Basic Theory and Methodology
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Basic Theory and Methodology

A central quantity in theories of the hydrophobic effect is the excess chemical potential of a solute molecule, tex2html_wrap_inline777, defined as the quasi-static work needed to bring this molecule from the gas phase into the solvent. The connection between tex2html_wrap_inline777 and the statistical properties of the neat solvent is provided by the potential distribution theorem. [25] For a rigid solute:


 equation56

where k is the Boltzmann constant, T is the temperature, tex2html_wrap_inline785, Usol is the solute-solvent potential energy and tex2html_wrap_inline789 abbreviates the thermal average over insertions of the solute into thermally representative configurations of the neat solvent. The extension of Eq. 1 to flexible solutes is available, [14] but it will not be needed here.

If the solute is an impenetrable volume the exponential in the right side of Eq. 1 is either one or zero, depending on whether the solute overlaps with van der Waals volumes assigned to solvent molecules. Then, tex2html_wrap_inline777 is expressed simply as:


 equation67

where p0 is the probability of successful insertion of the solute into the solvent.

Consider insertions of a hard sphere solute into a liquid represented as hard spheres or collections of hard spheres of one kind only. Their radius will be denoted as RS. The extension to spheres of different sizes is straightforward. [18] In practice, the calculated quantity is the probability distribution, tex2html_wrap_inline797, that a randomly chosen point is at a distance tex2html_wrap_inline799 from the center of the nearest solvent sphere. If tex2html_wrap_inline801 the radius of the largest solute that can be inserted at this site is tex2html_wrap_inline803. The probability p0(R) can be obtained from pm(R) using the relation:


 equation74

Two concrete methods for locating cavities of different sizes and calculating tex2html_wrap_inline797 were described previously. [18, 26] Note that once positions of suitable cavities are identified, they can be used as insertion sites for small spherical or nearly spherical solutes to evaluate efficiently tex2html_wrap_inline777 for these solutes directly from Eq. 1. [20, 27, ] An implementation of this approach is discussed in detail elsewhere. [26]

The third quantity of interest here is the density of solvent centers just outside the solute (contact density), tex2html_wrap_inline813, defined as: [19]


 equation83

where tex2html_wrap_inline815 is the density of solvent and tex2html_wrap_inline817 is called the contact correlation function. The last term in the right side of Eq. 4, tex2html_wrap_inline819, is the average compressive force exerted by the solvent on the cavity. Thus, the contact density is large in those solvents which squeeze out a non-polar solute.