1152
J . Phys. Chem. 1984,88, 1152-1156
A Theory for the Apparent “Negative Second Wien Effect” Observed in Electric-Field-Jump Studies of Suspensions‘ Z. A. Schelly* and R. D. Astumian Department of Chemistry, University of Texas at Arlington, Arlington, Texas 76019-0065 (Received: June 27, 1983; In Final Form: August 15, 1983)
Transient conductance decrease has recently been observed during E-jump relaxation kinetic studies of adsorption-desorption processes in several colloidal suspensions. A theoretical analysis of these observations is given and an interpretation is described, where electric-field-induced increased adsorption of ions is shown to be responsible for the apparent negative second Wien effect. A dipole model system is discussed, which exhibits a true negative second Wien effect. It is suggested that such a dipole mechanism may provide an interpretation for the hitherto unexplained negative second Wien effects in homogeneous solutions, which had been observed previously.
Introduction In relaxation kinetic studies2 of fast rate processes, an equilibrium system is perturbed by a small, rapid, stepwise (transient or jump methods) or periodic (stationary methods) variation of a thermodynamic variable, followed by the measurement of the rate and amplitude of the subsequent reequilibration, by monitoring a physical parameter that is proportional to the concentrations. During the perturbation, energy is introduced into the system. For a reversible exchange of energy between a system and its surrounding, the total differential of the Gibbs free enthalpy G, in general, is given by3 dG = COj dFj - A dC; J
(1)
where Oj and Fj are conjugate extensive (generalized displacements) and intensive (generalized forces) variables, respectively. A is the affinity, and C; is the extent of the reaction. Clearly, depending on the nature of the system, the change of any of the conjugate variables in (l), i.e. SdT
fidg
ni dpi
? dl?
can affect the shift of the equilibrium. In addition ot the familiar thermodynamic symkols (S, T, V, P , ni) in u2. (c) Illustration of the quantities relevant for surface. conductance us. (d) Bound and diffuse layer polarization of a positively charged particle at steady state in an external field. Charges in the double layer represent excess local charges.
as we shall show, can become a major factor, is the polarization of the particles of the system under investigation. All the secondary influences can be reduced to a negligible minumum by suitable difference measurements, the proper design of the instrument, and the choice of the experimental conditions. Details are given in ref 2 and 23. For conductivity detection, the essential part of the experimental setup is the Wheatstone bridge (Figure 1) that was used also in the studies of the colloidal syswhich we wish to analyze. In a typical E-jump experiment, the sample cell contains the weak electrolyte studied, and the reference cell is fitted with a solution of strong electrolyte that has the same specific conductivity as the weak electrolyte. Prior to the application of the high-voltage perturbation square pulse, the bridge is balanced to UDB= 0 with a small alternating potential applied across AC (see Figure 1). Under this condition the impedances of the arms of the bridge fulfill the relationship WZr =
(8) During the perturbation, the dissociation field effect increases the conductivity of the sample cell, resulting in a typical relaxation signal given in Figure 2b. Physical Model. To interpret the transient conductivity decrease (Figure 2c) observed in colloidal systems, let us consider one of the spherical particles of colloidal size (radius r > 100 A) carrying a free charge Q, suspended in a medium of conductance ul and dielectric constant t l between the plates of a capacitor (Figure 3a). The particle itself has a conductance of u2 and a dielectric constant c2. The medium is a dilute electrolyte, and the overall system is electrically neutral. In the absence of an electric field, the Coulombic interaction between the particle and the ions will be balanced by their thermal motions after the formation of an electrical double layer around the particle. At equilibrium, the net fluxes of both positive and negative ions toward the sphere are zero, J* = 0. If the charge Q of the particle is due to chemisorbed ions, the double layer may be broken down into two portions.25 The dense part at the sphere surface is the layer of tightly bound (chemisorbed) ions, which essentially determines Zl/ZZ
(25) Dukhin, S. S. Surf. Colloid Sci. 1973, 3, 83.
the surface potential of the particle. The outside diffuse part is made up of the supporting counterions, which are held by Coulombic attraction. Upon the application of a homogeneous, constant electric field ?!, between the plates of the capacitor, the particle undergoes rapid polarization, during which process work w
is done on the system (cf. (2)). Besides the negligibly small, very rapid electronic and atomic polarization of the particle, the relevant polarization mechanisms are all migratory in nature;26 Le., they involve the movement of ions over one or more molecular distances or lattice sites. The most important polarization mechanisms are the Maxwell-Wagner (interfacial) p o l a r i z a t i ~ npolarization ,~~~~~ by surface cond~ctivity,2~ and bound layer30 and diffuse layerz5 polarization. Maxwell-Wagner polarization occurs if the conductivity of the medium is different from that of the particle (cf. Figure 3b). If ul < a2,charges arrive from the particle interior faster than they can be carried away toward the electrodes through the solution. If u1> u2,charges arrive from the solution at the interface faster than they can be transported through the solid. In both cases a charged interface results, with charge accumulation at the two polar regions of the sphere facing the electrodes. For the systems under investigation a1 > az is the case, resulting in a situation depicted in Figure 3b. The relaxation time of the polarization process is given by3’ 2q 7Mw
= 2a1
+ €2 - v(t2 + a2 - v(a2 - a1) €1)
(10)
(where u denotes the volume fraction of the spheres), and the final effective induced dipole moment pMwbyz5
Surface conductivity is a frequency-independent mechanism by which charge is transferred from one side of the sphere to the other, along the surface,32within a thin shell of thickness d and conductivity a, (Figure 3c). Polarization results only if a, > ul. Surface conductivity can be treatedz9as a part of the total conductance u2/ of the particle ai = a2
+ 2u,d/r
(12)
If it is assumed that the outer surface of the conducting shell provides still another surface upon which charge accumulation can occur in the Maxwell-Wagner sense, for small volume fraction v, the relaxation time for the surface-conductance modified interfacial polarization is given by33 €2
rMws = a2 + 2A,/r
+ 2tl - U ( € 2 -
€1)
+ 2a1 - u(u2 + 2AJr
- al)
(13)
where A, = a,d is the effective surface conductance of the shell. The anticipated effective induced dipole moment p of a charged sphere with a surface conductance is found34to be
( ;+ ; :7)
p = po + p , = qr3E - -
-1
(14)
where po and ps are the induced dipole moment components of (26) Pohl, H. A. “Dielectrophoresis”;Cambridge University Press: Cam-
bridge, UK, 1978; and references therein.
(27) Maxwell, J. C. “A Treatise on Electricity and Magnetism”, 2nd ed.; Clarendon Press: Oxford, UK, 1881; Vol. 1 . (28) y g n e r , K. W. Arch. Electrotech. (Berlin) 1914, 2, 271. (29) 0 Konski, C. T. J . Phys. Chem. 1960,64, 605. (30) Schwarz, G. J . Phys. Chem. 1962, 66, 2636. (31) van Beek, L. Progr. Dielectr. 1970, 7, 69. (32) Miles, J. B.; Robertson, H. P. Phys. Rev. 1932, 40, 583. (33) Schwan, H. P.; Schwarz, G.; Maczuk, J.; Pauly, H. J . Phys. Chem. 1962, 66, 2626. (34) Dukhin, S.S.; Shilov, V. N. Ado. Colloid Znterface Sci. 1980, 13, 153.
The Journal of Physical Chemistry, Vol. 88, No. 6, 1984 1155
Apparent Negative Second Wien Effect the sphere uncharged and that due to charge flow on the surface, respectively, and y = u,/u,r. When y is very small and u1 > u2, the surface conductance component of the dipole is small and the particle polarizes by the Maxwell-Wagner mechanism, opposite to the field. When y >> 1, the charge component dominates and the particle polarizes in the direction of the field. At some value of y the induced dipole moment is zero, which is called the isopolarization state. Bound and diffuse layer polarization both represent mechanisms where ionic motion tangent to the surface is relatively unhindered, but diffusion away from the surface requires overcoming a considerable activation barrier. Thus, these mechanisms predict a sizeable polarization with little or no direct contribution to the dc conductivity, but they increase the surface capacitance. Several authors considered the possibility of each of the bound and diffuse layer polarizations occurring alone; however, Dukhin et ale’sanalysis34suggests that both must be present and that the latter is more important. They found the expression for the relaxation time of polarization as TBDL
= r2/2DJ4
(15)
where D, is the coefficient of bound ion diffusion and M denotes
hf = 1 + Z + z - ( Z +
+ Z-)Kpb/co
(16)
where z+ and z- are the ionic charges, K is the Debye-Hiickel reciprocal distance, & represents the surface charge equilibrium density of the bound ions, and co is the ion concentration in the bulk solution. For the induced dipole moment due to doublelayer polarization, the following expression was derived:34
where
3 = [e~p(e,+~/ZkT)+ 3m exp(eo{/2kT)]/~r
(18)
with the dimensionless parameter m being
m = el(kT)2/6~qDeo2
(19)
In eq 18 and 19, qd represents the potential at the outer Stern plane, {the zeta potential, 7 the viscosity of the medium, and D the mean diffusion coefficient of the counterions. Comparison of the several theories shows that simple bulk factors such as considered in the Maxwell-Wagner approach, even in the surface-conductance modified version, are inadequate to account for the experimental facts of the anomalous ”giant” polarization of suspensions in polar solvents.26 The more successful treatment^^^,^^ involve the consideration of the electrically flexible parts of the ionic double layer, with the that the major effect is the polarization of the double layer, where the contribution of the bound layer is muted by that of the diffuse portion (Figure 3d). The net result of the sudden (in less than 0.1 ps) application of the external dc field, at this point, is the production of giant dipoles aligned in the direction of the field. For particles of r < 80 nm, with the systems and experimental conditions of ref 18-18, all the mechanisms discussed above predict a relaxation time of polarization T~~ < 2 ps, and for r = 500 nm, T~~ < 50 fis. If the polarization of our model system was the only effect, to a good approximation, it would only increase the capacitance of the sample cell (Figure l ) , which would result in a pulse-shaped response (Figure 2d) of the Wheatstone bridge, with the rise time of the first pulse approximately equal to the relaxation time of polarization T ~ ~ . During the formation of the giant particle dipoles, the free ions in the solution are exposed not only to the external field but also to the local fields of the dipoles. The concerted effect of the fields is the preferred motion of the ions toward the oppositely charged ends of the dipoles where they adhere to the particles (Figurg 4). The details of the sorption process with respect to the depth of penetration of the double layer by the ions is less important than the fact that the mobility of a significant number of originally
Figure 4. Schematic representation of the sorption of ions on an induced-dipole particle (positively charged). Charges in the double layer represent excess local charges.
z I
Figure 5. Illustration of a permanent-dipole particle oriented by an external field. The charges within the equipotential surface re represent excess local charges. The two surfaces re are not spherical, in contrast to the simplified drawing given.
free ions is drastically reduced. Either because they actually adsorb at the bound layer or because they become incorporated in the diffuse layer, there is a decrease of free charge carriers. We propose this field-induced increased adsorption of ions to be the source of the transient conductivity decrease observed (Figure 2c). The amount of ions adsorbed should be proportional to the induced dipole moment p of the particles and the rate 7-l of their adsorption to the average bulk ion current I in the solution, if the polarization is very fast. Since both p and I , to a good approximation, linearly increase with the field E, one expects both the amplitude and the rate of the conductance decrease to do so too, as was borne out by experiments (see Figures 2 and 3 of ref 16). During the “binding” of ions, the strength of the dipoles is decreasing because of partial charge neutralization at the poles. If no other processes (second Wien effect) occurred simultaneously, the system would reach a steady state (in contrast to an equilibrium state in the absence of an external field) with respect to the thermal motions and fluxes, characterized by div J* = 0 for both positive and negative ions. The expected signal for this situation is depicted by the dashed line in Figure 2c. The “Binding” Constant of Free Ions to a Dipole. It is instructive to investigate how the number of ions associated with a dielectric spherical particle changes upon its polarization. Following Onsager, under associated we mean that the potential energy of interaction between the two moieties is greater than 2kT. For illustration, we shall investigate only the equilibrium case, Le. how a permanent dipole (without an external field) binds ions. Thus, in essence, we are comparing two systems: a spherical, uncharged particle with a spherical permanent-dipole particle, each suspended in a dilute electrolyte solution. In the first system, clearly, no ions are associated with the particle. In the second system, with reference to Figure 5 , if the dipole particle consists of two point charges *Q that are located at the surface, its dipole moment is p = 21Q. Since the ions cannot penetrate the sphere, we are interested only in the outside region, where the potential of the dipole is
+ = ( Q / 4 i ~ e ~ q ) ( r-~r2-l) -~
(20)
which according to the law of cosines can be written as
+
+ +
+(r,O) = (Q/4iT€o€l)[(r2 Iz - 21r cos 0)-1/2 - (r2 I2 21r cos ~ ) ] - l /(21) ~ Although, in the bulk solution, where r
>> I, eq 21 reduces to
+ = 2Q1 cos 0 / h r 2 e o q
(22)
in our case, close to the surface, this approximation cannot be used.
1156 The Journal of Physical Chemistry, Vol. 88, No. 6,1984
Since the potential energy of an i-type ion with charge zpo is just E, =
(23)
Z$O*
by setting E, = 2kT we can obtain an equipotential surface re(O, $ = constant), within which ions are considered bound. To find re, eq 21 must be expanded as a polynomial in r a n d solved, after setting J/ = 2kT/z,eo. Then, if the bulk number density of the i-type ions in the solution is oni = ,JVi/V,the number iN,l of the bound i-type ions per dipole can be obtained from iN,l = 8 s " s r e ~ r ' 2 0 nexp(-z,eoJ/(r,0)/kT)r2 i sin 0 d r d0 d 4 (24) 8=0
r=l
Schelly and Astumian signal (solid line in Figure 2c). It is important to note that this relaxation takes place between two steady states. In the colloidal systems studied,l6-l8 reaction 5 occurs on the surface of the suspended particles, involving ionic groups permanently attached to the surface. Thus, the particles S themselves are part of the weak-electrolyte equilibrium shifted by the second Wien effect. Hence, if all the possibly different types of cations and anions present are denoted by C+ and A-, respectively, eq 5 can be rewritten for the colloidal systems as S-C-A
k 2 (S-C+-A-) k-l
ion pair
S-C+
+ A-
(26)
k-2
$=O
Since the problem has cylindrical symmetry with respect to the y axis and planar symmetry with respect to the xz plane, the integration is carried out only for the first octant. The angle runs around the axis of the dipole. The integration limit Be corresponds to the point where the equipotential surface intersects the particle. If the total number of dipoles in the solution is Nd, the number of bound i-type ions is ii?b= NdiN
