Mass diffusion analysis

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 flux of concentration leaving S.

Using the divergence theorem,

Because the volume is arbitrary, this provides the pointwise equation

The equivalent weak form is

where $\delta \varphi$ is an arbitrary, suitably continuous, scalar field.

This statement can be rewritten as

Using the divergence theorem again yields

The diffusion is assumed to be driven by the gradient of a chemical potential, which gives the general 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 the temperature gradient; $\theta$ is the temperature; ${\theta }^Z$ is the absolute zero on the temperature scale 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, $p\stackrel{\mathrm{def}}{=}-\mathrm{trace}\left(\mathit{\sigma }\right)/3$; and $\mathbf{f}$ are any predefined field variables.

An example of a particular form of this constitutive model is the assumption made for hydrogen diffusion in a metal:

with the chemical potential, $\mu$, defined as

where ${\mu }^0$ is a fixed datum, R is the universal gas constant, and ${\overline{V}}_H$ is the partial molar volume of hydrogen in the solid solution. This form is similar to that used by Sofronis and McMeeking (1989) and results in a constitutive expression of the form

To implement this particular form, data for ${\kappa }_s$ and ${\kappa }_p$ must be calculated from the equations

Changing variables ($c=\varphi s$) and introducing the constitutive assumption of Equation 3 into Equation 2 yields

where

is the concentration flux entering the body across S.

Equilibrium in a finite element model is approximated by a finite set of equations through the introduction of appropriate interpolation functions. Discretized quantities are indicated by uppercase superscripts (for example, ${\varphi }^N$). The summation convention is adopted for the superscripts. These represent nodal variables, with nodes shared between adjacent elements and appropriate interpolation chosen to provide adequate continuity of the assumed variation. The interpolation is based on material coordinates $S_i$, $i=1$, 2, 3.

The virtual normalized concentration field is interpolated by

where ${\mathrm{N}}^N\left(S_i\right)$ are interpolation functions. Then, the discretized equations are written as

Time integration in transient problems utilizes the backward Euler method (the modified Crank-Nicholson operator). Adopting the convention that any quantity not explicitly associated with a point in time is taken at $t+\mathrm{\Delta }t$, we can drop the subscript $t+\mathrm{\Delta }t$ and write the integrated equations as

The Jacobian contribution from the conservation equation is obtained from the variation of Equation 6 with respect to $\varphi$ at time $t+\mathrm{\Delta }t$. This yields

Rearranging and using the interpolation $d\varphi ={\mathrm{N}}^{N}d{\varphi }^N$, we obtain

Inspecting the above equation, we observe that the Jacobian becomes unsymmetric whenever the diffusivity, $\mathbf{D}$; the temperature-driven diffusion coefficient, ${\kappa }_s$; or the pressure-driven diffusion coefficient, ${\kappa }_p$, is defined as a function of concentration.