Free energy of formation of clusters of sulphuric acid and water molecules determinedby guided disassembly

Jamie Y. Parkinsona, Gabriel V. Laub and Ian J. FordaaDepartment of Physics and Astronomy, University College London,

Gower Street, London WC1E 6BT, U.K.; b Department of Chemical Engineering,Imperial College London, South Kensington Campus, London SW7 2AZ, U.K.

We evaluate the grand potential of a cluster of two molecular species, equivalent to its free energyof formation from a binary vapour phase, using a nonequilibrium molecular dynamics techniquewhere guide particles, each tethered to a molecule by a harmonic force, move apart to disassemblea cluster into its components. The mechanical work performed in an ensemble of trajectories isanalysed using the Jarzynski equality to obtain a free energy of disassembly, a contribution to thecluster grand potential. We study clusters of sulphuric acid and water at 300 K, using a classicalinteraction scheme, and contrast two modes of guided disassembly. In one, the cluster is broken apartthrough simple pulling by the guide particles, but we find the trajectories tend to be mechanicallyirreversible. In the second approach, the guide motion and strength of tethering are modified in away that prises the cluster apart, a procedure that seems more reversible. We construct a surfacerepresenting the cluster grand potential, and identify a critical cluster for droplet nucleation undergiven vapour conditions. We compare the equilibrium populations of clusters with calculationsreported by Henschel et al. [J. Phys. Chem. A 118, 2599 (2014)] based on optimised quantumchemical structures.

I. INTRODUCTION

Discussions of the vapour phase often start with anideal gas approximation, but such a viewpoint entirelyignores the existence of molecular clusters that form anddissolve as a consequence of the weak interactions thatexist between the constituent particles. Were it notfor these ephemeral and often disordered structures, thevapour would not easily be able to transform itself intocondensed phases when prepared at lower temperaturesor higher densities. The nucleation of such phases inthese circumstances is determined by the kinetics of thegrowth and decay of molecular clusters, and considerableefforts over many years have been devoted to understand-ing these processes [1–4].

Direct numerical simulation of molecular clustering ina large computational cell, necessarily with the use ofempirical force fields, is increasingly being explored (e.g.[5–10]), though the expense is often very high and theapplication to complex species and realistic experimen-tal conditions is limited. A more thermodynamic pointof view is that even though nucleation is a nonequilib-rium process, the population kinetics can be framed interms of the equilibrium free energies of the molecularclusters that participate in the sequence of growth anddecay events, together with timescales for collisions be-tween clusters and monomers [11]. Various modellingapproaches have been used to compute cluster free en-ergies for species present in the atmosphere, rangingfrom highly detailed structural studies based on quan-tum chemistry [12, 13], to semiempirical descriptions em-ploying the continuum properties of the condensed phase[14–16]. Accurate modelling of molecular clusters is aparticularly difficult task, in view of their intrinsic insta-bility and typical lack of clear structural features such ascrystalline order. Small clusters can be liquid-like and

descriptions made on the basis of their resemblance tosolids might be questionable.

Thermodynamic techniques that make no assumptionof solid-like character exist, and they have often beenemployed in a Monte Carlo (MC) setting [17, 18], underconstraints introduced to define an equilibrium clusterstate [19]. Typically, MC methods involve comparisonsbetween ensembles of similar clusters, often differing insize by one molecule, for example. A sequence of suchcomparisons allows us to characterise the thermodynamicproperties of an arbitrary cluster, although constructingsuch a sequence can be laborious. Molecular dynamicsapproaches have also been developed [20–22] where fewer,or more natural limitations are placed upon the configu-rational freedom available to a cluster.

Recently, a nonequilibrium molecular dynamics (MD)method has been developed that examines the inverse ofa realistic process of cluster formation in order to identifyits thermodynamic stability [23]. The approach employsthe Jarzynski equality [24], according to which the me-chanical work performed on a system during a nonequi-librium process can be related to a change in equilibriumHelmholtz free energy. The method, denoted cluster dis-assembly, can be regarded as an MD version of thermo-dynamic integration [25] and it has some intuitively ap-pealing features [23]. External forces are used to pull acluster apart into its constituent molecules in a controlledor guided fashion. A variation, denoted cluster mitosis,has also been developed where a cluster is separated intotwo subclusters, again to characterise the change in freeenergy associated with the process [26]. The methodshave been successfully tested against calculations of freeenergies, obtained by other techniques, for model argonand water clusters.

In this paper we turn our attention to the disassemblyof binary clusters. We determine what is often referred to

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as the free energy of cluster formation, which controls theequilibrium cluster populations in a given mixed molecu-lar vapour. Technically, we compute the grand potentialof a cluster for a given temperature and chemical poten-tials of the species. We study clusters of sulphuric acidand water, since this mixture has received considerableattention in connection with the formation of aerosols inthe atmosphere [27–29].

In Section II we develop the theory of binary clusterdisassembly and describe how the free energy change as-sociated with such a process can be linked to equilibriumcluster populations. In Section III we discuss the MDsimulations and the Jarzynski analysis that allows us todetermine the free energy of disassembly. We explore twomodes of disassembly processing and show that a ‘pris-ing’ technique, where molecules are gently eased out ofthe cluster, has considerable advantages compared witha more straightforward separation by steady pulling thatcan give rise to mechanical ‘snapping’ or tearing of thecluster. In Section IV we compare our equilibrium clus-ter populations with calculations made by Henschel etal. [13] on the basis of minimum energy structures ob-tained from quantum chemistry. In Section V we giveour conclusions.

II. STATISTICAL MECHANICS OF TETHEREDBINARY CLUSTERS

The external forces used in the method of cluster dis-assembly take the form of harmonic tethers that attacheach constituent molecule of the cluster to a dedicated‘guide’ particle, initially placed at the origin. In themolecular dynamics, the guides separate in a prescribedmanner and carry their tethered molecules along withthem. The mechanical work exerted by the guides on thecluster may be related to a free energy change using theJarzynski equality. We therefore start the analysis of thedisassembly by considering the free energy of a binarycluster in the presence of a weak set of harmonic teth-ers. We relate this to the free energy of an untethered,or free cluster, and then to the equilibrium population ofthe cluster in a given vapour mixture.

A. Free energies of free and tethered clusters

We shall represent each molecule using a single positionand momentum and refer to it as a particle. The partitionfunction of a free cluster of N particles of species 1 andM particles of species 2, and hence its Helmholtz free

energy FF , are given byZF = exp[−FF /kBT ]

=1

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx1jdp1jdx2kdp2k

× exp[−H(x1j ,x2k,p1j ,p2k)/kBT ]. (1)The dependence of the Hamiltonian H on the momentawill not be noted explicitly in the following, for economyof notation. We introduce coordinates referring to thecluster centre of mass through the insertion of unity inthe form of

1=

∫δ

xc − 1

Nm1 +Mm2

N∑j=1

m1x1j +

M∑k=1

m2x2k

dxc,(2)

where m1 and m2 are the particle masses. We hence ex-tend the integration through the introduction of a clus-ter centre of mass variable xc, but insert a delta functionconstraint that categorises the molecular configurationsby their centre of mass position. This is followed by achange of variables to positions of particles with respectto xc, namely x′1j = x1j − xc and x′2k = x2k − xc. Thepartition function for the free cluster becomes

ZF =1

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx1jdp1jdx2kdp2kdxc

× exp[−H(x1j ,x2k)/kBT ]

×δ

xc −1

Nm1 +Mm2

N∑j=1

m1x1j +

M∑k=1

m2x2k

=

V

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx′1jdp1jdx′2kdp2k

× exp[−H(x′1j ,x′2k)/kBT ]

×δ

− 1

Nm1 +Mm2

N∑j=1

m1x′1j +

M∑k=1

m2x′2k

, (3)

which we can write as ZF = V ZcF , where V is the sys-tem volume and ZcF = exp(−F cF /kBT ) is the partitionfunction for a cluster with its centre of mass fixed at theorigin and F cF is its free energy.

For a tethered cluster, the Hamiltonian will includean additional set of harmonic potentials, each designedto hold its tethered particle in oscillation at a frequencyω, if isolated, irrespective of mass. We have a partitionfunction and free energy given by

ZT = exp[−FT /kBT ]

=1

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx1jdp1jdx2kdp2k

× exp

[−

(H(x1j ,x2k) +

N∑j=1

1

2m1ω

2x21j

+

M∑k=1

1

2m2ω

2x22k

)/kBT

], (4)

and we then perform a transformation to centre of massvariables. We first write

3

N∑j=1

1

2m1ω

2x21j+

M∑k=1

1

2m2ω

2x22k=

N∑j=1

1

2m1ω

2x′21j+

M∑k=1

1

2m2ω

2x′22k+1

2(Nm1+Mm2)ω2x2c+ω

2

N∑j=1

m1x′1j +

M∑k=1

m2x′2k

·xc,(5)

and noting that the final term can be ignored by virtue of the delta function constraint, we obtain

ZT =1

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx′1jdp1jdx′2kdp2kdxc exp

[−(Hx′1j ,x′2k) + ∆H

)/kBT

]

×δ

− 1

Nm1 +Mm2

N∑j=1

m1x′1j +

M∑k=1

m2x′2k

exp

(−1

2(Nm1 +Mm2)ω2x2c/kBT

)

=1

N !M !h3(N+M)

∫ N∏j=1

M∏k=1

dx′1jdp1jdx′2kdp2k exp

[−H(x′1j ,x′2k)/kBT

]

×[

2πkBT

(Nm1 +Mm2)ω2

]3/2exp(−∆H/kBT ) δ

− 1

Nm1 +Mm2

N∑j=1

m1x′1j +

M∑k=1

m2x′2k

=

[2πkBT

(Nm1 +Mm2)ω2

]3/2ZcT , (6)

where ∆H =∑Nj=1

12m1ω

2x′21j +∑Mk=1

12m2ω

2x′22k andZcT = exp(−F cT /kBT ) is the partition function of a clus-ter in the presence of tethering potentials and with itscentre of mass constrained to lie at the origin.

We shall treat the tethering terms in Eq. (6) as a per-turbation. We write F cT ≈ F cF + 〈∆H〉0 where F cF is thefree energy of the free cluster with its centre of mass con-strained to lie at the origin, and the suffix 0 indicatesthat the expectation value is to be taken in an ensembleof such clusters. We introduce single particle radial den-sity profiles ρ1NM (x′1) and ρ2NM (x′2) for each species in an(N,M) cluster according to such an ensemble and write

〈∆H〉0 =N

2

∫ρ1NM (x′1)m1ω

2x′21 dx′1

+M

2

∫ρ2NM (x′2)m2ω

2x′22 dx′2, (7)

so that the difference in free energy between the free andtethered cluster is:

FF − FT = ∆FT = −kBT ln[ρNMc (0)V ] (8)

−N2

∫ρ1NM (x′1)m1ω

2x′21 dx′1−

M

2

∫ρ2NM (x′2)m2ω

2x′22 dx′2,

where we have introduced ρNMc (0) = [(Nm1 +Mm2)ω2/(2πkBT )]3/2, which can be regarded as an in-verse volume associated with the motion of the centre ofmass of the tethered cluster about the origin. The firstterm on the right hand side of Eq. (8) is a correction tothe entropy of the cluster brought about by the tethering,while the remaining terms are corrections to the energy.

B. Grand potential and equilibrium clusterdensities

We now introduce the grand potential of a free (N,M)cluster, defined by

ΩNM (T, µ1, µ2) = FF (N,M)−Nµ1 −Mµ2, (9)

where µ1 and µ2 are the chemical potentials of the par-ticle bath for the two species. The equilibrium densi-ties of clusters in a binary vapour phase can be shown[11] to be given by nNM = V −1 exp(−ΩNM/kBT ). Wecould express nNM in terms of the density of a monomerof either species 1 or 2, so that, for example, nNM =n10 exp(−(ΩNM − Ω10)/kBT ) in terms of the grand po-tential difference ΩNM − Ω10 = FF (N,M) − FF (1, 0) −(N − 1)µ1 − Mµ2, but it is more straightforward toproceed without introducing the grand potential of amonomer.

Representing the particle bath as a mixture of idealgases with readily calculable chemical potentials, thecluster grand potential is

ΩNM = FF (N,M)−NkBT ln(Λ1n1)−MkBT ln(Λ2n2),(10)

where we introduce bath monomer densities n1 = n10 andn2 = n01, and where Λs = [h2/(2πmskBT )]3/2 for s =1, 2. Next, we consider the free energy difference ∆FD =Ff − FT between the disassembled, but still tethered,constituent particles and the tethered cluster. Ff is thecombined free energy of N harmonic oscillators of species1 and M harmonic oscillators of species 2, which can bewritten as:

Ff = −3kBT

[N ln

(kBT

~ωf1

)+M ln

(kBT

~ωf2

)], (11)

4

where ωfs = (κf/ms)1/2 is the oscillator frequency of

species s, written in terms of a final value of the tetheringstrength κf and the particle mass. In the MD procedure,the free energy of disassembly that we extract is actually∆FMD = Ff − F dist

T since particles are distinguishablein MD. The partition functions for distinguishable andindistinguishable particles are related through Zdist

T =N !M !ZT and thus

∆FMD = ∆FD + kBT lnN ! + kBT lnM !, (12)

so we can write the dimensionless grand potential of afree (N,M) cluster as

ΩNM/kBT = −3

[N ln

(kBT

~ωf1

)+M ln

(kBT

~ωf2

)]−∆FMD/kBT + ln(N !M !) + ∆FT /kBT

−N ln(Λ1n1)−M ln(Λ2n2). (13)

This gives

ΩNMkBT

= −N ln(n1vHO)−M ln(n2vHO)

−∆FMD/kBT + ln(N !M !)− ln[ρNMc (0)V

]− N

2kBT

∫ρ1NM (x′1)κi1x

′21 dx

′1

− M

2kBT

∫ρ2NM (x′2)κi2x

′22 dx

′2, (14)

where vHO = (2πkBT/κf )3/2 with κf = m1ω2f1 = m2ω

2f2

is a volume representing the freedom of motion of eachparticle about its guide after disassembly. The finaltether strengths are taken to be the same for both species,and the mass dependent initial tether strengths κi1 andκi2 have been inserted into the last two terms.

When M = 0, the expressions reduce to those pre-viously derived for the single species case [23]. Wefind that the grand potential, correctly, has no depen-dence on ~ and the equilibrium cluster density nNM =exp(−(ΩNM + kBT lnV )/kBT ) does not depend on thesystem volume V . Note that the equilibrium cluster den-sities can be expressed in terms of the chemical potentialof a saturated vapour mixture if so desired, which is thetraditional way to proceed in nucleation theory, but herewe avoid such a representation and consider their depen-dence on the monomer densities n1 and n2 rather thanthe saturated densities.

The analysis of binary clusters could easily be extendedto clusters of an arbitrary number of species, if such sys-tems are of interest. In the next section we turn ourattention to determining the free energy of disassembly∆FMD using MD simulation and the Jarzynski equality.

III. DETERMINING THE FREE ENERGY OFDISASSEMBLY OF BINARY CLUSTERS

A. Simulation details and data analysis using theJarzynski equality

We employ a method for cluster disassembly basedupon that developed in earlier work [23]. We study clus-ters of sulphuric acid and water, ranging in size fromdimers up to a cluster of twelve molecules. Each moleculeis harmonically tethered, initially weakly, to one of a setof guide particles located at the origin. The tetheringforce is applied to the heaviest atom in each molecule,namely the sulphur in sulphuric acid and the oxygen inwater. A preparatory MD run of 10.5 ns duration is car-ried out under NV T conditions in which the system isallowed to equilibrate at 300 K in the presence of teth-ering and intermolecular interactions. For the sulphuricacid we used a set of classical interaction potentials devel-oped by Loukonen et al. [30] based on a series of quantumchemistry calculations, and the water was described bythe SPC/E-F extended simple point charge model [31].

An ensemble of 1000 equilibrium configurations wasselected at intervals of 0.01 ns from the equilibrated tra-jectory, rejecting a very few where a water molecule hadtemporarily become detached from the cluster. A furtherset of simulations was then carried out, in which the equi-librated clusters were disassembled through programmedmotion of the guide particles along with variation in thestrength of the harmonic tethers. Such a trajectory overa time interval τ provides a work of disassembly given by

W =1

2

τ∫0

dκ1(t)

dt

N∑j=1

[x1j(t)−X1j(t)]2dt

−τ∫

0

κ1(t)

N∑j=1

[x1j(t)−X1j(t)] ·V1j(t) dt

+1

2

τ∫0

dκ2(t)

dt

M∑k=1

[x2k(t)−X2k(t)]2dt

−τ∫

0

κ2(t)

M∑k=1

[x2k(t)−X2k(t)] ·V2k(t) dt, (15)

where κs(t) is the time dependent strength of the tetherattached to species s, xsm is the position of the heavyatom in the mth molecule of species s, and Xsm andVsm are the position and velocity of the associated guideparticle.

We employ the Jarzynski equality to relate the workto the shift in free energy brought about by the changein Hamiltonian associated with evolution from the ini-tial to the final state of the system. This relationshipis ∆FMD = −kBT ln〈exp(−W/kBT )〉 where the angledbrackets represent an average over the ensemble of dis-assembly trajectories [24]. In principle, the free energy

5

Figure 1. Illustrations of intermediate stages in the disas-sembly of an NW = 4, NA = 4 cluster of sulphuric acidand water. In the image on the left, representing the sim-ple pulling scheme, the tethers between the guide particles(large spheres) and the molecules are significantly stretched,while on the right, taken from a prising simulation, moleculesare eased out of the cluster by guides situated at close rangewith increasing tether strength. Movies of these disassemblytrajectories are provided in the supplemental material.

change extracted in this way should not depend on theprotocol of disassembly, but in practice there can be aremnant dependence arising from limited statistical cov-erage of the range of trajectories, and we shall considerexamples of such dependence in Section III B.

Simulations were performed using a version of theDL_POLY molecular dynamics package [32], modifiedto implement the time dependent harmonic tethering po-tentials. The molecules, but not the guides, were coupledto a Langevin thermostat with a friction coefficient of0.1 ps-1. We carried out the disassembly procedure ona set of 46 clusters at 300 K, with the number of wa-ter molecules, NW , and the number of sulphuric acidmolecules, NA, ranging from 0 to 6, excluding the casesof a monomer of either species (these labels correspondto N and M , respectively, in the previous expressions).During the simulation, one of the guide particles remainsat the origin, while all others are distributed uniformlyon a sphere [33] of time dependent radius appropriate tothe disassembly protocol. The timestep employed in allcases was 1 fs.

The initial oscillator frequencies for each species arerequired to be the same, implying mass dependent ini-tial tether strengths satisfying κiW /κiA = mW /mA. Weemploy a reference initial tether strength κi of 0.09 kJmol-1Å-2. This is small enough that the elevation inenergy for a molecule at the edge of the cluster is lessthan kBT [23], and slight variations have been shownelsewhere [26] to have little effect on the extracted clus-ter free energy. We define κiW = 2κimW /MWA andκiA = 2κimA/MWA, where MWA = mW + mA. Ex-amples of cluster configurations during disassembly, to-gether with their guide particles, are shown in Figure 1.

B. Optimising the disassembly protocol

We considered two modes of cluster disassembly, andimplemented each for a range of separation times to ex-amine the convergence of the free energy of disassembly.We first carried out a simple pulling protocol in which theguide particles separate at a constant speed for the dura-tion of the simulation. The tethers start to tighten after20% of the total separation time and reach their maxi-mum strength at 80%, in a fashion that was successfullyemployed in the disassembly of argon clusters [23]. Theradius of the sphere on which all but one of the guideparticles are finally distributed is 25 Å, and the ultimatevalue of the tether strength is 0.60 kJ mol-1Å-2. The fi-nal guide particle separation was such that the tetheredmolecules did not interact significantly with one anotherand the mean work performed during the separation hadsaturated.

However it became apparent that simple pulling typi-cally disassembled a cluster through a sequence of abrupt‘snapping’ events. Molecules showed a reluctance to sep-arate from the cluster, indicated by an increase in thetether length as its guide particle moved away. This isillustrated in the left hand image in Figure 1. Whenthe force on the molecule was strong enough to removeit from the cluster, the guide particle had moved so faraway that the extracted molecule, after a short relax-ation period characteristic of the thermostat, had littlefurther interaction with the cluster. Mechanically, theseextractions were typically irreversible, which also implieda thermodynamic irreversibility, in the sense that a sig-nificant fraction of the exerted work was converted intoheat and passed to the heat bath. The contrast with ar-gon cluster separation in earlier work [23] was broughtabout by the stronger intermolecular interactions in thepresent system.

Consequently, a second protocol that we call ‘prising’was developed to overcome these problems. It consists ofthe following stages, where τ is the separation time:

• t ≤ 0.1 τ : Motion of the guide particles to a dis-tance of 7.5 Å from the origin, at constant initialtether strength.

• 0.1 τ < t ≤ 0.6 τ : Guide particles are held station-ary, while tethers tighten to κf = 3.80 kJ mol-1Å-2.

• 0.6 τ < t: Separation of the guide particles to finalpositions on a sphere of radius 10.5 Å, at constanttether strength.

The motivation for this procedure is that if the guide re-mains in position as the tethers tighten, a molecule canbe separated from the cluster but remains close enough toallow interaction and also re-attachment. The image onthe right hand side of Figure 1 illustrates a typical con-figuration from the middle stage in the prising sequence.Once the tethers are fully tightened and the moleculesprised or eased out of the cluster, having explored a vari-ety of ways of doing so, the guides resume their outward

6

0.0

5.0

10.0

15.0

0.0

5.0

10.0

15.0

Prising

Simple

0.0 1.0 2.0 3.0 4.0 5.0t / ns

Tet

her

Len

gth

/ Å

Sulphuric Acid

Water

Figure 2. The evolution of tether lengths, defined as the dis-tance between the tethered atom and its associated guide, forthe two disassembly schemes. The cluster consists of NW = 4,NA = 2 molecules and is separated over a time of 5 ns. Notethe sharp and irreversible ‘snapping’ of particles out of thecluster towards their guides in the simple pulling scheme, forexample at t ≈ 3.6 ns. The data have been smoothed using aGaussian filter of width 0.15 ns.

motion, although we found that little further work wasperformed, on average, in the third stage: the moleculeshad by then been sufficiently separated (in contrast tothe 25 Å employed in the simple separation scheme withmuch weaker tethering). The firm but more reversibleremoval of the molecules from the cluster would be ex-pected to lead to lower variance in the work of disassem-bly.

The difference between the simple pulling and the pris-ing protocols is illustrated in Figure 2 for the disassem-bly of a cluster of four waters and two sulphuric acidmolecules over a period of 5 ns. The lengths of a set oftethers, defined as the distance between the guide and theatom to which it is attached, smoothed using a Gaussianfilter of width 0.15 ns to remove some of the noise, areshown evolving in time for both protocols. The snappingbehaviour in the simple pulling protocol is evident in theform of an increasing extension of the tethers followed bya rapid decrease. For the prising case, the there is morehopping of molecules between the guide and the cluster,and the general shortening of the tether length with timeis a consequence of the progressive tether tightening. Amovie that shows the irreversible snapping of a NW = 4,NA = 4 cluster during the simple pulling mode of disas-sembly is available in the supplemental material, togetherwith a movie of the prising scheme.

Work distributions and the extracted free energies ofdisassembly are shown in Figures 3 and 4, respectively,for the NW = 4, NA = 2 cluster and covering a range ofseparation times for both disassembly protocols. There is

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

Prisin

gSimple

50 70 90 110 130

W kBT

P(W)

Separation Time / ns0.5 1.0 2.5 5.0 7.5 10.0 15.0 20.0 30.0

Figure 3. Kernel density estimates [34] for the distribution ofworkW (approximate probability density functions and hencedenoted P (W )) for the disassembly of a NW = 4, NA = 2cluster, for a range of separation times and for both disas-sembly schemes. Note the faster convergence and the lowervariance for the prising disassembly protocol.

a clear reduction of the variance in work and of the sim-ulation time required for convergence of the free energyof disassembly ∆FMD when using the prising protocol.Note that the final tether strength in the prising proto-col is higher than for simple pulling, so the convergedfree energy changes are not expected to be the same forthe two approaches. The most important feature of Fig-ure 4 is that converged values of ∆FMD are obtainedfor shorter simulations using the prising technique. Wepresent grand potentials of cluster formation in the nextsection based on a prising separation time of 15 ns, al-though shorter times could be used without significantloss of accuracy.

IV. COMPARISON WITH HARMONICQUANTUM CHEMICAL APPROACH

We now combine the theoretical development given inSection II (specifically Eq. (14)) with the numerical eval-uations of the free energy of disassembly discussed inSection III. We also require single particle radial densityprofiles ρsNM of a free cluster about its centre of mass, inorder to evaluate the integrals

∫ρsNM (x′s)x

′2s dx

′s in Eq.

(14). As an approximation, we computed these numer-ically using the initial configurations of clusters in thepresence of weak tethers prior to disassembly. In Fig-

7

60

70

80

90

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Separation time / ns

∆FM

DkBT

Disassembly Protocol

Prising

Simple

Figure 4. Estimates of the free energy of disassembly ∆FMD

for a NW = 4, NA = 2 cluster, for the two disassemblyschemes and for a range of separation times. Error bars arethe standard error in the mean based on 1000 independenttrajectories. The prising scheme offers better convergence andreduced errors. Note that we should not expect the values of∆FMD to be the same for each protocol, due to the differencein final tether strength.

ure 5 we present the grand potential of clusters contain-ing up to six molecules of each species. We employ sul-phuric acid and water monomer densities of 2.804×10−9

and 8.531 × 10−8 Å-3 respectively, (2.804 × 1015 and8.531× 1016 cm−3 in more commonly used units) whichwere chosen to make the deviations of the surface fromplanarity most apparent. The surface has a saddle pointand hence a critical cluster at NA ≈ 4 and NW ≈ 3 forthis case.

It should be emphasised that these monomer densi-ties do not correspond to conditions for observed particlenucleation [28, 35]. The acid monomer density is sev-eral orders of magnitude higher than the typical range of106−108 cm−3 for sulphuric acid in the atmosphere at 300K [35]. It is likely that atmospheric particle nucleationproceeds with the participation of additional molecularspecies, so we do not expect to find that our model re-produces such events. Furthermore, it is quite possiblethat the microscopic interactions used in this study are inneed of improvement: an extended model that permitsproton transfer has recently been developed and couldbe used to rectify some of its deficiencies [36]. A similarsituation was encountered in the earlier demonstrationof the cluster disassembly procedure for a single species,argon [23]. The extracted cluster thermodynamic prop-erties were consistent with other studies that used thesame Lennard-Jones interactions, but the implied cor-respondence with experimental nucleation rates for thatsubstance was known to be poor [3, 37]. Similarly, weplace emphasis here on the successful implementation ofthe disassembly procedure and the determination of clus-

Figure 5. A surface representing the grand potential forall clusters considered, based on sulphuric acid and watermonomer densities of 2.80 × 10−9 and 8.53 × 10−8 Å-3, re-spectively, and with T = 300 K. The constrained equilibriumdensity of (NW , NA) clusters is given by exp(−Ω/kBT ) inunits of Å-3. The mesh is interpolated between integer valuesof molecular numbers using a 3rd-order spline interpolation.Approximate lines of steepest descent and ascent are overlaidand the critical cluster lies at the saddle point where theycross. An interactive version of this plot is available as aCDF file in the supplemental material.

ter thermodynamic properties starting from a molecularinteraction scheme rather than a presentation of realisticcorrelation with data. Indeed, such is the sensitivity ofcluster populations to details of the interactions, we tendto regard a comparison with data to be informative ofthe molecular interactions rather than predictive of theexperimental outcomes.

The extracted thermodynamic information can be usedas the basis of a kinetic theory of binary nucleation[15, 38, 39] but our main objective is to compare theresults with a recent thermodynamic analysis based onoptimised cluster configurations and harmonic thermalfluctuations obtained from quantum chemistry [13]. Weexpect differences in outcomes since the force field weuse is a classical representation of quantum mechanicalinteractions and might lack detail such as a descriptionof molecular dissociation (though this can be remedied[36]), but equally, the harmonic approach might not cap-ture the correct entropic contributions to the cluster freeenergy since it is fundamentally based on a solid-like con-ception of each structure. We compare the approaches bycomputing the equilibrium populations of clusters. Tech-nically, these would be clusters in a constrained equilib-rium where detailed balance is artificially maintained be-tween the growth and decay of clusters of all sizes andcompositions. Under the correct kinetics, cluster popu-lations in a stationary state of steady nucleation may berelated to these equilibrium populations.

We evaluate populations normalised by the popula-tions of clusters with the same number of acid moleculesbut no waters, namely nNM/n0M , in order to make a di-

8

rect comparison with numerical data presented by Hen-schel et al. [13]. Since nNM = V −1 exp(−ΩNM/kBT ) wehave nNM/n0M = exp[−(ΩNM − Ω0M )/kBT ], which forM 6= 1 involves

ΩNM − Ω0M

kBT= −N ln(n1vHO)− ∆FMD(N,M)

kBT+ lnN !

− ln

[ρNMc (0)

ρ0Mc (0)

]− N

2kBT

∫ρ1NM (x′1)κi1x

′21 dx

′1

− M

2kBT

∫ρ2NM (x′2)κi2x

′22 dx

′2 +

∆FMD(0,M)

kBT

+M

2kBT

∫ρ20M (x′2)κi2x

′22 dx

′2, (16)

while for M = 1 we can write nN1/n01 =exp[−ΩN1/kBT − ln(n01V )] and use

ΩN1

kBT+ ln(n01V ) = −N ln(n01vHO)− ln(ρN1

c (0)vHO)

−∆FMD(N, 1)

kBT+ lnN !− N

2kBT

∫ρ1N1(x′1)κi1x

′21 dx

′1

− 1

2kBT

∫ρ2N1(x′2)κi2x

′22 dx

′2, (17)

noting that the acid monomer density does not appearin these expressions.

We use the normalised populations nNM/n0Mto construct relative populations x(NW , NA) =

nNWNA/∑5NW=0 nNWNA

to compare with those reportedby Henschel et al. [13] and we present these in Figure6 for two values of the water monomer density. As ex-pected, there are differences in detail, but the comparisonwith the quantum chemical calculations is quite reason-able, except for the acid tetramer where our approachcannot account for the striking dominance of the trihy-drated cluster in the quantum chemical case. It remainsto be seen whether the differences can be reduced by us-ing a better classical representation of the interactions, orby improving the estimation of cluster entropy, or both.

V. CONCLUSIONS

The key to understanding first order phase transitionsin complex gaseous mixtures of precursor molecules is tocompute the thermodynamic stability of molecular clus-ters of various sizes and compositions. Great strides havebeen made in recent years in extending quantum chem-ical methods to this area, but an accurate assessmentof the entropic contributions to the relevant thermody-namic potentials, for conditions where the clusters areliquid-like, requires an approach that goes beyond a con-sideration of harmonic fluctuations, currently implyinglengthy simulation times or a reduction in the level oftreatment from quantum to classical dynamics.

8.5×10−8 2.5×10−8

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

12

34

0 1 2 3 4 5 0 1 2 3 4 5NW

Rel

ativ

e po

pula

tion

Henschel et al This work

Water monomer density / Å−3

NA

Figure 6. Populations of a variety of clusters of sulphuric acidand water at two water monomer densities, normalised to sumto unity for each value of NA (and hence independent of sul-phuric acid monomer density), calculated using our approach(light grey) and taken from Henschel et al. [13] (dark grey).

Our approach is to employ a simplified force field forcomplex molecules fitted to quantum chemical compu-tations [30] and then to use a nonequilibrium classicalmolecular dynamics procedure to compute the free en-ergy associated with the disassembly of a cluster into itsconstituent molecules, avoiding a harmonic approxima-tion to the cluster entropy. The separation is broughtabout by the motion of guide particles, each of which istethered to one of the molecules in the cluster. The ap-proach offers advantages over typical Monte Carlo meth-ods since the assessment of the properties of a particularcluster is more direct. We do not have to go through alengthy comparison of ensembles of clusters differing byonly one molecule at a time; instead we immediately re-fer a cluster to a system of separated, though tethered,molecules. The approach is intrinsically a nonequilibrium

9

method, since it involves the mechanical manipulation ofa cluster in a finite period of time, driving the systemaway from equilibrium. We exploit the Jarzynski equal-ity [24] to extract equilibrium free energy changes from adistribution of nonequilibrium work, while taking care toensure that the outcomes are independent of the protocolof manipulation.

In this study, we have extended the method to clustersconsisting of two molecular species, noting that it couldeasily be generalised to an arbitrary number. The anal-ysis specifies the way in which the cluster constituentsshould initially be tethered, and how the free energyof disassembly should be combined with energy and en-tropy tethering corrections to produce the grand poten-tial that characterises clusters in a binary vapour. Whenpresented as a surface, the grand potential conveys theintuitive idea of a thermodynamic barrier with a criticalcluster and preferred path of formation. Our approachprovides a new way to construct such a surface in a con-trolled and well defined fashion, and furthermore, ourpresentation does not require input of the densities ofvapours in equilibrium with the condensed phase, whichcan complicate the formalism.

We have demonstrated the approach by studying theimportant binary system of sulphuric acid and water, andhave compared our results with those obtained recentlyusing optimised configurations obtained from quantumchemistry together with harmonic fluctuations. The forcefield employed is simplified, and in particular does notallow proton transfers, but more elaborate models haverecently been developed and could be employed in fur-ther studies [36]. We give particular attention to efficien-cies available through a suitable performance of the ma-nipulation. Instead of simply pulling the cluster apartwith soft harmonic forces, where disassembly proceeds

through a sequence of irreversible and violent snapping ortearing events, we prise the molecules apart using guideparticles that exert increasingly strong forces while po-sitioned at close range to the cluster. Such a protocolfavours molecular removal from the cluster in a man-ner that is mechanically more reversible, which meansthat the free energy of disassembly can be obtained fromshorter MD simulations. We required 2-3 days of timeon the multiprocessor Legion computing facility at UCLto study 46 clusters containing different numbers of thetwo molecular species.

Our computations do not capture all the dynamic sub-tleties of a treatment at electronic level, since they arebased on a fitted classical force field, but they are verymuch faster to perform than ab initio methods, andshould do better in assessing the entropy of a liquid-likecluster. The comparison between our results and thoseof Henschel et al. [13] is very reasonable and this studywill pave the way for further investigations based on im-proved force fields [36] and a wider variety of molecularspecies [30].

ACKNOWLEDGEMENTS

GVL was supported through a studentship in the Cen-tre for Doctoral Training on Theory and Simulation ofMaterials managed by the Department of Physics at Im-perial College, funded by the Engineering and PhysicalSciences Research Council (EPSRC) of the UK undergrant number EP/G036888, and JYP was funded byan EPSRC Undergraduate Vacation Bursary. We thankGeorge Jackson, Erich Müller and Patricia Hunt for theirsupport and we acknowledge the UCL Legion High Per-formance Computing Facility (Legion@UCL), and asso-ciated support services.

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