Prediction of Solubility for Large Molecules in Solvents by
Parallelized Molecular Simulation
1. Motivation
The prediction of the solubility of compounds in different solvents is an
important problem in many areas of chemistry. For example, volatile organic
compounds represent an important class of environmental contaminants. Knowledge
of their physico-chemical properties, such as the air-water distribution coefficient, the
octanol-water partition coefficient, and solubilities in water, is necessary for modeling
transport and distribution of these pollutants in the basic compartments of the
environment (water, air, soil), and for adopting rational remediation measures. A
related problem involves prediction of the solubilities of a range of solutes in
supercritical fluids such as carbon dioxide and water. These solubilities are required
in order to design equipment to extract them from other media such as soil by
dissolution in the solvent. A fundamental thermodynamic quantity in all these
solubility problems is either the solute excess chemical potential or the solute excess
chemical potential at infinite dilution related to the Henry's-law constant.
Although some macroscopic thermodynamic models exist for the prediction of
solubilities in bulk systems, their accuracy is generally poor [1]. Other macroscopic
approaches rely on empirical correlations, requiring a considerable amount of
experimental data for their implementation [2]. Experimental measurement is often
difficult and costly, as for example in situations involving organo-metallic
compounds. For these reasons, molecular-level simulation methods show considerable
promise as alternative predictive strategies.
2. State-of-the-Art
The key problem in the molecular simulation of solubility of organics in
solvents such as those considered here is that of efficiently calculating the excess
chemical potential of the solute, typically at very low concentrations. The limiting
case of infinite dilution allows the macroscopic-based Henry's law constant to be
calculated. The main method for calculating chemical potentials at the molecular
level involves the Widom test-particle-insertion method [3]. In the case of a
molecularly large solute molecule (as is the case here) inserted into a solvent fluid,
this method is very inefficient. A number of approaches has been devised to
circumvent this problem [4], although none are without difficulties in their
implementation and deficiencies in their accuracy. In the following we outline several
advanced simulation methods for calculating the chemical potential that are well
suited for parallelization.
The Kirkwood coupling parameter method [5] gradually turns on the
interaction between the extra particle and the system,  by changing the Kirkwood
coupling parameter  from 0 (no interaction between the extra particle and the
system) to 1 (the full interaction between the extra particle and the system). The
excess chemical potential is determined as an integral over of ensemble averages for
. The integral is evaluated numerically and it requires a series of simulations to
compute the ensemble averages for at different -values. The Kirkwood coupling
parameter method is useful for solutes formed by ring molecules e.g. naphthalene or
benzene.
The single-charging integral method [6] is a variant of the Kirkwood coupling
parameter method. The method relies on a number of separate simulations in which a
solute is slowly mutated from one form to another. Instead of a single coupling
parameter that varies from 0 to 1, the single-charging integral approach utilizes a
vector of coupling parameter, the elements of which correspond to simple functions of
each potential parameter that changes during the mutation. The single-charging
integral approach affords more flexibility in dealing with possible singularities,
numerical difficulties, or unwanted phase changes that can plague the Kirkwood
coupling parameter method. The single-charging integral method is useful for solutes
formed by ring molecules and also for solutes formed by shorter linear or branched
chain molecules.
The calculation of the excess chemical potential using the configurational-bias
method [7] is an approach ideally suited for solutes formed by chain molecules. To
generate samples with favourable statistics, the method introduces a configurational
bias that favours low-energy conformations. The position of the first two segments of
a test chain molecule in the host system and their orientation in space are chosen at
random. Subsequent segments are appended at the end of the test chain, one by one,
until a full test chain is grown. The bias introduced by the chain-growing scheme is
subsequently taken into account in a modified Widom expression of the excess
chemical potential. The efficiency of the method could be further improved by
introducing additional biasis for the position and orientation of the first two segments
of the test chain.
The test-segment-insertion method [8] decomposes the total solute excess
chemical potential into a sum of contributions from segments of a test particle. The
contributions are calculated in separate simulations. Here, a test segment is inserted
onto the end of “partially built” test particle and the contribution to the total solute
excess chemical potential is calculated using the Widom expression [3]. The testsegment-insertion method is useful for solutes formed by long chains including
polymers.
The expanded ensemble method [9] requires simulations of systems with
various degrees of coupling between the solute and the solvent, involving the gradual
turning on of the intermolecular potentials. The coupling ranges from completely
decoupled to fully interactive solute. The solute excess chemical potential is evaluated
using a histogram, describing the probability with which each of the sub-systems is
visited. Frequency of visiting the sub-systems is controlled by the pre-weighting
factors that are computed in the course of simulations. The expanded ensemble
method is useful for solutes formed by chain as well as ring molecules.
The above outlined advanced simulation methods are ideally suited for
implementation on parallel computers with distributed memory using the Message
Passing Interface library [10] since they rely on either the series of simulations (the
Kirkwood coupling parameter method, the single-charging integral method, the testsegment-insertion method and the expanded ensemble method) or the sampling in
various directions (the configurational-bias method). The simulations of particular
sub-systems or the sampling in particular directions can be straightforwardly
performed on separate computer nodes. Moreover, the methods require an interchange
of minimum information among the nodes. Thus, there is negligible time-loss due to
communication among the nodes. Another aspect of the simulations for large
molecules is an efficient performing of (rotational, translation and conformational)
moves for large molecules. This can be realized using the configurational-bias
technique that offers further parallelization opportunities for the simulations.
Finally, we have shown on prediction of chemical and vapour-liquid
equilibrium for mixtures [11] that an incorporation of experimental pure-component
vapour pressures into simulations leads to significant improvement in predictive
accuracy of simulation methods. A similar approach can be used in the prediction of
solubilities by simulations methods. In this case, the solvent excess chemical potential
(calculated from an accurate equation of state at given temperature and pressure) can
be fixed in the course of simulations instead of the system pressure.
3. Methodological Approach
The proposed work can be divided into two complementary parts:
development of simulation methodology for calculating the solute excess chemical
potential on parallel computers, and the application of the parallelized simulation
methods to predict the solubilities of real systems of practical interest. Throughout the
work we plan to interact with experimental studies wherever possible, both to validate
the parallelized simulation method and to aid in understanding the phenomena and
properties of the real systems of practical interest.
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1.