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Langmuir 2002, 18, 2036-2038
Role of Channel Wall Conductance in the Determination of ζ-Potential from Electrokinetic Measurements Andriy Yaroshchuk* and Volker Ribitsch Institut fu¨ r Chemie, Karl-Franzens-Universita¨ t, Graz, Austria Received August 30, 2001. In Final Form: November 27, 2001 The classical Smoluchowski formula for streaming potential is shown to be not applicable to the channels (pores) with conducting walls. The classical procedure of accounting for surface conductivity may also be ineffective in some cases. Measurements of streaming current instead of streaming potential are clearly beneficial in this case. Variation of channel height may also be used to extrapolate to true values of ζ-potential not complicated by the wall conductivity.
Introduction Recently, large discrepancies have been found experimentally between the values of ζ-potential calculated from streaming potential and streaming current measured in ultrathin channels made of conducting materials (ultrathin gold foils).1 Those discrepancies have been ascribed to the electrical conductance within the channel walls. The purpose of this paper is to provide a thermodynamic basis for this interpretation. It will also be shown that for certain wall materials, the conventional procedure of accounting for the surface conductance may not solve the problem. Theory Let us start from the phenomenological formulation of the problem within the scope of irreversible thermodynamics. If the concentration changes are negligible (which is always the case even in the thinnest channel of 5 µm used in the microslit setup), there are only two thermodynamic forces and conjugate flows, and within the linear domain they are related in this way:
JV ) LP∆P + LEP∆E I ) LEP∆P + LE∆E
(1)
where JV is the volume flux, I is the electric current, ∆P is the hydrostatic pressure difference, ∆E is the electric potential difference, LP is the hydraulic permeability, LE is the electric conductivity, and LEP is the nondiagonal phenomenological coefficient responsible for electrokinetic phenomena (sometimes called electroosmotic permeability). In eq 1, we have already taken into account Onsager’s symmetry relationship. Depending on the boundary and initial conditions, eq 1 can describe a variety of transport phenomena. Of interest for us are two of them: streaming current and streaming potential. By definition, the streaming current occurs when there is a pressure difference (and a volume flow) but no electric potential difference. Therefore, for its magnitude we obtain this:
Is ) LEP∆P
(2)
The streaming potential occurs when there is a pressure
difference (and a volume flow), but the electric current is zero instead of electric potential difference. By setting I ) 0 in eq 1, we obtain this:
∆Es ) -
LEP Is ∆P ≡ LE LE
(3)
It should be stressed that the effect of electric conductivity on the streaming potential is exclusively determined by the denominator in the right-hand side of eq 3. If the electric conductivity (whatever its nature) of the system has been measured, the streaming current can be calculated from the measured streaming potential (technically, it is often much easier to measure the streaming potential than the streaming current). As has been noticed by Hunter,2 the interpretation of streaming current is not complicated by the occurrence of surface conductance. However, to further interpret the streaming current in terms of ζ-potential one needs to know some characteristics of pore geometry that are usually not exactly known in the case of porous plugs (but are normally available for well-defined channels). At the same time, in certain circumstances those characteristics cancel out and do not appear in the formula for the streaming potential. That explains why the Smoluchowski formula for streaming potential is so much preferred to other versions of equations of electrokinetics. Indeed, it looks this way:
∆Es 0ζ ) ∆P ηλ
(4)
and apart from the sought-for ζ-potential, it contains only measurable (streaming potential and pressure difference) or reasonably known (solution viscosity, η, its relative dielectric constant, , and specific electric conductivity of bulk solution, λ) values. Moreover, a procedure has been developed of taking into account the effect of surface conductivity, which makes the formula appear almost universally applicable (at least, in not too narrow pores or channels). However, an (often) implicit assumption is made in its derivation, namely, that the pore (or channel) walls are nonconducting. That condition is not met in the case of channels whose walls are formed by conducting substrates. (They do not necessarily need to be metallic. Any ion-exchange or simply porous materials soaked with
* To whom correspondence should be addressed. (1) Schweiss, R.; Welzel, P.B.; Werner, C.; Knoll, W. Langmuir 2001, 17, 4304-4311.
(2) Hunter, R. J. Zeta Potential in Colloid Science. Principles and Applications; Academic Press: London, 1981.
10.1021/la015557m CCC: $22.00 © 2002 American Chemical Society Published on Web 01/24/2002
Effect of Channel Wall Conductance on ζ-Potential
Langmuir, Vol. 18, No. 6, 2002 2037
electrolyte solution will also do.) Accordingly, the Smoluchowski formula is not directly applicable to them. To demonstrate that, let us consider a rectangular channel of width H, length l, and height 2h. Let the x-axis be perpendicular to the channel walls and its origin be located at the plane of symmetry. Under the streaming current conditions, the electric field is zero. Therefore, the local streaming current density is equal to the product of local fluid velocity, v(x), and local electric charge density, F(x). The total streaming current is obtained by the integration over the channel cross section. Thus, by neglecting the fringe effects (the channel width is assumed to be much larger than its height) we obtain this:
Is ) 2H
∫0h dx v(x) F(x) ) -20H ∫0h dx v(x) φ′′(x)
(5)
where we have taken into account that the local electric charge density satisfies the Poisson equation, φ′′(x) ) -F(x)/0, and φ(x) is the local electrostatic potential. The fluid velocity satisfies the Stokes equation, ηv′′(x) ) ∆P/l, with these boundary conditions: v′(0) ) 0 and v(h) ) 0. By taking the integral in eq 5 two times by parts and by applying the boundary conditions, we obtain this:
Is ) -20H(v′(h) φ(h) +
∫0h dx v′′(x) φ(x))
(6)
By definition, φ(h) ≡ ζ. The Stokes equation can be easily solved to yield v′(h) ) (∆P/l)(h/η). Therefore, for the streaming current we finally obtain
0 S 1 ∆P ζ Is ) η l h
(
∫0
h
)
dx φ(x)
(7)
where we have introduced the channel cross section S ≡ 2Hh. In wide channels, the second term between parenthesis is very small (since in most of the channel the electric potential is equal to zero), so we finally obtain this Smoluchowski-like expression:
Is ) -
0ζ S ∆P η l
(8)
Until now, the reasoning has not been restricted to channels with nonconducting walls. To obtain the streaming potential, according to eq 3, we should divide the negative of streaming current by the electric conductivity. If the channel walls are nonconducting and the surface conductance is negligible, for the rectangular geometry by neglecting the fringe effects, one can write down this:
S LE ) λ l
(9)
By substituting eqs 8 and 9 into eq 3, we obtain the Smoluchowski formula of eq 4. The electric conductivity in eq 3 is a phenomenological quantity. Therefore, if the channel walls are conducting, a contribution due to that appears in LE, and the Smoluchowski formula in the form of eq 4 becomes inapplicable. Physically, that can be explained in this way. The streaming current has a convective nature and as such occurs at zero electric potential difference. If the net electric current is equal to zero because the external electrical circuit is open, a conventional (potential) electric current arises whose magnitude is exactly equal to that of the streaming current. However, by no means is that current obliged to take the same path as the convective current. Therefore, it flows wherever the electric conduc-
tivity is nonzero. By contrast, the convective current occurs only where the macroscopic liquid flow is possible (and, of course, where a space electric charge is available). The channel cross sections, S, appearing in eq 8 and eq 9 have different origins in each of them. In eq 8, that is the area where the fluid velocity is nonzero, while in eq 9 it is the area where the electric conductivity is nonzero. They do coincide in both equations only if it is assumed that the macroscopic liquid flow and electrical conductance occur within exactly the same domain. Procedures to Account for the Wall Conductivity. Now let us demonstrate that the conventional procedure of taking into account the effect of surface conductance may not eliminate the contribution of wall conductance. The basic assumption of the procedure is that the surface conductance is suppressed at sufficiently high electrolyte concentrations. Therefore, the so-called cell constant can be obtained from the high-concentration measurements: (0)
S LE ) (0) l λ
(10)
where the superscript (0) denotes the high-concentration values. By substituting eq 10 into eq 8 and the latter into eq 3, one obtains this version of the Smoluchowski formula:
∆Es 0ζ L(0) E ) (0) ∆P ηλ LE
(11)
Now the question is, what is the nature of wall conductance? If it is practically independent of salt concentration, then in principle it can be “suppressed” in the sense that the channel conductivity becomes much larger than the substrate one. (Of course, if the substrate conductivity is very high, that may happen only at unrealistically high concentrations. From the fact that in ref 1 the streaming potential could be observed at all, it follows that the conductivity of gold substrates used in ref 1 was not extremely high despite their metallic nature. That probably occurred due to the blockage of the metal/solution interface by the self-assembled monolayers of thiol derivatives.) However, if the wall conductance occurs due to the wall porosity or swelling and the pore size is not too small, it cannot be suppressed at any electrolyte concentration since it is roughly proportional to the conductance in the channel. Therefore, it makes an almost equal relative contribution into both L(0) E and LE, whose ratio proves proportional to λ(0)/λ, in this case. Therefore, the corrections just cancel out. A more effective way is to apply the well-known procedure of varying the channel height. Let us assume that the conducting wall layers have finite thicknesses. The electric conductivity of the “sandwich” made up by the channel and two conducting layers can be represented in this way:
Sm S LE ) λ + 2 λm l l
(12)
where Sm is the conducting layer cross section and λm is its specific conductivity. In the rectangular geometry, the cross sections are proportional to the thicknesses, so by substituting eq 12 into eqs 3 and 8 we obtain this:
∆Es 0ζ ) ∆P ηλ
1 λmhm 1+ λh
(13)
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where hm is the conducting layer thickness. From eq 13, it is seen that the reciprocal streaming potential coefficient is a linear function of reciprocal channel height. Therefore, if the measurements are performed at several channel heights, it is possible to carry out a linear extrapolation to infinitely large channel heights and a correct value of ζ-potential can be determined. From the slope of the
Yaroshchuk and Ribitsch
straight line, the relative contribution of channel walls to the sandwich conductivity can be determined. Acknowledgment. The financial support of Fonds zur Fo¨rderung der wissenschaftlichen Forschung (Austria) within the scope of Project P13978-CHE is gratefully acknowledged. LA015557M
