Fick's law of diffusion

# Fick's law of diffusion

Overview
Fick's laws of diffusion describe diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.

Fick's first law relates the diffusive flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

to the concentration, by postulating that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative).
Discussion
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Encyclopedia
Fick's laws of diffusion describe diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.

## Fick's first law

Fick's first law relates the diffusive flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

to the concentration, by postulating that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, this is

where
• is the "diffusion flux" [(amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

) per unit area per unit time], example . measures the amount of substance that will flow through a small area during a small time interval.

• is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example

• (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length−3], example

• is the position [length], example

is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

of the fluid and the size of the particles according to the Stokes-Einstein relation
Einstein relation (kinetic theory)
In physics the Einstein relation is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski in their papers on Brownian motion...

. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10−9 to 2x10−9 m2/s. For biological molecules the diffusion coefficients normally range from 10−11 to 10−10 m2/s.

In two or more dimensions we must use , the del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

operator, which generalises the first derivative, obtaining
.

The driving force for the one-dimensional diffusion is the quantity

which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

of this species. Then Fick's first law (one-dimensional case) can be written as:

where the index i denotes the ith species, c is the concentration (mol/m3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).

If the primary variable is mass fraction (, given, for example, in ), then the equation changes to:

where is the fluid density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

(for example, in ). Note that the density is outside the gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

operator.

## Fick's second law

Fick's second law predicts how diffusion causes the concentration to change with time:

Where
• is the concentration in dimensions of [(amount of substance) length−3], example
• is time [s]
• is the diffusion coefficient in dimensions of [length2 time−1], example
• is the position [length], example

It can be derived from Fick's First law and the mass balance
Mass balance
A mass balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique...

:

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiation and multiply by the constant:

and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions Fick's Second Law becomes

,

which is analogous to the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Fick's Second Law yields

An important example is the case where is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain

which is Laplace's equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, the solutions to which are called harmonic functions by mathematicians.

### Example solution in one dimension: diffusion length

A simple case of diffusion with time t in one dimension (taken as the x-axis) from a boundary located at position , where the concentration is maintained at a value is
.

where erfc is the complementary error function
Error function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...

. The length is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t.

As a quick approximation of the error function, the first 2 terms of the Taylor series can be used:

For more detail on diffusion length, see these examples.

## History

In 1855, physiologist Adolf Fick first reported his now-well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of Thomas Graham
Thomas Graham (chemist)
Thomas Graham FRS was a nineteenth-century Scottish chemist who is best-remembered today for his pioneering work in dialysis and the diffusion of gases.- Life and work :...

, but which fell short of proposing the fundamental laws for which Fick would become famous. The Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: Darcy's law
Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

(hydraulic flow), Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

(charge transport), and Fourier's Law (heat transport).

Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible. Today, Fick's Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does not follow Fick's laws (which does happen), we refer to such processes as non-Fickian, in that they are exceptions that "prove" the importance of the general rules that Fick outlined in 1855.

## Applications

Equations based on Fick's law have been commonly used to model transport processes
Passive transport
Passive transport means moving biochemicals and other atomic or molecular substances across membranes. Unlike active transport, this process does not involve chemical energy, because, unlike in an active transport, the transport across membrane is always coupled with the growth of entropy of the...

in foods, neuron
Neuron
A neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...

s, biopolymer
Biopolymer
Biopolymers are polymers produced by living organisms. Since they are polymers, Biopolymers contain monomeric units that are covalently bonded to form larger structures. There are three main classes of biopolymers based on the differing monomeric units used and the structure of the biopolymer formed...

s, pharmaceuticals
Pharmacology
Pharmacology is the branch of medicine and biology concerned with the study of drug action. More specifically, it is the study of the interactions that occur between a living organism and chemicals that affect normal or abnormal biochemical function...

, porous soil
Soil
Soil is a natural body consisting of layers of mineral constituents of variable thicknesses, which differ from the parent materials in their morphological, physical, chemical, and mineralogical characteristics...

s, population dynamics
Population dynamics
Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

, semiconductor doping
Doping (semiconductor)
In semiconductor production, doping intentionally introduces impurities into an extremely pure semiconductor for the purpose of modulating its electrical properties. The impurities are dependent upon the type of semiconductor. Lightly and moderately doped semiconductors are referred to as extrinsic...

process, etc. Theory of all voltammetric
Voltammetry
Voltammetry is a category of electroanalytical methods used in analytical chemistry and various industrial processes. In voltammetry, information about an analyte is obtained by measuring the current as the potential is varied.- Three electrode system :...

methods is based on solutions of Fick's equation. A large amount of experimental research in polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition
Glass transition
The liquid-glass transition is the reversible transition in amorphous materials from a hard and relatively brittle state into a molten or rubber-like state. An amorphous solid that exhibits a glass transition is called a glass...

. In the vicinity of glass transition the flow behavior becomes "non-Fickian". See also non-diagonal coupled transport processes (Onsager
Onsager
Onsager may refer to:* Lars Onsager, a Norwegian–American physical chemist and theoretical physicist* Onsager reciprocal relations, certain relations between flows and forces in thermodynamic systems...

relationship).

### Biological perspective

The first law gives rise to the following formula:

in which,
• is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
• is the surface area over which diffusion is taking place.
• is the difference in concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

of the gas across the membrane
Artificial membrane
An artificial membrane, or synthetic membrane, is a synthetically created membrane which is usually intended for separation purposes in laboratory or in industry. Synthetic membranes have been successfully used for small and large-scale industrial processes since the middle of twentieth century. A...

for the direction of flow (from to ).

Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter
Flux limiter
Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations...

.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law
Graham's law
Graham's law, known as Graham's law of effusion, was formulated by Scottish physical chemist Thomas Graham in 1846. Graham found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles...

.

## Fick's flow in liquids

When two miscible
Miscibility
Miscibility is the property of liquids to mix in all proportions, forming a homogeneous solution. In principle, the term applies also to other phases , but the main focus is usually on the solubility of one liquid in another...

liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration
evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and
molecular scale, where molecular random walks take place, fluctuations cannot be neglected.
Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.
In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with
hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative
approach, the zero order approximation is Fick's law. The first order gives the fluctuations, and it comes out that
fluctuations contribute to diffusion. This represents somehow a tautology
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...

, since the phenomena described by a lower order
approximation is the result of a higher approximation: this problem is solved only by renormalizing fluctuating hydrodynamics equations.

### Semiconductor fabrication applications

IC
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...

Fabrication technologies, model processes like CVD, Thermal Oxidation,
and Wet Oxidation, doping, etc. use diffusion equations obtained from Fick's law.

In certain cases, the solutions are obtained for boundary conditions such as constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate).

• Diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

• Osmosis
Osmosis
Osmosis is the movement of solvent molecules through a selectively permeable membrane into a region of higher solute concentration, aiming to equalize the solute concentrations on the two sides...

• Mass flux
Mass flux
Mass flux is the rate of mass flow across a unit area .-See also:*Flux*Fick's law*Darcy's law...

• Maxwell-Stefan diffusion
Maxwell-Stefan diffusion
The Maxwell–Stefan diffusion is a model for describing diffusion in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by James Clerk Maxwell for dilute gases and Josef Stefan for fluids...

• Churchill-Bernstein Equation
Churchill-Bernstein Equation
In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier–Stokes equations in the turbulent flow regime, even for...

• Nernst-Planck equation
Nernst-Planck equation
The Nernst–Planck equation is a conservation of mass equation used to describe the motion of chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient \nabla c and an electric field E=-\nabla \phi...

• Gas exchange
Gas exchange
Gas exchange is a process in biology where gases contained in an organism and atmosphere transfer or exchange. In human gas-exchange, gases contained in the blood of human bodies exchange with gases contained in the atmosphere. Human gas-exchange occurs in the lungs...