Diffusion in Solids

What is Diffusion?

Simply put, diffusion is the phenomenom of material transport by atomic motion. This unit discusses the atomic methods by which diffusion occurs, the maths behind it, and the influence of temperature and materials used, on the rate of diffusion. This unit will introduce the topic of diffusion, how it occurs, and some examples of its use in industry.


Introduction
If two pieces of different metal are joined together as shown here - for example, copper and nickel, and they are then heated for a long time (but below their melting points), the atoms from the metals migrate, or diffuse into the other.


Mechanisms of Diffusion
At an atomic level, atoms are arranged in a lattice pattern, e.g. as shown simply in the diagram. Diffusion is just the stepwise migration of atoms from lattice site to lattice site.

One type of diffusion involves the exchange of an atom from it's normal lattice position, to an adjacent vacant lattice site or vacancy. This is known as substitutional or vacancy diffusion. Of course, this process requires the presence of vacancies, and vacancy diffusion depends on the extent of vacancies in the material. It is represented in this animation.

The second type of diffusion involves atoms that migrate from an "interstitial" or "in - between" position, to a neighbouring one that is empty. This occurs with the infusion of impurities such as Hydrogen or Carbon, which have atoms that are small enough to fit into the interstitial positions. This process is called, as you might expect

Steady - State Diffusion

Diffusion is a time - dependent process, and often it is necessary to know how fast it occurs, or the rate of mass transfer. This rate is known as the diffusion flux, J, and is defined as the mass, M, diffusing through a unit cross - sectional area of solid, per unit of time. Therefore,

Where A is the area across which diffusion is occuring, and t is the elapsed diffusion time. If the diffusion flux does not change with time, a steady state condition exists, and this is called steady - state diffusion.

Non - Steady State Diffusion

In real life, most diffusion is non - steady state, i.e. the diffusion "flux", J, varies with time. Look back at the graph showing the concentration gradients between Nickel and Copper. It is how "harsh" this concentration gradient is, that determines this flux, which is how quickly diffusion is occuring. The concentration gradient drives diffusion: a high gradient means a high flux.

This means that the last equation we used is no longer valid. In these situations an equation known as Fick's Second Law is used:

where C is the concentration of the substance you're looking at (measured between 0 and 1). D is known as the diffusion coefficient, and is given in square metres per second.

In real life some simple boundary conditions can be applied to materials. These are that:

x is the distance from the interface you're looking at, and=0 at the surface or interface of the material.
The instant before diffusion starts, time is taken as zero, and
Before diffusion starts, all the atoms that will be diffusing are evenly distributed.

Because of these boundary conditions, Fick's Second Law can be simplified to give this simple equation:

Let's see an example of this law in use.

PROBLEM: FICK'S SECOND LAW

The diffusion coefficients for copper in aluminium at 500 and 600^oC are 4.8 x 10^-14 and 5.3 x 10^-13m²/s, respectively. What is the approximate time at 500^oC needed to produce the same diffusion result (in terms of Cu at some specific point in Al) as a 10-h heat treatment at 600^oC?