Diffusion Coefficient

The typical approach to characterizing diffusive transport of ionic species in porous materials begins with Fick's law of diffusion that relates the bulk diffusion flux j to an apparent bulk diffusion coefficient D_b for a species with concentration c [7]:

$${\bf j_b} = -D_b\,\nabla c$$ (5)

This equation is applied to each diffusing species. The typical approach is to then "fit" experimental data to this equation, yielding an apparent bulk diffusion coefficient D_b for the particular species being investigated. Fick's law does not account for interactions known to exist between, and among, diffusion ionic species. This has led some researchers to characterize the diffusion of ionic species through the use of Fick's law and a diffusion coefficient tensor [13]. This approach, however, yields a diffusion coefficient tensor that characterizes a particular experiment and cannot, in general, be applied to different scenarios.

The diffusion of ionic species in an electrolyte is better characterized by the electro-diffusion equation. For the i-th ionic species, the electro-diffusion equation relates the bulk flux j to the concentration c, the diffusion potential $\psi_{D}$_D, the bulk microstructural diffusion coefficient D_µ, and the bulk conventional mobility u [6]:

$${\bf j} = - D_\mu\left(1+ \frac{\partial \ln\gamma}{\partial \ln c}\right) \,\nabla c - z\,F\,u\,c\,\nabla\psi_D$$ (6)

The quantity $\gamma$ is the activity coefficient for the species. Although this equation neglects adsorption effects, it is otherwise a complete equilibrium thermodynamic description of nonreactive diffusive transport of charged species in concentrated electrolytes. Equation 6 still bears a resemblance to Fick's law of diffusion [7]. The quantity D_µ [1 + ln / lnc] is an agglomerated diffusion coefficient that includes the effects of both the microstructure and ion-ion interactions at hight concentration.

It should be noted that the agglomerated diffusion coefficient is not the apparent bulk diffusion coefficient D_b. Strictly speaking, the apparent diffusion coefficient also includes the effects of the diffusion potential $\psi_{D}$_D, which is related to the electrostatic interactions of the ions. In those cases where the self-diffusion coefficients of all the diffusing species are nearly identical, the diffusion potential will be nearly zero, and the apparent bulk diffusion coefficient will be nearly equal to the agglomerated diffusion coefficient. For cementitious systems, however, there are many species present with varying self-diffusion coefficients.

Ideally, one would like to distinguish between the effects due to microstructural changes and the effects due to changes in the pore solution electro-chemistry. By observation of Eqn. 6, the microstructural diffusion coefficient D_µ characterizes the solid microstructure, the quantity in parenthesis characterizes the thermodynamics, and the last term characterizes electrostatic interactions. The microstructural diffusion coefficient is itself independent of the pore solution chemistry. Demonstrating how to extract the value of microstructural diffusion coefficient from a diffusion experiment is accomplished through a detailed discussion of the formation factor.