Fluids in Extreme Confinement

For extremely confined fluids with a two-dimensional density $n$ in slit geometry of an accessible width $L$, we prove that in the limit $L\to 0$, the lateral and transversal degrees of freedom decouple, and the latter become ideal-gas-like. For a small wall separation, the transverse degrees of freedom can be integrated out and renormalize the interaction potential. We identify $n{L}^2$ as the hidden smallness parameter of the confinement problem and evaluate the effective two-body potential analytically, which allows calculating the leading correction to the free energy exactly. Explicitly, we map a fluid of hard spheres in extreme confinement onto a 2D fluid of disks with an effective hard-core diameter and a soft boundary layer. Two-dimensional phase transitions are robust and the transition point experiences a shift $\mathcal{O}(n{L}^2)$.

Figure 1

Illustration of the cluster expansion: The hard core with diameter ${\sigma }_L$ is shown in gray and the dashed circle with diameter $\sigma$ marks the range above which the excluded volume interactions vanish. 2D fluid of hard disks with diameter ${\sigma }_L$ (a), a 2-cluster (b), two irreducible 3-clusters and an irreducible 4-cluster (c) immersed in the hard-disk fluid.

Figure 2

The total two-body potential as function of distance $r$. The gray region represents the excluded volume region corresponding to the closest lateral distance ${\sigma }_L$ two spheres can assume in confinement.

Figure 3

Phase diagram of a HSHW taken from Refs. 13, 14. Symbols represent the MC data points, the dashed lines are guides for the eye, and the thin solid lines represent ${\phi }_{*}^{(3\mathrm{D})}(L=0)/{\phi }^{(3\mathrm{D})}(L)-1$ from Eq. (11). $1△$ and $b$ denote the triangular and buckling phase. Inset: Phase transition lines ${\phi }^{(2\mathrm{D})}$ as a function of $L/\sigma$.