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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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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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