Crystallization patterns of monomers with a nearest-neighbor interaction on a multilayered semi-infinite square lattice.

Abstract

The absorption of an external monomer gas in a square lattice of L layers, infinite in one direction and finite (M sites) in the other, is studied with a finite nearest-neighbor repulsion energy ${\mathit{V}}_{11}$0. In a physical system, the monomer chemical potential ${\mathrm{\ensuremath{\mu}}}_{1}$ is determined by the external gas pressure. For a temperature below (-${\mathit{V}}_{11}$/5${\mathit{k}}_{\mathit{B}}$)${\mathrm{log}}_{10}$e the curves of entropy S versus lattice coverage ${\mathrm{\ensuremath{\theta}}}_{1}$ exhibit two cusps for L=1 and 2, and three cusps for L=3, at S=0. Crystallization pattern analysis at the cusps leads, in terms of M and L, to exact expressions of ${\mathrm{\ensuremath{\theta}}}_{1}$ and the number ${\mathrm{\ensuremath{\theta}}}_{2}$ of nearest neighbors per site which are linear in 1/M. Setting M=\ensuremath{\infty} yields, for L=1 and 2, one cusp at (${\mathrm{\ensuremath{\theta}}}_{1}$=1/2, ${\mathrm{\ensuremath{\theta}}}_{2}$=0), and for L=3, two cusps at (${\mathrm{\ensuremath{\theta}}}_{1}$=1/2, ${\mathrm{\ensuremath{\theta}}}_{2}$=0) and at (${\mathrm{\ensuremath{\theta}}}_{1}$=5/6, ${\mathrm{\ensuremath{\theta}}}_{2}$=5/3). The change in ${\mathrm{\ensuremath{\mu}}}_{1}$ required to unlock the structural ordering is shown to be proportional to ${\mathit{V}}_{11}$. Other features of the system below the critical temperature reported above are also discussed.