Thermodynamic properties of quasi-one-dimensional fluids

We calculate thermodynamic and structural quantities of a fluid of hard spheres of diameter $\sigma$ in a quasi-one-dimensional pore with accessible pore width $W$ smaller than $\sigma$ by applying a perturbative method worked out earlier for a confined fluid in a slit pore [Phys. Rev. Lett. \textbf{109}, 240601 (2012)]. In a first step, we prove that the thermodynamic and a certain class of structural quantities of the hard-sphere fluid in the pore can be obtained from a purely one-dimensional fluid of rods of length $\sigma$ with a central hard core of size $\sigma_W =\sqrt{\sigma^2 - W^2}$ and a soft part at both ends of length $(\sigma-\sigma_W)/2$. These rods interact via effective $k$-body potentials $v^{(k)}_\text{eff}$ ($k \geq 2$) . The two- and the three-body potential will be calculated explicitly. In a second step, the free energy of this effective one-dimensional fluid is calculated up to leading order in $(W/\sigma)^2$. Explicit results for, e.g. the perpendicular pressure, surface tension, and the density profile as a function of density, temperature, and pore width are presented presented and partly compared with results from Monte-Carlo simulations and standard virial expansions. Despite the perturbative character of our approach it encompasses the singularity of the thermodynamic quantities at the jamming transition point.