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For a sphere of radius $a$ at separation $d\ll a$ from a half-space, pairwise integration of the attractive $r^{-6}$ molecular potential gives $F\simeq Aa/(6d^2)$. Equivalently, integrate the planar disjoining pressure magnitude $A/(6\pi h^3)$ over the local gap $h=d+r^2/(2a)$. Across a third medium the effective Hamaker constant can instead correspond to repulsion; state its sign convention explicitly.

Thermomolecular pressure is the force per area transmitted through a thin premelting liquid film between a solid and a substrate. Let $d$ be film thickness. For a repulsive, nonretarded dispersion interaction, define a positive interaction amplitude $A$ byThis sign convention assigns positive $p_T$ to repulsion; a signed Hamaker constant in another convention must be translated before using this formula. The strong thickness dependence permits a thin film to support appreciable load while remaining hydraulically connected to a liquid supply.

In the equal-phase-density approximation, the Clapeyron pressure relation for equal-density phases gives $p_s-p_l\simeq \rho L(T_m-T_i)/T_m$. A molecular film force can balance this phase-pressure difference and an applied load. With $\rho \simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, $L\simeq 3.34\times 10^5\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ and $T_m\simeq 273\mathrm{K}$, an undercooling of $0.1\mathrm{K}$ corresponds to a pressure scale about $0.12\mathrm{MPa}$. This is an equilibrium driving or limiting scale, not a guaranteed growth rate; liquid delivery through Darcy flow can limit frost heave.^[ref]

The Van der Waals force is a distance-dependent nonbonded interaction between atoms or molecules, including dispersion and dipolar contributions. In the attractive nonretarded pair model, an interaction energy $U(r)=-\beta /r^6$ gives force magnitude $6\beta /r^7$. Macroscopic integration introduces a Hamaker constant; it does not replace the sign change and molecular cutoff needed at very short distances.