Beyond Cavity Statistics - A Two-moment Model of the Hydrophobic Effect

For atomic or small molecule-sized cavities, p₀ can be calculated directly. Larger cavities, however, are so rare that determining p₀ with sufficient statistical precision becomes impractical. A different strategy is clearly needed. To this end, we observe that p₀ is just one element in the set of probabilities p_n of finding exactly n solvent centers in the volume excluded by the solute. These probabilities are subject to the constraint:

 

In practice, the summation extends only to n_max, the maximum number of solvent centers that can be placed in the excluded volume of the solute, v.

  
Figure 4: Probabilities p_n of finding n solvent centers in spherical volumes in water with radii 0.22, 0.30 and 0.38 nm. Numerical results are shown as symbols and the parabolas represent predictions of the two-moment model. Note the logarithmic scale for p_n.

An attempt to obtain p₀ directly from Eq. 7 by calculating the remaining is not promising. Instead, we will take a different approach. We will model all probabilities p_n simultaneously, such that the parameters in the model can be determined from those p_n which are known with the best statistical accuracy. As can be seen in Figure 4, the relation between p_n and n is very simple -- is almost parabolic. This means that the model has the functional form: [38]

 

The parameters depend on v and are determined by the constraint in Eq. 7 and the first two moments of the density distribution

 

where <n> is the average number of water molecules displaced by the solute with the excluded volume v. In this model

 

The proposed model can be readily applied to solutes of arbitrary shapes. No explicit simulations of the solute in the liquid are necessary. The only thing that needs to be calculated is <n> and <n²> inside ``imprints'' of the solute in the neat solvent. In spite of its conceptual and technical simplicity, the model is remarkably accurate. It reproduces practically exactly of spherical cavities in the full range accessible to direct calculations. For two hard sphere solutes at different separations, the model yields a solvent-separated minimum in at the distance of 0.7 nm between the cavity centers and another, shallow minimum at the distance of 1.1 nm. [38] This is in excellent agreement with much more involved simulations, in which the solutes were explicitly placed in water. [21] In another test, the chemical potential of rotating the central C-C bond of butane in water was calculated. [38] As shown in Figure 5, the agreement with explicit computer simulations [39] is again very good. In both cases, it is predicted that the chemical potential of placing in water a compact cavity, corresponding to the cis conformer, is lower than the chemical potential of placing the extended trans conformer.

  
Figure 5: The chemical potential of rotating the central C-C bond of butane modeled as four spheres. The solid line represents predictions of the two-moment model and the dotted line represents the results of computer simulations of Beglov and Roux [39].

The proposed model has been postulated rather than derived from statistical mechanical principles and, therefore, should be considered as heuristic. Nevertheless, it has strong connections to both experiment and theory. The first two moments in Eq. 10 can be directly related to two readily available properties of liquid water -- the density, , and the oxygen-oxygen radial distribution function,

 

 

By relying on the nearly parabolic behavior of the model is closely related to the Pratt-Chandler theory of the hydrophobic effect, [14] in which it is assumed that fluctuations of the solvent density field are Gaussian. [40] In the Gaussian field model the probability is a continuous function of the density and, therefore, can yield non-vanishing probabilities for n<0. In contrast, the present model avoids this problem by using a discrete representation of the probability with the values of p_n located on a Gaussian curve.

As the sizes of solutes increase, it is expected that the two-moment model will require extensions to describe correctly dewetting of large hydrophobic surfaces. One possibility is to increase the number of moments included in Eq. 8. This yields, however, only minor improvements unless almost all moments are considered. A more promising approach is to appeal to information theory, originally used for the derivation of the model. [38] An improved model for the probabilities can be obtained by maximizing information entropy using a ``default model'' that better captures physical effects responsible for dewetting. A default model provides an estimate of probabilities if no further information is supplied. Active work on discovering effective default models is underway.