+
+
Langmuir 1996, 12, 1331-1335
1331
Sogami Potential and Membrane Equilibria M. V. Smalley,*,† W. Scha¨rtl,† and T. Hashimoto†,‡ Hashimoto Polymer Phasing Project, ERATO, JRDC, 15 Morimoto-cho, Shimogamo, Sakyo-ku, Kyoto 606, Japan, and Division of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 606, Japan Received May 30, 1995. In Final Form: November 17, 1995X Membrane equilibria are investigated using the Sogami potential for plate macroions. Two methods are used to calculate the ratio s of the electrolyte concentrations in the two phases in the two-phase region of colloid stability. This ratio depends sensitively on the value of the electric potential function at the midplane between the plates, Φd. The approximation Φd ) 0, corresponding to the neglect of double-layer overlap, is found to lead from Sogami theory to the Donnan membrane equilibrium. An exponentially decaying electric potential function is used to estimate the effect of including the double-layer overlap. It is found that the intermacroionic interactions suppress the Donnan effect, leading to a reduced value of s that is roughly constant as a function of the surface potential, for any physically realistic surface potential greater than 40 mV. It is noted that this calculation of s may provide an explanation for the relative salinity of cells and sea water.
Introduction In the present paper, it will be shown that the Sogami potential for plate macroions leads to the Donnan membrane equilibrium as a limiting case, and can be used to generalize the Donnan effect to a system of interacting macroions. The Sogami potential for plate macroions1-3 has received considerable theoretical attention recently.4-9 The potential has been criticized,4-6 and replies to these criticisms have been published.7,8 One of the most powerful arguments in favor of the Sogami potential is the excellent account it gives for the properties of n-butylammonium vermiculite gels,9-13 including the distribution of salt between the two phases in the two-phase region of colloid stability illustrated in Figure 1a. This depicts a gel composed of a parallel stack of plate macroions with a well-defined interplate spacing (in the colloidal range 10100 nm) in equilibrium with a supernatant fluid. The boundary of the gel can be visualized as an effective membrane enclosing the macroions. Figure 1b is a typical illustration of the Donnan equilibrium. A membrane impermeable to macroions (Pn-) but permeable to small ions (M+, X-) and solvent molecules (S) divides a solution into two regions. The situation is a familiar one in colloid science, and the fact that the equilibrium salt concentration in region II (the simple * Corresponding author. Current address: Dept. of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, U.K. † Hashimoto Polymer Phasing Project. ‡ Division of Polymer Chemistry. X Abstract published in Advance ACS Abstracts, February 1, 1996. (1) Smalley, M. V. Mol. Phys. 1990, 71, 1251. (2) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1991, 74, 599. (3) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1992, 76, 1. (4) Levine, S.; Hall, D. G. Langmuir 1992, 8, 1090. (5) Ettelaie, R. Langmuir 1993, 9, 1888. (6) Overbeek, J. Th. G. Mol. Phys. 1993, 80, 685. (7) Smalley, M. V.; Sogami, I. S. Mol. Phys. 1995, 85, 869. (8) Smalley, M. V. Langmuir 1995, 11, 1813. (9) Smalley, M. V. Langmuir 1994, 10, 2884. (10) Smalley, M. V.; Thomas, R. K.; Braganza, L. F.; Matsuo, T. Clays Clay Miner. 1989, 37, 474. (11) Braganza, L. F.; Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Clays Clay Miner. 1990, 38, 90. (12) Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Adv. Colloid Interface Sci. 1991, 34, 537. (13) Williams, G. D.; Moody, K. R.; Smalley, M. V.; King, S. M. Clays Clay Miner. 1994, 42, 614.
Figure 1. Schematic illustration of the phenomenon: (a) a swollen n-butylammonium vermiculite gel and (b) the components present in the two phases, the gel (I) and the supernatant fluid (II). The dotted line represents an effective membrane enclosing the plate macroions Pn-. The symbols M+, X-, and S stand for the univalent counterions (n-butylammonium ions), univalent co-ions (chloride ions), and solvent (water) molecules, respectively.
electrolyte solution), [X-]II, is greater than that in region I (the region occupied by the macroions), [X-]I, has been used in countless dialysis experiments. It is also well known14 that equilibrium involves the establishment of not only a pressure difference but also an electrical potential difference across the membrane and that, in the simple case where the mobile ions behave as ideal solutes, the equilibrium condition is expressed as
[X-]II [X ]I -
(
) exp
)
eφDonnan ) exp(ΦDonnan) kBT
(1)
where e is the electronic charge, kB the Boltzmann constant, T the temperature, φDonnan the Donnan potential, or the difference in the average electric potential between the two regions, and ΦDonnan a dimensionless Donnan potential, defined by
ΦDonnan )
eφDonnan kBT
(2)
The case illustrated in Figure 1b is easily solved because it does not allow for interactions between the macroions. The solution given in eq 1 follows simply from equating the chemical potentials of the small ions in regions I and (14) Everett, D. H. Basic Principles of Colloid Science; Royal Society of Chemistry: London, 1988.
+
+
1332
Langmuir, Vol. 12, No. 5, 1996
Smalley et al.
II. The case illustrated in Figure 1a is not so straightforward, because it is the intermacroionic interactions themselves that create the effective membrane bounding the macroions. Since the formation of the two-state structure shown in Figure 1a is strong evidence for an attractive component in the intermacroionic interaction potential, it was natural to investigate this case with the Sogami potential,9 and excellent agreement was found with the experimental salt distribution in the n-butylammonium vermiculite system.13 However, the previous treatment9 did not bring out the relationship of the Sogami theory to the Donnan equilibrium. The main aims of this paper are to derive this relationship and to estimate the effect of the intermacroionic interactions on the Donnan potential. General Calculation For a simple uni-univalent electrolyte solution (the only case treated here), the inverse Debye screening length κ is defined by
( )
κ2 )n 8πλB
e2 4π0rkBT
(4)
[
]
sinh(2aκ) 1 4 + 2aκ κ 1 + cosh(2aκ)
(7)
where, for a negatively charged wall, Nde is the deficit of negative ions per unit area and Z0 is the total surface charge per unit area. The relationship between the surface charge, Z0, and the dimensionless surface potential, Φs, is given by3
Z0 )
( ) ( ) Φs κ sinh 2πλB 2
(8)
As g can also be calculated in terms of the surface potential (see below), Nde can be calculated as a function of Φs. When Nde is known, the average number density of the deficit, nde, can be calculated from eq 6; the position of the Sogami minimum is connected to the average electrolyte concentration in the gel phase because it defines the average volume occupied per macroion. The number density of the deficit of negative ions in the gel phase is given by
nde )
with 0 being the permittivity of free space and r the relative permittivity of the medium (dielectric constant). Such a definition of κ is only strictly valid in a simple electrolyte solution and is therefore, in the present case, strictly valid in region II, the infinite reservoir of macroionfree electrolyte. In the present paper, we do not attempt to define κ in the macroionic region I but work purely with this strict definition. In terms of the κ value so defined, when n in eq 3 is to be interpreted as the number density of simple ions in region II, the Sogami potential for the interaction of plate macroions1,9 has a minimum at the position xmin defined by
xmin )
Nde Z0
(3)
where n is the number density of the simple ions and λB is the Bjerrum length, defined by
λB )
g)
gZ0 l
(9)
where l is the half separation of the plates (this is because g is defined by an integral over a region of monotonically decaying electric potential, see below) and, according to the linearized Sogami theory of the plate interaction, l is given by (see eq 6)
l)
2 κ
(10)
Combining eqs 8-10 gives
nde )
( )
()
Φs κ2 2g sinh 8πλB 2
(11)
The prefactor is none other than the number density of the simple ions in the supernatant fluid, n (see eq 3). Equation 11 can therefore be rewritten as
( )
nde ) 2gn sinh
(5)
Φs 2
(12)
where 2a is the plate thickness. In dilute (n < 1019 cm-3) electrolyte solutions the position of the minimum in the pair potential between macroionic plates is given by the remarkably simple equation
Since the number density of the deficit of negative ions in the gel phase is defined by
κxmin ) 4
where ngel is the average number density of the negative ions in the fluid bounded by the macroions, the calculation yields ngel in terms of the experimentally measurable and controllable quantity n as
(6)
This localizes the plates at a distance of four Debye screening lengths and provides a qualitative explanation for the observed fact that in the n-butylammonium vermiculite system the d spacing along the swelling axis is approximately inversely proportional to the square root of the electrolyte concentration.10-13 Following ref 9, the d spacing is now assumed to be given by xmin according to eq 6 at all electrolyte concentrations. The distribution of small ions between regions I and II may be described as the expulsion of a certain amount of ions of the same sign as the colloid (co-ions).15 The result is encapsulated in a quantity g that expresses the ratio of the co-ion deficit to the total double-layer charge: (15) Klaarenbeek, F. W. Ph.D. Thesis, Utrecht, 1946.
nde ) n - ngel
[
(13)
( )]
ngel ) n 1 - 2g sinh
Φs 2
(14)
This in turn gives the ratio of these two quantities, the salt fractionation factor s, as
s)
n ) ngel
1
( )
Φs 1 - 2g sinh 2
(15)
and the result can be compared with eq 1, the equation
+
+
Sogami Potential and Membrane Equilibria
Langmuir, Vol. 12, No. 5, 1996 1333
for the Donnan equilibrium. In order to complete this calculation, we need the quantity g. Quantities g and s For a single flat double layer, g is given by the integral15
dx dΦ ∫0Φ (1 - e-Φ) dΦ s
g)
dx dΦ ∫0Φ (eΦ - e-Φ) dΦ
(16)
s
where Φs, the dimensionless surface potential, has been expressed as a positive definite quantity (this definition was not used in ref 9). In this expression the electric potential function is integrated from zero to its value at the surface, Φs. This is a strictly accurate procedure for an isolated plate macroion, since the potential decays to zero at an infinite distance from the plates. However, when the double layers overlap, the electric potential is a monotonically decaying function of distance from the surface in the half region between two macroionic plates and has some finite value, Φd, at the midplane. In this case, g should be calculated as
∫
dx (1 - e-Φ) dΦ Φd dΦ g) Φs Φ dx (e - e-Φ) dΦ Φd dΦ
Figure 2. Schematic illustration of the behavior of the electric potential function in the two models. The thin line is a sketch of Φ(x) for the approximation Φd ) 0, and the thick line is a plot of an exponentially decaying function, for which Φd ) 0.135Φs. The potential Φ(x) and the distance x are given in units of the dimensionless surface potential Φs and the Debye screening length 1/κ, respectively. The dashed line at x ) 2/κ designates the midplane.
g)
(1 - e-Φs/2) Φs/2
(e
-Φs/2
-e
) )
(1 - e-Φs/2) Φs 2 sinh 2
( )
(20)
Φs
∫
(17)
s)
In general, the integrals in eqs 16 and 17 cannot be solved analytically, so we adopt a series of brutal approximations in order to bring out the leading features of the salt fractionation effect between regions I and II. The crucial feature of our treatment is that the Sogami potential permits a calculation of the salt fractionation effect precisely because it defines the volume bounded by the macroions. Equations 7 and 9 tell us that we can always calculate the total number of co-ions expelled in terms of an electrical integral for g, but that we can only use this to calculate the number density of co-ions expelled, and hence the salt fractionation factor s, if we have a definite expression for the half separation of the plates l, which is taken to be 2/κ. We then estimate g within this fixed box by two methods, using (i) eq 16 and (ii) eq 17. (i) Non-overlapping Double Layers. Using eq 16 to estimate g in region I at first appears to be a strange procedure, because the case Φd ) 0 corresponds to no overlap of the double layers. The variation of electric potential with distance is then as sketched by the thin line in Figure 2. The thicker line in Figure 2 represents an exponentially decaying potential function (see method ii below). Let us first pursue the mathematics of the Φd ) 0 case. The crude approximation
dx s dx × Φs dΦ ) f( ) | ∫0Φ f(Φ) dΦ 2 dΦ Φ /2 Φ
s
s
(18)
can be used provided that the integrand f(Φ) is a smoothly varying function. Applying this approximation to both the numerator and denominator in eq 16 gives
dx × Φs | dΦ Φs/2 g) dx (eΦs/2 - e-Φs/2) × Φs | dΦ Φs/2
Substituting eq 20 into eq 15 we obtain
(1 - eΦs/2)
(19)
and the cancellation of terms reproduces Klaarenbeek’s result15 as
( )
Φs n ) exp ngel 2
(21)
Comparing eq 21 with eq 1, we see that the use of eq 20, which represents a crude approximation to the quantity g in region I, leads to a mathematical formula for s similar to the formula for the Donnan equilibrium. Indeed, if we identify ΦDonnan with Φs/2, the expressions are identical. Since the average value of the electric potential function is equal to zero in region II and the approximation used in eq 18 corresponds to taking the average potential in the gel to be Φs/2, the relationship ΦDonnan ) Φs/2 is consistent with the definition of the Donnan potential as the difference in the average electric potential between the two regions. We note that eq 1 is the form for the Donnan potential in the case where the activity coefficients of the simple ions are taken to be unity. Since this assumption is implicit in the use of the Poisson-Boltzmann equation which underlies the preceding calculations, eq 21 is the same as eq 1. Method i, the use of the position of the minimum of the Sogami potential with the Φd ) 0 value for g, leads to the same mathematical formula for s as that expressing the Donnan equilibrium, but we cannot say that this constitutes a derivation of the Donnan potential from the Sogami potential because it does not correspond to a physical limit. If Φd ) 0 were really the case, there would be no reason for the macroions to remain at the minimum position of the Sogami potential. Nevertheless, the identity of the expressions is an interesting result. We can view it in two ways. First, because eq 20 is derived in the case in which there is no double-layer overlap and eq 1 (the Donnan equilibrium) is likewise derived without reference to the overlap of the double layers, it is precisely in this limit that the calculation should reproduce the Donnan equilibrium. Second, if we were to take the standard electric integral for co-ion exclusion obtained from the Poisson-Boltzmann equation (eq 20)15 and the Donnan equilibrium (eq 1)14 as our starting points, then we would be forced to conclude that the equilibrium separation of the macroions is necessarily given by eq 6, the result of Sogami theory.
+
1334
+
Langmuir, Vol. 12, No. 5, 1996
Smalley et al.
Figure 3. (a) The quantity g and (b) the salt fractionation factor s as functions of the dimensionless surface potential Φs. In both (a) and (b), the upper and lower curves are obtained from the approximations Φd ) 0 and Φd ) 0.135Φs, respectively.
(ii) Overlapping Double Layers. Of more interest from the physical point of view is the case where the double layers overlap, represented by a finite value of Φd. The exact mean field potential for plate macroions3 in region I is expressed in terms of elliptic integrals, but it varies approximately exponentially with distance from the surface, and we use the simple function Φ(x) ) Φs exp(-κx) to represent it, as shown by the thicker line in Figure 2. For x ) 2/κ, the midplane potential is Φd ) e-2Φs ) 0.135Φs, and we take this as the lower limit of the integral in eq 17. Although the use of an exponentially decaying potential is itself a rough approximation to the electric potential function in region I (for example, it does not fulfill the symmetry requirement that the gradient of the potential goes to zero at the midplane), greater accuracy is not warranted in using the analog of eq 18 to solve for g. In keeping with this crude approximation, g is recalculated as9
g)
(1 - e-Φs/2) - (1 - e-Φd/2) (eΦs/2 - e-Φs/2) - (eΦd/2 - e-Φd/2)
(22)
The values for g obtained from eq 22 do not seem to be very different to those obtained from eq 20, as shown in Figure 3a. However, as shown in Figure 3b, the predicted salt fractionation effect obtained by substituting eq 22 into eq 15 is markedly different from the Donnan equilibrium. Discussion For low surface potentials (Φs < 1), the difference between the salt fractionation factors calculated (i) via the Sogami potential and the electric integral solved by eq 20 (i.e. via the Donnan equilibrium) and (ii) via the Sogami potential and the electric integral solved by eq 22
is practically indistinguishable, but for Φs > 2 (φs > 50 mV) the calculations diverge significantly, with the overlap of the double layers serving to suppress the salt fractionation factor. The reason for this suppression is connected to our limited approximation of calculating g and s at fixed d and Φs. Within this approximation, the electric potential becomes a more sharply varying function of position as Φd is reduced (see Figure 2); the average electric field strength in region I increases with decreasing Φd and, concomitantly, region I becomes a more hostile environment for co-ions. In the limit Φd ) 0, the average electric field strength in the macroionic gel is at a maximum (for fixed d and Φs) and the maximum salt fractionation occurs, giving us the Donnan equilibrium. For surface potentials greater than 104 mV, the behaviors are qualitatively different; s from method i always increases exponentially whereas s from method ii decreases as a function of Φs. It is particularly noteworthy that s from method ii is a very slowly varying function of Φs for Φs > 4, taking the values 3.1 and 2.3 at Φs ) 4 and 8, respectively; at high surface potentials, the salt fractionation effect is nearly constant both as a function of electrolyte concentration and surface potential. Indeed, s takes a value between 2.1 and 3.1 for any surface potential greater than 40 mV, up to unphysically high values. From the point of view of clay science, the near constancy of s for φs > 40 mV shows that great caution is needed in deriving surface potentials from co-ion exclusion effects.16 Independent experimental evidence for the values of the surface potential and salt fractionation factor have been obtained, to our knowledge, for only one experimental system, the n-butylammonium vermiculite gels. The value φs ) 70 mV was obtained from the effect of uniaxial stress on the gels,12 and the value s ) 2.6 ( 0.4 was obtained by standard chemical analysis of the fluids in regions I and II.13 Since the predicted values of s from methods i and ii are 4.0 and 2.8, respectively, this isolated fact suggests that method ii is more accurate, as expected for the realistic case of interacting macroions in the Sogami minimum. However, the quantitative agreement between the theoretical and experimental values of s in this case should also be treated with caution because of the severity of the approximations we have used in deriving the theoretical result. More accurate calculations using the exact mean field potential3 involving elliptic integrals would be desirable in making quantitative comparisons, but progress in our understanding of the salt fractionation effect is at present limited more by experimental than by theoretical inadequacies. It would be highly desirable to obtain further tests of our prediction for s in systems of interacting plate macroions, both in clay science and lamellar surfactant phases. A further intriguing consequence of the near constancy of s for φs > 40 mV may be obtained if we generalize the result to any system of interacting macroions in solution, including biological systems. It is well known that the average salinity of cells is approximately 1%, whereas the average salinity of sea water is about 3%. Since a cell can be viewed as a system of interacting macroions (a type of region I) and the origins of life are probably to be found in aggregates of charged macroions in solution, this interesting global phenomenon receives a natural explanation in terms of the calculations presented here. (16) Chan, D. Y. C.; Pashley, R. M.; Quirk, J. P. Clays Clay Miner. 1984, 32, 131.
+
+
Sogami Potential and Membrane Equilibria
Conclusion The Sogami potential is still not generally accepted in colloid science,4-6 and the debate continues.7,8 We have shown here that the Sogami theory not only is consistent with a well-known thermodynamic result for osmotic membrane equilibria as a limiting case but also permits
Langmuir, Vol. 12, No. 5, 1996 1335
us to extend our understanding of such equilibria to systems of interacting macroions. Acknowledgment. We wish to express our sincere thanks to Professors I. S. Sogami and N. Ise for stimulating discussion. LA950425C
