Fick’s Law of Diffusion

Fick’s law of diffusion tells that the  diffusion processes movement of molecules from higher concentration to lower concentration region. A diffusion process that obeys Fick’s laws is called normal diffusion or Fickian diffusion. A diffusion process that does not obey Fick’s laws is known as Anomalous diffusion or non-Fickian diffusion. 

There are two laws are semiconductors i.e. Fick’s first law is used to derive Fick’s second law which is similar to the diffusion equation. According to Fick’s law of diffusion, “The molar flux due to diffusion is proportional to the concentration gradient”. The rate of change of concentration of the solution at a point in space is proportional to the second derivative of concentration with space.

Fick’s First Law

Movement of solute from higher concentration to lower concentration across a concentration gradient.

\[ J=−D \frac{d\phi} {dx} \]

Where,

J: diffusion flux

D: diffusivity

\[\phi\] : concentration

x: position

Fick's Second Law of Diffusion

The second diffusion law of Fick is a linear equation with the dependent variable being the concentration of the chemical species under consideration. The spread of each chemical species occurs independently. These properties make it simple to simulate numerically the mass transport systems described in Fick 's second law.

When modeling diffusion, it is often a good idea to start with the assumption that all diffusion coefficients are equal and independent of temperature, pressure, etc. Such simplification guarantees the linearity of mass transport equations in the modeled domain and also allows for easier correlations with known analytical limits. This assumption can be relaxed once the behavior of the system with all the same coefficients of diffusion is well understood.

The dimensional study of Fick 's second law shows that there is a fundamental relationship in diffusive processes between the time elapsed and the square of the period over which the diffusion takes place. Understanding this relationship is very important for precise numerical simulation of diffusion.

Fick’s Second Law

Prediction of change in concentration along with time due to diffusion.

\[\frac{\partial{\phi}}{\partial{t}}\]= D \[\frac{\partial^{2}\phi}{\partial x^{2}} \]

Where,

D: diffusivity

t: time

x: position

\[\phi\] : concentration

Multi-Component Diffusion

In the case of condensed liquids or gas mixtures where more than one chemical species is present in large mass fractions, the coefficient of diffusion can no longer be viewed as constant or composition-independent. The interaction of molecules of different species with each other is too prevalent for a physical description to ignore these intermolecular dependencies. The coefficient of diffusion thus becomes a tensor and the equation for diffusion is modified to link the mass flux of one chemical species to the concentration gradients of all chemical species present. The requisite equations are formulated as the Maxwell-Stefan distribution description. They are often used to describe gas mixtures, such as syngas in a reactor, or the mixture of oxygen, nitrogen, and water in a fuel cell cathode.

In Maxwell-Stefan diffusion, the rational choice of dependent variables is not species concentrations, but species mole or mass fractions (xi and ωi respectively). 

The diffusive mass flux of each species is, in turn, expressed on the basis of mole gradients or mass fractions using Dik multi-component diffusion coefficients. They are symmetrical so that the n-component system requires n(n-1)/2 Independent coefficients to parameterize the diffusion rate of its components. Such amounts are often unknown for four-component or more complex mixtures. Simplifications can be applied to Maxwell-Stefan equations in order to use the equivalent Fick law diffusivity. Systems, most often, involving concentrated mixtures require convection and retention of momentum (fluid flow) to be resolved by diffusion.

In a material composed of two or more chemical species in which there are spatial inhomogeneities of the composition, there is a driving force for the interdiffusion of the various molecular species in order to make the composition of the material uniform. In a mixture of just two molecules, the diffusive flux of each molecular species is proportional to the gradient of its composition. This proportionality is known as Fick's Law of Diffusion and is, to a small degree, a mass transfer analogue of Newton's Law of Viscosity and Fourier 's Law of Heat Conduction

Bird(1960)

Bird(1960). The mathematical formulation of Fick's Law must be carried out with considerable caution, because there are a variety of ways in which the structure of the substance can be represented and because it is important to administer it.

Application of Fick’s Law

• Biological application:

flux=−P(c2−c1) (from Fick’s first law)

Where,

P: permeability

c2-c1: difference in concentration

• Liquids - Fick 's law refers to two miscible liquids when they come into contact and the diffusion takes place at a macroscopic point.

• Fabrication of semiconductor-Diffusion equations Fick's law is used for the manufacture of integrated circuits.

• Pharmaceutical application

• Applications in food industries.

Importance of  Fick's law

As we know, the gasses dissolved in liquids move randomly throughout the liquid in a thermodynamic process which is well described as diffusion. We know that the diffusion rates of a gas within a continuous body of liquid are constant, the presence of a barrier within the liquid can substantially affect the diffusion rate of the gas. 

The rate at which gasses can diffuse across membranes is an essential aspect of respiratory physiology as oxygen and carbon dioxide must cross the alveolar membrane during the gas exchange process. It is  important to know the physical laws which govern the diffusion of dissolved gas across membranes as they heavily inform our understanding of the gas exchange process at the alveolar membrane. Fick's Law describes the rate at which a dissolved gas diffuses across a membrane given certain properties of the membrane and gas.

Concept of  Fick’s Law of Diffusion

Adolf Fick first reported the laws governing the transport of mass through diffusive means. Fick's work was inspired by  Thomas Graham, who fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists. 

Fick's experiments always  dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is well observed that Fick's work primarily concerned diffusion in fluids because, at the time, diffusion in solids was not considered generally possible.

Now we can say that Fick's Laws form the main to understand diffusion in solids, liquids, and gasses.

When a diffusion process does not follow Fick's laws then it is referred to as non-Fickian.