# Fick's laws of diffusion
**Fick's laws of diffusion** are two mathematical statements, published by [Adolf Fick](https://www.edgepedia.org/adolf-fick) in 1855, that describe how a diffusing substance flows down a concentration gradient. The first law relates the diffusive flux to the concentration gradient; the second law, obtained by combining the first with conservation of mass, describes how the concentration profile itself changes with time.[1](https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.18551700105) • [2](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion)
| Key fact | Detail |
|---|---|
| First law | \( J = -D\,\partial c/\partial x \): flux is proportional to the concentration gradient, with the minus sign indicating transport from higher to lower concentration[2](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion) |
| Second law | \( \partial c/\partial t = D\,\partial^2 c/\partial x^2 \) for constant \( D \), derived by combining the first law with species conservation[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html) |
| Units | Flux \( J \) in mol m−2 s−1; diffusion coefficient \( D \) in m2 s−1 • [4](https://ucb-bioe.github.io/textbook/chapters/02_tissue/04_ficks_first.html) |
| Origin | Adolf Fick, "Ueber Diffusion", *Annalen der Physik* 170, pp. 59–86 (1855); an English version, "On liquid diffusion", appeared the same year in the *Philosophical Magazine*[1](https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.18551700105) • [5](https://www.tandfonline.com/doi/abs/10.1080/14786445508641925) |
| Typical magnitudes | Ions at room temperature: \( D \approx 0.6\times10^{-9} \) to \( 2\times10^{-9} \) m2/s; biological molecules: \( 10^{-11} \) to \( 10^{-10} \) m2/s[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Kinetics/09%3A_Diffusion) |
| Temperature dependence | Diffusion is thermally activated, \( D = D_0\,e^{-E_a/RT} \)[2](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion) |
| Characteristic time | A profile equilibrates over a distance \( L \) in a time of order \( \tau = L^2/(2D) \)[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html) |
## First law: flux and gradient
The diffusive flux \( J \) is the amount of substance crossing unit area per unit time, in units such as mol m−2 s−1. Fick's first law states that this flux is proportional to the concentration gradient, with the diffusion coefficient \( D \) as the constant of proportionality:
\[ J = -D\,\frac{\partial c}{\partial x} \]
The negative sign carries the physics: a species moves from where it is abundant toward where it is scarce, so wherever concentration falls as \( x \) increases, the gradient is negative and the flux is positive, meaning transport in the \( +x \) direction. Flux and gradient always point in opposite directions.[4](https://ucb-bioe.github.io/textbook/chapters/02_tissue/04_ficks_first.html) • [2](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion)
[Dimensional analysis](https://www.edgepedia.org/dimensional-analysis) fixes the units of \( D \): dividing mol m−2 s−1 by mol m−4 gives m2 s−1.[4](https://ucb-bioe.github.io/textbook/chapters/02_tissue/04_ficks_first.html) The law can be derived from counting alone. If molecules in two adjacent boxes of width \( \Delta x \) each have an equal chance of crossing the plane between them within a time \( \Delta t \), the net crossing rate is proportional to the difference in concentration, and in the limit \( \Delta x \to 0 \) the proportionality constant is \( D = (\Delta x)^2/(2\Delta t) \), the same combination of step size and step time that the random walk produces.[4](https://ucb-bioe.github.io/textbook/chapters/02_tissue/04_ficks_first.html)
**Steady state.** The first law in this form describes steady-state diffusion, in which the flux is constant in time and the concentration profile does not change: the mass entering one side of a slab equals the mass leaving the other, with no net accumulation.[7](https://uomustansiriyah.edu.iq/media/lectures/6/6_2025_01_29%2105_32_52_PM.pdf) For a slab of thickness \( L \) with concentrations held at \( c_0 \) and \( c_L \) on its two faces, the steady profile is a straight line and the flux is \( J = D(c_0 - c_L)/L \), proportional to the concentration difference and inversely proportional to the thickness; doubling the thickness halves the flux.[8](https://ucb-bioe.github.io/textbook/chapters/02_tissue/06_steady_state.html)
## Second law: the diffusion equation
The first law says what the flux is but not how concentration evolves. Combining it with conservation of a species in a control volume, \( \partial c/\partial t = -\partial J/\partial x - R \), where \( R \) is a volumetric consumption rate, gives the reaction–diffusion equation. With no reaction and constant \( D \), this is Fick's second law:
\[ \frac{\partial c}{\partial t} = D\,\frac{\partial^2 c}{\partial x^2} \]
The right-hand side is the curvature of the concentration profile. Where the profile curves upward (\( \partial^2 c/\partial x^2 > 0 \)) the concentration rises; where it curves downward it falls; and a straight-line profile does not change at all. Diffusion therefore smooths a profile out, and the larger \( D \) is, the faster it does so.[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html) In three dimensions the equation becomes \( \partial c/\partial t = D\,\nabla^2 c \), with \( \nabla^2 \) the Laplacian.[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html)
**Solutions.** Solutions depend on the boundary conditions, and comprehensive collections are given by Crank and by Carslaw and Jaeger.[9](https://ndl.ethernet.edu.et/bitstream/123456789/27623/1/487.pdf) Two standard cases for a semi-infinite solid are the thin source, a small fixed amount of solute diffusing in from one end, which gives a Gaussian profile
\[ C(x,t) = \frac{B}{\sqrt{\pi D t}}\,\exp\!\left(\frac{-x^2}{4Dt}\right), \]
and the infinite source, a surface held at constant concentration \( C_s \) against an initial uniform concentration \( C_0 \), which gives an error-function profile,[10](https://eng.libretexts.org/Workbench/Materials_Science_for_Electrical_Engineering/02%3A_Solids/2.02%3A_Diffusion/2.2.03%3A_Fick%27s_Second_Law_of_Diffusion)
\[ \frac{C_x - C_0}{C_s - C_0} = 1 - \operatorname{erf}\!\left(\frac{x}{2\sqrt{Dt}}\right). \]
A bar counts as semi-infinite when none of the diffusing atoms reaches its far end during the diffusion time, which holds when its length \( l > 10\sqrt{Dt} \).[9](https://ndl.ethernet.edu.et/bitstream/123456789/27623/1/487.pdf) For more complicated geometries or concentration-dependent diffusivity, no analytical solution exists and numerical methods are used.[10](https://eng.libretexts.org/Workbench/Materials_Science_for_Electrical_Engineering/02%3A_Solids/2.02%3A_Diffusion/2.2.03%3A_Fick%27s_Second_Law_of_Diffusion)
## The characteristic diffusion time
The second law implies a scaling that recurs throughout diffusion problems: the time to equilibrate over a distance \( L \) grows as the square of that distance. Setting the root-mean-square displacement \( \sqrt{\langle x^2\rangle} = \sqrt{2Dt} \) equal to \( L \) gives
\[ \tau = \frac{L^2}{2D}. \]
For a graft of thickness \( L = 1 \) mm with an oxygen diffusion coefficient \( D = 2\times10^{-9} \) m2 s−1, the estimate gives \( \tau \approx 250 \) s, about 4 minutes; a 200 µm epidermal construct equilibrates in about 10 s, and doubling the thickness quadruples the time.[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html) The same scaling explains why single cells can rely on diffusion while centimeter-scale tissues require a blood supply: diffusion across 10 µm takes milliseconds, while across 1 cm it takes hours.[3](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html)
## Diffusion coefficients
The diffusion coefficient varies over many orders of magnitude with the medium, the diffusing species, and temperature. Representative values include \( D \approx 0.2 \) cm2/s for O2 in air, \( 2\times10^{-5} \) cm2/s for O2 in water, and \( 0.15\times10^{-5} \) cm2/s for ATP in cytoplasm.[11](https://www.its.caltech.edu/~brokawc/Bi145/Diffusion.html) Ions at room temperature fall in the range \( 0.6\times10^{-9} \) to \( 2\times10^{-9} \) m2/s, and biological molecules in \( 10^{-11} \) to \( 10^{-10} \) m2/s; larger molecules have smaller coefficients.[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Kinetics/09%3A_Diffusion)
Like chemical reactions, diffusion is thermally activated, and the temperature dependence appears in the diffusivity as an Arrhenius-type equation,
\[ D = D_0\,e^{-E_a/RT}, \]
where \( D_0 \) includes factors such as the jump distance and vibrational frequency of the diffusing species.[2](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion) In liquids the coefficient also depends on viscosity, which the Stokes–Einstein relation \( D = kT/(6\pi\eta a) \) models in terms of the molecule radius \( a \) and viscosity \( \eta \).[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Kinetics/09%3A_Diffusion)
## Analogies and applications
Fick's first law is identical in form to Fourier's law for heat flow under a constant temperature gradient and to [Ohm's law](https://www.edgepedia.org/ohms-law) for current flow under a constant electric potential gradient; all three are linear phenomenological laws relating a flow to a conjugate gradient.[12](https://www.sciencedirect.com/topics/engineering/ficks-law) • [13](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.37%3A_Irreversible_Thermodynamics%3A_Onsager_Phenomenological_Equations) Historically, diffusion problems were first solved in the context of heat flow, which follows the same mathematics.[14](https://goldenberg.biology.utah.edu/courses/biol3550/courseMaterial/notes/3550_notes_ch1-7.pdf)
In physiology, the first law governs gas transport across the alveolar capillary membrane: the flux equals the permeability \( P \) times the transfer area \( A \) times the concentration gradient, with \( P = (D \times B)/X \), where \( B \) is the solubility (partition coefficient) and \( X \) the membrane thickness.[15](https://journals.physiology.org/doi/full/10.1152/advances.2001.25.2.15?doi=10.1152%2Fadvances.2001.25.2.15) For a uniform membrane separating two well-mixed compartments, the same law gives \( J = P(c_1 - c_2) \), and the homogeneous solubility-diffusion model links permeability to molecular properties as \( P = KD/h \), with \( K \) the partition coefficient and \( h \) the membrane thickness.[16](https://pmc.ncbi.nlm.nih.gov/articles/PMC6506413/)
## Limits of Fickian diffusion
Fick's laws assume that the concentration gradient is the driving force and that \( D \) is a constant. In nonideal mixtures, particularly those that can phase-separate, the true driving force is the gradient of chemical potential rather than of concentration, and conventional formulations of the chemical potential must be supplemented by a gradient term; the resulting nonlinear diffusion is described by the modified [Cahn–Hilliard equation](https://www.edgepedia.org/cahn-hilliard-equation).[17](https://www.sciencedirect.com/science/article/pii/S0009250901000057) In porous media, heterogeneity across broad scales often produces non-Fickian or anomalous transport in which the mean squared displacement does not scale linearly with time, so classical continuum models with constant coefficients do not predict the observed behavior well.[18](https://doi.org/10.1016/j.jconhyd.2020.103734) In such settings a concentration gradient and a scalar diffusivity are insufficient, and frameworks built on [electrochemical potential](https://www.edgepedia.org/electrochemical-potential) gradients are used instead.[19](https://www.sciencedirect.com/science/article/pii/S001346862601635X)
## History
Adolf Fick (1829–1901), a German physiologist who earned a medical degree at the University of Marburg in 1851, published "Ueber Diffusion" in *Annalen der Physik* in 1855, while a Demonstrator of Anatomy at Zurich; the paper has accumulated more than 2,000 citations.[1](https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.18551700105) • [12](https://www.sciencedirect.com/topics/engineering/ficks-law) • [20](https://www.scientific.net/DDF.143-147.-1.pdf) An English rendition, "On liquid diffusion", appeared in the *Philosophical Magazine* the same year.[5](https://www.tandfonline.com/doi/abs/10.1080/14786445508641925) The mechanistic theory of diffusive transport developed independently a few years later by Maxwell and Stefan implicitly recognized the chemical potential gradient as the driving force.[21](https://pubs.rsc.org/en/content/articlehtml/2016/nj/c5nj02836a)
## References
1. [Adolf Fick (1855). "Ueber Diffusion". Annalen der Physik 170 (1), 59–86. Wiley.](https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.18551700105)
2. ["1.9: Diffusion". Chemistry LibreTexts.](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Introduction_to_Solid_State_Chemistry/01%3A_Lectures/1.09%3A_Diffusion)
3. ["18: Fick's Second Law". Introduction to Bioengineering, UC Berkeley.](https://ucb-bioe.github.io/textbook/chapters/02_tissue/05_ficks_second.html)
4. ["17: Fick's 1st Law". Introduction to Bioengineering, UC Berkeley.](https://ucb-bioe.github.io/textbook/chapters/02_tissue/04_ficks_first.html)
5. [Adolph Fick (1855). "On liquid diffusion". London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10 (63).](https://www.tandfonline.com/doi/abs/10.1080/14786445508641925)
6. ["9: Diffusion". Chemistry LibreTexts.](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Kinetics/09%3A_Diffusion)
7. ["Fick's First Law" lecture notes. University of Mustansiriyah.](https://uomustansiriyah.edu.iq/media/lectures/6/6_2025_01_29%2105_32_52_PM.pdf)
8. ["19: Steady State". Introduction to Bioengineering, UC Berkeley.](https://ucb-bioe.github.io/textbook/chapters/02_tissue/06_steady_state.html)
9. [Materials science textbook chapter on Fick's second law (Callister-style excerpt).](https://ndl.ethernet.edu.et/bitstream/123456789/27623/1/487.pdf)
10. ["2.2.3: Fick's Second Law of Diffusion". Engineering LibreTexts (DoITPoMS).](https://eng.libretexts.org/Workbench/Materials_Science_for_Electrical_Engineering/02%3A_Solids/2.02%3A_Diffusion/2.2.03%3A_Fick%27s_Second_Law_of_Diffusion)
11. ["Diffusion". Brokaw, Caltech Bi145 course notes.](https://www.its.caltech.edu/~brokawc/Bi145/Diffusion.html)
12. ["Fick's Law: An Overview". ScienceDirect Topics.](https://www.sciencedirect.com/topics/engineering/ficks-law)
13. ["1.14.37: Irreversible Thermodynamics: Onsager Phenomenological Equations". Chemistry LibreTexts.](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.37%3A_Irreversible_Thermodynamics%3A_Onsager_Phenomenological_Equations)
14. ["4.2. Fick's Second Law". Course notes, University of Utah Biology 3550.](https://goldenberg.biology.utah.edu/courses/biol3550/courseMaterial/notes/3550_notes_ch1-7.pdf)
15. ["Pulmonary and Renal Pressure-Flow Relationships: What Should Be Taught?" Advances in Physiology Education 25 (2001).](https://journals.physiology.org/doi/full/10.1152/advances.2001.25.2.15?doi=10.1152%2Fadvances.2001.25.2.15)
16. [Venable, R. M., Krämer, A., Pastor, R. W. (2019). "Molecular Dynamics Simulations of Membrane Permeability". Chemical Reviews / PMC6506413.](https://pmc.ncbi.nlm.nih.gov/articles/PMC6506413/)
17. ["Nonlinear diffusion and phase separation". Chemical Engineering Science 56 (6), 1999–2018 (2001).](https://www.sciencedirect.com/science/article/pii/S0009250901000057)
18. ["A review of spatial Markov models for predicting pre-asymptotic and anomalous transport in porous and fractured media". Journal of Contaminant Hydrology (2020).](https://doi.org/10.1016/j.jconhyd.2020.103734)
19. ["Electrochemical-potential-driven transport in solid-state and hybrid ionic conductors". Electrochimica Acta (Critical Review).](https://www.sciencedirect.com/science/article/pii/S001346862601635X)
20. ["Preface" (History of Diffusion session). Defect and Diffusion Forum 143–147.](https://www.scientific.net/DDF.143-147.-1.pdf)
21. ["Diffusion in nanoporous materials: fundamental principles, insights and challenges". New Journal of Chemistry (2016).](https://pubs.rsc.org/en/content/articlehtml/2016/nj/c5nj02836a)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Continuum, solid, and fluid mechanics › Continuum mechanics foundations*
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