  An earlier one-dimensional lattice model of hydrophobic attraction is
extended to two and three dimensions and studied by Monte Carlo simulation. The
solvent-mediated contribution to the potential of mean force between
hydrophobic solute molecules and the solubility of the solute are determined.
As in the earlier model, an inverse relation is observed between the strength
and range of the hydrophobic attraction. The mean force no longer varies
monotonically with distance, as it does in one dimension, but has some
oscillations, reflecting the greater geometrical complexity of the lattice in
the higher dimensions. In addition to the strong attraction at short distances,
there is now also a local minimum in the potential of depth about $kT$ at a
distance of three lattice spacings in two dimensions and one of depth about
$2kT$ at a distance of two lattice spacings in three dimensions. The solubility
of the solute is found to decrease with increasing temperature at low
temperatures, which is another signature of the hydrophobic effect and also
agrees with what had been found in the one-dimensional model.
