Continuous condensation in nanogrooves

We consider condensation in a capillary groove of width $L$ and depth $D$, formed by walls that are completely wet (contact angle $\theta=0$), which is in a contact with a gas reservoir of the chemical potential $\mu$. On a mesoscopic level, the condensation process can be described in terms of the midpoint height $\ell$ of a meniscus formed at the liquid-gas interface. For macroscopically deep grooves ($D\to\infty$), and in the presence of long-range (dispersion) forces, the condensation corresponds to a second order phase transition, such that $\ell\sim (\mu_{cc}-\mu)^{-1/4}$ as $\mu\to\mu_{cc}^-$ where $\mu_{cc}$ is the chemical potential pertinent to capillary condensation in a slit pore of width $L$. For finite values of $D$, the transition becomes rounded and the groove becomes filled with liquid at a chemical potential higher than $\mu_{cc}$ with a difference of the order of $D^{-3}$. For sufficiently deep grooves, the meniscus growth initially follows the power-law $\ell\sim (\mu_{cc}-\mu)^{-1/4}$ but this behaviour eventually crosses over to $\ell\sim D-(\mu-\mu_{cc})^{-1/3}$ above $\mu_{cc}$, with a gap between the two regimes shown to be $\bar{\delta}\mu\sim D^{-3}$. Right at $\mu=\mu_{cc}$, when the groove is only partially filled with liquid, the height of the meniscus scales as $\ell^*\sim (D^3L)^{1/4}$. Moreover, the chemical potential (or pressure) at which the groove is half-filled with liquid exhibits a non-monotonic dependence on $D$ with a maximum at $D\approx 3L/2$ and coincides with $\mu_{cc}$ when $L\approx D$. Finally, we show that condensation in finite grooves can be mapped on the condensation in capillary slits formed by two asymmetric (competing) walls a distance $D$ apart with potential strengths depending on $L$.