What does flux mean physically

diffusion (from Latin: diffuse "Pouring, scattering, spreading") is a physical process that leads to an even distribution of particles and thus complete mixing of two substances.

Diffusion is based on the thermal movement of particles. The particles can be atoms, molecules, ions, but z. For example, other charge carriers such as electrons or defect electrons ("holes") can also be involved. In the case of an uneven distribution, statistically more particles move from areas of high to areas of low concentration or particle density than vice versa. This brings about a net macroscopic transport of substances. Diffusion is usually understood to mean this net transport. The term is also used for the underlying microscopic process.


In a closed system, diffusion causes the reduction of concentration differences to the point of complete mixing. The time required for this increases with the square of the distance. Diffusion is therefore mainly effective on nano to millimeter scales; On larger scales, mass transport by flow (convection) usually dominates in liquids and gases.

Diffusion can also take place through a porous wall or membrane. Osmosis is the diffusion of solvent through a membrane that is impermeable to the solute (semipermeable).


An experiment that is often cited to illustrate propagation by diffusion is the gradual coloring of lukewarm water with a drop of ink that is put in, but the water is neither stirred nor shaken. After a while, the ink color will be evenly distributed throughout the water. In addition to pure diffusion phenomena, differences in density and temperature also play a role in the spread of the ink in water. This can be avoided by overlaying a colored liquid with a higher density with a lower density and using very viscous liquids, e.g. B. colored syrup and honey. The gradual coloration of the honey then observed can be explained almost exclusively by diffusion, with both syrup diffusing into the honey and honey diffusing into the syrup.

Physical basics

Diffusion at a certain constant temperature takes place without any additional energy input and is passive in this sense; In biology in particular, a distinction is made between diffusion and active transport.

In theory, diffusion is an infinitely long process. In terms of measurability, however, it can often be viewed as completed in a finite time.

Thermal movement

The thermal movement on which the diffusion is based can have very different characters depending on the system under consideration. In gases it consists of linear motion interrupted by occasional impacts. The rapid thermal movement of liquid particles causes the much slower Brownian movement of mesoscopic objects, which can be observed under the microscope, due to frequent collisions. Occasional changes of location take place in solids, e.g. B. by exchanging the place of two neighboring particles or the "wandering" of empty spaces.

Probability and entropy

The direction of movement of a single particle is completely random. Due to the interaction with other particles, there are constant changes in direction. However, averaged over a longer period of time or over a large number of particles, transport in a certain direction can result, e.g. B. if a jump in a certain direction has a, perhaps only slightly, greater probability. This is the case when there is a concentration difference (also concentration gradient). A net flow of particles then arises until a steady state, the thermodynamic equilibrium, is established. Usually the state of equilibrium is the uniform distribution in which the concentration of all particles is equally high at every point in space.

Probability and Diffusion - an attempt to explain: Assume 1000 particles of a substance are only in the right half of a vessel and 10 particles in the left half; in addition, each particle moves a certain distance in a completely random direction due to Brownian molecular motion. Then it follows: The probability that one of the 1000 particles accidentally moves from the right to the left half is 100 times greater than the probability that one of the only 10 particles moves from left to right. So after a certain time there is a high probability that net particles will have migrated from right to left. As soon as the probability of wandering is the same on both sides, i.e. there are 505 particles each on the right and left, there will be no more net mass flow and the concentration will remain the same everywhere (within the framework of statistical fluctuations). Of course, particles still migrate from left to right and vice versa; But since there are now the same number of parts, no difference in concentration can be determined. If you now think of “right” and “left” as particularly small sub-spaces z. B. the ink test and all these subspaces at some point all have the same ink concentration, the ink has been evenly distributed.

Systems in which the particles are randomly distributed over the entire volume have a higher entropy than more ordered systems in which the particles prefer to stay in certain areas. Diffusion thus leads to an increase in entropy. According to the second law of thermodynamics, it is a voluntary process that cannot be reversed without external influence.

Analogy to heat conduction and conduction of electric current

Diffusion follows laws that are similar to those of heat conduction[1] are equivalent. Therefore, equations that describe one process can be adopted for the other.

Dissolved Particle Diffusion

At a fixed pressure (p) and a fixed temperature (T), from the point of view of chemical thermodynamics, the gradient of the chemical potential isµ the driving cause of the material flow. The flow is thus:

For simple applications, the concentration c be used. This is more easily accessible than the chemical potential of a substance.

In the case of very low concentrations (individual molecules), this consideration is no longer permissible without further ado, since classical thermodynamics regards solutions as a continuum. In the case of high concentrations, the particles influence each other so that the concentration equalization takes place more slowly when the interaction is attractive and faster when it is repulsive. In this case, the chemical potential is no longer logarithmically dependent on the concentration.

First Fick's law

According to Fick's First Law, the particle flux density (flux) is J (mol m−2 s−1) proportional to the concentration gradient opposite to the diffusion direction ∂c / ∂x (mol m−4). The constant of proportionality is the diffusion coefficient D. (m2 s−1).

The particle flux density makes a quantitative statement about the (in statistical mean) directed movement of particles, i. H. how many particles of an amount of substance move per unit of time through a unit of area that is perpendicular to the direction of diffusion, net. The equation given also applies to the general case that the diffusion coefficient is not constant, but depends on the concentration (but strictly speaking this is no longer the statement of Fick's First Law).

Fick's second law (diffusion equation)

Continuity equation and differential equation for the one-dimensional case

With the help of the continuity equation (conservation of mass)

the diffusion equation results from Fick's First Law

or, for constant diffusion coefficients, .

It represents a relationship between temporal and local concentration differences and is therefore suitable for representing unsteady diffusion, in contrast to Fick's 1st law, which describes a diffusion flow that is constant over time. There are numerous analytical and numerical solution approaches for this differential equation, which, however, depend heavily on the initial and boundary conditions. A possible solution is given in the article heat conduction equation.

Differential equation for the three-dimensional case

The case of three-dimensional diffusion can be described with Fick's Second Law in its most general form:

with the Nabla operator . The form of this parabolic partial differential equation is that of the heat conduction equation.

Solving this equation is usually time-consuming and, depending on the area under consideration, only possible numerically. In the stationary case

an elliptic partial differential equation results. In addition, if the diffusion coefficient is isotropic, a Laplace-type differential equation is obtained.

Types of diffusion

It is common to distinguish four types of diffusion[2]. The diffusion coefficients differ for different types of diffusion, even if the same particles diffuse under standard conditions.

Self diffusion

If there is no macroscopic gradient in a gas or solution, only true self-diffusion takes place. Self-diffusion is the transport of particles in a solution of the same substance, e.g. B. Sodium ions in a NaCl solution. Since this cannot be observed in this way, self-diffusion is approximated with isotopic tracers of the same substance, e.g. B. 22N / A+ for sodium ions. It is assumed that the gradient created by adding the tracer is negligibly small.

Self-diffusion is a model for describing Brownian molecular motion. The diffusion coefficients determined by means of tracers can be calculated using σ2 = 2D. Convert to the mean square displacement of a particle per unit of time[3].
Field gradient NMR is a particularly versatile method for measuring self-diffusion coefficients.

Tracer diffusion

Tracer diffusion is the diffusion of low concentrations of one substance in a solution of a second substance. Tracer diffusion differs from self diffusion in that a labeled particle of another substance is used as a tracer, e.g. B. 42K+ in NaCl solution. Radioactively or fluorescence-marked tracers are often used because they can be detected very well. At infinite dilution, the diffusion coefficients of self and tracer diffusion are identical.

Classic Fickian diffusion

This denotes the diffusion along a relatively strong gradient. With this type of diffusion, an approximation of the diffusion coefficient is best possible.

Counter diffusion

Counter-diffusion occurs when there are opposing gradients such that particles diffuse in opposite directions.

Diffusion of gases

In principle, the diffusion of particles in gases does not differ from the diffusion of dissolved particles in liquids with regard to their regularities. However, the speed of diffusion (with comparable gradients) is orders of magnitude higher here, since the movement of individual particles in gases is considerably faster. The diffusion of dilute gases in multicomponent systems can be described with the Maxwell-Stefan diffusion model.

Diffusion in solids

In a perfect crystal lattice, every lattice particle swings around its fixed lattice position, but cannot leave it. A necessary prerequisite for diffusion in a crystalline solid is therefore the presence of defects in the lattice. Only through lattice errors can atoms or ions change their position as a condition for mass transport. Various mechanisms are conceivable[4]:

  • The particles “jump” into voids in the grid, so that voids move through the grid and a net flow of particles takes place. This mechanism was demonstrated by the Kirkendall effect.
  • Smaller particles move through the spaces between the lattices. This mechanism has also been proven experimentally. Compared to diffusion via vacancies, it leads to very high diffusion coefficients.
  • Two particles change places or ring exchanges take place between several particles. This hypothetical mechanism could not be confirmed experimentally.

Diffusion in crystals can also be described by Fick's laws. However, diffusion coefficients can depend on the spatial direction (anisotropy). The diffusion coefficients, which are scalar in the isotropic case, then become a second-order tensor.

The diffusion in non-crystalline (amorphous) solids is mechanistically similar to that in crystals, although there is no distinction between regular and irregular lattice sites. Mathematically, such processes can be described as well as diffusion in liquids.

Fokker-Planck equation

An additional force from an existing potential means that the uniform distribution no longer corresponds to the steady state. The theory for this is provided by the Fokker-Planck equation.

Special case: facilitated diffusion (biology)

In biology, facilitated diffusion or permeability describes the possibility for certain substances to penetrate a biomembrane more easily than would actually be possible due to their size, charge, polarity, etc. Certain proteins, so-called tunnel proteins, form a tunnel through the cell membrane which, due to its diameter and / or charge distribution, allows certain substances to pass more easily than through the "closed" membrane (such as ion channels).

History of Diffusion Research

One of the first to systematically carry out diffusion experiments on a large scale was Thomas Graham. From his experiments on the diffusion of gases, he derived Graham's law, named after him:

  • "It is evident that the diffusiveness of the gases is inversely as some function of their density - apparently the square root of their density." (It is evident that the rate of diffusion of gases is the inverse of a function of their density - apparently the square root of their density.)[5]
  • "The diffusion or spontaneous intermixture of two gases in contact, is effected by an interchange in position of indefinitely minute volumes of the gases, wich volumes are not necessarily of equal magnitude, being, in the case of each gas, inversely proportional to the square root of the density of that gas. " (The diffusion or spontaneous mixing of two gases in contact is influenced by the exchange of the position of indefinitely small volumes of the gases, which do not necessarily have to be of the same order of magnitude and, in the case of any gas, inversely proportional to the square root of the density of the gas are.)[6]

With regard to diffusion in solutions, Graham was able to show that the diffusion rate is proportional to the concentration difference and dependent on the temperature (faster diffusion at higher temperatures).[7] Graham also showed the possibility of separating mixtures of solutions or gases by means of diffusion.[5][7]

Thomas Graham had not yet been able to determine the fundamental laws of diffusion. Adolf Fick succeeded in doing this only a few years later. He postulated that the law he was looking for must be analogous to the laws of heat conduction determined by Jean Baptiste Joseph Fourier: "The dissemination of a dissolved body in the solvent, as long as it takes place undisturbed under the exclusive influence of the molecular forces, proceeds according to the same law which Fourier set up for the dissemination of heat in a conductor, and which Ohm has already been so brilliantly successful on the The broadening of electricity (where, of course, it is known to be not strictly correct) has transmitted. "[8] Fick carried out experiments, the results of which proved the validity of Fick's First Law, which was later named after him. He could only deduce the validity of Fick's second law from the validity of the first. The direct proof failed because of its limited analytical and mathematical possibilities.

At the beginning of the 20th century, Albert Einstein succeeded in deriving Fick's laws from the laws of thermodynamics and thus giving diffusion a secure theoretical foundation.[3] He also derived the Stokes-Einstein relationship for calculating the diffusion coefficient: "The diffusion coefficient of the suspended substance depends on universal constants and the absolute temperature only on the coefficient of friction of the liquid and on the size of the suspended particles."


  • In steel production, diffusion plays a very important role in the coalescence of the powder components during sintering. You can also harden the surface of steel by diffusing in carbon and / or nitrogen, similar to doping semiconductor elements (wafers).
  • In diffusion furnaces, dopants are introduced into the semiconductor material at temperatures around 1000 ° C in order to specifically influence the electrical conductivity or mechanical properties of components in microsystem technology.
  • Diffusion plays a central role in technical chemistry. It often occurs here coupled with convection and chemical reactions. Typical applications are reactor and catalyst design.
  • In the building construction, the water vapor diffusion must be taken into account for moisture protection in order to avoid inadmissibly large amounts of condensation water. Vapor barriers and vapor retarders are used for this purpose.
  • In microbiology, diffusion is used in the agar diffusion test.

Wrong use of the word diffusion

The word diffusion is often used incorrectly for the German word diffusität. This is due to incorrect translations from English in the field of acoustics.


  • Peter W. Atkins: Physical chemistry. Wiley-VCH, ISBN 3527302360
  • E. L. Cussler: Diffusion - mass transfer in fluid systems. Cambridge University Press, Cambridge, New York, 1997, ISBN 0-521-56477-8
  • J. Crank: The Mathematics of Diffusion. Oxford University Press, 1980, ISBN 0-198-53411-6
  • W. Jost: Diffusion in solids, liquids, gases. Academic Press, New York 1970, 6th printing

Individual evidence

  1. H. S. Carslaw and J. C. Jaeger: Conduction of heat in solids, 2nd Ed., P. 28. Oxford University Press, London, 1959, ISBN 0198533683
  2. Yuan-Hui Li and Sandra Gregory (1974): Diffusion of ions in sea water and in deep-sea sediments, Geochimica et Cosmochimica Acta, 38, 703-714.
  3. ab Albert Einstein (1905): About the movement of particles suspended in liquids at rest, required by the molecular kinetic theory of heat, Annalen der Physik, 17, pp. 549-560. http://www.physik.uni-augsburg.de/annalen/history/papers/1905_17_549-560.pdf
  4. E. Bruce Watson and Ethan F. Baxter (2007): Diffusion in solid-earth systems, Earth and Planetary Science Letters, 253, 307-327.
  5. ab Thomas Graham (1829): A short Account of Experimental Researches on the Diffusion of Gases through each other, and their Separation by mechanical means, Quarterly Journal of Science, Literature and Art, 27, 74-83.
  6. Thomas Graham (1833): On the Law of the Diffusion of Gases, Philosophical Magazine, 2, 175-190.
  7. ab Thomas Graham (1850): The Bakerian Lecture - On the Diffusion of Liquids, Philosophical Transactions of the Royal Society of London, 140, 1-46.
  8. Adolf Fick (1855): About diffusion, Poggendorff’s Annalen der Physik, 94, 59–86.

See also

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