by the ratio of the penetrant partial pressure p and the component vapor pressure p0:

(20.4-7)

where x is the Flory-Huggins parameter that characterizes the polymer-penetrant interactions in the dis-solved state. For systems that show a pronounced deviation from Henry's Law, one can anticipate acorresponding tendency to exhibit plasticization in the D(Q versus C response, such as shown in Fig.20.3-6*.

Stern and coworkers have shown that deviations from constant diffusion coefficient behavior generallyoccur in rubbery polymers before the onset of deviations from Henry's Law solubility behavior.16 In eithercase, substitution of Eq. (20.4-6) along with the appropriate isotherm form into Eq. (20.3-2) permitsgraphical or analytical integration to obtain the permeability in terms of the upstream and downstreampenetrant volume fractions or concentrations. Even for the case in which Henry's Law applies, the resultingexpression is rather complicated and is not reproduced here in its general form.

To obtain an expression for permeability in terms of the actual upstream and downstream pressures,one must employ the appropriate isotherm equation. In the case of Henry's Law behavior, v = Kp, thefollowing expression can be qsed to a good approximation for the usual case in which the downstreampressure is negligible relative to the upstream pressure, that is, p, « ^2:13

(20.4-8)

where V* is the fractional free volume of the penetrant-free polymer at p = 0 and the temperature of thesystem. All other parameters were defined earlier. If the reference states mentioned earlier are used (1 atm

and the glass transition of the polymer), can be determined simply in terms of previously specifiedparameters:

(20.4-9)

Therefore, although rather cumbersome, the free-volume theory permits one to prepare theoretical plots ofpermeability as a function of temperature, penetrant pressure, and amorphous volume fraction in the rubberypolymer.

Vrentas and Duda also have approached the problem of penetrant sorption and diffusion in rubberypolymers in terms of the free-volume theory. Most of their work, however, focuses on the problem oflarger penetrant molecules such as benzene and the like. Their approach permits remarkably accuratedescriptions of diffusion coefficient behavior over a wide range of temperatures and penetrant activitiesusing a relatively small amount of information, such as the viscosity of the penetrant-free polymer and theeffective glass transition temperature of the penetrant. This approach is especially useful in dealing withhighly swelling vapors and liquids, but it is beyond the scope of the current discussion dealing with smallpenetrants. The interested reader is directed to several references describing their work.111217

20.4-3 Glassy Polymers

Glassy polymers are inherently more size and shape selective than rubbery materials. This additional"mobility selectivity" arises from the restrictive nature of transport in glassy environments, which arecharacterized by extremely limited segmental motions. Because of their large potential for use as monolithicmembranes and selective coatings on composite membranes, a fairly detailed discussion of the sorptionand transport behavior of glassy polymers is presented in the following section.

As noted earlier in the context of Tables 20.4-1, 20.4-2, and 20.4-3, the ratio of pure componentpermeabilities at an arbitrary pressure is a useful indication of the selectivity of a candidate membrane forthe chosen gas pair. Nevertheless, because of a variety of factors, the actual selectivity and productivityobserved in the mixed gas case may be somewhat different than predicted on the basis of the pure componentdata. One of these factors, illustrated in Fig. 20.4-4, is important for low- and intermediate-pressureapplications and is believed to be due to competition by mixture components for unrelaxed molecular-scalegaps between glassy-polymer chain segments. As shown in the left-hand side of the figure, the presenceof a second component B can depress the observed permeability of a component A relative to its purecomponent value at a given upstream driving pressure of component A. Hoehn and coworkers note thatthe permeability of a membrane to a component A may be reduced due to the sorption of a second componentB in the polymer which ". . . effectively reduces the microvoid content of the film and the availablediffusion paths for the nonreactive gases."18

At higher pressures, where microvoid saturation processes are essentially complete, additional sorptionsimply plasticizes the membrane, making it more rubberlike. As shown in Fig. 20.4-4, this higher activityrange corresponds to upswings in the observed permeability due to free-volume effects like those discussedpreviously for simple rubbery polymers and illustrated in Fig. 20.3-2^. Stern and Saxena have treated theseeffects theoretically for glassy polymers.19 Paulson et al.20 have reported such plasticizing effects for theCO2-CH4 system at high relative humidities for a variety of polymers. The effects become especiallynoticeable at higher temperatures.

simple dual mode

plasticizingby A or B

/permeability \\ of A /

increasing

upstream partial pressure of AFIGURE 20.4-4 Permeability of a glassy polymer to penetrant A in the presence of varying partialpressures of penetrant B.

The two preceding effects are due to true polymer-phase sorption and transport phenomena. At veryhigh pressures, an additional complexity related to nonideal gas-phase effects may arise and cause potentiallyincorrect conclusions about the transport phenomena involved. This confusion can be avoided by definingan alternative permeability P* in terms of the fugacity difference rather than the partial pressure differencedriving diffusion across the membrane. The benefit of using this "thermodynamic permeability" is illus-trated for the CO2-CH4 system at elevated pressures.

The pressure level at which the various nonidealities discussed above become important is dependentprimarily on the condensability of the penetrant. The most convenient measure of condensability is thepenetrant vapor pressure. Even for slightly supercritical components, a hypothetical vapor pressure can beobtained by extrapolation of a semilog plot of vapor pressure versus the inverse absolute temperature upto the system temperature. Water has a low vapor pressure at ambient temperatures and is quite condensable.Only a few mm Hg of water vapor may cause the permeability depression effect shown in the left-handside of Fig. 20.4-4 for typical less condensable penetrants such as CH4 and H2. Conversely, high partialpressures of supercritical components such as N2, CH4, CO, and H2 would be required to cause noticeabledepression of each other's permeabilities. Nevertheless, as noted earlier, measurable examples of this effecthave been observed even in the H2-CO and H2-CH4 cases where the faster permeating H2 appeared tohave been "slowed down" by the slower CO and CH4.

Many of the important applications for the separation of CO2 and CH4 correspond to CO2 partialpressures well above 20 atm (e.g., cases 1-3 in the previous discussion of enhanced oil recovery appli-cations). In such cases, competition for unrelaxed volume should be of second-order importance, since the"condensable" nature of CO2 causes it to dominate the microvoid environments in the polymer even inthe face of substantial methane partial pressures. At high CO2 partial pressures, plasticizing behavior suchas that illustrated in Fig. 20.4-4 may be observed. On the other hand, at relatively low CO2 pressures (20atm) (cases 4 and 5 in the discussion of enhanced oil recovery applications), competition effects would beanticipated for most glassy polymers. These effects could be masked in the case of cellulose acetate becauseof its strong tendency to interact with CO2 and exhibit plasticization at low partial pressures.

The magnitude of the partial pressure of a given penetrant required to cause strong plasticization,illustrated on the right-hand side of Fig. 20.4-4, depends on the magnitude of specific polymer-penetrantinteractions and is not currently predictable from first principles. Fortunately, the onset of nonideal gas-phase effects and their effects on the driving force for component permeation can be predicted from existingthermodynamic expressions such as the Soave modification of the Redlich-Kwong equation.15 This infor-mation permits one to decouple and analyze the polymer-related transport and sorption nonidealities withoutconfusion from simple gas-phase nonidealities.

UNRELAXED VOLUME A N D ITS RELATION TO SORPTION A N D TRANSPORT PROPERTIES OF GLASSYPOLYMERSThe concept of "unrelaxed volume", Vx- V1, illustrated in Fig. 20.4-5, may be used to interpret sorptionand transport data in glassy polymers exposed to pure and mixed penetrants. The extraordinarily longrelaxation times for segmental motion in the glassy state lead to trapping of nonequilibrium chain confor-mations in quenched glasses, thereby permitting minuscule gaps to exist between chain segments. Thesegaps can be redistributed by penetrant in so-called conditioning treatments during the initial exposure ofthe polymer to elevated pressures of the penetrant.21

Following such initial exposure to penetrant, settled isotherms characterized by concavity to the pressureaxis at low pressures and a tendency to approach linear high-pressure limits are observed as shown in Figs.20.4-6 and 20.3-3c. Reference to penetrant-induced conditioning effects have been made continuously sincethe very earliest investigations of penetrant-glassy polymer sorption behavior.21"23 If one "overswells" thepolymer with a highly sorbing penetrant, such as a high-activity vapor, and then removes the penetrant,the excess volume which is introduced will tend to relax quickly at first, followed by a very slow, long-term approach toward equilibrium.24

At extremely high gas conditioning pressures, substantial swelling of the polymer sample can occur,with a resultant increase in the value of VR — K,.23 After conditioning with gases such as CO2 at pressuresless than 30 atm, however, consolidation in the absence of penetrant is typically unmeasurable, sincepenetrant-induced swelling is not very extensive (less than 4-5% uptake by weight).23*24 As a result, insuch cases redistribution of the originally present intersegmental gaps may be the primary process occurringduring the first exposure of the polymer to high pressures of penetrant as shown in Fig. 20.4-6.21 At highpressures, sorption isotherms such as that shown in Fig. 20.3-3*/ are anticipated since Flory-Huggins-typebehavior typical of Fig. 2O.3-3& is superimposed on the simple dual-mode isotherm shape shown in Fig.20.3-3c. This rather complex topic is beyond the scope of the present discussion.

As noted previously, an interpretation of conditioning that occurs during the primary penetrant exposurein the absence of large swelling effects may be couched in terms of coalescence of packets of the originalintersegmental gaps. Redistribution of chain conformations, consistent with optimal accommodation of thepenetrant in the unrelaxed volume between chain segments, may permit this process during the conditioningtreatment. This rearrangement would tend to produce a more or less densified matrix with a small volumefraction of essentially uniformly distributed molecular-scale gaps or "holes" throughout the matrix. In such

T e m p e r a t u r e — >FIGURE 20.4-5 Schematic representation of the unrelaxed volume Vg — V1 in a glassy polymer. Notethat the unrelaxed volume disappears at the glass transition temperature T11.

a situation, one can appreciate the meaning of two slightly different molecular environments in the glassin which sorption of gas may occur.

Consider first the limiting case in which a highly annealed, truly equilibrium densified glass character-ized by V1 in Fig. 20.4-5 is exposed to a given pressure of a penetrant. In this hypothetical case, all gapsare missing, but clearly there still will be a certain characteristic sorption concentration (CD) typical of truemolecular dissolution in the densified glass. This sorption environment is believed to be similar to thatobserved in Iow-molecular-weight liquids or rubbers (above T1,).

Next, consider a corresponding conditioned nonequilibrium glass (illustrated by Vn in Fig. 20.4-5)containing unrelaxed volume in which the surrounding matrix has been densified more or less by thecoalescence of gaps to form molecular-scale berths for penetrant. A local equilibrium requirement leads toan average local concentration of penetrant held in the uniformly distributed molecular-scale gaps (CH) inequilibrium with the 4'dissolved" concentration (CD) at any given external penetrant pressure or activity.

This simple physical model can be described analytically up to reasonably high pressures (generally for

GLASS RUBBER

'actual glassy^ volume

volume ofdensi f ied glass

FIGURE 20.4-6 Schematic representation of the dual-mode concept. The sorption observed during the firstexposure to the penetrant is often considerably differentfrom that observed after the isotherm has "settled" atthe highest pressure of measurement. The conditioningprocess associated with the first exposure to the pene-trant is believed to be associated with redistribution ofunrelaxed volume elements in the polymer.

pressures less than or equal to the maximum conditioning pressure employed21) in terms of the sum ofHenry's Law for CD and a Langmuir isotherm for CH:

(20.4-10)

(20.4-11)

where kD is the coefficient, analogous to S in Eq. (20.1-4), that characterizes sorption of penetrant in thedensified regions that comprise most of the interchain gap in the polymer. The parameter C'H is the Langmuirsorption capacity of the glassy matrix and can be interpreted directly in terms of Fig. 20.4-5, in which theunrelaxed volume, Vg — V,t corresponds to the summation of all of the molecular-scale gaps in the glass.As shown in Fig. 20.4-7, the Langmuir capacity of glassy polymers tends to approach zero in the sameway that V8 — V1 approaches zero at the glass transition temperature. Since C'H> b, and kD all tend todecrease with increasing temperature, the sorption uptake at a given pressure tends to decrease with in-creasing temperature.

The preceding qualitative observations about the temperature dependence of CH and VR — V1 can beextended to a quantitative statement in cases for which the effective molecular volume of the penetrant inthe sorbed state can be estimated. As a first approximation, one may assume that the effective molecularvolume of a sorbed CO2 molecule is 80 A3 in the range of temperatures from 25 to 850C. This molecularvolume corresponds to an effective molar volume of 49 cm3/mol of CO2 molecules and is similar to thepartial molar volume of CO2 in various solvents, in several zeolite environments, and even as a puresubcritical liquid (see Tables 20.4-4 and 20.4-5).2I*25 The implication here is not that more than one CO2

molecule exists in each molecular-scale gap, but rather that the effective volume occupied by a CO2

molecule is roughly the same in the polymer sorbed state, in a saturated zeolite sorbed state, and even ina dissolved or liquidlike state since all these volume estimates tend to be similar for materials that are nottoo much above their critical temperatures. With the above approximation, the predictive expression givenbelow for CH can be compared to independently measured values for this parameter from sorption mea-surements:

(20.4-12)

where p* is the equivalent density of CO2 (^ mol/cm3) discussed above. The comparison of measured andpredicted C11 calculated from Eq. (20.4-12) using reported dilatometric parameters for various glassypolymers is shown in Fig. 20.4-8.2I-28-29 The correlation is clearly impressive.

Application of Eq. (20.4-12) to highly supercritical gases is somewhat ambiguous since the effectivemolecular volume of sorbed gases under these conditions is not estimated easily. A similar problem existsin a priori estimates of partial molar volumes of supercritical components even in low-molecular-weightliquids.30 The principle on which Eq. (20.4-12) is based remains valid, however, and while the total amountof unrelaxed volume may be available for a penetrant, the magnitude of CH depends strongly on howcondensable the penetrant is, since this factor determines the relative efficiency with which the componentcan use the available volume.

Dual-Mode Sorption

firstexposure

Temperature (0C)FIGURE 20.4-7 Langmuir sorption capacity C'H as a function of temperature for several polymer-penetrantsystems. Note that CH disappears near Tg in the same fashion as Vg — V1 does in Fig. 20.4-5.26

TRANSPORTA companion transport model that also acknowledges the fact that penetrant may execute diffusive jumpsinto and out of the two sorption environments expresses the local flux Af at any point in the polymer interms of a two-part contribution:32"34

(20.4-13)

where DD and DH refer to the mobility of the dissolved and Langmuir sorbed components, respectively. Itis typically found that DD is considerably larger than DH except for noncondensable gases such as he-lium.3235-36

The two transport coefficients obey Arrhenius expressions with the activation energy for DH tending tobe slightly larger than for DD.35 The above expression also can be written in terms of Fick's Law with aneffective diffusion coefficient Dcff(C) that is dependent on local concentration:

TABLE 20.4-5 Effective Molecular Size of CO2 in Various Dissolved and ZeoliticEnvironments

Effective Volume at 25 0C perEnvironment CO2 Molecule (A3)

Carbon tetrachloride 80.0Chlorobenzene 74.1Benzene 79.5Acetone 74.2Methyl acetate 73.94 or 5 A Zeolites at saturation of sorption capacity 86.9

Note: Average of above values = 78.1 A3 .Source: Barrer and Barrie22 and Chan and Paul.27

Polystyrene /C3H8PET/C6H6 'PET/CO2

Polycarbonate /CO2 Methacrylates/CO2

VPMMAPEMA

(CH ) predicted |_cm3(STP)/ cm3 polymer J

FIGURE 20.4-8 Quantitative comparison of experimentally measured values of C'H for CO2 in variouspolymers with the predictions of Eq. (20.4-12): • = polyethylene terephthalate) at 25, 35, 45, 55, 65,75, and 85°C. = poly(benzyl methacrylate) at 30°C. = poly(phenyl methaciylate) at 35, 50, and75°C. = poly(acrylonitrile) at 35, 55, and 65°C. = polycarbonate at 35, 55, and 75°C: the Oand points refer to predictions using dilatometric coefficients from different sources.29'31 = poly (ethylmethacrylate) at 30, 40, and 55°C. = poly(methyl methacrylate) at 35, 55, 80, and 1000C.

(20.4-14)

The dual-mobility model expresses the concentration dependency of Dcff(C) in terms of the localconcentration of dissolved penetrant, Q>, as shown in Eq. (20.4-15):

(20.4-15)

where F s DH/DDy K a CHb/kD, a n d a s b/kD. This model explains concentration dependency of thelocal diffusion coefficient such as that shown for CO2 in polyethylene terephthalate) (PET) in Figs. 20.3-6c and 20.4-9 in terms of a progressive increase in the concentration fraction that is present in the highermobility Henry's Law environment as the local Langmuir capacity saturates at higher pressures.

The points in Fig. 20.4-9 were evaluated from the permeability and sorption concentration data usingEq. (20.3-6), which do not depend on the dual-mode model in any way. The line through the data pointscorresponds to the predictions of £>cff(C) using Eq. (20.4-15) along with the independently determineddual-mode parameters for this system.35 It is also important to note that the form of data in this plot whichexhibit a tendency to form an asymptote at high pressures is not typical of plasticization.

Finally, if one considers the low-concentration region of Fig. 20.4-9, it is clear that the diffusioncoefficient is surprisingly concentration dependent. For example, £>cff(C) increases by more than 35% asthe local sorbed concentration rises from 0.142 cm3 (STP)/cm3 polymer (or 0.00020 wt. fraction) at 50mm Hg to 1.7 cm3 (STP)/cm3 polymer (or 0.0025 wt. fraction) at 760 mm Hg CO2 pressure. The aboveconcentrations are extremely dilute, corresponding to less than 1 CO2/1200 PET repeat units and 1 CO /̂98 repeat units, respectively.35

An even more dramatic case corresponds to CO2 in PVC in which Dcfr(C) rises by 86% as the localconcentrationjncreases from 1 CO2/4400 repeat units at 100 mm Hg to 1 CO2/1300 repeat units at 500mm Hg at 400C. The above values of CO2 solubility are based on measurements by Toi and by Tikhomorovand coworkers, which are in good agreement.38-39

At such extraordinarily low penetrant concentrations, plasticization of the overall matrix is not antici-pated. As noted by Michaels in the previous discussion concerning crystallinity, motions involving relativelyfew repeat units are believed to give rise to most short-term glassy-state properties such as diffusion. Inrubbery polymers, on the other hand, longer-chain concerted motions occur over relatively short timescales, and one expects plasticization to be easier to induce in these materials. Interestingly, no known

(C' H)

exp

erim

enta

l Lc

m3(S

TP) /

cm

3 pol

ymer

j

C, Concentration in polymer (cm3 (STP)/cm3 polymer)FIGURE 20.4-9 Local concentration-dependent diffusion coefficient for CO2 in poly(ethylene terephthal-ate).

transport studies in rubbers have indicated plasticization at the low sorption levels noted above for PVCand PET.

The discussion directly following Eq. (20.4-15) provides a simple, physically reasonable explanationfor the preceding observations of the marked concentration dependence of £>eff(C) at relatively low con-centrations. Clearly, at some point, the assumption of concentration independence of D0 and DH in Eq.(20.4-15) will fail; however, for work with most "conditioned" glassy polymers at CO2 pressures below300 psi, these plasticization effects are small. Even at a CO2 pressure of 10 atm, there still will be lessthan 1 CO2 molecule/20 PET repeat units at 35 0C.

In the case of cellulose acetate, however, it does appear that breakdown of the assumption of constantvalues of DH and DD occurs. Presumably, this hypersensitivity of cellulose acetate may derive from itsapparent special affinity for CO2 discussed in the context of Table 20.3-1. Stern and Saxena have describeda generalized form of the dual-mode transport model that permits handling situations in which nonconstancyof DD and DH manifest themselves.l9 It is reasonable to assume that the next generation of gas separationmembrane polymers will be even more resistant to plasticization than polysulfone and cellulose acetate, sothat the assumption of constancy of these transport parameters should be well justified.

Although not necessary in terms of phenomenological applications, it is interesting to consider possiblemolecular meanings of the coefficients DD and DH. If two penetrants exist in a polymer in the two respectivemodes designated by D and H to indicate the "dissolved" (Henry's Law) and the "hole" (Langmuir)environments, the molecules can execute diffusive movements within their respective modes, or they mayexecute intermode jumps.

D D dissolved to dissolved jump

H dissolved to hole jump

H H hole to hole jump

D hole to dissolved jump

Clearly, the true character (activation energy, entropy, and jump length) of the phenomenologicalIyobserved DD will be a weighted average of the relative frequency of D -> D and D -> H jumps, andlikewise for DH in terms of H -+ H and H -+ D jumps. Given the relatively dilute overall volume fractionassociated with the nonequilibrium gaps that comprise the H environment (less than 4-5% on a volumebasis), one may as a first approximation assume that most diffusive jumps of a penetrant from a D envi-ronment result in movement to another D environment, and most diffusive jumps from H environmentsresult in movement to a D environment. The observed activation energies, entropies, and jump lengthstherefore have fairly well defined meanings on a molecular scale.

The appropriate equation derived from the dual-mode sorption and transport models for the steady-statepermeability of a pure component in a glassy polymer is given by Eq. (20.4-16) when the downstream

FIGURE 20.4-10 Permeability of various gases in polycarbonate at 350C as a ftinction of the linearizingvariable, 1/[1 + bp], in Eq. (20.4-16).32 Similar good linear fits are observed for many other systems.

receiving pressure is effectively zero and the upstream driving pressure is p.35

(20.4-16)

The first term in Eq. (20.4-16) describes transport through the Henry's Law environment, while thesecond term is related to the Langmuir environment. The tendency shown in Fig. 30.3-2c for the perme-ability to approach a limiting value equal to kDDD at high pressures is clear from this equation. Physically,the asymptote is approached because after saturation of the upstream Langmuir capacity at high pressures,additional pressure increases result in additional flux contributions only from the term related to Henry'sLaw, which continues to increase as upstream pressure increases.

The remarkable efficacy of the dual-mode sorption and transport model for description of pure com-ponent data has been illustrated by plots of the linearized forms of Eq. (20.4-16) for a wide number ofpolymer-penetrant systems.21 •23<29-34-37-40 4I Typical examples of such data are shown in Fig. 20.4-10 forvarious gases in polycarbonate.32 These linearized plots are stringent tests of the ability of the proposedfunctional forms to describe the phenomenological data. Assink also has investigated the dual-mode modelusing a pulsed NMR technique and concluded the following:

We have been able to demonstrate the basic validity of the assumptions on which the dual modemodel is based and we have shown the usefulness of NMR relaxation techniques in the study ofthis model.42

MIXED COMPONENT SORPTION A N D TRANSPORTArguments similar to those presented above for pure components have been extended to generalize theexpressions given in Eq. (20.4-11) and Eq. (20.4-16) to account for the case of mixed penetrants.43'44 Theappropriate expressions are given below:

(2.4-17)

(20.4-18)

Per

mea

bilit

y x1

0,

(Bar

ters

)

He CO2

Ar CH4 N2

In the above expressions, A refers to the component of primary interest while B refers to a second,"competing" component.

As noted in the context of Eq. (20.3-8), the permeability of a polymer to a penetrant depends on themultiplicative contribution of a solubility and a mobility term. These two factors may be functions of localpenetrant concentration in the general case as indicated by the dual-mode model. Robeson has presenteddata for CO2 permeation in polycarbonate in which both solubility and diffusivity are reduced as a resultof antiplasticization caused by the presence of the strongly interacting 4,4'-dichloro diphenyl sulfone.45 Onthe other hand, sorption of a less-interacting penetrant, such as a hydrocarbon, may affect primarily onlythe solubility factor, without significantly changing the inherent mobility of the penetrant in either of thetwo modes. Flux reduction in this latter context occurs simply because the concentration driving force ofpenetrant A is reduced. This results from the exclusion of A by component B from Langmuir sorption sitesthat previously were available to penetrant A in the absence of penetrant B.

Consistent with the preceding discussion concerning sorption and flux reductions by relatively nonin-teracting penetrants, the data shown in Fig. 20.4-11 clearly illustrate the progressive exclusion of CO2

from Langmuir sorption sites in poiy(methyi methacrylate) (PMMA) as ethylene partial pressure (pB) isincreased in the presence of an essentially constant CO2 partial pressure ofp* - 5 O 5 T ° 1 3 atm.46 Thetendency of the CO2 sorption shown in Fig. 20.4-11 to decrease monotonically with ethylene pressureprovides impressive support for the "competition" concept on which Eqs. (20.4-17) and (20.4-18) arebased. Permeation data are not available for this system to determine if changes in the values of DD andDH occur in the mixed gas situation. If offsetting increases in these transport coefficients do not occur inthe presence of ethylene, the CO2 permeability will be depressed in the mixed gas permeation situations.

The data shown in Fig. 20.4-12 illustrate the reduction in permeability of polycarbonate to CO2 causedby competition between isopentane and CO2 for Langmuir sorption sites.47 The flux depression shown inFig. 20.4-12 was found to be reversible, with the permeability returning to the pure CO2 level after sufficientevacuation of the isopentane-contaminated membrane. Even at the low isopentane partial pressure consid-ered (117 mm Hg), the tendency is clear for the CO2 permeability to be depressed from its pure componentlevel toward the limiting value (PA = kDADDA) corresponding to complete exclusion of CO2 from theLangmuir environment. This indicates, for example, that the effective CO2 permeability at a CO2 partialpressure of 2 atm could be reduced by almost 40% as a result of small amounts of such hydrocarbons inthe feed streams. Further increases in the isopentane partial pressure in the feed eventually should completethe depression of CO2 permeability to its limiting value of 4.57 barrers and ultimately produce plasticizingeffects such as indicated in Fig. 20.4-4 at the higher isopentane levels.

The lines drawn through the data in Fig. 20.4-12 were calculated from Eqs. (20.4-16) and (20.4-18)for the pure and mixed penetrant feed situations, respectively, using the same CO2 model parameters inboth cases. The affinity constant of isopentane was estimated to be 13.8 atm"1.47 The excellent fit of thedata suggests that DD and DH for CO2 in the mixed feed case are not affected measurably by the presenceof the relatively noninteracting isopentane.

One can see that if the upstream pressure in Eq. (20.4-16) is sufficiently large for the pure component,the permeability will tend to form an asymptote to the value of kDDD. In such a case, further increases inthe denominator of Eq. (20.4-18) for the mixed gase case, because of contributions from the secondcomponent bBpB term, will not depress the magnitude of PA below the pure component kDDD value. Thistendency to form an asymptote is clear in Fig. 20.4-12 for CO2 in polycarbonate and explains why the

FIGURE 20.4-11 Demonstration of the depression ofCO2 sorption below the pure component CO2 sorptionlevel as a result of competition of C2H4 for unrelaxedvolume sorption sites in the polymer at increasing C2H4

partial pressures.46

35°CPMMA

PCO2 "" 5 0 5 ± 0.13 Otm

PURE CO2'

Depression ofCO2 Sorptiondue to C2H4Competition forUnrelaxed

Volume inPolymer

Pc2H4(atm>

C0Q

2 [jc

m3 (S

TP)/c

m3 p

olym

er J

Feed Pressure (atm)FIGURE 20.4-12 Demonstration of the flux-depressing effects of a small (117 mm Hg) partial pressureof isopentane in the upstream feedstream for CO2 permeating through polycarbonate at 35 0C.47

competition effect is more apparent at low values of CO2 partial pressure, where the permeability is stillconsiderably greater than kDDD and has "room" to be depressed. The magnitude of the competitionphenomenon in a given application appears to be dependent largely upon the fraction of the unrelaxedvolume (Vg - V1)IVx in Eq. (20.4-12) that a pure component occupies prior to introduction of the secondcompeting component.

An additional factor, concerning the effectiveness of the second component in competing for the un-relaxed volume, is determined largely by the condensability of the second component. A highly condensablematerial with a large affinity constant, such as water, is effective in saturation of the microvoid phase at amuch lower partial pressure than a supercritical, relatively noncondensable component. Because of theirhighly condensable nature, such components tend to sorb to substantial extents at low partial pressures inthe vapor phase. Therefore, at some point, such materials will begin to act as a flux enhancer rather thana flux depressor. This effect is complex and clearly dependent on polymer-penetrant interactions as wellas on simple volume filling, as is the case in the simple dual-mode regime of Fig. 20.4-4. The correlationshown in Fig. 20.4-13 suggests that components below their critical temperatures, that is, vapors, havemarkedly larger affinities for the microvoid environments in the polymer than corresponding supercriticalgaseous components.46 Carbon dioxide, which has a critical temperature of 303 K, is only slightly super-critical (777; = 1.01) at the 308 K measurement temperature shown in Fig. 20.4-13.

On the basis of the preceding discussion, fairly high system pressures are required to observe competitioneffects for highly supercritical components such as H2 and CH4. Pure component permeabilities for suchpenetrants also appear to be essentially independent of pressure because of the small value of the productbp appearing in the denominator of Eq. (20.4-16) for these low-affinity penetrants. The addition of a smallpartial pressure of water to the feed, however, may cause dramatic reductions in the accessibility of theunrelaxed pathways to both H2 and CH4 because of the large value of bHlO.

This finding is consistent with the earlier cited quote by Pye et al.18 It can be understood from Eq.(20.4-18), in which the second term effectively is suppressed, causing the permeability to approach thelimiting value of kDDD. If the polymer is water sensitive, of course, plasticization may occur as the relativehumidity is increased further, thereby causing increases in the value of DD with a resulting upswing in theobserved permeability as shown in Fig. 20.4-4.l9 Data for the series of superselective high-flux aromaticpolyimides studied by Pye and coworkers are shown in Table 20.4-6.l8 The table compares "dry" (0%RH) behavior to "wet" behavior (exposed to only 15 mm Hg of water vapor at 300C [50% RH]).

As illustrated in this table, exclusion of both H2 and CH4 by the water vapor produces a dramatic (16-59%) reduction in the flux of both components. Only a moderate reduction (3-9%) in selectivity wasobserved, since the dissolved-mode contribution to the permeability (kDiDDi) of each component stillexperiences the influence of the rigid size- and shape-selecting matrix. The Du Pont workers demonstrateda similar, but less dramatic, effect with H-C5, consistent with the lower affinity constant of pentane comparedto water (see Fig. 20.4-13). In both the water and W-C5 cases the effects were reversible after protracteddegassing followed by a return to contaminant-free feed.

A similar study of the effects of relative humidity on the CO2 permeability of Kapton® polyimideshowed only a 5% reduction compared to the pure component value at 250 psia. This finding is consistentwith the previous discussion, since the pure component CO2 permeability shown in Fig. 20.4-14 is ap-proaching its limiting value of kDDD at a CO2 pressure of 250 psia. This rapid saturation arises as a result

Polycarbonate35 0C

Barrers

PCO2

(Ba

rre

rs)

Pure CO2

Critical Temperature (0K)FIGURE 20.4-13 Relationship between observed affinity constants b for a series of penetrants in a varietyof polymers at 308 K. Note that the penetrants that are substantially subcritical (308/7;. < 1.0) tend tohave markedly larger affinity constants because of their higher condensability.48

of the high affinity constant of CO2 compared to the supercritical CH4 and H2 (see Fig. 20.4-13). Intro-duction of the high-affinity water simply completes the small remaining depression of the permeability ofthe limiting value of kDDD shown in Fig. 20.4-14. Hence, the observed flux depressions are rather undra-matic compared to the H2 and CH4 cases discussed previously.

NONIDEAL GAS-PHASE EFFECTSManifestations of external nonideal gas-phase effects on the apparent permeabilities and selectivities for aKapton® membrane are given in Tables 20.4-7 and 20.4-8 for the CO2-CH4 system at 600C. A CO2

TABLE 20.4-6 Permeabilities' and Permeability Rates at 300C for Various Stiff-Chain Polyimides

P H i Reduction />CH4 Reduction PHJPCH* Reduction: D u e to H 2 O D u e to H 2 O : Due to H 2 O

Polymer Dry* Wet' Contaminant Dry Wet Contaminant Dry Wet Contaminant

A d 3557 2 8 1 7 2 1 % 13.2 11.1 16% 270 254 6%B' 7637 6160 19% 68.2 57.0 16% 112 108 3%Cf 3168 1273 59% 7.5 3.3 56% 422 386 9%

"P = 10"12 cm3 (STP)-cm/cm2-s-cmHg.h Dry—The situation in which the feed is truly free of water.rWet—The situation in which the feed contains 15 mm Hg which corresponds roughly to 50% RH.dA—Prepared by reaction of 4,4'-hexafluoroisopropylidenediphthalic anhydride with 1,3-diamino benzene.'B—Prepared by reaction of 4,4'-hexafluoroisopropylidenediphthalic anhydride with 1,5-diaminonaphthal-ene.fC—Prepared by reaction of 4,4'-hexafluoroisopropylidenediphthalic anhydride with 3,5-diamino benzoicacid.

Lang

rnui

r A

ffin

ity

Con

stan

t, b

(atm

"1) Polymer

Ethyl CellulosePolysulfonePCPPOKapton*PMMAPEMAPSPANPET

FIGURE 20.4-14 Demonstration of the flux-depressing effect of low relative humidities on the CO2

permeability of Kapton® polyimide.48 Paulson et al.20 has shown that at high relative humidities, theobserved flux of CO2 in a number of polymers is higher than the pure component level. At much higherrelative humidities for Kapton®, it may be that this effect also will be observed, consistent with the right-hand side of Fig. 20.4-4.

pressure of 180 psia and a CH4 pressure of 380 psia were used in both the pure and mixed gas situations.48

The nonconventional permeabilities P* given in Table 20.4-7 are calculated by normalizing the observedsteady-state component fluxes with the actual thermodynamic fugacity driving force propelling transportacross the membrane rather than with the component partial pressure differences indicated in Eq. (20.1-3).The true driving force for permeation, of course, is the fugacity difference rather than the partial pressuredifference that usually is used for convenience. The component fugacities in this case were calculated usingthe Soave modification of the Redlich-Kwong equation.

Comparison of the differences between pure and mixed component cases for the permeabilities of eachcomponent calculated in the standard fashion and in the "thermodynamically normalized" fashion (P*) isrevealing. The difference between the pure and mixed gas cases for the P columns of each componentcorresponds to the apparent total depression in flux resulting from both true dual-mode competition effectsand nonideal gas-phase effects. The differences between the pure and mixed gas cases for the P* columnsare free of complications arising from nonideal gas-phase effects. Therefore, the differences between thesecolumns are manifestations of the rather small competition effect due to dual-mode sorption under theseconditions.

These two effects are summarized in Table 20.4-8. Moreover, the percentage reduction in flux for eachcomponent due to nonideal gas-phase effects is listed in Table 20.4-8. It is clear that the reduction infugacity driving force is very small (0.5%) for methane, which tends to be an ideal gas under theseconditions. For CO2, however, the effects amount to roughtly 7.5% and account for a roughly 6.9%reduction in the ideal separation factor. To reemphasize, these effects would be observed even if thepolymer-phase sorption and transport behavior did not show dual-mode effects and were perfectly ideal.

TABLE 20.4-7 Magnitude of the Nonideal Gas-Phase Effects for Both Pure and Mixed GasSituations with Kapton® at 600C for the CO2-CH4 System

Actual Mixed Gas Case (32.2% CO2-Pure Component Data 67.8% CH4)

CO2 Permeability CH4 PermeabilityCO2 Permeability CH4 Permeability at 180 psia CO2 at 380 psia CH4

at 180 psia CO2 at 380 psia CH4 Partial Pressure Partial PressurePressure (Barrers) Pressure (Barrers) (Barrers) (Bamers)

Standard pressure-based 0.382 0.00996 0.342 0.0097normalization P1

Fugacity-based 0.400 0.0102 0.388 0.0100normalization P?

Pur« CO2600CKapton

pco2

(Bar

rers

)

Standard pressure-basednormalization P1

Fugacity-basednormalization P?

Pure Component Data

CO2 Permeabilityat 180 psia CO2

Pressure (Barrers)

0.382

0.400

CH4 Permeabilityat 380 psia CH4

Pressure (Barrers)

0.00996

0.0102

Actual Mixed Gas Case (32.2% CO2-67.8% CH4)

CO2 Permeabilityat 180 psia CO2

Partial Pressure(Barrers)

0.342

0.388

CH4 Permeabilityat 380 psia CH4

Partial Pressure(Barrers)

0.0097

0.0100

TABLE 20.4-8 Magnitude of Apparent Permeability Reduction in the Mixtures of CO2 and CH4 in a Plasticization-Resistant Polymer, Kapton®, Under Conditions Discussed in TABLE 20.4-6

Reduction (%)in Ideal

SeparationFactor Owing

to FugacityEffects

F

Reduction (%)in Ideal

SeparationFactor Owingto Dual-Mode

EffectsE

Total Reduction (%)in Ideal SeparationFactor in MixtureCompared to Pure

Gas CaseD

Reduction (%) inPermeability

Owing toFugacity Effects

C

Reduction (%) inPermeability

Owing to Dual-Mode Effects

B

Total Reduction (%)in Permeability inMixture Comparedto Pure Gas Case

AComponent

6.917.97.50.6

32

10.52.6

CO2CH4

Note: Column A = Column B + C and Column D = Column E + F. The reductions (%) in the table refer to changes relative tothe values tabulated under the pure component permeability columns in Table 20.4-6. Separation factor reductions refer to changesrelative to the pure component permeability ratios of CO2 and CH4 tabulated in the pure component columns in Table 20.4-6. Asnoted earlier, for plasticization-prone polymers such as cellulose acetate, massive flux increases can be observed in the presence ofa strongly interacting gas such as CO2 in mixed gas situations. Such plasticization is generally undesirable because it produces lossesin selectivity.

These fugacity effects simply correspond to gas-phase driving force reductions resulting from nonideal gas-phase behavior. Such effects clearly will become more substantial as higher total system pressures areconsidered in CO2-containing systems. Conversely, such effects will be relatively unimportant for mixturescontaining only supercritical, relatively ideal components.

Fortunately, these phenomena can be predicted readily by using standard thermodynamic equations ofstate to calculate the fugacity of each component in the upstream and downstream gas phases. For plasti-cization-prone polymers such as cellulose acetate, the depression of CO2 fugacity in high CH4 pressuresituations may suppress large upswings in permeability noticed for pure CO2 (as in Fig. 20.3-9). The areaof nonideal gas-phase effects clearly requires considerably more careful investigation.

20.5 CHARACTERIZATION OF ASYMMETRIC MEMBRANES

Although models and techniques such as those described in preceding sections permit characterization ofdense-film transport properties, a need exists for improved characterization of the transport behavior ofasymmetric membrane properties. Currently, performance tests such as gas permeability and selectivityusing standardized feed streams provide useful tools for quality control but are ambiguous for fundamentalcharacterizations of the type possible for dense membranes. The following discussion presents informationpertinent to the characterization of porosity and morphology of asymmetric membranes and to analysis oftheir flux decline as a function of time in service. These are two of the most important additional charac-teristics that asymmetric structures introduce into the description of transport behavior of the separatormodule.

20.5-1 Porosity and Morphology Characterizations Void Fraction Determinations

Estimates of the overall void volume in an asymmetric membrane can be made by equilibrating a knowndry weight of sample, w,, in a suitable liquid which will penetrate into the membrane pores but will notswell the polymer substrate. After centrifugation to remove superficial fluid and fluid held in the bore ofhollow-fiber membranes, determination of the weight gain of the sample provides an estimate of the voidfraction present in the membrane using Eq. (20.5-1):'

Void volume (%) (20.5-1)

where p, refers to the density of the dry polymer and w2 and p2, respectively, refer to the weight anddensity of the liquid imbibed by capillary forces into the membrane pores.

This technique clearly assumes that the polymer is not swollen appreciably by the liquid. Care shouldbe exercised to verify this fact using a dense thin-film sample of the polymer immersed in the candidateliquid. In addition, the apparent value of w2, determined roughly as the difference between the centrifugedsample weight and the dry weight, should be corrected to account for actual molecular sorption into thepolymer mass. This sorption, which occurs in addition to the capillary uptake, can produce an overesti-mation of the void volume percent in the fiber by as much as 5% if unaccounted for. The simplest meansof accounting for such sorption involves adjustment of the value of w2 using sorption data for the liquid inthe dense film for which capillary effects are not present.

SCANNING ELECTRON MICROSCOPYScanning electron microscopy (SEM) characterizations of the surface of asymmetric membranes and supportstructure morphologies are invaluable for general qualitative monitoring of changes in the membrane struc-ture arising from changes in casting or spinning conditions. As a means of characterizing surface jx>rosity,however, the technique is rather poor, since gas molecules can "see" pores as small as 5-10 A, whichare on the borderline of resolution of most SEM equipment. Moreover, SEM often suffers from the problemthat variations along the membrane may be sufficiently great to make local characterizations of propertiesinsufficient to provide a representative picture of the overall membrane nature.

As a result of these shortcomings, porosity characterizations generally involve both gas permeabilitymeasurements to provide estimates of the average pore size and liquid displacement to measure maximumpore size in the membrane surface. The techniques rely on a number of assumptions for their results tohave physical meaning. Depending on the structure of the asymmetric membrane, some of the assumptionsmay be satisfied only marginally. In such a case, the characterizations are primarily useful as empiricalindices for comparison of different samples, and the fundamental meaning of the numbers derived fromsuch analyses is questionable.

AVERAGE PORE SIZE DETERMINATIONCharacterization of the average pore size in a porous solid can be performed using a technique describedby Yasuda and Tsai.2 The expression for the steady-state mass or molar flux of a gas through a porous

solid of thickness / in the presence of a gas pressure difference, Ap = p2 — P\, between the upstream anddownstream points is given by Eq. (20.5-2):2 3

Flux (20.5-2)

where K is the permeability coefficient for the gas.When the porosity properties of the medium are reasonably uniform, the permeability coefficient cal-

culated using observed fluxes by rearrangement of Eq. (20.5-2) has a well-defined meaning. For mediawith fine pores, slip flow in which gas molecules experience wall interactions frequently as well as gas-gas viscous interactions must be considered. In such cases, the observed permeability coefficient calculatedfrom Eq. (20.5-2) depends on the average pressure [p' = (/?, + /?2)/2] and may be represented by Eq.(20.5-3):3

(20.5-3)

where K0 is the Knudsen permeability, B0 is a term characteristic of the solid porosity, and t\ is the gasviscosity.

Clearly, plots of K versus/?' permit evaluation of K0 and B0 in such cases. Moreover, Yasuda and Tsaihave shown that the average pore size mx may be calculated as shown in Eq. (20.5-4):

(20.5-4)

where R is the gas constant, T is the absolute temperature, and M is the probe gas molecular weight.Cabasso and coworkers have noted that when nitrogen is used as the probe at 25 0C, a measure of the"effective porosity" can be derived as given below:1

(20.5-5)

The parameter e is the actual porosity and q is a measure of the tortuosity of the pores for complex porestructures. For the case of straight cylindrical pores, q is simply equal to unity.

Clearly, although one can obtain parameter values from such analyses in the case of nonhomogeneousporous materials, the parameters of K0, B0, and e/q1 have complex physical meanings that are averages ofall the local values extending from the dense skin through the support structure. While the above problemexists in a general sense, if the support structure is known to be sufficiently open to provide an order ofmagnitude or less resistance to flow than the dense surface layer, the parameters discussed above can begiven physical meaning in terms of properties of the skin.

In particular, if the pores in the surface are modeled reasonably as cylinders, the average pore diameterand fraction of surface area that is porous can be obtained from mx and e/^2, respectively. Clearly, iftortuosity is significant, the above technique will underestimate the actual surface porosity, and one willobtain an effective number of such idealized straight pores. Moreover, in cases where a gradation in denseskin porosity is apparent, one would again obtain a gross average of the pores in this thin dense layer.Nevertheless, the technique is useful for characterizing the average properties of a representative bundleof fibers or quantity of flat asymmetric membranes. The detailed procedures for carrying out the experi-ments, including designs for test cells, are discussed by Yasuda and Tsai2 and Cabasso et al.1

MAXIMUM PORE SIZE DETERMINATIONDetermination of the maximum pore size in a porous solid can be performed using the Kelvin equation:1

(20.5-6)

where 7 is the surface tension of a liquid filling the pores of the solid, and p* is the pressure that must beexerted to overcome the surface tension holding the liquid in a pore of diameter me.

As indicated by Eq. (20.5-6), if there is a distribution of pore sizes, the largest pore will be the firstto surrender its imbibed liquid as a steadily increasing gas pressure is applied to one side of the pores. Thefiber bundle or flat membrane first is allowed to equilibrate in a nonswelling liquid that wets the pores ofthe polymer, and then one applies a gradually increasing gas pressure to the porous side of the membraneor the bores of a fiber bundle. The pressure corresponding to the appearance of the first significant number

TABLE 20.5-1 Typical Uncoated Porous Polysulfone Membrane Properties

CalculatedTotal Wall Mean Maximum Porosity

Void Volume Thickness Pore Size Pore Size Parameter"Sample Number % (^m) 0*m) (/xm) (e/q1) x 102

1 75.6 143 0.038 0.09 4.62 74.2 116 0.040 0.11 1.53 70.6 100 0.040 0.08 1.14 73.2 90 0.040 0.11 2.8

"For straight pores, the tortuosity weighting parameter (^2) is unity, and e/q* provides an estimate of theequivalent area fraction of pores (e) in the membranes with a length equal to the membrane thickness. Fortortuous pores, it is not generally possible to separate decreases in porosity (e) from increases in tortuosity{q)y so that the composite parameter given in this table is reported.

Source: Cabasso et al.1

of bubbles on the membrane surface marks the value of p * and permits calculation of the maximum poresize from Eq. (20.5-6). In cases where the pores are not reasonably cylindrical or where the imbibed liquidswells the membrane surface appreciably, the measurement is compromised as a fundamental character-ization tool. As in the previous case for average pore size, even if the parameter estimated is not perfectlymeaningful as a well-defined physical property, it is useful for quality control and for comparisons amongtest lots of different membranes.

Typical data for asymmetric fibers for reverse osmosis applications are reported in Table 20.5-1.1 Theranges of these variables for as-spun and posttreated cellulose acetate and polysulfone membranes currentlyused in gas separation are proprietary. Nevertheless, the surface porosity for such membranes is undoubtedlylower than for those described in Table 20.5-1, since, as indicated in Table 20.1-2, in their posttreatedforms such membranes have selectivities approaching the values of dense films. Porosities as high as thoseshown in Table 20.5-1 would produce unacceptably low selectivities as a result of nondiscriminant poreflow.

20.5-2 Permeability Creep Characterization and Analysis

Time-dependent drifts in permeability, such as those shown in Fig. 20.5-1 for cellulose acetate, are im-portant practical features of asymmetric fibers that generally result in a loss in productivity over a periodof time.4 When fouling is known not to be a factor, the so-called permeability creep phenomenon is believedto arise from one or a combination of the following factors:

1. Compaction of the open-cell support structure to a point where the previously negligible resistanceto flow of permeated product through the support is no longer insignificant.

2. True glassy-state drifts in the dense selective membrane's properties resulting from a slow volu-metric consolidation or densification of this layer of the membrane.

CO2

PERM

EATI

ON

EFFI

CIEN

CY(P

/Po)

x 10

0%

t(doys)FIGURE 20.5-1 Demonstration of typical permeability creep behavior observed for asymmetric mem-branes. Both cellulose acetate and polysulfone display this behavior, which is believed to be related todensification of the dense separating layer or compaction of the support foam structure shown in Fig. 20.1-6.4

Cellulose AcetateAsymmetric Membrane

t (days)FIGURE 20.5-2 Demonstration of the usefulness of a log-log plot of the permeability creep data in Fig.20.5-1 to predict behavior over extended periods based on relatively short time behavior (20-60 days afterstart-up).

Plots of the product permeability versus time on a log-log coordinate system are often linear overrelatively long time periods, as shown in Fig. 20.5-2.4'5 Similar behavior is observed in asymmetric reverseosmosis membranes. The log-log plotting approach provides a simple and reasonably satisfactory meansof predicting the performance change of libers under long-term operation by extrapolation of short-termdata. Mechanical creep and volume recovery in glassy polymers after an initial perturbation also are knownto be reasonably represented on such log-log plots.6

Since the two explanations for permeability drifts are related, respectively, to creep and volume recoveryfollowing initial membrane fabrication, it is not surprising that the log-log permeability plots are reasonablylinear also. Both creep and volume recovery behavior are expected to be functions of temperature and thecomposition of the pressurizing medium, since a strongly sorbing permeant such as CO2 may affect boththe dense layer and the support structure properties at high pressures. Unfortunately, definitive data topermit comparisons of flux declines in the presence of CO2 and inert atmospheres are not available atpresent.

If the permeability creep occurs primarily as a result of support compaction, one may question thevalidity of the simple model generally offered for an asymmetric membrane. Specifically, the assumptionthat a dense surface layer or a composite surface layer constitutes the only significant resistance to flow ina dense-skinned asymmetric membrane underpressure may be incorrect. Instead, upon loading, the effectivevalue of K in Eq. (20.5-2) for the porous substructure may decrease significantly as a result of constrictionof flow paths and then slowly continue to decrease as time passes.

Evidence for clarifying whether the observed productivity declines are related to the support compactioneffect could be obtained by considering the membrane selectivity as a function of time during the perme-ability creep process. Compression of the support would cause the generation of a pressure drop throughthe porous support in order to move permeated material through the compressed porous structure. Dependingon the magnitude of the effect, a significant surface concentration of the fast gas could result at the interfacejust below the dense film at the porous support. This effect would produce a reduction in the concentrationdifference driving diffusion across the dense membrane and lead to a reduction in flux of the fast gas.Moreover, since the slow gas permeates less rapidly, restrictions to its removal from the downstream faceof the dense surface layer should be relatively less serious in the face of support porosity losses, and lesspolarization would be anticipated. Thus, the depression in the flux of the fast gas, coupled with a smallerdepression in the slow-gas flux upon compaction of the support foam, would be characterized by a reductionin selectivity as well as a drop in desired product permeability.

On the contrary, if the permeability creep of the fast gas is associated with the second factor introducedabove, that is, a true glassy-state consolidation of the dense selecting membrane, a different selectivityresponse is anticipated. Although permeation of both species will be restricted by the consolidation, thebulkier slow gas will tend to be retarded more because of its greater relative difficulty in moving throughthe more restrictive densified glass compared to the more streamlined fast gas. In this event, although onewould still expect a drop in fast-gas permeability, a corresponding rise in selectivity also would be antic-ipated.

Of course, these two contributions to permeability creep may occur simultaneously. To aid in deter-mining the relative magnitudes of the two contributions, it is wise to consider the time-dependent behaviorof totally dense membranes with a posttreatment history similar to that of the asymmetric membrane. Thisapproach, of course, eliminates the effects of the porous substructure. For example, the time dependenceof dense-membrane permeabilities following a conditioning treatment has been observed. As shown in Fig.20.5-3, the permeability of polycarbonate to pure methane is a decreasing function of time after exposureof both sides of the membrane to 300 psia of CO2 pressure for 24 h at 35°C. The slow relaxation inpermeability shown in Fig. 20.5-3 was still underway after 15 days, although it had slowed noticeablyfrom its original rate. The dissolved-mode concentration CD of CO2 in polycarbonate under the 300 psiaconditions amounts to almost 3% by weight. Rapid removal of the CO2 presumably is followed by a time-

CO

2 [P

ZP

Q)

K

100%

Cellulose Acetote Asymmetric Membrone

Time(X10"3min)FIGURE 20.5-3 Demonstration of time-dependent response of pure component CH4 permeation after priorexposure to uniform CO2 conditioning for 1440 min, followed by prolonged degassing (3780 min) of the5 mil film. Within 200 min of degassing, the CO2 had been removed effectively completely (less than0.01 % of the originally present CO2 remained). The relaxation is related apparently to redistribution ofpolymer segments in the absence of CO2.

dependent recovery of the slightly swollen polymer to its unswollen, but still nonequilibrium, state. In theabsence of CO2 this relaxing excess volume can serve as extra sorption and transport pathways for CH4.

A similar type of relaxation might be anticipated for a membrane which had been cast from a solventand the solvent quickly removed. Heat treatment at a slightly higher temperature would be expected tospeed the relaxation process considerably. In the context of the dual-mode model, one might explain sucheffects, and even some of the longer term drifts in permeability, in terms of steady reductions and/orredistributions of the excess volume in the glass, Vg — V1. Of course, data such as those in Fig. 20.5-3 donot prove that all or even most of the permeability creep behavior is associated with the dense layer. Noselectivity data are available for the above system to verify if the selectivity of the membrane rises as thestructure "tightened" during the relaxation of the excess volume. The subject of permeability creep requiresconsiderably more work to permit an unambiguous resolution of its origin and prediction of its magnitudein practical situations involving mixtures of penetrants.

20.6 MODELING AND DESIGN CONSIDERATIONS

20.6-1 General Discussion

As emphasized in the preceding sections, the successful application of membranes for separation purposesdepends primarily on discovery of economically competitive membranes with high permselectivities andpermeabilities. However, the ensuing considerations, such as membrane configurations and flow patternsof the feed and the permeant streams, are also very important in determining the performance of the finalseparator system.

The three major membrane configurations, flat sheet, spiral wound, and hollow fiber, each havingadvantages and limitations, will be reviewed briefly prior to considering the detailed analysis of thesedevices. Some of these issues include relative magnitudes of active membrane area per unit separatorvolume, minimization of pressure buildup in the permeant stream, membrane integrity, and the ease ofmodule manufacture and membrane replacement.

The permeability and selectivity of an asymmetric membrane are determined by its complex morphology

Polycarbonate/ 35 0C

Degassed3780 min after

' 1440 min x

exposure to CO,

?CH

A

(Ba

rre

rs)

Next Page