Mass Diffusion Analysis

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), $\varphi \stackrel{\mathrm{def}}{=}c/s$, where c is the mass concentration of the diffusing material and s 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: $c=s{(p/p_0)}^{\frac{1}{2}}$ , where p is the partial pressure of the diffusing gas, and the value of $p_0$ is customarily one atmosphere (for historical consistency). Combining Sievert's law with the definition of normalized concentration given earlier, $\varphi =c/s={(p/p_0)}^{\frac{1}{2}}$ . 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:

where V is any volume whose surface is S, $\mathbf{n}$ is the outward normal to S, $\mathbf{J}$ is the flux of concentration of the diffusing phase, and $\mathbf{n}\cdot \mathbf{J}$ is the concentration flux leaving S.

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

where $\mathbf{D}\left(c,\theta ,\mathbf{f}\right)$ is the diffusivity; $s\left(\theta ,\mathbf{f}\right)$ is the solubility; ${\kappa }_s\left(c,\theta ,\mathbf{f}\right)$ is the “Soret effect” factor, providing diffusion because of temperature gradient; $\theta$ is the temperature; ${\theta }^Z$ is the value of absolute zero on the temperature scale being used; ${\kappa }_p\left(c,\theta ,\mathbf{f}\right)$ is the pressure stress factor, providing diffusion driven by the gradient of the equivalent pressure stress, $p\stackrel{\mathrm{def}}{=}-\mathrm{trace}\left(\mathit{\sigma }\right)/3$; $\mathit{\sigma }$ is stress; and $\mathbf{f}$ are any predefined field variables.

Whenever D, ${\kappa }_s$ , or ${\kappa }_p$ 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 Defining an Analysis).

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

Fick's law is offered in Abaqus/Standard as a special case of the general chemical potential relation. To establish the relationship between Fick's law and the general chemical potential, we write Fick's law as

In most practical cases $s=s\left(\theta \right)$, and we can write

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

This conversion is done automatically in Abaqus/Standard when you request Fick's law (see Diffusivity).

An extended form of Fick's law can also be chosen by specifying a nonzero value for ${\kappa }_p$:

In this case Abaqus/Standard will still define ${\kappa }_s$ automatically as discussed earlier.

The units of concentration are commonly given as parts per million (P). Similarly, the units of solubility are P. The Soret effect factor is dimensionless. The units of the pressure stress factor are F⁻¹L², where F is force and L is length; and the units of equivalent pressure stress are FL⁻². The diffusivity, $\mathbf{D}$ , has units of L²T⁻¹, where T is time. The concentration flux, $\mathbf{J}$ , then has units of PLT⁻¹; and the concentration volumetric flux, $Q={\int }_S\mathbf{n}\cdot \mathbf{J}dS$ , has units of PL³T⁻¹.

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), 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 Defining an Analysis.

MASS DIFFUSION, STEADY STATE

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.