Permeation Breakthrough Models for Associating and Solvating

Ind. Eng. Chem. Res. 1997, 36, 483-492

483

Permeation Breakthrough Models for Associating and Solvating Penetrants in a Membrane Sameer S. Kasargod and Timothy A. Barbari* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218

Transport models are developed for diffusion with self-association or intersolute solvation through a membrane. Permeation breakthrough is defined as the time at which a specified amount of penetrant has accumulated downstream of the membrane. Numerical solutions for the transient portion of the cumulative flux expression are used to determine the effect of association or solvation on the breakthrough time. At low fractions of unassociated or unsolvated penetrant at the upstream surface, the breakthrough time increases with decreasing mobility of the associated or solvated species. However, the effect of finite kinetics results in breakthrough times that are either greater or less than those at local equilibrium depending on the mobility of the associated or solvated species. These results are discussed in terms of local sources and sinks of penetrant. For materials of equivalent penetrant solubilities, those in which the equilibrium strongly favors the formation of immobile clusters or complexes will have measurably longer breakthrough times. Introduction Polymeric films find wide ranging applications as barrier materials; examples include food packaging, noncorrosive coatings, and protective materials for hazardous environments. Typically, in these applications, it is not the steady-state flux that is of interest, but the transient portion during which a critical amount of penetrant has permeated across the membrane. The time required to reach this critical level is often less than the characteristic diffusion time lag of the system. The time lag is defined as the intercept on the time axis upon extrapolation of the steady state portion of the cumulative mass flux expression. For simple Fickian diffusion of a single penetrant across a membrane of thickness L with constant surface concentrations, one of which is maintained at zero, the time lag is given by L2/6D, where D is the diffusion coefficient (Crank, 1975). Expressions for the time lag also have been developed to account for immobilization of the penetrant. These include those developed by Goodknight and Fatt (1961) for diffusion in porous media with dead-end pore volume, by Paul (1969) for equilibrium dual-mode sorption in glassy polymers, by Tshudy and Von Frankenburg (1973) for reversible adsorption at fixed sites, and by Meldon et al. (1985) for an immobilizing reversible chemical reaction. Relaxing the assumption of total immobilization, Paul and Koros (1976) developed a time lag expression for equilibrium dual-mode sorption that was able to predict experimental data for gas molecules in glassy polymers. In the studies of transient permeation cited above, the emphasis was placed on single penetrant molecules undergoing immobilization or single penetrants of the same size with different mobilities. The effect of association to form clusters or solvation with a second penetrant to form complexes has not been treated in detail owing to the coupled nature of the equations and the need for a numerical solution. However, many barrier applications involve a penetrant, such as water, that is capable of self-association within a barrier polymer or able to solvate with a second penetrant. An * To whom correspondence should be addressed. Voice: (410) 516-5540. Fax: (410) 516-5510. E-mail: [email protected]. S0888-5885(96)00404-6 CCC: $14.00

understanding of how association and solvation affect permeation breakthrough could lead to the improved design of barrier materials. In this paper, transport models are developed for two different cases involving strong penetrant interactions. First, diffusion with molecular isodesmic association is considered. Isodesmic association and micellar association are two common frameworks for systems in which a penetrant self-associates to form clusters (Cussler, 1984). The former, based on stepwise addition, was used by McKeigue and Gulari (1989) to study the effect of association on diffusion in a liquid. They state that isodesmic association applies to many molecules that form hydrogen bonds or π-bonds. The latter framework, based on the formation of clusters of a favored size, is not considered here but may be applicable in some cases. Experiments are necessary to determine the appropriate association framework for a given system. Second, a model for diffusion with intersolute solvation is presented which takes into account the interactions between two different penetrants. The second case is similar to those that have been developed in the context of facilitated transport by several authors (Ward, 1970; Kutchai et al., 1970; Yung and Probstein, 1973; Way et al., 1982; Sasidhar and Ruckenstein, 1983; Folkner and Noble, 1983). General solutions to the transport equations are presented in terms of a permeation breakthrough time. Neither local equilibrium nor immobilization is assumed, but these limits will be presented and compared to the full solutions. The results are discussed in terms of the competing mechanisms of diffusion and either association or solvation as penetrant molecules are transported across the barrier film. Isodesmic Association Model Development. For a single penetrant capable of molecular association, the system can be modeled in terms of the diffusion of the various associated species (Cussler, 1983). The simplest approach to consider is that of isodesmic or linear association. In this case, the monomeric or unassociated molecules add in a stepwise manner. If A1 represents monomer or unassociated penetrant, then isodesmic association can © 1997 American Chemical Society

484 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997

Cn ) C2 + C3 + ... + CN-1 + CN

be written as

A1 + A1 h A2

(1)

A2 + A1 h A3

(2)

A3 + A1 h A4

(3)

∂2C1 ∂C1 ) D1 - kr[KC1(C1 + Cn) - Cn] ∂t ∂z2

(12)

(4)

∂Cn ∂2Cn kr ) Dn + [KC1(C1 + Cn) - Cn] ∂t n ∂z2

(13)

and N is very large, then the system of equations reduces to

l AN-1 + A1 h AN

where N is very large. The following continuity equations can then be written for each of the N species 2

∂ C1 ∂C1 ) D1 - kfC1(C1 + C2 + ... + CN-1) + kr(C2 + ∂t ∂z2 C3 + ... + CN) (5)

(

) (

∂C2 ∂2C2 C1 C2 ) D2 - C2 - kr - C3 + kfC1 2 ∂t 2 2 ∂z

)

where K is the equilibrium constant given by

K ) kf /kr

N

n) (6)

∂CN ∂2CN ) DN + krC1CN-1 - krCN ∂t ∂z2

where Ci is the concentration of component Ai, Di is its diffusion coefficient, and kf and kr are the forward and reverse rate constants. The assumptions made in eqs 5-9 are that the molar density in the membrane is constant and that the diffusion coefficients are independent of concentration. It is important to note that the forward and reverse rate constants are assumed to be independent of size (Stokes, 1965) and therefore constant. In this paper, eqs 5-9 are simplified by combining eqs 6-9 into a single continuity equation for 2 e i e N. This simplification is made for two reasons: (1) the unassociated penetrant is separated from all associated species or clusters to examine the case of immobile clusters, and (2) the unassociated penetrant can be separable from associated penetrant under certain conditions in Fourier transform infrared (FTIR) spectroscopy. Configured for attenuated total reflectance (ATR), FTIR spectroscopy is finding application as a tool for studying the diffusion of associating penetrants in polymers (Fieldson and Barbari, 1993; Fieldson and Barbari, 1995; Plunkett, 1995). Addition of eqs 6-9, appropriately weighted for the number of molecules in each cluster, gives

∂

∂

(15)

iDiCi ∑ i)2 N

(16)

iCi ∑ i)2 Here, n is the concentration-averaged size of the clusters, and Dn is an average concentration-weighted diffusion coefficient for the clusters. The total concentration, CT, is defined as

CT ) C1 + nCn

(17)

Initially, the membrane does not contain any penetrant; at t ) 0, for all z

C1 ) 0

(18)

Cn ) 0

(19)

The concentrations at the downstream surface are negligible relative to the upstream concentrations (zerosink condition); at z ) 0, for all t

C1 ) 0

(20)

Cn ) 0

(21)

At the upstream surface, equilibrium exists between the unassociated and associated species for an infinite reservoir of penetrant and constant partition coefficients; at z ) L, for all t

2 N

iDiCi + kfC1(C1 + C2 + ... + CN-1) ∑iCi ) 2∑ i)2

∂ti)2

N

N

Dn )

(9)

iCi ∑ i)2 Ci ∑ i)2

l ∂CN-1 ∂2CN-1 ) DN-1 + kfC1(CN-2 - CN-1) ∂t ∂z2 kr(CN-1 - CN) (8)

(14)

and n and Dn are given by

∂C3 ∂2C3 ) D3 + kfC1(C2 - C3) - kr(C3 - C4) (7) ∂t ∂z2

N

(11)

C1 ) C01

(22)

Cn ) C0n

(23)

and

∂z

kr(C2 + C3 + ... + CN) (10) If Cn is defined as

K)

C0n C01(C01 + C0n)

(24)

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 485

In this work, the total penetrant concentration at the upstream surface, C0T, is maintained at a constant value to focus on the effect of the distribution between unassociated and associated species on the transport process. The total amount permeated per unit area, Q, at the downstream surface after time t is given by

Q ) D1

∫0t

( ) ∂C1 ∂z

z)0

dt + nDn

∫0t

( ) ∂Cn ∂z

z)0

dt

µ)

∫0τ

( ) ∂θ1 ∂ζ

Cn ) (KC21)/(1 - KC1)

(26)

1

1

1

2nKDn

( )

∂C1 2 (1 - KC1)[(1 - KC1) + nKC1(2 - KC1)] ∂z

2

(27)

Numerical Solution Procedure. Owing to the coupled nature of the species continuity equations and the nonlinearity of the association terms, the model developed above was not amenable to an analytical solution and was therefore solved numerically. To facilitate a numerical solution, the species continuity equations were cast into dimensionless form using the following variables:

θ1 ) C1/C0T

(28)

Cn/C0T

(29)

θ2 )

2

τ ) (D1t)/L

(30)

ζ ) z/L

(31)

Equations 12 and 13 become

∂θ1 ∂2θ1 ) 2 - m2[m1θ1(θ1 + θ2) - θ2] ∂τ ∂ζ

∂θ2 ∂ζ

ζ)0

dτ

(37)

θ1 ) 0

(38)

θ2 ) 0

(39)

θ1 ) 0

(40)

θ2 ) 0

(41)

and those at the upstream surface (at ζ ) 1, for all τ) are transformed to

θ1 ) θ01

(42)

θ2 ) θ02

(43)

θ01 and θ02 are not independent parameters but are related through the equilibrium expression. In addition, the assumption of constant total concentration at the upstream surface results in another equation relating θ01 and θ02. These relations in dimensionless form are

m1 )

θ02

(44)

θ01(θ01 + θ02)

θ01 + nθ02 ) 1

(45)

Therefore, only one of the three parameters (m1, θ01, and θ02) can be specified independently. In dimensionless form, eq 27 is transformed to

∂θ1 (1 - m1θ1)2 + nm3m1θ1(2 - m1θ1) ∂2θ1 ) + ∂τ (1 - m θ )2 + nm θ (2 - m θ ) ∂ζ2 1 1

1 1

1 1

2nm1m3

( )

∂θ1 2 (1 - m1θ1)[(1 - m1θ1)2 + nm1θ1(2 - m1θ1)] ∂ζ (46) and the expression for local equilibrium becomes

(32)

∂θ2 ∂2θ2 m2 ) m3 2 + [m1θ1(θ1 + θ2) - θ2] ∂τ n ∂ζ

(33)

m1 ) KC0T

(34)

m2 ) (krL2)/D1

(35)

m3 ) Dn/D1

(36)

where

Equation 25 becomes

( )

The boundary conditions at the downstream surface (at ζ ) 0 for all τ) become

Adding eqs 12 and 13 and substituting eq 26 gives the following equation describing the transport of the unassociated penetrant for the special case of local equilibrium:

∂C1 D1(1 - KC1)2 + nKC1Dn(2 - KC1) ∂2C1 ) + ∂t (1 - KC )2 + nKC (2 - KC ) ∂z2

∫0τ

dτ + nm3

where µ is the dimensionless amount permeated. In dimensionless form, the initial conditions (at τ ) 0 for all ζ) become

(25)

in the most general case. If the clusters are taken as immobile, the second term is zero, and the cumulative amount permeated depends solely on the amount of monomer and rate at which it diffuses. If local equilibrium between the various species is assumed, association is instantaneous, and the cluster concentration at any position in the membrane, at any time, is determined by the equilibrium expression

ζ)0

θ2 ) (m1θ21)/(1 - m1θ1)

(47)

This system of equations was solved using the explicit forward-in-time, central-in-space finite difference algorithm. The space domain from the upstream surface to the downstream surface was divided into 41 node points, and a dimensionless time step of 0.0001 was used. Second-order accurate expressions were used for all spatial derivatives, and first-order accurate expressions were used for all time derivatives. The order of error for spatial and temporal derivatives was 6.25 × 10-4 and 1 × 10-4, respectively. The flux at the downstream surface was integrated over time in order to obtain the total amount permeated. The integration was performed using the trapezoidal rule for numerical integration with the same dimensionless time step.


Figure 1. Verification of the numerical solution for the total amount permeated, µ, in the absence of molecular association. The curved line represents the numerical solution, and the closed circles represent the analytical solution given by eq 48.

Results and Discussion. Since short time behavior of the system is of interest, it is essential to define a breakthrough time for comparative purposes. For diffusion of a single component, Crank (1975) derived an expression for the total amount permeated after time t which, in the dimensionless variables defined above, is given by

µ)τ-

2

1 6

π

∞

∑ j)1

2

(-1)j 2

exp(-j2π2τ)

(48)

j

As τ approaches infinity, the expression for the amount permeated becomes linear (steady state) and is given by the expression

µ)τ-

1 6

(49)

By definition, the time lag in dimensionless form is equal to 1/6. In this work, the breakthrough time is defined as the time required for the total amount permeated to reach 10% of the value at the time lag in the absence of association. Breakthrough for membrane permeation depends on the amount that can be tolerated in a barrier application and, therefore, depends on the actual penetrant. For many applications, this cumulative amount can be very small. Ten percent was chosen arbitrarily, but it also represents a lower bound in terms of numerical reliability for the model. The dimensionless amount permeated at this definition of breakthrough is 0.0039, and the breakthrough time in dimensionless form, τb, is equal to 0.08375 without association. This breakthrough time was treated as a reference to evaluate the effect of association in the system. The ratio τ/τb henceforth will be referred to as the breakthrough delay, a measure of the effect that association has on increasing the breakthrough time. To verify the finite difference algorithm, the numerical solution in the absence of association was compared to the analytical solution (eq 48). Excellent agreement between the two solutions is shown in Figure 1. Numerical solutions were used for all of the cases here, with or without association. In Figure 2, the breakthrough delay is plotted as a function of the fraction of unassociated penetrant at the upstream surface, θ01, for five different Dn/D1 values and n ) 4. The ratio krL2/D1 was set equal to 1

Figure 2. Effect of the fraction of unassociated penetrant (θ01) and the mobility of the associated species (Dn/D1) on the breakthrough delay for krL2/D1 ) 1 and n ) 4. Numbers on the curves correspond to values of Dn/D1.

arbitrarily to generate solutions representing comparable time scales for diffusion and association. The ordinate range used in this figure is the same as that in Figure 3 for the purpose of comparison. At high fractions of unassociated penetrant at the surface, association has very little effect on delaying breakthrough. At these conditions, the short time flux can be attributed, almost entirely, to that of the unassociated species owing to the lower expected diffusion coefficient for the clusters. The time scale is too short for the mobility of the clusters to have any effect on the breakthrough time. However, when the fraction of the unassociated penetrant at the surface is low, the equilibrium is shifted to the associated species (high K), and the breakthrough delay is much greater than unity. At low θ01, the clusters contribute significantly to the amount permeated, and their lower mobility becomes dominant. Eventually, if the surface concentration is composed entirely of clusters, the breakthrough time is determined completely by the transport of the clusters, and the breakthrough delay converges expectedly to a value close to D1/Dn. Consequently, at low values of θ01 (high K), the breakthrough delay increases as Dn/D1 decreases, while at high values of θ01 (low K), the breakthrough factor is independent of the Dn/D1 ratio. In Figure 3, the breakthrough delay is plotted as a function of θ01 for the case of local equilibrium at five different values of Dn/D1 and n ) 4. In this case, the association is effectively instantaneous relative to diffusion (krL2/D1 approaches infinity). The trends observed are similar to those in Figure 2. However, the breakthrough delay at low θ10 (high K) for the same value of Dn/D1 can be either higher or lower in the case of local equilibrium compared to the case for finite association above. When the clusters are immobile (Dn/ D1 ) 0), local equilibrium results in considerably higher breakthrough delays at low θ01. As Dn/D1 increases, this trend declines gradually. Ultimately, when Dn/D1 is greater than or equal to approximately 0.5, the breakthrough delays in Figure 3 are slightly lower than their corresponding values in Figure 2. This behavior can be better understood by examining the breakthrough delay plotted against krL2/D1 at different values of θ01 and Dn/D1. The parameter krL2/D1 represents the relative rate of monomer formation by dissociation to monomer


Figure 3. Effect of the fraction of unassociated penetrant (θ01) and the mobility of the associated species (Dn/D1) on the breakthrough delay for local equilibrium (krL2/D1 ) ∞) and n ) 4. Numbers on the curves correspond to values of Dn/D1.

Figure 4. Effect of association kinetics on the breakthrough delay for immobile clusters (Dn/D1 ) 0) and n ) 4. Numbers on the curves correspond to values of θ01.

transport by diffusion. Since the rate of monomer disappearance by association is related to the product of K and kr, increasing krL2/D1 also increases the rate of association. krL2/D1 can vary from zero (uncoupled, independent diffusion of monomer and cluster) to infinity (diffusion with local equilibrium). At any position in the membrane, molecular diffusion creates local differences in monomer and cluster concentrations. If the local concentration of monomer is higher than the equilibrium value, association will be favored and the clusters serve as a monomer sink. If the monomer concentration is lower than the equilibrium value, then the clusters serve as a monomer source in an attempt to establish equilibrium by dissociation. The effect of krL2/D1 on the parallel mechanisms of diffusion, association, and dissociation depends strongly on the value of the equilibrium constant K and the mobility of the clusters. Figure 4 demonstrates the effect of increasing the kinetics of association/dissociation on the breakthrough delay when the clusters are assumed to be immobile for three different values of monomer fraction at the upstream surface θ01. Since θ01 and the equilibrium constant K cannot be independently specified, low values of θ01 correspond to high values of K and vice versa. The condition of immobile clusters ensures that local concentrations of monomer are greater than or

Figure 5. Effect of association kinetics on the breakthrough delay for Dn/D1 ) 0.2 and n ) 4. Numbers on the curves correspond to values of θ01.

Figure 6. Effect of association kinetics on the breakthrough delay for Dn/D1 ) 0.5 and n ) 4. Numbers on the curves correspond to values of θ01.

equal to the equilibrium values. Therefore, the clusters always represent a net monomer sink that removes unassociated penetrant from the diffusion process. At θ01 ) 0.01 (high K), the breakthrough time increases significantly with an increase in association/dissociation kinetics. As krL2/D1 increases, unassociated penetrant molecules undergo association to form immobile clusters faster than they can diffuse and considerably faster than monomer can be formed by dissociation. Of course, the limit of very high krL2/D1 represents local equilibrium (instantaneous association/dissociation). As a result, fewer unassociated penetrant molecules are left to make diffusional jumps and longer breakthrough times result. At θ01 ) 0.1 (intermediate K) in Figure 4, the increase in the breakthrough delay with increasing krL2/D1 is much smaller. At these conditions, the relative rates of association and dissociation are more comparable, but the clusters continue to act as net sinks owing to the immobility of the clusters. Finally, when θ01 is equal to unity (K ) 0), there is no association, and the breakthrough delay is unity. The effect of cluster mobility on breakthrough delay is shown in Figures 5 and 6 by setting Dn/D1 equal to 0.2 and 0.5, respectively. These values were chosen arbitrarily for the purpose of demonstration. Again, breakthrough delay is plotted against krL2/D1 for three different values of θ01 in each figure. Contrary to the case of immobile clusters shown in Figure 4, cluster


mobility may create local concentrations of monomer that are less than the equilibrium value. Under certain conditions, the clusters may serve as net sources of monomer which in turn may lead to lower breakthrough times as krL2/D1 increases. For example, at θ01 ) 0.01 (high K) in Figure 5, the breakthrough time decreases as the kinetics of association/dissociation are increased. At these conditions, the concentration of clusters at the upstream surface is very high and their diffusion results in local cluster concentrations in the membrane that are greater than the equilibrium concentrations. As a result, increasing krL2/ D1 results in higher cluster dissociation rates that generate additional monomer, increase the local concentration gradient of monomer, and lower the breakthrough time. However, when θ01 is 0.1, the mobility of the clusters is not high enough to overcome the higher local monomer concentration that results from the lower K value and the higher diffusion coefficient. As a result, the clusters revert to being a local sink for monomer, and the breakthrough time increases. In Figure 6, the mobility of the clusters is high enough that, for any value of θ01 (or K), the equilibrium concentration of clusters is exceeded locally and faster kinetics cause a shift to unassociated penetrant resulting in a lower breakthrough time. Intersolute Solvation Model Development. In a binary penetrant mixture, if the two solutes A and B are capable of solvation resulting in the formation of a complex, AB, then the expression

A + B h AB

2

∂ CA ∂CA ) DA - kfCACB + krCAB ∂t ∂z2

(51)

∂2CB ∂CB ) DB - kfCACB + krCAB ∂t ∂z2

(52)

∂2CAB ∂CAB ) DAB + kfCACB - krCAB ∂t ∂z2

(53)

(57)

Similar to association, the concentrations are maintained at zero at the downstream face. At z ) 0, for all t

CA ) 0

(58)

CB ) 0

(59)

CAB ) 0

(60)

At the upstream surface, the three components are in equilibrium owing to the conditions of an infinite reservoir of penetrants and constant partition coefficients. At z ) L, for all t

CA ) C0A

(61)

CB ) C0B

(62)

CAB ) KeqC0A C0B

(63)

In this study, the total penetrant concentration at the upstream surface is maintained at a constant value to focus on the effect of the distribution between unsolvated and solvated species on the transport process. This condition is given by

C0A + C0B + KeqC0A C0B ) C0T

(64)

The accumulated amount of A that has permeated across the downstream surface after time t is given by

(50)

can be used to describe the equilibrium. Continuity equations can be written for each of the three species

Q ) DA

∫0t

( ) ∂CA ∂z

z)0

dt + DAB

∫0t

( ) ∂CAB ∂z

z)0

(54)

Equations 51-53 assume that the molar density of the membrane is constant and that the diffusion coefficients are independent of concentration. Initially, the membrane is considered to be free of either penetrant. At t ) 0, for all z

CA ) 0

(55)

CB ) 0

(56)

dt (65)

For the special case of local equilibrium, solvation is instantaneous and CAB can be eliminated from eqs 5153 using

CAB ) KeqCACB

where Ci is the concentration of component i, Di is its diffusion coefficient, and kf and kr are the forward and reverse rate constants, respectively, for solvation. The ratio of kf to kr is the equilibrium constant, Keq, which is given by

Keq ) CAB/(CACB)

CAB ) 0

(66)

to give

( ( ( (

) ( )( )( ) ( ) ( )( )( ) (

) ) ) )

DA + KeqCBDAB ∂2CA KeqCADAB ∂2CB ∂CA ) + + ∂t 1 + KeqCB 1 + KeqCB ∂z2 ∂z2 2KeqDAB ∂CA ∂CB KeqCA ∂CB (67) 1 + KeqCB ∂z ∂z 1 + KeqCB ∂t ∂CB DB + KeqCADAB ∂2CB KeqCBDAB ∂2CA ) + + ∂t 1 + KeqCA 1 + KeqCA ∂z2 ∂z2 2KeqDAB ∂CA ∂CB KeqCB ∂CA (68) 1 + KeqCA ∂z ∂z 1 + KeqCA ∂t Equations 67 and 68 are used to describe the transport of the two unsolvated penetrants, and the equilibrium expression is used to determine the concentration of the complex. Numerical Solution Procedure. Equations 5153 are coupled through the solvation terms. Consequently, for transport governed by both diffusion and solvation, an analytical solution is not available, and thus, the system of equations was solved numerically. To facilitate a solution, the system of equations was converted into dimensionless form using the following


variables:

θ1 ) CA/C0T

(69)

θ2 ) CB/C0T

(70)

θ3 ) CAB/C0T

(71)

τ ) (DAt)/L2

(72)

ζ ) z/L

(73)

Equations 51-53 become

∂θ1 ∂2θ1 ) 2 - m′2(m′1θ1θ2 - θ3) ∂τ ∂ζ

(74)

2

Figure 7. Effect of the fraction of unsolvated A (θ01) and the mobility of the solvation complex (DAB/DA) on the breakthrough delay for krL2/DA ) 1, DB/DA ) 1, and equal concentrations of unsolvated A and B at the upstream surface. Numbers on the curves correspond to values of DAB/DA.

∂ θ2 ∂θ2 ) m′3 2 - m′2(m′1θ1θ2 - θ3) ∂τ ∂ζ

(75)

∂θ3 ∂2θ3 ) m′4 2 + m′2(m′1θ1θ2 - θ3) ∂τ ∂ζ

(76)

m′1 ) KeqC0T

(77)

m′2 ) (krL2)/DA

(78)

For the case of local equilibrium, eqs 66-68 are transformed to

m′3 ) DB/DA

(79)

θ3 ) m′1θ1θ2

m′4 ) DAB/DA

(80)

In this paper, θ01 and θ02 are chosen as the independent parameters and m′1 is calculated. Equation 65 for the total mass flux is transformed to

where

The initial conditions (at τ ) 0 for all ζ) become

θ1 ) 0

(81)

θ2 ) 0

(82)

θ3 ) 0

(83)

The boundary conditions at the downstream surface (at ζ ) 0, at all τ) are transformed to

θ1 ) 0

(84)

θ2 ) 0

(85)

θ3 ) 0

(86)

and those at the upstream surface (at ζ ) 1, at all τ) become

θ1 ) θ01

(87)

θ2 ) θ02

(88)

θ3 ) m′1θ01θ02

(89)

From eq 64, the total dimensionless concentration at the upstream surface is given by

m′5 + m′6 + m′1m′5m′6 ) 1

(90)

The above equation implies that out of the three parameters, m′1, θ01, and θ02, only two are independent.

µ′ )

(

∫0τ

( ) ∂θ1 ∂ζ

)

ζ)0

(

∂θ2 m′3 + m′1m′4θ1 ) ∂τ 1 + m′1θ1 2m′1m′4 1 + m′1θ1

(

( ) ∂θ3 ∂ζ

ζ)0

dτ

(91)

(92)

( ) )( )( ) ( ) ) ( ) )( )( ) ( )

∂θ1 1 + m′1m′4θ2 ∂2θ1 ) + ∂τ 1 + m′1θ2 ∂ζ2 2m′1m′4 ∂θ1 1 + m′1θ2 ∂ζ

(

∫0τ

dτ + m′4

m′1m′4θ1 ∂2θ2 + 1 + m′1θ2 ∂ζ2 ∂θ2 m′1θ1 ∂θ2 (93) ∂ζ 1 + m′1θ2 ∂τ

∂2θ2

m′1m′4θ2 ∂2θ1 + + 1 + m′1θ1 ∂ζ2 ∂ζ2 ∂θ1 ∂θ2 m′1θ2 ∂θ1 (94) ∂ζ ∂ζ 1 + m′1θ1 ∂τ

As was the case with association, this system of equations was solved using the forward-in-time, centralin-space explicit algorithm, with the same time step and number of node points. For the case of local equilibrium, difference equations for eqs 93 and 94 had to be solved algebraically at each time step owing to coupling through the temporal derivatives. The numerical integration was performed using the trapezoidal rule with the same dimensionless time step. Results and Discussion. In the absence of solvation and for pure A, the system of equations reduces to the same result as above in the absence of association. Hence, the same dimensionless breakthrough time equal of 0.08375 was used as a reference. For all of the figures below, except for Figure 12, the diffusion coefficient of B is set equal to that of A and the fractions of A and B at the surface are held equal to each other. In Figure 7, the breakthrough delay is plotted as a function of the fraction of A at the upstream surface for five different values of DAB/DA. To examine solutions to this problem under the conditions of comparable time scales for diffusion and solvation, the parameter krL2/


Figure 8. Effect of the fraction of unsolvated A (θ01) and the mobility of the solvation complex (DAB/DA) on the breakthrough delay for local equilibrium (krL2/DA ) ∞), DB/DA ) 1, and equal concentrations of unsolvated A and B at the upstream surface. Numbers on the curves correspond to values of DAB/DA.

DA was set equal to 1 arbitrarily. The ordinate range used in this figure is the same as that in Figure 8 for the purpose of comparison. For a given value of θ01 (and hence Keq), the breakthrough delay increases as DAB/DA decreases as expected. Since the flux of AB is important only at small values of θ01, the effect of solvation on the enhancement of the breakthrough delay is significant only at these values. When DAB/DA ) 1, the breakthrough factor is not constant at a value of 1 as in the case of association because the sum of the concentrations of A and AB does not remain constant as θ01 is varied. In Figure 8, the breakthrough delay under the condition of local equilibrium is plotted for five different values of DAB/DA. The same trend in breakthrough delay with decreasing complex mobility is observed here as in Figure 7 at a fixed value of θ01. As in the case of association, local equilibrium (instantaneous solvation kinetics) yields greater breakthrough delays at all values of θ01 if the complex is immobile. However, when the complex is given mobility, the trend can be reversed at higher values of DAB/DA. Figures 9-11 illustrate the effect of solvation kinetics on breakthrough delay at different values of DAB/DA. In Figure 9, a relatively low value for the diffusion coefficient for the complex (DAB/DA ) 0.2) is used to plot the breakthrough delay versus krL2/DA at three different values of θ01. Under these conditions, the local concentration of A remains higher than the equilibrium value. As a result, the AB complex with its lower mobility represents a net sink for A as the kinetics increases. In other words, with increasing krL2/DA, solvation removes A from the transport process faster than A can diffuse and the breakthrough delay increases. This effect becomes less important at high fractions of A at the surface because the contribution of AB to the total amount permeated under these conditions is less. Similar results, with more pronounced changes in the breakthrough delay, would be observed for immobile AB. In Figure 10, DAB/DA is equal to 0.35 and the breakthrough delay is plotted again as a function of krL2/DA. At these conditions, increasing the kinetics of solvation over 5 orders of magnitude has a negligible effect on the breakthrough delay. The mobility of AB is such that the local concentrations of A and AB in the membrane remain close to the equilibrium concentra-

Figure 9. Effect of solvation kinetics on the breakthrough delay for DB/DA ) 1, DAB/DA ) 0.2, and equal concentrations of unsolvated A and B at the upstream surface. Numbers on the curves correspond to values of θ01 ) C0A/C0T.



tions. In Figure 11, DAB/DA is further increased to 0.5 and a similar plot is made. The mobility of the AB complex is now high enough to allow it to serve as a local source; the concentration of AB in the membrane is higher than the equilibrium concentration for all


Figure 12. Effect of the fraction of unsolvated A (θ01) and the mobility of B (DB/DA) on the breakthrough delay for krL2/DA ) 1, DAB/DA ) 0.1, and equal concentrations of unsolvated A and B at the upstream surface. Numbers on the curves correspond to values of DB/DA.

values of θ01. As a result, the breakthrough delay decreases as solvation becomes faster and more A is produced for diffusion. Figure 12 demonstrates the effect of the diffusion coefficient of B on the breakthrough delay at a fixed value of DAB/DA and krL2/DA. The breakthrough delay is plotted as a function of the fraction of A at the surface, θ01, for three values of DB/DA between DA/DA and DAB/ DA. When the mobility of B is less than that of A, there is less B present locally, causing more A to remain unsolvated. Therefore, the amount of A permeated after a given time will be more, and hence the breakthrough time will be lower. When DB/DA ) 1, solvation is not limited by either of the two penetrants, which leads to the maximum formation of AB, less A permeated, and hence a higher breakthrough delay. Conclusions To obtain the longest breakthrough times in the case of molecular association, the equilibrium constant should be high to favor the formation of clusters, and the clusters themselves should be formed instantaneously (local equilibrium) and be immobile. If the clusters are mobile, slow finite dissociation kinetics can result in longer breakthrough times than local equilibrium. Likewise, to obtain the highest breakthrough delays in the case of intersolute solvation, it is necessary to have a high equilibrium constant to favor immobile complexes that should be formed under the conditions of local equilibrium. Again, breakthrough times longer than those obtained under the condition of local equilibrium are possible if the complex is mobile and the solvation kinetics are slow. It is interesting to note that the kinetics of reversible association or solvation only plays a significant role under the conditions of immobile clusters or complexes. If the associated or solvated species are given mobility, the breakthrough times can vary only 10% as the kinetic rate constant is changed over 5 orders of magnitude. This lack of dependence on the kinetics of reversible processes may be the reason that the assumption of local equilibrium produces excellent fits of experimental transport data. As in most transport processes driven by a solutiondiffusion mechanism, permeation breakthrough times

are affected by both solubility and mobility. To be a good barrier material, the penetrant must have a low solubility in the polymer. This ensures a low concentration at the upstream face which leads to a lower driving force for transport. Therefore, the barrier material should be “penetrant-phobic”. In addition, two materials can have similar penetrant solubilities but very different breakthrough times owing to large differences in the diffusion coefficient. These same considerations apply to penetrants that are capable of association or solvation. In terms of solubility, the distribution of the penetrant between monomeric and associated or solvated states is important in determining the actual driving forces for diffusion. The relative rates of diffusion for the different species as well as the relative rates of association or solvation also play an important role in determining the breakthrough time. From the numerical results presented here, the following conclusions can be drawn for barrier materials in which the total solubility of penetrant is the same. First, the equilibrium at the upstream surface should be shifted as far as possible to the formation of clusters or complexes. This may be achieved in penetrant-phobic materials in which the penetrant would rather selfassociate or solvate with a second penetrant than interact with the polymer. An excellent example of this is the clustering of water in polymers. Second, the clusters or complexes should be immobile, or at least their mobility should be hindered greatly. This can be realized in polymers which have a free volume distribution that hinders the transport of large species. Lastly, for immobile clusters or complexes, the polymer matrix should be conducive to very rapid association or solvation kinetics (local equilibrium). Polymers whose chains relax rapidly in response to diffusing penetrants should meet this last criterion. The transport models presented in this paper comprehensively treat the cases of molecular isodesmic association and intersolute solvation, demonstrate the effects of various parameters on the effectiveness of a membrane as a barrier, and provide useful criteria for improved barrier materials. They also provide a framework for testing experimentally the assumptions of local equilibrium and cluster or complex immobility. Acknowledgment Support of this work from the U.S. Army Research Office through Grant DAAH04-95-1-0133 is gratefully acknowledged. Nomenclature Ci ) concentration C0i ) concentration at the upstream surface Di ) diffusion coefficient K ) isodesmic association equilibrium constant Keq ) intersolute solvation equilibrium constant kf ) forward rate constant for association or solvation kr ) reverse rate constant for association or solvation L ) membrane thickness Q ) cumulative amount permeated t ) time τb ) breakthrough time in the absence of association or solvation

492 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 τ/τb ) breakthrough delay z ) spatial position in the membrane

Literature Cited Crank, J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: London, 1975; p 51. Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 1st ed.; Cambridge University Press: Cambridge, 1984; p 167. Fieldson, G. T.; Barbari, T. A. The Use of FTIR-ATR Spectroscopy to Characterize Penetrant Diffusion in Polymers. Polymer 1993, 34, 1146. Fieldson, G. T.; Barbari, T. A. Analysis of Diffusion in Polymers Using Evanescent Field Spectroscopy. AIChE J. 1995, 41, 795. Folkner, C. A.; Noble, R. D. Transient Response of Facilitated Transport Membranes. J. Membr. Sci. 1983, 12, 303. Goodknight, R. C.; Fatt, I. The Diffusion Time-Lag in Porous Media with Dead-End Pore Volume. J. Phys. Chem. 1961, 65, 1709. Kutchai, H.; Jacquez, J. A.; Mather, F. J. Nonequilibrium Facilitated Oxygen Transport in Hemoglobin Solution. Biophys. J. 1970, 10, 38. McKeigue, K.; Gulari, E. Effect of Molecular Association on Diffusion in Binary Liquid Mixtures. AIChE J. 1989, 35, 300. Meldon, J. H.; Kang, Y. S.; Sung, N. H. Analysis of Transient Permeation through a Membrane with Immobilizing Chemical Reaction. Ind. Eng. Chem. Fundam. 1985, 24, 61. Paul, D. R. Effect of Immobilizing Adsorption on the Diffusion Time Lag. J. Polym. Sci., Part A-2 1969 7, 1811.

Paul, D. R.; Koros, W. J. Effect of Partially Immobilizing Sorption on Permeability and Diffusion Time Lag. J. Polym. Sci., Polym. Phys. Ed. 1976, 14, 675. Plunkett, J. J. M.S. Thesis in Chemical Engineering, The Johns Hopkins University, Baltimore, MD, 1995. Sasidhar, V.; Ruckenstein, E. Relaxation Method for Facilitated Transport. J. Membr. Sci. 1983, 13, 67. Tshudy, J. A.; Frankenburg, C. V. A Model Incorporating Reversible Immobilization for Sorption and Diffusion in Glassy Polymers. J. Polym. Sci., Polym. Phys. Ed. 1973, 11, 2027. Ward, W. J., III. Analytical and Experimental Studies of Facilitated Transport. AIChE J. 1970, 16, 405. Way, J. D.; Noble, R. D.; Flynn, T. M.; Sloan, E. D. Liquid Membrane Transport: A Survey. J. Membr. Sci. 1982, 12, 239. Yung, D.; Probstein, R. F. Similarity Considerations in Facilitated Transport. J. Phys. Chem. 1973, 77, 2201.

Received for review July 12, 1996 Revised manuscript received December 2, 1996 Accepted December 3, 1996X IE960404T

X Abstract published in Advance ACS Abstracts, January 15, 1997.

Permeation Breakthrough Models for Associating and Solvating

Recommend Documents