Fick's laws of diffusion

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 Thomas Graham, 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: Darcy's law (hydraulic flow), Ohm's law (charge transport), and Fourier's law (heat transport). 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 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 not 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 non-Fickian. == Fick's first law ==

'''Fick's first law''' relates the diffusive flux 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 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: J = -D \frac{d \varphi}{d x}, where • is the diffusion flux, of which the dimension is the amount of substance per unit area per unit time. measures the amount of substance that will flow through a unit area during a unit time interval, • is the diffusion coefficient or diffusivity. Its dimension is area per unit time, • \frac{d \varphi}{d x} is the concentration gradient, • (for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume, • is position, the dimension of which is length. is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes–Einstein relation. The modeling and prediction of Fick's diffusion coefficients is difficult. They can be estimated using the empirical Vignes correlation model or the physically motivated entropy scaling. In dilute aqueous solutions, the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of . For biological molecules the diffusion coefficients normally range from 10−10 to 10−11 m2/s. In two or more dimensions we must use , the del or gradient operator, which generalises the first derivative, obtaining \mathbf{J}=- D\nabla \varphi , where denotes the diffusion flux. The driving force for the one-dimensional diffusion is the quantity , which for ideal mixtures is the concentration gradient. Variations of the first law Another form for the first law is to write it with the primary variable as mass fraction (, given for example in kg/kg), then the equation changes to \mathbf{J}_i = -\frac{\rho D}{M_i}\nabla y_i , where • the index denotes the th species, • is the diffusion flux of the th species (for example in mol/m2/s), • is the molar mass of the th species, • is the mixture density (for example in kg/m3). The \rho is outside the gradient operator. This is because y_i = \frac{\rho_{si}}{\rho} , where is the partial density of the th species. Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for the diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x} , where • the index denotes the th species, • is the concentration (mol/m3), • is the universal gas constant (J/K/mol), • is the absolute temperature (K), • is the chemical potential (J/mol). The driving force of Fick's law can be expressed as a fugacity difference: J_i = - \frac{D}{RT} \frac{\partial f_i}{\partial x} , where f_i is the fugacity in Pa. f_i is a partial pressure of component in a vapor f_i^\text{G} or liquid f_i^\text{L} phase. At vapor liquid equilibrium the evaporation flux is zero because f_i^\text{G} = f_i^\text{L} . Derivation of Fick's first law for gases 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 diffusion equation from the kinetic theory of gases reduces to this version of Fick's law: \mathbf{V_i}=- D \, \nabla \ln y_i , where is the diffusion velocity of species . In terms of species flux this is \mathbf{J_i}=- \frac{\rho D}{M_i}\nabla y_i . If, additionally, \nabla \rho = 0, this reduces to the most common form of Fick's law, \mathbf{J_i}=- D\nabla \varphi . If (instead of or in addition to \nabla \rho = 0) both species have the same molar mass, Fick's law becomes \mathbf{J_i}=- \frac{\rho D}{M_i}\nabla x_i, where x_i is the mole fraction of species . == Fick's second law ==

'''Fick's second law''' predicts how diffusion causes the concentration to change with respect to time. It is a partial differential equation which in one dimension reads \frac{\partial \varphi}{\partial t} = D\,\frac{\partial^2 \varphi}{\partial x^2}, where • is the concentration in dimensions of [\mathsf{N}\mathsf{L}^{-3}], example mol/m3; is a function that depends on location and time , • is time, example s, • is the diffusion coefficient in dimensions of [\mathsf{L}^2\mathsf{T}^{-1}], example m2/s, • is the position, example m. In two or more dimensions we must use the Laplacian , which generalises the second derivative, obtaining the equation \frac{\partial \varphi}{\partial t} = D\Delta \varphi . Fick's second law has the same mathematical form as the Heat equation and its fundamental solution is the same as the Heat kernel, except switching thermal conductivity k with diffusion coefficient D: \varphi(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{x^2}{4Dt}\right). Derivation of Fick's second law -\tfrac{1}{2}\bigl[N(x + \Delta x, t) - N(x, t)\bigr] The flux, , is this net movement of particles across some area element of area , normal to the random walk during a time interval . Hence we may write: J = - \frac{1}{2} \left[\frac{ N(x + \Delta x, t)}{a \Delta t} - \frac{ N(x, t)}{a \Delta t}\right] Multiplying the top and bottom of the right hand side by and rewriting, one obtains: J = -\frac{\left(\Delta x\right)^2}{2 \Delta t}\left[\frac{N(x + \Delta x, t)}{a \left(\Delta x\right)^2} - \frac{N(x, t)}{a \left(\Delta x\right)^2}\right] Concentration is defined as particles per unit volume, and hence \varphi (x, t) = \frac{N(x, t)}{a \Delta x}. In addition, is the definition of the one-dimensional diffusion constant, . Thus our expression simplifies to: J = -D \left[\frac{\varphi (x + \Delta x, t)}{\Delta x} - \frac{\varphi (x , t)}{\Delta x}\right] In the limit where is infinitesimal, the right-hand side becomes a space derivative: J = - D \frac{\partial \varphi}{\partial x} This is only the case for the initial condition of a very of a initial gaussian distribution. Other problem geometries will lead to different solutions --> Fick's second law can be derived from Fick's first law and the mass conservation in absence of any chemical reactions: \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. Assuming the diffusion coefficient to be a constant, one can exchange the orders of the differentiation and multiply by the constant: \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}, and, thus, receive the form of the Fick's equations as was stated above. For the case of diffusion in two or more dimensions Fick's second law becomes \frac{\partial \varphi}{\partial t} = D\,\nabla^2\varphi, which is analogous to the heat equation. If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields \frac{\partial \varphi}{\partial t} = \nabla \cdot (D\,\nabla\varphi). An important example is the case where 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 , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain \nabla^2\varphi = 0, which is Laplace's equation, the solutions to which are referred to by mathematicians as harmonic functions. == Example solutions and generalization ==

Fick's second law is a special case of the convection–diffusion equation in which there is no advective flux and no net volumetric source. It can be derived from the continuity equation: \frac{\partial \varphi}{\partial t} + \nabla\cdot\mathbf{j} = R, where is the total flux and is a net volumetric source for . The only source of flux in this situation is assumed to be diffusive flux: \mathbf{j}_{\text{diffusion}} = -D \nabla \varphi . Plugging the definition of diffusive flux to the continuity equation and assuming there is no source (), we arrive at Fick's second law: \frac{\partial \varphi}{\partial t} = D\frac{\partial^2 \varphi}{\partial x^2} . If flux were the result of both diffusive flux and advective flux, the convection–diffusion equation is the result. Example solution 1: constant concentration source and diffusion length A simple case of diffusion with time in one dimension (taken as the -axis) from a boundary located at position , where the concentration is maintained at a value is n \left(x,t \right)=n_0 \operatorname{erfc} \left( \frac{x}{2\sqrt{Dt}}\right) , where is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space – that is, the corrosion product layer – is semi-infinite, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is infinite (lasting both through the layer with , and that with , ), then the solution is amended only with coefficient in front of (as the diffusion now occurs in both directions). This case is valid when some solution with concentration is put in contact with a layer of pure solvent. (Bokstein, 2005) The length 2\sqrt{Dt} is called the diffusion length and provides a measure of how far the concentration has propagated in the -direction by diffusion in time (Bird, 1976). As a quick approximation of the error function, the first two terms of the Taylor series can be used: n(x,t)=n_0 \left[ 1 - 2 \left(\frac{x}{2\sqrt{Dt\pi}}\right) \right] . If is time-dependent, the diffusion length becomes 2\sqrt{\int_0^t D( \tau ) \,d\tau}. This idea is useful for estimating a diffusion length over a heating and cooling cycle, where varies with temperature. Example solution 2: Brownian particle and mean squared displacement Another simple case of diffusion is the Brownian motion of one particle. The particle's Mean squared displacement from its original position is: \text{MSD} \equiv \left \langle \left(\mathbf{x} - \mathbf{x}_0\right)^2 \right \rangle=2nDt , where n is the dimension of the particle's Brownian motion. For example, the diffusion of a molecule across a cell membrane 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 eukaryotic cell is a 3-D diffusion. For a cylindrical cactus, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion. The square root of MSD, \sqrt{2nDt}, is often used as a characterization of how far the particle has moved after time t 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. Generalizations • In non-homogeneous media, the diffusion coefficient varies in space, . This dependence does not affect Fick's first law but the second law changes: \begin{align} \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}. \end{align} • In anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor . Fick's first law changes to J=-D \nabla \varphi , it is the product of a tensor and a vector: J_i=-\sum_{j=1}^3 D_{ij} \frac{\partial \varphi}{\partial x_j}. For the diffusion equation this formula gives \frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D \nabla \varphi(x,t)\bigr)=\sum_{i=1}^3\sum_{j=1}^3D_{ij} \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}. The symmetric matrix of diffusion coefficients should be positive definite. It is needed to make the right-hand side operator elliptic. • For inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in \begin{align} \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). \end{align} • The approach based on Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components: \frac{\partial \varphi_i}{\partial t} = \sum_j \nabla\cdot\left(D_{ij} \frac{\varphi_i}{\varphi_j} \nabla \, \varphi_j\right) , where are concentrations of the components and is the matrix of coefficients. Here, indices and are related to the various components and not to the space coordinates. The Chapman–Enskog formulae for diffusion in gases include exactly the same terms. These physical models of diffusion are different from the test models which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation. For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example , where refer to the components and correspond to the space coordinates. == Applications ==

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, nuclear materials, plasma physics, and semiconductor doping processes. The theory of voltammetric 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 polymer science and food science a more general approach is required to describe transport of components in materials undergoing a glass transition. One more general framework is the Maxwell–Stefan diffusion equations of multi-component mass transfer, from which Fick's law can be obtained as a limiting case, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell–Stefan equations are used. See also non-diagonal coupled transport processes (Onsager relationship). Fick's flow in liquids When two miscible 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 random walks 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. 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 law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a tautology, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing the fluctuating hydrodynamics equations. Sorption rate and collision frequency of diluted solute 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 when these interactions happen in diluted solutions. Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the Stokes–Einstein equation. In the ultrashort time limit, in the order of the diffusion time a2/D, where a is the particle radius, the diffusion is described by the Langevin equation. At a longer time, the Langevin equation merges into the Stokes–Einstein equation. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the fluctuation-dissipation theorem based on the Langevin equation in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is: D(t) = \mu \, k_{\rm B} T\left(1-e^{-t/(m\mu)}\right) , where (all in SI units) • kB is the Boltzmann constant, • T is the absolute temperature, • μ is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory), • m is the mass of the particle, • t is time. For a single molecule such as organic molecules or biomolecules (e.g. proteins) in water, the exponential term is negligible due to the small product of mμ in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute. The adsorption or absorption 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 the simulated molecular diffusion in the first section of this page: \Gamma= 2AC_b\sqrt{\frac{Dt}{\pi}}. • is the surface area (m2). • C_b is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m3). • is diffusion coefficient of the adsorber (m2/s). • is elapsed time (s). • \Gamma is the accumulated number of molecules in unit # molecules adsorbed during the time t. The equation is named after American chemists Irving Langmuir and Vincent Schaefer. Briefly as explained in, the concentration gradient profile near a newly created (from t=0) absorptive surface (placed at x=0) in a once uniform bulk solution is solved in the above sections from Fick's equation, \frac{\partial C}{\partial x} = \frac{C_b}{\sqrt{\pi Dt}}\text{exp} \left (-\frac{x^2}{4Dt} \right ) , where is the number concentration of adsorber molecules at x, t (#/m3). The concentration gradient at the subsurface at x = 0 is simplified to the pre-exponential factor of the distribution \left (\frac{\partial C}{\partial x} \right ) _{x = 0} = \frac{C_b}{\sqrt{\pi Dt}} . And the rate of diffusion (flux) across area A . of the plane is \left (\frac{\partial \Gamma }{\partial t} \right ) _{x = 0} = -\frac{DAC_b}{\sqrt{\pi Dt}} . Integrating over time, \Gamma = \int_0^t \left( \frac{\partial \Gamma}{\partial t} \right) _{x = 0} = 2AC_b\sqrt{\frac{Dt}{\pi}} . The Langmuir–Schaefer equation can be extended to the Ward–Tordai Equation to account for the "back-diffusion" of rejected molecules from the surface: L = \sqrt{2Dt} , where: • L~=C_b^{-1/3} (unit m) is the average nearest neighbor distance approximated as cubic packing, where C_b is the solute concentration in the bulk solution (unit # molecule / m3), • D is the diffusion coefficient defined by Fick's equation (unit m2/s), • t is the critical time (unit s). 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 actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem, \langle r \rangle = \frac{a}{t} = 2aC_b^{2/3}D , where a 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 A over solute nearest neighbor sphere surface area 4 \pi L^2 /4 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 neighbors. When the area of interest is the size of a molecule (specifically, a long cylindrical molecule 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 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. The above hitting rate equation is also useful to predict the kinetics of molecular self-assembly 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 into the equation one should be able to calculate the theoretical adsorption kinetic curve using the Langmuir adsorption model. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry. and Fick's laws of diffusion. Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target 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/m3: Z_{AB} = 4{\pi}RD_rC_AC_B, where: • R is the radius of the collision, • D_r = D_A + D_B is the relative diffusion constant between A and B (m2/s), • C_A and C_B are number concentrations of A and B respectively (#/m3). The reaction order of this bimolecular reaction is 2 which is the analogy to the result from collision theory 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. However, under a practical condition, the concentration gradient near the target molecule is evolving over time with the molecular flux evolving as well, Thus the concentration gradient evolution stops at the first nearest neighbor 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/m3: e.g., Z_{AB} = 4 VD_rC_AC_B(C_A+C_B)^{2/3}, • V is the volume of the collision 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 , which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules. 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 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. Biological perspective The first law gives rise to the following formula: \text{flux} = {-P \left(c_2 - c_1\right)} , where • is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature, • is the difference in concentration of the gas across the membrane for the direction of flow (from to ). 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 situation, one can use a flux limiter. The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law. 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, while biological molecules are not denser than water. Also, this equation assumes ideal concentration gradient forms near the membrane and evolves): By controlling the concentration gradient, the cooking time, and shape of the food, salting can be controlled. == See also ==