# Fick's laws of diffusion **Fick's laws of diffusion** are the two equations that turn [[Diffusion|diffusion]] from a description into a calculation. The first law says that matter flows down a concentration gradient in proportion to it, `J = −D·(dC/dx)`, where `J` is the flux in moles (or kilograms) per square metre per second, `C` the [[Molar_concentration|concentration]], `x` the distance and `D` the [[Mass_diffusivity|diffusion coefficient]] in square metres per second.[^fick1855][^crank] The second law follows by conserving matter, `∂C/∂t = D·(∂²C/∂x²)`, and predicts how a concentration profile flattens with time.[^crank] Adolf Fick published both in 1855 by direct analogy with [[Thermal_conduction|Fourier's law of heat conduction]].[^fick1855][^fourier1822] In the microsim below the reader carburizes a steel bar. The controls are the furnace [[Temperature|temperature]] `T` and the holding time `t`; the sim draws the profile `C(x,t) = C_s − (C_s − C_0)·erf(x/(2·sqrt(D·t)))` for a bar of 0.20 wt% carbon held in an atmosphere that fixes the surface at 1.00 wt%, with `D = D_0·exp(−Q/(R·T))` and the row's presets for carbon in face-centred cubic [[Austenite|γ-iron]], `D_0 = 2.3×10⁻⁵ m²/s` and `Q = 148 kJ/mol`.[^callister-ch5] The readout is the case depth, the position where the carbon reaches 0.4 wt%. A second panel runs a lattice [[Random_walk|random walk]] and plots its mean squared displacement against time, so the reader can watch `⟨x²⟩ = 2·D·t` assemble itself out of jumps.[^anag-ch13] On the [[Materials_science]] flagship this page serves Part VI, *Fundamentals › Kinetics*, where every later rate — [[Nucleation|nucleation]], [[Sintering|sintering]], [[Precipitation_hardening|precipitation]], [[Creep_(deformation)|creep]] — is ultimately a diffusion time in disguise. ## History Thomas Graham measured the spreading of salts in [[Water|water]] through the 1830s and 1840s and published his study of liquid diffusion in 1850, but he produced data rather than a differential equation.[^graham1850] Adolf Fick, then a 26-year-old demonstrator in anatomy at Zürich, supplied the equation in 1855. His argument was an analogy: Fourier had written heat flow as proportional to the temperature gradient in 1822, and Ohm had written current as proportional to the potential gradient in 1827, so "it was quite natural to suppose" that dissolved matter obeyed the same form.[^fick1855][^fourier1822] Fick tested it on salt diffusing from a reservoir into a funnel and a cylinder, and reported that the steady profiles matched the shapes his law demanded.[^fick1855] The laws are kinematic: they describe transport without saying what moves. Half a century later [[Albert_Einstein|Einstein]] closed that gap, deriving `⟨x²⟩ = 2·D·t` for a suspended particle from the [[Boltzmann_distribution|statistics]] of molecular kicks and identifying `D = k_B·T/(6·π·η·r)` for a sphere of radius `r` in a fluid of [[Viscosity|viscosity]] `η`.[^einstein1905][^likharev-sm5] In [[Metallurgy|metals]] the atomistic picture arrived later still: in 1947 Smigelskas and Kirkendall showed that the marker wires in a brass–copper couple moved, proving that zinc and copper diffuse at different rates by a [[Vacancy_defect|vacancy]] mechanism rather than by direct exchange.[^smigelskas1947][^kirkendall-note] Darken turned that observation into the interdiffusion coefficient in 1948.[^darken1948] ## Fick's first law The first law states that the flux is proportional to the negative of the gradient: in one dimension `J = −D·(∂C/∂x)`, and in three `J = −D·∇C`.[^crank] The minus sign is the whole physical content — matter moves from where there is more of it to where there is less, and a uniform solution has no flux. The constant of proportionality `D` has units of m²/s whatever units `C` is measured in, because `J` carries the same units as `C` times a velocity. `D` is not a property of one substance but of a pair, and it spans an enormous range. In gases at room conditions it is of order 10⁻⁵ m²/s, in liquids 10⁻⁹, and in solids it is both far smaller and violently temperature-dependent, following `D = D_0·exp(−Q/(R·T))` in the same [[Arrhenius_equation|Arrhenius]] form as a reaction rate.[^callister-ch5] With the sim's presets, carbon in γ-iron has `D = 8.3×10⁻¹² m²/s` at 927 °C and `4.3×10⁻¹³ m²/s` at 727 °C, a factor of nineteen for 200 degrees. Steady state is the one case needing no second law: a 1 mm iron sheet holding 1.2 kg/m³ of carbon on one face and 0.2 kg/m³ on the other at 927 °C passes `J = D·ΔC/Δx = 8.3×10⁻⁹ kg·m⁻²·s⁻¹`, about 0.72 grams per square metre per day. ### Variations of the first law Concentration is a convenient variable, not the correct one. The true driving force is the gradient of [[Chemical_potential|chemical potential]], and the first law is better written `J = −(D*·C/(R·T))·(∂µ/∂x)`, which reduces to the familiar form multiplied by a thermodynamic factor `∂lnα/∂lnC` in the activity `α`.[^porter] When that factor is negative — inside a [[Miscibility_gap|miscibility gap]] — the effective `D` is negative and matter climbs its own gradient, which is exactly what [[Spinodal_decomposition|spinodal decomposition]] does.[^porter] The reference frame matters too. In a binary [[Alloy|alloy]] the two species need not have equal intrinsic diffusivities, so the lattice itself drifts relative to the ends of the specimen; Darken's analysis separates the intrinsic coefficients from the single interdiffusion coefficient `D̃ = X_B·D_A + X_A·D_B` that an observer fixed to the specimen measures.[^darken1948][^porter] In electrolytes an electric field adds a drift term, giving the Nernst–Planck equation. And in anisotropic crystals `D` is a second-rank tensor; only in the [[Cubic_crystal_system|cubic]] system does it collapse to a single number.[^shewmon] ### Derivation of Fick's first law for gases For a dilute gas the first law can be obtained by counting. Molecules crossing a plane last collided about one mean free path `λ` away, so they carry the concentration prevailing there; adding the traffic in both directions, with a mean speed `v̄`, leaves a net transfer proportional to the gradient, `D ≈ (1/3)·λ·v̄`.[^averill-gases] For nitrogen at room conditions `v̄ = sqrt(8·R·T/(π·M)) = 475 m/s`, and the path between collisions is about 60 nm, so `D ≈ 9×10⁻⁶ m²/s` — the right order for a gas.[^averill-gases] The estimate also explains the scaling: `λ` falls as the pressure rises and `v̄` grows as `sqrt(T)`, so gas diffusivity goes as `T^(3/2)/P`, nothing like the exponential of a solid. ## Fick's second law The second law describes the unsteady case, in which `C` changes with time: `∂C/∂t = D·(∂²C/∂x²)` for constant `D`, and `∂C/∂t = ∇·(D·∇C)` when `D` varies with composition.[^crank] Read it as a statement about curvature. Where the profile is concave up the concentration rises, where it is concave down it falls, and a straight profile — the steady state of the first law — does not move at all. Dimensional analysis extracts almost everything before any solution is attempted. The equation contains only `D` (m²/s) and `t` (s), so the only length available is `sqrt(D·t)`. Every diffusion problem therefore has a **diffusion length** of that order, and every diffusion time scales as the square of the distance: the [[Boundary_layer|penetration]] doubles when the time quadruples. That single result decides why case-hardening is measured in hours, why capillaries are tens of micrometres apart, and why a [[Semiconductor_device_fabrication|semiconductor]] process engineer counts the thermal budget in seconds. ### Derivation of Fick's second law Take a slab between `x` and `x + dx` of unit area. Whatever enters at `x` and does not leave at `x + dx` accumulates, so `∂C/∂t = −∂J/∂x`, the continuity equation for a species with no source. Substituting the first law gives `∂C/∂t = ∂/∂x(D·∂C/∂x)`, and if `D` is independent of position, `∂C/∂t = D·(∂²C/∂x²)`.[^crank] The second law contains no new physics whatever: it is the first law plus [[Conservation_law|conservation of matter]]. The same [[Partial_differential_equation|partial differential equation]] governs heat conduction, with the thermal diffusivity in place of `D`, which is why solutions are interchangeable between the two subjects.[^crank] Solving it numerically is equally standard: discretising on a grid gives the explicit update `C_i ← C_i + (D·Δt/Δx²)·(C_{i+1} − 2·C_i + C_{i−1})`, stable only while `D·Δt/Δx² ≤ 1/2`, a constraint that forces the time step down as the square of the grid spacing.[^anag-ch10] ## Example solutions and generalization Two closed-form solutions carry most of the practical load, and the microsim shows both at once: the error-function profile that the carburizing panel draws, and the mean squared displacement that the random-walk panel accumulates. ### Example solution 1: constant concentration source and diffusion length For a semi-infinite solid initially uniform at `C_0` whose surface is held at `C_s` from `t = 0`, the solution of the second law is `C(x,t) = C_s − (C_s − C_0)·erf(x/(2·sqrt(D·t)))`, with `erf` the Gaussian error function.[^crank][^callister-ch5] This is the sim's governing equation, and carburizing is its classic use: a low-carbon [[Steel|steel]] part is held in a carbon-rich atmosphere at [[Austenite|austenitising]] temperature so that a hard, high-carbon case grows inward over a tough core.[^carburizing-link] Work the sim's default. At `T = 1200 K` the presets give `D = 2.3×10⁻⁵·exp(−148000/(8.314·1200)) = 8.3×10⁻¹² m²/s`. With `C_0 = 0.20` and `C_s = 1.00 wt%`, the case depth is the `x` at which `C = 0.40 wt%`, so `erf(z) = 0.75` and `z = 0.814`. After `t = 4 h = 14400 s`, `sqrt(D·t) = 3.46×10⁻⁴ m` and the case depth is `x = 2·z·sqrt(D·t) = 0.56 mm`. Dropping the furnace to 1000 K cuts `D` by a factor of nineteen and the depth to 0.13 mm; reaching 1.0 mm at 1200 K takes about 12.6 hours, because depth grows only as `sqrt(t)`. Everything the reader can change enters through the single group `D·t`, so an hour at high temperature is worth many at low — the reason [[Heat_treating|heat treatment]] schedules trade time against temperature the way they do. ### Example solution 2: Brownian particle and mean squared displacement The second solution starts from one particle rather than a profile. A walker taking `N` independent steps of length `a` along a line has zero mean displacement but `⟨x²⟩ = N·a²`, because the cross terms average away; with a jump frequency `Γ = N/t` this is `⟨x²⟩ = Γ·a²·t`, and comparison with the spreading Gaussian that solves the second law identifies `D = Γ·a²/2` in one dimension and `⟨r²⟩ = 6·D·t` in three.[^anag-ch13][^shewmon] The sim's second panel is precisely this construction: many walkers on a lattice, with `⟨x²⟩` plotted against `t` as a straight line whose slope is `2·D`. The result converts a diffusion coefficient into atomic traffic. Taking the jump distance in γ-iron as roughly 0.18 nm, `D = 8.3×10⁻¹² m²/s` implies `Γ = 6·D/a² ≈ 1.5×10⁹` jumps per second per carbon atom — ILLUSTRATIVE, since the jump distance is a lattice estimate rather than a measured value.[^shewmon] In a liquid the same relation is measured directly: Einstein's `D = k_B·T/(6·π·η·r)` gives `4.3×10⁻¹³ m²/s` for a 1 µm sphere in water at 20 °C, so in one second it wanders `sqrt(2·D·t) = 0.93 µm` along any axis, about its own diameter.[^einstein1905][^likharev-sm5] ### Generalizations Fick's laws are a first approximation with well-mapped edges. When `D` depends on composition the profile is no longer an error function and the Boltzmann–Matano construction recovers `D(C)` from the measured shape.[^shewmon] When diffusion is fast along [[Grain_boundary|grain boundaries]], [[Dislocation|dislocations]] or a free surface, the bulk equation must be supplemented by short-circuit paths whose coefficients can exceed the lattice value by many orders of magnitude at low temperature.[^porter] In glassy [[Polymer|polymers]] relaxation of the matrix competes with transport and produces non-Fickian "case II" fronts that advance linearly in time rather than as `sqrt(t)`. And in [[Reaction–diffusion_system|reaction–diffusion systems]] a source term is added to the second law, after which the same operator that only ever smooths a profile can sharpen one into a pattern. ## Applications The laws appear wherever a composition has to be changed without melting anything, and the square-root law is what makes them memorable. ### Fick's flow in liquids Liquid diffusivities cluster near 10⁻⁹ m²/s, which makes unaided mixing hopeless at human scales. Salt spreading through still water covers 1 cm in `t ≈ x²/(2·D) ≈ 9` hours and 1 m in about eleven years. [[Convection|Convection]] and stirring exist because advection and [[Turbulence|turbulence]] beat diffusion at every length above a fraction of a millimetre; diffusion wins only inside the thin films that stirring cannot remove, which is why it still sets the rate at an electrode or a [[Catalysis|catalyst]] surface. ### Sorption rate and collision frequency of diluted solute For a spherical sink of radius `R` in a solution of concentration `C`, solving the second law in the steady state gives a capture rate `I = 4·π·D·R·C`, the Smoluchowski result. With `D = 10⁻⁹ m²/s` and `R = 1 nm` the corresponding rate constant is `4·π·D·R·N_A ≈ 7.6×10⁹` litres per mole per second — the diffusion-limited ceiling on any bimolecular reaction in water, and the number against which a fast [[Enzyme|enzyme]] is judged in [[Chemical_kinetics|chemical kinetics]]. ### Biological perspective Diffusion is a superb delivery system over a micrometre and useless over a centimetre, and cell biology is organised around that fact. With `D ≈ 10⁻⁹ m²/s` oxygen crosses a 0.5 µm alveolar barrier in under a millisecond, a 50 µm gap between capillaries in about a second, and 1 cm in some 14 hours. No animal thicker than a millimetre can therefore be supplied by diffusion alone, and every larger one carries a circulatory system pumping [[Oxygen|oxygen]]-bearing fluid, which reduces the diffusion problem back to micrometres. ### Semiconductor fabrication applications The same two solutions govern doping. A predeposition step holding the wafer surface at a fixed [[Doping_(semiconductor)|dopant]] concentration gives the error-function profile; a drive-in of a fixed dose `Q` per unit area gives the Gaussian `C(x,t) = (Q/sqrt(π·D·t))·exp(−x²/(4·D·t))`, and the junction sits where either profile crosses the background doping.[^crank] Because the depth scales as `sqrt(D·t)`, the shrinking of [[Integrated_circuit|integrated circuits]] has been a long campaign against the thermal budget: modern nodes implant ions cold and anneal for seconds precisely to keep `sqrt(D·t)` down to nanometres. ### Food production and cooking Curing, brining and marinating are diffusion problems whose square-root law is often ignored. Doubling the thickness of a ham quadruples the time for salt to reach its centre, so a cure is scaled by dimension rather than by weight. Heat, meanwhile, obeys the same equation with the thermal diffusivity, about `1.4×10⁻⁷ m²/s` for water — three orders of magnitude larger than the salt's — which is why a roast comes to temperature in an hour and needs a week to cure. ## See also - [[Diffusion]] - [[Carburizing]] - [[Random_walk]] - [[Mass_diffusivity]] - [[Kirkendall_effect]] - [[Sintering]] - [[Arrhenius_equation]], the same exponential in `D(T)` - [[Spinodal_decomposition]], the case in which the effective `D` is negative - [[Thermal_conduction]], the equation Fick copied ## Citations [^fick1855]: Fick, A. (1855). "Ueber Diffusion." *Annalen der Physik* 170 (1): 59–86. https://doi.org/10.1002/andp.18551700105 An English version appeared the same year in the *Philosophical Magazine*, series 4, volume 10. [^graham1850]: Graham, T. (1850). "On the diffusion of liquids." *Philosophical Transactions of the Royal Society of London* 140 (page to pin). [^fourier1822]: Fourier, J. (1822). *Théorie analytique de la chaleur*. Paris: Firmin Didot (the heat-conduction law Fick argued by analogy from) (page to pin). [^einstein1905]: Einstein, A. (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." *Annalen der Physik* 322 (8): 549–560. https://doi.org/10.1002/andp.19053220806 [^smigelskas1947]: Smigelskas, A. D.; Kirkendall, E. O. (1947). "Zinc Diffusion in Alpha Brass." *Transactions of the AIME* 171: 130–142. [^darken1948]: Darken, L. S. (1948). "Diffusion, Mobility and Their Interrelation through Free Energy in Binary Metallic Systems." *Transactions of the AIME* 175: 184–201. [^crank]: Crank, J. *The Mathematics of Diffusion*, 2nd ed. (1975), Oxford University Press, Ch. 1 (the laws and the continuity derivation), Ch. 2 (methods of solution) and Ch. 3 (infinite and semi-infinite media: the error-function and Gaussian solutions, and the heat-conduction analogy) (page to pin). [^shewmon]: Shewmon, P. *Diffusion in Solids*, 2nd ed. (1989), TMS, Ch. 1 (Fick's laws and the error-function solution), Ch. 2 (the random-walk derivation of `D`, jump frequencies, the diffusivity tensor) and the Boltzmann–Matano analysis (page to pin). [^callister-ch5]: Callister, W. D.; Rethwisch, D. G. *Materials Science and Engineering: An Introduction*, 9th ed. (2014), Ch. 5 Diffusion: steady-state and non-steady-state diffusion, the error-function solution and carburizing, and `D = D_0·exp(−Q_d/(R·T))` with its table of `D_0` and `Q_d` for common couples (page to pin). The `D_0 = 2.3×10⁻⁵ m²/s` and `Q = 148 kJ/mol` used here for carbon in γ-iron are the M30 row's presets; the tabulated source (Callister's table, or an ASM handbook entry) still needs pinning. [^porter]: Porter, D. A.; Easterling, K. E.; Sherif, M. Y. *Phase Transformations in Metals and Alloys*, 3rd ed. (2009), Ch. 2 Diffusion: interstitial and substitutional mechanisms, the chemical-potential form of the first law and the thermodynamic factor, uphill diffusion, Darken's interdiffusion coefficient, and high-diffusivity paths (page to pin). [^anag-ch10]: Anagnostopoulos, K. *Computational Physics: A Practical Introduction to Computational Physics and Scientific Computing (using C++)*, 2nd ed. (2016), Ch. 10 The Diffusion Equation, pp. 372–391 (the finite-difference scheme and its stability limit). https://open.umn.edu/opentextbooks/textbooks/computational-physics-a-practical-introduction-to-computational-physics-and-scientific-computing-using-c [^anag-ch13]: Anagnostopoulos (2016), Ch. 13 The Random Walker, pp. 466–499 (lattice random walks, the linear growth of the mean squared displacement, and the pseudorandom generators the sim's walk panel needs). [^likharev-sm5]: Likharev, K. *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Ch. 5 Fluctuations, pp. 143–186 (Brownian motion, the Langevin and Smoluchowski equations, and the Einstein relation between diffusion and mobility) (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^averill-gases]: Averill, B.; Eldredge, P. *General Chemistry: Principles, Patterns, and Applications* (2011), Gases: kinetic molecular theory, the mean speed `sqrt(8·R·T/(π·M))`, and the statement that a molecule's real path at room conditions is a zig-zag of roughly 60 nm segments, p. 951. https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^kirkendall-note]: The marker-motion experiment is usually called the Kirkendall effect after the second author; Smigelskas was the graduate student who ran it. [^carburizing-link]: Callister & Rethwisch (2014), Ch. 5, treats carburizing as the worked case of non-steady-state diffusion; the case-depth numbers quoted here are computed in this article from the row's presets, not taken from the book (page to pin). ## Further reading - Crank, J. *The Mathematics of Diffusion*, 2nd ed. (1975) — the standard solution catalogue for the second law. - Shewmon, P. *Diffusion in Solids*, 2nd ed. (1989) — the atomistic side: mechanisms, jump frequencies, short-circuit paths. - Porter, D. A.; Easterling, K. E.; Sherif, M. Y. *Phase Transformations in Metals and Alloys*, 3rd ed. (2009), Ch. 2. - Anagnostopoulos, K. *Computational Physics*, 2nd ed. (2016), Ch. 10 and Ch. 13 — Portal Book, on the [[PORTAL_Materials_science]] Kinetics shelf. - Likharev, K. *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Ch. 5 — Portal Book, the fluctuation route to `D`. ## External links - The Wikipedia pair's *External links* section lists the current diffusion calculators and lecture notes; none is reproduced here until its URL has been checked. - Anagnostopoulos, *Computational Physics*, and Likharev, *Part SM*, are on the Open Textbook Library (links in the citations above). <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Fick's_laws_of_diffusion.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Fick's laws of diffusion* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Fick's_laws_of_diffusion.html" data-title="Fick's laws of diffusion"></div> *Built from `MICROSIM_GUIDE/specs/sims/Fick's_laws_of_diffusion.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Fick's_laws_of_diffusion) : [Wikitube](https://en.wikitube.io/wiki/Fick's_laws_of_diffusion) · pinned revision [1368840984](https://en.wikipedia.org/w/index.php?oldid=1368840984) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]], [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M30 · sim pending (matter/Fick's_laws_of_diffusion).*