Chapter 5: Solutions to the Diffusion Equation

Address methods to solve the diffusion equation for a variety of initial and boundary conditions when D is constant and therefore of the simple form below

dc / dt = D grad^2 c

This equation is a second-rder linear partial-differential equation.

In a large class of initial and boundary conditions, there are theorems of and existence of solutions and theorems of maximum and minimum values

Solutions of many boundary-value problems can be adopted as solutions to corresponding diffusion problems.

It can be difficult to find a closed-form solution in problems with highly specific and complicated boundary conditions.  Numerical methods can be employed.

 The spreading Gaussian distribution describes the diffusion out into an infinite medium of from various instantaneouw localized sources

When the initial conditions can be represented by a distribution of sources, one simply superposes the solutions of individual sources by integration

Solutions can be obtained by the separation-of-variables method

Laplace transforms can be used to derive many results.

The difficult part of using the Laplace transform is back-transforming to the desired solution, which usually involves integration on the complex domain.

5.1 Steady-State Solutions

A simple case occurs when the diffusion is in steady state and the composition profile is not a function of time.

Steady-state conditions are often achieved for constant boundary conditions in finite samples at very long times.

The diffusion equation reduces to the Laplace equation.

grad^2 c = 0

Solutions to the Laplace equation are called harmonic functions

5.1.1 One Dimension

Consider diffusion through an infinite plate of thickness L with constant concentration at boundaries

The concentration varies linearly across the plate.  The flux is constant and proportional to the slope.

5.1.2 Cylindrical Shell

Because boundary conditions are independent of theta and z, the solution is independent of these variables

The total current of particles entering the inner surface per unit length of cylinder is the same as the total current leaving the outer surface, which is a requirement of steady state.

5.1.3 Spherical Shell

Consider the Laplacian operator operating on c(r, theta, phi) 

The steady-state solution for diffusion through spherical shells with boundary conditions dependent only on r may be obtained by integrating twice and determining the two constants of integration by fitting the solution to the boundary conditions

5.1.4 Variable Diffusivity

When steady-state conditions prevail and D varies with position, the diffusion equation can readily be integrated.

5.2 Non-Steady-State (Time-Dependent) Diffusion

When the diffusion profile is time-dependent, solutions require considerably more effort and familiarity with applied mathematical methods for solving partial-differential equations.

Build up solutions to more complicated situations by means of superposition

5.2.1 Instantaneous Localized Sources in Infinite Media

Solutions of two- and three-dimensional diffusion can easily be obtained by using products of the one-dimensional solution

The general form of the solution in the infinite domain is valid in the semi-infinite domain with an initial thin source of diffusant at x = 0.

In the semi-infinite case, the initial concentration diffuses into one side rather than two and the concentration is larger by a factor of two.

5.2.2 Solutions Involving the Error Function

The instantaneous local-source solutions can be used to build up solutions for general initial distributions of diffusant by using the method of superposition.

The same solution can be obtained by superposing the one-dimensional diffusion from a distribution of instantaneous local sources arrayed to simulate the initial step function

Summations over point-, line-, or planar-source solutions are useful examples of the general method of Green's functions

The solution of two-dimensional diffusion from a line source lying along z in three dimensions can be obtained by integrating over a distribution of point sources lying along the z-axis.

5.2.3 Method of Superposition

The method of superposition can be applied to finite systems

5.2.4 Method of Separation of Variables: Diffusion on a Finite Domain

A standard method to solve many partial-differential equations is to assume that the solution can be written as a product of functions, each function of one of the independent variables.

Because the left side depends on t and the right side depends only on x, each side must be equal to the same constant.

A variable must take on values appropriate to the boundary conditions

Find the linear differential equation's eigenvalues of the boundary conditions and the eigenfunctions of the boundary conditions.

The general solution is a sum of the products of the eigenfunction solutions

The concentration decays exponentially with characteristic time

The method of separation of variables can be applied in the same manner to other initial distributions of diffusant.  The effort lies in determining the Fourier coefficients, which can often be looked up in tables.

Cylindrical Coordinates The separation-of-variables method also applies when the boundary conditions and initial conditions demonstrate cylindrical symmetry

5.2.5 Method of Laplace Transforms

The Laplace transform method is a powerful technique to solve a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain.  After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved.  The transformed solution is then back-solved to obtain the desired solution.

The key utility of the Laplace transform involves its operation on time derivatives.

The Laplace transform of a spatial derivative of f is seen to be equal to the spatial derivative of f

The Laplace transform removes the t-dependence and turns the partial-differential equation into an ordinary-differential equation.

Boundary conditions must be transformed

erfc(z) is known as the complementary error function

This solution could have been deduced directly from a previous equation since the plane x = 0 always maintains a constant composition

Example with Time-Dependent Boundary Conditions

Consider the case where a constant flux is imposed on the surface of a semi-infinite sample

The boundary condition might apply for solute absorption with its rate moderated by some thin passive surface layer

5.2.6 Estimating the Diffusion Depth and Time to Approach Steady State

A rough estimate of the diffusion penetration distance from a point source is the location where the concentration has fallen off by about 1/e of the concentration at x=0

An estimate of the penetration distance of the error-function solution is the distance where c(x,t) = c0 / 8

A reasonable estimate of the penetration depth is s sqrt(Dt)

To estimate the time at which steady-state conditions are expected, the required penetration distance is set equal to the largest characteristic length over which diffusion can take place in the system.