Fick’s Law of Diffusion

Fick’s Law Of Diffusion

Fick 's Law, discovered by Adolf Fick, is a commonly used principle when looking at Diffusion whether it is determining the rate of diffusion or the result of the concentration of the particle in a substance. The equations founded by Fick are applied to modern pharmaceuticals, models for understanding transport processes, biopolymers, etc. The simplest description of diffusion is given by the laws of Fick, developed by Adolf Fick in the 19th century:

1.  The molar flow due to diffusion is proportional to the concentration gradient.

2.  The rate of change of concentration at a point in space is proportional to the second derivative of concentration in space.

Fick's First Law of Diffusion

Writing the first law in a modern mathematical form:

Ni = -Di ∇ ci

where for species i, Ni is the molar flux (mol m-2 s-1), Di is the diffusion coefficient (m2 s-1), and ci is the concentration (mol m-3).

From the continuity equation for mass:

\[\frac{\partial c_{i}}{\partial t} + \bigtriangledown . N_{i} = 0\]

we can derive Fick's second law directly:

\[\frac{\partial c_{i}}{\partial t} = D_{i} \bigtriangledown^{2} c_{i}\]

This means that Di is a constant, which is valid even for dilute solutions. This is typically a good assumption for diffusion in solids; diffusion of chemicals in dilute solution, water or other common liquid solvents; and diffusion of dilute (trace) species during the gas process, such as carbon dioxide in the air.

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 modelling 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 modelled domain and also allows for easier correlations with known analytical limits. This assumption can be relaxed once the behaviour 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.

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)]. 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)


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.

FAQ (Frequently Asked Questions)

1. What is Fick’s Law of Diffusion?

Fick 's law of diffusion describes the mechanism of diffusion (movement of molecules from higher concentration to lower concentration) in order to solve the coefficient of diffusion developed by Adolf Fick in 1855.

There are two laws that are interrelated, ie; Fick's first law is used to derive Fick 's second law, which is similar to the distribution equation.

2. What is Ficks First Law Diffusion? What is Fick's Second Law of Diffusion?

Fick 's laws of diffusion describe diffusion and were developed by Adolf Fick in 1855. They can be used to solve the coefficient of diffusion, D. Fick's first law can be used to derive his second law which, in effect, is similar to the distribution equation. Fick’s first law

Moving the solute from higher concentration to lower concentration through the concentration gradient.



J: diffusion flux

D: diffusivity

φ: concentration

x: position

Fick’s Second Law

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



D: diffusivity

t: time

x: position

Φ: concentration