Fick's laws of diffusion

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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,

5830: 4332: 6386: 4011: 3597: 4642:, 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 ( 2075: 2227: 3982: 3356: 8189:

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.

4327:{\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).} 6406:

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:

33: 6370:

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

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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,

6632:

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

6369:

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

4902:

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

4469: 152:

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

8162:

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

6915:

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

5845:

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

5825:

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

4684:

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

1081:

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

6377:

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

6200:

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

5750: 1937: 128:

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

7604:

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

7267:

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

5841:

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

7306:

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

7302:

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

7284:

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

6026:

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

4667:

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

8179:

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

8171:

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

7074:

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

6636:

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

6405:

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

6030:

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

2090: 4663:

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

4889:

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

5614: 5224: 4658:

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

8172:

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

7412: 1539: 2691: 4347: 3592:{\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}}}} 2841:. 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 2828: 7307:

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.

1918: 5495: 8448: 55:: 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. 6628:

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.

7714: 3230: 2374: 5374: 3760: 2070:{\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} 7280:

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.

6750: 5626: 4625:

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

2527: 765: 2296: 2222:{\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}}}} 3977:{\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}}}.} 894: 6916:

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

1757: 7268:

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.

1412: 1230: 8261: 8079: 506: 4822: 153:

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:

2601: 5915: 6895:

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

7600: 3324:

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.

7010: 129:

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

7880: 4960: 3128: 7313: 7164: 4886: 1144: 1308: 8506: 8180:

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.

1071: 654: 113:, 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: 6027:

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.

8153: 6481: 377: 6633:

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.

6274: 5506: 203: 5125: 109:

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

1817: 8176:

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.

284: 2433: 3663: 7621: 3156: 6833: 6562: 5944: 3057: 1466: 1021: 989: 6123: 6068: 3668: 2619: 3302: 1339: 1261: 7462: 6358: 5837:

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

2742: 7229: 6778: 5405: 7923: 7435: 5030: 2385:

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

7993: 7954: 7802: 7773: 7744: 7521: 7499: 7197: 6889: 6862: 6618: 6591: 6150: 5995: 5820: 5780: 4996: 1439: 951: 924: 597: 5287: 5257: 5117: 5091: 8159:

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.

6299:

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

8102: 7903: 6507: 6317: 6297: 6194: 6172: 6017: 5966: 5800: 5397: 5050: 3322: 3250: 2314: 1812: 1792: 4464:{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right).} 1688: 1596: 7090:

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

5295: 5745:{\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 } 8352: 1344: 1162: 8000: 6643: 2472: 43:

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.

5620:

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

683: 133:

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

7294:

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

2241: 6637:

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:

6201:

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

7542: 8981: 824: 313: 7814: 316:. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 3260:

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

1713: 673:

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

4724:

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

441: 88:: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient. 4730: 1089: 1266: 4498: 4672:, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by 2562: 5852: 4668:

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

6393:

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

4716:. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the 3268:, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion. 6949: 4909: 3072: 8540: 8866:

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

7096: 4630:

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

2308:

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

6935:, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules. 3630: 8953:

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

8759:

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

9178: 4856: 2845:, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is 1026: 609: 8556:

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

5802:

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

8475: 6394: 5609:{\displaystyle \Gamma =\int _{0}^{t}({\frac {\partial \Gamma }{\partial t}})_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}} 2701: 2455: 8109: 6412: 4688:

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

345: 9183: 8206: 5219:{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}(-{\frac {x^{2}}{4Dt}})} 4535:

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

6207: 8367: 4717: 4337: 159: 9193: 8192:

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

7407:{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }}} 1457: 252: 7808:

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

2399: 51:: With more molecules, there is a clear trend where the solute fills the container more and more uniformly. 9148: 8399: 8231: 4882:

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

4713: 4689: 4643: 4635: 4536: 2916:(as the diffusion now occurs in both directions). This case is valid when some solution with concentration 2463: 7811:

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

1534:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}}} 97:

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

9188: 2686:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}} 8236: 6785: 6514: 5923: 2962: 7605:

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

4501:

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

2606:

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

994: 962: 308:

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

8989: 7082:

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

6379: 6378:

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

6076: 6037: 3150: 425: 3274: 2823:{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right).} 1315: 1237: 7442: 3994: 2084:

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

295:(for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume. 6322: 1913:{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).} 6019:

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

5490:{\displaystyle ({\frac {\partial \Gamma }{\partial t}})_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}} 1083: 110: 8467:

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

7608:

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

7204: 4542:

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

231:

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

9082: 7023: 6761: 4899: 4618: 795: 8402:, Sargsyan HP, Wahab HA (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". 3133:

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

7908: 7420: 6624:

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

6382:. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry. 5015: 2439: 324:. For biological molecules the diffusion coefficients normally range from 10 to 10 m/s. 9074: 7963: 7929: 5946:

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

5842:

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

8962: 8909: 8822: 8768: 8630: 8575: 8300: 7780: 7751: 7722: 7709:{\displaystyle D=D_{0}\left({\frac {P_{0}}{P}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2}} 7506: 7484: 7231:

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

7175: 6867: 6840: 6596: 6569: 6128: 5973: 5805: 5758: 4974: 4847: 3225:{\displaystyle {\text{MSD}}\equiv \langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\rangle =2nDt} 1767: 1417: 929: 902: 582: 220: 4998:

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

2369:{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).} 78:. Fick's first law can be used to derive his second law which in turn is identical to the 8: 8729:

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

7049: 6398: 5266: 5236: 5096: 5070: 4878:(e.g. proteins) in water, the exponential term is negligible due to the small product of 4610: 2707: 224: 98: 36: 8966: 8913: 8826: 8772: 8634: 8579: 8304: 5369:{\displaystyle ({\frac {\partial C}{\partial x}})_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}} 5119:) in a once uniform bulk solution is solved in the above sections from Fick's equation, 3755:{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.} 1456:

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

70:

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

9173: 9032: 8930: 8897: 8843: 8810: 8703: 8678: 8651: 8618: 8599: 8565: 8429: 8411: 8087: 7888: 7291: 6492: 6302: 6282: 6179: 6157: 6125:(unit m) is the average nearest neighbor distance approximated as cubic packing, where 6002: 5951: 5785: 5382: 5067:

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

5035: 4855:

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

4838: 3307: 3235: 2443: 1797: 1777: 674: 79: 3140: 1639: 1554: 9124: 9075: 9024: 8935: 8848: 8708: 8656: 8591: 8536: 8471: 4721: 4709: 4705: 4669: 4587: 3998: 3256:

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

2389:, the solution for the concentration will be a linear change of concentrations along 1928: 8433: 6745:{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt{C_{A}+C_{B}}}} 6373:

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

5032:

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

2522:{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,} 9120: 9036: 9016: 8925: 8917: 8871: 8838: 8830: 8776: 8738: 8698: 8690: 8646: 8638: 8603: 8583: 8421: 8331: 8308: 6625: 5061: 4631: 3304:, is often used as a characterization of how far has the particle moved after time 243: 122: 71: 760:{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}}} 8465: 8216: 7083: 6402: 5057: 4885: 4673: 3261: 3146: 9004: 8898:"Why Should the Reaction Order of a Bimolecular Reaction be 2.33 Instead of 2?" 8587: 8324:

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

7232: 5968:

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

4614: 3353:. This dependence does not affect Fick's first law but the second law changes: 2838: 2291:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,} 114: 24: 20: 8679:"Photobleaching of YOYO-1 in super-resolution single DNA fluorescence imaging" 8335: 2235:

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

9167: 9020: 8921: 8875: 8313: 8288: 7277: 7045: 6374: 5829: 4639: 3257: 2953: 2302: 1763: 677:

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

94:: Prediction of change in concentration gradient with time due to diffusion. 8811:"Simulating stochastic adsorption of diluted solute molecules at interfaces" 8535:. Wiley Series in Chemical Engineering. Vol. 2. John Wiley & Sons. 8425: 2925:

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

9028: 8939: 8852: 8712: 8660: 8595: 8221: 7288:

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

7076: 4599: 2952:

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

424:

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

118: 8353:"One and a Half Centuries of Diffusion: Fick, Einstein, before and beyond" 4679: 2941:

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

889:{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}}} 8694: 8570: 8173: 5997:

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

4875: 4660: 4655: 4622: 3603: 2232:

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

1771: 9005:"Understanding diffusion theory and Fick's law through food and cooking" 8742: 3608:, the diffusion coefficient depends on the direction. It is a symmetric 1752:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi } 1023:

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

8642: 6152:

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

4895: 4595: 4493:

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

558: 67: 39:

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

8834: 8780: 2708:

Example solution 1: constant concentration source and diffusion length

1407:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},} 1225:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.} 32: 9154: 8226: 8211: 8201: 8074:{\displaystyle J=-D_{i}\left({\frac {P_{i}-P_{0}}{\delta RT}}\right)} 6365:

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

4903:

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

3253: 2697: 2459: 1701: 501:{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i}} 309: 63: 4817:{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right)} 391:

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

9158: 6174:

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

4692:. In the ultrashort time limit, in the order of the diffusion time 3141:

Example solution 2: Brownian particle and mean squared displacement

815: 600: 336: 8416: 8163:

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

1707:, which generalises the second derivative, obtaining the equation 8241: 7995:, the concentration of the gas is expressed by partial pressure. 5838: 4627: 2596:{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi } 572: 9077:

Thermodynamics and Kinetics in Materials Science: A Short Course

7263:

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

6385: 5910:{\displaystyle \langle r\rangle ={\frac {4}{\pi }}Ac_{b}^{4/3}D} 5289:

is simplified to the pre-exponential factor of the distribution

4008:

these two forms of the diffusion equation should be combined in

4603: 4591: 4586:

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

3609: 3265: 1927:

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

40: 4340:

gives the following generalization of Fick's equation for the

2442:, the solutions to which are referred to by mathematicians as 2552:. The only source of flux in this situation is assumed to be 435:, given for example in kg/kg), then the equation changes to: 7022:

is the permeability, an experimentally determined membrane "

1341:) both species have the same molar mass, Fick's law becomes 339:

operator, which generalises the first derivative, obtaining

312:

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

8724: 8722: 7595:{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}} 4606: 2539: 1076: 149: 9072: 8262:"Molecular Diffusion - an overview | ScienceDirect Topics" 7005:{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)}} 6389:

Comparing collision theory and diffusive collision theory.

4704:

is the particle radius, the diffusion is described by the

416:, which for ideal mixtures is the concentration gradient. 7875:{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right)} 7271: 6835:

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

4955:{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}} 3123:{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.} 2462:

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

332: 8804: 8802: 8800: 8798: 8796: 8794: 8792: 8790: 8719: 8507:"Fickian Diffusion - an overview | ScienceDirect Topics" 7297: 7199:

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

6891:

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

6620:

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

6564:

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

5833:

A brief history of the theories on diffusive adsorption.

1762:

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

4680:

Sorption rate and collision frequency of diluted solute

2449: 9073:

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

6780:

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

2891:), then the solution is amended only with coefficient 1263:, this reduces to the most common form of Fick's law, 814:

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

8787: 8112: 8090: 8003: 7966: 7932: 7911: 7891: 7817: 7783: 7754: 7725: 7624: 7545: 7509: 7487: 7445: 7423: 7316: 7207: 7178: 7159:{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {D/(\pi t)}}} 7099: 6952: 6870: 6843: 6788: 6764: 6646: 6599: 6572: 6517: 6495: 6415: 6325: 6305: 6285: 6210: 6182: 6160: 6131: 6079: 6040: 6005: 5976: 5954: 5926: 5855: 5808: 5788: 5761: 5629: 5509: 5408: 5385: 5298: 5269: 5239: 5233:

is the number concentration of adsorber molecules at

5128: 5099: 5073: 5038: 5018: 4977: 4912: 4733: 4685:

when these interactions happen in diluted solutions.

4350: 4014: 3768: 3671: 3633: 3359: 3310: 3277: 3238: 3159: 3075: 2965: 2745: 2622: 2565: 2475: 2402: 2317: 2244: 2093: 1940: 1820: 1800: 1780: 1716: 1642: 1557: 1469: 1420: 1347: 1318: 1269: 1240: 1165: 1092: 1029: 997: 965: 932: 905: 827: 686: 612: 585: 444: 348: 255: 162: 9063: 3997:. It is needed to make the right hand side operator 1139:{\displaystyle \mathbf {V_{i}} =-D\nabla \ln y_{i},} 8398: 6943:The first law gives rise to the following formula: 4874:For a single molecule such as organic molecules or 2727:, where the concentration is maintained at a value 2696:If flux were the result of both diffusive flux and 1303:{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .} 9149:Fick's equations, Boltzmann's transformation, etc. 8616: 8166: 8147: 8096: 8073: 7987: 7948: 7917: 7897: 7874: 7796: 7767: 7738: 7708: 7594: 7515: 7493: 7456: 7429: 7406: 7223: 7191: 7158: 7004: 6883: 6856: 6827: 6772: 6744: 6612: 6585: 6556: 6501: 6475: 6352: 6311: 6291: 6268: 6188: 6166: 6144: 6117: 6062: 6011: 5989: 5960: 5938: 5909: 5814: 5794: 5774: 5744: 5608: 5489: 5391: 5368: 5281: 5251: 5218: 5111: 5085: 5044: 5024: 4990: 4954: 4816: 4463: 4326: 3976: 3754: 3657: 3591: 3316: 3296: 3244: 3224: 3122: 3051: 2822: 2685: 2595: 2521: 2427: 2368: 2290: 2221: 2069: 1922: 1912: 1806: 1786: 1751: 1682: 1590: 1533: 1433: 1406: 1333: 1302: 1255: 1224: 1138: 1065: 1015: 983: 945: 918: 888: 759: 648: 591: 500: 371: 278: 197: 8754: 8752: 8384:Vázquez JL (2006). "The Porous Medium Equation". 7241:is the diffusion constant of the solute unit m⋅s. 6319:over solute nearest neighbor sphere surface area 6022:Dimensional analysis of these units is satisfied. 1066:{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}} 649:{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }}} 9165: 9134:Foundations of Materials Science and Engineering 8946: 8891: 8889: 8887: 8885: 8865: 8728: 5263:The concentration gradient at the subsurface at 3066:is time-dependent, the diffusion length becomes 16:Mathematical descriptions of molecular diffusion 9002: 8859: 8555: 5004:is diffusion coefficient of the adsorber (m/s). 3984:The symmetric matrix of diffusion coefficients 8749: 8672: 8670: 8446: 8183: 8148:{\displaystyle {\frac {P_{i}-P_{0}}{\delta }}} 6476:{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B}} 5056:The equation is named after American chemists 4499:Chapman–Enskog formulae for diffusion in gases 3762:For the diffusion equation this formula gives 1636:is the diffusion coefficient in dimensions of 419: 372:{\displaystyle \mathbf {J} =-D\nabla \varphi } 301:is position, the dimension of which is length. 9003:Zhou L, Nyberg K, Rowat AC (September 2015). 8882: 8617:Bian X, Kim C, Karniadakis GE (August 2016). 8530: 8392: 5379:And the rate of diffusion (flux) across area 4485:is the matrix of coefficients. Here, indices 4101: 4061: 3846: 3815: 3446: 3406: 3338:, the diffusion coefficient varies in space, 9111:Fick, Adolph (1995). "On liquid diffusion". 9081:. Oxford: Oxford University Press. pp. 8952: 6401:in a seminal 1916 publication, derived from 6269:{\displaystyle <r>=a/t=2aC_{b}^{2/3}D} 5933: 5927: 5862: 5856: 3665:it is the product of a tensor and a vector: 3204: 3168: 8667: 8404:Mathematical Modelling of Natural Phenomena 2720:-axis) from a boundary located at position 2454:Fick's second law is a special case of the 198:{\displaystyle J=-D{\frac {d\varphi }{dx}}} 47:: A single molecule moves around randomly. 9064:Bird RB, Stewart WE, Lightfoot EN (1976). 8758: 8470:. Cambridge University Press. p. 43. 1700:In two or more dimensions we must use the 8929: 8842: 8702: 8650: 8569: 8415: 8350: 8312: 7026:" for a given gas at a given temperature. 6938: 5735: 4752: 4676:the fluctuating hydrodynamics equations. 4649: 4478:are concentrations of the components and 4442: 3108: 2353: 2271: 2060: 1496: 8731:Journal of the American Chemical Society 8676: 8490: 6384: 5828: 4884: 3145:Another simple case of diffusion is the 1774:, except switching thermal conductivity 1077:Derivation of Fick's first law for gases 31: 8463: 8450:Physical Chemistry for the Life Science 8383: 4338:Einstein's mobility and Teorell formula 2379:An important example is the case where 1617:is a function that depends on location 1086:reduces to this version of Fick's law: 279:{\displaystyle {\frac {d\varphi }{dx}}} 9166: 9099:Fick A (1855). "On liquid diffusion". 8322:Fick A (1855). "On liquid diffusion". 7956:is the first reactant's concentration. 7310:The velocity profile for gas flow is: 7272:Semiconductor fabrication applications 3264:is a 3-D diffusion. For a cylindrical 2428:{\displaystyle \nabla ^{2}\varphi =0,} 2393:. In two or more dimensions we obtain 1931:in absence of any chemical reactions: 1663: 1649: 1571: 1563: 1551:is the concentration in dimensions of 327:In two or more dimensions we must use 247:. Its dimension is area per unit time. 9131: 9089: 7616:) in a gas, diffusion is determined. 7298:CVD method of fabricate semiconductor 4578:correspond to the space coordinates. 2849:(lasting both through the layer with 2712:A simple case of diffusion with time 1155:is the diffusion velocity of species 9110: 9098: 9054: 8895: 8808: 8321: 8286: 4634:. One more general framework is the 3658:{\displaystyle J=-D\nabla \varphi ,} 2450:Example solutions and generalization 1448: 544:th species (for example in mol/m-s), 8902:The Journal of Physical Chemistry A 8683:Beilstein Journal of Nanotechnology 8677:Pyle JR, Chen J (2 November 2017). 2613:), we arrive at Fick's second law: 2080:Assuming the diffusion coefficient 1159:. In terms of species flux this is 953:is a partial pressure of component 388:denotes the diffusion flux vector. 140: 13: 9047: 8491:Williams FA (1985). "Appendix E". 7571: 7568: 7539:, it gives the average thickness: 7450: 7447: 7379: 7376: 7351: 7348: 5630: 5545: 5540: 5537: 5510: 5423: 5418: 5415: 5313: 5305: 5140: 5132: 5019: 4913: 4857:Einstein relation (kinetic theory) 4759: 4439: 4391: 4369: 4354: 4300: 4277: 4258: 4231: 4209: 4196: 4167: 4078: 4053: 4041: 4018: 3955: 3942: 3913: 3823: 3807: 3795: 3772: 3733: 3725: 3646: 3573: 3550: 3531: 3514: 3466: 3423: 3398: 3386: 3363: 3328: 2667: 2653: 2634: 2626: 2587: 2499: 2487: 2479: 2404: 2354: 2341: 2329: 2321: 2273: 2256: 2248: 2203: 2189: 2167: 2163: 2152: 2148: 2123: 2119: 2100: 2096: 2043: 2039: 2020: 2016: 2002: 1994: 1970: 1966: 1952: 1944: 1743: 1728: 1720: 1515: 1501: 1481: 1473: 1388: 1319: 1291: 1241: 1206: 1114: 877: 862: 810:is the chemical potential (J/mol). 748: 733: 485: 363: 14: 9205: 9142: 8979: 8279: 8155:is the partial pressure gradient. 6828:{\displaystyle D_{r}=D_{A}+D_{B}} 6557:{\displaystyle D_{r}=D_{A}+D_{B}} 6395:Smoluchowski coagulation equation 5939:{\displaystyle \langle r\rangle } 3052:{\displaystyle n(x,t)=n_{0}\left} 1312:If (instead of or in addition to 9009:Advances in Physiology Education 7052:for the direction of flow (from 5093:) absorptive surface (placed at 3188: 3184: 3175: 3149:of one particle. The particle's 2945:-direction by diffusion in time 2568: 2506: 1441:is the mole fraction of species 1354: 1350: 1276: 1272: 1172: 1168: 1099: 1095: 1016:{\displaystyle f_{i}^{\text{L}}} 984:{\displaystyle f_{i}^{\text{G}}} 447: 350: 8996: 8973: 8868:Journal of Physical Chemistry B 8610: 8549: 8524: 8167:Invalidity of Fickian diffusion 6118:{\displaystyle L~=C_{b}^{-1/3}} 6063:{\displaystyle L={\sqrt {2Dt}}} 4718:fluctuation-dissipation theorem 4581: 4006:inhomogeneous anisotropic media 3153:from its original position is: 2716:in one dimension (taken as the 2546:is a net volumetric source for 1923:Derivation of Fick's second law 804:is the absolute temperature (K) 19:For the technique of measuring 8986:Essentials of Human Physiology 8619:"111 years of Brownian motion" 8499: 8484: 8457: 8440: 8377: 8344: 8254: 7804:is the standard diffusitivity. 7470:is the length of the substrate 7326: 7320: 7151: 7142: 6509:is the radius of the collision 6196:is the critical time (unit s). 5718: 5712: 5555: 5531: 5433: 5409: 5323: 5299: 5213: 5182: 4804: 4795: 4743: 4737: 4295: 4283: 4253: 4247: 4191: 4179: 4160: 4154: 4096: 4084: 4075: 4069: 4036: 4024: 3937: 3925: 3841: 3829: 3790: 3778: 3627:. Fick's first law changes to 3568: 3556: 3526: 3520: 3484: 3472: 3463: 3457: 3441: 3429: 3420: 3414: 3381: 3369: 3297:{\displaystyle {\sqrt {2nDt}}} 3195: 3171: 3105: 3099: 2981: 2969: 2360: 2347: 1988: 1836: 1824: 1677: 1643: 1585: 1558: 1460:which in one dimension reads: 1334:{\displaystyle \nabla \rho =0} 1256:{\displaystyle \nabla \rho =0} 666:is the partial density of the 1: 9151:(with figures and animations) 9101:Annalen der Physik und Chemie 8447:Atkins P, de Paula J (2006). 7457:{\displaystyle \mathrm {Re} } 7285:second through mathematics. 5826:convection are significant. 4638:equations of multi-component 2702:convection–diffusion equation 2456:convection–diffusion equation 1458:partial differential equation 227:per unit area per unit time. 9136:(3rd ed.). McGraw-Hill. 9125:10.1016/0376-7388(94)00230-v 9092:The Mathematics of Diffusion 8533:Multicomponent mass transfer 8531:Taylor R, Krishna R (1993). 8247: 8207:Churchill–Bernstein equation 6353:{\displaystyle 4\pi L^{2}/4} 4567:refer to the components and 1082:diffusion equation from the 788:is the concentration (mol/m) 7: 9113:Journal of Membrane Science 8761:Journal of Chemical Physics 8195: 8184:Food production and cooking 7775:is the standard temperature 5782:is the bulk concentration, 4864:is the mass of the particle 4344:of the perfect components: 1794:with diffusion coefficient 603:operator. This is because: 420:Variations of the first law 10: 9210: 9094:. Oxford University Press. 8588:10.1103/PhysRevE.63.012105 8366:: 1.1–1.10. Archived from 7224:{\displaystyle \eta _{tm}} 2301:which is analogous to the 1696:is the position, example m 104: 101:or non-Fickian diffusion. 66:and were first posited by 18: 9179:Eponymous laws of physics 8336:10.1080/14786445508641925 7075:situation, one can use a 6773:{\displaystyle {\sigma }} 6380:Langmuir adsorption model 6367:long cylindrical molecule 4621:processes. The theory of 3151:Mean squared displacement 3137:varies with temperature. 9068:. John Wiley & Sons. 9021:10.1152/advan.00133.2014 8982:"Section 3/3ch9/s3ch9_2" 8922:10.1021/acs.jpca.2c07500 8896:Chen J (December 2022). 8876:10.1021/acs.jpcb.4c03359 8314:10.1002/andp.18551700105 8232:Maxwell–Stefan diffusion 7746:is the standard pressure 4969:is the surface area (m). 4827:where (all in SI units) 4714:Stokes–Einstein equation 4708:. At a longer time, the 4690:Stokes–Einstein equation 4636:Maxwell–Stefan diffusion 4537:Maxwell–Stefan diffusion 4342:multicomponent diffusion 3271:The square root of MSD, 314:Stokes–Einstein relation 121:(charge transport), and 60:Fick's laws of diffusion 9184:Mathematics in medicine 9057:Random Walks in Biology 8809:Chen J (January 2022). 7918:{\displaystyle \delta } 7430:{\displaystyle \delta } 5500:Integrating over time, 5025:{\displaystyle \Gamma } 1084:kinetic theory of gases 8955:Zeitschrift für Physik 8360:Diffusion Fundamentals 8237:Nernst–Planck equation 8149: 8098: 8075: 7989: 7988:{\displaystyle PV=nRT} 7950: 7949:{\displaystyle dc_{i}} 7919: 7899: 7876: 7798: 7769: 7740: 7710: 7596: 7517: 7495: 7464:is the Reynolds number 7458: 7431: 7408: 7225: 7193: 7160: 7048:of the gas across the 7006: 6939:Biological perspective 6885: 6858: 6829: 6774: 6746: 6614: 6587: 6558: 6503: 6477: 6390: 6354: 6313: 6293: 6270: 6190: 6168: 6146: 6119: 6064: 6013: 5991: 5962: 5940: 5911: 5834: 5822:is a dummy variable. 5816: 5796: 5776: 5746: 5610: 5491: 5393: 5370: 5283: 5253: 5220: 5113: 5087: 5046: 5026: 4992: 4956: 4891: 4818: 4650:Fick's flow in liquids 4465: 4336:The approach based on 4328: 4135: 3978: 3895: 3874: 3756: 3708: 3659: 3593: 3510: 3318: 3298: 3246: 3226: 3124: 3053: 2824: 2687: 2597: 2523: 2429: 2370: 2292: 2223: 2071: 1914: 1808: 1788: 1753: 1684: 1592: 1535: 1435: 1408: 1335: 1304: 1257: 1226: 1140: 1067: 1017: 985: 947: 920: 890: 796:universal gas constant 761: 650: 593: 575:(for example in kg/m). 502: 373: 288:concentration gradient 280: 199: 148:relates the diffusive 56: 9194:Statistical mechanics 8511:www.sciencedirect.com 8426:10.1051/mmnp/20116509 8388:. Oxford Univ. Press. 8266:www.sciencedirect.com 8150: 8099: 8076: 7990: 7951: 7920: 7900: 7877: 7799: 7797:{\displaystyle D_{0}} 7770: 7768:{\displaystyle T_{0}} 7741: 7739:{\displaystyle P_{0}} 7711: 7597: 7518: 7516:{\displaystyle \rho } 7496: 7494:{\displaystyle \eta } 7459: 7432: 7409: 7226: 7194: 7192:{\displaystyle A_{P}} 7161: 7044:is the difference in 7007: 6886: 6884:{\displaystyle C_{B}} 6859: 6857:{\displaystyle C_{A}} 6830: 6775: 6747: 6615: 6613:{\displaystyle C_{B}} 6588: 6586:{\displaystyle C_{A}} 6559: 6504: 6478: 6388: 6355: 6314: 6294: 6271: 6191: 6169: 6147: 6145:{\displaystyle C_{b}} 6120: 6065: 6014: 5992: 5990:{\displaystyle C_{b}} 5963: 5941: 5912: 5832: 5817: 5815:{\displaystyle \tau } 5797: 5777: 5775:{\displaystyle C_{b}} 5747: 5611: 5492: 5394: 5371: 5284: 5254: 5221: 5114: 5088: 5047: 5027: 4993: 4991:{\displaystyle C_{b}} 4957: 4888: 4819: 4613:, nuclear materials, 4466: 4329: 4109: 3979: 3875: 3854: 3757: 3688: 3660: 3594: 3490: 3336:non-homogeneous media 3319: 3299: 3247: 3227: 3125: 3054: 2837:is the complementary 2825: 2688: 2598: 2524: 2458:in which there is no 2430: 2371: 2293: 2224: 2072: 1915: 1809: 1789: 1754: 1685: 1593: 1536: 1436: 1434:{\displaystyle x_{i}} 1409: 1336: 1305: 1258: 1227: 1141: 1068: 1018: 986: 948: 946:{\displaystyle f_{i}} 921: 919:{\displaystyle f_{i}} 891: 762: 651: 594: 592:{\displaystyle \rho } 538:diffusion flux vector 503: 374: 281: 239:diffusion coefficient 200: 72:diffusion coefficient 35: 8961:: 557–571, 585–599. 8695:10.3762/bjnano.8.229 8495:. Benjamin/Cummings. 8351:Philibert J (2005). 8110: 8088: 8001: 7964: 7930: 7909: 7889: 7815: 7781: 7752: 7723: 7622: 7543: 7507: 7485: 7443: 7421: 7314: 7303:create thin films. 7205: 7176: 7097: 6950: 6868: 6841: 6786: 6762: 6644: 6597: 6570: 6515: 6493: 6413: 6323: 6303: 6283: 6208: 6180: 6158: 6129: 6077: 6038: 6003: 5974: 5952: 5924: 5853: 5806: 5786: 5759: 5627: 5507: 5406: 5383: 5296: 5267: 5237: 5126: 5097: 5071: 5036: 5016: 5010:is elapsed time (s). 4975: 4910: 4848:absolute temperature 4731: 4619:semiconductor doping 4348: 4012: 3766: 3669: 3631: 3357: 3308: 3275: 3236: 3157: 3073: 2963: 2743: 2620: 2563: 2473: 2400: 2315: 2242: 2091: 1938: 1818: 1798: 1778: 1768:fundamental solution 1714: 1640: 1555: 1467: 1418: 1345: 1316: 1267: 1238: 1163: 1090: 1027: 995: 963: 930: 903: 825: 684: 610: 583: 442: 346: 253: 160: 9066:Transport Phenomena 8967:1916ZPhy...17..557S 8914:2022JPCA..126.9719C 8827:2022AIPA...12a5318C 8773:1946JChPh..14..453W 8743:10.1021/ja01290a091 8635:2016SMat...12.6331B 8580:2000PhRvE..63a2105B 8464:Conlisk AT (2013). 8386:Mathematical Theory 8373:on 5 February 2009. 8305:1855AnP...170...59F 8104:is the gas constant 7612:) and temperature ( 6399:Marian Smoluchowski 6262: 6114: 5903: 5705: 5530: 5282:{\displaystyle x=0} 5252:{\displaystyle x,t} 5112:{\displaystyle x=0} 5086:{\displaystyle t=0} 4611:population dynamics 4588:transport processes 3095: 2464:continuity equation 1770:is the same as the 1062: 1044: 1012: 980: 225:amount of substance 99:anomalous diffusion 37:Molecular diffusion 9189:Physical chemistry 8643:10.1039/c6sm01153e 8564:(1 Pt 1): 012105. 8293:Annalen der Physik 8145: 8094: 8071: 7985: 7946: 7915: 7895: 7872: 7794: 7765: 7736: 7706: 7592: 7513: 7491: 7454: 7427: 7404: 7292:Integrated circuit 7221: 7189: 7156: 7002: 6881: 6854: 6825: 6770: 6742: 6610: 6583: 6554: 6499: 6473: 6391: 6350: 6309: 6289: 6266: 6240: 6186: 6164: 6142: 6115: 6089: 6060: 6009: 5987: 5958: 5936: 5907: 5881: 5835: 5812: 5792: 5772: 5742: 5689: 5606: 5516: 5487: 5389: 5366: 5279: 5249: 5216: 5109: 5083: 5042: 5022: 4988: 4952: 4892: 4839:Boltzmann constant 4814: 4461: 4390: 4324: 3974: 3752: 3655: 3589: 3314: 3294: 3242: 3222: 3120: 3081: 3049: 2820: 2683: 2593: 2519: 2444:harmonic functions 2440:Laplace's equation 2425: 2366: 2288: 2219: 2067: 1910: 1804: 1784: 1749: 1680: 1630:is time, example s 1588: 1531: 1431: 1404: 1331: 1300: 1253: 1234:If, additionally, 1222: 1136: 1063: 1048: 1030: 1013: 998: 981: 966: 943: 916: 886: 757: 675:chemical potential 646: 589: 498: 369: 276: 195: 125:(heat transport). 117:(hydraulic flow), 80:diffusion equation 57: 9155:Fick's Second Law 9132:Smith WF (2004). 8992:on 24 March 2016. 8908:(51): 9719–9725. 8835:10.1063/5.0064140 8781:10.1063/1.1724167 8737:(11): 2400–2414. 8629:(30): 6331–6346. 8558:Physical Review E 8542:978-0-471-57417-0 8493:Combustion Theory 8289:"Ueber Diffusion" 8143: 8097:{\displaystyle R} 8065: 7960:In ideal gas law 7905:is the thickness 7898:{\displaystyle x} 7866: 7686: 7660: 7590: 7402: 7369: 7154: 6956: 6740: 6671: 6502:{\displaystyle R} 6312:{\displaystyle A} 6292:{\displaystyle a} 6189:{\displaystyle t} 6167:{\displaystyle D} 6085: 6058: 6012:{\displaystyle D} 5961:{\displaystyle A} 5876: 5795:{\displaystyle C} 5733: 5732: 5703: 5687: 5686: 5669: 5668: 5650: 5604: 5603: 5552: 5485: 5484: 5430: 5392:{\displaystyle A} 5364: 5363: 5320: 5211: 5180: 5175: 5174: 5147: 5045:{\displaystyle t} 4950: 4949: 4722:Langevin equation 4710:Langevin equation 4706:Langevin equation 4437: 4381: 4376: 4314: 4272: 4223: 4048: 3995:positive definite 3969: 3802: 3747: 3587: 3545: 3393: 3317:{\displaystyle t} 3292: 3245:{\displaystyle n} 3163: 3115: 3038: 3035: 2811: 2808: 2681: 2641: 2575: 2494: 2336: 2263: 2217: 2174: 2159: 2130: 2107: 2050: 2027: 2009: 1977: 1959: 1929:mass conservation 1900: 1861: 1860: 1807:{\displaystyle D} 1787:{\displaystyle k} 1735: 1598:, example mol/m; 1529: 1488: 1454:Fick's second law 1449:Fick's second law 1386: 1204: 1060: 1042: 1010: 978: 884: 857: 755: 728: 644: 483: 274: 193: 92:Fick's second law 9201: 9137: 9128: 9108: 9095: 9090:Crank J (1980). 9086: 9080: 9069: 9060: 9055:Berg HC (1977). 9041: 9040: 9000: 8994: 8993: 8988:. Archived from 8977: 8971: 8970: 8950: 8944: 8943: 8933: 8893: 8880: 8879: 8863: 8857: 8856: 8846: 8806: 8785: 8784: 8756: 8747: 8746: 8726: 8717: 8716: 8706: 8674: 8665: 8664: 8654: 8614: 8608: 8607: 8573: 8571:cond-mat/0006163 8553: 8547: 8546: 8528: 8522: 8521: 8519: 8517: 8503: 8497: 8496: 8488: 8482: 8481: 8461: 8455: 8454: 8444: 8438: 8437: 8419: 8396: 8390: 8389: 8381: 8375: 8374: 8372: 8357: 8348: 8342: 8339: 8318: 8316: 8283: 8277: 8276: 8274: 8272: 8258: 8154: 8152: 8151: 8146: 8144: 8139: 8138: 8137: 8125: 8124: 8114: 8103: 8101: 8100: 8095: 8080: 8078: 8077: 8072: 8070: 8066: 8064: 8053: 8052: 8051: 8039: 8038: 8028: 8022: 8021: 7994: 7992: 7991: 7986: 7955: 7953: 7952: 7947: 7945: 7944: 7924: 7922: 7921: 7916: 7904: 7902: 7901: 7896: 7881: 7879: 7878: 7873: 7871: 7867: 7865: 7857: 7856: 7855: 7842: 7836: 7835: 7803: 7801: 7800: 7795: 7793: 7792: 7774: 7772: 7771: 7766: 7764: 7763: 7745: 7743: 7742: 7737: 7735: 7734: 7715: 7713: 7712: 7707: 7705: 7704: 7700: 7691: 7687: 7685: 7684: 7672: 7665: 7661: 7656: 7655: 7646: 7640: 7639: 7601: 7599: 7598: 7593: 7591: 7589: 7588: 7587: 7583: 7574: 7561: 7553: 7538: 7534: 7530: 7522: 7520: 7519: 7514: 7500: 7498: 7497: 7492: 7478: 7469: 7463: 7461: 7460: 7455: 7453: 7437:is the thickness 7436: 7434: 7433: 7428: 7413: 7411: 7410: 7405: 7403: 7398: 7387: 7382: 7374: 7370: 7368: 7367: 7363: 7354: 7345: 7337: 7230: 7228: 7227: 7222: 7220: 7219: 7198: 7196: 7195: 7190: 7188: 7187: 7165: 7163: 7162: 7157: 7155: 7141: 7133: 7131: 7130: 7118: 7117: 7069: 7060: 7043: 7021: 7011: 7009: 7008: 7003: 7001: 7000: 6996: 6995: 6994: 6982: 6981: 6957: 6954: 6934: 6932: 6931: 6928: 6925: 6921: 6914: 6912: 6911: 6908: 6905: 6901: 6890: 6888: 6887: 6882: 6880: 6879: 6863: 6861: 6860: 6855: 6853: 6852: 6834: 6832: 6831: 6826: 6824: 6823: 6811: 6810: 6798: 6797: 6779: 6777: 6776: 6771: 6769: 6751: 6749: 6748: 6743: 6741: 6739: 6734: 6733: 6732: 6720: 6719: 6709: 6707: 6706: 6697: 6696: 6687: 6686: 6677: 6672: 6664: 6659: 6658: 6626:collision theory 6619: 6617: 6616: 6611: 6609: 6608: 6592: 6590: 6589: 6584: 6582: 6581: 6563: 6561: 6560: 6555: 6553: 6552: 6540: 6539: 6527: 6526: 6508: 6506: 6505: 6500: 6482: 6480: 6479: 6474: 6472: 6471: 6462: 6461: 6452: 6451: 6439: 6428: 6427: 6359: 6357: 6356: 6351: 6346: 6341: 6340: 6318: 6316: 6315: 6310: 6298: 6296: 6295: 6290: 6275: 6273: 6272: 6267: 6261: 6257: 6248: 6227: 6195: 6193: 6192: 6187: 6173: 6171: 6170: 6165: 6151: 6149: 6148: 6143: 6141: 6140: 6124: 6122: 6121: 6116: 6113: 6109: 6097: 6083: 6069: 6067: 6066: 6061: 6059: 6048: 6018: 6016: 6015: 6010: 5996: 5994: 5993: 5988: 5986: 5985: 5967: 5965: 5964: 5959: 5945: 5943: 5942: 5937: 5916: 5914: 5913: 5908: 5902: 5898: 5889: 5877: 5869: 5821: 5819: 5818: 5813: 5801: 5799: 5798: 5793: 5781: 5779: 5778: 5773: 5771: 5770: 5751: 5749: 5748: 5743: 5734: 5722: 5721: 5707: 5704: 5699: 5697: 5688: 5679: 5678: 5670: 5664: 5656: 5655: 5653: 5652: 5651: 5648: 5615: 5613: 5612: 5607: 5605: 5599: 5591: 5590: 5588: 5587: 5569: 5568: 5553: 5551: 5543: 5535: 5529: 5524: 5496: 5494: 5493: 5488: 5486: 5474: 5473: 5472: 5471: 5455: 5447: 5446: 5431: 5429: 5421: 5413: 5399:of the plane is 5398: 5396: 5395: 5390: 5375: 5373: 5372: 5367: 5365: 5353: 5352: 5351: 5342: 5337: 5336: 5321: 5319: 5311: 5303: 5288: 5286: 5285: 5280: 5258: 5256: 5255: 5250: 5232: 5225: 5223: 5222: 5217: 5212: 5210: 5199: 5198: 5189: 5181: 5178: 5176: 5164: 5163: 5162: 5153: 5148: 5146: 5138: 5130: 5118: 5116: 5115: 5110: 5092: 5090: 5089: 5084: 5062:Vincent Schaefer 5051: 5049: 5048: 5043: 5031: 5029: 5028: 5023: 5009: 5003: 4997: 4995: 4994: 4989: 4987: 4986: 4968: 4961: 4959: 4958: 4953: 4951: 4945: 4937: 4936: 4934: 4933: 4823: 4821: 4820: 4815: 4813: 4809: 4808: 4807: 4794: 4764: 4763: 4762: 4712:merges into the 4646:relationship). 4632:glass transition 4577: 4566: 4556: 4534: 4492: 4488: 4484: 4477: 4470: 4468: 4467: 4462: 4457: 4453: 4452: 4451: 4438: 4436: 4435: 4426: 4425: 4416: 4414: 4413: 4389: 4377: 4375: 4367: 4366: 4365: 4352: 4333: 4331: 4330: 4325: 4320: 4316: 4315: 4313: 4312: 4311: 4298: 4275: 4273: 4271: 4270: 4269: 4256: 4246: 4245: 4229: 4224: 4222: 4221: 4220: 4208: 4207: 4194: 4175: 4174: 4164: 4153: 4152: 4134: 4129: 4105: 4104: 4065: 4064: 4049: 4047: 4039: 4016: 3992: 3983: 3981: 3980: 3975: 3970: 3968: 3967: 3966: 3954: 3953: 3940: 3921: 3920: 3910: 3908: 3907: 3894: 3889: 3873: 3868: 3850: 3849: 3819: 3818: 3803: 3801: 3793: 3770: 3761: 3759: 3758: 3753: 3748: 3746: 3745: 3744: 3731: 3723: 3721: 3720: 3707: 3702: 3681: 3680: 3664: 3662: 3661: 3656: 3626: 3598: 3596: 3595: 3590: 3588: 3586: 3585: 3584: 3571: 3548: 3546: 3544: 3543: 3542: 3529: 3512: 3509: 3504: 3450: 3449: 3410: 3409: 3394: 3392: 3384: 3361: 3352: 3323: 3321: 3320: 3315: 3303: 3301: 3300: 3295: 3293: 3279: 3251: 3249: 3248: 3243: 3231: 3229: 3228: 3223: 3203: 3202: 3193: 3192: 3191: 3178: 3164: 3161: 3136: 3129: 3127: 3126: 3121: 3116: 3094: 3089: 3080: 3065: 3058: 3056: 3055: 3050: 3048: 3044: 3043: 3039: 3037: 3036: 3025: 3016: 2996: 2995: 2948: 2944: 2939:diffusion length 2936: 2935: 2934: 2924: 2915: 2906: 2904: 2903: 2900: 2897: 2890: 2883: 2866: 2859: 2836: 2829: 2827: 2826: 2821: 2816: 2812: 2810: 2809: 2801: 2792: 2780: 2779: 2767: 2763: 2735: 2726: 2719: 2715: 2692: 2690: 2689: 2684: 2682: 2680: 2679: 2678: 2665: 2661: 2660: 2650: 2642: 2640: 2632: 2624: 2612: 2602: 2600: 2599: 2594: 2577: 2576: 2573: 2571: 2551: 2545: 2537: 2528: 2526: 2525: 2520: 2509: 2495: 2493: 2485: 2477: 2434: 2432: 2431: 2426: 2412: 2411: 2392: 2388: 2384: 2375: 2373: 2372: 2367: 2337: 2335: 2327: 2319: 2297: 2295: 2294: 2289: 2281: 2280: 2264: 2262: 2254: 2246: 2228: 2226: 2225: 2220: 2218: 2216: 2215: 2214: 2201: 2197: 2196: 2186: 2175: 2173: 2162: 2160: 2158: 2147: 2139: 2135: 2131: 2129: 2118: 2108: 2106: 2095: 2083: 2076: 2074: 2073: 2068: 2059: 2055: 2051: 2049: 2038: 2028: 2026: 2015: 2010: 2008: 2000: 1992: 1978: 1976: 1965: 1960: 1958: 1950: 1942: 1919: 1917: 1916: 1911: 1906: 1902: 1901: 1899: 1888: 1887: 1878: 1862: 1847: 1843: 1813: 1811: 1810: 1805: 1793: 1791: 1790: 1785: 1758: 1756: 1755: 1750: 1736: 1734: 1726: 1718: 1706: 1695: 1689: 1687: 1686: 1683:{\displaystyle } 1681: 1676: 1675: 1667: 1666: 1659: 1658: 1653: 1652: 1635: 1629: 1624: 1620: 1616: 1597: 1595: 1594: 1591:{\displaystyle } 1589: 1584: 1583: 1575: 1574: 1567: 1566: 1550: 1540: 1538: 1537: 1532: 1530: 1528: 1527: 1526: 1513: 1509: 1508: 1498: 1489: 1487: 1479: 1471: 1444: 1440: 1438: 1437: 1432: 1430: 1429: 1413: 1411: 1410: 1405: 1400: 1399: 1387: 1385: 1384: 1375: 1367: 1359: 1358: 1357: 1340: 1338: 1337: 1332: 1309: 1307: 1306: 1301: 1281: 1280: 1279: 1262: 1260: 1259: 1254: 1231: 1229: 1228: 1223: 1218: 1217: 1205: 1203: 1202: 1193: 1185: 1177: 1176: 1175: 1158: 1154: 1145: 1143: 1142: 1137: 1132: 1131: 1104: 1103: 1102: 1072: 1070: 1069: 1064: 1061: 1058: 1056: 1043: 1040: 1038: 1022: 1020: 1019: 1014: 1011: 1008: 1006: 990: 988: 987: 982: 979: 976: 974: 958: 952: 950: 949: 944: 942: 941: 925: 923: 922: 917: 915: 914: 895: 893: 892: 887: 885: 883: 875: 874: 873: 860: 858: 856: 845: 837: 836: 809: 803: 793: 787: 781: 777: 766: 764: 763: 758: 756: 754: 746: 745: 744: 731: 729: 727: 719: 718: 717: 704: 696: 695: 669: 665: 655: 653: 652: 647: 645: 640: 639: 627: 622: 621: 598: 596: 595: 590: 570: 564: 556: 543: 535: 522: 518: 507: 505: 504: 499: 497: 496: 484: 482: 481: 472: 464: 456: 455: 450: 434: 415: 414: 412: 411: 405: 402: 387: 378: 376: 375: 370: 353: 330: 323: 321: 307: 300: 294: 285: 283: 282: 277: 275: 273: 265: 257: 236: 230: 214: 204: 202: 201: 196: 194: 192: 184: 176: 146:Fick's first law 141:Fick's first law 86:Fick's first law 77: 9209: 9208: 9204: 9203: 9202: 9200: 9199: 9198: 9164: 9163: 9145: 9140: 9109:– reprinted in 9050: 9048:Further reading 9045: 9044: 9001: 8997: 8978: 8974: 8951: 8947: 8894: 8883: 8864: 8860: 8807: 8788: 8757: 8750: 8727: 8720: 8675: 8668: 8615: 8611: 8554: 8550: 8543: 8529: 8525: 8515: 8513: 8505: 8504: 8500: 8489: 8485: 8478: 8462: 8458: 8445: 8441: 8397: 8393: 8382: 8378: 8370: 8355: 8349: 8345: 8287:Fick A (1855). 8284: 8280: 8270: 8268: 8260: 8259: 8255: 8250: 8217:False diffusion 8198: 8186: 8169: 8133: 8129: 8120: 8116: 8115: 8113: 8111: 8108: 8107: 8089: 8086: 8085: 8054: 8047: 8043: 8034: 8030: 8029: 8027: 8023: 8017: 8013: 8002: 7999: 7998: 7965: 7962: 7961: 7940: 7936: 7931: 7928: 7927: 7910: 7907: 7906: 7890: 7887: 7886: 7858: 7851: 7847: 7843: 7841: 7837: 7831: 7827: 7816: 7813: 7812: 7788: 7784: 7782: 7779: 7778: 7759: 7755: 7753: 7750: 7749: 7730: 7726: 7724: 7721: 7720: 7696: 7692: 7680: 7676: 7671: 7667: 7666: 7651: 7647: 7645: 7641: 7635: 7631: 7623: 7620: 7619: 7579: 7575: 7567: 7566: 7562: 7554: 7552: 7544: 7541: 7540: 7536: 7532: 7528: 7527:Integrated the 7508: 7505: 7504: 7486: 7483: 7482: 7473: 7467: 7446: 7444: 7441: 7440: 7422: 7419: 7418: 7388: 7386: 7375: 7359: 7355: 7347: 7346: 7338: 7336: 7332: 7315: 7312: 7311: 7300: 7274: 7262: 7255: 7247:is time unit s. 7212: 7208: 7206: 7203: 7202: 7183: 7179: 7177: 7174: 7173: 7137: 7132: 7123: 7119: 7113: 7109: 7098: 7095: 7094: 7068: 7062: 7059: 7053: 7042: 7035: 7029: 7019: 6990: 6986: 6977: 6973: 6972: 6968: 6961: 6953: 6951: 6948: 6947: 6941: 6929: 6926: 6923: 6922: 6919: 6917: 6909: 6906: 6903: 6902: 6899: 6897: 6875: 6871: 6869: 6866: 6865: 6848: 6844: 6842: 6839: 6838: 6819: 6815: 6806: 6802: 6793: 6789: 6787: 6784: 6783: 6765: 6763: 6760: 6759: 6735: 6728: 6724: 6715: 6711: 6710: 6708: 6702: 6698: 6692: 6688: 6682: 6678: 6673: 6663: 6651: 6647: 6645: 6642: 6641: 6604: 6600: 6598: 6595: 6594: 6577: 6573: 6571: 6568: 6567: 6548: 6544: 6535: 6531: 6522: 6518: 6516: 6513: 6512: 6494: 6491: 6490: 6467: 6463: 6457: 6453: 6447: 6443: 6435: 6420: 6416: 6414: 6411: 6410: 6403:Brownian motion 6342: 6336: 6332: 6324: 6321: 6320: 6304: 6301: 6300: 6284: 6281: 6280: 6253: 6249: 6244: 6223: 6209: 6206: 6205: 6181: 6178: 6177: 6159: 6156: 6155: 6136: 6132: 6130: 6127: 6126: 6105: 6098: 6093: 6078: 6075: 6074: 6047: 6039: 6036: 6035: 6004: 6001: 6000: 5981: 5977: 5975: 5972: 5971: 5953: 5950: 5949: 5925: 5922: 5921: 5894: 5890: 5885: 5868: 5854: 5851: 5850: 5807: 5804: 5803: 5787: 5784: 5783: 5766: 5762: 5760: 5757: 5756: 5708: 5706: 5698: 5693: 5677: 5657: 5654: 5647: 5643: 5642: 5628: 5625: 5624: 5619: 5592: 5589: 5583: 5579: 5558: 5554: 5544: 5536: 5534: 5525: 5520: 5508: 5505: 5504: 5467: 5463: 5456: 5454: 5436: 5432: 5422: 5414: 5412: 5407: 5404: 5403: 5384: 5381: 5380: 5347: 5343: 5341: 5326: 5322: 5312: 5304: 5302: 5297: 5294: 5293: 5268: 5265: 5264: 5238: 5235: 5234: 5230: 5200: 5194: 5190: 5188: 5177: 5158: 5154: 5152: 5139: 5131: 5129: 5127: 5124: 5123: 5098: 5095: 5094: 5072: 5069: 5068: 5058:Irving Langmuir 5037: 5034: 5033: 5017: 5014: 5013: 5007: 5001: 4982: 4978: 4976: 4973: 4972: 4966: 4938: 4935: 4929: 4925: 4911: 4908: 4907: 4836: 4790: 4783: 4779: 4772: 4768: 4758: 4757: 4753: 4732: 4729: 4728: 4682: 4652: 4600:pharmaceuticals 4584: 4568: 4558: 4555: 4543: 4532: 4525: 4520: 4513: 4508: 4502: 4490: 4486: 4483: 4479: 4476: 4472: 4447: 4443: 4431: 4427: 4421: 4417: 4415: 4406: 4402: 4401: 4397: 4385: 4368: 4361: 4357: 4353: 4351: 4349: 4346: 4345: 4307: 4303: 4299: 4276: 4274: 4265: 4261: 4257: 4238: 4234: 4230: 4228: 4216: 4212: 4203: 4199: 4195: 4170: 4166: 4165: 4163: 4145: 4141: 4140: 4136: 4130: 4113: 4100: 4099: 4060: 4059: 4040: 4017: 4015: 4013: 4010: 4009: 3990: 3985: 3962: 3958: 3949: 3945: 3941: 3916: 3912: 3911: 3909: 3900: 3896: 3890: 3879: 3869: 3858: 3845: 3844: 3814: 3813: 3794: 3771: 3769: 3767: 3764: 3763: 3740: 3736: 3732: 3724: 3722: 3713: 3709: 3703: 3692: 3676: 3672: 3670: 3667: 3666: 3632: 3629: 3628: 3624: 3617: 3612: 3580: 3576: 3572: 3549: 3547: 3538: 3534: 3530: 3513: 3511: 3505: 3494: 3445: 3444: 3405: 3404: 3385: 3362: 3360: 3358: 3355: 3354: 3339: 3331: 3329:Generalizations 3309: 3306: 3305: 3278: 3276: 3273: 3272: 3262:eukaryotic cell 3237: 3234: 3233: 3198: 3194: 3187: 3183: 3182: 3174: 3160: 3158: 3155: 3154: 3147:Brownian motion 3143: 3134: 3090: 3085: 3079: 3074: 3071: 3070: 3063: 3024: 3020: 3015: 3011: 3001: 2997: 2991: 2987: 2964: 2961: 2960: 2946: 2942: 2930: 2928: 2926: 2923: 2917: 2914: 2908: 2901: 2898: 2895: 2894: 2892: 2885: 2882: 2868: 2861: 2850: 2834: 2800: 2796: 2791: 2787: 2775: 2771: 2753: 2749: 2744: 2741: 2740: 2734: 2728: 2721: 2717: 2713: 2710: 2704:is the result. 2674: 2670: 2666: 2656: 2652: 2651: 2649: 2633: 2625: 2623: 2621: 2618: 2617: 2607: 2572: 2567: 2566: 2564: 2561: 2560: 2547: 2543: 2533: 2505: 2486: 2478: 2476: 2474: 2471: 2470: 2452: 2407: 2403: 2401: 2398: 2397: 2390: 2386: 2380: 2328: 2320: 2318: 2316: 2313: 2312: 2276: 2272: 2255: 2247: 2245: 2243: 2240: 2239: 2210: 2206: 2202: 2192: 2188: 2187: 2185: 2166: 2161: 2151: 2146: 2122: 2117: 2113: 2109: 2099: 2094: 2092: 2089: 2088: 2081: 2042: 2037: 2033: 2029: 2019: 2014: 2001: 1993: 1991: 1969: 1964: 1951: 1943: 1941: 1939: 1936: 1935: 1925: 1889: 1883: 1879: 1877: 1873: 1869: 1842: 1819: 1816: 1815: 1799: 1796: 1795: 1779: 1776: 1775: 1727: 1719: 1717: 1715: 1712: 1711: 1704: 1693: 1668: 1662: 1661: 1660: 1654: 1648: 1647: 1646: 1641: 1638: 1637: 1633: 1627: 1622: 1618: 1599: 1576: 1570: 1569: 1568: 1562: 1561: 1556: 1553: 1552: 1548: 1522: 1518: 1514: 1504: 1500: 1499: 1497: 1480: 1472: 1470: 1468: 1465: 1464: 1451: 1442: 1425: 1421: 1419: 1416: 1415: 1395: 1391: 1380: 1376: 1368: 1366: 1353: 1349: 1348: 1346: 1343: 1342: 1317: 1314: 1313: 1275: 1271: 1270: 1268: 1265: 1264: 1239: 1236: 1235: 1213: 1209: 1198: 1194: 1186: 1184: 1171: 1167: 1166: 1164: 1161: 1160: 1156: 1152: 1147: 1127: 1123: 1098: 1094: 1093: 1091: 1088: 1087: 1079: 1057: 1052: 1039: 1034: 1028: 1025: 1024: 1007: 1002: 996: 993: 992: 975: 970: 964: 961: 960: 954: 937: 933: 931: 928: 927: 910: 906: 904: 901: 900: 876: 869: 865: 861: 859: 849: 844: 832: 828: 826: 823: 822: 807: 801: 791: 785: 779: 775: 747: 740: 736: 732: 730: 720: 713: 709: 705: 703: 691: 687: 685: 682: 681: 667: 664: 660: 632: 628: 626: 617: 613: 611: 608: 607: 599:is outside the 584: 581: 580: 571:is the mixture 568: 565:th species, and 562: 555: 547: 541: 534: 526: 520: 516: 492: 488: 477: 473: 465: 463: 451: 446: 445: 443: 440: 439: 433: 429: 422: 406: 403: 397: 396: 394: 392: 383: 349: 347: 344: 343: 328: 319: 317: 305: 298: 292: 266: 258: 256: 254: 251: 250: 234: 228: 219:, of which the 212: 185: 177: 175: 161: 158: 157: 143: 107: 75: 28: 17: 12: 11: 5: 9207: 9197: 9196: 9191: 9186: 9181: 9176: 9162: 9161: 9152: 9144: 9143:External links 9141: 9139: 9138: 9129: 9096: 9087: 9070: 9061: 9051: 9049: 9046: 9043: 9042: 9015:(3): 192–197. 8995: 8972: 8945: 8881: 8858: 8786: 8767:(7): 453–461. 8748: 8718: 8666: 8609: 8548: 8541: 8523: 8498: 8483: 8476: 8456: 8439: 8410:(5): 184–262. 8391: 8376: 8343: 8341: 8340: 8278: 8252: 8251: 8249: 8246: 8245: 8244: 8239: 8234: 8229: 8224: 8219: 8214: 8209: 8204: 8197: 8194: 8185: 8182: 8168: 8165: 8157: 8156: 8142: 8136: 8132: 8128: 8123: 8119: 8105: 8093: 8069: 8063: 8060: 8057: 8050: 8046: 8042: 8037: 8033: 8026: 8020: 8016: 8012: 8009: 8006: 7984: 7981: 7978: 7975: 7972: 7969: 7958: 7957: 7943: 7939: 7935: 7925: 7914: 7894: 7870: 7864: 7861: 7854: 7850: 7846: 7840: 7834: 7830: 7826: 7823: 7820: 7806: 7805: 7791: 7787: 7776: 7762: 7758: 7747: 7733: 7729: 7703: 7699: 7695: 7690: 7683: 7679: 7675: 7670: 7664: 7659: 7654: 7650: 7644: 7638: 7634: 7630: 7627: 7586: 7582: 7578: 7573: 7570: 7565: 7560: 7557: 7551: 7548: 7525: 7524: 7512: 7502: 7490: 7480: 7479:at any surface 7471: 7465: 7452: 7449: 7438: 7426: 7401: 7397: 7394: 7391: 7385: 7381: 7378: 7373: 7366: 7362: 7358: 7353: 7350: 7344: 7341: 7335: 7331: 7328: 7325: 7322: 7319: 7299: 7296: 7273: 7270: 7265: 7264: 7260: 7253: 7248: 7242: 7236: 7233:chromatography 7218: 7215: 7211: 7200: 7186: 7182: 7167: 7166: 7153: 7150: 7147: 7144: 7140: 7136: 7129: 7126: 7122: 7116: 7112: 7108: 7105: 7102: 7072: 7071: 7066: 7057: 7040: 7033: 7027: 7013: 7012: 6999: 6993: 6989: 6985: 6980: 6976: 6971: 6967: 6964: 6960: 6940: 6937: 6893: 6892: 6878: 6874: 6851: 6847: 6836: 6822: 6818: 6814: 6809: 6805: 6801: 6796: 6792: 6781: 6768: 6753: 6752: 6738: 6731: 6727: 6723: 6718: 6714: 6705: 6701: 6695: 6691: 6685: 6681: 6676: 6670: 6667: 6662: 6657: 6654: 6650: 6622: 6621: 6607: 6603: 6580: 6576: 6565: 6551: 6547: 6543: 6538: 6534: 6530: 6525: 6521: 6510: 6498: 6484: 6483: 6470: 6466: 6460: 6456: 6450: 6446: 6442: 6438: 6434: 6431: 6426: 6423: 6419: 6363: 6362: 6349: 6345: 6339: 6335: 6331: 6328: 6308: 6288: 6277: 6276: 6265: 6260: 6256: 6252: 6247: 6243: 6239: 6236: 6233: 6230: 6226: 6222: 6219: 6216: 6213: 6198: 6197: 6185: 6175: 6163: 6153: 6139: 6135: 6112: 6108: 6104: 6101: 6096: 6092: 6088: 6082: 6071: 6070: 6057: 6054: 6051: 6046: 6043: 6024: 6023: 6020: 6008: 5998: 5984: 5980: 5969: 5957: 5947: 5935: 5932: 5929: 5918: 5917: 5906: 5901: 5897: 5893: 5888: 5884: 5880: 5875: 5872: 5867: 5864: 5861: 5858: 5811: 5791: 5769: 5765: 5753: 5752: 5741: 5738: 5731: 5728: 5725: 5720: 5717: 5714: 5711: 5702: 5696: 5692: 5685: 5682: 5676: 5673: 5667: 5663: 5660: 5646: 5641: 5638: 5635: 5632: 5617: 5616: 5602: 5598: 5595: 5586: 5582: 5578: 5575: 5572: 5567: 5564: 5561: 5557: 5550: 5547: 5542: 5539: 5533: 5528: 5523: 5519: 5515: 5512: 5498: 5497: 5483: 5480: 5477: 5470: 5466: 5462: 5459: 5453: 5450: 5445: 5442: 5439: 5435: 5428: 5425: 5420: 5417: 5411: 5388: 5377: 5376: 5362: 5359: 5356: 5350: 5346: 5340: 5335: 5332: 5329: 5325: 5318: 5315: 5310: 5307: 5301: 5278: 5275: 5272: 5261: 5260: 5248: 5245: 5242: 5227: 5226: 5215: 5209: 5206: 5203: 5197: 5193: 5187: 5184: 5173: 5170: 5167: 5161: 5157: 5151: 5145: 5142: 5137: 5134: 5108: 5105: 5102: 5082: 5079: 5076: 5054: 5053: 5041: 5021: 5011: 5005: 4999: 4985: 4981: 4970: 4963: 4962: 4948: 4944: 4941: 4932: 4928: 4924: 4921: 4918: 4915: 4872: 4871: 4865: 4859: 4850: 4841: 4834: 4825: 4824: 4812: 4806: 4803: 4800: 4797: 4793: 4789: 4786: 4782: 4778: 4775: 4771: 4767: 4761: 4756: 4751: 4748: 4745: 4742: 4739: 4736: 4681: 4678: 4651: 4648: 4615:plasma physics 4583: 4580: 4547: 4530: 4523: 4516: 4511: 4504: 4495: 4494: 4481: 4474: 4460: 4456: 4450: 4446: 4441: 4434: 4430: 4424: 4420: 4412: 4409: 4405: 4400: 4396: 4393: 4388: 4384: 4380: 4374: 4371: 4364: 4360: 4356: 4334: 4323: 4319: 4310: 4306: 4302: 4297: 4294: 4291: 4288: 4285: 4282: 4279: 4268: 4264: 4260: 4255: 4252: 4249: 4244: 4241: 4237: 4233: 4227: 4219: 4215: 4211: 4206: 4202: 4198: 4193: 4190: 4187: 4184: 4181: 4178: 4173: 4169: 4162: 4159: 4156: 4151: 4148: 4144: 4139: 4133: 4128: 4125: 4122: 4119: 4116: 4112: 4108: 4103: 4098: 4095: 4092: 4089: 4086: 4083: 4080: 4077: 4074: 4071: 4068: 4063: 4058: 4055: 4052: 4046: 4043: 4038: 4035: 4032: 4029: 4026: 4023: 4020: 4002: 3988: 3973: 3965: 3961: 3957: 3952: 3948: 3944: 3939: 3936: 3933: 3930: 3927: 3924: 3919: 3915: 3906: 3903: 3899: 3893: 3888: 3885: 3882: 3878: 3872: 3867: 3864: 3861: 3857: 3853: 3848: 3843: 3840: 3837: 3834: 3831: 3828: 3825: 3822: 3817: 3812: 3809: 3806: 3800: 3797: 3792: 3789: 3786: 3783: 3780: 3777: 3774: 3751: 3743: 3739: 3735: 3730: 3727: 3719: 3716: 3712: 3706: 3701: 3698: 3695: 3691: 3687: 3684: 3679: 3675: 3654: 3651: 3648: 3645: 3642: 3639: 3636: 3622: 3615: 3599: 3583: 3579: 3575: 3570: 3567: 3564: 3561: 3558: 3555: 3552: 3541: 3537: 3533: 3528: 3525: 3522: 3519: 3516: 3508: 3503: 3500: 3497: 3493: 3489: 3486: 3483: 3480: 3477: 3474: 3471: 3468: 3465: 3462: 3459: 3456: 3453: 3448: 3443: 3440: 3437: 3434: 3431: 3428: 3425: 3422: 3419: 3416: 3413: 3408: 3403: 3400: 3397: 3391: 3388: 3383: 3380: 3377: 3374: 3371: 3368: 3365: 3330: 3327: 3313: 3291: 3288: 3285: 3282: 3241: 3221: 3218: 3215: 3212: 3209: 3206: 3201: 3197: 3190: 3186: 3181: 3177: 3173: 3170: 3167: 3142: 3139: 3131: 3130: 3119: 3114: 3111: 3107: 3104: 3101: 3098: 3093: 3088: 3084: 3078: 3060: 3059: 3047: 3042: 3034: 3031: 3028: 3023: 3019: 3014: 3010: 3007: 3004: 3000: 2994: 2990: 2986: 2983: 2980: 2977: 2974: 2971: 2968: 2949:(Bird, 1976). 2937:is called the 2921: 2912: 2880: 2867:and that with 2839:error function 2831: 2830: 2819: 2815: 2807: 2804: 2799: 2795: 2790: 2786: 2783: 2778: 2774: 2770: 2766: 2762: 2759: 2756: 2752: 2748: 2732: 2709: 2706: 2698:advective flux 2694: 2693: 2677: 2673: 2669: 2664: 2659: 2655: 2648: 2645: 2639: 2636: 2631: 2628: 2604: 2603: 2592: 2589: 2586: 2583: 2580: 2570: 2554:diffusive flux 2530: 2529: 2518: 2515: 2512: 2508: 2504: 2501: 2498: 2492: 2489: 2484: 2481: 2460:advective flux 2451: 2448: 2436: 2435: 2424: 2421: 2418: 2415: 2410: 2406: 2377: 2376: 2365: 2362: 2359: 2356: 2352: 2349: 2346: 2343: 2340: 2334: 2331: 2326: 2323: 2299: 2298: 2287: 2284: 2279: 2275: 2270: 2267: 2261: 2258: 2253: 2250: 2230: 2229: 2213: 2209: 2205: 2200: 2195: 2191: 2184: 2181: 2178: 2172: 2169: 2165: 2157: 2154: 2150: 2145: 2142: 2138: 2134: 2128: 2125: 2121: 2116: 2112: 2105: 2102: 2098: 2078: 2077: 2066: 2063: 2058: 2054: 2048: 2045: 2041: 2036: 2032: 2025: 2022: 2018: 2013: 2007: 2004: 1999: 1996: 1990: 1987: 1984: 1981: 1975: 1972: 1968: 1963: 1957: 1954: 1949: 1946: 1924: 1921: 1909: 1905: 1898: 1895: 1892: 1886: 1882: 1876: 1872: 1868: 1865: 1859: 1856: 1853: 1850: 1846: 1841: 1838: 1835: 1832: 1829: 1826: 1823: 1803: 1783: 1760: 1759: 1748: 1745: 1742: 1739: 1733: 1730: 1725: 1722: 1698: 1697: 1691: 1679: 1674: 1671: 1665: 1657: 1651: 1645: 1631: 1625: 1587: 1582: 1579: 1573: 1565: 1560: 1542: 1541: 1525: 1521: 1517: 1512: 1507: 1503: 1495: 1492: 1486: 1483: 1478: 1475: 1450: 1447: 1428: 1424: 1403: 1398: 1394: 1390: 1383: 1379: 1374: 1371: 1365: 1362: 1356: 1352: 1330: 1327: 1324: 1321: 1299: 1296: 1293: 1290: 1287: 1284: 1278: 1274: 1252: 1249: 1246: 1243: 1221: 1216: 1212: 1208: 1201: 1197: 1192: 1189: 1183: 1180: 1174: 1170: 1150: 1135: 1130: 1126: 1122: 1119: 1116: 1113: 1110: 1107: 1101: 1097: 1078: 1075: 1055: 1051: 1047: 1037: 1033: 1005: 1001: 973: 969: 940: 936: 926:has Pa units. 913: 909: 897: 896: 882: 879: 872: 868: 864: 855: 852: 848: 843: 840: 835: 831: 812: 811: 805: 799: 789: 783: 768: 767: 753: 750: 743: 739: 735: 726: 723: 716: 712: 708: 702: 699: 694: 690: 662: 657: 656: 643: 638: 635: 631: 625: 620: 616: 588: 577: 576: 566: 551: 545: 530: 524: 509: 508: 495: 491: 487: 480: 476: 471: 468: 462: 459: 454: 449: 431: 421: 418: 380: 379: 368: 365: 362: 359: 356: 352: 303: 302: 296: 290: 272: 269: 264: 261: 248: 232: 217:diffusion flux 206: 205: 191: 188: 183: 180: 174: 171: 168: 165: 142: 139: 106: 103: 25:Fick principle 21:cardiac output 15: 9: 6: 4: 3: 2: 9206: 9195: 9192: 9190: 9187: 9185: 9182: 9180: 9177: 9175: 9172: 9171: 9169: 9160: 9156: 9153: 9150: 9147: 9146: 9135: 9130: 9126: 9122: 9118: 9114: 9106: 9102: 9097: 9093: 9088: 9084: 9079: 9078: 9071: 9067: 9062: 9058: 9053: 9052: 9038: 9034: 9030: 9026: 9022: 9018: 9014: 9010: 9006: 8999: 8991: 8987: 8983: 8976: 8968: 8964: 8960: 8957:(in German). 8956: 8949: 8941: 8937: 8932: 8927: 8923: 8919: 8915: 8911: 8907: 8903: 8899: 8892: 8890: 8888: 8886: 8877: 8873: 8869: 8862: 8854: 8850: 8845: 8840: 8836: 8832: 8828: 8824: 8821:(1): 015318. 8820: 8816: 8812: 8805: 8803: 8801: 8799: 8797: 8795: 8793: 8791: 8782: 8778: 8774: 8770: 8766: 8762: 8755: 8753: 8744: 8740: 8736: 8732: 8725: 8723: 8714: 8710: 8705: 8700: 8696: 8692: 8689:: 2296–2306. 8688: 8684: 8680: 8673: 8671: 8662: 8658: 8653: 8648: 8644: 8640: 8636: 8632: 8628: 8624: 8620: 8613: 8605: 8601: 8597: 8593: 8589: 8585: 8581: 8577: 8572: 8567: 8563: 8559: 8552: 8544: 8538: 8534: 8527: 8512: 8508: 8502: 8494: 8487: 8479: 8477:9780521881685 8473: 8469: 8468: 8460: 8452: 8451: 8443: 8435: 8431: 8427: 8423: 8418: 8413: 8409: 8405: 8401: 8395: 8387: 8380: 8369: 8365: 8361: 8354: 8347: 8337: 8333: 8330:(63): 30–39. 8329: 8325: 8320: 8319: 8315: 8310: 8306: 8302: 8298: 8295:(in German). 8294: 8290: 8282: 8267: 8263: 8257: 8253: 8243: 8240: 8238: 8235: 8233: 8230: 8228: 8225: 8223: 8220: 8218: 8215: 8213: 8210: 8208: 8205: 8203: 8200: 8199: 8193: 8190: 8181: 8177: 8175: 8164: 8160: 8140: 8134: 8130: 8126: 8121: 8117: 8106: 8091: 8084: 8083: 8082: 8067: 8061: 8058: 8055: 8048: 8044: 8040: 8035: 8031: 8024: 8018: 8014: 8010: 8007: 8004: 7996: 7982: 7979: 7976: 7973: 7970: 7967: 7941: 7937: 7933: 7926: 7912: 7892: 7885: 7884: 7883: 7868: 7862: 7859: 7852: 7848: 7844: 7838: 7832: 7828: 7824: 7821: 7818: 7809: 7789: 7785: 7777: 7760: 7756: 7748: 7731: 7727: 7719: 7718: 7717: 7701: 7697: 7693: 7688: 7681: 7677: 7673: 7668: 7662: 7657: 7652: 7648: 7642: 7636: 7632: 7628: 7625: 7617: 7615: 7611: 7606: 7602: 7584: 7580: 7576: 7563: 7558: 7555: 7549: 7546: 7510: 7503: 7488: 7481: 7476: 7472: 7466: 7439: 7424: 7417: 7416: 7415: 7399: 7395: 7392: 7389: 7383: 7371: 7364: 7360: 7356: 7342: 7339: 7333: 7329: 7323: 7317: 7308: 7304: 7295: 7293: 7289: 7286: 7282: 7279: 7278:semiconductor 7269: 7259: 7252: 7249: 7246: 7243: 7240: 7237: 7234: 7216: 7213: 7209: 7201: 7184: 7180: 7172: 7171: 7170: 7148: 7145: 7138: 7134: 7127: 7124: 7120: 7114: 7110: 7106: 7103: 7100: 7093: 7092: 7091: 7087: 7085: 7080: 7078: 7065: 7056: 7051: 7047: 7046:concentration 7039: 7032: 7028: 7025: 7018: 7017: 7016: 6997: 6991: 6987: 6983: 6978: 6974: 6969: 6965: 6962: 6958: 6946: 6945: 6944: 6936: 6876: 6872: 6849: 6845: 6837: 6820: 6816: 6812: 6807: 6803: 6799: 6794: 6790: 6782: 6766: 6758: 6757: 6756: 6736: 6729: 6725: 6721: 6716: 6712: 6703: 6699: 6693: 6689: 6683: 6679: 6674: 6668: 6665: 6660: 6655: 6652: 6648: 6640: 6639: 6638: 6634: 6630: 6627: 6605: 6601: 6578: 6574: 6566: 6549: 6545: 6541: 6536: 6532: 6528: 6523: 6519: 6511: 6496: 6489: 6488: 6487: 6468: 6464: 6458: 6454: 6448: 6444: 6440: 6436: 6432: 6429: 6424: 6421: 6417: 6409: 6408: 6407: 6404: 6400: 6396: 6387: 6383: 6381: 6376: 6375:self-assembly 6371: 6368: 6347: 6343: 6337: 6333: 6329: 6326: 6306: 6286: 6279: 6278: 6263: 6258: 6254: 6250: 6245: 6241: 6237: 6234: 6231: 6228: 6224: 6220: 6217: 6214: 6211: 6204: 6203: 6202: 6183: 6176: 6161: 6154: 6137: 6133: 6110: 6106: 6102: 6099: 6094: 6090: 6086: 6080: 6073: 6072: 6055: 6052: 6049: 6044: 6041: 6034: 6033: 6032: 6028: 6021: 6006: 5999: 5982: 5978: 5970: 5955: 5948: 5930: 5920: 5919: 5904: 5899: 5895: 5891: 5886: 5882: 5878: 5873: 5870: 5865: 5859: 5849: 5848: 5847: 5843: 5840: 5831: 5827: 5823: 5809: 5789: 5767: 5763: 5739: 5736: 5729: 5726: 5723: 5715: 5709: 5700: 5694: 5690: 5683: 5680: 5674: 5671: 5665: 5661: 5658: 5644: 5639: 5636: 5633: 5623: 5622: 5621: 5600: 5596: 5593: 5584: 5580: 5576: 5573: 5570: 5565: 5562: 5559: 5548: 5526: 5521: 5517: 5513: 5503: 5502: 5501: 5481: 5478: 5475: 5468: 5464: 5460: 5457: 5451: 5448: 5443: 5440: 5437: 5426: 5402: 5401: 5400: 5386: 5360: 5357: 5354: 5348: 5344: 5338: 5333: 5330: 5327: 5316: 5308: 5292: 5291: 5290: 5276: 5273: 5270: 5246: 5243: 5240: 5229: 5228: 5207: 5204: 5201: 5195: 5191: 5185: 5171: 5168: 5165: 5159: 5155: 5149: 5143: 5135: 5122: 5121: 5120: 5106: 5103: 5100: 5080: 5077: 5074: 5065: 5063: 5059: 5039: 5012: 5006: 5000: 4983: 4979: 4971: 4965: 4964: 4946: 4942: 4939: 4930: 4926: 4922: 4919: 4916: 4906: 4905: 4904: 4901: 4897: 4887: 4883: 4881: 4877: 4869: 4866: 4863: 4860: 4858: 4854: 4851: 4849: 4845: 4842: 4840: 4833: 4830: 4829: 4828: 4810: 4801: 4798: 4791: 4787: 4784: 4780: 4776: 4773: 4769: 4765: 4754: 4749: 4746: 4740: 4734: 4727: 4726: 4725: 4723: 4720:based on the 4719: 4715: 4711: 4707: 4703: 4699: 4695: 4691: 4686: 4677: 4675: 4674:renormalizing 4671: 4665: 4662: 4657: 4647: 4645: 4641: 4640:mass transfer 4637: 4633: 4629: 4624: 4620: 4616: 4612: 4608: 4605: 4601: 4597: 4593: 4589: 4579: 4575: 4571: 4565: 4561: 4554: 4550: 4546: 4540: 4538: 4533: 4526: 4519: 4514: 4507: 4500: 4458: 4454: 4448: 4444: 4432: 4428: 4422: 4418: 4410: 4407: 4403: 4398: 4394: 4386: 4382: 4378: 4372: 4362: 4358: 4343: 4339: 4335: 4321: 4317: 4308: 4304: 4292: 4289: 4286: 4280: 4266: 4262: 4250: 4242: 4239: 4235: 4225: 4217: 4213: 4204: 4200: 4188: 4185: 4182: 4176: 4171: 4157: 4149: 4146: 4142: 4137: 4131: 4126: 4123: 4120: 4117: 4114: 4110: 4106: 4093: 4090: 4087: 4081: 4072: 4066: 4056: 4050: 4044: 4033: 4030: 4027: 4021: 4007: 4003: 4000: 3996: 3991: 3971: 3963: 3959: 3950: 3946: 3934: 3931: 3928: 3922: 3917: 3904: 3901: 3897: 3891: 3886: 3883: 3880: 3876: 3870: 3865: 3862: 3859: 3855: 3851: 3838: 3835: 3832: 3826: 3820: 3810: 3804: 3798: 3787: 3784: 3781: 3775: 3749: 3741: 3737: 3728: 3717: 3714: 3710: 3704: 3699: 3696: 3693: 3689: 3685: 3682: 3677: 3673: 3652: 3649: 3643: 3640: 3637: 3634: 3625: 3618: 3611: 3607: 3605: 3600: 3581: 3577: 3565: 3562: 3559: 3553: 3539: 3535: 3523: 3517: 3506: 3501: 3498: 3495: 3491: 3487: 3481: 3478: 3475: 3469: 3460: 3454: 3451: 3438: 3435: 3432: 3426: 3417: 3411: 3401: 3395: 3389: 3378: 3375: 3372: 3366: 3350: 3346: 3342: 3337: 3333: 3332: 3326: 3311: 3289: 3286: 3283: 3280: 3269: 3267: 3263: 3259: 3258:cell membrane 3255: 3239: 3219: 3216: 3213: 3210: 3207: 3199: 3179: 3165: 3152: 3148: 3138: 3117: 3112: 3109: 3102: 3096: 3091: 3086: 3082: 3076: 3069: 3068: 3067: 3045: 3040: 3032: 3029: 3026: 3021: 3017: 3012: 3008: 3005: 3002: 2998: 2992: 2988: 2984: 2978: 2975: 2972: 2966: 2959: 2958: 2957: 2956:can be used: 2955: 2954:Taylor series 2950: 2940: 2933: 2920: 2911: 2888: 2879: 2875: 2871: 2864: 2857: 2853: 2848: 2844: 2843:semi-infinite 2840: 2817: 2813: 2805: 2802: 2797: 2793: 2788: 2784: 2781: 2776: 2772: 2768: 2764: 2760: 2757: 2754: 2750: 2746: 2739: 2738: 2737: 2731: 2724: 2705: 2703: 2699: 2675: 2671: 2662: 2657: 2646: 2643: 2637: 2629: 2616: 2615: 2614: 2610: 2590: 2584: 2581: 2578: 2559: 2558: 2557: 2555: 2550: 2541: 2538:is the total 2536: 2516: 2513: 2510: 2502: 2496: 2490: 2482: 2469: 2468: 2467: 2465: 2461: 2457: 2447: 2445: 2441: 2422: 2419: 2416: 2413: 2408: 2396: 2395: 2394: 2383: 2363: 2357: 2350: 2344: 2338: 2332: 2324: 2311: 2310: 2309: 2306: 2304: 2303:heat equation 2285: 2282: 2277: 2268: 2265: 2259: 2251: 2238: 2237: 2236: 2233: 2211: 2207: 2198: 2193: 2182: 2179: 2176: 2170: 2155: 2143: 2140: 2136: 2132: 2126: 2114: 2110: 2103: 2087: 2086: 2085: 2064: 2061: 2056: 2052: 2046: 2034: 2030: 2023: 2011: 2005: 1997: 1985: 1982: 1979: 1973: 1961: 1955: 1947: 1934: 1933: 1932: 1930: 1920: 1907: 1903: 1896: 1893: 1890: 1884: 1880: 1874: 1870: 1866: 1863: 1857: 1854: 1851: 1848: 1844: 1839: 1833: 1830: 1827: 1821: 1801: 1781: 1773: 1769: 1765: 1764:Heat equation 1746: 1740: 1737: 1731: 1723: 1710: 1709: 1708: 1703: 1692: 1690:, example m/s 1672: 1669: 1655: 1632: 1626: 1614: 1610: 1606: 1602: 1580: 1577: 1547: 1546: 1545: 1523: 1519: 1510: 1505: 1493: 1490: 1484: 1476: 1463: 1462: 1461: 1459: 1455: 1446: 1426: 1422: 1401: 1396: 1392: 1381: 1377: 1372: 1369: 1363: 1360: 1328: 1325: 1322: 1310: 1297: 1294: 1288: 1285: 1282: 1250: 1247: 1244: 1232: 1219: 1214: 1210: 1199: 1195: 1190: 1187: 1181: 1178: 1153: 1133: 1128: 1124: 1120: 1117: 1111: 1108: 1105: 1085: 1074: 1053: 1049: 1045: 1035: 1031: 1003: 999: 971: 967: 957: 938: 934: 911: 907: 880: 870: 866: 853: 850: 846: 841: 838: 833: 829: 821: 820: 819: 817: 806: 800: 797: 790: 784: 773: 772: 771: 751: 741: 737: 724: 721: 714: 710: 706: 700: 697: 692: 688: 680: 679: 678: 676: 671: 641: 636: 633: 629: 623: 618: 614: 606: 605: 604: 602: 586: 574: 567: 560: 554: 550: 546: 539: 533: 529: 525: 514: 513: 512: 493: 489: 478: 474: 469: 466: 460: 457: 452: 438: 437: 436: 427: 426:mass fraction 417: 410: 401: 389: 386: 366: 360: 357: 354: 342: 341: 340: 338: 334: 325: 315: 311: 297: 291: 289: 270: 267: 262: 259: 249: 246: 245: 240: 233: 226: 222: 218: 211: 210: 209: 189: 186: 181: 178: 172: 169: 166: 163: 156: 155: 154: 151: 147: 138: 136: 132: 126: 124: 123:Fourier's Law 120: 116: 112: 111:Thomas Graham 102: 100: 95: 93: 89: 87: 83: 81: 73: 69: 65: 61: 54: 50: 46: 42: 38: 34: 30: 26: 22: 9133: 9116: 9112: 9104: 9100: 9091: 9076: 9065: 9059:. Princeton. 9056: 9012: 9008: 8998: 8990:the original 8985: 8975: 8958: 8954: 8948: 8905: 8901: 8867: 8861: 8818: 8815:AIP Advances 8814: 8764: 8760: 8734: 8730: 8686: 8682: 8626: 8622: 8612: 8561: 8557: 8551: 8532: 8526: 8514:. Retrieved 8510: 8501: 8492: 8486: 8466: 8459: 8449: 8442: 8407: 8403: 8394: 8385: 8379: 8368:the original 8363: 8359: 8346: 8327: 8323: 8299:(1): 59–86. 8296: 8292: 8281: 8269:. Retrieved 8265: 8256: 8222:Gas exchange 8191: 8187: 8178: 8170: 8161: 8158: 7997: 7959: 7810: 7807: 7618: 7613: 7609: 7607: 7603: 7526: 7501:is viscosity 7474: 7309: 7305: 7301: 7290: 7287: 7283: 7275: 7266: 7257: 7250: 7244: 7238: 7168: 7088: 7084:Graham's law 7081: 7077:flux limiter 7073: 7063: 7054: 7037: 7030: 7014: 6942: 6894: 6754: 6635: 6631: 6623: 6485: 6397:proposed by 6392: 6372: 6366: 6364: 6199: 6029: 6025: 5844: 5836: 5824: 5754: 5618: 5499: 5378: 5262: 5066: 5055: 4893: 4879: 4876:biomolecules 4873: 4867: 4861: 4852: 4843: 4831: 4826: 4701: 4697: 4693: 4687: 4683: 4666: 4661:random walks 4653: 4623:voltammetric 4585: 4582:Applications 4573: 4569: 4563: 4559: 4552: 4548: 4544: 4541: 4528: 4521: 4517: 4509: 4505: 4496: 4341: 4005: 3986: 3620: 3613: 3602: 3348: 3344: 3340: 3335: 3270: 3144: 3132: 3061: 2951: 2938: 2931: 2918: 2909: 2907:in front of 2886: 2877: 2873: 2869: 2862: 2855: 2851: 2846: 2842: 2832: 2729: 2722: 2711: 2695: 2608: 2605: 2553: 2548: 2534: 2531: 2453: 2437: 2381: 2378: 2307: 2300: 2234: 2231: 2079: 1926: 1761: 1699: 1612: 1608: 1604: 1600: 1543: 1453: 1452: 1311: 1233: 1148: 1080: 955: 898: 818:difference: 813: 778:denotes the 769: 672: 670:th species. 658: 578: 552: 548: 537: 531: 527: 519:denotes the 510: 423: 408: 399: 390: 384: 381: 326: 304: 287: 242: 238: 216: 207: 145: 144: 134: 130: 127: 108: 96: 91: 90: 85: 84: 59: 58: 52: 48: 44: 29: 8623:Soft Matter 8271:26 February 8174:random walk 7523:is density. 7024:conductance 4596:biopolymers 3604:anisotropic 1772:Heat kernel 959:in a vapor 523:th species, 322:10 m/s 244:diffusivity 135:non-Fickian 115:Darcy's law 9168:Categories 8980:Nosek TM. 6361:neighbors. 4900:absorption 4896:adsorption 4890:diffusion. 4590:in foods, 4539:equation. 3993:should be 991:or liquid 782:th species 774:the index 559:molar mass 515:the index 68:Adolf Fick 9174:Diffusion 9119:: 33–38. 8417:1012.2908 8400:Gorban AN 8248:Citations 8227:Mass flux 8212:Diffusion 8202:Advection 8141:δ 8127:− 8056:δ 8041:− 8011:− 7913:δ 7825:− 7547:δ 7511:ρ 7489:η 7425:δ 7400:η 7393:ρ 7318:δ 7210:η 7146:π 7121:η 7015:in which 6984:− 6963:− 6767:σ 6675:σ 6669:π 6437:π 6330:π 6100:− 5934:⟩ 5928:⟨ 5874:π 5863:⟩ 5857:⟨ 5810:τ 5740:τ 5730:τ 5727:− 5716:τ 5691:∫ 5684:π 5672:− 5666:π 5631:Γ 5601:π 5546:∂ 5541:Γ 5538:∂ 5518:∫ 5511:Γ 5476:π 5452:− 5424:∂ 5419:Γ 5416:∂ 5355:π 5314:∂ 5306:∂ 5186:− 5166:π 5141:∂ 5133:∂ 5020:Γ 4947:π 4914:Γ 4802:μ 4785:− 4777:− 4750:μ 4670:tautology 4654:When two 4576:= 1, 2, 3 4445:φ 4440:∇ 4429:φ 4419:φ 4395:⋅ 4392:∇ 4383:∑ 4370:∂ 4359:φ 4355:∂ 4301:∂ 4281:φ 4278:∂ 4259:∂ 4232:∂ 4210:∂ 4197:∂ 4177:φ 4168:∂ 4111:∑ 4082:φ 4079:∇ 4057:⋅ 4054:∇ 4042:∂ 4022:φ 4019:∂ 3956:∂ 3943:∂ 3923:φ 3914:∂ 3877:∑ 3856:∑ 3827:φ 3824:∇ 3811:⋅ 3808:∇ 3796:∂ 3776:φ 3773:∂ 3734:∂ 3729:φ 3726:∂ 3690:∑ 3686:− 3650:φ 3647:∇ 3641:− 3574:∂ 3554:φ 3551:∂ 3532:∂ 3515:∂ 3492:∑ 3470:φ 3467:Δ 3427:φ 3424:∇ 3402:⋅ 3399:∇ 3387:∂ 3367:φ 3364:∂ 3254:dimension 3205:⟩ 3180:− 3169:⟨ 3166:≡ 3113:τ 3103:τ 3083:∫ 3033:π 3006:− 2785:⁡ 2668:∂ 2663:φ 2654:∂ 2635:∂ 2630:φ 2627:∂ 2591:φ 2588:∇ 2582:− 2574:diffusion 2503:⋅ 2500:∇ 2488:∂ 2483:φ 2480:∂ 2438:which is 2414:φ 2405:∇ 2358:φ 2355:∇ 2345:⋅ 2342:∇ 2330:∂ 2325:φ 2322:∂ 2283:φ 2274:∇ 2257:∂ 2252:φ 2249:∂ 2204:∂ 2199:φ 2190:∂ 2177:φ 2168:∂ 2164:∂ 2153:∂ 2149:∂ 2133:φ 2124:∂ 2120:∂ 2101:∂ 2097:∂ 2053:φ 2044:∂ 2040:∂ 2021:∂ 2017:∂ 2012:− 2003:∂ 1998:φ 1995:∂ 1989:⇒ 1971:∂ 1967:∂ 1953:∂ 1948:φ 1945:∂ 1875:− 1867:⁡ 1852:π 1822:φ 1747:φ 1744:Δ 1729:∂ 1724:φ 1721:∂ 1702:Laplacian 1670:− 1621:and time 1578:− 1516:∂ 1511:φ 1502:∂ 1482:∂ 1477:φ 1474:∂ 1389:∇ 1370:ρ 1364:− 1323:ρ 1320:∇ 1295:φ 1292:∇ 1286:− 1245:ρ 1242:∇ 1207:∇ 1188:ρ 1182:− 1121:⁡ 1115:∇ 1109:− 899:Fugacity 878:∂ 863:∂ 842:− 798:(J/K/mol) 749:∂ 738:μ 734:∂ 701:− 642:ρ 630:ρ 587:ρ 486:∇ 467:ρ 461:− 367:φ 364:∇ 358:− 310:viscosity 263:φ 221:dimension 182:φ 170:− 119:Ohm's law 64:diffusion 62:describe 9159:OpenStax 9029:26330037 8940:36520427 8853:35070490 8713:29181286 8661:27396746 8596:11304296 8434:18961678 8196:See also 7050:membrane 6918:⁠2 6898:⁠2 4870:is time. 4700:, where 4656:miscible 4557:, where 3999:elliptic 2858:, 0) = 0 2847:infinite 1766:and its 816:fugacity 601:gradient 337:gradient 9037:3921833 8963:Bibcode 8931:9805503 8910:Bibcode 8844:8758205 8823:Bibcode 8769:Bibcode 8704:5687005 8652:5476231 8631:Bibcode 8604:1302913 8576:Bibcode 8301:Bibcode 8242:Osmosis 6933:⁠ 6913:⁠ 6755:where, 6486:where, 5839:fractal 4846:is the 4837:is the 4644:Onsager 4628:polymer 4592:neurons 3252:is the 2929:√ 2905:⁠ 2893:⁠ 2876:, 0) = 794:is the 573:density 561:of the 557:is the 540:of the 536:is the 413:⁠ 395:⁠ 318:(0.6–2) 286:is the 237:is the 223:is the 215:is the 105:History 9035: 9027: 8938: 8928: 8851: 8841: 8711: 8701: 8659: 8649: 8602: 8594: 8539: 8516:11 May 8474: 8432: 8081:where 7882:where 7716:where 7414:where 7169:where 6084: 5755:where 5259:(#/m). 4617:, and 4604:porous 4471:where 3610:tensor 3266:cactus 3232:where 2865:> 0 2833:where 2700:, the 2532:where 1544:where 1414:where 1146:where 770:where 659:where 511:where 382:where 331:, the 208:where 53:Bottom 49:Middle 41:solute 23:, see 9107:: 59. 9085:–171. 9033:S2CID 8600:S2CID 8566:arXiv 8430:S2CID 8412:arXiv 8371:(PDF) 8356:(PDF) 7531:from 6218:>= 4607:soils 3606:media 1705:Δ = ∇ 9025:PMID 8936:PMID 8849:PMID 8709:PMID 8657:PMID 8592:PMID 8537:ISBN 8518:2022 8472:ISBN 8273:2024 7276:The 6955:flux 6864:and 6593:and 6212:< 5060:and 4894:The 4497:The 4489:and 4004:For 2835:erfc 2782:erfc 2542:and 2540:flux 579:The 150:flux 9157:on 9121:doi 9117:100 9083:167 9017:doi 8926:PMC 8918:doi 8906:126 8872:doi 8839:PMC 8831:doi 8777:doi 8739:doi 8699:PMC 8691:doi 8647:PMC 8639:doi 8584:doi 8422:doi 8332:doi 8309:doi 7535:to 7477:= 0 7061:to 5179:exp 4898:or 4515:= Σ 3601:In 3334:In 3162:MSD 3062:If 2889:≤ 0 2736:is 2725:= 0 2611:= 0 1864:exp 335:or 333:del 241:or 131:not 45:Top 9170:: 9115:. 9105:94 9103:. 9031:. 9023:. 9013:39 9011:. 9007:. 8984:. 8959:17 8934:. 8924:. 8916:. 8904:. 8900:. 8884:^ 8870:. 8847:. 8837:. 8829:. 8819:12 8817:. 8813:. 8789:^ 8775:. 8765:14 8763:. 8751:^ 8735:29 8733:. 8721:^ 8707:. 8697:. 8685:. 8681:. 8669:^ 8655:. 8645:. 8637:. 8627:12 8625:. 8621:. 8598:. 8590:. 8582:. 8574:. 8562:63 8560:. 8509:. 8428:. 8420:. 8406:. 8362:. 8358:. 8328:10 8326:. 8307:. 8297:94 8291:. 8285:* 8264:. 7556:10 7256:, 7086:. 7079:. 7070:). 7036:− 5064:. 4880:mμ 4609:, 4602:, 4598:, 4594:, 4572:, 4562:, 4553:αβ 4549:ij 4524:ij 4482:ij 3989:ij 3623:ij 3619:= 3616:ji 3343:= 2932:Dt 2884:, 2860:, 2556:: 2466:: 2446:. 2305:. 1814:: 1603:= 1445:. 1118:ln 1073:. 663:si 137:. 82:. 74:, 9127:. 9123:: 9039:. 9019:: 8969:. 8965:: 8942:. 8920:: 8912:: 8878:. 8874:: 8855:. 8833:: 8825:: 8783:. 8779:: 8771:: 8745:. 8741:: 8715:. 8693:: 8687:8 8663:. 8641:: 8633:: 8606:. 8586:: 8578:: 8568:: 8545:. 8520:. 8480:. 8453:. 8436:. 8424:: 8414:: 8408:6 8364:2 8338:. 8334:: 8317:. 8311:: 8303:: 8275:. 8135:0 8131:P 8122:i 8118:P 8092:R 8068:) 8062:T 8059:R 8049:0 8045:P 8036:i 8032:P 8025:( 8019:i 8015:D 8008:= 8005:J 7983:T 7980:R 7977:n 7974:= 7971:V 7968:P 7942:i 7938:c 7934:d 7893:x 7869:) 7863:x 7860:d 7853:i 7849:c 7845:d 7839:( 7833:i 7829:D 7822:= 7819:J 7790:0 7786:D 7761:0 7757:T 7732:0 7728:P 7702:2 7698:/ 7694:3 7689:) 7682:0 7678:T 7674:T 7669:( 7663:) 7658:P 7653:0 7649:P 7643:( 7637:0 7633:D 7629:= 7626:D 7614:T 7610:P 7585:2 7581:/ 7577:1 7572:e 7569:R 7564:3 7559:L 7550:= 7537:L 7533:0 7529:x 7475:v 7468:x 7451:e 7448:R 7396:L 7390:v 7384:= 7380:e 7377:R 7372:) 7365:2 7361:/ 7357:1 7352:e 7349:R 7343:x 7340:5 7334:( 7330:= 7327:) 7324:x 7321:( 7261:1 7258:c 7254:2 7251:c 7245:t 7239:D 7235:. 7217:m 7214:t 7185:P 7181:A 7152:) 7149:t 7143:( 7139:/ 7135:D 7128:m 7125:t 7115:p 7111:A 7107:2 7104:= 7101:P 7067:2 7064:c 7058:1 7055:c 7041:1 7038:c 7034:2 7031:c 7020:P 6998:) 6992:1 6988:c 6979:2 6975:c 6970:( 6966:P 6959:= 6930:3 6927:/ 6924:1 6920:+ 6910:3 6907:/ 6904:1 6900:+ 6877:B 6873:C 6850:A 6846:C 6821:B 6817:D 6813:+ 6808:A 6804:D 6800:= 6795:r 6791:D 6737:3 6730:B 6726:C 6722:+ 6717:A 6713:C 6704:B 6700:C 6694:A 6690:C 6684:r 6680:D 6666:8 6661:= 6656:B 6653:A 6649:Z 6606:B 6602:C 6579:A 6575:C 6550:B 6546:D 6542:+ 6537:A 6533:D 6529:= 6524:r 6520:D 6497:R 6469:B 6465:C 6459:A 6455:C 6449:r 6445:D 6441:R 6433:4 6430:= 6425:B 6422:A 6418:Z 6348:4 6344:/ 6338:2 6334:L 6327:4 6307:A 6287:a 6264:D 6259:3 6255:/ 6251:2 6246:b 6242:C 6238:a 6235:2 6232:= 6229:t 6225:/ 6221:a 6215:r 6184:t 6162:D 6138:b 6134:C 6111:3 6107:/ 6103:1 6095:b 6091:C 6087:= 6081:L 6056:t 6053:D 6050:2 6045:= 6042:L 6007:D 5983:b 5979:C 5956:A 5931:r 5905:D 5900:3 5896:/ 5892:4 5887:b 5883:c 5879:A 5871:4 5866:= 5860:r 5790:C 5768:b 5764:C 5737:d 5724:t 5719:) 5713:( 5710:C 5701:t 5695:0 5681:D 5675:A 5662:t 5659:D 5649:b 5645:C 5640:A 5637:2 5634:= 5597:t 5594:D 5585:b 5581:C 5577:A 5574:2 5571:= 5566:0 5563:= 5560:x 5556:) 5549:t 5532:( 5527:t 5522:0 5514:= 5482:t 5479:D 5469:b 5465:C 5461:A 5458:D 5449:= 5444:0 5441:= 5438:x 5434:) 5427:t 5410:( 5387:A 5361:t 5358:D 5349:b 5345:C 5339:= 5334:0 5331:= 5328:x 5324:) 5317:x 5309:C 5300:( 5277:0 5274:= 5271:x 5247:t 5244:, 5241:x 5231:C 5214:) 5208:t 5205:D 5202:4 5196:2 5192:x 5183:( 5172:t 5169:D 5160:b 5156:C 5150:= 5144:x 5136:C 5107:0 5104:= 5101:x 5081:0 5078:= 5075:t 5052:. 5040:t 5008:t 5002:D 4984:b 4980:C 4967:A 4943:t 4940:D 4931:b 4927:C 4923:A 4920:2 4917:= 4868:t 4862:m 4853:μ 4844:T 4835:B 4832:k 4811:) 4805:) 4799:m 4796:( 4792:/ 4788:t 4781:e 4774:1 4770:( 4766:T 4760:B 4755:k 4747:= 4744:) 4741:t 4738:( 4735:D 4702:a 4698:D 4696:/ 4694:a 4574:β 4570:α 4564:j 4560:i 4551:, 4545:D 4531:j 4529:φ 4527:Δ 4522:D 4518:j 4512:i 4510:φ 4506:t 4503:∂ 4491:j 4487:i 4480:D 4475:i 4473:φ 4459:. 4455:) 4449:j 4433:j 4423:i 4411:j 4408:i 4404:D 4399:( 4387:j 4379:= 4373:t 4363:i 4322:. 4318:) 4309:i 4305:x 4296:) 4293:t 4290:, 4287:x 4284:( 4267:i 4263:x 4254:) 4251:x 4248:( 4243:j 4240:i 4236:D 4226:+ 4218:j 4214:x 4205:i 4201:x 4192:) 4189:t 4186:, 4183:x 4180:( 4172:2 4161:) 4158:x 4155:( 4150:j 4147:i 4143:D 4138:( 4132:3 4127:1 4124:= 4121:j 4118:, 4115:i 4107:= 4102:) 4097:) 4094:t 4091:, 4088:x 4085:( 4076:) 4073:x 4070:( 4067:D 4062:( 4051:= 4045:t 4037:) 4034:t 4031:, 4028:x 4025:( 4001:. 3987:D 3972:. 3964:j 3960:x 3951:i 3947:x 3938:) 3935:t 3932:, 3929:x 3926:( 3918:2 3905:j 3902:i 3898:D 3892:3 3887:1 3884:= 3881:j 3871:3 3866:1 3863:= 3860:i 3852:= 3847:) 3842:) 3839:t 3836:, 3833:x 3830:( 3821:D 3816:( 3805:= 3799:t 3791:) 3788:t 3785:, 3782:x 3779:( 3750:. 3742:j 3738:x 3718:j 3715:i 3711:D 3705:3 3700:1 3697:= 3694:j 3683:= 3678:i 3674:J 3653:, 3644:D 3638:= 3635:J 3621:D 3614:D 3582:i 3578:x 3569:) 3566:t 3563:, 3560:x 3557:( 3540:i 3536:x 3527:) 3524:x 3521:( 3518:D 3507:3 3502:1 3499:= 3496:i 3488:+ 3485:) 3482:t 3479:, 3476:x 3473:( 3464:) 3461:x 3458:( 3455:D 3452:= 3447:) 3442:) 3439:t 3436:, 3433:x 3430:( 3421:) 3418:x 3415:( 3412:D 3407:( 3396:= 3390:t 3382:) 3379:t 3376:, 3373:x 3370:( 3351:) 3349:x 3347:( 3345:D 3341:D 3312:t 3290:t 3287:D 3284:n 3281:2 3240:n 3220:t 3217:D 3214:n 3211:2 3208:= 3200:2 3196:) 3189:0 3185:x 3176:x 3172:( 3135:D 3118:. 3110:d 3106:) 3100:( 3097:D 3092:t 3087:0 3077:2 3064:D 3046:] 3041:) 3030:t 3027:D 3022:2 3018:x 3013:( 3009:2 3003:1 2999:[ 2993:0 2989:n 2985:= 2982:) 2979:t 2976:, 2973:x 2970:( 2967:n 2947:t 2943:x 2927:2 2922:0 2919:n 2913:0 2910:n 2902:2 2899:/ 2896:1 2887:x 2881:0 2878:n 2874:x 2872:( 2870:n 2863:x 2856:x 2854:( 2852:n 2818:. 2814:) 2806:t 2803:D 2798:2 2794:x 2789:( 2777:0 2773:n 2769:= 2765:) 2761:t 2758:, 2755:x 2751:( 2747:n 2733:0 2730:n 2723:x 2718:x 2714:t 2676:2 2672:x 2658:2 2647:D 2644:= 2638:t 2609:R 2585:D 2579:= 2569:j 2549:φ 2544:R 2535:j 2517:, 2514:R 2511:= 2507:j 2497:+ 2491:t 2423:, 2420:0 2417:= 2409:2 2391:x 2387:D 2382:φ 2364:. 2361:) 2351:D 2348:( 2339:= 2333:t 2286:, 2278:2 2269:D 2266:= 2260:t 2212:2 2208:x 2194:2 2183:D 2180:= 2171:x 2156:x 2144:D 2141:= 2137:) 2127:x 2115:D 2111:( 2104:x 2082:D 2065:0 2062:= 2057:) 2047:x 2035:D 2031:( 2024:x 2006:t 1986:0 1983:= 1980:J 1974:x 1962:+ 1956:t 1908:. 1904:) 1897:t 1894:D 1891:4 1885:2 1881:x 1871:( 1858:t 1855:D 1849:4 1845:1 1840:= 1837:) 1834:t 1831:, 1828:x 1825:( 1802:D 1782:k 1741:D 1738:= 1732:t 1694:x 1678:] 1673:1 1664:T 1656:2 1650:L 1644:[ 1634:D 1628:t 1623:t 1619:x 1615:) 1613:t 1611:, 1609:x 1607:( 1605:φ 1601:φ 1586:] 1581:3 1572:L 1564:N 1559:[ 1549:φ 1524:2 1520:x 1506:2 1494:D 1491:= 1485:t 1443:i 1427:i 1423:x 1402:, 1397:i 1393:x 1382:i 1378:M 1373:D 1361:= 1355:i 1351:J 1329:0 1326:= 1298:. 1289:D 1283:= 1277:i 1273:J 1251:0 1248:= 1220:. 1215:i 1211:y 1200:i 1196:M 1191:D 1179:= 1173:i 1169:J 1157:i 1151:i 1149:V 1134:, 1129:i 1125:y 1112:D 1106:= 1100:i 1096:V 1059:L 1054:i 1050:f 1046:= 1041:G 1036:i 1032:f 1009:L 1004:i 1000:f 977:G 972:i 968:f 956:i 939:i 935:f 912:i 908:f 881:x 871:i 867:f 854:T 851:R 847:D 839:= 834:i 830:J 808:μ 802:T 792:R 786:c 780:i 776:i 752:x 742:i 725:T 722:R 715:i 711:c 707:D 698:= 693:i 689:J 668:i 661:ρ 637:i 634:s 624:= 619:i 615:y 569:ρ 563:i 553:i 549:M 542:i 532:i 528:J 521:i 517:i 494:i 490:y 479:i 475:M 470:D 458:= 453:i 448:J 432:i 430:y 428:( 409:x 407:∂ 404:/ 400:φ 398:∂ 393:− 385:J 361:D 355:= 351:J 329:∇ 320:× 306:D 299:x 293:φ 271:x 268:d 260:d 235:D 229:J 213:J 190:x 187:d 179:d 173:D 167:= 164:J 76:D 27:.

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Knowledge

Fick's laws of diffusion

Index