Electrical Fluctuations in Colloid and Ionic Solutions

Abstract

A method is developed in order to determine the natural electrical thermal fluctuations and its spectral distribution across two points of a solution of ions or spherical charged particles immersed in an ionic solution. The electrical equivalent between two points of a solution is considered as a capacitor and a resistor in parallel. The method is applied within the Debye–Hückel approximation (linearized Poisson–Boltzmann equation), although it is valid in general. Among the results is the diminution of electrical fluctuations as particle sizes increase, as a consequence large particles produce electrical stabilization in their neighbourhood. Also can be observed that fluctuations are not quite sensitive to ionic concentrations for large particles. When the size of the particles becomes negligible we obtain similar results with the already obtained using the method of the mode expansion.

Reprinted from [José A. Fornés, J. Colloid Interface Sci. 186, 90, (1997)] Copyright (1997), with permission from Elsevier.

Appendix:Theoretical Calculation of the Electrical Resistivity

When we have highly charged particles or polyelectrolytes immersed in a symmetrical electrolyte solution, another path of electric conduction can be open through this particles or polyelectrolytes and the electrical conductivity $\sigma =\rho ^{-1}$ of the solution can be written as:

$$\displaystyle{ \rho ^{-1} = \rho _{ i}^{-1} + \rho _{ p}^{-1} }$$

(A2.1)

where ρ _i and ρ _p are the contributions to the total electrical resistivity of the ions and particles, respectively. The relation between the electrical resistivity, ρ _i , and the equivalent conductance $\Lambda$ is given by:

$$\displaystyle{ \rho _{i} = \frac{N_{A}} {nz\Lambda } }$$

(A2.2)

According to Debye and Hückel [2] and Onsager [10] interionic attractions and repulsions lead to two effects both of which result in the lowering of the equivalent conductance with increasing ion concentrations, correspondingly it can be decomposed into three terms (see [1] for a good treatise on this subject):

$$\displaystyle{ \Lambda = \Lambda _{0} - \Lambda _{e} - \Lambda _{\tau } }$$

(A2.3)

where $\Lambda _{0}$ is the equivalent conductance at infinite dilution, and is given by:

$$\displaystyle{ \Lambda _{0} = \frac{ze_{0}^{2}N_{A}} {kT} (D_{0}^{+} + D_{ 0}^{-}) }$$

(A2.4)

where D ₀ ^± are the diffusion constants.

$\Lambda _{e}$ is the contribution of the electrophoretic effect and tends to diminish $\Lambda _{0}$, is given by:

$$\displaystyle{ \Lambda _{e} = \frac{2ze_{0}^{2}\kappa N_{A}} {6\pi \eta (1 +\kappa a_{i})} }$$

(A2.5)

where η is the viscosity of the solution and a _i is the mean ions radius.

$\Lambda _{\tau }$ is called the time of relaxation effect and is the other mechanism tending to decrease the equivalent conductance, namely:

$$\displaystyle{ \Lambda _{\tau } = \frac{(e_{0}z)^{2}\kappa } {24\pi \epsilon \epsilon _{0}kT} \frac{\sqrt{2}} {1 + \sqrt{2}}\Lambda _{0} }$$

(A2.6)

From Eqs. A2.2–A2.6, we get for the ions electrical resistivity:

$$\displaystyle{ \rho _{i} = \frac{1} {n(ze_{0})^{2}\left [\left [1 -\frac{(ze_{0})^{2}\kappa } {24\pi \epsilon \epsilon _{0}kT} \frac{\sqrt{2}} {1+\sqrt{2}}\right ]\frac{(D_{0}^{+}+D_{0}^{-})} {kT} - \frac{\kappa } {3\pi \eta (1+\kappa a_{i})}\right ]} }$$

(A2.7)

In Fig. 2.11 is represented Eq. A2.7 for a KCl solution as a function of concentration.

Fig. 2.11

Representation of Eq. A2.7 for ρ _i as a function of concentration for a KCl solution (Eq. 2.3 for κ was also used)

The electrical resistivity corresponding to the particles, ρ _p , is given by:

$$\displaystyle{ \rho _{p} = \frac{6\pi \eta a_{p}(1 +\kappa a_{p})} {n_{p}Q^{2}(1 + \frac{K_{s}\rho _{i}} {a_{p}} )f(\kappa a_{p})} }$$

(A2.8)

where n _p is the number of particles per m³, Q is the net charge on the particle, a _p the radius of the particle and f(κ a _p ) is called Henry’s function [4]; it varies between 1.0 and 1.5 as κ a _p goes from zero to infinity and K _s is the surface conductance of the particle.

In deriving Eq. A2.8 we have used the relation between the current density J (A/m²) and the external applied field E, namely:

$$\displaystyle{ J = n_{p}Qv = \frac{1} {\rho _{p}}E }$$

(A2.9)

where v is the velocity of the particles and is given by Henry’s equation, Henry [4]:

$$\displaystyle{ v = \frac{\xi 4\pi \epsilon \epsilon _{0}} {6\pi \eta } f(\kappa a_{p})E }$$

(A2.10)

where the $\xi$ potential is given by:

$$\displaystyle{ \xi = \frac{Q} {4\pi \epsilon \epsilon _{0}a_{p}} \frac{1} {1 +\kappa a_{p}} }$$

(A2.11)

Henry [5] introduced a correction for the surface conductance, K _s , considering that the mobility of the particle would be reduced on account of the distortion of the spherical symmetry of the electrical double layer, relaxation effect. Also, the applied field would be modified in the vicinity of the particle by the electrical conductivity of the double layer.

$$\displaystyle{ \xi _{\mathrm{corr}} =\xi (1 + \frac{K_{s}\rho _{i}} {a_{p}} ) }$$

(A2.12)

The surface conductance of the particle can be evaluated using equations due to Street [18]. The relaxation effect may be neglected when (a) the values for $\xi$ potential are far below 25 mV and (b) values for κ a _p are small (less than 1) or when κ a _p  ≫ 1 (Overbeek [14]).