Fick's laws of diffusion

5857:

surface. But real adsorption is often done much faster than this infinite time limit, i.e., the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have predicted for low or none energy barrier adsorption (unless there is a significant adsorption energy barrier that slows down the absorption significantly), for example, thousands to millions time faster in the self-assembly of monolayers at the water-air or water-substrate interfaces. As such, it is necessary to calculate the evolution of the concentration gradient near the surface and find out a proper time to stop the imagined infinite evolution for practical applications. While it is hard to predict when to stop but it is reasonably easy to calculate the shortest time that matters, the critical time when the first nearest neighbor from the substrate surface feels the building-up of the concentration gradient. This yields the upper limit of the adsorption rate under an ideal situation when there are no other factors than diffusion that affect the absorber dynamics:

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passenger usually comes from many layers of neighbors away from the target, thus its arriving time is significantly longer than the nearest neighbor diffusion time. Using the mean free path time plus the Langmuir equation will cause an artificial concentration gradient between the initial location of the first passenger and the target surface because the other neighbor layers have no change yet, thus significantly lower estimate the actual binding time, i.e., the actual first passenger arriving time itself, the inverse of the above rate, is difficult to calculate. If the system can be simplified to 1D diffusion, then the average first passenger time can be calculated using the same nearest neighbor critical diffusion time for the first neighbor distance to be the MSD,

5841: 4343: 6397: 4022: 3608: 4653:, from which Fick's law can be obtained as a limiting case, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell–Stefan equations are used. See also non-diagonal coupled transport processes ( 2086: 2238: 3993: 3367: 8200:

first law can also be used to predict the changing moisture profiles across a spaghetti noodle as it hydrates during cooking. These phenomena are all about the spontaneous movement of particles of solutes driven by the concentration gradient. In different situations, there is different diffusivity which is a constant.

4338:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).} 6417:

molecule, e.g. B is the target molecule holding fixed relatively, and A is the moving molecule that creates a concentration gradient near the target molecule B due to the coagulation reaction between A and B. Smoluchowski calculated the collision frequency between A and B in the solution with unit #/s/m:

44: 6381:

relative diffusion constant between two diffusing molecules. This estimation is especially useful in studying the interaction between a small molecule and a larger molecule such as a protein. The effective diffusion constant is dominated by the smaller one whose diffusion constant can be used instead.

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The formulation of Fick's first law can explain a variety of complex phenomena in the context of food and cooking: Diffusion of molecules such as ethylene promotes plant growth and ripening, salt and sugar molecules promotes meat brining and marinating, and water molecules promote dehydration. Fick's

7100:

Under the condition of a diluted solution when diffusion takes control, the membrane permeability mentioned in the above section can be theoretically calculated for the solute using the equation mentioned in the last section (use with particular care because the equation is derived for dense solutes,

6643:

However, under a practical condition, the concentration gradient near the target molecule is evolving over time with the molecular flux evolving as well, and on average the flux is much bigger than the infinite time limit flux Smoluchowski has proposed. Before the first passenger arrival time, Fick's

6380:

such as DNA), the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule (random orientation) hit one side of the other. The diffusion constant need to be updated to the

4913:

rate of a dilute solute to a surface or interface in a (gas or liquid) solution can be calculated using Fick's laws of diffusion. The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in

4480: 163:

to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region

8173:

In many realistic situations, the simple Fick's law is not an adequate formulation for the semiconductor problem. It only applies to certain conditions, for example, given the semiconductor boundary conditions: constant source concentration diffusion, limited source concentration, or moving boundary

6926:

instead of 2. Both the Smoluchowski equation and the JChen equation satisfy dimensional checks with SI units. But the former is dependent on the radius and the latter is on the area of the collision sphere. From dimensional analysis, there will be an equation dependent on the volume of the collision

5856:

A more problematic result of the above equations is they predict the lower limit of adsorption under ideal situations but is very difficult to predict the actual adsorption rates. The equations are derived at the long-time-limit condition when a stable concentration gradient has been formed near the

5836:

Monte Carlo simulations show that these two equations work to predict the adsorption rate of systems that form predictable concentration gradients near the surface but have troubles for systems without or with unpredictable concentration gradients, such as typical biosensing systems or when flow and

4695:

Adsorption, absorption, and collision of molecules, particles, and surfaces are important problems in many fields. These fundamental processes regulate chemical, biological, and environmental reactions. Their rate can be calculated using the diffusion constant and Fick's laws of diffusion especially

1092:

Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref. shows in detail how the

6388:

on a surface. Molecules are randomly oriented in the bulk solution. Assuming 1/6 of the molecules has the right orientation to the surface binding sites, i.e. 1/2 of the z-direction in x, y, z three dimensions, thus the concentration of interest is just 1/6 of the bulk concentration. Put this value

6211:

In this critical time, it is unlikely the first passenger has arrived and adsorbed. But it sets the speed of the layers of neighbors to arrive. At this speed with a concentration gradient that stops around the first neighbor layer, the gradient does not project virtually in the longer time when the

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Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally

7615:

To keep the reaction balanced, reactants must diffuse through the stagnant boundary layer to reach the substrate. So a thin boundary layer is desirable. According to the equations, increasing vo would result in more wasted reactants. The reactants will not reach the substrate uniformly if the flow

7278:

The flux is decay over the square root of time because a concentration gradient builds up near the membrane over time under ideal conditions. When there is flow and convection, the flux can be significantly different than the equation predicts and show an effective time t with a fixed value, which

5852:

nature of diffusion. Most computer simulations pick a time step for diffusion which ignores the fact that there are self-similar finer diffusion events (fractal) within each step. Simulating the fractal diffusion shows that a factor of two corrections should be introduced for the result of a fixed

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The viscous flow regime of CVD is driven by a pressure gradient. CVD also includes a diffusion component distinct from the surface diffusion of adatoms. In CVD, reactants and products must also diffuse through a boundary layer of stagnant gas that exists next to the substrate. The total number of

7313:

The wafer is a kind of semiconductor whose silicon substrate is coated with a layer of CVD-created polymer chain and films. This film contains n-type and p-type dopants and takes responsibility for dopant conductions. The principle of CVD relies on the gas phase and gas-solid chemical reaction to

7295:

The relationship between Fick's law and semiconductors: the principle of the semiconductor is transferring chemicals or dopants from a layer to a layer. Fick's law can be used to control and predict the diffusion by knowing how much the concentration of the dopants or chemicals move per meter and

6037:

This equation can be used to predict the initial adsorption rate of any system; It can be used to predict the steady-state adsorption rate of a typical biosensing system when the binding site is just a very small fraction of the substrate surface and a near-surface concentration gradient is never

4678:

In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's

8190:

In such a process, the movements of diffusing species (atoms, molecules, plasma etc.) are treated as a discrete entity, following a random walk through the CVD reactor, boundary layer, material structures etc. Sometimes, the movements might follow a biased-random walk depending on the processing

8182:

Even though Fickian diffusion has been used to model diffusion processes in semiconductor manufacturing (including CVD reactors) in early days, it often fails to validate the diffusion in advanced semiconductor nodes (< 90 nm). This mostly stems from the inability of Fickian diffusion to

7085:

Fick's first law is also important in radiation transfer equations. However, in this context, it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this

6647:

In 2022, Chen calculates the upper limit of the collision frequency between A and B in a solution assuming the bulk concentration of the moving molecule is fixed after the first nearest neighbor of the target molecule. Thus the concentration gradient evolution stops at the first nearest neighbor

6416:

and Fick's laws of diffusion. Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target

6041:

This critical time is significantly different from the first passenger arriving time or the mean free-path time. Using the average first-passenger time and Fick's law of diffusion to estimate the average binding rate will significantly over-estimate the concentration gradient because the first

2101: 4674:

take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.

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assuming cubic packing each unit has 8 neighbors shared with other units. This example fraction converges the result to the 3D diffusive adsorption solution shown above with a slight difference in pre-factor due to different packing assumptions and ignoring other

4900:

Scheme of molecular diffusion in the solution. Orange dots are solute molecules, solvent molecules are not drawn, black arrow is an example random walk trajectory, and the red curve is the diffusive Gaussian broadening probability function from the Fick's law of

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liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular

8183:

model diffusion processes accurately at molecular level and smaller. In advanced semiconductor manufacturing, it is important to understand the movement at atomic scales, which is failed by continuum diffusion. Today, most semiconductor manufacturers use

7423: 1550: 2702: 4358: 3603:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}} 2852:. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space – that is, the corrosion product layer – is 2839: 7318:

steps required for CVD film growth are gas phase diffusion of reactants through the boundary layer, adsorption and surface diffusion of adatoms, reactions on the substrate, and gas phase diffusion of products away through the boundary layer.

1929: 5506: 8459: 66:: With an enormous number of solute molecules, randomness becomes undetectable: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Fick's laws. 6639:

by replacing the moving speed of the molecule with diffusive flux. In the collision theory, the traveling time between A and B is proportional to the distance which is a similar relationship for the diffusion case if the flux is fixed.

7725: 3241: 2385: 5385: 3771: 2081:{\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0} 7291:

is a collective term for a series of devices. It mainly includes three categories：two-terminal devices, three-terminal devices, and four-terminal devices. The combination of the semiconductors is called an integrated circuit.

6761: 5637: 4636:

methods is based on solutions of Fick's equation. On the other hand, in some cases a "Fickian (another common approximation of the transport equation is that of the diffusion theory)" description is inadequate. For example, in

2538: 776: 2307: 2233:{\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}} 3988:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.} 905: 6927:

sphere but eventually, all equations should converge to the same numerical rate of the collision that can be measured experimentally. The actual reaction order for a bimolecular unit reaction could be between 2 and

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makes the flux stable instead of decay over time. A critical time has been estimated under idealized flow conditions when there is no gradient formed. This strategy is adopted in biology such as blood circulation.

1423: 1241: 8272: 8090: 517: 4833: 164:

of high concentration to a region of low concentration across a concentration gradient. In one (spatial) dimension, the law can be written in various forms, where the most common form (see) is in a molar basis:

2612: 5926: 6906:

This equation assumes the upper limit of a diffusive collision frequency between A and B is when the first neighbor layer starts to feel the evolution of the concentration gradient, whose reaction order is

7611: 3335:

has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is a Maxwell-Boltzmann distribution.

7021: 140:

possible. Today, Fick's Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does

7891: 4971: 3139: 7324: 7175: 4897: 1155: 1319: 8517: 8191:

conditions. Statistical analysis is done to understand variation/stochasticity arising from the random walk of the species, which in-turn affects the overall process and electrical variations.

1082: 665: 124:, which fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: 6038:

formed; It can also be used to predict the adsorption rate of molecules on the surface when there is a significant flow to push the concentration gradient very shallowly in the sub-surface.

8164: 6492: 388: 6644:

equation predicts a concentration gradient over time which does not build up yet in reality. Thus, this Smoluchowski frequency represents the lower limit of the real collision frequency.

6285: 5517: 214: 5136: 120:

In 1855, physiologist Adolf Fick first reported his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of

1828: 8187:

to study and model diffusion processes. This allows us to study the effects of diffusion in a discrete manner to understand the movement of individual atoms, molecules, plasma etc.

295: 2444: 3674: 7632: 3167: 6844: 6573: 5955: 3068: 1477: 1032: 1000: 6134: 6079: 3679: 2630: 3313: 1350: 1272: 7473: 6369: 5848:

A brief history of diffusive adsorption is shown in the right figure. A noticeable challenge of understanding the diffusive adsorption at the single-molecule level is the

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is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant

8004: 7965: 7813: 7784: 7755: 7532: 7510: 7208: 6900: 6873: 6629: 6602: 6161: 6006: 5831: 5791: 5007: 1450: 962: 935: 608: 5298: 5268: 5128: 5102: 8170:

As a result, Fick's first law tells us we can use a partial pressure gradient to control the diffusivity and control the growth of thin films of semiconductors.

6310:

is a factor of converting the 3D diffusive adsorption problem into a 1D diffusion problem whose value depends on the system, e.g., a fraction of adsorption area

8113: 7914: 6518: 6328: 6308: 6205: 6183: 6028: 5977: 5811: 5408: 5061: 3333: 3261: 2325: 1823: 1803: 4475:{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right).} 1699: 1607: 7101:

while biological molecules are not denser than water. Also this equation assumes ideal concentration gradient forms near the membrane and evolves over time):

5306: 5756:{\displaystyle \Gamma =2A{C_{\text{b}}}{\sqrt {\frac {Dt}{\pi }}}-A{\sqrt {\frac {D}{\pi }}}\int _{0}^{\sqrt {t}}{\frac {C(\tau )}{\sqrt {t-\tau }}}\,d\tau } 8363: 1355: 1173: 8011: 6654: 2483: 54:

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.

5631:

The Langmuir–Schaefer equation can be extended to the Ward–Tordai Equation to account for the "back-diffusion" of rejected molecules from the surface:

694: 144:

follow Fick's laws (which happens in cases of diffusion through porous media and diffusion of swelling penetrants, among others), it is referred to as

7305:

fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law.

2252: 6648:

layer given a stop-time to calculate the actual flux. He named this the critical time and derived the diffusive collision frequency in unit #/s/m:

6212:

actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem,

7553: 8992: 835: 324: 7825: 327:. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 3271:

8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a

1724: 684:

Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of

4735:

in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:

452: 99:: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient. 4741: 1100: 1277: 4509: 17: 4683:, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by 2573: 5863: 4679:

law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a

6404:

The bimolecular collision frequency related to many reactions including protein coagulation/aggregation is initially described by

4727:. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the 3279:, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion. 6960: 4920: 3083: 8551: 8877:

Pandey S, Gautam D, Chen J (16 July 2024). "Measuring the Adsorption Cross Section of YOYO-1 to Immobilized DNA Molecules".

7107: 4641:

science and food science a more general approach is required to describe transport of components in materials undergoing a

2319:

If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields

6946:, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules. 3641: 8964:

Smoluchowski M (1916). "Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen".

8770:

Ward AF, Tordai L (1946). "Time-dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time-effects".

9189: 4867: 2856:, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is 1037: 620: 8567:

Brogioli D, Vailati A (January 2001). "Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited".

5813:

is the sub-surface concentration (which is a function of time depending on the reaction model of the adsorption), and

8486: 6405: 5620:{\displaystyle \Gamma =\int _{0}^{t}({\frac {\partial \Gamma }{\partial t}})_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}} 2712: 2466: 8120: 6423: 4699:

Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the

356: 9194: 8217: 5230:{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}(-{\frac {x^{2}}{4Dt}})} 4546:

which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the

6218: 8378: 4728: 4348: 170: 9204: 8203:

By controlling the concentration gradient, the cooking time, shape of the food, and salting can be controlled.

7418:{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }}} 1468: 263: 7819:

The equation tells that increasing the temperature or decreasing the pressure can increase the diffusivity.

2410: 62:: With more molecules, there is a clear trend where the solute fills the container more and more uniformly. 9159: 8410: 8242: 4893:

in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute.

4724: 4700: 4654: 4646: 4547: 2927:(as the diffusion now occurs in both directions). This case is valid when some solution with concentration 2474: 7822:

Fick's first law predicts the flux of the reactants to the substrate and product away from the substrate:

1545:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}}} 108:

A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called

9199: 2697:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}} 8247: 6796: 6525: 5934: 2973: 7616:

becomes turbulent. Another option is to switch to a new carrier gas with lower viscosity or density.

4512:

include exactly the same terms. These physical models of diffusion are different from the test models

2617:

Plugging the definition of diffusive flux to the continuity equation and assuming there is no source (

1005: 973: 319:

is proportional to the squared velocity of the diffusing particles, which depends on the temperature,

9000: 7093:

The exchange rate of a gas across a fluid membrane can be determined by using this law together with

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into the equation one should be able to calculate the theoretical adsorption kinetic curve using the

6087: 6048: 3161: 436: 3285: 2834:{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right).} 1326: 1248: 7453: 4005: 2095:

to be a constant, one can exchange the orders of the differentiation and multiply by the constant:

306:(for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume. 6333: 1924:{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).} 6030:

is the diffusion constant of the absorber (solute) in the solution (m/s) defined with Fick's law.

5501:{\displaystyle ({\frac {\partial \Gamma }{\partial t}})_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}} 1094: 121: 8478:

Essentials of Micro- and Nanofluidics: With Applications to the Biological and Chemical Sciences

7619:

The Fick's first law describes diffusion through the boundary layer. As a function of pressure (

7215: 4553:

For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example

242:

measures the amount of substance that will flow through a unit area during a unit time interval.

9093: 7034: 6772: 4910: 4629: 806: 8413:, Sargsyan HP, Wahab HA (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". 3144:

This idea is useful for estimating a diffusion length over a heating and cooling cycle, where

7919: 7431: 6635:

The reaction order of this bimolecular reaction is 2 which is the analogy to the result from

6393:. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry. 5026: 2450: 335:. For biological molecules the diffusion coefficients normally range from 10 to 10 m/s. 9085: 7974: 7940: 5957:

is the adsorption rate assuming under adsorption energy barrier-free situation, in unit #/s.

5853:

time-step adsorption simulation, bringing it to be consistent with the above two equations.

8973: 8920: 8833: 8779: 8641: 8586: 8311: 7791: 7762: 7733: 7720:{\displaystyle D=D_{0}\left({\frac {P_{0}}{P}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2}} 7517: 7495: 7242:

transmembrane efficiency (unitless), which can be calculated from the stochastic theory of

7186: 6878: 6851: 6607: 6580: 6139: 5984: 5816: 5769: 4985: 4858: 3236:{\displaystyle {\text{MSD}}\equiv \langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\rangle =2nDt} 1778: 1428: 940: 913: 593: 231: 5009:

is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m).

2380:{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).} 89:. Fick's first law can be used to derive his second law which in turn is identical to the 8: 8740:

Langmuir I, Schaefer VJ (1937). "The Effect of Dissolved Salts on Insoluble Monolayers".

7060: 6409: 5277: 5247: 5107: 5081: 4889:(e.g. proteins) in water, the exponential term is negligible due to the small product of 4621: 2718: 235: 109: 47: 8977: 8924: 8837: 8783: 8645: 8590: 8315: 5380:{\displaystyle ({\frac {\partial C}{\partial x}})_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}} 5130:) in a once uniform bulk solution is solved in the above sections from Fick's equation, 3766:{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.} 1467:

predicts how diffusion causes the concentration to change with respect to time. It is a

81:

in 1855 on the basis of largely experimental results. They can be used to solve for the

9184: 9043: 8941: 8908: 8854: 8821: 8714: 8689: 8662: 8629: 8610: 8576: 8440: 8422: 8098: 7899: 7302: 6503: 6313: 6293: 6190: 6168: 6136:(unit m) is the average nearest neighbor distance approximated as cubic packing, where 6013: 5962: 5796: 5393: 5078:

Briefly as explained in, the concentration gradient profile near a newly created (from

5046: 4866:

is the mobility of the particle in the fluid or gas, which can be calculated using the

4849: 3318: 3246: 2454: 1808: 1788: 685: 90: 3151: 1650: 1565: 9135: 9086: 9035: 8946: 8859: 8719: 8667: 8602: 8547: 8482: 4732: 4720: 4716: 4680: 4598: 4009: 3267:

of the particle's Brownian motion. For example, the diffusion of a molecule across a

2400:, the solution for the concentration will be a linear change of concentrations along 1939: 8444: 6756:{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt{C_{A}+C_{B}}}} 6384:

The above hitting rate equation is also useful to predict the kinetics of molecular

5043:

is the accumulated number of molecules in unit # molecules adsorbed during the time

2533:{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,} 9131: 9047: 9027: 8936: 8928: 8882: 8849: 8841: 8787: 8749: 8709: 8701: 8657: 8649: 8614: 8594: 8432: 8342: 8319: 6636: 5072: 4642: 3315:, is often used as a characterization of how far has the particle moved after time 254: 133: 82: 771:{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}}} 8476: 8227: 7094: 6413: 5068: 4896: 4684: 3272: 3157: 9015: 8909:"Why Should the Reaction Order of a Bimolecular Reaction be 2.33 Instead of 2?" 8598: 8335:

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science

7243: 5979:

is the area of the surface of interest on an "infinite and flat" substrate (m).

4625: 3364:. This dependence does not affect Fick's first law but the second law changes: 2849: 2302:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,} 125: 35: 31: 8690:"Photobleaching of YOYO-1 in super-resolution single DNA fluorescence imaging" 8346: 2246:

For the case of diffusion in two or more dimensions Fick's second law becomes

9178: 9031: 8932: 8886: 8324: 8299: 7288: 7056: 6385: 5840: 4650: 3268: 2964: 2313: 1774: 688:

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

105:: Prediction of change in concentration gradient with time due to diffusion. 8822:"Simulating stochastic adsorption of diluted solute molecules at interfaces" 8546:. Wiley Series in Chemical Engineering. Vol. 2. John Wiley & Sons. 8436: 2936:

is put in contact with a layer of pure solvent. (Bokstein, 2005) The length

9039: 8950: 8863: 8723: 8671: 8606: 8232: 7299:

Therefore, different types and levels of semiconductors can be fabricated.

7087: 4610: 2963:

As a quick approximation of the error function, the first two terms of the

435:

Another form for the first law is to write it with the primary variable as

129: 8364:"One and a Half Centuries of Diffusion: Fick, Einstein, before and beyond" 4690: 2952:

and provides a measure of how far the concentration has propagated in the

900:{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}}} 8705: 8581: 8184: 6008:

is the concentration of the absorber molecule in the bulk solution (#/m).

4886: 4671: 4666: 4633: 3614: 2243:

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

1782: 9016:"Understanding diffusion theory and Fick's law through food and cooking" 8753: 3619:, the diffusion coefficient depends on the direction. It is a symmetric 1763:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi } 1034:

phase. At vapor liquid equilibrium the evaporation flux is zero because

8653: 6163:

is the solute concentration in the bulk solution (unit # molecule / m).

4906: 4606: 4504:

are related to the various components and not to the space coordinates.

569: 78: 50:

from a microscopic and macroscopic point of view. Initially, there are

8845: 8791: 2719:

Example solution 1: constant concentration source and diffusion length

1418:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},} 1236:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.} 43: 9165: 8237: 8222: 8212: 8085:{\displaystyle J=-D_{i}\left({\frac {P_{i}-P_{0}}{\delta RT}}\right)} 6376:

When the area of interest is the size of a molecule (specifically, a

4914:

the simulated molecular diffusion in the first section of this page:

3264: 2708: 2470: 1712: 512:{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i}} 320: 74: 4828:{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right)} 402:

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

9169: 6185:

is the diffusion coefficient defined by Fick's equation (unit m/s).

4703:. In the ultrashort time limit, in the order of the diffusion time 3152:

Example solution 2: Brownian particle and mean squared displacement

826: 611: 347: 8427: 8174:

diffusion (where junction depth keeps moving into the substrate).

1718:, which generalises the second derivative, obtaining the equation 8252: 8006:, the concentration of the gas is expressed by partial pressure. 5849: 4638: 2607:{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi } 583: 9088:

Thermodynamics and Kinetics in Materials Science: A Short Course

7274:

concentration should use unit mol m, so flux unit becomes mol s.

6396: 5921:{\displaystyle \langle r\rangle ={\frac {4}{\pi }}Ac_{b}^{4/3}D} 5300:

is simplified to the pre-exponential factor of the distribution

4019:

these two forms of the diffusion equation should be combined in

4614: 4602: 4597:

Equations based on Fick's law have been commonly used to model

3620: 3276: 1938:

Fick's second law can be derived from Fick's first law and the

51: 4351:

gives the following generalization of Fick's equation for the

2453:, the solutions to which are referred to by mathematicians as 2563:. The only source of flux in this situation is assumed to be 446:, given for example in kg/kg), then the equation changes to: 7033:

is the permeability, an experimentally determined membrane "

1352:) both species have the same molar mass, Fick's law becomes 350:

operator, which generalises the first derivative, obtaining

323:

of the fluid and the size of the particles according to the

8735: 8733: 7606:{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}} 4617: 2550: 1087: 160: 9083: 8273:"Molecular Diffusion - an overview | ScienceDirect Topics" 7016:{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)}} 6400:

Comparing collision theory and diffusive collision theory.

4715:

is the particle radius, the diffusion is described by the

427:, which for ideal mixtures is the concentration gradient. 7886:{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right)} 7282: 6846:

is the relative diffusion constant between A and B (m/s).

4966:{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}} 3134:{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.} 2473:

and no net volumetric source. It can be derived from the

343: 8815: 8813: 8811: 8809: 8807: 8805: 8803: 8801: 8730: 8518:"Fickian Diffusion - an overview | ScienceDirect Topics" 7308: 7210:

is the total area of the pores on the membrane (unit m).

6902:

are number concentrations of A and B respectively (#/m).

6631:

are number concentrations of A and B respectively (#/m).

6575:

is the relative diffusion constant between A and B (m/s)

5844:

A brief history of the theories on diffusive adsorption.

1773:

Fick's second law has the same mathematical form as the

4691:

Sorption rate and collision frequency of diluted solute

2460: 9084:

Bokshtein BS, Mendelev MI, Srolovitz DJ, eds. (2005).

6791:

is the area of the cross-section of the collision (m).

2902:), then the solution is amended only with coefficient 1274:, this reduces to the most common form of Fick's law, 825:

The driving force of Fick's law can be expressed as a

8798: 8123: 8101: 8014: 7977: 7943: 7922: 7902: 7828: 7794: 7765: 7736: 7635: 7556: 7520: 7498: 7456: 7434: 7327: 7218: 7189: 7170:{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {D/(\pi t)}}} 7110: 6963: 6881: 6854: 6799: 6775: 6657: 6610: 6583: 6528: 6506: 6426: 6336: 6316: 6296: 6221: 6193: 6171: 6142: 6090: 6051: 6016: 5987: 5965: 5937: 5866: 5819: 5799: 5772: 5640: 5520: 5419: 5396: 5309: 5280: 5250: 5244:

is the number concentration of adsorber molecules at

5139: 5110: 5084: 5049: 5029: 4988: 4923: 4744: 4696:

when these interactions happen in diluted solutions.

4361: 4025: 3779: 3682: 3644: 3370: 3321: 3288: 3249: 3170: 3086: 2976: 2756: 2633: 2576: 2486: 2413: 2328: 2255: 2104: 1951: 1831: 1811: 1791: 1727: 1653: 1568: 1480: 1431: 1358: 1329: 1280: 1251: 1176: 1103: 1040: 1008: 976: 943: 916: 838: 697: 623: 596: 455: 359: 266: 173: 9074: 4008:. It is needed to make the right hand side operator 1150:{\displaystyle \mathbf {V_{i}} =-D\nabla \ln y_{i},} 8409: 6954:The first law gives rise to the following formula: 4885:For a single molecule such as organic molecules or 2738:, where the concentration is maintained at a value 2707:If flux were the result of both diffusive flux and 1314:{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .} 9160:Fick's equations, Boltzmann's transformation, etc. 8627: 8177: 8158: 8107: 8084: 7998: 7959: 7928: 7908: 7885: 7807: 7778: 7749: 7719: 7605: 7526: 7504: 7467: 7440: 7417: 7234: 7202: 7169: 7015: 6894: 6867: 6838: 6783: 6755: 6623: 6596: 6567: 6512: 6486: 6363: 6322: 6302: 6279: 6199: 6177: 6155: 6128: 6073: 6022: 6000: 5971: 5949: 5920: 5825: 5805: 5785: 5755: 5619: 5500: 5402: 5379: 5292: 5262: 5229: 5122: 5096: 5055: 5035: 5001: 4965: 4827: 4474: 4337: 3987: 3765: 3668: 3602: 3327: 3307: 3255: 3235: 3133: 3062: 2833: 2696: 2606: 2532: 2438: 2379: 2301: 2232: 2080: 1933: 1923: 1817: 1797: 1762: 1693: 1601: 1544: 1444: 1417: 1344: 1313: 1266: 1235: 1149: 1076: 1026: 994: 956: 929: 899: 770: 659: 602: 511: 382: 289: 208: 8765: 8763: 8395:Vázquez JL (2006). "The Porous Medium Equation". 7252:is the diffusion constant of the solute unit m⋅s. 6330:over solute nearest neighbor sphere surface area 6033:Dimensional analysis of these units is satisfied. 1077:{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}} 660:{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }}} 9176: 9145:Foundations of Materials Science and Engineering 8957: 8902: 8900: 8898: 8896: 8876: 8739: 5274:The concentration gradient at the subsurface at 3077:is time-dependent, the diffusion length becomes 27:Mathematical descriptions of molecular diffusion 9013: 8870: 8566: 5015:is diffusion coefficient of the adsorber (m/s). 3995:The symmetric matrix of diffusion coefficients 8760: 8683: 8681: 8457: 8194: 8159:{\displaystyle {\frac {P_{i}-P_{0}}{\delta }}} 6487:{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B}} 5067:The equation is named after American chemists 4510:Chapman–Enskog formulae for diffusion in gases 3773:For the diffusion equation this formula gives 1647:is the diffusion coefficient in dimensions of 430: 383:{\displaystyle \mathbf {J} =-D\nabla \varphi } 312:is position, the dimension of which is length. 9014:Zhou L, Nyberg K, Rowat AC (September 2015). 8893: 8628:Bian X, Kim C, Karniadakis GE (August 2016). 8541: 8403: 5390:And the rate of diffusion (flux) across area 4496:is the matrix of coefficients. Here, indices 4112: 4072: 3857: 3826: 3457: 3417: 3349:, the diffusion coefficient varies in space, 9122:Fick, Adolph (1995). "On liquid diffusion". 9092:. Oxford: Oxford University Press. pp. 8963: 6412:in a seminal 1916 publication, derived from 6280:{\displaystyle <r>=a/t=2aC_{b}^{2/3}D} 5944: 5938: 5873: 5867: 3676:it is the product of a tensor and a vector: 3215: 3179: 8678: 8415:Mathematical Modelling of Natural Phenomena 2731:-axis) from a boundary located at position 2465:Fick's second law is a special case of the 209:{\displaystyle J=-D{\frac {d\varphi }{dx}}} 58:: A single molecule moves around randomly. 9075:Bird RB, Stewart WE, Lightfoot EN (1976). 8769: 8481:. Cambridge University Press. p. 43. 1711:In two or more dimensions we must use the 8940: 8853: 8713: 8661: 8580: 8426: 8361: 8323: 7037:" for a given gas at a given temperature. 6949: 5746: 4763: 4687:the fluctuating hydrodynamics equations. 4660: 4489:are concentrations of the components and 4453: 3119: 2364: 2282: 2071: 1507: 8742:Journal of the American Chemical Society 8687: 8501: 6395: 5839: 4895: 3156:Another simple case of diffusion is the 1785:, except switching thermal conductivity 1088:Derivation of Fick's first law for gases 42: 8474: 8461:Physical Chemistry for the Life Science 8394: 4349:Einstein's mobility and Teorell formula 2390:An important example is the case where 1628:is a function that depends on location 1097:reduces to this version of Fick's law: 290:{\displaystyle {\frac {d\varphi }{dx}}} 14: 9177: 9110:Fick A (1855). "On liquid diffusion". 8333:Fick A (1855). "On liquid diffusion". 7967:is the first reactant's concentration. 7321:The velocity profile for gas flow is: 7283:Semiconductor fabrication applications 3275:is a 3-D diffusion. For a cylindrical 2439:{\displaystyle \nabla ^{2}\varphi =0,} 2404:. In two or more dimensions we obtain 1942:in absence of any chemical reactions: 1674: 1660: 1582: 1574: 1562:is the concentration in dimensions of 338:In two or more dimensions we must use 258:. Its dimension is area per unit time. 9142: 9100: 7627:) in a gas, diffusion is determined. 7309:CVD method of fabricate semiconductor 4589:correspond to the space coordinates. 2860:(lasting both through the layer with 2723:A simple case of diffusion with time 1166:is the diffusion velocity of species 9121: 9109: 9065: 8906: 8819: 8332: 8297: 4645:. One more general framework is the 3669:{\displaystyle J=-D\nabla \varphi ,} 2461:Example solutions and generalization 1459: 555:th species (for example in mol/m-s), 8913:The Journal of Physical Chemistry A 8694:Beilstein Journal of Nanotechnology 8688:Pyle JR, Chen J (2 November 2017). 2624:), we arrive at Fick's second law: 2091:Assuming the diffusion coefficient 1170:. In terms of species flux this is 964:is a partial pressure of component 399:denotes the diffusion flux vector. 151: 24: 9058: 8502:Williams FA (1985). "Appendix E". 7582: 7579: 7550:, it gives the average thickness: 7461: 7458: 7390: 7387: 7362: 7359: 5641: 5556: 5551: 5548: 5521: 5434: 5429: 5426: 5324: 5316: 5151: 5143: 5030: 4924: 4868:Einstein relation (kinetic theory) 4770: 4450: 4402: 4380: 4365: 4311: 4288: 4269: 4242: 4220: 4207: 4178: 4089: 4064: 4052: 4029: 3966: 3953: 3924: 3834: 3818: 3806: 3783: 3744: 3736: 3657: 3584: 3561: 3542: 3525: 3477: 3434: 3409: 3397: 3374: 3339: 2678: 2664: 2645: 2637: 2598: 2510: 2498: 2490: 2415: 2365: 2352: 2340: 2332: 2284: 2267: 2259: 2214: 2200: 2178: 2174: 2163: 2159: 2134: 2130: 2111: 2107: 2054: 2050: 2031: 2027: 2013: 2005: 1981: 1977: 1963: 1955: 1754: 1739: 1731: 1526: 1512: 1492: 1484: 1399: 1330: 1302: 1252: 1217: 1125: 888: 873: 821:is the chemical potential (J/mol). 759: 744: 496: 374: 25: 9216: 9153: 8990: 8290: 8166:is the partial pressure gradient. 6839:{\displaystyle D_{r}=D_{A}+D_{B}} 6568:{\displaystyle D_{r}=D_{A}+D_{B}} 6406:Smoluchowski coagulation equation 5950:{\displaystyle \langle r\rangle } 3063:{\displaystyle n(x,t)=n_{0}\left} 1323:If (instead of or in addition to 9020:Advances in Physiology Education 7063:for the direction of flow (from 5104:) absorptive surface (placed at 3199: 3195: 3186: 3160:of one particle. The particle's 2956:-direction by diffusion in time 2579: 2517: 1452:is the mole fraction of species 1365: 1361: 1287: 1283: 1183: 1179: 1110: 1106: 1027:{\displaystyle f_{i}^{\text{L}}} 995:{\displaystyle f_{i}^{\text{G}}} 458: 361: 9007: 8984: 8879:Journal of Physical Chemistry B 8621: 8560: 8535: 8178:Invalidity of Fickian diffusion 6129:{\displaystyle L~=C_{b}^{-1/3}} 6074:{\displaystyle L={\sqrt {2Dt}}} 4729:fluctuation-dissipation theorem 4592: 4017:inhomogeneous anisotropic media 3164:from its original position is: 2727:in one dimension (taken as the 2557:is a net volumetric source for 1934:Derivation of Fick's second law 815:is the absolute temperature (K) 30:For the technique of measuring 8997:Essentials of Human Physiology 8630:"111 years of Brownian motion" 8510: 8495: 8468: 8451: 8388: 8355: 8265: 7815:is the standard diffusitivity. 7481:is the length of the substrate 7337: 7331: 7162: 7153: 6520:is the radius of the collision 6207:is the critical time (unit s). 5729: 5723: 5566: 5542: 5444: 5420: 5334: 5310: 5224: 5193: 4815: 4806: 4754: 4748: 4306: 4294: 4264: 4258: 4202: 4190: 4171: 4165: 4107: 4095: 4086: 4080: 4047: 4035: 3948: 3936: 3852: 3840: 3801: 3789: 3638:. Fick's first law changes to 3579: 3567: 3537: 3531: 3495: 3483: 3474: 3468: 3452: 3440: 3431: 3425: 3392: 3380: 3308:{\displaystyle {\sqrt {2nDt}}} 3206: 3182: 3116: 3110: 2992: 2980: 2371: 2358: 1999: 1847: 1835: 1688: 1654: 1596: 1569: 1471:which in one dimension reads: 1345:{\displaystyle \nabla \rho =0} 1267:{\displaystyle \nabla \rho =0} 677:is the partial density of the 13: 1: 9162:(with figures and animations) 9112:Annalen der Physik und Chemie 8458:Atkins P, de Paula J (2006). 7468:{\displaystyle \mathrm {Re} } 7296:second through mathematics. 5837:convection are significant. 4649:equations of multi-component 2713:convection–diffusion equation 2467:convection–diffusion equation 1469:partial differential equation 238:per unit area per unit time. 9147:(3rd ed.). McGraw-Hill. 9136:10.1016/0376-7388(94)00230-v 9103:The Mathematics of Diffusion 8544:Multicomponent mass transfer 8542:Taylor R, Krishna R (1993). 8258: 8218:Churchill–Bernstein equation 6364:{\displaystyle 4\pi L^{2}/4} 4578:refer to the components and 1093:diffusion equation from the 799:is the concentration (mol/m) 18:Fick's law of diffusion 7: 9124:Journal of Membrane Science 8772:Journal of Chemical Physics 8206: 8195:Food production and cooking 7786:is the standard temperature 5793:is the bulk concentration, 4875:is the mass of the particle 4355:of the perfect components: 1805:with diffusion coefficient 614:operator. This is because: 431:Variations of the first law 10: 9221: 9105:. Oxford University Press. 8599:10.1103/PhysRevE.63.012105 8377:: 1.1–1.10. Archived from 7235:{\displaystyle \eta _{tm}} 2312:which is analogous to the 1707:is the position, example m 115: 112:or non-Fickian diffusion. 77:and were first posited by 29: 9190:Eponymous laws of physics 8347:10.1080/14786445508641925 7086:situation, one can use a 6784:{\displaystyle {\sigma }} 6391:Langmuir adsorption model 6378:long cylindrical molecule 4632:processes. The theory of 3162:Mean squared displacement 3148:varies with temperature. 9079:. John Wiley & Sons. 9032:10.1152/advan.00133.2014 8993:"Section 3/3ch9/s3ch9_2" 8933:10.1021/acs.jpca.2c07500 8907:Chen J (December 2022). 8887:10.1021/acs.jpcb.4c03359 8325:10.1002/andp.18551700105 8243:Maxwell–Stefan diffusion 7757:is the standard pressure 4980:is the surface area (m). 4838:where (all in SI units) 4725:Stokes–Einstein equation 4719:. At a longer time, the 4701:Stokes–Einstein equation 4647:Maxwell–Stefan diffusion 4548:Maxwell–Stefan diffusion 4353:multicomponent diffusion 3282:The square root of MSD, 325:Stokes–Einstein relation 132:(charge transport), and 71:Fick's laws of diffusion 9195:Mathematics in medicine 9068:Random Walks in Biology 8820:Chen J (January 2022). 7929:{\displaystyle \delta } 7441:{\displaystyle \delta } 5511:Integrating over time, 5036:{\displaystyle \Gamma } 1095:kinetic theory of gases 8966:Zeitschrift für Physik 8371:Diffusion Fundamentals 8248:Nernst–Planck equation 8160: 8109: 8086: 8000: 7999:{\displaystyle PV=nRT} 7961: 7960:{\displaystyle dc_{i}} 7930: 7910: 7887: 7809: 7780: 7751: 7721: 7607: 7528: 7506: 7475:is the Reynolds number 7469: 7442: 7419: 7236: 7204: 7171: 7059:of the gas across the 7017: 6950:Biological perspective 6896: 6869: 6840: 6785: 6757: 6625: 6598: 6569: 6514: 6488: 6401: 6365: 6324: 6304: 6281: 6201: 6179: 6157: 6130: 6075: 6024: 6002: 5973: 5951: 5922: 5845: 5833:is a dummy variable. 5827: 5807: 5787: 5757: 5621: 5502: 5404: 5381: 5294: 5264: 5231: 5124: 5098: 5057: 5037: 5003: 4967: 4902: 4829: 4661:Fick's flow in liquids 4476: 4347:The approach based on 4339: 4146: 3989: 3906: 3885: 3767: 3719: 3670: 3604: 3521: 3329: 3309: 3257: 3237: 3135: 3064: 2835: 2698: 2608: 2534: 2440: 2381: 2303: 2234: 2082: 1925: 1819: 1799: 1764: 1695: 1603: 1546: 1446: 1419: 1346: 1315: 1268: 1237: 1151: 1078: 1028: 996: 958: 931: 901: 807:universal gas constant 772: 661: 604: 586:(for example in kg/m). 513: 384: 299:concentration gradient 291: 210: 159:relates the diffusive 67: 9205:Statistical mechanics 8522:www.sciencedirect.com 8437:10.1051/mmnp/20116509 8399:. Oxford Univ. Press. 8277:www.sciencedirect.com 8161: 8110: 8087: 8001: 7962: 7931: 7911: 7888: 7810: 7808:{\displaystyle D_{0}} 7781: 7779:{\displaystyle T_{0}} 7752: 7750:{\displaystyle P_{0}} 7722: 7608: 7529: 7527:{\displaystyle \rho } 7507: 7505:{\displaystyle \eta } 7470: 7443: 7420: 7237: 7205: 7203:{\displaystyle A_{P}} 7172: 7055:is the difference in 7018: 6897: 6895:{\displaystyle C_{B}} 6870: 6868:{\displaystyle C_{A}} 6841: 6786: 6758: 6626: 6624:{\displaystyle C_{B}} 6599: 6597:{\displaystyle C_{A}} 6570: 6515: 6489: 6399: 6366: 6325: 6305: 6282: 6202: 6180: 6158: 6156:{\displaystyle C_{b}} 6131: 6076: 6025: 6003: 6001:{\displaystyle C_{b}} 5974: 5952: 5923: 5843: 5828: 5826:{\displaystyle \tau } 5808: 5788: 5786:{\displaystyle C_{b}} 5758: 5622: 5503: 5405: 5382: 5295: 5265: 5232: 5125: 5099: 5058: 5038: 5004: 5002:{\displaystyle C_{b}} 4968: 4899: 4830: 4624:, nuclear materials, 4477: 4340: 4120: 3990: 3886: 3865: 3768: 3699: 3671: 3605: 3501: 3347:non-homogeneous media 3330: 3310: 3258: 3238: 3136: 3065: 2848:is the complementary 2836: 2699: 2609: 2535: 2469:in which there is no 2441: 2382: 2304: 2235: 2083: 1926: 1820: 1800: 1765: 1696: 1604: 1547: 1447: 1445:{\displaystyle x_{i}} 1420: 1347: 1316: 1269: 1238: 1152: 1079: 1029: 997: 959: 957:{\displaystyle f_{i}} 932: 930:{\displaystyle f_{i}} 902: 773: 662: 605: 603:{\displaystyle \rho } 549:diffusion flux vector 514: 385: 292: 250:diffusion coefficient 211: 83:diffusion coefficient 46: 8972:: 557–571, 585–599. 8706:10.3762/bjnano.8.229 8506:. Benjamin/Cummings. 8362:Philibert J (2005). 8121: 8099: 8012: 7975: 7941: 7920: 7900: 7826: 7792: 7763: 7734: 7633: 7554: 7518: 7496: 7454: 7432: 7325: 7314:create thin films. 7216: 7187: 7108: 6961: 6879: 6852: 6797: 6773: 6655: 6608: 6581: 6526: 6504: 6424: 6334: 6314: 6294: 6219: 6191: 6169: 6140: 6088: 6049: 6014: 5985: 5963: 5935: 5864: 5817: 5797: 5770: 5638: 5518: 5417: 5394: 5307: 5278: 5248: 5137: 5108: 5082: 5047: 5027: 5021:is elapsed time (s). 4986: 4921: 4859:absolute temperature 4742: 4630:semiconductor doping 4359: 4023: 3777: 3680: 3642: 3368: 3319: 3286: 3247: 3168: 3084: 2974: 2754: 2631: 2574: 2484: 2411: 2326: 2253: 2102: 1949: 1829: 1809: 1789: 1779:fundamental solution 1725: 1651: 1566: 1478: 1429: 1356: 1327: 1278: 1249: 1174: 1101: 1038: 1006: 974: 941: 914: 836: 695: 621: 594: 453: 357: 264: 171: 9077:Transport Phenomena 8978:1916ZPhy...17..557S 8925:2022JPCA..126.9719C 8838:2022AIPA...12a5318C 8784:1946JChPh..14..453W 8754:10.1021/ja01290a091 8646:2016SMat...12.6331B 8591:2000PhRvE..63a2105B 8475:Conlisk AT (2013). 8397:Mathematical Theory 8384:on 5 February 2009. 8316:1855AnP...170...59F 8115:is the gas constant 7623:) and temperature ( 6410:Marian Smoluchowski 6273: 6125: 5914: 5716: 5541: 5293:{\displaystyle x=0} 5263:{\displaystyle x,t} 5123:{\displaystyle x=0} 5097:{\displaystyle t=0} 4622:population dynamics 4599:transport processes 3106: 2475:continuity equation 1781:is the same as the 1073: 1055: 1023: 991: 236:amount of substance 110:anomalous diffusion 48:Molecular diffusion 9200:Physical chemistry 8654:10.1039/c6sm01153e 8575:(1 Pt 1): 012105. 8304:Annalen der Physik 8156: 8105: 8082: 7996: 7957: 7926: 7906: 7883: 7805: 7776: 7747: 7717: 7603: 7524: 7502: 7465: 7438: 7415: 7303:Integrated circuit 7232: 7200: 7167: 7013: 6892: 6865: 6836: 6781: 6753: 6621: 6594: 6565: 6510: 6484: 6402: 6361: 6320: 6300: 6277: 6251: 6197: 6175: 6153: 6126: 6100: 6071: 6020: 5998: 5969: 5947: 5918: 5892: 5846: 5823: 5803: 5783: 5753: 5700: 5617: 5527: 5498: 5400: 5377: 5290: 5260: 5227: 5120: 5094: 5053: 5033: 4999: 4963: 4903: 4850:Boltzmann constant 4825: 4472: 4401: 4335: 3985: 3763: 3666: 3600: 3325: 3305: 3253: 3233: 3131: 3092: 3060: 2831: 2694: 2604: 2530: 2455:harmonic functions 2451:Laplace's equation 2436: 2377: 2299: 2230: 2078: 1921: 1815: 1795: 1760: 1691: 1641:is time, example s 1599: 1542: 1442: 1415: 1342: 1311: 1264: 1245:If, additionally, 1233: 1147: 1074: 1059: 1041: 1024: 1009: 992: 977: 954: 927: 897: 768: 686:chemical potential 657: 600: 509: 380: 287: 206: 136:(heat transport). 128:(hydraulic flow), 91:diffusion equation 68: 9166:Fick's Second Law 9143:Smith WF (2004). 9003:on 24 March 2016. 8919:(51): 9719–9725. 8846:10.1063/5.0064140 8792:10.1063/1.1724167 8748:(11): 2400–2414. 8640:(30): 6331–6346. 8569:Physical Review E 8553:978-0-471-57417-0 8504:Combustion Theory 8300:"Ueber Diffusion" 8154: 8108:{\displaystyle R} 8076: 7971:In ideal gas law 7916:is the thickness 7909:{\displaystyle x} 7877: 7697: 7671: 7601: 7413: 7380: 7165: 6967: 6751: 6682: 6513:{\displaystyle R} 6323:{\displaystyle A} 6303:{\displaystyle a} 6200:{\displaystyle t} 6178:{\displaystyle D} 6096: 6069: 6023:{\displaystyle D} 5972:{\displaystyle A} 5887: 5806:{\displaystyle C} 5744: 5743: 5714: 5698: 5697: 5680: 5679: 5661: 5615: 5614: 5563: 5496: 5495: 5441: 5403:{\displaystyle A} 5375: 5374: 5331: 5222: 5191: 5186: 5185: 5158: 5056:{\displaystyle t} 4961: 4960: 4733:Langevin equation 4721:Langevin equation 4717:Langevin equation 4448: 4392: 4387: 4325: 4283: 4234: 4059: 4006:positive definite 3980: 3813: 3758: 3598: 3556: 3404: 3328:{\displaystyle t} 3303: 3256:{\displaystyle n} 3174: 3126: 3049: 3046: 2822: 2819: 2692: 2652: 2586: 2505: 2347: 2274: 2228: 2185: 2170: 2141: 2118: 2061: 2038: 2020: 1988: 1970: 1940:mass conservation 1911: 1872: 1871: 1818:{\displaystyle D} 1798:{\displaystyle k} 1746: 1609:, example mol/m; 1540: 1499: 1465:Fick's second law 1460:Fick's second law 1397: 1215: 1071: 1053: 1021: 989: 895: 868: 766: 739: 655: 494: 285: 204: 103:Fick's second law 16:(Redirected from 9212: 9148: 9139: 9119: 9106: 9101:Crank J (1980). 9097: 9091: 9080: 9071: 9066:Berg HC (1977). 9052: 9051: 9011: 9005: 9004: 8999:. Archived from 8988: 8982: 8981: 8961: 8955: 8954: 8944: 8904: 8891: 8890: 8874: 8868: 8867: 8857: 8817: 8796: 8795: 8767: 8758: 8757: 8737: 8728: 8727: 8717: 8685: 8676: 8675: 8665: 8625: 8619: 8618: 8584: 8582:cond-mat/0006163 8564: 8558: 8557: 8539: 8533: 8532: 8530: 8528: 8514: 8508: 8507: 8499: 8493: 8492: 8472: 8466: 8465: 8455: 8449: 8448: 8430: 8407: 8401: 8400: 8392: 8386: 8385: 8383: 8368: 8359: 8353: 8350: 8329: 8327: 8294: 8288: 8287: 8285: 8283: 8269: 8165: 8163: 8162: 8157: 8155: 8150: 8149: 8148: 8136: 8135: 8125: 8114: 8112: 8111: 8106: 8091: 8089: 8088: 8083: 8081: 8077: 8075: 8064: 8063: 8062: 8050: 8049: 8039: 8033: 8032: 8005: 8003: 8002: 7997: 7966: 7964: 7963: 7958: 7956: 7955: 7935: 7933: 7932: 7927: 7915: 7913: 7912: 7907: 7892: 7890: 7889: 7884: 7882: 7878: 7876: 7868: 7867: 7866: 7853: 7847: 7846: 7814: 7812: 7811: 7806: 7804: 7803: 7785: 7783: 7782: 7777: 7775: 7774: 7756: 7754: 7753: 7748: 7746: 7745: 7726: 7724: 7723: 7718: 7716: 7715: 7711: 7702: 7698: 7696: 7695: 7683: 7676: 7672: 7667: 7666: 7657: 7651: 7650: 7612: 7610: 7609: 7604: 7602: 7600: 7599: 7598: 7594: 7585: 7572: 7564: 7549: 7545: 7541: 7533: 7531: 7530: 7525: 7511: 7509: 7508: 7503: 7489: 7480: 7474: 7472: 7471: 7466: 7464: 7448:is the thickness 7447: 7445: 7444: 7439: 7424: 7422: 7421: 7416: 7414: 7409: 7398: 7393: 7385: 7381: 7379: 7378: 7374: 7365: 7356: 7348: 7241: 7239: 7238: 7233: 7231: 7230: 7209: 7207: 7206: 7201: 7199: 7198: 7176: 7174: 7173: 7168: 7166: 7152: 7144: 7142: 7141: 7129: 7128: 7080: 7071: 7054: 7032: 7022: 7020: 7019: 7014: 7012: 7011: 7007: 7006: 7005: 6993: 6992: 6968: 6965: 6945: 6943: 6942: 6939: 6936: 6932: 6925: 6923: 6922: 6919: 6916: 6912: 6901: 6899: 6898: 6893: 6891: 6890: 6874: 6872: 6871: 6866: 6864: 6863: 6845: 6843: 6842: 6837: 6835: 6834: 6822: 6821: 6809: 6808: 6790: 6788: 6787: 6782: 6780: 6762: 6760: 6759: 6754: 6752: 6750: 6745: 6744: 6743: 6731: 6730: 6720: 6718: 6717: 6708: 6707: 6698: 6697: 6688: 6683: 6675: 6670: 6669: 6637:collision theory 6630: 6628: 6627: 6622: 6620: 6619: 6603: 6601: 6600: 6595: 6593: 6592: 6574: 6572: 6571: 6566: 6564: 6563: 6551: 6550: 6538: 6537: 6519: 6517: 6516: 6511: 6493: 6491: 6490: 6485: 6483: 6482: 6473: 6472: 6463: 6462: 6450: 6439: 6438: 6370: 6368: 6367: 6362: 6357: 6352: 6351: 6329: 6327: 6326: 6321: 6309: 6307: 6306: 6301: 6286: 6284: 6283: 6278: 6272: 6268: 6259: 6238: 6206: 6204: 6203: 6198: 6184: 6182: 6181: 6176: 6162: 6160: 6159: 6154: 6152: 6151: 6135: 6133: 6132: 6127: 6124: 6120: 6108: 6094: 6080: 6078: 6077: 6072: 6070: 6059: 6029: 6027: 6026: 6021: 6007: 6005: 6004: 5999: 5997: 5996: 5978: 5976: 5975: 5970: 5956: 5954: 5953: 5948: 5927: 5925: 5924: 5919: 5913: 5909: 5900: 5888: 5880: 5832: 5830: 5829: 5824: 5812: 5810: 5809: 5804: 5792: 5790: 5789: 5784: 5782: 5781: 5762: 5760: 5759: 5754: 5745: 5733: 5732: 5718: 5715: 5710: 5708: 5699: 5690: 5689: 5681: 5675: 5667: 5666: 5664: 5663: 5662: 5659: 5626: 5624: 5623: 5618: 5616: 5610: 5602: 5601: 5599: 5598: 5580: 5579: 5564: 5562: 5554: 5546: 5540: 5535: 5507: 5505: 5504: 5499: 5497: 5485: 5484: 5483: 5482: 5466: 5458: 5457: 5442: 5440: 5432: 5424: 5410:of the plane is 5409: 5407: 5406: 5401: 5386: 5384: 5383: 5378: 5376: 5364: 5363: 5362: 5353: 5348: 5347: 5332: 5330: 5322: 5314: 5299: 5297: 5296: 5291: 5269: 5267: 5266: 5261: 5243: 5236: 5234: 5233: 5228: 5223: 5221: 5210: 5209: 5200: 5192: 5189: 5187: 5175: 5174: 5173: 5164: 5159: 5157: 5149: 5141: 5129: 5127: 5126: 5121: 5103: 5101: 5100: 5095: 5073:Vincent Schaefer 5062: 5060: 5059: 5054: 5042: 5040: 5039: 5034: 5020: 5014: 5008: 5006: 5005: 5000: 4998: 4997: 4979: 4972: 4970: 4969: 4964: 4962: 4956: 4948: 4947: 4945: 4944: 4834: 4832: 4831: 4826: 4824: 4820: 4819: 4818: 4805: 4775: 4774: 4773: 4723:merges into the 4657:relationship). 4643:glass transition 4588: 4577: 4567: 4545: 4503: 4499: 4495: 4488: 4481: 4479: 4478: 4473: 4468: 4464: 4463: 4462: 4449: 4447: 4446: 4437: 4436: 4427: 4425: 4424: 4400: 4388: 4386: 4378: 4377: 4376: 4363: 4344: 4342: 4341: 4336: 4331: 4327: 4326: 4324: 4323: 4322: 4309: 4286: 4284: 4282: 4281: 4280: 4267: 4257: 4256: 4240: 4235: 4233: 4232: 4231: 4219: 4218: 4205: 4186: 4185: 4175: 4164: 4163: 4145: 4140: 4116: 4115: 4076: 4075: 4060: 4058: 4050: 4027: 4003: 3994: 3992: 3991: 3986: 3981: 3979: 3978: 3977: 3965: 3964: 3951: 3932: 3931: 3921: 3919: 3918: 3905: 3900: 3884: 3879: 3861: 3860: 3830: 3829: 3814: 3812: 3804: 3781: 3772: 3770: 3769: 3764: 3759: 3757: 3756: 3755: 3742: 3734: 3732: 3731: 3718: 3713: 3692: 3691: 3675: 3673: 3672: 3667: 3637: 3609: 3607: 3606: 3601: 3599: 3597: 3596: 3595: 3582: 3559: 3557: 3555: 3554: 3553: 3540: 3523: 3520: 3515: 3461: 3460: 3421: 3420: 3405: 3403: 3395: 3372: 3363: 3334: 3332: 3331: 3326: 3314: 3312: 3311: 3306: 3304: 3290: 3262: 3260: 3259: 3254: 3242: 3240: 3239: 3234: 3214: 3213: 3204: 3203: 3202: 3189: 3175: 3172: 3147: 3140: 3138: 3137: 3132: 3127: 3105: 3100: 3091: 3076: 3069: 3067: 3066: 3061: 3059: 3055: 3054: 3050: 3048: 3047: 3036: 3027: 3007: 3006: 2959: 2955: 2950:diffusion length 2947: 2946: 2945: 2935: 2926: 2917: 2915: 2914: 2911: 2908: 2901: 2894: 2877: 2870: 2847: 2840: 2838: 2837: 2832: 2827: 2823: 2821: 2820: 2812: 2803: 2791: 2790: 2778: 2774: 2746: 2737: 2730: 2726: 2703: 2701: 2700: 2695: 2693: 2691: 2690: 2689: 2676: 2672: 2671: 2661: 2653: 2651: 2643: 2635: 2623: 2613: 2611: 2610: 2605: 2588: 2587: 2584: 2582: 2562: 2556: 2548: 2539: 2537: 2536: 2531: 2520: 2506: 2504: 2496: 2488: 2445: 2443: 2442: 2437: 2423: 2422: 2403: 2399: 2395: 2386: 2384: 2383: 2378: 2348: 2346: 2338: 2330: 2308: 2306: 2305: 2300: 2292: 2291: 2275: 2273: 2265: 2257: 2239: 2237: 2236: 2231: 2229: 2227: 2226: 2225: 2212: 2208: 2207: 2197: 2186: 2184: 2173: 2171: 2169: 2158: 2150: 2146: 2142: 2140: 2129: 2119: 2117: 2106: 2094: 2087: 2085: 2084: 2079: 2070: 2066: 2062: 2060: 2049: 2039: 2037: 2026: 2021: 2019: 2011: 2003: 1989: 1987: 1976: 1971: 1969: 1961: 1953: 1930: 1928: 1927: 1922: 1917: 1913: 1912: 1910: 1899: 1898: 1889: 1873: 1858: 1854: 1824: 1822: 1821: 1816: 1804: 1802: 1801: 1796: 1769: 1767: 1766: 1761: 1747: 1745: 1737: 1729: 1717: 1706: 1700: 1698: 1697: 1694:{\displaystyle } 1692: 1687: 1686: 1678: 1677: 1670: 1669: 1664: 1663: 1646: 1640: 1635: 1631: 1627: 1608: 1606: 1605: 1602:{\displaystyle } 1600: 1595: 1594: 1586: 1585: 1578: 1577: 1561: 1551: 1549: 1548: 1543: 1541: 1539: 1538: 1537: 1524: 1520: 1519: 1509: 1500: 1498: 1490: 1482: 1455: 1451: 1449: 1448: 1443: 1441: 1440: 1424: 1422: 1421: 1416: 1411: 1410: 1398: 1396: 1395: 1386: 1378: 1370: 1369: 1368: 1351: 1349: 1348: 1343: 1320: 1318: 1317: 1312: 1292: 1291: 1290: 1273: 1271: 1270: 1265: 1242: 1240: 1239: 1234: 1229: 1228: 1216: 1214: 1213: 1204: 1196: 1188: 1187: 1186: 1169: 1165: 1156: 1154: 1153: 1148: 1143: 1142: 1115: 1114: 1113: 1083: 1081: 1080: 1075: 1072: 1069: 1067: 1054: 1051: 1049: 1033: 1031: 1030: 1025: 1022: 1019: 1017: 1001: 999: 998: 993: 990: 987: 985: 969: 963: 961: 960: 955: 953: 952: 936: 934: 933: 928: 926: 925: 906: 904: 903: 898: 896: 894: 886: 885: 884: 871: 869: 867: 856: 848: 847: 820: 814: 804: 798: 792: 788: 777: 775: 774: 769: 767: 765: 757: 756: 755: 742: 740: 738: 730: 729: 728: 715: 707: 706: 680: 676: 666: 664: 663: 658: 656: 651: 650: 638: 633: 632: 609: 607: 606: 601: 581: 575: 567: 554: 546: 533: 529: 518: 516: 515: 510: 508: 507: 495: 493: 492: 483: 475: 467: 466: 461: 445: 426: 425: 423: 422: 416: 413: 398: 389: 387: 386: 381: 364: 341: 334: 332: 318: 311: 305: 296: 294: 293: 288: 286: 284: 276: 268: 247: 241: 225: 215: 213: 212: 207: 205: 203: 195: 187: 157:Fick's first law 152:Fick's first law 97:Fick's first law 88: 21: 9220: 9219: 9215: 9214: 9213: 9211: 9210: 9209: 9175: 9174: 9156: 9151: 9120:– reprinted in 9061: 9059:Further reading 9056: 9055: 9012: 9008: 8989: 8985: 8962: 8958: 8905: 8894: 8875: 8871: 8818: 8799: 8768: 8761: 8738: 8731: 8686: 8679: 8626: 8622: 8565: 8561: 8554: 8540: 8536: 8526: 8524: 8516: 8515: 8511: 8500: 8496: 8489: 8473: 8469: 8456: 8452: 8408: 8404: 8393: 8389: 8381: 8366: 8360: 8356: 8298:Fick A (1855). 8295: 8291: 8281: 8279: 8271: 8270: 8266: 8261: 8228:False diffusion 8209: 8197: 8180: 8144: 8140: 8131: 8127: 8126: 8124: 8122: 8119: 8118: 8100: 8097: 8096: 8065: 8058: 8054: 8045: 8041: 8040: 8038: 8034: 8028: 8024: 8013: 8010: 8009: 7976: 7973: 7972: 7951: 7947: 7942: 7939: 7938: 7921: 7918: 7917: 7901: 7898: 7897: 7869: 7862: 7858: 7854: 7852: 7848: 7842: 7838: 7827: 7824: 7823: 7799: 7795: 7793: 7790: 7789: 7770: 7766: 7764: 7761: 7760: 7741: 7737: 7735: 7732: 7731: 7707: 7703: 7691: 7687: 7682: 7678: 7677: 7662: 7658: 7656: 7652: 7646: 7642: 7634: 7631: 7630: 7590: 7586: 7578: 7577: 7573: 7565: 7563: 7555: 7552: 7551: 7547: 7543: 7539: 7538:Integrated the 7519: 7516: 7515: 7497: 7494: 7493: 7484: 7478: 7457: 7455: 7452: 7451: 7433: 7430: 7429: 7399: 7397: 7386: 7370: 7366: 7358: 7357: 7349: 7347: 7343: 7326: 7323: 7322: 7311: 7285: 7273: 7266: 7258:is time unit s. 7223: 7219: 7217: 7214: 7213: 7194: 7190: 7188: 7185: 7184: 7148: 7143: 7134: 7130: 7124: 7120: 7109: 7106: 7105: 7079: 7073: 7070: 7064: 7053: 7046: 7040: 7030: 7001: 6997: 6988: 6984: 6983: 6979: 6972: 6964: 6962: 6959: 6958: 6952: 6940: 6937: 6934: 6933: 6930: 6928: 6920: 6917: 6914: 6913: 6910: 6908: 6886: 6882: 6880: 6877: 6876: 6859: 6855: 6853: 6850: 6849: 6830: 6826: 6817: 6813: 6804: 6800: 6798: 6795: 6794: 6776: 6774: 6771: 6770: 6746: 6739: 6735: 6726: 6722: 6721: 6719: 6713: 6709: 6703: 6699: 6693: 6689: 6684: 6674: 6662: 6658: 6656: 6653: 6652: 6615: 6611: 6609: 6606: 6605: 6588: 6584: 6582: 6579: 6578: 6559: 6555: 6546: 6542: 6533: 6529: 6527: 6524: 6523: 6505: 6502: 6501: 6478: 6474: 6468: 6464: 6458: 6454: 6446: 6431: 6427: 6425: 6422: 6421: 6414:Brownian motion 6353: 6347: 6343: 6335: 6332: 6331: 6315: 6312: 6311: 6295: 6292: 6291: 6264: 6260: 6255: 6234: 6220: 6217: 6216: 6192: 6189: 6188: 6170: 6167: 6166: 6147: 6143: 6141: 6138: 6137: 6116: 6109: 6104: 6089: 6086: 6085: 6058: 6050: 6047: 6046: 6015: 6012: 6011: 5992: 5988: 5986: 5983: 5982: 5964: 5961: 5960: 5936: 5933: 5932: 5905: 5901: 5896: 5879: 5865: 5862: 5861: 5818: 5815: 5814: 5798: 5795: 5794: 5777: 5773: 5771: 5768: 5767: 5719: 5717: 5709: 5704: 5688: 5668: 5665: 5658: 5654: 5653: 5639: 5636: 5635: 5630: 5603: 5600: 5594: 5590: 5569: 5565: 5555: 5547: 5545: 5536: 5531: 5519: 5516: 5515: 5478: 5474: 5467: 5465: 5447: 5443: 5433: 5425: 5423: 5418: 5415: 5414: 5395: 5392: 5391: 5358: 5354: 5352: 5337: 5333: 5323: 5315: 5313: 5308: 5305: 5304: 5279: 5276: 5275: 5249: 5246: 5245: 5241: 5211: 5205: 5201: 5199: 5188: 5169: 5165: 5163: 5150: 5142: 5140: 5138: 5135: 5134: 5109: 5106: 5105: 5083: 5080: 5079: 5069:Irving Langmuir 5048: 5045: 5044: 5028: 5025: 5024: 5018: 5012: 4993: 4989: 4987: 4984: 4983: 4977: 4949: 4946: 4940: 4936: 4922: 4919: 4918: 4847: 4801: 4794: 4790: 4783: 4779: 4769: 4768: 4764: 4743: 4740: 4739: 4693: 4663: 4611:pharmaceuticals 4595: 4579: 4569: 4566: 4554: 4543: 4536: 4531: 4524: 4519: 4513: 4501: 4497: 4494: 4490: 4487: 4483: 4458: 4454: 4442: 4438: 4432: 4428: 4426: 4417: 4413: 4412: 4408: 4396: 4379: 4372: 4368: 4364: 4362: 4360: 4357: 4356: 4318: 4314: 4310: 4287: 4285: 4276: 4272: 4268: 4249: 4245: 4241: 4239: 4227: 4223: 4214: 4210: 4206: 4181: 4177: 4176: 4174: 4156: 4152: 4151: 4147: 4141: 4124: 4111: 4110: 4071: 4070: 4051: 4028: 4026: 4024: 4021: 4020: 4001: 3996: 3973: 3969: 3960: 3956: 3952: 3927: 3923: 3922: 3920: 3911: 3907: 3901: 3890: 3880: 3869: 3856: 3855: 3825: 3824: 3805: 3782: 3780: 3778: 3775: 3774: 3751: 3747: 3743: 3735: 3733: 3724: 3720: 3714: 3703: 3687: 3683: 3681: 3678: 3677: 3643: 3640: 3639: 3635: 3628: 3623: 3591: 3587: 3583: 3560: 3558: 3549: 3545: 3541: 3524: 3522: 3516: 3505: 3456: 3455: 3416: 3415: 3396: 3373: 3371: 3369: 3366: 3365: 3350: 3342: 3340:Generalizations 3320: 3317: 3316: 3289: 3287: 3284: 3283: 3273:eukaryotic cell 3248: 3245: 3244: 3209: 3205: 3198: 3194: 3193: 3185: 3171: 3169: 3166: 3165: 3158:Brownian motion 3154: 3145: 3101: 3096: 3090: 3085: 3082: 3081: 3074: 3035: 3031: 3026: 3022: 3012: 3008: 3002: 2998: 2975: 2972: 2971: 2957: 2953: 2941: 2939: 2937: 2934: 2928: 2925: 2919: 2912: 2909: 2906: 2905: 2903: 2896: 2893: 2879: 2872: 2861: 2845: 2811: 2807: 2802: 2798: 2786: 2782: 2764: 2760: 2755: 2752: 2751: 2745: 2739: 2732: 2728: 2724: 2721: 2715:is the result. 2685: 2681: 2677: 2667: 2663: 2662: 2660: 2644: 2636: 2634: 2632: 2629: 2628: 2618: 2583: 2578: 2577: 2575: 2572: 2571: 2558: 2554: 2544: 2516: 2497: 2489: 2487: 2485: 2482: 2481: 2463: 2418: 2414: 2412: 2409: 2408: 2401: 2397: 2391: 2339: 2331: 2329: 2327: 2324: 2323: 2287: 2283: 2266: 2258: 2256: 2254: 2251: 2250: 2221: 2217: 2213: 2203: 2199: 2198: 2196: 2177: 2172: 2162: 2157: 2133: 2128: 2124: 2120: 2110: 2105: 2103: 2100: 2099: 2092: 2053: 2048: 2044: 2040: 2030: 2025: 2012: 2004: 2002: 1980: 1975: 1962: 1954: 1952: 1950: 1947: 1946: 1936: 1900: 1894: 1890: 1888: 1884: 1880: 1853: 1830: 1827: 1826: 1810: 1807: 1806: 1790: 1787: 1786: 1738: 1730: 1728: 1726: 1723: 1722: 1715: 1704: 1679: 1673: 1672: 1671: 1665: 1659: 1658: 1657: 1652: 1649: 1648: 1644: 1638: 1633: 1629: 1610: 1587: 1581: 1580: 1579: 1573: 1572: 1567: 1564: 1563: 1559: 1533: 1529: 1525: 1515: 1511: 1510: 1508: 1491: 1483: 1481: 1479: 1476: 1475: 1462: 1453: 1436: 1432: 1430: 1427: 1426: 1406: 1402: 1391: 1387: 1379: 1377: 1364: 1360: 1359: 1357: 1354: 1353: 1328: 1325: 1324: 1286: 1282: 1281: 1279: 1276: 1275: 1250: 1247: 1246: 1224: 1220: 1209: 1205: 1197: 1195: 1182: 1178: 1177: 1175: 1172: 1171: 1167: 1163: 1158: 1138: 1134: 1109: 1105: 1104: 1102: 1099: 1098: 1090: 1068: 1063: 1050: 1045: 1039: 1036: 1035: 1018: 1013: 1007: 1004: 1003: 986: 981: 975: 972: 971: 965: 948: 944: 942: 939: 938: 921: 917: 915: 912: 911: 887: 880: 876: 872: 870: 860: 855: 843: 839: 837: 834: 833: 818: 812: 802: 796: 790: 786: 758: 751: 747: 743: 741: 731: 724: 720: 716: 714: 702: 698: 696: 693: 692: 678: 675: 671: 643: 639: 637: 628: 624: 622: 619: 618: 610:is outside the 595: 592: 591: 582:is the mixture 579: 576:th species, and 573: 566: 558: 552: 545: 537: 531: 527: 503: 499: 488: 484: 476: 474: 462: 457: 456: 454: 451: 450: 444: 440: 433: 417: 414: 408: 407: 405: 403: 394: 360: 358: 355: 354: 339: 330: 328: 316: 309: 303: 277: 269: 267: 265: 262: 261: 245: 239: 230:, of which the 223: 196: 188: 186: 172: 169: 168: 154: 118: 86: 39: 28: 23: 22: 15: 12: 11: 5: 9218: 9208: 9207: 9202: 9197: 9192: 9187: 9173: 9172: 9163: 9155: 9154:External links 9152: 9150: 9149: 9140: 9107: 9098: 9081: 9072: 9062: 9060: 9057: 9054: 9053: 9026:(3): 192–197. 9006: 8983: 8956: 8892: 8869: 8797: 8778:(7): 453–461. 8759: 8729: 8677: 8620: 8559: 8552: 8534: 8509: 8494: 8487: 8467: 8450: 8421:(5): 184–262. 8402: 8387: 8354: 8352: 8351: 8289: 8263: 8262: 8260: 8257: 8256: 8255: 8250: 8245: 8240: 8235: 8230: 8225: 8220: 8215: 8208: 8205: 8196: 8193: 8179: 8176: 8168: 8167: 8153: 8147: 8143: 8139: 8134: 8130: 8116: 8104: 8080: 8074: 8071: 8068: 8061: 8057: 8053: 8048: 8044: 8037: 8031: 8027: 8023: 8020: 8017: 7995: 7992: 7989: 7986: 7983: 7980: 7969: 7968: 7954: 7950: 7946: 7936: 7925: 7905: 7881: 7875: 7872: 7865: 7861: 7857: 7851: 7845: 7841: 7837: 7834: 7831: 7817: 7816: 7802: 7798: 7787: 7773: 7769: 7758: 7744: 7740: 7714: 7710: 7706: 7701: 7694: 7690: 7686: 7681: 7675: 7670: 7665: 7661: 7655: 7649: 7645: 7641: 7638: 7597: 7593: 7589: 7584: 7581: 7576: 7571: 7568: 7562: 7559: 7536: 7535: 7523: 7513: 7501: 7491: 7490:at any surface 7482: 7476: 7463: 7460: 7449: 7437: 7412: 7408: 7405: 7402: 7396: 7392: 7389: 7384: 7377: 7373: 7369: 7364: 7361: 7355: 7352: 7346: 7342: 7339: 7336: 7333: 7330: 7310: 7307: 7284: 7281: 7276: 7275: 7271: 7264: 7259: 7253: 7247: 7244:chromatography 7229: 7226: 7222: 7211: 7197: 7193: 7178: 7177: 7164: 7161: 7158: 7155: 7151: 7147: 7140: 7137: 7133: 7127: 7123: 7119: 7116: 7113: 7083: 7082: 7077: 7068: 7051: 7044: 7038: 7024: 7023: 7010: 7004: 7000: 6996: 6991: 6987: 6982: 6978: 6975: 6971: 6951: 6948: 6904: 6903: 6889: 6885: 6862: 6858: 6847: 6833: 6829: 6825: 6820: 6816: 6812: 6807: 6803: 6792: 6779: 6764: 6763: 6749: 6742: 6738: 6734: 6729: 6725: 6716: 6712: 6706: 6702: 6696: 6692: 6687: 6681: 6678: 6673: 6668: 6665: 6661: 6633: 6632: 6618: 6614: 6591: 6587: 6576: 6562: 6558: 6554: 6549: 6545: 6541: 6536: 6532: 6521: 6509: 6495: 6494: 6481: 6477: 6471: 6467: 6461: 6457: 6453: 6449: 6445: 6442: 6437: 6434: 6430: 6374: 6373: 6360: 6356: 6350: 6346: 6342: 6339: 6319: 6299: 6288: 6287: 6276: 6271: 6267: 6263: 6258: 6254: 6250: 6247: 6244: 6241: 6237: 6233: 6230: 6227: 6224: 6209: 6208: 6196: 6186: 6174: 6164: 6150: 6146: 6123: 6119: 6115: 6112: 6107: 6103: 6099: 6093: 6082: 6081: 6068: 6065: 6062: 6057: 6054: 6035: 6034: 6031: 6019: 6009: 5995: 5991: 5980: 5968: 5958: 5946: 5943: 5940: 5929: 5928: 5917: 5912: 5908: 5904: 5899: 5895: 5891: 5886: 5883: 5878: 5875: 5872: 5869: 5822: 5802: 5780: 5776: 5764: 5763: 5752: 5749: 5742: 5739: 5736: 5731: 5728: 5725: 5722: 5713: 5707: 5703: 5696: 5693: 5687: 5684: 5678: 5674: 5671: 5657: 5652: 5649: 5646: 5643: 5628: 5627: 5613: 5609: 5606: 5597: 5593: 5589: 5586: 5583: 5578: 5575: 5572: 5568: 5561: 5558: 5553: 5550: 5544: 5539: 5534: 5530: 5526: 5523: 5509: 5508: 5494: 5491: 5488: 5481: 5477: 5473: 5470: 5464: 5461: 5456: 5453: 5450: 5446: 5439: 5436: 5431: 5428: 5422: 5399: 5388: 5387: 5373: 5370: 5367: 5361: 5357: 5351: 5346: 5343: 5340: 5336: 5329: 5326: 5321: 5318: 5312: 5289: 5286: 5283: 5272: 5271: 5259: 5256: 5253: 5238: 5237: 5226: 5220: 5217: 5214: 5208: 5204: 5198: 5195: 5184: 5181: 5178: 5172: 5168: 5162: 5156: 5153: 5148: 5145: 5119: 5116: 5113: 5093: 5090: 5087: 5065: 5064: 5052: 5032: 5022: 5016: 5010: 4996: 4992: 4981: 4974: 4973: 4959: 4955: 4952: 4943: 4939: 4935: 4932: 4929: 4926: 4883: 4882: 4876: 4870: 4861: 4852: 4845: 4836: 4835: 4823: 4817: 4814: 4811: 4808: 4804: 4800: 4797: 4793: 4789: 4786: 4782: 4778: 4772: 4767: 4762: 4759: 4756: 4753: 4750: 4747: 4692: 4689: 4662: 4659: 4626:plasma physics 4594: 4591: 4558: 4541: 4534: 4527: 4522: 4515: 4506: 4505: 4492: 4485: 4471: 4467: 4461: 4457: 4452: 4445: 4441: 4435: 4431: 4423: 4420: 4416: 4411: 4407: 4404: 4399: 4395: 4391: 4385: 4382: 4375: 4371: 4367: 4345: 4334: 4330: 4321: 4317: 4313: 4308: 4305: 4302: 4299: 4296: 4293: 4290: 4279: 4275: 4271: 4266: 4263: 4260: 4255: 4252: 4248: 4244: 4238: 4230: 4226: 4222: 4217: 4213: 4209: 4204: 4201: 4198: 4195: 4192: 4189: 4184: 4180: 4173: 4170: 4167: 4162: 4159: 4155: 4150: 4144: 4139: 4136: 4133: 4130: 4127: 4123: 4119: 4114: 4109: 4106: 4103: 4100: 4097: 4094: 4091: 4088: 4085: 4082: 4079: 4074: 4069: 4066: 4063: 4057: 4054: 4049: 4046: 4043: 4040: 4037: 4034: 4031: 4013: 3999: 3984: 3976: 3972: 3968: 3963: 3959: 3955: 3950: 3947: 3944: 3941: 3938: 3935: 3930: 3926: 3917: 3914: 3910: 3904: 3899: 3896: 3893: 3889: 3883: 3878: 3875: 3872: 3868: 3864: 3859: 3854: 3851: 3848: 3845: 3842: 3839: 3836: 3833: 3828: 3823: 3820: 3817: 3811: 3808: 3803: 3800: 3797: 3794: 3791: 3788: 3785: 3762: 3754: 3750: 3746: 3741: 3738: 3730: 3727: 3723: 3717: 3712: 3709: 3706: 3702: 3698: 3695: 3690: 3686: 3665: 3662: 3659: 3656: 3653: 3650: 3647: 3633: 3626: 3610: 3594: 3590: 3586: 3581: 3578: 3575: 3572: 3569: 3566: 3563: 3552: 3548: 3544: 3539: 3536: 3533: 3530: 3527: 3519: 3514: 3511: 3508: 3504: 3500: 3497: 3494: 3491: 3488: 3485: 3482: 3479: 3476: 3473: 3470: 3467: 3464: 3459: 3454: 3451: 3448: 3445: 3442: 3439: 3436: 3433: 3430: 3427: 3424: 3419: 3414: 3411: 3408: 3402: 3399: 3394: 3391: 3388: 3385: 3382: 3379: 3376: 3341: 3338: 3324: 3302: 3299: 3296: 3293: 3252: 3232: 3229: 3226: 3223: 3220: 3217: 3212: 3208: 3201: 3197: 3192: 3188: 3184: 3181: 3178: 3153: 3150: 3142: 3141: 3130: 3125: 3122: 3118: 3115: 3112: 3109: 3104: 3099: 3095: 3089: 3071: 3070: 3058: 3053: 3045: 3042: 3039: 3034: 3030: 3025: 3021: 3018: 3015: 3011: 3005: 3001: 2997: 2994: 2991: 2988: 2985: 2982: 2979: 2960:(Bird, 1976). 2948:is called the 2932: 2923: 2891: 2878:and that with 2850:error function 2842: 2841: 2830: 2826: 2818: 2815: 2810: 2806: 2801: 2797: 2794: 2789: 2785: 2781: 2777: 2773: 2770: 2767: 2763: 2759: 2743: 2720: 2717: 2709:advective flux 2705: 2704: 2688: 2684: 2680: 2675: 2670: 2666: 2659: 2656: 2650: 2647: 2642: 2639: 2615: 2614: 2603: 2600: 2597: 2594: 2591: 2581: 2565:diffusive flux 2541: 2540: 2529: 2526: 2523: 2519: 2515: 2512: 2509: 2503: 2500: 2495: 2492: 2471:advective flux 2462: 2459: 2447: 2446: 2435: 2432: 2429: 2426: 2421: 2417: 2388: 2387: 2376: 2373: 2370: 2367: 2363: 2360: 2357: 2354: 2351: 2345: 2342: 2337: 2334: 2310: 2309: 2298: 2295: 2290: 2286: 2281: 2278: 2272: 2269: 2264: 2261: 2241: 2240: 2224: 2220: 2216: 2211: 2206: 2202: 2195: 2192: 2189: 2183: 2180: 2176: 2168: 2165: 2161: 2156: 2153: 2149: 2145: 2139: 2136: 2132: 2127: 2123: 2116: 2113: 2109: 2089: 2088: 2077: 2074: 2069: 2065: 2059: 2056: 2052: 2047: 2043: 2036: 2033: 2029: 2024: 2018: 2015: 2010: 2007: 2001: 1998: 1995: 1992: 1986: 1983: 1979: 1974: 1968: 1965: 1960: 1957: 1935: 1932: 1920: 1916: 1909: 1906: 1903: 1897: 1893: 1887: 1883: 1879: 1876: 1870: 1867: 1864: 1861: 1857: 1852: 1849: 1846: 1843: 1840: 1837: 1834: 1814: 1794: 1771: 1770: 1759: 1756: 1753: 1750: 1744: 1741: 1736: 1733: 1709: 1708: 1702: 1690: 1685: 1682: 1676: 1668: 1662: 1656: 1642: 1636: 1598: 1593: 1590: 1584: 1576: 1571: 1553: 1552: 1536: 1532: 1528: 1523: 1518: 1514: 1506: 1503: 1497: 1494: 1489: 1486: 1461: 1458: 1439: 1435: 1414: 1409: 1405: 1401: 1394: 1390: 1385: 1382: 1376: 1373: 1367: 1363: 1341: 1338: 1335: 1332: 1310: 1307: 1304: 1301: 1298: 1295: 1289: 1285: 1263: 1260: 1257: 1254: 1232: 1227: 1223: 1219: 1212: 1208: 1203: 1200: 1194: 1191: 1185: 1181: 1161: 1146: 1141: 1137: 1133: 1130: 1127: 1124: 1121: 1118: 1112: 1108: 1089: 1086: 1066: 1062: 1058: 1048: 1044: 1016: 1012: 984: 980: 951: 947: 937:has Pa units. 924: 920: 908: 907: 893: 890: 883: 879: 875: 866: 863: 859: 854: 851: 846: 842: 823: 822: 816: 810: 800: 794: 779: 778: 764: 761: 754: 750: 746: 737: 734: 727: 723: 719: 713: 710: 705: 701: 673: 668: 667: 654: 649: 646: 642: 636: 631: 627: 599: 588: 587: 577: 562: 556: 541: 535: 520: 519: 506: 502: 498: 491: 487: 482: 479: 473: 470: 465: 460: 442: 432: 429: 391: 390: 379: 376: 373: 370: 367: 363: 314: 313: 307: 301: 283: 280: 275: 272: 259: 243: 228:diffusion flux 217: 216: 202: 199: 194: 191: 185: 182: 179: 176: 153: 150: 117: 114: 36:Fick principle 32:cardiac output 26: 9: 6: 4: 3: 2: 9217: 9206: 9203: 9201: 9198: 9196: 9193: 9191: 9188: 9186: 9183: 9182: 9180: 9171: 9167: 9164: 9161: 9158: 9157: 9146: 9141: 9137: 9133: 9129: 9125: 9117: 9113: 9108: 9104: 9099: 9095: 9090: 9089: 9082: 9078: 9073: 9069: 9064: 9063: 9049: 9045: 9041: 9037: 9033: 9029: 9025: 9021: 9017: 9010: 9002: 8998: 8994: 8987: 8979: 8975: 8971: 8968:(in German). 8967: 8960: 8952: 8948: 8943: 8938: 8934: 8930: 8926: 8922: 8918: 8914: 8910: 8903: 8901: 8899: 8897: 8888: 8884: 8880: 8873: 8865: 8861: 8856: 8851: 8847: 8843: 8839: 8835: 8832:(1): 015318. 8831: 8827: 8823: 8816: 8814: 8812: 8810: 8808: 8806: 8804: 8802: 8793: 8789: 8785: 8781: 8777: 8773: 8766: 8764: 8755: 8751: 8747: 8743: 8736: 8734: 8725: 8721: 8716: 8711: 8707: 8703: 8700:: 2296–2306. 8699: 8695: 8691: 8684: 8682: 8673: 8669: 8664: 8659: 8655: 8651: 8647: 8643: 8639: 8635: 8631: 8624: 8616: 8612: 8608: 8604: 8600: 8596: 8592: 8588: 8583: 8578: 8574: 8570: 8563: 8555: 8549: 8545: 8538: 8523: 8519: 8513: 8505: 8498: 8490: 8488:9780521881685 8484: 8480: 8479: 8471: 8463: 8462: 8454: 8446: 8442: 8438: 8434: 8429: 8424: 8420: 8416: 8412: 8406: 8398: 8391: 8380: 8376: 8372: 8365: 8358: 8348: 8344: 8341:(63): 30–39. 8340: 8336: 8331: 8330: 8326: 8321: 8317: 8313: 8309: 8306:(in German). 8305: 8301: 8293: 8278: 8274: 8268: 8264: 8254: 8251: 8249: 8246: 8244: 8241: 8239: 8236: 8234: 8231: 8229: 8226: 8224: 8221: 8219: 8216: 8214: 8211: 8210: 8204: 8201: 8192: 8188: 8186: 8175: 8171: 8151: 8145: 8141: 8137: 8132: 8128: 8117: 8102: 8095: 8094: 8093: 8078: 8072: 8069: 8066: 8059: 8055: 8051: 8046: 8042: 8035: 8029: 8025: 8021: 8018: 8015: 8007: 7993: 7990: 7987: 7984: 7981: 7978: 7952: 7948: 7944: 7937: 7923: 7903: 7896: 7895: 7894: 7879: 7873: 7870: 7863: 7859: 7855: 7849: 7843: 7839: 7835: 7832: 7829: 7820: 7800: 7796: 7788: 7771: 7767: 7759: 7742: 7738: 7730: 7729: 7728: 7712: 7708: 7704: 7699: 7692: 7688: 7684: 7679: 7673: 7668: 7663: 7659: 7653: 7647: 7643: 7639: 7636: 7628: 7626: 7622: 7617: 7613: 7595: 7591: 7587: 7574: 7569: 7566: 7560: 7557: 7521: 7514: 7499: 7492: 7487: 7483: 7477: 7450: 7435: 7428: 7427: 7426: 7410: 7406: 7403: 7400: 7394: 7382: 7375: 7371: 7367: 7353: 7350: 7344: 7340: 7334: 7328: 7319: 7315: 7306: 7304: 7300: 7297: 7293: 7290: 7289:semiconductor 7280: 7270: 7263: 7260: 7257: 7254: 7251: 7248: 7245: 7227: 7224: 7220: 7212: 7195: 7191: 7183: 7182: 7181: 7159: 7156: 7149: 7145: 7138: 7135: 7131: 7125: 7121: 7117: 7114: 7111: 7104: 7103: 7102: 7098: 7096: 7091: 7089: 7076: 7067: 7062: 7058: 7057:concentration 7050: 7043: 7039: 7036: 7029: 7028: 7027: 7008: 7002: 6998: 6994: 6989: 6985: 6980: 6976: 6973: 6969: 6957: 6956: 6955: 6947: 6887: 6883: 6860: 6856: 6848: 6831: 6827: 6823: 6818: 6814: 6810: 6805: 6801: 6793: 6777: 6769: 6768: 6767: 6747: 6740: 6736: 6732: 6727: 6723: 6714: 6710: 6704: 6700: 6694: 6690: 6685: 6679: 6676: 6671: 6666: 6663: 6659: 6651: 6650: 6649: 6645: 6641: 6638: 6616: 6612: 6589: 6585: 6577: 6560: 6556: 6552: 6547: 6543: 6539: 6534: 6530: 6522: 6507: 6500: 6499: 6498: 6479: 6475: 6469: 6465: 6459: 6455: 6451: 6447: 6443: 6440: 6435: 6432: 6428: 6420: 6419: 6418: 6415: 6411: 6407: 6398: 6394: 6392: 6387: 6386:self-assembly 6382: 6379: 6358: 6354: 6348: 6344: 6340: 6337: 6317: 6297: 6290: 6289: 6274: 6269: 6265: 6261: 6256: 6252: 6248: 6245: 6242: 6239: 6235: 6231: 6228: 6225: 6222: 6215: 6214: 6213: 6194: 6187: 6172: 6165: 6148: 6144: 6121: 6117: 6113: 6110: 6105: 6101: 6097: 6091: 6084: 6083: 6066: 6063: 6060: 6055: 6052: 6045: 6044: 6043: 6039: 6032: 6017: 6010: 5993: 5989: 5981: 5966: 5959: 5941: 5931: 5930: 5915: 5910: 5906: 5902: 5897: 5893: 5889: 5884: 5881: 5876: 5870: 5860: 5859: 5858: 5854: 5851: 5842: 5838: 5834: 5820: 5800: 5778: 5774: 5750: 5747: 5740: 5737: 5734: 5726: 5720: 5711: 5705: 5701: 5694: 5691: 5685: 5682: 5676: 5672: 5669: 5655: 5650: 5647: 5644: 5634: 5633: 5632: 5611: 5607: 5604: 5595: 5591: 5587: 5584: 5581: 5576: 5573: 5570: 5559: 5537: 5532: 5528: 5524: 5514: 5513: 5512: 5492: 5489: 5486: 5479: 5475: 5471: 5468: 5462: 5459: 5454: 5451: 5448: 5437: 5413: 5412: 5411: 5397: 5371: 5368: 5365: 5359: 5355: 5349: 5344: 5341: 5338: 5327: 5319: 5303: 5302: 5301: 5287: 5284: 5281: 5257: 5254: 5251: 5240: 5239: 5218: 5215: 5212: 5206: 5202: 5196: 5182: 5179: 5176: 5170: 5166: 5160: 5154: 5146: 5133: 5132: 5131: 5117: 5114: 5111: 5091: 5088: 5085: 5076: 5074: 5070: 5050: 5023: 5017: 5011: 4994: 4990: 4982: 4976: 4975: 4957: 4953: 4950: 4941: 4937: 4933: 4930: 4927: 4917: 4916: 4915: 4912: 4908: 4898: 4894: 4892: 4888: 4880: 4877: 4874: 4871: 4869: 4865: 4862: 4860: 4856: 4853: 4851: 4844: 4841: 4840: 4839: 4821: 4812: 4809: 4802: 4798: 4795: 4791: 4787: 4784: 4780: 4776: 4765: 4760: 4757: 4751: 4745: 4738: 4737: 4736: 4734: 4731:based on the 4730: 4726: 4722: 4718: 4714: 4710: 4706: 4702: 4697: 4688: 4686: 4685:renormalizing 4682: 4676: 4673: 4668: 4658: 4656: 4652: 4651:mass transfer 4648: 4644: 4640: 4635: 4631: 4627: 4623: 4619: 4616: 4612: 4608: 4604: 4600: 4590: 4586: 4582: 4576: 4572: 4565: 4561: 4557: 4551: 4549: 4544: 4537: 4530: 4525: 4518: 4511: 4469: 4465: 4459: 4455: 4443: 4439: 4433: 4429: 4421: 4418: 4414: 4409: 4405: 4397: 4393: 4389: 4383: 4373: 4369: 4354: 4350: 4346: 4332: 4328: 4319: 4315: 4303: 4300: 4297: 4291: 4277: 4273: 4261: 4253: 4250: 4246: 4236: 4228: 4224: 4215: 4211: 4199: 4196: 4193: 4187: 4182: 4168: 4160: 4157: 4153: 4148: 4142: 4137: 4134: 4131: 4128: 4125: 4121: 4117: 4104: 4101: 4098: 4092: 4083: 4077: 4067: 4061: 4055: 4044: 4041: 4038: 4032: 4018: 4014: 4011: 4007: 4002: 3982: 3974: 3970: 3961: 3957: 3945: 3942: 3939: 3933: 3928: 3915: 3912: 3908: 3902: 3897: 3894: 3891: 3887: 3881: 3876: 3873: 3870: 3866: 3862: 3849: 3846: 3843: 3837: 3831: 3821: 3815: 3809: 3798: 3795: 3792: 3786: 3760: 3752: 3748: 3739: 3728: 3725: 3721: 3715: 3710: 3707: 3704: 3700: 3696: 3693: 3688: 3684: 3663: 3660: 3654: 3651: 3648: 3645: 3636: 3629: 3622: 3618: 3616: 3611: 3592: 3588: 3576: 3573: 3570: 3564: 3550: 3546: 3534: 3528: 3517: 3512: 3509: 3506: 3502: 3498: 3492: 3489: 3486: 3480: 3471: 3465: 3462: 3449: 3446: 3443: 3437: 3428: 3422: 3412: 3406: 3400: 3389: 3386: 3383: 3377: 3361: 3357: 3353: 3348: 3344: 3343: 3337: 3322: 3300: 3297: 3294: 3291: 3280: 3278: 3274: 3270: 3269:cell membrane 3266: 3250: 3230: 3227: 3224: 3221: 3218: 3210: 3190: 3176: 3163: 3159: 3149: 3128: 3123: 3120: 3113: 3107: 3102: 3097: 3093: 3087: 3080: 3079: 3078: 3056: 3051: 3043: 3040: 3037: 3032: 3028: 3023: 3019: 3016: 3013: 3009: 3003: 2999: 2995: 2989: 2986: 2983: 2977: 2970: 2969: 2968: 2967:can be used: 2966: 2965:Taylor series 2961: 2951: 2944: 2931: 2922: 2899: 2890: 2886: 2882: 2875: 2868: 2864: 2859: 2855: 2854:semi-infinite 2851: 2828: 2824: 2816: 2813: 2808: 2804: 2799: 2795: 2792: 2787: 2783: 2779: 2775: 2771: 2768: 2765: 2761: 2757: 2750: 2749: 2748: 2742: 2735: 2716: 2714: 2710: 2686: 2682: 2673: 2668: 2657: 2654: 2648: 2640: 2627: 2626: 2625: 2621: 2601: 2595: 2592: 2589: 2570: 2569: 2568: 2566: 2561: 2552: 2549:is the total 2547: 2527: 2524: 2521: 2513: 2507: 2501: 2493: 2480: 2479: 2478: 2476: 2472: 2468: 2458: 2456: 2452: 2433: 2430: 2427: 2424: 2419: 2407: 2406: 2405: 2394: 2374: 2368: 2361: 2355: 2349: 2343: 2335: 2322: 2321: 2320: 2317: 2315: 2314:heat equation 2296: 2293: 2288: 2279: 2276: 2270: 2262: 2249: 2248: 2247: 2244: 2222: 2218: 2209: 2204: 2193: 2190: 2187: 2181: 2166: 2154: 2151: 2147: 2143: 2137: 2125: 2121: 2114: 2098: 2097: 2096: 2075: 2072: 2067: 2063: 2057: 2045: 2041: 2034: 2022: 2016: 2008: 1996: 1993: 1990: 1984: 1972: 1966: 1958: 1945: 1944: 1943: 1941: 1931: 1918: 1914: 1907: 1904: 1901: 1895: 1891: 1885: 1881: 1877: 1874: 1868: 1865: 1862: 1859: 1855: 1850: 1844: 1841: 1838: 1832: 1812: 1792: 1784: 1780: 1776: 1775:Heat equation 1757: 1751: 1748: 1742: 1734: 1721: 1720: 1719: 1714: 1703: 1701:, example m/s 1683: 1680: 1666: 1643: 1637: 1625: 1621: 1617: 1613: 1591: 1588: 1558: 1557: 1556: 1534: 1530: 1521: 1516: 1504: 1501: 1495: 1487: 1474: 1473: 1472: 1470: 1466: 1457: 1437: 1433: 1412: 1407: 1403: 1392: 1388: 1383: 1380: 1374: 1371: 1339: 1336: 1333: 1321: 1308: 1305: 1299: 1296: 1293: 1261: 1258: 1255: 1243: 1230: 1225: 1221: 1210: 1206: 1201: 1198: 1192: 1189: 1164: 1144: 1139: 1135: 1131: 1128: 1122: 1119: 1116: 1096: 1085: 1064: 1060: 1056: 1046: 1042: 1014: 1010: 982: 978: 968: 949: 945: 922: 918: 891: 881: 877: 864: 861: 857: 852: 849: 844: 840: 832: 831: 830: 828: 817: 811: 808: 801: 795: 784: 783: 782: 762: 752: 748: 735: 732: 725: 721: 717: 711: 708: 703: 699: 691: 690: 689: 687: 682: 652: 647: 644: 640: 634: 629: 625: 617: 616: 615: 613: 597: 585: 578: 571: 565: 561: 557: 550: 544: 540: 536: 525: 524: 523: 504: 500: 489: 485: 480: 477: 471: 468: 463: 449: 448: 447: 438: 437:mass fraction 428: 421: 412: 400: 397: 377: 371: 368: 365: 353: 352: 351: 349: 345: 336: 326: 322: 308: 302: 300: 281: 278: 273: 270: 260: 257: 256: 251: 244: 237: 233: 229: 222: 221: 220: 200: 197: 192: 189: 183: 180: 177: 174: 167: 166: 165: 162: 158: 149: 147: 143: 137: 135: 134:Fourier's Law 131: 127: 123: 122:Thomas Graham 113: 111: 106: 104: 100: 98: 94: 92: 84: 80: 76: 72: 65: 61: 57: 53: 49: 45: 41: 37: 33: 19: 9144: 9127: 9123: 9115: 9111: 9102: 9087: 9076: 9070:. Princeton. 9067: 9023: 9019: 9009: 9001:the original 8996: 8986: 8969: 8965: 8959: 8916: 8912: 8878: 8872: 8829: 8826:AIP Advances 8825: 8775: 8771: 8745: 8741: 8697: 8693: 8637: 8633: 8623: 8572: 8568: 8562: 8543: 8537: 8525:. Retrieved 8521: 8512: 8503: 8497: 8477: 8470: 8460: 8453: 8418: 8414: 8405: 8396: 8390: 8379:the original 8374: 8370: 8357: 8338: 8334: 8310:(1): 59–86. 8307: 8303: 8292: 8280:. Retrieved 8276: 8267: 8233:Gas exchange 8202: 8198: 8189: 8181: 8172: 8169: 8008: 7970: 7821: 7818: 7629: 7624: 7620: 7618: 7614: 7537: 7512:is viscosity 7485: 7320: 7316: 7312: 7301: 7298: 7294: 7286: 7277: 7268: 7261: 7255: 7249: 7179: 7099: 7095:Graham's law 7092: 7088:flux limiter 7084: 7074: 7065: 7048: 7041: 7025: 6953: 6905: 6765: 6646: 6642: 6634: 6496: 6408:proposed by 6403: 6383: 6377: 6375: 6210: 6040: 6036: 5855: 5847: 5835: 5765: 5629: 5510: 5389: 5273: 5077: 5066: 4904: 4890: 4887:biomolecules 4884: 4878: 4872: 4863: 4854: 4842: 4837: 4712: 4708: 4704: 4698: 4694: 4677: 4672:random walks 4664: 4634:voltammetric 4596: 4593:Applications 4584: 4580: 4574: 4570: 4563: 4559: 4555: 4552: 4539: 4532: 4528: 4520: 4516: 4507: 4352: 4016: 3997: 3631: 3624: 3613: 3359: 3355: 3351: 3346: 3281: 3155: 3143: 3072: 2962: 2949: 2942: 2929: 2920: 2918:in front of 2897: 2888: 2884: 2880: 2873: 2866: 2862: 2857: 2853: 2843: 2740: 2733: 2722: 2706: 2619: 2616: 2564: 2559: 2545: 2542: 2464: 2448: 2392: 2389: 2318: 2311: 2245: 2242: 2090: 1937: 1772: 1710: 1623: 1619: 1615: 1611: 1554: 1464: 1463: 1322: 1244: 1159: 1091: 966: 909: 829:difference: 824: 789:denotes the 780: 683: 681:th species. 669: 589: 563: 559: 548: 542: 538: 530:denotes the 521: 434: 419: 410: 401: 395: 392: 337: 315: 298: 253: 249: 227: 218: 156: 155: 145: 141: 138: 119: 107: 102: 101: 96: 95: 70: 69: 63: 59: 55: 40: 8634:Soft Matter 8282:26 February 8185:random walk 7534:is density. 7035:conductance 4607:biopolymers 3615:anisotropic 1783:Heat kernel 970:in a vapor 534:th species, 333:10 m/s 255:diffusivity 146:non-Fickian 126:Darcy's law 9179:Categories 8991:Nosek TM. 6372:neighbors. 4911:absorption 4907:adsorption 4901:diffusion. 4601:in foods, 4550:equation. 4004:should be 1002:or liquid 793:th species 785:the index 570:molar mass 526:the index 79:Adolf Fick 9185:Diffusion 9130:: 33–38. 8428:1012.2908 8411:Gorban AN 8259:Citations 8238:Mass flux 8223:Diffusion 8213:Advection 8152:δ 8138:− 8067:δ 8052:− 8022:− 7924:δ 7836:− 7558:δ 7522:ρ 7500:η 7436:δ 7411:η 7404:ρ 7329:δ 7221:η 7157:π 7132:η 7026:in which 6995:− 6974:− 6778:σ 6686:σ 6680:π 6448:π 6341:π 6111:− 5945:⟩ 5939:⟨ 5885:π 5874:⟩ 5868:⟨ 5821:τ 5751:τ 5741:τ 5738:− 5727:τ 5702:∫ 5695:π 5683:− 5677:π 5642:Γ 5612:π 5557:∂ 5552:Γ 5549:∂ 5529:∫ 5522:Γ 5487:π 5463:− 5435:∂ 5430:Γ 5427:∂ 5366:π 5325:∂ 5317:∂ 5197:− 5177:π 5152:∂ 5144:∂ 5031:Γ 4958:π 4925:Γ 4813:μ 4796:− 4788:− 4761:μ 4681:tautology 4665:When two 4587:= 1, 2, 3 4456:φ 4451:∇ 4440:φ 4430:φ 4406:⋅ 4403:∇ 4394:∑ 4381:∂ 4370:φ 4366:∂ 4312:∂ 4292:φ 4289:∂ 4270:∂ 4243:∂ 4221:∂ 4208:∂ 4188:φ 4179:∂ 4122:∑ 4093:φ 4090:∇ 4068:⋅ 4065:∇ 4053:∂ 4033:φ 4030:∂ 3967:∂ 3954:∂ 3934:φ 3925:∂ 3888:∑ 3867:∑ 3838:φ 3835:∇ 3822:⋅ 3819:∇ 3807:∂ 3787:φ 3784:∂ 3745:∂ 3740:φ 3737:∂ 3701:∑ 3697:− 3661:φ 3658:∇ 3652:− 3585:∂ 3565:φ 3562:∂ 3543:∂ 3526:∂ 3503:∑ 3481:φ 3478:Δ 3438:φ 3435:∇ 3413:⋅ 3410:∇ 3398:∂ 3378:φ 3375:∂ 3265:dimension 3216:⟩ 3191:− 3180:⟨ 3177:≡ 3124:τ 3114:τ 3094:∫ 3044:π 3017:− 2796:⁡ 2679:∂ 2674:φ 2665:∂ 2646:∂ 2641:φ 2638:∂ 2602:φ 2599:∇ 2593:− 2585:diffusion 2514:⋅ 2511:∇ 2499:∂ 2494:φ 2491:∂ 2449:which is 2425:φ 2416:∇ 2369:φ 2366:∇ 2356:⋅ 2353:∇ 2341:∂ 2336:φ 2333:∂ 2294:φ 2285:∇ 2268:∂ 2263:φ 2260:∂ 2215:∂ 2210:φ 2201:∂ 2188:φ 2179:∂ 2175:∂ 2164:∂ 2160:∂ 2144:φ 2135:∂ 2131:∂ 2112:∂ 2108:∂ 2064:φ 2055:∂ 2051:∂ 2032:∂ 2028:∂ 2023:− 2014:∂ 2009:φ 2006:∂ 2000:⇒ 1982:∂ 1978:∂ 1964:∂ 1959:φ 1956:∂ 1886:− 1878:⁡ 1863:π 1833:φ 1758:φ 1755:Δ 1740:∂ 1735:φ 1732:∂ 1713:Laplacian 1681:− 1632:and time 1589:− 1527:∂ 1522:φ 1513:∂ 1493:∂ 1488:φ 1485:∂ 1400:∇ 1381:ρ 1375:− 1334:ρ 1331:∇ 1306:φ 1303:∇ 1297:− 1256:ρ 1253:∇ 1218:∇ 1199:ρ 1193:− 1132:⁡ 1126:∇ 1120:− 910:Fugacity 889:∂ 874:∂ 853:− 809:(J/K/mol) 760:∂ 749:μ 745:∂ 712:− 653:ρ 641:ρ 598:ρ 497:∇ 478:ρ 472:− 378:φ 375:∇ 369:− 321:viscosity 274:φ 232:dimension 193:φ 181:− 130:Ohm's law 75:diffusion 73:describe 9170:OpenStax 9040:26330037 8951:36520427 8864:35070490 8724:29181286 8672:27396746 8607:11304296 8445:18961678 8207:See also 7061:membrane 6929:⁠2 6909:⁠2 4881:is time. 4711:, where 4667:miscible 4568:, where 4010:elliptic 2869:, 0) = 0 2858:infinite 1777:and its 827:fugacity 612:gradient 348:gradient 9048:3921833 8974:Bibcode 8942:9805503 8921:Bibcode 8855:8758205 8834:Bibcode 8780:Bibcode 8715:5687005 8663:5476231 8642:Bibcode 8615:1302913 8587:Bibcode 8312:Bibcode 8253:Osmosis 6944:⁠ 6924:⁠ 6766:where, 6497:where, 5850:fractal 4857:is the 4848:is the 4655:Onsager 4639:polymer 4603:neurons 3263:is the 2940:√ 2916:⁠ 2904:⁠ 2887:, 0) = 805:is the 584:density 572:of the 568:is the 551:of the 547:is the 424:⁠ 406:⁠ 329:(0.6–2) 297:is the 248:is the 234:is the 226:is the 116:History 9046: 9038: 8949: 8939: 8862: 8852: 8722: 8712: 8670: 8660: 8613: 8605: 8550: 8527:11 May 8485: 8443: 8092:where 7893:where 7727:where 7425:where 7180:where 6095: 5766:where 5270:(#/m). 4628:, and 4615:porous 4482:where 3621:tensor 3277:cactus 3243:where 2876:> 0 2844:where 2711:, the 2543:where 1555:where 1425:where 1157:where 781:where 670:where 522:where 393:where 342:, the 219:where 64:Bottom 60:Middle 52:solute 34:, see 9118:: 59. 9096:–171. 9044:S2CID 8611:S2CID 8577:arXiv 8441:S2CID 8423:arXiv 8382:(PDF) 8367:(PDF) 7542:from 6229:>= 4618:soils 3617:media 1716:Δ = ∇ 9036:PMID 8947:PMID 8860:PMID 8720:PMID 8668:PMID 8603:PMID 8548:ISBN 8529:2022 8483:ISBN 8284:2024 7287:The 6966:flux 6875:and 6604:and 6223:< 5071:and 4905:The 4508:The 4500:and 4015:For 2846:erfc 2793:erfc 2553:and 2551:flux 590:The 161:flux 9168:on 9132:doi 9128:100 9094:167 9028:doi 8937:PMC 8929:doi 8917:126 8883:doi 8850:PMC 8842:doi 8788:doi 8750:doi 8710:PMC 8702:doi 8658:PMC 8650:doi 8595:doi 8433:doi 8343:doi 8320:doi 7546:to 7488:= 0 7072:to 5190:exp 4909:or 4526:= Σ 3612:In 3345:In 3173:MSD 3073:If 2900:≤ 0 2747:is 2736:= 0 2622:= 0 1875:exp 346:or 344:del 252:or 142:not 56:Top 9181:: 9126:. 9116:94 9114:. 9042:. 9034:. 9024:39 9022:. 9018:. 8995:. 8970:17 8945:. 8935:. 8927:. 8915:. 8911:. 8895:^ 8881:. 8858:. 8848:. 8840:. 8830:12 8828:. 8824:. 8800:^ 8786:. 8776:14 8774:. 8762:^ 8746:29 8744:. 8732:^ 8718:. 8708:. 8696:. 8692:. 8680:^ 8666:. 8656:. 8648:. 8638:12 8636:. 8632:. 8609:. 8601:. 8593:. 8585:. 8573:63 8571:. 8520:. 8439:. 8431:. 8417:. 8373:. 8369:. 8339:10 8337:. 8318:. 8308:94 8302:. 8296:* 8275:. 7567:10 7267:, 7097:. 7090:. 7081:). 7047:− 5075:. 4891:mμ 4620:, 4613:, 4609:, 4605:, 4583:, 4573:, 4564:αβ 4560:ij 4535:ij 4493:ij 4000:ij 3634:ij 3630:= 3627:ji 3354:= 2943:Dt 2895:, 2871:, 2567:: 2477:: 2457:. 2316:. 1825:: 1614:= 1456:. 1129:ln 1084:. 674:si 148:. 93:. 85:, 9138:. 9134:: 9050:. 9030:: 8980:. 8976:: 8953:. 8931:: 8923:: 8889:. 8885:: 8866:. 8844:: 8836:: 8794:. 8790:: 8782:: 8756:. 8752:: 8726:. 8704:: 8698:8 8674:. 8652:: 8644:: 8617:. 8597:: 8589:: 8579:: 8556:. 8531:. 8491:. 8464:. 8447:. 8435:: 8425:: 8419:6 8375:2 8349:. 8345:: 8328:. 8322:: 8314:: 8286:. 8146:0 8142:P 8133:i 8129:P 8103:R 8079:) 8073:T 8070:R 8060:0 8056:P 8047:i 8043:P 8036:( 8030:i 8026:D 8019:= 8016:J 7994:T 7991:R 7988:n 7985:= 7982:V 7979:P 7953:i 7949:c 7945:d 7904:x 7880:) 7874:x 7871:d 7864:i 7860:c 7856:d 7850:( 7844:i 7840:D 7833:= 7830:J 7801:0 7797:D 7772:0 7768:T 7743:0 7739:P 7713:2 7709:/ 7705:3 7700:) 7693:0 7689:T 7685:T 7680:( 7674:) 7669:P 7664:0 7660:P 7654:( 7648:0 7644:D 7640:= 7637:D 7625:T 7621:P 7596:2 7592:/ 7588:1 7583:e 7580:R 7575:3 7570:L 7561:= 7548:L 7544:0 7540:x 7486:v 7479:x 7462:e 7459:R 7407:L 7401:v 7395:= 7391:e 7388:R 7383:) 7376:2 7372:/ 7368:1 7363:e 7360:R 7354:x 7351:5 7345:( 7341:= 7338:) 7335:x 7332:( 7272:1 7269:c 7265:2 7262:c 7256:t 7250:D 7246:. 7228:m 7225:t 7196:P 7192:A 7163:) 7160:t 7154:( 7150:/ 7146:D 7139:m 7136:t 7126:p 7122:A 7118:2 7115:= 7112:P 7078:2 7075:c 7069:1 7066:c 7052:1 7049:c 7045:2 7042:c 7031:P 7009:) 7003:1 6999:c 6990:2 6986:c 6981:( 6977:P 6970:= 6941:3 6938:/ 6935:1 6931:+ 6921:3 6918:/ 6915:1 6911:+ 6888:B 6884:C 6861:A 6857:C 6832:B 6828:D 6824:+ 6819:A 6815:D 6811:= 6806:r 6802:D 6748:3 6741:B 6737:C 6733:+ 6728:A 6724:C 6715:B 6711:C 6705:A 6701:C 6695:r 6691:D 6677:8 6672:= 6667:B 6664:A 6660:Z 6617:B 6613:C 6590:A 6586:C 6561:B 6557:D 6553:+ 6548:A 6544:D 6540:= 6535:r 6531:D 6508:R 6480:B 6476:C 6470:A 6466:C 6460:r 6456:D 6452:R 6444:4 6441:= 6436:B 6433:A 6429:Z 6359:4 6355:/ 6349:2 6345:L 6338:4 6318:A 6298:a 6275:D 6270:3 6266:/ 6262:2 6257:b 6253:C 6249:a 6246:2 6243:= 6240:t 6236:/ 6232:a 6226:r 6195:t 6173:D 6149:b 6145:C 6122:3 6118:/ 6114:1 6106:b 6102:C 6098:= 6092:L 6067:t 6064:D 6061:2 6056:= 6053:L 6018:D 5994:b 5990:C 5967:A 5942:r 5916:D 5911:3 5907:/ 5903:4 5898:b 5894:c 5890:A 5882:4 5877:= 5871:r 5801:C 5779:b 5775:C 5748:d 5735:t 5730:) 5724:( 5721:C 5712:t 5706:0 5692:D 5686:A 5673:t 5670:D 5660:b 5656:C 5651:A 5648:2 5645:= 5608:t 5605:D 5596:b 5592:C 5588:A 5585:2 5582:= 5577:0 5574:= 5571:x 5567:) 5560:t 5543:( 5538:t 5533:0 5525:= 5493:t 5490:D 5480:b 5476:C 5472:A 5469:D 5460:= 5455:0 5452:= 5449:x 5445:) 5438:t 5421:( 5398:A 5372:t 5369:D 5360:b 5356:C 5350:= 5345:0 5342:= 5339:x 5335:) 5328:x 5320:C 5311:( 5288:0 5285:= 5282:x 5258:t 5255:, 5252:x 5242:C 5225:) 5219:t 5216:D 5213:4 5207:2 5203:x 5194:( 5183:t 5180:D 5171:b 5167:C 5161:= 5155:x 5147:C 5118:0 5115:= 5112:x 5092:0 5089:= 5086:t 5063:. 5051:t 5019:t 5013:D 4995:b 4991:C 4978:A 4954:t 4951:D 4942:b 4938:C 4934:A 4931:2 4928:= 4879:t 4873:m 4864:μ 4855:T 4846:B 4843:k 4822:) 4816:) 4810:m 4807:( 4803:/ 4799:t 4792:e 4785:1 4781:( 4777:T 4771:B 4766:k 4758:= 4755:) 4752:t 4749:( 4746:D 4713:a 4709:D 4707:/ 4705:a 4585:β 4581:α 4575:j 4571:i 4562:, 4556:D 4542:j 4540:φ 4538:Δ 4533:D 4529:j 4523:i 4521:φ 4517:t 4514:∂ 4502:j 4498:i 4491:D 4486:i 4484:φ 4470:. 4466:) 4460:j 4444:j 4434:i 4422:j 4419:i 4415:D 4410:( 4398:j 4390:= 4384:t 4374:i 4333:. 4329:) 4320:i 4316:x 4307:) 4304:t 4301:, 4298:x 4295:( 4278:i 4274:x 4265:) 4262:x 4259:( 4254:j 4251:i 4247:D 4237:+ 4229:j 4225:x 4216:i 4212:x 4203:) 4200:t 4197:, 4194:x 4191:( 4183:2 4172:) 4169:x 4166:( 4161:j 4158:i 4154:D 4149:( 4143:3 4138:1 4135:= 4132:j 4129:, 4126:i 4118:= 4113:) 4108:) 4105:t 4102:, 4099:x 4096:( 4087:) 4084:x 4081:( 4078:D 4073:( 4062:= 4056:t 4048:) 4045:t 4042:, 4039:x 4036:( 4012:. 3998:D 3983:. 3975:j 3971:x 3962:i 3958:x 3949:) 3946:t 3943:, 3940:x 3937:( 3929:2 3916:j 3913:i 3909:D 3903:3 3898:1 3895:= 3892:j 3882:3 3877:1 3874:= 3871:i 3863:= 3858:) 3853:) 3850:t 3847:, 3844:x 3841:( 3832:D 3827:( 3816:= 3810:t 3802:) 3799:t 3796:, 3793:x 3790:( 3761:. 3753:j 3749:x 3729:j 3726:i 3722:D 3716:3 3711:1 3708:= 3705:j 3694:= 3689:i 3685:J 3664:, 3655:D 3649:= 3646:J 3632:D 3625:D 3593:i 3589:x 3580:) 3577:t 3574:, 3571:x 3568:( 3551:i 3547:x 3538:) 3535:x 3532:( 3529:D 3518:3 3513:1 3510:= 3507:i 3499:+ 3496:) 3493:t 3490:, 3487:x 3484:( 3475:) 3472:x 3469:( 3466:D 3463:= 3458:) 3453:) 3450:t 3447:, 3444:x 3441:( 3432:) 3429:x 3426:( 3423:D 3418:( 3407:= 3401:t 3393:) 3390:t 3387:, 3384:x 3381:( 3362:) 3360:x 3358:( 3356:D 3352:D 3323:t 3301:t 3298:D 3295:n 3292:2 3251:n 3231:t 3228:D 3225:n 3222:2 3219:= 3211:2 3207:) 3200:0 3196:x 3187:x 3183:( 3146:D 3129:. 3121:d 3117:) 3111:( 3108:D 3103:t 3098:0 3088:2 3075:D 3057:] 3052:) 3041:t 3038:D 3033:2 3029:x 3024:( 3020:2 3014:1 3010:[ 3004:0 3000:n 2996:= 2993:) 2990:t 2987:, 2984:x 2981:( 2978:n 2958:t 2954:x 2938:2 2933:0 2930:n 2924:0 2921:n 2913:2 2910:/ 2907:1 2898:x 2892:0 2889:n 2885:x 2883:( 2881:n 2874:x 2867:x 2865:( 2863:n 2829:. 2825:) 2817:t 2814:D 2809:2 2805:x 2800:( 2788:0 2784:n 2780:= 2776:) 2772:t 2769:, 2766:x 2762:( 2758:n 2744:0 2741:n 2734:x 2729:x 2725:t 2687:2 2683:x 2669:2 2658:D 2655:= 2649:t 2620:R 2596:D 2590:= 2580:j 2560:φ 2555:R 2546:j 2528:, 2525:R 2522:= 2518:j 2508:+ 2502:t 2434:, 2431:0 2428:= 2420:2 2402:x 2398:D 2393:φ 2375:. 2372:) 2362:D 2359:( 2350:= 2344:t 2297:, 2289:2 2280:D 2277:= 2271:t 2223:2 2219:x 2205:2 2194:D 2191:= 2182:x 2167:x 2155:D 2152:= 2148:) 2138:x 2126:D 2122:( 2115:x 2093:D 2076:0 2073:= 2068:) 2058:x 2046:D 2042:( 2035:x 2017:t 1997:0 1994:= 1991:J 1985:x 1973:+ 1967:t 1919:. 1915:) 1908:t 1905:D 1902:4 1896:2 1892:x 1882:( 1869:t 1866:D 1860:4 1856:1 1851:= 1848:) 1845:t 1842:, 1839:x 1836:( 1813:D 1793:k 1752:D 1749:= 1743:t 1705:x 1689:] 1684:1 1675:T 1667:2 1661:L 1655:[ 1645:D 1639:t 1634:t 1630:x 1626:) 1624:t 1622:, 1620:x 1618:( 1616:φ 1612:φ 1597:] 1592:3 1583:L 1575:N 1570:[ 1560:φ 1535:2 1531:x 1517:2 1505:D 1502:= 1496:t 1454:i 1438:i 1434:x 1413:, 1408:i 1404:x 1393:i 1389:M 1384:D 1372:= 1366:i 1362:J 1340:0 1337:= 1309:. 1300:D 1294:= 1288:i 1284:J 1262:0 1259:= 1231:. 1226:i 1222:y 1211:i 1207:M 1202:D 1190:= 1184:i 1180:J 1168:i 1162:i 1160:V 1145:, 1140:i 1136:y 1123:D 1117:= 1111:i 1107:V 1070:L 1065:i 1061:f 1057:= 1052:G 1047:i 1043:f 1020:L 1015:i 1011:f 988:G 983:i 979:f 967:i 950:i 946:f 923:i 919:f 892:x 882:i 878:f 865:T 862:R 858:D 850:= 845:i 841:J 819:μ 813:T 803:R 797:c 791:i 787:i 763:x 753:i 736:T 733:R 726:i 722:c 718:D 709:= 704:i 700:J 679:i 672:ρ 648:i 645:s 635:= 630:i 626:y 580:ρ 574:i 564:i 560:M 553:i 543:i 539:J 532:i 528:i 505:i 501:y 490:i 486:M 481:D 469:= 464:i 459:J 443:i 441:y 439:( 420:x 418:∂ 415:/ 411:φ 409:∂ 404:− 396:J 372:D 366:= 362:J 340:∇ 331:× 317:D 310:x 304:φ 282:x 279:d 271:d 246:D 240:J 224:J 201:x 198:d 190:d 184:D 178:= 175:J 87:D 38:. 20:)

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Knowledge

Fick's laws of diffusion

Index