Diffusion Coefficient and Laws: Fick’s Laws | Metallurgy

Fick proposed laws governing the diffusion of atoms and molecules, which can be applied to the diffusion processes in metals and alloys. He proposed two laws, the first for steady-state condition and unidirectional flow of atoms and the second law which deals with time dependence of concentration gradient; and the flow of atoms in all directions.

Fick’s First Law:

Pick’s first law describes the rate at which diffusion occurs under steady-state conditions.

This states that-

Where, dm = The amount of diffusing element that migrates in time dt across a surface of cross-section A lying between two points and being normal to the axis of the bars,

D = Diffusion coefficient,

(The diffusion coefficient; D, may be defined as the amount of substance diffusing in unit time across a unit area through a unit concentration gradient and is generally expressed in cm²/s or m²/s)

A = Area of plane across which diffusion tables place,

dC / dx = The concentration gradient in the X-direction, and

dt = Duration of diffusion.

The negative sign indicates that the flow of matter occurs down the concentration gradient.

The first law entails the following assumptions:

(i) The flux of diffusing atoms is constant throughout in the given direction and is independent of time.

(ii) The jump length is constant, equal to the Burger’s vector of the structure.

(iii) There is only one jump per atom at a time which means that jump frequency is constant.

By definition, the flux J is flow per unit cross-sectional area per unit time so that, Pick’s first law can also be written as-

Under steady-state flow, the flux is independent of time and remains the same at any cross- sectional plane along the diffusion direction.

Fig. 4.6 shows that the concentration gradient varies with x. A large negative slope corresponds to a high diffusion rate. The B atoms will diffuse from the left side in accordance with Pick’s first law. The net migration of B atoms to the right side means the concentration will decrease on the left side of the solid and increase on the right side as diffusion progresses.

Fick’s law is identical in form to Fourier’s law for heat flow under a constant temperature gradient and Ohm’s law for current flow under a constant electric field gradient.

Diffusion Profiles under Steady-State of Flow:

Under steady-state of flow, the flux of atoms flowing is independent of time and remains constant at all cross-sections along the diffusion direction. The profiles are illustrated in Fig. 4.7.

If D = f (C), the profiles will be such that the product D. dC/ dx remains constant. In both the cases the profile does not change with time.

In both the cases the profile does not change with time.

Practical Example of Steady-State Diffusion:

One practical example of steady-state diffusion is found in purification of hydrogen gas. One side of a thin sheet of palladium metal is exposed to the impure gas composed of hydrogen and other gaseous species such as nitrogen, oxygen, and water vapour. The hydrogen selectively diffuses through the sheet to the opposite side, which is maintained at a constant and lower hydrogen pressure.

Fick’s Second Law-Time Dependence:

Fick’s first law permits the calculation of instantaneous mass flow rate (flux) past any plane in a solid but gives no information about the time dependence of the concentration. The time dependence is contained in Fick’s second law, which can be derived using Fick’s first law and second law of conservation of mass.

Fick’s second law states that:

For non-steady state processes, at the same cross- section, the flux is not the same at different times. Hence the concentration-distance profile (Fig. 4.8) changes with time (t).

The differential form of Fick’s second law is the basic equation for the study of isothermal diffusion.

Derivation of Fick’s Second Law:

Consider an elemental slab of thickness Ax along the diffusion distance x. Let the slab X- section be perpendicular to x and its area be unity. The volume of the slab is then ∆x x 1 = ∆x.

Let, J / x = Rate of change of flux with distance, and

C / t = Rate of change of concentration per unit volume in the elementary slab.

Under non-steady state condition, the flux entering into the slab J_x is not equal to the flux coming out of the slab, J_{(x + ∆x)}.

Hence there will be net accumulation (or depletion) of flux within the volume of the elementary slab which is given by- C / t x volume of elementary slab = C / t . ∆x

Solution to this expression (concentration is terms of both position and time) is possible when physically meaningful boundary conditions are specified. Most practical diffusion situations are non-steady ones. That is, the diffusion flux and the concentration gradient at some particular point in a solid vary with time, with a net accumulation or depletion of the diffusing species resulting.

Solution to Fick’s Second Law of Diffusion:

One practially important solution is for semi-infinite solid (A bar of solid is considered to be semi-infinite if none of the diffusing atoms reaches the bar end during the time over which the diffusion takes place. A bar of length I is considered to be semi-infinite when I > 10 √Dt), in which the surface concentration is held constant. Frequently the source of the diffusing species is a gas phase, the partial pressure of which is maintained at constant value.

Furthermore, the following assumptions are made:

(i) Prior to diffusion, any of the diffusing atoms in the solid are uniformly distributed with concentration of C₀.

(ii) At the surface the value of x is zero and it increases with distance into the solid.

(iii) At the instant before the diffusion process begins, the time in taken to be zero.

The boundary conditions are simply stated as:

For t = 0, C = C₀ at 0 ≤ x ≤ ∞

For t > 0, C = C_s (the constant surface concentration) at x = 0’

C = C₀ at x = ∞

Applying these boundary conditions to eqn. (4.6), yields the solution,

Applications of Fick’s Second Law:

The following are the applications of Fick’s second law:

1. Experimental determination of diffusion coefficient.

2. Carburising, gavalnising, nitriding processes.

3. Doping of semiconductors.

4. Study of isothermal diffusion.

Experimental Determination of Diffusion Coefficient (D):

The diffusion coefficient (D) can be determined by using a diffusion couple. A diffusion couple (Fig. 4.11) consists of two long solid bars (metal A and metal B) welded face to face. The rate of diffusion of A into B and B into A for any given value of x can be determined by Fick’s first law. The concentration-distance profiles at different lengths of diffusion time are shown in Fig. 4.11.

Time t₀ is the instant at which diffusion begins and concentration profiles show a steep change at the contact surface. A and B diffuse into each other and concentration profiles change with time as shown by curves t₁ > 0 and t₂ > t₁. The concentration of either A or B, as a function of, x and t, can be obtained from the solution of Fick’s second law as applied to the diffusion couple.

The diffusion couple provides one method of experimentally determining the diffusion coefficient.

Casehardening-Diffusion with Constant Concentration:

Casehardening is a process in which one element (usually in gaseous form) is diffused into another (a solid), the diffusing being limited to a small region near the surface. Consequently the properties of this region are changed. Usually surface regions become harder and brittle where the core remains ductile.

This process is governed by Fick’s second law. Diffusion of gas atoms into the solid takes place by one of the diffusion mechanisms, usually the interstitial mechanism. The depth to which the atoms of gas penetrate increases with time. Thus as the diffusion progresses, the depth of interstitial alloy increases.

Examples:

I. Nitrogen can be dissolved in the interstitial sites of an iron crystal and the result is Fe-N alloy is stronger, harder and more brittle than the original iron. The interstitial atoms inhibit dislocation motion and the nitrogen concentration would be highest near the surface in accordance with concentration profiles. This diffusion of nitrogen into iron is called nitriding.

II. The process of introducing carbon atoms to low carbon steels, in order to give it a hard surface is called carburising. The opposite of carburisation is “decarburisation”. Here the carbon is lost from the surface layers of steel, due to an oxidising atmosphere that reacts with carbon to produce CO or CO₂. The fatigue resistance of steels is lowered due to decarburisation and, therefore, it should be avoided by using a protective atmosphere during the heat treatment of steel.

Doping of Semiconductors:

The pure semiconductor surface is exposed to the atmosphere containing doping species which will contain unlimited source of diffusing atoms. The solution equation of Fick’s second law can be utilised for the doping process.