Nonstationary diffusion through membranes. 1. Transient diffusion

The Journal of Physical Chemistry, Vol. 82, No. 11, 1978

Nonstationary Diffusion through Membranes

We are indebted to the University of Hawaii Computing Center.

Supplementary Material Available: Listings of the observed and calculated structure factors (Supplementary Table I) (2 pages). Ordering information is available on any current masthead page. References and Notes (1) A discussion of zeolite nomenclature is available: (a) R. Y. Yanagida, A. A. Amaro, and K. Seff, J . Pbys. Chem., 77, 805 (1973); (b) K. Seff, Acc. Cbem. Res., 9, 121 (1976). (2) Y. Kim and K. Seff, J . Am. Chem. SOC.,99, 7055 (1977). (3) Y. Kim and K. Seff, J . Am. Cbem. Soc., 100, 178 (1978). (4) Y. Kim and K. Seff, J. Am. Chem. SOC.,submitted for publication. (5) This nomenclature refers to the contents of the unit cell. For example, Ag6Tle-Arepresents Ag,Ti,Al,Sil,Oe, exclusive of guest molecules which may be present, such as water. (6) J. F. Charnell, J . Cwst. Growth, 8, 291 (1971). (7) H. S. Sherry, J . Pbys. Chem., 71, 1457 (1967). (8) K. Seff, J . Phys. Cbem., 76, 2601 (1972). (9) R. Y. Yanagida and K. Seff, J . Phys. Chem., 76, 2597 (1972). (10) P. E. Riley and K. Seff, J . Am. Cbem. Soc., 95, 8180 (1973). (11) Principal computer programs used in this study: T. Ottersen, COMPARE data reduction program, University of Hawaii, 1973; full-matrix

1311

least-sqwres, P. K. Gantzel, R. A. Sparks, and K. N. Truebid, UCLA LW, American Crystallographic Association Program Library (old) No. 317 (revised 1976); Fourier program, C. R. Hubbard, C. 0. Quicksall, and R. A. Jacobson, Ames Laboratory Fast Fourier, Iowa State University, 1971; C. K. Johnson, ORTEP, Report No. ORNL-3794, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1965. “Handbook of Chemistry and Physics”, 55th ed, The Chemical Rubber Co., Cleveland, Ohio, 1974, p F-198. P. E. Riley and K. Seff, J . Phys. Cbem., 76, 2593 (1972). R. L. Flror and K. Seff, J . Am. Cbem. Soc., 99, 4039 (1977). D. W.J. Cruickshank, Acta Cwstallagr., 2, 65 (1949). “International Tables for X-ray Crystallography”, Vol. IV, Kynoch Press, Birmingham, England, 1974, pp 73-87. “International Tables for X-ray Crystallography”, Vd. IV, Kynoch Press, Birmingham, England, 1974, pp 149-150. Y. Kim and K. Seff, J. Phys. Chem., 82, 1071 (1978). Y. Klm and K. Seff, J . Phys. Cbem., 82, 925 (1978). Y. Kim and K. Seff, J. Am. Cbem. Soc., in press. P. C. W.Leung, K. B. Kunz, I. E. Maxwell, and K. Seff, J. phys. chem., 79, 2157 (1975). R. L. Firor and K. Seff, J . Am. Chem. Soc., 98, 5031 (1976). R. b. Firor and K. Seff, J . Am. Cbem. Soc., 99, 1112 (1977). V. Subramanian and K. Seff, J . Phys. Chem., 81, 2249 (1977). Y. Kim and K. Seff, J . Phys. Chem., 82, 921 (1978). G. R. Eulenberger, D. P. Shoemaker, and J. G. Keil, J. Pbys. Cbem., 71, 1812 (1967). K. Tsutsumi and H, Takahashi, Bull. Cbem. SOC.Jpn., 45, 2332 (1972).

Nonstationary Diffusion through Membranes. 1. Transient Diffusion through a Membrane Separating Two Unequal Volumes of Well-Stirred Solutions C. J. B. Hoogervorst, J. A. P. P.

van Dijk, and J. A. M. Smit”

Gorlaeus Laboratories of the State University of Leiden, LeMen, The Netherlands (Received June 6, 1977; Revised Manuscrlpt Received December 5, 1977)

From the theory of thermodynamics of irreversible processes, equations are derived describing the nonstationary diffusion of nonelectrolytic permeants through membranes under conditions of vanishing volume flow with the bathing solutions being well-stirred. A series of experiments is decribed with glass membranes (Vycor) and aqueous solutions of sucrose, mannitol, and pentaerythritol. The measured relations between solute concentration and time enabled us, with the developed theory, to determine two relevant parameters: the partition coefficient K and the membrane diffusion coefficient D. The value of K obtained here especially by studying the initial nonstationary states of the diffusing system agrees with values resulting from other methods (mass balance and investigation of stationary processes). Results are discussed in terms of frictional interactions. Moreover, it is pointed out how experimental conditions including the geometry of the osmotic cell can be optimalized such that the transport parameters are determined in a straightforward way.

1. Introduction Nonstationary diffusion in membranes in contact with solutions has been studied by many investigators. The numerous papers can roughly be divided into two categories with respect to the role of the membrane. On the one hand the membranes constitute a medium allowing sorption or desorption of solute. Usually a plane sheet is suspended in a volume of solution which is stirred or unstirred. Crank1 has given an extensive survey of this part. On the other hand membranes may separate two solutions and form part of a real osmotic system. Contrary to the former case concentrations may be varied on both sides and osmotic pressures are established correspondingly. Treatments of this type are scattered in the lite r a t ~ r e . ~In - ~three subsequent articles we shall study certain aspects of the latter category. In this paper (part 1) we allow the concentrations of both stirred bathing solutions to vary with time, whereas their finite volumes may be unequal. In the following papers we shall consider

analogous cases now referring to unstirred bathing solutions, having semi-infinite (part 2) or finite volumes. By the standard method of Laplace transformation6 diffusion equations are solved. The treatment is restricted to one-dimensional flow, the volume flow being zero a t every point in the membrane. The diffusion coefficient is assumed to be constant, which means that this quantity does not depend upon the solute concentration and the membrane may be considered as a homogeneous medium. Experimental results are presented in terms of friction coefficients which reflect the pairwise interaction in the ternary system membrane-solute-solvent. Our general results cover more specific results obtained previously by others. 3p4,6

2. The Transport Equations at Zero Volume Flow The system to be discussed here consists of three phases: two external ones denoted by a and P separated by a membrane phase. The phases a! and are dilute binary 0 1978 Amerlcan Chemical Society

1312


solutions consisting of the same solvent (index 1) and solute (index 2) but differing in composition. The intermediate medium consists of the diffusing species 1and 2 and a nondiffusing component 3, the membrane, which together form the "membrane phase". This is considered as a homogeneous medium bounded by two planes normal to the x axis at x = 0 and x = 1. The membrane is assumed to be dispersed in space at a molecular level in which case the application of the gel concept7 would seem a realistic approach. For one-dimensional flow (x direction) in a homogeneous membrane phase the phenomenological equations read'

C.J. P. Hoogervorst, J. A. P. P. van Dijk, and J. A. M. Smit

(7) where r13c3has been abbreviated as f13. Eliminating the fluxes from eq 3 and 7 we see that when J , = 0, a relation between vp1 and vp2must hold:

Equations 7 and 8 can be written as Fick's equation with the same diffusion coefficient D for the solvent and the solute. They become wherein vpi is the gradient of the chemical potential of species i, ci the local molar concentration of species i, pi the local velocity of species i with respect to a fixed point in space, and rikthe friction coefficient accounting for the pairwise interaction between species i and k. The coefficiencts rik are subjected to the Onsager reciprocal relations

rik = rki

i,k = 1,2,3 i#k

which have been tested experimentally for the system investigated here.8i9 Equations 1 result from Kirkwood and Bearman's theorylOJ1provided that inertial terms and viscous contributions are neglected. The latter neglect is usually made12though arguments against it have been put forward.13 Since in our systems locally the volume flow vanishes (J, = 0) the barycentric velocity can be considered vanishingly small everywhere in the membrane, annihilating the viscous contributions. The condition of zero volume flow ( J , = 0) may be written, with u, the partial molar volume of species i

(3) with the local flux Jidefined as

J;:=qui

i = 1,2

(4)

The membrane component does not appear in eq 3 and

J;: = -Doci

i = 1, 2

(9)

with

So far the equations derived bear a general character except that neither volume changes on mixing of the pore liquid nor changes in membrane swelling are allowed. To elaborate eq 10 pz is simply taken equal to the chemical potential in a dilute ternary solution. Neglecting the minor pressure term8 and choosing the mole ratio as unit of composition14 we write dp2 = R T d In ( c 2 / c l ) (11) in which R T has its usual meaning. Combination of eq 10 and 11 leads after approximating clul by (1 - 43) to a simple expression for D in terms of friction coefficients

This result, in essence derived by Spiegler,15has been arrived at more explicitly by others too.13J6 Much complexity is avoided by proceeding with constant friction coefficients r12,f23, and f13. Then eq 8 can be integrated from x = 0 to x = 1. If it is assumed that at the interfaces p1 is continuous but that the concentrations show a discontinuity expressed by a partition coefficientK according to

4 as,due to its immobility, u3is set equal to zero. However,

it does appear in the relation between the volume fractions 2 i= 1

q u i = 1- $3

(5)

in which 43 represents the volume fraction of the membrane and 1- $9 the pore volume fraction. The quantities u, and 43 are assumed to remain constant. So far the equations describe equally the stationary regime and the nonstationary regime always assuming that the mass flows (Ji) and the composition of.the membrane phase (ci)depend on position ( x ) and time ( t ) in the latter regime, whereas they are constant with respect to the position ( x ) in the former. Hence under nonstationary conditions we have to supplement the equations with the relation

aci/at = -aqlax

i = I, 2

(6)

which expresses the conservation of masses of the permeating components. Setting u3 = 0 in eq 1and using eq 2-5 we find simple relations between the fluxes and the thermodynamic forces, reading

the integration combined with eq 11 and 13 yields

The symbol A refers to the difference between the respec?ive values in the dilute bathing solutions a and /3 at the interfaces. A trivial development now is to use the classical re1ationsl4

A p 1 = Vl"(AP - A n )

RT

(15)

):(

An = a A Ul

with the superscript 0 referring to the value in the bathing solutions and to insert them into eq 14. The result, which is valid for J , = 0, reads

AP(t) = a A n ( t ) (17) with u called the reflection coefficient introduced by Staverman17 and given by the equation



where C,, C,(x), and

1313

respectively, represent

T,,

of which the first derivation is due to Kedem and Spiegler.l8 It is worth noting that under the assumption made above (T is independent of t contrary to AF' and AT (eq 17). Only in that case the same transient behavior for aP and AT or A(c,/c,) may be expected. 3. The Solution of the Differential Equation of Diffusion Referring to the three-phase system defined above we shall now consider the membrane with cross sectional area S and thickness 1 separating two finite voumes V" and Vp. Outside the membrane the solute is present in the respective concentrations cz*(t) and c$(t), whereas the solute concentration inside the membrane at coordinate x and time t is defined as c z ( x , t ) . At the interfaces (x = 0 and x = I ) a coefficient k2 describes the partitioning of the solute

AQ, sin

(FI]

(27)

r, = lz/Qn2D

This coefficent k2 is related t o K as it follows from eq 13 by kz = K ( 1 -

U1°

(20)

031, 1

where we have neglected czu2 with respect to clul in all phases. The diffusion process is described by the diffusion equation

by the boundary conditions x

Va k2Sl

c 2 a ( 0 )= a ;

C,P(O) = X

=o

kzC(4 - P$J

b ; c&,

0) =

- 6)+ b 1

(24)

The diffusion eq 21 follows from eq 6 with J2 given by eq 9 with constant D. In the initial conditions (24) p and q represent adjustable parameters which can describe any linear solute concentration profile. The boundary conditions (22) relate the change in the solute concentration in the external volumes to the rate at which the solute penetrates or leaves the interfaces. In fact they express the continuity of fluxes together with the solute partition equilibrium at the interfaces. We have solved the system eq 21-24 using the Laplace transformation method (see paragraph at end of text regarding supplementary material). The solution can be expressed as

cZ(x,t ) = k z {C-

(29)

Thus the local concentration cz(x, t ) changes with time till a uniform limiting value k 2 C , has been reached. With respect to the coefficients C, and the characteristic times T,, it may be noted that the former parameters are dependent of the initial conditions (24) imposed on the membrane, while the latter are not. Equations for the measurable concentrations c2"(t) and czb(t) can be derived in a simple manner by substituting the expression for kz (19) into the solution (25) of the diffusion equation. They become

c- + W5lc,,(o) exp(- t / r n ) czP(t)= c- + 5 C,(Q exp(-t/.r,) n=l

VP k $1 and by the initial conditions

B=-

(AB&' - 1)tg Q = & ( A+ B )

c,a(t) =

with the abbreviations

A=-

(28) and where the Q, are the nonzero, positive roots of the transcendental equation

+ n= 1Cn(x)exp(-t/rn) }

(25)

(30)

So far we have not yet specified the initial conditions by assigning values to the parameters p and q in eq 24. For illustrative purposes we shall now consider three cases designated by I, 11, and 111, which are readily accessible experimentally. I. The membrane is first equilibrated with the bathing solution /3 to establish a uniform concentration. This condition corresponds to p = 0 and q = 0 in eq 24. After substitution of these values into eq 30 and rearrangements with the use of eq 29, we find for the difference of the solute concentrations in the bathing solutions

5

A C , ( ~ )= AC,(O) n = lA,I

exp ( - Q , ~ D ~ / I ~ )

(31)

in which

An1 = ( A { ( l+ B2Qn2) [(l+ AzQn2)(1 + B2Qn2)]1'2 1 ) / ( ' / 2 { ( 1 + A2Qn2)(1 + B2Qn2)+ ( A + B)(1 + AB&,') I ) The plus sign in the equation for A,' applies if the Q, lie in the first and the second quadrant, whereas the minus sign applies if the &, lie in the third and the fourth quadrant. A great simplification is obtained if the volumes of the bathing solutions are equal ( A = B). In that case eq 29 and A,' in eq 31 reduce, respectively, to

A& tg '/zQ = 1;An1= 4A(1 + A'&,'

+ 2A)-'

(32)

1314

The Journal of Physical Chemistty, Vol. 82, No. I 1. 1978

A. P. P. van Dijk, and J. A. M. Smlt

C.J. P. Hoogervorst, J.

whose solution has been reported by K r ~ i s s i n k .SpaEek ~ and Kubin3 have given an approximate solution which is valid if the volumes V" and Va are close together ( A N B). Their result is equivalent with Kruissink's if the A is replaced by a harmonic average of A and B. Our treatment, however, involving unequal outer volumes, shows that such approximations are unnecessary. Finally, it may be noted that the experimental situation b = 0, meaning that the membrane is initially free of solute, is also covered by the equations given for case I. 11. The membrane is first equilibrated with the bathing solution a to establish a uniform concentration throughout. This condition corresponds to p = 0 and q = 1 in eq 24. In an analogous manner as before we derive for the difference of the outer solute concentrations

A c , ( t ) = Acz(0) n= 51An11e~p(-Qf12Dt/12)

(33)

where A:I directly follows from A I by interchanging A and B according to A;'(A, B ) = A,,f( B ,A ) . Especially we have investigated this case experimentally. 111. The membrane is first equilibrated with on both sides bathing solutions a and p which differ in composition to establish a linear concentration profile within the membrane. This condition corresponds to p = 1 and q = 1in eq 24. Here, along the same lines as before, we arrive at

(34) in which A;'I

20

0

LO

60

80 A

Figure 1. The roots Q, of eq 29 as function of the parameter A at dffferent ratlos B I A , indicated by the numbers under the curves.

010

0.00

-010

can be represented by -0 20

Equations 31-34 are of interest because of the relationship of Ac&) to the experimental osmotic pressure hP(t). As it is evident from eq 16 and 17 the following proportionality exists

A P ( t ) = oRT A c , ( t )

(35)

where in the derivation the term c l u t has been taken equal to 1 in conformity with the dilute solution hypothesis. Applying eq 35 to eq 31,33, and 34 one may conclude that the decay of the measured osmotic pressure with time is described by

A P ( ~= ) AP(O)

5 ~~j

n=l

exp(-Qn2Dt/12)

(36)

j = I, 11, I11

4. The Quasistationary Region Generally the sum in eq 25 represents a rapidly converging series to which all terms contribute in early stages of the diffusion process. However, after a certain period only the leading term (n = 1) is of importance and the diffusion process is said to be quasistationary. Under this condition eq 25,31,33, 34, and 36 reduce with eq 28, respectively, to c2(3c,t ) = k 2 {C.. iC1exp(-t/.rl)3. (37) Ac,(t) = A c ~ ( O )exp(-(t - e i ) / T , } (38) A P ( t ) = A P ( 0 ) exp{-(t in which the

- Oi)/T1 }

(39)

are given by . In Alj 1' 01 = .r1 In A,J = -- j = I, 11, I11 &I2 D and represent the time lags with which a region of simple dj

-0 30

1 1

0 25

0.50

075

100

0:

Flgure 2. Relation between In A , and Q, at various ratios B I A indicated by the numbers beside the curves. Between brackets in Roman numerals the case, for which the curve is calculated, is indicated (cases I and 11).

exponential decay sets ina6J9The last member of eq 40 follows directly from the application of eq 28. The root Q1 can be found by solving eq 29 numerically by a computer. It has been done for a number of adopted values of A at various ratios B / A (= VJ/V"). The results are shown in Figure 1. With the use of eq 31,33, and 34 the correspondingvalues of In A? 0' = I, 11,111) have been calculated and plotted against Q;. Cases I and I1 are illustrated in Figure 2 (cf. paragraph at the end of text regarding case 111). From Figure 1 it is seen that the roots Q1 decrease sharply with increasing A and more so as the ratio VP/V" increases. For low values of Q1 or corre-, sponding large values of A and Val Vuthe coefficients A? 0' = I, 11,111) center around unity (Figure 2 in this paper and Figure I in the supplementary material). This occurs when the volume occupied by the membrane phase is small compared to the volumes of the external phases. Furthermore it should be noted that the curves represented in Figure 2 do not deviate much from,straight lines. As follows from eq 40 this means the 8, are roughly proportional to 12/D. However, the coefficients of proportionality between 8 j and 12/D do depend on the ratio V@/ Va. Some typical limits of their values at vanishingly small Q t are shown in Table I. The values -113 and +1/6 found in cases I and 11, respectively, when VP >> V"

The Journal of Physical Chemistty, Vol. 82,No. 11, 1978 1315


TABLE I: Some Limits at Vanishingly Small Q1 Related to Different Ratios of @/v" lim (eID/z2) lim (e11D/z2) lim (eInD/z2) VfiIP

Q12-+0

Q,'+O

1 2 5 > 300

-1112 - 1/6 - 114 - 113

- 1/12 0

+ 1/12 + 116

QI2+O

0 0

0 0

confirm the results reported recently by Barrie et a1.6 who described diffusion phenomena involving a membrane separating a finite volume from a semiinfinite volume. Explicit expressions for Q1 and T~ can be formulated in case the quantities A and B are very large compared to unity. So the approximation tg Q1 = Q1 applied to eq 29 yields, with eq 23 QI2

= k2Sl

VCY + vp va vp

Substitution of this equation into eq 28 leads to

'

vaVP12 = k2S1(VQ+ Vp)L)

+

Thus T~ is proportional to 12/D by a factor A B / ( A B ) (eq 23) which is much larger than unity under the adopted conditions. Comparing this factor with the values mentioned in Table I, it is immediately seen the T~ >> JB1'), which means in relation to eq 38 and 39 that rather soon a simple exponential decay sets in. This entrance of the quasistationary state is quite analogous to cases discussed by Barrer.lg Evidently the small values of Q1 lead to linear concentration profiles in the membrane (eq 37 and 27). 5. Determination of Transport Parameters

Equations 23-30 provide a means of measuring D and k2. An experimental requirement is the measurement of the solute concentration in the a-bathing solution at the beginning (t = 0) and at the end (t = t,) of the experiment, yielding c2"(0) and cZa(te),and the measurement of the solute concentration in the 0-bathing solution as a continuous function of time yielding cz@(t)for 0 -< t -< t,. Then the calculation to be performed may be the following: estimate a value for h2 from, e.g., a mass balance after time t,; solve eq 29 for the roots Q,. For this purpose the geometry of the system (V", V@, S, and I ) is supposed to be known. According to eq 28 the T, values are related to T~ by a factor Q12/Q,2. Hence T~ can be calculated from eq 30 which is specified to t = t, since all other quantities appearing in this equation are known. Consequently the values of T, are calculated and substituted into eq 30 with the estimated value of k2 This yields a calculated function c2(t) which can be compared to the measured function c2(t). Finally a least-squares procedure is set up by which k2 is adjusted in such a way that the Calculated function approximates the measured function. The diffusion coefficient D is calculated from the value found for T~ according to eq 28. The foregoing procedure is based on the set of eq 28-30 which are valid for the whole range of times covering t = 0 to t = t,. Another starting point for the determination of transport parameters may be eq 38 in combination with eq 28 and 29. The validity of this set of equations is restricted to the quasistationary region. Procedures for calculating transport parameters following principles of this kind are well known and have been put forward especially by Barrer.lg The following procedure whieh can be considered as an extension of his treatment seems useful and is il-

lustrated in Figures 1 and 2. First a linear plot of In (Ac2(t)/Ac2(0)]vs., t is prepared. Its slope and intercept yield T~ and In A? (j= I, 11),respectively. Furthermore, in Figure 2 the curve with the corresponding ratio Val V" is observed q d the value of Q t belonging to the measwed value of In A? can be found. Then the diffusioh coefficient D is calculated according to eq 28. Finally, once Q1 is known, the corresponding value of A can be read from Figure 1leading by means of eq 23 to the determination of k2.

6. Experimental Section Experimentally we have observed nonstationary diffusion through a Vycor glass membrane in which the respective solutes pentaerythritol, mannitol, and sucrose and the solvent water were involved. A description of the osmotic cell has been given in detail*and recently also in a more comprehensive way.I The osmotic cell possesses a concentric construction in which the a compartment with a volume of 2 X m3 is envelopped by the 0compartment with a volume of 2 5 X m3. As the membrane forms a whole with the a compartment, the investigation of case I1 especially was found useful. During the experiments a hydrostatic pressure was applied consisting of a major constant gas pressure and a minor variable liquid head which was observed by following a moving meniscus in a capillary. The initial pressure was such that the meniscus run through a smooth maximum equal to a A ~ ( 0 (see ) eq 17). The maximum was estimated previously by a dynamic measuring methods7 In this manner the volume flow J, was kept practically equal. to zero. The composition of the bathing solutions was measured at t = 0 and t = t, by taking samples and by analyzing them with a Rayleigh interferometer (Zeiss). Values for cZ"(O), c2"(te),cza(0), and c2P(t,) have been tabulated (see paragraph at the end of the text regarding tabulated concentration values). Additionally, the concentration of the @-bathingsolution was measured as a continuous function of time. This is illustrated in Figure 3 for the solute pentaerythritol (analogous curves are presented for the solutes sucrose + mannitol, see paragraph at the end of the text). For that purpose a peristaltic pump (L. K. B.)circulated the 0-bathing solution with a flow rate of 1mL/min through a differential refractometer (Waters R4) connected in a circuit with the 0compartment. The effect of solvent evaporation was eliminated as much as possible by performing the measurement with respect to a reference solution placed in a dummy cell with practically identical circumstances regarding evaporation. 7. Results and Interpretation In section 5 we have described two procedures which lead to the determination of the transport parameters T~ and k2. The first method, utilizing the experimental data measured in the total period from t = 0 to t = t,, proved the more useful one for evaluation of T~ and K , the latter calculated according to eq 20 under the assumption that ul0 equals ul. The relevant data are mentioned in Table 11. The parameters T~ and K have not been found to depend in a significant way on the concentration of the permeating solute (first and second column of Table 11). For comparison values of K are added to Table I1 (third column) which have been calculated from a mass balance after time t , under the assumption of a linear solute concentration profile in the membrane. Their values agree with those of the second column of Table 11. As pointed out in section 5 the coefficient D is directly derivable from the characteristic time (last column Table

1316

The Journal of Physical Chemistry, Vol. 82, No. 11, 1978 I

C. J. P. Hoogervorst, J. A. P. P. van Dijk, and J. A. M. Smit

TABLE 111:

F r i c t i o n Coefficients Calculated from Observed Quantities a t 25 C in a Glass Membrane

PENTA -ERYTHRITOL

I

10"D

10-9r12

io-if,,

Nsm-'

Solute

U

m2 s-l

Nsm2 mol-,

mol-'

K

Pentaerythritol Mannitol Sucrose

0.082

6.38

1.32

1.32

1.39

0.120 0.186

6.51 4.04

2.02 2.90

0.35

0.97

1.19

1.01

0 00-

-005-

- 0 10-

-015-

,

0 004 0

36

10

,

72

5L

90

103 t

-020-

Is)

Figure 3. The concentration of the fl bathing solution as a function of time in the case of pentaerythritol for different concentration differences between the a and fl compartment. Ac, = 9.722 kg/m3; Ac, = 12.729 kg/m3; Ac, = 15.877 kg/m3; Ac, = 19.081 kg/m3;

Ac, = 22.127 kg/m3.

TABLE 11:

D i f f u s i o n Parameters for Polyols at 25 C

in a Glass Membrane K(mass Solute

T],

Pentaerythritol Mannitol Sucrose

3.32

i

4.76 7.40

i

s

i

K

balance)

10"D m2s-l

0.06 1.39 i 0.21 1.31 * 0.25

6.38

0.08 0.06

6.51 4.04

0.97 i 0.11 0.85 1.01 f 0.20 0.96

i i

0.19 0.22

11). The relatively large value of K and attendant small value of D in the case of pentaerythritol suggest a strong interaction between the membrane and this solute. In this respect it is of interest to compare the values of the friction coefficients fz3 for the diffusing solutes. Starting from the experimental quantities, u,' D, and K friction coefficients can be calculated from eq 12 and 18, where, as has been pointed fI3u2may be neglected with respect to rlz(l - p3) in the system discussed here. The same calculation yields a value for r12 The values of the different quantities are shown in Table 111. The large value of f23 in the case of pentaerythritol compared with the molecular dimensions emerges and reflects a stronger interaction with the membrane. The values of r12correspond to values reported and interpreted earlier.7,8 The values of lz are not significantly correlated to the values of f23 in glass membranes contrary to the corresponding findings in cellophane membranes.20 The second method (section 5 ) for determining T~ and k 2 has also been applied by us. The characteristic times 71,thus found, agree with the values mentioned in Table 11. The values of Al showed deviations of less than 5% from unity which fall completely within the range of experimental error. Therefore it makes a determination of k2 impossible. In fact, this approach fails in cases where the volume occupied by the membrane is small compared

\

-""iI 0

18

36

5L

72

90 t [si

Figure 4. The decrease of the concentration difference vs. time. The discrete points represent experimental data: (+) sucrose; (A) mannitol; (0)pentaerythritol. The solid lines are calculated according to eq 33 with values of 7, and Kmentioned in Table 11.

to the volumes of the external solutions. Finally we will broach the question of a possible simple exponential decay of the measured osmotic pressure in the course of time. From the discussion around eq 35 and 36 it is evident that the transient behavior of the experimental osmotic pressure concurs with the transient behavior of the solute concentration difference between the bathing solutions. Hence, it is of interest to prepare curves with the use of eq 28-33 and to inspect their behavior from the very beginning to the end of the experiment. The results are compiled in Figure 4. The solid lines have been calculated with the uniform values of 71 and K mentioned in Table 11. The discrete points have been obtained directly from experimental data. All curves refer to case 11. It may be noticed that over a period of about 1-1.5 h the curves in Figure 4 are flattened but thereupon they become linear in the quasistationary range. In that range the slopes are proportional to the product k2D in agreement with eq 38 and 42. If extrapolations of experimental data to zero time are concerned, it should be realized that it may be of interest that the initial conditions and other experimental conditions are chosen in such a way as to make the time lags as small as possible. In Figure 5 an example is given which illustrates cases I, 11, and 111, which agree with each other with respect to the initial concentrations outside the membrane but which differ in initial concentration inside the membrane (eq 21).



1317

initial solute concentration of the a solution, p solution, mol m-3 concentration of species i in the membrane, mol m-3 Ci concentrations of species i in the stirred bathing Cia, Cip solutions facing the membrane at x = 0 ( a )and x = 1 (p), mol m-3 solute concentration in the membrane after infinite C, time, mol m-3 diffusion coefficient in the membrane, m2 s-l D abbreviation for T i k C k , N s m-l mol-' fik flux of species, i, equal to qui, mol my2s-l Ji total volume flow per unit area of membrane, m s-l J" K partition coefficient relating concentrations inside and outside the membrane (eq 13) partition coefficient relating solute concentrations k2 inside and outside the membrane (eq 19) thickness of the membrane, m 1 pressure, N m-2 P P , 4 adjustable dimensionless parameters describing the concentration profile in the membrane (eq 24) roots of trancedental eq 29 Qfl friction coefficients coupling the driving force -vwi r ik to the diffusional flow ck (ui - uk),N m2 s molP R universal gas constant, N m mol-' K-l S membrane surface, m2 time, s t T absolute temperature, K local velocitv of mecies i. m U; q;,u t partial mol& vol&es of species i in the membrane, in free solution (superscript O), m3 mol-l V", volumes of the bathing solutions facing the V@ membrane at x = 0 ( a ) and x = 1 (p), m3 x space coordinate in the transport direction, m 4i volume fraction of species i wi chemical potential of species i, N m mol-l T osmotic pressure, N mm2 B reflection coefficient (eq 17) characteristic times, s (eq 28) time lag for case j 0' = I, 11, 1111, s Operators local gradient of y in the transport direction: v y vy = ay(x)/a(x) difference of y between the external phases Ay = Ay Y" - Y P Indices 1,2,3 in the ternary system 1refers to the solvent, i, k 2 refers to the solute, and 3 refers to the membrane species

a, b

0

-010

-020

-0.30

Figure 5. The decay of the concentration difference across the membrane for the solute pentaerythritol in the cases indicated by Roman numerals calculated from eq 31, 33, and 34 with values of r1 and K mentioned In Table 11.

Extrapolations from the linear region lead in cases I, 11, 111,respectively, to a negative, positive, or zero intercept. This behavior will be found if V@/Va > 2. Under these circumstances initial conditions corresponding to case I11 are to be preferred to those where extrapolations to zero time are involved. When Va/Vais equal to 1 / 2 or 2, in case I and case 11, respectively, the time lags vanish. Moreover if Va and V@are large with respect to the volume occupied by the membrane, time lags disappear in all cases (small values of Q1 in Figures 1 and 2). These criteria may be applied to the construction of osmometers with stirring devices especially where easy extrapolation to zero time is wanted. However it must be realized that the determination of the partition coefficient K just requires the presence of a time lag and that if the conditions of the system are such that time lags fail to occur, only the reflection coefficient u and the diffusion coefficient D can be determined by observing the osmotic pressure as a function of time. Apart from the dimensions of the osmometer and the membrane, also the filling conditions leading to the initial concentration profile according to eq 24 are of interest. Strictly speaking this concentration profile ought to be established under the condition J , = 0. It can be realized by applying such a pressure difference that the volume flow J , stops. This pressure difference is prescribed by eq 17. If the initial concentration profile is built up under the condition AP = 0, it will depend on J,. For case I11 such a profile has been calculated (see eq 29, ref 7). Actually the effect of J , on the initial concentration profile could be ignored completely in our experiments as the occurring volume flows were rather small.

Symbols A, B dimensionless quantities representing the ratio of the volumes V", V@, and the volume k2Sl

2

Supplementary Material Available: Figures analogous to Figure 2 of the text referring to case 111and to Figure 3 of the text referring to sucrose and mannitol, a table showing the solute concentrations and expiration times (to), and moreover an Appendix in which the solution of the differential equation has been given explicitly (12 pages). Ordering information is available on any current masthead page. References and Notes (1) J. Crank in "The Mathematics of Diffusion", 2nd ed,Clarendon Press, Oxford, 1975. (2) H. Coll, Macromol. Rev., 5 , 541 (1972). (3) P. SpaEek and M. Kubin, J . folym, Sci., C-16, 705 (1967). (4) Ch. A. Kruissink, IUPAC Symposium Macromolecules, Wiesbaden; Short Communications I1 B 11 (1959). (5) H. S.Carslaw and J. C. Jaeger in "Conduction of Heat in Solids", 2nd ed, Clarendon Press, Oxford, 1973, Chapter 12. (6) J. A. Barrie, H. G. Spencer, and A. J. Staverman, J . Chem. Soc., Faraday Trans. 7 , 71, 2459 (1975). (7) J. A. M. Smit, J. C. Eijsermans, and A. J. Staverman, J. Phys. Chem., 79, 2168 (1975). (8) J. A. M. Smit, Thesis, Leiden, 1970.

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J. A. M. Smit and A. J. Staverman, J. Phys. Chem.,74, 966 (1970). R. J. Bearman and P. F. Jones, J . Chem. Phys., 33, 1432 (1960). R. J. Bearman and J. G. Kirkwood, J. Chem. Phys., 28, 136 (1958). D. C. Mikulecky and S. R. Caplan, J. Phys. Chem., 70, 4089 (1966). G. S. Manning, J . Phys. Chem., 76, 393 (1972). E. A. Guggenheim in “Thermodynamics”, North Holland Publishing Co., Amsterdam, 1959. K. S. Spiegler, Trans. Faraday Soc., 54, 1417 (1958).

Smit et al. (16) A. J. Staverman, Symposium on Biological and Artificial Membranes, Rome, 1975; Pontificiae Academiae Scientiarium Scripla Varia, Citta del Vaticano, 1976. (17) A. J. Staverman, Trans. faraday Soc., 48, 623 (1952). (18) K. S. Spiegler and 0. Kedem, Desalination, 1, 311 (1966). (19) R. M. Barrer in “Diffusion in and through Solis”, Cambridge University Press, London, 1941. (20) T. G. Kaufmann and E. F. Leonard, AIChE J . 14, 110 (1968).

Nonstationary Diffusion through Membranes. 2. Transient Diffusion through a Membrane Separating Two Semiinfinite Volumes of Unstirred Solutions C. J. P. Hoogervorst, J. de Goede,+ C. W. Versluijs, and J. A. M. Smit” Gorlaeus Laboratories and the Laboratory for Physiology of the State University of Leiden, Leiden, The Netherlands (Received June 6, 1977; Revised Manuscript Received December 5, 1977)

In this paper measurement of the osmotic pressure with a membrane permeable to the solute is treated both theoretically and experimentally. In the case of a membrane permeable to the solute the experimental pressure, AP,the pressure to make the volume flow vanish, differs from the theoretical osmotic pressure difference, Ax, between the cells of the osmometer. The ratio of these two quantities is called the reflection coefficient, CT, and is a measure of the permeability of the membrane to the solute. Owing to the permeation, both A x and hp decrease during an osmotic experiment. Theoretically we have derived expressions for hp as a function of time for the case of unstirred solutions in both cells for different boundary conditions with some simplifying assumptions. It follows that AP(t) can be expressed in two parameters, the reflection coefficient CT and the diffusion coefficient D of the solute in the membrane. The course of AI’ as a function of t and also the value of hp extrapolated toward t +O is different for case I, in which the membrane is free of solute at t = 0, and for case 111, in which at t = 0 the membrane is equilibrated with the two cell solutions. The ratio of limt,+,, U ( t )for the two cases is expressed in a quantity r, containing the partition coefficient k and the ratio of the diffusion coefficients D in the membrane and Dfin free solution. Experimentally AP(t) has been measured for a number of polystyrene fractions in toluene in a high-speed automatic osmometer with cellophane membranes permeable to the solute. From the experimental curves values for D and u can be calculated by curve fitting. The experimentally determined ratio of the values limt,+o AP(t) agrees very well with the theoretical value. -+

1. Introduction In part 1of this work’ we have studied the nonstationary

diffusion in a membrane placed between two well-stirred solutions. Here, however, we consider the situation where the bathing solutions are not stirred; as in the stirred case the volume flow is zero. Such a situation arises in an automatic osmometer (Hallikainen), which is equipped with a balancing mechanism keeping the volume flow zero during the experiment. The membrane is placed in a horizontal position separating the upper solution of lower density from the lower solution of higher density. In this way crucial conditions are fulfilled to assure that the solutions remain unstirred. With an automatic osmometer (Mechrolab) Hoffmann and Unbehend2B have measured osmotic pressures as a function of time. They have interpreted their results in terms of nonstationary diffusion of the permeating solute. Their approach involves the solution of the diffusion equation applied to a system consisting of two semiinfinite media.4 In fact they consider in the transport direction ( x ) a regicn x < 0 in which the bathing solution is present and a region x > 0 occupied by the membrane. In the membrane region, the solute concentration has been calculated as a function of position (x) and time ( t )for the range 0 < x < 1 where 1 corresponds to the thickness of the membrane. The osmotic pressure difference Ax is simply taken proportional to the solute concentration difference between x = 0 and x = 1. +Laboratoryfor Physiology. 0022-365417812082-1318$01 .OO/O

In spite of its simplicity Hoffmann and Unbehend’s approach has the serious drawback of accounting only for the interface a t x = 0 and not for the interface at x = 1 between the membrane and the solutions. However, expressions can be obtained which describe the solute concentration as a function of x and t in the whole system by adopting a system consisting of three regions: the slab 0 C x < 1 occupied by the membrane, which is in contact at x = 0 with the solution region ( x C 0), and at x = 1 with the solvent region (x > I). This requires the solutions of three differential equations referring to the three regions together with appropriate boundary conditions. From these solutions the external solute concentrations at x = 0 and x = 1 can thus be evaluated. The theoretical osmotic pressure may therefore be calculated and, multiplied by the reflection coefficient u, yields the measurable experimental osmotic pressure AP( t ) . It turns out that the function aP(t)is characterized by the two transport parameters CT and D, and a quantity r defined by

r = h(D/Df)1’2

(1)

in which k is the solute partition coefficient5 and Dfthe solute diffusion coefficient in free solution. Due to the slow diffusion of the macromolecules used in our experiments the measuring times could be chosen large enough for the accurate determination of the parameters on the one hand and small enough to neglect the influence of the finite dimensions of the osmotic cells on the other. Hence the use of the solution for a system consisting of a membrane 0 1978 American Chemical Society

Nonstationary diffusion through membranes. 1. Transient diffusion

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