Chapter 5. Diffusion

### 5.1 Introduction

Many important reactions and processes in materials occur by the motion of atoms in the solid (transport), which happens by diffusion.

Inhomogeneous materials can become homogeneous by diffusion, if the temperature is high enough (temperature is needed to overcome energy barriers to atomic motion.

### 5.2 Diffusion Mechanisms

Atom diffusion can occur by the motion of vacancies (vacancy diffusion) or impurities (impurity diffusion). The energy barrier is that due to nearby atoms which need to move to let the atoms go by. This is more easily achieved when the atoms vibrate strongly, that is, at high temperatures.

There is a difference between diffusion and net diffusion. In a homogeneous material, atoms also diffuse but this motion is hard to detect. This is because atoms move randomly and there will be an equal number of atoms moving in one direction than in another. In inhomogeneous materials, the effect of diffusion is readily seen by a change in concentration with time. In this case there is a net diffusion. Net diffusion occurs because, although all atoms are moving randomly, there are more atoms moving in regions where their concentration is higher.

### 5.3 Steady-State Diffusion

The flux of diffusing atoms, J, is expressed either in number of atoms per unit area and per unit time (e.g., atoms/m2-second) or in terms of mass flux (e.g., kg/m2-second).

Steady state diffusion means that J does not depend on time. In this case, Fick’s first law holds that the flux along direction x is:

J = – D dC/dx

Where dC/dx is the gradient of the concentration C, and D is the diffusion constant. The concentration gradient is often called the driving force in diffusion (but it is not a force in the mechanistic sense). The minus sign in the equation means that diffusion is down the concentration gradient.

### 5.4 Nonsteady-State Diffusion

This is the case when the diffusion flux depends on time, which means that a type of atoms accumulates in a region or that it is depleted from a region (which may cause them to accumulate in another region).

### 5.5 Factors That Influence Diffusion

As stated above, there is a barrier to diffusion created by neighboring atoms that need to move to let the diffusing atom pass. Thus, atomic vibrations created by temperature assist diffusion. Also, smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions. Similar to the case of vacancy formation, the effect of temperature in diffusion is given by a Boltzmann factor: D = D0 × exp(–Qd/kT).

### 5.6 Other Diffusion Paths

Diffusion occurs more easily along surfaces, and voids in the material (short circuits like dislocations and grain boundaries) because less atoms need to move to let the diffusing atom pass. Short circuits are often unimportant because they constitute a negligible part of the total area of the material normal to the diffusion flux. .

### Terms:

• Activation energy
• Diffusion
• Diffusion coefficient
• Diffusion flux
• Driving force
• Fick’s first and second laws
• Interdiffusion
• Interstitial diffusion
• Self-diffusion
• Vacancy diffusion
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