6156
Langmuir 1999, 15, 6156-6162
Gibbs’ Dividing Surface between a Fixed-Charge Membrane and an Electrolyte Solution. Application to Electrokinetic Phenomena in Charged Pores Vicente M. Aguilella,*,† Julio Pellicer,‡ and Marcelo Aguilella-Arzo† Departamento de Ciencias Experimentales, Universitat Jaume I, Ap.224, 12080 Castello´ n, Spain, and Departamento de Termodina´ mica, Universitat de Vale` ncia, 46100 Burjassot, Valencia, Spain Received June 26, 1998. In Final Form: May 13, 1999 The Gibbs model for the boundary between two phases consists of replacing the finite interfacial region, where the properties of the system change gradually, by a dividing surface which acts as a third phase of zero volume in which some magnitudes change abruptly. This thermodynamic concept was recently applied to a planar interface between a fixed charge membrane and an electrolyte solution.1 The continuous decrease of counterions with the distance from the charged surface is replaced by a step function, so that the diffuse double layer is ideally represented by a charged region depleted of all co-ions. Here the cylindrical geometry is analyzed, and the planar case is revisited by proposing a slightly different step function for ion concentration. By starting from Boltzmann distributions of ions in solution, simple formulas are obtained for the distance between the charged interface and the dividing surface in both geometries. For the purpose of applying this model to charged, microporous membranes, expressions are given for some of the coefficients Lij of the Onsager phenomenological relations in a model membrane composed of identical, parallel, and cylindrical pores. A comparison is made between the values for conductance and streaming potential in a homogeneously charged pore predicted by the Gibbs model and the classical space charge model. Despite the obvious simplifications inherent to the model, good agreement is found for conductivity, while only qualitative agreement is found between streaming potential predictions.
Introduction A common feature of a number of biological and synthetic membranes is the presence of structural charge due to charged groups. When such membranes are immersed in electrolyte solutions, the surface charges attract oppositely charged counterions from the solution and repel co-ions of the same charge. Therefore, a layer of space charge (compensated by fixed charge) due to the unequal positive and negative ion concentrations in thermal equilibrium is formed at the membrane-solution interface. This layer is usually denoted as the diffuse double layer. There is strong experimental evidence of the influence of this charge distribution on the properties of the membrane itself and also on the transport of charged species across the membrane. In addition, the coupling of lateral charge and mass transport near the charged surface of the membrane gives rise to a number of interesting electrokinetic phenomena. Thus, for instance, streaming potential or electroosmotic coefficient measurements have been used to estimate the size of membrane pores,2 the membrane fixed-charge surface density,3 or even the number of ions simultaneously occupying a narrow ionic channel.4,5 Modeling such kind of interfaces usually involves assuming that charged groups are smeared over the * To whom correspondence may be addressed: Tel: 34-964728045. Fax: 34-964-728066. E-mail:
[email protected]. † Universitat Jaume I. ‡ Universitat de Vale ` ncia. (1) Rostovtseva, T. K.; Aguilella, V. M.; Bezrukov, S. M.; Vodyanoy, I.; Parsegian, V. A. Biophys. J. 1998, 32, 1783. (2) Aguilella, V. M.; Kontturi, K.; Murtoma¨ki, L.; Ramı´rez, P. J. Controlled Release 1994, 32, 249. (3) Martı´nez-Dı´ez, L.; Martı´nez-Villa, F.; Herna´ndez-Gime´nez, A.; Tejerina, F. J. Colloid Interface Sci. 1989, 132, 27. (4) Levitt, D. G.; Elias S. R.; Hautman, J. M. Biochim. Biophys. Acta 1978, 512, 436. (5) Rosenberg, P. A.; Finkelstein, A. J. Gen. Physiol. 1978, 72, 327.
membrane surface or discretely distributed in a twodimensional lattice. Other models have taken into account that structural charges do not lie on a plane but extend into the solution6,7 and also that the fixed charge groups may have a nonzero dipole moment.8 These refinements are particularly important in lipid membranes where the distribution of charge perpendicular to the membrane surface may occur either due to out-of-plane fluctuations of lipid molecules or due to thermal motions of the hydrophilic part of the molecules. Despite these attempts to model the charged membrane/solution interface little is known about the structure of that region. The situation becomes still worse when there is lateral diffusion of the fixed-charge bearing molecules, as happens in the lipid membranes. From a thermodynamic point of view, the system of charged membrane plus electrolyte solution can be regarded as a two-phase heterogeneous system separated by a sharp boundary, although a more realistic, microscopic, view should include a thin region (less than 100 Å) where the properties of the whole system change continuously. For many purposes, however, the above assumption of considering the boundary as an ideal surface of zero thickness across which the properties change abruptly is an acceptable simplification. Gibbs proposed a prescriptionsthe dividing surfacesfor evaluating the contribution due to this surface in terms of the measurable properties of the whole system.9 The application of the concept of Gibbs dividing surface is common in problems of adsorption, surface tension in binary mixtures of (6) Cevc, G.; Svetina S.; Zeks, B. J. Phys. Chem. 1981, 85, 1762. (7) Aguilella, V. M.; Aguilella-Arzo, M.; Ramı´rez, P. J. Membr. Sci. 1996, 113, 191. (8) Aguilella, V. M.; Belaya M.; Levadny V. Thin Solid Films 1996, 272, 10. (9) Gibbs, J. W. Collected Works; Longmans: London, 1906; Vol. 1 (Dover: New York, 1961).
10.1021/la980773p CCC: $18.00 © 1999 American Chemical Society Published on Web 07/16/1999
Electrokinetic Phenomena in Charged Pores
immiscible liquids, and related topics but rather unusual in electrochemical systems. Recently,1 this concept was applied to the planar interface between a charged membrane and an electrolyte solution and it was shown that it can be regarded as a valid and more simple alternative to other descriptions based on the Gouy-Chapman model for the diffuse part of the electric double layer (EDL). Furthermore, it was shown that the interpretation of conductance measurements across a gramicidin A channel embedded in a lipid membrane was better by modeling that interface according to the Gibbs prescription than using the Gouy-Chapman model. Here two different geometries are considered for the charged surface: planar and cylindrical. The first case could be that of a planar charged membrane bathed by an electrolyte solution or that of a charged, big, colloidal particle immersed in an electrolyte solution (big in the sense that its radius be much greater than the size of its associated EDL). The second case could correspond to an electrolyte filling a pore in a microporous cation-exchange membrane or filling a large ion channel, with fixed charge groups, embedded in a lipid bilayer (large in the sense that the interaction between the mobile ions and the fixed-charge groups is purely electrostatic). The Gibbs Dividing Surface Model
Langmuir, Vol. 15, No. 19, 1999 6157
Figure 1. Sketch of Gibbs’ representation for a transition region between two homogeneous phases R and β. (a) Variation of the volume density of a thermodynamic property y through the physical surface of transition. (b) The corresponding hypothetical system with the dividing surface (modified from DeHoff10).
To describe the macroscopic properties of a transition region between two homogeneous phases, Gibbs introduced the concept of a dividing surface. His approach consists of redefining the whole systemscontaining the two phases plus the interfacial regionsso that the boundary region is regarded as an ideal surface. The extensive properties of the whole system and the intensive properties of each phase remain unchanged. Therefore, the whole system can be considered composed of three fictitious subsystems. All of them have intensive properties identical to the original phases, and the volume of each subsystem may be arbitrarily modified with the only constraint that the overall volume remain unchanged. The third subsystem, Gibbs’ dividing surface, is artificially introduced to ensure that the total extensive properties are not changed. Hence, it has zero volume. The location of the dividing surface depends on a given application. Once the location is chosen in conjunction with one of the properties, it must be retained for all of the others. Although there is plenty of literature about Gibbs’ dividing surface at an introductory level, we will follow here DeHoff’s treatment10 for a brief outline of the model. Figure 1a illustrates a hypothetical system composed of two phases R and β and the transition zone σ between them. The continuous line represents the real variation of the volume density y of some extensive thermodynamic property Y with position through the actual interface. Suppose the system under consideration has a constant cross-sectional area. Then the total value of Y for the system is proportional to the shaded area under the curve in Figure 1a. The surface excess of Y, that is, the part of the total value of Y for the system that may be associated with the presence of interface, is assigned by imagining a hypothetical system in which the densities yR and yβ in R and β are kept without variation up to the dividing surface, where there is a discontinuity in y (see Figure 1b). The total value of Y for this hypothetical system is proportional to the shaded area under the density curve in Figure 1b. This quantity can be calculated from the known densities yR and yβ in the parts of the system where these quantities are constant and from the location of the
Let us apply Gibbs’ approach to a charged membrane immersed in an electrolyte solution. Unlike the metalsolution interface, fixed charges in the membranesolution interface are usually distributed over a thin region
(10) DeHoff, R. T. Thermodynamics in Materials Science; McGrawHill: New York. 1993; Chapter 12.
(11) Jaycock, M. J.; Parfitt, G. D. Chemistry of Interfaces; Ellis Horwood: Chichester, U.K., 1986; Chapter 2.
dividing surface. The surface excess of Y for the system as a whole is defined as
Yσ ≡ Y - (VRyR + Vβyβ)
(1)
where the volumes VR and Vβ are defined by the position of the dividing surface and by the conservation equation
V R + Vβ ) V
(2)
That is, Yσ is the difference in Y evaluated for the actual system and that computed from the intensive properties yR and yβ and the location of the dividing surface. This quantity is proportional to the difference between the area under the curve in Figure 1a and under the step function in Figure 1b. Let A be the area of the interface in the system. Then the specific surface excess of Y is given by
yσ ≡ Yσ/A
(3)
This quantity, which has units of Y per unit area, can be defined for any extensive thermodynamic property. The last equations can be combined to give
Y ) VRyR +Vβyβ +Ayσ
(4)
Since the location chosen in the model for the surface dividing the two bulk phases determines the relative magnitudes of VR and Vβ, this location also controls the values of the surface excess quantities yσ.11 Usually a dividing surface is defined by putting it where yσ ≡ 0. Theory
6158 Langmuir, Vol. 15, No. 19, 1999
Aguilella et al.
Figure 2. Idealized picture of the interface between a fixed charge membrane and an electrolyte solution. Fixed charges from ionizable groups on the membrane surface are generally distributed over a thin layer (usually regarded as a surface) which is shown here enlarged for better visualization. Counterion and co-ion concentration can be considered constant in this region. The interplay between charge screening and thermal motion determines the concentration of mobile charges near the membrane surface (solid line curves). Far away from the membrane, the solution is electroneutral and the concentration of ions of both signs is constant.
filled with ionizable groups, water molecules, and mobile charges (see Figure 2). This small layer can be considered as an electroneutral zone and homogeneous from the point of view of the density of mobile charges. The neutral salt solution far from the membrane can also be regarded as a homogeneous phase. There is an interfacial region between these two phases where the electric potential, the ion concentration, and the space charge density change. The whole system “charged membrane + solution” is replaced by a three-phase system: (1) charged surface with its counterions; (2) Gibbs dividing surface; and (3) neutral bulk solution. According to Gibbs, the overall value of the extensive variables for the whole system should remain unaltered. Here we intend to define the dividing surface in such a way that the surface excess of positive ions and negative ions be zero. This implies that the total number of positive, n+, and negative ions, n-, must be the same after the introduction of this fictitious subsystem of zero volume (the dividing surface), i.e.
n+ ) VRc+R +Vβc+β
(5a)
n- ) VRc-R +Vβc-β
(5b)
Figure 3 shows the step functions that replace the continuous decrease of counterions and increase of coions with the distance from a negatively charged surface (there is no loss of generality in the arbitrary choice of negative fixed charge). The distance from the charged wall to the dividing surface, lG, is unambiguously determined by the above conditions (5) provided the concentrations c(R and c(β are known. For the present discussion let us denote by c(G the ion concentrations in the region closer to the membrane and c will stand for the bulk solution concentration (the same for cations and anions). There is no doubt about the meaning of c since it can be measured by experiment. However, the case of c(G is not the same and a choice must be made between different assumptions depending on the microscopic structure of the fixed charge layer. The simplest assumption (A) is considering this region totally depleted of co-ions,1 that is, c-G ) 0, with
Figure 3. Ion concentration profiles near a negatively charged planar surface (fixed charge membrane). Dashed lines denote the approximate cation and anion profile, and solid lines their idealization according to the Gibbs model (case A; see text). Gibbs dividing surface is represented by the vertical dot line.
the necessary amount of cations to counterbalance the fixed charge. (Thus c+G comes out from the electroneutrality requirement.) Another assumption (B) is choosing as c+G or c-G the values predicted by the PoissonBoltzmann equation for ion concentrations on a uniformly charged surface bathed by an electrolyte solution. (If c-G is fixed, c+G is obtained from the electroneutrality condition.) According to this hypothesis the co-ion concentration c-G is small but nonzero. In any case, the electrolyte solution is divided into two regions, the electroneutral salt solution, far from the charged surface, and a charged layer of solution of uniform concentration in contact with the fixed-charge groups. Let us analyze in detail the expressions for lG which arise for a planar and cylindrical negatively charged surface according to the assumptions (A) and (B). Charged Planar Surface. Case A. The constraint of zero surface excess of ions of each sign can be expressed either referred to the excess of cations (counterions) or to the deficit of anions (co-ions). For anions it reads
∫0∞ [c-(x) - c] dx ) -lGc
(6)
where c-(x) is the anion concentration at distance x from the charged surface and c is the bulk concentration. Let us first use for the ion concentration profile the GouyChapman equation, the exact result for the nonlinear Poisson-Boltzmann (PB) equation for a 1:1 electrolyte bathing a surface of uniform charge density σ. We have
c-(x) ) c exp[Fφ(x)/RT]
(7)
where F is Faraday’s constant, R is the universal gas constant, T is the absolute temperature, φ(x) is the electric potential given by
φ(x) )
2RT 1 - γ exp(-x/λD) ln F 1 + γ exp(-x/λD)
(8)
λD is the Debye length of the solution (λD ≡ [RTo/2F2c]1/2) and γ is the positive root of the quadratic equation
γ2 +
(
)
8RTco σ
2
1/2
γ-1)0
(9)
Electrokinetic Phenomena in Charged Pores
Langmuir, Vol. 15, No. 19, 1999 6159
In the above expressions and o denote the dielectric constant and the electric permittivity, respectively. Substitution of eqs 7 and 8 into 6 gives1
c
∫0∞
[(
) ]
1 - γ exp(-x/λD)
1 + γ exp(-x/λD)
2
- 1 dx ) -lGc
(10)
c+G ) c
γ 1+γ
(11)
|σ| 1 + γ 4FλD γ
(12)
In addition, very often the low value of the electric potential near the charged surface (φ(x) , RT/F) justifies the linearization of the PB equation and the use of a much simpler expression for φ(x)
φ(x) )
σλD exp(- x/λD) o
(13)
and also for c-(x)
(
c-(x) ) c 1 +
FσλD exp(-x/λD) RTo
)
(14)
By substituting eq 14 into eq 6, we get
FσλDc RTo
∫0
∞
exp(-x/λD) dx ) -lGc
(15)
σ 2Fc
c+
σ )) 2c FlG
(17)
i.e., within the linear approximation, the counterion concentration for 0 < x < lG is twice its bulk concentration. Case B. The balance equation for the deficit of co-ions, analogous to eq 6, is now
∫0∞ [c-(x) - c] dx ) -lG [c - c-G]
(11 +- γγ)
2
)
(21)
)
σλDF σ )c+ oRT 2FλD
(22)
And after integration, as in eq 18, we get
l G ) λD
(23)
c+G ) c
(24)
and
Charged Cylindrical Surface. Let us consider now a membrane with cylindrical pores of radius a and length d in which fixed charges are homogeneously distributed over their wall. The membrane separates two solutions of a 1:1 electrolyte of concentration c. We choose the Gibbs dividing surface as a cylindrical surface which divides the pore into an inner cylinder of undisturbed, neutral solution (0 < r < lG) and a hollow cylinder of solution (lG < r < a) with ion concentrations c(G. See Figure 4. The volume of the outer cylinder is πd(2a - lG)lG. The same cases A and B as above will be considered here; i.e., assumption (A) means taking c-G ) 0, and assumption (B) means obtaining c-G from the solution of the linearized PB equation in cylindrical coordinates at the wall of a uniformly charged pore.12 Case A. The constraint of unchanged total number of ions of each sign referred to the deficit of anions (co-ions) is analogous to eq 6
∫0a ∫0d [c-(r) - c]2πr dr dz ) -πd (2a - lG)lGc
[
c-(r) ) c 1 +
(25)
σλDFI0(r/λD) RToI1(a/λD)
]
(26)
where I0 and I1 are modified Bessel functions of zero and first order. The integration with respect the radial coordinate is greatly simplified by noting that
∫0a I0 (r/λD)r dr ) aλDI1 (a/λD)
(27)
and by taking into account that λD2 ≡ RTo/2Fc, we get a quadratic equation in lG
(18)
where c-G ≡ c-(0) given by eqs 7 and 8 for x ) 0:
c-G ≡ c-(0) ) c
(
(
(16)
Since |σ| ) -σ, the electroneutrality of the system leads to G
1+γ2 σ σ )c FlG 1-γ FλD(1 + γ)
where we have used cylindrical coordinates. c denotes the concentration in the neutral inner cylinder. The anion concentration profile according to the linearized PB equation in cylindrical coordinates12 is
which gives, after integration
lG ) -
-
c-G ) c 1 +
The electroneutrality of the system “charged surface plus its counterions” implies that the total number of counterions between x ) 0 and x ) lG must be equal to the number of fixed charges on the membrane surface, i.e., Fc+G lG ) |σ|. Therefore
c+G )
2
By using the linearized PB equation for the coion profile c-(x) eq 14 gives
which leads to
lG ) 4λD
(11 -+ γγ)
lG2 - 2alG -
σa )0 Fc
(28)
σ Fac
(29)
which yields
(19)
Substitution of (7) and (8) into (18) leads to
[ (
lG ) a 1 - 1 +
1/2
) ]
(20)
Note that the other solution of the quadratic equation is physically meaningless since lG cannot be greater than the pore radius, a. The condition of electroneutrality of
and the counterion concentration near the membrane is, according to the electroneutrality requirement
(12) Sørensen, T. S.; Koefoed, J. J. Chem. Soc., Faraday Trans. 2 1974, 70, 665.
lG ) λD (1 + γ)
6160 Langmuir, Vol. 15, No. 19, 1999
Aguilella et al. Table 1. Summary of the Expressions Derived for Locating Gibbs’ Dividing Surface According to Assumptions A and B (See Main Text) for Planar and Cylindrical Geometries planar surface nonlinear PB equation case A
lG ) 4λD
(c-G ) 0) case B
γ 1+γ
lG ) λD (1 + γ)
linear PB equation
lG ) -
σ 2Fc
l G ) λD
cylindrical surface linear PB equation
the whole pore, expressed in the form
Fc+G π d(2a - lG)lG ) -σ 2πad
(30)
gives, after substituting lG from eq 29
c+G ) 2c
(31)
Case B. Anion concentration in the outer cylinder is assumed to be constant and equal to their concentration on the surface of the charged pore, that is, c-G ) c-(a) given by eq 26. Thus the balance equation is
∫0a ∫0d [c-(r) - c]2πr dr dz )
σλDF I0(a/λD)
- πd(2a - lG)lGc
RToI1(a/λD)
I0(a/λD) I1(a/λD)
)0
(32)
(33)
When the pore radius is much larger than Debye’s length, I0(a/λD) = I1(a/λD) and the above expression can be simplified to give
[ (
lG ) a 1 - 1 -
)]
2λD a
[ (
2λD a
1/2
) ]
)] 1/2
to the dependence of lG on the surface charge density, σ, which is only included in the expressions obtained for case B by using the linearized PB equation. In principle, it could be expected that the Gibbs approach would be a reasonable idealization when λD and lG have similar values. Table 2 shows a brief survey of λD/lG values for typical solution concentrations (e.g., 0.01-1 M) and surface charge densities (from 0.3 C/m2 in a fully charged phospholipid bilayer down to 5 mC/m2 in a slightly charged microporous membrane). These values were obtained by using nonlinear PB equations for cases A and B. Table 2. Ratio λD/lG for a Charged Planar Membrane with Different Surface Charge Densities Bathed by Electrolyte Solutions of Different Concentrations 0.005 C/m2
0.05 C/m2
0.3 C/m2
0.6 0.8 1.5
0.5 0.5 0.6
0.6 0.7 0.8
0.5 0.5 0.6
Case A 0.01 M 0.1 M 1M
1.5 4.0 12.1
0.01 M 0.1 M 1M
0.8 0.9 1.0
Case B
which leads to a quadratic equation in lG analogous to eq 28
lG2 - 2alG + 2aλD
σ Fac
lG ) a 1 - 1 -
(c-G ) c-surface)
Figure 4. Schematic representation of the two regions of the pore: the inner neutral cylinder and the charged cylindrical shell. The Gibbs dividing surface is represented by the dashed line.
[ (
lG ) a 1 - 1 +
1/2
(34)
Note that in the limit λD , a we get lG ) λD as expected when approaching a planar geometry. The similarity between the counterion concentration near the membrane obtained for both geometries in case A (when using the linearized Poisson-Boltzmann equation) as shown in eqs 17 and 31 has no particular physical meaning. Mathematically, it follows from the fact that the results of the integrals in eqs 15 and 25 are both proportional to λD. Results and Discussion Classically, the extent of the space charge zone in an EDL has been associated with the Debye length of the solution, λD, i.e., the distance at which the electric potential is approximately 1/e times its value at the charged surface. According to Gibbs’ approach, lG is also a measure of the thickness of the EDL. Table 1 shows a summary of the expressions found for lG. Here we see that lG is related to λD in all cases, either explicitly or indirectly via ion concentration c. Still, some differences appear with regard
It is shown that except for poorly charged membranes bathed by concentrated solutions, λD and lG are of the same order of magnitude. Even under such conditions (high c and low σ) case B prediction for lG is similar to λD. This similarity, together with the widely recognized approximate nature of the Gouy-Chapman theory,13,14 allows one to expect that this simple two-region model be a fairly good representation of the real system. Actually, Rostovtseva et al.1 used this approach together with the classical one from Gouy-Chapman and found better agreement with experiment by using Gibbs’ construction when calculating the counterion concentration near a phosphatidylserine bilayer membrane. The “analogous” parameter to λD for a cylindrical charged surface is the distance at which the electric potential decays at 1/e of its value at the surface. This critical distance is approximately the value of rc, which satisfies the equation I0((a - rc)/λD)/I0(a/λD) ) 1/e. At high ionic strength, rc = λD, but at low ionic strength, rc > λD and even rc > a. A similar reasoning to that made for a planar surface suggests exploring the relative values of lG and rc under different surface charge and ionic environments, according to the above-mentioned assumptions A and B for a co-ion concentration near the charged surface. The results are shown in Table 3. Equations 29 and 34 were used for estimating lG. It turns out that using Gibbs’ prescription with the assumption c-G ) c-surface (case B) leads to a more accurate representation of a cylindrical EDL. (13) McLaughlin, S. Annu. Rev. Biophys. Biophys. Chem. 1989, 18, 113. (14) Peitzsch, R. M.; Eisenberg, M.; Sharp, K. A.; McLaughlin, S. Biophys. J. 1995, 68, 729.
Electrokinetic Phenomena in Charged Pores
Langmuir, Vol. 15, No. 19, 1999 6161
Table 3. Ratio rc/lG for a Charged Cylindrical Pore with Different Surface Charge Densities Bathed by Electrolyte Solutions of Different Concentrationsa case A c (M)
σ ) 0.005 C/m2
σ ) 0.05 C/m2
σ ) 0.1 C/m2
case B
0.01 0.1 1
1.2 3.7 11.8
0.1 0.4 1.2
0.06 0.2 0.6
1.0 1.0 1.0
a Large pores have been considered (a . λ ) in order to avoid D dependence of this ratio on a. Results are shown for the two cases A and B discussed in the text.
The data shown in Tables 2 and 3 give a general indication of the range of applicability of Gibbs’ model. However, one should not forget that this construction is essentially a thermodynamic approach and as such its validity should be tested not at the microscopic level but in the calculation of measurable, macroscopic properties of the system. In a series of phenomena known collectively as electrokinetic phenomena, the main feature is the lateral relative motion of the ions and the charged surfaces. There are a number of properties of membranes that can be determined by electrokinetic measurements. Furthermore, the study of electrokinetic phenomena across charged membranes has been often carried out by modeling the membrane as an array of identical parallel cylindrical pores, homogeneously charged. The combination of the cylindrical geometry with the Gouy-Chapman model for the EDL gave rise to the so-called space charge model. From the seminal work by Helmholtz15 up to recent developments,12,16 different approaches have been used for the derivation of the phenomenological coefficients Lij as they appear in Onsager’s relations:
J ) L21
dφ dc dp + L12 + L13 dx dx dx
[ ] [ ] [ ] dc dp + L [- ] + L [- ] [- dφ dx ] dx dx
I ) L11 -
22
Figure 5. Pore conductivity per pore cross-sectional area, κ/πa2 (κ ≡ -I/(dφ/dx)). The solid line shows the value calculated according to the space charge model (eq 36). The dot line and the dashed line correspond to ν values obtained by using the model of two regions separated by Gibbs’ dividing surface in cases A (eq A1) and B (eq A5), respectively.
brane as composed of a parallel array of identical cylindrical pores, homogeneously charged, it is easy to derive expressions for pore conductivity, κ ) I/(dφ/dx), and for the streaming potential, ν ) -(dφ/dp)I)0, by using the space charge model and the approximation of pore radius much larger than the Debye length of the solution17,18
o(D+ + D-) (D+ - D-)Fσ κ + ) 2 RTa πa 2λD2 σ2 [1 + Q(2λD/a - Q)] (36) η
(35) and
23
The above equations show only those corresponding to the electric current and the ionic flux due to gradients of electric potential, concentration, and pressure. Here, the expressions of some of the coefficients Lij derived according to Gibbs’ model for a charged capillary are given in the Appendix. Having divided the pore into a neutral region and a charged cylindrical one, the phenomenological coefficients can be derived in a straightforward manner by using Nernst-Planck and Navier-Stokes equations according to assumptions A and B for counterion concentration c-G. We have considered here the case of equal solution concentrations on both sides of the pore because a gradient of concentration involves great problems in the location of the dividing surface since the central part of the pore is no longer a homogeneous region. Thus, only the expressions for coefficients L11, L13, L21 and L13 are given in the Appendix. The aim of this paper is not exhausting the possibilities of the model but showing that Gibbs’ concept of dividing surface can be applied successfully to the EDL near a charged membrane. For this purpose, we have chosen two well-known examples of measurable parameters in charged porous membranes, the conductivity and the streaming potential. As is known, the latter is the electric potential difference necessary to keep zero electric current when there is charge separation as a consequence of the solvent drag produced by a pressure gradient across the membrane. If we consider the mem(15) Helmholtz, H. L. F. Ann. Phys. 1879, 7, 337. (16) Westermann-Clark, G. B.; Anderson, J. L. J. Electrochem. Soc. 1983, 130, 839.
ν)
σλD(Q - 2λD/a)πa2 κη
(37)
where
Q≡
I0(a/λD) I1(a/λD)
(38)
D+ and D- are the cation and anion diffusion coefficients, and η is the solution viscosity. Following a similar procedure, by dividing the pore into two regions separated by Gibbs’ dividing surface, we can obtain other expressions for κG ≡ L11 and νG ≡ -L13 /L11. It is worth mentioning that no assumption is made about the radial profile of the electric potential in the pore in this derivation of the coefficients Lij according to Gibbs’ approach. Just Nernst-Planck and Navier-Stokes equations have been applied to the two-region model. Figure 5 shows the change of pore conductivity (per cross-sectional area) with solution concentration. A “large” pore (a ) 10-6 m, λD = 3-14 Å, so that a . λD), moderately charged (σ ) -0.05 C/m2), is considered. Comparison of κ values obtained according to the space charge model (eq 37) and Gibbs model (eq A1 for case A and eq A5 for case B) shows excellent quantitative agreement. Only at low concentration, in case A, are slight deviations from the (17) Mafe´, S.; Manzanares J. A.; Pellicer, J. J. Membr. Sci. 1990, 51, 161. (18) Newman, J. S. Electrochemical Systems; Prentice Hall: Englewood Cliffs, NJ, 1973.
6162 Langmuir, Vol. 15, No. 19, 1999
Aguilella et al.
hand, the Gouy-Chapman theory is in itself an approximation. It is expected that this approach may be particularly useful when the geometry of the system does not allow a closed form solution of the equations for the EDL as happened in the experimental work already mentioned,1 where the concept of Gibbs’ dividing surface was first applied to estimate the counterion concentration in the vicinity of the mouth of a protein ion channel embedded in a charged lipid bilayer.1 In conclusion, though a careful experimental test of the validity of Gibbs model in a wide range of experimental situations is still needed, the results presented give an idea of the power of Gibbs’ concept of the dividing surface. In addition, the example of conductivity and streaming potential demonstrates that once located the dividing surface, not all extensive variables of the system, can have zero surface excess value simultaneously. Figure 6. Streaming potential in a charged cylindrical pore filled with NaCl solutions of different concentrations. The solid line shows the value for ν calculated according to the space charge model (eq 37). The dot line and the dashed line correspond to ν values obtained by using the model of two regions separated by Gibbs’ dividing surface in cases A (eqs A1 and A2) and B (eqs A5 and A6), respectively. The pore surface charge density is σ ) -0.05 C/m2. The pore radius is much greater than the solution Debye length for all concentrations.
conductivity predicted by the space charge model found. This means that the redistribution of ions following Gibbs’ prescription does not alter the overall result of κ in the pore. Figure 6 shows a comparison of values for the streaming potential, ν, obtained following the two procedures: Gibbs’ model (cases A and B) and the space charge model. Pore parameters are the same as those used in the κ calculations. The results show qualitative agreement between the two models for a concentration range that covers most typical values. Although the order of magnitude of ν obtained from different procedures is the same, we cannot speak of quantitative agreement. This fact shows one of the obvious limitations of the model. The dividing surface was defined so that the surface excess of ion density was zero. However, streaming potential is not simply dependent on ion concentration but on the product of ion concentration and solvent velocity which is not constant along the radial direction. This means that a redistribution of mobile charges as that performed by Gibbs’ prescription is expected to cause discrepancies between the space charge model and Gibbs model predictions for streaming potential. In case A, the dependence of lG on solution concentration (see eq 29) causes that lG to be smaller or greater than rc (i.e., the typical extent of the EDL) depending on the value of c. On the other hand, in case B, lG is always lower than λD as seen in eq 34 (λD = rc for a . λD as considered here), so that mobile charges are distributed over a region which is closer to the charged surface than in a typical EDL, that is, over a region where solvent velocity is small. Thus, streaming potential is always smaller than that predicted by the Gouy-Chapman model, as seen in Figure 6. It seems then that the requirement of zero surface excess in counterion and coion concentration leads to nonzero surface excess in the streaming potential when using the Gibbs dividing surface to model this nonhomogeneous system. Throughout the discussion, comparison has been made again with the classical space charge model based on the Gouy-Chapman description of the EDL. This fact is based on the assumption that the latter provides a more accurate description of the EDL. However, we must not forget that there are situations in which the hypotheses underlying the Gouy-Chapman theory are not met and, on the other
Acknowledgment. Financial support from DGICYT (Ministry of Education and Culture of Spain, Project PB950018) and Fundacio´ Caixa Castello´-Bancaixa (Project P1B98-12) is gratefully acknowledged. V.A. wishes to thank Adrian Parsegian for having introduced him to the Gibbs’ concept of dividing surface and for helpful discussions and suggestions. Appendix Case A.
πF2c2 [1.5lG4 - 6alG3 + 7a2lG2 - 2a3lG η πF2c [D+(a2 + 2alG - lG2) + 2(a - lG)4 ln(1 - lG/a)] + RT D-(a - lG)2] (A1)
L11 )
L13 ) πFclG2(lG - 2a)2/4η L21 )
(A2)
πFc2 2 πFc l (l - 2a)2 + [D+(a2 + 2alG - lG2) 2η G G RT D-(a - lG)2] (A3) L23 ) πca4/4η
(4A)
Case B.
{π(lG2 - 2alG)[2a2cF2λDη(D+ + D-) (lG2 - 2alG) F(lG2 - 2alG)2 ησ(D+ + D-) - 4aD+λDησF (lG2 2alG) + RTλDσ2 (2a4 - 6a3lG + 3a2lG2)] - 4πa2(a lG)4RTλDσ2 ln(1 - lG/a)}/[2(lG2 - 2alG)2RTλDη] (A5) L13 ) πσalG(lG - 2a)/4η
(A6)
L21 ) {π[(lG2 - 2alG) (-4a2c(D- - D+)F2(lG2 2alG)λDη + 2F(lG2 - 2alG)2 (D- - D+)ησ + 2F(lG2 2alG)aλD (c(lG2 - 2alG) RT - 4D+η)σ + (2a3 6a2lG + 3alG2) RT ((lG2 - 2alG) + 2aλD)σ2) + 4a(a lG)4RT(- (lG2 - 2alG) + 2aλD)σ2 ln(1 - lG/a)]}/
L23 )
(
[4F(lG2 - 2alG)2RTλDη] (A7)
)
(2a - lG)2lG2 πσ a4c - (2a - lG)lG + 4Fη σ 2λD
LA980773P
(A8)
