The basic laws of diffusion. The first law states that the flux is proportional to the concentration gradient, J = −D(dC/dx); the second gives the change of concentration with time, ∂C/∂t = D(∂²C/∂x²).
Japanese name: フィックの法則
Fick's laws appears in 1 practice question across 1 topic, including Crystal defects and diffusion.
Fick's first law, J = −D(dC/dx), states that the flux of atoms J is proportional to the concentration gradient and flows from higher to lower concentration; the minus sign indicates this direction. In a steady state where the concentration profile does not change with time, this equation alone gives the flow rate. Permeation through a plate whose two faces are held at constant concentrations is an example, used in hydrogen permeation tests, hydrogen purification with palladium membranes and the evaluation of gas-barrier films. If the membrane is made thicker, the flux decreases in inverse proportion to the thickness.
When the concentration changes with time, you solve the second law, ∂C/∂t = D(∂²C/∂x²), derived from the first law and conservation of mass. For diffusion into a sufficiently thick material (initial concentration C₀) with a constant surface concentration Cs, the solution is (C − C₀)/(Cs − C₀) = 1 − erf(x/(2√(Dt))). For example, with D = 1.0×10⁻¹¹ m²/s and a 10-hour treatment, √(Dt) is about 0.6 mm, and the depth at which the concentration lies exactly halfway between the surface and interior values is about 0.57 mm. This is the basis for estimating carburizing and nitriding depths.
Fick's laws treat the concentration gradient as the driving force, but strictly it is the gradient of chemical potential that moves atoms. Usually the two point the same way, so this causes no trouble, but when other elements make the chemical potential gradient point the opposite way, atoms move from lower to higher concentration in uphill diffusion. Darken's experiment is well known: when steels with and without Si are joined and heated, carbon escapes from the Si-containing side. Spinodal decomposition is another example, in which the apparent diffusion coefficient is negative.
In impurity diffusion in semiconductors, predeposition with a constant surface concentration gives an error-function profile, while drive-in with a fixed amount of dopant gives a Gaussian profile, C = Q/√(πDt)·exp(−x²/(4Dt)). In both cases the diffusion depth is proportional to √(Dt), so doubling the depth needs four times the time, or a temperature rise that quadruples D. In practice D may depend on concentration, and the Boltzmann–Matano method is used to obtain D from a measured concentration profile.
Answer: C. About 2.4×10⁻⁹ kg/(m²·s)
At steady state, Fick's first law J = −D(dC/dx) applies. The concentration gradient is (0.8 − 1.2)/(5.0×10⁻³ m) = −80 kg/m⁴, so J = 3.0×10⁻¹¹ × 80 = 2.4×10⁻⁹ kg/(m²·s), flowing from the high-concentration side to the low-concentration side. About 2.4×10⁻¹² is a unit-conversion error from dividing by the thickness left as 5.0 in mm, and about 7.2×10⁻⁹ uses the concentration on one side, 1.2 kg/m³, instead of the concentration difference. It is the concentration difference that drives the flow, so getting the gradient right matters. The same approach is used in hydrogen permeation tests and in evaluating gas separation membranes.
The flow rate in a steady state, where the concentration profile does not change with time, is found with the first law. For non-steady problems in which the profile changes with time, such as carburizing or doping, the second law is solved under the boundary conditions.
When the surface concentration is constant, the material is thick enough compared with the diffusion depth to be treated as semi-infinite, and the diffusion coefficient can be treated as constant regardless of concentration. Thin plates or treatments in which the surface concentration changes need other solutions.
It means the flow goes from the higher-concentration side to the lower. When concentration decreases with x (dC/dx < 0), J is positive and the flow is in the +x direction.
At steady state, Fick's first law J = −D(dC/dx) applies.
Carbon diffuses at steady state through a 5.0 mm thick stee…