Chemical potential gradient diffusion and uphill transport calculator

H. K. D. H. Bhadeshia & H. Yan, Queen Mary University of London

Derivation of diffusion flux from chemical potential gradient

Fick’s empirical first law asserts that mass flux is proportional to a negative concentration gradient ( $J = - D d c / d z$ ). However, concentration gradients are not fundamental driving forces. According to linear non-equilibrium thermodynamics (Onsager reciprocal relations), spontaneous mass transport is driven by the spatial gradient of molar Gibbs free energy—the chemical potential gradient ( $d μ / d z$ ).

Thermodynamic driving force and atom mobility

The chemical force acting per atom of species $i$ is the negative spatial gradient of its chemical potential. The thermodynamic force on an atom due to its negative chemical potential gradient causes a flux:

J i = - L i d μ i d z

where $L i > 0$ is the always positive mobility since matter always diffuses down its chemical potential gradient ( $J i$ has the opposite sign to $d μ i / d z$ ), with zero exceptions. Applying the chain rule to the chemical potential gradient with respect to mole fraction $x B$ :

d μ B d z = \partial μ B \partial c B d c B d z

Since chemical potential is per unit concentration, in the following we multiply it by concentration, and compare against Fick's first law to obtain the diffusion coefficient in terms of mobility and the dependence of chemical potential on concentration:

J B = - L B c B d μ B d z = - L B c B d μ B d c B d c B d z so that D B \equiv c B L B \partial μ B \partial c B

making the diffusion coefficient concentration dependent both through how the chemical potential varies with composition and the concentration multiplier.

Interactive exploration of normal and uphill diffusion

In regular solutions with interaction parameter $Ω$ , the Darken factor is $Φ = 1 - 2 Ω x A x B / (R T)$ . Adjust the temperature and interaction parameter below to transition across the spinodal boundary ( $Φ < 0$ ), reversing the sign of $D inter$ and creating spontaneous uphill diffusion against the concentration gradient.

Thermodynamic parameters

Operating regime preset

Solute mole fraction (

x B

) 0.50

Interaction parameter (

Ω

) +18.0 kJ/mol

Absolute temperature (

T

) 800 K

Imposed concentration gradient (

d x B / d z

) +2.00 m⁻¹

Spinodal critical temperature (T_c = Ω / 2R) 1082.5 K

Thermodynamic factor (Φ) -0.353 (Spinodal unstable)

Chemical potential gradient (dμ_B/dz) -9.39 × 10³ J/(mol·m)

Net diffusion flux (J_B) +1.41 × 10⁻⁷ mol/(m²·s) (Uphill)

Thermodynamic factor Φ across composition

Spatial profile of concentration vs chemical potential

Dynamic simulation of uphill composition wave development

Beginning with a nominally uniform alloy ( $c 0 = 0.50$ ) perturbed only by microscopic thermal fluctuations, the simulation below integrates the one-dimensional Cahn–Hilliard equation along distance coordinate $z$ :

\partial c \partial t = M̃ \partial 2 μ \partial z 2 = M̃ \partial 2 \partial z 2 [\partial f 0 \partial c - 2 κ \partial 2 c \partial z 2]

Inside the spinodal region ( $\partial 2 f 0 /\partial c 2 < 0$ ), conventional diffusion would lead to an ill-posed negative diffusivity. The gradient energy coefficient $κ$ provides capillary regularisation, penalising sharp composition gradients. As a result, composition waves with wavelengths greater than the critical wavelength ( $λ > λ c$ ) grow spontaneously in amplitude, phase-separating the initially homogeneous solid into alternating A-rich and B-rich domains.

One-dimensional Cahn–Hilliard wave solver Step: 0 | Time: 0.00 arbitrary units

Gradient energy penalty (κ): 1.5 × 10⁻¹⁰ J·m²/mol

Initial noise amplitude: ±0.010

Physical mechanism of uphill diffusion

The sign of the interdiffusion coefficient $D inter = D * Φ$ is entirely determined by the second derivative of the molar Gibbs energy with respect to composition:

Downhill diffusion ( $Φ > 0$ , $\partial 2 Δ G m /\partial x B 2 > 0$ ): In ideal, ordering, or high-temperature clustering solutions outside the spinodal, chemical potential increases monotonically with concentration ( $d μ B / d z$ has the same sign as $d x B / d z$ ). Diffusing down the chemical potential gradient simultaneously moves solute down the concentration gradient, dissipating concentration differences until the mixture is uniform.
Uphill diffusion ( $Φ < 0$ , $\partial 2 Δ G m /\partial x B 2 < 0$ ): Inside the miscibility gap below the spinodal temperature, like atoms prefer to cluster together because $Ω > 0$ . Thermodynamically, adding solute into an already concentrated region lowers its free energy relative to the matrix. Consequently, the chemical potential gradient has the opposite sign to the concentration gradient. Atoms still obey thermodynamics and move strictly down the chemical potential gradient ( $J B \propto - d μ B / d z$ ), but this drives them into the region of higher concentration ( $J B$ has the same sign as $d x B / d z$ ). This is the fundamental mechanism of spinodal decomposition.