Electrostatic model for enhancement of membrane transport by a

Electrostatic model for enhancement of membrane transport by a nonuniform ... Control of the Fixed Charge Distribution in an Ion-Exchange Membrane via...
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J . Phys. Chem. 1988, 92, 6113-6120

6113

Electrostatic Model for Enhancement of Membrane Transport by a Nonuniform Electric Field Curtis G. Steinmetz and Raima Larter* Department of Chemistry, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46223 (Received: February 26, 1988)

Previous experimentalwork in our group has shown that membrane transport of ions can be enhanced by a nonuniform electric field. In this paper, an electrostatic model for the applied field used in the experimental studies is developed. We calculate the flux two ways in the presence of this field: (1) by using the Goldman equation and (2) by solving the Nernst-Planck flux equation numerically. Results from both methods confirm that a nonuniform electric field is the source of the enhancement of flux observed experimentally. In addition, we conclude that the Nernst-Planck equation is inherently a nonlinear function of the field and this nonlinearity must lead to enhancement of transport by nonuniformity. This phenomenon may be related to the high degree of selectivity displayed by certain nonuniform ionomer membranes, of which Nafion is the best-known example.

I. Introduction Transport in spatially inhomogeneous systems is not well understood; however, some groups have recently begun to study the influence of inhomogeneity on transport. The increased interest in inhomogeneous systems is partly due to the observation that certain ionomers (polymers composed of ionic monomers), known to be spatially inhomogeneous, have unusual transport properties. As an example, Nafion membranes, made from a perfluorosulfonic acid polymer, have an unusually high selectivity for cations over anions as well as a high selectivity for certain cations over others.’ These properties have been linked to Nafion’s inhomogeneous structure* (morphology), and several mechanisms have been proposed3” for the influence of Nafion’s morphology on its transport properties. We recently reported an experimental study7 that was designed to probe the effect of nonuniformity on transport efficiency. We specifically looked at the influence of a nonuniform electric field on the transport of ions through ion-permselective membranes. Our general observation was that the nonuniformity of the imposed field can enhance the transport of ions over that of a correspondingly more uniform system. Since nonuniformity in Nafion is the result of clustering of charged side chains on the polymer backbone, the electric field profile within the membrane will be nonuniform. Hence, our experimental observation may be a partial explanation for the high selectivity observed in Nafion. Here, we report on an improved theoretical model of our experimental system. The nonuniform field profile is calculated via a simple electrostatic approach. The flux of ions in the presence of this field is modeled via the Goldman equation? an analytically integrated form of the Nernst-Planck flux equation, and via numerical (finite difference) integration of the flux equation. In our previous experimental study, the nonuniformity of the field was controlled by certain geometrical parameters; here we study the influence of these geometrical parameters on the flux of ions and compare the predictions of the model to experimental observations. The experimental studies were conducted by measuring the changes in ion concentration in either side of the cell shown schematically in Figure 1. The electric field is applied via two (1) Ions in Polymers; Eisenberg, A,, Ed.; Advances in Chemistry Series 187; American Chemical Society: Washington DC, 1980. (2) Cutler, S. G. In ref 1 , p 307. (3) Hsu,W. Y.; Gierke, T. D. J . Membr. Sci. 1983, 13, 307. (4) Al-Jishi, R.; Datye, V. K.; Taylor, P. L.;Hopfinger, A. J. Mucromolecules 1985, 18, 297. ( 5 ) Reiss, H.; Bassignana, I. C. J. Membr. Sci. 1982, 11, 219. (6) Larter, R. J . Membr. Sci. 1986, 28, 165. (7) Kuntz, W. H.; Larter, R.; Uhegbu, C. E. J . Am. Chem. Soc. 1987,109, 2582. (8) Goldman, D. E. J . Gen. Physiol. 1943, 26, 37.

0022-3654/88/2092-6113$01.50/0

platinum wire electrodes, bent into the grid configuration shown, and transport of ions occurs through the ion-permselective membrane separating the two cells. We find a different transport rate when the platinum electrodes are in a “staggered” configuration than when they are in an “eclipsed” configuration. In the eclipsed configuration, the wires on one side of the membrane are directly across from those on the other side, while in the staggered configuration they are shifted. In both configurations, the resulting field is nonuniform. A higher flux was observed when the staggered configuration was used. In a previous theoretical study,7 we proposed that the flux enhancement could be attributed to a greater degree of nonuniformity of the applied electric field. In the earlier study, the nonuniform electric field was modeled by a sum of Gaussian functions and the Goldman equation was used to calculate the flux in the presence of this field. A perturbation theoretic approach indicated that any functional form for the nonuniformity should lead to a difference in fluxes for the staggered and eclipsed configurations but that this would occur only for passive transport, Le., transport in which the electric field carries ions down a concentration gradient. In the present work, we expand on these conclusions by considering a more physically realistic expression for the applied electric field. We find that the conclusions of the perturbation theory approach in our previous study are partially correct: the flux in the presence of a nonuniform field is greater than that in a uniform field, but here we find this to be true for both passive and nonpassive (electrically facilitated) transport. Our results show unambiguously that the inherent nonlinearity of the Nernst-Planck flux equation is responsible for this effect. The nonlinearity arises from the term in the Nernst-Planck equation in which the concentration profile of the transported ion is multiplied by the field; the concentration profile, in turn, depends on the field profile because of local charge neutrality constraints, so the second term in the Nernst-Planck equation is always inherently nonlinear in the electric field. We have also extended our previous theoretical approach by computing a flux enhancement factor that is more closely related to that measured experimentally. We find that the Goldman equation confirms the experimental observations qualitatively but does not always agree quantitatively with the more accurate numerical approach. We conclude with a proposed physical explanation of the origin of the flux enhancement effect. The local electric field in the region of the membrane is found to be higher in the staggered configuration than in the eclipsed configuration. The higher field leads to a greater flux in these important local regions, and hence the flux in the staggered configuration is observed to be e n h a n d over that of the eclipsed configuration. This field enhancement effect is only partially responsible for the observed flux enhancement, however; nonlinear effects are also quite important. 0 1988 American Chemical Society

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The Journal of Physical Chemistry. Vol. 92. No. 21, 1988

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a

Figure 1. Schematic diagram of cell. The electrodialysis cell used experimentally consists of two compartments containing salt solutions and separated by an ion-permselective membrane. The field is applied via platinum wire electrodes, bent into the configuration shown. The arrangement of the electrodes with reSpect to one another is discussed in the text and shown schematically in Figure 2.

11. Model

The transport of ions in the bulk of a system such as that shown in Figure 1 will he governed by the Nernst-Planck equation9

j, = - D , ~ c-~M&

v

(1)

where j,is the flux of species i, c, its concentration, D, its diffusion coefficient, M iits mobility ( M , = D , R T j 3 , according to the Einstein relation), and Vthe electrical potential. Equation I gives the vector flux in all three dimensions; we anticipate that the component normal to the membrane (J,) will dominate the dynamics since transport in the directions tangential to the membrane surface (JiXand Jjy)should not contribute much to the measured current. The validity of this assumption is considered in a later section by studying the numerically integrated flux equations from which we can calculate both the normal component of the flux as well as a tangential component. The time-dependent equation (analogous to Fick‘s law) that we consider, then, contains only the gradient of Jjz:

In the following section, we derive an expression for V, the electric potential, by developing an electrastatic model of the applied field in our experimental system. A. Electrostatic Model. In our experiment, the electric field was applied via an array of wires: to derive an electrostatic expression for the field due to this array, we first make the simplifying assumption that the wires are infinitely long. This will mean that the field is constant in one of the perpendicular directions (y, for example) but will vary in the other direction (x). Hence, the potential in eq 2 is a function of x and z hut independent of y . The field due to one wire can he determined by Gauss’ law. It is a radial one and is given a t any position r from the wire by the expression E(r) = 2X’/r, where A’ = XJ4m,6 and Xis the linear charge density on the wire, i.e., X has units of coulombs per length. Since the field is the negative derivative of the potential, the corresponding potential a t this position is, then, V(r) = -2X‘ In ( r ) + C, where Cis the potential at some arbitrary reference point, such as r - m. Since we will consider only potential differences, we can set C = 0, without loss of generality. For an array of electrodes such as that shown in Figure 2, the total field is a superposition of the fields due to each wire, Thus the field is a vector field and is given by (3) for N pairs of wires, where i,is a unit vector from the ith positively charged wire to an arbitrary point ( X J ) and SI is a unit vector from the ith negatively charged wire to the point (x,z). The (9) Labhminarayanaiah, N. T~onrporlPhenomena in Membranes;Academic: New York, 1969.

1

0.1027

@-.I027

0.1247

@ ,1055

@-.lo55

Q ,1192

Figure 2. Eclipsed and staggered electrode configurations. The positions of the vertical platinum wires in the electrodes of Figure I can be (a) eclipsed or (b) staggered. The distance between the cathode and anode is 8. while the distance between the wires in each array is a. The parameter 6 = 6‘/a gives the degree of ‘staggering“ of one array with respect to the other. When the cathode and anode are held at a fixed potential difference of 8 = I V with respect to one another, with other parameters as given in Table I, the charge densities A, (see the Appendix) are as shown. If the potential difference 6 is something other than I V, the value of each A, increases, but the ratios A,/Ai stay the same.

electrodes are numbered (see Figure 2) such that Xi is the charge density of a positive wire and -A, is its negative counterpart in the other array. The electrical potential is, similarly, given by N

V(x,r) = 2 Z X ; In ( s l / r l )

(4)

,=I

To consider the effect of “staggering” the electrodes, Le., shifting one array of wires with respect to the other, as in Figure 2b, we must introduce a shift parameter and transform eq 3 and 4 into Cartesian coordinates. The vectors Fi and ?j in Cartesian coordinates are given by 7, = (x - ( i - ~ ) a ) l z]+ 3, = ( i - I + 6)a - x)? + (0 - z)] (5) where a is the spacing between the wires within an array, 0 is the spacing between the parallel arrays of wires, and 6 is the shift parameter given by 6 = 6’/a,as shown in Figure 2. Substituting eq 5 into eq 3 and 4 yields the final expression for the applied field. Letting n = i - I simplifies the expression to E(x,z) = (x - na)? + z j ( ( n + 6)a - x)l + (0 - z ) j n=o (x + z2 ( ( n + 6)a - x)’ + (0 - z ) ~ ( 6 )

i.q

]

+

The corresponding expression for the applied potential is V(x,z) = -X A‘” In “=O

+ ( ( n + 6)a - x)’ + (0 - z)’ (x - na)2

2’

Again, the sums in eq 6 and 7 run over pairs of wires, one positive and one negative wire in each pair. The geometric parameters a. 0, and 6 are known from the experimental apparatus. The values of XI are the charge densities on the wires that arise when each parallel array (which is really a single electrode in the experiment) is held a t a fixed potential (such as 1 V) relative to the other array. The charge densities on the wires within a single array will not necessarily he equal. We have approached the determination of XI as follows. The first step is to note from the symmetry of the electrode configurations that it is necessary to find only one charge density for each pair of wires, so long as the pairing is done properly (see Figure 2b). The pairing that simplifies the problem is to number the wires from top to bottom on one side but from bottom to top on the other. In this way, the numbering is symmetric in both the eclipsed and staggered cases. To find the four unknown charge densities requires four equations; these are derived by imposing an electrostatic equilibrium condition on the electrodes (see the Appendix). Since the electrodes of like sign are connected, it is reasonable to assume

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The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6115

that they will quickly come to electrostatic equilibrium when the power is turned on. With this approach it is possible to generate equations for the potentials at the positions of all of the positive (or negative) electrodes, with these potentials all equal to half the voltage readout of the power source-so that the potential drop between any two oppositely charged electrodes will be the instrument reading (such as 1 V). The unknowns in these equations are precisely the unknown charge densities Xi in terms of the known parameters a , p , and 6. Solving these equations for the Xi's and substituting the result into eq 7 give the final expression for the applied electrical potential. The applied potential, eq 7, will be taken to be the total potential V(x,z)in the Nernst-Planck equation, (2). In doing so, we neglect contributions to Vfrom fixed charges (which may be present in the membrane) and from the reorganization of the electrolyte after the field is imposed on the system. The latter effect is neglected at the point in the derivation in which we use Laplace's equation (in which electroneutrality is assumed) rather than Poisson's equation for the applied potential. Contributions to Vfrom the fixed charges also essentially neglect the contributions from the double layer at the interphase region. All of these effects may lead to considerable potentials, but these will generally be the same in the two-electrode configurations. Since we are concerned only with differences between the two configurations, we choose to neglect these other possible contributions to I/. B . Goldman Equation. In the first part of our study, we will not solve eq 2 rigorously but will consider an approximate integrated form of the steady-state equation. Goldman showed many years agos that the steady state (dci/dt = 0) form of eq 2 can be integrated if thefield is constant in the z direction to give

where a and b are the positions of the boundaries in the x direction; these are taken to be just beyond the array of wires on either end. The spatially averaged membrane potential is similarly defined:

(AV) = L a -l b d x

AV(x)

(9b)

To determine if the average flux of ions according to (9a) depends on the staggering of the wires in the two arrays, we computed ( J i ) for 6 = 0 and for a nonzero value of 6, the staggering parameter, and formed the flux enhancement factor, y, which tells how the degree of staggering changes the flux through the membrane:

Note that the flux enhancement factor y is not the same as that considered in our previous papers?,' The flux enhancement factor y gives the relative change in the spatially averaged flux due to a nonzero staggering parameter and is equivalent to the flux enhancement factor measured experimentally. A negative value of y means that the spatially averaged flux in the staggered case is less than that in the eclipsed case, while a positive value of y means that the staggered case has a greater flux. The flux enhancement factor that we introduced in our earlier work (denoted a in those papers) gives the relative difference between the flux in a spatially nonuniform system (which could, in principle, correspond to a zero staggering parameter) and a spatially uniform system. This previously computed flux enhancement factor is, thus, defined as y' =

(Ji)

- Ji((AV)) Ji(

( A V )1

(1 1)

where the integration has been made over zL < z < zR and the boundary conditions are ci(zL)= c?, ci(zR)= c?, and A Y = V(zR) - V(zL). Here, hi is the permeability of the membrane to an ion of valence zi and is simply the ratio of the diffusion coefficient Di to the membrane thickness. If zL and zR are taken to be the positions of the left and right surfaces of the membrane, A Y i n eq 8 is then the membrane potential. Equation 8 strictly applies, of course, only to a one-dimensional membrane, i.e., one in which the vector flux has only one component Jiz = Ji. To use this approach, we assume Jix and Jiy are small, in some sense, and that (8) describes the flux in the z direction. Any dependence of the electrical potential on the perpendicular directions x and y will be included after the Nernst-Planck equations have been integrated to get (8). The validity of this assumption is considered in a following section. The Goldman equation rigorously applies only to a very special situation in which any fixed charges are distributed in such a way that the total field is constant. We do not expect it to be valid in most cases but include it as an interesting theoretical model. Results from the study of the Goldman equation are, in addition, crucial to our interpretation of a more rigorous and widely applicable model. C . Flux Enhancement Factors. To compute the expected flux through the membrane, we use eq 7 and evaluate the membrane potential AV(x) from eq 7 as AV(x) = V(x,zR)- V(x,zL).The membrane potential AV is a function of the direction parallel to the membrane surface, x. This is true in both the eclipsed and staggered cases because the field is applied with an array of wires rather than with parallel plate electrodes. The membrane potential generates a flux that is also dependent on the direction parallel to the membrane. This flux, Ji(x),can be calculated from either eq 8 or a numerical approach described in the following section. Because we are considering the steady-state case, Ji should be independent of z although it will depend on x. The observed flux will actually be a spatially averaged flux, not one that depends on x. We define, then, the spatially averaged flux6 as

where the value of the uniform membrane potential is taken to be. equal to the spatial average of the nonuniform potential, ( A n . Note that the first term in (1 1) is the spatially averaged flux due to the nonuniform potential. A positive value of y' means that the observed flux due to the nonuniform potential is greater than the flux in the presence of a uniform potential of the same magnitude. As we show in the next section, the electric field is not uniform everywhere even in the eclipsed case, when the staggering parameter is zero, so the comparison of fluxes made in our experiment actually yields the flux enhancement factor y(6=0.5), eq 10, rather than y', eq 11. To compare to our previous theoretical studies, though, we will consider both of these definitions of a flux enhancement factor in this paper. It is also instructive, as we will see, to look at a field enhancement factor. For a given applied voltage we cannot assume that ( AV) will be the same for different values of 6. Hence, we define the field enhancement factor, K:

(9a)

(10) Botha, J. F.; Pinder, G . F.Fundamental Concepts in the Numerical Solution of Differential Equations; Wiley: New York, 1983.

K(6)

=

(AV(8)) - (AV(6=0)) (AV(6=0))

(12)

D. Finite Difference Method. In addition to calculations using the Goldman equation, we also used the finite difference methodlo to numerically integrate the steady-state Nernst-Planck equation. In this portion of the study, we did not make any assumptions about the relative magnitudes of the x, y , and z components of the vector flux. We did, though, assume that the system was at steady state; the partial differential equation that was solved was then

(Note that Jiy.does not appear in (1 3) because we still assume the wires are infinitely long in the y direction.) The electrical potential V(x,z)in eq 13 is that due to the wire arrays and is given

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The Journal of Physical Chemistry, Vol. 92, No. 21, 1988

Steinmetz and Larter

by eq 7. The steady-state equation was made discrete in a rectangular region corresponding to the position of the membrane and defined by a Ix Ib, zL Iz IzR. Central finite differences were used to make discrete the derivatives in the governing equation. Boundary conditions must be specified to compute a solution to eq 13. W e chose to take the concentration c, as the constant values ck and ciR at z = zLand z = zR, respectively, as two of the specified boundary conditions. The physical interpretation of these boundary conditions is that the left and right surfaces of the membrane are bathed with well-stirred solutions of fixed, but different, concentrations. Since the physical system is bounded by impermeable walls a t x = a and x = b, we took the flux to be zero as the boundary conditions at these positions. Since the flux expression involves a derivative, we used forward and backward finite differences to make discrete these conditions. The resulting system of inhomogeneous, linear equations for ci at the nodes of the grid was solved by using a standardized package (routine LEQTZB from the International Mathematics and Statistics Library, IMSL) that arrives at an initial solution to the system of equations with row equilibration and partial pivoting and then uses an iterative algorithm to improve the solution. W e computed the solution c,(x,z) for a variety of grid spacings and substituted the result into the Nernst-Planck equation to compute the x and z components of the flux:

As discussed above, the spatially averaged flux components, ( J l x ) and ( J I z ) are , the quantities of interest here; the integration over x was computed by using a trapezoidal rule algorithm. The value of ( J,x) is highly dependent on the electrode configuration; in fact, we found that ( J l x )was zero for the eclipsed (6 = 0) case but was quite large for the staggered (6 =0.5) case. Since the quantity ( J l x ) is the net flux across the membrane, not through it, the measured current must be due only to ( J l z ) . The results for ( J l z ) were nearly independent of z (the standard deviation was at least 3-4 orders of magnitude smaller than the mean). So, the spatially averaged z component ( J l z )was averaged over the range zL < z < z R and the result set equal to ( J l ) ;this was used to compute the flux enhancement factors y and 7’. The computed values of ( J , ) depend on the grid size but converge slowly to an asymptotic value as the number of grid points is increased. W e extrapolated the results at finite grid sizes to a grid with an infinite number of points by the method of repeated Richardson extrapolation.” This was done by using four different grids such that the number of nodes in the grids ( n ) formed a geometric series: (15)

no, q2no, q4nO,4%

for some values of no and q. In most cases, the values of no and q that we used were 1024 (322) and 1.285 640795, respectively (the latter is the q value needed for @no = 682). The quantity ( J , ) / D , was considered since D, has no effect on the computed flux enhancement factors. It was assumed that the value of ( J , ) / D , obtained from the finite difference method is a function of the number of grid points: (Ji)

-(n) Di

= ao+

a1

a2

a3

n2

n4

n6

-+-+-+0

-

where a. is the (in general, unknown) limit as n m . If we label the four grids 0-3 and label the corresponding values of ( J i ) / D i ao,o-a3,0then we can extrapolate three new ( J i ) / D ivalues, al,i-a3,1 with the formula am,k

- am.k-I +

am.k-l

qk

am-l,k-l

-1

(17)

(11) Dahlquist, G.; Bjork, A. Numerical Methods; Prentice Hall: Englewood Cliffs, NJ, 1974; pp 269-273.

Figure 3. Electrical potential contours. Isopotential curves, computed from eq 7, are shown for (a) 6 = 0, (b) 6 = 0.167, (c) 6 = 0.333 and (d) 6 = 0.5 for the region -1.0 cm < z < +1.0 cm. The values of X i used are given in Figure 2; all other parameter values are given in the footnote caption to Table I. The zero-potential contour is the first solid line from the positions of the wires, and each successive contour corresponds to 0.0375 V. Closeup views of the region covered by the membrane, zL = 0.18 cm to zR = 0.22 cm, are shown for (e) 6 = 0 (contours separated by 0.00333 Volts) and (f) 6 = 0.5 (contours separated by 0.0122 V).

By application of (17) again to al,l-a3,1two more values ( u ~and , ~ u ~ ,can ~ ) be extrapolated. The last value, ~ 3 . 2is the value reported for ( J , ) / D ,and the uncertainty in the final result is la3,2- a2,21. It is possible to assess the accuracy of the Richardson extrapolation procedure by considering a limiting case for which we can calculate the flux exactly, namely, that of a uniform field. In this case, the Goldman eq 8 is exact. The difference between the extrapolated value ~ 3 . 2and the exact value can then be compared to the uncertainty lq2- q21. 111. Results and Discussion A . Electric Potential. Equation 7 was used to compute the electric potential profile for the two electrode configurations shown in Figure 2; the resulting contours are shown in Figure 3. The profile V(x,z)due to the eclipsed electrodes, Figure 3, parts a and e, is seen to be nearly uniform (Le., nearly independent of x ) around the midline between the electrodes. If the membrane is

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The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6117

TABLE I

6,v

103(J ~/ D~ ) ( 6 = 0)

103( J ~/ D~ ) (6 = 5)

102y(6 = 0.5)

103~'(6 = 0)

103~'(6 = 0.5)

(a) Goldman Equation Results for Passive Transport" 0.025 0.065 0.25 0.50 1.o 2.0 3 .O 4.0 5.0 6.0 7.0 8.0 10.0 13.0 16.0 25.0 0.01 0.03 0.05 0.065 0.08 0.10 0.25 0.50 1.o 2.0 3.0 4.0 5 .O 6.0 7.0 8.0 10 13 16 25

0.345595 0.498704 1.20934 2.17605 4.12936 8.09248 12.0930 16.1069 20.1257 24.1465 28.1682 32.1905 40.2360 52.3055 64.3755 100.586 -0.21 1783 -0.135386 -0.0590373 -0.0180752 0.0553950 0.1 31623 0.701799 1.64623 3.51520 7.19665 10.8407 14.4713 18.097 1 21.7209 25.3438 28.9660 36.2096 47.0739 57.9376 90.5277

0.351993 0.515365 1.27388 2.30630 4.39381 8.63001 12.9027 17.1867 21.4745 25.7640 30.054 5 34.3454 42.9285 55.8048 68.68 18 107.315

1.851 30 3.34080 5.33638 5.985 31 6.40400 6.642 30 6.695 73 6.703 99 6.702 13 6.698 94 6.696 24 6.694 21 6.691 74 6.687 78 6.689 41 6.689 02

0.01 1 959 0 0.056 035 3 0.337 551 0.7 19 417 1.288 28 1.490 01 1.069 87 0.654 678 0.386 331 0.230 589 0.141 031 0.088 332 5 0.036 574 1 0.010 574 4 0.003 203 02 0.000 043 136 9

(b) Goldman Equation Results for Facilitated Transportb 1.206 76 0.003 141 36 -0.209228 -0.127725 5.658 77 0.044 090 6 0.280 690 -0.0462790 21.6107 -0.014808 181.709 15.7240 0.0757851 36.808 5 -0.764 086 0.157091 19.349 2 -0.502 121 0.765100 9.019 82 -0.581 337 1.77166 7.619 14 -0.949 540 3.762 10 7.023 75 -1.509 30 -1.67036 7.68181 6.741 47 -1.19091 11.5650 6.68 1 46 6.672 29 -0.727 823 15.4369 6.702 13 -0.429 441 19.3050 23.17 14 6.677 92 -0.256 356 27.0370 6.680 92 -0.156850 -0.098 303 8 30.9019 6.683 35 38.6306 6.685 91 -0.040 792 2 50.2221 6.690 06 -0.01 1 900 3 6.688 51 -0.003 715 53 61.81 27 96.5831 6.689 02 -0.000 196 300

0.016 158 8 0.074 649 1 0.440 992 0.932519 1.653 0 1.88400 1.357 17 0.851 519 0.326 0.522 925 956 0.209 681 0.137613 0.062 405 0 0.020 636 8 0.007 180 06 0.002 473 0.004 385 24 0.064 395 5 0.493 298 -2.602 49 -0.769 156 -0.579 302 -0.733 726 -1.21 162 -1.92405 -2.108 38 -1.5 10 09 -0.946613 -0.581 345 -0.363 524 -0.233 242 -0.153 177 -0.069 598 4 -0.023 205 2 -0.008 250 87 -0.000 551 792

"Spatidly averaged reduced flux ( J i ) / D ifor 6 = 0 and 6 = 0.5 and flux enhancement factors y and y', calculated from the Goldman equation for the case of passive transport. The parameter values used in these calculations are zi = +1, T = 298.15 K, N = 4, (Y = j3 = 0.4 cm, zR - zL = 0.04 cm, a - b = 2.0 cm, and cL = 0.10 M, cR = 0.09 M. The charge densities Xi are given in Figure 2 for 6 = 1 V and X,(e) = 6Xi(6=1). All results are given to six significant figures. The error associated with the spatial averaging procedure is about lo4 mol/cm4. bSpatially averaged reduced flux and flux enhancement factors calculated from the Goldman equation for the case of facilitated transport. Parameter values and significant figures as given in part a except cL = 0.09 M and cR = 0.10 M. very thin and is located at this midline, we would expect the membrane potential to be nearly uniform when the electrodes are in the eclipsed configuration. In the staggered configuration, shown in Figure 3, parts d and f, the electrical potential depends strongly on x everywhere in the region between the electrodes, so even a very thin membrane would be subject to a nonuniform field when the staggered configuration is used. The variation of uniformity with staggering parameter 6 is clearly seen in parts a-d of Figure 3. The situation for a membrane that is not very thin (Le., whose thickness is some substantial fraction of the spacing between the wire arrays) will be somewhat different. As can be seen in Figure 3a, the electrical potential V(x,z)varies strongly with x for values of z not near the midline; hence, a thick membrane would be subject to a nonuniform membrane potential even if the eclipsed configuration is used. The significance of this fact will be explored further in the following two sections. The profile depends, of course, on the values of the charge densities Xi in eq 7 . These were computed by the electrostatic equilibrium method described in section 1I.A and in the Appendix. Typical values of the Xi are shown in Figure 2 for a fixed potential difference, 6, of 1 V between the electrode arrays. For the eclipsed configuration, the charge densities are seen to be symmetric (see Figure 2a) with respect to the plane midway between the top and bottom. For the staggered configuration, the t o p b o t t o m symmetry is broken, and charge flows through each array from the midplane to the periphery. For example, XI for the staggered

arrays is larger than XI for the eclipsed arrays, while X4 is smaller. This variation in charge density contributes to the flux enhancement effect but is probably not entirely responsible for it. The quantity ( A V ) = (V(x,zR)- V ( x , z L ) )is used in the Goldman calculations, the results of which are discussed in the following section. We found that ( A V ) depends on 6, as anticipated. The value of ~(6=0.5),the field enhancement factor given by eq 12, was found to be a constant 6.69%, independent of the applied field. As we will see in the following section, K should be the asymptotic value (high-voltage limit) of the flux enhancement factor y when the Goldman equation is used. B. Results from the Goldman Equation. Equation 8 was used to calculate the flux Jiof a single univalent cation (zi = 1) between baths of different concentration. As in our previous paper, we considered both the case in which the ions move against a concentration gradient (electrically facilitated transport) and the case in which they move down the concentration gradient (passive transport). Facilitated transport is associated with bath concentrations c,'- = 0.10 M and c? = 0.09 M while passive transport occurs with c: = 0.09 M and ciR = 0.10 M. Using eq 7, we first computed AV(x) = V(x,zR)- V(x,zL).This spatially dependent membrane potential is, thus, associated with a spatially dependent flux J i ( x ) . Equation 9a is used to compute values of ( J i ) for 6 = 0 and 6 = 0.5; results for the passive case are given in Table Ia and for the facilitated case in Table Ib. For convenience, the quantity ( J i ) / D iis reported since D, has no effect on the subsequently computed flux enhancement factors.

6118

The Journal of Physical Chemistry, Vol. 92, No. 21, 1988

Steinmetz and Larter

TABLE I1

6,v

103( J ~/oi ) (6 = 0)

lo3(J I )/Di (6 = 0.5)

102y(6 = 0.5)

103y(6 = 0)

103y(6 = 0.5)

(a) Finite Difference Results for Passive Transport“ 0.025 0.25 0.5 1 .o 2.0 5.0 6.0 7.0 8.0 10.0 13.0 16.0

0.350623 (0.000039) 1.259944 (0.000036) 2.27800 (0.00059) 4.3358 (0.001 1) 8.5121 (0.0018) 21.1812 (0.0012) 25.4237 (0.0061) 29.6582 (0.0064) 33.8872 (0.001 1) 42.3635 (0.0077) 55.0654 (0.0027) 67.77217 (0.00041)

0.353348 (0.000036) 1.28730 (0.00029) 2.33289 (0.00057) 4.4464 (0.001 1) 8.7375 (0.0061) 21.699 (0.026) 26.046 (0.022) 30.409 (0.010) 34.741 (0.027) 43.4655 (0.0018) 56.500 (0.01 1) 69.54682 (0.0008 1)

0.777 (0.022) 2.17 1 (0.052) 2.410 (0.051) 2.551 (0.050) 2.648 (0.093) 2.45 (0.13) 2.45 (0.1 1) 2.533 (0.057) 2.571 (0.036) 2.602 (0.023) 2.605 (0.024) 2.6190 (0.0018)

1.46 (0.01 1) 4.219 (0.030) 4.760 (0.027) 5.135 (0.026) 5.342 (0.022) 5.2851 (0.0062) 5.314 (0.025) 5.305 (0.023) 5.2801 (0.0034) 5.291 (0.019) 5.2777 (0.0051) 5.27668 (0.00065)

0.025 0.05 0.065 0.1 0.5 1 .o 2.0 5.0 6.0 7.0 8.0 10.0 13.0 16.0

-0,14945 (0.000018) -0.049003 (0.000053) 0.01 1235 (0.000070) 0.151681 (0.000075) 1.74531 (0.00056) 3.71 10 (0.0010) 7.5826 (0.0015) 19.0510 (0.0017) 22.8742 (0.0051) 26.6822 (0.0027) 30.5021 1 (0.0056) 38.1283 (0.0062) 49.5664 (0.0059) 61.0055 (0.0070)

(b) Finite Difference Results for Facilitated Transportb -0.146736 (0.000015) 1.820 (0.022) 3.250 (0.012) -0.043558 (0.000041) 11.11 (0.18) 16.973 (0.089) 0.018309 (0.000057) 63.0 (1.2) 731.4 (3.9) 0.16254 (0.00010) 7.16 (0.12) 15.181 (0.057) 1.79934 (0.00054) 3.096 (0.064) 5.918 (0.034) 3.8181 (0.0011) 2.887 (0.058) 5.410 (0.029) 7.7951 (0.0060) 2.80 (0.10) 5.187 (0.021) 19.521 (0.025) 2.46 (0.14) 5.2255 (0.0094) 23.436 (0.021) 2.45 (0.12) 5.283 (0.023) 27.3334 (0.0039) 2.441 (0.025) 5.264 (0.01 1) 31.28703 (0.00055) 2.573 (0.037) 5.2927 (0.0019) 39.1206 (0.0021) 2.603 (0.022) 5.294 (0.017) 50.8653 (0.0012) 2.620 (0.015) 5.294 (0.013) 62.6091 (0.0010) 2.629 (0.013) 5.295 (0.012)

0.387 (0.010) 1.098 (0.023) 1.247 (0.025) 1.364 (0.025) 1.436 (0.071) 1.10 (0.12) 1.203 (0.035) 1.203 (0.035) 1.164 (0.078) 1.2573 (0.0041) 1.248 (0.019) 1.2601 (0.0012) 0.9138 (0.0099) 5.832 (0.089) 23.65 (0.38) 3.409 (0.066) 1.439 (0.030) 1.293 (0.029) 1.260 (0.078) 1.06 (0.13) 1.103 (0.091) 1.073 (0.014) 1.2309 (0.0018) 1.2615 (0.0055) 1.2784 (0.0024) 1.2876 (0.001 5)

“Spatially averaged reduced flux and flux enhancement factors calculated with the finite difference method for the case of passive transport. All parameter values are as given in Table Ia. Usually, a grid of 68 X 68 points was used, and the result extrapolated as described in the text. The uncertainty is calculated by the repeated Richardson extrapolation technique and is given in parentheses for each result. Two significant figures are retained in each uncertainty, and the computed flux or flux enhancement factor is reported out to the last significant digit. bSpatially averaged reduced flux and flux enhancement factors calculated with the finite difference method for the case of facilitated transport. Ail parameter values are as given in Table Ib. Uncertainties and significant figures are handled as per Table IIa. In Table I we also list computed values of three types of flux enhancement factors. The first, y(6=0.5), can be compared directly to the flux enhancement factor obtained experimentally. Figure 4 shows the variation of y(6=0.5) with applied voltage, 6,for both the facilitated and passive cases. The flux enhancement factor is positive for all voltages for both cases, indicating that the flux in the staggered configuration is larger than that in the eclipsed, as we observed experimentally at an applied potential of 2 V. Experimental data are not yet available to compare with all these results; however, a flux enhancement of about 10% was observed in our previously published experimental investigation. A somewhat surprising result of these calculations can be seen in Figure 4a, which shows the low-voltage behavior of y ( t ~ O . 5 ) for both the facilitated and passive cases. Both quantities must go to zero at zero voltage since the transport is purely diffusive in that case and no distinction between “eclipsed” and “staggered” can be made. The flux enhancement factor for passive transport is close to zero for small applied voltage and rises nearly monotonically to a high of about 6.7% (see Figure 4b). The facilitated transport case seems to have the same asymptotic value for high voltages. In fact, the high-voltage limit should be given by eq 12, since taking the limit of eq 8 as the membrane potential AV yields a flux that is proportional to AV. The flux enhancement effect at high voltages as computed from the Goldman equation, then, can be completely explained in terms of the enhancement of the average membrane potential. The flux enhancement factor for facilitated transport is quite large for low voltages and at first glance seems to have a maximum near 0.1 V. Actually, the flux enhancement factor is anomalously large in this region, and we do not mean to suggest that flux enhancements of 200% are to be expected. The flux for the eclipsed configuration is changing sign in this region. Since it is the denominator of y,the value of the flux enhancement factor becomes quite large near this voltage. (Because of this sign change, it is necessary to take the absolute value of the denominator for facilitated transport for small voltages to have a quantity that

-

corresponds to the actual enhancement of flux.) The voltage at which the singularity occurs is just the value of AV that satisfies qL = A singularity does not occur in the passive transport case because we consider only negative values of AV. The last two columns in Table I list results for the enhancement of flux by a nonuniform field over a corresponding uniform field of the same magnitude, Le., values of y’ defined by eq 11. The first term in (1 l), ( J i ) ,is the spatially averaged flux in the presence of a nonuniform field. Here, we really have two different nonuniform fields to consider since both the eclipsed and the staggered configurations produce spatially nonuniform potential profiles, as shown in Figure 3. Note that, in either case, the magnitude of the uniform field in y’ is taken to be the spatial average of the nonuniform field but that this value differs for the eclipsed and staggered cases. The values of y’are positive for the passive case and vary from positive to negative for the facilitated case but are always quite small except near the singularity. In a previous paper,’ we predicted that all spatially nonuniform fields should, to a first approximation, lead to a higher flux for passive transport and a lower flux for facilitated transport over corresponding uniform fields of the same magnitude. Here we find that passive transport is slightly enhanced, as predicted, but facilitated transport follows the prediction only at high voltages. In both cases, the numbers, whether positive or negative, are quite small. As shown in the following section, this result is highly dependent upon the assumptions inherent in the Goldman equation and is not a general result. C. Results from the Finite Difference Calculations. Most of the results of the finite difference calculations are qualitatively the same as for the Goldman calculations (see Table 11). The fluxes are of about the same magnitude and follow the same trend with applied voltage, and values of y follow qualitatively the same trend as in the Goldman results; the exceptions to this qualitative agreement are the values of y’. The low-voltage behavior of y is qualitatively the same as that found in the Goldman calculation-the flux enhancement factor

The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6119

Model for Enhancement of Membrane Transport Law Voltage

Law Voltage

(a)

(e)

3.65 cJ>/D

xlooo 3.55

351

OW

025

0.50 (volts)

High Voltage

075

100

0.W

13

(volts)

25

3.45 3.35 0.25

0.50

E (Volts) High Voltage

(b)

2.25

I

i~ 0.75

1.00

(d)

1

9

(volts)

36

45

Grid Size

56

66

Figure 5. Convergence of finite difference results. The reduced flux ( J , ) / D ,is shown as a function of the grid size (number of points along either the x or z direction) for the case of a uniform field ( A V )of -0.041 348 929 V (which corresponds to 6 = 1 V). The data can be extrapolated by the repeated Richardson method (discussed in text) and are shown along with the exact result (computed from the Goldman equation for the case of a uniform field). The difference between the Goldman result and the Richardson extrapolation is only 0.0226%.

17

Figure 4. Flux enhancement factor versus applied potential. The (a) low-voltage and (b) high-voltage behavior of r(6=0.5) as a function of G is shown for the cases of passive and facilitated transport as calculated from the Goldman equation. Parameter values are given in Table I. Finite difference calculations for (c) low-voltage values and (d) highvoltage values are also shown. Error bars are given by the uncertainties (numbers in parentheses) in Table 11.

for the facilitated case goes through an apparent maximum a t about 0.1 V, although the absolute value of the “maximum” is not meaningful since this point is really a singularity. The flux goes to zero for small voltages in the electrically facilitated case because the magnitude of the diffusive terms and electric migration terms become comparable in this region; since they are of opposite sign for the facilitated case, a voltage at which the flux becomes zero is inevitable. For our parameter values this occurs near 0.1 V. The high-voltage behavior of y is also qualitatively the same except for a shallow minimum observed near 6 V for both the passive and facilitated transport cases. Lower values of y are found for the finite difference calculations throughout the range of voltages considered. Except for the shallow minimum, which is, in fact, associated with the greatest amount of error, the results follow the same trend as those from the Goldman equation. Note however that a shallow minimum in the facilitated case is seen even in the Goldman results (Figure 4b) and that it occurs approximately the same voltage, which implies that the minimum in Figure 4d is real and not simply a function of the computational method. These results differ from the Goldman results when they are considered quantitatively; the difference accounts for as much as 4.5% out of about 6.5% at high voltages. Some rather strong assumptions were used when applying the Goldman equation, and it may be that these assumptions are partially invalid and are the source of the discrepancy. To fully investigate this difference, we have performed extensive convergence checks of the finite difference results. These are described below. The accuracy of the Richardson extrapolation can be assessed by considering a uniform applied field, in which case the Goldman eq 8 is exact. The extrapolated result a3,2 for this case agreed with the exact result to within fla3:p - a2,*1for all voltages considered. An example is illustrated in Figure 5, which shows the convergence of ( J i ) / D ias a function of n for a uniform (constant) field of ( A V ) = -0.041 348 929 V/cm (which corresponds to G = -1 V). Also shown are the extrapolated value ( J i ) / D i= 3.5197 X mol/cm4 and the value given by the Goldman equation, 3.5205 X lo4 mol/cm4. These values differ by 0.0008 X mol/cm4, while the value of 1a3,2 - a2,21is 0.0012 X IO4 mol/cm4. Therefore, the quantity la3,2 - a2,21gives a reliable measure of the uncertainty in ~ 3 , as 2 an estimate of the limiting value of eq 16 a. as n Because of the stringent requirements on the validity of the Goldman equation, we conclude that the quantitative difference

-

3.25 26

in results between the two methods is due to inaccuracies in the Goldman equation approach. The error investigation of the finite difference results lends support to this point of view. Important qualitative differences exist between the Goldman results and the finite difference results when we consider y’ values. The last two columns of Table I1 show that y’ varies qualitatively with applied voltage in the same way as y does for either the Goldman equation or the finite difference calculations but d i f ferently than y‘ does for the Goldman equation. The facilitated flux undergoes a change of sign at some small applied voltage, leading to a singularity; the high-voltage behavior is essentially constant as it is for y and is nearly independent of the concentration gradient. For y’(6=0) the asymptotic value is 5.27% for the passive case and 5.30% for the facilitated case. The asymptotic value of y’(6=0.5) is 1.26% for the passive case and 1.29% for the facilitated case. So, the finite difference calculations imply that both types of nonuniform field (6 = 0 or 6 = 0.5) enhance the flux over that of a corresponding uniform field. Compare the high-field results for y’ from the finite difference calculations to those calculated from the Goldman equation: the Goldman equation predicts that y’ 0 at high voltages for either the passive or the facilitated case. This is not surprising in view of the fact that ( J i ) a ( A V ) for the Goldman equation at high ) high fields for the Goldman fields; this is why y(0.5) = ~ ( 0 . 5 at results. The flux enhancement factor y’, eq 1 1, is defined as the fractional difference between the spatially averaged flux in the presence of a nonuniform field and the flux in the uniform field whose magnitude is the spatial average of the nonuniform field. If the relationship between the flux and the field is linear, as it is for the Goldman equation at high fields, it makes no difference whether the spatial average is taken first of the field and then the flux evaluated or whether the flux is first evaluated and then the spatial average taken; the same result will be obtained regardless of the order of operation, so y’ must go to zero. The finite difference results are that y’ is not zero, even for high voltages. According to the above discussion, we would conclude, then, that the flux must depend on the voltage in a nonlinear way. The Nernst-Planck flux equation, (14), gives the relationship between the flux and the applied field; the field comes in through the second term, which also involves the concentration profile ci(x,z). This concentration profile is, in turn, determined by the applied potential V(x,z) (eq 7) through the steady-state eq 13. Hence, ci(x,z)is really a function of V(x,z);therefore, the flux depends on the applied field in a fairly complicated fashion. It is this nonlinear dependence that leads to the nonzero value of y’ at high fields.

-

IV. Conclusions To study the effect of spatial nonuniformity on transport, we have developed an electrostatic model for an electrochemical cell like the one used in experimental studies in our lab. In this model, the field is applied via two arrays of parallel wires, modeled as infinitely long line charges. By imposing the condition that the

6120 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988

potential difference between the arrays is fixed, we derived expressions for the linear charge density on each wire in each array. With the superposition principle, expressions for the electric field and potential at any point between the arrays can be derived. These expressions constitute the electrostatic model. We have studied the flux of ions in the presence of this field by two methods: (1) by using an approximate, analytical expression, the Goldman equation, generalized to a two-dimensional system; (2) by numerically solving the Nernst-Planck flux equations. From these calculations, we find that either method gives the same qualitative result: a staggered electrode configuration yields a higher spatially averaged flux than that due to the same voltage applied via electrodes in a n eclipsed configuration. This agrees with our experiment. In addition, the finite difference calculations show that the flux due to either the staggered or the eclipsed configuration is larger than that in a corresponding uniform field of equal magnitude for either passive or facilitated transport. This result is due to the nonlinear dependence of the flux on the field strength. The results of the Goldman calculations do not agree quantitatively with those of the finite difference method and even disagree qualitatively in some cases. The assumptions made in applying the Goldman calculation are probably the cause of this discrepancy. The electric field should be uniform to use the Goldman equation method, as this is the assumption made in deriving the Goldman equation itself. (Indeed, when a uniform electric field profile is used, the Goldman equation results agree well with those of the finite difference approach.) Therefore, we believe that the finite difference method yields better results for this system, although even the Goldman equation results agree qualitatively with experimental observations. Experimental results are not extensive enough to make comments regarding quantitative agreement between experiment and theory. The results reported here also give us the opportunity to speculate about the reason that transport is enhanced by the staggering of one array of wires with respect to another. The spatially averaged membrane potential (AV) for electrode configurations with 6 # 0 is larger than for 6 = 0. The flux, is therefore, larger partly because the local electric field is larger. In the real (experimental) system, nonlinear effects may also be important, since the field exerts its influence on the distribution of ions, which also determine the magnitude of the flux. Our theoretical results suggest an experiment in which the flux due to a nonuniform field is compared to the flux in the presence of a field that is euerywhere uniform, such as that between two large parallel plate electrodes. A flux enhancement observed in such an experiment would be a measure of the nonlinear effect. We have also found from our calculations that the magnitude of the local electric field is partially due to the variation of charge densities X i from one electrode configuration to the other, but the results are not entirely dependent on this effect. To determine the extent of the dependence of flux enhancement on the variation of Xi, calculations involving arrays of electrically disconnected wires are necessary. In such arrays, the wires can be independently charged while still controlling the total applied potential. Experimental investigation of this effect may also be possible by using an array of power supplies attached to an array of electrodes. The results of our investigation may also have some bearing on the high selectivity and special transport properties displayed by nonuniform ionomer membranes such as Nafion. Selvey and Reissi2and, more recently, Kuehl and Sanderson13 investigated the magnitude of the flux in the presence of a periodic charge (12) Selvey, C.; Reiss, H. J . Membr. Sci. 1985, 23, 1 1 . (13) Kuehl, S. A,; Sanderson, R. D. J . Phys. Chem. 1988, 92, 517.

Steinmetz and Larter distribution in a one-dimensional membrane. Both groups found that the flux is enhanced by nonuniform charge distributions of the proper symmetry. Our results are in qualitative agreement with theirs but extend the analysis to a two-dimensional membrane and a more physically relevant potential profile for our experimental investigation. Note that the charge density distribution is associated with a nonuniform electrical potential profile and vice versa, so the physical interpretation of our model is not fundamentally different from theirs. Hence, our results also have implications for the mechanism of selectivity enhancement in nonuniform ionomer membranes. Our approach is somewhat different from those of ref 12 and 13 in that we neglect the effect of fixed charges in the membrane and, in addition, assume charge neutrality everywhere. A relaxation of these assumptions would involve solving Poisson's equation for the potential rather than Laplace's equation and would introduce an even higher degree of nonlinearity through the interdependence of the electric field profile and the concentration profile. We would expect to find similar, but more elaborate, effects in such a model. Another effect of our assumption of zero fixed charge is the neglect of double-layer effects. To rigorously include such effects, one must invoke a multiple length scale e ~ p a n s i o n , ' ~which J ~ complicates the analysis but may lead to interesting results. Acknowledgment. We gratefully acknowledge support of this work by the donors of the Petroleum Research Fund, administered by the American Chemical Society, and by the National Science Foundation under Grant No. CHE-8616408.

Appendix The charge densities Xi in eq 6 and 7 are calculated by an electrostatic equilibrium approach. Each array is held at a fixed potential & relative to the other array and the wires are numbered as shown in Figure 2. The potential at the position of wire i in the positive array is taken to be = &/2, for convenience. The corresponding negative wires are held at a potential of -6/2. The potential at the wire i is calculated from the superposition of potentials due to all the surrounding wires, those in the same array as wire i and those in the oppositely charged array. Those in the same array are electrically connected and so come to equilibrium fairly quickly as the relative positions of the arrays are shifted. The charge densities Xi on each wire may change when this occurs. The potential at each positive wire is always fixed at = G / 2 , however, so we may write the following expression for the constant potential:

-

xi

In [p2

1

+ ( N + 6 - 2i)2a2]1/2

Here, Xi is the linear charge density on the jth electrode. The first term in the above equation is the superposition of potentials due to both the wires in the positive array (numerator) and those in the negative array (denominator). T h e j = i term is not included in the sum since only the contribution from the negative wire is needed; the second term in the above equation is the potential due to this single negative wire. Four equations, one for each value of i , are formed, and each is taken to be &/2. The resulting system of equations for the Xi are solved simultaneously. Results for G = 1 V are shown in Figure 2. (14) MacGillivray, A. D. J . Chem. Phys. 1968, 48, 2903. (15) Larter, R.; Ortoleva, P. J . Theor. Biol. 1982, 96, 175.