Why does the interface stay at the average concentration?

For equal diffusivities and an infinite couple, symmetry of the error-function solution keeps c(0,t) = (c_L + c_R)/2 for all t > 0.

What does 2√(Dt) mean?

It is a standard diffusion-length scale: most of the concentration change between the two ends sits within a few multiples of √(Dt) of the weld plane.

Diffusion Couple (Fick's Second Law)

Infinite A|B diffusion couple: error-function concentration profiles, diffusivity D, time t, and diffusion length 2√(Dt).

About this model

Fick's second law ∂c/∂t = D ∂²c/∂x² governs unsteady diffusion. For an infinite diffusion couple with a sharp initial step between concentrations c_L and c_R, the exact solution is an error-function profile whose characteristic width grows as 2√(Dt). The simulator plots c(x,t), marks ±2√(Dt), and shows a color bar of the couple so students can connect diffusivity and annealing time to how far the interface blurs.

Who it's for: Materials science, metallurgy, physical chemistry, solid-state diffusion, and heat/mass transfer courses.

Key terms

Fick's second law
Diffusion couple
Error function
Diffusion length
Diffusivity D
Concentration profile

How it works

Classic infinite diffusion couple for Fick's second law: an error-function concentration profile spreads from a sharp initial step. Tune diffusivity D and time t to see the diffusion length 2√(Dt) grow.

Key equations

∂c/∂t = D ∂²c/∂x²

c(x,t) = (c_L+c_R)/2 − (c_L−c_R)/2 · erf( x / (2√(Dt)) )

Characteristic width ∼ 2√(Dt) (amber markers)

Frequently asked questions

Why does the interface stay at the average concentration?: For equal diffusivities and an infinite couple, symmetry of the error-function solution keeps c(0,t) = (c_L + c_R)/2 for all t > 0.
What does 2√(Dt) mean?: It is a standard diffusion-length scale: most of the concentration change between the two ends sits within a few multiples of √(Dt) of the weld plane.

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Diffusion Couple (Fick's Second Law)

About this model

How it works

Key equations

Frequently asked questions

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How it works

Key equations

Frequently asked questions