The capability to perform a coupled electrical-diffusion analysis exists in the following elements:

These elements support electromigration (mass transport of charged particles due to an electric field) when used in coupled-field analyses having electrical and diffusion DOFs.

Constitutive Equations

In a coupled electric-diffusion analysis, the diffusion flux density {J} is coupled to the electric potential as follows:

(10–81)

where:

= diffusivity matrix

Dxx, Dyy, and Dzz = diffusivity coefficients in the element's X, Y, and Z directions, respectively (input as DXX, DYY, DZZ on MP command)

C = concentration;

= normalized concentration (input/output as CONC)

= saturated concentration (input as MP,CSAT)

V = electric potential (input/output as VOLT)

T = absolute temperature = Tc + Toff

Tc = current temperature (input/output as TEMP on D or BF commands)

Toff = offset temperature from absolute zero to zero (input on TOFFST command)

Ze/k = charge constant (input as C4 on TBDATA command with TB,MIGR)

Z = charge number

e = elementary charge

k = Boltzmann constant

For more information on Equation 10–81 and related material constant input, see Migration Model in the Material Reference.

For more information on diffusion analysis, see Diffusion.

Derivation of Electric-Diffusion Matrices

Applying the variational principle with respect to electrical and diffusion DOFs to the electric charge (Equation 5–5) and mass (Equation 9–6) continuity equations coupled by the constitutive equation (Equation 10–81), we obtain the following finite element matrix equation for the electric-diffusion analysis:

(10–82)

where:

{V} = nodal electric scalar potential (input/output as VOLT)

{C} = nodal concentration (input/output as CONC)

[Cv] = element electric damping matrix (defined by Equation 5–115)

[Cd] = element diffusion damping matrix (defined by Equation 9–9)

[Kv] = element electric conductivity matrix (defined by Equation 5–115)

[Kd] = element diffusion conductivity matrix (defined by Equation 9–9)

= element transport conductivity matrix

= nonlinear part of the element diffusion conductivity matrix

{E} = electric field (output as EF)

{I} = nodal current vector (input/output as AMPS)

{R} = nodal diffusion flow rate vector (input/output as RATE)

{N} = element shape functions

The finite element equation Equation 10–82 is unsymmetric. If a symmetric equation is desirable, the electromigration coupling can be applied as a load vector by setting KEYOPT(2) = 1 for the coupled-field elements (PLANE223, SOLID226, SOLID227).