INVITED ARTICLE

PCCPThe hydrophobic eﬀect
B. Widom, P. Bhimalapuram and Kenichiro Kogay
Department of Chemistry, Baker Laboratory, Cornell University, Ithaca,
New York 14853-1301, USA
Received 10th April 2003, Accepted 4th June 2003
First published as an Advance Article on the web 24th June 2003

The thermodynamics of the hydrophobic eﬀect, as measured primarily through the temperaturedependence of solubility, is reviewed, and then a class of models that incorporate the basic mechanism ofhydrophobicity is described. These models predict a quantitative relation between the free energy ofhydrophobic hydration and the strength of the solvent-mediated attraction between pairs of solute molecules. It
is remarked that the free energy of attraction being just of the order of the thermal energy kT may be important
for the eﬀective operation of the hydrophobic eﬀect in proteins. Deviations from pairwise additivity ofhydrophobic forces are also brieﬂy discussed.

I.

Introduction

Hydrocarbons are only slightly soluble in water: they are hydrophobic. The accommodation of a hydrocarbon molecule inwater is accompanied by an increase in an associated freeenergy. Hydrocarbons are not the only hydrophobes but they
are typical of the class. Characteristically, their solubility decreases with increasing temperature at low temperatures, which
provides an important clue to the mechanism of hydrophobicity. At higher temperatures the solubility, after reaching a minimum, often then increases with further increase of temperature.
These eﬀects are illustrated in Fig. 1, which shows the Ostwald absorption coeﬃcient S of methane in water (the ratioof the number density of dissolved methane to that in the equilibrium vapor), as a function of temperature.1 This is at a partial pressure of methane of 1 atm, although there is almost no
dependence on the pressure as long as the concentration of
methane in both phases is low (Henry’s law).
The unfavorable free energy change accompanying the dissolution of the hydrocarbon results from structural changes

Fig. 1 Ostwald absorption coeﬃcient S of methane as a function oftemperature T (from compilation of Battino1).
y Permanent address: Department of Chemistry, Okayama University,
3-1-1 Tsushimanaka, Okayama 700-8530, Japan.

DOI: 10.1039/b304038k

in the solvent around each solute molecule. This is the phenomenon of hydrophobic hydration. The total volume ofsolvent so aﬀected by a pair of solute molecules is less when
the two are close together than when they are far apart, as illustrated schematically in Fig. 2. The result is an eﬀective,
solvent-mediated attraction between the two. This is thehydrophobic attraction.
These eﬀects have long been recognized to be important inphysical chemistry and biochemistry. The subject thus has anenormous literature, ranging from works that are now classic2–17 to those more nearly current,18–85 many of these quite
sophisticated. A recent authoritative assessment of the statusof the ﬁeld with emphases diﬀerent from those in the present
account is in a review by Pratt.29 A beautiful earlier review
by Scheraga86 with an account of experimental results andemphasis on the role of hydrophobicity in biochemistry,should also be noted.
The thermodynamics of transfer of a molecule from one
phase to another is outlined in Section II, and then the thermodynamics of hydrophobic hydration as inferred from solubility
measurements such as those in Fig. 1 is presented in Section
III. What is seen there, among other principles, is that the dissolution of a hydrophobe in water is energetically favorable,

Fig. 2 Two hydrophobic molecules, (a) far apart, and (b) close
together. The regions within the dashed curves represent schematicallythe volumes of solvent that are signiﬁcantly aﬀected by the presence ofthe solutes. The total volume so aﬀected by the pair is smaller in (b)than in (a).

Phys. Chem. Chem. Phys., 2003, 5, 3085–3093
This journal is # The Owner Societies 20033085

perhaps contrary to naı̈ve expectation, but is entropically unfavorable to such an extent that the equilibrium solubility is very
low. This was the historic observation that led to our present
understanding of hydrophobicity and its implications.2 Those
two sections are an expanded version of an earlier treatment;87
the principles are mostly well known,11 but it is useful to bringthem together here for reference.
Section IV describes a class of models that incorporate themain mechanism of hydrophobicity as revealed by the thermodynamic studies. The models make detailed predictions abouthydrophobic attraction including a quantitative connection
between the strength of the attraction and the free energy
of hydrophobic hydration. These models, too, have been
described earlier;87 the present account includes some more
recent results.
In Section V the geometric idea expressed in Fig. 2 isextended to the simultaneous interaction of three solute
molecules, from which it is possible to make some speculative
inferences about the three-body contribution to the solventmediated interaction energy.
The main results are brieﬂy summarized in the concluding
Section VI

II. Thermodynamics of transfer
We ask here for the changes in various thermodynamic quantities accompanying the transfer of a molecule A from a phasea to a phase b. Each phase separately is in complete internal
equilibrium both before and after the transfer but the twophases need not be in equilibrium with each other. Later A will
be a hydrophobic molecule, a some reference phase containing
A, and b water or an aqueous solution, but for now A, a, and b
are general.
The cases of most interest are those in which the temperatures T of the two phases are equal and are the same before
and after the transfer (isothermal transfer). We will specify
that, and then contemplate either of two further circumstancesin which the transfer is made: that in which the volume of eachphase separately is the same before and after the transfer, orthat in which the pressure of each phase separately is the same
before and after.
Let maA and mbA be the chemical potentials of A in the respective phases. In the transfer with the volumes ﬁxed, the diﬀerence mbA  maA is the net change in the Helmholtz free energyof the two phases together; call it DFV :DF V ¼ mbA  maA :ð1Þeither phase is dilute in A the subtracted term accounts for thewhole of the dependence of DFV and DGp on the concentrationof A in that phase, and the resulting DFV and DGp are thenindependent of that concentration. This remains true, andDFV and DGp remain ﬁnite, even when cbA ! 0; i.e., in the limit
in which there is no A in the b phase prior to the transfer; while
the original DFV and DGp diverge to 1 in that limit. Whenthe phases are not dilute in A there is a residual dependenceof DFV and DGp on the concentrations, reﬂecting the fact thatthe other A molecules are then a signiﬁcant part of the environment from or to which the contemplated transfer of the A
molecule occurs.
Let DSV and DSp be the corresponding changes in the composite system’s entropy in the two diﬀerent transfer processes,
and deﬁneð2ÞNote that DFV ¼ DGp , but they are the DF and DG of diﬀerent
processes.
Signiﬁcant parts of DFV and DGp result merely from the differing concentrations of A in the two phases, but we wish toknow how much of DFV and DGp reﬂect only the intrinsic energetic and structural changes that occur in the phases as a resultof the transfer of the A molecule from one to the other. We
thus deﬁnecb
DFV ¼ DFV  kT ln A
caA

ð3Þ

cb
DGp ¼ DGp  kT ln A
caA

ð4Þ

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Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

cb
DSp ¼ DSp þ k ln A
:
caA

ð6Þ

DSV ¼ @DFV;
@T

DSV ¼ 

@DFV
@T

ð7Þ

where the temperature diﬀerentiations are at ﬁxed numbers ofmolecules of all the substances in each phase and at ﬁxed
volumes of both phases. The change in the composite system’senergy, DEV , accompanying the constant-volume transfer, is
DE V ¼ DF V þ TDS V ¼ DF V þ TDSV

ð8Þ

@DFV =T @DFV =T
¼
:
@1=T
@1=T

ð9Þ

¼

The constant-volume heat capacity CV of the two phases
together also changes as a result of the transfer, the net change
beingðDCV ÞV ¼

@DEV
@DSV
@DSV
¼T
¼T
@T
@T
@T

ð10Þ

@ 2 DFV@ 2 DFV
¼ T
;
@T 2
@T 2

ð11Þ

¼ T

where the second subscript V in (DCV)V speciﬁes ﬁxed-volumetransfer.
In the transfer at ﬁxed pressure the quantities of most interest, besides DGp and DSp , are the change DHp in the composite
system’s enthalpy and the change (DCp)p in its constantpressure heat capacity; but now the relations analogous to(7)–(11) are slightly more complicated because the volumes V
in cA ¼ NA/V are no longer ﬁxed when the temperaturechanges at ﬁxed pressure and composition. Let ea and eb be
the coeﬃcients of thermal expansion of the two phases. Thenfrom (1), (4), and (6), the analog of (7) is
DSp ¼ where k is Boltzmann’s constant and caA and cbA are the number
densities of A in the respective phases; and now DFV ¼ DGp . Ifð5ÞIf either phase is dilute in A there is no dependence of DSV andDSp on the concentration of A in that phase, but otherwise
there is. Again, these remain ﬁnite in the limit cbA ! 0, while
DSV and DSp would be inﬁnite.
Now consider ﬁrst the ﬁxed-volume transfer. Each number
density cA is NA/V with NA the number of A molecules in
the phase and V its volume. During the transfer, NA increases
or decreases by 1 (in the b or a phase, respectively), while V is
ﬁxed. Then from (1), (3), and (5),In the transfer with ﬁxed pressures, the same diﬀerencembA  maA is the change in the composite system’s Gibbs freeenergy; call it DGp :DGp ¼ mbA  maA :cb
DSV ¼ DSV þ k ln A
caA

@DGp
;
@T

DSp ¼ @DGpþ kTðeb  ea Þ;
@T

ð12Þ

where now the temperature diﬀerentiations are at ﬁxed numbers of molecules of all the substances in each phase and at
ﬁxed pressure of each phase. With the same understandingof what is now held ﬁxed in the temperature diﬀerentiations,the analogs of (8)–(11) are
DH p ¼ DGp þ TDS p ¼ DGp þ TDSp¼@DGp =T @DGp =Tþ kT 2 ðeb  ea Þ
¼
@1=T
@1=T

@ 2 DGp@T 2@ 2 DGp
@ðeb  ea Þ
¼ Tþ 2kTðeb  ea Þ þ kT 2
:
2
@T
@T

ð13Þ
ð14Þ

ðDCp Þp ¼ T

ð15ÞA special case of the transfer process is that in which thetwo phases are in equilibrium with respect to the transfer of
A. This is when maA ¼ mbA , so that DFV ¼ DGp ¼ 0. Then from
(1) and (2),DF V ¼ DG p ¼ kT ln Sðequilibrium transferÞ;are those in which b is a condensed phase dilute in A (or fromwhich A is absent), while (i) a is a dilute gas containing A; or(ii) a is a condensed phase dilute in A; or (iii) a is a condensed
phase that is pure or nearly pure A.
In these situations the temperature derivatives of S may be
written as total derivatives, d/dT, and the thermodynamic
quantities of interest are obtainable from S, i.e., from the equilibrium partition coeﬃcient, by the formulas
DGp
DFV
¼
¼  ln S
ð18Þ
kT
kT
DSp
DSV
d
d
ðT ln SÞ;ðT ln SÞ þ Tðeb  ea Þ ð19Þ
¼
¼
dT
dT
k
k
DHp
DEV
d
d
ð20Þ
¼T
¼T
ln S;
ln S þ Tðeb  ea Þ
kT
kT
dT
dT

ð16Þ

ðDCV ÞV
d2 ðT ln SÞ
¼T
dT 2
k

where
S ¼ ðcbA =caA Þeq ;ð17Þ

the ratio of the number densities of A in the two phases when
they are in equilibrium with each other. This is the key toobtaining the thermodynamics of transfer from measurementsof equilibrium solubility.
When b is a liquid or solid solution dilute in A and a is adilute gas, S is the Ostwald absorption coeﬃcient, a common
measure of gas solubility, as illustrated for methane in Fig. 1.
Since b is then a hardly compressible condensed phase its properties as a solvent are insensitive to the pressure to moderatepressures; and the equilibrium ratio (cbA /caA )eq of the concentrations of A in these dilute phases is independent of their separate values (Henry’s law); so S is practically a function of
the temperature alone (as remarked in Section I). This circumstance, in which both phases are dilute in A, is also, we saw,
that in which DFV , DGp , DSV , and DSp , and so also DEV ,DHp , (DCV)V , and (DCp)p , are independent of the concentrations. Thus, their values are independent of whether or not
the transfer of the A molecule from a to b occurs with the ratio
cbA /caA equal to that for which the two phases are in equilibrium. In this case, then, the signiﬁcance of (16) and (17) isthat DFV , DGp , and the quantities derivable from them are
the same as when the transfer occurs with the two phases inequilibrium, even when it does not; and these quantities arethen obtainable from the equilibrium solubility S, which is virtually a function of the temperature alone.
When a and b are both condensed phases, but both again
dilute in A, the same considerations apply. The equilibrium
ratio S in (17) is then the partition coeﬃcient of A between
the phases at equilibrium, and is virtually independent of thepressure and of the separate values of cbA and caA ; it is practically a function of the temperature alone; and DFV and DGp ,as given by (16) and (17), and the quantities obtainable fromthem, are the same as when the transfer occurs with the twophases in equilibrium even when it does not.
Another case of practical interest is that in which the aphase is pure or nearly pure liquid or solid A while b is again
a condensed phase dilute in A. We continue to deﬁne DFVand DGp by (3) and (4) and DSV and DSp by (5) and (6),so the remaining relations (7)–(17), which follow from (3)–(6), still all hold. Now caA in the denominator of the argumentof the logarithms in (3)–(6) is practically a constant, and,since we are supposing b to be dilute in A, both DFV and
DGp are independent of the remaining variable cbA . Thus,(16) with (17) hold in the case, too.
Summarizing, there are three common situations in which
all the relations (7)–(16) hold, with S, deﬁned by (17), practically a function of the temperature alone; i.e., practically independent of pressure to moderate pressures and independent of
the concentration of A in the dilute phase(s). These situationsðDCp Þp
k

¼T

ð21Þ

b
a
d2 ðT ln SÞba
2 @ðe  e Þ
þ
2Tðe

e
Þ
þ
T
dT 2
@T

ð22Þ

as now follow from the more general formulas stated earlier.
Applications of these formulas to the transfer of gaseoushydrocarbons into water are described in the next section,
where they illustrate the principles of hydrophobic hydration.

III. Hydrophobic hydration
In Table 1 we give the results for the transfer of a molecule ofmethane, ethane, propane, or n-butane from the gas phase into
water, at 300 K, as derived from (18)–(22), with S taken fromthe compilations of Battino.1,88–90 The b phase is pure or practically pure water. For eb and its temperature derivative,
required for the correction terms in (19), (20) and (22), we took
the data from the compilation by Rowlinson and Swinton.91
Those data are for water as it coexists with its vapor, so not
at 1 atm, whereas the data for S are at 1 atm, but the distinction is of no importance since neither the coeﬃcient of thermal
expansion of water nor the Ostwald absorption coeﬃcient S
varies signiﬁcantly with the pressure over this range. Whatmakes the pressure variation of eb even smaller is that it is proportional to the temperature variation of the isothermal compressibility,92 and 300 K is not too far from 320 K, where thecompressibility of water has a temperature minimum.91 The aphase, meanwhile, is a dilute, practically ideal gas, for whichea ¼ 1/T and dea/dT ¼  1/T2. The correction terms in (19),(20), and (22) are small; the diﬀerence between DSV /k andDSp /k and between DEV/kT and DHp/kT at 300 K is only
0.9. By coincidence, the two correction terms on the right-handside of (22), which are opposite in sign, nearly cancel eachother, with the result that (DCV)V and (DCp)p are nearly equal
at 300 K.
Table 1 has many important lessons. The ﬁrst and mostobvious is that the free energy of transfer is positive and isseveral multiples of the thermal energy kT, which, via (18), justreﬂects the low solubility of the hydrocarbons in water. It isthe next entries in the table that tell the hydrophobicity story:
that the low solubility is a consequence of an unfavorable
Table 1 Thermodynamics of transfer of a hydrocarbon moleculefrom the gas phase into water at 300 K

CH4
C2H6
C3H8
n-C4H10

DGp
DFV
¼
kT
kT

DSV
k

DSp

DEV
kT

DHp
kT

ðDCV ÞVkðDCp Þp

k

3.4
3.1
3.4
3.6

7.5
9.7
11.0
12.5

8.4
10.6
11.9
13.4

4.1
6.6
7.6
8.9

5.0
7.5
8.5
9.8

29
36
41
40

29
36
41
40

Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

k

3087

(negative) entropy change that overweighs a favorable (negative) energy change. The diﬀerence between DSV /k andDEV/kT (or, equivalently, between DSp /k and DHp/kT ), which is DFV /kT ( ¼ DGp /kT ), is nevertheless smaller
than either one separately; i.e., DEV/kT, while less than
DSV /k, is nevertheless more than half of it (DHp/kT is
more than half of DSp /k), so there is considerable cancellation between them. That the entropy of transfer is negative
at 300 K means, from (7), that at this temperature the freeenergy of transfer becomes more positive (more unfavorable)with increasing temperature; and that the energy of transfer
is negative at 300 K means, from (20), that at this temperature
the solubility decreases with increasing temperature. Thus, at
300 K, these hydrocarbons—and hydrophobes generally—
become more hydrophobic with increasing temperature.
The heat capacity changes on transfer, (DCV)V and (DCp)p ,are positive and huge: many multiples of the Boltzmann
constant k. This was one of the earliest observations of a‘‘ hydrophobic eﬀect ’’.13,93 This and the negative energetic
and entropic changes are ascribable to structural changes inthe solvent, generally thought to be the strengthening of (orformation of additional) hydrogen bonds among the watermolecules neighboring the solute.2
A further observation from the table is the manifestation of‘‘ enthalpy–entropy compensation ’’,15,94,95 according to which
free-energy changes (free energies of transfer in this instance)
are nearly constant through a homologous series, although
the changes in the separate energetic and entropic componentsof the free energy vary substantially through the series.
Often when a solute can itself form hydrogen bonds withwater the result is a closed-loop solubility curve,96 illustrated
here in Fig. 3. The ﬁgure shows the temperature dependenceof the mutual solubility of two hypothetical substances A
and B at a ﬁxed pressure (or while simultaneously in equilibrium with vapor). The tilt of the solubility curve is exaggerated in the ﬁgure to make clear that in general thetemperature T1 of minimum solubility of A in B is not thesame as the temperature T4 of minimum solubility of B in A
(although in theoretical models with an artiﬁcial symmetry it
may be97). One may think of the A-rich side of the loop as
the phase earlier called a and the B-rich side as the aqueous
phase b. Since on the lower part of the loop the solubility of
A in B (as well as that of B in A) is decreasing with increasing
temperature, it is manifesting the hydrophobic eﬀect.23 That

Fig. 3 Closed-loop solubility curve with components A and B. Concentration variable xB is the mole or mass fraction of B. Points C andC0 mark the lower and upper critical solution points. Temperatures T1
and T4 are those of the minimum solubility of A in B and of B in A,
respectively, while the intermediate temperature T2 is that at which
DHA and DSA vanish, and T3 that at which DHB and DSB vanish.

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Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

the solubility passes through a minimum with increasing temperature and thereafter increases is a characteristic also of thesolubility of most hydrocarbons in water1,88–90(as seen formethane in Fig. 1).
For the transfer of a molecule between coexisting phases
at temperature T, the enthalpy and entropy changes DH
and DS (6¼ DS*) are related by DH ¼ TDS (6¼ TDS*). Referring still to Fig. 3, we will for now use the notation DHA( ¼ TDSA) for the enthalpy of transfer of an A moleculefrom the phase a relatively rich in A to the phase b poorer
in A, and DHB ( ¼ TDSB) for the enthalpy of transfer of amolecule B from b to a. One may then show that DHA andDSA are negative for temperatures below some temperature
T2 between T1 and T4 , they vanish at T2 , and are positive
above T2 ; while DHB and DSB are negative below some temperature T3 between T1 and T4 , they vanish at T3 , and are
positive above T3 ; with T3 < T2 if the coexistence curve isoriented as in the ﬁgure.
For the transfer of an A molecule from the phase a richer in
A to the phase b poorer in A, the modiﬁed entropy change
DSA , deﬁned analogously to DSp in (6), is algebraically smaller(more negative or less positive) than DSA itself, and so is stillnegative when DSA vanishes at the temperature T2 .
Where the b phase is very dilute in A it hardly matters to thevalues of the DH and DS* of transfer whether the transfer ofthe A molecule from the a phase is to the coexisting b phase(where they are then DHA and DSA ) or to pure B (pure water,say, if b is an aqueous phase). In the latter case the unmodiﬁed
DS of transfer would be inﬁnite. Likewise, if the a phase isnearly pure A it hardly matters to the values of the DH and
DS of transfer whether the transfer of the A molecule to b isfrom the coexisting a phase (when they are then DHA andDSA) or from pure A, and now in either case there is hardly
any diﬀerence between DS and DS*.
This completes our review of the thermodynamics of hydrophobic hydration. We turn next to microscopic models and to
the phenomenon of hydrophobic attraction.

IV.

Microscopic models

There are many and diverse microscopic models of solutions ofhydrophobes in water (see refs. 9, 19, 21, 29, 35, 40, 77, 78,
98 among many others), but we shall here describe only those
we have ourselves been studying, which is a class of latticemodels.87,99–102
Fig. 4 is an example in two dimensions with lattice coordination number Z ¼ 4, but the same basic model may be in any
number of dimensions with a variety of lattice types. Solvent
molecules are represented by the larger circles, one at each lattice site. Each may be in any of q ( 2) possible orientations
or internal states, of which one (state number 1, say) is distinguished from the q  1 others: neighboring solvent moleculesinteract with an energy w when both are in that special state
1 but with a higher energy u ( > w) otherwise. Solute molecules
are represented by the smaller circles in Fig. 4. They can beaccommodated only on the bonds between lattice sites, i.e.,
between neighboring pairs of solvent molecules, and only thenif both solvent molecules of the neighboring pair are in thespecial state or orientation 1. Thus, accommodation of a solute
restricts the possible orientations of the neighboring solvent
molecules and at the same time puts that solvent pair in its
low-energy conﬁguration. Thus, the model has built into it
the basic mechanism of hydrophobicity: the forced accommodation of the solute is energetically favorable but entropicallyunfavorable.
Lest the models seem more artiﬁcial than they are, it shouldbe emphasized that, as in any lattice ﬂuid model,97,103 themolecules are not to be thought of as conﬁned to lattice sites
or bond centers. One should think of the volume divided into

cubic lattices, respectively, in three dimensions, etc.). In theequivalent one-component lattice gas106 in which the cells areof volume v0 , the activity is z, and the near-neighbor interaction energy is e, the equivalences aree ¼ u  w; v0 z ¼ 1=ðq  1Þ:Fig. 4 The lattice model. Large open circles at the lattice sites represent the solvent molecules, small ﬁlled circles on the bonds betweensites represent the solute molecules.

cells (only as a coordinate system, not with impenetrable
walls), so that every molecule has access to any point in thewhole volume. The only artiﬁciality is that the interactions
between molecules now depend on discrete rather than continuous distance variables.
The two parameters u  w and q  1 are properties of themodel solvent alone. The model has a third parameter, v,
which is the energy of interaction between an accommodated
solute molecule and the two solvent molecules that neighbor
it. All three parameters enter into the solubility S, deﬁned in(17), while the solvent-mediated part of the potential of mean
force between pairs of solutes is determined by the solvent
parameters u  w and q  1 alone.
These follow from the expressions87,99,100
S ¼ P11 ev=kT
W ðr1 ; r2 Þ ¼ kT ln

Pðr1 ; r2 Þ
;
P211

ð23Þ

1
2H ¼ Zðu  wÞ  kT lnðq  1Þ2

The probability P1 that any given solvent molecule in thehydrophobic-interaction model be in the special state 1 rather
than in any of the q  1 other states is equivalently the probability that any given cell in the lattice gas model be occupied
rather than empty or that any given spin in the Ising model bein the direction of the ﬁeld rather than opposed.
The model in any number of dimensions and for any
coordination number may be treated in Bethe–Guggenheim
approximation107,108 as well as by simulation. (For the application of the Bethe–Guggenheim approximation to diﬀerent lattice models of water and hydrophobic solvation see Besselingand Lyklema35 and Eads.78) The approximation is exact on a
Bethe lattice (Cayley tree), on which there are no closed loops,
so that, as in one dimension, between any two sites there
is only a single, unique path. Then W(r1 ,r2) may be written
W(r), where r is the distance between bond centers measured
as a number of lattice steps; and then via (24),W ðrÞ ¼ kT ln gðr  1Þ

ðr  2Þ

ð27Þ

where g(r) is the pair distribution function (radial distributionfunction) of the equivalent one-component lattice gas in one
dimension. For r ¼ 1,
W ð1Þ ¼ kT ln y;ð28Þwith y (0  y  1) the dimensionless density of the one-component lattice gas. As remarked earlier, this lattice-gas density isalso the probability P1 that any solvent molecule in our original model be in the special state 1. Further, the solubility S in
our model, via (23), is given in Bethe-Guggenheim approximation by
S ¼ y2 gð1Þev=kT :

ð29Þ

ð24Þ

for the ﬁrst of which it is assumed that the phase that is inequilibrium with the saturated solution is a dilute gas. HereP11 is the probability that any neighboring pair of solvent
molecules in the pure solvent will both be found to be in
the special orientation 1; P(r1 ,r2) is the probability thatthe solvent molecules that are the neighbors of the bonds centered at r1 and r2 will all be found to be in that special orientation; and W(r1 , r2) is the solvent-mediated part of thepotential of mean force between a pair of solute molecules
at r1 and r2 in the inﬁnitely dilute solution; i.e., the potentialof mean force from which the direct solute–solute interaction,
whatever that might be, has been subtracted. The parametersu  w and q  1 are implicit in P11 and P(r1 , r2) while v
appears explicitly in (23). The nearest-neighbor pair correlation P11 and the pair–pair correlation function P(r1 ,r2) are
obtained analytically in the one-dimensional version of themodel99 and with suﬃcient accuracy by Monte Carlo simulation in two or three dimensions.87,100
For the simulations it is convenient to recognize the equivalence of the model to a related Ising model or one-component
lattice gas.102 With conventional Ising-model notation ,104,105
the spin–spin interaction-energy parameter J and magnetic
ﬁeld H in the equivalent Ising model are related to u  w andq  1 by1
J ¼ ðu  wÞ;
4

ð26Þ

ð25Þ

where Z is the coordination number of the lattice (Z ¼ 2 in one
dimension, Z ¼ 6 or 8 for the simple cubic or body-centered

In this approximation, the lattice-gas g(r) and density yrequired in (27)–(29) are expressible in terms of the solvent
parameters u  w and q  1 of the hydrophobic-interaction
model by



1
Q1 r
gðrÞ ¼ 1 þ
1ðr  1Þ
ð30Þ
y
Qþ1
1
1þa
1¼
y
að1 þ caÞð31Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ 4ðc  1Þð1  yÞy

ð32Þ

where
Q¼

c ¼ eðuwÞ=kT
1 þ ca Z1¼ ðq  1Þa;
1þa

ð33Þ
ð34Þ

with Z again the coordination number of the lattice. The lattice-gas density y is thus given parametrically (parameter a)in terms of the model parameters q  1 and u  w (and Z
and the temperature T ) by (31) and (34), with c deﬁned by (33).
With Z ¼ 2 the Bethe–Guggenheim approximation becomesthe exact one-dimensional version of the model.99 In the limit
Z ! 1 and u  w ! 0 with ﬁxed f ¼ Z(u  w) it becomes themean-ﬁeld approximation, in which g(r)1 for all r  1 andfor which y is given implicitly in terms of q  1 and f by q  1 ¼(1/y  1)exp(yf/kT ).
One object in the study of these models is to ﬁnd a connection between the free energy of hydrophobic hydration (DFV
Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

3089

or DGp , Section III), and the solvent-mediated part, W(r1 , r2),of the potential of mean force between hydrophobic solutes.
To this end we choose the parameters u  w and q  1 to matchthe experimental solubility S of methane as closely as possible
over the 60 temperature interval 273 K  T  333 K, and thenwith these values of the parameters calculate W.
We continue to deﬁne f ¼ Z(u  w). In Fig. 5 we show thenearly perfect ﬁt to the experimental S obtained with themodel in Bethe–Guggenheim approximation with Z ¼ 8 andthe model parameters
f=k ¼ 1008 K; q  1 ¼ 6:16; v=k ¼ 464:3 K:

ð35Þ

The corresponding plot when the properties of the model on
the body-centered cubic lattice (Z ¼ 8) are obtained by Monte
Carlo simulation87 and the solubility then ﬁt with the parameter values
f=k ¼ 1026 K; q  1 ¼ 6:7; v=k ¼ 406:7 K

ð36Þ

is almost indistinguishable from that in Fig. 5.
It is noteworthy that the parameter values in (35) and (36),where Z ¼ 8 in both, are not very diﬀerent. It should also be
noted that it is not a forgone conclusion that the experimental
S even over this restricted temperature range could have been
ﬁt by these three-parameter functions. Only the two parameters u  w and q  1 occur in the factor P11 in (23), so theslope and curvature and thus the whole shape of a plot ofkT ln S as a function of T have to be ﬁt by those alone, while
the third parameter, v, corresponds merely to a constant displacement of such a plot without a change in its shape. In
Bethe–Guggenheim approximation the ﬁt is better the greater
the coordination number Z, and so, although already nearly
perfect at Z ¼ 8, is even better in mean-ﬁeld approximation(Z ! 1), where the ﬁt is with the parameter values
f=k ¼ 912:4 K; q  1 ¼ 4:95; v=k ¼ 498:2 K;

ð37Þ

while the ﬁt is poorest, although still acceptable, in one dimension (Z ¼ 2).87
We note in (35)–(37) that the model parameter v is positive.
One must make a distinction between the interaction energy
between solute and solvent, which here is v, and the energy
or enthalpy change DEV or DHp of the whole system that
accompanies the forced accommodation of the solute. The latter are negative here, as well as in reality, because they follow
from the experimental S, which we have ﬁt nearly exactly withthe parameter values in (35)–(37). But we note also in (35)–(37)that f, which is Z(u  w), is greater than v; so the favorable
energy and enthalpy of hydration is a consequence of theFig. 5 Fit of the model S in Bethe–Guggenheim approximation withZ ¼ 8 to the experimental S for methane. The points are the theoretical values with the parameters given in (35); the curve is from Battino’s
interpolation formula ﬁtting the experimental results.1

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Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

Fig. 6 W(r)/kT for the model in Bethe–Guggenheim approximation
with Z ¼ 8.

energetically favorable change in the solvent–solvent interaction that accompanies the dissolution of the solute, which
overweighs what in this model (not necessarily in reality) is
the unfavorable solute–solvent interaction.
With the parameters in (35) or (36) that ﬁt the experimental
S, one may now calculate the solvent-mediated part of thepotential of mean force, W, between methane molecules in
the inﬁnitely dilute solution. This follows from (24) in generalfor the models of this class, or in particular from (27) and (28)with (30)–(34) in Bethe–Guggenheim approximation. In Fig. 6
we show W(r)/kT in this approximation with Z ¼ 8, as a function of r at each of the three temperatures 273 K, 298 K, and
333 K. These curves are strongly reminiscent of the theoreticalcurves in Marčelja et al.,17 although those are indexed bysolute size rather than temperature. In Fig. 7, from ref. 87,
we show the corresponding results from the Monte Carlosimulations of the model on the body-centered cubic lattice.
In the latter plot W(r1 ,r2) is again called W(r), with r again
in multiples of the shortest distance between bond centers,pbut this shortest distance is now the fraction 1/ 3 ¼
0.577   of the bond length itself. In both ﬁgures, then, ‘‘ r ¼ 1 ’’is the closest the centers of two hydrophobic molecules may
come if both molecules are imagined located at the lattice’s
bond centers.
Not visible in Fig. 6 on the scale of that plot, and scarcely
visible in Fig. 7, is the crossing of the curves for the various
temperatures: the higher the temperature the stronger thehydrophobic attraction as measured by W(1) but the shorter
is its range as measured by its rate of decay as r ! 1. For the
body-centered cubic lattice, and generally in three dimensions

Fig. 7 W(r)/kT by Monte Carlo simulation of the model on thebody-centered cubic lattice.

with the direct interactions of short range, W(r) decays proportionally to (1/r)exp(r/x), and in Bethe–Guggenheim approximation proportionally to exp(r/x), where the decay lengths
x decrease with increasing temperature. This may not be an
important feature of hydrophobic forces, however, because
W(r) is already so small when the curves cross: the three curves
in Fig. 6 taken in pairs cross at r ¼ 2.7, 4.0, and 5.9.
This attraction, which is largely entropic in origin, has beentermed the ‘‘ hydrophobic bond ’’.3,6 In both ﬁgures it is seenthat the strength of the attraction, W(1), is only about kT.
The direct van der Waals attraction between pairs of solutes
at this distance is only about 15 or 20% of the solvent-mediatedattraction,19,109 so the total is still only about kT. This means,for example, that of the ‘‘ hydrophobic bonds ’’ in the native
structure of a protein, at any moment approximately the fraction 1/(1 + e) ﬃ 1/4 of them are ‘‘ broken ’’. This may be
important for the dynamical functioning of the protein.110
This has illustrated the calculation of the strength W(1) of
the hydrophobic attraction in the model, using parameters thatclosely ﬁt the experimental S; or, equivalently, from (16), thatﬁt the experimental hydration free energy DFV or DGp . Fig. 8
now shows how W(1)/kT varies with DGp /kT. Shown are theBethe–Guggenheim approximation with Z ¼ 2, 3, 8, and 1(Z ¼ 2 is the one-dimensional model and Z ¼ 1 the meanﬁeld approximation), and the results of the simulation on thebody-centered cubic lattice. We see a steady progression withZ. Over this restricted temperature range all the plots arenearly linear: almost exactly linear from the simulations butwith slight positive curvature for Z ¼ 2 and 3 and slight negative curvature for Z ¼ 8 and 1 in the Bethe–Guggenheim
approximation. They are nearly parallel with a common slope
near 0.7.
These plots answer the question, at least for this class ofmodels, of how the strength of the solvent-mediated attraction
between a pair of hydrophobic solute molecules depends on
the hydration free energy of a single one; i.e., they are a quantitative answer to the question implicit in Fig. 2. We note againthat the hydrophobic eﬀect, i.e., DGp > 0 and W(1) < 0,becomes stronger with increasing temperature. We also see
again that W(1) is only about kT, while DGp , from the ﬁgure
or from Table 1, is about 3kT. One can understand why
W(1) < DGp .87 Suppose in Fig. 2 that the volume of solvent
structurally and energetically aﬀected by the presence of an
isolated solute molecule is v1 and that the total volume so
aﬀected by a closely spaced pair of solute molecules is v2 .

Fig. 8 Variation of W(1)/kT with DGp /kT; W(1) is the strength of
the hydrophobic attraction between a pair of solute molecules while
DGp is the hydration free energy of a single one. The solid curves are
the Bethe–Guggenheim approximation with the indicated coordination
numbers Z; the dashed curve is the result of Monte Carlo simulation of
the model on the body-centered cubic lattice, for which Z ¼ 8.

Let the resulting unfavorable free-energy density in thoseaﬀected regions be some f > 0. Then DGp ¼ v1f > 0; while
the solvent-mediated potential energy, W(1) in the notationof our model, is W(1) ¼ (v2  2v1)f, which is negative (attractive) because v2 < 2v1 . But also v2 > v1 , so v2  2v1 >  v1 ;i.e., W(1) < DGp . In this argument we have assumed thatthe free energy density (what we have called f) in the lensshaped overlap region and in the non-overlap region are notvery diﬀerent.
A related argument allows one to guess the sign of the threebody contribution to the hydrophobic attraction. This is the
subject of the next section.

V. Three-body potential
In an analysis of the one-dimensional version of this latticemodel101 it was found that the solvent-mediated part of thepotential of mean force among n solute molecules on the bonds
centered at x1 , x2 ,. . ., xn with x1 < x2 <   < xn is the sum of
the pair potentials between nearest neighbors only:W ðr12 ; r23 ;    ; rn1;n Þ ¼ W ðr12 Þ þ W ðr23 Þ þ    þ W ðrn1;n Þð38Þ
where r12 ¼ x2  x1 , etc. With three solute molecules at x1 , x2 ,x3 , this isW ðr12 ; r23 Þ ¼ W ðr12 Þ þ W ðr23 Þ:ð39ÞThis is greater than the sum of the three pair potentials,
W ðr12 ; r23 Þ  W ðr12 Þ  W ðr23 Þ  W ðr13 Þ ¼ W ðr13 Þ > 0
ð40Þ
because the pair potential W(r) is negative (attractive); i.e.,
the three-body contribution to the energy of interaction of
three such solute molecules is positive: the three-body force
is repulsive.
One may guess that the same will be true in near linear conﬁgurations of three hydrophobic solutes in two or three dimensions as well, while an extension of the argument at the end of
Section IV indicates that it may also be true in triangular conﬁgurations of the three. An equilateral conﬁguration of three isshown schematically in Fig. 9. The volume of solvent aﬀectedby the presence of a single solute molecule is called u1 ; it is thesame as what was called v1 in the argument at the end of Section IV. The volume of the lens-shaped region of overlap of
two such regions is u2 , previously called 2v1  v2 . The volumeof the curvilinear region of triple overlap is u3 .
The total volume of solvent aﬀected by the presence of thesolutes in this conﬁguration is 3u1  3u2 + u3 . But with W1 ,W2 , and W3 the one-, two-, and three-body eﬀective potentials,
the total excess free energy due to the presence of the threesolute molecules is 3W1 + 3W2 + W3 , because there are three
distinct single bodies, three distinct pairs, and one triple. The

Fig. 9 Three solute molecules in an equilateral conﬁguration. The
volume of solvent aﬀected by the presence of a single solute molecule
is u1 , the volume of the lens-shaped region of overlap of two suchregions is u2 , and the volume of the curvilinear region of triple overlap
is u3 .

Phys. Chem. Chem. Phys., 2003, 5, 3085–3093

3091

one-body potential W1 is the earlier hydration free energy DGp ,which was v1f, now called u1f; the pair potential W2 is what
was earlier called W(1), the eﬀective potential between a closely
spaced pair of solutes, which was (v2  2v1)f, now called u2 f.
Thus, the three-body potential W3 , which is our present
object, is here seen to be W3 ¼ u3f, which is positive. Thus,in this conﬁguration, as in the linear conﬁguration in the
one-dimensional model, the three-body force is repulsive,
which is what we wished to determine.
But for application to real hydrophobic interactions thisargument can be no more than suggestive; it is not a substitutefor detailed calculation with realistic models.53,76

VI. Summary and conclusions
The basic result of the thermodynamics as outlined in Section
II and illustrated in Section III is contained in eqns. (18)–(22),where the thermodynamics of transfer of a hydrophobic molecule into water are seen to follow from measurements of theequilibrium partition coeﬃcient or Ostwald absorption coeﬃcient. That the suitably deﬁned free energy of transfer [denotedwith an asterisk in eqns. (3) and (4)] is positive and equal to afew multiples of the thermal energy kT, and that the energy (orenthalpy) and suitably deﬁned entropy of transfer are both
negative, are characteristic of, and essentially deﬁne, hydrophobic solutes. Hydrophobicity increases with increasing temperature, reﬂecting the corresponding decrease in solubility.
Many of the features of the hydrophobic eﬀect are realistically rendered by the lattice model described in Section IV.
The model was designed to make the accommodation of asolute in the solvent energetically favorable but entropicallyunfavorable, and thus to incorporate the basic mechanism ofhydrophobicity as revealed by the thermodynamics. Once theparameters in the model are chosen to reproduce the experimentally measured solubility of methane and its temperaturedependence in the temperature interval 273 to 333 K, themodel leads to a realistic picture of the solvent-mediated part,W(r), of the potential of mean force between a pair of solute
molecules as a function of their separation r. It is found that‘‘ W(1) ’’, the potential at close approach of the pair, is negative, which is the hydrophobic attraction, and becomes morenegative with increasing temperature, in accord with the conclusion from the thermodynamic analysis that hydrophobicity
increases with increasing temperature. The strength W(1) of
the hydrophobic attraction increases nearly linearly with thefree energy of hydrophobic hydration, but is less than thelatter: only about kT as compared with about 3kT. Thatthe solvent-mediated attraction is as small as kT means thathydrophobic ‘‘ bonds ’’ are easily ‘‘ broken ’’, which may have
implications for dynamical eﬀects in protein structure.
It is remarked in Section V that in the one-dimensional version of the lattice model the three-body force in the solventmediated interaction among hydrophobic solutes is always
repulsive. A simple argument is presented according to whichthat may still be so in various conﬁgurations in two and three
dimensions as well; but one is warned that the argument, whilesuggestive, is not deﬁnitive.

Acknowledgements
We thank Jonathan Widom and Harold Scheraga for helpful
discussions. KK acknowledges the award of a fellowship fromthe Japan Society for the Promotion of Science, which madepossible his participation in this work. The work was supported by the U.S. National Science Foundation and theCornell Center for Materials Research.
3092

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