
ELEMENTARY STRUCTURAL TRANSITIONS IN A SIMPLE AMORPHOUS SOLID
USING MULTIDIMENSIONAL TRANSITIONSTATE THEORY
Nikolaos P. Kopsias and Doros N. Theodorou
National Research Centre for the Physical Sciences "Demokritos", Institute
of Physical Chemistry; and University of Patras, Department of Chemical
Engineering, Patras, GREECE.
In configuration space, a glassy amorphous system can be viewed as a point
spending most of its time vibrating about local minima of the energy
hypersurface. The energy associated with thermal fluctuations around a local
minimum is small compared to the height of the surrounding energy barriers,
and therefore transitions between adjacent minima are infrequent.
The above picture leads to a distinction between two different
contributions to the system's free energy: (a) the potential energy of the
"underlying" structure at the minimum (Stillinger and Weber's "inherent
structure"); and (b) the contribution from vibrational motion about the
inherent structure. For each configuration of atoms, the calculation of the
first of the two free energy contributions mentioned above is performed by
constantvolume steepest descent minimization of the potential energy with
respect to the Cartesian coordinates of the atoms. On the other hand, a
valuable tool for the determination of the vibrational contribution is the
quasiharmonic approximation (QHA). According to the QHA, diagonalization
of the Hessian matrix of each inherent structure of a system of N atoms
with periodic boundary conditions yields the 3N3 normal modes of vibration
(eigenvectors, the three subtracted degrees of freedom corresponding to rigid
translation) along with the associated frequencies (square roots of the
eigenvalues), and hence an estimate of the vibrational free energy at the
volume V under consideration. At given temperature T and pressure P, the
configurationally arrested system will assume that volume which minimizes
the Gibbs energy G, calculated as a sum of the inherent structure energy,
the vibrational free energy, and PV.
We have applied the above procedure for the generation and analysis of
glassy configurations of a LennardJones system. Starting from classically
quenched glassy configurations, we varied for each one of them the specific
volume V at constant temperature T and pressure P and mapped G as a function
of V. The G(V) curve for each configuration exhibits a clear minimum at some
volume representative of the "equilibrium" state of the configurationally
arrested glassy structure. The average values of the specific volume
obtained in this way for a series of different temperatures and pressures
are in close agreement with the exact molecular dynamics results, leading to
the conclusion that the QHA is very reasonable for our system.
Having constructed "equilibrated" glassy configurations ("states" or
"wells" around which the system spends long time intervals vibrating), our
main interest was to analyze the rare events whereby the system escapes
from each well into another, neighboring well. Knowing that the escape paths
pass through transition states (i.e. firstorder saddle points of the
energy in (3N3)dimensional configuration space), we used Baker's algorithm
to locate such transition states on the ridges of the energy walls
surrounding each arrested configuration. This saddle point determination
was carried out at given T and P, allowing for changes in volume between
the original state and the transition state. Steepest descent energy
minimization, starting from the transition state and leading to a new
minimum, followed by volume relaxation of that minimum, completed the
determination of each escape path. By repeating this calculation many
times, we accumulated a large number of pairs of adjacent free energy minima
and the transition states inbetween under given T and P. Each
minimumtominimum elementary transition is characterized by a volume
and a free energy difference between the states it connects as well as by
a free energy barrier at the saddle point it traverses. For each
transition we calculated a rate constant based on the QHA.
The distribution of the free energy barriers in a system of 198 particles
was found to be extremely broad and strongly asymmetric, spanning more than
100kT [1]. This behavior is mainly due to the enthalpic contribution to each
barrier; the distribution of activation entropies is quite narrow, spanning
ca. 10k. Moreover, a strong positive correlation between the volume change
and the free energy change accompanying each transition was detected. Since
the structural relaxation of real amorphous solids can be envisioned as a
series of such elementary relaxational events, this correlation indicates
that movement towards more stable states will be accompanied by reduction
of the specific volume, as observed in actual physical ageing experiments.
A kinetic Monte Carlo strategy has been designed for tracking sequences
of elementary transitions in configuration space and thereby following
the process of structural relaxation in the model glass.
[1] N.P.Kopsias and D.N.Theodorou, J.Chem.Phys., in press (1998)
http://calypso.nrcps.ariadnet.gr/public.html
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* Professor Doros N. Theodorou Tel (+3061) 997 398 *
* Department of Chemical Engineering Fax (+3061) 993 255 *
* University of Patras doros@sequoia.chemeng.upatras.gr
*
* GR 26500 Patras, GREECE http://tahoe.chemeng.upatras.gr
*
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