Note on the energy density in the solvent induced by a solute

Abstract

The question of how far the effect of the presence of a solute molecule propagates into the solvent is studied in a lattice model that had been used earlier to describe hydrophobic interactions. The local energy density in the model solvent is obtained as an explicit function of distance from the solute and is found to decay to its bulk-phase value with the same decay length as that of the solvent-mediated part of the potential of mean force between a pair of solute molecules. The integrated deviation of the energy density from its bulk-phase value is evaluated in Bethe-Guggenheim approximation and shown to be identical to the energy change accompanying the dissolution of the solute as obtained from the temperature dependence of its solubility.

A solute molecule induces structural and energetic changes in the neighboring solvent that propagate away from the solute over some characteristic distance, which is the correlation length in the solvent. The question of how far these effects propagate into the solvent before decaying to negligible magnitude has been of particular interest for hydrophobic solutes in water (1–9). It may be anticipated that this decay length characterizing the effect of a single solute molecule on the structure and energetics of the solvent is then also the range of the solvent-mediated component of the potential of mean force between two such hydrophobic solute molecules. The latter, particularly in its dependence on the distance between the two solute molecules, is primarily a property of the solvent alone, although in general (but not in the model to be discussed here) it may depend also on the solute–solvent interaction potential.

The main result obtained here, from an analysis of a particular model, is indeed an explicit connection between the deviation of the local energy density in the solvent from its bulk-phase value and the solvent-mediated part of the potential of mean force between two solute molecules. In both cases we contemplate the limit of infinite dilution, so, in effect, a single, isolated solute molecule or an isolated pair. The relation we obtain is in Eq. 6. It is suggested there that some aspects of the form of the derived relation may have a certain generality, not restricted to the present model.

These questions are treated here in the context of a model that has been applied in the past in the study of hydrophobic hydration and the solvent-mediated hydrophobic interaction (10). It is a lattice model, defined as follows.

Associated with each lattice site is a solvent molecule that may be in any of q(>2) internal states or orientations. Among these q is a special state or orientation, call it number 1, differing from the other q − 1 states. When a pair of neighboring solvent molecules are both in that special state 1 they interact with energy w; when they are not both in that special state 1 they interact with energy u. The difference u − w, along with the positive q − 1, are two model parameters.

Solute molecules may be accommodated only on bonds of the lattice, and then only if the solvent molecules associated with the two sites at the ends of the bond are both in the special state 1. The accommodation of the solute molecule on the lattice is energetically favorable when w < u and entropically unfavorable because 1 < q − 1. The model may thus incorporate the essence of the hydrophobic effect. An accommodated solute molecule interacts with its solvent neighbors with a direct interaction energy υ; this is the third parameter in the model.

The solvent-mediated part of the potential of mean force between two solute molecules at the bond centers located at r₁ and r₂ is denoted W(r₁, r₂). It is the difference that remains after the direct interaction potential between the two solute molecules, which plays no role in the model, is subtracted from the total potential of mean force. It is expressible in terms of two quantities, P₁₁ and P(r₁, r₂), which are properties of the pure solvent. Of these, P₁₁ is the probability that a pair of neighboring solvent molecules be both in the special state 1 (so that the bond between them is accessible to a solute molecule), and P(r₁, r₂) is the probability that the solvent molecules at the ends of the two bonds centered at r₁ and r₂ are all in state 1 (so that both bonds are simultaneously accessible to solute molecules). Then,

with k being Boltzman's constant and T the absolute temperature (10).

The solubility of the solute in the solvent, measured as its Ostwald absorption coefficient, is denoted Σ. It is the ratio of the number density of the solute in the dilute saturated solution to its number density in a dilute gas phase in equilibrium with the solution. It is expressible in terms of P₁₁ and the parameter υ (10),

From the definitions of P₁₁ and P(r₁, r₂), their ratio P(r₁, r₂)/P₁₁ is the probability that the solvent molecules neighboring the bond centered at r₂ are both in the special state 1, given that those neighboring the bond centered at r₁ are, and 1 − P(r₁, r₂)/P₁₁ is then the probability that the solvent molecules neighboring the bond centered at r₂ are not both in that special state when those neighboring the bond centered at r₁ are. In the first case the solvent–solvent interaction energy at r₂ is w, in the second case it is u. We now associate a unit volume with each bond, so the mean potential energy density ε(r₁, r₂) in the solvent at r₂ when there is a solute molecule at r₁ (≠ r₂), is then

At an infinite distance from the solute molecule the mean potential energy density ε(∞) in the solvent is

so the local excess energy density in the solvent at r₂ due to a solute molecule at r₁ is

From 1 and 5 we have our main result,

The excess energy density in the solvent at r₂ due to the presence of a solute molecule at r₁ is thus the product of a factor that is independent of r₁ and r₂ and depends only on the thermodynamic state of the solvent, and the factor 1 − exp[−W(r₁, r₂)/kT], with W the solvent-mediated part of the potential of mean force between solute molecules at r₁ and r₂. Stated in these terms, this relation between the solute-induced deviation of the local energy density in the solvent from its value in the bulk-phase solvent far from the solute molecule, and the solvent-mediated effective interaction between solutes, may have a certain degree of generality, and so may not be restricted to the present model. First, at large separations r₂ − r₁ the solvent-mediated potential W becomes much less than kT, so ε(r₁, r₂) − ε(∞) becomes proportional to W(r₁, r₂)/kT, showing that at large separations the two quantities vanish with the same decay length. As an asymptotic result for large separations this connection may well prove to be universal. It is a proposition that could in principle be tested by experiment and computer simulations of realistic models. That the particular relation between the two effects expressed in Eq. 6 could hold generally and literally also at smaller separations is doubtful (J. D. Weeks, personal communication), but may nevertheless prove to be nearly true for small solutes. This supposition, too, is susceptible to test by experiment or simulation.

With a solute molecule at r₁ the potential energy density ε(0) at that point is υ + w, so its excess over the mean of that in the pure solvent far away is, from 4,

The total excess potential energy ΔE in the system due to a solute molecule at r₁ is obtained by summing the right side of 5 over all bond centers r₂(≠ r₁), and adding the right side of 7:

This ΔE should be the same as the energy change, ΔE_th, accompanying the transfer of the solute molecule from the dilute gas into the lattice, as obtained from the temperature dependence of the Ostwald absorption coefficient Σ by the thermodynamic relation ΔE_th = −d lnΣ/d(1/kT), or, from 2,

ΔE − υ and ΔE_th − υ are properties of the pure solvent and are thus independent of υ. The identity ΔE − υ ≡ ΔE_th − υ is illustrated below.

The quantity ΔE − υ, where the solute–solvent interaction energy υ has been subtracted from the total potential energy difference ΔE, is essentially the “solvent reorganization energy” (6, 8, 9), i.e., the change in total solvent–solvent interaction energy resulting from the change in solvent structure induced by the solute. This solvent reorganization energy depends, in general, on whether the solute has been introduced at fixed volume or fixed pressure (6, 8, 9), but in the context of the present lattice model that distinction cannot be made, just as the ΔE_th in 9 is not distinguishable as an energy or enthalpy difference.

One may evaluate ΔE − υ from 8 and ΔE_th − υ from 9 once P₁₁ and P(r₁, r₂) are known. They may be obtained analytically and explicitly in Bethe-Guggenheim (“quasi-chemical”) approximation (10–12). This approximation has also been applied in the analysis of other lattice models of water and hydrophobic solvation (13, 14). The “approximation” is exact on a Bethe lattice (Cayley tree), in which there are no closed loops, so we now take this to be our lattice. It may have any coordination number Z, which is the number of bonds emanating from each site, and so is also the number of lattice sites neighboring each site. The special case Z = 2 is the linear chain.

The Bethe lattice is illustrated in Fig. 1 for the case Z = 3. It can be embedded in any number of dimensions. There is no significance in the metrical distances or angles, only in the number r of lattice steps from one site to another or from one bond center to another. Thus, P(r₁, r₂) is now taken to be a function P(r) of that number r (≥1). Because there are no closed loops on the lattice, the path from one site to another is unique. For that reason the various correlation functions, such as P(r), vary with r just as in one dimension (Z = 2), although such correlation functions depend also on the thermodynamic state, which in turn depends on Z.

Fig. 1.

Bethe lattice (Cayley tree) of coordination number Z = 3. The small circles represent solvent molecules at the lattice sites. The large circle represents a solute molecule on the bond between two lattice sites. The numbers 1, 2, and 3 denote bond centers distant r = 1, 2, and 3 lattice steps, respectively, from the solute molecule, taken to be at r = 0.

The bond center r = 0, where the solute molecule is located, divides the Bethe lattice into two equivalent sublattices, and in each of them the number of bond centers that are r steps away from r = 0 is (Z − 1)^r. Thus, for the Bethe lattice, 8 becomes

The quantities P₁₁ and P(r) required in 8 and 10 are now obtained in Bethe-Guggenheim approximation (10). We introduce the abbreviation c,

Then a quantity y is obtained in terms of the model's physical parameters c, q − 1, and Z − 1 upon eliminating a second quantity α from the pair of equations

We next introduce the abbreviation Q for the quantity

which is thus also in principle obtained in terms of the model's physical parameters c, q − 1, and Z − 1. Then (10),

These are properties of the model solvent alone. The parameter y, which is obtained from 12 when α is the solution of 13, is the probability that any solvent molecule be in the special state 1. The parameter α, in turn, is ω/(1 − ω), where ω is the probability that any chosen one of the Z neighbors of a solvent molecule that is in one of the q − 1 nonspecial states, is itself in the special state. These interpretations of y and α may be traced to the equivalence between the present model solvent and the lattice-gas model of a one-component fluid treated in Bethe-Guggenheim approximation (10–12, 15).

In terms of Q and y, which are obtained as functions of (u − w)/kT, q − 1, and Z − 1 via 11- 14, we now have from 10, 15, and 16,

while from 9 and 15,

On evaluating the right side of 18, with y and Q implicit functions of (u − w)/kT for fixed q − 1 and Z − 1 via 11–14, one obtains an inordinately complicated expression that does not at first sight seem anything like the right side of 17. Nevertheless, after much labor, and by what almost seems an algebraic miracle, they prove to be identical for all (u − w)/kT, q − 1, and Z − 1. Thus, ΔE ≡ ΔE_th.

The identity is probably most expeditiously verified with the help of a computer algebra program, but there is a simple special case for which the calculation is readily done “by hand.” That is the mean-field limit Z → ∞ and u − w → 0 with φ ≡ Z(u − w) fixed. In this limit α satisfies exp{[α/(1 + α)]φ/kT} = (q − 1)α, while 1/y − 1 = 1/α, so that y satisfies (1/y − 1)exp(yφ/kT) = q − 1; and, further, Q ∼ 1 while Q − 1 ∼ 2y(1 − y)(u − w)/kT; and P₁₁ ∼ y². Then one finds explicitly from 17 and 18 that in this limit,

The identity ΔE ≡ ΔE_th may also be demonstrated by hand straightforwardly, but still laboriously, for the special case Z = 2 (the linear chain).

The common value of (ΔE − υ)/kT and (ΔE_th − υ)/kT is plotted vs. T over the interval 285 K ≤ T ≤ 330 K, for Z = 8 and Z = ∞ (mean-field limit), in Fig. 2. The parameters q − 1 and u − w for the plot with Z = 8 and the parameters q − 1 and φ for the plot with Z = ∞ were separately chosen to reproduce as well as possible the shape of the experimentally measured kT lnΣ of methane in water as a function of temperature over that temperature interval (10). It is only the shape of that function, not its absolute magnitude, that matters for the choice of parameters for these plots, as one sees from 2 and 9 or from 8, because υ cancels in the difference ΔE − υ. When, within this model, one needs to fit the solubility of methane in its entirety, so not merely its shape as a function of temperature but also its absolute magnitude, it is best fit with a positive value of υ, which is recognized to be an artificiality of the model (10). One can, indeed, fit the solubility data also with a negative (or zero) υ, although such fits are not quite as good as those with positive υ. It is also noteworthy that fits obtained with values of υ that are more negative than about −5 kT require negative rather than positive values of u − w. Such fits would imply that the hydrophobic solute is constrained to occupy a location in which two neighboring lattice sites are both in the one special state (orientation) in which they interact with the higher energy w than when one or both are in any of the remaining q − 1 orientations, when they interact with energy u. Although such fits are not as good as those with positive u − w, the resulting Ostwald coefficient has the same magnitude and general temperature dependence as that of methane in water. Moreover, both the solute–solvent interaction energy υ and solvent reorganization energy ΔE − υ (the sign of which is invariably opposite that of u − w) obtained from the latter fits are within about ±kT of those inferred from the corresponding experimental and simulation results (which imply a solute–solvent interaction energy of about −5.6 kT and a water-reorganization energy of about +1.2 kT for methane in water at 298 K and 0.1 MPa) (6, 8, 9, 16, 17). But for our present purpose, as remarked above, the value of υ is irrelevant, and the methane solubility data were fit with positive u − w.

Fig. 2.

Solvent reorganization energy ΔE − v, plotted as (ΔE − v)/kT ≡ (ΔE_th − v)/kT vs. T in degrees K, for Z = 8 and Z = ∞.

That the two curves in Fig. 2 nearly coincide over the interval 293 K ≤ T ≤300 K is simply because the parameters for each plot were chosen to reproduce the same experimental data. That they are close over the whole temperature interval of the plots, with the Z = ∞ curve lying only slightly above that for Z = 8 at both extremes, is because the dimensionless 8 is already so large as to be practically ∞.