Diffusion in solids

Ahmed Samy

4 Nov, 2023

Diffusion can be defined as the mechanism by which one matter is transported through another matter. Atoms in gases, liquids, and solids are in constant motion and migrate over a period of time. In gases, atomic movement is relatively rapid, as indicated by the rapid movement of cooking odors or smoke particles. Atomic movements in liquids are in general slower than in gases, as evidenced by the movement of colored dye in liquid water.

Can diffusion occur in solids?

The atomic movements are restricted due to bonding to equilibrium positions so, the diffusion in solids is very slow. However, thermal vibrations occurring in solids to allow some atoms to move. Diffusion of atoms in metals and alloys is particularly important since most solid-state reactions involve atomic movements. Examples of solid-state reactions are the precipitation of a second phase from solid solution and the nucleation and growth of new grains in the recrystallization of a cold-worked metal.

What are the types of diffusion?

The types of diffusion are self diffusion, impurity diffusion, grain boundary diffusion, surface diffusion, volume diffusion.

1. Self-Diffusion

Self-diffusion occurs when atoms of the same element within a pure solid material migrate through the lattice, a particular atom does not remain at one equilibrium site indefinitely. It is a fundamental process in materials and is often studied to understand diffusion mechanisms and diffusion coefficients.

2. Interstitial Diffusion

Interstitial diffusion, also known as solute diffusion or impurity diffusion, involves the movement of foreign atoms (impurities) within a solid material. It occurs in binary system such as Cu- Ni system. This type of diffusion is essential in processes like alloying, where small amounts of different elements are intentionally added to alter the material's properties.

3. Grain Boundary Diffusion

Grain boundary diffusion occurs at the boundaries between individual crystalline grains in a polycrystalline material. This type of diffusion can significantly affect material properties and is particularly relevant in ceramics and other polycrystalline materials.

4. Surface Diffusion

Surface diffusion refers to the movement of atoms or molecules along the surface of a solid material. It plays a crucial role in processes like thin film growth, surface reactions, and surface diffusion-controlled phenomena.

5. Volume Diffusion

Volume diffusion encompasses diffusion that occurs within the bulk of the material, away from surfaces or interfaces.

Note that:

The activation energy for grain boundary diffusion is lower than that of volume diffusion.

Diffusion Mechanisms

There are two main mechanisms of diffusion of atoms in a crystalline lattice: vacancy or substitutional mechanism and interstitial mechanism.

Vacancy Diffusion Mechanism

In vacancy diffusion atoms can move in crystal lattices from one atomic site to another if there is enough activation energy provided by the thermal vibration of the atoms and if there are vacancies or other crystal defects in the lattice for atoms to move into.

Vacancies in metals and alloys are equilibrium defects, and therefore some are always present to enable vacancy diffusion of atoms to take place.

Note that:

As the temperature of the metal increases, more vacancies are present and more thermal energy is available, and so the diffusion rate is higher at higher temperatures.

Vacancies are continually being created and destroyed at surface, grain boundaries and dislocations.

The vacancy mechanism of diffusion in substitution solid solution is dominant mechanism in FCC metals and alloys and has been shown to be operative in both BCC and HCP metals .

Consider the example of vacancy diffusion shown in Figure (1) on a (111) plane of copper atoms in a copper crystal lattice. If an atom next to the vacancy has sufficient activation energy, it can move into the vacant site and thereby contribute to the self-diffusion of copper atoms in the lattice.

During self-diffusion or substitutional solid-state diffusion, atoms must break the original bonds among atoms and replace these with new bonds. This process is assisted by having vacancies present, and thus it can occur at lower activation energies.

Note that:

In general as the melting point of the metal is increased, the activation energy is also increase. This relationship exists because the higher melting temperature metals tend to have stronger bonding energies between their atoms.

The activation energy for self-diffusion is equal to the sum of the activation energy to form a vacancy and the activation energy to move the vacancy.

The activation energy of self-diffusion process is constant for each pure metal.

In order for this process to occur in alloys, there must exist solid solubility of one type of atom in another. Thus, this process is dependent on the rules of solid solubility, which are listed in crystalline imperfections article. Because of these differences in chemical bonding and solid solubility and other factors, substitutional diffusion data must be obtained experimentally. With time, these measurements are made more precisely, and hence these data may change with time.

Figure (1)

One of the major breakthroughs in diffusion measurements occurred in the 1940s when the Kirkendall effect was discovered. This effect showed that the markers at the diffusion interface moved slightly in the opposite direction to the most rapidly moving (faster diffusing) species of a binary diffusion couple figure (2). After much discussion, it was concluded that the presence of vacancies allowed this phenomenon to occur.

Figure (2)

Experiment to illustrate the Kirkendall effect. (a) At start of diffusion experiment (t = 0). (b) After time t, markers move in the direction opposite the most rapidly diffusing species, B.

Note that:

Diffusion can also occur by the vacancy mechanism in solid solutions.

Atomic size differences and bonding energy differences between the atoms are factors that affect the diffusion rate.

Interstitial Diffusion Mechanism

The interstitial diffusion of atoms in crystal lattices takes place when atoms move from one interstitial site to another neighboring interstitial site without permanently displacing any of the atoms in the matrix crystal lattice.

The larger atoms occupy the lattice sites while the small ones fit into the voids created by the large one. The voids are named interstices.

For the interstitial mechanism to be operative, the size of the diffusing atoms must be relatively small compared to the matrix atoms. Small atoms such as hydrogen, oxygen, nitrogen, and carbon can diffuse interstitially in some metallic crystal lattices. For example, carbon can diffuse interstitially in BCC α iron and FCC γ iron. In the interstitial diffusion of carbon in iron, the carbon atoms must squeeze between the iron matrix atoms.

Note that:

The presence of small atoms greatly affect the mechanical properties of metals (O2 , N 2 ,and H2 can be diffused in metals easily at low temperature).

Figure (3)

Interstitial diffusion VS Vacancy diffusion

Interstitial diffusion involves the movement of species through interstitial sites, while vacancy diffusion involves the movement of species by jumping between vacant lattice sites. Both mechanisms are significant in the field of materials science and engineering.

The choice of which diffusion mechanism dominates in a particular material depends on factors such as crystal structure, temperature, the size of the diffusing species, and the presence of defects in the lattice. Both interstitial and vacancy diffusion play essential roles in various materials processing and properties, including heat treatment, phase transformations, and creep in metals and ceramics.

Steady-State Diffusion

Steady-state diffusion refers to the process by which particles, such as atoms, molecules, or ions, move through a medium or material under constant conditions, leading to a stable and unchanging concentration gradient. This type of diffusion occurs when there is a continuous flow of particles from an area of higher concentration to an area of lower concentration, but the rate of diffusion remains constant over time.

Note that:

The steady diffusion do not depend on time. (No change in system with time).

Consider the diffusion of solute atoms in the x direction between two parallel atomic planes perpendicular to the paper separated by a distance x as shown in figure (4). We will assume that over a period of time the concentration of atoms at plane 1 is C1 and that of plane 2 is C2. That is, there is no change in the concentration of solute atoms at these planes for the system with time. Such diffusion conditions are said to be steady-state conditions.

This type of diffusion takes place when a nonreacting gas diffuses through a metal foil. For example, steady-state diffusion conditions are attained when hydrogen gas diffuses through a foil of palladium if the hydrogen gas is at high pressure on one side and low pressure on the other.

Figure (4)

If in the diffusion system shown in Figure (4) no chemical interaction occurs between the solute and solvent atoms, because there is a concentration difference between planes 1 and 2, there will be a net flow of atoms from the higher concentration to the lower concentration. The flux or flow of atoms in this type of system can be represented by the Fick’s first law of diffusion:

Where:

J = flux or net flow of atoms.
D = proportionality constant called the diffusivity (atomic conductivity) or diffusion coefficient.
dC/dx = concentration gradient.

Note that:

A negative sign is used because the diffusion is from a higher to a lower concentration; that is, there is a negative diffusion gradient.

The net flow of atoms by atomic diffusion is equal to the diffusivity D times the diffusion gradient dC/dx. The SI units for this equation are:

The diffusivity depends on many variables, of which the following are important:

1. The type of diffusion mechanism

Whether the diffusion is interstitial or substitutional will affect the diffusivity. Small atoms can diffuse interstitially in the crystal lattice of larger solvent atoms. For example, carbon diffuses interstitially in the BCC or FCC iron lattices. Copper atoms diffuse substitutionally in an aluminum solvent lattice since both the copper and the aluminum atoms are about the same size.

2. The temperature

At which the diffusion takes place greatly affects the value of the diffusivity. As the temperature is increased, the diffusivity also increases for all the systems.

3. The type of crystal structure of the solvent lattice

For example, the diffusivity of carbon in BCC iron is 10^-12 m^2/s at 500°C, which is much greater than 5 × 10^-15 m^2/s, the value for the diffusivity of carbon in FCC iron at the same temperature. The reason for this difference is that the BCC crystal structure has a lower atomic packing factor of 0.68 as compared to that of the FCC crystal structure, which is 0.74. Also, the interatomic spaces between the iron atoms are wider in the BCC crystal structure than in the FCC one, and so the carbon atoms can diffuse between the iron atoms in the BCC structure more easily than in the FCC one.

4. The type of crystal imperfections present in the region of solid-state diffusion

More open structures allow for more rapid diffusion of atoms. For example, diffusion takes place more rapidly along grain boundaries than in the grain matrix in metals and ceramics. Excess vacancies will increase diffusion rates in metals and alloys.

5. The concentration of the diffusing species

In that higher concentrations of diffusing solute atoms will affect the diffusivity. This aspect of solid-state diffusion is very complex.

Non Steady-State Diffusion

Non steady-state diffusion, also known as unsteady-state diffusion or transient diffusion, refers to the process of mass transfer where the concentration of a substance varies with time in a system. Unlike steady-state diffusion, where the concentration remains constant with time after an initial period, non-steady-state diffusion describes the changing concentration profile as a substance diffuses through a medium over time.

Note that:

Is not commonly encountered with engineering materials.
The non steady-state diffusion depends on time. (There is a change in system with time)

In most cases, non steady-state diffusion in which the concentration of solute atoms at any point in the material changes with time takes place. For example, if carbon is being diffused into the surface of a steel camshaft to harden its surface, the concentration of carbon under the surface at any point will change with time as the diffusion process progresses. For cases of non-steady-state diffusion in which the diffusivity is independent of time, Fick’s second law of diffusion applies, which is :

This law states that the rate of compositional change is equal to the diffusivity times the rate of change of the concentration gradient. However, the particular solution to this equation in which a gas is diffusing into a solid is of great importance for some engineering diffusion processes and will be used to solve some practical industrial diffusion problems.

Let us consider the case of a gas A diffusing into a solid B, as illustrated in Figure(5a). As the time of diffusion increases, the concentration of solute atoms at any point in the x direction will also increase, as indicated for times t1 and t2 in Figure(5b). If the diffusivity of gas A in solid B is independent of position, then the solution to Fick’s second law is

Where:

Cs = surface concentration of element in gas diffusing into the surface.
C0 = initial uniform concentration of element in solid.
Cx = concentration of element at distance x from surface at time t.
x = distance from surface.
D = diffusivity of diffusing solute element.
t = time.
erf : is a mathematical function called error function.

Figure (5)

Note that:

The error function, erf, is a mathematical function existing by agreed definition and is used in some solutions of Fick’s second law. The error function can be found in standard tables in the same way as sines and cosines. Table below is an abbreviated table of the error function.

Application of diffusion in industry

Diffusion plays a vital role in various industrial processes, enabling the controlled movement of substances within materials. It has a wide range of applications in industries include:

Heat treatment of metals.
Powder metallurgy.
Welding, brazing, soldering, and galvanizing.
Oxidation of metals.
Doping of semiconductors.
Recovery and recrystallization.
Surface treatment of steel (case hardening).
Alloying.

Heat Treatment of Metals

Diffusion is fundamental in heat treatment processes like annealing, where the controlled heating and cooling of metals enhance their properties. This involves the migration of atoms within the crystal lattice, leading to the desired microstructure and improved mechanical properties.

Powder Metallurgy

In the production of powdered metal components, diffusion facilitates the sintering process. Powder particles diffuse at elevated temperatures, allowing them to bond and form a cohesive structure. This results in the creation of dense and durable metal parts.

Welding, Brazing, Soldering, and Galvanizing

These processes involve the joining of materials, and diffusion is essential for ensuring a strong bond between the joined materials. In welding, for example, diffusion helps in the establishment of a metallurgical bond between the welded surfaces.

Oxidation of Metals

Oxidation involves the reaction of metals with oxygen, and diffusion is critical for the movement of oxygen atoms into the metal lattice. This process is vital in the formation of oxide layers on the metal surface, providing protection against further corrosion.

Doping of Semiconductors

In the semiconductor industry, controlled diffusion is used to introduce specific impurities (dopants) into semiconductor materials. This process modifies the electrical properties of the semiconductor, enabling the creation of transistors, diodes, and other electronic components.

Recovery and Recrystallization

Diffusion is involved in the recovery and recrystallization processes of metals after deformation. During annealing, atoms diffuse to remove defects and rearrange, resulting in a more ordered and refined microstructure.

Surface Treatment of Steel (Case Hardening)

Case hardening involves diffusing carbon or nitrogen into the surface of steel to enhance its hardness and wear resistance. This is achieved through controlled heat treatment where carbon or nitrogen atoms diffuse into the steel lattice.

Alloying

Diffusion is central to the alloying process, where different metals are combined to create alloys with specific properties. The diffusion of atoms between the alloying elements results in a homogeneous mixture, influencing the alloy's mechanical, thermal, and chemical characteristics.

Examples on industrial applications of diffusion

In this article we will focus on 2 examples on industrial applications of diffusion process:

Case hardening of steel by gas carburizing.
The impurity doping of silicon wafers for integrated electronic circuits.

1. Case Hardening of Steel by Gas Carburizing

Many rotating or sliding steel parts such as gears and shafts must have a hard outside case for wear resistance and a tough inner core for fracture resistance. In the manufacture of a carburized steel part, usually the part is machined first in the soft condition, and then, after machining, the outer layer is hardened by some case-hardening treatment such as gas carburizing. Carburized steels are low-carbon steels that have about 0.10% to 0.25% C. However, the alloy content of the carburized steels can vary considerably, depending on the application for which the steel will be used. Some typical gas-carburized parts are shown in Figure (6) .

Figure (6)

In the first part of the gas-carburizing process, the steel parts are placed in a furnace in contact with gases containing methane or other hydrocarbon gases at about 927°C (1700°F). The carbon from the atmosphere diffuses into the surface of the component so that after subsequent heat treatments, the gears are left with high-carbon hard cases as indicated, for example, by the darkened surface areas of the macrosection of the steel components shown in Figure (7). In Figure (7a), the thickness of the hardened surface (depth of penetration appears as a darker shade) in a sprocket is represented by the discolorations around the teeth. In Figure (7b), the cross section of a single gear tooth showing the depth of penetration of carbon and the hardened layer.

Figure (7)

(a) Cross section of a steel sprocket showing showing hardened layers around the teeth. (b) A single tooth of a

case hardened gear showing the unaffected material inside and the hardened surface layer (the dark spots are

produced as a result of hardness measurements).

Figure (8) shows some typical carbon gradients in test bars of AISI 1022 (0.22% C) plain-carbon steel carburized at 1685°F (918°C) by a carburizing atmosphere with 20% CO. Notice how the carburizing time greatly affects the carbon content versus distance-below-the-surface profile.

Figure (8)

2. Impurity Diffusion into Silicon Wafers for Integrated Circuits

Impurity diffusion into silicon wafers to change their electrical conducting characteristics is an important phase in the production of modern integrated electronic circuits. In one method of impurity diffusion into silicon wafers, the silicon surface is exposed to the vapor of an appropriate impurity at a temperature above about 1100°C in a quartz tube furnace, as shown schematically in The part of the silicon surface not to be exposed to the impurity diffusion must be masked off so that the impurities diffuse into the parts selected by the design engineer for conductivity change.

As in the case of the gas carburizing of a steel surface, the concentration of impurities diffused into the silicon surface decreases as the depth of penetration increases. Changing the time of diffusion will also change the concentration of impurities versus depth-of-penetration profile.

Note that:

Typical diffusion depths in silicon wafers are of the order of a few micrometers (i.e., about 10−6 m), while the wafer is usually several hundred micrometers thick.

Effect of Temperature on Diffusion in Solids

Since atomic diffusion involves atomic movements, it is to be expected that increasing the temperature of a diffusion system will increase the diffusion rate. By experiment, it has been found that the temperature dependence of the diffusion rate of many diffusion systems can be expressed by the following Arrhenius-type equation: