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ms2: A molecular simulation tool for thermodynamic

properties, release 4.0

Robin Fingerhuta, Gabriela Guevara-Carriona, Isabel Nitzkea, Denis Sarica,

Joshua Marxb, Kai Langenbachb, Sergei Prokopevc, David Celn´yd, Martin

Bernreuthere, Simon Stephanb, Maximilian Kohnsb, Hans Hasseb, Jadran

Vrabeca,∗

aThermodynamics and Process Engineering, Technical University Berlin, 10587 Berlin,

Germany

bLaboratory of Engineering Thermodynamics, University Kaiserslautern, 67653

Kaiserslautern, Germany

cComputational Fluid Dynamics Laboratory, Institute of Continuous Media Mechanics

UB RAS, 614013 Perm, Russia

dNuclear Sciences and Physical Engineering, Czech Technical University in Prague,

11519 Prague, Czech Republic

eHigh Performance Computing Center Stuttgart (HLRS), 70550 Stuttgart, Germany

Abstract

A new version release (4.0) of the molecular simulation tool ms2 (Deublein

et al., 2011; Glass et al., 2014; Rutkai et al., 2017) is presented. Version 4.0

of ms2 features two additional potential functions to address the repulsive

and dispersive interactions in a more versatile way, i.e. the Mie potential and

the Tang-Toennies potential. This version further introduces Kirkwood-Buﬀ

integrals based on radial distribution functions, which allow the sampling of

the thermodynamic factor of mixtures with up to four components, orien-

tational distribution functions to elucidate mutual conﬁgurations of neigh-

boring molecules, thermal diﬀusion coeﬃcients of binary mixtures for heat,

mass as well as coupled heat and mass transport, Einstein relations to sample

transport properties with an alternative to the Green-Kubo formalism, dielec-

tric constant of non-polarizable ﬂuid models, vapor-liquid equilibria relying

on the second virial coeﬃcient and cluster criteria to identify nucleation.

Keywords: Molecular simulation; Molecular dynamics; Monte Carlo

∗Corresponding author.

E-mail address: vrabec@tu-berlin.de

Preprint submitted to Computer Physics Communications January 25, 2021

New version programm summary

Program Title: ms2

Program Files doi: http://dx.doi.org/10.17632/nsfj67wydx.3

Licensing provisions: CC by NC 3.0

Programming language: Fortran95

Supplemental material: A detailed description of the parameter setup for the in-

troduced methods, properties, functionalities etc. is given in the supplemental

material. Furthermore, all molecular force ﬁeld models developed by our group

are provided by the MolMod Database: Stephan et al., Mol. Sim. 45 (2019) 806

Journal reference of previous version: Deublein et al., Comput. Phys. Commun.

182 (2011) 2350 and Glass et al., Comput. Phys. Commun. 185 (2014) 3302 and

Rutkai et al., Comput. Phys. Commun. 221 (2017) 343

Does the new version supersede the previous version?: Yes

Reasons for the new version: Introduction of new features as well as enhancement

of computational eﬃciency

Summary of revisions: Two new potential functions to address repulsive and dis-

persive interactions (Mie and Tang-Toennies potential), new properties (Helmholtz

energy, Kirkwood-Buﬀ integrals, thermodynamic factor, thermal diﬀusion coeﬃ-

cients, dielectric constant, mean-squared displacement and non-Gaussian param-

eter), new functionalities (Kirkwood-Buﬀ integration with extrapolation to the

thermodynamic limit, van der Vegt correction for the radial distribution function,

orientational distribution function, Einstein relations, vapor-liquid equilibria esti-

mations, cluster criteria to identify nucleation).

Nature of problem: Calculation of application-oriented thermodynamic proper-

ties: vapor-liquid equilibria of pure ﬂuids and multi-component mixtures, thermal,

caloric and entropic data as well as transport properties and data on microscopic

structure

Solution method: Molecular dynamics, Monte Carlo, various ensembles, Grand

equilibrium method, Green-Kubo formalism, Einstein formalism, Lustig formal-

ism, OPAS method, Smooth-particle mesh Ewald summation

1. Introduction

Signiﬁcant increases in computing power have led to a broader usage of molec-

ular modeling and simulation, which simultaneously widens the ability to tackle

challenges in physics, chemistry and engineering in a sound and detailed man-

ner. Over the last decades, it has often been shown that these computer-based

methods may predict physical reality very successfully. Thus, the long-standing

2

obstacle of sparse or lacking experimental information on thermophysical data can

be overcome by trustworthy and rapid predictions with massively-parallel high

performance computing (HPC) hardware and scalable codes.

The program ms2 (molecular simulation 2) was developed to compute ther-

mophysical equilibrium properties of pure ﬂuids and mixtures with Monte Carlo

(MC) or molecular dynamics (MD) simulations that are both implemented in a

single source code. Licenses are freely available for all purposes which concern

academic research and teaching under www.ms-2.de together with a substantial

set of molecular force ﬁeld models [1]. ms2 [2, 3, 4] supports the microcanonical

(NV E), canonical (N V T ), isobaric-isenthalpic (NpH), isobaric-isothermal (N pT )

and grand canonical (µV T ) ensembles as well as the simulation of vapor-liquid

equilibria (VLE) with the Grand equilibrium method. Moreover, ms2 facilitates

the sampling of numerous thermodynamic bulk properties, including transport

data, like Maxwell-Stefan (MS) and Fick diﬀusion coeﬃcients, for molecular mod-

els consisting of Lennard-Jones (LJ) interaction sites, point charges, point dipoles

and point quadrupoles. It allows for the sampling of the chemical potential with

Widom’s particle insertion and thermodynamic integration as well as osmotic pres-

sure, hydrogen bond statistics and other features. Next to these thermophysical

properties, it was focused on an eﬃcient parallelization of ms2 using the message

passing interface (MPI), open multi-processing (OpenMP) and its hybrid form

(MPI+OpenMP).

There is a series of molecular simulation tools, such as CHARMM, DL POLY,

ESPResSo, GIBBS, GROMACS, IMD, LAMMPS, ls1 mardyn, NAMD, TINKER

or Towhee, that is being developed for a range of communities. Both industrial

and academic users are addressed by ms2 with a focus on applications of molecular

modeling and simulation in process and energy engineering. In contrast to most

of the tools listed above, ms2 is limited to rigid force ﬁeld models which are

appropriate for small molecular species only. However, the implementation of the

internal degrees of freedom into ms2 is underway for some time in an unpublished

version of ms2 [5].

Aiming at high accuracy and short response time, ms 2 is characterized by

the large variety of properties that are sampled on the ﬂy. This user-friendly

design was extended by the ability to concurrently sample an arbitrary number

of state points in one program execution. Concurrent sampling was optimized in

the present ms2 version such that communication between ensembles was removed

entirely. Combining this with its dedication to generate large sets of Helmholtz

energy derivative data for the development of equations of state [6], ms2 is very

much suited to be executed on HPC infrastructure.

A more versatile molecular model development was prioritized in this version

release 4.0 such that the traditional LJ 12-6 potential was generalized to the Mie

3

potential and the more complex Tang-Toennies potential [7] was introduced. Con-

sequently, the basis of molecular modeling and simulation may be improved by a

more accurate description of the repulsive and dispersive interactions. Moreover,

ms2 is now able to yield the Fick diﬀusion coeﬃcient of mixtures constituted by up

to four components due to the concurrent sampling of the MS diﬀusion coeﬃcient

and Kirkwood-Buﬀ integrals (KBI) [8] that give access to the thermodynamic fac-

tor [9, 10, 11]. Additionally, with this release, more rapid VLE estimations can

be made by carrying out a single NpT ensemble simulation sampling the chemical

potential of the liquid and using the second virial coeﬃcient for the vapor. These

and further new features were implemented in the source code and the toolset

provided at www.ms-2.de. The present work discusses the fourth major release

of ms2 and its most important innovations, which are presented in the following

sections.

2. Mie potential

Addressing repulsive and dispersive interactions in a more versatile way with

ms2, the standard LJ 12-6 potential function was generalized with the Mie po-

tential function [12]. The pairwise interaction between diﬀerent sites iand jin a

distance rij is modeled by

uij(rij) = n

n−mn

mm/(n−m)·εσ

rij n

−σ

rij m,(1)

where σand εare the Mie size and energy parameters, respectively, and n,mare

the repulsive and dispersive exponents.

In ms2, the interactions between two diﬀerent Mie sites are described by the

Lorentz-Berthelot combining rules for pure components, while for mixtures the

modiﬁed Lorentz-Berthelot rules are applied [2]. The unlike repulsive and disper-

sive exponents are determined according to Laﬁtte et al. [13] by

kij = 3 + q(ki−3)(kj−3) for k=n, m . (2)

Long-range interactions beyond the cutoﬀ radius rcare considered analytically

with the angle averaging formalism derived for the Mie potential by Lustig [14].

The generalization from LJ 12-6 to Mie was introduced throughout the entire

code so that all properties and functionalities are accessible with it.

3. Tang-Toennies potential

To describe the intermolecular interactions, an additional potential function

based on the work of Tang and Toennies (TT) [7] was introduced into ms2. Con-

4

siderations for its use are outlined in the following, whereas proceedings which are

the same as for the Mie potential are not discussed in detail.

The pairwise interaction between diﬀerent sites iand jin a distance rij is

modeled by

uij(rij) = Aij exp(−αij rij)−f6(rij)C6, ij

r6

ij −f8(rij)C8, ij

r8

ij

,(3)

where fn(rij) denotes a damping function of the form

fn(rij) = 1 −exp(−bij rij )

n

X

k=0

(bijrij)k

k!for n= 6,8.(4)

At ﬁrst glance, the reader might have the impression that this functional form

entails high computational costs, but this is not the case. In fact, despite its

greater complexity, the TT potential is computed equally fast as the Mie potential

and scales very well for larger ensembles, cf. Fig. 1.

The TT potential contains ﬁve parameters Aij, αij , bij, C6, ij, C8, ij . In case

of mixtures or pure ﬂuid molecular models constituted by diﬀerent TT sites, the

parameters of the interactions between unlike sites are determined by the following

combination rules [15, 16]

αij = 2 αi·αj

αi+αj

,(5)

Aij =1

αij (Aiαi)1/αi·(Ajαj)1/αjαij /2,(6)

bij = 2 bi·bj

bi+bj

,(7)

Cn, ij =C1/bi

n, i ·C1/bj

n, j bij /2for n= 6,8,(8)

sij =si+sj

2.(9)

Subscripts iand jdenote the parameters for the like interactions, whereas ij

indicates the unlike interactions. In the remainder of this section, the indices ij

will be omitted for brevity. The parameter srefers to the shielding for the short

range correction that is discussed below.

Long-range interactions beyond the speciﬁed cutoﬀ radius rcare not calcu-

lated explicitly. Instead, analytical correction terms are used, which result in the

necessity to compute

Z∞

rc

r2u(r)dr. (10)

5

0

20

40

60

speed-up

TT

Mie

N=2048

0

20

40

60

speed-up

TT

Mie

N=4000

0 20 40 60 80 100 120

0

20

40

60

# cores

speed-up

TT

Mie

N=10976

1

Figure 1: Performance of the TT and Mie potentials for 5000 MD simulation steps with

varying particle number Nusing MPI parallelization.

6

To this end, the application of angle averaging according to Lustig [14] is desired,

which reduces the computational eﬀort to one logical operation per pair of inter-

acting molecules to verify whether r < rc. Hence, the potential function needs to

be expressed in terms of r2m. Note that m < −3/2 must be satisﬁed to guaran-

tee convergence of Eq. (10). For simplicity, the repulsive term Aexp(−αr) was

neglected. Further, since fn(r)→1 as r→ ∞, it is reasonable to assume

fn(r)Cn

rn≈fn(rc)Cn

rnfor n= 6,8.(11)

Numerical experiments suggest that this is a suitable approximation as long as rc

exceeds three to four times the distance of the potential well minimum.

Widom’s test particle insertion and thermodynamic integration [2, 4] are meth-

ods to determine the chemical potential of a given component. However, in com-

bination with the TT potential, both techniques require some caution. For the

application of Widom’s insertion method, a short range correction has to be ap-

plied. Numerical inaccuracies associated with the calculation of the exponential

terms on computer hardware via Taylor series expansion cause the damping func-

tions to exhibit oscillations in their entire domain. Their amplitude is negligible,

except for small values of r. In case of r→0 the oscillations may lead to a change

of sign of the otherwise strictly positive functions, which in turn produces erratic

results for the chemical potential.

This can be prevented by choosing a representation of the dispersive interac-

tions which avoids the exponential term

fn(r)Cn

rn=−Cnbn

∞

X

k=n+1

ηn,k(br)k−nfor n= 6,8,(12)

where the coeﬃcients ηn,k are given by

ηn,k =1

k!

n

X

m=0

(−1)k+mk

mfor n= 6,8; k > n. (13)

In ms2, series (12) is computed up to k= 18. With regard to the decreasing

absolute value of the summands and therefore fading contribution of higher order

terms, this proved to be an appropriate and suﬃciently accurate choice.

To determine which representation of the dispersive terms is used, the addi-

tional parameter shielding was introduced into ms2. It denotes a lower bound of r

up to which the usual form of the TT potential (3) is used. Note that the required

shielding depends solely on parameter b.

Thermodynamic integration [4] is another technique to compute the chemical

potential. In ms2, a non-linear scaling was implemented, i.e. u(λ) = λdufor

7

λ∈[0 ,1], with dbeing an input parameter. The default value is set to d= 4 in

order to prevent the occurrence of singularities at λ= 0 or 1 when the LJ potential

is used [17].

However, dhas to be chosen under consideration of the shape of the repulsive

interaction. Clearly, the slope of the repulsive term exp(−αr) of the TT potential

(3) depends on α. A comparison to the slope of r−12 as used in the LJ potential,

yields

d exp(−αr)

dr

=αexp(−αr)12r−13 =

dr−12

dr

,(14)

for α < 5 and suﬃciently small values of r, where each potential function is domi-

nated by its repulsive part. Hence, it is recommended to adjust daccordingly, since

the statistical uncertainty of the obtained chemical potential rises as dincreases.

4. Thermodynamic factor through Kirkwood-Buﬀ integration

Both the Fick diﬀusion coeﬃcient Dand the MS diﬀusion coeﬃcient Dmatri-

ces are of central importance when describing mass transport in liquid mixtures.

While the former can be measured in the lab due to its composition dependence,

the latter cannot be acquired by experiments because it is related to the chem-

ical potential. The thermodynamic factor Γconnects these diﬀusion coeﬃcient

matrices and is given for a multi-component mixture by

Γij =xi

kBT∂µi

∂xjT ,p,Σ

,(15)

where µiis the chemical potential of component i,xjthe mole fraction of compo-

nent j,Tthe temperature and kBthe Boltzmann constant. This equation applies

at constant temperature and pressure as well as Pn

i=1 xi= 1, with number of

components n[18].

For mixtures containing three or more components, D=B−1Γ, where matrix

Bis determined by the MS diﬀusion coeﬃcient matrix D[18]. Describing mix-

tures constituted of many components, matrix Dasymptotically requires about

twice as much information than matrix D, i.e. Γconnects nine Fick with six MS

diﬀusion coeﬃcient elements in case of a quaternary mixture [10]. This is an im-

portant advantage of MS theory, but when mass transport needs to be accessed

experimentally, the Fick approach can be applied directly. Taking advantage of

both, the thermodynamic factor Γis indispensable. Since it is a derivative of the

chemical potential, Γcannot be measured experimentally. Instead, excess Gibbs

energy models or equations of state are usually employed to extract Γfrom phase

equilibrium data. As an alternative route, molecular simulation combined with

KBI [8] allows for the sampling of Γij [9, 10, 18].

8

KBI based on molecules’ center of mass radial distribution functions (RDF)

was implemented in the NV T ensemble both for MC or MD simulations. In ms2,

RDF are sampled in the entire cubic simulation volume L3up to √3L/2, i.e.

beyond the cutoﬀ radius that is independently speciﬁed for explicitly evaluating

the intermolecular interactions. Thus, extended schemes may be applied [19]. In

the context of KBI, RDF corrections are required. The correction proposed by

Ganguly and van der Vegt [20], referred to as vdV, was implemented into ms2

because it was found to be the most adequate [9]. It takes the excess or depletion

of molecular species jaround a given molecule iat the distance rinto account such

that the asymptotic behavior of the RDF should yield an improved convergence

to unity. The vdV correction is given by Ganguly and van der Vegt [20]

gvdV

ij (r) = gij(r)Nj(1 −V(r)/L3)

Nj(1 −V(r)/L3)−∆Nij(r)−δij

,(16)

where gij(r) is the RDF between components iand j,Njis the number of molecules

j,δij the Kronecker delta and V(r)=4πr3/3. Excess or depletion is determined

by ∆Nij(r) = Rr

04πr02ρj[gij (r0)−1]dr0with the partial density ρj.

KBI are strictly deﬁned in the µV T ensemble only, which is challenging to

impose for dense liquid states. In order to apply KBI to N V T ensemble simulation

data, an integral truncation and correction by Kr¨uger et al. [21] was implemented,

such that KBI are calculated by

Gij(R) = Z2R

0

4πr2(1 −3x/2 + x3/2)[gij (r)−1]dr , (17)

with x=r/2R. Its success for ﬁnite system sizes was discussed recently [9]. How-

ever, the extrapolation to the thermodynamic limit V→ ∞, where all ensemble

types converge, is essential [9, 19]. Thus, for V→ ∞ the following KBI approxi-

mation [19] was implemented

G∞

ij (R)≈Z2R

0

4πr2(1 −23x3/8+3x4/4+9x5/8)[gij(r)−1]dr . (18)

Eqs. (17) and (18) were implemented for both standard and vdV corrected RDF

in ms2. Fig. 2 exemplarily shows Gij over inverse radius R−1for a binary LJ

mixture. An almost linear behavior is produced by Eq. (17) and extrapolations to

the thermodynamic limit are well presented by Eq. (18), whereas standard KBI is

of little use for extrapolation purposes.

Expressions of the (n−1)×(n−1) matrix Γbased on KBI Gij can be found in

the literature [8, 18, 22] for binary and ternary mixtures. Ben-Naim [22] outlined

a general formalism to obtain chemical potential derivatives from KBI. Employing

these, Γexpressions for quaternary mixtures were recently derived and evaluated

9

Figure 2: Gij and G∞

ij over inverse radius R−1for a liquid binary LJ mixture (σj/σi= 1.5,

εj/εi= 0.75). RDF were sampled by MD every time step over a production period of

1.5·107time steps in the NV T ensemble containing N= 4000 molecules; dashed lines:

standard Gij = 4πRR

0[gij (r)−1]r2dr; solid lines: Gij from Eq. (17); squares: G∞

ij from

Eq. (18); black: standard RDF; red: vdV corrected RDF.

by our group [10]. Fig. 3 shows results for Γfor a quaternary state point. Taking

into consideration that there are no other standard simulation methods available

for the direct sampling of Γand due to the satisfactory performance shown in [11],

Γexpressions for mixtures with up to four components were implemented into

ms2 such that cumbersome post-processing is omitted. Invoking KBI leads to an

increase of execution time by about 4 % only for realistic molecules because of

eﬃcient coding and parallelization [9].

10

Figure 3: Thermodynamic factor Γof a liquid-like supercritical quaternary LJ mixture.

Black circles: Γbased on numerical chemical potential derivatives sampled with Widom’s

test particle insertion; red circles/triangles: Γcalculated with the expressions for quater-

nary mixtures based on vdV corrected RDF and Eq. (18)/Eq. (17) (statistical uncertainties

are within symbol size).

5. Orientational distribution function

The sampling of the orientational distribution function (ODF) of dipolar ﬂuids

with MD simulation was implemented as a new feature in ms2. The ODF quantiﬁes

how neighboring molecules are mutually oriented. Such information on the ﬂuid

structure is useful for the parametrization of equations of state that give access to

the relative permittivity [23, 24].

The ODF Oij for molecules of species jaround central molecules of species i

can be deﬁned implicitly by the two particle density

nij(r, ϕi, ϕj, γ) = ρiρjgij(r)Oij (r, ϕi, ϕj, γ),(19)

which quantiﬁes how many molecules of species iand jat a distance rhave a

mutual orientation given by the angles ϕi,ϕjand γ.nij can be separated into the

bulk partial densities ρiand ρj, the RDF gij and the ODF Oij , which considers

mutual orientation. The distance and the three angles characterizing the ODF are

depicted in Fig. 4.

11

Figure 4: Distance rand angles of the coordinate system for ODF calculation. µiand

µjdenote the eﬀective dipole vectors of molecules iand j, which may arise either from

point dipoles, a distribution of charges or a combination of the two. ϕiand ϕjare their

inclination angles with respect to the intermolecular distance vector r. The torsion angle γ

describes the angle diﬀerence between the molecules’ orientation around the intermolecular

distance vector.

These three angles fully describe the mutual orientation of two molecules with

two rotational degrees of freedom, such as hydrogen chloride or Stockmayer mod-

els, which do not rotate around their main axis. The characterization of the mu-

tual orientation of two molecules with three rotational degrees of freedom would

necessitate the introduction of two additional angles to describe each molecule’s

orientation around their main axis, which is not yet supported by ms2. The ODF

may nonetheless be sampled for molecules with three rotational degrees of freedom,

but only the three angles depicted in Fig. 4 will be evaluated.

The ODF is sampled as an average quantity by discretizing the distance and

orientation space with a classic binning scheme. The cosines of the three angles can

be computed directly from the direction vectors of the dipoles through algebraic

operations. When sampling the ODF, cos(ϕi), cos(ϕj), γand rare discretized

into bins of constant size. Sampling ϕiand ϕjin terms of their respective cosines

avoids two numerically costly arccosine operations per molecular pair. It also

homogenizes the quality of the data across the orientation space, as the volumes

around the molecules spanned by the discrete angular elements are equally large

when a constant angular increment ∆cos(ϕi) is chosen instead of constant ∆ϕi.

The increment size may be speciﬁed by the user. The total number of sampling

points is the product of the number of grid points in the four relevant dimensions.

Langenbach [23] used 40 grid points for cos(ϕi) and cos(ϕj) and 36 grid points

for the torsion angle γ, while sampling the ODF within the ﬁrst coordination

shell without applying further discretization to the intermolecular distance, which

resulted in a total of 57,600 sampling points. This led to a satisfactory resolution of

the ODF, while keeping memory demand and data output within reasonable limits.

The present implementation always samples the ODF within the cutoﬀ radius. It

is recommended to specify the element size ∆ron the same order of magnitude

as the size of the molecular models under investigation. For spherical molecular

12

models, it thus makes sense to choose ∆rto be the molecular radius. In this case,

the ﬁrst radial element corresponds to the volume occupied by the central molecule,

which carries little information. Beyond the ﬁrst radial element, then each two

consecutive radial elements cover one spherical shell that roughly corresponds to

one of the central molecule’s coordination shells. With post-processing, data for

two radial elements can be merged to characterize the respective coordination shell.

The ODF is normalized by ms2 so that the average of all sampling points within

the same radial sampling element is unity. The normalization value of each shell

is provided in the output.

The present implementation samples mutual orientations on the basis of the

total dipole moment vectors of the molecules and works for both point dipoles

and dipole moments arising from a distribution of partial point charges. The

implementation is compatible with Ewald summation and the reaction ﬁeld method

for treating electrostatic long-range interactions. Any number of components may

be chosen as long as at least one dipolar species is present (which may have a

vanishing dipole moment). In this case, ODF are recorded for every like and unlike

pair of dipolar species. In case of a binary mixture of the dipolar components

iand j, this entails four ODF: the two like ones Oii and Ojj , which describe

mutual orientations between molecules of the same species, Oij , which describes

how molecules of species jorient themselves around central molecules of species i

and Oji describing the reverse case. The unlike ODF are not identical, but adhere

to symmetry conditions. Thus, Oji is not sampled directly by ms2. Instead, it is

computed upon output from the data for Oij .

The ODF can be visualized in terms of isosurfaces that represent equal prob-

abilities within orientation space. Fig. 5 exemplarily shows the ODF of a pure

Stockmayer ﬂuid sampled with MD simulation in the NV T ensemble.

13

Figure 5: Isosurfaces of the ODF within the ﬁrst coordination shell 0.5< r/σ < 1.5

of the Stockmayer ﬂuid with a dipole moment µ/(4πε0εσ3)1/2= 1.5 at kBT /ε = 3 and

ρσ3= 1. Red surfaces represent states with a probability that is 40 % higher than

random orientation, green surfaces indicate random orientation and blue surfaces depict

a probability that is 40 % lower than random orientation.

6. Thermal diﬀusion in binary mixtures

The phenomenological coeﬃcients for heat, mass as well as coupled heat and

mass transport, as deﬁned by the framework of irreversible thermodynamics [25],

can be sampled with equilibrium MD simulation employing the Green-Kubo for-

malism. Considering the Soret and Dufour eﬀects, the equations for heat ﬂux JQ

and mass ﬂux of component 1 Jm

1in a binary mixture are [25]

JQ=−LQQ ∇T

T2−LQ1∂µ1

∂w1T ,p

∇w1

(1 −w1)T,(20)

Jm

1=−L1Q∇T

T2−L11 ∂µ1

∂w1T ,p

∇w1

(1 −w1)T,(21)

where Lab are the so-called phenomenological Onsager coeﬃcients, describing the

proportionality between the thermodynamic forces and the ﬂuxes that they induce.

µ1and w1stand for the chemical potential and mass fraction of component 1,

respectively. The coupled phenomenological transport coeﬃcients follow Onsager’s

reciprocity relations (ORR), i.e. L1Q=LQ1.

The phenomenological coeﬃcient for mass transport L11 is

14

L11 =V

3kBZ∞

0hJm

1(0)Jm

1(t)idt, (22)

with the mass ﬂux Jm

i

Jm

i(t) = 1

V

Ni

X

k=1

mi(vk

i(t)− hvi).(23)

Therein, Vis the volume, mithe molecular mass of component i,vk

i(t) the center

of mass velocity vector of molecule kof component iat some time tand Nithe

number of molecules of component i. The brackets h...idenote the NV T ensemble

average. If initialized accordingly, hvideviates from zero during simulation only

within machine error so that Jm

1=−Jm

2.

The phenomenological coeﬃcient for heat transport LQQ is

LQQ =V

3kBZ∞

0hJQ(0)JQ(t)idt. (24)

However, in equilibrium MD simulation, only the internal energy ﬂux JEand

not the heat ﬂux JQcan be accessed directly, but both quantities are related by

Perronace et al. [26]

JQ=JE−h1

m1−h2

m2Jm

1,(25)

where hiis the partial molar enthalpy of component iand

JE

V=1

2

2

X

i=1

Ni

X

k=1

mk

ivk

i2·vk

i−1

2

2

X

i=1

2

X

j=1

Ni

X

k=1

Nj

X

l6=k"rkl

ij :∂ukl

ij

∂rkl

ij −I·ukl

ij #·vk

i.(26)

Therein, ukl

ij is the intermolecular potential energy, rkl

ij the distance vector between

molecules kand l, while the indices iand jdenote the molecular species. The

second term in the brackets is a dyadic product (denoted by a colon) with the

unitary tensor I.

The phenomenological coeﬃcient for internal energy transport LEE is

LEE =V

3kBZ∞

0hJE(0)JE(t)idt, (27)

and is related to LQQ by [27]

LQQ =LEE −2L1Eh1

m1−h2

m2+L11 h1

m1−h2

m22

.(28)

15

The phenomenological coeﬃcient for coupled heat and mass transport L1Qis

L1Q=V

3kBZ∞

0hJm

1(0)JQ(t)idt, (29)

and the phenomenological coeﬃcient for coupled internal energy and mass trans-

port L1Eis

L1E=V

3kBZ∞

0hJm

1(0)JE(t)idt. (30)

Because of the ORR, phenomenological cross-coeﬃcients are symmetric, i.e. L1E=

LE1and L1Q=LQ1, but statistically independent. Exemplarly, Fig. 6 shows

the cross-correlation functions underlying to the phenomenological coeﬃcients for

coupled heat and mass transport L1Qand LQ1of the mixture of argon + krypton

in its liquid state. As can be seen, both functions oscillate around the same values

and can therefore be averaged to improve statistics. The resulting averaged cross-

correlation function is then integrated to obtain the coupled phenomenological

coeﬃcient, cf. Fig. 6.

If the partial molar enthalpy hiof all components is known, the according set of

values should be speciﬁed in the *.par ﬁle and ms2 will calculate the phenomeno-

logical coeﬃcients LQQ and L1Q. If this is not the case, the phenomenological

coeﬃcients LEE and L1Ewill be calculated instead. L1Qand L1Eare related

by [27]

L1Q=L1E−h1

m1−h2

m2L11.(31)

An example of how to obtain partial molar enthalpy values is given in the

supplemental material. Finally, the thermal diﬀusion coeﬃcient DTcan be ac-

cessed by comparing the phenomenological Eqs. (20) and (21) with the equations

for the heat and mass ﬂuxes according to Fourier and Fick considering the Soret

and Dufour eﬀects

JQ=−λ∇T−∂µ1

∂w1T ,p

ρw1T DD

T∇w1,(32)

J1=−ρw1w2DS

T∇T−ρD∇w1,(33)

(34)

where ρis the density, λthe thermal conductivity and Dthe Fick diﬀusion coeﬃ-

cient. DS

Tand DD

Tare the thermal diﬀusion coeﬃcients of Soret- and Dufour-type,

respectively [26]

16

Figure 6: Cross-correlation functions (top) hJ1(0)JQ(t)i(black line) and hJQ(0)J1(t)i

(blue line) are shown together with the integral of their average (bottom) as a function of

time for the liquid mixture argon + krypton at T= 95.2 K, p= 0.1 MPa and x1= 0.6759

mol·mol−1sampled with N= 1000 molecules.

DS

T=L1Q

ρw1w2T2, DD

T=LQ1

ρw1w2T2.(35)

It thus also follows from the ORR that DS

T=DD

T=DT.ms2 calculates the

thermal diﬀusion coeﬃcient DTon the basis of the average of the sampled phe-

nomenological cross-coeﬃcients for coupled heat and mass transport L1Qand LQ1.

The thermal diﬀusion coeﬃcient DTis strongly dependent on the enthalpic contri-

bution to the heat ﬂow so that it is only calculated if the partial molar enthalpy of

all components is speciﬁed in the *.par ﬁle. Note that Eq. (35) is valid for binary

mixtures only.

17

7. Einstein relations

The Green-Kubo formalism was adopted in previous versions of ms2 to sample

transport properties, such as diﬀusion coeﬃcients, viscosity or thermal conductiv-

ity. An alternative is oﬀered by the Einstein relations, which can be understood

as an integral form for determining these properties. For diﬀusion coeﬃcients,

the Einstein relations deal with molecular displacements, while the Green-Kubo

formalism operates with correlation functions of velocities. Both approaches are

equivalent, but they show diﬀerent statistics in practice. For example, long time

tails may be encountered with the Green-Kubo formalism, while the Einstein re-

lations do not suﬀer from this problem.

The expressions for self-diﬀusion or intra-diﬀusion Diand Onsager coeﬃcients

Λij take the form [28]

Di=1

6Ni

lim

∆t→∞

1

∆tNi

X

k=1

[rk

i(t+ ∆t)−rk

i(t)]2,(36)

Λij =1

6Nlim

∆t→∞

1

∆tNi

X

k=1

[rk

i(t+ ∆t)−rk

i(t)]

Nj

X

l=1

[rl

j(t+ ∆t)−rl

j(t)].(37)

Therein, Niand Njstand for the number of molecules of components iand j,

Nis the total number of molecules, rk

idenotes the Cartesian coordinate vector

of molecule kbelonging to component iand the brackets h...iindicate ensemble

averaging. These relations are analogous to those of the Green-Kubo formalism,

except that molecular displacements are considered instead of correlation functions

of velocities. Onsager coeﬃcients from both approaches are associated with MS

diﬀusion coeﬃcients in the same way [2, 3, 4].

The Einstein relation for the shear viscosity has the form

η=1

2V kBTlim

∆t→∞ [G(t+ ∆t)−G(t)]2,(38)

where

G(t) =

N

X

i=1

mirα

i(t)vβ

i(t).(39)

Eq. (38) cannot be directly applied because G(t) is not continuous and introduces

unphysical behavior under periodic boundary conditions [29]. However, this prob-

lem can be avoided by substituting the diﬀerence in Eq. (38) with the integral

[30]

18

∆G=G(t+ ∆t)−G(t) =

t+∆t

Z

t

Jαβ

p(τ)dτ. (40)

Consequently, the expression for shear viscosity reads

η=1

2V kBTlim

∆t→∞ t+∆t

Z

t

Jαβ

p(τ)dτ2,(41)

where Jαβ

pis a stress tensor element, which is exactly the same as that in the

Green-Kubo formalism (see Eq. (11) in Ref. [2] for details)

Jαβ

p=

N

X

l=1

mivα

ivβ

i−

N−1

X

i=1

N

X

j=i+1

n

X

k=1

n

X

l=1

rα

ij

∂uij

∂rβ

kl

.(42)

Therein, α, β =x, y, z are Cartesian coordinates, miand vα

iare mass and velocity

of molecule i,kand lare the indices of the ninteraction sites constituting a

molecular model, rα

ij is the site-site distance and uij the potential energy of the site-

site interaction. The present implementation averages over the three oﬀ-diagonal

elements Jxy

p,Jxz

p,Jyz

pof the stress tensor (42).

Procedures for sampling the transport properties can be employed concurrently

with both approaches. Invoking the Einstein formalism leads to little additional

computational eﬀort. On average, switching on the Einstein procedure increases

the total execution time by less than 2%.

Fig. 7 shows a comparison between data sampled with the Green-Kubo for-

malism and the Einstein relations. For all considered transport properties, an

excellent agreement between both approaches was reached.

Based on Eq. (36) it is straightforward to analyze system dynamics, e.g. solid-

ﬂuid phase transitions by the mean-squared displacement (MSD) h∆r2(t)i[31]. In

this context, the closely related Non-Gaussian parameter is given by [32]

α2(t) = 3h∆r4(t)i

5h∆r2(t)i2−1.(43)

This property can be applied to study dynamic heterogeneity in terms of mobile

and immobile particles [31]. Both MSD and α2were implemented into ms2 and

can be sampled independently of the other transport properties.

19

Figure 7: Intra-diﬀusion coeﬃcient of argon (top), Onsager coeﬃcient Λ11 (center) and

shear viscosity (bottom) determined with the Green-Kubo formalism (blue) and the Ein-

stein relations (black) for the liquid mixture argon (1) + krypton (2) at T= 95.25 K,

p= 0.1 MPa and x1= 0.6759 mol·mol−1sampled with N= 1000 molecules.

20

8. Dielectric constant

The sampling of the static dielectric constant εof non-polarizable ﬂuid mod-

els, also known as relative permittivity, was implemented in ms2. In the NV T

ensemble, it is computed from Kirkwood’s ﬂuctuation formula [33]

ε−1 = 4π

3kBT V hM2i−hMi2,(44)

where all symbols have their usual meaning and Mis the total dipole moment of

the simulation volume

M=

N

X

i=1

µi,(45)

that is constituted by the sum of the dipole moment vectors µiof all molecules i.

In the NpT ensemble, the volume Vin Eq. (44) has to be replaced by the ensemble

average hVi. In case of isotropic and non-ferroelectric ﬂuids, the second term hMi2

should vanish when suﬃciently long sampling is carried out. Nevertheless, that

term is preserved in the present implementation to allow for convergence checks.

In ms2, Eq. (44) can be sampled both with MC and MD simulations. How-

ever, MD simulations are recommended as long individual series of samples are

needed for the term hM2ito converge (see below). The present implementation is

compatible with both the reaction ﬁeld method and Ewald summation for treat-

ing the long-range electrostatic interactions. For molecular models containing a

distribution of partial charges, in case of the reaction ﬁeld method, their dipole

moment vectors µiare readily available. In case of Ewald summation, the sum-

mation of partial charges to a molecular dipole moment µiis carried out for each

molecule. If a mixture contains ions, which carry a permanent charge, they aﬀect

the dielectric constant only through their interactions that alter the orientation of

solvent molecules [34].

Fig. 8 shows the running averages of the two terms involved in Eq. (44) from MD

simulations of SPC/E water. The reaction ﬁeld method with conducting bound-

ary conditions was used to treat long-range electrostatics. It can be seen that the

second term hMi2indeed quickly vanishes, but a long simulation is needed for the

ﬁrst term hM2ito converge.

In addition to previous validations of the ms2 implementation [24, 34, 35] by

comparison to literature data, the dielectric constant of two water models and

two methanol models at ambient conditions was computed with ms2. Again, the

reaction ﬁeld method with conducting boundary conditions was used to treat long-

range electrostatics. Good agreement with literature data was found, as Tab. 1

shows.

21

0

0.5

1.0

1.5

2.0

2.5

hM2i,hMi2/ 10−57 (Cm)2

hM2i

hMi2

0 2 4 6 8 10 12

0

0.5

1.0

1.5

2.0

2.5

t/ ns

hM2i,hMi2/ 10−57 (Cm)2

hM2i

hMi2

Figure 8: Convergence of running averages of the total dipole moment of SPC/E water at

T= 298.15 K and p= 0.1 MPa. Top: N pT simulation, bottom: N V T simulation.

Table 1: Dielectric constant εof four pure component models at T= 298.15 K and p=

0.1 MPa calculated with ms2 in comparison with benchmark values from the literature.

Uncertainties of the last speciﬁed digit of the ms2 results are given in parentheses. Un-

certainties of the literature values are not available.

Model ms2 Literature Ref.

Water SPC/E 71(1) 70 [36]

Water TIP4P/2005 57(1) 59 [36]

Methanol OPLS/2016 26.6(7) 26.4 [37]

Methanol Schnabel et al. 21.2(6) 21.2 [37]

22

9. Vapor-liquid equilibria with the NpT plus second virial coeﬃ-

cient method

VLE of ﬂuids constituted by any number of components can be calculated by

ms2 with the Grand equilibrium method [38] through two subsequent simulations

of the coexisting phases. However, the vapor simulation can often be substituted

with an equation of state, which shortens the process, saves computational ef-

fort and avoids the diﬃculty of sampling low density states. For instance, the

Haar-Shenker-Kohler equation has been applied in concert with the NpT plus test

particle method to pure ﬂuids [39] and mixtures [40, 41].

An alternative route to VLE at low pressure is the NpT plus second virial

coeﬃcient method (NpT + SVC). This approach was used for VLE calculations

of phenol, aniline and cyclohexylamine as well as their mixtures before [42] and

was fully described elsewhere [39].

One liquid phase simulation run in the NpT ensemble at speciﬁed temperature

T, composition xand some pressure p0gives the chemical potential as a function of

pressure pof all components in the liquid as a ﬁrst-order Taylor expansion around

p0

µl

i(T, x, p) = µl

i0(T, x, p0)+(∂µi/∂p)T ,x·(p−p0),(46)

where (∂µi/∂p)T,x=vl

iis the partial molar volume of component i. Sampling of

the liquid phase with Widom’s test particle method yields values for all coeﬃcients

of Eq. (46) [38, 39]. The chemical potential of the vapor can be expressed on the

basis of the virial equation of state p=ρkBT(1 + Bρ) by

µv

i(T, y, p) = kBTln yi+kBTln ρ+ 2kBT ρ

n

X

j=1

yjBij,(47)

with the second virial coeﬃcient B=Pn

i=1 Pn

j=1 yiyjBij and the vapor density

ρ= (p4Bp/kBT+ 1 −1)/(2B). The second virial coeﬃcient is evaluated in ms2

by numerical integration of Mayer’s f-function. The phase equilibrium conditions

are then employed to identify the saturated vapor pressure pand the saturated

vapor composition ythrough the nonlinear system of equations

µl

i(T, x, p) = µv

i(T, y, p) for i= 1, . . . , n. (48)

A modiﬁed Newton method was implemented into the present version of ms2 to

solve Eq. (48).

To evaluate the performance of the NpT + SVC method, the saturated vapor

density and compressibility factor of the pure LJ ﬂuid are compared to the EOS

by Thol et al. [6] in Fig. 9. The NpT + SVC method excellently reproduces both

of these properties up to kBT /ε = 1.05, with maximum deviations of 0.6 %. In

23

fact, an evaluation of 45 diﬀerent VLE data sets for the LJ ﬂuid showed that the

systematic simulation errors of the saturated vapor density and the compressibility

factor are ±1.0 % and ±1.25 %, respectively [43]. Therefore, the N pT + SVC

method should not be used for the LJ ﬂuid outside this systematic error span, i.e.

above kBT/ε = 1.05, cf. Fig. 9. Moreover, this method can be applied to all VLE

state points of the binary mixture N2+ O2between 80 and 120 K, as shown in

Fig. 10. However, it fails for higher temperatures and thus higher vapor densities

due to the limitations of the virial expansion, while the Grand equilibrium method

also operates under such conditions.

The range of applicability and precision of the NpT + SVC method can be

estimated before the start of a VLE calculation, if data for the compressibility

factor, the SVC and the saturated vapor density are available. The vapor density

can only be positive, if p(4Bp/kBT+ 1) <1 and (4Bp/kBT+ 1) ≥0, since the

SVC is negative up to the Boyle temperature. The combination of these terms

with z=p/(ρkBT) yields the limiting compressibility factor zlim =−1/4Bρ.

Thus, Eq. (48) has real solutions only for z < zlim. The closer the ratio z/zlim is

to unity, the less accurate the NpT + SVC method becomes, cf. Tab. 2. However,

at low saturated vapor densities, where z/zlim <0.7, the deviations for zand ρ00

remain under 1.6 % for both the LJ ﬂuid and the mixture N2+ O2.

The NpT + SVC method can replace simulations in the low density regime

and leads to lower statistical uncertainties of the VLE properties compared to

methods entirely based on simulations. However, it fails to yield VLE near the

critical point because the SVC is insuﬃcient in this region. Consequently, the

Grand equilibrium method in its classic form with a vapor simulation run should

be used under such conditions.

Table 2: Ratio of the calculated to the limiting compressibility factor for the LJ ﬂuid.

Results from the NpT + SVC method and the Grand equilibrium method are presented,

where the former does not converge above kBT/ε = 1.20 as indicated by an asterisk.

kBT/ε zGE zSVC zlim,GE zlim,SVC zGE

zlim,GE

zSVC

zlim,SVC

1.05 0.81 0.80 1.27 1.26 0.63 0.64

1.15 0.71 0.68 0.83 0.77 0.86 0.88

1.25 0.58 * 0.52 * 1.10 *

24

Figure 9: Relative error of saturated vapor density (top) and compressibility factor (bot-

tom) for the LJ ﬂuid determined with the NpT + SVC method (red triangles) and the

Grand equilibrium method (blue triangles) in comparison to the EOS by Thol et al. [6].

The dotted lines represent the systematic error of the saturated vapor density (±1%) and

the compressibility factor (±1.25%) evaluated from 45 diﬀerent VLE data sets for the LJ

ﬂuid [43]. Statistical uncertainties are within symbol size.

Figure 10: VLE phase diagram of the binary mixture N2+ O2from the NpT + SVC

method (red triangles) and the Grand equilibrium method (blue triangles) compared to

the Peng-Robinson EOS (black line) and experimental data (cross symbols) [44].

25

10. Cluster criteria for nucleation

A method in the context of metastable states and homogeneous nucleation

[45] was implemented in ms2. Its focus lies on the identiﬁcation of an ongoing

vapor-liquid transition, considering both droplet and bubble formation. The pri-

mary requirement of such methods is a deﬁnition of clusters and voids to identify

emerging phases, which typically translates into the evaluation of intermolecular

distances, as opposed to methods relying on the chemical potential or multiple

metrics.

Instead of comparing distances between molecules and constructing logical

structures that are similar to neighbor lists, the present method utilizes an in-

dependent grid. Distance checks between molecular positions and grid points of

a regular cubic lattice are performed. This route is robust, can be parallelized

eﬃciently and its feasibility has been tested for droplet and bubble nucleation

[46].

Phase transitions are associated with spontaneous and signiﬁcant changes of

the local density [47]. However, when the interest lies in the sampling of thermody-

namic properties of metastable states, trajectories with an ongoing phase change

should be avoided [48]. A criterion has to be set up to consider microstates only

that are still consistent with the initial phase and eventually terminates sampling

if this is not wanted.

By introducing a regular cubic grid, the instantaneous local density is sampled

with ms2 by assigning every particle to its surrounding grid points. This opera-

tion has a complexity of O(N), where Nis the number of particles and the spatial

density distribution is evaluated on that grid at user-speciﬁed time instances. This

evaluation should only be carried out when the molecular conﬁguration had suﬃ-

cient time to signiﬁcantly change its structure.

Processing is done directly as a molecule contribution to the according grid

points, avoiding expensive iterations over the entire grid. This contribution is

calculated by division of the molecular position by the grid constant ∆L. Fig. 11

provides an illustration of the inner workings of the method. For each molecule

a primary grid point is evaluated by indexx=INT(rx/∆L). All other grid points

surrounding that molecule are subsequently assigned with that molecule as well.

This simple design works straightforwardly for grids that ﬁll the entire simulation

volume.

However, this principle was extended to grids that do not ﬁll the entire simu-

lation volume. This is a mandatory requirement when the grid constant ∆Lis an

input parameter and L/∆Lis not an integer. The remaining tripod (in 3D) with

a thickness below ∆Lwas treated in a periodic boundary fashion to preserve same

surrounding volume of each grid point.

26

0110

0121

0231

0121

ΔL

Figure 11: Illustration of the regular grid example (in 2D) with associated values at the

grid points marked with +, the simulation volume is delimited by the dashed line, molecule

contributions to the grid points are denoted as arrows and grid constant ∆L.

The present approach is rounded up with a decision procedure that evaluates

the grid data and terminates sampling, if a speciﬁed percentage of grid points sig-

nals a local density below or above a speciﬁed density threshold. Here, priority was

given to robustness across molecular species, investigated phase transition direc-

tion, number of molecules etc. The multi-ensemble feature of ms2 [4] requires two

levels of termination. Once a given ensemble has reached its termination criterion,

it is not further sampled. This approach easily extends into simultaneous simu-

lation of multiple ensembles. The entire multi-ensemble simulation is ended only

when all ensembles have either reached their termination criterion, the speciﬁed

number of time steps or wall time.

The desired outcome is to sample the properties of metastable systems before

the onset of a phase transition, which can be achieved by properly specifying

the parameters of the present approach. The criteria can also be used for other

purposes, such as explorations subsequent to nucleation processes and cluster/void

precursor evolution. An example for such an observation is presented in Fig. 12

for a bubble nucleation situation, depicting a system before the liquid-to-vapor

transition with emerging voids in the system volume. This simpliﬁed ﬁgure depicts

only grid points reporting low molecule neighbor counts, corresponding to emerging

voids at the beginning of the phase transition.

27

Figure 12: Visualization of grid points that report less than three molecule neighbors in

a system with bubble formation. Grid points shown as + are color coded based on the

reported number of molecule neighbors. Figure axes correspond to the spatial placement

of the grid points within the simulation volume.

Acknowledgments

The authors gratefully acknowledge ﬁnancial support by German Federal Min-

istry of Education and Research (BMBF) under the grant 01IH16008 “TaLPas:

Task-basierte Lastverteilung und Auto-Tuning in der Partikelsimulation”. We

gratefully acknowledge the Paderborn Center for Parallel Computing (PC2) for

the generous allocation of computer time on the OCuLUS and Noctua clusters, the

High Performance Computing Center Stuttgart (HLRS) under the grant MMHBF2

as well as the Boltzmann-Zuse Society (BZS).

Appendix A. Supplemental material

Supplemental material related to this article can be found online at ...

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31

Supplemental Material to:

ms2: A molecular simulation tool for thermodynamic properties,

release 4.0

Robin Fingerhuta, Gabriela Guevara-Carriona, Isabel Nitzkea, Denis Sarica, Joshua

Marxb, Kai Langenbachb, Sergei Prokopevc, David Celn´yd, Martin Bernreuthere, Simon

Stephanb, Maximilian Kohnsb, Hans Hasseb, Jadran Vrabeca,∗

aThermodynamics and Process Engineering, Technical University Berlin, 10587 Berlin, Germany

bLaboratory of Engineering Thermodynamics, University Kaiserslautern, 67653 Kaiserslautern,

Germany

cComputational Fluid Dynamics Laboratory, Institute of Continuous Media Mechanics UB RAS,

614013 Perm, Russia

dNuclear Sciences and Physical Engineering, Czech Technical University in Prague, 11519 Prague,

Czech Republic

eHigh Performance Computing Center Stuttgart (HLRS), 70550 Stuttgart, Germany

This supplemental material contains additional information on the new features of ms2

as well as example parameter and potential model ﬁles for execution.

∗Corresponding author.

E-mail address: vrabec@tu-berlin.de

Preprint submitted to Elsevier January 25, 2021

1. Mie potential

In ms2, the potential model type for repulsive and dispersive interactions is selected

in the *.pm ﬁle that speciﬁes the molecular model, cf. Listing 1. To select the Mie poten-

tial [1], the user has to set the keyword to SiteType = MIE which has to appear below

the keyword NSiteTypes. For each repulsive/dispersive site of the molecular model the

keywords MIE n = ’#mie parameter n’and MIE m = ’#mie parameter m’have to be

set to real (or integer) values #mie parameter nand #mie parameter maccording to

the Mie potential parameters nand m. Both keywords have to appear above the coordi-

nates x, y, z of each according site, whereas the repulsion parameter nappears above

the dispersion parameter m.

Listing 1: Example ﬁle (*.pm) of a two site Mie potential model.

NSiteTypes = 1

Sit eT yp e = MIE

NSites = 2

MIE n = 1 2 . 2 5

MIE m = 6 . 0

x = 0 . 0

y = 0 . 0

z = 0 . 0

si g ma = 1 . 0

e p s i l o n = 1 . 0

ma ss = 1 . 0

MIE n = 1 1 . 9

MIE m = 6 . 0

x = 0 . 8

y = 0 . 0

z = 0 . 0

si g ma = 1 . 0

e p s i l o n = 1 . 0

ma ss = 1 . 0

NRotAxes = auto

2

2. Tang-Toennies potential

In ms2, the potential model type for repulsive and dispersive interactions is selected in

the *.pm ﬁle that speciﬁes the molecular model. In case of the Tang-Toennies potential [2],

the user has to set the keyword to SiteType = TT68 which has to appear below the key-

word NSiteTypes, cf. Listing 2. For each repulsive/dispersive site of the molecular model,

the keywords A=’#tt68 parameter A’,b=’#tt68 parameter b’,alpha =’#tt68

parameter alpha ’,C6 =’#tt68 parameter C6 ’and C8 =’#tt68 parameter C8 ’

have to appear in the given order below the coordinates x, y, z of the site. They have

to be set to real values #tt68 parameter A,#tt68 parameter b,#tt68 parameter

alpha ,#tt68 parameter C6 and #tt68 parameter C8 according to the Tang-Toennies

potential parameters A,b,α,C6and C8. Further, the keyword shielding = ’#tt68

parameter shielding ’has to be set to the real value #tt68 parameter shielding to

specify the minimal distance up to which the standard representation of the potential

is used. The keyword has to appear below the parameter mass of each repulsion and

dispersion site.

For the calculation of the chemical potential via thermodynamic integration [3], the

user has to set ChemPotMethod = ThermoInt in the ensemble section in the *.par ﬁle.

Next, the keyword LambdaExponent = ’#TI exponent of lambda’ has to be set to the

real (or integer) value #TI exponent of lambda. This value speciﬁes the parameter d

for non-linear scaling, i.e. u(λ) = λdufor λ∈[0,1].

Listing 2: Example ﬁle (*.pm) of a single site Tang-Toennies potential model.

NSiteTypes = 1

Sit eT yp e = TT68

NSites = 1

x = 0 . 0

y = 0 . 0

z = 0 . 0

A = 4466710

b = 4 . 1 8

a lp h a = 2 . 4 8

C6 = 756300

C8 = 1 00 00 00 0

ma ss = 3 9 . 9 4 8

s h i e l d i n g = 0 .0 2

NRotAxes = auto

3

3. Thermodynamic factor through Kirkwood-Buﬀ integration

Kirkwood-Buﬀ integration (KBI) [4] as well as the calculation of the thermodynamic

factor matrix Γ[4, 5, 6, 7, 8] can be invoked by the keyword KBIFreq = ’#KBI frequency’

that has to appear under the simulation section in the *.par ﬁle, cf. Listing 3. The user

has to set #KBI frequency to an integer value. This value speciﬁes the sampling frequency

for center of mass radial distribution functions (RDF) gij which are the essential input

for KBI. The computational eﬀort for KBI is minor [9], sampling each step is thus rec-

ommended for highly accurate RDF. Next, the keyword KBIResetFreq = ’#KBI reset

frequency’ has to be set to the integer value #KBI reset frequency according to a

block length in simulation steps. In these blocks, RDF are independently sampled and

KBI Gij are determined with the approach given by Kr¨uger et al. [10]. In this procedure,

KBI Gij are accumulated block-wise over simulation and their statistical uncertainties

are determined. For applying KBI G∞

ij in the thermodynamic limit V→ ∞ according to

Kr¨uger et al. [11], averaged RDF are required and calculated block-wise. The keyword

KBINumShells = ’#RDF number shells’ has to be set to the integer value #RDF number

shells according to the number of shells for the RDF. Invoking KBI, RDF are sampled

in the entire cubic simulation volume up to √3L/2, i.e. beyond the cutoﬀ radius that is

independently speciﬁed for explicitly evaluating the intermolecular interactions. Half of

the edge length of the simulation volume L/2 is divided into the chosen number of RDF

shells so that the number of shells is extended automatically by a factor of √3.

The *.kbirdf output ﬁle contains the block-wise averaged RDF over the entire cubic

simulation volume. Therein, standard RDF as well as corrected RDF according to Gan-

guly and van der Vegt [12] are written with the #KBI reset frequency. The *.kbirav

output ﬁle contains the block-wise running averages of KBI Gij ,G∞

ij and their statis-

tical uncertainties for standard KBI as well as the expressions developed by Kr¨uger et

al. [10, 11]. Moreover, each KBI is given for standard RDF and corrected RDF [12]. The

simulation result ﬁle *.res contains the thermodynamic factor matrix Γ[4, 5, 6, 7, 8]

for each RDF type and KBI type with statistical uncertainties according to the error

propagation law.

4

Listing 3: Example simulation parameter ﬁle (*.par) of a binary Lennard-Jones mixture for executing

KBI.

Uni ts = R edu ced

Le n g th U ni t = 1 . 0

EnergyUnit = 1.0

Ma ss Un i t = 1 . 0

S i m u l a t i o n = MD

Integrator = Gear

Ti me St ep = 3 . 0E - 4

En sembl e = NVT

MCORSteps = 0

NVTSteps = 80 0000

Ru nStep s = 1 50000 00

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

KBI Freq = 1

KB IR esetF re q = 1 00 00

KBINumShells = 500

CutoffMode = COM

NEnsembles = 1

Te m pe r at ur e = 0 . 8 5

P r e s s u r e = 0 . 0 3

Density = 0.199734749

N P a r t i c l e s = 40 0 0

NComponents = 2

Po tM ode l = LJA .pm

Mo l e Fr a ct = 0 . 0 5

ChemPotMethod= no ne

Po tM ode l = LJB .pm

Mo l e Fr a ct = 0 . 9 5

ChemPotMethod= no ne

e t a = 1 . 0

x i = 1 . 0

C u t o f f = 5 . 0

E p s i l o n = 1 . 0 E 10

5

4. Orientational distribution function

Sampling of the orientational distribution function (ODF) [13] is enabled by enter-

ing the keyword ODFRecordingFreq = ’#ODF recording frequency’ in the ensemble

section of the *.par ﬁle, cf. Listing 4. By choosing #ODF recording frequency as an

integer value greater than zero, the user speciﬁes the frequency with which the ODF is

sampled. Due to the high dimensionality of the ODF, it is recommended to sample it

every time step to achieve an adequate data quality within a reasonable time frame. The

frequency of output generation is speciﬁed with the keyword ODFOutputFreq = ’#ODF

output frequency’. A single output ﬁle is written for the ODF, which is updated with

a frequency speciﬁed by the value of #ODF output frequency. The output ﬁle does not

list the data for all sampling blocks individually and the ODF is not reset. Instead, when-

ever output is generated, the existing output ﬁle is overwritten with the data covering the

entire production run up until the last completed time step. The ODF is sampled with a

classic binning scheme. By specifying the value NShellsODF = ’#number of shells for

ODF sampling’ the user chooses into how many segments the sampling radius of the ODF

is divided. The sampling radius always equals the cutoﬀ radius. The number of segments

for the cosines of the angles ϕiand ϕjis speciﬁed with the keywords nPhiODF = ’#number

of segments of cos(ϕi)and cos(ϕj)for ODF sampling’ and the number of segments

for the angle γis speciﬁed with nGammaODF = ’#number of segments of γfor ODF

sampling’.

The output ﬁle ﬁrst lists the normalization values for each radial shell of each component

pair. Subsequently, the ODF values for each component pair and each bin are listed in a

table. Each bin is referred to by its coordinate values at the center of the segment. I.e.,

if a cutoﬀ radius of 4.5σis chosen and divided into three radial segments, the length of

each segment is 1.5σ. The values for the intermolecular radius rlisted in the output ﬁle

are then 0.75 σ, 2.25 σand 3.75 σ, referring to the value of rin the center of the ﬁrst,

second and third radial segment, respectively.

6

Listing 4: Example simulation parameter ﬁle (*.par) of the pure ﬂuid R32 for sampling ODF.

Uni ts = S I

Le n g th U ni t = 3 . 5

EnergyUnit = 100.0

Ma ss U ni t = 4 0 . 0

S i m u l a t i o n = MD

Ti meS tep = 5E- 4

En sembl e = NVT

NVTSteps = 5 00 00

Ru nStep s = 1 00 00 00

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

CutoffMode = COM

NEnsembles = 1

Te mp er at ure = 300

Den si t y = 20

P is t on M as s = 1 . 0 E - 4

N P a r t i c l e s = 1 00 0

NComponents = 1

Po tMo del = R32 . pm

M ol a rF r a ct = 1 . 0

ODFRecordingFreq= 1

ODFOutputFreq = 1 00000 0

NShell sOD F = 3

nPhiODF = 40

nGammaODF = 36

C u t o f f = 4 . 5

E p s i l o n = 1 . 0 E1 0

7

5. Thermal diﬀusion in binary mixtures

In ms2, the thermal diﬀusion coeﬃcient is calculated with the Green-Kubo formalism,

which requires the same input parameters in the *.par ﬁle from the user as required for

other transport properties in the previous implementation, e.g. diﬀusion coeﬃcients, shear

viscosity, thermal and electrical conductivity. Therefore, the keyword CorrfunMode= YES

has to appear under the simulation section in the *.par ﬁle and the ensemble-speciﬁc key-

words StepsCorrfun, Corrlength, SpanCorrfun, ViewCorrfun and ResultFreqCF as

implemented in previous versions of ms2. Additionally, the user has to specify the partial

molar enthalpy for each component in reduced units employing the keyword PartMolEnt=

’#partial molar enthalpy’ in the *.par ﬁle, cf. Listing 5. Note that if the keyword

PartMolEnt is not given in the *.par ﬁle or its value is set to zero, ms 2 will not calculate

the thermal diﬀusion coeﬃcient. However, the values of the phenomenological cross-

coeﬃcients L1Eand LE1as well as the thermal conductivity will be calculated, neglecting

the eﬀect of the partial molar enthalpy, and the resulting values will be written into the

*.res ﬁle with the corresponding remark for the users.

The *.rtr output ﬁle, containing the averaged correlation functions and their in-

tegrals, was extended with the corresponding values for the phenomenological cross-

coeﬃcients. It should be noted that, unlike other transport properties, these correlation

functions are not normalized with their initial value, therefore they do not start at unity.

In the following, an example of how to obtain the partial molar enthalpy is given.

In the case of binary mixtures, it is determined in two steps. First, the residual molar

enthalpy of the mixture hres is calculated in the isobaric-isothermal (NpT ) ensemble over

a wide composition range around the required state point. Thereupon, the total enthalpy

his calculated by adding the ideal part hid to the residual enthalpy hres . An appropriate

function h=f(xi) is ﬁtted by a least square optimization to the resulting data for the

composition dependence of the total molar enthalpy. The partial molar enthalpy is then

calculated by

hi=h+xj∂h

∂xi.

In case of that hid is the same for both mixture components, the partial molar enthalpy can

directly be determined from the values of the residual enthalpy. For mixtures consisting

of more than two components, more simulations are required to obtain the appropriate

function for the total enthalpy.

8

Listing 5: Example simulation parameter ﬁle (*.par) of the binary mixture argon + krypton for sampling

the thermal diﬀusion coeﬃcient.

Uni ts = S I

Le n g th U n it = 3. 4 0 5

EnergyUnit = 119.8

Ma s sU ni t = 3 9 . 9 4 4

S i m u l a t i o n = MD

Integrator = Gear

Ti me S te p = 6 . 9 5 6 5 8 E - 4

En sembl e = NVT

MCORSteps = 100

NVTSteps = 6000000

NPTSteps = 1 00 00 0

Ru nStep s = 1 20000 00

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

CutoffMode = COM

NEnsembles = 1

CorrfunMode = y es

Te m pe r at ur e = 9 5 . 2

P r e s s u r e = 0 . 1

D e n s i t y = 3 1 . 0 6 9 6 4 9 1 6 5

N P a r t i c l e s = 10 0 0

NComponents = 2

StepsCorrfun = 2

C o r r l e n g t h = 4 0 0 0

SpanCorrfun = 200

Vi ew Co rr fu n = 1 0

ResultFreqCF = 1

Po tMo del = Ar . pm

Mo l e F ra c t = 0 . 6 7 5 9

Pa rt M ol En t = - 3 . 8 2 3 7 8 7 4 5 8

ChemPotMethod= no ne

Po tMo del = Kr . pm

Mo l e F ra c t = 0 . 3 2 4 1

Pa rt M ol En t = - 6 . 9 7 9 8 5 8 8 8 6

ChemPotMethod= no ne

e t a = 1 . 0

x i = 1 . 0

C u t o f f = 4 . 0

E p s i l o n = 1 . 0 E 10

9

6. Einstein relations

The procedure for sampling transport properties with the Einstein relations, as well

as with the Green-Kubo formalism, requires the compiler ﬂag TRANS=1. To select the em-

ployed calculation procedure, the keyword TransMethod = ’#Einstein/GKEinstein/GK’

was introduced with following options:

-Einstein: Einstein procedure is switched on (Green-Kubo oﬀ),

-GKEinstein: Both, Green-Kubo and Einstein procedure are switched on,

-GK: Green-Kubo procedure is switched on (Einstein oﬀ), which is the default setting,

cf. *.par ﬁle (Listing 6). No other options are needed because the Einstein procedure

uses the same parameters as the Green-Kubo formalism, i.e. Corrlength, SpanCorrfun

and StepsCorrfun.

The meaning of these parameters is outlined in the following. Simultaneously, av-

eraging over several correlation function samples is performed, cf. Fig. S1. The ﬁrst

correlation function is sampled when the equilibration process has terminated. After

SpanCorrfun time steps, a new correlation function is started. Each correlation function

has the length Corrlength (in time steps) and after the end of the averaging length, a

new correlation function is sampled. Averaging is made over all samples of correlation

functions. The parameter StepsCorrfun stands for the frequency, which determines

how often correlation functions are called. As an example, these parameters could be

set as Corrlength = 30000, SpanCorrfun = 1000 and StepsCorrfun = 2. It is rec-

ommended to set them as multiples of each other.

The result of averaging transport properties over time is written to the ﬁle with the

extension *.ecoef (in analogy to the *.rtr ﬁles of the Green-Kubo formalism). The ﬁnal

results are written to the *.res ﬁle.

10

0 4 8 12 16

-0.8

-0.4

0

0.4

0.8

1.2

Corrlength

SpanCorrfun

entire computation time

arbitrary quantity F

5 8 11 14 ...

2

6 9 12 15 ...

3

4 7 10 13 ...

1

Figure S1: An arbitrary quantity F as a function of time.

Listing 6: Example simulation parameter ﬁle (*.par) of the binary mixture argon + krypton for applying

the Einstein relations.

Uni ts = S I

Le n g th U n it = 3 . 4 0 5

EnergyUnit = 119.8

Ma s sU ni t = 3 9 . 9 4 4

S i m u l a t i o n = MD

Integrator = Gear

Ti me S te p = 6 . 9 5 6 5 8 E - 4

En sembl e = NVT

MCORSteps = 20 0

NVTSteps = 1 00 00 00

NPTSteps = 10 00 00

Ru nStep s = 2 00 00 00 0

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

CutoffMode = COM

NEnsembles = 1

CorrfunMode = y es

Tr an sMe th od = E i n s t e i n

Te m pe r at ur e = 9 5 . 2

P r e s s u r e = 0 . 1

D e n s i t y = 3 1 . 0 6 9 6 4 9 1 6 5

N P a r t i c l e s = 1 00 0

NComponents = 2

StepsCorrfun = 1

C o r r l e n g t h = 2 00 0 0

SpanCorrfun = 200

Vi ew Co rr fu n = 1 00

ResultFreqCF = 1

11

Po tMo del = Ar . pm

Mo l e F ra c t = 0 . 6 7 5 9

Pa rt M ol En t = - 3 . 8 2 3 7 8 7 4 5 8

ChemPotMethod = n one

Po tMo del = Kr . pm

Mo l e F ra c t = 0 . 3 2 4 1

Pa rt M ol En t = - 6 . 9 7 9 8 5 8 8 8 6

ChemPotMethod = n one

e t a = 1 . 0

x i = 1 . 0

C u t o f f = 4 . 0

E p s i l o n = 1 . 0 E1 0

12

The mean-squared displacement (MSD) h∆r2(t)iand the Non-Gaussion parameter

α2(t) can be invoked in MD simulations with the keyword ALPHA2Freq = ’#Alpha2

frequency’ that has to appear under the simulation section in the *.par ﬁle, cf. List-

ing 7. The user has to set #Alpha2 frequency to an integer value which speciﬁes the

sampling frequency of both time correlation functions. Next, the keyword ALPHA2Length

= ’#Alpha2 length’ has to be set to the integer value #Alpha2 length according to

the chosen time correlation function length in simulation steps. Moreover, the keyword

ALPHA2Span = ’#Alpha2 span’ has to be set to the integer value #Alpha2 span in sim-

ulation steps. Time correlation functions are subsequently started when #Alpha2 span

is reached.

The *.a2rav output ﬁle contains ensemble averaged time correlation functions h∆r2(t)i

and α2(t) over the chosen time correlation function length #Alpha2 length. Time is given

in reduced units and SI units (in fs) according to the chosen time step ∆tin the *.par and

the sampling frequency #Alpha2 frequency. Moreover, the number of averaged functions

is given in this output ﬁle and is written with the frequency of the result ﬁle *.res.

13

Listing 7: Example simulation parameter ﬁle (*.par) of a pure LJ fcc solid for sampling time correlation

functions h∆r2(t)iand α2(t).

Uni ts = R edu ced

Le n g th U ni t = 1 . 0

EnergyUnit = 1 .0

Ma ss Un i t = 1 . 0

S i m u l a t i o n = MD

Integrator = Gear

Ti me St ep = 0 . 0 0 1

En sembl e = NVT

MCORSteps = 0

NVTSteps = 20 0000

Ru nStep s = 1 000000

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

ALPHA2Freq = 1

ALPHA2Length = 1 00 00

ALPHA2Span = 100

CutoffMode = COM

NEnsembles = 1

Te m pe ra t ur e = 1 . 0

D en s i t y = 1 . 8

N P a r t i c l e s = 10 9 76

NComponents = 1

Po tMo del = LJ 126 . pm

M ol a rF r a ct = 1 . 0

ChemPotMethod= no ne

NTest = 2 00 0

C u t o f f = 9 . 1

E p s i l o n = 1 . 0 E 10

14

7. Vapor-liquid equilibria with the NpT plus second virial coeﬃcient method

The more rapid calculation of vapor-liquid equilibria taking advantage of the second

virial coeﬃcient (NpT + SVC method) is speciﬁed in the *.par ﬁle under the ensemble

section, cf. Listing 8. Speciﬁcally, the user has to deﬁne Ensemble = NPTSVC. Note that

this method only works if the chemical potential calculation is turned on in the *.par

ﬁle, i.e. ChemPotMethod 6=None.

The simulation result ﬁle *.res contains results of the liquid simulation run in the

NpT ensemble, the second virial coeﬃcient and all vapor-liquid equilibrium properties.

Furthermore, the compressibility factor ratio for the assessment of the accuracy of the

NpT + SVC method is also written to the *.res ﬁle.

Listing 8: Example simulation parameter ﬁle (*.par) of the binary mixture nitrogen + oxygen with the

NpT + SVC method.

Uni ts = S I

Le n g th U ni t = 3 . 0

EnergyUnit = 100.0

Ma ss U ni t = 5 0 . 0

S i m u l a t i o n = MC

A cc e pt a n ce = 0 . 5

En sembl e = NPTSVC

NVTSteps = 20 00 0

NPTSteps = 2 00 00

Ru nStep s = 5 00 000

ResultFreq = 1000

ErrorsFreq = 5000

VisualFreq = 0

CutoffMode = COM

NEnsembles = 1

Te m pe r at u re = 8 0 . 0 0

P r e s s u r e = 0 . 0 3 3

D e ns i t y = 3 5 . 0 0

N P a r t i c l e s = 8 6 4

NComponents = 2

Po tM ode l = N2 .pm

M ol a r Fr a c t = 0 . 0 5 0

ChemPotMethod= Widom

NTest = 3 45 6

Po tM od el = O2 . pm

M ol a r Fr a c t = 0 . 9 5 0

ChemPotMethod= Widom

NTest = 3 45 6

e t a = 1 . 0

x i = 1 . 0 0 7

C u t o f f = 4 . 0

E p s i l o n = 1 . 0 E 10

15

8. Cluster criteria for nucleation

The identiﬁcation of clusters/voids is enabled with the option ClusterIsCriteria=yes

in the simulation section of the *.par ﬁle, cf. Listing 9. Although it can be applied to

MC and MD simulations, it is recommended to use MD only to sample the thermody-

namic properties of metastable states. The identiﬁcation should be carried out in suf-

ﬁciently spaced time intervals that can be speciﬁed for each ensemble individually with

ClusterCriteriaFreq. Reasonable values are multiples of 100 time steps. The identiﬁca-

tion operates in two directions, i.e. in vapor→liquid or in liquid→vapor, which are selected

with the option ClusterCriteriaType=’#gridvap / gridliq’. This determines how

ClusterMoleculeCount is utilized. In the vapor→liquid case, where clusters may emerge,

the grid points are checked for greater than or equal to ClusterMoleculeCount. Al-

ternatively, in the liquid→vapor case, grid points are checked for voids with less than

or equal to ClusterMoleculeCount.

The parameter ClusterCriteriaDistance speciﬁes the grid constant ∆L(in σref )

and thus the volume attributed to each grid point Vgridpoint = (2∆L)3. Of course, ∆L

implicitly places an upper limit on the maximum count of molecules that may be assigned

to a grid point. It is not required that the edge length of the simulation volume has to

be an integer multiple of the grid constant ∆Las the algorithm can deal with this case.

To account for small clusters/voids that can temporarily emerge even in stable systems,

the parameter ClusterMaximumAllowed was included. It is a threshold percentage of grid

points of the entire grid and determines how many grid points have to report a density

ﬂuctuation for the ensemble to be terminated. Experience shows that 5 to 10% is a good

choice for smaller systems with <2000 grid points.

The identiﬁcation procedure generates two ﬁle types. A ﬁle with the extension *.grid

contains the positions of the grid points and additional grid properties in human readable

format. A second ﬁle with the extension *.clust contains neighbor counts for all grid

points that were sampled, assigning one time instance to a line. Depending on the grid

size, this ﬁle can become large. Files are created for all ensembles individually and are

updated with the frequency ClusterCriteriaFreq.

Fig. S2 explains the parameters ClusterMoleculeCount and ClusterMaximumAllowed.

A multi-ensemble simulation was carried out for many densities along one isotherm with

non-terminating criteria settings, while returning grid point neighbor data. Fig. S2 shows

the results after 106time steps.

A percentage of grid points is speciﬁed with ClusterMaximumAllowed that termi-

nates the sampling of the ensemble, when the corresponding number of grid points

signals that the threshold ClusterMoleculeCount was reached. If the parameter pair

ClusterMaximumAllowed=2.0% and ClusterMoleculeCount=4 would have been speciﬁed

for the vapor states depicted in Fig. S2, only the stable and ﬁrst two metastable ensembles

16

Figure S2: Percentage of grid points reporting a speciﬁed number of neighboring molecules for multiple

densities along one isotherm, sampled over 106time steps without termination. For orientation, the

corresponding regions of the phase diagram are color-coded.

would have been sampled over 106time steps, while the other metastable vapor ensem-

bles would have been terminated earlier. Similar considerations apply to the liquid side.

With the parameter pair ClusterMaximumAllowed=2.0% and ClusterMoleculeCount=1,

the two most supersaturated ensembles in Fig. S2 would have been terminated before

reaching 106time steps.

The present approach was designed to be simple and provide insight into the metastable

region, while ensuring that phase identity of the system is preserved. The current imple-

mentation considers only pure component systems, but can be extended to mixtures.

17

Listing 9: Example simulation parameter ﬁle (*.par) of the single ensemble simulation of the Lennard-

Jones ﬂuid with cluster criteria enabled. The simulation in this conﬁguration is stopped after 100 time

steps.

Uni ts = Re duc ed

Le n gt h Un i t = 1 . 0

EnergyUnit = 1.0

Ma ss Un i t = 1 . 0

S i m u l a t i o n = MD

Integrator = Gear

Ti me St e p = 0 . 0 0 2 5

En sembl e = NVT

MCORSteps = 10

NVTSteps = 10 0

Ru nStep s = 10 00

ResultFreq = 100

ErrorsFreq = 100

CutoffMode = COM

NEnsembles = 1

ClusterIsCriteria = Yes

Te m pe r at u r e = 1 . 2 1 9 6 7 7

D e n si t y = 0 . 1 4 2 3 3 2

N P a r t i c l e s = 1 3 72

NComponents = 1

ClusterCriteriaFreq = 100

C l u s t e r C r i t e r i a T y p e = g r id v a p

C l u s t e r C r i t e r i a D i s t a n c e= 1 . 0

Clu ste rM o le cul eCo un t = 5

ClusterMaximumAllowed = 0. 0

C l u s t e r I s C v i m = No

Po tMo del = LJ12 6 .pm

M ol a rF r a ct = 1 . 0

ChemPotMethod = none

NTest = 10

C u t o f f = 5 . 0

E p s i l o n = 1 . 0 E 10

18

9. Minor changes with version release 4.0

Pressure calculation with Monte Carlo simulations

MC simulations require potential energy calculations for the translation and rotation

acceptance criteria. Thus, force calculations are not necessary in contrast to MD. How-

ever, these intense force computations must be invoked when pressure is sampled with

MC. Therefore, the keyword OptPressure = ’#yes/no’ under the ensemble section in

the *.par ﬁle was set to logicals #yes or #no when pressure sampling was turned on or

oﬀ. Due to structural changes and optimizations in the MC code, the force computations

are done most eﬃciently now. Thus, such as with MD, pressure is throughout sampled

with MC up from this version release and the keyword OptPressure = ’#yes/no’ was

removed.

Helmholtz energy A00 calculation with NV T and N V E ensemble simulations

Residual Helmholtz energy derivatives Ar

mn are determined on the ﬂy with the Lustig

formalism [14] when applying NV T or NV E ensemble simulations in ms2 [15]. Addi-

tionally, for these ensemble types the residual Helmholtz energy Ar

00 is determined by

Ar

00 =−Ar

01 +

n

X

i=1

xiµr

i,(1)

if the chemical potential is set to ChemPotMethod 6=None in the *.par ﬁle. It was

implemented for pure ﬂuids and mixtures. Results for Ar

00 and its statistical uncertainty

are given in the *.res ﬁle.

19

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