The governing equations for mass diffusion are an extension of Fick's equations: they allow for nonuniform solubility of the diffusing substance in the base material and for mass diffusion driven by gradients of temperature and pressure. The basic solution variable (used as the degree of freedom at the nodes of the mesh) is the “normalized concentration” (often also referred to as the “activity” of the diffusing material), , where is the mass concentration of the diffusing material and is its solubility in the base material. Therefore, when the mesh includes dissimilar materials that share nodes, the normalized concentration is continuous across the interface between the different materials.

For example, a diatomic gas that dissociates during diffusion can be described using Sievert's law: , where is the partial pressure of the diffusing gas. Combining Sievert's law with the definition of normalized concentration given earlier, . Equilibrium requires the partial pressure to be continuous across an interface, so normalized concentration will be continuous as well. If an expression other than Sievert's law defines the relationship between concentration and partial pressure for a diffusing material, solubility should be defined accordingly.

The diffusion problem is defined from the requirement of mass conservation for the diffusing phase:

Diffusion is assumed to be driven by the gradient of a general chemical potential, which gives the behavior

Whenever , , or depends on concentration, the problem becomes nonlinear and the system of equations becomes nonsymmetric. In practical cases the dependence on concentration is quite strong, so the nonsymmetric matrix storage and solution scheme is invoked automatically when a mass diffusion analysis is performed (see “Procedures: overview,” Section 6.1.1).

Mass diffusion behavior is often described by Fick's law (Crank, 1956):

The two terms in this equation describe the normalized concentration and temperature-driven diffusion, respectively. The normalized concentration-driven diffusion term is identical to that given in the general relation. The temperature-driven diffusion term in Fick's law is recovered in the general relation if

An extended form of Fick's law can also be chosen by specifying a nonzero value for :

The units of concentration are commonly given as parts per million (). On the basis of the applicability of Sievert's law to the mass diffusion, the units of solubility are , where is force and is length. The units of the Soret effect factor are . The units of the pressure stress factor are , and the units of equivalent pressure stress are . The diffusivity, , has units of , where is time. The concentration flux, , then has units of ; and the concentration volumetric flux, , has units of .

Steady-state mass diffusion analysis provides the steady-state solution directly: the rate of change of concentration with respect to time is omitted from the governing diffusion equation in steady-state analysis. In nonlinear cases iteration may be necessary to achieve a converged solution.

Since the rate term is removed from the governing equations, the steady-state problem has no intrinsic physically meaningful time scale; nevertheless, you may assign a “time” scale to the analysis step. This time scale is often convenient for output identification and for specifying prescribed normalized concentrations and fluxes with varying magnitudes. Thus, when steady-state analysis is chosen, you specify a “time” increment and a “time” period for the step; ABAQUS/Standard then increments through the step accordingly. If a steady-state analysis step is to be followed by a transient analysis step and total time is used in amplitude definitions (“Amplitude curves,” Section 19.1.2), the time period should be defined to be negligibly small in the steady-state step. For more details on time scales and time stepping, see “Procedures: overview,” Section 6.1.1.

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Time integration in transient diffusion analysis is done with the backward Euler method (also referred to as the modified Crank-Nicholson operator). This method is unconditionally stable for linear problems.

Automatic or fixed time incrementation can be used for transient analysis. The automatic time incrementation scheme is generally preferred because the response is usually simple diffusion: the rate of change of normalized concentration varies widely during the step and requires different time increments to maintain accuracy in the time integration.

Spurious oscillations due to small time increments

In transient mass diffusion analysis with second-order elements there is a relationship between the minimum usable time step and the element size. A simple guideline is

where is the time increment, is the diffusivity, and is a typical element dimension (such as the length of a side of an element). If time increments smaller than this value are used, spurious oscillations can appear in the solution. ABAQUS/Standard provides no check on the initial time increment defined for a mass diffusion analysis; you must ensure that the given value does not violate the above criterion.

In transient analysis using first-order elements the solubility terms are lumped, which eliminates such oscillations but can lead to locally inaccurate solutions for small time increments. If smaller time increments are required, a finer mesh should be used in regions where the normalized concentration changes occur.

Generally there is no upper limit on the time increment because the integration procedure is unconditionally stable unless nonlinearities cause numerical problems.

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