Fick's Laws: The Physics of How Things Spread Out

The First Law: Flux Follows the Slope

Fick's first law states that the diffusive flux — the amount of substance crossing unit area per unit time — points down the concentration gradient and grows in proportion to how steep that gradient is:

• J = −D ∇c

Here J is the flux vector (mol·m⁻²·s⁻¹), c is the concentration (mol·m⁻³), ∇c is the spatial gradient of concentration (mol·m⁻⁴), and D is the diffusion coefficient or diffusivity (m²/s). In one dimension it collapses to J = −D (∂c/∂x). The minus sign is the whole story: matter flows from high concentration to low, opposing the gradient, which is why the law is a statement of irreversible transport rather than a conservative force.

Dimensionally, D has units of m²/s — length squared over time — and that is not a coincidence. It is the signature of anything that spreads as the square root of time, which we will see is the deep fingerprint of a random walk. A small molecule in water has D ≈ 2×10⁻⁹ m²/s; a gas molecule in air is ~10⁴ times faster at ~2×10⁻⁵ m²/s; a carbon atom hopping through solid iron near 900 °C crawls at ~10⁻¹² m²/s.

The Second Law: The Diffusion Equation

Fick's first law tells you the flux for a given gradient, but gradients change as material moves. To track how a concentration profile evolves, combine the first law with conservation of mass. The continuity equation says that whatever accumulates in a small volume equals the net flux into it: ∂c/∂t = −∇·J. Substituting J = −D ∇c gives Fick's second law:

• ∂c/∂t = −∇·J = ∇·(D ∇c)
• For constant D: ∂c/∂t = D ∇²c

This is the diffusion equation, mathematically identical to the heat equation Fourier wrote in 1822 — temperature and concentration obey the same physics, because both are conserved scalars carried by a gradient-driven flux. The Laplacian ∇²c measures the local curvature of the profile: where concentration is a peak (∇²c < 0) it falls, where it is a valley (∇²c > 0) it fills in. Diffusion always smooths, never sharpens.

A canonical solution shows the character vividly. Release N moles at a single point in 1D at t = 0; the profile is a spreading Gaussian:

• c(x,t) = [N / √(4πDt)] · exp(−x² / 4Dt)

Its width grows as √(2Dt): the mean-squared displacement is ⟨x²⟩ = 2Dt (in three dimensions, ⟨r²⟩ = 6Dt). The peak height falls as 1/√t. Nothing moves ballistically — the front never travels at constant speed; it decelerates forever as √t.

Why √t? The Random Walk Underneath

The macroscopic law hides a microscopic truth: diffusion is the ensemble average of countless random walks. A molecule in a liquid is battered by ~10¹³ collisions per second, each nudging it a tiny, uncorrelated step. Model this as N steps of length ℓ in random directions. The average displacement is zero — the molecule is equally likely to go left or right — but the mean-squared displacement is not: ⟨x²⟩ = N ℓ². Since the number of steps grows linearly with time (N = t/τ for step time τ), we get ⟨x²⟩ ∝ t, and the root-mean-square distance ∝ √t.

Comparing to ⟨x²⟩ = 2Dt fixes the diffusion coefficient as D = ℓ²/2τ — diffusivity is just the microscopic step size squared over step time. This connection was made rigorous by Einstein in 1905 and independently by Smoluchowski, the same analysis that turned Brownian motion into a measurement of Avogadro's number.

The √t scaling is why diffusion is superb over microns and hopeless over meters. Reaching rms distance L takes t ≈ L²/2D. For a small molecule in water (D ≈ 10⁻⁹ m²/s):

• Cross a 10 μm cell: t ≈ (10⁻⁵)² / (2×10⁻⁹) ≈ 0.05 s — fast.
• Cross a 1 mm capillary bed: t ≈ (10⁻³)² / (2×10⁻⁹) ≈ 500 s ≈ 8 minutes.
• Cross a 1 m room by diffusion alone: t ≈ 1 / (2×10⁻⁹) ≈ 5×10⁸ s ≈ 16 years.

Perfume crosses a room in seconds not by diffusion but by convection — air currents. Diffusion only wins at the smallest scales, which is exactly why life is built from micron-sized cells.

What Sets D: Temperature, Size, and the Medium

The diffusion coefficient is where the microscopic physics lives. For a spherical particle of radius r drifting through a fluid of viscosity η, the Stokes–Einstein relation pins D down:

• D = k_B T / (6πηr)

where k_B = 1.38×10⁻²³ J/K is Boltzmann's constant and T the absolute temperature. Every term is intuitive: hotter fluids (larger T) diffuse faster because thermal kicks are stronger; more viscous fluids (larger η) and bigger particles (larger r) diffuse slower because drag resists the walk. For a molecule ~0.3 nm in radius in water (η ≈ 1.0×10⁻³ Pa·s) at 298 K, this predicts D ≈ (1.38×10⁻²³ × 298)/(6π × 10⁻³ × 3×10⁻¹⁰) ≈ 7×10⁻¹⁰ m²/s — right in the measured range.

In solids, diffusion happens by atoms hopping between lattice sites past an energy barrier, so D is fiercely temperature-dependent, following an Arrhenius law D = D₀ exp(−E_a / k_B T). For carbon in iron E_a ≈ 0.8 eV; raising the temperature from 700 °C to 900 °C can boost D by more than an order of magnitude, which is precisely why steel is carburized hot. In gases, kinetic theory gives D ≈ ⅓ v̄ λ, the product of mean molecular speed and mean free path, scaling as T^(3/2)/P.

Steady State and the Diffusion Length

When the profile stops changing (∂c/∂t = 0), Fick's second law reduces to Laplace's equation ∇²c = 0, and in 1D the concentration is simply linear: c(x) = c₁ + (c₂ − c₁)(x/L). This is the workhorse of membrane physics. Across a membrane of thickness L with concentrations c₁ and c₂ on either side, the steady flux is J = D(c₁ − c₂)/L, or with a partition coefficient K, the permeability P = DK/L governs how fast a drug crosses a cell wall.

When diffusion competes with a reaction or decay that consumes the species at rate 1/τ, a natural length scale emerges — the diffusion length L_D = √(Dτ). Beyond L_D the species is gone before it can diffuse further. In a semiconductor solar cell, minority-carrier diffusion length (often tens to hundreds of μm in silicon) sets how far a photo-generated electron travels before recombining, and therefore how thick the absorber can usefully be. In electrochemistry the Nernst diffusion layer (~10–100 μm thick) limits current at an electrode. The same √(Dτ) appears everywhere diffusion races a clock.

Where Fick's Laws Rule — and Where They Break

Fick's laws are the quiet engine behind an enormous range of technology and biology:

• Physiology: Oxygen crossing the ~1 μm alveolar membrane, CO₂ leaving, neurotransmitters crossing the ~20 nm synaptic cleft in ~1 μs — all Fickian, and all fast only because the distances are tiny.
• Semiconductors: Thermal diffusion of boron and phosphorus dopants into silicon wafers at ~1000 °C creates p–n junctions; the junction depth is set by √(Dt) and controlled to nanometers.
• Metallurgy: Case-hardening, carburizing, and nitriding all rely on Fick's second law to compute how deep carbon or nitrogen penetrates steel.
• Chemical engineering: Catalysis, membrane separation, drug-release polymers, and CO₂ capture are all diffusion-limited processes designed around D.

But the laws have limits. Fick's laws assume dilute, ideal mixtures with no cross-coupling; in concentrated or charged systems the true driving force is the gradient of chemical potential, not concentration, and one must use the Maxwell–Stefan formulation — sometimes yielding 'uphill' diffusion against the concentration gradient. Charged ions feel electric fields (the Nernst–Planck equation adds a drift term). And on scales below the mean free path, or over times shorter than the collision time, the infinite propagation speed implied by ∂c/∂t = D ∇²c is unphysical — hyperbolic corrections (Cattaneo's equation) restore a finite front speed. Within its dilute, macroscopic regime, though, Fick's 1855 laws remain astonishingly accurate.

Fick's first vs. second law, and a sense of diffusion coefficients across media.

Aspect	Fick's First Law	Fick's Second Law
Equation	J = −D ∇c	∂c/∂t = D ∇²c
Describes	Steady flux from a fixed gradient	How concentration evolves in time
Regime	Steady state (∂c/∂t = 0)	Transient / relaxing profiles
Typical D — gas in air	~2×10⁻⁵ m²/s (O₂ in air)	spreads ~4 mm in 1 s
Typical D — ion in water	~2×10⁻⁹ m²/s (Na⁺)	spreads ~60 μm in 1 s
Typical D — atom in solid	~10⁻¹² m²/s (C in Fe, ~900 °C)	spreads ~30 μm in ~10³ s