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J. Phys. Chem. 1983, 87, 136-149
ratios were unchanged when C,Hlo was admitted, as expected. It is proposed that reaction 6 does indeed scavenge S atoms. It was also observed that the S2+/COS+ratio decreased significantly in the presence of 2-butyne or butane. One of several explanations for this observation is that one or more of the precursors to S2+are efficientlyscavenged, thus decreasing its abundance. Reaction 6 of S with 2-butyne could serve this purpose. In summary, carbonyl sulfide in the plasma primarily forms carbon monoxide. Several reactions are proposed to be involved in forming this product. Sulfur atoms are also important constituents of the plasma zone. These atoms are reactive and form other products. They can be scavenged by 2-butyne, but not by butane. The ionic composition of the plasma zone is expectedly complex. With ICR data some of this complexity can be resolved. Emission from electronicallyexcited CO and CS dominates
the emission spectrum. Quenching studies with SF, suggest that electronically excited CS arises in part from dissociative recombination of low-energy electrons with COS’. The complexity of mechanistic analyses is emphasized by the experiments which demonstrated sputtering of ions from the solid product deposited on the reactor wall. Acknowledgment. This research was supported by the Office of Naval Research (Contract No. N00014-80-C0244). Dr. D. Ernie’s efforts in constructing the apparatus are gratefully acknowledged. The ICR experiments were performed by Mr. David Weil and Dr. David Dixon. A full explication of these results will be reported. Registry No. COS, 463-58-1; S,7704-34-9;COS’, 12169-37-8; S’, 14701-12-3;Sz’, 14127-58-3;S3+,22541-72-6; CO’, 12144-04-6 CS2+,12539-80-9; SF6,2551-62-4;butane, 106-97-8;CO, 630-08-0; CS, 2944-05-0; S p , 23550-45-0; 2-butyne, 503-17-3.
Nonequilibrium Effects due to Ion Transport at the Forward Biased Interface between an Electrolyte Solution and an Infinitely Thick Ion-Exchange Membrane I . C. Basslgnanat and H. Reiss‘ Department of Chemistry and Biochemistry, University of California,Los Angeles, California 90024 (Received: February 2, 1982, I n final Form: August 12, 1982)
I n this paper we study ion transport through one of the interfaces (electrical double layer) between an infinitely thick ion-exchange membrane and a n electrolyte solution. The interface is biased in the forward direction. The assumption of electroneutrality is not made, and we solve the Nernst-Planck equations of transport and the Poisson equation simultaneously. For this purpose a quasianalytical method of some generality is developed and discussed at length in an appendix. Our principal goal is to investigate the applicability of the popular assumption of “local”equilibrium during transport, at the interface. It turns out that the assumption is reliable in the case of “film” control but not “membrane”control. The reasons for this are several. The most important reason is that, in “film”control ions can readily accumulate in the Nernst layer, and thereby change the effectiue electrolyte concentration so as to restore local equilibrium, repairing the “stress”induced by transport. In contrast, in the case of “membrane” control ions do not readily accumulate in the Nernst layer, and the system departs from “local”equilibrium. Several experimentalists have noted what appears to be a departure from “local” equilibrium.
Introduction In spite of the respectable age of the field of ion exchange, and the corresponding voluminous literature,’V2 certain important problems seem to have been neglected. These generally involve situations in which transport is occurring in a region where there is finite space charge density. The relevant spacial dimensions of such regions are quite small (of the order of a Debye length), and this by itself invites neglect. More important is the fact that the thorough treatment of the problem involves the simultaneous solution of the equations of transport and the Poisson equation of electrostatics. This is a highly nonlinear coupled system of equations whose solution is exceedingly difficult. Some examples of situations of this type are (1)transport in the double layer between an electrolyte solution and an ion-exchange membrane,3v4(2) transport in an ion Work performed in partial fulfillment of the requirements for the Ph.D. degree at the University of California, Los Angeles.
exchanger where the density of immobile ions fluctuates significantly within distances comparable to a Deybe length,4v5and (3) transport and carrier generation in the double layer at a bipolar junction.6~‘ There are many other examples. Various devices have been used to compensate for the neglect involved in a nondetailed treatment of this region, and in many instances the devices lead to results which are essentially correct.g12 These devices frequently (1) F. Helfferich, “Ion Exchange”, McGraw-Hill, New York, 1962. (2) N. Lakshminarayanaiah, “TransportPhenomena in Membranes”, Academic Press, New York, 1969. (3) P. Lauger, Angew. Chem., Int. Edit.Engl. 8, 42 (1969). (4) T. D. Gierke, “PerfluorocarbonIon Exchange Membranes”, Symposium, Electrochemical Society Meeting, Atlanta, GA, Oct 1977. (5) H. Reiss, I. C. Bassignana, J. Membr. Sci., accepted for publication. (6) S. Ohki, J.Phys. SOC. Jpn., 20, 1674 (1965). (7) P. Lauger, Ber. Bumenges. Phys. Chem., 68, 534 (1964). (8) M. Planck, Ann. Phys. Chem. N . F.,39,161 (1890); 40,561 (1890). (9) D. E. Goldman, J. Gen. Physiol., 27, 37 (1943). (IO) T. Teorell, 2.Electrochem. Angew. Chem., 55, 460 (1951). (11) R. Schlogl, Ber. Bunsenges. Phys. Chem., 82, 225 (1978).
0022-3654/83/2087-0 136$0 1.50/0 0 1983 American Chemical Society
Nonequilibrium Effects due to Ion Transport
involve the assumption that the distribution of both ions and potential in space charge regions are those characteristic of local equilibrium. For example, at the interface between an electrolyte solution and an ion-exchange membrane it is common to assume that a particular ion is partitioned between the phases in the same ratio as at equilibrium. In many instances this is a valid assumption, but as we shall show there are also practical situations in which this assumption is poor. Most of the published theory concerning ion exchange and ion-exchange membranes has relied on the assumption of electroneutrality (zero space charge) in the regions where transport is confronted and treated. The coupling between ions is then accounted for by the familiar diffusion potential. At the boundaries where space charge may be important, the situation is handled by the above-mentioned assumption of local equilibrium. Some experimental evidence suggests that the equilibrium assumption does not adequately describe the situation during trans~ 0 r t . l ~There ’ ~ are a few examples in the literature where transport and space charge are treated simultaneously.1*25 With one exception, the above-mentioned investigations are concerned with electrodes rather than membranes and, for the most part, directed toward nonsteady (time dependent) situations in which the problems of frequency dependence are of primary interest. Most workers in the field of ion exchange and ion-exchange membranes have been concerned with the practical problems of technology and have, therefore, attempted to deal with systems which bear some resemblance to the real ones. This means that multicomponent electrolytes are considered in which the dissolved ions may not only have differing mobilities, but will have different valences as well. It also means that, in the membrane, the effect of pore structure, channel distribution, ion pairing or association, swelling and osmotic pressure, and thermodynamic nonideality26in general may be treated. Convection and electroosmosis may also have to be considered. Theoretical treatments have usually assumed that “local equilibrium” adequately describes the state of the interface for the case of small f l ~ ~ e s . ~If~the ~ mechanistic J ~ * ~ ~ problem , ~ ~ on the molecular level becomes too difficult the methods of irreversible thermodynamics may be invoked.28 Such a phenomenological approach may provide a useful framework within which to tabulate experimental data at the expense of obscuring the details of mechanism. However, (12)H. U. Demisch and W. Pusch, J. Colloid Interface Sci., 69,247 (1979). (13)V. I. Zabolotskii, N. P. Gnusin, S. L. Reprintseva, and V. V. Nikonenko, Elektrokhimiya, 15,1124 (1979). (14)V. V. Nikonenko, V. I. Zabolotskii, and N. P. Gnusin, Elektrokhimiya, 15,1494 (1979). (15)V. V. Nikonenko, V. I. Zabolotakii, and N. P. Gnusin, Elektrokhimiya, 16,556 (1980). (16)Y. Oren and A. Litan, J. Phys. Chem., 78, 1805 (1974). (17)A. T. DBenedetto and E. N. Lightfoot, Znd. Eng. Chem., 50,691 (1958). (18)Y. Onoue, Y. Misutani, R. Yamane, and Y. Takasaki, J . Electrochem. SOC.Jpn., 29,E-155(1961);29,E-229(1961). (19)J. Newman, Trans. Faraday Soc., 61, 2229 (1965). (20)L. Bass, Trans. Faraday SOC., 60, 1656 (1964). (21)T. R. Brumleve and R. P. Buck, J. Electroanal. Chem., 90, 1 (1978). (22)A. 0.MacGillivray, J. Chem. Phys., 52,3126 (1970). (23)J. R. MacDonald, J.Ck” Phys., 54,2026(1971);58,4982(1973). (24)L. J. Bruner, Biophys. J., 5, 867,887 (1965). (25)J. E. Hall and L. J. Bruner, J. Chem. Phys., 50, 1596 (1969). (26)N. Kamo, Y. Toyoshima, H. Nozzaki, and Y. Kobatake, Kolloid. 2.2.Polvm.. 248. 914-21 (1971). (27)R P. Buck, F. S. Stover, A d D. E. Mathis, J. Electroanal. Chem., 82, 345 (1977). (28)A. Katchalsky and 0. Kedem, Menbr. Biophys., 2, 53 (1962).
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the validity of irreversible thermodynamics is called into question when dealing with nonlinear effects such as the rectification which occurs at interfaces where there is space charge. Although it is important, for practical purposes, to attempt the treatment of systems which are not stripped of so many features that they no longer bear a resemblance to reality, the simultaneous treatment of a reasonable fraction of these features makes the mathematical problem so complex that there is a danger of obscuring the real problem. Surely, if one is interested in a single phenomenon such as,for example, the validity of local equilibrium at an interface, this problem, difficult in itself, is made far more difficult if it is treated as part of a “real” system in which many other effects are dealt with simultaneously. We will therefore pursue a strategy in which we examine only one or a few features at a time, employing a highly idealized, essentially schematic, and, in this sense, unreal system. Our goal will be to understand the trend in that phenomenon alone. When the trends in a sufficient number of such phenomena are definitively understood, in the sense that the mathematical results are understood in terms of the physical phenomena, we will be in a better position to exercise our intuition in the synthesis of the behavior of a more realistic system incorporating a large number of such features. Ultimately, the problem of synthesis is made more difficult by the fact that almost every situation in the ion exchange and membrane field is dramatically different from every other situation in the field. Thus, each theoretical analysis (and each synthesis) tends to stand alone. The boundary value problem is indeed a boundary value problem and the precise combination of physical elements (types and number of membranes, and types and number of electrolyte solutions) are of central importance. There is an “action at a distance” character to such problems, and the precise specification of boundary conditions is important. This paper is the first of a series of investigations aimed at examining only one or a few effects at a time in idealized, schematic situations following the strategy outlined above. The present paper deals primarily with the question of the validity of the local equilibrium assumption during transport at a single interface between an electrolyte solution and an ion-exchange membrane. Formulation of the Forward Biased Interface Problem As indicated in the last section, in this paper we concentrate on transport through an interface between an electrolyte solution and an ion-exchange membrane. To have something concrete in mind we will, without loss of generality, consider the membrane to be cation selective so that its fixed ions will be anions, i.e., the cations are mobile counterions while the coions are mobile anions. In line with our policy of dealing with the simplest possible situation capable of exposing the significant trends, we will consider the membrane to be both homogeneous and ideal. This means, for example, that diffusion and electromigration can take place uniformly (except for restrictions imposed by the boundary value problem itself) throughout the bulk of the membrane. The details of channel and pore structure are ignored, as are possible fluctuations in fixed ion concentration. Furthermore, we ignore electroosmosis and convection as well as swelling and osmotic pressure. All ions, including those fixed to the membranes, will be univalent, and the membrane will be thermodynamically ideal. Among other things thermodynamic ideality requires that ion pairing or association between counterions
138
Bassignana and Reiss
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983 BULK ELECTROLYTE
n = p
SOLUTION DIFFUSE SPACE-CidARGE LAYER
NERNST LAY E R
I
MEMBRANE SPACE-CHARGE LAYER
BULK MEMBRANE
n = p
G- H =
pin
Ll
1
a41
-#
L
0
Figure 1. Plan of the calculation. The equations for concentration,charge density, and electrical potential are shown in
boundary conditions are also indicated.
and fixed ions be forbidden. We also assume that the electrolyte solution bathing the membrane is thermodynamically ideal. Thus, in all instances we can use the concentrations in place of activities, both in the solution and in the membrane. Any membrane has, of course, two interfaces each of which is in contact with an electrolyte solution. We will be focusing on only one of the interfaces. However, since the currents through the interfaces are in series, they cannot be strictly isolated from one another. Furthermore, as is well-known, each interface behaves as a rectifier. The direction of easy current flow (the direction of forward bias or open direction) corresponds to the flow of coions into the membrane, while the closed direction (reverse bias) corresponds to the flow of coions out of the membrane. There are situations in which the transport process can deplete the system of both counter- and coions in the vicinity of the interface. In such cases current carriers are usually supplied to the system in the form of H+ and OH-, by the thermal dissociation or field-assisted thermal dissociation of water molecules. We do not consider this process of "water splitting"'S2 in our model. For reasons which will become clear later, our analysis will be restricted to the situation in which the single interface under consideration is forward biased. The remaining membrane interface will therefore be reuerse biased. Furthermore we consider that interface to be infinitely far removed from the forward biased one under consideration. I t is extremely important to bear i n mind that our analysis (ut least in this paper) is restricted to t h e single forward biased interface. We consider the two-sided membrane in a succeeding paper, Since we will deal, without loss of generality, with a cation-selective membrane the chosen interface, forward biased, is the one a t which cations move from the membrane to the solution. The electrolyte solution is taken to be uni-univalent and of bulk concentration, Co. The counterions are the cation species of the external salt. The fixed ions in the membrane are assumed to be uniformly distributed and of constant density. The situ-
each region; the appropriate
ation is one dimensional with X , the direction of transport. The membrane-electrolyte interface is defined to lie at X = 0. The solution extends infinitely to the left of this interface while the membrane extends infinitely to the right (see Figure 1). Convective flow of solution through the membrane is neglected. In view of all our idealizations the equations of transport (for the particle flux densities) of the ions in either phase will be the Nernst-Planck equations. These have the form
for cations, and J , = -D, aA
ax
+[$]A
av
ax
(2)
for anions. In these equations C represents the concentration of cations and A the concentration of anions. V is the electrical potential while D, and D, are the diffusivities of cations and anions, respectively. The quantity q is the electronic charge, k the Boltzmann constant, and T the temperature. Finally J , and J , are the particle flux densities of cations and anions, respectively. The diffusivities will, in general, be different in the two phases, accounting for the well-known limiting cases of "film" and "membrane" control. In our analysis we allow for this difference. The potential, V, and the concentrations are coupled by the Poisson equation, namely (in Gaussian units) 4 w d2V --(C(X) - A(X)- Q(X)) (3) dX2 K in which K represents the dielectric constant of the medium, and Q the concentration of the fixed ions. Obviously, in the electrolyte solution Q = 0, but it has a finite value, determining exchange capacity, in the membrane. The simultaneous solution of eq 1-3 usually presents a formidable problem, not only because of the simultaneity -=
Nonequilibrium Effects due to Ion Transport
The Journal of Physical Chemistry, Vol. 87,
but also because of the implicit nonlinearity of the equations. In almost all of the literature dealing with ion-exchange membranes, the Poisson equation is replaced by the condition of zero space charge,*15 namely IC(X) - A ( X ) - Q(X)l = 0
(4)
The boundary condition commonly used at X = 0 is based on the assumption of local equilibrium, so that the ions are partitioned between the phases in the same ratio as in the Donnan e q u i l i b r i ~ m . In ~ ~the case of an ideal system, if the concentration of cations in the bulk solution at X = 0 is denoted by C,(O), then the Donnan theory requires the concentration of cations in the membrane at X = 0 to be
No. 1,
1983
139
must contend, in detail, with five separate regions. From left to right, beginning in the solution, these are the bulk solution, the Nernst layer, the diffuse space charge layer, the membrane space charge layer, and the bulk membrane (see Figure 1). It is useful to introduce some reduced variables which will simplify the various equations. As indicated earlier we will assume that, in each phase, the diffusivities of mobile cations and anions are the same. Thus the diffusivity will be donated by a single D. The transformations we now have in mind are the following: 4 = qV/(kT)
(7)
c/co
(8)
n = A/Co
(9)
P =
Q/co
(10)
5 = x/LD
(12)
h = JJD/(DCO)
(13)
g = JaLD/(DcO)
(14)
N = The concentration of anions in the solution at X = 0 is also C,(O) and the concentration of anions in the membrane at X = 0 will be given by
where V* is the Donnan potential. Assuming our idealized model, we find that these equations hold exactly at equilibrium (J= 0). Our goal, in this paper, is to investigate their validities when transport is occurring. For this purpose it will be necessary to solve eq 1 and 2 in the double layer, along with eq 3. This layer which is the seat of space charge is so narrow (sometimes only of the order of a hundred angstroms) and the gradients of both concentration and potential are so large that the Nernst-Planck equations may themselves be called into question, i.e., transport equations based more on molecular theory might be required. Although such equations may be formulated, in this beginning study we do not attempt to go beyond the Nernst-Planck equations. A feature common to solution-electrode interfaces in general, and to the solution-membrane system in particular, is the existence of an unstirred liquid film (Nernst layer) adhering to the interface which affects the overall rate of ion p e r m e a t i ~ n . ~ J ~The , ~ ' Nernst layer is considered to extend from some position in the electrolyte solution to the surface of the membrane. In our treatment we must have it extending from a position in the solution to the beginning of the space charge layer (diffuse layer) in the solution. The validity of this separation has been investigated by Newman.lg The diffwe layer extends from the Nernst layer to the surface of the membrane. The Nernst layer is commonly considered to be a domain of zero space charge where the electroneutrality condition eq 4 (with Q = 0) holds. On the other hand, in the diffuse layer we cannot use eq 4 but must rely on eq 3. Within the membrane, directly to the right of the interface is another space charge layer (membrane layer) in which, also, both the Nernst-Planck equations and the Poisson eq 3 must be dealt with simultaneously. However, here Q has a finite value. Finally, to the right of the space charge layer in the membrane, we have the bulk membrane in which eq 4, the condition of electroneutrality, once more holds. Thus, we (29) F. G. Donnan, Z. Phys. Chem. (Liepzig), A168, 369 (1934); Z. Electrochem., 17, 572 (1911). (30) J. F. Brady and J. C. R. Turner, J. Chem. Soc., Faraday Trans. 1, 74, 2839 (1978). (31) A. J. Makai and J. C. R. Turner, J. Chem. Soc., Faraday Trans. 1, 74, 2850 (1978).
In these equations Co denotes the concentration of cations and anions in the bulk electrolyte solution to the left of the Nernst layer. All other symbols, aside from those defined by the equations themselves, have been defined previously. With these transformations eq 1, 2, and 3 become
d24/dE2 = p - n - N ( [ )
(17)
Outside the membrane, N , appearing in eq 17 may be set equal to zero. The analysis will be sufficiently involved so that it is appropriate to provide some sense of the plan of calculation. In particular it is important to demonstrate that enough conditions are available to determine all the unknown quantities. We do this in Appendix B. There are, however, several features of the calculation, and, in particular, some of the conditions used whose physical bases require separate discussion. We attempt this here. We need to discuss the solution of eq 15 and 16 subject ot the assumption of electroneutrality, p - n - N = 0, for the case of the infinitely thick membrane. This solution, expressed in terms of p @ ) ,may be shown to be
The potential 4 satisfies the condition d4 dt
g-h 2p - N
- =-
(19)
as can be seen by merely substituting the requirement of
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The Journal of Physical Chemistry, Vol. 87, No. I, 1983
electroneutrality into eq 15 and 16. That eq 18 is indeed a solution can be proved by direct substitution of it together with eq 19 into eq 15 and 16. The membrane is considered to be infinitely thick. If the interface at 5 = 0 is reverse biased, then the interface at 5 = a will be forward biased. This means that, at t = a,the cations move from the membrane to the solution on the right. In other words, h > 0 and g < 0. Because of these signs the denominator in the last exponent on the right of eq 18 is positive, and (h g ) 2 in the numerator must always be positive, so that the last exponential factor must vanish as 5 m. Since the exponential on the left cannot vanish, the solution represented by eq 18, at very large values o f t , must correspond to setting the factor in curly brackets, on the left, equal to zero. But this requires the relation g/h = - n ( t ) / p ( 5 ) (20)
-
+
-
to hold with increasing accuracy as 6 a. This means that in the bulk membrane, just outside of the space charge region, belonging to the forward biased junction, eq 20 will hold. In fact, since g , h, and N are constant eq 20 shows that p , and therefore n, in this region will be constant and have zero gradient. Equation 20 is very useful for describing the bulk membrane, but this condition must be used with caution. For example, it will not be a good approximation for very thin membranes (of the type encountered in biological systems) where the width of the electrically neutral region is of the same order of magnitude as the charged interface. For these membranes the more complete form must be used. Equation 20, however, is certainly correct for the case we are treating here of an infinite membrane and will be adequate for studying the very thick membranes of the type encountered in industrial ion-exchange processes. When we confine our attention to the single forward biased interface and use eq 20, we are of necessity considering the other interface to be infinitely distant. The potential which would have to be applied across the resulting membrane to establish a finite current would then also have to be infinite. However, we do not let this bother us since we are not interested in that potential. We regard, as the independent variable motivating transport, the cation flux density h, and do not ask how its value is established by external means. We are only interested in how other phenomena at the forward junction respond to h, however it is established. There is, however, a subtle point in all of this which is not so self-evident, and which has hardly ever been discussed in the membrane literature. This concerns the question of how it is possible to have a nonzero local gradient of 4 in the absence of space charge, i.e., how consistent is the assumption of electroneutrality. This question was addressed by Jackson.32 Jackson was able to find a suitable ”small” parameter in terms of which the zero space charge solution could be exhibited as a firstorder perturbation to the zero field case, and he was able to give rigorous justification to the approximation. It should now be apparent that eq 20 does not hold in the neighborhood of a reverse biased junction. Without eq 20 the analysis undertaken below becomes considerably more difficult. As a result we have restricted our first study to the forward biased case. This is the most important case since it proves to be the one in which coions tend to violate the selectivity of the membrane. Our focus on a single interface is consistent with the strategy of looking, first, at the simplest meaningful parts (32) J. L. Jackson, J. Phys. Chem., 78, 2060 (1974).
Bassignana and Reiss
of the system. However, as this paper demonstrates, even this contraction of view leaves a not so simple problem. Figure 1 illustrates the five regions of the problem, mentioned above, together with the equations which hold within each region, and the boundary conditions which connect the regions. The figure deals with a case where the diffusivities are not the same in the solution and in the membrane. The changes required to accommodate differing diffusivities are fairly minor (as will be apparent later) and add nothing to the conceptual structure of the plan. Beginning a t the far left of the figure we have the bulk electrolyte solution which extends as far to the right as E = 4 1 . Next we have the Nernst layer which extends from t = -L1 to -W. Beyond f = -W, and extending to 5 = 0, we have a diffuse space charge layer. Between = 0 and 5 = L we have the membrane space charge layer, and, finally, beyond = L we have the bulk membrane which extends to t = m. In the bulk solution we have no unidirectional transport; the concentrations are maintained uniform by stirring, and there is no space charge so that n = p . In the Nernst layer, as indicated in the figure, the Nernst-Planck equations, eq 15 and 16, apply, but electroneutrality is still maintained and n = p , All this is indicated in the figure, the Nernst-Planck equations being inscribed at the top of the region designated as the Nemst layer. In the diffuse charge layer the Nernst-Planck equations still apply. We have inscribed them in an easily obtained integral form at the top of the region designated as the diffuse layer. We use this form because we need both p and n in the Poisson equation which must be used in the diffuse layer, and which is also inscribed, there, below the Nernst-Planck equations. Substitutions of the integral forms of the Nernst-Planck equations in the Poisson equation yields an equation on the variable cp alone. This equation which is the analog of the Poisson-Boltzmann equation, and to which it reduces when the currents, i.e., g and h, vanish, has been called the Nernst-Planck-Poisson equation.21,23,27,33 As indicated in the figure, the same situation applies in the membrane space charge layer. Here, also, it is convenient to use the Nernst-Planck equations in their integral forms. Finally, at the extreme right we have the bulk membrane. Once again, we have electroneutrality as indicated by the inscription N + n”’ = p”’. In the bulk membrane where there are no gradients (as long as the interface is forward biased), the Nernst-Planck equations require that the flux densities stand in the ratio of the coion and counterion concentrations. This accounts for the inscription glh = - n / p (eq 20). The various boundary conditions are shown, straddling the various interfaces. Thus at the interface between the bulk electrolyte solution and the Nernst layer, we have p = 1 reflecting the continuity of the concentrations of cations and anions across the boundary, and the fact that these concentrations, in reduced form, are unity in the bulk solution. The potential in the bulk solution is referenced at 0, and this, together with the requirement of continuity, gives rise to the condition 9 = 0 which also straddles the indicated boundary. A t the boundary between the Nernst layer and the diffuse layer, we have continuity of p , n, 4, and the gradient of 4 (the electric field). These conditions are indicated by the equations straddling this boundary. The last equation does represent the continuity of the field (a condition which evolves from the fact that there is no (33) H.Cohen and J. W. Cooley, Biophys. J . , 5, 145 (1965).
Nonequilibrium Effects due to Ion Transport surface charge density at the boundary), but the term (G
- H ) / ( p + n) is an expression for the gradient of the po-
tential obtained from the Nernst-Planck equations in the Nernst layer together with the condition of electroneutrality. It is easily obtained by subtracting the second of the Nernst-Planck equations in the Nernst layer from the first. A t the boundary between the diffuse layer and the membrane layer we also require the continuity of the ion concentrations, the potential, and the field. The four equations straddling this boundary in the figure reflect these conditions. Finally, in the membrane layer and the bulk membrane we again have continuity. The four equations straddling this boundary reflect these conditions also. The term - h / p on the right of the last equation again comes from the Nernst-Planck equation for the cations under the condition that there is no gradient of ion concentration in the bulk membrane. The most difficult part of the solution connected with this plan is the solution of the Nernst-Planck-Poisson equation in both the diffuse layer and the membrane layer. In principle, with sufficient computer capacity, we could perform the entire numerical solution exactly (i.e., to any degree of accuracy). However, this procedure is extremely difficult. For example, if one wishes to integrate the Nernst-Planck-Poisson equations using, say, a standard Runge-Kutta technique, it would be necessary to know g as well as h, to say nothing about the location of the boundaries (i-e.,the value, say, of W). But g is not known anymore than W until the last boundary conditions on the right of Figure 1 have been applied. Thus the RungeKutta method is confronted with a differential equation whose parameters have not been fixed. Such a procedure is therefore uneconomic and we have developed an approximate technique for dealing with the Nernst-Planck-Poisson equations which requires relatively little computer capacity and yet is fairly accurate. This method, as a general procedure, is discussed in Appendix A. In the body of this paper we will simply use it without justification, referring the reader to Appendix A for that purpose. The method involves the use of a “guessed”initial parameterized form of the space charge density. However, it must be emphasized that it does not therefore amount to inserting the solution into the problem at the outset. The parameters in the guessed function are eventually determined self-consistently as the solution proceeds. Discussion of the Solution In actual practice the steps in the solution are performed in an order different from that in Appendix B. A value of h is chosen. This allows the determination of the concentration and potential profiles, eq B7 and B8 in the Nernst layer and the diffuse layer boundary to within a parameter. Then the Poisson equation, incorporating eq B9 in the diffuse layer, is integrated subject to the boundary conditions at 5 = -W. The Poisson equation in the membrane layer eq B14 and its derivative are set equal to the corresponding quantities from the diffuse layer at 5 = 0. The integrated forms of the Nernst-Planck equations in both layers are then subjected to the boundary conditions of Figure 1. Next, we use the condition (part of the approximative method) that the total (integrated) space charge based on the “guessed” form be equal to that derived from the Nernst-Planck equations based on the potentials derived from the ”guessed” forms. Finally, we use the electroneutrality condition in the bulk membrane. This procedure provides a set of six conditions on the six unknowns, CY, W, pl, L , p”’(L), #(I,). The conditions are nonlinear
The Journal of Physical Chemistry, Vol. 87,
No. 1, 1983 141
equations whose simultaneous solution is performed by initially “guessing” at values of the unknowns, followed by an application of a modified Newton’s method iteration,34 until the system solves self-consistently for the six parameters which best fit the six equations. The computation of the final values of the six parameters for one value of h requires about 20 min of CPU time on a DEC VAX 11-780. Note that L1 and N are not unknown quantities. Several attempts to measure the thickness of the Nernst layer have been reported and a review2of the data indicates that this thickness is of the order of to m. This distance is large compared to the electrical double layer at the surface of the membrane. For the purpose of this calculation, L1 was taken to be one thousand times the Debye length, a typical value for the thickness of the Nernst layer. For N , the ratio of fixed ion concentration inside the membrane to the bulk electrolyte concentration, we chose the values 10,100, and lo00 for the cases studied. This means that we deal with membranes having fixed ion concentrations 10,100,and 1000 times as large as the ion concentration in the electrolyte solution. A membrane with N equal to 10 will have a rather poor exchange capacity in practical terms, while that with N = 1000 will be an exceptionally good membrane. N = 100 is good. Real membranes usually have N between 10 and 100. As indicated earlier, because of the rich nonlinearity of the governing equations, the variety of solutions is large, even when dealing with only a single configuration, such as the electrolyte-membrane interface under consideration in this paper. As a result, it is necessary to choose conditions, and the correspondingsolutions, carefully in order to display the important trends in the most economic manner. One problem that must be dealt with at the outset concerns the question of “film” or “membrane” control. This is a well-known subject and has been discussed extensively elsewhere.’ It is especially important in the present study since it appears, as we discuss later, that appreciable departures from the assumption of local interface equilibrium are experienced in the “membrane” control case but not in “film” control. As is also wellknown, the cases of membrane and film control are not sharply delineated, although some approximate rules of thumb have been advanced to help one decide which case is invo1ved.l In our discussion, we shall adopt an operational approach to the definitions of film and membrane control. In general, film control implies that, in the electrolyte solution, the slow step occurs in the Nernst layer; i.e., that the diffusivities or mobilities of the ions are much smaller in the electrolyte solution than in the membrane. membrane control refers to the inverse situation. If the diffusivities or mobilities in the electrolyte solution are large enough, no appreciable concentration gradients can be supported in the Nernst layer. The boundary condition at E = -W is essentially that of the bulk solution eq B7 in which both G and H becomes vanishingly small (the diffusivity in the electrolyte solution is very large y a). Here H and G are the ion currents h and g multiplied by y, where y is the ratio of the ion diffusion constant in the solution D to that in the meribrane D’ (see Appendix B). We define this situation, in which there is no appreciable concentration gradient in the Nernst layer, as the case of membrane control. By default, all other situations represent film control. We will see below that even in the case
-
(34) K. M. Brown, Commun. ACM, 10, 728 (1967).
142
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983
Bassignana and Reiss
25 7
h--O. 10
20
hm-0.05
0' z
l l i 12
7
c h--3.01
I
-1000
-800
-600
I
I
-400
-200
h-0. U 1
O
200
0
i
-2
-3
-1
E
1
0
E
Figure 2. Anion concentration profile in the Nernst layer in reduced units. For the case of C o = 0.001 M, N = 100,L 1 = lOOOL,, y =
1.
Concentration profiles in the interface space charge region in reduced units. For the case of C , = 0.001 M, N = 100, L 1 =
lOOOL,, y = 1. 15. 0 1 12.54 1
10.0 8
j
h--0.05 h--0.05
7.5-1
-
h--0.01
2. 0.0
;y/
h--0.01
I
I
I
-1000
-800
-600
I
-400
I
I
I
-200
0
200
-4
-3
-2
-1
E Figure 3. Potential profiles in the Nernst layer in reduced units. For the case, C, = 0.001 M, N = 100,L 1 = lOOOL,, y = 1.
where the diffusivities in the electrolyte solution and the membrane are equal, very appreciable gradients develop in the Nernst layer at levels of current typical of experimental situations. This merely indicates that there are other factors associated with the electrodynamics of the space charge layer, besides the relative magnitudes of the diffusivities, that act to establish f ' i i or membrane control. These factors are associated with, among other things, overall current, concentration of fixed ions in the membrane, stirring rate, and whether the interface is biased in the forward or reverse direction. We first treat the case in which the diffusivities in the solution and in the membrane are equal. Under our definition this will prove to be a case of film control. Figures 2 and 3 deal with the Nernst layer for this case. Figure 2 exhibits anion concentrations in the Nernst layer plotted vs. position, 6. In terms of the reduced variable 6, the Nernst layer extends from -1000 (-151) to 0. The ordinate corresponds to the reduced concentration n. Four plots are shown for four different values of reduced current H. These are H = -0.1,-0.05, -0.01, and 0. As expected, the anion concentration increases toward the membrane. The results exhibited in these figures can be obtained directly (analytically) from equations such as (B7) and (B8) once g is known. According to our definition, since an appreciable concentration gradient can be supported in the Nemst layer, this case must be counted one of film control. Figure 3 shows how the reduced potential varies through the Nernst layer for the same membrane and the same four values of cation current. It, too, increases toward the membrane. It should be noted that, in this case, the membrane will, in effect, be interfaced with an electrolyte in which the relative anion and cation concentrations will be much increased over the bulk values.
1
0
E Figure 5. Potential profiles in the interface space charge region in reduced units. For the case of Go = 0.001 M, N = 100,L 1 = 1000L,,
y = 1. O.OO1
r\
cn +,
-0. 10
-i
0
>
-0. 15
0
-0.20
N-1
NO -lO
-I
i
4
I
-0.25
I
-2.0
-1. 5
-1.0
1
-0.5
I
I
0.0
0.5
CURRENT (AMP CM-') Figure 6. Current-vottage curves at the solution membrane interface for the case of G o = 0.01 M, L 1 = 1000,N = 10, 100,and 1000,
y = 1.
Figures 4 and 5 explore the situation within the electrical double layer (diffuse layer and membrane layer) surrounding the interface. The same values of N and H are involved. For the cases shown the region under investigation extends from W = -3 to L = 1. The thickness is very much less than the extent of the Nernst layer. For this reason, in Figures 2 and 3 we can show the Nernst layer extending as far as E = 0 without introducing any appreciable error. In Figure 4, we plot the concentration n' of anions against 6 within the double layer. This concentration drops precipitously toward the membrane as would be expected for coions. The ultimate concentration in the membrane becomes higher with forward bias; it also begins at higher
Nonequilibrium Effects due to
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Ion Transport
O2
Z
1
FILM CONTROL
1 -1
Q
I-
cn Z 0
c (
0 0
x
I: A\ 0-2.0 1 -2.0
I -1.5 -1.5
I N-1000 -1.0
-1.0
\I0 . 0
I -0.5 -0.5
CURRENT (AMP
0.0
I 0.5 0.5
-‘)
Figure 7. Concentration of anions in the bulk membrane at X = L calculated as percent of total mobile ions for the case of C o = 0.001 M, L 1 = lOOOLD, N = 10, 100, and 1000, y = 1.
concentration, as a result of the buildup at 5 = - W, due to the gradient in the Nernst layer. Figure 5 shows how the reduced potential 4 varies as a function of 5 through the double layer. It can be seen that the negative potential difference is much reduced at high forward currents. This, again, is expected. Figures 6 and 7 are somewhat different. Figure 6 contains plots of the potential across the space charge region in volts vs. current density. The three plots are for N = 10, 100, and 1000, respectively. Again, we deal with the case where the diffusivities are the same in the solution as in the membrane. Here, too, zero current density corresponds to equilibrium. As expected, the plotted potential (membrane potential) increases with increasing forward current. So that the data are presented in terms of absolute, rather than reduced, current, it is necessary to assign a value to the diffusivities. In the case of Figure 6 this value, common to both the solution and the membrane, is cm2 s-l and is typical of many electrolyte solutions. The potential remains more negative when N is larger. This is a straightforward consequence of the fact that, for small values of N , e.g., N = 10, the concentration gradient developed in the Nernst layer can bring the effective concentration of the electrolyte solution up to the level of the fixed ion concentration in the membrane, thereby almost eliminating the double layer and its corresponding potential. Figure 7 contains plots of the coion accumulation at 5 = L , for the same three membranes, again as a function of current density in absolute units. The accumulation of coions (anions) is indicated in the plots in terms of the fraction, at 5 = L , of all mobile ions that are coions. In the case of N = 10, and at sufficiently negative (forward) currents, coions actually outnumber counterions. For the membranes having higher exchange capacities, this is not the case; nevertheless, the concentration of coions is appreciably increased over that expected at equilibrium. Thus, especially for membranes of modest exchange capacity, the coion concentration in the membrane will be markedly increased when transport is occurring. We are now ready for the first time to investigate the effects upon which the title of this paper is based, namely, nonequilibrium behavior, or the failure of “local equilibrium” at the solution-membrane interface. For this purpose, it is most convenient to investigate how the potential across the double layer differs from its equilibrium value when transport is occurring. Equation 5 gives the Donnan equilibrium value of the counterion concentration in the membrane. This concentration can also be specified in terms of the equilibrium potential across the interface
143
_]
0 -0.04
4
2
a
t
-0.06 -0.08
-0.10
f -2.4
I
I
-1. 8
I
I
-1.2
I
I
-0.6
I
I
I
0. 0
CURRENT (AMP CM-‘) Figure 8. Current-voltage curve for the solution-membrane interface in the case of film control. The solid line indicates our calculation, the circles represent the potential calculated according to the equilibrium condition eq 21. For the case of Co = 0.001 M, N = 10, L 1 = lOOOL,, y = 1.
double layer. In fact, in reduced notation, the counterion concentration may be expressed as p m ( 0 ) = ps(0)e-+*
(21)
in which 4* is the potential across the double layer, pm(0) the concentration of cations inside the membrane at X = 0, and p,(O) the cation concentration in the solution at X = 0. Thus utilizing eq 5 in order to obtain p,(O), we may use eq 21 to evaluate the equilibrium potential 4* across the double layer. We can then compare this with the actual potential across the double layer evaluated, for example, as the difference between the potential at -W and at L. We must also keep in mind that the effective solution concentration is dependent on the current in the Nernst layer. The curves in Figure 8, corresponding to different currents, also correspond to different effective electrolyte concentrations, in view of the Nernst-layer gradients exhibited in Figure 2. Thus, for each comparison, we must use the effective electrolyte concentrations of eq B7 in evaluating the equilibrium $* by means of eq 21. This is why the equilibrium potential appears to depend on current. Under these conditions, if the potential difference obtained from eq 21 is indistinguishable from the equilibrium potential evaluated by subtracting the value of 4(-w) from @(L)then it follows that the assumption of local equilibrium is valid. In other words, the potential difference across the double layer is the same in the presence of transport (nonzero H ) as the potential difference evaluated from the Donnan equilibrium, based on the “local” or effective electrolyte concentration. Figure 8 illustrates just such a comparison. The figure is a plot of the potential (in volts) across the double layer as a function of total current in A cm-2. The membrane is one in which N = 10, and the system is one in which film control exists (the diffusivities are the same in the solution and membrane). Again, we have chosen the common diffusivity to be cm2s-l, a value typical for electrolyte solutions. The solid line in Figure 8 is a plot of the potential across the double layer vs. the total current, while the open circles represent the equilibrium Donnan potential across the same double layer as calculated from eq 21. Since the open circles fall on the curve, at least within the current range investigated, we must conclude that, in this case at least, the assumption of local equilibrium is valid. Since practical current densities seldom exceed 0.5 A cm-2,it appears that local equilibrium remains valid, for this case of film control, throughout the entire practical range of current density. In essence, in the film control case, changing the current has the same net effect as
144
Bassignana and Reiss
The Journal of Physical Chemistry, Vol. 87, No. I , 1983 O‘ 33
0.01
1
PARTIAL FILM CONTROL 7 = l o 0 0
0.01
4
t-
-0. 31 j
0
k-
0
_J
_I
C
>
PARTIAL FILM CONTROL 7 - i o 4
-0.01
A
tn
1
0
2
1 -0.034
\
-0.05
-0.03 0
-e-0. 05
tN-10
-0.07 1 -2.0
I
-1.5
I
-1.0
CURRENT
I
-0.5
I
I
0.0
0.5
changing the concentration of the bathing electrolyte, so that equilibrium tends to be reestablished. The effect of the current on the potential and coion concentration is seen entirely in the Nernst layer, and not across the solutionmembrane interface. We now turn to membrane control and examine the assumption of local equilibrium by constructing plots similar to the one in Figure 8, Le., plots of voltage across the double layer as a function of cation current density in A cm-2. The degree of membrane control will be characterized by the quantity y, defined in eq B3 and above. Large values of y correspond to high degrees of membrane control. Thus complete membrane control would correa. Figure 9 illustrates a case of complete spond to y membrane control. It refers to a situation in which N = 10 and y = lo7 which, for all practical purposes, is infinite. Again, the solid line is a plot of potential (in volts) across the double layer as a function of current density in A cm-2, while the open circles represent the Donnan potential calculated from equilibrium considerations. Here, the equilibrium potential proves to be independent of current because no concentration gradients can develop in the Nernst layer, and the effective concentration of the electrolyte remains constant. We note that there is considerable disparity between the potential, in the presence of transport, and the equilibrium potential. In this case, the assumption of local equilibrium is very bad. In fact, it is so bad that, at forward current densities in excess of 0.5 A cm-2, the potential across the double layer is positive while the equilibrium potential is negative. As a necessary qualification, it should be born in mind that a current of 0.5 A cm-2 is representative of the limit of those typical of either experiment or practical application. Nevertheless, there is, even without inversion of the sign of the potential, an appreciable disparity between the equilibrium and nonequilibrium potentials at forward current densities in the neighborhood of 0.2 A cm-’. Figure 10 shows the case for which y = lo4. Again, the relative concentration of fixed ions in the membrane is 10. Here, we deal with a case of partial membrane control, as is evidenced by the fact that the loci of the open circles, representing the equilibrium potential, is a curve that depends on current, indicating a buildup of concentration in the Nernst layer. Nevertheless, the disparity between the solid nonequilibrium curve and the equilibrium open circles indicates, once again, that the assumption of local equilibrium is invalid. Finally, Figure 11 shows the same curves for the case of y = 1O00, a more realistic value of y. Again N = 10. The
I
-1.5
1
I
I
I
-1.0
-0. 5
0. 0
0. 5
CURRENT (AMP CM-‘)
(AMP C M - * )
Flgure 9. Current-voltage curves for the solution membrane interface in the case of membrane control. The solid line indicates our calculation, the circles represent the potential calculated according to the equilibrium conditions, eq 21. For the case of C , = 0.001, N = 10, L 1 = 1000L,, y = a.
-
-0.07 -2.0
Figure I O . Current-voltage curve for the solution-membrane interface in the case of partial film control, C , = 0.001 M, N = 10, 11 = lOOOL,, y = 10. O‘ 070
1
COMPLETE MEMBRANE CONTROL
_1
0
-0.014-
a -0. 042 0
00 0
0 0 0
NP1o
The Journal of Physical Chemistry, Vol. 87,
Nonequilibrium Effects due to Ion Transport
0.
0
-a
0.
\
0.
0. 0.
€ Flgure 12. Reduced electric potential function 4 14,, (4 , = 0.06) in a diffuse charge layer of an electrical double layer according to the theory of Gouy-Chapman (solid line) and approximation of eq A23: X = 3.306, r = 1 (dotted line) and A = 9.83, r = 2 (triangles). 1.0 0. 8
0. 6 8 \
0.4 0.2 0. 0
0. 00
0. 25
0. 50
0. 75
1. 00
1. 25
1. 50
€ Figure 13. Reduced electric potential function 4 /$ (4 = 3) in a diffuse charge layer of an electrical double layer according to the theory of Gouy-Chapman (solid line) and approximation of eq A23: = 43.57, r = 1.9 (dotted line) and X = 2.22, r = 1 (broken line). 0.070
0.042 -
+
Appendix A In this appendix, we discuss the logic of the approximation used in this paper in which the Poisson equation is first solved using a “guessed” form for the dependence of space charge on position. We provide no absolute proof; our aim is to demonstrate the plausibility of the method. However, we do compare, for a particular case, the potential derived in the approximate manner with that derived exactly, analytically. Unfortunately, the available analytical solution pertains to a diffuse space charge layer (Gouy-Chapman problem)35extending throughout the positive semiinfinite domain. For such a semiinfinite system, the approximation is likely to be strained. Nevertheless, the results will appear to be reasonable. In a sense, our approximation is an extension of a method used repeatedly in semiconductor theory.36 In semiconductors mobile carriers are assumed to be approximately absent in the depletion layer. Where Schottky depletion layers are encountered, one avoids dealing with the Poisson-Boltzmann equation by assuming that the space charge term in the Poisson equation is a known density of charge due to the fixed ions. The question arises as to whether an extension of this approach (i.e., beginning with an assumed form of space charge density) can be used to avoid the direct use of the Poisson-Boltzmann equation in the case of the general double layer, including the cases of diffuse space charge layers, in which all of the charge density may be due to mobile carriers. We investigate this question by focusing on one-dimensional plane parallel problems, but the same method will be applicable to multidimensional situations. In Gaussian units, the one-dimensional Poisson equation is
d2V dX2
K
d X ) = EViCi(X)
0.014-
i
1 0
(A2)
where C i ( X )is the concentration of the ith species, vi, the valence of the ith species (negative for anions), and the sum goes over all species i. For simplicity, we consider a uni-univalent electrolyte such that we have only
-0.0148
-0.042 -
v+ = 1 -0.070
4 w
-= --7(X) where K is the dielectric constant of the medium, q the charge on the electron, V(X)the electrical potential, X is the position coordinate, and v(X)is the local net ion concentration
-
r\
cn
No. 1, 1983 145
I
I
I
I
1
v- = -1
(A3)
and there are no other species of ion. Then d X ) = C + ( X )- a x )
(A41
We introduce the transformations utilized in the body of the text, namely
P(X)= d X ) / C O moves, from the system, the mechanism for the shift which could remove the stress. Hence the system remains out of equilibrium. In summary, we must conclude that there are either practical or experimental situations in which the assumption of local equilibrium can fail, and the idea of “local equilibrium” should be used with care. Acknowledgment. This work has been supported by NSF Grant CHE-81 12658 and Occidental Research Corporation.
(A5)
where Co is some constant concentration. In addition, the LD, and 5 will retain the same meanings quantities $(X), as in eq 7, 11, and 12, respectively. Substitution of eq A5 into eq A1 yields d2@ / dt2
-p
(A6)
In the situation where the ion concentrations at a point (35) E. J. W. Verwey and J. Th. G . Overbeek, “Theory of the Stability of Lyophobic Colloids”, Elsevier, Amsterdam, 1948. (36) Shockley, Bell System Tech. J.,28,435 (1949).
148
The Journal of Physical Chemistry, Vol. 87,No. 1, 1983
are calculated the density becomes p = -2 sinh 4
(A7)
When this charge density is substituted into eq A6, it yields the well-known Poisson-Boltzmann equation. This equation is highly nonlinear, and analytical solutions are available in only the rarest of circumstances. One such circumstance involves the semiinfinite domain (A81
O S t 5 m
in which @(O) = @o
(a constant)
(A9)
4(m) = 0 [d#/d[]pm = 0 P(m)
(All) (-412)
=0
In this case, we generally choose C, to be the concentration of electrolyte in the bulk solution at 4 = m. The known solution35is then given by 4=21n
[
e40/2+ 1 + (eW2 - 1)e-(2)''*E
1
e4012 + 1 - (e@o/z- 1)e-(2)"%
(A131
We will return to this solution later. For the moment, we concentrate on eq A6. To be concrete, we will consider it under the boundary conditions specified by eq A10 and eq A l l . If we integrate eq A6 twice, subject to these boundary conditions, we obtain
Bassignana and Reiss
boundary conditions that may be imposed on p ( t ) , and we then substitute this approximate form into eq 16, the single free parameter will allow that equation to be satisfied at only one value of [ and, in turn, the value of the parameter will depend on the value of [ chosen. Of course, if we happen, fortuitously, to have guessed the exact solution, p ( l ) , eq A16 would be satisfied everywhere. Failing this, we are left with the situation just described. Since we have only one free parameter, our form has already lost some of the information contained in the exact solution, and it makes sense to confront it with a condition that is designed down to the information level of the form itself. Equation A17 fulfills this requirement, and is furthermore a global condition, placing some restriction on the function at every value of 5. Thus, it is well suited to the purpose of adjusting an approximate choice to the requirements of the Poisson-Boltzmann equation. To illustrate how well the approximate procedure works, we first use it to develop an approximate solution to the situation constrained by the boundary conditions eq A9A l l . An intuitively acceptable approximation to p is one that decays exponentially toward zero as t increases. For example, we might choose the form
0 I[ IX; p = 0,[ > X (A181 in which a,and X are parameters, but where r is to be determined at our convenience rather than by the method described above. We include r only for the convenience of later discussion. An unbiased choice for r at the outset might, for example, be unity. In the case where X is m, this function becomes
An integration by parts converts this into @ = -Jm(t"-8 ~ ( f " ) dt"
p
(A151
If we wish to determine p ( [ ) , the density used in the Poisson-Boltzmann equation and specified by eq A7, we merely substitute eq A15 into A7 and obtain p(t) =
[
2 sinh Jm(t" - t ) p(E") d Y ]
= LYe-rE
(A191
and represents a pure exponential decay. However, rather than assume the form of eq A19, we permit maximum flexibility by allowing the value of X to be chosen by the system itself. Nevertheless, eq A19 remains a possible special case of eq A18. Substitution of eq A18 into eq A17 gives
(A161
This is merely an integral equation form of the PoissonBoltzmann equation that allows us to solve for p rather than [. Note that it provides for a self-consistentp at every value of 5. Equation A7 brings thermodynamics into the Poisson equation. If we approximate the potential of mean force by V ( X ) (or @), it introduces the requirement that the distribution of ions be canonical and correspond to thermodynamic equilibrium. Substitution of eq 15, to obtain eq A16, then assures that the distribution also satisfies the laws of electrostatics. We can replace the integral equation A16 by an integral condition by integrating eq A16 over the semiinfinite domain to which it applies. Thus, we may write
(A171
This is still a condition on the form of p ( t ) but contains much less information than eq A16 since it is a global rather than a local (pointwise) condition like eq A16. Suppose that we now choose an approximate form of p ( t ) , exercising our intuition. If the form has one free parameter, beyond those necessary to satisfy various
Equation A15 supplies a boundary condition when we apply it to 5 = 0; this yields
do = -xmE"~(E") dE" where we have defined eq A18, this becomes
@ = @o
-
at 1
(A211
= 0. For the form in
-
+ -(1 + rX - erh)
At the edge of the interface, the value of the potential @o is a quantity that varies with the amount of charge in the double layer. Equation A22 is substituted into eq A20 and solved numerically for a value of X. From equation A15, using eq A18, we are able to derive the functional form of 4. This is
Nonequilibrium Effects due to
Ion Transport
since a and A in this expression have already been determined in terms of $o, all that remains is to choose a value for r. r is varied to see if the fit between the approximate expression eq A23 and the analytical solution eq A13 can be improved. As indicated above, r is not fixed by the approximation scheme, but, in some special cases the best value for r is apparent. For example, in the limit @o
