Watch solute atoms diffuse through a crystal lattice via a vacancy mechanism. The diffusion coefficient follows the Arrhenius law D = D₀·exp(−Eₐ/kT). Observe how temperature, activation energy, and time shape the concentration profile.
D: - m²/s
D/D₀: -
Sim time: 0 s
√(Dt): - nm
Peak C: 1.000
How it works
Vacancy mechanism: Diffusion in crystalline solids primarily occurs when an atom jumps into an adjacent vacant site. The probability of a successful jump depends on the thermal energy available relative to the migration barrier Ea.
Arrhenius diffusivity: D(T) = D0 · exp(−Ea / kBT), where kB = 8.617×10⁻⁵ eV/K is the Boltzmann constant. Doubling T dramatically increases D.
Fick's 2nd law: ∂C/∂t = D · ∂²C/∂x². The concentration profile C(x,t) evolves from an initial spike (delta function) into a Gaussian of width σ = 2√(Dt).
Analytical solution: For a thin initial layer: C(x,t) = C0 / [2√(πDt)] · exp(−x²/4Dt). The half-width scales as √(Dt), called the diffusion length.
Crystal lattice view: The top panel shows a 2-D slice of the crystal; atoms colored by local solute concentration. The bottom panel plots C(x,t) with the analytical Gaussian overlay.
About Crystal Diffusion
This simulation models how solute atoms spread through a crystalline solid by the vacancy mechanism, where an atom jumps into a neighbouring empty lattice site. A thin source of atoms is placed at the centre and its concentration profile C(x,t) is advanced in time by numerically solving Fick’s second law, ∂C/∂t = D·∂²C/∂x², using an explicit finite-difference scheme on a 120-cell one-dimensional grid.
The diffusion coefficient is set by the Arrhenius law D = D₀·exp(−Eₐ/kₕT). The four sliders adjust temperature T (300–1800 K), activation energy Eₐ (0.3–3.0 eV), the pre-exponential factor D₀, and a time-step scale. A dashed Gaussian shows the analytical solution. Such diffusion governs real processes like carburising steel, doping semiconductors, and creep in turbine alloys.
Frequently Asked Questions
What does this simulation show?
It shows solute atoms diffusing outward from a thin central source through a crystal lattice. The top panel colours atoms by local concentration, while the lower panel plots the evolving concentration profile C(x,t) against position in nanometres, alongside the analytical Gaussian prediction.
What is the vacancy mechanism?
In most crystalline solids an atom moves by hopping into an adjacent vacant lattice site rather than squeezing between atoms. Each jump must overcome a migration energy barrier, so diffusion depends strongly on both the supply of vacancies and the available thermal energy.
Which equation governs the profile?
The profile follows Fick’s second law, ∂C/∂t = D·∂²C/∂x². The simulation solves it with an explicit finite-difference update, automatically sub-dividing each step so the stability ratio D·dt/dx² stays at or below 0.4.
How is the diffusion coefficient calculated?
The diffusivity uses the Arrhenius relation D = D₀·exp(−Eₐ/kₕT), with the Boltzmann constant kₕ = 8.617×10⁻⁵ eV/K. Raising temperature or lowering the activation energy increases D sharply because it sits inside an exponential.
What do the four sliders control?
Temperature T (300–1800 K) and activation energy Eₐ (0.3–3.0 eV) together set the diffusivity through the Arrhenius law. The pre-exponential factor D₀ rescales the overall jump rate, and the time-step scale speeds up or slows down how quickly simulated time advances.
Why is the profile a Gaussian?
A point or thin-layer source diffusing under Fick’s second law evolves into a Gaussian whose width grows with time. The analytical solution is C(x,t) ∝ exp(−x²/4Dt), so the spread is set entirely by the product Dt. The dashed green curve overlays this exact result.
What does √(Dt) mean?
The quantity √(Dt) is the diffusion length, a measure of how far atoms have typically migrated. The Gaussian half-width scales with it, so doubling the time only increases the spread by about 1.4 times rather than doubling it.
Is the simulation physically accurate?
The Arrhenius diffusivity and Fick’s second law are textbook-accurate and the finite-difference solver matches the analytical Gaussian closely. However, it is a simplified one-dimensional, continuum model: it ignores discrete atomic jumps, defect clustering, grain boundaries, and concentration-dependent diffusivity.
Why does heating speed diffusion so dramatically?
Because temperature appears inside the exponential of the Arrhenius law, a modest rise in T produces a large rise in D. More atoms gain enough thermal energy to clear the migration barrier, so a few hundred kelvin can change the diffusivity by orders of magnitude.
Where does crystal diffusion matter in industry?
It underlies many materials processes: carburising and nitriding to harden steel surfaces, doping silicon wafers to make transistors, homogenising cast alloys, sintering ceramics, and the slow creep deformation that limits the life of jet-engine turbine blades at high temperature.