= Solution
In the quasisteady <diffusion> approximation, the vapour-<density> disturbance outside a small stationary particle satisfies <Laplace's equation>, with a specified equilibrium value on its surface and a uniform far-field value. This has the same mathematical boundary-value problem as the electrostatic potential outside a conductor held at fixed potential. The integrated normal gradient is proportional to the conductor's geometrical <diffusion-capacitance analogy>[capacitance].
Here $C$ is a \i[length]: a sphere of <radius> $r$ has $C=r$. The corresponding electrical <capacitance> in SI units would be $4\pi\epsilon_0C$, not $C$ farads. Replacing the spherical diffusive conductance $4\pi r$ by $4\pi C$ accounts for the crystal's non-spherical geometry. Coupling the analogous heat-conduction problem to vapour <diffusion> supplies the ice-specific thermal and diffusional resistances.
The approximation assumes continuum <diffusion>, equilibration of the surface vapour concentration, an approximately isothermal particle, a slowly changing shape, weak perturbations and a far field sufficiently distant relative to the crystal size. It neglects ventilation by falling motion, interactions with neighbouring particles and facet-dependent attachment kinetics. Shape affects diffusive growth through <diffusion-capacitance analogy>[capacitance], not merely through projected collecting area as in a collision calculation.
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