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. |  |
 | Concentrations of both metals vary with position, as shown in the graph. This process is known as interdiffusion, and is a time - dependent process. Click below to see this |
 | 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, interstitial diffusion. |  |
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:
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x is the distance from the interface you're looking at, and=0 at the surface or interface of the material.
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The instant before diffusion starts, time is taken as zero, and
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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 600oC are 4.8 x 10-14 and 5.3 x 10-13m2/s, respectively. What is the approximate time at 500oC needed to produce the same diffusion result (in terms of Cu at some specific point in Al) as a 10-h heat treatment at 600oC?
Or,
| t500= | (Dt)600 | = (5.3 x 10-13 m2/s) (10 hours) | = 110.4 hours |
| D500 | 4.8 x 10-14 m2/s |
If you had a problem with following any of this, talk to your supervisor.
Diffusing Species
The magnitude of the diffusion coeffecient, D, is a measure of the rate at which atoms diffuse. The diffusing species, as well as the host material, influences the diffusion coefficient. For example, if a diffusing species has smaller atoms, it will interstitially diffuse through a host material more easily. Also substitutional diffusion is made easier if the host material has lots of vacancies to start with.
Temperature
Temperature has the most profound influence on the coefficients and diffusion rates. For example, for the self - diffusion of Fe in alpha Fe, the diffusion coefficient increases about five times, after the temperature is raised from 500 to 900oC. This is because diffusion is a thermally activated process - i.e. there is an energy barrier (an activation energy) that has to be overcome, in order for the atoms to move from one lattice site to the other.
The more thermal energy there is around, the easier it is for this energy barrier to be overcome. Therefore, there is an exponential relationship between Temperature (T), and Diffusion Coefficient (D), and this is:
| D = A exp-Q/RT |
Where Q=activation energy, T=temperature in Kelvin, R=Universal Gas Constant, and A is a constant. Taking natural logarithms of this equation,
| In D = In A -Q/RT |
And since A, Q and R are all constants, this expression is similar to the equation of a straight line:
y = mx + c
So x and y here are equivalent to ln D and 1/T. So, if ln D is plotted against 1/T, the result will be a straight line. Its gradient will be -Q/R, and its y-intercept will be ln A. This is how the values of A and Q are found experimentally, and an example is shown below.
So here is how an example graph would look - this one is for C in alpha-Fe.
Applications
Case Hardening
The first common application of diffusion is case hardening. For some applications, it is necessary to harden the surface of a steel above that of its interior. One way of doing this is by increasing the concentration of carbon in the surface in a process called carburizing. In this process, a piece of steel is exposed, at an elevated temperature, to an atmosphere rich in a hydrocarbon gas, such as methane (CH4).
Now try this example question:
A piece of iron is placed in a carburizing atmosphere on one side and a decarburizing atmosphere on the other side at 700oC. Assuming a steady-state condition, calculate the diffusion flux of carbon through the plate, if the concentrations of carbon at positions of 5 and 10mm beneath the carburizing surface are 1.2 and 0.8kg/m3 respectively. Assume a diffusion coefficient of 3 x 10-11m2/s at this temperature.
If you had any problems with this, talk to your supervisor.
Gas Purification
Another application of diffusion is in the purification of hydrogen gas: 
As shown above, a thin sheet of palladium metal separates the two gases. On one side is a gas composed of hydrogen plus (for example) nitrogen, oxygen, or water vapour. From this, the hydrogen selectively diffuses through the sheet to the other side, which has been kept at a constant, lower hydrogen pressure.
Agustin Egui
Caf
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