Fig.1: Graph of the specific heat as function
of temperatureof monospecied LD liquid
Fig.2: Graph of thespecific heat as function of temperatureof
double-specied LD liquid |
Abstract
The characterization of the specific heat is essential to investigate
the fundamental issues of phase transitions and to determine the order
of the phase transitions. In this work, molecular dynamics
(MD) simulations were perfromed to study the behaviors of specific heat
at constant pressure (Cp) versus temperature of two Lennard-Jones
(LD) liquids including one specie atom and two specie atoms, respectively.
The simulations demonstrated that the specfic heat increases monotonically
as function of decreasing temperature for the monospecie LD liquid while
the specific heat displays a sudden drop at a characteristic temperature
for the double-specie LD liquid. These results indicate that a first-ordered
liquid-solid phase transformation occurs in the monospecie LD liquid, while
a second-ordered liquid-glass phase transformation occurs in the double-specie
LD liquid as the temperature is reduced.
Introduction
Phase transformation is a generally interested topic in the
mateial research field because it has profound impact on the application
of the materials.
Thermodynamic properties are primary for exploring the physical issues
of Lennard-Jones liquids and the related phase transformations (solid-liquid
transformation and glass-liquid transformation). Specific heat, a
such parameter, is especially important for exploiting the order of the
phase transformation because it is the second derivative of the Gibbs free
energy. For nth phase transformation, the (n - 1)th derivatives of
the Gibbs free energy should be continuous while the nth derivatives of
the Gibbs free energy should be discontinuous at the transformation point.
More specifically, entropy and volume are discontinuous as the system undergoes
a first order phase transformation. In contrast, entropy and volume
are continuous but the specific heat becomes discontinuous for the second
order phase transformation. Thus, the specific heat anomalies are
indeed the criteria to evaluate the order of phase transformations. In
addition, to detect the anomaly in the specfic heat is key to determine
the phase transition temperature for the second-ordered phase transition
because the volume behaves monotonically as function of temperature.
Experimentally, the specific heat is difficult to measure especially
for the systems undergoing a transformation with large rate. Molecular
dynamics simulations, on the other hand, offer us an unparalleled opportunity
to explore this problem. The similar studies have been rarely reported
in literature.
Simulations
The system studied is Lennard-Jones liquid. The potential
between two atoms is Lennard-Jones pair potential. [1] The code used is
developed by Prof. M. Falk at the University of Michigan. To extract the
specific heat of the system, the simulation at the constant pressure and
temperature (using a Nose-Hoover thermostat) was performed for 2 x 104
steps and all the thermodynamic data were recorded every 103
steps. To obtain the specific heat as function of temperature, the
simulation has been performed at different temperatures.
Two types of LD liquid were studied, both containing 512 atoms. One
is the mono-specie liquid involving one type atom. The other involves
256 atoms of type A and 256 atoms of type B with the given mass ratio of
mB/mA=2/1. The potential was cut off at 2.5.
Data Analysis
The specfic heat at constant pressure and temperature was attained
based on the following procedures. The specific heat at constant pressure
is Cp=(dE/dT). In canonical ensemble, Cp=1/kT2<(E-<E>)2>,
where k is Boltzman constant, T is the temperature. E is the total energy
of the system at a certain time, <E> is the mean total energy during
a certain period of time. [2]
Fig.1 is the plot of the specific heat as function of the temperature
of the monospecie LD liquid. It is seen that the specific heat behaves
monotonically increased as the temperature is reduced, indicating that
a first-ordered phase transition from liquid to solid occurs in the LD
liquid of one specie.
In contrast, Fig.2 shows the plot of the specific heat as function of
the temperature of the LD liquid of two species. A sharp drop is
observed at a characteristic temperature. The discontinuity in the specific
heat behavior indicates that a second-ordered phase transtion occurs at
this characteristic temperature. This second-ordered phase transformation
is related to the liquid-glass transition of the double-specie LD liquid.
[3-4] The phase transition temperature Tg is determined
from the plot, which is ~1.15.
References
[1] J. E. Lennard-Jones, Physcia 4, 941 (1937).
[2] Mse693 course note.
[3] 1. J. Lu and D. W. Qi, Phys. Lett. A 157 (1991) 283.
[4] 3. C. Marshall, B. B. Laird, and A. D. J. Haymet, Chem. Phys. Lett.
122 (1985) 320.
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