Diffusion

Diffusion ( diffundere Latin, pour ',' scatter ',' spread ') is a naturally flowing, physical process. He leads in time to complete mixing of two or more substances by the uniform distribution of the particles involved. The particles can be either atoms, molecules or carriers. When the substances are mostly gases and liquids to solids and less plasmas.

Diffusion is based on the undirected random movement of particles due to their thermal energy. Uneven distribution statistically more particles move from areas of high concentration to areas smaller or particle density than vice versa. Thus, a macroscopic mass transport is effected net. Diffusion is usually this net transport. The term is also used for the underlying microscopic process.

In a closed system, diffusion causes the degradation of concentration differences to the complete mixing. The time that is required for growing in -dimensional space with the- th power of the distance. Diffusion is therefore particularly effective on the nanometer to millimeter scales; on larger scales dominates in liquids and gases generally solute transport by flow ( convection).

Diffusion can also be a porous wall or membrane carried out therethrough, and plays an important role in the osmosis.

• 3.7.1 Self- diffusion
• 3.7.2 Tracer diffusion
• 3.7.3 Classical Fick's diffusion
• 3.7.4 counter diffusion

History

One of the first to systematically conducted diffusion experiments on a larger scale, Thomas Graham was. From his experiments on the diffusion of gases, he headed the starting named after him Graham'sche Act:

"It is evident did 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 inversely as one of its density function -. Apparently to the square root of its density "

"The diffusion or spontaneous intermixture of two gases in contact, is Effected by at interchange in position of indefinitely minute volumes of the gases, Which volumes are not Necessarily of equal magnitude, being, in the case of each gas, inversely proportional to the square root of the density of gas did. "

" The diffusion or spontaneous mixing of two located ends in contact gases is influenced by the exchange of the position of the undetermined small volumes of gases which do not necessarily have to be of the same order and, in case of each gas, inversely proportional to the square root of the density of the gas have. "

With respect to the diffusion in Graham solutions showed that the rate of diffusion is proportional to the concentration difference and dependent on the temperature (faster diffusion at higher temperatures). Graham further pointed to the ability to separate mixtures of gases by means of solutions or diffusion.

Thomas Graham had the basic laws of diffusion can not yet be determined. This was only a few years later, Adolf Fick. He postulated that the law must be sought with the laws of heat conduction, the Jean Baptiste Joseph Fourier had determined his analogy:

" The distribution of a dissolved substance in the solvent goes, they wofern undisturbed under the exclusive influence of the molecular forces takes place, according to the same law before him, which Fourier has established for the distribution of heat in a conductor, and which ohms worked with such brilliant success on the broadening of electricity (where it is known not strictly correct, of course ) has transferred. "

Fuck conducted experiments whose results confirmed the validity of the later named after him Fick's First Law. The validity of the second Fick's law he could only derive from the validity of the First. The direct detection failed due to its limited analytical and mathematical possibilities.

Albert Einstein succeeded in the early 20th century, derive the Fick's laws from the laws of thermodynamics and to give as the diffusion a secure theoretical foundation. He also led the Stokes - Einstein relation to calculate the diffusion coefficient ago:

" The diffusion coefficient of the suspended substance that is dependent only on the friction coefficient of the liquid and the size of the suspended particles, from out of the universal constant and the absolute temperature. "

Einstein also showed how to detect the movement of an individual diffusing particle and thus the Brownian motion can be understood as a fluctuation phenomenon. He averaged to the mean square displacement of a particle in the time τ for the one-dimensional case. Shortly after Einstein also came Smoluchowski another way to practically the same relation, and therefore this equation is now called the Einstein - Smoluchowski equation.

Illustration

An often called experiment to illustrate the spread by diffusion is the gradual coloring of lukewarm water with a drop of ink, which one is into the water but neither stirs nor the container shakes. After some time, the ink color is evenly distributed throughout the water. The spread of ink in water, however, can also be favored by density and temperature differences. These effects can be reduced by layered a colored liquid with a higher density with a liquid of lower density and very viscous liquids used, eg colored syrup and honey. The then observed gradual coloring of the honey can be explained almost entirely by diffusion, with both syrup into the honey and honey diffuses into the syrup.

Physical Basics

The diffusion at a certain constant temperature is achieved without further energy input and is passive in this sense; in particular, the diffusion in biology is distinguished from active transport.

Theoretically, an infinitely long lasting process diffusion. However, in the context of measurability can be considered complete in a finite time frequently.

Thermal motion

The thermal motion on which the diffusion is based, may, depending on the considered system have very different character. In gases it consists of rectilinear motion, interrupted by occasional shocks. The rapid thermal motion of fluid particles caused by frequent collisions the much slower, observable under the microscope Brownian motion of mesoscopic objects. In solids carried occasional change of location, eg through the exchange of positions of two adjacent particles, or the " wandering" of vacancies. With carriers (e.g., ions, electrons, holes ) of the heat movement, however, a drift due to the electrostatic forces is superimposed.

Probability and entropy

The direction of motion of a single particle is completely random. Due to the interaction with other particles carried continuous direction changes. Averaged over a longer period of time or over many particles transport in a certain direction may still arise, such as when a jump in a certain direction one, maybe only slightly greater likelihood has. This is the case where a difference in concentration ( and concentration gradient ) is present. It then produces a net flow of particles, until a steady state, the thermodynamic equilibrium is established. Usually, the equilibrium state is the uniform distribution, wherein the concentration of all the particles at any point in space is the same.

Probability and diffusion - an explanation: Assume 1,000 particles of a substance would only be in the right half of a vessel, and 10 particles in the left half; In addition, each particle moves through the Brownian motion a certain distance in a completely random direction. Then follows: the probability that one of the 1000 particles happens to be moved from the right to the left half 100 times greater than the probability that moves one of only 10 particles from left to right. So will be after a certain time with high probability net particles moved from right to left. Once the probability of walking on both sides is equal to right and left hand are per 505 particles, no more mass flow is net take place and the concentration remains everywhere ( within statistical fluctuations ) the same size. Of course, as the particles migrate to from left to right and vice versa; but as it is now are the same number of parts, there was no difference in the concentration can be observed. If you now "right" and "left" as a particularly small compartments such as the ink experiment, imagine and all at some point all these subspaces have the same concentration of ink, the ink has spread evenly.

Systems in which the particles are randomly distributed over the whole volume, have a higher entropy than orderly systems in which the particles are preferably resident in certain areas. Diffusion leads to a entropy. It is after the Second Law of Thermodynamics is a voluntary process running, which can not be reversed without any external influence.

Analogy to heat conduction and conduction of electric current

The diffusion follows laws that are similar to those of heat conduction equivalent. Therefore, one can assume equations that describe the a process for the other.

Diffusion of dissolved particles

At specified pressure p and temperature T is defined from the viewpoint of thermodynamics, the gradient of the chemical potential μ is the driving source of the material stream. The flow is therefore given as:

• J: particle current density (flux) in mol · m -2 · s- 1
• : Chemical potential in J · mol -1
• X: length in m
• K: a coefficient in mol2 · s · kg -1 · m-3

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

It is the chemical potential at standard pressure. Depends on the temperature is not explicitly on the location from, then:

Substituting this into the above equation, we obtain Fick's first law:

Here, the diffusion coefficient D was introduced. The relationship between the coefficient K and D is

In which

• D: diffusion coefficient in m2 s -1
• C: molar concentration in mol · m-3
• T: temperature in K
• R: Universal gas constant in J · K-1 · mol -1

At very low concentrations ( individual molecules ), this observation is not allowed more readily, because the classical thermodynamics solutions considered as a continuum. At high concentrations, the particles interact with each other, so that slower at attractive interaction of concentration equalization, is faster at repelling. The chemical potential is not logarithmically dependent on the concentration in such cases.

First Fick's law

By the first Fick law, the particle current density is (flux) J (mol m-2 s-1 ) is proportional to the concentration gradient opposite to the direction of diffusion ∂ c / ∂ x (mol · m -4). The proportionality constant D is the diffusion coefficient ( m2 s -1).

The particle current density makes a quantitative assessment of the directed ( statistical average ) motion of particles, ie how many particles of a material amount of move per unit time through a unit area perpendicular to the direction of diffusion, net. The equation stated is also valid for the general case that the diffusion coefficient is not constant, but depends on the concentration ( it is, however, strictly speaking, not the message of the first Fick's Law).

As an extension of Fick's law, the Nernst -Planck equation can be viewed.

Second Fick's law (diffusion equation)

Equation of continuity and equation of the one-dimensional case,

Using the continuity equation ( conservation of mass )

Results from the first Fick's law of diffusion equation

For constant diffusion coefficients obtained from this

It represents a relationship between temporal and spatial differences in concentration and thus suitable for the representation of unsteady diffusion, in contrast to the first Fick's law, which describes a time-constant diffusion flux. There are numerous analytical and numerical approaches, however, depend strongly on the initial and boundary conditions for this differential equation.

Mathematically, the diffusion equation is identical to the heat conduction equation, and its mathematical properties and solutions are treated in the local product.

Differential equation for the three-dimensional case

The case of the three-dimensional diffusion can be described by Fick's law, the second in its most general form:

With the nabla operator. Mathematically, these diffusion equation is identical to the ( three-dimensional ) heat conduction equation, and its mathematical properties and solutions are treated in the local product. The solution of this equation is usually only possible numerically expensive and depending on the considered area.

In the stationary case, that is, for

Results in the elliptic partial differential equation

Now, when the diffusion coefficient is also isotropic, one obtains a differential equation of the Laplacian - type.

Types of diffusion

It is customary to distinguish between four types of diffusion. The diffusion coefficients are different 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, a pure liquid or a solution is only true self-diffusion (English: self diffusion ) instead. The self-diffusion ( also often referred to as an intra- diffusion ) of the transport of particles in the same substance, for example water molecules in pure water and sodium ions in a sodium chloride solution. Since this because of the difficult differentiation physically and chemically identical particles is at best to watch with great effort, self-diffusion approaches is often with isotopic tracers of the same substance at, for example, 22 Na for sodium ions. In this case, it is assumed that the gradient resulting from the addition of the tracer, is negligibly small. The self-diffusion is a model for the description of the Brownian movement. The measured diffusion coefficients can be converted into the mean square displacement of a particle per unit time.

An especially suitable method for measurement of self- diffusion coefficient, the field gradient NMR where no dar. isotopic tracers are needed because the same particles are physically and chemically distinguishable by means of the nuclear spin situated in Präzessionsphase a particle nucleus. This NMR technique both self-diffusion coefficients can be determined very precisely in pure liquids as well as in complex fluid mixtures. The self-diffusion coefficient of pure water was measured with extreme precision and therefore often used as a reference value. It amounts to 2.299 · 10-9 m · s- 1 at 25 ° C and 1.261 · 10-9 m · s- 1 at 4 ° C.

Tracer diffusion

Tracer diffusion is the diffusion of low concentrations of a substance in a solution of a second substance. Tracer diffusion differs from the self-diffusion, that a labeled particles of another substance is used as a tracer, for example, 42K in the NaCl solution. Frequently radioactively or fluorescently labeled tracers are used, because you can detect this very well. At infinite dilution, the diffusion coefficients of self-and tracer diffusion are identical.

Classic Fick's diffusion

This refers to the diffusion along a relatively strong gradient. In this way an approximation to the diffusion of the diffusion coefficient is the best possible.

Counter diffusion

Counter diffusion (English: counter diffusion ) occurs when opposing gradients are present, so that particles diffuse in opposite directions.

Diffusion of gases

In principle, the diffusion of particles in gases not differ with respect to their laws from the diffusion of solutes in liquids. However, the speed of diffusion (in comparable gradient ) here is orders of magnitude higher, as well as the motion of individual particles is significantly faster in gases. The diffusion of dilute gases in multi -component systems can be described with the model of Maxwell-Stefan diffusion.

Diffusion in solids

In a perfect crystal lattice each lattice oscillates around its fixed lattice site, but this can not leave. Therefore, a necessary condition for diffusion in a crystalline solid state, the presence of defects in the lattice. Only by lattice defects, exchange between atoms or ions can take place as a condition of mass transport. Different mechanisms are conceivable:

• The particles " jump" into spaces of the grid, so move through the lattice vacancies and held a net flux of particles. This mechanism has been demonstrated by the Kirkendall effect.
• Smaller particles move through the lattice interstices. This mechanism has also been demonstrated experimentally. He performs in comparison to the diffusion through vacancies at very high diffusion coefficient.
• Two particles exchange places or find it ring Swap between multiple particles instead. This hypothetical mechanism could not be confirmed experimentally.
• If free charge carriers sufficiently experienced much scattering in semiconductors (eg by phonons, electrons, and impurities ), they also propagate diffusively.

The diffusion in crystals can be described by the Fick's laws. However, diffusion coefficients here depend on the spatial direction ( anisotropy ). The isotropic case in the scalar diffusion coefficients are then to a second order tensor. Therefore, the diffusion is an important parameter for the description of diffusion processes in solids.

The diffusion in non-crystalline ( amorphous ) solids is similar in mechanistic terms, the in crystals, although eliminates the distinction between regular and irregular lattice sites. Mathematically, such processes are well described as the diffusion in liquids.

Fokker -Planck equation

An additional force by an existing potential leads to the uniform distribution no longer corresponds to the steady state. The theory provides to the Fokker -Planck equation.

Special case: Anomalous diffusion

In the above described diffusion process, which may be described by the Fick's diffusion equation, the mean square displacement of the diffusing particles (ie, the average distance of the particles to its starting point after the time t ) increases in proportion to time to:

This law follows from the theory of Brownian motion. In cells as well as other principles can be observed, for example in the movement of macromolecules through the cytoplasm of the cell. This densely populated with organelles and (macro) molecules medium leads to a braked diffusion movement that follows a power law. We then have:

For these braked movement that subdiffusion is called, is considered. There also exist diffusion processes where α > 1, that are accelerated. These are called superdiffusion.

Special case: Facilitated diffusion (biology)

The facilitated diffusion or permeability describes in biology, the possibility for certain substances, a biomembrane easier to penetrate than would be possible actually because of their size, charge, polarity, etc.. Certain proteins, so-called channel proteins which form a tunnel through the cell membrane which can pass certain substances easily by its diameter and / or charge distributions as a "closed" by the membrane ( as ions channels).

Applications

• During sintering, the diffusion plays a very important role in the convergence of the powder components.
• Steel can be surface hardened by diffusion of carbon and / or nitrogen.
• In diffusion furnaces dopants are introduced into the semiconductor material to directly influence the electrical conductivity or mechanical properties for components of microsystems technology there at temperatures around 1000 ° C.
• The diffusion plays a central role in industrial chemistry. Frequently it occurs here coupled with on convection and chemical reactions. Typical applications are reactor and catalyst design. In chemical process technology, the selective separation of mixtures of substances is often by means of molecular sieves and / or membrane technology. Both methods are based on " kinetic separation ", where the differences in the diffusion of the individual substances in nanopores play an essential role and open up control.
• In the building construction for protection from humidity must be considered, the water vapor diffusion to avoid unacceptably large dew condensation. These vapor barriers and vapor retarders are used with a defined water vapor diffusion resistance.
• In microbiology, diffusion is used in the agar diffusion test.
• In the lung breathing, gas exchange between alveoli and blood takes place by diffusion.

Wrong word application of diffusion

Frequently, the word diffusion is used incorrectly for the German word diffusivity. This is due to the incorrect translation from English in the field of acoustics.