Diffusivity

Diffusivity:

defines the diffusion or movement of one material through another, such as the diffusion of hydrogen through a metal;
must always be defined for mass diffusion analysis;
must be defined in conjunction with Solubility;
can be defined as a function of concentration, temperature, and/or predefined field variables;
can be used in conjunction with a “Soret effect” factor to introduce mass diffusion caused by temperature gradients;
can be used in conjunction with a pressure stress factor to introduce mass diffusion caused by gradients of equivalent pressure stress (hydrostatic pressure); and
can produce a nonlinear mass diffusion analysis when dependence on concentration is included (the same can be said for the Soret effect factor and the pressure stress factor).

The following topics are discussed:

Defining diffusivity
Directional dependence of diffusivity
Temperature-driven mass diffusion
Pressure stress-driven mass diffusion
Mass diffusion driven by both temperature and pressure stress
Specifying the value of absolute zero
Elements

Defining diffusivity

Diffusivity is the relationship between the concentration flux, $J$ , of the diffusing material and the gradient of the chemical potential that is assumed to drive the mass diffusion process. Either general mass diffusion behavior or Fick's diffusion law can be used to define diffusivity, as discussed below.

General chemical potential

Diffusive behavior provides the following general chemical potential:

J = - s D \cdot [\frac{\partial ϕ}{\partial x} + κ_{s} \frac{\partial}{\partial x} (\ln (θ - θ^{Z})) + κ_{p} \frac{\partial p}{\partial x}],

where

$D (c, θ, f_{i})$: is the diffusivity;
$s (θ, f_{i})$: is the solubility (see Solubility);
$κ_{s} (c, θ, f_{i})$: is the Soret effect factor, providing diffusion because of temperature gradient (see below);
$κ_{p} (c, θ, f_{i})$: is the pressure stress factor, providing diffusion because of the gradient of the equivalent pressure stress (see below);
$ϕ \overset{def}{=} c / s$: is the normalized concentration;
c: is the concentration of the diffusing material;
$θ$: is the temperature;
$θ^{Z}$: is the temperature at absolute zero (see below);
$p \overset{def}{=} - trace (σ) / 3$: is the equivalent pressure stress; and
$f_{i}$: are any predefined field variables.

Input File Usage

DIFFUSIVITY, LAW=GENERAL (default)

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: 
Law: General

Fick's law

An extended form of Fick's law can be used as an alternative to the general chemical potential:

J = - D \cdot (\frac{\partial c}{\partial x} + s κ_{p} \frac{\partial p}{\partial x}) .

Input File Usage

DIFFUSIVITY, LAW=FICK

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: 
Law: Fick

Directional dependence of diffusivity

Isotropic, orthotropic, or fully anisotropic diffusivity can be defined. For non-isotropic diffusivity a local orientation of the material directions must be specified (see Orientations).

Isotropic diffusivity

For isotropic diffusivity only one value of diffusivity is needed at each concentration, temperature, and field variable value.

Input File Usage

DIFFUSIVITY, TYPE=ISO

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: Type: Isotropic

Orthotropic diffusivity

For orthotropic diffusivity three values of diffusivity ( $D_{11}$ , $D_{22}$ , $D_{33}$ ) are needed at each concentration, temperature, and field variable value.

Input File Usage

DIFFUSIVITY, TYPE=ORTHO

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: Type: Orthotropic

Anisotropic diffusivity

For fully anisotropic diffusivity six values of diffusivity ( $D_{11}$ , $D_{12}$ , $D_{22}$ , $D_{13}$ , $D_{23}$ , $D_{33}$ ) are needed at each concentration, temperature, and field variable value.

Input File Usage

DIFFUSIVITY, TYPE=ANISO

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: Type: Anisotropic

Temperature-driven mass diffusion

The Soret effect factor, $κ_{s}$ , governs temperature-driven mass diffusion. It can be defined as a function of concentration, temperature, and/or field variables in the context of the constitutive equation presented above. The Soret effect factor cannot be specified in conjunction with Fick's law since it is calculated automatically in this case (see Mass diffusion analysis).

Input File Usage

Use both of the following options to specify general temperature-driven mass diffusion:

DIFFUSIVITY, LAW=GENERAL
KAPPA, TYPE=TEMP

Use the following option to specify temperature-driven diffusion governed by Fick's law:

DIFFUSIVITY, LAW=FICK

Abaqus/CAE Usage

Use the following options to specify general temperature-driven mass diffusion:

Property module: material editor: OtherMass DiffusionDiffusivity: Law: General: SuboptionsSoret Effect

Use the following option to specify temperature-driven diffusion governed by Fick's law:

Property module: material editor: OtherMass DiffusionDiffusivity: 
Law: Fick

Pressure stress-driven mass diffusion

The pressure stress factor, $κ_{p}$ , governs mass diffusion driven by the gradient of the equivalent pressure stress. It can be defined as a function of concentration, temperature, and/or field variables in the context of the constitutive equation presented above.

Input File Usage

Use both of the following options:

DIFFUSIVITY, LAW=GENERAL
KAPPA, TYPE=PRESS

Abaqus/CAE Usage

Property module: material editor: OtherMass DiffusionDiffusivity: Law: General: SuboptionsPressure Effect

Mass diffusion driven by both temperature and pressure stress

Specifying both $κ_{s}$ and $κ_{p}$ causes gradients of temperature and equivalent pressure stress to drive mass diffusion.

Input File Usage

Use all of the following options to specify general diffusion driven by gradients of temperature and pressure stress:

DIFFUSIVITY, LAW=GENERAL
KAPPA, TYPE=TEMP
KAPPA, TYPE=PRESS

Use both of the following options to specify diffusion driven by the extended form of Fick's law:

DIFFUSIVITY, LAW=FICK
KAPPA, TYPE=PRESS

Abaqus/CAE Usage

Use the following options to specify general diffusion driven by gradients of temperature and pressure stress:

Property module: material editor: OtherMass DiffusionDiffusivity: 
Law: General: SuboptionsSoret Effect and SuboptionsPressure Effect

Use the following options to specify diffusion driven by the extended form of Fick's law:

Property module: material editor: OtherMass DiffusionDiffusivity: Law: Fick: SuboptionsPressure Effect

Specifying the value of absolute zero

You can specify the value of absolute zero as a physical constant.

Input File Usage

PHYSICAL CONSTANTS, ABSOLUTE ZERO= $θ^{Z}$

Abaqus/CAE Usage

Any module: ModelEdit Attributesmodel_name: Absolute zero temperature

Elements

The mass diffusion law can be used only with the two-dimensional, three-dimensional, and axisymmetric solid elements that are included in the heat transfer/mass diffusion element library.