Diffusion

Diffusion is the net movement of molecules from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in chemical potential of the diffusing species.

A gradient is the change in the value of a quantity e.g. concentration, pressure, or temperature with the change in another variable, usually distance. A change in concentration over a distance is called a concentration gradient, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is called a temperature gradient.

The word diffusion derives from the Latin word, diffundere, which means "to spread way out.”

A distinguishing feature of diffusion is that it depends on particle random walk, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of advection.[1] The term convection is used to describe the combination of both transport phenomena.

Diffusion
A diffusion is a process in physics. Some particles are dissolved in a glass of water. At first, the particles are all near one top corner of the glass. If the particles randomly move around ("diffuse") in the water, they eventually become distributed randomly and uniformly from an area of high concentration to an area of low concentration, and organized (diffusion continues, but with no net flux).
Time lapse video of diffusion of a dye dissolved in water into a gel!
DiffusionMicroMacro
Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a statistical trend that the solute fills the container more and more uniformly. Bottom: With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic force pushing molecules rightward, but there appears to be one in the bottom panel. This apparent force is called an entropic force.
Blausen 0315 Diffusion
Three-dimensional rendering of diffusion of purple dye in water.

Diffusion vs. bulk flow

An example of a situation in which bulk motion and diffusion can be differentiated is the mechanism by which oxygen enters the body during external respiration known as breathing. The lungs are located in the thoracic cavity, which expands as the first step in external respiration. This expansion leads to an increase in volume of the alveoli in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air outside the body at relatively high pressure and the alveoli at relatively low pressure. The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal i.e. the movement of air by bulk flow stops once there is no longer a pressure gradient.

The air arriving in the alveoli has a higher concentration of oxygen than the “stale” air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the capillaries that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of carbon dioxide in the alveoli decreases. This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli, as fresh air has a very low concentration of carbon dioxide compared to the blood in the body.

The pumping action of the heart then transports the blood around the body. As the left ventricle of the heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through blood vessels by bulk flow down the pressure gradient. As the thoracic cavity contracts during expiration, the volume of the alveoli decreases and creates a pressure gradient between the alveoli and the air outside the body, and air moves by bulk flow down the pressure gradient.

Diffusion in the context of different disciplines

Centrotherm diffusion furnaces at LAAS 0481
Diffusion furnaces used for thermal oxidation

The concept of diffusion is widely used in: physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas and of price values). However, in each case, the object (e.g., atom, idea, etc.) that is undergoing diffusion is “spreading out” from a point or location at which there is a higher concentration of that object.

There are two ways to introduce the notion of diffusion: either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or a physical and atomistic one, by considering the random walk of the diffusing particles.[2]

In the phenomenological approach, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion. According to Fick's laws, the diffusion flux is proportional to the negative gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in the frame of thermodynamics and non-equilibrium thermodynamics.[3]

From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein.[4] The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals.

Biologists often use the terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the probability that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there is a net movement of oxygen molecules down the concentration gradient.

History of diffusion in physics

In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, Pliny the Elder had previously described the cementation process, which produces steel from the element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of stained glass or earthenware and Chinese ceramics.

In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:[5]

"...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time.”

The measurements of Graham contributed to James Clerk Maxwell deriving, in 1867, the coefficient of diffusion for CO2 in the air. The error rate is less than 5%.

In 1855, Adolf Fick, the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion. He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating a formalism that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827).

Robert Boyle demonstrated diffusion in solids in the 17th century[6] by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. William Chandler Roberts-Austen, the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on the example of gold in lead in 1896. :[7]

"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals."

In 1858, Rudolf Clausius introduced the concept of the mean free path. In the same year, James Clerk Maxwell developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion was developed by Albert Einstein, Marian Smoluchowski and Jean-Baptiste Perrin. Ludwig Boltzmann, in the development of the atomistic backgrounds of the macroscopic transport processes, introduced the Boltzmann equation, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.[8]

In 1920–1921, George de Hevesy measured self-diffusion using radioisotopes. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead.

Yakov Frenkel (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data.

Sometime later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.[7]

Henry Eyring, with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion.[9] The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.[10]

Basic models of diffusion

Diffusion flux

Each model of diffusion expresses the diffusion flux through concentrations, densities and their derivatives. Flux is a vector . The transfer of a physical quantity through a small area with normal per time is

where is the inner product and is the little-o notation. If we use the notation of vector area then

The dimension of the diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, , the diffusion equation has the form

where is intensity of any local source of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the no-flux boundary conditions can be formulated as on the boundary, where is the normal to the boundary at point .

Fick's law and equations

Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient:

The corresponding diffusion equation (Fick's second law) is

where is the Laplace operator,

Onsager's equations for multicomponent diffusion and thermodiffusion

Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, .

In 1931, Lars Onsager[11] included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport,

where is the flux of the ith physical quantity (component) and is the jth thermodynamic force.

The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s (he used the term "force" in quotation marks or "driving force"):

where are the "thermodynamic coordinates". For the heat and mass transfer one can take (the density of internal energy) and is the concentration of the ith component. The corresponding driving forces are the space vectors

because

where T is the absolute temperature and is the chemical potential of the ith component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.

For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:

where the derivatives of s are calculated at equilibrium n*. The matrix of the kinetic coefficients should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth).

The transport equations are

Here, all the indexes i, j, k = 0, 1, 2, ... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix of the diffusion (i,k > 0), thermodiffusion (i > 0, k = 0 or k > 0, i = 0) and thermal conductivity (i = k = 0) coefficients.

Under isothermal conditions T = constant. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, , and the matrix of diffusion coefficients is

(i,k > 0).

There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations can be measured. For example, in the original work of Onsager[11] the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics[12] this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not affect the measurable quantities.

Nondiagonal diffusion must be nonlinear

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form

If the matrix of diffusion coefficients is diagonal, then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, , and consider the state with . At this state, . If at some points, then becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.[10]

Einstein's mobility and Teorell formula

The Einstein relation (kinetic theory) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)[13]

where D is the diffusion constant, μ is the "mobility", kB is Boltzmann's constant, T is the absolute temperature.

Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation .

The mobility-based approach was further applied by T. Teorell.[14] In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula:

the flux is equal to mobility × concentration × force per gram-ion.

This is the so-called Teorell formula. The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains Avogadro's number of ions (particles). The common modern term is mole.

The force under isothermal conditions consists of two parts:

  1. Diffusion force caused by concentration gradient: .
  2. Electrostatic force caused by electric potential gradient: .

Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential.

The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, If for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule.

The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is[10]

where μ is the chemical potential, μ0 is the standard value of the chemical potential. The expression is the so-called activity. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form[10]

The standard derivation of the activity includes a normalization factor and for small concentrations , where is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity :

Teorell formula for multicomponent diffusion

The Teorell formula with combination of Onsager's definition of the diffusion force gives

where is the mobility of the ith component, is its activity, is the matrix of the coefficients, is the thermodynamic diffusion force, . For the isothermal perfect systems, . Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:

where is the matrix of coefficients. The Chapman–Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.

Jumps on the surface and in solids

Chemical surface diffusion slow
Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.

Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.

The system includes several reagents on the surface. Their surface concentrations are The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free places is . The sum of all (including free places) is constant, the density of adsorption places b.

The jump model gives for the diffusion flux of (i = 1, ..., n):

The corresponding diffusion equation is:[10]

Due to the conservation law, and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because . For two and more components the equations are nonlinear.

If all particles can exchange their positions with their closest neighbours then a simple generalization gives

where is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration .

Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.

Diffusion in porous media

For diffusion in porous media the basic equations are:[15]

where D is the diffusion coefficient, n is the concentration, m > 0 (usually m > 1, the case m = 1 corresponds to Fick's law).

For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is

where k is the permeability of the medium, μ is the viscosity and p is the pressure. The flux J = nv and for Darcy's law gives the equation of diffusion in porous media with m = γ + 1.

For underground water infiltration, the Boussinesq approximation gives the same equation with m = 2.

For plasma with the high level of radiation, the Zeldovich–Raizer equation gives m > 4 for the heat transfer.

Diffusion in physics

Elementary theory of diffusion coefficient in gases

Translational motion
Random collisions of particles in a gas.

The diffusion coefficient is the coefficient in the Fick's first law , where J is the diffusion flux (amount of substance) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length].

Let us consider two gases with molecules of the same diameter d and mass m (self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient

where kB is the Boltzmann constant, T is the temperature, P is the pressure, is the mean free path, and vT is the mean thermal speed:

We can see that the diffusion coefficient in the mean free path approximation grows with T as T3/2 and decreases with P as 1/P. If we use for P the ideal gas law P = RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T1/2 and for given temperature it decreases with the total concentration as 1/n.

For two different gases, A and B, with molecular masses mA, mB and molecular diameters dA, dB, the mean free path estimate of the diffusion coefficient of A in B and B in A is:

The theory of diffusion in gases based on Boltzmann's equation

In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, , where t is the time moment, x is position and c is velocity of molecule of the ith component of the mixture. Each component has its mean velocity . If the velocities do not coincide then there exists diffusion.

In the Chapman–Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:[8]

  • individual concentrations of particles, (particles per volume),
  • density of momentum (mi is the ith particle mass),
  • density of kinetic energy

The kinetic temperature T and pressure P are defined in 3D space as

where is the total density.

For two gases, the difference between velocities, is given by the expression:[8]

where is the force applied to the molecules of the ith component and is the thermodiffusion ratio.

The coefficient D12 is positive. This is the diffusion coefficient. Four terms in the formula for C1-C2 describe four main effects in the diffusion of gases:

  1. describes the flux of the first component from the areas with the high ratio n1/n to the areas with lower values of this ratio (and, analogously the flux of the second component from high n2/n to low n2/n because n2/n = 1 – n1/n);
  2. describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion;
  3. describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.
  4. describes thermodiffusion, the diffusion flux caused by the temperature gradient.

All these effects are called diffusion because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a bulk transport and differ from advection or convection.

In the first approximation,[8]

  • for rigid spheres;
  • for repulsing force

The number is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book[8])

We can see that the dependence on T for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration n for a given temperature has always the same character, 1/n.

In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity V is the mass average velocity. It is defined through the momentum density and the mass concentrations:

where is the mass concentration of the ith species, is the mass density.

By definition, the diffusion velocity of the ith component is , . The mass transfer of the ith component is described by the continuity equation

where is the net mass production rate in chemical reactions, .

In these equations, the term describes advection of the ith component and the term represents diffusion of this component.

In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.[16] For the diffusion velocities in multicomponent gases (N components) they used

Here, is the diffusion coefficient matrix, is the thermal diffusion coefficient, is the body force per unite mass acting on the ith species, is the partial pressure fraction of the ith species (and is the partial pressure), is the mass fraction of the ith species, and

Diffusion center
As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes.

Diffusion of electrons in solids

When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electron diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as diffusion current.

Diffusion current can also be described by Fick's first law

where J is the diffusion current density (amount of substance) per unit area per unit time, n (for ideal mixtures) is the electron density, x is the position [length].

Diffusion in geophysics

Analytical and numerical models that solve the diffusion equation for different initial and boundary conditions have been popular for studying a wide variety of changes to the Earth's surface. Diffusion has been used extensively in erosion studies of hillslope retreat, bluff erosion, fault scarp degradation, wave-cut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation.[17] Although the Earth's surface is not literally diffusing in many of these cases, the process of diffusion effectively mimics the holistic changes that occur over decades to millennia. Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.[18]

Random walk (random motion)

The apparent random motion of atoms, ions or molecules explained. Substances appear to move randomly due to collisions with other substances. From the iBook "Cell Membrane Transport", free license granted by IS3D, LLC, 2014.

One common misconception is that individual atoms, ions or molecules move randomly, which they do not. In the animation on the right, the ion on in the left panel has a “random” motion, but this motion is not random as it is the result of “collisions” with other ions. As such, the movement of a single atom, ion, or molecule within a mixture just appears random when viewed in isolation. The movement of a substance within a mixture by “random walk” is governed by the kinetic energy within the system that can be affected by changes in concentration, pressure or temperature.

Separation of diffusion from convection in gases

While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task.

Under normal conditions, molecular diffusion dominates only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection, and to study diffusion on the larger scale, special efforts are needed.

Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it can soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because of the temperature [inhomogeneity]. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).

In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

Other types of diffusion

See also

References

  1. ^ J.G. Kirkwood, R.L. Baldwin, P.J. Dunlop, L.J. Gosting, G. Kegeles (1960)Flow equations and frames of reference for isothermal diffusion in liquids. The Journal of Chemical Physics 33(5):1505–13.
  2. ^ J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Archived 2013-12-13 at the Wayback Machine Diffusion Fundamentals, 2, 1.1–1.10.
  3. ^ S.R. De Groot, P. Mazur (1962). Non-equilibrium Thermodynamics. North-Holland, Amsterdam.
  4. ^ A. Einstein (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (PDF). Ann. Phys. 17 (8): 549–60. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806.
  5. ^ Diffusion Processes, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971.
  6. ^ L.W. Barr (1997), In: Diffusion in Materials, DIMAT 96, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9.
  7. ^ a b H. Mehrer; N.A. Stolwijk (2009). "Heroes and Highlights in the History of Diffusion" (PDF). Diffusion Fundamentals. 11 (1): 1–32.
  8. ^ a b c d e S. Chapman, T. G. Cowling (1970) The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press (3rd edition), ISBN 052140844X.
  9. ^ J.F. Kincaid; H. Eyring; A.E. Stearn (1941). "The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State". Chem. Rev. 28 (2): 301–65. doi:10.1021/cr60090a005.
  10. ^ a b c d e A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". Mathematical Modelling of Natural Phenomena. 6 (5): 184–262. arXiv:1012.2908. doi:10.1051/mmnp/20116509.
  11. ^ a b Onsager, L. (1931). "Reciprocal Relations in Irreversible Processes. I". Physical Review. 37 (4): 405–26. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
  12. ^ L.D. Landau, E.M. Lifshitz (1980). Statistical Physics. Vol. 5 (3rd ed.). Butterworth-Heinemann. ISBN 978-0-7506-3372-7.
  13. ^ S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, ISBN 0815320515.
  14. ^ T. Teorell (1935). "Studies on the "Diffusion Effect" upon Ionic Distribution. Some Theoretical Considerations". Proceedings of the National Academy of Sciences of the United States of America. 21 (3): 152–61. Bibcode:1935PNAS...21..152T. doi:10.1073/pnas.21.3.152. PMC 1076553. PMID 16587950.
  15. ^ J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033.
  16. ^ S. H. Lam (2006). "Multicomponent diffusion revisited" (PDF). Physics of Fluids. 18 (7): 073101–073101–8. Bibcode:2006PhFl...18g3101L. doi:10.1063/1.2221312.
  17. ^ Pasternack, Gregory B.; Brush, Grace S.; Hilgartner, William B. (2001-04-01). "Impact of historic land-use change on sediment delivery to a Chesapeake Bay subestuarine delta". Earth Surface Processes and Landforms. 26 (4): 409–27. Bibcode:2001ESPL...26..409P. doi:10.1002/esp.189. ISSN 1096-9837.
  18. ^ Gregory B. Pasternack. "Watershed Hydrology, Geomorphology, and Ecohydraulics :: TFD Modeling". pasternack.ucdavis.edu. Retrieved 2017-06-12.
  19. ^ D. Ben-Avraham and S. Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems (PDF). Cambridge University Press. ISBN 978-0521622783.
  20. ^ Weiss, G. (1994). Aspects and Applications of the Random Walk. North-Holland. ISBN 978-0444816061.
Brownian motion

Brownian motion or pedesis (from Ancient Greek: πήδησις /pέːdεːsis/ "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid.This pattern of motion typically alternates random fluctuations in a particle's position inside a fluid sub-domain with a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such fluid there exists no preferential direction of flow as in transport phenomena. More specifically the fluid's overall linear and angular momenta remain null over time. It is important also to note that the kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations sum up to the caloric component of a fluid's internal energy.

This motion is named after the botanist Robert Brown, who was the most eminent microscopist of his time. In 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water, the triangular shaped pollen burst at the corners, emitting particles which he noted jiggled around in the water in random fashion. He was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules, making one of his first big contributions to science. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist, and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter". The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion.

The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist both simpler and more complicated stochastic processes which in extreme ("taken to the limit") may describe the Brownian Motion (see random walk and Donsker's theorem).

Cell membrane

The cell membrane (also known as the plasma membrane (PM) or cytoplasmic membrane, and historically referred to as the plasmalemma) is a biological membrane that separates the interior of all cells from the outside environment (the extracellular space) which protects the cell from its environment consisting of a lipid bilayer with embedded proteins. The cell membrane controls the movement of substances in and out of cells and organelles. In this way, it is selectively permeable to ions and organic molecules. In addition, cell membranes are involved in a variety of cellular processes such as cell adhesion, ion conductivity and cell signalling and serve as the attachment surface for several extracellular structures, including the cell wall, the carbohydrate layer called the glycocalyx, and the intracellular network of protein fibers called the cytoskeleton. In the field of synthetic biology, cell membranes can be artificially reassembled.

Cloud chamber

A cloud chamber, also known as a Wilson cloud chamber, is a particle detector used for visualizing the passage of ionizing radiation.

A cloud chamber consists of a sealed environment containing a supersaturated vapor of water or alcohol. An energetic charged particle (for example, an alpha or beta particle) interacts with the gaseous mixture by knocking electrons off gas molecules via electrostatic forces during collisions, resulting in a trail of ionized gas particles. The resulting ions act as condensation centers around which a mist-like trail of small droplets form if the gas mixture is at the point of condensation. These droplets are visible as a "cloud" track that persist for several seconds while the droplets fall through the vapor. These tracks have characteristic shapes. For example, an alpha particle track is thick and straight, while an electron track is wispy and shows more evidence of deflections by collisions.

Cloud chambers played a prominent role in the experimental particle physics from the 1920s to the 1950s, until the advent of the bubble chamber. In particular, the discoveries of the positron in 1932 (see Fig. 1) and the muon in 1936, both by Carl Anderson (awarded a Nobel Prize in Physics in 1936), used cloud chambers. Discovery of the kaon by George Rochester and Clifford Charles Butler in 1947, also was made using a cloud chamber as the detector.. In each case, cosmic rays were the source of ionizing radiation.

Convection

Convection is the heat transfer due to the bulk movement of molecules within fluids such as gases and liquids, including molten rock (rheid). Convection includes sub-mechanisms of advection (directional bulk-flow transfer of heat), and diffusion (non-directional transfer of energy or mass particles along a concentration gradient).

Convection cannot take place in most solids because neither bulk current flows nor significant diffusion of matter can take place. Diffusion of heat takes place in rigid solids, but that is called heat conduction. Convection, additionally may take place in soft solids or mixtures where solid particles can move past each other.

Thermal convection can be demonstrated by placing a heat source (e.g. a Bunsen burner) at the side of a glass filled with a liquid, and observing the changes in temperature in the glass caused by the warmer fluid circulating into cooler areas.

Convective heat transfer is one of the major types of heat transfer, and convection is also a major mode of mass transfer in fluids. Convective heat and mass transfer takes place both by diffusion – the random Brownian motion of individual particles in the fluid – and by advection, in which matter or heat is transported by the larger-scale motion of currents in the fluid. In the context of heat and mass transfer, the term "convection" is used to refer to the combined effects of advective and diffusive transfer. Sometimes the term "convection" is used to refer specifically to "free heat convection" (natural heat convection) where bulk-flow in a fluid is due to temperature-induced differences in buoyancy, as opposed to "forced heat convection" where forces other than buoyancy (such as pump or fan) move the fluid. However, in mechanics, the correct use of the word "convection" is the more general sense, and different types of convection should be further qualified, for clarity.

Convection can be qualified in terms of being natural, forced, gravitational, granular, or thermomagnetic. It may also be said to be due to combustion, capillary action, or Marangoni and Weissenberg effects. Heat transfer by natural convection plays a role in the structure of Earth's atmosphere, its oceans, and its mantle. Discrete convective cells in the atmosphere can be seen as clouds, with stronger convection resulting in thunderstorms. Natural convection also plays a role in stellar physics.

DIC Entertainment

DiC Entertainment was an international film and television production company that was also known as The Incredible World of DiC, DiC Audiovisuel, DiC Enterprises, DiC Animation City and DiC Productions at various times in its history. In 2008, DiC was acquired by the Cookie Jar Group and was folded into it. Most of the DiC library is currently owned by DHX Media after DHX acquired the Cookie Jar Group on October 22, 2012.

In addition to animated and live-action television shows, while under Disney, DiC produced live-action feature films, including Meet the Deedles (1998) and Inspector Gadget (1999), and licensed various anime series such as Sailor Moon, Saint Seiya and Speed Racer X.

Diaphragm (optics)

In optics, a diaphragm is a thin opaque structure with an opening (aperture) at its center. The role of the diaphragm is to stop the passage of light, except for the light passing through the aperture. Thus it is also called a stop (an aperture stop, if it limits the brightness of light reaching the focal plane, or a field stop or flare stop for other uses of diaphragms in lenses). The diaphragm is placed in the light path of a lens or objective, and the size of the aperture regulates the amount of light that passes through the lens. The centre of the diaphragm's aperture coincides with the optical axis of the lens system.

Most modern cameras use a type of adjustable diaphragm known as an iris diaphragm, and often referred to simply as an iris.

See the articles on aperture and f-number for the photographic effect and system of quantification of varying the opening in the diaphragm.

Diffusion MRI

Diffusion-weighted magnetic resonance imaging (DWI or DW-MRI) is the use of specific MRI sequences as well as software that generates images from the resulting data, that uses the diffusion of water molecules to generate contrast in MR images. It allows the mapping of the diffusion process of molecules, mainly water, in biological tissues, in vivo and non-invasively. Molecular diffusion in tissues is not free, but reflects interactions with many obstacles, such as macromolecules, fibers, and membranes. Water molecule diffusion patterns can therefore reveal microscopic details about tissue architecture, either normal or in a diseased state. A special kind of DWI, diffusion tensor imaging (DTI), has been used extensively to map white matter tractography in the brain.

Diffusion limited enzyme

A Diffusion limited enzyme is an enzyme which catalyses a reaction so efficiently that the rate limiting step is that of substrate diffusion into the active site, or product diffusion out. This is also known as kinetic perfection or catalytic perfection. Since the rate of catalysis of such enzymes is set by the diffusion-controlled reaction, it therefore represents an intrinsic, physical constraint on evolution (a maximum peak height in the fitness landscape). Diffusion limited perfect enzymes are very rare. Most enzymes catalyse their reactions to a rate that is 1,000-10,000 times slower than this limit. This is due to both the chemical limitations of difficult reactions, and the evolutionary limitations that such high reaction rates do not confer any extra fitness.

Diffusion of innovations

Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Everett Rogers, a professor of communication studies, popularized the theory in his book Diffusion of Innovations; the book was first published in 1962, and is now in its fifth edition (2003). Rogers argues that diffusion is the process by which an innovation is communicated over time among the participants in a social system. The origins of the diffusion of innovations theory are varied and span multiple disciplines.

Rogers proposes that four main elements influence the spread of a new idea: the innovation itself, communication channels, time, and a social system. This process relies heavily on human capital. The innovation must be widely adopted in order to self-sustain. Within the rate of adoption, there is a point at which an innovation reaches critical mass.

The categories of adopters are innovators, early adopters, early majority, late majority, and laggards. Diffusion manifests itself in different ways and is highly subject to the type of adopters and innovation-decision process. The criterion for the adopter categorization is innovativeness, defined as the degree to which an individual adopts a new idea.

Enriched uranium

Enriched uranium is a type of uranium in which the percent composition of uranium-235 has been increased through the process of isotope separation. Natural uranium is 99.284% 238U isotope, with 235U only constituting about 0.711% of its mass. 235U is the only nuclide existing in nature (in any appreciable amount) that is fissile with thermal neutrons.Enriched uranium is a critical component for both civil nuclear power generation and military nuclear weapons. The International Atomic Energy Agency attempts to monitor and control enriched uranium supplies and processes in its efforts to ensure nuclear power generation safety and curb nuclear weapons proliferation.

During the Manhattan Project enriched uranium was given the codename oralloy, a shortened version of Oak Ridge alloy, after the location of the plants where the uranium was enriched. The term oralloy is still occasionally used to refer to enriched uranium. There are about 2,000 tonnes (t, Mg) of highly enriched uranium in the world, produced mostly for nuclear power, nuclear weapons, naval propulsion, and smaller quantities for research reactors.

The 238U remaining after enrichment is known as depleted uranium (DU), and is considerably less radioactive than even natural uranium, though still very dense and extremely hazardous in granulated form – such granules are a natural by-product of the shearing action that makes it useful for armor-penetrating weapons and radiation shielding. At present, 95 percent of the world's stocks of depleted uranium remain in secure storage.

Fick's laws of diffusion

Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

Membrane transport protein

A membrane transport protein (or simply transporter) is a membrane protein involved in the movement of ions, small molecules, or macromolecules, such as another protein, across a biological membrane. Transport proteins are integral transmembrane proteins; that is they exist permanently within and span the membrane across which they transport substances. The proteins may assist in the movement of substances by facilitated diffusion or active transport. The two main types of proteins involved in such transport are broadly categorized as either channels or carriers. The solute carriers and atypical SLCs are secondary active or facilitative transporters in humans.

Molecular diffusion

Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. Once the concentrations are equal the molecules continue to move, but since there is no concentration gradient the process of molecular diffusion has ceased and is instead governed by the process of self-diffusion, originating from the random motion of the molecules. The result of diffusion is a gradual mixing of material such that the distribution of molecules is uniform. Since the molecules are still in motion, but an equilibrium has been established, the end result of molecular diffusion is called a "dynamic equilibrium". In a phase with uniform temperature, absent external net forces acting on the particles, the diffusion process will eventually result in complete mixing.

Consider two systems; S1 and S2 at the same temperature and capable of exchanging particles. If there is a change in the potential energy of a system; for example μ1>μ2 (μ is Chemical potential) an energy flow will occur from S1 to S2, because nature always prefers low energy and maximum entropy.

Molecular diffusion is typically described mathematically using Fick's laws of diffusion.

Osmosis

Osmosis () is the spontaneous net movement of solvent molecules through a selectively permeable membrane into a region of higher solute concentration, in the direction that tends to equalize the solute concentrations on the two sides. It may also be used to describe a physical process in which any solvent moves across a selectively permeable membrane (permeable to the solvent, but not the solute) separating two solutions of different concentrations. Osmosis can be made to do work. Osmotic pressure is defined as the external pressure required to be applied so that there is no net movement of solvent across the membrane. Osmotic pressure is a colligative property, meaning that the osmotic pressure depends on the molar concentration of the solute but not on its identity.

Osmosis is a vital process in biological systems, as biological membranes are semipermeable. In general, these membranes are impermeable to large and polar molecules, such as ions, proteins, and polysaccharides, while being permeable to non-polar or hydrophobic molecules like lipids as well as to small molecules like oxygen, carbon dioxide, nitrogen, and nitric oxide. Permeability depends on solubility, charge, or chemistry, as well as solute size. Water molecules travel through the plasma membrane, tonoplast membrane (vacuole) or protoplast by diffusing across the phospholipid bilayer via aquaporins (small transmembrane proteins similar to those responsible for facilitated diffusion and ion channels). Osmosis provides the primary means by which water is transported into and out of cells. The turgor pressure of a cell is largely maintained by osmosis across the cell membrane between the cell interior and its relatively hypotonic environment.

Passive transport

Passive transport is a movement of ions and other atomic or molecular substances across cell membranes without need of energy input. Unlike active transport, it does not require an input of cellular energy because it is instead driven by the tendency of the system to grow in entropy. The rate of passive transport depends on the permeability of the cell membrane, which, in turn, depends on the organization and characteristics of the membrane lipids and proteins. The four main kinds of passive transport are simple diffusion, facilitated diffusion, filtration, and/or osmosis.

Siphoviridae

Siphoviridae is a family of double-stranded DNA viruses in the order Caudovirales. Bacteria and archaea serve as natural hosts. There are currently 313 species in this family, divided among 47 genera. The characteristic structural features of this family are a nonenveloped head and noncontractile tail.

Technological change

Technological change (TC), technological development, technological achievement, or technological progress is the overall process of invention, innovation and diffusion of technology or processes. In essence, technological change covers the invention of technologies (including processes) and their commercialization or release as open source via research and development (producing emerging technologies), the continual improvement of technologies (in which they often become less expensive), and the diffusion of technologies throughout industry or society (which sometimes involves disruption and convergence). In short, technological change is based on both better and more technology.

Trans-cultural diffusion

In cultural anthropology and cultural geography, cultural diffusion, as conceptualized by Leo Frobenius in his 1897/98 publication Der westafrikanische Kulturkreis, is the spread of cultural items—such as ideas, styles, religions, technologies, languages—between individuals, whether within a single culture or from one culture to another. It is distinct from the diffusion of innovations within a specific culture. Examples of diffusion include the spread of the war chariot and iron smelting in ancient times, and the use of automobiles and Western business suits in the 20th century.

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