- Email: [email protected]
Plasticization effects on the gas permeability and permselectivity of polymer membranes
Journal of Membrane Science, 75 (1992) 47-59 Elsevier Science Publishers B.V., Amsterdam
47
Plasticization effects on the gas permeability and permselectivity of polymer membranes J.H. Petropoulos Physical Chemistry Institute, Democritos National Research Centre, 153 10 Aghia Paraskevi, Athens (Greece) (Received February 15,1992; accepted in revised form July 21,1992)
Abstract Plasticization effects may substantially modify the performance of polymer membranes used for gas separation in practice. The present paper is concerned with the development of simplified but physically meaningful formulations to promote better understanding of these effects (particularly below T,), on one hand, and to facilitate systematic analysis of pertinent data, on the other hand. Keywords: gas separation; plasticization;
theory
tical application. The present paper constitutes an attempt to advance along these lines.
Introduction Plasticization effects on vapor transport in polymeric media have been discussed extensively (e.g. [l-3 ] ) but are hardly touched upon in most treatments on gas sorption and diffusion (e.g. [ 1,4-61). This is, of course, justified in the case of simple or low-pressure gaseous penetrants. However, plasticization phenomena become increasingly significant for heavier gases and higher pressures (e.g. [ 7-131) and are, thus, import in practical gas separation. It is, therefore, of considerable practical as well as theoretical interest to develop appropriate theoretical treatments (e.g. [ 14,151). It is of no less importance, however, that such treatments combine, as far as possible, a sound physical basis with simplicity conducive to useful pracCorrespondence to: J.H. Petropoulos, Physical Chemistry Institute, Democritos National Research Centre, 153 10 Aghia Paraskevi, Athens, Greece.
0376-7388/92/$05.00
Formulation
of pure penetrant
transport
As discussed in Refs. [6,7], the unidimensional (molar) flux density JA of a micromolecular penetrant A at a position 3c(O
J A=-DTASA(dfA/dX)'-PA(dfA/dX)
(1)
where &A, SA= CA/fA and PAlDTASA are the relevant “thermodynamic” diffusion, solubility (sorption) and permeability coefficients; f.& is the gas-phase fugacity of the penetrant, which would be at equilibrium with the molar concentration CAof the penetrant in the membrane at position X. For sufficiently dilute (or otherwise nearly ideal) gaseous penetrants, fA may, of course, be replaced by the corresponding gas
0 1992 Elsevier Science Publishers B.V. All rights reserved.
J.H. Petropoulos/J.
48
pressure PA. DrA iSconveniently defined on the
so-called zero-bulk-flow frame of reference (“intrinsic diffusion coefficient”) [ 161. Integration of eqn. (1) between the upstream (X= 0, fA =fAO, CA= CA,) and downstream (x=1, fA=fAl, CA=CAL)faces of the membrane under steady state permeation conditions yields JAs =I-1
Membrane
virtual quantity in the case of a permanent gas ); vA,vp denote partial molar volume of penetrant and polymer respectively (assumed constant); VA= VACAis the fractional volume of the penetrant in the membrane; andXA is the FloryHuggins interaction parameter [ 71. Given that VA<< Vr and neglecting terms in ui ( uA<< 1 ), eqn. (4) may be conveniently reformulated as
[7,17,181 PAdfA=pA(fAO-fAl)/l
(2) sA=KAexp(bACA)
where JAS is the (constant) permeation flux density normally measured in practice and PA is the integral permeability coefficient deduced therefrom. Note that the Fick diffusion coefticient DA normally used in practice is related to DTA~' DA'DTASA(~~A/~CA)'PA(~~A/~CA)
(3)
Equation (3) yields DAsDTA when SA=dCJ dfA= constant. Pure penetrant transport in rubbery polymers Permeation of weakly interactingpenetrants (simple gases) through polymers above the glass transition temperature is normally characterized by SA= & = const. (Henry’s law) and DTA = DA = const. (Fick’s law). Strongly interacting penetrants (heavy gases at relatively high pressure, vapors) cause plasticization of the polymer and deviate to a greater or lesser extent from the aforesaid simple laws. In systems characterized by non-specific (non-polar or weakly polar) interactions, the behavior of SA, as PA is increased beyond the Henry law range, may be described by the well known ’ simple Flory-Huggins treatment, namely
ln(fA/fk)=ln~A+(l-~A/~~)(l-U~) +xA(~-uA)~
where f
Sci. 75 (1992) 47-59
(4)
Ft denotes saturation vapor fugacity (a
=KAexp(bAKAfA)
(da) (4b)
where
~A=(~Af~t)-lexp[-(l+~A)];
(5a)
bA= (l+&&)vA
(5b)
For examples of the application of eqn. (4a,b) the reader is referred to [ 17,181. The most successful way of modelling the plasticization effects leading to deviation of DTA from Fick’s law, relates DTA to the fractional free volume of the system uf; which, in turn, increases linearly with VA,i.e. [ 21 U~=U~,+~J~UA=U~~+P~CA
(6)
In eqn. (6)) ufs is the fractional free volume of the pure polymer and yi or j?i =ya VA are constants indicating the “plasticizing power” of the particular penetrant for the particular polymer. The simplest [ 21 (though less rigorous [ 31) version of this approach yields
exp[BdAPaCA/U~~(1+PaCA/Ufs)l
(7)
where D TAO= DTA (CA = 0) and Bd is a material constant strongly correlated with the molecular size of the penetrant. For pACA << ufs, eqn. (7) reduces to a simple exponential relation, as is often assumed in treatments of pervaporation [ 19,201. Note, however, that this assumption is only a rather crude approximation for strongly interacting penetrants, as
J.H. PetropouloslJ.
Membrane
PA IcmHg) 4
3-
Sci. 75 (1992) 47-59
-8
6
10
BA=&A/VW PA=PaIvfs
(8)
Equations (7a,b) indicate that DTA(fA)should be substantially closer to a simple exponential relation (due to partial cancellation of the term responsible for the deviation) than DTA(CA). This is confirmed by the relevant plots in Fig. 1, using experiment data [21,22] which were made available to us in numerical form. Equations (4b) and (7b) lead to the following first-order expression for the permeability
(a)
h
PA =~AOew(GAfd
(cm%)--
(9)
where PACI=KADTAO;
(loa)
GA=
(lob)
(BAPA+~A)KA
The corresponding integral permeability for follows from eqn. (2 )
fAo> fAl= 0 then
& @Jexp(GAfAO)
,O'
0
0.02
1
I
0.06
0.10
I
0.14
"A-
Fig. 1. Demonstration of the point that IAn+ and PA(fa) (Fig. la) are much closer to simple exponential functions than DTA(C,) and PA ( CA) (Fig. lb), using experimental diffusivity ( DTA in cmx/sec; 0, 0 ) and permeability (DTAvA/pA=DTAvACA/pA in cmx/sec-cmHg; 0, n ) data for polymethyl acrylate-benzene [21] (open points; K= K’ = - 11) , and polyvinyl acetate-methanol [ 22 ] (filled points; K= -8, K’ = -6), at 25°C. Straight lines (solid for permeability, and broken for diffusivity, data) have been drawn between the first and last experimental points in each case to facilitate assessment of the deviation of the relevant plots from linearity.
shown in Ref. [ 21 (cf. also Fig. 1 here). Upon substitution from eqn. (4b), eqn. (7) reduces, for small PACA and bACA,to &d&~
z ewPdL&
fA[ 1+ lb*
-PA)KAfAl}NNexp(BAPAKAfA) W,W where
-WZfAo
(11)
For a more detailed examination of the behaviour of PA(fA), we obtain, upon expansion to second order in fA within the exponential PA rPAOexp(GAfA +GXf X)
(12)
where
Equation (12 ) predicts that when plasticization effects are weak (small PA), as would be the case with very flexible-chain polymers, it is likely that Gi > 0. In accordance with this prediction, the logpA (fAo) vs. fA,,plots for organic vapors permeating through silicone rubber membranes shown in Ref. [ 181 deviate positively from linearity [implying still larger positive deviations of the corresponding logPA (fA) vs. fA plots; in this particular case one may further conclude that bA > PA, because the relevant sorption data indicate that bi FZ01. For stifferchain polymers the reverse situation (GX < 0)
J.H. Petropoulos/J.
50
is more likely, as illustrated by the negative deviation from linearity of the relevant plots in Fig. 1. Formulation of mixed penetrant transport
of the appropriate theories for pure penetrant transport. Thus, the counterpart of eqn. (4) for the ternary polymer-penetrant system considered here is eqn. (15) below [191 h(fA/f
The steady-state permeation of a binary penetrant mixture AB is described by [ 71 J AMS
=
-PAM(dfA/h)
= pAM
J BMS
=
/l
-PBM(dfB/h)
(13b)
-fBl)/l
where PAM=SAM&‘AM;
PBM’SBMDTBM
The subscripts M are necessary, because, in general, the presence of penetrant B affects the value of SA and DTAand vice versa. This means, incidentally, that PAM, PBM must, in general, be derived by simultaneous solution of eqn. (13a,b ). A measure of the “intrinsic permselectivity” of the membrane for penetrant A over B, which is convenient from the theoretical point of view, is given by eqn. (14) [ 71 (x~=~AM(~Ao,~AI=O)I~BM(~BO,~B~=O)
=JAMs fdhms =~KP(YBcIIYAo)~
“A”t)=huA+l-uA -
-
(UB
+
(XAB~B
-
(XBPUBUP
VA/v,)
-
(UP
+XAP~P)
VA/VP) (UB
+UP)
(13a)
(fA0 -fAl)
=pBM(fBO
Membrane Sci. 75 (1992) 47-59
fAo
(14)
(14a)
(15)
VA/~,)
where UA -I- UB + UP = 1; &pzX~ in eqn. (4) and Xsp~Xs, XABare the corresponding parameterS characterizing the interaction between B and polymer and between A and B respectively. Since VA<< rp, vn << vp and neglecting terms in vi, U; and UAl&(UA<
sAM=sAexp(hBCB)
(15b)
=sAexp@ABKBfB)
where (16)
bAB=[XA-XAB+(l+XB)~A/~Bl~B
The counterparts of eqns. (6)- (7b) are eqns. (17)- (18b) below, respectively [23] Uf =ufs =ufs
+hvA
+h3uB
+&CA
+bcB
07
)
(18
)
(14b)
In eqn. (14b), CC&~ represents the usual practical measure of intrinsic permselectivity (mole fraction of penetrants emerging into an evacuated downstream reservoir divided by the corresponding mole fraction in the upstream reservoir) and YAo,j&o are the gas-phase activity coefficients applicable to the upstream mixture. Mixed penetrant transport in rubbery polymers The dependence of the behavior of penetrant A on B may be described by suitable extension
= expkA(hCA+hCB)/
&AM/%AO d(l+hcA/ufs
&AM/&A0 +
(bA
+&cB/ufsl
=ex@APAKAfA -bA)KAfA
-fb)KBfBI -/h)&sfB =exp[BA(PA&fA
[1
+ (bAB +BAhiiKBfB[1+
+
(bB (18a)
&A--PA)KAfAl} +pBKBfB)
1
(18b)
where DTAOsDTA(CA = 0, CB= 0) ; and KB, bg, bBAand /& are the counterparts of KA, bA, bAB
J.H. Petropoulos/J.
51
Membrane Sci. 75 (1992) 47-59
and PA respectively [cf. eqns. (5a,b), (8) and
(16)l.
The resulting permeabilities to be inserted in eqn. (13a,b) are then given (to first order in fA,fBin the exponent) by PAM
=PAedGAfA
P Em=&WP(‘%h
+GAsfB)
(19a)
+Gmf.J
(19b)
where PBO=KBDTB,,,GBare the exact analogues of PAo, GA [cf. eqn. (lOa,b) ] respectively and GAB
=
GB.4=
@Ah
+bAB&;
U&PA
+ &A
)KA
The solution of eqn. (13a,b) proceeds along the lines indicated in Refs. [20,24] yielding a relation between JAMSand JBMS,on one hand, and between fA(x) and fB(x), on the other hand. For fAl=0, we obtain JAMS/JBMS
=PA~kBhdkAfAo)
-11
/hwkA[exP(bfm)--1
(21)
[exP(kAfA)--l//eXP(kAfAo)-11 = [eXP(kBfB)-1l/[eXP(kBfBo)-11
(22)
where [cf. eqns. (lob) and (20a,b) ] kA=GA-GBA
@a)
=(BA-BB)&i+bA-bm Ftg=GB -GAB
(23b)
=(BB-Bdh+bB-bm The implicit nature of eqn. (22) means that no explicit expressions can be given for PAM and PnM [cf. Ref. [24] also]. On the other hand, eqn. (21) leads directly to an explicit result for the intrinsic permselectivity, in conjunction with eqn. (14a). For kAfA0-4C1, kBfB0-SC1 (relatively weak plasticization effects), eqns. (14a) and (21) yield the first-order result a&= (PA&&
]I+ f
(kAfAo-ksfBo)1
(24)
which shows clearly that the effect of plastici-
zation on permselectivity depends on conditions. Thus, kAfAo> kBfBoleads to a& > P& PBo, which implies an increase or decrease in permselectivity, according to whether PA0> PB, or PA0 < PB@ Two examples of practical significance are considered below. An AB mixture where A represents a light non-interacting gas (like He) and B a fairly strongly interacting one (like CH, or COZ), would be characterized by &A0 > DTBO,KA < KB [cf. Ref. [7]], bAZ0, bBA=O, PAwO, BB>BA and bB> bAB.Hence [cf. eqn. (23a,b)], kAwO, kg> 1. Separations of this kind are usually effected in practice by membranes which are selective towards the fast light gas [7], i.e. PA0> PBO.Hence, on the basis of the above conclusions, the permselectivity is expected to be adversely affected by plasticization. An interesting situation arises in the separation of COZ(A) from CH, (B ), which is of considerable practical importance. Here, the heavier gas A is also the faster one (in line with its smaller effective molecular size indicated by penetration of molecular sieving crystals [ 71) , i.e. KA> KB and DTAo> I&no. Studies of the sorption and permeation of these gases in polyethylene [8,9] show that BA < BB (as expected, in view of the fact that these parameters should vary in line with the effective molecular size of the penetrant). No significant deviations from Henry’s law were noticeable in the experimental pressure range, i.e. bAz b z bm z b&4z 0. It follows that PA0> PBO, while kA< 0, kB> 0. Hence, here too, plasticization leads to reduction in permselectivity. The above predictions are in accord with the general experience that, in practice, gas separation is adversely affected by plasticization. Equation (24) may, with the aid of analogous first-order expansions for pA(fAo) and pB(fB,,) [cf. eqn. (ll)], be recast in the form ao, “N[~A(fAO)/~B(fBd [I+
1
%fm
(24a)
-&fAo)
1
52
J.H. Petropoulos/J.
where kk=GA
+GBA
(254
= @A+&3)P.4+b, +b3* k;3 =G,
+GAB
(25b)
= WA+&3)&i +b, +b*B For the He ( A)-CH, (B ) example considered above, we have k$, z 0, kb > 0. Hence, the permselectivity in mixed gas flow tends to be better than indicated by the ratio of the effective puregas permeabilities for the same fAOand fsO values. In the example of CO, ( A)-CH, (B ) , on the other hand, the data of Refs. [8,9] indicate PA> /3n (i.e. CO, is a more effective plasticizer than CH,). Hence, eqn. (25) yields ka > kb and eqn. (24a) shows that the outcome depends on the composition of the mixture AB: for f,,, wfnO, a$$ will be less than PA (fAO)/& (feO).
Application of the treatment of plasticization given above to glassy polymer membranes is subject to considerable complications due to the microheterogeneity of the polymer. The simplest reasonably successful approach to sorption and transport in such systems so far developed, is to regard the glassy polymer as consisting of (i) a uniform dense matrix, wherein the penetrant molecules “dissolve” exactly as above T, and (ii) semipermanent microcavities interspersed in the said matrix (and collectively constituting the “excess free volume” of the polymer occupying a volume fraction u? ), which act as adsorption sites (concentration sA2) for the penetrant molecules (e.g. [ 5,6] ). The concentration, and the resulting solubility coefficient, of penetrant molecules “dissolved” or “adsorbed” in the polymer in the above manner are denoted by CAl,SAl and CA2,SA2respectively; and their mobility is char-
Sci. 75 (1992) 47-59
acterized by thermodynamic diffusion coefficients &A1 and &A2 respectively [6,7]. According to the simple dual-mode sorption and transport models [5,25], we have, in the absence of plasticization, sA=cA/fA=sA,+sA2 (26) =KAl
PA=pAl
+h&2/(1+&fA) +PA~=SAIDTAI+SA~DTA:!
(27)
where sA2 has been formulated in terms of a simple Langmuir adsorption isotherm, characterized by capacity and affinity constants sA2 and KA2 respectively; DTA1, DTA2 are constant. Equations (26) and (27) lead to the following expression for the integral permeability (with fAl=o) PA
=PAl +PA2 =KAlDTAl +
Pure penetrant transport in glassy polymers
Membrane
(2&b)
SA~DTAP~~(~+KA~~AO)/~AO
On the basis of the above models, the treatments of plasticization previously given is immediately applicable to the dissolved penetrant molecules (i.e. to PAI ), except for the fact that, in general, absorbed molecules may also exercise some plasticizing effect. In an attempt to allow for the latter effect, an “effective” concentration CA, has been defined by means of eqn. (29) below CAe
=cAl
+&CA%
(29)
where &,= const. < 1 has been evaluated from correlations of gas sorption and polymer dilation measurements (e.g. [ 121). As far as PA2 is concerned, two main opposing plasticization effects are to be expected, namely (a) a decrease in sA2, in line with the and (b) expected decrease in uF as T,-+T[6] an increase of DTA2. The former effect has been formulated as [ 121 SA2(CAe)=SA20(1-CAe/CAG)
(30)
53
J.H. PetropoulosjJ. Membrane Sci. 75 (1992) 47-59
where sA~O~SA~ (CA = CA,= 0) ; CAo represents the value of CAat which glass transition occurs, at the given experimental temperature, i.e. T,(CA=CAo)=T, and sA~(CA>CA~)=O. An alternative formulation is SA2(CA)=SA20[Tg(CA)-Tl
(31)
/[T,(CA=Ob-Tl
where Tg (CA) is given by an appropriate theory andSA2(CA)=OforTg(CA)~T[13]. Empirical formulations of effect (b) have been given [ 14,151. The simpler approach of Ref. [ 141 is equivalent to (i) assuming &As/ D TAl- -const. and thereby transforming eqn. (27) to
tiguous adsorption sites being sufficiently rare to be neglected [ 261. In the simple dual mode transport model approach [ 25,26,28-301, the aforementioned kinds of activated diffusion jump give rise to corresponding additive molecular flux components. Thus, the total (molar) flux density in the forward ( +x) direction is given by [ 261 IA=
(~-SAZ/SA~)U~A~~AI
+
(32)
(sA~/sA~)u:A~CAI+CA~U~A~
In eqn. (32), u,Al, u:A1 and &.A2are the effective molecular velocities pertaining to the aforementioned kinds of diffusion jump, respectively, and are given by
(27a)
where Sk, is an effective solubility coefficient and (ii) setting $,,=DTAS/DTA1 in eqn. (29). The treatment of Ref. [ 151 on the other hand, is equivalent to assuming that DTAl and DTA2 are independent functions of CA1 and of CA, respectively. In order to establish a physical basis from which to proceed in this matter, we refer to the molecular aspects of the dual mode transport model set out in detail in previous papers [ 26,271. According to this treatment, both dissolved and adsorbed penetrant molecules in their ground state occupy sites between the polymer chains; the difference between “solution” and “absorption” sites being that the potential energy wells (cf. Fig. 2 ) representing the former are shallower and much more numerous (concentration aA ) than those representing the latter (i.e. s& << sA1) . The penetrant molecular flux is made up of elementary activated molecular jumps over three kinds of potential energy barrier, namely E&+ (solution site to adjacent solution site), EA1+’ (solution site to adjacent adsorption site) and EAz+ (adsorption site to adjacent solution site); the occurrence of con-
u,A2=
u:A2yA2
/ Y:2
(33c)
where uz denotes the effective molecular velocity at infinite dilution; y is the activity coefficient describing deviation from thermodynamic ideality; the superscript ( # ) denotes the activated state; and 0 uzA=dAe(kBT/h)exp(ASAf/R) Xexp(
-E;/RT)
(34)
=u”,exp( -E,f/RT) In eqn. (34), kB and h are the Boltzmann and Planck constants respectively AA is the mean effective jump length (which may for simplicity be assumed to be the same for all kinds of molecular jump considered above); K is a geometrical factor (equal to l/6 for a simple threedimensional random walk); and AS,Z , E 2 denote (molar) activation entropy and energy respectively ). E 2 plays the dominant role in eqn. (34) (note also that there is a high degree of correlation between AS: and Ez, cf. Fig. 12 in chap. 2 of Ref. [l] ). In accordance with the
._-P J.H. Petropoulos/J.
54
Tf
Membrane
Sci. 75 (1992) 47-59
I -f
En1
-A2 I .___
L
SITE
t ADSORPTION SITE
Fig. 2. Schematic representation of the variation of potential energy along the path of a diffusing penetrant molecule, aciording to the dualmode transport model.
simple dual mode sorption model of eqn. (26)
[261, YAl=Yil=l;
(35a)
YA~=Y~~=(~-CA~/SA~)-~=~+KA~~A
(35b)
It is considered that local equilibrium between adsorbed and dissolved molecules is maintained at all times, i.e. (36)
v:A1CA1/SAI=vxA2CA2/SA2
Hence, the last two terms in eqn. (32) are equal. In Ref. [ 261, the final result ior the net unidimensional flux density JA= JA- JA was derived in terms of V,A$A~ [cf. eqn. (13) therein], because it leads to the simplest molecular interpretation of the standard macroscopic treatment of eqns. (1) and (27), namely D TAl=
D TA~~~(SA~/SAIKA~)DTA~
(1-SA2/SA1bxA1AA
=(l_SA2/SA1)U~lexp(--EAZ,/RT)
D TA2=2v,A2lA
=2u&exp(
(37)
-E,f,/RT)
(38)
However, an entirely equivalent result is obviously obtainable in terms of vLA1 and CA,. This alternative formulation modifies eqn. (38) to [cf. eqn. (36) and (26) also] D TA2=2(~A2SAl/SA1YA2SA2)U~1 xexp(-KdW (3W =2(~Al/~Al~A,h& xexp(--E3RT)
Note, incidentally, that in Ref. [29] separate diffusion coefficients were defined for the last two terms of eqn. (32 ). This practice, which has been adopted in a few other cases also, is not physically meaningful, in the opinion of the present author; in view of eqn. (36) and of the additional consideration that the corresponding net flux components are not separable in this way (since the forward flux component due to solution-site-to-adsorption-site jumps is associated with the backward flux component due to adsorption-site-to-solution-site jumps and vice versa). Equation (38a) is useful in the present context; because, in conjunction with eqn. (37)) it shows immediately that, if EA’; = EL (uZ1 zugl ), then (39)
=2pA1/sAlK~2
Equation (39) shows that, under the specified conditions, the concentration dependence of D TA2due to plasticization will follow that of DTAl (Or, more precisely, of PA1), since KA2and sA1are presumed to remain constant. This provides physical justification for the treatment based on eqn. (27a) with Sk given by sa=SAl[1+2SA2/S~l(1+K~2f~)l
(46)
The question now is whether the condition Ez{ z Ezl can plausibly be expected to hold in
J.H. PetropoulosjJ.
55
Membrane Sci. 75 (1992) 47-59
practice. The analysis of dual mode sorption and transport data given in Refs. [ 271 and [ 281 indicates that the answer to this question is most likely negative. This conclusion is important, because the aforesaid condition constitutes the basis on which the validity of the simple dual mode transport model in its various forms [ 25,26,28-301 rests [ 26,271. A more general treatment [ 271, which is not subject to this restriction, shows that simple additive dual mode transport formulations (eqn. 27) result in the following cases: (i) uiAl << u,_& (E;f; s-E&)); (ii) &&I =u,Al (E&‘=E&),i.e. the case considered above; and (iii) u:_&r>> u,Ar (E&’ <
=
P(1+Q)2/Q1
(sAl/sAlKA2)
(41)
x uL exp ( - E;f,/RT) Equation (41) is identical with eqn. (39)) except for an additional numerical factor [which carries over into eqn. (40) also]. The data analysis referred to above suggests that the conditions prevailing in typical real systems lie between cases (ii ) and (iii ) and probably close to (iii) [27]. We conclude, therefore, that within the model framework of eqn. (27)) treatment of plasticization effects on the basis of eqn. (27a) appears to be justified on physical grounds. On the other hand, no comparable justification can be found at the present time for the assumption
Of $,,= &&&.A~ Of Ref. [ 141 Or for the formulation of DTAZas a function of CA2adopted inRef. [15]. In the light of the above discussion and in accord with the treatment of Ref. [12], an expression for pA(fA) may be developed on the basis of eqn. (27) by substituting CA in eqn. (4a) and (7) by CA, defined in eqn. (29). The resulting expression is rather complex, but simplifies in the case of &,x 0 to pA
=
[PAlO
(42)
+PAZO(l--fA/fAG)
/(1+&fA)]exp(GA&) where PA~OGPA~ (f~ = 0 ), PA20cPA2(fA = 0 ) [Cf. eqns. (26 ), (27 ) ] ; GA1is given by the analogue of eqn. (lob); and eqn. (30) has been rewritten in terms of fA with for convenience sA2(fA >.fAo) = 0. In practice, @,, may be nearly zero [31] but may also attain substantially higher values [ 121. It is not expected, however, that the behavior of PA in the latter case will be substantially different. Equation (42 ) shows that, in the absence of plasticization, PA will tend to decline with increasing fA; in keeping with experimental findings which show that PA(fAo,fAl=O) is typically a decreasing function off_&, [6,11]. In the complete absence of plasticization (GA1= 0) , a limiting value is attained at sufficiently high pressure, namely pA(fA+d+~A(fAO+~,
fAl=o)*pAKl.
How-
ever, in the presence of even very weak plasticization, the exponential factor in eqn. (42 ) will ultimately lead to a reversal of the above trend, thus causing PA (fA) and PA (f_&Al= 0) to pass through a minimum; again in keeping with what is observed in practice [lo]. In some cases, no initial decline of PA (f_&,,)with increasing fAOis found [ 111. The condition for this to happen follows from eqn. (42) upon expansion to first order in f_&and integI%tiOn yielding (for fAl= 0) PA
=PAIo(~+
f (GAI~Ao)
+pA20
56
J.H. Petropoulos/J. Membrane Sci. 75 (1992) 47-59
x{ln(l+KA2fAo)lKA2fA0+ (GA1-l/fAd X t1-ln(l+~A2fAO)I~A2fADll~A2}
(43)
which for fAO-+Oreduces to WP40 = I+ t [GA*- (P~,/p~io) X (KU + l/fAG)I
(43a)
H(fAO,fBO)= (1 -fAObfAG
(~+PA~O/PAIO)I~AO
According to eqn. (31a), PA increases with f&, throughout, if the coefficient of fAOis positive. Mixed penetrant transport in glassy polymers In the presence of a second penetrant B, eqn. (42) may be extended in the following way
P AM=
where
/(I+
;&,fAo
-fBO/zfBG)
(48)
+ #mfBo)
and &, kg1 are given by the analogues of eqn. (23). The above simplified treatment is justifiable, at least for practical purposes, if the dualmode transport effect on cu0,is not unduly large. The fact that this is normally so follows from our previous study [24], which shows that, in the absence of plasticization ( kA1M0, Kmz 0, CX&’is given to a good apf AG-+GQ, fsc+co), proximation in practice by
[PAIO+PA~O(~-~A/~AG-~B/~BG)
(44)
/(~+KA~~A+KB~~B)I
(49)
xexP(GA1fA+GABlfB) on the basis of eqn. (19a) and extensions of eqns. (26) (cf. Refs. [6,32] ) and (30) [as modified in connection with eqn. (42 ) ] to eqns. (45) and (46)) respectively, below SA,=SA,KA,/(~+KA~~A
+K,,f,)
SA2=SA20(1-fA/fAG-fB/fBG)
(45) (46)
where s&.o=s& (fA = O,fB = 0) and aA = 0 when fA/fAG+fB/fBG>l.
In eqn. (44), GABI is given
by the analogue of eqn. (20a) and &, w 0 is assumed as before. Further progress by analytical methods does not seem possible, unless the second term within the brackets in eqn. (44) can be replaced by a “mean value”; in which case the intrinsic permselectivity will be given by the analogue of the combined eqns. (14a) and (21)) namely
where F(fAO,fBO)
=ln(l+K~,fAO
+KmfBo)/
(KA~~Ao +KmfBo)
(50)
A somewhat different formulation of F(fAo9fBo) follows from the treatment of Ref. [ 331; but the general predicted behavior is the same, with F(O,O)=l andF(fAo,fBo)bOas (fAO+fBO)+m. It follows that the effect on a$ is determined by the relative magnitude of PA20/PA10and PBzo IPmo- The general tendency is for Pzo/Plo to be higher for the heavier, more strongly interacting gases [ 24,261. Hence, permselectivity would, in general, be expected to be enhanced (reduced), if the basic selectivity of the membrane, namely PAlo/Pelo, favors the heavier (lighter) more strongly (less strongly) interacting gas. As pointed out in Ref. [ 241, how-
J.H. PetropoulosjJ.
Membrane
Sci. 75 (1992) 47-59
57
TABLE 1 Calculated values of ff& (P,,,/P,,,) for the case of P&P,,,=O.97, P,,,/P,,,=O.17, KA2=0.215 atm-‘, K,,=O.O74 atm-r, in the absence of plasticization (Case 2 of Ref. [ 241): (a) “exact” values obtained by the numerical procedure of Ref. [ 241; (b) from eqns. (49) and (50); (c) from eqns. (49) and (50a)
fdfm
0
2 6 12 20
= 0.20
fAolfLul= 0.50
fdfm
= 0.80
(a)
(b)
(cl
(a)
(b)
(cl
(a)
(b)
(c)
1.684 1.630 1.548 1.464 1.390
1.684 1.630 1.551 1.470 1.399
1.684 1.631 1.547 1.456 1.373
1.684 1.611 1.510 1.417 1.341
1.684 1.611 1.513 1.422 1.348
1.684 1.620 1.523 1.422 1.336
1.684 1.593 1.479 1.381 1.306
1.684 1.594 1.482 1.385 1.311
1.684 1.612 1.506 1.402 1.315
ever, the observed differences in P2,,/P10are not very pronounced, because of partial compensation of the opposing tendencies of the sorption and diffusion parameters contained therein. Their practical effect will be further substantially diminished by the low values of F(f.&s,,) corresponding to the high pressures used in normal gas separation operations. As a practical illustration of these points we have chosen Case 2 studied in Ref. [ 241, because it may be considered an extreme example of what may be encountered in practice. The results of model calculations in the (total) gas pressure range of O-20 atm are given in Table 1. The results of column (a) are included in the table to illustrate the point made earlier that eqns. (49) and (50) yield excellent approximations to the “exact” numerical values; whereas column (c ) serves to show that a mean value approximation of the type made in eqns. (47), (48), namely GOa) also gives very good results for practical purposes in this case. It will be noted that, even in the extreme example under consideration here, the dual mode transport effect on permselectivity remaining at 20 atm is not large. In the presence of plasticization, further reduction of
this effect, due to the numerator appearing in eqn. (48)) would be expected. Conclusion An attempt has been made, in the present paper, to approach the problem of modelling the effect of plasticization on gas permeation and separation in a reasonably thorough and systematic way. Previous extensive work on vapor transport has led to a reasonably well established physical theory for this purpose above T,. We have shown, however, that some useful simplification is possible (at the expense of relatively little loss in rigor), which constitutes a significant improvement over comparable formulations of a more empirical character currently employed in the treatment of pervaporation. Some further simplification is possible in the case of gas separation, as a result of the relatively weak plasticization effects involved. On this basis, the effects to be expected in particular cases can be immediately deduced and the precise reasons why practical gas separation is commonly found to be adversely affected by plasticization, understood. Below Tg, the situation is complicated by the microheterogeneity of the polymeric medium, but we have shown that the dual mode trans-
J.H. PetropoulosjJ. Membrane Sci. 75 (1992) 47-59
58
model, properly understood, provides a useful physical basis permitting a relatively simple extension of the treatment of plasticization above Tg. Further simplifications are needed, however, in order to keep within the bounds of a readily applicable analytical treatment. It is, nevertheless, expected that the results obtained here will prove useful, at least as a first step, for the analysis and interpretation of pertinent experimental data. port
13
14
15
16 17
References 1 2 3
4
5 6
I
8
9
10
11
12
J. Crank and G.S. Park, Diffusion in Polymers, Academic Press, New York, NY, 1968. H. Fujita, Diffusion in polymer-diluent systems, Fortschr. Hochpolym. Forsch., 3 (1961) 1. J.S. Vrentas and J.L. Duda, Diffusion in polymer-solvent systems. I. Reexamination of the free-volume theory; II. A predictive theory for the dependence of diffusion coefficients on temperature, concentration and molecular weight, J. Polym. Sci., Polym. Phys. Ed., 15 (197’7) 403,417. V.T. Stannett, The transport of gases in synthetic polymeric membranes-An historic perspective, J. Membrane Sci., 3 (1978) 97. W.R. Vieth, J.H. Howell and J.H. Hsieh, Dual sorption theory, J. Membrane Sci., 1 (1976) 177. J.H. Petropoulos, Membranes with non-homogeneous sorption and transport properties, Adv. Polym. Sci., 64 (1985) 93. J.H. Petropoulos, Some fundamental approaches to membrane gas permeability and permselectivity, J. Membrane Sci., 53 (1990) 229. S.S. Kulkarni and S.A. Stern, The diffusion of COz, CH,, C&H, and f&H, in polyethylene at elevated pressures, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 441. S.A. Stern and S.S. Kulkarni, Tests of a “free-volume” model of gas permeation through polymer membranes. I. Pure COz, CHI, C,H, and C&H, in polyethylene, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 467. SM. Jordan, W.J. Koros and G.K. Fleming, The effects of COz exposure on pure and mixed gas permeation behavior: comparison of glassy polycarbonate and silicone rubber, J. Membrane Sci., 30 (1987) 191. J.S. Chiou and D.R. Paul, Effects of CO2 exposure on gas transport properties of glassy polymers, J. Membrane Sci., 32 (1987) 195. Y. Kamiya, K. Mizoguchi, T. Hirose and Y. Naito, Sorption and dilation in poly (ethyl methacrylate ) carbon dioxide system, J. Polym. Sci., Part B, 27 (1989) 879.
18
19
20
21 22
23
24
25
26
27
28
29
J.S. Chiou and D.R. Paul, Gas sorption and permeation in poly (ethyl methacrylate), J. Membrane Sci., 45 (1989) 167. S.A. Stern and V. Saxena, Concentration-dependent transport of gases and vapors in glassy polymers, J. Membrane Sci., 7 (1980) 47. S. Zhou and S.A. Stern, The effect of plasticization on the transport of gases in and through glassy polymers, J. Polym. Sci., Part B, 27 (1989) 205. J. Crank, The Mathematics of Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, p. 209. V.M. Shah, B.J. Hardy and S.A. Stern, Solubility of carbon dioxide, methane and propane in silicone polymers: effect of polymer side chains, J. Polym. Sci., Part B, 24 (1986) 2033. M.S. Suwandi and S.A. Stern, Transport of heavy organic vapors through silicone rubber, J. Polym. Sci., Polym. Phys. Ed., 11 (1973) 663. M.H.V. Mulder and C.A. Smolders, On the mechanism of separation of ethanol/water mixtures by pervaporation. I. Calculation of concentration profiles, J. Membrane Sci., 17 (1984) 289. J.-P. Brun, C. Larchet, G. Bulvestre and B. Auclair, Sorption and pervaporation of dilute aqueous solutions of organic compounds through polymer membranes, J. Membrane Sci., 25 (1985) 55. A.C. Newns, Diffusion of benzene in polymethyl acrylate, Trans. Faraday Sot., 59 (1963) 2150. A. Kishimoto, Diffusion and viscosity of polyvinyl acetate-diluent systems, J. Polym. Sci., Part A, 2 (1964) 1421. S.A. Stern, G.R. Mauze and H.L. Frisch, Test of a free-volume model for the permeation of gas mixtures through polymer membranes. C02-&HI, COz-CaHs and C,H,-C&H, mixtures in polyethylene, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 1275. J.H. Petropoulos, Formulation of dual-mode mixed gas transport in glassy polymers, J. Membrane Sci., 48 (1990) 79. J.H. Petropoulos, Quantitative analysis of gaseous diffusion in glassy polymers, J. Polym. Sci., Part A2, 8( 1970) 1797. J.H. Petropoulos, On the dual-mode gas transport model for glassy polymers, J. Polym. Sci., Part B, 26 (1988) 1009. J.H. Petropoulos, A generalized, topologically consistent, dual-mode transport model for glassy polymergas systems, J. Polym. Sci., Part B, 27 (1989) 603. D.R. Paul and W.J. Koros, Effect of partially immobilizing sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed., 14 (1976) 675. R.M. Barrer, Diffusivities in glassy polymers for the dual-mode sorption model, J. Membrane Sci., 18 (1984) 25.
J.H. Petropoulos/J. 30
31
Membrane Sci. 75 (1992) 47-59
G.H. Fredrickson and E. Helfand, Dual-mode transport of penetrants in glassy polymers, Macromolecules, 18 (1985) 2201. G.K. Fleming and W.J. Koros, Dilation of polymers by sorption of carbon dioxide at elevated pressures. 1. Silicone rubber and unconditioned polycarbonate, Macromolecules, 19 (1986) 2285.
59
32
33
W.J. Koros, Model for sorption of mixed gases in glassy polymers, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 981. W.J. Koros, R.T. Chern, V. Stannett and H.B. Hopfenberg, A model for permeation of mixed gases and vapors in glassypolymers, J. Polym. Sci., Polym. Phys. Ed., 19 (1981) 1513.
47
Plasticization effects on the gas permeability and permselectivity of polymer membranes J.H. Petropoulos Physical Chemistry Institute, Democritos National Research Centre, 153 10 Aghia Paraskevi, Athens (Greece) (Received February 15,1992; accepted in revised form July 21,1992)
Abstract Plasticization effects may substantially modify the performance of polymer membranes used for gas separation in practice. The present paper is concerned with the development of simplified but physically meaningful formulations to promote better understanding of these effects (particularly below T,), on one hand, and to facilitate systematic analysis of pertinent data, on the other hand. Keywords: gas separation; plasticization;
theory
tical application. The present paper constitutes an attempt to advance along these lines.
Introduction Plasticization effects on vapor transport in polymeric media have been discussed extensively (e.g. [l-3 ] ) but are hardly touched upon in most treatments on gas sorption and diffusion (e.g. [ 1,4-61). This is, of course, justified in the case of simple or low-pressure gaseous penetrants. However, plasticization phenomena become increasingly significant for heavier gases and higher pressures (e.g. [ 7-131) and are, thus, import in practical gas separation. It is, therefore, of considerable practical as well as theoretical interest to develop appropriate theoretical treatments (e.g. [ 14,151). It is of no less importance, however, that such treatments combine, as far as possible, a sound physical basis with simplicity conducive to useful pracCorrespondence to: J.H. Petropoulos, Physical Chemistry Institute, Democritos National Research Centre, 153 10 Aghia Paraskevi, Athens, Greece.
0376-7388/92/$05.00
Formulation
of pure penetrant
transport
As discussed in Refs. [6,7], the unidimensional (molar) flux density JA of a micromolecular penetrant A at a position 3c(O
J A=-DTASA(dfA/dX)'-PA(dfA/dX)
(1)
where &A, SA= CA/fA and PAlDTASA are the relevant “thermodynamic” diffusion, solubility (sorption) and permeability coefficients; f.& is the gas-phase fugacity of the penetrant, which would be at equilibrium with the molar concentration CAof the penetrant in the membrane at position X. For sufficiently dilute (or otherwise nearly ideal) gaseous penetrants, fA may, of course, be replaced by the corresponding gas
0 1992 Elsevier Science Publishers B.V. All rights reserved.
J.H. Petropoulos/J.
48
pressure PA. DrA iSconveniently defined on the
so-called zero-bulk-flow frame of reference (“intrinsic diffusion coefficient”) [ 161. Integration of eqn. (1) between the upstream (X= 0, fA =fAO, CA= CA,) and downstream (x=1, fA=fAl, CA=CAL)faces of the membrane under steady state permeation conditions yields JAs =I-1
Membrane
virtual quantity in the case of a permanent gas ); vA,vp denote partial molar volume of penetrant and polymer respectively (assumed constant); VA= VACAis the fractional volume of the penetrant in the membrane; andXA is the FloryHuggins interaction parameter [ 71. Given that VA<< Vr and neglecting terms in ui ( uA<< 1 ), eqn. (4) may be conveniently reformulated as
[7,17,181 PAdfA=pA(fAO-fAl)/l
(2) sA=KAexp(bACA)
where JAS is the (constant) permeation flux density normally measured in practice and PA is the integral permeability coefficient deduced therefrom. Note that the Fick diffusion coefticient DA normally used in practice is related to DTA~' DA'DTASA(~~A/~CA)'PA(~~A/~CA)
(3)
Equation (3) yields DAsDTA when SA=dCJ dfA= constant. Pure penetrant transport in rubbery polymers Permeation of weakly interactingpenetrants (simple gases) through polymers above the glass transition temperature is normally characterized by SA= & = const. (Henry’s law) and DTA = DA = const. (Fick’s law). Strongly interacting penetrants (heavy gases at relatively high pressure, vapors) cause plasticization of the polymer and deviate to a greater or lesser extent from the aforesaid simple laws. In systems characterized by non-specific (non-polar or weakly polar) interactions, the behavior of SA, as PA is increased beyond the Henry law range, may be described by the well known ’ simple Flory-Huggins treatment, namely
ln(fA/fk)=ln~A+(l-~A/~~)(l-U~) +xA(~-uA)~
where f
Sci. 75 (1992) 47-59
(4)
Ft denotes saturation vapor fugacity (a
=KAexp(bAKAfA)
(da) (4b)
where
~A=(~Af~t)-lexp[-(l+~A)];
(5a)
bA= (l+&&)vA
(5b)
For examples of the application of eqn. (4a,b) the reader is referred to [ 17,181. The most successful way of modelling the plasticization effects leading to deviation of DTA from Fick’s law, relates DTA to the fractional free volume of the system uf; which, in turn, increases linearly with VA,i.e. [ 21 U~=U~,+~J~UA=U~~+P~CA
(6)
In eqn. (6)) ufs is the fractional free volume of the pure polymer and yi or j?i =ya VA are constants indicating the “plasticizing power” of the particular penetrant for the particular polymer. The simplest [ 21 (though less rigorous [ 31) version of this approach yields
exp[BdAPaCA/U~~(1+PaCA/Ufs)l
(7)
where D TAO= DTA (CA = 0) and Bd is a material constant strongly correlated with the molecular size of the penetrant. For pACA << ufs, eqn. (7) reduces to a simple exponential relation, as is often assumed in treatments of pervaporation [ 19,201. Note, however, that this assumption is only a rather crude approximation for strongly interacting penetrants, as
J.H. PetropouloslJ.
Membrane
PA IcmHg) 4
3-
Sci. 75 (1992) 47-59
-8
6
10
BA=&A/VW PA=PaIvfs
(8)
Equations (7a,b) indicate that DTA(fA)should be substantially closer to a simple exponential relation (due to partial cancellation of the term responsible for the deviation) than DTA(CA). This is confirmed by the relevant plots in Fig. 1, using experiment data [21,22] which were made available to us in numerical form. Equations (4b) and (7b) lead to the following first-order expression for the permeability
(a)
h
PA =~AOew(GAfd
(cm%)--
(9)
where PACI=KADTAO;
(loa)
GA=
(lob)
(BAPA+~A)KA
The corresponding integral permeability for follows from eqn. (2 )
fAo> fAl= 0 then
& @Jexp(GAfAO)
,O'
0
0.02
1
I
0.06
0.10
I
0.14
"A-
Fig. 1. Demonstration of the point that IAn+ and PA(fa) (Fig. la) are much closer to simple exponential functions than DTA(C,) and PA ( CA) (Fig. lb), using experimental diffusivity ( DTA in cmx/sec; 0, 0 ) and permeability (DTAvA/pA=DTAvACA/pA in cmx/sec-cmHg; 0, n ) data for polymethyl acrylate-benzene [21] (open points; K= K’ = - 11) , and polyvinyl acetate-methanol [ 22 ] (filled points; K= -8, K’ = -6), at 25°C. Straight lines (solid for permeability, and broken for diffusivity, data) have been drawn between the first and last experimental points in each case to facilitate assessment of the deviation of the relevant plots from linearity.
shown in Ref. [ 21 (cf. also Fig. 1 here). Upon substitution from eqn. (4b), eqn. (7) reduces, for small PACA and bACA,to &d&~
z ewPdL&
fA[ 1+ lb*
-PA)KAfAl}NNexp(BAPAKAfA) W,W where
-WZfAo
(11)
For a more detailed examination of the behaviour of PA(fA), we obtain, upon expansion to second order in fA within the exponential PA rPAOexp(GAfA +GXf X)
(12)
where
Equation (12 ) predicts that when plasticization effects are weak (small PA), as would be the case with very flexible-chain polymers, it is likely that Gi > 0. In accordance with this prediction, the logpA (fAo) vs. fA,,plots for organic vapors permeating through silicone rubber membranes shown in Ref. [ 181 deviate positively from linearity [implying still larger positive deviations of the corresponding logPA (fA) vs. fA plots; in this particular case one may further conclude that bA > PA, because the relevant sorption data indicate that bi FZ01. For stifferchain polymers the reverse situation (GX < 0)
J.H. Petropoulos/J.
50
is more likely, as illustrated by the negative deviation from linearity of the relevant plots in Fig. 1. Formulation of mixed penetrant transport
of the appropriate theories for pure penetrant transport. Thus, the counterpart of eqn. (4) for the ternary polymer-penetrant system considered here is eqn. (15) below [191 h(fA/f
The steady-state permeation of a binary penetrant mixture AB is described by [ 71 J AMS
=
-PAM(dfA/h)
= pAM
J BMS
=
/l
-PBM(dfB/h)
(13b)
-fBl)/l
where PAM=SAM&‘AM;
PBM’SBMDTBM
The subscripts M are necessary, because, in general, the presence of penetrant B affects the value of SA and DTAand vice versa. This means, incidentally, that PAM, PBM must, in general, be derived by simultaneous solution of eqn. (13a,b ). A measure of the “intrinsic permselectivity” of the membrane for penetrant A over B, which is convenient from the theoretical point of view, is given by eqn. (14) [ 71 (x~=~AM(~Ao,~AI=O)I~BM(~BO,~B~=O)
=JAMs fdhms =~KP(YBcIIYAo)~
“A”t)=huA+l-uA -
-
(UB
+
(XAB~B
-
(XBPUBUP
VA/v,)
-
(UP
+XAP~P)
VA/VP) (UB
+UP)
(13a)
(fA0 -fAl)
=pBM(fBO
Membrane Sci. 75 (1992) 47-59
fAo
(14)
(14a)
(15)
VA/~,)
where UA -I- UB + UP = 1; &pzX~ in eqn. (4) and Xsp~Xs, XABare the corresponding parameterS characterizing the interaction between B and polymer and between A and B respectively. Since VA<< rp, vn << vp and neglecting terms in vi, U; and UAl&(UA<
sAM=sAexp(hBCB)
(15b)
=sAexp@ABKBfB)
where (16)
bAB=[XA-XAB+(l+XB)~A/~Bl~B
The counterparts of eqns. (6)- (7b) are eqns. (17)- (18b) below, respectively [23] Uf =ufs =ufs
+hvA
+h3uB
+&CA
+bcB
07
)
(18
)
(14b)
In eqn. (14b), CC&~ represents the usual practical measure of intrinsic permselectivity (mole fraction of penetrants emerging into an evacuated downstream reservoir divided by the corresponding mole fraction in the upstream reservoir) and YAo,j&o are the gas-phase activity coefficients applicable to the upstream mixture. Mixed penetrant transport in rubbery polymers The dependence of the behavior of penetrant A on B may be described by suitable extension
= expkA(hCA+hCB)/
&AM/%AO d(l+hcA/ufs
&AM/&A0 +
(bA
+&cB/ufsl
=ex@APAKAfA -bA)KAfA
-fb)KBfBI -/h)&sfB =exp[BA(PA&fA
[1
+ (bAB +BAhiiKBfB[1+
+
(bB (18a)
&A--PA)KAfAl} +pBKBfB)
1
(18b)
where DTAOsDTA(CA = 0, CB= 0) ; and KB, bg, bBAand /& are the counterparts of KA, bA, bAB
J.H. Petropoulos/J.
51
Membrane Sci. 75 (1992) 47-59
and PA respectively [cf. eqns. (5a,b), (8) and
(16)l.
The resulting permeabilities to be inserted in eqn. (13a,b) are then given (to first order in fA,fBin the exponent) by PAM
=PAedGAfA
P Em=&WP(‘%h
+GAsfB)
(19a)
+Gmf.J
(19b)
where PBO=KBDTB,,,GBare the exact analogues of PAo, GA [cf. eqn. (lOa,b) ] respectively and GAB
=
GB.4=
@Ah
+bAB&;
U&PA
+ &A
)KA
The solution of eqn. (13a,b) proceeds along the lines indicated in Refs. [20,24] yielding a relation between JAMSand JBMS,on one hand, and between fA(x) and fB(x), on the other hand. For fAl=0, we obtain JAMS/JBMS
=PA~kBhdkAfAo)
-11
/hwkA[exP(bfm)--1
(21)
[exP(kAfA)--l//eXP(kAfAo)-11 = [eXP(kBfB)-1l/[eXP(kBfBo)-11
(22)
where [cf. eqns. (lob) and (20a,b) ] kA=GA-GBA
@a)
=(BA-BB)&i+bA-bm Ftg=GB -GAB
(23b)
=(BB-Bdh+bB-bm The implicit nature of eqn. (22) means that no explicit expressions can be given for PAM and PnM [cf. Ref. [24] also]. On the other hand, eqn. (21) leads directly to an explicit result for the intrinsic permselectivity, in conjunction with eqn. (14a). For kAfA0-4C1, kBfB0-SC1 (relatively weak plasticization effects), eqns. (14a) and (21) yield the first-order result a&= (PA&&
]I+ f
(kAfAo-ksfBo)1
(24)
which shows clearly that the effect of plastici-
zation on permselectivity depends on conditions. Thus, kAfAo> kBfBoleads to a& > P& PBo, which implies an increase or decrease in permselectivity, according to whether PA0> PB, or PA0 < PB@ Two examples of practical significance are considered below. An AB mixture where A represents a light non-interacting gas (like He) and B a fairly strongly interacting one (like CH, or COZ), would be characterized by &A0 > DTBO,KA < KB [cf. Ref. [7]], bAZ0, bBA=O, PAwO, BB>BA and bB> bAB.Hence [cf. eqn. (23a,b)], kAwO, kg> 1. Separations of this kind are usually effected in practice by membranes which are selective towards the fast light gas [7], i.e. PA0> PBO.Hence, on the basis of the above conclusions, the permselectivity is expected to be adversely affected by plasticization. An interesting situation arises in the separation of COZ(A) from CH, (B ), which is of considerable practical importance. Here, the heavier gas A is also the faster one (in line with its smaller effective molecular size indicated by penetration of molecular sieving crystals [ 71) , i.e. KA> KB and DTAo> I&no. Studies of the sorption and permeation of these gases in polyethylene [8,9] show that BA < BB (as expected, in view of the fact that these parameters should vary in line with the effective molecular size of the penetrant). No significant deviations from Henry’s law were noticeable in the experimental pressure range, i.e. bAz b z bm z b&4z 0. It follows that PA0> PBO, while kA< 0, kB> 0. Hence, here too, plasticization leads to reduction in permselectivity. The above predictions are in accord with the general experience that, in practice, gas separation is adversely affected by plasticization. Equation (24) may, with the aid of analogous first-order expansions for pA(fAo) and pB(fB,,) [cf. eqn. (ll)], be recast in the form ao, “N[~A(fAO)/~B(fBd [I+
1
%fm
(24a)
-&fAo)
1
52
J.H. Petropoulos/J.
where kk=GA
+GBA
(254
= @A+&3)P.4+b, +b3* k;3 =G,
+GAB
(25b)
= WA+&3)&i +b, +b*B For the He ( A)-CH, (B ) example considered above, we have k$, z 0, kb > 0. Hence, the permselectivity in mixed gas flow tends to be better than indicated by the ratio of the effective puregas permeabilities for the same fAOand fsO values. In the example of CO, ( A)-CH, (B ) , on the other hand, the data of Refs. [8,9] indicate PA> /3n (i.e. CO, is a more effective plasticizer than CH,). Hence, eqn. (25) yields ka > kb and eqn. (24a) shows that the outcome depends on the composition of the mixture AB: for f,,, wfnO, a$$ will be less than PA (fAO)/& (feO).
Application of the treatment of plasticization given above to glassy polymer membranes is subject to considerable complications due to the microheterogeneity of the polymer. The simplest reasonably successful approach to sorption and transport in such systems so far developed, is to regard the glassy polymer as consisting of (i) a uniform dense matrix, wherein the penetrant molecules “dissolve” exactly as above T, and (ii) semipermanent microcavities interspersed in the said matrix (and collectively constituting the “excess free volume” of the polymer occupying a volume fraction u? ), which act as adsorption sites (concentration sA2) for the penetrant molecules (e.g. [ 5,6] ). The concentration, and the resulting solubility coefficient, of penetrant molecules “dissolved” or “adsorbed” in the polymer in the above manner are denoted by CAl,SAl and CA2,SA2respectively; and their mobility is char-
Sci. 75 (1992) 47-59
acterized by thermodynamic diffusion coefficients &A1 and &A2 respectively [6,7]. According to the simple dual-mode sorption and transport models [5,25], we have, in the absence of plasticization, sA=cA/fA=sA,+sA2 (26) =KAl
PA=pAl
+h&2/(1+&fA) +PA~=SAIDTAI+SA~DTA:!
(27)
where sA2 has been formulated in terms of a simple Langmuir adsorption isotherm, characterized by capacity and affinity constants sA2 and KA2 respectively; DTA1, DTA2 are constant. Equations (26) and (27) lead to the following expression for the integral permeability (with fAl=o) PA
=PAl +PA2 =KAlDTAl +
Pure penetrant transport in glassy polymers
Membrane
(2&b)
SA~DTAP~~(~+KA~~AO)/~AO
On the basis of the above models, the treatments of plasticization previously given is immediately applicable to the dissolved penetrant molecules (i.e. to PAI ), except for the fact that, in general, absorbed molecules may also exercise some plasticizing effect. In an attempt to allow for the latter effect, an “effective” concentration CA, has been defined by means of eqn. (29) below CAe
=cAl
+&CA%
(29)
where &,= const. < 1 has been evaluated from correlations of gas sorption and polymer dilation measurements (e.g. [ 121). As far as PA2 is concerned, two main opposing plasticization effects are to be expected, namely (a) a decrease in sA2, in line with the and (b) expected decrease in uF as T,-+T[6] an increase of DTA2. The former effect has been formulated as [ 121 SA2(CAe)=SA20(1-CAe/CAG)
(30)
53
J.H. PetropoulosjJ. Membrane Sci. 75 (1992) 47-59
where sA~O~SA~ (CA = CA,= 0) ; CAo represents the value of CAat which glass transition occurs, at the given experimental temperature, i.e. T,(CA=CAo)=T, and sA~(CA>CA~)=O. An alternative formulation is SA2(CA)=SA20[Tg(CA)-Tl
(31)
/[T,(CA=Ob-Tl
where Tg (CA) is given by an appropriate theory andSA2(CA)=OforTg(CA)~T[13]. Empirical formulations of effect (b) have been given [ 14,151. The simpler approach of Ref. [ 141 is equivalent to (i) assuming &As/ D TAl- -const. and thereby transforming eqn. (27) to
tiguous adsorption sites being sufficiently rare to be neglected [ 261. In the simple dual mode transport model approach [ 25,26,28-301, the aforementioned kinds of activated diffusion jump give rise to corresponding additive molecular flux components. Thus, the total (molar) flux density in the forward ( +x) direction is given by [ 261 IA=
(~-SAZ/SA~)U~A~~AI
+
(32)
(sA~/sA~)u:A~CAI+CA~U~A~
In eqn. (32), u,Al, u:A1 and &.A2are the effective molecular velocities pertaining to the aforementioned kinds of diffusion jump, respectively, and are given by
(27a)
where Sk, is an effective solubility coefficient and (ii) setting $,,=DTAS/DTA1 in eqn. (29). The treatment of Ref. [ 151 on the other hand, is equivalent to assuming that DTAl and DTA2 are independent functions of CA1 and of CA, respectively. In order to establish a physical basis from which to proceed in this matter, we refer to the molecular aspects of the dual mode transport model set out in detail in previous papers [ 26,271. According to this treatment, both dissolved and adsorbed penetrant molecules in their ground state occupy sites between the polymer chains; the difference between “solution” and “absorption” sites being that the potential energy wells (cf. Fig. 2 ) representing the former are shallower and much more numerous (concentration aA ) than those representing the latter (i.e. s& << sA1) . The penetrant molecular flux is made up of elementary activated molecular jumps over three kinds of potential energy barrier, namely E&+ (solution site to adjacent solution site), EA1+’ (solution site to adjacent adsorption site) and EAz+ (adsorption site to adjacent solution site); the occurrence of con-
u,A2=
u:A2yA2
/ Y:2
(33c)
where uz denotes the effective molecular velocity at infinite dilution; y is the activity coefficient describing deviation from thermodynamic ideality; the superscript ( # ) denotes the activated state; and 0 uzA=dAe(kBT/h)exp(ASAf/R) Xexp(
-E;/RT)
(34)
=u”,exp( -E,f/RT) In eqn. (34), kB and h are the Boltzmann and Planck constants respectively AA is the mean effective jump length (which may for simplicity be assumed to be the same for all kinds of molecular jump considered above); K is a geometrical factor (equal to l/6 for a simple threedimensional random walk); and AS,Z , E 2 denote (molar) activation entropy and energy respectively ). E 2 plays the dominant role in eqn. (34) (note also that there is a high degree of correlation between AS: and Ez, cf. Fig. 12 in chap. 2 of Ref. [l] ). In accordance with the
._-P J.H. Petropoulos/J.
54
Tf
Membrane
Sci. 75 (1992) 47-59
I -f
En1
-A2 I .___
L
SITE
t ADSORPTION SITE
Fig. 2. Schematic representation of the variation of potential energy along the path of a diffusing penetrant molecule, aciording to the dualmode transport model.
simple dual mode sorption model of eqn. (26)
[261, YAl=Yil=l;
(35a)
YA~=Y~~=(~-CA~/SA~)-~=~+KA~~A
(35b)
It is considered that local equilibrium between adsorbed and dissolved molecules is maintained at all times, i.e. (36)
v:A1CA1/SAI=vxA2CA2/SA2
Hence, the last two terms in eqn. (32) are equal. In Ref. [ 261, the final result ior the net unidimensional flux density JA= JA- JA was derived in terms of V,A$A~ [cf. eqn. (13) therein], because it leads to the simplest molecular interpretation of the standard macroscopic treatment of eqns. (1) and (27), namely D TAl=
D TA~~~(SA~/SAIKA~)DTA~
(1-SA2/SA1bxA1AA
=(l_SA2/SA1)U~lexp(--EAZ,/RT)
D TA2=2v,A2lA
=2u&exp(
(37)
-E,f,/RT)
(38)
However, an entirely equivalent result is obviously obtainable in terms of vLA1 and CA,. This alternative formulation modifies eqn. (38) to [cf. eqn. (36) and (26) also] D TA2=2(~A2SAl/SA1YA2SA2)U~1 xexp(-KdW (3W =2(~Al/~Al~A,h& xexp(--E3RT)
Note, incidentally, that in Ref. [29] separate diffusion coefficients were defined for the last two terms of eqn. (32 ). This practice, which has been adopted in a few other cases also, is not physically meaningful, in the opinion of the present author; in view of eqn. (36) and of the additional consideration that the corresponding net flux components are not separable in this way (since the forward flux component due to solution-site-to-adsorption-site jumps is associated with the backward flux component due to adsorption-site-to-solution-site jumps and vice versa). Equation (38a) is useful in the present context; because, in conjunction with eqn. (37)) it shows immediately that, if EA’; = EL (uZ1 zugl ), then (39)
=2pA1/sAlK~2
Equation (39) shows that, under the specified conditions, the concentration dependence of D TA2due to plasticization will follow that of DTAl (Or, more precisely, of PA1), since KA2and sA1are presumed to remain constant. This provides physical justification for the treatment based on eqn. (27a) with Sk given by sa=SAl[1+2SA2/S~l(1+K~2f~)l
(46)
The question now is whether the condition Ez{ z Ezl can plausibly be expected to hold in
J.H. PetropoulosjJ.
55
Membrane Sci. 75 (1992) 47-59
practice. The analysis of dual mode sorption and transport data given in Refs. [ 271 and [ 281 indicates that the answer to this question is most likely negative. This conclusion is important, because the aforesaid condition constitutes the basis on which the validity of the simple dual mode transport model in its various forms [ 25,26,28-301 rests [ 26,271. A more general treatment [ 271, which is not subject to this restriction, shows that simple additive dual mode transport formulations (eqn. 27) result in the following cases: (i) uiAl << u,_& (E;f; s-E&)); (ii) &&I =u,Al (E&‘=E&),i.e. the case considered above; and (iii) u:_&r>> u,Ar (E&’ <
=
P(1+Q)2/Q1
(sAl/sAlKA2)
(41)
x uL exp ( - E;f,/RT) Equation (41) is identical with eqn. (39)) except for an additional numerical factor [which carries over into eqn. (40) also]. The data analysis referred to above suggests that the conditions prevailing in typical real systems lie between cases (ii ) and (iii ) and probably close to (iii) [27]. We conclude, therefore, that within the model framework of eqn. (27)) treatment of plasticization effects on the basis of eqn. (27a) appears to be justified on physical grounds. On the other hand, no comparable justification can be found at the present time for the assumption
Of $,,= &&&.A~ Of Ref. [ 141 Or for the formulation of DTAZas a function of CA2adopted inRef. [15]. In the light of the above discussion and in accord with the treatment of Ref. [12], an expression for pA(fA) may be developed on the basis of eqn. (27) by substituting CA in eqn. (4a) and (7) by CA, defined in eqn. (29). The resulting expression is rather complex, but simplifies in the case of &,x 0 to pA
=
[PAlO
(42)
+PAZO(l--fA/fAG)
/(1+&fA)]exp(GA&) where PA~OGPA~ (f~ = 0 ), PA20cPA2(fA = 0 ) [Cf. eqns. (26 ), (27 ) ] ; GA1is given by the analogue of eqn. (lob); and eqn. (30) has been rewritten in terms of fA with for convenience sA2(fA >.fAo) = 0. In practice, @,, may be nearly zero [31] but may also attain substantially higher values [ 121. It is not expected, however, that the behavior of PA in the latter case will be substantially different. Equation (42 ) shows that, in the absence of plasticization, PA will tend to decline with increasing fA; in keeping with experimental findings which show that PA(fAo,fAl=O) is typically a decreasing function off_&, [6,11]. In the complete absence of plasticization (GA1= 0) , a limiting value is attained at sufficiently high pressure, namely pA(fA+d+~A(fAO+~,
fAl=o)*pAKl.
How-
ever, in the presence of even very weak plasticization, the exponential factor in eqn. (42 ) will ultimately lead to a reversal of the above trend, thus causing PA (fA) and PA (f_&Al= 0) to pass through a minimum; again in keeping with what is observed in practice [lo]. In some cases, no initial decline of PA (f_&,,)with increasing fAOis found [ 111. The condition for this to happen follows from eqn. (42) upon expansion to first order in f_&and integI%tiOn yielding (for fAl= 0) PA
=PAIo(~+
f (GAI~Ao)
+pA20
56
J.H. Petropoulos/J. Membrane Sci. 75 (1992) 47-59
x{ln(l+KA2fAo)lKA2fA0+ (GA1-l/fAd X t1-ln(l+~A2fAO)I~A2fADll~A2}
(43)
which for fAO-+Oreduces to WP40 = I+ t [GA*- (P~,/p~io) X (KU + l/fAG)I
(43a)
H(fAO,fBO)= (1 -fAObfAG
(~+PA~O/PAIO)I~AO
According to eqn. (31a), PA increases with f&, throughout, if the coefficient of fAOis positive. Mixed penetrant transport in glassy polymers In the presence of a second penetrant B, eqn. (42) may be extended in the following way
P AM=
where
/(I+
;&,fAo
-fBO/zfBG)
(48)
+ #mfBo)
and &, kg1 are given by the analogues of eqn. (23). The above simplified treatment is justifiable, at least for practical purposes, if the dualmode transport effect on cu0,is not unduly large. The fact that this is normally so follows from our previous study [24], which shows that, in the absence of plasticization ( kA1M0, Kmz 0, CX&’is given to a good apf AG-+GQ, fsc+co), proximation in practice by
[PAIO+PA~O(~-~A/~AG-~B/~BG)
(44)
/(~+KA~~A+KB~~B)I
(49)
xexP(GA1fA+GABlfB) on the basis of eqn. (19a) and extensions of eqns. (26) (cf. Refs. [6,32] ) and (30) [as modified in connection with eqn. (42 ) ] to eqns. (45) and (46)) respectively, below SA,=SA,KA,/(~+KA~~A
+K,,f,)
SA2=SA20(1-fA/fAG-fB/fBG)
(45) (46)
where s&.o=s& (fA = O,fB = 0) and aA = 0 when fA/fAG+fB/fBG>l.
In eqn. (44), GABI is given
by the analogue of eqn. (20a) and &, w 0 is assumed as before. Further progress by analytical methods does not seem possible, unless the second term within the brackets in eqn. (44) can be replaced by a “mean value”; in which case the intrinsic permselectivity will be given by the analogue of the combined eqns. (14a) and (21)) namely
where F(fAO,fBO)
=ln(l+K~,fAO
+KmfBo)/
(KA~~Ao +KmfBo)
(50)
A somewhat different formulation of F(fAo9fBo) follows from the treatment of Ref. [ 331; but the general predicted behavior is the same, with F(O,O)=l andF(fAo,fBo)bOas (fAO+fBO)+m. It follows that the effect on a$ is determined by the relative magnitude of PA20/PA10and PBzo IPmo- The general tendency is for Pzo/Plo to be higher for the heavier, more strongly interacting gases [ 24,261. Hence, permselectivity would, in general, be expected to be enhanced (reduced), if the basic selectivity of the membrane, namely PAlo/Pelo, favors the heavier (lighter) more strongly (less strongly) interacting gas. As pointed out in Ref. [ 241, how-
J.H. PetropoulosjJ.
Membrane
Sci. 75 (1992) 47-59
57
TABLE 1 Calculated values of ff& (P,,,/P,,,) for the case of P&P,,,=O.97, P,,,/P,,,=O.17, KA2=0.215 atm-‘, K,,=O.O74 atm-r, in the absence of plasticization (Case 2 of Ref. [ 241): (a) “exact” values obtained by the numerical procedure of Ref. [ 241; (b) from eqns. (49) and (50); (c) from eqns. (49) and (50a)
fdfm
0
2 6 12 20
= 0.20
fAolfLul= 0.50
fdfm
= 0.80
(a)
(b)
(cl
(a)
(b)
(cl
(a)
(b)
(c)
1.684 1.630 1.548 1.464 1.390
1.684 1.630 1.551 1.470 1.399
1.684 1.631 1.547 1.456 1.373
1.684 1.611 1.510 1.417 1.341
1.684 1.611 1.513 1.422 1.348
1.684 1.620 1.523 1.422 1.336
1.684 1.593 1.479 1.381 1.306
1.684 1.594 1.482 1.385 1.311
1.684 1.612 1.506 1.402 1.315
ever, the observed differences in P2,,/P10are not very pronounced, because of partial compensation of the opposing tendencies of the sorption and diffusion parameters contained therein. Their practical effect will be further substantially diminished by the low values of F(f.&s,,) corresponding to the high pressures used in normal gas separation operations. As a practical illustration of these points we have chosen Case 2 studied in Ref. [ 241, because it may be considered an extreme example of what may be encountered in practice. The results of model calculations in the (total) gas pressure range of O-20 atm are given in Table 1. The results of column (a) are included in the table to illustrate the point made earlier that eqns. (49) and (50) yield excellent approximations to the “exact” numerical values; whereas column (c ) serves to show that a mean value approximation of the type made in eqns. (47), (48), namely GOa) also gives very good results for practical purposes in this case. It will be noted that, even in the extreme example under consideration here, the dual mode transport effect on permselectivity remaining at 20 atm is not large. In the presence of plasticization, further reduction of
this effect, due to the numerator appearing in eqn. (48)) would be expected. Conclusion An attempt has been made, in the present paper, to approach the problem of modelling the effect of plasticization on gas permeation and separation in a reasonably thorough and systematic way. Previous extensive work on vapor transport has led to a reasonably well established physical theory for this purpose above T,. We have shown, however, that some useful simplification is possible (at the expense of relatively little loss in rigor), which constitutes a significant improvement over comparable formulations of a more empirical character currently employed in the treatment of pervaporation. Some further simplification is possible in the case of gas separation, as a result of the relatively weak plasticization effects involved. On this basis, the effects to be expected in particular cases can be immediately deduced and the precise reasons why practical gas separation is commonly found to be adversely affected by plasticization, understood. Below Tg, the situation is complicated by the microheterogeneity of the polymeric medium, but we have shown that the dual mode trans-
J.H. PetropoulosjJ. Membrane Sci. 75 (1992) 47-59
58
model, properly understood, provides a useful physical basis permitting a relatively simple extension of the treatment of plasticization above Tg. Further simplifications are needed, however, in order to keep within the bounds of a readily applicable analytical treatment. It is, nevertheless, expected that the results obtained here will prove useful, at least as a first step, for the analysis and interpretation of pertinent experimental data. port
13
14
15
16 17
References 1 2 3
4
5 6
I
8
9
10
11
12
J. Crank and G.S. Park, Diffusion in Polymers, Academic Press, New York, NY, 1968. H. Fujita, Diffusion in polymer-diluent systems, Fortschr. Hochpolym. Forsch., 3 (1961) 1. J.S. Vrentas and J.L. Duda, Diffusion in polymer-solvent systems. I. Reexamination of the free-volume theory; II. A predictive theory for the dependence of diffusion coefficients on temperature, concentration and molecular weight, J. Polym. Sci., Polym. Phys. Ed., 15 (197’7) 403,417. V.T. Stannett, The transport of gases in synthetic polymeric membranes-An historic perspective, J. Membrane Sci., 3 (1978) 97. W.R. Vieth, J.H. Howell and J.H. Hsieh, Dual sorption theory, J. Membrane Sci., 1 (1976) 177. J.H. Petropoulos, Membranes with non-homogeneous sorption and transport properties, Adv. Polym. Sci., 64 (1985) 93. J.H. Petropoulos, Some fundamental approaches to membrane gas permeability and permselectivity, J. Membrane Sci., 53 (1990) 229. S.S. Kulkarni and S.A. Stern, The diffusion of COz, CH,, C&H, and f&H, in polyethylene at elevated pressures, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 441. S.A. Stern and S.S. Kulkarni, Tests of a “free-volume” model of gas permeation through polymer membranes. I. Pure COz, CHI, C,H, and C&H, in polyethylene, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 467. SM. Jordan, W.J. Koros and G.K. Fleming, The effects of COz exposure on pure and mixed gas permeation behavior: comparison of glassy polycarbonate and silicone rubber, J. Membrane Sci., 30 (1987) 191. J.S. Chiou and D.R. Paul, Effects of CO2 exposure on gas transport properties of glassy polymers, J. Membrane Sci., 32 (1987) 195. Y. Kamiya, K. Mizoguchi, T. Hirose and Y. Naito, Sorption and dilation in poly (ethyl methacrylate ) carbon dioxide system, J. Polym. Sci., Part B, 27 (1989) 879.
18
19
20
21 22
23
24
25
26
27
28
29
J.S. Chiou and D.R. Paul, Gas sorption and permeation in poly (ethyl methacrylate), J. Membrane Sci., 45 (1989) 167. S.A. Stern and V. Saxena, Concentration-dependent transport of gases and vapors in glassy polymers, J. Membrane Sci., 7 (1980) 47. S. Zhou and S.A. Stern, The effect of plasticization on the transport of gases in and through glassy polymers, J. Polym. Sci., Part B, 27 (1989) 205. J. Crank, The Mathematics of Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, p. 209. V.M. Shah, B.J. Hardy and S.A. Stern, Solubility of carbon dioxide, methane and propane in silicone polymers: effect of polymer side chains, J. Polym. Sci., Part B, 24 (1986) 2033. M.S. Suwandi and S.A. Stern, Transport of heavy organic vapors through silicone rubber, J. Polym. Sci., Polym. Phys. Ed., 11 (1973) 663. M.H.V. Mulder and C.A. Smolders, On the mechanism of separation of ethanol/water mixtures by pervaporation. I. Calculation of concentration profiles, J. Membrane Sci., 17 (1984) 289. J.-P. Brun, C. Larchet, G. Bulvestre and B. Auclair, Sorption and pervaporation of dilute aqueous solutions of organic compounds through polymer membranes, J. Membrane Sci., 25 (1985) 55. A.C. Newns, Diffusion of benzene in polymethyl acrylate, Trans. Faraday Sot., 59 (1963) 2150. A. Kishimoto, Diffusion and viscosity of polyvinyl acetate-diluent systems, J. Polym. Sci., Part A, 2 (1964) 1421. S.A. Stern, G.R. Mauze and H.L. Frisch, Test of a free-volume model for the permeation of gas mixtures through polymer membranes. C02-&HI, COz-CaHs and C,H,-C&H, mixtures in polyethylene, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 1275. J.H. Petropoulos, Formulation of dual-mode mixed gas transport in glassy polymers, J. Membrane Sci., 48 (1990) 79. J.H. Petropoulos, Quantitative analysis of gaseous diffusion in glassy polymers, J. Polym. Sci., Part A2, 8( 1970) 1797. J.H. Petropoulos, On the dual-mode gas transport model for glassy polymers, J. Polym. Sci., Part B, 26 (1988) 1009. J.H. Petropoulos, A generalized, topologically consistent, dual-mode transport model for glassy polymergas systems, J. Polym. Sci., Part B, 27 (1989) 603. D.R. Paul and W.J. Koros, Effect of partially immobilizing sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed., 14 (1976) 675. R.M. Barrer, Diffusivities in glassy polymers for the dual-mode sorption model, J. Membrane Sci., 18 (1984) 25.
J.H. Petropoulos/J. 30
31
Membrane Sci. 75 (1992) 47-59
G.H. Fredrickson and E. Helfand, Dual-mode transport of penetrants in glassy polymers, Macromolecules, 18 (1985) 2201. G.K. Fleming and W.J. Koros, Dilation of polymers by sorption of carbon dioxide at elevated pressures. 1. Silicone rubber and unconditioned polycarbonate, Macromolecules, 19 (1986) 2285.
59
32
33
W.J. Koros, Model for sorption of mixed gases in glassy polymers, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 981. W.J. Koros, R.T. Chern, V. Stannett and H.B. Hopfenberg, A model for permeation of mixed gases and vapors in glassypolymers, J. Polym. Sci., Polym. Phys. Ed., 19 (1981) 1513.