SUPPLEMENTARY MATERIAL
The Lee & Graziano theory of hydrophobicity
Solvation refers to the transfer of a solute molecule from a fixed position in the ideal
gas phase to a fixed position in a solvent at constant temperature and pressure.1 The process
can be regarded as the insertion of an external perturbing potential (X), where X is a vector
representing a single configuration of the N molecules of the liquid solvent.2 On the basis of
the Widom’s potential distribution theorem,3 the Ben-Naim standard Gibbs energy change is:
G = -RTln<exp[-(X)/RT]>p
(S1)
where the subscript p means that the ensemble average is performed over the pure solvent
configurations, assuming an NPT ensemble (i.e., constant temperature, pressure and number
of molecules). The average is taken using the probability density function of the pure solvent:
p(X) = exp[-H(X)/RT] /  exp[-H(X)/RT]dX
(S2)
where H(X) = U(X) + PV(X) is the enthalpy function of a configuration, U(X) and V(X) are
the corresponding internal energy and volume, and the denominator is the isobaric-isothermal
configurational partition function of the pure liquid. It is important to realize that, in
performing the ensemble average in Eq. (S1), one has to consider the configurations of the
liquid before the insertion of the potential, so that the Boltzmann weights in the average do
not include (X), that acts as a ghost.
Since a liquid is a condensed state of the matter, the insertion of a solute molecule
requires the exclusion of solvent molecules from the region of space that will be occupied by
the solute. This basic physical consideration suggests that the perturbing potential has to be
factorized in the following manner:2
exp[-(X)/RT] = (X)exp[-a(X)/RT]
(S3)
1
where (X) can assume only the values of one or zero depending on whether or not there is a
cavity suitable to host the solute molecule in the given solvent configuration; and a(X)
represents the attractive potential between the solute molecule and surrounding solvent
molecules. Insertion of Eq. (S3) into Eq. (S1) leads to the following exact relationship:
G = -RTln<(X)>p - RTln<exp[-a(X)/RT]>c
(S4)
where the subscript c means that the ensemble average is performed over the solvent
configurations possessing a cavity suitable to host the solute whose probability density
function is:
c(X) = (X)exp[-H(X)/RT] /  (X)exp[-H(X)/RT]dX
(S5)
Such configurations are only a very small fraction of the total solvent configurations (i.e.,
those for which the function (X) is equal to one). Equation (S4) indicates that G is the sum
of two terms: the reversible work to create the cavity, Gc, and the reversible work to turn on
the attractive solute-solvent potential, Ga; but there is no additivity of independent
contributions because the second step is conditional to the first. The two sub-processes are
now analysed in detail.
Cavity creation. The reversible work of cavity creation is:
Gc = -RTln<(X)>p
(S6)
and is the work to select the configurations containing the cavity from the ensemble of pure
solvent configurations (i.e., the exact position of the cavity is not important due to the spatial
homogeneity of the liquid, except for surface regions). This is an exact result of statistical
mechanics.4 Direct application of equilibrium statistical mechanics provides expressions for
Hc and Sc that can readily be worked out leading to:5
2
Hc =  H(X)[c(X) - p(X)]dX = <H(X)>c - <H(X)>p
(S7)
Sc = -R [c(X)lnc(X) - p(X)lnp(X)]dX =
= Rln<(X)>p + [(<H(X)>c - <H(X)>p)/T] = Sx + (Hc/T)
(S8)
The difference in the ensemble average enthalpy between the liquid configurations possessing
the desired cavity and the total liquid configurations gives rise to Hc. Two different
contributions for Sc emerge: (a) a reduction in the number of configurations because the
liquid configurations possessing the desired cavity are a very small fraction of the total liquid
configurations; this reduction of configurations provides a negative contribution to the
entropy in all liquids and is the solvent-excluded volume contribution, Sx; (b) a qualitative
difference between the two sets of liquid configurations because those possessing the cavity
would have a different distribution of energy levels; this contribution is distinct from the
solvent-excluded volume contribution, and is given by Hc/T because cavity creation is a
special process: the liquid configurations possessing the desired cavity are a sub-ensemble of
the total liquid configurations.6 Equations (S7) and (S8) show that Hc is exactly
compensated for by the non-solvent-excluded volume entropy contribution upon cavity
creation.5,6
Therefore, Gc is entirely entropic and due to the solvent-excluded volume effect
associated with the reduction in the size of the configuration space accessible to solvent
molecules (note that this result has nothing to do with classic SPT):
Gc = -TSx
(S9)
It is a measure of the loss in configurational-translational entropy of solvent molecules upon
cavity creation.
Turning on the attractive potential. The reversible work to turn on the attractive
solute-solvent potential is:
3
Ga = -RTln<exp[-a(X)/RT]>c
(S10)
Application of the Gibbs-Helmholtz equation leads to:7
Ha = <a(X)>c + [<H(X) + a(X)>s - <H(X) + a(X)>c] = <a(X)>c + Har
(S11)
where the subscript s means that the ensemble average is performed over the solution
configurations whose probability density function is:
s(X) = (X)exp-[a(X) + H(X)]/RT /  (X)exp-[a(X) + H(X)]/RTdX
(S12)
The solution configurations are the fraction of solvent configurations possessing the cavity
occupied by the solute molecule that interacts with the surrounding solvent molecules.
According to Eq. (S11), the enthalpy change consists of two parts: the first is the average
solute-solvent interaction energy excluding the effect from any solvent reorganization; the
other is the enthalpic contribution arising from the solvent reorganization upon turning on the
attractive solute-solvent potential. The entropy change is:
Sa = Rln<exp[-a(X)/RT]>c + [<a(X)>c/T] + (Har/T)
(S13)
By setting  = a(X) - <a(X)>c, expanding in power series the exponential function, and
keeping in mind that <>c  0, one obtains:8,9
Rln<exp[-a(X)/RT]>c  - [<a(X)>c/T] - [<2>c/2RT2]
(S14)
When the attractive solute-solvent potential a(X) is weak in comparison to the strength of
the solvent-solvent interactions, the fluctuations in the value of <a(X)>c are small, and the
second term on the right-hand side of Eq. (S14) can be neglected. Thus:
4
Sa  (Har/T)
(S15)
and the solvent reorganization associated with the process of turning on the attractive solutesolvent potential proves to be a compensating process. The condition of small fluctuations in
the <a(X)>c value is verified for nonpolar solutes in water because the attractive solutewater van der Waals interactions are weak in comparison to water-water H-bonds.7-9 Thus,
the Gibbs energy change is simply given by the average solute-solvent interaction energy:
Ga = <a(X)>c  Ea
(S16)
By combining Eqs. (S9) and (S16), one obtains:
G = Gc + Ea = -TSx + Ea
(S17)
that corresponds to Eq. (1) of the main text.
For polar and charged solutes the situation might be different because the water
molecules in the hydration shell make direct H-bonds with the solute molecule. Such direct
solute-water H-bonds have to be correctly accounted for in the partitioning of the total
enthalpy change: half of their energy has to be considered as part of the water-water H-bond
reorganization.10 Adopting this procedure, enthalpy-entropy compensation proved to hold
also for the water-water reorganization associated with the hydration of n-alcohols.10
More on the entropic nature of Gc
Cavity creation can be considered as a filtering process wherein all possible solvent
configurations of a pure liquid system are screened for those that happen to have a cavity of
the given size.5,11 Two different sources of entropy change can then be identified for the
process. First, since the entropy depends on the total size of the ensemble, the simple fact that
there is less number of configurations after the filtering produces a decrease in entropy. This
entropy loss is denoted as Sx (x for solvent-exclusion). It arises from the direct perturbation
5
(the act of filtering) and represents the solvent-excluded volume effect. The second source is
more intrinsic. If the filtering were done randomly, the ensemble of configurations after the
filtering would have the same character as the original ensemble and will have the same
entropy except for Sx due to the reduction in the size of the ensemble. However, if the
filtering is done on some non-random criteria (i.e., the configurations with the desired cavity),
the important configurations represented by the filtered ensemble will be significantly
different from those in the original ensemble. Since these different representative
configurations will generally have different degeneracy, this produces an additional entropy
difference over and above that due to the ensemble-size reduction alone. Thus, one obtains:
Sc = Sx + Scr
(S18)
where Sc is the entropy change due to the cavity creation, and Scr is the entropy change
due to the change in the dominant configurations before and after cavity creation. The Scr
term arises from the reorganization of the solvent as a result of the filtering process; it is a
solvent response term upon the constraint imposed by the presence of the cavity. It was shown
that, for all liquids, there is perfect compensation between Scr and the enthalpy change due
to cavity creation:5
Scr = Hc/T
(S19)
and that, therefore, the reversible work of cavity creation is an exact measure of the solventexcluded volume entropy loss:
Gc = -TSx
(S20)
that corresponds to Eq. (S9) and confirms the entropic nature of Gc.
6
References
1. A. Ben-Naim, Solvation Thermodynamics (Plenum Press, New York, 1987).
2. B. Lee, Biopolymers 31, 993-1008 (1991).
3. B. Widom, J.Phys.Chem. 86, 869-872 (1982).
4. R.C. Tolman, The Principles of Statistical Mechanics (Oxford University Press,
London, 1938).
5. B. Lee, J.Chem.Phys. 83, 2421-2425 (1985).
6. G. Graziano, J.Phys.Chem.B 110, 11421-11426 (2006).
7. G. Graziano and B. Lee, J.Phys.Chem.B 105, 10367-10372 (2001).
8. B. Lee, Biophys.Chem. 51, 271-278 (1994).
9. G. Graziano, Chem.Phys.Lett. 429, 114-118 (2006).
10. G. Graziano, Phys.Chem.Chem.Phys. 1, 3567-3576 (1999).
11. G. Graziano, J.Chem.Phys. 120, 4467-4471 (2004).
7