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{title2=$F(h)=2\pi R_{\rm eff}W(h)$}

For gently curved interacting surfaces whose range and closest gap are small compared with their radii, integrate the local planar interaction over nearly parallel surface elements. With $W(s)$ the planar interaction <free energy> per area and $\Pi(s)=-W'(s)$, the local gap for two spheres is $s\simeq h+r^2/(2R_{\rm eff})$, where $R_{\rm eff}^{-1}=a_1^{-1}+a_2^{-1}$. Thus $F=2\pi\int_0^\infty r\Pi(s)dr=2\pi R_{\rm eff}\int_h^\infty\Pi(s)ds=2\pi R_{\rm eff}W(h)$ when $W(\infty)=0$. Equal spheres have $R_{\rm eff}=a/2$; a sphere and plane have $R_{\rm eff}=a$. Use consistent surface boundary conditions when supplying the planar interaction.

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Derjaguin approximation ( $F (h) = 2 π R_{eff} W (h)$ )

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Derjaguin approximation (F(h)=2πReff​W(h))

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Derjaguin approximation ( $F (h) = 2 π R_{eff} W (h)$ )

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