Hybrid MC−DFT Method for Studying Multidimensional Entropic Forces

Published: January 20, 2011

r 2011 American Chemical Society 1450 dx.doi.org/10.1021/jp110066z | J. Phys. Chem. B 2011, 115, 1450–1460

ARTICLE

pubs.acs.org/JPCB

Hybrid MC-DFT Method for Studying MultidimensionalEntropic ForcesZhehui Jin and Jianzhong Wu*

Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521-0444, United States

ABSTRACT: Entropic force has been a focus of many recenttheoretical studies because of its fundamental importance insolution thermodynamics and its close relevance to a broadrange of practical applications. Whereas previous investigationsare mostly concerned with the potential energy as a one-dimensional function of the separation, here we propose ahybrid method for studying multidimensional systems by com-bining Monte Carlo simulation for the microscopic configura-tions of the solvent and the density functional theory for the freeenergy. We demonstrate that the hybrid method predicts the potential of mean force between a test particle and various concave objectsin a hard-sphere solvent in excellent agreement with the results from alternative but more expensive computational methods. Inparticular, the hybridmethod captures the entropic force between asymmetric particles and its dependence on the particle size and shapethat underlies the “lock and key” interactions. Because the samemolecularmodel is used for the theory and simulation, we expect that thehybrid method provides a new avenue to efficient computation of entropic forces in complex molecular systems.

1. INTRODUCTION

An entropic force arises in a solution or a colloidal dispersionwhen the local distribution of solvent (or cosolvent) molecules isdisrupted due to the presence of the solute. Because it is affiliatedwith the solvent configuration rather than a specific microscopicenergy, themicroscopic origin of an entropic force appears obscure,and sometimes it is misinterpreted. Even in its simplest forminduced by the excluded-volume effects, current understanding ismostly limited to interaction between colloidal particles in thepresence of polymer chains dissolved in a good solvent. The firstattempt to quantify such interaction was made by Asakura andOosawa (AO),1 who developed an analytical expression for thepotential between colloidal particles in a polymer solution bytreating polymer chains as “ideal-gas” molecules that are depletedfrom the particle surface. According to the AO theory, two colloidalparticles experience an entropic attraction when there is an overlapof the polymer depletion layers. The attraction is introduced by anincrease of the system entropy or accessible volume for thepolymer. Despite its simplicity, the AO theory gives a semiquanti-tative prediction of the polymer-mediated entropic force at low andintermediate polymer concentrations. However, its performance isunsatisfactory at high polymer concentration when the interchaininteractions become non-negligible. Besides, the AO theory isinadequate if the size of the colloidal particles is smaller orcomparable to the radius gyration of the polymer chains.2 Towardan improvement of the AO theory, many theoretical investigationsand molecular simulations have been reported over the past fewdecades. Common computational methods include the hyper-netted-chain (HNC) equation,3 the polymer self-consistent-fieldtheory,4 the classical density functional theory (DFT),5,6 and

various forms of computer simulations.7,8 These investigationsaremostly focused on the entropic potential between two sphericalparticles, or a spherical particle and a hard wall, in the presence ofhard-sphere-like solvent.

Whereas existing theoretical methods are adequate to quantifyentropic force in colloidal systems with relatively simple geometry,their extension to multidimensional interactions such as thoseimportant for biological systems is challenging. On the one hand,simulation of the entropic potential entails free-energy calculationsthat are computationally demanding. The power of analyticalmethods, on the other hand, is often compromised by numericalsolution of the multidimensional density distribution functions.Unlike a typical colloidal potential, interaction between biomacro-molecules often depends on the molecular geometry and orienta-tion, in addition to the center-to-center distance. While in generalthe entropic force is closely affiliated with the self-organization ofwater molecules around the solute, the geometry or shape ofbiomacromolecules plays a central role in receptor-ligand or “lockand key” interactions, and its contribution to the overall potential isoften manifested through the excluded-volume effects of thesurrounding water molecules.3,9,10 Recently, colloidal lock andkey systems have been developed to mimic the geometry affinityof biointeractions.11-16 Such model systems have generated ex-tensive interest not only because they resemble interactions inbiological processes, including host and guest interactions17 andrecognition of drug compounds,18 but also because the affinity

Received: October 20, 2010Revised: January 5, 2011

1451 dx.doi.org/10.1021/jp110066z |J. Phys. Chem. B 2011, 115, 1450–1460

The Journal of Physical Chemistry B ARTICLE

force empowers a precise control of the microscopic structureduring colloidal self-assembly.11 These model systems have alsoinspired a number of theoretical and simulation studies. Forexample, Kinoshita19 used a three-dimensional HNC equation toexamine interactions between a substrate with a hemisphere cavityand a testing spherical particle in a hard-sphere solvent. Konig andco-workers9 used theWhite Bear20 version of fundamentalmeasuretheory (FMT)21 and a curvature expansion technique22 to studythe depletion potential between a smooth substrate and a biaxialelliptical testing particle. On the simulation side, Odriozola and co-workers23 investigated the effective interaction between a spherewith an open cavity using Monte Carlo (MC) simulation.

In this work, we propose a hybrid method for computation ofentropy forces by using MC simulation to predict the densitydistributions of solvent molecules and the fundamental measuretheory for free-energy calculations.Whereas the DFT is formulatedin terms of the density profiles, direct minimization of the free-energy functional with respect to a three-dimensional densityis unrealistic for most anisotropic systems. This is because theDFT calculation often entails lengthy iterations and requires an ex-tremely small grid (∼0.002 molecular diameter) to capture thedensity profiles. On the other hand, simulation of the solvation freeenergy is in general very time-consuming, but simulation of thesolvent density at a fixed solute configuration is not a computa-tionally demanding task. For calculating the solvent density,simulation on the time scale comparable to that correspondingto the relaxation dynamics is often sufficient. Besides, simulationis not limited by the dimensionality, and much larger grids (∼0.1molecular diameter) can be used to sample the solvent densityprofile.While extension ofmolecular simulation for simple colloidalsystems to those with solutes of more complicated shape requiresessentially no increase of the computational cost, DFT provides anefficient link between the microscopic structure and thermody-namic potentials. A combination of simulation and DFT takesadvantage of the good features of both methods.

With an analytical expression from DFT for the free-energy as afunctional of the solute and solvent density profiles, we cancalculate the entropic force by using the potential distributiontheorem (PDT).24 This method has been previously used tocalculate the colloidal force between two big particles and particle-wall interactions5,6 in the presence of a hard-sphere solvent ormodel polymers.6 Alternatively, entropic force can be calculatedfrom the free energy of the solvent with the colloidal particlestreated as an external potential. In comparison with PDT, compu-tation of the free energy at explicit colloidal configurations is moretime-consuming, but the latter method is useful when an accurateexpression for the free-energy functional of the solute-solventmixture is not attainable.

The remainder of this article is organized as follows. Section 2introduces the molecular theory and simulation methods. Insection 3, we define the thermodynamic systems and molecularmodels studied in this work and validate the proposed methods bycomparing with previous computer simulations and theoreticalinvestigations on the colloidal forces for the targeted systems. Insection 4, we summarize the key conclusions and discuss furtherimplications of the proposed methods.

2. THEORETICAL APPROACHES

The main idea behind our new method for calculating theentropic force in multidimensional systems is as follows: First, weuse MC simulation to generate the three-dimensional density

profiles of the solvent required in analytical calculations. Then,we use the DFT to calculate the free energy and subsequently theentropic forces. The hybrid method can be implemented either byexplicit consideration of the colloidal configuration or by using thepotential distribution theorem (PDT). The first procedure isreferred to as the MC-DFT and the second as MC-PDT.Because PDT requires only the density profile of the solventmolecules around a single solute, the numerical advantage ofMC-PDT is self-evident. However, MC-PDT hinges on a free-energy functional that is accurate for the solute-solvent mixture,and the numerical efficiency is often compromised by approxima-tions introduced in the DFT. In this work, we will validateboth procedures by considering entropic forces (1) between twospherical particles, (2) between a spherical particle and a planarhard wall, (3) between a spherical particle and a substrate with ahemispherical cavitymimicking the “key and lock” systems, and (4)between a hard rod and a hard wall. In all cases, the molecularexcluded volume effect is represented by using a hard-spheresolvent, and the theoretical predictionswill be validatedwith resultsfrom direct simulations of the colloidal forces.a. MC-DFT. In MC-DFT, the density profile of solvent

molecules is simulated at a given solute configuration. For interac-tion between two colloidal particles in a hard-sphere solvent, thepotential of mean force corresponds to the reversible work toseparate them or the difference in the free energy when the systemis at a given colloidal configuration and that when the particles areinfinitely apart. If both particles are spherical, the potential of meanforce depends only on the center-to-center distance, designatedas r. The potential of mean force,W(r), can be calculated from

WðrÞ ¼ ΩðrÞ-Ωð¥Þ ð1ÞwhereΩ(r) stands for the grand potential of the system when theparticles are separated at r andΩ(¥) is that for the same systembutwhen the particles are “far” apart, i.e., r f ¥. Equation 1 can beextended to many-body interactions and to multidimensionalsystems by augmenting r with the orientational angles and/oradditional solute configurations.In a one-component hard-sphere system, the grand potential

can be related to the intrinsic Helmholtz energy functionalF[F(r)] via the Legendre transform:

Ω½FðrÞ� ¼ F½FðrÞ� þZ

½VðrÞ- μ�FðrÞ dr ð2Þ

In eq 2, dr denotes a differential volume, F(r) is the densitydistribution of hard spheres, μ is the chemical potential, and V(r)stands for the external potential. For studying the interactionpotential between a pair of colloidal particles, V(r) refers to thepotential due to the colloidal particles. The intrinsic Helmholtz freeenergy functional can be further divided into two parts: an ideal-gasterm and an excess free energy. The former is exactly known

Fid½FðrÞ� ¼ kBTZ

FðrÞfln½FðrÞλ3�- 1g dr ð3Þ

where kB represents the Boltzmann constant, T is the thermaltemperature, and λ is the thermal wavelength. For hard-spheresystems, the excess Helmholtz free energy functional can beaccurately represented by the modified fundamental measuretheory20,25

Fex ¼ kBTZ

Φ½nRðrÞ� dr ð4Þ



whereΦ(r) is the reduced excess Helmholtz free energy density

Φ½nRðrÞ� ¼ - n0 lnð1- n3Þþ n1n2 - nV1nV21- n3

þ 136π

n3 lnð1- n3Þþ n23ð1- n3Þ2

" #� n32 - 3n2nV2nV2

n33

ð5ÞIn eqs 4 and 5, nR(r), R = 0, 1, 2, 3, and V1 and V2 are weighteddensities.26 The expressions for weighted densities are given inprevious publications.25

In contrast to typical DFT calculations, in this work the three-dimensional (3D) density profiles are not calculated fromminimization of the grand potential but from grand-canonicalMonte Carlo simulations. The two procedures are equivalent ifthe free-energy functional is exact. While minimization of thegrand potential is computationally efficient if the density profilevaries only in a single dimension, its extension to higher dimensionbecomes numerically cumbersome. To implement the hybridmethod, we first simulate hard-sphere systems in the presence oftwo colloidal particles within the grand-canonical (μVT) en-semble. The number of hard spheres simulated varies with thecolloidal separation. The colloidal particles are fixed along theX-axis of the simulation box, and the periodic boundary condi-tions are used in all directions. In each MC cycle, a trial randomdisplacement is applied to all solvent spheres (but not colloidalparticles), and a solvent particle is randomly removed from orinserted into the simulation box at approximately equal prob-ability. TheMCmoves are implemented by using theMetropolisalgorithm.27 The simulation consists of one million MC cyclesfor equilibrium and fivemillionMC cycles for sampling the three-dimensional density profiles.b. MC-PDT. In the MC-PDT method, the potential of mean

force is obtained from the one-body direct correlation function. Forexample, the solvent-mediated potential between two sphericalparticles is related to the solvent density profile around a singleparticle and the one-body direct correlation of the second particle inthe presence of the solvent5

βWðrÞ ¼ cð1Þð¥Þ- cð1ÞðrÞ ð6Þwhere β = 1/(kBT). The one-body direct correlation function ofthe second particle c(1)(r) is obtained from the excess intrinsicHelmholtz free energyFex for the particle and solventmixture in thelimit of zero particle concentration Fp(r) f 0

cð1ÞðrÞ ¼ -δβFex

δFpðrÞ

��FpðrÞ f 0

ð7Þ

When two particles are far apart, c(1)(¥) is related to the reducedexcess chemical potential of the particle in the bulk solvent, i.e.

cð1Þð¥Þ ¼ - βμexp ð8ÞWith the density profile of the solvent molecules around asingle particle obtained fromMC simulation, PDT allows us tocalculate the potential of mean force between two particles atall colloidal separations. Like MC-DFT, MC-PDT can alsobe extended to multidimensional systems. The key differencebetween the MC-DFT and the MC-PDT is whether simulation

includes a single particle or two particles at different separations. Forsome relatively simple systems (e.g., hard-sphere mixtures), wemayderive the density profile of the solvent by directminimization of thegrand potential and use the PDT to calculate the PMF. The resultscalculated from this procedure are labeled as DFT-PDT.

3. RESULTS AND DISCUSSION

For calibration of the proposed computational procedures, weconsider fourmodel systemswhere the entropic potentials have beenstudied before and the underlying physics is relatively well under-stood.Thepotential between two identical spheres or between a hardsphere and a planar surface depends only on the surface-to-surfaceseparation. On the other hand, the interaction in a colloidal key andlock system or between a hard rod and a hard wall depends on boththe separation and orientation. We will compare the numerical per-formances of MC-DFT andMC-PDTwith full simulation resultsandwith alternative analyticalmethods. Table 1 summarizes differenttheoretical methods discussed in this work.a. Two Identical Spheres. Entropic interaction between two

big particles (B) immersed in a sea of small hard spheres (S) isaffiliated with the sphere and particle interactions

jSBðrÞ ¼¥, r < ðσþDBÞ=20, rg ðσþDBÞ=2

8<: ð9Þ

and with the pair interaction among small hard spheres

jSSðrÞ ¼ ¥, r < σ

0, rg σ

(ð10Þ

In eqs 9 and 10,σ stands for the diameter of the solvent hard spheres,DB is the diameter of big particles, and r is the center-to-centerdistance. For comparison with results from direct simulation,8 we setDB = 5σ. In ourMC-DFT calculations, two colloidal particles wereplaced at (x, y, z) = (-(DBþH)/2, 0, 0) and (x, y, z) = ((DBþH)/2, 0, 0), respectively, where H represents the contact distancebetween two big spheres. The simulation box is a rectangular prismwith sides Lx = 22σ and Ly = Lz = 14σ. In MC-PDT calculations, asingle particle is placed at the center of a simulation box withdimensions Lx = 20σ and Ly = Lz = 16σ.Figure 1 presents the depletion potentials calculated from direct

MC simulations and from three different semianalytical methods.Figure 2 shows the contour plots of the density profiles of the hard-sphere solvent at different colloidal separations. Here η = πF0σ3/6stands for the packing fraction of the hard-sphere solvent in thebulk and F0 is the bulk density. In the DFT-PDT calculations,28

the density profiles of the solvent molecules near a single particleare calculated from minimization of the grand potential. At lowdensity (η = 0.116), all methods give virtually the same results.Noticeable differences are observed, however, at high density (η =0.229). The PDT methods overpredict the repulsive barrierregardless of whether the density profile is from the DFT or from

Table 1. Computational Methods for the Entropic ForceDiscussed in This Work

methods MC-DFT MC-PDT DFT-PDT MC

density profile MC MC DFT direct sampling

free energy

calculation

DFT PDT PDT integration of

the mean force



simulation. On the other hand, MC-DFT provides good agree-ment with the simulation results.8

As mentioned before, the attraction between colloidalparticles arises from an overlap of the solvent depletion layers(see Figure 2). The depletion potential is most significantwhen the colloidal particles are at contact. Because the twoparticles are aligned in the X-axis (y = 0, z = 0), the solventdensity profiles are cylindrically symmetric and can be repre-sented in terms of F(x, r) where r = (y2 þ z2)1/2. Whereas allDFT-based methods are able to reproduce the simulationresults quantitatively, the numerical performance of the AOtheory is unsatisfactory. While it captures the surface deple-tion, the AO theory fails to describe accumulation of thesolvent molecules at the interstice of colloidal spheres (asseen in Figure2a) that is responsible for the repulsivebarrier.b. A Sphere near a Hard Wall. We now consider the PMF

between a spherical particle and a hard wall in a hard-spheresolvent. The system can be understood as a limit case for theinteractions between two asymmetric hard spheres. Similar to thecase of two identical spheres, we again use MC-DFT and MC-PDT to calculate the entropic forces. InMC simulations, the wall isplaced at x = 0 of the simulation box, and a spherical particle ofdiameter DB = 5σ is placed at (Hþ (DB/2), 0, 0) whereH standsfor the surface-to-surface distance between the hardwall and the bigparticle. The dimensions of the simulation box are set as Lx = 24σand Ly = Lz = 14σ for MC-DFT calculations and Lx = 16σ andLy = Lz = 14σ for MC-PDT. To apply the periodic boundaryconditions to all three directions, we place the flat wall in the centerof the simulation box, and the boundaries are selected such that thesolvent density approaches that corresponding to the bulk.Figure 3 shows the depletion potential at two representative

solvent packing fractions. In this case, both MC-DFT andMC-PDT agree well with the simulation results. At bothsolvent densities, the entropic potential shows a global mini-mum at the contact point and a repulsive barrier at intermedi-ate separation, similar to the interaction between identical

Figure 1. (Color online) Depletion potential between two big particles ina solvent of small particles at different solvent densities. The size ratiobetween the big and small particles is s= 5 . The symbols areMC simulationresults by Dickman,8 the red lines are from theMC-DFT calculations, thegreen lines are from MC-PDT, and the blue lines are from DFT-PDTcalculations.5,6 The solid lines are for bulk solvent packing fractionη= 0.116and dashed lines for bulk solvent packing fraction η = 0.229.

Figure 2. (Color online) (a) Contour plots of the solvent density neartwo big spheres at contact (H = 0). Here the packing fraction of smallspheres is η = 0.116. The centers of two big particles are placed along thex-axis (y = 0, z = 0). (b) H = 1.5σ. (c) H = 2.5σ.



spheres. In comparison to the results shown in Figure 1,the performance of MC-PDT is slightly improved mainlybecause, comparing to that between spherical surfaces, numer-ical integrations in 3D Cartesian grid is more easily formulatednear the flat wall.

c. Colloidal Lock and Key Interactions. We now investigatea model lock and key system where the lock is represented by aplanar substrate with a hemispherical cavity and the key is

Figure 3. (Color online) Depletion potentials between a big sphere(s = 5) and a flat hard wall immersed in a sea of small hard spheres atdifferent bulk densities. The symbols are for MC results by Dickman,8

the red lines are from theMC-DFT calculations, and the green lines arefrom the MC-PDT. The solid lines correspond to bulk solvent packingfraction η = 0.1, and dashed lines are for bulk solvent packing fractionη = 0.2.

Figure 4. (Color online) Geometry of a key and lock systemconsidered in this work (y = 0 plane cut). The key is representedby a big spherical particle of diameter Dkey, and the lock is representedby a planar substrate with a hemispherical pocket of diameter Dlock.The separation between the key and the lock is measured by thedistance between the centers of the key at (x, 0, 0) and the lock pocketat (x0, 0, 0).

Figure 5. (Color online) (a) Depletion potential between a key ofdiameter Dkey = 4σ and a lock with cavity diameter Dlock = 5σ in ahard-sphere solvent at bulk packing fraction η = 0.367. We comparethe MC-PDT (blue line) and MC-DFT (magenta line) methodswith the simulation data (symbols), HNC results (red line) fromKinoshita,19 and DFT results (green line) from Konig et al.9 (b) Sameas (a) except Dkey = 5σ. (c) Same as (a) except Dkey = 6σ.



represented by a spherical testing particle. Again the solventmolecules are represented by hard spheres. In contrast to thesimple systems discussed above, the PMF for the lock-and-keyinteraction depends on the three-dimensional distribution of thesolvent molecules.

To facilitate numerical comparison between simulation andDFT calculations, we consider a symmetric configuration ofthe lock and key system as shown in Figure 4. In our MCsimulations, the center of the hemispherical cavity is placed at(x0, 0, 0) of the simulation box with x0 = 3σ, and the center of

Figure 6. (Color online) Contour plots for the solvent density near a “key-lock” system. The key diameter isDkey = 5σ, and the lock has a cavityof diameter Dlock = 5σ. The packaging fraction of hard spheres in the bulk is η = 0.367. The centers of the key and the lock pocket are alignedin the x-direction, and the figure is plotted in a way similar to Figure 2. (a) x - x0 = 0, (b) x - x0 = 2σ, and (c) x - x0 = 4σ. (d) Change inthe excluded volume ΔVex(x-x0) = Vex(x-x0)- Vex(¥) versus the separation between the lock and the key complex. Here Vex(x-x0) standsfor the total excluded volume of the lock and key complex. The solvent accessible volume is negatively proportional to the total excludedvolume.



the key is located at (x, 0, 0) with varying x. The radius of thecavity is fixed at Rlock = 2.5σ. We examine the depletionpotential W(x-x0) for various key particles with radius Rkeyalong the symmetric axis. At all situations, the dimensions ofthe simulation box are Ly = Lz = 14σ and Lx = 30σ. TheMC-DFT and MC-PDT calculations utilize the three-di-mensional density profiles of the solvent molecules, with orwithout the key particles, respectively. Besides, we also use adirect sampling method originally proposed by Wu et al.29 to

obtain the entropic force f(x-x0) at different lock and keyseparations. The potential of mean force can be derived fromthe integration of the mean force.30

Figure 5 presents the entropic potentials corresponding to acavity lock and three different spherical keys. Again bothMC-DFT and MC-PDT methods are in quantitative agreementwith the direct sampling method. They all faithfully predictthat the lock and key interaction is highly specific at shortseparations. When the key is either too small or too large, the

Figure 7. (Color online) Same as Figure 6 except Dkey = 4σ and (a) x - x0 = -0.5σ.



entropic potential resembles that between two identicalspheres or that between a spherical particle and a hard wall.In those cases, the entropic potential oscillates with separationand exhibits a maximum attraction at the contact. When thekey geometry matches that of the substrate, the contactpotential is drastically magnified, and, less intuitively, therepulsion barrier disappears. For comparison, also shown inFigure 5 are predictions from the three-dimensional HNC19

and from the DFT and curvature expansion method.9 Whilethe HNC overestimates the maximum attraction between thelock and a perfectly matched key, the opposite is true for theresults from the curvature expansion method. Furthermore,application of the curvature expansion is ambiguous for con-cave objects, especially for systems with a discontinuouscurvature.9,31

Similar to interactions between two identical spheres, theattraction between the lock and the key is also related to thelocal distributions of small particles. Figure 6 shows the solventdensity distributions obtained from MC simulation. When thesize of the testing particle (key) matches perfectly that of thecavity at the substrate (lock), the system reaches the maximumentropy when the key fits into the lock. In this case, the solventmolecules have the maximum accessible volume. Figure 6dindicates that the solvent accessible volume and the key-lockpotential decrease monotonically as the separation betweenthe key and the lock increases. When the separation betweenthe lock and the key is sufficiently large, we expect that thedistributions of the solvent molecules around on the lock andthe key become independent and thus the entropic potentialdiminishes.Figure 7 shows similar solvent density profiles when the lock

and the key are not in a perfect match. When the key is too small,the solvent enters into the unfilled hemispherical pocket com-promising the attraction at contact. Furthermore, at certainlock-key separation, the solvent is highly concentrated in thepocket (Figure 7b). The high local concentration results in amaximum repulsion in the lock-key interaction. The depen-dence of the PMF on separation for the oversized key is similar tothat for the case of the perfect match case (Figure 5), and thus it isnot shown here.d. Angle-Dependent Entropic Potential. To test the applic-

ability of the proposed method for nonspherical systems, westudy also the angle dependence of the entropic potentialbetween a spherocylinder and a hard wall. Because the modifiedfundamental measure theory (MFMT) is not directly applicableto rod-sphere mixtures, we calculate the depletion potentialonly from MC-DFT and compare the results with the simula-tion data and with predictions of an analytical method by Rothet al.32Whereas theMC-PDTmethod hinges on the free energyfunctional of the solute-solvent mixture, only the free energy ofthe solvent is needed in the MC-DFT calculation, which makesit more generic in particular for systems containing solutes ofcomplicated shape.Figure 8 shows a schematic representation of the cylinder-

wall system. In simulation of the density profile of solventmolecules at a given rod configuration, we consider a hard wallplaced at x = 0 and a hard spherocylinder of diameter σ andcylinder length L = 10σ with its center placed at distance x fromthe wall. The orientation of the spherocylinder, relative to thedirection perpendicular to the wall, varies from θ∈ [0,π/2]. Thesimulation box is filled with a hard-sphere solvent, and theperiodic boundary conditions are applied to the y- and

z-directions.32 Because the focus here is on the orientationdependence of the entropic potential, the rod center is eitherfixed at the smallest distance from the wall (i.e., with one end incontact with the surface, xmin(θ) = (σ þ |cosθ|)/2) or at x =5.5σ. For all cases, the sides of the simulation box are Ly = Lz =20σ and Lx = 16σ.Figure 9 presents the reduced depletion potential βW(xmin,

θ) as a function of θ. The agreement between MC-DFT andthe DFT calculation by Roth et al. is excellent.32 When the rodis parallel to the wall, i.e., θ = 0, the DFT calculations yield acontact potential significantly greater than that from the AOtheory. The trend is consistent with their performance for the

Figure 8. Schematic of a hard spherocylinder near a planar wall (y = 0plane cut). The spherocylinder is placed at (x, 0, 0) while the wall is at x=0. At a fixed orientation, the minimum possible separation between thespherocylinder and the planar wall is xmin(θ) = (σ þ L|cosθ|)/2.

Figure 9. Contact potential βW(xmin, θ) as a function of angle θbetween the spherocylinder and the hard wall shown in Figure 8. Herethe length of the spherocylinder is L = 10σ, and the packing fraction ofthe hard-sphere solvent in the bulk is η = 0.2239. The solid line is theDFT prediction by Roth et al.,32 the dashed line is from the AO theory,and symbols represent the MC-DFT predictions.



sphere-sphere and sphere-wall interactions.8 The AO theoryunderestimates the contact potential primarily due to itsnegligence of the solvent-solvent interactions. Such depletioneffect is due to the solvent density distribution (see Figure 10).Because of the inhomogeneity of the depletion potential on

the rod surface, the entropic force introduces a torque M(x, θ)that varies with the rod position and angle θ.32 From thedepletion potential W(x, θ), we can obtain the torque M(x, θ)around an axis passing through the center of the rod and parallel

to the wall32

Mðx, θÞ ¼ -∂Wðx, θÞ

∂θnθ ð11Þ

where nθ is the unit vector normal to the symmetry plane shown inFigure 8. Figure 11 shows the dependence of torque on orientationwhen the rod center is fixed at x = 5.5σ. The line denotes resultsfrom theMC-DFT and the symbols are frommolecular dynamicssimulations.32 The hybrid method and simulation agree well.

Figure 10. (Color online) Contour plots (in y = 0 plane) for the density distribution of small hard spheres in the presence of a hardrod at (5.5σ, 0, 0) and a flat wall at x = 0. The packing fraction of the bulk state is η = 0.2239. (a) θ = 90, (b) θ = 60, and(c) θ = 0.



4. CONCLUSION

We introduced two complementary methods for studying theentropic forces between colloidal particles in hard-sphere solventsby a combination ofMonte Carlo (MC) simulation and the densityfunctional theory (DFT). In the first method (MC-DFT), MCsimulation is used to sample the solvent density profiles at differentcolloidal configurations, and the entropic force is obtained fromthe difference in the grand potential. In the second method(MC-PDT), MC simulation is used to calculate the solventdensity around a single particle, and the potential of mean forceis obtained by using the potential distribution theorem (PDT). Forhard-sphere systems, the free energy functional has been calibratedin a number of previous publications20,25 and has been broadly usedby others.33 Because the fundamental measure theory gives anextremely accurate prediction of the free-energy functional, thedensity profile obtained from minimization of the free energy isexpected to be similar to that from simulation; thus, the latter can beused as an input for the DFT calculations. We have tested theperformance of the hybridMC-DFTmethods by comparing withresults from direct MC/MD simulations for the potential of meanforce between a pair of hard spheres, between a hard sphere and ahard wall, in colloidal “lock and key” systems, and between a hardrod and a hard surface. In addition, comparison was made betweenthe hybrid methods and direct DFT calculations. The theoreticalpredictions are all in good agreement with results from fullsimulations. Because MC-DFT requires only an accurate DFTfor the solvent, its performance is in general superior toMC-PDT,which relies on the density profile of the solvent near a singleparticle and a free energy function for the solute-solvent mixture.Nevertheless, MC-PDT avoids simulation at different colloidalconfigurations, and therefore it is numerically much more efficient.In comparison with alternative methods, both MC-DFT andMC-PDT take the advantages of simulation and analyticalcalculations for multidimensional structure and thermodynamicproperties, and both are computationally more efficient than directsimulations.

Whereas for numerical calibration this work is focused onrelatively simple systems, we expect that, with a good expressionof the free-energy functional to account for various components of

intermolecular interactions and correlation effects, a similar proce-dure can be extended to more realistic systems including to thoseunderlying biological and physical processes of direct interest forpractical applications.

’AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

’ACKNOWLEDGMENT

For financial support, the authors are grateful to the U.S.Department of Energy, the National Institute of Health, and theNational Science Foundation. This work utilizes supercomputersfrom the National Energy Research Scientific Computing Center(NERSC).

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Figure 11. Torque βM(x, θ) between a hard rod and a flat wall in ahard-sphere solvent. Here x = 5.5σ, the length of the spherocylinder is L= 10σ, and the packing fraction of the bulk state is η = 0.2239. The solidline is from the MC-DFT calculation, and symbols are from MDsimulation by Roth et al.32.



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Hybrid MC−DFT Method for Studying Multidimensional Entropic Forces

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