computer simulation of liquid/liquid interfaces. i. theory and application to octane/water

Computer simulation of liquid/liquid interfaces. I. Theory and application tooctane/waterYuhong Zhang, Scott E. Feller, Bernard R. Brooks, and Richard W. Pastor

Citation: The Journal of Chemical Physics 103, 10252 (1995); doi: 10.1063/1.469927 View online: http://dx.doi.org/10.1063/1.469927 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular structure and hydrophobic solvation thermodynamics at an octane–water interface J. Chem. Phys. 119, 9199 (2003); 10.1063/1.1605942 Adsorption of apolar molecules at the water liquid–vapor interface: A Monte Carlo simulations study of the water-n-octane system J. Chem. Phys. 119, 1731 (2003); 10.1063/1.1581848 Molecular dynamics computer simulations of solvation dynamics at liquid/liquid interfaces J. Chem. Phys. 114, 2817 (2001); 10.1063/1.1334902 Computer simulation studies of liquid lenses at a liquid–liquid interface J. Chem. Phys. 112, 5985 (2000); 10.1063/1.481171 Monte Carlo simulation of liquid–liquid benzene–water interface J. Chem. Phys. 86, 4177 (1987); 10.1063/1.451877

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Computer simulation of liquid/liquid interfaces. I. Theory and applicationto octane/water

Yuhong Zhang and Scott E. FellerBiophysics Laboratory, Center for Biologics Evaluation and Research, Food and Drug Administration,1401 Rockville Pike, Rockville, Maryland 20852-1448

Bernard R. BrooksLaboratory of Structural Biology, Division of Computer Research and Technology, National Institutesof Health, Bethesda, Maryland 20892

Richard W. PastorBiophysics Laboratory, Center for Biologics Evaluation and Research, Food and Drug Administration,1401 Rockville Pike, Rockville, Maryland 20852-1448

~Received 10 May 1995; accepted 13 September 1995!

Statistical ensembles for simulating liquid interfaces at constant pressure and/or surface tension areexamined, and equations of motion for molecular dynamics are obtained by various extensions ofthe Andersen extended system approach. Valid ensembles include: constant normal pressure andsurface area; constant tangential pressure and length normal to the interface; constant volume andsurface tension; and constant normal pressure and surface tension. Simulations at 293 K and 1 atmnormal pressure show consistent results with each other and with a simulation carried out at constantvolume and energy. Calculated surface tensions for octane/water~61.5 dyn/cm!, octane/vacuum~20.4 dyn/cm! and water/vacuum~70.2 dyn/cm! are in very good agreement with experiment~51.6,21.7, and 72.8 dyn/cm, respectively!. The practical consequences of simulating with two otherapproaches commonly used for isotropic systems are demonstrated on octane/water: applying equalnormal and tangential pressures leads to an instability; and applying a constant isotropic pressure of1 atm leads to a large positive normal pressure. Both results are expected for a system of nonzerosurface tension. Mass density and water polarization profiles in the liquid/liquid and liquid/vaporinterfaces are also compared.

I. INTRODUCTION

The structure and dynamics of liquid interfaces are chal-lenging problems of great interest to physicists, chemists andbiologists. For example, electrochemists are concerned withthe chemical reactions taking place at an electrode, chemistswith the mechanisms by which reactions are catalyzed at asolid surface, and biochemists with how a protein inserts in amembrane. These studies all require knowledge of the struc-ture and dynamics of liquid interfaces on the molecular level.However, molecules in an interface typically form only atiny fraction of a fluid, and, despite recent advances in ex-perimental techniques,1–3 perturbations from bulk structureand dynamics are difficult to measure and interpret. Conse-quently, the study of liquid interfaces has been an active areafor theoreticians. Mean field theories for the distribution ofions at a electrified interface date back over 85 years.4 Thesetheories based on the Poisson–Boltzmann equation havebeen shown by both experiment and more rigorous theory toyield incorrect results under certain conditions~e.g., highsurface charge and/or divalent electrolytes!.5 Later, integralequation theories for the structure of simple fluids were ex-tended to interfaces.6 These theoretical treatments of surfacesoften incorporate drastic simplifying assumptions regardingquantities such as the particle pair distribution function ordielectric constant, which can lead to incorrect predictions,7

or even serious inconsistencies.8 More recently, computersimulation techniques that can treat the interface in fullatomic detail have been applied to systems from simple

liquids9 to films10 and biological membranes.11 By providingmolecular details, simulations are a valuable complement toboth experiment and formal theory.

Simulations of liquid/liquid interfaces under a normalpressurePn of 1 atm present a difficulty not encountered inliquid/vapor systems. In the latter@Fig. 1~a!#, simulations aretypically carried out at constant particle number, volume andenergy~the NVE, or microcanonical, ensemble! with peri-odic boundary conditions.12–22 Because there is explicitvacuum, the density in the interfacial region can readily ad-just to its equilibrium value. Consequently, a slab taken froma previous simulation of the bulk fluid provides an adequateinitial condition. The dynamical equations for the microca-nonical ensemble are also straightforward.

Now consider anNVE simulation of the liquid/liquidinterface sketched in Fig. 1~b!. An initial condition com-prised of two abutting slabs of bulk fluid would most likelyrelax to a system withPn significantly different from 1 atm.In general, prior knowledge of the interfacial density profileor related quantities is required. Otherwise, the volume of thesimulation cell must be adjusted manually during the equili-bration until the normal pressure equals the target value orthe densities far from the interface approach their bulkvalues.23 One solution to this problem is to add layers ofvacuum@Figs. 1~c! and 1~d!#.7,24 This approach has the ad-vantage of enabling simulations at constantNVE while al-lowing for relaxation of the density. Two drawbacks to thisapproach are: the analysis is complicated by the presence ofseveral different interfaces@three for Fig. 1~c! and two for

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Fig. 1~d!#; the normal pressure cannot be varied~rather, itsaverage value is 0 atm!. An alternative and perhaps morenatural approach is to maintain the geometry of Fig. 1~b! ~asingle type of interface!, but to allow volume or shape fluc-tuations by setting thermodynamic quantities such as the nor-mal pressure, surface area or surface tension to appropriatetarget values. Techniques for carrying out simulations of ho-mogeneous systems at constant pressure are now standard,25

and some have already been applied to liquid/liquidinterfaces.26

It is the purpose of this paper to present general methodsfor simulating liquid/liquid interfaces. We consider in detailfive adiabatic ensembles, all with constant particle number:

~1! microcanonical (NVE);~2! constant normal pressure and surface area (NPnAH);~3! constant tangential pressure and length normal to the sur-

face (NPthzH);~4! constant volume and surface tension (NVgH);~5! constant normal pressure and surface tension (NPngH).

As we demonstrate here and in the following paper27 ~hence-forth denoted Paper II!, different applications call for differ-ent ensembles. For example, an oil/water system is generallybest simulated with~2!, while ~3! is appropriate for expand-ing a lipid monolayer when calculating a pressure/area iso-therm;~5! is the probably the method of choice when insert-

ing a peptide in a lipid bilayer, but~4! might be used whenthe volume and surface tension are already known~or rea-sonably estimated!, but the area is uncertain.

By way of outline, Sec. II defines the ensembles listedand derives conserved enthalpy functions for~2!–~5!. Sec-tion III specifies the equations of motion which are based onthe extended system method of Andersen.28 Dynamical equa-tions for ~2! and~3! can be obtained by simple extensions ofthe Parrinello–Rahman29 or Nose–Klein equations30 ~whichwere developed for anisotropicsolids!; new equations arederived for the last two. The extension to isothermal en-sembles is also discussed. Sections IV and V present theprotocol and results, respectively, for simulations of octane/water, octane/vacuum, and water/vacuum for selected en-sembles; even though some of the ensembles would not typi-cally be used to simulate an oil/water interface, they provideinteresting test cases. It is also demonstrated that approachesusing an isotropic pressure tensor, though correct for isotro-pic systems, can lead to difficulties when simulating systemswith interfaces. The final part of Sec. V compares the inter-facial density and polarization profiles of octane/water withthose of water/vacuum and octane/vacuum interfaces. This isfollowed by a summary in Sec. VI. The important applicationof these methods to a dipalmitoylphosphatidylcholine~DPPC! lipid bilayer and monolayer is presented in Paper II.

II. THERMODYNAMICS

A. Definitions

Consider a system of two immiscible liquids forming aplanar interface normal to thez direction with areaA. Fromthe condition of hydrostatic stability, the pressure normal tothe interfacePn is equal to the bulk pressureP. Then, fromthe first law of thermodynamics,31,32

dE5TdS2PndV1gdA1(i51

2

m idNi , ~2.1!

where E is the internal energy,T the temperature,S theentropy,V the volume,g the interfacial tension,Ni the num-ber of particles of liquidi , and mi its chemical potential.Here we consider only simulations carried out with constantparticle number~i.e., a closed system!, so

m1dN15m2dN250. ~2.2!

If the simulation is also carried out adiabatically~i.e., no heatexchange between the system and surroundings!,

TdS50 ~2.3!

and Eq.~2.1! becomes

dE52PndV1gdA. ~2.4!

To evaluate the surface tension, we begin with the pres-sure tensor of the system

P5S Pxx

00

0Pyy

0

00Pzz

D , ~2.5!

wherePzz is the normal pressurePn, and (Pxx1Pyy)/2 is thetangential pressurePt(z). For a planar interface,Pn is

FIG. 1. Simulation geometries for:~a! liquid/vacuum interface;~b! liquid/liquid with a single type of interface;~c! liquid/liquid interface with twodifferent liquid/vacuum interfaces;~d! liquid/liquid interface with a singletype of liquid/vacuum interface.

10253Zhang et al.: Computer simulation of liquid/liquid interfaces. I

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uniform; however,Pt depends strongly on position: far fromthe interface,Pt5Pn5P, but in the interfacial region it be-comes large and negative~see Fig. 2!. The surface tension isgiven by the well known expression32,33

g5E2`

1`

dz@Pn2Pt~z!#. ~2.6!

Note that only the region near the interface contributes to theintegral ~Fig. 2!.

Assume now that the fluid is contained in an orthorhom-bic box with sides of lengthshx , hy andhz ; hence,A5hxhyandV5Ahz . If we define theaveragedtangential pressureover the box

Pt51

2hzE

21/2hz

11/2hzdz@Pxx~z!1Pyy~z!# ~2.7!

then the surface tension can be rewritten as

g5hz~Pn2Pt!. ~2.8a!

Although it is possible to simulate an interfacial system withperiodic boundary conditions including only one interface, itis more straightforward to employ the geometries sketched inFig. 1 and enforce periodicity by simple translations of theunit cell. Hence, Eq.~2.8a! must be modified to reflect thepresence of the additional interfaces. We will only considerthe geometries shown in Figs. 1~a! and 1~b!, where there aretwo instances of one type of interface. In this case,

g51

2hz~Pn2Pt! ~two interfaces!. ~2.8b!

This point is discussed further in Paper II when consideringsimulations of monolayers and bilayers.

B. Ensembles

We now develop several useful ensembles for an isolatedsystem~i.e., no exchange of particles or heat!, beginning, forcompleteness, with the microcanonical. Each ensemble ischaracterized by a set of thermodynamic variables that de-scribe the system and thus remain constant over the course ofa computer simulation; each of these thermodynamic vari-ables is associated with a conjugate variable whose valuefluctuates and may be determined from a simulation by av-eraging over a trajectory of sufficient length. Issues related tothe extension to isothermal systems are discussed in Secs.II C and III B 5.

1. Constant volume, energy and surface area (NVAE)

If the volume and area of the box are fixed it is clear that

PndV5gdA50, ~2.9!

and we recover the simple result that

dE50, ~2.10!

i.e., the total energy of the system is a constant. Thus in asimulation carried out at constantNVAE, the average normalpressure and surface tension~two intensive quantities! couldbe evaluated because their conjugate extensive variables,volume and surface area, respectively, are fixed. Becausesimulations in the microcanonical ensemble are invariablycarried out at fixed volumeandshape, we maintain the stan-dard terminology and useNVE when referring toNVAE.This point is discussed further in subsection C.

2. Constant normal pressure and surface area(NPnAHpn )

Consider now the system sketched in Fig. 3~a!: ~i! fixedsurface area but variablehz ~so thatdV is nonzero! and ~ii !constantPn . Then, from condition~i! and Eq.~2.4!

dE1PndV50, ~2.11a!

and from condition~ii !

d~E1PnV!50. ~2.11b!

The enthalpy function of the system

Hpn5E1PnV ~2.12!

is therefore a constant.Hpn could alternatively be derived asthe Legendre transform34,35 from E5E(N,S,V,A) toHpn(N,S,Pn ,A).

3. Constant tangential pressure and height(NPth zHpt )

If hz and Pt are fixed@Fig. 3~b!#, then Eqs.~2.4! and~2.8a! can be combined to obtain

dE52PndV1PnhzdA2PthzdA52PthzdA52PtdV~2.13a!

or

d~E1PtV!50. ~2.13b!

This implies that the new enthalpy function

FIG. 2. A sketch of the density~top! and pressure~bottom! profiles in theregion of a planar interface between two liquids with bulk densitiesr1 andr2. Pn and Pt are the normal and tangential components of the pressuretensor, respectively;d is the approximate thickness of the interface.

10254 Zhang et al.: Computer simulation of liquid/liquid interfaces. I


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Hpt5E1PtV ~2.14!

is a constant.

4. Constant volume and surface tension (NVgHng)

For fixed volume and surface tension@Fig. 3~c!#, Eq.~2.4! becomes,

dE2gdA5d~E2gA!50 ~2.15!

and

Hvg5E2gA ~2.16!

is constant.

5. Constant normal pressure and surface tension(NPngHpg)

Lastly, if Pn andg are held constant@Fig. 3~d!#, we havesimply

dE1PndV2gdA5d~E1PnV2gA!50 ~2.17!

and constant

Hpg5E1PnV2gA. ~2.18!

As for Hpn , the conserved enthalpy functions of the preced-ing ensembles are derivable as a Legendre transforms.

C. Remarks

Though a bulk fluid can be described by only three ther-modynamic variables~e.g.,N,V,T!, the addition of a fourthvariable is required to specify the state of an interfacial sys-

tem due to its inherent anisotropy.31,32As an example, con-sider a one component fluid forming a liquid/vapor interface,and construct two systems with identicalN, V, andT whichdiffer only in their surface areas. From Eq.~2.1!, the totalenergies~and hence the thermodynamic states! of these sys-tems are different because of the termgDA. Alternatively, ifthe systems are prepared at identicalN, V andE, the poten-tial energies~and therefore the temperatures! will be differ-ent. We considered the pairs~m,N!, (T,S), (Pn ,V) and~g ,A!. The microcanonical ensemble corresponds to thespecification of all extensive variables. We obtained, by suit-able transforms, four additional ensembles for an isolatedsystem where one or both of the intensive variables~Pn andg! are specified in place of their corresponding extensivevariablesV andA.

There are limits, however, on the number of intensivevariables which are capable of independent variation,34 i.e.,which can be specified. From the Gibbs phase rule, thisequalsc2p12, wherec is the number of components andpis the number of phases. Thus, for a one component, onephase system we can set (N,P,T) but not~m,P,T!. For a onecomponent, two phase system~e.g. liquid/vapor!, only asingle intensive variable may be specified. This can be un-derstood physically by recalling that at a given temperaturethere is only one equilibrium vapor pressure above a liquid,i.e., Pn5 f (T); thus setting both pressure and temperature isredundant or inconsistent. For a liquid/liquid interface withtwo components, each in a single phase, two intensive vari-ables may be specified. The most natural choice is (T,Pn),although (Pn ,g! is permissible; (N,Pn ,g,T), however, is notallowed becauseg5f (Pn ,T). It is useful to consider whathappens when a system with surface tensiong0 is simulatedat (N,Pn ,g,H) and g is incorrectly specified~i.e., gÞg0!.Assumingg,g0, the surface will contract, which decreasesthe potential energy, increases the temperature and therebylowersg0. Consequently, if the appliedg is not too far off,the system will reach equilibrium during the simulation.Equilibrium will not be attained if the simulation is carriedout at (N,Pn ,g,T), but, as noted above, setting three inten-sive variables has already been ruled out. We do not expectthat simulation at (N,Pn ,g ,H) will be be the method ofchoice for liquid/liquid interfaces~though an example is pro-vided in Sec. IV!; as shown in Paper II, however, it is veryuseful for simulating surfactant monolayers and bilayerswhere g5f (Pn ,T,A). Importantly, a generalization of thephase rule for surface phases36,37permits the specification ofmore than two intensive variables for these systems, allow-ing simulations to be carried out at (N,Pn ,g,T).

The pairs of thermodynamic variables presented here arenot the only possible sets, though they are natural in that theyappear directly in the statement of the first law of thermody-namics, and the ensembles are simple extensions of the en-sembles most commonly used in molecular simulations.Other acceptable sets of variables should be obtainable bysuitable transformation. Some combinations that might ap-pear to be acceptable are not necessarily correct. For ex-ample, it seems reasonable to specify an anisotropic pressuretensor which implicitly takes into account the surface tensionof the system38 ~and its implementation into existing constant

FIG. 3. Schematic of a liquid/liquid system with a single [email protected]~b!# as simulated by the following ensembles:~a! constantNPnAH; ~b!constantNhzPtH; ~c! constantNVgH; ~d! constantNPngH.



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pressure tensor molecular dynamics algorithms is straightfor-ward!. We are not aware of a transformation of variablesfrom E(N,S,V,A) toH(N,S,Pn ,Pt), though it is possible tofind transformations which lead to intensive variables con-tainingPn andPt as products of several variables. Of course,it may be that in practice there is little difference between thespecification of~Pn ,g! and (Pn ,Pt). However, the validityof an ensemble specifyingPn and Pt remains to be estab-lished.

While it may seem possible in some cases to describe aninterfacial system with only three thermodynamic variables,the system is well specified only when an additional implicitconstraint is in place. For example, the isotropic pressure,~the trace of the pressure tensorPiso5

13(Pxx1Pyy1Pzz)! can

also be used to specify an interfacial system, and, in fact amolecular dynamics~MD! method has already been devel-oped which achieves a constant isotropic pressure by allow-ing uniform expansions and contractions of the systemvolume.28 In this case, the implicit constraint is the ratio ofsurface area to system volume; this is analogous to the earlierobservation that simulations of interfaces carried out at(N,V,E) are implicitly at (N,V,A,E). It should be recog-nized that sincePt,Pn for a system with nonzero surfacetension, settingPiso51 atm results inPn.1 atm; this is dem-onstrated in Sec. IV for a simulation of octane/water. An-other way of simulating an interfacial system with only threeapparent variables is to use an isotropic pressuretensor, i.e.,Pxx5Pyy5Pzz. In this case, the area and volume can adjustand/or fluctuate separately. This constraint, however, isequivalent to specifying the normal pressure and implicitlysettingg50. The simulation in Sec. IV of octane/water underthe conditionPn51 atm andg50 demonstrates the responseof an oil/water interface to an applied isotropic pressure ten-sor. A simulation of a lipid bilayer under these constraints isreported in Paper II.

III. EQUATIONS OF MOTION

This section specifies equations of motion consistentwith the ensembles just defined. For the microcanonical,these are simply Newton’s equations. For the others, the con-straint conditions are applied using theextended systemapproach.28 This method introduces additional degrees offreedom~often referred to as ‘‘pistons’’! that couple dynami-cally to the rest of the system, thereby imposing the con-straint~e.g.,Pn51 atm! on average; it is to be distinguishedfrom constraint dynamics,25 where the constraint is rigor-ously satisfied at each time step of the simulation.

Various extended system algorithms have been proposedfor performing constant pressure molecular dynamics com-puter simulations for ahomogeneousmolecular system. Twoof them have been widely used. The first, proposed byAndersen28 and later generalized to the constant pressure ten-sor by Parrinello and Rahman29 and Nose and Klein,30 de-scribes the piston with second order differential equations.The second is the weak coupling algorithm of Berendsen andco-workers39 in which the equations of motion for the pres-sure pistons are first order diffusivelike. We recently demon-strated that the weak coupling algorithm can induce undesir-

able artifacts when simulating liquid/liquid interfaces40 and,therefore, adopted Andersen’s pressure piston model.

A. Adiabatic ensembles

1. Constant NP nAHpn

For this ensemble, the box lengthshx andhy are fixed;hz fluctuates, in keeping with the condition that its conjugatevariablePn is constrained. Therefore, we introduce one newdynamical variable~or piston! hz with a massMz into thedynamical system. The Lagrangian is

L~r ,hz!5(i51

N F12 mixi21

1

2miyi

211

2mi S zi2 hz

hzzi D 2G

2U~r !11

2Mzhz

22Pn0hxhyhz , ~3.1!

whereN5N11N2 is the number of atoms,mi is the mass ofthe i th atom,U~r ! is the interparticle potential energy, andPn0 is the referencenormal pressure. The canonical mo-menta of thei th particle are

pxi5]L

] xi5mixi , pyi5

]L

] yi5miyi ,

pzi5]L

] zi5mi S zi2 hz

hzzi D , ~3.2a!

and the canonical momentum for the piston is

Pz5]L

]hz5Mzhz2(

i

N

miS zi2 hz

hzzi D zi

hz. ~3.2b!

Thez component of the particle canonical momentum is nowthe sum of the linear momentummizi and a contributionfrom the scaling of the simulation cell. The following equa-tions of motion are derived from the Lagrangian@Eq. ~3.1!#

xi5pximi

, yi5pyimi

, zi5pzimi

1hzhz

zi ,

pxi5 f xi , pyi5 f yi , pzi5 f zi2hzhz

pzi ,

hx5constant, ~3.3!

hy5constant,

Mzhz5hxhy~Pzz2Pn0!,

where the forcef xi52]U/]xi ~and similarly forf yi and f zi!,andPzz is the component of the pressure tensor defined asfollows

Pab51

V (i51

N S pa i pb i

mi1xa i f b i D ~3.4!

with xa i5xi ,yi ,zi for i51,2,3. The conserved Hamiltonianis



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H5(i

N

~ xi pxi1 yi pyi1 zi pzi!1hzPz2L

5(i

N S pxi22mi1

pyi2

2mi1

pzi2

2miD 1U~r !

11

2MzS Pz1

1

hz(i

N

zipziD 21Pn0hxhyhz . ~3.5!

Substituting Eq.~3.2! into Eq. ~3.5! yields

H5(i51

N F12 mixi21

1

2miyi

211

2mi S zi2 hz

hzzi D 2G1U~r !

11

2Mzhz

21Pn0hxhyhz . ~3.6!

This Hamiltonian has the form of an enthalpy~i.e., it con-tains aPV term!. In the limit of infinite piston mass,hz50~i.e., hz is constant! and Newton’s equations of motion arerecovered.

Formally, the equations of motion@Eq. ~3.3!# can be con-sidered a limiting case of the Nose–Klein equations30 of mo-tion for constant pressure tensor dynamics, where the massesfor the piston variableshx and hy are set to infinity. How-ever, their original application was to elastic solids, andhence, corresponded to a different ensemble.

2. Constant NP th zHpt

For this ensemble,hz is fixed, andhx andhy are the newdynamical variables. The Lagrangian is

L~r ,hz!5(i51

N H 12 miF xi2 hxhx

xi G21 1

2miF yi2 hy

hyyi G2

11

2mizi

2J 2U~r !11

2Mxhx

211

2Myhy

2

2Pt0hxhyhz , ~3.7!

wherePt0 is the reference tangential pressure. The equationsof motion are

xi5pximi

1hxhx

xi , yi5pyimi

1hyhy

yi , zi5pzimi

,

pxi5 f xi2hxhx

pxi , pyi5 f yi2hyhy

pyi , pzi5 f zi ,

Mxhx5hyhz~Pxx2Pt0!, ~3.8!

Myhy5hxhz~Pyy2Pt0!,

hz5constant,

where the components of the pressure tensorPxx andPyy aregiven in Eq.~3.4!, and the conserved Hamiltonian is

H5(i51

N H 12 miF xi2 hxhx

xi G21 1

2miF yi2 hy

hyyi G2

11

2mizi

2J 1U~r !11

2Mxhx

211

2Myhy

2

1Pt0hxhyhz . ~3.9!

Again, the equations of motion@Eq. ~3.8!# can be consideredformally as the limiting caseMz51` of the Nose–Kleinequations.

The special case of constant surface shape~i.e., hx/hy5constant! is described in Sec. III B.

3. Constant NV gHvg

Since the volume of the simulation box is constant inthis ensemble,

V5hxhyhz1hxhyhz1hxhyhz50, ~3.10a!

and, dividing byV,

hzhz

52F hxhx 1hyhy

G . ~3.10b!

Hence, not all three box lengths are independent dynamicalvariables. Let us choosehx andhy as independent variables.Using the constraint condition,@Eq. ~3.10b!#, the Lagrangianis

L~r ,hx ,hy!5(i51

N H 12 miF xi2 hxhx

xi G21 1

2miF yi2 hy

hyyi G2

11

2miF zi1S hxhx 1

hyhy

D zi G2J 2U~r !11

2Mxhx

2

11

2Myhy

21g0hxhy , ~3.11!

whereg0 is the reference surface tension. The equations ofmotion are

xi5pximi

1hxhx

xi , yi5pyimi

1hyhy

yi ,

zi5pzimi

2S hxhx 1hyhy

D zi ,pxi5 f xi2

hxhx


pyi ,

pzi5 f zi1S hxhx 1hyhy

D pzi ,~3.12!

Mxhx5hy~g02gxx!,

Myhy5hx~g02gyy!,

where

gxx5hz~Pzz2Pxx!,~3.13!

gyy5hz~Pzz2Pyy!.

The conserved Hamiltonian is



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H5(i51

N H 12 miF xi2 hxhx

xi G21 1

2miF yi2 hy

hyyi G2

11

2miF zi1S hxhx 1

hyhy

D zi G2J 1U~r !11

2Mxhx

2

11

2Myhy

22g0hxhy ~3.14!

4. Constant NP ngHpg

Since the box lengthshx , hy andhz all can vary, they areall dynamical variables. The Lagrangian is

L~r ,hx ,hy ,hz!5(i51

N H 12 miF xi2 hxhx

xi G2

11

2miF yi2 hy

hyyi G21 1

2miF zi2 hz

hzzi G2J

2U~r !11

2Mxhx

211

2Myhy

211

2Mzhz

2

2Pn0hxhyhz1g0hxhy , ~3.15!

wherePn0 andg0 are the reference normal pressure and sur-face tension, respectively. The equations of motion are

xi5pximi

1hxhx

xi , yi5pyimi

1hyhy

yi , zi5pzimi

1hzhz

zi ,

pxi5 f xi2hxhx


pyi ,

pzi5 f zi2hxhx

pzi ,

Mxhx5hy~g02gxx!, ~3.16!

Myhy5hx~g02gyy!,

Mzhz5hxhy~Pzz2Pn0!,

where

gxx5hz~Pn02Pxx!,~3.17!

gyy5hz~Pn02Pyy!.

Note thatgxx and gyy are different fromgxx andgyy in Eq.~3.13!. The conserved Hamiltonian is

H5(i51

N F12 mi S xi2 hxhx

xi D 21 1

2mi S yi2 hy

hyyi D 2

11

2mi S zi1 hz

hzzi D 2G1U~r !1

1

2Mxhx

211

2Myhy

2

11

2Mzhz

21Pn0hxhyhz2g0hxhy . ~3.18!

When g050, Eq. ~3.16! reduces to the Nose–Klein equa-tions.

B. Remarks

It is worthwhile pointing out some common features inthe equations of motion for the preceding ensembles.

1. Equations of motion

The general form of the dynamical equations for the at-oms is

xi5pximi

1F (a5x,y,z

sag~ha!Gxi~3.19a!

pxi5 f xi2F (a5x,y,z

sag~ha!Gpxi ,wheresa can be 0 or61. The coordinate and momentum arescaled simultaneously by the same scaling factors

g~ha!5ha

ha~3.19b!

but with different sign.The scaling factor is determined by the dynamical equa-

tions of the pistons which display the same structure for eachensemble

Maha;@Q~ t !2Q0#, ~3.20!

whereQ(t) is the instantaneous value of the physical quan-tity which is constrained to a constantQ0, such as normal ortangential pressure, or surface tension. The dynamics of thepiston is much like a damped anharmonic oscillator, with aneffective damping force arising from the internal energy dis-sipation and fluctuation~rather than from explicit frictionterms!. The restoring force;[Q(t)2Q0] drives the pistonto the target valueQ0 from some initial value ofQ(t). Thepiston then oscillates aroundQ0 with an amplitude indepen-dent ofMa . Dynamic quantities~e.g., the decay of pressurefluctuations! can show dependence on piston mass; artifactssuch as ‘‘piston ringing’’ can be eliminated by introducingexplicit friction into the algorithm~i.e., as in the Langevinpiston method!.40

2. Conserved Hamiltonians

The Hamiltonian in each dynamical system is alwaysconserved. For theNVE ensemble, the Hamiltonian is theinternal energy. For the other four ensembles, itapproxi-mately equals one of the various enthalpy functions de-scribed in the previous section. The difference is the kineticenergy for each piston~12kbT on average!. Since the numberof atoms is much larger than the number of pistons,

H.(i51

N1

2mi@pxi

2 1pyi2 1pzi

2 #1U~r !1W, ~3.21a!

where

W5PnV for NPnAHpt ,

W5PtV for NhzPtHpn ,~3.21b!

W52gA for NVgHVg ,

W5PnV2gA for NPngHpg



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are the work terms associated with the normal pressure, tan-gential pressure or surface tension. Monitoring the conserva-tion of the Hamiltonian is a useful technical check of a mo-lecular dynamics simulation, in theCHARMM ~Chemistry atHARvard Macromolecular Mechanics! implementation ofthese methods the full Hamiltonian is calculated at each timestep.

3. Special case: Preserved shape of interface

A special case in which the shape of the interface ispreserved is particularly useful. The appropriate equations ofmotion can be derived straightforwardly from Eqs.~3.8!,~3.12! and~3.16!. As an example, let us consider dynamics atconstantNPng . First define the constant

hyhx

5k. ~3.22!

Hence, there is only one independent variable amonghx andhy . Choosinghx as the dynamical variable and substitutingEq. ~3.22! into Eq. ~3.16!, the equations of motion become

xi5pximi

1hxhx

xi , yi5pyimi

1hxhx

yi , zi5pzimi

1hzhz

zi ,

pxi5 f xi2hxhx

pxi , pyi5 f yi2hxhx

pyi ,

pzi5 f zi2hxhx

pzi ,

~3.23!Mxhx5khx~g02g !,

Mzhz5khx2~Pzz2Pn0!,

where

g51

2~ gxx1gyy!5khzFPn02

1

2~Pxx1Pyy!G . ~3.24!

The conserved Hamiltonian is

H5(i51

N F12 mi S xi2 hxhx

xi D 21 1

2mi S yi2 hx

hxyi D 2

11

2mi S zi1 hz

hzzi D 2G1U~r !1

1

2~2Mx!hx

2

11

2Mzhz

21Pn0khx2hz2g0khx

2. ~3.25!

4. Correspondence between molecular dynamics andstatistical ensembles

The proofs that the equations of motion derived abovecorrespond to the appropriate statistical ensembles are simi-lar to the one given by Andersen in his original paper,28 andwill not be reproduced in detail. They begin with the obser-vation that since the modified Hamiltonians@Eqs. ~3.4!,~3.8!, ~3.12!, and~3.16!# are conserved, according to the er-godic hypothesis, the dynamical trajectories produce micro-canonical probability distributions of thecombined systems~molecules and pistons!. After the dynamical variables of thepistons are averaged away, the microcanonical distribution of

the combined system reduces to the probability distributionof themolecular systemin the correct statistical ensemble.

5. Isothermal-isobaric ensembles

For simplicity, we have restricted discussion to adiabaticensembles up to this point. While in many cases simulationat constant energy or constant enthalpy is satisfactory whenallowing a system to change volume or shape, it is oftendesirable to impose conditions of constant temperature si-multaneously~keeping in mind restrictions imposed by theGibbs phase rule!.

When the interfacial system is closed but nonadiabatic~i.e., heat exchange is allowed!, Eq. ~2.3! no longer holds.Following the procedures in Sec. II B, we can define the fivecorresponding isothermal-isobaric ensembles and their freeenergies for a liquid/liquid interface:

~1! constantNVT, F5E2TS,

~2! constantNPnAT, Fpn5E2TS1PnV,

~3! constantNPthzT, Fpt5E2TS1PtV,

~4! constantNVgT, Fvg5E2TS2gA,

~5! constantNPngT, Fpn5E2TS1PnV2gA.

Isothermal ensembles have been simulated with a varietyof algorithms including stochastic~collisional and Lange-vin!, extended and constrained.25 Constant temperature andpressure methods can be combined with41 or without28,42,43

coupling of the thermostat and barostat to allow simulationof isothermal-isobaric systems. Here we use the extendedsystem method described by Hoover,43 where a temperaturepiston is added to the system and is uncoupled from thepressure pistons. In contrast to the extended isobaric method,the isothermal extended system Hamiltonian does not corre-spond to the characteristic thermodynamic function of theensemble.

IV. SIMULATION METHODOLOGY

A. Model and data analysis

This section describes molecular dynamics simulationsof three interfacial systems at 293 K: 62 octanes/560 waters;560 waters/vacuum; and 62 octanes/vacuum. The octane wasmodeled by the fully flexible all-atom parameter setCHARMM PARM22b4b44 ~i.e., hydrogens are included, and bondstretching and angle bending are taking into account alongwith torsional, Lennard-Jones and electrostaticinteractions.! The water was described with modifiedTIP3Pparameters45,46 with a rigid geometry imposed bySHAKE.47

Electrostatic interactions were shifted to zero by 12 Å, andLennard-Jones~LJ! interactions were switched to zero overthe range 10–12 Å; LJ parameters between unlike atomswere calculated using standard combining rules~additive forradii and geometric for well depths!. A complete specifica-tion of the potential and parameters is provided in thesupplementary material for Paper II. Simulations were car-ried out using a version ofCHARMM48 which incorporated thealgorithms presented in the previous section withhx5hy~i.e., constant surface shape!. Orthorhombic periodic bound-



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ary conditions in three dimensions were imposed for allsimulations, and the mass of all pressure pistons was setequal to 500 amu. The time step was 0.001 ps. The nonbondlist included neighbors up to 14 Å, and was updated every 20time steps.

Coordinate sets and velocities were saved every 100time steps for later analysis~e.g., calculation of density pro-files!. Rapidly fluctuating quantities such as pressure and sur-face tension were accumulated during the run and saved as1.0 ps averages. Note that when the height of the boxhz isnot fixed during the simulation,

g51

2^hz~Pn2Pt!&

Þ1

2^hz&~^Pn&2^Pt&!, ~4.1!

where the angular brackets denote the average over trajec-tory.

Values reported here preceded by6 denote standard er-rors, i.e.,s/ANb, wheres is the standard deviation amongindependent blocks of averaged data, andNb the number ofindependent blocks. An analysis of the ‘‘statisticalinefficiency’’25 yielded the value 0.2 ps for the smallest in-dependent block length for surface tensions and pressures.Hence, the 1 ps block sizes used here and the 0.5 ps sizesadopted in Paper II for these quantities are sufficiently large.Block sizes for system dimensions were 5 ps, and the analy-sis was adjusted accordingly. The statistical significance ofdifferences in averages~e.g.,^V&, ^g&! obtained from differ-ent simulations was determined by the Student’st-test,49 us-ing programs fromNumerical recipes.50 The result of thistest is ap value. This quantity indicates the probability thatthe observed difference in means occurred by chance~i.e.,the distributions were the same, and a combination of naturalfluctuations and small sample size could explain the result!.Here we adoptp,0.01 as a cutoff for statistical significance;e.g., two means whose difference was associated with ap50.04 would be considered statistically equivalent. A dis-cussion of hypothesis testing aimed toward analysis of mo-lecular dynamics simulations is contained in the appendix ofRef. 51.

Most simulations were run either on a single HP 9000/735 workstation or a cluster of four HP 9000/735 worksta-tions; approximate timings for 1 ps of dynamics on a singleprocessor were: 1.2 h~water/octane!; 0.4 h~water/vacuum oroctane/vacuum!; simulations run approximately three timesfaster using four processors.52 Some simulations were run onan Intel Hypercube Gamma utilizing 32 iPSC/860 proces-sors; approximate timings for 1 ps with this configurationwere: 0.5 h~octane/water!; 0.18 h ~water/vacuum!; 0.14 h~octane/vacuum!.

For notational simplicity, subscripts on the different con-served enthalpy functions are dropped when referring tothem in the text or figures; e.g.,NVgHVg becomesNVgH.

B. Octane/water

1. Initial conditions and equilibration

An octane/water system was constructed by combiningthe coordinate frames from separate MD simulations of neatoctane and neat water previously equilibrated at 293 K uti-lizing a developmental parameter set,PARM22b2.53 Octanewas placed at the center of the box and the water above andbelow @cf. Fig. 1~b!#. The box dimensions were 25.6 Å inxandy directions and 51.1548 Å inz ~a volume correspondingto the volume of 560 water and 62 octane molecules at bulkdensity!. The system was then equilibrated by the followingprocedure:~i! 400 steps of adapted-based Newton–Raphsonenergy minimization48 ~to eliminate initial bad contacts be-tween the water and octane molecules!; ~ii ! 20 ps of dynam-ics with velocity reassignment and 30 ps with velocity res-caling at constantNVE; ~iii ! 200 ps of dynamics with novelocity adjustment at constantNVE; ~iv! upon changing toPARM22b4b, another phase of equilibration consisting of 10 psof velocity reassignment and 60 ps of velocity rescaling;~v!120 ps of dynamics at constantNPnAH, with Pn0 ~the ref-erence normal pressure! equal to 1 atm. This last simulation~at constantNPnAH!, denotedSimulation I, provided initialconditions for the production runs listed in the followingsubsection. The average box height^hz& over the interval of20–120 ps equaled 52.1460.11 Å, indicating a slight expan-sion of the system.

The initial condition for the first production simulation, aconstantNPnAH ensemble, was simply the last trajectoryframe of Simulation I~i.e., t5120 ps,hz551.09 Å!. Becausehz at this frame differs from its average, such a frame is notnecessarily suitable for starting simulations with fixed vol-ume or box height. To enable a satisfactory comparison ofensembles, simulations at constantNVE and NVgH werecarried out at a volume equal to the 20–120 average volumeof Simulation I. To ensure smooth trajectories, the frame att5117.5 ps of Simulation I, wherehz552.16 Å, was used forinitial conditions for the rest of the production runs.

2. Production runs

The seven simulations listed below were carried out for100 ps each. For those run at constant normal pressurePn051 atm, and for those at constant area,hx5hy525.6 Å.

~1! ConstantNPnAH.~2! ConstantNVE, with V05A352.1453.4173104 Å3.~3! ConstantNVgH, with V053.4173104 Å3, andg0551.7

dyn/cm ~the experimental value!.51

~4! ConstantNPngH, with g0551.7 dyn/cm.~5! ConstantNPngH, with g050 @from Eq.~2.8!, these con-

straint conditions are equivalent to settingPn5Pt51atm#.

~6! ConstantNPisoH, with Piso51 atm.~7! ConstantNPnAT, with T05293 K and a thermal piston

mass of 1500 kcal ps2.

Simulations of systems~5! and~6! were carried out to deter-mine the practical consequences of the alternative ap-proaches discussed in Sec. II C.



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C. Water/vacuum interface

A coordinate frame from the octane/water system at theend of equilibration procedure was adopted for the initialcondition for this and the octane/vacuum simulation de-scribed in Sec. IV D. Following removal of the octane, thewater molecules were placed at the center of the new simu-lation box with the same area~hx5hy525.6 Å! andhz575.6Å @as in Fig. 1~a!#; the volume of the empty space above andbelow the liquid is sufficiently large so that the two water-vacuum interfaces do not interact with each other eventhough periodic boundary conditions are employed. The sys-tem was equilibrated for 50 ps by reassigning the velocitiesfor 20 ps and then rescaling velocities for 30 ps. A constantNVE production run of 300 ps was then carried out. For aliquid/vacuum system carried out at constantNVE, the aver-age normal pressure is zero rather than 1 atm. However,because the instantaneous fluctuations inPn are on the orderof several hundred atm, the two are essentially identical.

D. Octane/vacuum interface

The setup of this simulation was analogous to the water/vacuum system just described, and the box dimensions wereidentical. The only difference was that after following theequilibration scheme described in the previous section, a 100ps simulation was carried out withPARM22b2. Upon changingto PARM22b4b, a second equilibration phase consisting of 20ps of velocity reassignment and 30 ps of velocity rescalingwas carried out.~This additional equilibration was not nec-essary for the water/vacuum system because the water modelwas unchanged in the new parameter set.! A 100 ps produc-tion run was then carried out.

E. Alternative equilibration and simulation of octane/water

To check the octane/water simulations described in Sec.IV B. and to explore an alternative method of equilibrationof a liquid/liquid interface, coordinate frames at 100 ps fromthe preceding water/vacuum and octane/vacuum simulationswere combined in a periodic box withhx5hy525.6 Å andhz552.2 Å. Fifty steps of steepest descent energy minimiza-tion were carried out to eliminate bad contacts, followed by50 ps of simulation at constantNVE ~20 ps with velocityreassignment and 30 ps with velocity rescaling!. The result-ing configuration was used as initial conditions for a 200 pssimulation at constantNPnAH.

V. SIMULATION RESULTS

To demonstrate the stability and numerical accuracy ofthe simulations, we first present results pertaining to boxsizes and pressures. We then proceed to surface tensions: forthe constant area simulations, the comparison of the calcu-lated surface tension with experiment provides an excellenttest of the potential energy parameters; for constant surfacetension simulations, the response of the system to differentapplied surface tensions is an important methodological issue~and will be considered further in Paper II!. Finally, the in-terfacial density and polarization profiles for the liquid/liquidand liquid/vacuum systems are described.

A. Box sizes

Box dimensions, pressure components and surface ten-sions for all of the octane/water simulations are listed inTable I. Figure 4 plots the 1 ps block averages of box dimen-sions from the first, third and fourth octane/water simulationslisted in Sec. IV B 2. As consistent with theNPnAH en-

TABLE I. Thermodynamic quantities for octane/water interface.

NPnAH NPnAT NVENVgH

~g0551.7!NPngH

~g0551.7!NPngH~g050! NPisoH

hx ~Å! 25.6 25.6 25.6 26.9860.08 26.8060.06 25.4860.02hz ~Å! 52.1260.11 52.1860.13 52.14 46.9560.27 47.5260.20 51.8960.03V ~103 Å3! 34.1760.03 34.2060.08 34.17 34.17 34.1260.03 33.8260.01 33.6860.09Pn ~atm! 0615 1616 26617 0614 1613 2614 169612Pt ~atm! 2230613 2232615 2197615 2218612 221468 267 28169Piso ~atm! 262g ~dyn/cm! 60.563.6 61.664.2 58.763.6 51.361.3 51.163.3 20.765.7 65.664.5

FIG. 4. Dimensions of the octane/water system versus time for the simula-tions at: constantNPnAH, with Pn051 atm ~left panel!; constantNVgH,with g0551.7 dyn/cm~center!; constantNPngH, with Pn051 atm andg0551.7 dyn/cm~right!. Results for this and Figs. 5 and 6 are averages over1 ps blocks.



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semble,hx is constant whilehz and V fluctuate over thecourse of the simulation, withhz&552.1260.11 Å; thisvalue is almost identical to the average from Simulation I,indicating a satisfactory equilibration. During the first 20 psof the constantNVgH andNPngH simulations there is anadjustment in the area and height of approximately 10% andin opposite directions. This compensation preserves the vol-ume exactly for the constantNVgH system~middle columnof Fig. 4!, and approximately for the constantNPngH sys-tem ~a statistically insignificant difference of less than 0.11%from the average value at constantNPnAH!.

The box dimensions for the constantNPngH simulationwith g050 are plotted in the left column of Fig. 5. The boxshape rapidly undergoes a dramatic shape change~note thedifference in scale from Fig. 4!, and appears to be unstable.This behavior is expected for a fluid interface under an ap-plied surface tension of zero: the system minimizes its sur-face area by becoming long and thin.

The simulation cell for the constantNPisoH simulationwith Piso51 atm is stable~Fig. 5, right column!. However, itsaverage volume was 1.5% smaller than that of the constantNPnAH system; this difference is statistically significant.

B. Pressures

The pressure components and surface tension for theoctane/water system at constantNPnAH are shown in Fig. 6~left column!. Even though the fluctuations are substantial,there is no noticeable drift. From Table I, the average normalpressures for this simulation and those at constantNVgH~g0551.7 dyn/cm! andNPngH ~Pn051 atm andg0551.7dyn/cm! are all statistically indistinguishable from 1 atm,indicating that the pressure piston mechanism is effective.^Pn&526617 atm from the simulation at constantNVE,

which is somewhat higher though still within statistical errorof the preceding three. A small increase in box height wouldbring this value closer to 1 atm.

The averaged isotropic pressure for theNPisoH simula-tion is 263 atm, which is close to the reference value;^Pn&,however, is approximately 169 atm, which differs signifi-cantly from the others. As already discussed in Sec. II C, tomeet conditionsPiso51 atm ~the constraint! andPn2Pt.0~nonzero surface tension!, Pn cannot equal 1 atm.

The normal pressures for the water/vacuum and octane/vacuum simulations were 166 and 064 atm, respectively.This is as expected: unless there is significant evaporation ofthe liquid to the vacuum region, the normal pressure shouldfluctuate about 0 atm.

C. Surface tensions

The average surface tensions from the simulations atconstantNVgH and the two at constantNPngH are all veryclose to the reference values, indicating healthy algorithms.

The surface tension at constantNPnAH is 60.563.6dyn/cm, which is reasonably close~although not statisticallyequal! to the experimental value 51.68 dyn/cm. The secondNPnAH simulation~Sec. IV E!, and the constantNPATandNVE systems also provide values in this range~Table I!. Thesurface tension obtained from the average of these four simu-lations is included in Table II, along with the water/vacuumand octane/vacuum results.

FIG. 5. Dimensions of the octane/water system versus time for the simula-tions at: constantNPngH, with Pn051 atm andg050 dyn/cm~left column!;constantNPisoH, with Piso51 atm ~right!.

FIG. 6. The normal, tangential, isotropic pressures and surface tensions asfunctions of time for the octane/water at: constantNPngH ~left column!;constantNPisoH ~right!.



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Interestingly, although Pn& was significantly greaterthan 1 atm in theNPisoH system,g565.664.5 dyn/cm,which is not significantly different from the preceding ones.While it is possible that a statistically significant differencecould be discerned from longer trajectories, this result im-plies that the surface tension of octane/water is not very sen-sitive to a 170 atm increase in the normal pressure.

By way of comparison with previous work, errors fordecane/water surface tensions from molecular dynamicssimulations at constantNVE and NPA vary from 6% to72%, depending on the L–J parameters.26 The surface ten-sion of benzene/water, determined by Monte Carlo~MC!simulation at 293 K, was in error by 57%.54

FromNVE simulations at 300 K using the united atomOPLS model,55 Harris20 obtained a decane/vapor surface ten-sion of 30.9 dyn/cm~33% error!. Given that the presentoctane/vacuum value is only 6% in error~but also withinstatistical error limits!, it is possible that an all-atom param-eter set is necessary to model the surface properties of al-kanes.

The surface tension of the water/vapor interface has beenevaluated using different water models, different treatmentsof the long ranged force, and by both the pressure [email protected]., as in Eq.~2.7!# and capillary wave56,57 methods. Asreviewed in Ref. 9, reported values at 300 K range from 30.5to 63.5 dyn/cm~after correcting for errors in some originalpublications!. Alejandre, Tildesley, and Chapela,22 using bothEwald sums and the long range tail correction on theLennard-Jones, obtained 71.5 dyn/cm at 316 K. The water/vacuum surface tension reported here is in good agreementwith experiment~within 3.5%! though Ewald sums were notemployed.

In addition to the use of different potential parametersets, it is probable that some of the spread in reported surfacetensions can be explained by the relatively short cutoffs~7–9Å! often employed. With this in mind, we examined the ef-fects of shorter cutoff~9 Å and 6 Å! of nonbond force fieldsusing parameter setPARM22b2. We found relatively small dif-ferences~9%! in going from 12 to 9 Å, but very significantones upon further reduction from 9 to 6 Å~25% for theoctane/water, 50% for the water/vacuum, and the octane‘‘boiled’’ at the lower cutoff!. Additionally, we observed thatthe liquid/vacuum systems tended to be more sensitive tocutoff effects than octane/water. Whether this is generallytrue remains to be determined.

D. Density profiles

Figure 7 shows the mass density profiles for water~dot-ted line!, octane~dashed line!, water and octane combined~solid line! for the octane/water simulation at constantNPnAH. The profiles were obtained by calculating the den-sity in 103 slabs parallel to thexy plane, with a thickness of0.5 Å, using the coordinates of all atoms. The octane/waterdensity profile from the constantNVE simulation is almostindistinguishable from the profile shown in Fig. 7, which isfurther evidence of the equivalence of results obtained withdifferent ensembles. From Fig. 7, one sees a relativelysmooth interface with a widthd of approximately 3 Å~de-fined as the distance over which the water density or octanedensity changes from 90% to 10%, sometimes denoted the10%–90% width!.9 The octane bulk density~the averageddensity in the region not containing the interface! is almostidentical to the experimental value, while the bulk density ofwater is 2.5% higher than experiment. This is consistent withconstantNVE simulations of neatTIP3P water at its bulkdensity, where the pressure is approximately2600 atm;when simulated at a constant pressure of 1 atm, the densityincreases several percent. There are significant fluctuations inboth octane and water densities. Such density oscillationswere also observed in MD simulations~at constantNVE! ofhexane/water,58 nonane/water,24 water/1,2-dichloroethane7

interfaces and a MC simulation of benzene/water.54 Whetheroil/water and other organic/water interfaces indeed have lay-ered structures or whether the oscillations are artifacts of thesmall simulation cell and/or short simulation length is stillunclear. As would be expected, the total density of octaneand water in the interfacial region is higher than the bulkdensity of octane; similar observations were made fromsimulations of decane/water26 and nonane/water24 interfaces.Anomalous results were obtained from simulations ofhexane/water,56 where the density in the interface was higherthanbothbulk phases; apparently several hexanes dissolvedin the water phase, indicating probable deficiencies in theparametrization.

The top two panels of Fig. 8 compare the density profiles

TABLE II. Surface tensions at 293 K. The value for octane/water is theaverage of simulations at constantNPnAH, NPnT andNVE ~a total of 500ps!.

Simulation~dyn/cm!

Experiment~Ref. 37! ~dyn/cm!

Water/octane 61.561.9 51.68Water/vacuum 70.261.7 72.75Octane/vacuum 20.461.8 21.69

FIG. 7. The density profiles of water~dotted line!, octane~dashed!, waterand octane combined~solid! from the octane/water simulation at constantNPnAH. The stars show the experimental bulk densities of water and oc-tane.



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of octane/water with octane/vacuum and water/vacuum. Thedensity of octane in the octane/vacuum system~d55 Å! de-creases more slowly than in octane/water~d53 Å!. The wa-ter profiles in the water/vacuum and octane/water interfacesare similar~d53 Å!.

Lastly, we estimate capillary wave broadening of the in-terface. A simple measure of these effects is the mean squareamplitude of capillary waves,~dj!2&, which is given by56,57

^~dj!2&5kbT

2pglnL

l~5.1!

whereg is the surface tension,L is the upper wavelengthlimit determined by the size of the surface, andl is the lowerwavelength limit which is of the order of a molecular diam-eter. The mean square amplitude of capillary waves can bedirectly computed from the simulation trajectories; alterna-tively, we can use the above formula to estimate its valuesinceg has already been calculated. Assuming a value of 3 Åfor l ~the approximate diameter of water! and 25.6 Å forL~the value ofhx!, we obtaindj'1.4 Å for the octane/waterand water/vacuum interfaces;dj'2.7 Å for octane/vacuum,where a diameter of 6.5 Å is assumed for the diameter ofoctane. These results qualitatively agree with the density pro-

files show in Fig. 8, although the effects of surface rough-ness, capillary waves and hydrophobic effects are difficult todistinguish for octane/water.

E. Water ordering

We now consider the orientation of water molecules nearthe interface, as characterized by the cosine of the anglebetween the direction of the water dipole moment and thenormal to the interface. The water layer is defined here to belocated in the regionz,0, and we follow the convention ofdefining the dipole vector as pointing from a negative topositive charge. The comparison of octane/water and water/vacuum is shown in the bottom two panels of Fig. 8. Figure9 shows the orientational distributions of water dipoles inthree representative slabs. The important features are as fol-lows:

~1! There are two distinct layers near the octane/waterinterface. In the outer layer~z.0 Å!, the water dipoles pointtoward the octane phase, while the next layer~25 Å,z,0Å!, the dipoles point toward the bulk water. The bulk region~i.e., where there is no net polarization! begins at approxi-matelyz,25 Å. The preceding features have also been ob-served from simulations of decane/water.26 Note that al-though^cosu& is larger in the outer layer~third panel of Fig.8!, when normalized for water density, the total polarizationis larger in the inner layer~bottom panel of Fig. 8!. The

FIG. 8. Density profiles of octane and water~top two panels!, water polar-ization ~third panel! and density weighted water polarization~bottom panel!from the simulations of octane/water~solid lines!, octane/vacuum~dashed!and water/vacuum~dashed!. To obtain better statistics, all data were aver-aged over both interfaces, and data for octane/water were also averaged overNPnAH, NPnAT andNVE simulations~for a total of 300 ps!. The centersof the interfaces have been shifted to the origin~for octane/water this isdefined as the point of equal densities, and for octane/vacuum and water/vacuum as where the densities equal half their bulk values!.

FIG. 9. Distribution functions for water polarization at three representativeslabs from the simulations of octane/water~solid lines! and water/vacuum~dashed!.



128.6.218.72 On: Thu, 03 Jul 2014 11:44:32

tywater polarization for water/vacuum is qualitatively similarto octane/water, although the fluctuations appear to be larger.

~2! The orientational distributions of the water dipole~shown in Fig. 9! are quite broad and almost identical foroctane/water~solid line! and water/vacuum~dashed line!.While the most probable orientation is parallel to the inter-face, there is an asymmetry toward the positive cosine in theouter layer~top panel of Fig. 9!, and toward the negativecosine in the next layer~middle panel!. The distributions areuniform in the region of bulk water as expected~bottompanel!. Matsumoto and Kataoka17 obtained similar distribu-tions in their water/vacuum simulations, while Wilson, Po-horille, and Pratt14 did not report polarization in the outerlayer.

~3! The interfacial thickness measured by the length ofperturbation of water polarization in octane/water and water/vacuum appears to be approximately 6 Å, or about twice aslarge as the thickness defined by density variation. It is prob-able that the polarization profiles are much more sensitive toColoumb effects~including the method of truncation! thanthe density profiles.

VI. SUMMARY

We have investigated in detail five adiabatic statisticalensembles for simulating liquid/liquid interfaces. Each en-semble is defined by appropriate combinations of thermody-namic intensive and extensive variables, and a conserved en-thalpy ~or internal energy for the microcanonical!. Althoughnot demonstrated explicitly, the equations of motion speci-fied in Sec. III generate the correct probability distributionsin the appropriate statistical ensemble. Such correspondencebetween the equations of motion in molecular dynamics andthe ensemble in statistical mechanics establishes the neces-sary foundation for molecular dynamics computer simula-tion.

Many different valid ensembles can describe one physi-cal system, and, in principle, all are equivalent when thenumber of particles is large. However, the equivalence ofsimulation results~typically obtained from less than 10,000particles over a short time period! is not apparenta priori.The consistent results of the octane/water simulations~TableI, Sec. V! demonstrate that the assumption of equivalence isreasonable.

Surface tensions for octane/water, octane/vacuum andwater/vacuum obtained with the potential energy parametersfrom the setCHARMM PARM22b4b are in good to excellentagreement with the experiment~Table II!. It was also dem-onstrated that the density and polarization of water mol-ecules in the octane/water and water/vacuum interfaces arevery similar; the density profile of octane in the octane/vacuum interface is broader than in octane/water, in keepingwith the lower surface tension of the alkane/vapor system.Density profiles and water ordering near the interface are ingeneral agreement with those found in previous studies.

The ramifications of two other approaches commonlyemployed for isotropic systems were explicitly demon-strated. In the first, the conditionPn5Pt51 atm was im-posed by setting the applied surface tension equal to zero inthe NPngH ensemble. This effectively applies a large tan-

gential pressure gradient to the system, and because theoctane/water interface is fluid, the simulation cell becameprogressively longer and thinner.~The behavior of a lipidbilayer, which is more elastic than octane/water, is consid-ered in the following paper.! In the second demonstration, asimulation of octane/water was carried out at a constant iso-tropic pressure of 1 atm. Although deformations of the boxwere minor, the average normal pressure equaled 169 atm. Ineach of the preceding cases, the intrinsic anisotropy of thepressure tensor in the interfacial region manifested itself, al-though in different ways.

In closing, we expect that the ensembles discussed herewill be particularly useful for computer simulations of com-plex liquid/liquid interfaces. As an illustration, the surfacetension of a surfactant system such as a lipid bilayer is astrong function of surface area; the calculated surface tensionis also very sensitive to details of the potential energy func-tion and other details of the simulation. Hence, it is impor-tant in simulation studies to evaluate the surface tension atdifferent surface areas. TheNPnAH ensemble is appropriatefor doing this. Suppose now that the calculated surface ten-sion at a given surface area per lipid equalsg0. As Paper IIshows in detail, the surface area can be compressed by simu-lating the system in theNPngH ensemble withg,g0, orexpanded by settingg.g0. Consequently, it may be useful toemploy several ensembles when studying complex systems.

ACKNOWLEDGMENTS

We thank Dr. Milan Hodoscek for implementing the par-allel CHARMM code used on the cluster of HP workstationsand the Intel Hypercube. We are also grateful to the Compu-tational Biosciences and Engineering Laboratory, DCRT,NIH for computing time on the Intel.

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computer simulation of liquid/liquid interfaces. i. theory and application to octane/water

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