5856:
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:
6041:
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,
6949:
probability of them overlapping at time zero, thus the rate of that two molecules association is infinity. It does not matter that other millions of molecules have to wait for their first mate to diffuse and arrive. The average rate is thus infinity. But statistically this argument is meaningless. The maximum rate of a molecule in a period of time larger than zero is 1, either meet or not, thus the infinite rate at time zero for that molecule pair really should just be one, making the average rate 1/millions or more and statistically negligible. This does not even count in reality no two molecules can magically meet at time zero.
5840:
4343:
6396:
4022:
3608:
4653:, from which Fick's law can be obtained as a limiting case, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the MaxwellâStefan equations are used. See also non-diagonal coupled transport processes (
2086:
2238:
3993:
3367:
8203:
first law can also be used to predict the changing moisture profiles across a spaghetti noodle as it hydrates during cooking. These phenomena are all about the spontaneous movement of particles of solutes driven by the concentration gradient. In different situations, there is different diffusivity which is a constant.
4338:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).}
6416:
molecule, e.g. B is the target molecule holding fixed relatively, and A is the moving molecule that creates a concentration gradient near the target molecule B due to the coagulation reaction between A and B. Smoluchowski calculated the collision frequency between A and B in the solution with unit #/s/m:
44:
6380:
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.
3776:
8202:
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
7103:
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,
6642:
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
6379:
such as DNA), the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule (random orientation) hit one side of the other. The diffusion constant need to be updated to the
4913:
rate of a dilute solute to a surface or interface in a (gas or liquid) solution can be calculated using Fick's laws of diffusion. The accumulated number of molecules adsorbed on the surface is expressed by the
Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in
4480:
163:
to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region
8176:
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
6925:
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
5855:
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
5835:
Monte Carlo simulations show that these two equations work to predict the adsorption rate of systems that form predictable concentration gradients near the surface but have troubles for systems without or with unpredictable concentration gradients, such as typical biosensing systems or when flow and
4695:
Adsorption, absorption, and collision of molecules, particles, and surfaces are important problems in many fields. These fundamental processes regulate chemical, biological, and environmental reactions. Their rate can be calculated using the diffusion constant and Fick's laws of diffusion especially
1092:
Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref. shows in detail how the
6387:
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
6210:
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
5760:
1948:
139:
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
7618:
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
7281:
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
5851:
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
7320:
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
7316:
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
7298:
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
6036:
This equation can be used to predict the initial adsorption rate of any system; It can be used to predict the steady-state adsorption rate of a typical biosensing system when the binding site is just a very small fraction of the substrate surface and a near-surface concentration gradient is never
4678:
In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's
8193:
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
8185:
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
7088:
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
6646:
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
6415:
and Fick's laws of diffusion. Under an idealized reaction condition for A + B â 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
6040:
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
6948:
These new equations also avoid the singularity on the adsorption rate at time zero for the
Langmuir-Schaefer equation. The infinity rate is justifiable under ideal conditions because when you introduce target molecules magically in a solution of probe molecule or vice versa, there always be a
2101:
4674:
take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.
6370:
assuming cubic packing each unit has 8 neighbors shared with other units. This example fraction converges the result to the 3D diffusive adsorption solution shown above with a slight difference in pre-factor due to different packing assumptions and ignoring other
4900:
Scheme of molecular diffusion in the solution. Orange dots are solute molecules, solvent molecules are not drawn, black arrow is an example random walk trajectory, and the red curve is the diffusive
Gaussian broadening probability function from the Fick's law of
5625:
5235:
4669:
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
8186:
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
7426:
1550:
2702:
4358:
3603:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}}
2852:. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space â that is, the corrosion product layer â is
2839:
7321:
steps required for CVD film growth are gas phase diffusion of reactants through the boundary layer, adsorption and surface diffusion of adatoms, reactions on the substrate, and gas phase diffusion of products away through the boundary layer.
1929:
5506:
8478:
66:: With an enormous number of solute molecules, randomness becomes undetectable: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Fick's laws.
6638:
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.
7728:
3241:
2385:
5385:
3771:
2081:{\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0}
6760:
5636:
4636:
methods is based on solutions of Fick's equation. On the other hand, in some cases a "Fickian (another common approximation of the transport equation is that of the diffusion theory)" description is inadequate. For example, in
2538:
7294:
is a collective term for a series of devices. It mainly includes three categoriesďźtwo-terminal devices, three-terminal devices, and four-terminal devices. The combination of the semiconductors is called an integrated circuit.
776:
2307:
2233:{\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}}
3988:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.}
905:
6926:
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
1768:
7282:
makes the flux stable instead of decay over time. A critical time has been estimated under idealized flow conditions when there is no gradient formed. This strategy is adopted in biology such as blood circulation.
1423:
1241:
8093:
517:
4833:
164:
of high concentration to a region of low concentration across a concentration gradient. In one (spatial) dimension, the law can be written in various forms, where the most common form (see) is in a molar basis:
2612:
5925:
6905:
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
7614:
3335:
has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is
Gaussian and 3D is a Maxwell-Boltzmann distribution.
7024:
140:
possible. Today, Fick's laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does
7894:
4971:
3139:
7327:
7178:
4897:
1155:
1319:
8194:
conditions. Statistical analysis is done to understand variation/stochasticity arising from the random walk of the species, which in-turn affects the overall process and electrical variations.
1082:
665:
124:, which fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists:
6037:
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.
8167:
6491:
388:
6643:
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.
6284:
5517:
214:
5136:
120:
In 1855, physiologist Adolf Fick first reported his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of
1828:
295:
8190:
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.
2444:
3674:
7635:
3167:
6843:
6572:
5954:
3068:
1477:
1032:
1000:
6133:
6078:
3679:
2630:
3313:
1350:
1272:
7476:
6368:
5847:
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
2753:
7243:
6788:
5416:
7937:
7449:
5041:
2396:
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
8007:
7968:
7816:
7787:
7758:
7535:
7513:
7211:
6899:
6872:
6628:
6601:
6160:
6005:
5830:
5790:
5007:
1450:
962:
935:
608:
5298:
5268:
5128:
5102:
8173:
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.
6309:
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
8116:
7917:
6517:
6327:
6307:
6204:
6182:
6027:
5976:
5810:
5408:
5061:
3333:
3261:
2325:
1823:
1803:
4475:{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right).}
1699:
1607:
7104:
while biological molecules are not denser than water. Also this equation assumes ideal concentration gradient forms near the membrane and evolves over time):
5306:
5755:{\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 }
8374:
1355:
1173:
8014:
6653:
2483:
54:
molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.
5630:
The
LangmuirâSchaefer equation can be extended to the WardâTordai Equation to account for the "back-diffusion" of rejected molecules from the surface:
694:
144:
follow Fick's laws (which happens in cases of diffusion through porous media and diffusion of swelling penetrants, among others), it is referred to as
7308:
fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law.
2252:
6647:
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:
6211:
actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem,
7556:
9008:
835:
324:
7828:
327:. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of
3271:
8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a
1724:
684:
Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of
4735:
in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:
452:
99:: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient.
4741:
1100:
1277:
4509:
4683:, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by
2573:
5862:
4679:
law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a
6403:
The bimolecular collision frequency related to many reactions including protein coagulation/aggregation is initially described by
4727:. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the
3279:, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion.
6963:
4920:
3083:
17:
8551:
8511:
8292:
8877:
Pandey S, Gautam D, Chen J (16 July 2024). "Measuring the Adsorption Cross Section of YOYO-1 to Immobilized DNA Molecules".
7110:
4641:
science and food science a more general approach is required to describe transport of components in materials undergoing a
2319:
If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields
6945:, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules.
3641:
8980:
Smoluchowski M (1916). "Drei Vorträge ßber Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen".
8770:
Ward AF, Tordai L (1946). "Time-dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time-effects".
9205:
4867:
2856:, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is
1037:
620:
8567:
Brogioli D, Vailati A (January 2001). "Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited".
5812:
is the sub-surface concentration (which is a function of time depending on the reaction model of the adsorption), and
6404:
5620:{\displaystyle \Gamma =\int _{0}^{t}({\frac {\partial \Gamma }{\partial t}})_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}}
2712:
2466:
8123:
6422:
4699:
Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the
356:
9210:
8220:
5230:{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}(-{\frac {x^{2}}{4Dt}})}
4546:
which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the
6217:
8397:
4728:
4348:
170:
9220:
8206:
By controlling the concentration gradient, the cooking time, shape of the food, and salting can be controlled.
7421:{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }}}
1468:
263:
7822:
The equation tells that increasing the temperature or decreasing the pressure can increase the diffusivity.
2410:
62:: With more molecules, there is a clear trend where the solute fills the container more and more uniformly.
9175:
8429:
8245:
4893:
in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute.
4724:
4700:
4654:
4646:
4547:
2927:(as the diffusion now occurs in both directions). This case is valid when some solution with concentration
2474:
7825:
Fick's first law predicts the flux of the reactants to the substrate and product away from the substrate:
1545:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}}}
108:
A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called
9215:
2697:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}}
8250:
6795:
6524:
5933:
2973:
4512:
include exactly the same terms. These physical models of diffusion are different from the test models
2617:
Plugging the definition of diffusive flux to the continuity equation and assuming there is no source (
1005:
973:
319:
is proportional to the squared velocity of the diffusing particles, which depends on the temperature,
9016:
7619:
becomes turbulent. Another option is to switch to a new carrier gas with lower viscosity or density.
7096:
The exchange rate of a gas across a fluid membrane can be determined by using this law together with
6389:
6388:
into the equation one should be able to calculate the theoretical adsorption kinetic curve using the
6086:
6047:
3161:
436:
3285:
2834:{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right).}
1326:
1248:
7456:
4005:
2095:
to be a constant, one can exchange the orders of the differentiation and multiply by the constant:
306:(for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume.
6332:
1924:{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).}
6029:
is the diffusion constant of the absorber (solute) in the solution (m/s) defined with Fick's law.
5501:{\displaystyle ({\frac {\partial \Gamma }{\partial t}})_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}}
1094:
121:
7622:
The Fick's first law describes diffusion through the boundary layer. As a function of pressure (
7218:
4553:
For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example
242:
measures the amount of substance that will flow through a unit area during a unit time interval.
9109:
7037:
6771:
4910:
4629:
806:
8432:, Sargsyan HP, Wahab HA (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion".
3144:
This idea is useful for estimating a diffusion length over a heating and cooling cycle, where
7922:
7434:
6634:
The reaction order of this bimolecular reaction is 2 which is the analogy to the result from
6392:. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry.
5026:
2450:
335:. For biological molecules the diffusion coefficients normally range from 10 to 10 m/s.
9101:
7977:
7943:
5956:
is the adsorption rate assuming under adsorption energy barrier-free situation, in unit #/s.
5852:
time-step adsorption simulation, bringing it to be consistent with the above two equations.
8989:
8936:
8833:
8779:
8641:
8586:
8322:
7794:
7765:
7736:
7723:{\displaystyle D=D_{0}\left({\frac {P_{0}}{P}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2}}
7520:
7498:
7245:
transmembrane efficiency (unitless), which can be calculated from the stochastic theory of
7189:
6877:
6850:
6606:
6579:
6138:
5983:
5815:
5768:
4985:
4858:
3236:{\displaystyle {\text{MSD}}\equiv \langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\rangle =2nDt}
1778:
1428:
940:
913:
593:
231:
8393:
5009:
is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m).
2380:{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).}
89:. Fick's first law can be used to derive his second law which in turn is identical to the
8:
8740:
Langmuir I, Schaefer VJ (1937). "The Effect of Dissolved Salts on Insoluble Monolayers".
7063:
6408:
5277:
5247:
5107:
5081:
4889:(e.g. proteins) in water, the exponential term is negligible due to the small product of
4621:
2718:
235:
109:
47:
8993:
8940:
8837:
8783:
8645:
8590:
8326:
5380:{\displaystyle ({\frac {\partial C}{\partial x}})_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}}
5130:) in a once uniform bulk solution is solved in the above sections from Fick's equation,
3766:{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.}
1467:
predicts how diffusion causes the concentration to change with respect to time. It is a
81:
in 1855 on the basis of largely experimental results. They can be used to solve for the
9200:
9059:
8957:
8924:
8854:
8821:
8714:
8689:
8662:
8629:
8610:
8576:
8459:
8441:
8284:
8101:
7902:
7305:
6502:
6312:
6292:
6189:
6167:
6135:(unit m) is the average nearest neighbor distance approximated as cubic packing, where
6012:
5961:
5795:
5393:
5078:
Briefly as explained in, the concentration gradient profile near a newly created (from
5046:
4866:
is the mobility of the particle in the fluid or gas, which can be calculated using the
4849:
3318:
3246:
2454:
1808:
1788:
685:
90:
3151:
1650:
1565:
9151:
9102:
9051:
8962:
8898:
8859:
8719:
8667:
8602:
8547:
8507:
8288:
4732:
4720:
4716:
4680:
4598:
4009:
3267:
of the particle's Brownian motion. For example, the diffusion of a molecule across a
2400:, the solution for the concentration will be a linear change of concentrations along
1939:
8463:
6755:{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt{C_{A}+C_{B}}}}
6383:
The above hitting rate equation is also useful to predict the kinetics of molecular
5043:
is the accumulated number of molecules in unit # molecules adsorbed during the time
2533:{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,}
9147:
9063:
9043:
8952:
8944:
8894:
8886:
8849:
8841:
8787:
8749:
8709:
8701:
8657:
8649:
8614:
8594:
8499:
8451:
8389:
8353:
8330:
8280:
6635:
5072:
4642:
3315:, is often used as a characterization of how far the particle has moved after time
254:
133:
82:
771:{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}}}
8230:
7097:
6412:
5068:
4896:
4684:
3272:
3157:
9031:
8925:"Why Should the Reaction Order of a Bimolecular Reaction be 2.33 Instead of 2?"
8598:
8346:
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
7246:
5978:
is the area of the surface of interest on an "infinite and flat" substrate (m).
4625:
3364:. This dependence does not affect Fick's first law but the second law changes:
2849:
2302:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,}
125:
35:
31:
8690:"Photobleaching of YOYO-1 in super-resolution single DNA fluorescence imaging"
8357:
2246:
For the case of diffusion in two or more dimensions Fick's second law becomes
9194:
9047:
8948:
8890:
8503:
8335:
8310:
7291:
7059:
6384:
5839:
4650:
3268:
2964:
2313:
1774:
688:
of this species. Then Fick's first law (one-dimensional case) can be written
105:: Prediction of change in concentration gradient with time due to diffusion.
8822:"Simulating stochastic adsorption of diluted solute molecules at interfaces"
8546:. Wiley Series in Chemical Engineering. Vol. 2. John Wiley & Sons.
8455:
2936:
is put in contact with a layer of pure solvent. (Bokstein, 2005) The length
9055:
8966:
8902:
8863:
8723:
8671:
8606:
8235:
7302:
Therefore, different types and levels of semiconductors can be fabricated.
7090:
4610:
2963:
As a quick approximation of the error function, the first two terms of the
435:
Another form for the first law is to write it with the primary variable as
129:
8375:"One and a Half Centuries of Diffusion: Fick, Einstein, before and beyond"
4690:
2952:
and provides a measure of how far the concentration has propagated in the
900:{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}}}
8705:
8581:
8187:
6007:
is the concentration of the absorber molecule in the bulk solution (#/m).
4886:
4671:
4666:
4633:
3614:
2243:
and, thus, receive the form of the Fick's equations as was stated above.
1782:
9032:"Understanding diffusion theory and Fick's law through food and cooking"
8753:
3619:, the diffusion coefficient depends on the direction. It is a symmetric
1763:{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi }
1034:
phase. At vapor liquid equilibrium the evaporation flux is zero because
8653:
6162:
is the solute concentration in the bulk solution (unit # molecule / m).
4906:
4606:
4504:
are related to the various components and not to the space coordinates.
569:
78:
50:
from a microscopic and macroscopic point of view. Initially, there are
8845:
8791:
2719:
Example solution 1: constant concentration source and diffusion length
1418:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},}
1236:{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.}
43:
9181:
8240:
8225:
8215:
8088:{\displaystyle J=-D_{i}\left({\frac {P_{i}-P_{0}}{\delta RT}}\right)}
6375:
When the area of interest is the size of a molecule (specifically, a
4914:
the simulated molecular diffusion in the first section of this page:
3264:
2708:
2470:
1712:
512:{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i}}
320:
74:
4828:{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right)}
402:
The driving force for the one-dimensional diffusion is the quantity
9185:
8275:
Vallero, Daniel A. (2024). "Physical transport of air pollutants".
6184:
is the diffusion coefficient defined by Fick's equation (unit m/s).
4703:. In the ultrashort time limit, in the order of the diffusion time
3152:
Example solution 2: Brownian particle and mean squared displacement
826:
611:
347:
8446:
8177:
diffusion (where junction depth keeps moving into the substrate).
1718:, which generalises the second derivative, obtaining the equation
8255:
8009:, the concentration of the gas is expressed by partial pressure.
5848:
4638:
2607:{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi }
583:
9104:
Thermodynamics and Kinetics in Materials Science: A Short Course
7277:
concentration should use unit mol m, so flux unit becomes mol s.
6395:
5920:{\displaystyle \langle r\rangle ={\frac {4}{\pi }}Ac_{b}^{4/3}D}
5300:
is simplified to the pre-exponential factor of the distribution
4019:
these two forms of the diffusion equation should be combined in
4614:
4602:
4597:
Equations based on Fick's law have been commonly used to model
3620:
3276:
1938:
Fick's second law can be derived from Fick's first law and the
51:
4351:
gives the following generalization of Fick's equation for the
2453:, the solutions to which are referred to by mathematicians as
2563:. The only source of flux in this situation is assumed to be
446:, given for example in kg/kg), then the equation changes to:
7036:
is the permeability, an experimentally determined membrane "
1352:) both species have the same molar mass, Fick's law becomes
350:
operator, which generalises the first derivative, obtaining
323:
of the fluid and the size of the particles according to the
8735:
8733:
7609:{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}}
4617:
2550:
1087:
160:
9099:
7019:{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)}}
6399:
Comparing collision theory and diffusive collision theory.
4715:
is the particle radius, the diffusion is described by the
427:, which for ideal mixtures is the concentration gradient.
7889:{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right)}
7285:
6845:
is the relative diffusion constant between A and B (m/s).
4966:{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}}
3134:{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.}
2473:
and no net volumetric source. It can be derived from the
343:
8815:
8813:
8811:
8809:
8807:
8805:
8803:
8801:
8730:
7311:
7213:
is the total area of the pores on the membrane (unit m).
6901:
are number concentrations of A and B respectively (#/m).
6630:
are number concentrations of A and B respectively (#/m).
6574:
is the relative diffusion constant between A and B (m/s)
5843:
A brief history of the theories on diffusive adsorption.
1773:
Fick's second law has the same mathematical form as the
4691:
Sorption rate and collision frequency of diluted solute
2460:
9100:
Bokshtein BS, Mendelev MI, Srolovitz DJ, eds. (2005).
6790:
is the area of the cross-section of the collision (m).
2902:), then the solution is amended only with coefficient
1274:, this reduces to the most common form of Fick's law,
825:
The driving force of Fick's law can be expressed as a
8798:
8126:
8104:
8017:
7980:
7946:
7925:
7905:
7831:
7797:
7768:
7739:
7638:
7559:
7523:
7501:
7459:
7437:
7330:
7221:
7192:
7173:{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {D/(\pi t)}}}
7113:
6966:
6880:
6853:
6798:
6774:
6656:
6609:
6582:
6527:
6505:
6425:
6335:
6315:
6295:
6220:
6192:
6170:
6141:
6089:
6050:
6015:
5986:
5964:
5936:
5865:
5818:
5798:
5771:
5639:
5520:
5419:
5396:
5309:
5280:
5250:
5244:
is the number concentration of adsorber molecules at
5139:
5110:
5084:
5049:
5029:
4988:
4923:
4744:
4696:
when these interactions happen in diluted solutions.
4361:
4025:
3779:
3682:
3644:
3370:
3321:
3288:
3249:
3170:
3086:
2976:
2756:
2633:
2576:
2486:
2413:
2328:
2255:
2104:
1951:
1831:
1811:
1791:
1727:
1653:
1568:
1480:
1431:
1358:
1329:
1280:
1251:
1176:
1103:
1040:
1008:
976:
943:
916:
838:
697:
623:
596:
455:
359:
266:
173:
9090:
4008:. It is needed to make the right hand side operator
1150:{\displaystyle \mathbf {V_{i}} =-D\nabla \ln y_{i},}
8428:
6957:The first law gives rise to the following formula:
4885:For a single molecule such as organic molecules or
2738:, where the concentration is maintained at a value
2707:If flux were the result of both diffusive flux and
1314:{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .}
9176:Fick's equations, Boltzmann's transformation, etc.
8627:
8180:
8161:
8110:
8087:
8001:
7962:
7931:
7911:
7888:
7810:
7781:
7752:
7722:
7608:
7529:
7507:
7470:
7443:
7420:
7237:
7205:
7172:
7018:
6893:
6866:
6837:
6782:
6754:
6622:
6595:
6566:
6511:
6485:
6362:
6321:
6301:
6278:
6198:
6176:
6154:
6127:
6072:
6021:
5999:
5970:
5948:
5919:
5824:
5804:
5784:
5754:
5619:
5500:
5402:
5379:
5292:
5262:
5229:
5122:
5096:
5055:
5035:
5001:
4965:
4827:
4474:
4337:
3987:
3765:
3668:
3602:
3327:
3307:
3255:
3235:
3133:
3062:
2833:
2696:
2606:
2532:
2438:
2379:
2301:
2232:
2080:
1933:
1923:
1817:
1797:
1762:
1693:
1601:
1544:
1444:
1417:
1344:
1313:
1266:
1235:
1149:
1076:
1026:
994:
956:
929:
899:
770:
659:
602:
511:
382:
289:
208:
8765:
8763:
8414:VĂĄzquez JL (2006). "The Porous Medium Equation".
7255:is the diffusion constant of the solute unit mâ
s.
6329:over solute nearest neighbor sphere surface area
6032:Dimensional analysis of these units is satisfied.
1077:{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}}
660:{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }}}
9192:
9161:Foundations of Materials Science and Engineering
8973:
8918:
8916:
8914:
8912:
8876:
8739:
5274:The concentration gradient at the subsurface at
3077:is time-dependent, the diffusion length becomes
27:Mathematical descriptions of molecular diffusion
9029:
8870:
8566:
5015:is diffusion coefficient of the adsorber (m/s).
3995:The symmetric matrix of diffusion coefficients
8760:
8683:
8681:
8476:
8197:
8162:{\displaystyle {\frac {P_{i}-P_{0}}{\delta }}}
6486:{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B}}
5067:The equation is named after American chemists
4510:ChapmanâEnskog formulae for diffusion in gases
3773:For the diffusion equation this formula gives
1647:is the diffusion coefficient in dimensions of
430:
383:{\displaystyle \mathbf {J} =-D\nabla \varphi }
312:is position, the dimension of which is length.
9030:Zhou L, Nyberg K, Rowat AC (September 2015).
8909:
8628:Bian X, Kim C, Karniadakis GE (August 2016).
8541:
8422:
5390:And the rate of diffusion (flux) across area
4496:is the matrix of coefficients. Here, indices
4112:
4072:
3857:
3826:
3457:
3417:
3349:, the diffusion coefficient varies in space,
9138:Fick, Adolph (1995). "On liquid diffusion".
9108:. Oxford: Oxford University Press. pp.
8979:
6411:in a seminal 1916 publication, derived from
6279:{\displaystyle <r>=a/t=2aC_{b}^{2/3}D}
5943:
5937:
5872:
5866:
3676:it is the product of a tensor and a vector:
3215:
3179:
8678:
8434:Mathematical Modelling of Natural Phenomena
2731:-axis) from a boundary located at position
2465:Fick's second law is a special case of the
209:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
58:: A single molecule moves around randomly.
9091:Bird RB, Stewart WE, Lightfoot EN (1976).
8769:
1711:In two or more dimensions we must use the
8956:
8853:
8713:
8661:
8580:
8445:
8372:
8334:
7040:" for a given gas at a given temperature.
6952:
5745:
4763:
4687:the fluctuating hydrodynamics equations.
4660:
4489:are concentrations of the components and
4453:
3119:
2364:
2282:
2071:
1507:
8742:Journal of the American Chemical Society
8687:
8526:
6394:
5838:
4895:
3156:Another simple case of diffusion is the
1785:, except switching thermal conductivity
1088:Derivation of Fick's first law for gases
42:
8493:
8480:Physical Chemistry for the Life Science
8413:
8274:
4349:Einstein's mobility and Teorell formula
2390:An important example is the case where
1628:is a function that depends on location
1097:reduces to this version of Fick's law:
290:{\displaystyle {\frac {d\varphi }{dx}}}
14:
9193:
9126:Fick A (1855). "On liquid diffusion".
8344:Fick A (1855). "On liquid diffusion".
7970:is the first reactant's concentration.
7324:The velocity profile for gas flow is:
7286:Semiconductor fabrication applications
3275:is a 3-D diffusion. For a cylindrical
2439:{\displaystyle \nabla ^{2}\varphi =0,}
2404:. In two or more dimensions we obtain
1942:in absence of any chemical reactions:
1674:
1660:
1582:
1574:
1562:is the concentration in dimensions of
338:In two or more dimensions we must use
258:. Its dimension is area per unit time.
9158:
9116:
8496:Essentials of Micro- and Nanofluidics
8394:10.62721/diffusion-fundamentals.2.187
7630:) in a gas, diffusion is determined.
7312:CVD method of fabricate semiconductor
4589:correspond to the space coordinates.
2860:(lasting both through the layer with
2723:A simple case of diffusion with time
1166:is the diffusion velocity of species
9137:
9125:
9081:
8922:
8819:
8343:
8308:
4645:. One more general framework is the
3669:{\displaystyle J=-D\nabla \varphi ,}
2461:Example solutions and generalization
1459:
555:th species (for example in mol/m-s),
8929:The Journal of Physical Chemistry A
8694:Beilstein Journal of Nanotechnology
8688:Pyle JR, Chen J (2 November 2017).
2624:), we arrive at Fick's second law:
2091:Assuming the diffusion coefficient
1170:. In terms of species flux this is
964:is a partial pressure of component
399:denotes the diffusion flux vector.
151:
24:
9074:
8527:Williams FA (1985). "Appendix E".
8285:10.1016/B978-0-443-13987-1.00017-X
7585:
7582:
7553:, it gives the average thickness:
7464:
7461:
7393:
7390:
7365:
7362:
5640:
5556:
5551:
5548:
5521:
5434:
5429:
5426:
5324:
5316:
5151:
5143:
5030:
4924:
4868:Einstein relation (kinetic theory)
4770:
4450:
4402:
4380:
4365:
4311:
4288:
4269:
4242:
4220:
4207:
4178:
4089:
4064:
4052:
4029:
3966:
3953:
3924:
3834:
3818:
3806:
3783:
3744:
3736:
3657:
3584:
3561:
3542:
3525:
3477:
3434:
3409:
3397:
3374:
3339:
2678:
2664:
2645:
2637:
2598:
2510:
2498:
2490:
2415:
2365:
2352:
2340:
2332:
2284:
2267:
2259:
2214:
2200:
2178:
2174:
2163:
2159:
2134:
2130:
2111:
2107:
2054:
2050:
2031:
2027:
2013:
2005:
1981:
1977:
1963:
1955:
1754:
1739:
1731:
1526:
1512:
1492:
1484:
1399:
1330:
1302:
1252:
1217:
1125:
888:
873:
821:is the chemical potential (J/mol).
759:
744:
496:
374:
25:
9232:
9169:
9006:
8301:
8169:is the partial pressure gradient.
6838:{\displaystyle D_{r}=D_{A}+D_{B}}
6567:{\displaystyle D_{r}=D_{A}+D_{B}}
6405:Smoluchowski coagulation equation
5949:{\displaystyle \langle r\rangle }
3063:{\displaystyle n(x,t)=n_{0}\left}
1323:If (instead of or in addition to
9036:Advances in Physiology Education
7066:for the direction of flow (from
5104:) absorptive surface (placed at
3199:
3195:
3186:
3160:of one particle. The particle's
2956:-direction by diffusion in time
2579:
2517:
1452:is the mole fraction of species
1365:
1361:
1287:
1283:
1183:
1179:
1110:
1106:
1027:{\displaystyle f_{i}^{\text{L}}}
995:{\displaystyle f_{i}^{\text{G}}}
458:
361:
9023:
9000:
8879:Journal of Physical Chemistry B
8621:
8560:
8181:Invalidity of Fickian diffusion
6128:{\displaystyle L~=C_{b}^{-1/3}}
6073:{\displaystyle L={\sqrt {2Dt}}}
4729:fluctuation-dissipation theorem
4592:
4017:inhomogeneous anisotropic media
3164:from its original position is:
2727:in one dimension (taken as the
2557:is a net volumetric source for
1934:Derivation of Fick's second law
815:is the absolute temperature (K)
30:For the technique of measuring
9013:Essentials of Human Physiology
8630:"111 years of Brownian motion"
8535:
8520:
8487:
8470:
8407:
8366:
8268:
7818:is the standard diffusitivity.
7484:is the length of the substrate
7340:
7334:
7165:
7156:
6519:is the radius of the collision
6206:is the critical time (unit s).
5728:
5722:
5566:
5542:
5444:
5420:
5334:
5310:
5224:
5193:
4815:
4806:
4754:
4748:
4306:
4294:
4264:
4258:
4202:
4190:
4171:
4165:
4107:
4095:
4086:
4080:
4047:
4035:
3948:
3936:
3852:
3840:
3801:
3789:
3638:. Fick's first law changes to
3579:
3567:
3537:
3531:
3495:
3483:
3474:
3468:
3452:
3440:
3431:
3425:
3392:
3380:
3308:{\displaystyle {\sqrt {2nDt}}}
3206:
3182:
3116:
3110:
2992:
2980:
2371:
2358:
1999:
1847:
1835:
1688:
1654:
1596:
1569:
1471:which in one dimension reads:
1345:{\displaystyle \nabla \rho =0}
1267:{\displaystyle \nabla \rho =0}
677:is the partial density of the
13:
1:
9178:(with figures and animations)
9128:Annalen der Physik und Chemie
8494:Conlisk, A. Terrence (2013).
8477:Atkins P, de Paula J (2006).
7471:{\displaystyle \mathrm {Re} }
4649:equations of multi-component
2713:convectionâdiffusion equation
2467:convectionâdiffusion equation
1469:partial differential equation
238:per unit area per unit time.
9163:(3rd ed.). McGraw-Hill.
9152:10.1016/0376-7388(94)00230-v
9119:The Mathematics of Diffusion
8544:Multicomponent mass transfer
8542:Taylor R, Krishna R (1993).
8261:
8221:ChurchillâBernstein equation
7299:second through mathematics.
6363:{\displaystyle 4\pi L^{2}/4}
5836:convection are significant.
4578:refer to the components and
1093:diffusion equation from the
799:is the concentration (mol/m)
7:
9140:Journal of Membrane Science
8772:Journal of Chemical Physics
8209:
8198:Food production and cooking
7789:is the standard temperature
5792:is the bulk concentration,
4875:is the mass of the particle
4355:of the perfect components:
1805:with diffusion coefficient
614:operator. This is because:
431:Variations of the first law
10:
9237:
9121:. Oxford University Press.
8599:10.1103/PhysRevE.63.012105
8277:Air Pollution Calculations
7238:{\displaystyle \eta _{tm}}
2312:which is analogous to the
1707:is the position, example m
115:
112:or non-Fickian diffusion.
77:and were first posited by
29:
9206:Eponymous laws of physics
8358:10.1080/14786445508641925
7089:situation, one can use a
6783:{\displaystyle {\sigma }}
6390:Langmuir adsorption model
6377:long cylindrical molecule
4632:processes. The theory of
3162:Mean squared displacement
3148:varies with temperature.
9095:. John Wiley & Sons.
9048:10.1152/advan.00133.2014
9009:"Section 3/3ch9/s3ch9_2"
8949:10.1021/acs.jpca.2c07500
8923:Chen J (December 2022).
8891:10.1021/acs.jpcb.4c03359
8504:10.1017/CBO9781139025614
8336:10.1002/andp.18551700105
8246:MaxwellâStefan diffusion
7760:is the standard pressure
4980:is the surface area (m).
4838:where (all in SI units)
4725:StokesâEinstein equation
4719:. At a longer time, the
4701:StokesâEinstein equation
4647:MaxwellâStefan diffusion
4548:MaxwellâStefan diffusion
4353:multicomponent diffusion
3282:The square root of MSD,
325:StokesâEinstein relation
132:(charge transport), and
71:Fick's laws of diffusion
9211:Mathematics in medicine
9084:Random Walks in Biology
8820:Chen J (January 2022).
7932:{\displaystyle \delta }
7444:{\displaystyle \delta }
5511:Integrating over time,
5036:{\displaystyle \Gamma }
1095:kinetic theory of gases
8982:Zeitschrift fĂźr Physik
8382:Diffusion Fundamentals
8251:NernstâPlanck equation
8163:
8112:
8089:
8003:
8002:{\displaystyle PV=nRT}
7964:
7963:{\displaystyle dc_{i}}
7933:
7913:
7890:
7812:
7783:
7754:
7724:
7610:
7531:
7509:
7478:is the Reynolds number
7472:
7445:
7422:
7239:
7207:
7174:
7062:of the gas across the
7020:
6953:Biological perspective
6895:
6868:
6839:
6784:
6756:
6624:
6597:
6568:
6513:
6487:
6400:
6364:
6323:
6303:
6280:
6200:
6178:
6156:
6129:
6074:
6023:
6001:
5972:
5950:
5921:
5844:
5826:
5806:
5786:
5756:
5621:
5502:
5404:
5381:
5294:
5264:
5231:
5124:
5098:
5057:
5037:
5003:
4967:
4902:
4829:
4661:Fick's flow in liquids
4476:
4347:The approach based on
4339:
4146:
3989:
3906:
3885:
3767:
3719:
3670:
3604:
3521:
3329:
3309:
3257:
3237:
3135:
3064:
2835:
2698:
2608:
2534:
2440:
2381:
2303:
2234:
2082:
1925:
1819:
1799:
1764:
1695:
1603:
1546:
1446:
1419:
1346:
1315:
1268:
1237:
1151:
1078:
1028:
996:
958:
931:
901:
807:universal gas constant
772:
661:
604:
586:(for example in kg/m).
513:
384:
299:concentration gradient
291:
210:
159:relates the diffusive
67:
18:Concentration gradient
9221:Statistical mechanics
8456:10.1051/mmnp/20116509
8418:. Oxford Univ. Press.
8164:
8113:
8090:
8004:
7965:
7934:
7914:
7891:
7813:
7811:{\displaystyle D_{0}}
7784:
7782:{\displaystyle T_{0}}
7755:
7753:{\displaystyle P_{0}}
7725:
7611:
7532:
7530:{\displaystyle \rho }
7510:
7508:{\displaystyle \eta }
7473:
7446:
7423:
7240:
7208:
7206:{\displaystyle A_{P}}
7175:
7058:is the difference in
7021:
6896:
6894:{\displaystyle C_{B}}
6869:
6867:{\displaystyle C_{A}}
6840:
6785:
6757:
6625:
6623:{\displaystyle C_{B}}
6598:
6596:{\displaystyle C_{A}}
6569:
6514:
6488:
6398:
6365:
6324:
6304:
6281:
6201:
6179:
6157:
6155:{\displaystyle C_{b}}
6130:
6075:
6024:
6002:
6000:{\displaystyle C_{b}}
5973:
5951:
5922:
5842:
5832:is a dummy variable.
5827:
5825:{\displaystyle \tau }
5807:
5787:
5785:{\displaystyle C_{b}}
5757:
5622:
5503:
5405:
5382:
5295:
5265:
5232:
5125:
5099:
5058:
5038:
5004:
5002:{\displaystyle C_{b}}
4968:
4899:
4830:
4624:, nuclear materials,
4477:
4340:
4120:
3990:
3886:
3865:
3768:
3699:
3671:
3605:
3501:
3347:non-homogeneous media
3330:
3310:
3258:
3238:
3136:
3065:
2848:is the complementary
2836:
2699:
2609:
2535:
2469:in which there is no
2441:
2382:
2304:
2235:
2083:
1926:
1820:
1800:
1765:
1696:
1604:
1547:
1447:
1445:{\displaystyle x_{i}}
1420:
1347:
1316:
1269:
1238:
1152:
1079:
1029:
997:
959:
957:{\displaystyle f_{i}}
932:
930:{\displaystyle f_{i}}
902:
773:
662:
605:
603:{\displaystyle \rho }
549:diffusion flux vector
514:
385:
292:
250:diffusion coefficient
211:
83:diffusion coefficient
46:
8988:: 557â571, 585â599.
8706:10.3762/bjnano.8.229
8531:. Benjamin/Cummings.
8373:Philibert J (2005).
8279:. pp. 163â190.
8124:
8102:
8015:
7978:
7944:
7923:
7903:
7829:
7795:
7766:
7737:
7636:
7557:
7521:
7499:
7457:
7435:
7328:
7219:
7190:
7111:
6964:
6878:
6851:
6796:
6772:
6654:
6607:
6580:
6525:
6503:
6423:
6333:
6313:
6293:
6218:
6190:
6168:
6139:
6087:
6048:
6013:
5984:
5962:
5934:
5863:
5816:
5796:
5769:
5637:
5518:
5417:
5394:
5307:
5278:
5248:
5137:
5108:
5082:
5047:
5027:
5021:is elapsed time (s).
4986:
4921:
4859:absolute temperature
4742:
4630:semiconductor doping
4359:
4023:
3777:
3680:
3642:
3368:
3319:
3286:
3247:
3168:
3084:
2974:
2754:
2631:
2574:
2484:
2411:
2326:
2253:
2102:
1949:
1829:
1809:
1789:
1779:fundamental solution
1725:
1651:
1566:
1478:
1429:
1356:
1327:
1278:
1249:
1174:
1101:
1038:
1006:
974:
941:
914:
836:
695:
621:
594:
453:
357:
264:
171:
9093:Transport Phenomena
8994:1916ZPhy...17..557S
8941:2022JPCA..126.9719C
8838:2022AIPA...12a5318C
8784:1946JChPh..14..453W
8754:10.1021/ja01290a091
8646:2016SMat...12.6331B
8591:2000PhRvE..63a2105B
8416:Mathematical Theory
8403:on 5 February 2009.
8327:1855AnP...170...59F
8118:is the gas constant
7626:) and temperature (
7317:create thin films.
6409:Marian Smoluchowski
6272:
6124:
5913:
5715:
5541:
5293:{\displaystyle x=0}
5263:{\displaystyle x,t}
5123:{\displaystyle x=0}
5097:{\displaystyle t=0}
4622:population dynamics
4599:transport processes
3106:
2475:continuity equation
1781:is the same as the
1073:
1055:
1023:
991:
236:amount of substance
110:anomalous diffusion
48:Molecular diffusion
9216:Physical chemistry
8654:10.1039/c6sm01153e
8575:(1 Pt 1): 012105.
8315:Annalen der Physik
8159:
8108:
8085:
7999:
7960:
7929:
7909:
7886:
7808:
7779:
7750:
7720:
7606:
7527:
7505:
7468:
7441:
7418:
7306:Integrated circuit
7235:
7203:
7170:
7016:
6891:
6864:
6835:
6780:
6752:
6620:
6593:
6564:
6509:
6483:
6401:
6360:
6319:
6299:
6276:
6250:
6196:
6174:
6152:
6125:
6099:
6070:
6019:
5997:
5968:
5946:
5917:
5891:
5845:
5822:
5802:
5782:
5752:
5699:
5617:
5527:
5498:
5400:
5377:
5290:
5260:
5227:
5120:
5094:
5053:
5033:
4999:
4963:
4903:
4850:Boltzmann constant
4825:
4472:
4401:
4335:
3985:
3763:
3666:
3600:
3325:
3305:
3253:
3233:
3131:
3092:
3060:
2831:
2694:
2604:
2530:
2455:harmonic functions
2451:Laplace's equation
2436:
2377:
2299:
2230:
2078:
1921:
1815:
1795:
1760:
1691:
1641:is time, example s
1599:
1542:
1442:
1415:
1342:
1311:
1264:
1245:If, additionally,
1233:
1147:
1074:
1059:
1041:
1024:
1009:
992:
977:
954:
927:
897:
768:
686:chemical potential
657:
600:
509:
380:
287:
206:
136:(heat transport).
128:(hydraulic flow),
91:diffusion equation
68:
9182:Fick's Second Law
9159:Smith WF (2004).
9019:on 24 March 2016.
8935:(51): 9719â9725.
8885:(29): 7254â7262.
8846:10.1063/5.0064140
8792:10.1063/1.1724167
8748:(11): 2400â2414.
8640:(30): 6331â6346.
8569:Physical Review E
8553:978-0-471-57417-0
8529:Combustion Theory
8513:978-0-521-88168-5
8311:"Ueber Diffusion"
8294:978-0-443-13987-1
8157:
8111:{\displaystyle R}
8079:
7974:In ideal gas law
7919:is the thickness
7912:{\displaystyle x}
7880:
7700:
7674:
7604:
7416:
7383:
7168:
6970:
6750:
6681:
6512:{\displaystyle R}
6322:{\displaystyle A}
6302:{\displaystyle a}
6199:{\displaystyle t}
6177:{\displaystyle D}
6095:
6068:
6022:{\displaystyle D}
5971:{\displaystyle A}
5886:
5805:{\displaystyle C}
5743:
5742:
5713:
5697:
5696:
5679:
5678:
5660:
5615:
5614:
5563:
5496:
5495:
5441:
5403:{\displaystyle A}
5375:
5374:
5331:
5222:
5191:
5186:
5185:
5158:
5056:{\displaystyle t}
4961:
4960:
4733:Langevin equation
4721:Langevin equation
4717:Langevin equation
4448:
4392:
4387:
4325:
4283:
4234:
4059:
4006:positive definite
3980:
3813:
3758:
3598:
3556:
3404:
3328:{\displaystyle t}
3303:
3256:{\displaystyle n}
3174:
3126:
3049:
3046:
2822:
2819:
2692:
2652:
2586:
2505:
2347:
2274:
2228:
2185:
2170:
2141:
2118:
2061:
2038:
2020:
1988:
1970:
1940:mass conservation
1911:
1872:
1871:
1818:{\displaystyle D}
1798:{\displaystyle k}
1746:
1609:, example mol/m;
1540:
1499:
1465:Fick's second law
1460:Fick's second law
1397:
1215:
1071:
1053:
1021:
989:
895:
868:
766:
739:
655:
494:
285:
204:
103:Fick's second law
16:(Redirected from
9228:
9164:
9155:
9135:
9122:
9117:Crank J (1980).
9113:
9107:
9096:
9087:
9082:Berg HC (1977).
9068:
9067:
9027:
9021:
9020:
9015:. Archived from
9004:
8998:
8997:
8977:
8971:
8970:
8960:
8920:
8907:
8906:
8897: 11286311.
8874:
8868:
8867:
8857:
8817:
8796:
8795:
8767:
8758:
8757:
8737:
8728:
8727:
8717:
8685:
8676:
8675:
8665:
8625:
8619:
8618:
8584:
8582:cond-mat/0006163
8564:
8558:
8557:
8539:
8533:
8532:
8524:
8518:
8517:
8491:
8485:
8484:
8474:
8468:
8467:
8449:
8426:
8420:
8419:
8411:
8405:
8404:
8402:
8396:. Archived from
8379:
8370:
8364:
8361:
8340:
8338:
8305:
8299:
8298:
8272:
8168:
8166:
8165:
8160:
8158:
8153:
8152:
8151:
8139:
8138:
8128:
8117:
8115:
8114:
8109:
8094:
8092:
8091:
8086:
8084:
8080:
8078:
8067:
8066:
8065:
8053:
8052:
8042:
8036:
8035:
8008:
8006:
8005:
8000:
7969:
7967:
7966:
7961:
7959:
7958:
7938:
7936:
7935:
7930:
7918:
7916:
7915:
7910:
7895:
7893:
7892:
7887:
7885:
7881:
7879:
7871:
7870:
7869:
7856:
7850:
7849:
7817:
7815:
7814:
7809:
7807:
7806:
7788:
7786:
7785:
7780:
7778:
7777:
7759:
7757:
7756:
7751:
7749:
7748:
7729:
7727:
7726:
7721:
7719:
7718:
7714:
7705:
7701:
7699:
7698:
7686:
7679:
7675:
7670:
7669:
7660:
7654:
7653:
7615:
7613:
7612:
7607:
7605:
7603:
7602:
7601:
7597:
7588:
7575:
7567:
7552:
7548:
7544:
7536:
7534:
7533:
7528:
7514:
7512:
7511:
7506:
7492:
7483:
7477:
7475:
7474:
7469:
7467:
7451:is the thickness
7450:
7448:
7447:
7442:
7427:
7425:
7424:
7419:
7417:
7412:
7401:
7396:
7388:
7384:
7382:
7381:
7377:
7368:
7359:
7351:
7244:
7242:
7241:
7236:
7234:
7233:
7212:
7210:
7209:
7204:
7202:
7201:
7179:
7177:
7176:
7171:
7169:
7155:
7147:
7145:
7144:
7132:
7131:
7083:
7074:
7057:
7035:
7025:
7023:
7022:
7017:
7015:
7014:
7010:
7009:
7008:
6996:
6995:
6971:
6968:
6944:
6942:
6941:
6938:
6935:
6931:
6924:
6922:
6921:
6918:
6915:
6911:
6900:
6898:
6897:
6892:
6890:
6889:
6873:
6871:
6870:
6865:
6863:
6862:
6844:
6842:
6841:
6836:
6834:
6833:
6821:
6820:
6808:
6807:
6789:
6787:
6786:
6781:
6779:
6761:
6759:
6758:
6753:
6751:
6749:
6744:
6743:
6742:
6730:
6729:
6719:
6717:
6716:
6707:
6706:
6697:
6696:
6687:
6682:
6674:
6669:
6668:
6636:collision theory
6629:
6627:
6626:
6621:
6619:
6618:
6602:
6600:
6599:
6594:
6592:
6591:
6573:
6571:
6570:
6565:
6563:
6562:
6550:
6549:
6537:
6536:
6518:
6516:
6515:
6510:
6492:
6490:
6489:
6484:
6482:
6481:
6472:
6471:
6462:
6461:
6449:
6438:
6437:
6369:
6367:
6366:
6361:
6356:
6351:
6350:
6328:
6326:
6325:
6320:
6308:
6306:
6305:
6300:
6285:
6283:
6282:
6277:
6271:
6267:
6258:
6237:
6205:
6203:
6202:
6197:
6183:
6181:
6180:
6175:
6161:
6159:
6158:
6153:
6151:
6150:
6134:
6132:
6131:
6126:
6123:
6119:
6107:
6093:
6079:
6077:
6076:
6071:
6069:
6058:
6028:
6026:
6025:
6020:
6006:
6004:
6003:
5998:
5996:
5995:
5977:
5975:
5974:
5969:
5955:
5953:
5952:
5947:
5926:
5924:
5923:
5918:
5912:
5908:
5899:
5887:
5879:
5831:
5829:
5828:
5823:
5811:
5809:
5808:
5803:
5791:
5789:
5788:
5783:
5781:
5780:
5761:
5759:
5758:
5753:
5744:
5732:
5731:
5717:
5714:
5709:
5707:
5698:
5689:
5688:
5680:
5674:
5666:
5665:
5663:
5662:
5661:
5658:
5626:
5624:
5623:
5618:
5616:
5610:
5602:
5601:
5599:
5598:
5580:
5579:
5564:
5562:
5554:
5546:
5540:
5535:
5507:
5505:
5504:
5499:
5497:
5485:
5484:
5483:
5482:
5466:
5458:
5457:
5442:
5440:
5432:
5424:
5410:of the plane is
5409:
5407:
5406:
5401:
5386:
5384:
5383:
5378:
5376:
5364:
5363:
5362:
5353:
5348:
5347:
5332:
5330:
5322:
5314:
5299:
5297:
5296:
5291:
5269:
5267:
5266:
5261:
5243:
5236:
5234:
5233:
5228:
5223:
5221:
5210:
5209:
5200:
5192:
5189:
5187:
5175:
5174:
5173:
5164:
5159:
5157:
5149:
5141:
5129:
5127:
5126:
5121:
5103:
5101:
5100:
5095:
5073:Vincent Schaefer
5062:
5060:
5059:
5054:
5042:
5040:
5039:
5034:
5020:
5014:
5008:
5006:
5005:
5000:
4998:
4997:
4979:
4972:
4970:
4969:
4964:
4962:
4956:
4948:
4947:
4945:
4944:
4834:
4832:
4831:
4826:
4824:
4820:
4819:
4818:
4805:
4775:
4774:
4773:
4723:merges into the
4657:relationship).
4643:glass transition
4588:
4577:
4567:
4545:
4503:
4499:
4495:
4488:
4481:
4479:
4478:
4473:
4468:
4464:
4463:
4462:
4449:
4447:
4446:
4437:
4436:
4427:
4425:
4424:
4400:
4388:
4386:
4378:
4377:
4376:
4363:
4344:
4342:
4341:
4336:
4331:
4327:
4326:
4324:
4323:
4322:
4309:
4286:
4284:
4282:
4281:
4280:
4267:
4257:
4256:
4240:
4235:
4233:
4232:
4231:
4219:
4218:
4205:
4186:
4185:
4175:
4164:
4163:
4145:
4140:
4116:
4115:
4076:
4075:
4060:
4058:
4050:
4027:
4003:
3994:
3992:
3991:
3986:
3981:
3979:
3978:
3977:
3965:
3964:
3951:
3932:
3931:
3921:
3919:
3918:
3905:
3900:
3884:
3879:
3861:
3860:
3830:
3829:
3814:
3812:
3804:
3781:
3772:
3770:
3769:
3764:
3759:
3757:
3756:
3755:
3742:
3734:
3732:
3731:
3718:
3713:
3692:
3691:
3675:
3673:
3672:
3667:
3637:
3609:
3607:
3606:
3601:
3599:
3597:
3596:
3595:
3582:
3559:
3557:
3555:
3554:
3553:
3540:
3523:
3520:
3515:
3461:
3460:
3421:
3420:
3405:
3403:
3395:
3372:
3363:
3334:
3332:
3331:
3326:
3314:
3312:
3311:
3306:
3304:
3290:
3262:
3260:
3259:
3254:
3242:
3240:
3239:
3234:
3214:
3213:
3204:
3203:
3202:
3189:
3175:
3172:
3147:
3140:
3138:
3137:
3132:
3127:
3105:
3100:
3091:
3076:
3069:
3067:
3066:
3061:
3059:
3055:
3054:
3050:
3048:
3047:
3036:
3027:
3007:
3006:
2959:
2955:
2950:diffusion length
2947:
2946:
2945:
2935:
2926:
2917:
2915:
2914:
2911:
2908:
2901:
2894:
2877:
2870:
2847:
2840:
2838:
2837:
2832:
2827:
2823:
2821:
2820:
2812:
2803:
2791:
2790:
2778:
2774:
2746:
2737:
2730:
2726:
2703:
2701:
2700:
2695:
2693:
2691:
2690:
2689:
2676:
2672:
2671:
2661:
2653:
2651:
2643:
2635:
2623:
2613:
2611:
2610:
2605:
2588:
2587:
2584:
2582:
2562:
2556:
2548:
2539:
2537:
2536:
2531:
2520:
2506:
2504:
2496:
2488:
2445:
2443:
2442:
2437:
2423:
2422:
2403:
2399:
2395:
2386:
2384:
2383:
2378:
2348:
2346:
2338:
2330:
2308:
2306:
2305:
2300:
2292:
2291:
2275:
2273:
2265:
2257:
2239:
2237:
2236:
2231:
2229:
2227:
2226:
2225:
2212:
2208:
2207:
2197:
2186:
2184:
2173:
2171:
2169:
2158:
2150:
2146:
2142:
2140:
2129:
2119:
2117:
2106:
2094:
2087:
2085:
2084:
2079:
2070:
2066:
2062:
2060:
2049:
2039:
2037:
2026:
2021:
2019:
2011:
2003:
1989:
1987:
1976:
1971:
1969:
1961:
1953:
1930:
1928:
1927:
1922:
1917:
1913:
1912:
1910:
1899:
1898:
1889:
1873:
1858:
1854:
1824:
1822:
1821:
1816:
1804:
1802:
1801:
1796:
1769:
1767:
1766:
1761:
1747:
1745:
1737:
1729:
1717:
1706:
1700:
1698:
1697:
1694:{\displaystyle }
1692:
1687:
1686:
1678:
1677:
1670:
1669:
1664:
1663:
1646:
1640:
1635:
1631:
1627:
1608:
1606:
1605:
1602:{\displaystyle }
1600:
1595:
1594:
1586:
1585:
1578:
1577:
1561:
1551:
1549:
1548:
1543:
1541:
1539:
1538:
1537:
1524:
1520:
1519:
1509:
1500:
1498:
1490:
1482:
1455:
1451:
1449:
1448:
1443:
1441:
1440:
1424:
1422:
1421:
1416:
1411:
1410:
1398:
1396:
1395:
1386:
1378:
1370:
1369:
1368:
1351:
1349:
1348:
1343:
1320:
1318:
1317:
1312:
1292:
1291:
1290:
1273:
1271:
1270:
1265:
1242:
1240:
1239:
1234:
1229:
1228:
1216:
1214:
1213:
1204:
1196:
1188:
1187:
1186:
1169:
1165:
1156:
1154:
1153:
1148:
1143:
1142:
1115:
1114:
1113:
1083:
1081:
1080:
1075:
1072:
1069:
1067:
1054:
1051:
1049:
1033:
1031:
1030:
1025:
1022:
1019:
1017:
1001:
999:
998:
993:
990:
987:
985:
969:
963:
961:
960:
955:
953:
952:
936:
934:
933:
928:
926:
925:
906:
904:
903:
898:
896:
894:
886:
885:
884:
871:
869:
867:
856:
848:
847:
820:
814:
804:
798:
792:
788:
777:
775:
774:
769:
767:
765:
757:
756:
755:
742:
740:
738:
730:
729:
728:
715:
707:
706:
680:
676:
666:
664:
663:
658:
656:
651:
650:
638:
633:
632:
609:
607:
606:
601:
581:
575:
567:
554:
546:
533:
529:
518:
516:
515:
510:
508:
507:
495:
493:
492:
483:
475:
467:
466:
461:
445:
426:
425:
423:
422:
416:
413:
398:
389:
387:
386:
381:
364:
341:
334:
332:
318:
311:
305:
296:
294:
293:
288:
286:
284:
276:
268:
247:
241:
225:
215:
213:
212:
207:
205:
203:
195:
187:
157:Fick's first law
152:Fick's first law
97:Fick's first law
88:
21:
9236:
9235:
9231:
9230:
9229:
9227:
9226:
9225:
9191:
9190:
9172:
9167:
9136:â reprinted in
9077:
9075:Further reading
9072:
9071:
9028:
9024:
9005:
9001:
8978:
8974:
8921:
8910:
8875:
8871:
8818:
8799:
8768:
8761:
8738:
8731:
8686:
8679:
8626:
8622:
8565:
8561:
8554:
8540:
8536:
8525:
8521:
8514:
8492:
8488:
8475:
8471:
8427:
8423:
8412:
8408:
8400:
8377:
8371:
8367:
8309:Fick A (1855).
8306:
8302:
8295:
8273:
8269:
8264:
8231:False diffusion
8212:
8200:
8183:
8147:
8143:
8134:
8130:
8129:
8127:
8125:
8122:
8121:
8103:
8100:
8099:
8068:
8061:
8057:
8048:
8044:
8043:
8041:
8037:
8031:
8027:
8016:
8013:
8012:
7979:
7976:
7975:
7954:
7950:
7945:
7942:
7941:
7924:
7921:
7920:
7904:
7901:
7900:
7872:
7865:
7861:
7857:
7855:
7851:
7845:
7841:
7830:
7827:
7826:
7802:
7798:
7796:
7793:
7792:
7773:
7769:
7767:
7764:
7763:
7744:
7740:
7738:
7735:
7734:
7710:
7706:
7694:
7690:
7685:
7681:
7680:
7665:
7661:
7659:
7655:
7649:
7645:
7637:
7634:
7633:
7593:
7589:
7581:
7580:
7576:
7568:
7566:
7558:
7555:
7554:
7550:
7546:
7542:
7541:Integrated the
7522:
7519:
7518:
7500:
7497:
7496:
7487:
7481:
7460:
7458:
7455:
7454:
7436:
7433:
7432:
7402:
7400:
7389:
7373:
7369:
7361:
7360:
7352:
7350:
7346:
7329:
7326:
7325:
7314:
7288:
7276:
7269:
7261:is time unit s.
7226:
7222:
7220:
7217:
7216:
7197:
7193:
7191:
7188:
7187:
7151:
7146:
7137:
7133:
7127:
7123:
7112:
7109:
7108:
7082:
7076:
7073:
7067:
7056:
7049:
7043:
7033:
7004:
7000:
6991:
6987:
6986:
6982:
6975:
6967:
6965:
6962:
6961:
6955:
6939:
6936:
6933:
6932:
6929:
6927:
6919:
6916:
6913:
6912:
6909:
6907:
6885:
6881:
6879:
6876:
6875:
6858:
6854:
6852:
6849:
6848:
6829:
6825:
6816:
6812:
6803:
6799:
6797:
6794:
6793:
6775:
6773:
6770:
6769:
6745:
6738:
6734:
6725:
6721:
6720:
6718:
6712:
6708:
6702:
6698:
6692:
6688:
6683:
6673:
6661:
6657:
6655:
6652:
6651:
6614:
6610:
6608:
6605:
6604:
6587:
6583:
6581:
6578:
6577:
6558:
6554:
6545:
6541:
6532:
6528:
6526:
6523:
6522:
6504:
6501:
6500:
6477:
6473:
6467:
6463:
6457:
6453:
6445:
6430:
6426:
6424:
6421:
6420:
6413:Brownian motion
6352:
6346:
6342:
6334:
6331:
6330:
6314:
6311:
6310:
6294:
6291:
6290:
6263:
6259:
6254:
6233:
6219:
6216:
6215:
6191:
6188:
6187:
6169:
6166:
6165:
6146:
6142:
6140:
6137:
6136:
6115:
6108:
6103:
6088:
6085:
6084:
6057:
6049:
6046:
6045:
6014:
6011:
6010:
5991:
5987:
5985:
5982:
5981:
5963:
5960:
5959:
5935:
5932:
5931:
5904:
5900:
5895:
5878:
5864:
5861:
5860:
5817:
5814:
5813:
5797:
5794:
5793:
5776:
5772:
5770:
5767:
5766:
5718:
5716:
5708:
5703:
5687:
5667:
5664:
5657:
5653:
5652:
5638:
5635:
5634:
5603:
5600:
5594:
5590:
5569:
5565:
5555:
5547:
5545:
5536:
5531:
5519:
5516:
5515:
5478:
5474:
5467:
5465:
5447:
5443:
5433:
5425:
5423:
5418:
5415:
5414:
5395:
5392:
5391:
5358:
5354:
5352:
5337:
5333:
5323:
5315:
5313:
5308:
5305:
5304:
5279:
5276:
5275:
5249:
5246:
5245:
5241:
5211:
5205:
5201:
5199:
5188:
5169:
5165:
5163:
5150:
5142:
5140:
5138:
5135:
5134:
5109:
5106:
5105:
5083:
5080:
5079:
5069:Irving Langmuir
5048:
5045:
5044:
5028:
5025:
5024:
5018:
5012:
4993:
4989:
4987:
4984:
4983:
4977:
4949:
4946:
4940:
4936:
4922:
4919:
4918:
4847:
4801:
4794:
4790:
4783:
4779:
4769:
4768:
4764:
4743:
4740:
4739:
4693:
4663:
4611:pharmaceuticals
4595:
4579:
4569:
4566:
4554:
4543:
4536:
4531:
4524:
4519:
4513:
4501:
4497:
4494:
4490:
4487:
4483:
4458:
4454:
4442:
4438:
4432:
4428:
4426:
4417:
4413:
4412:
4408:
4396:
4379:
4372:
4368:
4364:
4362:
4360:
4357:
4356:
4318:
4314:
4310:
4287:
4285:
4276:
4272:
4268:
4249:
4245:
4241:
4239:
4227:
4223:
4214:
4210:
4206:
4181:
4177:
4176:
4174:
4156:
4152:
4151:
4147:
4141:
4124:
4111:
4110:
4071:
4070:
4051:
4028:
4026:
4024:
4021:
4020:
4001:
3996:
3973:
3969:
3960:
3956:
3952:
3927:
3923:
3922:
3920:
3911:
3907:
3901:
3890:
3880:
3869:
3856:
3855:
3825:
3824:
3805:
3782:
3780:
3778:
3775:
3774:
3751:
3747:
3743:
3735:
3733:
3724:
3720:
3714:
3703:
3687:
3683:
3681:
3678:
3677:
3643:
3640:
3639:
3635:
3628:
3623:
3591:
3587:
3583:
3560:
3558:
3549:
3545:
3541:
3524:
3522:
3516:
3505:
3456:
3455:
3416:
3415:
3396:
3373:
3371:
3369:
3366:
3365:
3350:
3342:
3340:Generalizations
3320:
3317:
3316:
3289:
3287:
3284:
3283:
3273:eukaryotic cell
3248:
3245:
3244:
3209:
3205:
3198:
3194:
3193:
3185:
3171:
3169:
3166:
3165:
3158:Brownian motion
3154:
3145:
3101:
3096:
3090:
3085:
3082:
3081:
3074:
3035:
3031:
3026:
3022:
3012:
3008:
3002:
2998:
2975:
2972:
2971:
2957:
2953:
2941:
2939:
2937:
2934:
2928:
2925:
2919:
2912:
2909:
2906:
2905:
2903:
2896:
2893:
2879:
2872:
2861:
2845:
2811:
2807:
2802:
2798:
2786:
2782:
2764:
2760:
2755:
2752:
2751:
2745:
2739:
2732:
2728:
2724:
2721:
2715:is the result.
2685:
2681:
2677:
2667:
2663:
2662:
2660:
2644:
2636:
2634:
2632:
2629:
2628:
2618:
2583:
2578:
2577:
2575:
2572:
2571:
2558:
2554:
2544:
2516:
2497:
2489:
2487:
2485:
2482:
2481:
2463:
2418:
2414:
2412:
2409:
2408:
2401:
2397:
2391:
2339:
2331:
2329:
2327:
2324:
2323:
2287:
2283:
2266:
2258:
2256:
2254:
2251:
2250:
2221:
2217:
2213:
2203:
2199:
2198:
2196:
2177:
2172:
2162:
2157:
2133:
2128:
2124:
2120:
2110:
2105:
2103:
2100:
2099:
2092:
2053:
2048:
2044:
2040:
2030:
2025:
2012:
2004:
2002:
1980:
1975:
1962:
1954:
1952:
1950:
1947:
1946:
1936:
1900:
1894:
1890:
1888:
1884:
1880:
1853:
1830:
1827:
1826:
1810:
1807:
1806:
1790:
1787:
1786:
1738:
1730:
1728:
1726:
1723:
1722:
1715:
1704:
1679:
1673:
1672:
1671:
1665:
1659:
1658:
1657:
1652:
1649:
1648:
1644:
1638:
1633:
1629:
1610:
1587:
1581:
1580:
1579:
1573:
1572:
1567:
1564:
1563:
1559:
1533:
1529:
1525:
1515:
1511:
1510:
1508:
1491:
1483:
1481:
1479:
1476:
1475:
1462:
1453:
1436:
1432:
1430:
1427:
1426:
1406:
1402:
1391:
1387:
1379:
1377:
1364:
1360:
1359:
1357:
1354:
1353:
1328:
1325:
1324:
1286:
1282:
1281:
1279:
1276:
1275:
1250:
1247:
1246:
1224:
1220:
1209:
1205:
1197:
1195:
1182:
1178:
1177:
1175:
1172:
1171:
1167:
1163:
1158:
1138:
1134:
1109:
1105:
1104:
1102:
1099:
1098:
1090:
1068:
1063:
1050:
1045:
1039:
1036:
1035:
1018:
1013:
1007:
1004:
1003:
986:
981:
975:
972:
971:
965:
948:
944:
942:
939:
938:
921:
917:
915:
912:
911:
887:
880:
876:
872:
870:
860:
855:
843:
839:
837:
834:
833:
818:
812:
802:
796:
790:
786:
758:
751:
747:
743:
741:
731:
724:
720:
716:
714:
702:
698:
696:
693:
692:
678:
675:
671:
643:
639:
637:
628:
624:
622:
619:
618:
610:is outside the
595:
592:
591:
582:is the mixture
579:
576:th species, and
573:
566:
558:
552:
545:
537:
531:
527:
503:
499:
488:
484:
476:
474:
462:
457:
456:
454:
451:
450:
444:
440:
433:
417:
414:
408:
407:
405:
403:
394:
360:
358:
355:
354:
339:
330:
328:
316:
309:
303:
277:
269:
267:
265:
262:
261:
245:
239:
230:, of which the
223:
196:
188:
186:
172:
169:
168:
154:
118:
86:
39:
28:
23:
22:
15:
12:
11:
5:
9234:
9224:
9223:
9218:
9213:
9208:
9203:
9189:
9188:
9179:
9171:
9170:External links
9168:
9166:
9165:
9156:
9123:
9114:
9097:
9088:
9078:
9076:
9073:
9070:
9069:
9042:(3): 192â197.
9022:
8999:
8972:
8908:
8869:
8797:
8778:(7): 453â461.
8759:
8729:
8677:
8620:
8559:
8552:
8534:
8519:
8512:
8498:. p. 43.
8486:
8469:
8440:(5): 184â262.
8421:
8406:
8365:
8363:
8362:
8300:
8293:
8266:
8265:
8263:
8260:
8259:
8258:
8253:
8248:
8243:
8238:
8233:
8228:
8223:
8218:
8211:
8208:
8199:
8196:
8182:
8179:
8171:
8170:
8156:
8150:
8146:
8142:
8137:
8133:
8119:
8107:
8083:
8077:
8074:
8071:
8064:
8060:
8056:
8051:
8047:
8040:
8034:
8030:
8026:
8023:
8020:
7998:
7995:
7992:
7989:
7986:
7983:
7972:
7971:
7957:
7953:
7949:
7939:
7928:
7908:
7884:
7878:
7875:
7868:
7864:
7860:
7854:
7848:
7844:
7840:
7837:
7834:
7820:
7819:
7805:
7801:
7790:
7776:
7772:
7761:
7747:
7743:
7717:
7713:
7709:
7704:
7697:
7693:
7689:
7684:
7678:
7673:
7668:
7664:
7658:
7652:
7648:
7644:
7641:
7600:
7596:
7592:
7587:
7584:
7579:
7574:
7571:
7565:
7562:
7539:
7538:
7526:
7516:
7504:
7494:
7493:at any surface
7485:
7479:
7466:
7463:
7452:
7440:
7415:
7411:
7408:
7405:
7399:
7395:
7392:
7387:
7380:
7376:
7372:
7367:
7364:
7358:
7355:
7349:
7345:
7342:
7339:
7336:
7333:
7313:
7310:
7287:
7284:
7279:
7278:
7274:
7267:
7262:
7256:
7250:
7247:chromatography
7232:
7229:
7225:
7214:
7200:
7196:
7181:
7180:
7167:
7164:
7161:
7158:
7154:
7150:
7143:
7140:
7136:
7130:
7126:
7122:
7119:
7116:
7086:
7085:
7080:
7071:
7054:
7047:
7041:
7027:
7026:
7013:
7007:
7003:
6999:
6994:
6990:
6985:
6981:
6978:
6974:
6954:
6951:
6903:
6902:
6888:
6884:
6861:
6857:
6846:
6832:
6828:
6824:
6819:
6815:
6811:
6806:
6802:
6791:
6778:
6763:
6762:
6748:
6741:
6737:
6733:
6728:
6724:
6715:
6711:
6705:
6701:
6695:
6691:
6686:
6680:
6677:
6672:
6667:
6664:
6660:
6632:
6631:
6617:
6613:
6590:
6586:
6575:
6561:
6557:
6553:
6548:
6544:
6540:
6535:
6531:
6520:
6508:
6494:
6493:
6480:
6476:
6470:
6466:
6460:
6456:
6452:
6448:
6444:
6441:
6436:
6433:
6429:
6373:
6372:
6359:
6355:
6349:
6345:
6341:
6338:
6318:
6298:
6287:
6286:
6275:
6270:
6266:
6262:
6257:
6253:
6249:
6246:
6243:
6240:
6236:
6232:
6229:
6226:
6223:
6208:
6207:
6195:
6185:
6173:
6163:
6149:
6145:
6122:
6118:
6114:
6111:
6106:
6102:
6098:
6092:
6081:
6080:
6067:
6064:
6061:
6056:
6053:
6034:
6033:
6030:
6018:
6008:
5994:
5990:
5979:
5967:
5957:
5945:
5942:
5939:
5928:
5927:
5916:
5911:
5907:
5903:
5898:
5894:
5890:
5885:
5882:
5877:
5874:
5871:
5868:
5821:
5801:
5779:
5775:
5763:
5762:
5751:
5748:
5741:
5738:
5735:
5730:
5727:
5724:
5721:
5712:
5706:
5702:
5695:
5692:
5686:
5683:
5677:
5673:
5670:
5656:
5651:
5648:
5645:
5642:
5628:
5627:
5613:
5609:
5606:
5597:
5593:
5589:
5586:
5583:
5578:
5575:
5572:
5568:
5561:
5558:
5553:
5550:
5544:
5539:
5534:
5530:
5526:
5523:
5509:
5508:
5494:
5491:
5488:
5481:
5477:
5473:
5470:
5464:
5461:
5456:
5453:
5450:
5446:
5439:
5436:
5431:
5428:
5422:
5399:
5388:
5387:
5373:
5370:
5367:
5361:
5357:
5351:
5346:
5343:
5340:
5336:
5329:
5326:
5321:
5318:
5312:
5289:
5286:
5283:
5272:
5271:
5259:
5256:
5253:
5238:
5237:
5226:
5220:
5217:
5214:
5208:
5204:
5198:
5195:
5184:
5181:
5178:
5172:
5168:
5162:
5156:
5153:
5148:
5145:
5119:
5116:
5113:
5093:
5090:
5087:
5065:
5064:
5052:
5032:
5022:
5016:
5010:
4996:
4992:
4981:
4974:
4973:
4959:
4955:
4952:
4943:
4939:
4935:
4932:
4929:
4926:
4883:
4882:
4876:
4870:
4861:
4852:
4845:
4836:
4835:
4823:
4817:
4814:
4811:
4808:
4804:
4800:
4797:
4793:
4789:
4786:
4782:
4778:
4772:
4767:
4762:
4759:
4756:
4753:
4750:
4747:
4692:
4689:
4662:
4659:
4626:plasma physics
4594:
4591:
4558:
4541:
4534:
4527:
4522:
4515:
4506:
4505:
4492:
4485:
4471:
4467:
4461:
4457:
4452:
4445:
4441:
4435:
4431:
4423:
4420:
4416:
4411:
4407:
4404:
4399:
4395:
4391:
4385:
4382:
4375:
4371:
4367:
4345:
4334:
4330:
4321:
4317:
4313:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4279:
4275:
4271:
4266:
4263:
4260:
4255:
4252:
4248:
4244:
4238:
4230:
4226:
4222:
4217:
4213:
4209:
4204:
4201:
4198:
4195:
4192:
4189:
4184:
4180:
4173:
4170:
4167:
4162:
4159:
4155:
4150:
4144:
4139:
4136:
4133:
4130:
4127:
4123:
4119:
4114:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4074:
4069:
4066:
4063:
4057:
4054:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4013:
3999:
3984:
3976:
3972:
3968:
3963:
3959:
3955:
3950:
3947:
3944:
3941:
3938:
3935:
3930:
3926:
3917:
3914:
3910:
3904:
3899:
3896:
3893:
3889:
3883:
3878:
3875:
3872:
3868:
3864:
3859:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3828:
3823:
3820:
3817:
3811:
3808:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3762:
3754:
3750:
3746:
3741:
3738:
3730:
3727:
3723:
3717:
3712:
3709:
3706:
3702:
3698:
3695:
3690:
3686:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3633:
3626:
3610:
3594:
3590:
3586:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3552:
3548:
3544:
3539:
3536:
3533:
3530:
3527:
3519:
3514:
3511:
3508:
3504:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3459:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3419:
3414:
3411:
3408:
3402:
3399:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3341:
3338:
3324:
3302:
3299:
3296:
3293:
3252:
3232:
3229:
3226:
3223:
3220:
3217:
3212:
3208:
3201:
3197:
3192:
3188:
3184:
3181:
3178:
3153:
3150:
3142:
3141:
3130:
3125:
3122:
3118:
3115:
3112:
3109:
3104:
3099:
3095:
3089:
3071:
3070:
3058:
3053:
3045:
3042:
3039:
3034:
3030:
3025:
3021:
3018:
3015:
3011:
3005:
3001:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2960:(Bird, 1976).
2948:is called the
2932:
2923:
2891:
2878:and that with
2850:error function
2842:
2841:
2830:
2826:
2818:
2815:
2810:
2806:
2801:
2797:
2794:
2789:
2785:
2781:
2777:
2773:
2770:
2767:
2763:
2759:
2743:
2720:
2717:
2709:advective flux
2705:
2704:
2688:
2684:
2680:
2675:
2670:
2666:
2659:
2656:
2650:
2647:
2642:
2639:
2615:
2614:
2603:
2600:
2597:
2594:
2591:
2581:
2565:diffusive flux
2541:
2540:
2529:
2526:
2523:
2519:
2515:
2512:
2509:
2503:
2500:
2495:
2492:
2471:advective flux
2462:
2459:
2447:
2446:
2435:
2432:
2429:
2426:
2421:
2417:
2388:
2387:
2376:
2373:
2370:
2367:
2363:
2360:
2357:
2354:
2351:
2345:
2342:
2337:
2334:
2310:
2309:
2298:
2295:
2290:
2286:
2281:
2278:
2272:
2269:
2264:
2261:
2241:
2240:
2224:
2220:
2216:
2211:
2206:
2202:
2195:
2192:
2189:
2183:
2180:
2176:
2168:
2165:
2161:
2156:
2153:
2149:
2145:
2139:
2136:
2132:
2127:
2123:
2116:
2113:
2109:
2089:
2088:
2077:
2074:
2069:
2065:
2059:
2056:
2052:
2047:
2043:
2036:
2033:
2029:
2024:
2018:
2015:
2010:
2007:
2001:
1998:
1995:
1992:
1986:
1983:
1979:
1974:
1968:
1965:
1960:
1957:
1935:
1932:
1920:
1916:
1909:
1906:
1903:
1897:
1893:
1887:
1883:
1879:
1876:
1870:
1867:
1864:
1861:
1857:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1814:
1794:
1771:
1770:
1759:
1756:
1753:
1750:
1744:
1741:
1736:
1733:
1709:
1708:
1702:
1690:
1685:
1682:
1676:
1668:
1662:
1656:
1642:
1636:
1598:
1593:
1590:
1584:
1576:
1571:
1553:
1552:
1536:
1532:
1528:
1523:
1518:
1514:
1506:
1503:
1497:
1494:
1489:
1486:
1461:
1458:
1439:
1435:
1414:
1409:
1405:
1401:
1394:
1390:
1385:
1382:
1376:
1373:
1367:
1363:
1341:
1338:
1335:
1332:
1310:
1307:
1304:
1301:
1298:
1295:
1289:
1285:
1263:
1260:
1257:
1254:
1232:
1227:
1223:
1219:
1212:
1208:
1203:
1200:
1194:
1191:
1185:
1181:
1161:
1146:
1141:
1137:
1133:
1130:
1127:
1124:
1121:
1118:
1112:
1108:
1089:
1086:
1066:
1062:
1058:
1048:
1044:
1016:
1012:
984:
980:
951:
947:
937:has Pa units.
924:
920:
908:
907:
893:
890:
883:
879:
875:
866:
863:
859:
854:
851:
846:
842:
823:
822:
816:
810:
800:
794:
779:
778:
764:
761:
754:
750:
746:
737:
734:
727:
723:
719:
713:
710:
705:
701:
673:
668:
667:
654:
649:
646:
642:
636:
631:
627:
599:
588:
587:
577:
562:
556:
541:
535:
520:
519:
506:
502:
498:
491:
487:
482:
479:
473:
470:
465:
460:
442:
432:
429:
391:
390:
379:
376:
373:
370:
367:
363:
314:
313:
307:
301:
283:
280:
275:
272:
259:
243:
228:diffusion flux
217:
216:
202:
199:
194:
191:
185:
182:
179:
176:
153:
150:
117:
114:
36:Fick principle
32:cardiac output
26:
9:
6:
4:
3:
2:
9233:
9222:
9219:
9217:
9214:
9212:
9209:
9207:
9204:
9202:
9199:
9198:
9196:
9187:
9183:
9180:
9177:
9174:
9173:
9162:
9157:
9153:
9149:
9145:
9141:
9133:
9129:
9124:
9120:
9115:
9111:
9106:
9105:
9098:
9094:
9089:
9085:
9080:
9079:
9065:
9061:
9057:
9053:
9049:
9045:
9041:
9037:
9033:
9026:
9018:
9014:
9010:
9003:
8995:
8991:
8987:
8984:(in German).
8983:
8976:
8968:
8964:
8959:
8954:
8950:
8946:
8942:
8938:
8934:
8930:
8926:
8919:
8917:
8915:
8913:
8904:
8900:
8896:
8892:
8888:
8884:
8880:
8873:
8865:
8861:
8856:
8851:
8847:
8843:
8839:
8835:
8832:(1): 015318.
8831:
8827:
8823:
8816:
8814:
8812:
8810:
8808:
8806:
8804:
8802:
8793:
8789:
8785:
8781:
8777:
8773:
8766:
8764:
8755:
8751:
8747:
8743:
8736:
8734:
8725:
8721:
8716:
8711:
8707:
8703:
8700:: 2296â2306.
8699:
8695:
8691:
8684:
8682:
8673:
8669:
8664:
8659:
8655:
8651:
8647:
8643:
8639:
8635:
8631:
8624:
8616:
8612:
8608:
8604:
8600:
8596:
8592:
8588:
8583:
8578:
8574:
8570:
8563:
8555:
8549:
8545:
8538:
8530:
8523:
8515:
8509:
8505:
8501:
8497:
8490:
8482:
8481:
8473:
8465:
8461:
8457:
8453:
8448:
8443:
8439:
8435:
8431:
8425:
8417:
8410:
8399:
8395:
8391:
8387:
8383:
8376:
8369:
8359:
8355:
8352:(63): 30â39.
8351:
8347:
8342:
8341:
8337:
8332:
8328:
8324:
8320:
8317:(in German).
8316:
8312:
8304:
8296:
8290:
8286:
8282:
8278:
8271:
8267:
8257:
8254:
8252:
8249:
8247:
8244:
8242:
8239:
8237:
8234:
8232:
8229:
8227:
8224:
8222:
8219:
8217:
8214:
8213:
8207:
8204:
8195:
8191:
8189:
8178:
8174:
8154:
8148:
8144:
8140:
8135:
8131:
8120:
8105:
8098:
8097:
8096:
8081:
8075:
8072:
8069:
8062:
8058:
8054:
8049:
8045:
8038:
8032:
8028:
8024:
8021:
8018:
8010:
7996:
7993:
7990:
7987:
7984:
7981:
7955:
7951:
7947:
7940:
7926:
7906:
7899:
7898:
7897:
7882:
7876:
7873:
7866:
7862:
7858:
7852:
7846:
7842:
7838:
7835:
7832:
7823:
7803:
7799:
7791:
7774:
7770:
7762:
7745:
7741:
7733:
7732:
7731:
7715:
7711:
7707:
7702:
7695:
7691:
7687:
7682:
7676:
7671:
7666:
7662:
7656:
7650:
7646:
7642:
7639:
7631:
7629:
7625:
7620:
7616:
7598:
7594:
7590:
7577:
7572:
7569:
7563:
7560:
7524:
7517:
7502:
7495:
7490:
7486:
7480:
7453:
7438:
7431:
7430:
7429:
7413:
7409:
7406:
7403:
7397:
7385:
7378:
7374:
7370:
7356:
7353:
7347:
7343:
7337:
7331:
7322:
7318:
7309:
7307:
7303:
7300:
7296:
7293:
7292:semiconductor
7283:
7273:
7266:
7263:
7260:
7257:
7254:
7251:
7248:
7230:
7227:
7223:
7215:
7198:
7194:
7186:
7185:
7184:
7162:
7159:
7152:
7148:
7141:
7138:
7134:
7128:
7124:
7120:
7117:
7114:
7107:
7106:
7105:
7101:
7099:
7094:
7092:
7079:
7070:
7065:
7061:
7060:concentration
7053:
7046:
7042:
7039:
7032:
7031:
7030:
7011:
7005:
7001:
6997:
6992:
6988:
6983:
6979:
6976:
6972:
6960:
6959:
6958:
6950:
6946:
6886:
6882:
6859:
6855:
6847:
6830:
6826:
6822:
6817:
6813:
6809:
6804:
6800:
6792:
6776:
6768:
6767:
6766:
6746:
6739:
6735:
6731:
6726:
6722:
6713:
6709:
6703:
6699:
6693:
6689:
6684:
6678:
6675:
6670:
6665:
6662:
6658:
6650:
6649:
6648:
6644:
6640:
6637:
6615:
6611:
6588:
6584:
6576:
6559:
6555:
6551:
6546:
6542:
6538:
6533:
6529:
6521:
6506:
6499:
6498:
6497:
6478:
6474:
6468:
6464:
6458:
6454:
6450:
6446:
6442:
6439:
6434:
6431:
6427:
6419:
6418:
6417:
6414:
6410:
6406:
6397:
6393:
6391:
6386:
6385:self-assembly
6381:
6378:
6357:
6353:
6347:
6343:
6339:
6336:
6316:
6296:
6289:
6288:
6273:
6268:
6264:
6260:
6255:
6251:
6247:
6244:
6241:
6238:
6234:
6230:
6227:
6224:
6221:
6214:
6213:
6212:
6193:
6186:
6171:
6164:
6147:
6143:
6120:
6116:
6112:
6109:
6104:
6100:
6096:
6090:
6083:
6082:
6065:
6062:
6059:
6054:
6051:
6044:
6043:
6042:
6038:
6031:
6016:
6009:
5992:
5988:
5980:
5965:
5958:
5940:
5930:
5929:
5914:
5909:
5905:
5901:
5896:
5892:
5888:
5883:
5880:
5875:
5869:
5859:
5858:
5857:
5853:
5850:
5841:
5837:
5833:
5819:
5799:
5777:
5773:
5749:
5746:
5739:
5736:
5733:
5725:
5719:
5710:
5704:
5700:
5693:
5690:
5684:
5681:
5675:
5671:
5668:
5654:
5649:
5646:
5643:
5633:
5632:
5631:
5611:
5607:
5604:
5595:
5591:
5587:
5584:
5581:
5576:
5573:
5570:
5559:
5537:
5532:
5528:
5524:
5514:
5513:
5512:
5492:
5489:
5486:
5479:
5475:
5471:
5468:
5462:
5459:
5454:
5451:
5448:
5437:
5413:
5412:
5411:
5397:
5371:
5368:
5365:
5359:
5355:
5349:
5344:
5341:
5338:
5327:
5319:
5303:
5302:
5301:
5287:
5284:
5281:
5257:
5254:
5251:
5240:
5239:
5218:
5215:
5212:
5206:
5202:
5196:
5182:
5179:
5176:
5170:
5166:
5160:
5154:
5146:
5133:
5132:
5131:
5117:
5114:
5111:
5091:
5088:
5085:
5076:
5074:
5070:
5050:
5023:
5017:
5011:
4994:
4990:
4982:
4976:
4975:
4957:
4953:
4950:
4941:
4937:
4933:
4930:
4927:
4917:
4916:
4915:
4912:
4908:
4898:
4894:
4892:
4888:
4880:
4877:
4874:
4871:
4869:
4865:
4862:
4860:
4856:
4853:
4851:
4844:
4841:
4840:
4839:
4821:
4812:
4809:
4802:
4798:
4795:
4791:
4787:
4784:
4780:
4776:
4765:
4760:
4757:
4751:
4745:
4738:
4737:
4736:
4734:
4731:based on the
4730:
4726:
4722:
4718:
4714:
4710:
4706:
4702:
4697:
4688:
4686:
4685:renormalizing
4682:
4676:
4673:
4668:
4658:
4656:
4652:
4651:mass transfer
4648:
4644:
4640:
4635:
4631:
4627:
4623:
4619:
4616:
4612:
4608:
4604:
4600:
4590:
4586:
4582:
4576:
4572:
4565:
4561:
4557:
4551:
4549:
4544:
4537:
4530:
4525:
4518:
4511:
4469:
4465:
4459:
4455:
4443:
4439:
4433:
4429:
4421:
4418:
4414:
4409:
4405:
4397:
4393:
4389:
4383:
4373:
4369:
4354:
4350:
4346:
4332:
4328:
4319:
4315:
4303:
4300:
4297:
4291:
4277:
4273:
4261:
4253:
4250:
4246:
4236:
4228:
4224:
4215:
4211:
4199:
4196:
4193:
4187:
4182:
4168:
4160:
4157:
4153:
4148:
4142:
4137:
4134:
4131:
4128:
4125:
4121:
4117:
4104:
4101:
4098:
4092:
4083:
4077:
4067:
4061:
4055:
4044:
4041:
4038:
4032:
4018:
4014:
4011:
4007:
4002:
3982:
3974:
3970:
3961:
3957:
3945:
3942:
3939:
3933:
3928:
3915:
3912:
3908:
3902:
3897:
3894:
3891:
3887:
3881:
3876:
3873:
3870:
3866:
3862:
3849:
3846:
3843:
3837:
3831:
3821:
3815:
3809:
3798:
3795:
3792:
3786:
3760:
3752:
3748:
3739:
3728:
3725:
3721:
3715:
3710:
3707:
3704:
3700:
3696:
3693:
3688:
3684:
3663:
3660:
3654:
3651:
3648:
3645:
3636:
3629:
3622:
3618:
3616:
3611:
3592:
3588:
3576:
3573:
3570:
3564:
3550:
3546:
3534:
3528:
3517:
3512:
3509:
3506:
3502:
3498:
3492:
3489:
3486:
3480:
3471:
3465:
3462:
3449:
3446:
3443:
3437:
3428:
3422:
3412:
3406:
3400:
3389:
3386:
3383:
3377:
3361:
3357:
3353:
3348:
3344:
3343:
3337:
3322:
3300:
3297:
3294:
3291:
3280:
3278:
3274:
3270:
3269:cell membrane
3266:
3250:
3230:
3227:
3224:
3221:
3218:
3210:
3190:
3176:
3163:
3159:
3149:
3128:
3123:
3120:
3113:
3107:
3102:
3097:
3093:
3087:
3080:
3079:
3078:
3056:
3051:
3043:
3040:
3037:
3032:
3028:
3023:
3019:
3016:
3013:
3009:
3003:
2999:
2995:
2989:
2986:
2983:
2977:
2970:
2969:
2968:
2967:can be used:
2966:
2965:Taylor series
2961:
2951:
2944:
2931:
2922:
2899:
2890:
2886:
2882:
2875:
2868:
2864:
2859:
2855:
2854:semi-infinite
2851:
2828:
2824:
2816:
2813:
2808:
2804:
2799:
2795:
2792:
2787:
2783:
2779:
2775:
2771:
2768:
2765:
2761:
2757:
2750:
2749:
2748:
2742:
2735:
2716:
2714:
2710:
2686:
2682:
2673:
2668:
2657:
2654:
2648:
2640:
2627:
2626:
2625:
2621:
2601:
2595:
2592:
2589:
2570:
2569:
2568:
2566:
2561:
2552:
2549:is the total
2547:
2527:
2524:
2521:
2513:
2507:
2501:
2493:
2480:
2479:
2478:
2476:
2472:
2468:
2458:
2456:
2452:
2433:
2430:
2427:
2424:
2419:
2407:
2406:
2405:
2394:
2374:
2368:
2361:
2355:
2349:
2343:
2335:
2322:
2321:
2320:
2317:
2315:
2314:heat equation
2296:
2293:
2288:
2279:
2276:
2270:
2262:
2249:
2248:
2247:
2244:
2222:
2218:
2209:
2204:
2193:
2190:
2187:
2181:
2166:
2154:
2151:
2147:
2143:
2137:
2125:
2121:
2114:
2098:
2097:
2096:
2075:
2072:
2067:
2063:
2057:
2045:
2041:
2034:
2022:
2016:
2008:
1996:
1993:
1990:
1984:
1972:
1966:
1958:
1945:
1944:
1943:
1941:
1931:
1918:
1914:
1907:
1904:
1901:
1895:
1891:
1885:
1881:
1877:
1874:
1868:
1865:
1862:
1859:
1855:
1850:
1844:
1841:
1838:
1832:
1812:
1792:
1784:
1780:
1776:
1775:Heat equation
1757:
1751:
1748:
1742:
1734:
1721:
1720:
1719:
1714:
1703:
1701:, example m/s
1683:
1680:
1666:
1643:
1637:
1625:
1621:
1617:
1613:
1591:
1588:
1558:
1557:
1556:
1534:
1530:
1521:
1516:
1504:
1501:
1495:
1487:
1474:
1473:
1472:
1470:
1466:
1457:
1437:
1433:
1412:
1407:
1403:
1392:
1388:
1383:
1380:
1374:
1371:
1339:
1336:
1333:
1321:
1308:
1305:
1299:
1296:
1293:
1261:
1258:
1255:
1243:
1230:
1225:
1221:
1210:
1206:
1201:
1198:
1192:
1189:
1164:
1144:
1139:
1135:
1131:
1128:
1122:
1119:
1116:
1096:
1085:
1064:
1060:
1056:
1046:
1042:
1014:
1010:
982:
978:
968:
949:
945:
922:
918:
891:
881:
877:
864:
861:
857:
852:
849:
844:
840:
832:
831:
830:
828:
817:
811:
808:
801:
795:
784:
783:
782:
762:
752:
748:
735:
732:
725:
721:
717:
711:
708:
703:
699:
691:
690:
689:
687:
682:
652:
647:
644:
640:
634:
629:
625:
617:
616:
615:
613:
597:
585:
578:
571:
565:
561:
557:
550:
544:
540:
536:
525:
524:
523:
504:
500:
489:
485:
480:
477:
471:
468:
463:
449:
448:
447:
438:
437:mass fraction
428:
421:
412:
400:
397:
377:
371:
368:
365:
353:
352:
351:
349:
345:
336:
326:
322:
308:
302:
300:
281:
278:
273:
270:
260:
257:
256:
251:
244:
237:
233:
229:
222:
221:
220:
200:
197:
192:
189:
183:
180:
177:
174:
167:
166:
165:
162:
158:
149:
147:
143:
137:
135:
134:Fourier's law
131:
127:
123:
122:Thomas Graham
113:
111:
106:
104:
100:
98:
94:
92:
84:
80:
76:
72:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
9160:
9143:
9139:
9131:
9127:
9118:
9103:
9092:
9086:. Princeton.
9083:
9039:
9035:
9025:
9017:the original
9012:
9002:
8985:
8981:
8975:
8932:
8928:
8882:
8878:
8872:
8829:
8826:AIP Advances
8825:
8775:
8771:
8745:
8741:
8697:
8693:
8637:
8633:
8623:
8572:
8568:
8562:
8543:
8537:
8528:
8522:
8495:
8489:
8479:
8472:
8437:
8433:
8424:
8415:
8409:
8398:the original
8388:: 1.1â1.10.
8385:
8381:
8368:
8349:
8345:
8321:(1): 59â86.
8318:
8314:
8303:
8276:
8270:
8236:Gas exchange
8205:
8201:
8192:
8184:
8175:
8172:
8011:
7973:
7824:
7821:
7632:
7627:
7623:
7621:
7617:
7540:
7515:is viscosity
7488:
7323:
7319:
7315:
7304:
7301:
7297:
7289:
7280:
7271:
7264:
7258:
7252:
7182:
7102:
7098:Graham's law
7095:
7091:flux limiter
7087:
7077:
7068:
7051:
7044:
7028:
6956:
6947:
6904:
6764:
6645:
6641:
6633:
6495:
6407:proposed by
6402:
6382:
6376:
6374:
6209:
6039:
6035:
5854:
5846:
5834:
5764:
5629:
5510:
5389:
5273:
5077:
5066:
4904:
4890:
4887:biomolecules
4884:
4878:
4872:
4863:
4854:
4842:
4837:
4712:
4708:
4704:
4698:
4694:
4677:
4672:random walks
4664:
4634:voltammetric
4596:
4593:Applications
4584:
4580:
4574:
4570:
4563:
4559:
4555:
4552:
4539:
4532:
4528:
4520:
4516:
4507:
4352:
4016:
3997:
3631:
3624:
3613:
3359:
3355:
3351:
3346:
3281:
3155:
3143:
3072:
2962:
2949:
2942:
2929:
2920:
2918:in front of
2897:
2888:
2884:
2880:
2873:
2866:
2862:
2857:
2853:
2843:
2740:
2733:
2722:
2706:
2619:
2616:
2564:
2559:
2545:
2542:
2464:
2448:
2392:
2389:
2318:
2311:
2245:
2242:
2090:
1937:
1772:
1710:
1623:
1619:
1615:
1611:
1554:
1464:
1463:
1322:
1244:
1159:
1091:
966:
909:
829:difference:
824:
789:denotes the
780:
683:
681:th species.
669:
589:
563:
559:
548:
542:
538:
530:denotes the
521:
434:
419:
410:
401:
395:
392:
337:
315:
298:
253:
249:
227:
218:
156:
155:
145:
141:
138:
119:
107:
102:
101:
96:
95:
70:
69:
63:
59:
55:
40:
8634:Soft Matter
8188:random walk
7537:is density.
7038:conductance
4607:biopolymers
3615:anisotropic
1783:Heat kernel
970:in a vapor
534:th species,
333:10 m/s
255:diffusivity
146:non-Fickian
126:Darcy's law
9195:Categories
9007:Nosek TM.
6371:neighbors.
4911:absorption
4907:adsorption
4901:diffusion.
4601:in foods,
4550:equation.
4004:should be
1002:or liquid
793:th species
785:the index
570:molar mass
526:the index
79:Adolf Fick
9201:Diffusion
9146:: 33â38.
8447:1012.2908
8430:Gorban AN
8262:Citations
8241:Mass flux
8226:Diffusion
8216:Advection
8155:δ
8141:−
8070:δ
8055:−
8025:−
7927:δ
7839:−
7561:δ
7525:ρ
7503:η
7439:δ
7414:η
7407:ρ
7332:δ
7224:η
7160:π
7135:η
7029:in which
6998:−
6977:−
6777:σ
6685:σ
6679:π
6447:π
6340:π
6110:−
5944:⟩
5938:⟨
5884:π
5873:⟩
5867:⟨
5820:τ
5750:τ
5740:τ
5737:−
5726:τ
5701:∫
5694:π
5682:−
5676:π
5641:Γ
5612:π
5557:∂
5552:Γ
5549:∂
5529:∫
5522:Γ
5487:π
5463:−
5435:∂
5430:Γ
5427:∂
5366:π
5325:∂
5317:∂
5197:−
5177:π
5152:∂
5144:∂
5031:Γ
4958:π
4925:Γ
4813:μ
4796:−
4788:−
4761:μ
4681:tautology
4665:When two
4587:= 1, 2, 3
4456:φ
4451:∇
4440:φ
4430:φ
4406:⋅
4403:∇
4394:∑
4381:∂
4370:φ
4366:∂
4312:∂
4292:φ
4289:∂
4270:∂
4243:∂
4221:∂
4208:∂
4188:φ
4179:∂
4122:∑
4093:φ
4090:∇
4068:⋅
4065:∇
4053:∂
4033:φ
4030:∂
3967:∂
3954:∂
3934:φ
3925:∂
3888:∑
3867:∑
3838:φ
3835:∇
3822:⋅
3819:∇
3807:∂
3787:φ
3784:∂
3745:∂
3740:φ
3737:∂
3701:∑
3697:−
3661:φ
3658:∇
3652:−
3585:∂
3565:φ
3562:∂
3543:∂
3526:∂
3503:∑
3481:φ
3478:Δ
3438:φ
3435:∇
3413:⋅
3410:∇
3398:∂
3378:φ
3375:∂
3265:dimension
3216:⟩
3191:−
3180:⟨
3177:≡
3124:τ
3114:τ
3094:∫
3044:π
3017:−
2796:
2679:∂
2674:φ
2665:∂
2646:∂
2641:φ
2638:∂
2602:φ
2599:∇
2593:−
2585:diffusion
2514:⋅
2511:∇
2499:∂
2494:φ
2491:∂
2449:which is
2425:φ
2416:∇
2369:φ
2366:∇
2356:⋅
2353:∇
2341:∂
2336:φ
2333:∂
2294:φ
2285:∇
2268:∂
2263:φ
2260:∂
2215:∂
2210:φ
2201:∂
2188:φ
2179:∂
2175:∂
2164:∂
2160:∂
2144:φ
2135:∂
2131:∂
2112:∂
2108:∂
2064:φ
2055:∂
2051:∂
2032:∂
2028:∂
2023:−
2014:∂
2009:φ
2006:∂
2000:⇒
1982:∂
1978:∂
1964:∂
1959:φ
1956:∂
1886:−
1878:
1863:π
1833:φ
1758:φ
1755:Δ
1740:∂
1735:φ
1732:∂
1713:Laplacian
1681:−
1632:and time
1589:−
1527:∂
1522:φ
1513:∂
1493:∂
1488:φ
1485:∂
1400:∇
1381:ρ
1375:−
1334:ρ
1331:∇
1306:φ
1303:∇
1297:−
1256:ρ
1253:∇
1218:∇
1199:ρ
1193:−
1132:
1126:∇
1120:−
910:Fugacity
889:∂
874:∂
853:−
809:(J/K/mol)
760:∂
749:μ
745:∂
712:−
653:ρ
641:ρ
598:ρ
497:∇
478:ρ
472:−
378:φ
375:∇
369:−
321:viscosity
274:φ
232:dimension
193:φ
181:−
130:Ohm's law
75:diffusion
73:describe
9186:OpenStax
9056:26330037
8967:36520427
8903:39014882
8864:35070490
8724:29181286
8672:27396746
8607:11304296
8464:18961678
8210:See also
7064:membrane
6928:2
6908:2
4881:is time.
4711:, where
4667:miscible
4568:, where
4010:elliptic
2869:, 0) = 0
2858:infinite
1777:and its
827:fugacity
612:gradient
348:gradient
9064:3921833
8990:Bibcode
8958:9805503
8937:Bibcode
8855:8758205
8834:Bibcode
8780:Bibcode
8715:5687005
8663:5476231
8642:Bibcode
8615:1302913
8587:Bibcode
8323:Bibcode
8256:Osmosis
6943:
6923:
6765:where,
6496:where,
5849:fractal
4857:is the
4848:is the
4655:Onsager
4639:polymer
4603:neurons
3263:is the
2940:√
2916:
2904:
2887:, 0) =
805:is the
584:density
572:of the
568:is the
551:of the
547:is the
424:
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329:(0.6â2)
297:is the
248:is the
234:is the
226:is the
116:History
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6094:
5765:where
5270:(#/m).
4628:, and
4615:porous
4482:where
3621:tensor
3277:cactus
3243:where
2876:> 0
2844:where
2711:, the
2543:where
1555:where
1425:where
1157:where
781:where
670:where
522:where
393:where
342:, the
219:where
64:Bottom
60:Middle
52:solute
34:, see
9134:: 59.
9112:â171.
9060:S2CID
8611:S2CID
8577:arXiv
8460:S2CID
8442:arXiv
8401:(PDF)
8378:(PDF)
7545:from
6228:>=
4618:soils
3617:media
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9052:PMID
8963:PMID
8899:PMID
8860:PMID
8720:PMID
8668:PMID
8603:PMID
8548:ISBN
8508:ISBN
8289:ISBN
7290:The
6969:flux
6874:and
6603:and
6222:<
5071:and
4905:The
4508:The
4500:and
4015:For
2846:erfc
2793:erfc
2553:and
2551:flux
590:The
161:flux
9184:on
9148:doi
9144:100
9110:167
9044:doi
8953:PMC
8945:doi
8933:126
8895:PMC
8887:doi
8883:128
8850:PMC
8842:doi
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8750:doi
8710:PMC
8702:doi
8658:PMC
8650:doi
8595:doi
8500:doi
8452:doi
8390:doi
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8331:doi
8281:doi
7549:to
7491:= 0
7075:to
5190:exp
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344:del
252:or
142:not
56:Top
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6671:=
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178:=
175:J
87:D
38:.
20:)
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