Chapter 4: The Diffusion Equation

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux.

The solution to this equation gives the time- and spatial-dependence of the concentration.

Examine cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction

4.1 Fick's Second Law

Consider a diffusion equation in the general form with diffusive flux in a system, J

dc/dt = n - grad J

There are frequently no sources or sinks, and n = 0.

Fick's second law results, and it is the consequence of the conservation of the diffusing species.

dc / dt = - grad J

dc / dt = grad ( d grad c)

Equations imply a conservation constraint for the entire concentration field

The equation of Fick's scond law contains one time and two spatial derivatives, and its solution requires three independent conditions: an initial condition and two independent boundary conditions.

Development of the diffusion equation in this chapter depends only on the form of Fick's law, J = -D grad c.

D may be a function of concentration or time.

4.1.1 Linearization of the Diffusion Equation

If Dtilda does not vary rapidly with composition, it can be replaced by successive approximations of a uniform diffusivity and results in a linearization of the diffusion equation.

A diffusion equation of constant diffusivity is found.

4.1.2 Relation of Fick's Second Law to the Heat Equation

An equation of the same form as Fick's law arises in describing the evolution of a temperature field during heat flow.

4.1.3 Variational Interpretation of the Diffusion Equation

Consider the rate of entropy production

Localized changes in c(x,t) affect the rate of total entropy production.  How changes in the evolution of a field affect a functional (such as an integral quantity like total entropy production) is a topic of calculus of variations.

The rate of the total entropy production is a functional of the concentration field c

In evolution that follows the diffusion equation, the concentration, c(x,t), changes so that the total entropy "acceleration" is most negative.  Entropy production decreases in time as rapidly as possible when dc/dt prop d2c / dx2

4.1 Constant Diffusivity

When the diffusivity is constant, Fick's law is a linear second-order partial differential equation

4.2.1 Geometric Interpretation of the Diffusion Equation when Diffusivity is Constant

When the curvature is negative, the concentration must decrease at a rate proportional to the magnitude of the curvature.  Conversely, the concentration must increae where the curvature is positive.

4.2.2 Scaling of the Diffusion Equation

Under certain conditions, boundary-value diffusion problems can be solved conveniently by scaling.  Introduce a dimensionless variable, n, into the diffusion equation.

Suppose that for the particular boundar-value problem under consideration, the initial and boundary conditions are unchanged by scale change

The concentration, c, becomes a function of the single variable, n.

Consider the one-dimensional step-function diffusion problem

Scaling as Means to Compare Similar Systems  Equal values of n can be used to determine relationships between length, time, and the value of their diffusivity

4.2.3 Superposition

The superposition of two solutions also solves the diffusion equation with superposed boundary and initial conditions.

4.3 Diffusivity as a Function of Concentration

When the diffusivity is a function of concentration, the differential equation becomes nonlinear and solutions can be obtained analytically only in certain special cases.

When Fick's law applies, the concentration profile generally contains information about the concentration dependence of diffusivity.

When diffusivity is a function of local concentration, the concentration profile tends to be relatively flat at a concentration where D(c) is large and relatively steep where D(c) is small.

Asymmetry of the diffusion profile in a diffusion couple is an indicator of a concentration-dependent diffusivity

The boundary condition determines the position of the original interface

The integral equation can be used to determine Dtilda(c1) by a graphical construction or numerical solution

It's preferable to obtain diffusion profiles of various assumed diffusivities as a function of concentration by computation.

Dtilda(c) could be deduced by fitting calculated results for a parametric representation of Dtilda(c) to an experimentally determined diffusion profile.

4.4 Diffusivity as a Function of Time

If the boundary conditions for a time-dependent diffusivity problem are invariant under this change of variable, solutions from known constant-D problems can be applied to the time-dependent D case.

4.5 Diffusivity as a Function of Direction

In the expression for Fourier's law of heat conductivity and Fick's law for mass flux, it has been assumed that the flux vector is always parallel to the driving force vector.  However, these vectors are not parallel in general materials.

It is possible to generalize the isotropic relations between driving forces and fluxes to account for anisotropy.

In the anisotropic case, there is a linear relation between flux and gradient vectors.

D is called the diffusivity tensor and acts as an object that connects one vector to another.

Because Dij is symmetric, it is always possible to find a linear coordinate transformation that will make the Dij diagonal with real components.

The diagonal elements of Dhat are the eigenvalues of Dhat, and the coordinate system of Dhat defines the principal axes x1, x2, x3.

If R is the matrix that rotates the original coordinate system into the principle coordinate system, Dhat, RDR-1

Cartesian space can be stretched or contracted along the principal axes by scaling.

Known solutions to the diffusion equation for constant isotropic diffusivity can be used to find solutions for anisotropic constant diffusivities by a simple algorithm.

The diffusivity tensor has special forms for particular choices of coordinate axes if the diffusing body itself has special symmetry.

Neumann's principle states: