Diffusion in a medium with a constant diffusion coefficient is often described using the equation (Carslaw and Jaeger, 1967; Crank, 1956;Kirkham and Powers, 1976)

where c = c(x,t) is the concentration of the substance at position x and time t and D is the diffusion coefficient. The problem solved in this example is for a diffusing substance at an initial concentration C_o in the medium from position zero to position h (called the mixing depth in this model) with none of the diffusing substance present at other positions. That is, the initial condition is given by

and

The solution to this equation with these initial conditions is

where erf is the error function. This solution is for chemicals which are not adsorbed on the soil surfaces. When sorption occurs, it reduces the effective diffusion coefficient. In that case, the diffusion coefficient D is replaced in the above solution by the effective diffusion coefficient, D_E , where

and r is the bulk density of the soil, K_d is the partition coefficient of the chemical, and q is the volumetric water content of the soil. The partition coefficient K_d can be approximated by the product of the organic carbon partition coefficient and the organic carbon content of the soil.

1. Diffusion coefficient may vary with the concentration of a diffusing substance and the medium in which a substance diffuses. In dilute solutions, such as soil-water solutions, the behavior of each diffusing molecule or ion is expected to be independent of the behavior of the others. The diffusion coefficient of a solute in a dilute solution can reasonably be taken as constant. In other systems, such as the inter-diffusion of metals or the diffusion of organic vapors in high polymer substances, the diffusion coefficient depends on the concentration of diffusing substance. In this program, we assumed that the diffusion coefficients are independent of the concentration of diffusing substance.

2. Diffusion coefficient may vary with location if the diffusing medium is not homogeneous. The simulated diffusing medium in this program is considered homogeneous.

3. In isotropic media, the structure and diffusion properties in the neighborhood of any point are the same relative to all directions. The general differential equation of diffusion in a Cartesian coordinate system is expressed as

where C is the concentration of the diffusing substance; D, the diffusion coefficient, t, time; x, y, and z are the Cartesian coordinates. If the diffusion is one-dimensional, i.e., a concentration gradient exists only along the direction of x, equation [1] reduces to

In an anisotropic medium, diffusion properties depend on the direction in which they are measured. Some common examples of anisotropic media are crystals, textile fibres, and polymer films in which the molecules have a preferential direction of orientation. The general differential equation of diffusion in an anisotropic medium is expressed as

where D_ij (i=x,y,z; j=x, y, z) are elements of the diffusion coefficient matrix for the coordinate system selected. If the diffusion is strictly one-dimensional, i.e., a concentration gradient exists only along the direction of x and both C and ¶ C/¶ x are everywhere independent of y and z, equation [3] reduces to

Both equation [2] and [4] are mathematically equivalent to the governing equation for the program. Therefore, the third assumption of the program is that THE DIFFUSION IS STRICTLY ONE-DIMENSIONAL.