In dense simple liquids particles are crushed close
together, and important density correlations arise from the requirements
that the repulsive molecular cores cannot overlap. Moreover, in *uniform* liquids the net force on a given molecule arising from the longer-ranged and more slowly varying attractive interactions essentially cancels in most typical
configurations, leaving only the repulsive force correlations. This explains
the success of the hard sphere model in describing the structure of uniform
simple fluids. However, in *nonuniform* fluids, clearly this cancellation
does not occur. The resulting "unbalanced"
attractive force can have important structural effects, leading, e.g.,
to the existence of a smooth liquid-vapor interface under appropriate conditions.
Standard integral equation methods, which accurately describe correlations
in uniform fluids, fail completely in applications such as these, where
both attractive and repulsive forces can induce significant correlations.

We have shown that in many cases it is possible to obtain accurate results using a generalized mean field description of the attractive interactions self-consistently combined with a new and accurate treatment of repulsive force correlations we have developed. The basic idea is first to determine the relatively slowly varying component of the nonuniform density that is induced by the unbalanced attractive interactions alone. This can generally be done very simply by expansion about a uniform system evaluated at the local density. Under appropriate conditions of two phase coexistence, this component can describe the free liquid-vapor interface. In a second step, we take account of possible fluctuations about this slowly varying density field arising from the repulsive intermolecular interactions, using a generalized linear response or Gaussian approximation that we show can accurately describe harshly repulsive interactions. When the unbalanced attractions have little effect on the structure, the slowly varying density component is just the uniform fluid density. In this special case the second step is equivalent to the accurate Percus-Yevick equation for hard spheres. The equations are iterated until self-consistency is found, when both components are accurately described. This approach provides a natural interpolation from the smooth liquid-vapor interface to the rapidly-varying and oscillating profiles seen for hard spheres near a hard wall.

The figure above gives predictions of the theory
for the full density profile (solid line) of a Lennard-Jones fluid near
a hard wall for two states of different density near the critical temperature.
Also shown is the smooth component (dashed line). Circles give the results
of computer simulations. We see that the theory is capable of reproducing
very accurately both the oscillatory profile with a density *maximum*
at the wall seen at the higher density state as well as the relatively
featureless profile with a density *minimum* at the wall seen at the
lower density. At lower temperatures close to the coexistence the theory
correctly predicts the formation of a vapor layer (a "drying transition")
near the wall.

One very important application of these ideas is to the study of hydrophobic interactions in water. In collaboration with Professor David Chandler and his student Ka Lum at Berkeley, we have developed a simplified version of the theory discussed above that can be applied to more complicated liquids such as water, where the idea of a hard sphere reference fluid is not applicable. Nevertheless, associated with a density gradient in water are unbalanced attractive interactions which can lead to the formation of interfaces, just as for the LJ fluid. The density response to a hydrophobic object (modeled by an excluded volume region of a certain size) again has two components, one arising from the harshly repulsive forces expelling the water density from the excluded volume region, and another arising from the unbalanced attractive interactions. Again our theory treats both components in a self-consistent way. For more infromation click here.

We show that the nature of the hydrophobic interaction changes dramatically depending on which component of the density response dominates; this in turn depends on the size and geometric arrangements of the hydrophobic residues. Small molecular-sized species produce relatively small changes in the hydrogen-bond network, which can simply go around such molecules. However, a larger extended hydrophobic interface requires that some hydrogen bonds be broken. This favors the existence of a thin vapor-like "drying" layer of depleted hydrogen bonding at the interface. The most successful molecular based theory of hydrophobicity, the Pratt-Chandler theory, successfully describes the small molecule regime, but fails when applied to large hydrophobic objects. Our new theory can describe both regimes and the transition between them, which occurs on the biologically relevant length scale of nanometers. We argue that the multifaceted nature of the hydrophobic interaction may have important implications for theories of protein folding. This process is believed to be driven by hydrophobic interactions, but current theories do not take into account the possibility of phase transitions involving the solvent (water), which could profoundly affect the dynamics.

The Figure above gives predictions of the LCW
theory for the relative density of water at STP conditions near idealized
hydrophobic objects of different sizes (modeled by hard spheres of varying
radius *R*) as a function of the distance *r* from the sphere
boundary. A transition from wetting to drying behavior is predicted; the
crossover regime is in the biologically relevant length scale of nanometers.
The inset gives the relative density at contact (solid lines) compared
to simulations (circles); the dashed lines gives the result of the Pratt-Chandler
theory. The Figure below gives the predicted water density near a cylindrical
object, where half the surface is hydrophobic and the other half hydrophilic.

The basic idea we exploit in all this work --- that the complicated correlations in disordered systems may often be decomposed into simpler parts arising from slowly varying and rapidly varying components --- is quite general. We expect this physical picture and the associated mathematical techniques we have developed to have impact in a number of diverse areas where one attempts to combine results from a continuum theory with more rapidly varying molecular scale correlations. Examples include dielectric continuum theory, the theory of ionic solutions, and treatments of dislocation cores.

This material is based upon work supported by the National Science Foundation under Grants No. CHE-0111104 and CHE-0517818. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.