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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

constant-volume 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

quasi-harmonic 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 3N-3 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 Lennard-Jones 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. first-order saddle points of the

energy in (3N-3)-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 in-between under given T and P. Each

minimum-to-minimum 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)




* Professor Doros N. Theodorou Tel (+3061) 997 398 *

* Department of Chemical Engineering Fax (+3061) 993 255 *

* University of Patras


* GR 26500 Patras, GREECE



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