WORK OF FORMATION OF CAESIUM HYDROXIDE CLUSTERS DETERMINED BY
GUIDED MITOSIS
J. THOMPSON1,2, J.C. BARRETT2 and I.J. FORD1
1
Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT,
United Kingdom.
2
Nuclear Department, Defence Academy, HMS Sultan, Military Road, Gosport, PO12 3BY,
United Kingdom.
Keywords: Free energy computation, Molecular dynamics, Molecular clusters, Nuclear aerosols.
INTRODUCTION
Metal hydroxides are of technological significance in many applications (Chroneos et al., 2006). In
particular, a key hazard from a severe accident in a water-cooled nuclear reactor is the release of
radioactive caesium which may react with steam to form high temperature caesium hydroxide vapour. As
this vapour flows away from the molten core it will cool and condense. This study is concerned with the
the formation of caesium hydroxide aerosol by nucleation from this vapour phase. We employ the
Jarzynski equality to extract the nucleation barrier from nonequilibrium molecular dynamics simulations,
using the code LAMMPS (Plimpton, 1995).
METHODS
The first stage in molecular dynamics modelling is the selection of an appropriate potential. We followed
Chroneos et al. (2006) in representing the interatomic potentials of metal hydroxides by Coulomb
interactions supplemented by short-range Buckingham terms. The parameters in the potential were found
by fitting simulations of the crystal structure to the known lattice parameters for CsOH (Jacobs et al.,
1987). CsOH has an orthorhombic phase at temperatures below 500 K and a cubic phase at higher
temperatures. The simplicity of our potential did not allow a match of the lattice parameters of both phases
simultaneously but our attention is focussed on modelling the high temperature cubic phase, and we
adjusted the potential to reproduce the lattice parameters for this phase to within 2%. The model material
had a melting point of around 650 K, slightly higher than the experimental value for CsOH of 620 K.
The potential was then used to calculate the free energy change, or thermodynamic work, of cluster
formation using the mitosis method developed by Ford and co-workers (Tang and Ford, 2015; Lau et al.,
2015; Parkinson et al., 2016). The method relies on the Jarzynski equality:
exp  W / kT   exp( F / kT ) ,
(1)
where W is the mechanical work performed during the manipulation of the Hamiltonian of a system and
F is the associated change in free energy between the initial and final equilibrium states. The angled
brackets indicate an average over many realisations of the process.
Equation (1) was applied as follows. An ensemble of equilibrium clusters of 2n molecules at temperature T
was set up with the constituent atoms attached to two fictitious “guide particles” by weak springs (half the
atoms attached to each guide particle). The spring constants were sufficiently small that the equilibrium
structure of the cluster was not significantly disturbed. The temperature was maintained by a Langevin
thermostat. The cluster was then separated into two clusters, each of size 2n/2, by the following 5-stage
procedure:
Stage 1: the guide particles were moved apart at constant velocity through a distance d;
Stage 2: the guide particles were held stationary and the springs strengthened, pulling the cluster
apart;
Stage 3: the guide particles were moved further away with the same velocity as before;
Stage 4: with the guide particles still moving, the spring constants were reduced to their initial
value;
Stage 5: the guide particles continue to move, with the spring constants equal to their initial value.
This procedure was adopted in preference to simpler protocols since it tends to avoid abrupt irreversible
“snapping” events which lead to poor statistics (Parkinson et al., 2016). The procedure was repeated many
times (typically 1000), and the total work done recorded for each trajectory. Four such realisations are
shown in figure 1. All simulations were performed at a temperature of 1000 K.
Figure 1: variation of work done against time elapsed during the mitosis of four CsOH octomers. The
dashed lines indicate the 5 stages in the procedure, as discussed in the text. The performance of work in
stages 1 and 2 corresponds to the breaking of intermolecular forces and strengthening of tethers, while the
return of work in stage 4 arises from tether loosening.
Figure 2: histogram of work performed during the mitosis of an octomer (1000 trials) showing a twopeaked distribution.
Figure 3: total free energy change in the splitting of an octomer into two tetramers by mitosis, as a
function of total simulation time. The error bars are estimated by dividing the 1000 values of W into 10
equal groups and applying Eq. (1) to each group.
The point of using Eq. (1) is that a broad distribution of W can be reduced to a specific value of the
associated free energy change F. Nevertheless, accuracy in computing the free energy change from
limited simulation data is improved if the work distribution can be narrowed. However, for some ranges of
parameters, anomalously high values of W were found, e.g. see figure 2. The reason for their appearance
was investigated. It was found that these cases related to situations where a hydrogen and an oxygen atom
attached to different guide particles happened to form a hydroxyl group which was later pulled apart when
the spring constants increased, releasing a large amount of stored energy as heat. The method works best
when the mechanical separation of the cluster is conducted as gently as possible, so we restricted our
simulations to ranges of parameters which avoided generating these costly trajectories.
The resulting values of W were then used in Eq. (1) to determine Fmit, the free energy difference between
a tethered cluster containing N=2n molecules and two tethered clusters each of size N/2. This free energy
difference is not simply F(N)-2F(N/2), where F is the total free energy of an untethered molecular cluster,
as there are additional terms relating to the indistinguishability of the molecules and the fixing of the
centres of mass in the simulations, see Lau et al. (2015); however, these terms can be calculated in a
straightforward manner. It is then possible to determine the effective surface free energy Fs(N) of dimers,
tetramers, octomers, etc. For large droplets, this quantity should converge to the planar surface tension
multiplied by a notional cluster surface area, and it is a key ingredient in nucleation theory (Ford (1997)).
Furthermore, a formula can be derived for “uneven mitosis” to relate, for example, the surface free energy
of a 12-cluster to that of the octomer and tetramer. Values obtained depend on the time over which the
separation takes place, although the dependence is weak (provided it is not too short), as can be seen in
figure 3.
The thermodynamic potential associated with cluster growth and evaporation (N), which can also be
referred to as the thermodynamic work of cluster formation, is related to the surface free energy Fs(N)
according to
 ( N )  Fs( N )  NkT ln v / vs ,
(2)
where v is the molecular number density of the precursor vapour (assumed to behave as a perfect gas),
and vs is the saturated vapour density. The difference  = (N)-(1) is commonly (if loosely) referred to
as the nucleation free energy barrier that controls the kinetics of the emergence of droplets from the
vapour. It should be noted that the nucleation barrier  (N) does not depend on saturated vapour density:
the latter is only required for the evaluation of the surface free energy Fs(N).
Figure 4: nucleation barrier for CsOH cluster formation, in units of kT, vs size in CsOH molecules.
For v >vs ,  (N) has a peak at N*, which can be interpreted as the critical size above which growth
becomes increasingly likely. Figure 4 illustrates such a case with v=1.451025 m-3, corresponding to a
highly supersaturated vapour (the experimental saturated vapour density for CsOH at 1000 K is around
1.451023 m-3) and consequently a fairly low nucleation barrier.
CONCLUSIONS
The mitosis method is a promising approach for determining the nucleation free energy barrier for liquidlike clusters of metal hydroxides such as CsOH. It is based on nonequilibrium molecular dynamics
simulations of cluster separation, interpreted using the Jarzynski equality. We have identified a number of
aspects that need careful attention for these systems, particularly the need to avoid costly realisations
where the external work is performed against intramolecular rather than intermolecular forces. We have
noted that identifying the nucleation barrier need not require the evaluation of the saturated vapour
density. Current work is focussed on extending the calculations to larger clusters, more representative of
those likely to be important in practical situations.
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