**Electromigration**

**Anisotropy and Onsager Coefficients**

**The Diffusion Equation**

**The Diffusion Equation for Uniform Diffusivity**

**The Steady-State Condition**

__Scaling and the Diffusion Equation__

Note that the diffusivity has units (number cm)/s; therefore, is dimensionless.

Introduce a dimensionless variable :

(07-1) |

*Suppose for a particular problem*, that the boundary and
initial conditions are also invariant to this scaling
implied by , in other words, if the length scale is tripled (yards
instead of feet) then the time scale for the boundary conditions is
multiplied by a factor of nine.
Or, under the scaling,

then: | (07-2) |

the boundary conditions can be written entirely in terms of the one dimensionless variable .

An example of such a problem would be the ``step-function'' initial conditions with either Dirichlet or zero-flux Neumann boundary conditions at :

Initial conditions:

or

(07-4) |

with boundary conditions:

or | (07-5) |

and

or | (07-6) |

Using the definition for :

(07-7) |

The diffusion equation becomes:

(07-8) |

Because everything in the above equation depends only on , the partial differential equation becomes an ordinary differential equation ( ) that can be integrated without too much difficulty.

Let

(07-9) |

so that

(07-10) |

which can be integrated

(07-11) |

where is an integration constant and this can be integrated again

(07-12) |

Let and with the step-function initial conditions, ,

(07-13) |

or, by introducing another constant for ,

(07-14) |

The integral with an argument is just some function with a peculiar name:

(07-15) |

that has the properties: , (this is where the comes from),

So, the solution to the one-dimensional step function IC uniform diffusion equation is:

(07-16) |

or

(07-17) |

__Short Distances, Long Times__

An additional and useful property of the error function is that for

Because the scaling parameter
is small for
long times
where is the characteristic
diffusion time:^{1}

(07-18) |

or for

(07-19) |

the solution of the step-function IC becomes:

(07-20) |

The flux at the origin varies as,

(07-21) |

The total amount of material that flows past the origin (per unit area) up to a time goes as :

(07-22) |

__Superposition Example__

The simple solution that was developed above can be used with the method of superposition to develop a large number of solutions.

For example, suppose the initial conditions are changed so that
there is a finite source of material diffusing into an infinite
material and the finite source is confined to slab of with
in the -direction but extending infinitely in
the - and -directions.^{2}

These initial conditions can be obtained by summing two known solutions, i.e. those for the initial conditions:

(07-23) |

and

(07-24) |

The solution is just the sum:

(07-25) |

It will be beneficial to reflect on what kinds of problems (i.e. which initial and boundary conditions) will admit solutions from scaling and then subsequent summing of solutions. First for the scaling solution, the boundary and initial conditions had to be invariant under the scale factor --this is usually the case for infinite domains with zero flux conditions at . Second, the summing method works when each solution that is to be summed also satisfies scaling--therefore the initial conditions for the particular problem must also be invariant. Typically, these methods are useful in the infinite domain.

__The One-Dimensional Fundamental Solution__

The solution for ``point-source'' initial conditions will prove to
be quite useful for cases when the addition of a large number of
solutions generate the initial conditions for a particular
problem.
The solution obtained in Eq. 7-26 can be used
to obtain the fundamental solution.
Expanding in powers of small
:^{3}

(07-27) |

This is the fundamental solution for a point source in a 1D infinite domain.

Many other solutions by summing point source solutions,

In general,^{4}for almost any initial conditions
on the 1D infinite domain with zero
flux at
,

(07-28) |

Otherwise known as the the Green's function solution.

__Other Fundamental Solutions__

The same method can be used to find fundamental solutions for cases where the finite source is a object of a given dimensionality embedded in an infinite space of larger dimension. This is called the ``co-dimension'' and it is the dimensionality of the infinite region minus the dimensionality of the embedded object.

All the pertinent examples are contained in the following table:

Co-dimension | Example | Symmetric Part of | Fundamental Solution |

Point on Line | |||

1 | Line on Plane | ||

Plane in 3D | |||

2 | Point in Plane | ||

Line in 3D | |||

3 | Point in 3D |