Relaxation of Charge Density

Let us impose a temporary fluctuation on charge density, say pir,t), at some location r in a uniform electrolyte solution. The time dependence of pir, t) is given by the conservation equation (Equation 8.1) as 

Substituting Equation 8.28 into Equation 8.27, and with the use of the Poisson equation (Equation 3.9), we get 

The solution of this linear differential equation is that a fluctuation pir,t) decays exponentially with time from its initial value pir, o). 

Furthermore, assuming that the diffusion coefficients of the cation and the anion are the same, D, = D, the relaxation rate can be rewritten for a z z-type electrolyte of concentration c as 

Using the definition of the electrostatic screening length in Section 3.1.3.1, the relaxation time r is simply the ratio of the square of the Debye length to the diffusion coefficient of the ions, 

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