Modified Kelvin equations for capillary condensation in narrow and wide grooves

We consider the location and order of capillary condensation transitions occurring in deep grooves of width $L$ and depth $D$. For walls that are completely wet by liquid (contact angle $\theta=0$) the transition is continuous and its location is not sensitive to the depth of the groove. However for walls which are partially wet by liquid, where the transition is first-order, we show that the pressure at which it occurs is determined by a modified Kelvin equation characterized by an edge contact angle $\theta_E$ describing the shape of the meniscus formed at the top of the groove. The dependence of $\theta_E$ on the groove depth $D$ relies, in turn, on whether corner menisci are formed at the bottom of the groove in the low density gas-like phase. While for macroscopically wide grooves these are always present when $\theta<45\degree$ we argue that their formation is inhibited in narrow grooves. This has a number of implications including that the local pining of the meniscus and location of the condensation transition is different depending on whether the contact angle is greater or less than a universal value $\theta^*\approx 31\degree$. Our arguments are supported by detailed microscopic density functional theory calculations which show that the modified Kelvin equation remains highly accurate even when $L$ and $D$ are of the order of tens of molecular diameters.