Evolutive Ensemble Thermodynamics of Nanoclusters.


Oct 27, 2013 (3 years and 5 months ago)


Evolutive Ensemble Thermodynamics of Nanoclusters

J. Mulia,

A. Tame



Facultad de Ciencias, Universidad Autónoma del Estado de México, Instituto


100 Toluca

Estado de México,

México. C.P. 50000



Thermodynamic properties of nanoclusters are derived through a statistical determination of their
potential energy surface using a genetic algorithm combined
with a conjugate gradient local
minimization. Stationary points, their interconnections, and a mea
sure of their contribution to the
configurational density of states are determined. This information, combined with knowledge of the
kinetic degrees of freedom, is sufficient to yield the total density of states, allowing the calculation of
thermodynamic p
roperties. There is no calculation of the forces, Boltzmann

weights, nor are
Newton’s equations of motion solved. Problems arrising from potential energy barriers are non
existent and ergodicity is greatly improved over

the mo
lecular dynamics or Monte Carl
approaches. Since evaporation is a natural part of this evolutive ensemble, the thermodynamics
can be extended into the liquid
gas transition region. Application is
made to the Lennard
clusters of size 7, 13 and 38 atoms.

A great diversity of

unusual behavior has been observed

for nanoclusters; solid
solid phase

[1], pre
melting, a continuous “melting” transition

region in which solid and liquid
phases coexist

[2], ebullition, negative heat capacities in the microcanonical

ensemble [3, 4
] as well
as a nontrivial

dependence of most properties on cluster size

]. This flavor in the dynamics is a
result of the

unique energy distributions of the lowest lying minima

and their saddle points for each
particular type

and size of cl
uster. This, in turn, is a direct consequence

of the interplay between
atomic interaction

range and the small finite size of nanoclusters.

A common error in many of
the analyses has been

to confuse the initial condition dependent dynamic

behavior wit
h true
thermodynamic behavior of nanoclusters. Nanoclusters are small systems which

may have little
total energy compared to the often

large energy barriers separating different regions of

energetically available phase space. Nanoclusters,

either in ex
periment or in simulation, may,

easily become trapped and thus prevented from visiting

all of the energetically allowed
phase space. Convergence

in simulation or experiment does not imply

thermodynamic equilibrium
On raising the energy

ntly, suddenly new regions of phase space

become accessible. In
have shown how these

problems can lead to a particular dynamical behavior

that, if confused with
thermodynamics, can lead to

the erroneous determination of negative heat capacity


Therefore, unlike in the case of the bulk, where

the initial state morphological
symmetry can be controlled

and where extremely high energy barriers pre
vent the observation of
more than one symmetry in

the solid, accurate modeling of the thermodyna

behavior of
nanoclusters requires sampling the contribution

of many different symmetry configurational

structures relevant to the energetically allowed phase

space. Much of this phase space may not be

isoenergetically but may be isolated by h

energy barriers, frustrating the approach to

in the traditional molecular dynamics or Monte

Carlo approac
hes. The purpose of this


describe an alternative approach to determining the

thermodynamics of nanoclusters based on a

determination of the potential energy surface using

a genetic algorithm combined with a

gradient local optimization which is immune to the

problems associated with trapping.

Both the molecular dynamic and Monte Carlo approaches to the thermodyn
amics of nanoclusters
sufferfrom problems of ergodicity and quasi
ergodicity. Ensuring ergodicity requires large duration
trajectories, making large systems not amenable, because of CPU time limitations, to the molecular
dynamic or Monte Carlo approaches.
A further problem is that large duration trajectories increase
the probability of evaporation, rendering the trajectory useless. Avoiding quasi
ergodicity due to
potential energy barriers and evaporation in the molecular dynamics and Monte Carlo approaches
requires special considerations which complicate the simulation and further reduce efficiencies and
reliability. Since the evolutive statistical approach described here concentrates on mapping the
potential energy surface and not simulating a single traj
ectory over the surface, problems of
ergodicity are reduced and quasi
ergodicity is non
existent. The natural inclusion of evaporation in
the evolutive ensemble allows the determination of thermodynamic properties into the liquid to gas
transition region.
No forces nor Boltzmann weights are calculated, neither are Newton’s equations
of motion solved. The resulting efficiency of the approach facilitates larger systems and more
realistic modeling (for example ab initio calculations).



J. P.
Neirotti, F. Calvo, D. L. Freeman, J. D.Doll, J. Chem. Phys. 112 (2000) 10340.


D. J. Wales, Mol. Phys. 78 (1993) 151.

J.P.K. Doye and D.J.Wales, J. Chem. Phys. 102

(1995) 9659.


K. Michaelian, A. Taméz, I. L. Garzón, Chem.

Phys. Lett. 370
(2003) 654.


E. M. Pearson, T. Halicioglu, and W. A. Tiller,

Phys. Rev. A 32 (1985) 3030.

I. L. Garzón, K. Michaelian, M. R. Beltrán, A.

Amarillas, P. Ordejón, E. Artacho, D.

Portal, and J. M. Soler, Phys.
Rev. Lett.

(1998) 1600.