Cavity Statistics in Water and Other Organic Liquids

One surprising observation related to the solubilities of non-polar solutes is that liquid water at room temperature has more fractional free volume than many organic liquids. Fractional free volume is defined as the fraction of the total volume accessible to solutes of any, even subatomic, size, and is equal to p₀(0). If there is more accessible volume for a hard sphere in water than in an organic liquid, why then are the solubilities of non-polar solutes in water so low? The comparisons of p₀(R), and for cavities in water and hexane, obtained by analyzing configurations of these two liquids generated in molecular dynamics simulations, reveals the answer to this question. As can be seen in Figure 1, for small cavities is lower in water than in hexane, but the two curves intersect at R=1.3 Å, close to the radius of the smallest atomic solute. This means that the excess chemical potential of inserting small, subatomic-sized cavities is lower in water but the the excess chemical potential of inserting atomic or molecular-sized cavities is lower in hexane. Thus, the fractional free volume of water is distributed in smaller packets than the free volume in hexane. The same conclusion follows from the comparison between water and other organic liquids. [18]

  
Figure 1: The excess chemical potential, , for inserting a cavity of radius R in bulk water (green solid line) and hexane (blue dashed line) at T = 300 K.

The finding that the excess chemical potential for cavities of atomic sizes is larger in water than in organic liquids raises a question: what property of water is primarily responsible for this sharpness and, consequently, for the low solubilities of non-polar solutes in water? One view holds that this property is the pattern of hydrogen bonds between water molecules. [28] To accommodate a non-polar molecule, the network of hydrogen bonds has to be disrupted and rearranged around this molecule. This is an unfavorable transition, not compensated fully by favorable solute-solvent interactions. Alternatively, it has been proposed that the behavior of non-polar species characteristic of the hydrophobic effect is not directly determined by the hydrogen bonding structure of liquid water, but rather by the efficient packing of water molecules, mostly due to their small size. [29, 30] This argument follows the ideas underlying the scaled particle model of solubilities of non-polar solutes in molecular liquids. [12, 31] This model considers solubilities of hard spheres as functions of the size of solvent molecules, the packing density of the solvent, and its equation of state. Its basic idea is to connect smoothly the functional form of such properties as or , which are known for small , with the anticipated functional form for large in the region of atomic and small molecular-size cavities, where a simple functional form is not known. The small functional form is determined by the size and density of solvent molecules, whereas the large form is defined by macroscopic parameters: the pressure and the surface tension, The accuracy of the small term has been improved and extended for aqueous solutions by incorporating the experimentally known oxygen-oxygen radial distribution function of liquid water, yielding the revised scaled particle model (RSPM). [13]

Since the scaled particle model has been remarkably, although perhaps somewhat fortuitously, [32, 13] successful in predicting solubilities of small non-polar solvents in molecular liquids, it is of interest to compare its predictions with accurate results from computer simulations. Additional insight into the role of hydrogen bonding can be obtained by considering a hypothetical, reference liquid, composed of spherical molecules interacting only via the Lennard-Jones potential. This liquid has the density, pressure and the distance of the closest approach between molecules identical to liquid water at a given temperature. The parameters and , needed to define intermolecular interactions in the liquid, can be uniquely determined from these conditions [19] using our advanced knowledge of Lennard-Jones fluids. [33] Note, that and are temperature dependent, i.e. somewhat different reference liquids correspond to water at different temperatures.

Probably the most sensitive function of is , which can be reliably determined from simulations of liquid water at T=300 K for nm. This quantity is compared in Figure 2 with the prediction of RSPM and for the reference liquid. The numerical results for in hexane are also shown.

  
Figure 2: obtained from computer simulations of bulk water (solid green line), the reference, Lennard-Jones liquid (dotted magenta line), and hexane (cyan dot-dashed line) are compared with predictions of the scaled particle model (blue long-dashed line) and the revised scale particle model (red short-dashed line).

obtained from computer simulations of water increases with for small cavity sizes and reaches a maximum near = 0.24 nm. This value of can be considered as separating small and large cavity regimes. For cavities corresponding to larger than 0.24 nm, decreases with . This reflects a tendency of water molecules to ``pull away'' from hydrophobic surfaces. This dewetting behavior was discussed by Stillinger. [13] It is, however, not certain whether the decrease in is monotonic, especially for cavities of atomic and small molecular sizes. There are some indications that may exhibit local maxima in this range of sizes which correspond to preferred, clathrate-like structures of water around hydrophobic solutes. [34]

for hexane is quite different than for water. For all values of it is smaller and remains fairy close to 1. This is consistent with the interpretation [35] that water, but not organic liquids, squeezes out non-polar solutes.

As can be seen from Figure 2, RSPM provides a fair description of for water over the range of cavity sizes studied here. This can be contrasted with the result of the scaled particle model which predicts a considerably smaller maximum in located at a lower value of . [19] Perhaps more interestingly, for the Lennard-Jones, reference liquid is also qualitatively similar to that for water. At = 0.3 nm, the two curves differ by approximately 20%. This indicates that a significant part of the hydrophobic effect can be captured without considering explicitly the hydrogen-bonded structure of water.

The relative importance of different properties of water for describing the hydrophobic effect can be further examined by calculating the solubility of a model non-polar solute, methane, in water and in the reference liquids as a function of temperature. To do so we return to Eq. 1. can be calculated efficiently by using the known p₀(R) as the probability density from which insertion sites are sampled. Then

 

where <...>_R represents a statistical average over insertions into cavities with radii between R and R+dR and R_min is the radius of the smallest cavities that are still suitable insertion sites. A detailed description of the implementation of this method can be found elsewhere. [26] Other authors [20, 27] used similar approaches. Once is known, the mole-fraction solubility, s, can be readily calculated from [36]

 

where is the density of solute in the gas phase at its partial pressure of 1 atm and is the number density of solute in the solution.

As shown in Figure 3, the calculated and experimental solubilities of methane in water at different temperatures are in close agreement. Consistent with expectations arising from the hydrophobic nature of the solute, the solubilities decrease in the range of temperatures studied here at a pressure of 1 atm. The solubilities in the reference liquids are consistently lower that those in water, but their temperature dependence is very similar. This, again, indicates that properly chosen Lennard-Jones reference liquids can exhibit properties considered as characteristic signatures of the hydrophobic effect.

  
Figure 3: Experimental mole fraction solubilities of methane in water [37](green solid line) as a function of temperature are compared with calculated solubilities in water (blue circles) and in the reference liquid (red triangles).