The aqueous diffusion coefficient describes how individual molecules and ions migrate, or randomly walk, through subsurface soils and rocks, independent of the water or carrier fluid (Figure 1). Diffusion coefficients often control the ultimate source of a contaminant, i.e., how it comes out of the waste form or how it gets through some containment barrier or liner. Once a contaminant gets out into the environment, advection or flow usually becomes the dominant transport process. Diffusion coefficients control many remediation processes, such as adsorption or chemical filtering, Diffusion becomes important in unsaturated environments where the flow of water is very slow.

Figure 1.

The nature of ionic diffusion and electrical conductivity for water films in subsurface materials. In this example, the complexation of hydrated sodium ions on the exposed edge oxygens of a mineral surface alters the dielectric constant of the water, lowering the electrical conductivity. In an analogous way, weak bonds between the hydrated ions, edge oxygens and water molecules inhibit ionic diffusion and lowers the overall diffusion coefficient of the water film. Higher valent ions have a greater effect on the system. If the water film is more than about 20 angstroms thick then the diffusion coefficient will be unaffected by complexation and will be determined only by the volumetric water content and ionic strength of the solution. In natural systems, aqueous diffusion is always through bulk water. Direct and indirect techniques have been used to determine diffusion coefficients in porous media. Both techniques employ steady-state and transient-state techniques. Fick's first law forms the basis of steady-state experiments, while Fick's second law describes the transient-state. Unfortunately, it has been extremely difficult to measure diffusion coefficients below 10^-8 cm²/s, using direct steady-state techniques (tracer diffusion tests) because it is difficult to maintain proper boundary conditions. This necessitates an indirect method to measure diffusion coefficients in unsaturated or relatively impermeable soils and rocks. The most widely used indirect method is to measure electrical conductivity in a potentiostatic mode. During the run, the degree of saturation and other physical conditions can be fixed in the sample using the Unsaturated Flow Apparatus (UFA).(Conca, J. L. and J. V. Wright. 1992. Applied Hydrogeology 1:5-24.) Electrical conductivity cells designed for use in the UFA have two stainless steel electrodes in contact with the sample. Once electrical conductivity is measured, it is related to the diffusion coefficient through the Nernst-Einstein equation(Jurinak, J. J., S. S. Sandhu, and L. M. Dudley. 1987. Soil Science Society of American Journal 51:625-630.) which is given by:

where D_i is the diffusion coefficient of the ith ion (cm²/s), R is the gas constant (J/deg mol), T is absolute temperature (Kelvin), F is Faraday's constant (coul/mol), is equivalent conductance of the ith ion (cm² S/mol), Z_i is the charge number on the ith ion (dimensionless), t_i is the transference number of the ith ion, C_i is the molar concentration of the ith ion, is the cell constant for the conductivity cell sample holder (cm^-1), and G is the measured conductance on a conductivity bridge (S). All of these parameters are known or easily measured. This method is the foundation of diffusion and upon which all subsequent methods are based. The ease with which molecules diffuse through the water is exactly related to the ease with which the water molecules can align their dipoles along the electric field vector (Figure 1). There are different representations of diffusion, describing various physicochemical properties of the media:

where D_a is the apparent diffusion coefficient, D_e is the effective diffusion coefficient which is usually equal to the simple diffusion coefficient, D_s measured by electrical conductivity methods, D_p is the diffusion coefficient in the pore water, D_v is the diffusion coefficient in free water, is the porosity, is the density, K_d is the distribution coefficient, is a geometric factor, is the constrictivity, and is the tortuosity. The distribution coefficient and the extent of retardation used in transport models need to be determined separately for each specific ionic species, media, and fluid composition. Retardation can significantly affect the migration of species in porous media, resulting in an apparent diffusion coefficient that can be markedly different from the simple diffusion coefficient. However, this apparent diffusion coefficient is transient and will approach the simple diffusion coefficient as the sorption sites are filled or as the system otherwise tends toward chemical equilibrium. The diffusion coefficient for most aqueous species, including organics, are similar, differing by, at most, a factor of two from the self-diffusion coefficient of water which is 2.4 x 10^-5 cm²/s at 25 degrees C. Diffusion coefficients in porous media that are less than 10^-5 cm²/s result from mechanisms or conditions other than the inherent mobility differences between the species themselves. These mechanisms include path length differences due to water content differences and retardation due to sorption and transient chemical effects. Figure 2 shows over 300 diffusion coefficients for various geologic media as a function of volumetric water content determined using the Nersnt-Einstein method. The UFA was used to fix the target water content with 0.01 M KCl or NaCl solutions. It can be seen that the simple, or effective, diffusion coefficient is primarily a function of the water content and not of the material type.