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Monte Carlo methods in Materials Studio
Reinier L.C. Akkermansa, Neil A. Spenleya & Struan H. Robertsona
a
Accelrys Ltd, 334 Cambridge Science Park, Cambridge, CB4 0WN, UK
Published online: 14 Oct 2013.
To cite this article: Reinier L.C. Akkermans, Neil A. Spenley & Struan H. Robertson (2013) Monte Carlo methods in Materials
Studio, Molecular Simulation, 39:14-15, 1153-1164, DOI: 10.1080/08927022.2013.843775
To link to this article: http://dx.doi.org/10.1080/08927022.2013.843775
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Molecular Simulation, 2013
Vol. 39, Nos. 14 – 15, 1153–1164, http://dx.doi.org/10.1080/08927022.2013.843775
MONTE CARLO CODES, TOOLS AND ALGORITHMS
Monte Carlo methods in Materials Studio
Reinier L.C. Akkermans*, Neil A. Spenley and Struan H. Robertson
Accelrys Ltd, 334 Cambridge Science Park, Cambridge CB4 0WN, UK
Downloaded by [Moskow State Univ Bibliote] at 12:26 14 November 2013
(Received 20 March 2013; final version received 18 June 2013)
We survey the use of the Monte Carlo method within the Materials Studio application, which integrates a large number of
modules for molecular simulation. Several of these modules work by generating configurations of a system at random,
which can then be used to calculate averages of interest – for instance, interaction energies of contacting pairs of molecules
(Blends module) and properties of a flexible polymer chain (Conformers). A different technique is used to sample an
appropriate physical distribution (which in practice is that for the canonical ensemble) using the Metropolis or
configurational bias method. This is done by the Sorption module (which calculates the thermodynamic properties of small
molecules in a matrix) and Amorphous Cell (which constructs periodic simulation cells). Lastly, certain other modules use
simulated annealing and related methods to optimise a function, with application to crystal structure prediction from
molecular structure (Polymorph Predictor), to crystal structure prediction from X-ray powder diffraction data (Powder
Solve) and to find preferential sites for adsorption (Adsorption Locator).
Keywords: Monte Carlo; Materials Studio; simulated annealing; Sorption; morphology
1. Introduction
It is hard to exaggerate the impact that Monte Carlo (MC)
methods have made in the arena of atomic and molecular
modelling. An exhaustive survey would run into many
volumes. One of the principal problems with such a survey
is to identify the scope of MC methods, as any method that
uses computer-generated (i.e. pseudo) random numbers
tends to be referred to as an MC method. In this study, we
vastly reduce the scope of the methods surveyed to those
that are currently implemented in the Materials Studio
application.[1]
The Materials Studio application is an integrated
molecular modelling environment. In addition to offering
users a number of tools with which to build and display
atomistic models, Materials Studio presents a number of
modules that perform simulations on a variety of length
and timescales. At the shorter end of the length and
timescale spectrum, modules are available which calculate
electronic properties based on well-known quantum
mechanical approaches. At larger length and timescales,
there are a number of modules that simulate and exploit
the statistics associated with molecular configuration.
These are divided into two groups: those that generate
states of a system evolving in time by solving Newton’s
laws and those that generate new states at random. It is the
latter that are the focus of this study.
*Corresponding author. Email: reinier.akkermans@accelrys.com
q 2013 Taylor & Francis
The methods surveyed fall very roughly into the
following areas:
(1) General configuration sampling as employed in
the Blends and Conformers applications.
(2) Traditional Metropolis and biased sampling, the
basis of the Sorption and Amorphous Cell
applications.
(3) MC as an optimisation device, deployed in the
Polymorph Predictor, Powder Solve and Adsorption Locator applications.
In the following, a brief discussion of the theory
behind each of these general approaches is given, followed
by a more detailed discussion of the applications that use
these approaches within Materials Studio.
2.
Theory
2.1 MC as a configurational sampling method
Often considerable insight into the characteristics of a
system can be obtained by simply exploring the phase space
spanned by the coordinates defining the system. A simple
example is the conformer search algorithm in which a set of
N torsional angles is specified, thus defining an Ndimensional phase space, each point of which maps to a
different configuration of the molecule. This space can be
explored in a number of ways, the simplest being a
1154
R.L.C. Akkermans et al.
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systematic scan of all the angles. However, this can be
expensive because if each angle range is sampled on m
points this will generate m N configurations, a potentially
huge number of configurations which would require further
analysis to extract quantities of interest. Of more interest
are the properties of typical conformations, and the
generation of these configurations can often be effected
more economically by simply generating sets of random
angles and using these to generate structures.
With G a point in configuration space, the average of
property A can be written as an integral over the same space,
ð
kAl ¼ dG AðGÞrðGÞ;
ð1Þ
where r ðGÞ is the probability density of the configuration G,
normalised to unity. Since G comprises many variables, this
integral is highly dimensional and expensive to evaluate by
quadrature. As noted above, evaluating an N-dimensional
integral using regular sampling requires M ¼ m N points.
With an order-k integration method, the error in the integral
would be proportional to M 2k=N.
weight, r ðG Þ ¼ 0. Straightforward MC sampling is then
very inefficient since the majority of random configurations will have negligible weight, and not contribute to
the average. It would be much more efficient if it were
possible to draw independent configurations straight from
the distribution r ðGÞ (akin to the Box– Muller transform
for normally distributed random variables), but this is
difficult in general. Fortunately, non-uniform sampling
can be realised straightforwardly using Markov chains at
the expense of introducing correlation between the
samples. This is discussed in the next section.
2.2 MC as importance sampling method
The MC method used as importance sampling aims to
sample an ensemble with a given density r ðG Þ . This MC
method is based on a sequence of correlated samples. The
sequence is defined by the transition probability T ðG; G0Þ to
change a state G into another state G0 . Defining the density
at step n as rðG; nÞ, it follows that
X
r G; n þ 1 2 r G; n ¼
T G0 ; G r G0 ; n
0G
Instead of integrating over a regular array of points, the
MC method integrates over a random sampling of points,[2]
kAl <P
AðGi ÞrðGi Þ ;
P rðGj Þ
M
i¼1
M
j¼1
ð2Þ
where G1; .. . ; GM represent M random configurations of the
system generated by a computer. The error in such estimate
is proportional to M 21=2; thus, the MC method becomes
more efficient than an order-k algorithm if N . 2k.
This approach works well if the density rðGÞ is
uniform. For instance for an ideal polymer chain, all
configurations are equally probable, and the average radius
of gyration of such a chain can be obtained simply by
drawing random torsion angles and averaging the radius of
gyration of the resulting conformations. The Conformers
module in Materials Studio described in Section 3.1 offers
the random sampling method of torsion space as well as a
systematic grid scan.
In general, each random configuration must be weighted
by its probability r ðGÞ . The Blends module (Section 3.2) uses
such an approach. Here, configurations of molecular pairs
are generated uniformly, subject to a contact constraint. The
average energy of interaction at a temperature of interest is
then obtained by weighting each configuration by its
probability r at that temperature. The advantage of this
approach is that one sampling is sufficient to obtain the
average over an entire temperature range. This allows
prediction of a phase diagram from a single experiment.
The reweighting becomes problematic as particle
density increases because of the likelihood of atomic
overlap in a random distribution of molecules. When
atoms overlap the energy is infinite, leading to zero
2 T G; G0 r G; n Þ:
ð3Þ
When sampling equilibrium densities, the step
dependency vanishes, rðG; nÞ ¼ rðGÞ, such that
X
0¼
T G0 ; G r G0 2 T G; G0 rðGÞ :
ð4Þ
0G
This condition is satisfied if the transfer probabilities
TðG; G0 Þ obey detailed balance,
TðG; G0 Þ
rðG0 Þ
:
¼
rðGÞ
T G0 ; G
ð5Þ
As first put forward by Metropolis et al. [3], choosing
rðG0 Þ
0
TðG; G Þ ¼ min 1;
rðGÞ
ð6Þ
satisfies detailed balance. Consequently, steps that
increase the density (i.e. those that are more important to
the ensemble, r ðG0 Þ . r ðGÞ ) are always accepted, whereas
steps that decrease the density (steps to a less important
state) are accepted with a lower probability.
The probability density function of most interest in
molecular simulation is that corresponding to the
canonical ensemble,
rð
Þ¼Ð
expð2bE ðGÞÞ
G
0
dG0 expð2bEðG
;
ð7Þ ÞÞ
where EðGÞ is the potential energy of system in state G and
b ¼ 1=ðkBTÞ, with kB the Boltzmann constant and T the
Molecular Simulation
temperature. Kinetic energy is not included in this definition,
since integrating momentum space can be carried out
analytically and sampling is not required. The acceptance
probability follows from Equation (6) as follows:
(
0
TðG; G Þ ¼
expð2bðE 0 2 EÞÞ;
E0 . E;
1;
E0 # E;
ð8Þ
where E0 ¼ EðG0 Þ.
Changing a state into another is usually implemented as
a two-stage process. First, a trial step is proposed, which is
then accepted or rejected. Let vðG; G0 Þ be the attempt
probability and aðG; G0 Þ be the acceptance probability, then
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TðG; G0 Þ ¼ vðG; G0 ÞaðG; G0 Þ:
ð9Þ
In the traditional MC method, trials are made with equal
probability in either direction, v ðG; G0 Þ ¼ vðG0 ; G Þ. For
example, as many attempts are made to change the x
coordinate of an atom from 5 to 5.5 Å, as moving it from 5.5
to 5 Å. The acceptance probability thus reads a ðG; G0 Þ ¼
minð1; rðG0Þ=rð GÞÞ . This approach is implemented in the
Sorption module in Materials Studio as the Metropolis MC
method (Section 3.3).
It is sometimes useful to bias the attempts to achieve a
higher acceptance rate. In that case, v ðG; G0 Þ – v ðG0 ; G Þ.
Typically, more attempts are made to step to configurations of higher density (lower energy) to avoid
unnecessary evaluations of high energy. By tracking the
bias v, the original ensemble can still be obtained by
accepting with the modified probability a ðG; G0 Þ ¼
minð1; ðvðG0 ; G Þ=vðG; G0 ÞÞðr ðG0 Þ=rðGÞÞÞ. This approach is
implemented as the configurational bias MC method in the
Sorption module of Materials Studio (Section 3.3.1). A
similar biased insertion method (but without evolution) is
used in Amorphous Cell (Section 3.5).
2.3 MC as an optimisation procedure
The MC method can also be used as a heuristic
optimisation procedure in which the aim is to find (an
approximation to) the minimum of the energy of the
system. The Metropolis algorithm is used to generate
successive points in the configuration space. The
‘temperature’ is usually without physical significance; it
is simply a parameter which controls the calculation.
It would be possible simply to record the lowest energy
state visited so far, but in practice the search is more
efficient if a more elaborate procedure is used. A commonly
used method is simulated annealing.[4] The temperature
parameter is initially high; this means that the acceptance
probability for new states is high. The configuration space
can be traversed rapidly, so the whole space can be
explored. The temperature is then lowered progressively as
the computation proceeds. This causes the probability
1155
density of states visited to be concentrated on the lower
energy regions, so these are now sampled thoroughly.
The simulated annealing method is used in the
modules Adsorption Locator (Section 3.4) and Polymorph
Predictor (Section 3.7) of Materials Studio. The technique
is quite generic, and can be used to obtain the minimum of
any function. It is often used for functions that have
multiple minima and/or for which the phase space is
simply too large to sample systematically. In the Powder
Solve module (Section 3.6) of Materials Studio, this is
exploited by defining a figure of merit; the negative of this
corresponds to the energy.
3.
Materials Studio modules
3.1 Conformers
The Conformers module in Materials Studio can be used to
explore the conformer space of molecules, which exhibit
internal rotation about a bond, in particular polymers. The
preparation for a conformer search requires the torsion
degrees of freedom that will be the basis of the search to be
specified. These can be defined in the visualiser using
Torsion Twister objects as illustrated in Figure 1. Conformers offer automated tools to find rotatable bonds in a
polymer, which can be modified manually when needed.
The module offers three algorithms with which to
explore configuration space: systematic grid scan, random
sampling and Boltzmann jump. The systematic grid scan
explores the configuration space on a regular grid. As
indicated in Section 2.1, generating configurations on a
regular grid can be expensive, mainly because of the large
number of configurations generated. In order to reduce the
number, various parameters can be set to restrict the search.
Figure 1. (Colour online) Polymer model with torsion twisters
defining, and monitoring, the torsional degrees of freedom.
Torsion twisters are visualised as a wireframe labelled by the
current torsion angle. Torsion twisters are used to specify torsion
degrees of freedom in the modules Conformers, Sorption,
Amorphous Cell and Polymorph Predictor.
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1156
R.L.C. Akkermans et al.
For example, certain configurations can be eliminated on
the grounds that there is significant overlap of atomic radii
– such configurations are likely to have a very high energy
and are, therefore, very unlikely to occur. The radii used
are specified as a percentage of van der Waals radii.
The random sampling search generates configurations
by altering torsion angles by a random amount. The
amount by which a torsion angle is altered, du, is generated
from a uniform distribution of width w, i.e. from the range
of 2w=2 # du # þw=2, the maximum value of w being
1808. The number of configurations generated is a
specified parameter.
The Boltzmann jump search also generates configurations through random sampling, the main difference
being that it is the Boltzmann distribution from which
samples are drawn. The sampling is generated as a Markov
chain using the Metropolis algorithm as described in
Section 2.2, and as a consequence, one of the input
parameters for this sampling is temperature. The specified
temperature will determine the width of the distribution –
the higher the temperature the broader the width.
3.1.1
Applications of Conformers
A polysaccharide molecule (pullulan) was studied by Liu
et al. [5]. They used molecular mechanics methods,
including a conformer search based on the torsion angles
of the glycosidic linkage, to construct rotational isomeric
state models. Peters [6] examined tricyclic terpanes, and
did a conformer search calculation to identify low energy
structures, which were then optimised using a semiempirical method. Surin et al. [7] used conformer search to
find the stable configuration of a photoluminescent
fluorene-based monomer, followed by energy minimisation calculations on a stack of oligomers.
3.2 Blends
The Blends module in Materials Studio is designed to
estimate the miscibility behaviour of binary mixtures, such
as solvent– solvent, polymer– solvent and polymer–
polymer mixtures. Blends predicts the thermodynamics
of mixing directly from the chemical structures of the two
components and, therefore, requires only their molecular
structures and a force field as input.
Blends provides three tasks to the user for calculating
binding energy, cluster size and mixing energy. The output
consists of a study table (the spreadsheet document in
Materials Studio) containing the average energies of
molecular pairs at the specified temperature, the cluster
sizes and mixing energies. The study table can be further
analysed to obtain mixing energies at other temperatures,
as well as abstract thermodynamic data such as free energy
isotherms, the binodal and spinodal, and critical point,
based on an off-lattice generalisation of the Flory–
Huggins theory.[8,9]
The main calculation in Blends consists of estimating
the average energy between two molecules in contact at a
given temperature. Two molecules are said to be in contact
if their van der Waals surfaces touch (are tangent). The
sampling is a two-stage process. First, a large number of
random pair configurations are drawn. In each configuration, two atoms are required to be in contact, but
otherwise no temperature is imposed on the MC sampling.
In this ensemble, the energy distribution (or ‘density of
states’) is evaluated, based on the user-specified force
field. The histogram is then reweighted to obtain the
distribution at a given temperature, from which finally the
average energies are obtained.
The binding energy calculation is repeated for all
unique combinations of base and screen molecules. A
cluster size calculation for all four combinations is also
made to estimate the average number of interactions in the
condensed state. The cluster size is analogous to the
coordination number in lattice theories, but can vary
depending on the combination of species. A cluster
consists of a molecule of one species, surrounded by as
many molecules of another species as will fit within the
contact condition. This means that the van der Waals
surface of the central molecule is in contact with the
surfaces of all the coordinating molecules, approximating
the first solvation shell of the molecule in solution. From
the binding energies and cluster sizes, the mixing energy
can be obtained, which leads to the Flory– Huggins
interaction parameter, x.
By retaining the density of state data, the calculation of
the x-parameters can be repeated at different temperatures
without additional MC sampling. This temperaturedependent interaction parameter can be used in the
Flory– Huggins theory of miscibility to predict non-ideal
phase diagrams.
3.2.1 Applications of Blends
Blends is often used to screen a number of molecules for
binding capacity on one or more base materials. Another
application of Blends is aimed at calculating x-parameters
to be used in the parameterisation of mesoscale models
such as dissipative particle dynamics and dynamic density
functional theory.
An example of the latter is found in the mesoscale
simulation studies by Wu et al. [10] on predicting the
morphology of perfluorosulphonic acid fuel cell membranes. The input parameters in this case were determined
from Blends calculations in conjunction with the COMPASS force field. A similar workflow was used by Lin
et al. [11] to generate input parameters for a mesoscale
simulation of unilamellar vesicles.
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Molecular Simulation
Pajula et al. [12] exploited Blends as a screening tool
to find potential stabilisers needed to produce a stable
amorphous binary mixture. They applied the module to a
test set of 39 drug molecules, and found a good correlation
between the predicted Flory– Huggins parameter and the
crystallisation tendency as measured experimentally with
differential scanning calorimetry and polarised light
microscopy. Ren et al. [13] have used Blends to
investigate the solubility of a series of cyclohexanone
formaldehyde resins in a variety of solvents (tetrachloromethane, alcohol, toluene and ethyl ether).
Modifications to the Blends algorithm to include
entropy of mixing and other correlation effects were
discussed by Akkermans [9]. Here, it was shown that the
inclusion of excess entropy term in the Flory– Huggins
parameter can provide a better description of the critical
point of the nitrobenzene– hexane mixture, as well as
predict more accurately the interaction parameters for
polymer blends.
3.3 Sorption
The Sorption module in Materials Studio allows
simulation of a framework– sorbate system using MC
sampling. The input consists of a framework structure,
typically a unit cell (or super cell) of a microporous
crystal, and the structures of one or more sorbate
molecules. The framework is always considered rigid,
thus providing a static external field to the sorbate
molecules. Sorbate molecules can also be treated as rigid
bodies, in which case the degrees of freedom of the system
are specified by the centre-of-mass position (Ri) and
orientation (Vi) of the molecules, G ¼ ðR1; V1; . . . ; R N ;
V N Þ. It is also possible to specify intramolecular torsional
degrees of freedom, as described in more detail below.
Bond lengths and valence angles are always held fixed.
Two sampling methods are supported: the Metropolis
and the configurational bias MC methods. Each method can
be used to sample two ensembles: the Fixed Loading task
samples the canonical ensemble, whereas the Fixed
Pressure task samples the grand canonical ensemble. The
former task is described by the loading of each component
N ¼ ðN1; . . . ; NMÞ, whereas the latter requires the
fugacities f ¼ ðf 1; .. . ; f MÞ of each component. The
fugacity is related to the chemical potential m by
f ¼ f + expðbðm 2 m +ÞÞ, where + denotes a reference state,
for instance the ideal gas. The fugacity reduces to the
partial pressure for an ideal gas.
The density of the canonical ensemble is given by
Equation (7). All terms that are independent of G, such as
the energy of the fixed framework and the intramolecular
energy of the sorbate molecules (when treated as rigid),
may be excluded and incorporated in a constant
multiplication factor.
1157
In the grand canonical ensemble, the system has
additional degrees of freedom N ¼ N
ð 1; .. . ; NM Þ, specifying the loading of each component. The density is
rðG; NÞ ¼ CFðNÞexpð2bEðGÞÞ;
ð10Þ
where E is the potential energy and FðNÞ is the product of
FðN Þ ¼
i
ðbf i VÞ Ni
Ni!
exp 2bN m
i
;
ð11Þ
intra;i
where f i is the fugacity of component i and V is the
constant volume of the system. The term mintra;i is the
intramolecular chemical potential. The latter is defined as
2kBT lnkexpð 2buintra;i Þ l, where uintra;i is the intramolecular potential of a molecule of component i, obtained at
constant temperature in vacuum. Without torsional degrees
of freedom this energy is constant, in which case it can be
subtracted from the total energy E in Equation (10),
leaving the intermolecular energy.
Substituting the ensemble density in the acceptance
rule, Equation (6) gives
a G; N; G 0; N 0 ¼ min 1;
F
expð2bðE 0 2 EÞÞ :
F0
ð12Þ
The Fixed Loading task supports the following step
types: Translation, Rotation, Regrowth and Conformer. All
moves are applied to a random sorbate molecule of a
random component in the system. The relative probability
of each move can be specified in the input.
The Translation move corresponds to moving the
centre-of-mass of the selected sorbate molecule over a
distance dr along axis A. The distance is drawn from a
uniform distribution between 0 and Dt. Axis A is the vector
from a random point on a sphere to the origin of the sphere.
Care is taken to achieve a uniform distribution on the
sphere surface. The maximum displacement Dt can be
specified in the input, or selected automatically by the
algorithm to achieve an acceptance rate of 50%.
The Rotation step rotates the molecule about the
centre-of-mass, by an angle du about axis A. The angle is
þ Dr .
drawn from a uniform distribution between 2Dr and
Axis A is again the vector from a random point on a sphere
to the origin of the sphere. As for Translation, the
maximum displacement can be fixed in the input, or
optimised by the algorithm to achieve an acceptance rate
of 50%.
In the Regrowth move, a sorbate molecule is removed
from the system, and reintroduced with a random position
and orientation. Such moves can be used to transfer
sorbate molecules between pores in the framework.
The Conformer move is used when the input of a
sorbate consists of multiple conformations for a collection
of trans and gauche conformations. The move will attempt
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1158
R.L.C. Akkermans et al.
to replace the current conformation by another randomly
chosen from the trajectory.
The Fixed Pressure task supports, in addition, the
Exchange step type, comprising creation of a new sorbate
and deletion of an existing molecule. Care is taken to
always attempt creation and deletion with equal
probability, even if the system is empty. If the system is
empty, deletion steps are simply rejected, adding the
empty state as an additional sample to the ensemble
average. This can be important when calculating the
adsorption isotherm at low fugacities.
The output of Sorption consists of reported values for
the average loading, isosteric heat per component, as well
as average total energy of the system. The distribution of
the insertion energy can be obtained as a histogram and as
a three-dimensional (3D) field. A density field may also be
obtained, as illustrated in Figure 2.
3.3.1
Configuration bias MC
In the configurational bias MC method of Sorption, the
attempt rates v ðG; G0Þ are constructed as follows. Instead
of generating one trial configuration G0 , a total of K trial
configurations G0 1 ; . . . ; G0 K are generated. Each configuration G0 k is given a weight wk . One configuration is then
selected with a probability proportional to its weight. The
attempt rate of a configuration n is thus given by
v G; G0 n ¼
1=K
wn
PK
k¼1
wk
:
ð13Þ
Consequently, configurations with higher than average
weight are more likely to be attempted, that is v . 1. Of
course, it is still possible, but less likely, to select a trial
configuration with a lower than average weight, v , 1.
To determine the acceptance probability, the attempt
rate of the reverse step, v ðG0 n ; GÞ , is also required. This is
the attempt rate for transforming a configuration G0 n into
the known configuration G. It is calculated as before, with
the selected configuration G being given; hence, only K 2
1 trial configurations have to be generated to determine
this attempt rate.
The above procedure applies to one degree of freedom.
The configurational bias method is typically applied to
chain molecules with multiple rotational degrees of
freedom. In this case, the procedure is simply repeated
for each degree of freedom. For each degree of freedom, K
states are generated and their weights are determined. One
state is drawn from the weighted distribution, and the
attempt rate for this selection is calculated as above. After
all degrees of freedom have been assigned a value, the
resulting configuration is the candidate for the acceptance
step. The total attempt rate of this configuration is the
product of the attempt rates for each degree of freedom.
The attempt rate of the reverse step is calculated in the
same way.
In Materials Studio, rotational degrees of freedom are
defined, and visualised, using torsion monitors, as
illustrated in Figure 1. Torsion monitors can be defined
manually on the model, or automatically, taking into
account constraints due to rings, double bonds or terminal
hydrogens. The Sorption module processes the torsion
monitors on the input structure, building up a graph of
atom groups, or segments, mutually connected by
rotatable bonds. The segments are treated as rigid bodies,
equivalent to Motion Groups in Materials Studio. As such,
the state of a chain molecule is completely defined by
the centre-of-mass position and orientation of the first
(head) segment, and the values of the torsions,
G ¼R;ð V; f1; :::; fNÞ . By assigning fewer, or more,
torsions, the size of the segments can be controlled.
To define the weights used in the selection of trial
configurations, Sorption calculates the segment energy Vi,
defined as the sum of the (non-electrostatic) intermolecular energy between segment i and the framework and all
other sorbate molecules,
Vi ¼ VFi þ VSi:
ð14Þ
The electrostatic energy is excluded in the bias to allow
for charged segments, and only used in the acceptance step.
The first segment of a chain is placed by drawing Kh
positions and orientations and by evaluating the weight
wH ¼ exp ð2bV 1 Þ
Figure 2. (Colour online) Density of CO2 in the zeolite MFI at
10 kPa and 298 K as output by Sorption. Pores are visualised
using a solvent-accessible Connolly surface.
ð15Þ
of each of them. The next segments are defined through the
associated torsions. To position segment t, Kt torsion
Molecular Simulation
values are drawn, and the weights
wT;t ¼ expð2bðV tþ1 þ V tþ2 þ uintra ÞÞ
ð16Þ
are evaluated for each resulting configuration. In this case,
the intermolecular energy of the two segments succeeding
segment t is included. If a torsion has only one successive
segment, the term Vtþ2 is excluded.
The step types in the configurational bias method are
similar to those described above with Translation and
Rotation applying to the head segment. In addition, a Twist
step type is available, which will change the angle of a
randomly chosen torsion by an amount df, chosen
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uniformly between 2Dr and þDr.
3.3.2
1159
Sorption has also been applied to metal– organic
frameworks, focusing typically on adsorption and storage
of small molecules, such as carbon dioxide and methane.
Greathouse et al. [19] also considered more complicated
organic molecules relevant to chemical sensing and
detection, such as aromatics, explosives and certain
chemical warfare agents. They calculated the isosteric
heats of these materials on zinc, chromium and copper
organic frameworks, and found that the chromiumcontaining framework CrMIL-53lp had the highest
adsorption energy for all analytes, suggesting that this
material may be suitable for detection of low-level
organics. Their results demonstrated that Sorption can be
used as an efficient first step in the screening of metal–
organic frameworks for detection of large molecules.
Applications of Sorption
Sorption has been applied to a variety of materials, ranging
from traditional zeolites used as molecular sieves for
separation purpose to metal– organic frameworks and
polymers for storage of gases. The sorbate species are
typically small molecules such as hydrogen, methane,
carbon dioxide and water. Adsorbent structures can be
constructed using the visualiser tools in Materials Studio,
or imported from a structure database; an extensive library
of zeolite structures is available with the product.
Wood et al. studied microporous organic polymer
networks for storage of hydrogen [14] and methane.[15]
They calculated the adsorption isotherms and isosteric
heats in hypercrosslinked poly( p-dichloroxylene) with
Sorption using the Universal force field to complement
their solid-state NMR, gas sorption measurements and
pycnometry. The isosteric heat for hydrogen was found to
be in the range of 6 – 7.5 kJ/mol (at 77.3 and 87.2 K), in
good agreement with experiments. Hydrogen storage in
polymers (in this case lithium-doped conjugated microporous polymers) was also subject of a simulation study by
Li et al. [16], who found an isosteric heat of 8.1 kJ/mol,
and hydrogen adsorption up to 6.1 wt% at 77 K under an
ambient pressure of 1 bar.
Yang et al. [17] used Sorption to simulate the
adsorption of ethanol and water vapour in a silicalite
crystal. They calculated the adsorption of pure ethanol and
water on silicalite at 303 K as a function of the sorbate
activity and found good agreement with experimental data
with a reasonable fit on a Langmuir model. They also
studied the permeation of ethanol– water mixtures through
silicalite membrane, and investigated the influence of
ethanol concentration in the feed on the separation factor.
Silica in its mesoporous form was studied by Huiyong
et al. [18]. These novel nanocomposites combine
microporous silica with mesopore channels and can be
used as highly functional adsorbents, catalyst supports and
nanoreactors. The authors used Sorption to study the
adsorption of toluene on ZSM-5 – MCM-41.
3.4 Adsorption Locator
Adsorption Locator simulates a substrate loaded with an
adsorbate or an adsorbate mixture of a fixed composition.
Adsorption Locator is designed for the study of individual
systems to find low energy adsorption sites on both
periodic and non-periodic substrates or to investigate the
preferential adsorption of mixtures of adsorbate components, for example. The input to Adsorption Locator
consists of a substrate structure, for example a metal
surface, and the structure of one or more adsorbate
molecules.
The simulation technique is similar to Sorption, in that
it performs a MC simulation of a substrate– adsorbate
system. However, in Adsorption Locator, the temperature
is modified externally to simulate the annealing of the
system. The temperature is first increased steeply to a high
value, then it is slowly decreased to the final value,
allowing the system to settle to a state of minimal energy.
The process can be repeated in a number of cycles to allow
the system to explore states of still lower energies.
The output of Adsorption Locator consists of a study
table collecting all lowest energy configurations found in
the sampling. These can be processed further, for instance
as input to a quantum mechanical optimisation. The
energy distribution can be calculated as part of an
adsorption calculation. An example output of adsorption
of quinoxaline on the (110) surface of Fe2O3 is shown in
Figure 3.
3.4.1 Applications of Adsorption Locator
Adsorption Locator is a relatively new module in
Materials Studio, and has already been used in a variety
of cases to identify binding sites and study their energetics,
for instance, of organic molecules on metal surfaces and
nanoparticles.
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1160
R.L.C. Akkermans et al.
Figure 3. (Colour online) Adsorption of quinoxaline on the
(110) surface of Fe2O3.
Khaled has used Adsorption Locator to determine the
adsorption and corrosion inhibition behaviour of selected
thiosemicarbazone molecules on a Ni(111) substrate.[20]
Low energy adsorption sites were located providing a way
to rank the thiosemicarbazone molecules on their
inhibition efficiencies using the adsorption and binding
energies. In another study,[21] Khaled used the module to
study thiourea derivatives on iron (110).
A similar study was conducted by Musa et al. [22] to
identify the efficiency of phthalazine derivatives to inhibit
corrosion of mild steel. The low energy adsorption sites, as
identified by Adsorption Locator, were subject to a
quantum chemical calculation, identifying phthalazone as
the best inhibitor on the Fe2O3(110) surface.
Liang et al. [23] used Adsorption Locator in their work
of biomolecule-mediated ZnO formation. They studied the
adsorption of peptides on the (0001) and ð101̄0 Þ planes of
ZnO to investigate their ability to mediate the relatively
growth rate of those surfaces. They calculated the
adsorption modes and energies of G-12 and GT-16
peptides on the ZnO planes and identified the adsorbing
moieties.
3.5 Amorphous Cell
The Amorphous Cell module in Materials Studio allows
construction of 3D periodic disordered molecular and
polymer structures. The technique used has similarities to
the configurational bias MC method of Sorption as
described in Section 3.3.1. Like Sorption, a polymer is
divided into segments by assigning torsion twisters to
rotatable bonds, as illustrated in Figure 1. The latter is
largely automated using the same assignment as
implemented in the Conformer module (Section 3.1).
Rotational degrees of freedom can be defined in both
backbone and side chains. Segments themselves are
treated as rigid. A system of N chains is then constructed
segment by segment, starting from an arbitrary end of each
chain.[24]
The segments are positioned using the biasing technique
as described above by drawing a given number of Kh
positions and orientations of the head segment, evaluating
their (Boltzmann) weight and drawing one from the
weighted distribution. Once all first segments have been
positioned, the algorithm proceeds by positioning the
second segment of all molecules, if present. To this end, a
given number of Kt torsion angles are drawn from the
associated Boltzmann distribution, and the associated
weights are evaluated. One torsion angle is selected, thus
fixing the position of the second segment. This procedure is
repeated until all segments have been positioned. Typically
after the raw construction is completed, a geometry
optimisation is carried out to eliminate any bad contacts
from the structure. The structure is then ready to be
equilibrated using molecular dynamics, e.g. with the Forcite
module in Materials Studio.
During the construction process, a number of
constraints can be imposed. Configurations that do not
obey the constraints are removed from the MC draw. An
example is the ring spearing constraint. Trial configurations that lead to ‘spearing’ (i.e. to a bond passing through
the centre of a ring or to a pair of interlinked rings) are
rejected. This is vital when building ring-containing
polymers, as well as when packing into ring-containing
host materials, such as carbon nanotubes. Other constraints
include close contact penalties and isosurfaces.
In addition to the Construction task, Amorphous Cell
offers a Packing task and a Confined Layer task. The
Packing task can be used to pack into existing structures,
for instance to solvate a protein or drug molecule. It is
possible to confine the packing volume using isosurfaces,
such as van der Waals, solvent accessible and Connolly
surfaces. This can be powerful when combined with the
Field Segregation tools in Materials Studio, making it
possible for example to pack into the interior of a
nanotube. In all cases, the algorithm is based on the
configurational bias MC method as described earlier.
The Confined Layer task is designed to build structures
that are bound in one direction, i.e. without molecules
crossing either side of the box. This can be useful to fold
polymers in a box. Such confined cells are typically used
to build layered composites using the Layer Builder in
Materials Studio. In turn, such layered structures can be
used to run a confined shear simulation, in which the top
and bottom layers are displaced in opposite direction
applying a shear stress to the middle layer. The Confined
Layer construction algorithm works as described above
with an external potential used to confine the material.
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Molecular Simulation
1161
3.5.1 Applications of Amorphous Cell
Amorphous Cell is typically used as the start of a
modelling study to generate the input structures, to be
further simulated using molecular dynamics.
Liu et al. [25] studied the spreading of water
nanodroplets on amorphous cellulose and polypropylene
surfaces. They used the Confined Layer option in
Amorphous Cell to build initial structures of the polymers,
and further compacted and optimised those using a layered
structure with xenon crystals. This procedure gave a degree
of molecular surface roughness that corresponds well to
experimental values. In addition, calculated properties such
as density, cohesive energy density, coefficient of thermal
expansion and the surface energy agreed with experimental
values. It was found that a water nanodroplet spreads on the
amorphous cellulose surfaces, but there was no significant
change in the dimension of the droplet on the polypropylene
surface; the resulting water contact angles on polypropylene
and amorphous cellulose surfaces were determined to be
1068 and 338, respectively.
Peng et al. [26] used Amorphous Cell to prepare
nanocomposite membranes by incorporating chitosanwrapped multiwalled carbon nanotubes into poly(vinyl
alcohol). They used these structures to explore the intrinsic
correlation between pervaporation performance and free
volume characteristics. They found that including 1%
nanotube material gave a threefold increase in the
permeation flux. They compared with positron annihilation lifetime spectroscopy.
Lin et al. [27] investigated moisture diffusion in epoxy
resins. They constructed epoxy-water boxes using Amorphous Cell, and calculated the mean-square displacement of
water in those structures as a function of temperature. They
compared the desorption activation energies with those
abstracted from moisture uptake experiments.
Pan et al. [28] studied the effects of graphite particles
on the diffusion behaviour of benzene and cyclohexane in
poly(vinyl alcohol)-graphite hybrid membranes. Amorphous Cell construction was used to create the structures of
polymer and graphite particles with different amounts of
hydroxyl and carboxyl groups. Polymer chain mobility
was analysed by mean-square displacement and glass
transition temperature, and it was found that incorporation
of graphite particles into poly(vinyl alcohol) increased the
polymer chain mobility.
An example in which the Packing task was used is
found in the work of Hölck et al. [29]. They developed
procedures to construct 3D networked epoxy moulding
compounds using a cross-linking scheme.
addition to the diffraction pattern, are the atoms in the unit
cell along with the space group and the cell parameters.
(Materials Studio has other tools to allow these to be
estimated from the diffraction pattern.) A crystal structure
is defined by the positions of the atoms (or rigid groups of
atoms) in the unit cell. The diffraction pattern of a
candidate structure is calculated and compared with the
experimental data, and the space of possible structures is
searched for good matches.
The basic measure of similarity (figure of merit)
between a simulated and an experimental diffraction pattern
is the weighted profile R-factor Rwp, a weighted sum of
squares of the difference between the intensities of the two
patterns. In addition to this, it is possible to make use of
some chemical information. Under some circumstances, for
instance if the quality of the experimental data is poor, or if
the structure is of high symmetry and has few peaks, then
the Powder Solve procedure is likely to identify a number of
possible structures. In these cases, it is useful to be able to
eliminate structures that contain overlaps between atoms.
This is done by calculating an energetic figure of merit
Renergy which penalises close contacts between atoms. The
search is done using a combined figure of merit, which is
calculated as a linear combination of Rwp and Renergy.
The search can be done either by a simulated annealing
or by a parallel tempering procedure. Simulated annealing
uses a series of MC moves. Each move is an attempted
change of one degree of freedom of the structure (either a
translation or rotation of a group of atoms) after which the
new combined figure of merit is calculated. (The size of
the attempted move, i.e. the distance or the angle, is
regulated adaptively so that the acceptance probability
remains around 50%.) The move is accepted or rejected
according to the Metropolis criterion (with the figure of
merit playing the role of the energy). The temperature is
progressively lowered from a high to a low value.
Parallel tempering uses several different structures at
different temperatures, which evolve simultaneously using
the same MC procedure. However, there is an additional
move type which swaps a pair of structures between the
temperatures. The high temperature allows rapid movement
across the configuration space, while a low temperature
allows the vicinity of a local minimum to be explored.
In each case, the final output is a set of structures with
their associated figures of merit. These can be inspected
and further analysed. Figure 4 shows an example of a
predicted structure; also shown is the expected X-ray
powder diffraction result together with the experimental
data used as input to the calculation.
3.6 Powder Solve
Powder Solve can calculate the structure of a crystal from
its powder diffraction pattern. The inputs required, in
3.6.1 Applications of Powder Solve
Dinnebier et al. [30] synthesised a ferrocene-based
macrocycle compound, carried out X-ray powder diffrac-
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Figure 4. (Colour online) Predicted structure for (E)-2-(4,6-difluoroindan-1-ylidene)acetamide; comparison between experimental
(points) and predicted (line) X-ray powder diffraction data.
tion on it and analysed the data using simulated annealing
in Powder Solve. This generated a structure which they
then further optimised using density functional theory.
Neumann et al. [31] were able to identify the hitherto
unknown structure of phase III of solid methane. Neutron
diffraction was needed for this difficult material, and was
again analysed by Powder Solve using simulated
annealing. It had previously been thought that the crystal
had a tetragonal space group; they were able to show that
this is not the case (and found that the correct space group
is C m c a).
Corma et al. [32] examined a zeolite, ITQ-40, with an
unusually open structure. They used several approaches
(using electron diffraction, X-ray powder and X-ray single
crystal data) to determine the structure. Powder Solve was
able to provide a partial structure from the powder data.
Luneau et al. [33] synthesised various compounds
based on manganese, all with layered structures, most of
which had ferromagnetic behaviour. Again, Powder Solve
was used for structure solution.
Dova et al. [34] worked on an iron-based compound
with complex spin crossover behaviour. Powder diffraction data were obtained using synchrotron radiation and
were analysed using parallel tempering in Powder Solve,
as well as a rival technique (a genetic algorithm).
3.7 Polymorph Predictor
The Polymorph Predictor module of Materials Studio
attempts to predict the most favourable structures for a
molecular crystal, starting only from the molecule itself.
The basic assumption is that the structures that are most
likely to occur tend to be those of lowest energy – entropy
is neglected. Whilst the energy of a crystal can be readily
computed using force field methods, the entropy is much
more difficult to obtain, and is therefore ignored.
A workflow diagram is given in Figure 5. The main
part of the procedure is the Packing phase, in which the
space of possible crystal structures is searched for
candidates of low energy. A crystal structure is specified
by its crystal lattice parameters (cell lengths and angles)
and the position and orientation of each molecule inside
it. This yields a rugged energy landscape with many
local minima, so a simulated annealing process is used.
The user provides the structure of the molecule of
interest (or a structure containing multiple molecules),
together with a force field (any of the usual force fields
available in Materials Studio can be used). In addition,
the space groups of interest must be provided (usually a
small subset is required, not the full set of 230 space
groups).
A trial MC move consists of reorienting and moving
each molecule as a rigid body. Flexible molecules can also
be handled: the user can specify which torsion angles are
active, and these angles will also be changed during the
MC procedure. Then, the size of the lattice cell is adjusted
so that the molecules are just in contact but not
overlapping. The energy is calculated, and the move is
accepted or rejected according to the usual Metropolis
criterion. The simulated annealing regime involves an
initial heating phase, followed by cooling, and the whole
process is repeated for each space group.
The set of structures generated by the Packing
procedure can then be subjected to a Cluster analysis,
which reduces it by removing similar structures. The
criterion for similarity is based on the radial distribution
function; a similarity measure is calculated from the
difference between the radial distribution functions of the
two structures, and if this lies within a specified tolerance,
the structures are deemed to be similar, and the one of
highest energy is eliminated.
The candidate structures are usually optimised at this
point and again passed through the Cluster analysis.
Finally, they are ranked in order of energy. This is the
output from Polymorph Predictor.
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Molecular Simulation
1163
Figure 5. (Colour online) Polymorph Predictor workflow starting with a MC simulated annealing (MC-SA) step.
3.7.1 Applications of Polymorph Predictor
Neumann and Perrin [35] studied the crystal structure of
several small molecules (ethane, ethylene, acetylene,
methanol, acetic acid and urea). Candidate crystal packings
for these molecules were produced using Polymorph
Predictor, then optimised using energies calculated by a
combination of density functional theory and an empirical
correction for the van der Waals interaction.
Day et al. [36] examined a set of 50 small organic
molecules and compared predicted with known experimental crystal structures; they found that about half of the
known structures had an energy that was very close in
energy to the calculated minimum (and about a third of the
known crystal structures in fact were the calculated
minimum energy structure).
Cross et al. [37] did a combined computational and
experimental study of diflunisal, which is a fluorinated
aromatic carboxylic acid. They did structure predictions
using Polymorph Predictor (on five space groups) and used
the results to guide their choice of crystallisation solvents.
Leusen [38] did structure prediction of a crystal of two
different molecules, a cyclic phosphoric acid and an
enantiomer of ephedrine. This system is complicated by
both molecular flexibility and chirality.
Sarma and Desiraju [39] augmented Polymorph
Predictor results using experimental data. The computational procedure generated lists of low energy structures,
and then they reordered these using known structures of
similar molecules. This has the effect of supplementing the
enthalpic considerations on which Polymorph Predictor is
based, with kinetic information implicit in the experimental results.
4. Conclusions
Materials Studio contains a comprehensive set of modules
for molecular simulation, and many of these are based on
the MC method. This technique is exploited in a variety of
ways, ranging from traditional configuration sampling to
structure determination from experimental diffraction
data, through to simulated annealing. Where energy
calculations are required, all modules support a choice of
force fields, including COMPASS, pcff, cvff, Universal
and Dreiding; in addition, new force fields can be created
using the force field editing facilities, and used in place of
those supplied with the application.
As all modules share a common infrastructure, they
can be brought together to support complex workflows.
For instance, crystallisation, quantum and atomistic
modelling can be combined in a single simulation study.
These workflows can be scripted using the Perl-based
MaterialsScript API, which also allows access to the entire
data model. In addition, workflow protocols can be created
from Materials Studio components in the Pipeline Pilot
product, and combined with any other component from the
vast selection offered. Materials Studio has now been used
effectively over more than a decade for numerous different
applications across a range of physical and chemical
sciences and almost every industrial sector.
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