Crystal growth.
Classical nucleation rate
Energy barrier of the critical cluster and nucleus radius in classical nucleation theory.
Nucleation rate $J$ (per volume, per time):
$$J = A\,\exp\!\Big(-\frac{\Delta G^*}{k_B T}\Big)$$with the homogeneous nucleation barrier
$$\Delta G^* = \frac{16\pi\,\sigma^3}{3(\Delta G_v)^2}$$where $\sigma$ is the surface tension and $\Delta G_v \approx \tfrac{R T}{V_m}\ln S$ for a supersaturation ratio $S$. Critical radius: $r_c = \tfrac{2\sigma}{\Delta G_v}$.
Growth rate — diffusion vs. surface
Two limiting regimes for linear crystal growth.
For supersaturation ratio $S$:
$$G_{\text{diff}} = k_d\,(S - 1), \qquad G_{\text{surf}} = k_r\,(S - 1)$$$k_d$ reflects mass transport ($D/\delta$) and $k_r$ the surface attachment rate.
Avrami equation (JMAK)
Crystallization kinetics — transformed fraction and mean size.
$Y$ is the crystallised fraction at time $t$, $n$ the Avrami exponent, $K$ the rate constant. Optional fields estimate the mean crystal diameter assuming spherical crystals.
Supersaturation ratio
$S = C / C^*$ — ratio of actual concentration to equilibrium solubility.
van't Hoff equation
Temperature dependence of solubility, assuming constant $\Delta H_{\text{sol}}$.
$R = 8{,}314\ \text{J/(mol·K)}$. ΔH > 0 corresponds to endothermic dissolution (solubility increases with $T$).
Dimensionless numbers
Reynolds, Péclet and Richardson — characterise the flow regime.
- Re — inertial vs. viscous forces: $\text{Re} = \rho v L / \mu$.
- Pe — advection vs. diffusion: $\text{Pe} = v L / D$.
- Ri — buoyancy vs. shear: $\text{Ri} = g \, (\Delta\rho/\rho) \, L / v^2$.
If $\rho_2$ is left blank or equal to $\rho_1$, Ri treats $\Delta\rho = 0$.
Flow regime (Navier–Stokes)
Simplified forms dominating fluid dynamics in crystallisation.
The Navier–Stokes equation for incompressible flow with gravity:
$$\rho\Big(\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v}\Big) = -\nabla p + \mu\,\nabla^2 \mathbf{v} + \rho\,\mathbf{g}$$