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[ Visualization of the Orientational Order-Disorder Phase Transitions in Plastic Molecular Clusters ]

Table 1 Results for the time evalution a 89 molecule cluster at different temperatures. The transition temperature for this cluster size is in the vicinity of 75 K. For a larger size, this temperature is higher. Compare with the results shown in Table 2.

Temperature (K) Number of Molecules Java Animation GIF Image
110 89 Animation 1.1 Figure 1.1
80 89 Animation 1.2 Figure 1.2
70 89 Animation 1.3 Figure 1.3
20 89 Animation 1.4 Figure 1.4



Table 2 Results for the time evolution of a 137 molecule cluster at different temperatures. The transition temperature is in the vicinity of 80 K.

Temperature (K) Number of Molecules Java Animation GIF Image
110 137 Animation 2.1 Figure 2.1
80 137 Animation 2.2 Figure 2.2
70 137 Animation 2.3 Figure 2.3
20 137 Animation 2.4 Figure 2.4


II. Computational procedure

The procedure used for arranging clusters of rigid octahedral molecules and computing molecular trajectories is described in 2. For highly symmetrical molecules, it is reasonable to neglect the intramolecular force and to deal only with the intermolecular potential that is a sum of an atom-atom Lennard-Jones term and a Coulomb term 2. Initially, the cluster was arranged in a base-centered cubic lattice at elevated temperatures (below the melting point). The classical equations of motion for a cluster of N molecules interacting via this potential are written in a Hamiltonian form. Computing center-of-mass trajectories requires solving a set of 6N first-order differential equations. These equations are integrated using the velocity Verlet algorithm 8. Cluster's phase diagrams are obtained by successive decrease and increase of the temperature. The cluster is thermalized at a particular temperature using the velocity scaling algorithm which has been proved 9 to give the correct canonical distribution in the coordinate space with an accuracy of order if the scaling is carried out in every time step during the thermalization 3. The time step= 10 fs is chosen to keep the total energy constant during the run. After the thermalization is switched off, the system's equilibration is monitored by recording instantaneous values of the potential and kinetic energy during this period. These are taken every 0.1 ps, in order to avoid collecting correlated values of the quantities. Adequate equilibration is especially important when the initial configuration is a lattice with one type of symmetry and the state point of interest is in another type of symmetry. Examples of a 89 molecule Tef6 cluster at two different temperatures are shown in Figure 2 and Figure 3.

In our computations, the orientation of the molecules relatively to the cluster center-of-mass (space system) is defined by the four quaternions suggested by Evans 10 to solve the problem of divergence in the orientation equations for the three Euler angles when transforming between the space and the body system. The orientation of the rigid molecule specifies the relation between an axis system fixed in the space and one fixed with respect to the body. We have chosen the body system of each molecule to have its origin at the center of mass of that molecule so the inertia tensor is diagonal. Any unit vector e can be expressed in terms of components in the body or space coordinate system. Conversions from the body-fixed to space-fixed systems is handled by the Equation 1.

, Equation 1

where denotes the rotational matrix expressed in terms of quaternions.

The quaternions of each molecules are recorded during the simulations of molecular trajectories. These quaternions are used to project the three dimensional orientational distributions 3. The quaternion is a set of four scalar quantities which are subjects of a constraint: .

We have proposed 3 a novel method for representation of the molecular orientation in order to study its time dependence and to compute the density of molecules oriented in a given direction. The orientations of the molecules in the 3D space are projected onto a 2D spherical surface. Each projection point is represented with an oriented cross. In such a way we can preserve the complete information.

In a rigid octahedral molecule TeF6, Figure 1, the six fluorine atoms are at the vertices of a octahedron and the Te atom is at the center. In an instant configuration the orientation of the molecule is defined by the three orthogonal F-Te-F directions in the body frame. At a given temperature these three directions differ from one configuration to the other. If we take a snap shot of one configuration, Table 1, Figure 1.1, we have information about the instant orientation of the molecules in the cluster. If we project many configurations, we get the time evolution of the cluster, Table 1, Animation 1.1.

The step-by-step procedure to project a configuration is given in 3. Briefly: a unit vector from the molecule is converted in the space system using , where is the transpose matrix. The vector located either in the octant (x, y, z) or in the octant (-x,-y,-z) is taken as the unit vector of the projection. When it is in the first octant, it is not changed in the further calculations. If the projected vector belongs to the complementary octant (-x,-y,-z), then it is rotated. Two of the spherical coordinates in the 2D spherical surface are determined by the position of this vector. In order to keep the information about the third angle, we use the second unit vector of the molecule. Its relative orientation with respect to the first one (in the space system) determines the third angle . This last angle is taken as modulus of due to the molecular symmetry. In such a way we obtain three coordinates that are plotted onto 2D surface as follows: the coordinate is along the angular axis and is along the radial axis. Thus we obtain a point with coordinates . In this point we draw a cross which is rotated at an angle with respect to the radius-vector of the point.

This method makes possible to follow the dynamics of each molecule during the phase transformations when cluster is cooling or heating when the projections are used for animation. Table 1 and Table 2 show two clusters containing different number of molecules.


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Last Modified: 15 Mar 2001 10:25:03  by R.Radev00013654  hits 

 
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