Molecular theory and the effects of solute attractive forces on hydrophobic interactions

0.92 MB·English

by  M. I. Chaudhari

Molecular theory and the effects of solute attractive forces on hydrophobic
interactions
Mangesh I. Chaudhari∗ and Susan B. Rempe†
Center for Biological and Material Sciences, Sandia National Laboratories, Albuquerque, NM 87185
D. Asthagiri‡
Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005
L. Tan§ and L. R. Pratt¶
Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, LA 70118
5
(Dated: October 28, 2015)
1
0 Theroleofsoluteattractiveforcesonhydrophobicinteractionsisstudiedbycoordinateddevelop-
2
mentoftheoryandsimulationresultsforAratomsinwater. Wepresentaconcisederivationofthe

t localmolecularfield(LMF)theoryfortheeffectsofsoluteattractiveforcesonhydrophobicinterac-
c
tions,aderivationthatclarifiesthecloserelationofLMFtheorytotheEXPapproximationapplied
O
to this problem long ago. The simulation results show that change from purely repulsive atomic
soluteinteractionstoincluderealisticattractiveinteractionsdiminishes thestrengthofhydrophobic
7
bonds. FortheAr-Arrdfsconsideredpointwise,thenumericalresultsfortheeffectsofsoluteattrac-
2
tiveforcesonhydrophobicinteractionsareofoppositesignandlargerinmagnitudethanpredicted

] byLMFtheory. Thatcomparisonisdiscussedfromthepointofviewofquasi-chemicaltheory,andit
h issuggestedthatthefirstreasonforthisdifferenceistheincompleteevaluationwithinLMFtheory
p of the hydration energy of the Ar pair. With a recent suggestion for the system-size extrapolation
- of the required correlation function integrals, the Ar-Ar rdfs permit evaluation of osmotic second
m
virialcoefficientsB . ThoseB alsoshowthatincorporationofattractiveinteractionsleadstomore
2 2
e positive (repulsive) values. With attractive interactions in play, B can change from positive to
2
h negativevalueswithincreasingtemperatures. ThisisconsistentwiththehistoricalworkofWatan-
c abe, et al., that B ≈0 for intermediate cases. In all cases here, B becomes more attractive with
. 2 2
s increasing temperature.
c
i
s
y I. INTRODUCTION From early days, the involvement of the hydration en-
h tropyhasbeenconceptualizedbyimaginingicebergs sur-
p Theconceptofahydrophobicinteractionisfirmlyem- roundingsimplehydrophobicsolutes,suchasinertgases,
[
bedded in general views of the folding of water-soluble e.g., Ar below. Tanford [3] attributed the original “ice-

2 proteinmolecules. Kauzmann[1]clearlyarticulatedthat berg” language to G. N. Lewis. Silverstein [4], decades
v idea: “Thus,proteinsarestabilizedbythesamephysical later, provides a modern view of the relevance of the ice-
5 forces as those that keep oil and water from mixing ...” berg concept to hydrophobic phenomena. “Iceberg” is a
9
Akeyphenomenologicalpointisthatenthalpichydration widely recognized figure of speech, but has not been the
4
2 contributions to the thermodynamics of protein unfold- basis of defensible statistical mechanical theory of these
0 ing decrease, even vanish at moderate temperatures [1]. entropy effects. In fact, the statistical mechanical the-
. Hydrophobic interactions are then entropy dominated. orythateventuallydoesexplaintheentropyconvergence
1
The enthalpy-entropy balance depends importantly on phenomenadoesnotdefineorcharacterizeicebergstruc-
0
5 temperature, and switches at higher temperatures [1]. tures [2].
1 The entropic hydration contributions to the thermody-
Becausetheicebergparlanceisvagueandprovocative,
: namicsofproteinunfoldingcanvanishathighertemper-
v direct experimental demonstrations of so-called inverse-
i atures [1], and that condition has been called the “en-
X tropyconvergence”point[2]. Nevertheless,belowsuchan temperature behaviors are particularly helpful. Aggre-
gation of sickle hemoglobin is a standard example [5].
r entropy convergence temperature, i.e., where hydropho-
a Well-known aqueous polymers that separate with tem-
biclow-solubilityisassociatedanegativeentropychange,
perature increases, i.e., systems that exhibit a lower
hydrophobic interactions get stronger with temperature
critical solution temperature (LCST), also provide ex-
increases though with reduced rate of strengthening.
amples. Elastin-like peptides (ELPs) are probably the
best known cases [6–10]. Substantial molecular simula-
tion work is available describing ELP collapse [11–18]
without addressing the statistical mechanical theory of
∗ [email protected]
† [email protected] hydrophobic interactions. Those descriptive simulation
‡ [email protected] efforts are largely consistent with the traditional idea of
§ [email protected] thefoldingofelastin-likepeptidesuponheating,andwith
¶ [email protected] each other, but not entirely [17].
2
Aqueous solutions of poly(N-isopropylacrylamide)s
5
provide other examples of LCSTs [19, 20]. Polyethylene
glycols (PEGs) in water also exhibit LCSTs [21]. The r) 360K
(
polymers noted are water-soluble below their LCSTs. HS 4 340K
Thustheyaresubstantiallyhydrophilic. Butthoughthey y i 320K
)
arecomplicatedmolecules,theydirectlydemonstratethe r( 3 300K
ienffveecrtsse.-temperature phenomena of classic hydrophobic (0)uArAr
β 2
−
The successful statistical mechanical theory for the
h
entropy convergence behavior [22–29] developed over p
x 1
decades from counter-intuitive initial steps [30–38]. The e
statisticalmechanicaltheoryofhydrophobicinteractions
[35, 39] was formulated for hard sphere hydrophobic so-
0.2 0.4 0.6 0.8 1.0 1.2
lutes in water, and theoretical progress has been as-
r (nm)
sociated with attention to detail for such simple cases
[24, 35, 39]. That methodical analysis strategy per-
FIG. 1. Modeled radial distribution functions for WCA
mits clarity in isolating the features that are the ul-
repulsive-force Ar solutes, based on the hard-sphere cavity
timate interest. An important accomplishment of re-
distribution functions [39].
cent work [39] was then to prove numerically that rigor-
ously defined hydrophobic interactions between atomic-
sized hard sphere solutes in water also exhibit inverse-
temperature behavior. Independently, new results for 3
broader solute models arrived at consistent conclusions 360K
for those cases [40]. Building from those important ac- 340K
complishments, the present work investigates the theory ) 320K
for adding attractive inter-atomic forces to those primi- r( 2 300K
tive cases. ArAr 280K
g
The counter-intuitive ingredients of the statistical me-
chanicaltheory[36]togetherwithapparentdisagreement 1
withsomeexperiments[37,41–43]thatdoinvolveattrac-
tive forces, lead promptly to questions about the conse-
quencesofsoluteattractiveforcesassociatedwithsimple
hydrophobic solutes [44]. That issue has been broadly 0.2 0.4 0.6 0.8 1.0 1.2
discussedseveraltimesovertheinterveningyears[42,44– r (nm)
48] without achieving a definitive solution. That situa-
tion can now change on the basis of the new results for FIG. 2. Ar-Ar radial distribution functions reconstructed
hydrophobic interactions noted above. from stratified (window) calculations. Notice (compare
FIG. 1) that contact hydrophobic interactions are weaker
Distinctions [49] deriving from inclusion of solute at-
when solute attractive forces are included. In contrast,
tractive forces are exemplified in FIG. 1 and FIG. 2. In-
solvent-separated correlations are more strongly structured
clusionofsoluteattractiveforcesdiminishes thestrength
with inclusion of atomic attractive forces.
of hydrophobic bonding: solvent attraction to the so-
lutetendstopullthesolutespeciesapart. Thisbehavior
could be expected from sensitive appreciation [44, 48] II. LOCAL MOLECULAR FIELD THEORY
of preceding results. The local molecular field theory
(LMF)discussedbelowisasimple, persuasivetheoryfor
The LMF idea is to study the inhomogeneous density
these effects of attractive interactions [50]. Clarifying
of a fluid subject to an external field. We focus on the
and testing that theory is the goal of this work.
density structure resulting from the placement of an Ar
Though the substantially the same theory we test be- atom at a specific location. That distinguished atom ex-
low was known and used [44, 51, 52] long ago, the in- ertsanexternalfieldonthesurroundingfluidanddistorts
sight underlying recent discussions of LMF theory, e.g., the density. With U the intermolecular potential energy
[50], have considerably strengthened it. We here give a function for the system and Φ the external field exerted
concise derivation with a clear analogy to a thermody- bythedistinguishedatom,theresultingdistorteddensity
namic van der Waals picture, and is therefore unusually is ρ (r;U,Φ) at position r of α atoms of a molecule
αM
compelling. In the next section we outline the LMF the- of type M. The goal of the LMF theory is to analyze
ory. Numerical results, and conclusions are identified in ρ (r;U,Φ) on the basis of the characteristics of the
αM
Sec.IV.Methodsfortheseveralcomputationalstepsare interactions U and Φ.
collected in Sec. III. We assume that a reference potential energy, de-
3
noted by U(0), has been identified to help in analyzing TheformsEqs.(3)and(5)allowsustoexpressthematch
ρ (r;U,Φ). Specifically, our goal is the match Eq. (1) as
αM
ραM(r;U,Φ)=ραM(r;U(0),Φ(0)) , (1) ϕ(0) (r)=ϕ (r)
αM αM
achieved for the reference system with interactions U(0), + µ(ex)(r;ρ,βU) µ(ex)(r;ρ,βU(0))
andaneffectiveexternalfieldΦ(0). Thateffectivefieldis αM − αM
(cid:104) +consta(cid:105)nt . (7)
the objective of the analysis below. A successful match
Eq. (1) establishes aspects of U that can be treated as
The bracketed terms in Eq. (7) depend functionally on
molecular mean-fields, thus offering a molecular mecha-
the densities, not on the external field. The constant
nism for ρ (r;U,Φ).
αM in Eq. (7) involves the chemical potentials of the two
Identification of a reference potential energy function
systems.
U(0) thus requires physical insight. One suggestion for
The approximation
the inter-atomic force fields of current simulation calcu-
lations corresponds to Gaussian-truncated electrostatic
interactions associated with the partial charges of simu- µ(ex)(r;ρ,βU) µ(ex)(r;ρ,βU(0))
αM ≈ αM
lation models [50]. In that case, the crucial difference
+ ρ (r(cid:48))u(1) (r(cid:48) r )dr(cid:48) (8)
γM(cid:48) γM(cid:48)αM | − |
U −U(0) = u(α1M)γM(cid:48)(|rαM−rγM(cid:48)|) , (2) γ(cid:88)M(cid:48)(cid:90)
αM(cid:88),γM(cid:48) is then transparently analogous to van der Waals theory
[54]fortheinclusionofattractiveinteractions,i.e.,∆µ
isatom-pairdecomposable. Forthecaseofinteresthere, ≈
2aρ, with a the van der Waals parameter describing
u(α1M)γM(cid:48)(|rαM−rγM(cid:48)|)istheWCA-attractivepartofthe −attractive intermolecular interactions. Transcribing to
Lennard-JonesinteractionsassociatedwiththeAratoms
the case of Ar(aq) at infinite dilution produces
[51].
InseekingthematchEq.(1), weadoptanatom-based
perspective, and focus on the chemical potential [53], ϕ(A0r)(r)≈ϕAr(r)
+ [ρ (r(cid:48)) ρ ]u(1) (r(cid:48) r )dr(cid:48) . (9)
µ =β−1ln ρ (r;U,Φ)Λ 3 O − O OAr | − |
αM αM αM (cid:90)
(cid:2) +ϕαM(r)+µ(cid:3)(αeMx)(r;ρ,βU) , (3) The fields vanish far from their source, and therefore the
constant contribution of Eq. (7) is accommodated ex-
of αM atoms, which decomposes plicitly in Eq. (9). This argument matches the results
of Rodgers and Weeks [55] in the several instances they
Φ= ϕ (r ) . (4) considered. Derivationsthatemphasizealternative(elec-
αM αM
trostatic) interactions are available elsewhere [56–58].
αM
(cid:88)
Though the statistical mechanical theory of Eq. (9) is
Here the temperature is T = (k β)−1; the thermal de- simple, the field sought depends on the density, which
B
Broglie wavelength Λ depends only on T and on fun- depends on the field. A linear statistical mechanical ap-
αM
damental parameters associated with α atoms. As in- proximation Eq. (8) produces the non-linear Eq. (9) to
dicated, the excess contribution µ(ex)(r;ρ,βU) depends solve. The non-linearity is not an obstacle here because
αM
functionally on (ρ,βU), not on the external field. thedensitiesontherightofEq.(9)arestraightforwardly
obtained from routine simulation (FIG. 3, see also [59]).
For some simulation models, the atom-based µ
αM
Notice (FIG. 3) that the effects of attractive ArO inter-
(Eq. (3)) may raise questions regarding the operational
actions on ArO correlations are modest, as was argued
status of atom chemical potentials. But this perspective
long ago [44].
would be satisfactory for ab initio descriptions of the so-
Now consider ρ (r;U(0),Φ(0)), the density of Ar
lution, and is sufficiently basic that we do not further Ar
side-trackthisdiscussion. Similarlyforthereferencecase atoms without attractive interactions βu(1) (r) but ex-
OAr
periencing the effective field βϕ(0)(r). We approximate
Ar
µ(0) =β−1ln ρ(0) r;U(0),Φ(0) Λ 3 [62]
αM αM αM
+(cid:104) ϕ(0)(cid:16)(r)+µ(ex) (cid:17)r;ρ(0),(cid:105)βU(0) , (5) lnρ (r;U(0),Φ(0))/ρ βϕ(0)(r)+lny (r)
αM αM Ar Ar ≈− Ar HS
(cid:16) (cid:17) = β ϕ(0)(r) u(0) (r) +lng(0) (r) , (10)
with − Ar − ArAr ArAr
(cid:16) (cid:17)
Φ(0) = ϕ(0) (r ) . (6) adoptingtherepulsive-forcesoluteresultsofFIG.1. The
αM j field ϕ(0)(r) incorporates aspects of the intermolecular
(cid:88)j Ar
4
ory for h (r) is
ArO
1.0
300K
320K ln gArO(r) βu(1) (r)
0.5 340K − (cid:34)g(0) (r)(cid:35)≈ OAr
ArO
) 360K
r( + h (r(cid:48))ρ βu(1) (r(cid:48) r )dr(cid:48), (14)
ArO 0.0 (cid:90) OO O OAr | − |
h
where h (r) is the observed OO correlation function
OO
for pure water. Acknowledging closure approximations
0.5
− specific to traditional implementations, this is just the
EXPapproximation[51]appliedtothiscorrelationprob-
lem long ago [44, 52]. This observation serves further to
1.0
− 0.3 0.6 0.9 1.2 identify Eq. (13) as a relative of the EXP approximation
r (nm) also. Nevertheless, the distinction between the theory of
Ref. [44], with its specific implementation details, from
FIG. 3. Observed radial correlation of O atoms with an Eq. (13) should be kept in mind. The most prominent
Ar atom, T = 300 K, p = 1 atm (heavy curve). Correlation distinction is that Eq. (13) exploits hArO(r) evaluated
functions (fainter, background curves) for hard-sphere model self-consistently or, here, the numerically exact result.
soluteswithdistancesofclosestapproach0.31nm(FIG.1)on Note further that the Eq. (13) offers additional pos-
thebasisofcavitymethods[39,59],fromChaudhari[49]. The sibilities compared to Eq. (14) for variety of outcomes
PCtheory[35]predictionsforthemaximaofthehardsphere because of possibilities from imbalance of u(1) (r) and
correlation functions would be close to 2, larger than these OAr
numericalresults[35,60]. Forthesoft-spherecase,attractive u(1) (r).
ArAr
van der Waals interactions draw-in near-neighbor O-atoms
slightly[39,61]. Sinceattractionsdraw-in,ratherthandraw-
up, the attractive interactions case is not wetter than the B. Perspective from quasi-chemical theory
reference case. [44, 48, 63]
Quasi-chemical theory (QCT) provides insight into
attractions as mean-field effects according to Eq. (9). the LMF approximation Eq. (13). Since QCT is de-
The match Eq. (1) pairs this with signed to evaluate interaction contributions to chemi-
cal potentials,[39, 61, 64] Eq. (7) is the relevant starting
lnρ (r;U,Φ)/ρ =lng (r) . (11) point. From the QCT formulation [48, 63], the packing
Ar Ar ArAr
contributions to those two chemical potentials are iden-
Combining with Eq. (10) tical, and cancel each other. Next to be considered [48]
is the mean hydration energy, denoted by εr,n =0 ,
λ
(cid:104) | (cid:105)
of the Ar appearing at r. That previous QCT effort [48]
lngArAr(r)=lngA(0r)Ar(r)−β ϕAr(r)−u(A0r)Ar(r) observed that the outer-shell QCT fluctuation contribu-
tion was comparatively small. Thus εr,n =0 is the
(cid:16) (cid:17) λ
[ρ (r(cid:48)) ρ ]βu(1) (r(cid:48) r )dr(cid:48) . (12) leading factor in describing the effec(cid:104)t|of attract(cid:105)ive in-
− O − O OAr | − |
(cid:90) teractions being added [44, 48]. In the QCT study,[48]
εr,n =0 was evaluated from molecular simulation
λ
Finally noting ϕAr = u(A0r)Ar + u(A1r)Ar and rearranging (cid:104)da|ta. The P(cid:105)C modeling of long-ago [44] recognized the
yields importance of εr,n =0 , and used a RISM approxi-
λ
(cid:104) | (cid:105)
mation to incorporate the specific structure of the Ar
2
diatom. Returning to the LMF theory, the right-most of
g (r)
−ln(cid:34)gA(0r)Ar(r)(cid:35)≈βu(A1r)Ar(r) Efoqr.t(h8e)daedtdarielesdseAs r(cid:104)ε|rg,enoλm=etr0y(cid:105).,Ibnuctomdopelsetneoetvcaalulcautliaotneoitf
ArAr 2
εr,n =0 is thus the chief neglect of the present LMF
+ h (r(cid:48))ρ βu(1) (r(cid:48) r )dr(cid:48). (13) (cid:104) | λ (cid:105)
ArO O OAr | − | theory.
(cid:90)
III. METHODS
A. Comparison to EXP theory [44]
A. Simulations
As noted above, the approximate theory Eq. (13) re-
quires h (r), and we can conveniently take that from Thesimulationswerecarried-outwiththeGROMACS
ArO
routine simulation. The corresponding approximate the- package [65], the SPC/E model of the water molecules
5
0.4 1000 water molecules. Initial configurations were con-
ln g (r)/g(0) (r) structedwithPACKMOL[69]toconstructasystemclose
− ArAr ArAr to the density of interest. The solute-solute separation
h i
0.2 spanning 0.33 nm to 1.23 nm was stratified using a stan-
LMFwatercontribution dardwindowingapproachandtheresultscombinedusing
the weighted histogram analysis method (WHAM) [70].
0.0 This involved 19 windows (and simulations) for window
separations r ranging from 0.33 nm to 1.23 nm.
netLMFapproximation
0.2
−
B. Implementation of LMF theory
βu(1) (r) T =300K
0.4 ArAr
− With the information of FIG. 3, the LMF approxi-
0.3 0.5 0.7 0.9
mation Eq. (13) depends only linearly on the attractive
r (nm) interactions. We evaluated Eq. (13) standardly, intro-
ducing the spatial Fourier transforms
FIG. 4. Test of the LMF theory, Eq (13). The net re-
sult for the LMF approximation (black, dashed, right-side uˆ(1) (k)= u(1) (r) sinkr dr , (15)
of Eq. (13)) is the sum of the direct interaction (blue, dot- OAr OAr kr
tedcurve)andthewatercontribution(red,dotdashedcurve, (cid:90) (cid:18) (cid:19)
Eq. (17)). −ln(cid:104)g (r)/g(0) (r)(cid:105) differs from the net LMF and
ArAr ArAr
approximation both in contact and solvent-separated config- sinkr
hˆ (k)= h (r) dr . (16)
urations. OAr OAr kr
(cid:90) (cid:18) (cid:19)
Then
4
Eq. (13) sinkr dk
hˆ (k)ρ βuˆ(1) (k)
3 gArAr(r) (cid:90) ArO O OAr (cid:18) kr (cid:19)(2π)3
gA(0r)Ar(r) = hArO(r(cid:48))ρOβu(O1A)r(|r(cid:48)−r|)dr(cid:48). (17)
(cid:90)
2
The parameters for this application are ρ = 33.8/nm3,
O
ε = 0.798 kJ/mol, σ = 0.328 nm, ε =
OAr OAr ArAr
1 0.978 kJ/mol, and σArAr =0.340 nm.
C. Osmotic B and Infinite Size Extrapolation
2
0.2 0.4 0.6 0.8
r (nm) Thedistributionfunctiong (r)=h (r)+1pro-
ArAr ArAr
vides access to the osmotic second virial coefficient,
FIG. 5. Comparison of LMF approximation Eq. (13)
1
with g(0) (r) (reference system, fainter, dotted curve) and B = lim h (r)d3r . (18)
gArAr(Ar)r.ArNotethesignificantlydifferentbehaviorofgA(0r)Ar(r) 2 −2ρAr→0(cid:90) ArAr
and gArAr(r) in the second shell, not addressed by this ap- We utilize the extrapolation procedure of Kru¨ger, et al.
proximation.
[71, 72]
2R
2B = lim 4π h (r)w(r/2R)r2dr ,(19)
[66], and the OPLS force field. GROMACS selects SET- − 2 R→∞ (cid:90)0 ArAr
TLE [67] constraint algorithm for rigid SPC/E water
with
molecules. The same constraint algorithm was used in
previous simulations involving water [39, 68]. Standard 3 1
w(x)=1 x+ x3 . (20)
periodicboundaryconditionswereemployed,withparti-
− 2 2
cle mesh Ewald utilizing a cutoff of 1 nm and long-range (cid:18) (cid:19) (cid:18) (cid:19)
dispersion corrections applied to energy and pressure. Computed values for 1/2R > 0 were least-squares fitted
The Parrinello-Rahman barostat controlled the pressure with a polynomial quadratic order in 1/2R, then extrap-
at 1 atm, and the Nose-Hoover thermostat was used to olatedto1/2R=0. Thisprocedurehasbeensuccessfully
maintain the temperature. The simulation cell for the tested[68,73,74]anddoesnotrequirefurtherstatistical
Ar(aq) system consisted of two (2) argon molecules and mechanical theory.
6
tractive interactions. The earlier application [44] used
the EXP approximation to analyze the available Monte
50
Carlo calculations on atomic LJ solutes in water [75].
−
) That theoretical modelling found modest effects of at-
mol tractive interactions, and encouraging comparison with
/ the Monte Carlo results. This application of the LMF
3 100
m
− theory(Eq.13)againpredictsmodesteffectsofattractive
c
( 300K interactions,butthenetcomparisonfromthesimulation
(0)B2 320K results shows big differences. The outcome alternative
150 340K to the historical work is due to the fact that the earlier
−
360K theoryusedthePCapproximateresultsforthereference
system g(0) (r), and we now know that approximation
ArAr
isnotaccurateforthisapplication[39],despitebeingthe
0.0 0.2 0.4 0.6 0.8 1.0
only theory available. Here the LMF theory (Eq. (13))
1/2R (nm)
predictsmodest-sizedchanges,thoughinadditionitpre-
dicts changes opposite in sign to the observed changes.
FIG. 6. Extrapolation to evaluate the osmotic second virial Note further that g(0) (r) and g (r) differ distinc-
coefficient, B(0), for the WCA repulsive-force Ar solutes ArAr ArAr
2 tivelyinthesecondhydrationshell,andthosedifferences
(FIG. 1). The symbol at 1/2R = 0 is the extrapolated
suggest more basic structural changes driven by attrac-
value;seeSecIIIC.HydrophobicinteractionsgaugedbyB(0)
2 tive interactions.
become more attractive with increasing temperature in this
range.
The earliest study of these effects [44] went further to
analyze a Lennard-Jones model of CH -CH (aq) with
75 4 4
much larger attractive interactions. The theory devel-
280K
opedforthatapplicationwassuccessfulforthecasestud-
300K
50 ied[47],butthecorrespondenceofthatLJmodeltoCH
320K 4
solutes was not accurate enough [47] to warrant further
ol) 340K interest.
m 25 360K
/
3
m Thatearliertheoryfeaturedstudyof εr,n =0 that
(c 0 has acquired a central role in QCT stud(cid:104)y|of tλhe pr(cid:105)esent
2
B problem [48]. A more accurate evaluation would involve
25 n-body(n>2)correlations,perhapseventreatedbynat-
− ural superposition approximations [76]. Detailed treat-
ment of the Ar diatom geometry is the most prominent
2
50
difference between that QCT approach and the present
− 0.5 1.0 1.5 2.0
LMF theory (Eq. (13)). Nevertheless, a full QCT anal-
1/2R (nm)
ysis of these differences is clearly warranted and should
be the subject of subsequent study.
FIG. 7. Extrapolation (Sec IIIC) to evaluate the osmotic
second virial coefficient, B , for the full attractions case of
2 These changes due to attractive interatomic interac-
FIG. 2. Comparing with FIG. 6, we see that inclusion of
tions are directly reflected in the values of B (FIGs. 6
solute attractive-forces makes these B more positive (repul- 2
2
sive). Hydrophobic interactions gauged by B become more and 7). Slight curvature of the extrapolation (FIGs. 6
2
attractive with increasing temperature in this range. B and 7) is evident but, in view of the previous testing
2
changesfrompositivetonegativevaluesinT =(320K,340K). [68,73,74],notconcerning. Inallcaseshere,B2becomes
ThusB ≈0inthatinterval,qualitativelyconsistentwiththe more attractive with increasing temperature below T =
2
work of Watanabe, et al., [45]. 360K.Thisbehaviorisconsistentwithaccumulatedexpe-
rienceandrecentlyobtainedresults[39,40,73]. Withat-
tractiveinteractionsinplay,B canchangefrompositive
2
IV. RESULTS AND DISCUSSION to negative values with increasing temperatures. This is
consistent with the historical work of Watanabe, et al.,
Changing purely repulsive atomic interactions to in- [45] that B 0 for intermediate cases.
2
≈
clude realistic attractions diminishes the strength of hy-
drophobic bonds (FIGs. 1 and 2). Within this LMF the- Finally,weemphasizethatsinceattractionsmakesub-
ory, and also the earliest theories for this [44, 48], the stantial contributions, precise tests of the PC theory
hydration environment competes with direct Ar-Ar at- [2,75]withresultsoncaseswithrealisticattractiveinter-
tractiveinteractions(FIG.4). Theoutcomeofthatcom- actions should specifically address the role of attractive
petition is sensitive to the differing strengths of the at- interactions that were not included in the PC theory.
7
V. ACKNOWLEDGEMENT tional Nuclear Security Administration under Contract
No. DE-AC04-94AL8500. The financial support of San-
We thank J. D. Weeks (University of Maryland) for dia’s LDRD program and the Gulf of Mexico Research
helpful discussions. Sandia is a multiprogram labora- Initiative (Consortium for Ocean Leadership Grant SA
tory operated by Sandia Corporation, a Lockheed Mar- 12-05/GoMRI-002) is gratefully acknowledged.
tin Company, for the U.S. Department of Energy’s Na-
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