Transferable Coarse-Grained Models of Liquid Liquid ...ABSTRACT: Bottom-up coarse-grained (CG) models are now regularly pursued to enable large length and time scale molecular simulations

Transferable Coarse-Grained Models of Liquid−Liquid EquilibriumUsing Local Density Potentials Optimized with the Relative EntropyTanmoy Sanyal and M. Scott Shell*

Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California, United States

*S Supporting Information

ABSTRACT: Bottom-up coarse-grained (CG) models arenow regularly pursued to enable large length and time scalemolecular simulations of complex, often macromolecularsystems. However, predicting fluid phase equilibria usingsuch models remains fundamentally challenging. A majorproblem stems from the typically low transferability of CGmodels beyond the densities and/or compositions at whichthey are parametrized, which is necessary if they are to describedistinct structural and thermodynamic properties unique toeach phase. CG model transferability is compounded by therepresentation of the inherently multibody coarse interactionsusing pair potentials that neglect higher order effects. Here, wepropose to construct transferable single site CG models ofliquid mixtures by supplementing traditional CG pair interactions with local density potentials, which constitute acomputationally inexpensive mean-field approach to describe many-body effects, in that site energies are modulated by thelocal solution environment. To illustrate the approach, we use intra- and interspecies local density potentials to develop CGmodels of benzene−water solutions that show impressive transferability in structural metrics (pair correlation functions, densityprofiles) throughout composition space, in contrast to pair-only CG representations. While further refinement may be necessaryto represent more complex thermodynamic properties, like the liquid−liquid interfacial tension, the generality and improvementoffered by the local density approach are highly encouraging for enabling complex phase equilibrium modeling using CG models.

I. INTRODUCTION

Equilibrium fluid phase transitions play a pivotal role in manytechnologies, ranging from complex fluids in consumerproducts to separation strategies in large-scale chemicalprocessing. Particularly for liquid mixtures, phase transitionforms the basis of liquid−liquid extraction, a unit operationwith widespread applications in food processing, organicsynthesis, petroleum refineries, renewable energy, nuclearreprocessing, and biotechnology, for example. Molecularsimulation methods for predicting phase equilibria for small,relatively rigid molecular species are now well established,1,2

and typically require clever sampling Monte Carlo (MC)techniques3,4 such as Gibbs-ensemble MC,5 multiple histogramreweighting,6,7 or transition-matrix MC.8 Tackling the phaseequilibrium problem for large, flexible, or asymmetrically sizedspecies remains a critical challenge and a major research effort.Coarse-grained (CG) models potentially offer a promisingroute for complex phase equilibrium calculations throughsimpler representations that are dramatically easier to sample,particularly for MC algorithms that involve particle insertionsbut also for direct interfacial simulations using moleculardynamics (MD) that require long equilibration run times.Indeed, the past decade has seen significant progress in CG

model development, with a particular effort directed towardbiomacromolecular systems (e.g., polypeptides and polynucleo-

tides) and their folding and self-assembly. CG descriptions ofbiomolecules have been motivated both by bottom-up methodsthat parametrize on the basis of small representative all-atommodels and by top-down approaches that tune CG interactionsto match macroscopic thermophysical properties. A number ofthese approaches have proven successful in capturing first-order-type structural and thermodynamic transitions. A fewexamples include morphological phase transitions in mem-branes and bilayers9−12 (e.g., using the MARTINI CGmodel13), and studies of folding−unfolding transitions inproteins and peptides.14−21 Despite the burgeoning success inthe biomolecular realm, CG models do not yet seem widelycapable of capturing f luid phase equilibria (although there arenotable efforts along this direction, as we shortly describe),which has limited their routine application to chemicalthermodynamics problems.The main roadblock in developing CG models of equilibrium

fluid phase behavior stems from their lack of transferability, i.e.,their limited ability to translate to thermodynamic states(density, composition, etc.) different from the one at which

Special Issue: Ken A. Dill Festschrift

Received: December 18, 2017Revised: February 13, 2018Published: February 21, 2018

Article

pubs.acs.org/JPCBCite This: J. Phys. Chem. B 2018, 122, 5678−5693

© 2018 American Chemical Society 5678 DOI: 10.1021/acs.jpcb.7b12446J. Phys. Chem. B 2018, 122, 5678−5693

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pubs.acs.org/JPCBhttp://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.jpcb.7b12446http://dx.doi.org/10.1021/acs.jpcb.7b12446

they were parametrized. This prevents accurate realization ofphase behavior that inherently spans the range of statesencompassing the phases of interest, and is a particular problemfor bottom-up CG strategies. One contributor to transferabilityissues is neglect of the strong coupling between the reducedCG degrees of freedom in the model. In principle, an “ideal”bottom-up CG force field is represented by a highlymultidimensional free energy function W(R), constructed byprojecting the all-atom (AA) potential UAA(r) on the CGdegrees of freedom R:22

∫ β δ= − − −W k T UR r r R M r( ) ln d exp[ ( )] ( ( ))V

B AA

(1)

Here, M is a “mapping function” that translates an AAconfiguration r to a CG one R, and β = 1/kBT. Unfortunately,W(R) is highly multibody and difficult to implementpractically; instead, CG force fields typically contain pairwisenonbonded interactions between CG sites (in addition toconventional bonded terms), effectively ignoring true higherorder many-body correlations in the underlying potential ofmean force.23−28 In turn, this tends to make the CG modelsensitive to the thermodynamic state (density and/orcomposition) at which it is developed, and limits not onlythe model’s capability to scale to other states but also its abilityto simultaneously reproduce different properties even at thereference state, as discussed extensively in the works of Louis,Head-Gordon, and co-workers.29,30

One resolution that has been particularly successful for CGwater models has been to include explicit three-bodyterms.31−33 However, such approaches may become computa-tionally expensive to parametrize and use in general, particularlyif the form and order of the physically relevant interactions arenot known. A second approach has been to include informationabout the local environment around a pair of CG particles tomodulate the pair potential between them.34−37 Recently, Vothand co-workers developed a theory of ultra-coarse-graining(UCG) that produces low resolution CG models where CGsites embed discrete internal states that are in a state of localquasi-equilibrium.38 They used this technique to mix separateCG force fields for different phases in a phase-separated liquidmixture, based on the location of the phase interface as well as alocal-density parameter that can discriminate between the twophases. These authors applied the approach to vapor−liquidequilibrium in a Lennard-Jones fluid and the cooperativeaggregation of neopentane in methanol. Noid and co-workersalso developed transferable CG models of heptane−toluenemixtures by expanding the force field with global volume-basedcorrections, such that the model reproduces correct NPTfluctuations in density and pressure and consequently thepressure−volume equation of state.39Here, we take a distinct approach and use so-called “local

density” potentials to expand the CG force field with mean-fieldrepresentations of multibody effects that in turn improvetransferability. In this case, the CG potential is inspired bymean-field embedded-atom models of metals,40 in which siteshave energies that directly depend on the local density ofneighboring CG sites (within a cutoff radius), and thesepotentials serve as an additive correction over traditional pairinteractions.41 Such potentials are mean-field in nature, whichallows them to remain inexpensive, but they account for higher-order interactions beyond pair in a manner modulated by thelocal environment of a particle, such as the local coordination

number or composition. Allen and Rutledge initially explored asimilar approach in which they supplemented conventionalsolute−solvent pair interactions with effective solvent-mediatedintersolute interactions.34,35,42 They expressed these additionalinteractions in terms of the excess chemical potential associatedwith transferring a solute from a solvent-exposed to acompletely solute-locked state, and parametrized the excesschemical potential as functions of both the global and localsolute density.In a previous article, we introduced a general approach to CG

local density potentials that we tested in model aqueoussolutions, where water was coarse-grained away to an implicitdescription with only effective intersolute interactions remain-ing.41 In that effort, we found that both the fidelity of the CGmodel and its transferability significantly improved with theaddition of local density potentials. Voth and co-workerssubsequently employed local density dependent interactions toimprove the characterization of vapor−liquid interfaces inmethanol and acetronitrile.43 Noid and co-workers also foundthat local density dependent potentials improved CG models ofmethanol, allowing parametrization in the bulk liquid thattransfer well to the vapor−liquid coexistence.44Here, we extend our previous work on local density

potentials to outline a general strategy for transferable CGmodels suitable for phase equilibria, using a coarse-grainingtheoretical framework based on the relative entropy.26,45 Aswith conventional bottom-up coarse-graining, we use anunderlying AA reference simulation to parametrize CG pairinteractions between species, but we also simultaneouslyparametrize intra- and interspecies local-density potentials.While our previous work showed that local density potentialsaid in capturing cooperative folding and association likebehaviors for solutes in implicit solvent,41 the present workexplores the generality of local density potentials in mixturescapable of macroscopic phase separation into chemicallydistinct environments.As a case study, we investigate macroscopic phase separation

in benzene−water mixtures, which provides an excellent andchallenging test system because of the water−benzene sizeasymmetry and the intrinsically multibody nature of water-mediated hydrophobic interactions.46−51 An earlier study of thissystem by Villa et al.52 examined the transferability of single-siteCG representations of benzene and water parametrized fromvery dilute benzene solutions (0.1 M), and using an advancedimplementation of the iterative Boltzmann inversion technique.These CG models demonstrated transferability in describingstructural and thermodynamic metrics like pair correlations andthe chemical potential at low concentrations but producedqualitatively incorrect behavior at high benzene concentrations(∼9.5 M) that missed the macroscopic liquid−liquid phaseseparation, illustrating inherent challenges in capturing thissystem’s composition transferability.In this paper, we develop force fields for single-site CG

models of benzene−water mixtures that augment CG pairinteractions with local density potentials. Here, we add fourdistinct such potentials that modulate the energy of a CGmolecule based on its identity (benzene or water) and itsaverage (mean-field) concentration of either species within ashort-range distance cutoff. We parametrize CG models fromAA reference systems at several distinct compositions toinvestigate the effect of AA reference on model quality.Subsequently, we benchmark the structural and thermodynamictransferability of the CG models across composition space,

The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.7b12446J. Phys. Chem. B 2018, 122, 5678−5693

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http://dx.doi.org/10.1021/acs.jpcb.7b12446

spanning both sides of the phase transition point predicted bythe reference AA force field. We discuss both the improvementsthe local density strategy enables and the limitations that wefind.

II. METHODSII.A. CG Model and Force Field Design. The CG model,

as illustrated in Figure 1, maps benzene and water molecules tosingle sites and, in that sense, is an explicit water CG modelunlike our previous test of local-density CG interactions.41 Inthe present case, the local density is essentially a localcoordination number, and is given for a central CG site i oftype α with neighboring sites j of type β by

∑ρ φ=αββ

≠∈

r( )ij ij

ij

(2)

where φ(rij) is an indicator function based on the pair distancerij, that sharply but smoothly interpolates to zero at a cutoffradius rc. We use the computationally efficient form

φ =

≤

+ + + ∈

≥

⎧⎨⎪⎪

⎩⎪⎪

r

r r

c c r c r c r r r r

r r

( )

1

( , )

0

0

0 22

44

66

0 c

c (3)

The coefficients c0, c2, c4, and c6 are determined41 by imposing

continuity of φ and its first derivatives at the cutoff rc and aslightly smaller inner-cutoff r0, which we fix as rc − 1 Å for thethis work. The LD potential of type αβ due to all central CGsites of type α and neighbors of type β then follows

∑ ρ=αβα

αβ αβ

∈

U f ( )i

iLD(4)

where the summation proceeds only over atoms of type α. Thefunctions fαβ are yet unknown and are determined as splines byoptimizing the CG model. Although ULD

αβ is a mean-field many-body potential, it gives rise to a pair-additive force

∑ ρρ

ρ

ρ

φ= − +

−αβ

α β

αβ αβ βα βα

≠∈ ∈

⎡

⎣⎢⎢

⎤

⎦⎥⎥

f f r

r rf

r rd ( )

d

d ( )

d

d ( )

di j ii j

i j ij i j

ij,

(5)

The reader is referred to ref 41 for further details on theformulation of LD potentials, including expressions for thecoefficients in eq 3.

We consider several illustrative cases for CG force fields.Each contains three distinct CG pair interactions modeled bysplines, in addition to potentially four LD interactions betweenall possible species and environment types. For convenience,the CG pair potentials are referred to as simply pair-αβ or αβpair potentials, while the LD potentials are annotated as LD-αβ,where α, β = B (benzene) or W (water). Note that pair-αβ andpair-βα denote the same potential, while LD-αβ and LD-βα donot, given the asymmetric role between central and neighboringsite types. Figure 2 schematically shows the different types of

possible LD-αβ potentials. If all possible LD potentials areincluded with the three pair potentials, the overall CGHamiltonian reads as

∑ ∑ ∑

∑

∑

ρ ρ

ρ ρ

= + +

+ +

+ +

<∈

<∈

<∈∈

∈

∈

U u r u r u r

f f

f f

( ) ( ) ( )

[ ( ) ( )]

[ ( ) ( )]

i ji j

iji j

i j

iji jij

ij

ii i

ii i

CG

, B

pairBB

, W

pairWW

BW

pairBW

B

BB BB BW BW

W

WW WW WB WB

(6)

Figure 1. Atomistic system (left) and single site CG model (right) of a 10% mole fraction benzene/90% water solution, with a total of 500molecules. The cubic box length of 27.45 Å is tuned to achieve 1 atm pressure in the AA system at 300 K. In the CG model, molecular orientationaldegrees of freedom are coarse-grained away and effective intermolecular interactions are determined by relative entropy minimization.

Figure 2. Schematic representations of central (α) and neighboring(β) species configurations for the different types of LD potentialsoutlined in Table 1. Yellow CG sites denote benzene, and blue CGsites are water.



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We study a total of six distinct force fields, which aresummarized in Table 1. This includes the control with only pairpotentials, the full potential of eq 6 involving all four localdensity potentials, and four subsets in which only one of thelocal density terms is used. The cutoffs for determining localdensities for the like LD potentials (LD-BB and LD-WW) arechosen to include the first coordination shell from their firstminimum in the respective radial distribution functions; thesedistances are arithmetically averaged to determine the cutoffsfor the unlike potentials (LD-BW and LD-WB).Admittedly, including all four LD potentials (the pair + LD-

all force field) may contain redundant information, since localdensities essentially measure coordination numbers and theseare subject to geometric constraints in condensed, incompres-sible fluids. To illustrate this point using simple lattice statistics,consider zBB, zWW, zBW, and zWB as the average nearest-neighborcoordination numbers of the different types, for N total CGparticles (benzene and water) with xB benzene mole fraction(and xW = 1 − xB water mole fraction) on a lattice with N sites.One might think of zαβ = ραβ, where ⟨ ⟩ denotes ensembleaveraging. If each molecule occupies a single lattice site and thetotal lattice nearest neighbor coordination number is z (e.g., z =6 for a cubic lattice), balances on connections between sitesproduce z = zBB + zBW = zWW + zWB and xBz

BW = xWzWB. These

three constraint equations show that, at most, one of the fourpossible local densities are independent once the mole fractionis specified. Of course, the off-lattice nature of the actual CGmodel means that the local coordination shell is not trulyconstrained to a fixed number of neighbors, and of course, thebenzene−water size asymmetry introduces additional compli-cations. Still, this analysis illustrates that the addition of all fourlocal density potentials may not be necessary to cover thefunctional space or basis that these interactions provide beyondpair potentials, as long as the system remains in a condensedphase. In particular, this analysis also motivates consideration ofCG force fields using a single LD potential, namely, the second,third, and fourth rows of Table 1.In our approach, we represent all CG interactionsboth pair

and local densityby cubic B-splines. The spline knots formthe optimizable parameters of the model and are determined byminimizing the relative entropy between the model and itsatomistic reference.26,45 The relative entropy provides a way toquantitatively compare the quality of a CG model by measuringthe loss in information upon coarse-graining the AA system to areduced set of CG degrees of freedom with a particular CGforce field. Minimizing the relative entropy increases theoverlap between the ensemble microstate probability distribu-tions of the AA and CG models, and provides a natural strategyfor parametrizing CG force fields. In the canonical ensemble,the relative entropy takes the form26

λ λβ β= ⟨ − ⟩ − − +S U U A A S( ) ( ( ) )rel CG AA AA CG AA map(7)

where UX and AX are the potential and Helmholtz free energiesin ensemble X = AA or CG, which are functions of the CGforce field parameters λ, namely, in this case, the spline knots ofall component potentials. Here, Smap is a mapping entropy thatmeasures the degeneracy associated with the AA to CGmapping; it is independent of the CG potential and thus doesnot depend on these parameters. The derivative of the relativeentropy in λ space (for the full potential with all LD terns) canbe written as

∑ ∑

∑ ∑

λ λ λ

λ

λ

β

βρ

ρ

∂∂

=∂

∂−

∂∂

+∂

∂

−∂

∂

α β

αβ αβ

α β α

αβ αβ

αβ αβ

∈<

∈∈

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

S u r u r

f

f

( ) ( )

( )

( )

i j

ij ij

i

i

i

rel

( , )

(BB,WW,BW)

pair

AA

pair

CG

( , )

(BB,WW,BW,WB)AA

CG (8)

where the set of local density potentials given by atom-typecombinations (α, β) would differ across the different cases inTable 1. The coarse-graining algorithm proceeds by locatingthe minimum of Srel in λ space, i.e., the zeros of eq 8, using acombination of nonlinear conjugate gradient and quasi-Newtonmethods. Details are provided in our earlier work.41,45

II.B. Simulation Details. Atomistic simulations of aqueousbenzene solutions are carried out at 300 K and 1 atm pressure,with 500 molecules, spanning a range of benzene mole fractionsfrom xB = 10 to 90%. For these, we use the MD engineGROMACS (version 4.6.5)53,54 and employ the GROMOS53a6 force field55 for benzene and the SPC/E model of water.56

All bonds are constrained using the LINCS algorithm.57,58 Thesystem is first equilibrated in an NPT ensemble using theParinello−Rahman barostat59 and Nose−́Hoover thermo-stat60,61 for 44 ns to determine the equilibrium bulk density(data from the last 4 ns is used to estimate average equilibriumbox volume) followed by a further 60−80 ns of equilibrationunder NVT conditions, the last 40 ns of which is used to collecttrajectory data. It should be mentioned that a previous studyinferred that the AA force field overpredicts the solubility ofbenzene in water by almost an order of magnitude;52 while theexperimental solubility is 0.02 M,62 benzene and water were stillfound to be miscible at 0.5 M, which likely stems from thereported underestimation of benzene hydration free energy bythe GROMOS 53a6 force field.63 However, the present study isonly interested in the ability of optimized CG systems to

Table 1. Case Studies for Local Density Potentials Examined in This Worka

CG force field case pair potentials LD potentials

pair-only pair-BB, pair-WW, pair-BW nonepair + LD-BB pair-BB, pair-WW, pair-BW LD-BBpair + LD-WW pair-BB, pair-WW, pair-BW LD-WWpair + LD-BW pair-BB, pair-WW, pair-BW LD-BWpair + LD-WB pair-BB, pair-WW, pair-BW LD-WBpair + LD-all pair-BB, pair-WW, pair-BW LD-BB, LD-WW, LD-BW, LD-WB

aWe consider cases involving zero, one, or all possible (total of four) local density potentials that differ in the central and neighboring moleculetypes.



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recapitulate correct AA properties, no matter how accuratethese may be relative to reality; thus, this particular AAreference serves as an instructive model system. It is also worthnoting that the AA systems are at concentrations higher thaneither the experimental or the AA solubility limit, so that thereference solutions are already phase separated. However,because of the small system size, the interfaces encompass asignificant fraction of molecules, which allows the references tosample cross-interactions between the demixed species.To study the effect of reference state point on the

transferability, CG models are parametrized from 10, 50, and90% benzene solutions, respectively. All CG pair potentials andBB, BW, and WB LD potentials are developed using cubic B-splines with 30 knot points, while the LD WW potential uses60 knots (chosen higher initially to accommodate potentiallycomplex water−water interactions). A cutoff of 10 Å is used forthe pair potentials. Outer cutoffs for LD-BB and LD-WWpotentials are estimated from the first solvation shell radii of B−B and W−W radial distribution functions, respectively (7.5 Åfor B−B and 3.5 Å for W−W), while those for LD-BW and LD-WB potentials are computed as the arithmetic average of theabove two (5.5 Å). The inner cutoff for all LD potentials is 1 Åless than these. MD simulations of the CG models are carriedout in the NVT ensemble using the LAMMPS MD code,64

modified to include local density potentials. We find that theLD-augmented CG systems provide at least a 15-fold speedupover the AA simulations; this might be further increased withoptimization of the LD code.

III. RESULTS AND DISCUSSION

III.A. CG Force Fields.We parametrize three versions of thesix force fields introduced in Table 1, involving distinctatomistic references at compositions of 10, 50, and 90%benzene mole fraction, respectively. Figure 3 compares the

potentials for the pair-only and pair + LD-all force fields, asdescribed in Table 1. For all of the references, the BB pairpotentials with and without the use of LD potentials havesimilar forms and are largely repulsive; only for the dilute 10%reference solution do they exhibit a slightly attractive (∼−0.2kcal/mol) interaction around 6.5 Å. It is instructive to note thatthis attractive well is not as significant as the value −0.6 kcal/mol reached for a pair-only CG BB potential reportedpreviously that was developed from an even more dilutebenzene solution (xB ≈ 0.002).52 On the other hand, the BBLD potentials are largely all attractive, and the versionparametrized from the 50% solution shows the largest decreaseof around 2.5 kcal/mol over the entire range of the BB localdensity.In contrast to the benzene self-interactions, the WW pair

potentials show significant attractive interactions; many have atypical double-well form. This outer well near 6.5 Å is weaklyattractive (∼−0.4 kcal/mol) and remains similar in magnituderegardless of the presence of LD potentials and parametrizationreference. On the other hand, the depth of the inner wellincreases (to ∼−0.5 kcal/mol) for the 50 and 90% referenceswithout LD potentials. When reoptimized with all LD terms,the inner core converts into a repulsive shoulder for the 90%reference but becomes very attractive (∼−0.8 kcal/mol) forthe 10% solution. The LD-WW potential seems to compensatefor these changes: it exhibits attractive behavior around a localdensity of 4 (ΔULD−WW ≈ −2 kcal/mol) at 10 and 50%reference compositions and becomes the strongest of all fourLD potentials for the 90% solution, decreasing by 12 kcal/mol.It is interesting to note that the minimum in the LD potentialof the former two cases occurs near a coordination of four,consistent with stabilization of water’s tetrahedral coordination.The BW pair potential is strictly positive but nonetheless

contains a high-energy inner minimum within the core-repulsive region at 3 Å and a shallow outer minimum or

Figure 3. Intra- and intermolecular local density potentials (top row) and pair potentials (bottom row) optimized with the relative entropy fromreference atomistic benzene solutions at 10% (orange and blue), 50% (red and purple), and 90% (black and green) benzene mole fraction. The BBand WW pair potentials show the most significant differences when optimized simultaneously with all four LD potentials. Unlike LD potentials,potentials of types BW and WB are likely modulated by the small number of intermolecular multibody correlations at the interface, characterized bythe low range of local density (∼2) over which they change.



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shoulder near 4.5 Å. The pair distances of these features remainconstant across all reference solutions and are agnostic to theinclusion of LD potentials. When parametrized without LDpotentials, the inner minimum in the BW pair potentialdecreases (by ∼0.5 kcal/mol) when moving from the 10 to90% reference systems, but this variation vanishes when LD

potentials are included. The corresponding optimized inter-species LD potentials depend on reference compositions, butboth the LD-BW and LD-WB interactions are relatively small inmagnitude. The LD-WB potential is always completelyrepulsive, increasing up to 1.2 kcal/mol over a short range ofthe W−B local density. The LD-BW potential is overall the

Figure 4. Transferability of B−W RDF (top panel) and LD distribution (bottom panel) between AA (represented by black dots) and CG models forthe different CG force fields in Table 1. CG models for various combinations of these LD potentials are parametrized from reference compositions of10, 50, and 90% benzene mole fraction (dark framed plot marked “ref”, along the rows). CG models constructed at a particular AA reference aresimulated at compositions spanning 10−90% benzene (along the columns), which are used to calculate and compare RDFs and LD distributionswith corresponding AA MD simulations at these compositions. The pair + LD-all CG model is nearly quantitatively transferable for both metrics atevery composition.



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weakest, and manifests both attractive and repulsive forms thatseem specific to the inclusion of LD potentials and theparticular reference composition.All three pair potentials exhibit either multiple wells or a

well-and-shoulder structure, which is commonly seen in CGmodels that coarse-grain away directional interactions likehydrogen bonds into spherically symmetric interactions.65−67

The balance between the inner and outer wells in the WWinteraction is well-known for water and, on the basis of the ratioof these characteristic distances, promotes tetrahedrallyenriched liquid-phase correlations.67,68 The shoulder featurein the BB potentials likely also promotes specific geometriesthat result from the significant size asymmetry and packing dueto benzene’s aspect ratio. The addition of LD potentials, whichextend only until the first coordination shell of thecorresponding types, rather naturally then seems only tomodify the inner well or shoulder of the corresponding pairpotential.It is instructive to note the possibility of interaction

redundancies between the CG pair and LD potentials, whichis distinct from redundancies within the set of LD potentialsdiscussed in section II. In particular, if the LD potential is linearin the coordination number, then it can mimic the behavior of ashort-range pair potential (i.e., with an energy that scales withthe number of nearest neighbors), leading to the possibility thateffective interactions might shift between the LD and pairpotentials. We see this in the pair + LD-all CG models from the90% reference solution case, where the WW interactionsinvolve an entirely repulsive pair potential such that all short-range attractions shift into an almost linear LD WW potential.This may be due to the small number of water molecules in thatcase and consequently insufficient local crowding and reduced

multibody correlations that weaken the role of the LDpotential.It may be noted that each LD potential acts at relatively low

local density values, corresponding to small changes in localcoordination number, before saturating to constant values forlarger variations. The LD-BB and LD-WW potentials saturatenear 5−6 benzene and water neighbors, respectively, while theLD-BW and LD-WB potentials show variation only largelybetween zero and two neighbors (except for the LD-BWpotential from the 10% solution). These results may suggestthat most of the needed multibody correction needed occurs atlow local densities, while the optimized pair potentials are ableto capture remaining interactions. Indeed, the calculated localdensity distributions (discussed shortly) show that the biggestcorrections occur at low coordination numbers. Interestingly,the unlike LD potentials (LD-BW and LD-WB) have muchsmaller magnitudes compared to the like-species potentials,perhaps because B−W and W−B interactions are limited to thethin interfaces in the phase separated AA reference solutions.Ultimately, the same-species local density interactions, i.e., LDBB and LD-WW, seem to capture the most importantmultibody effects in this system.It should be mentioned, however, that the choice of knot

density in the cubic B-splines that represent the LD potentialscan slightly influence the magnitudes and shapes of all of theLD potentials. In principle, a very large number of knots willapproach a limiting form for each LD potential; however,because some of them vary significantly over small ranges oflocal density, even a relatively dense number of knots may notquite reach this limit. In the present investigation, we use 0.3Å/knot for the pair splines, 0.66 local density per knot for theBB, BW, and WB potentials, and a finer spline of 0.10 localdensity per knot for the LD-WW interactions. Even with this

Figure 5. Composition weighted RMS overlap for (a) pair correlations and (b) local density distributions, between AA systems and CG modelsparametrized from 10, 50, and 90% reference compositions (along the vertical subpanels). The LD-WW potential is arguably the most importantmany-body interaction, and CG model transferability is significantly improved when it alone is included. CG models with only pair potentials or withintrabenzene LD potentials have poor transferability. The 50% reference CG model with all LD potentials performs best, being transferable to bothmore dilute and more concentrated solutions for both structural metrics.



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relatively high knot resolution, we still detect some minorchallenges in modeling sharp peaks that appear in the LDdistributions at low coordination numbers. However, furtherincreases in resolution make the relative entropy minimizationmore difficult, and thus, we remain with the presentparametrization as a balance of accuracy and efficiency.III.B. Structural Transferability. We first evaluate the

relative transferabilities of the CG models in Table 1 bycomparing structural properties across the entire compositionspectrum, beyond the specific compositions at which they areparametrized. We consider the radial distribution functions(RDFs or gαβ(r), where αβ = BB, WW, and BW) as well as thelocal-density distributions (PLD(ραβ), where αβ = BB, WW,BW, and WB). For the sake of brevity, two examples ofstructural correlation functions are presented here; theremainder have qualitatively similar behavior and are given inthe Supporting Information (Figures S1−S5).Figure 4 shows the transferability of B−W pair correlations

and local density distributions across benzene compositions forthe different CG model. For the RDFs, all CG models showgood representability at the composition at which they weredeveloped, as expected from the relative entropy strategy,which should match pair correlations when finely discretizedand flexible spline pair potentials are used.69 At othercompositions, the pair-only models fare poorly: the one from10 and 50% solutions overestimates the RDF peaks at highercompositions, while that developed from the 90% solutionunderpredicts the RDF peak at 10% composition. The pair +LD-BB and pair + LD-WB models seem equally limited in theirability to capture states beyond the reference composition. Onthe other hand, the models with LD-WW and LD-BW

potentials perform much better and the pair + LD-all CGmodel is nearly quantitatively transferable to all of thecompositions in terms of RDF reproduction. A similar trendemerges in the LD distributions, where the pair-only modelshows limited fidelity, even at the reference composition, andgrossly overpredicts both the peak and the spread of thedistribution at other compositions. Once again, better trans-ferability in the LD distribution results from the inclusion of theLD-WW potential, and the pair + LD-all model scalesreasonably well across the entire composition space.Figure 5 summarizes the overlap of all three RDFs and four

LD distributions for the different classes of CG force fieldsacross composition space and for different parametrizationreferences. Here we compute the root-mean-square (RMS)differences between the AA and CG versions of these metrics,averaged over the distinct species combinations and weighed bycomposition

ρ

= ⟨Δ ⟩ + ⟨Δ ⟩ + ⟨Δ ⟩

= ⟨Δ ⟩ + ⟨Δ ⟩ +⟨Δ ⟩ + ⟨Δ ⟩

g rx g x g x x g

P x P x P xx P P

RMS[ ( )]( 2 )

RMS[ ( )] [( )]

B2

BB2

W2

WW2

B W BW2 1/2

LD B2

LD,BB2

W2

LD,WW2

B

W LD,BW2

LD,BW2 1/2

(9)

where ⟨Δ ⟩ = ∑ −αβ αβ αβX X X( )N2 1 AA CG 2 for Xαβ = gαβ(r) or

PLD(ραβ) and N numbers the discrete values of the arguments(r or ρ).Figure 5a demonstrates that all of the CG models are

accurate in predicting the RDFs at the reference composition atwhich they are parametrized, but the pair-only and pair + LD-

Figure 6. Fractional errors between AA and CG predictions of excess chemical potential for inserting a hard sphere with diameters ranging from 0 to5 Å, for the different CG model force fields per Table 1. CG models are parametrized from reference compositions of 10, 50, and 90% benzene molefractions (dark framed plot marked “ref” along the rows). Excess chemical potentials are calculated using the Widom test particle insertion method.Including W−W multibody interactions is essential for CG model transferability, as all models without this interaction predict insertion free energies(for R ≥ 1.5 Å) incorrectly modulated by composition.



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BB force fields worsen significantly at other compositions. Forcalibration, a RMS error of 1.2 for the pair-only CG model at10% composition corresponds to an overprediction of the WWRDF peak by 300% and BW peak by 113%, on average. On theother hand, the pair + LD-WW and pair + LD-all CG modelshave very low RMS errors across all of composition space.Remarkably, the high transferability of these models is notaffected by the reference composition. Particularly for the pair +LD-all case, models parametrized at either very high or lowbenzene concentrations perform well. The transferabilities inLD distributions in Figure 5b have a similar trend, but the RMSerror is nonzero everywhere including the reference composi-tion. This is likely because the absolute RMS metric used in eq9 is sensitive to the large variation in the range of the(normalized) LD distributions across BB, WW, BW, and WBtypes, and very high-resolution splines for LD interactionswould be needed to capture some sharp features of the LDdistributions (which pose numerical challenges in the relativeentropy minimization). Figure S6 in the Supporting Informa-tion presents an alternate version of Figure 5 where the RMSerrors are normalized relative to the AA distribution for bothpair and LD.The poor performance of the pair + LD-BB force field and

the high transferability of CG models that include the LD-WWpotential show that, although the BB and WW LD potentialsare comparable in magnitude (as observed in Figure 3), theLD-WW potential is the more important multibody contribu-tion. Admittedly, the reference solutions used in this study arealready phase-separated, unlike previous efforts to parametrizebenzene−water CG models from much more dilute con-ditions.52 Intrawater local many-body correlations are likelyparticularly important to hydrophobic-mediated interactionsand phase separated conditions, which we believe underlie thesuccess of the LD-WW potential.

It should be mentioned that it is not clear if the LD strategyimproves the structural transferability of the CG models acrossbulk density (or pressure). Figure S7 in the SupportingInformation shows that LD potentials sometimes improvereproduction of pair correlation functions at slightly higher orlower densities, compared to pair-only models, but the effect isnot uniform.

III.C. Thermodynamic Transferability. We also considerthe extent to which the CG models reproduce selectedthermodynamic properties in addition to structural correlationfunctions. As a first example, we calculate the excess chemicalpotential due to inserting a hard sphere of radius R in benzenesolutions of composition 10−90%, μex(R). Since a hard sphereinteracts simply by excluding other particles within a thresholdvolume, μex(R) = −kBT ln Pcav is calculated from the probabilityof finding a cavity of radius R in the equilibrium solution withinthe simulation box. For pure solvents, the hard sphere solvationchemical potential is directly related to the equilibriumfluctuations in local density of the solvent around solutes,which is an important measure of solute hydrophobicity inaqueous solutions.70 In this case, we introduce hard sphereswithin the entire aqueous benzene solution and thus theresulting excess chemical potential is related to the species-averaged density fluctuations.Figure 6 compares the fractional error between AA and CG

predictions of μex(R), i.e., − μμ( )1 CGex

AAex , for the different CG

models. Direct comparison between AA and CG excesschemical potentials can be found in the Supporting Information(Figure S8). The AA and CG predictions match nearly exactlyfor hard spheres of radius ≤1.5 Å for all force fields. For R ≥ 1.5Å, the pair + LD-WW and pair + LD-all models produce ∼10−15% deviation from the AA predictions at the referencecompositions. This error is magnified at compositions away

Figure 7.Macroscopic phase separation in large systems and transferability of density profiles of benzene (top panel) and water (bottom panel) at anoverall composition of 27.5% benzene mole fraction in a long rectangular box, using the different CG models parametrized from 10, 50, and 90%reference compositions (across the columns). CG models without LD contribution of water (pair-only, pair + LD-BB) underpredict the equilibriumdensity of benzene and overestimate that of water, with the deviations increasing when the parameterization reference is more extreme incomposition. Dotted lines are the experimental bulk densities of benzene (top panel: ∼0.89 g/cm3) and water (bottom panel: ∼1.0 g/cm3).



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from the reference for the 10 and 50% reference solutions butinterestingly decreases to 8% when the pair + LD-all force fieldfrom the 90% reference is used at lower compositions. Thepair-only, pair + LD-BB, pair + LD-BW, and pair + LD-WBforce fields have higher error (∼20−30%) regardless of thecomposition. In most cases, μCG

ex ≤ μAAex , as can be intuitivelyexpected, since coarse-graining multiple atoms into single,isotropic CG point molecules opens more effective free volumein the solution. However, the pair-only model from the 50%reference and several models from the 90% reference predictpositive errors, i.e., μCG

ex > μAAex , when applied to lower

compositions.It is well-known from scaled-particle theory that μex for a

hard sphere solute of radius R is related to the solute contactdensity71,72

μ πβ

ρ=R

R R G Rd

d( )

4[ ( )]ex 2 solvent (10)

where G(R) is the first-peak value of the hard-particle−solventRDF. Although this result strictly holds for pure solvents, itremains conceptually similar for liquid mixtures.71 Therefore,hydrocarbon solvents, with typically lower contact densities forhard-particle solutes, are expected to exhibit smaller excesschemical potentials.72 For most of the benzene solutionsupward of 50% composition, it is likely that cavity volumes aremuch more frequent in the benzene phase, thus decreasing thefree energy of insertion. CG force fields that do not includeW−W multibody interactions (namely, pair-only, pair + LD-BB, pair + LD-BW, and pair + LD-WB) are sensitive to thesystem composition: when parametrized from the 10%reference and transferred to higher benzene compositions,they predict lower μex, and when constructed from the 90%reference and transferred to dilute solutions, they producesystematically higher μex. Thus, it appears that capturing W−Wself-interactions accurately is crucial to model transferability.To investigate the macroscopic phase separation between

benzene and water and the associated equilibrium interfacialbehavior, we perform MD simulations with 380 benzene and1000 water molecules in a periodic box that is extended (∼129Å) along the z axis. The volume of this system is nearly 5 timeslarger than those of the parametrization references. Thesolution composition was chosen such that the equilibriumbox volume determined through an initial (40 ns) NPTsimulation produces an overall concentration of ∼7 M, which isclose to the highest concentration limit (∼9.5 M) at which pair-only CG models studied in ref 52 fail to capture themacroscopic phase separation. Bulk density profiles aresubsequently computed from sampling equilibrated trajectories(∼60 ns) in the constant volume NVT ensemble, along the zdirection.Figure 7 compares the bulk density profiles of benzene and

water for the different CG models from the 10, 50, and 90%reference compositions. The AA system and all of the CGmodels demonstrate macroscopic phase segregation in whichthe benzene and water phases move to opposite sides of thebox along the longest dimension. The pair-only, pair + LD-BB,pair + LD-BW, and pair + LD-WB models systematicallypredict up to 10% lower bulk densities for benzene and up to40% higher ones for water, when parametrized from the 90%solution. Consequently, these force fields also produce a largersize of the benzene phase. While CG models that include theLD-WW potential reproduce the bulk densities nearly correctly,the pair + LD-all model from the 90% reference does not

replicate the sharp segregation along the interface seen in theAA system, and leads to a longer benzene phase and a slightnonzero concentration of benzene in the water phase. It isworth noting that this particular model displayed a significantlydifferent character in the water−water pair and LD potentials,as shown in Figure 3. It may be difficult for this model tocapture the sharp local density gradients across the phaseinterface.The behavior of the benzene−water interface can be further

quantified by calculating the interfacial surface tension (γ),calculated by the Kirkwood−Buff formula73

∫γ = −+

−∞

∞ ⎛⎝⎜

⎞⎠⎟z p

p pd

2zzxx yy

(11)

where pii is the diagonal component of the pressure tensoralong the i axis (i = x, y, z) and ±∞ represents the benzene andwater bulk phases. Our AA simulations and most of the CGmodels reveal two nearly sharp planar interfaces (as shown inFigure 7) so that the difference between the pressure normal to

the interface (pzz) and the total tangential pressure+( )p p2xx yy is

nearly zero everywhere except the interface, and pzz remainsroughly constant along the z axis. For a finite box of length Lzin the z direction, the discrete version of eq 11 gives theinterfacial tension (averaged over two interfaces due to periodicboundary conditions) as74−76

γ = ⟨ ⟩ −⟨ ⟩ + ⟨ ⟩⎛

⎝⎜⎜

⎞⎠⎟⎟

Lp

p p

2 2z

zzxx yy

(12)

where ⟨ ⟩ denotes spatial averages along the z coordinate.Equations 11 and 12 can be sensitive to the cutoff fortruncating pair potentials.77 We perform AA simulations withnonbonded pair cutoffs of 10, 11, and 12 Å, and find that theinterfacial tension does not change within statistical certaintybetween the last two cutoffs, with a value of 43.7 mN/m at 12Å. Figure 8 compares the AA surface tension with that of thedifferent CG models in Table 1 and from the threeparametrization references. The AA surface tension is ∼43.7mN/m which overestimates by 36% the experimental value (32mN/m);78 this deviation is likely due to the overprediction ofthe solubility of benzene in water by the GROMOS 53a6 AAforce field, as discussed in section II.B. However, we note thatour focus here is the ability of the CG models to capture thegiven reference, regardless of their absolute accuracy.All CG models in Figure 8 embed significant transferability

errors such that predicted γ values systematically increase fromlow to high benzene concentrations. The absolute errors in theinterfacial tension are lowest for the 50% reference solution;under these conditions, the pair-only and pair + LD-WWmodels come closest to matching the AA value, under-estimating it by 1%. CG models from the 10 and 90%references give values off by approximately 37 and 175%,respectively. It should be noted that the pressure tensorcomponents for calculating the surface tension are obtainedfrom constant volume (NVT) CG MD simulations (boxdimensions are taken from the equilibrated AA simulation). Weverified that the ensemble averaged pressure has no correlationwith the surface tension, so that the differences in γ values aredue to differences in the CG force fields and not differences instate.



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Although the differences in bulk density profiles predicted bythe LD-augmented CG models are small, they perform poorlyin reproducing the thermodynamic pressure and consequentlythe interfacial tension. The average CG pressure varies between

4500 and 7000 atm with a 3% average (RMS) fluctuation fromthe mean. This is not terribly surprising, as bottom-up CGmodels are well-known to have difficulty capturing correctpressure, as has been explored in a number of pa-pers.24,29,30,79−82 Das and Andersen83 and more recentlyNoid and co-workers44,84 have suggested that, to predict thepressure correctly, CG models might incorporate additionalvolume-dependent terms in the interaction potential; by thensampling AA simulations under constant pressure conditions,where volume fluctuates, the CG volume-dependent energyterm can be parametrized in a bottom-up fashion.As a further investigation of thermodynamic properties of the

liquid mixtures, we consider Kirkwood−Buff integrals. Thetheory of solutions pioneered by Kirkwood and Buff and laterworked out in detail by Ben-Naim provides an important linkbetween microscopic structure and macroscopic thermody-namic observables of solutions.85,86 The central elements ofKirkwood−Buff (KB) theory are integrals that depend on theRDFs as

∫ π= −αβ αβG R r r g r( ) d 4 [ ( ) 1]R

0

2(13)

where gαβ(r) is the RDF between species α and β (αβ = BB,WW, BW) and R is a correlation distance beyond which theRDF becomes flat and the integrand in eq 13 vanishes. TheKirkwood−Buff integral (KBI) is related to the excesscoordination number around a particle relative to a flat densityprofile. Thus, a positive value of Gαβ signifies a higherpropensity of molecule type β around α (within a correlationradius of R), whereas a negative value indicates low

Figure 8. Comparison of benzene−water interfacial tension betweenthe AA system and the different CG models from 10, 50, and 90%reference solutions, for a 27.5% benzene solution in a long rectangularbox. The AA system overpredicts the experimental surface tension(∼32 mN/m) of the benzene−water interface,78 but regardless of itsabsolute accuracy, all CG models embed different amounts oftransferability errors relative to it. The pair-only and pair + LD-WWmodels parametrized from the 50% solution come closest toreproducing the AA surface tension. CG models from moreconcentrated reference solutions predict increasingly high surfacetensions.

Figure 9. Comparison of the Kirkwood−Buff integrals (KBIs) of types B−B, W−W, and B−W (along the vertical subpanels) shown as a function ofcorrelation length between AA and CG models for a very dilute solution at 0.2% mole fraction of benzene (left panel) and a superconcentratedsolution at 99.8% (right panel). The CG models are parametrized from a 50% reference solution. All CG models have poor transferability for the BBKBI at dilute composition and for the WW KBI at very high composition. CG models with all LD potentials come closest to reproducing the WWand BW KBIs at low composition.



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intermolecular affinity. Solute−solute, solute−solvent, andsolvent−solvent KBIs can be additively combined into apreferential solvation parameter

Δ = + −G G G2BB WW BW (14)

which quantifies the mutual affinity between the solute andsolvent and provides a gateway between microscopic structureand macroscopic thermodynamic properties through itsconnection to chemical potentials and activity coefficients. Inparticular, the composition derivatives of the benzene chemicalpotential (μB) and activity coefficient (γB) at temperature T andpressure P, for a solution with benzene and waterconcentrations (in M) ρB, ρW and mole fractions xB, xW,respectively, follow

μβ ρ

∂∂

= + Δ −⎛⎝⎜

⎞⎠⎟x x x[ (1 )]

P T

B

B ,B W B BW

1

(15)

and

γ ρρ

∂∂

= −Δ

+ Δ

⎛⎝⎜

⎞⎠⎟x

x

x

ln

ln 1P T

B

B ,

W B BW

W B BW (16)

It should be noted that, for small values of r and/or smallsystem size, the RDFs are insufficiently sampled at lowinterparticle distances, which can lead to convergence issuesin KBIs and impart oscillatory character to these integrals whenconsidered as a function of R. The convergence properties ofKBIs and corrections to adapt them for small system sizes havebeen of interest in the recent literature and have been studiedby van der Vegt and co-workers87,88 and Schnell and co-workers.89

We evaluate the transferability of KBIs calculated using CGmodels developed from the 50% benzene solution, underextreme composition conditions at 0.2 and 99.8% benzenemole fraction that lie outside of the phase separation envelope.It should be noted that eq 13 is exact only for MD simulationsin the grand canonical ensemble, and holds approximately forclosed systems in a manner that becomes increasingly accuratewith larger system sizes. Therefore, we carry out atomisticsimulations with 5000 molecules and larger equilibrium cubicbox dimensions (≈53 Å for the 0.2% solution and 89 Å for the99.8% solution).Figure 9 compares the B−B, W−W, and B−W types of KBIs

as a function of the correlation length from among the CGmodels of Table 1 for xB = 0.002 (left panel) and xB = 0.998(right panel). In the 0.2% solution, the BB KBI fails to convergeat large R for the pair-only CG model, and when LD potentialsare used, it converges on values that are different from the AAsystem (black line). For the WW and BW KBIs, CG modelswith only pair potentials and/or lacking the LD-WW potential

also perform worse. The pair + LD-all force field comes closestto predicting all three KBIs, reproducing the converged valuesto within 41, 1.8, and 1.6% accuracy, for BB, WW, and BW,respectively. On the other hand, for the 99.8% solution, the BBKBI barely converges and is captured similarly by all of the CGmodels. Both the WW and BW KBIs do not converge for thiscase. KBI values for the AA, pair-only, and pair + LD-allmodels, averaged over the correlation length for the differentCG force fields, are compared in Table 2.The BB KBIs for all of the CG models at 0.2% mole fraction

have high positive values pointing toward higher benzeneaggregation at very dilute benzene composition than predictedby the AA model. This is likely due to transferability errorsstemming from the choice of the parametrization reference of50% mole fraction, which is significantly more concentratedthan the phase separation point of the system (xB = 0.0095, aspredicted by the AA force field55). It should be kept in mindthat converged KBIs calculated according to eq 13 are highlysensitive to small variations in the RDFs, especially over thecorrelation lengths that are chosen to report the averageconverged value. While Figure 5 shows that the transferabilityerrors for RDFs using local density-assisted force fields are low,it may be that such small variations are more difficult to capturequantitatively through the approach used here, and additionalconstraints may be useful to help optimize CG models tocorrectly predict the KBIs.90

Arguably, this test of transferability is somewhat extremebecause it attempts to bridge both sides of the phase transition.Previous studies of this system have sought the opposite route,i.e., constructing CG models at very dilute compositions andobserving their transferability to extremely high concentra-tions.52 Intra- and interspecies many-body effects are typicallyabsent in dilute solutions, so perhaps mean-field approacheslike the LD potential necessitate the use of a reference AAcomposition that is concentrated enough to demonstratesufficient multiparticle interactions through macroscopicaggregation and/or consequent phase separation behavior.

■ CONCLUSIONSIn this work, we used local density (LD) potentials as a simpleand computationally inexpensive way to develop transferableCG models of molecular liquid solutions and hence theirequilibrium phase behavior. Specifically, we used relativeentropy minimization to develop CG models of aqueousbenzene solutions that map benzene (B) and water (W)molecules to single CG sites, thus coarse-graining awayorientational degrees of freedom that heavily mediate theunique geometric and hydrogen bonded interactions governingliquid phase properties. We explored the ability of LDpotentials to capture the multibody impact of these interactionsand their mediation by the local solution environment. In

Table 2. Comparison of the Values of the KBIs (Averaged over Correlation Lengths in the Interval 14−18 Å for the 0.2%Solution and 35−40 Å for the 99.8% Solution) between the AA System, Pair-Only, and Pair + LD-All CG Modelsa

xB force field type GBB (Å3) GWW (Å

3) GBW (Å3)

0.002 AA 305 ± 66 −23.96 ± 0.02 −124.58 ± 0.850.002 pair-only 102100 ± 2400* −8.12 ± 0.29 −1310 ± 250.002 pair + LD-all 179 ± 14 −23.55 ± 0.03 −122.5 ± 1.10.998 AA −97.55 ± 0.17 2000 ± 1100* −24.0 ± 3.3*0.998 pair-only −93.92 ± 0.17 −4600 ± 1200* −32.3 ± 2.1*0.998 pair + LD-all −86.92 ± 0.26 4050 ± 620* −209.8 ± 2.7*

aValues marked with an asterisk (*) have not converged.



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combination with three CG pair potentials (BB, WW, BW), wetested the relative roles of the four possible LD potentials (BB,WW, BW, WB) by combining them zero, one, and all at a time,and we evaluated the sensitivity of the models to differentreference all-atom system compositions spanning low to highbenzene mole fractions. Here, the relative entropy minimizationstrategy provided a straightforward and simple way toparametrize both the conventional pair interactions and themore novel local-density potentials, all modeled by flexiblesplines.Our results show that CG models built entirely out of pair

potentials have limited transferability and poorly reproducestructural and thermodynamic properties at compositions farfrom the parametrization conditions. On the other hand, wefind that CG models including LD potentials show improvedstructural transferability, in some cases dramatically so:including all four LD interactions allows quantitativereproduction of radial distribution functions and coordinationnumber distributions across all of the composition space. Of thedifferent LD potentials, the LD-BB interaction appears the leastimportant to improving transferability, while the LD-WWpotential has the strongest effect, suggesting that water−waterinteractions comprise the dominant multibody force in thesystem. This highlights the role of water’s unique structuralcorrelations and its tetrahedral hydrogen bonded net-work31,67,91 in mediating its interactions with other watermolecules as well as hydrophobic interactions with the muchlarger benzene molecule. Ultimately, the unlike LD potentialshave an important role too in characterizing the B−Wsegregation under phase separated conditions, so that the CGforce field including all four LD potentials is always the mosttransferable in terms of both structure and thermodynamics,regardless of the AA reference.In terms of other thermodynamic properties, we also found

that the LD strategy improved macroscopic simulations ofphase separations, allowing individual phases to equilibrate atcorrect bulk compositions. However, even with the LD strategy,CG models produced more diffuse benzene−water interfacesthan expected from all-atom results, likely due to the significantgradients in local composition that are not addressed by LDpotentials. All CG models showed difficulty in capturing correctbulk pressures and interfacial tensions, likely connected to thedifficulty in resolving the interfacial profile but also to the well-known problem of recapitulating pressure with bottom-up CGmodels.24,29,30,79−82 Moreover, we found that the LD strategygenerally improved the ability of CG models to captureKirkwood−Buff integrals at very dilute compositions, althoughno one model was able to completely reproduce thecorresponding AA results.This work illustrates a proof of principle for using LD

potentials in bottom-up CG models capable of improvingtransferability and thus capturing phase separated systems. It isinteresting to note that all models were developed fromrelatively small reference systems, yet translated well to larger-scale simulations. Indeed, here the small system size magnifiedthe effect of interfacial and cross-species interactions, whichprovided sampling of local composition fluctuations importantfor model parametrization by relative entropy minimization.The determined forms of the local density interactions werealso informative: the self-LD interactions (LD-BB, LD-WW)showed the most significant energy variations and impact. Thismay suggest the possibility of tabulating self LD interactions fordifferent species independent of any particular mixture, such

that these might be combined in arbitrary mixtures to createtransferable force fields; we leave this idea for future directions.However, one surprising result was the significant variation inthe form of the LD potentials determined at distinct referencecompositions, even though each model still showed excellenttransferability in structural properties across composition(particularly for the LD-all case). Perhaps even with LDpotentials, the CG force field still admits some flexibility thatcould be adjusted to target reproduction of other properties, forexample, those relevant to solvation thermodynamics.In a previous work, we used a single type of LD potential

between CG monomers of a superhydrophobic polymer toconstruct implicit-water solvent models that led to a much-improved description of the conformational space sampled bythe polymer.41 Hydrophobic polymer collapse and benzeneclustering in water both share common themes of higher-ordercooperativity inherent to hydrophobic interactions, which canbe modulated by the size asymmetry of the involved species.51

It seems that, by accounting for local structure, even if simply interms of coordination numbers without explicitly includingorientational degrees of freedom, LD potentials can capture thedelicate balance between entropy-driven hydrophobic forcesaround small solutes and enthalpy-driven hydrophobicinteractions around large macromolecules and macroscopicinterfaces. Thus, the corrective effect of the LD potential overtraditional CG pair interaction may be broadly useful forimproving a wide variety of CG models of fluid phase physicscharacterized by short ranged forces. In turn, the use of CGmodels may significantly enable the application of rigorousMonte Carlo phase equilibrium calculations5,8 that involveinsertion and deletion moves. While the minimal set of LDpotentials needed for good transferability in a CG model maybe difficult to predict a priori, it is useful to note that water is avery common solvent and retaining its local structuralcorrelations in CG models is arguably important for a largeclass of fluid mixtures.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcb.7b12446.

A detailed comparison of radial distribution functionsand local density distributions for different force fieldsand reference compositions (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected]. Scott Shell: 0000-0002-0439-1534NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by the National Science Foundation(Award No. CHEM-1300770). We gratefully acknowledge thehelp from Prof. Dr. Christine Peter at the Department ofChemistry, Universitat Konstanz, for sharing importantinformation about the GROMACS implementation of theatomistic systems. We are also grateful to Prof. Dr. Nico vander Vegt and David Rosenberger, of the Department of



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Computational and Physical Chemistry, T.U. Darmstadt, forhelpful comments and insight.

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