Behavior of confined fluids in nanoslit pores: the normal pressure tensor

The aim of our research is to develop a theory, which can predict the behavior of confined fluids in nanoslit pores. The nanoslit pores studied in this work consist of two structureless and parallel walls in the xy plane located at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a given uniform bulk density. We have derived the following general equation for prediction of the normal pressure tensor Pzz of confined inhomogeneous fluids in nanoslit pores:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_{zz} = kT\rho \left( {r_{1z} } \right)\left[ {1 + \frac{1}{kT}\frac{{\partial \phi_{\text{ext}} }}{{\partial r_{1z} }}{\text{d}}r_{1z} } \right] - \frac{1}{2}\int\limits_{v} {\varphi^{\prime}(\vec{r}_{12} )\rho^{(2)} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{12} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{1} } \right)} \frac{{(r_{12z} )^{2} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{12} }}{\text{d}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{12} , $$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \vec{r}_{12} \equiv \vec{r}_{1} - \vec{r}_{2} $$\end{document} is the intermolecular position vector of molecule 2 with respect to molecule 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r_{12z} = |\vec{r}_{12} |_{z} $$\end{document} is the projection of distance of molecule 1 from molecule 2 in the z-direction. This equation may be solved for any fluid possessing a defined intermolecular pair-potential energy function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varphi \left( {\vec{r}_{12} } \right) ,$$\end{document} confined in a nanoslit pore and with a given fluid molecules—wall interaction potential function ϕext. As an important example of its application we have solved this equation for the hard-sphere fluid confined between two parallel–structureless hard walls with different nanometer distances and at various uniform bulk densities. Our results indicate the oscillatory form of the normal pressure tensor versus distance from the wall at high densities. As the density of the nanoconfined fluid decreases, the height and depth of the normal pressure tensor oscillations are reduced.