Molecular Scale

At the molecular scale, the behavior of subatomic particles, such as electrons or protons, is no longer relevant, but the interaction between individual atoms and/or molecules is considered as a whole. There are two main types of simulation methods at the molecular scale: (1) stochastic (Monte Carlo) and (2) deterministic (molecular dynamics).

Monte Carlo Simulation

The Monte Carlo (MC) simulation is apurelystochasticmethod, which generates random configurations of the system for determining the lowest energy configuration, and thus obtaining the most probable thermodynamic properties of the system. The different configurations of the system can be generated in a successive or a parallel way or using a combination of both. Parallel generation allows a fast identification of the low-energy configuration region, whereas successive generation allows a more precise identification of the minimum energy configuration of the system. In the successive generation approach, each new configuration of the system is generated by randomly changing the position of individual atoms or molecules. The total energy of the new configuration (Unew) is determined as the sum of the potential interaction energy between all pairs of atoms or molecules. Notice that a prerequisite for performing reliable MC simulations lies in the determination of accurate intermolecular interaction potentials. The choice of the best interaction potential functions depends upon the system investigated and on the quality of the force fields available, such as ММ2, ММ3, and AMBER [27]. Particularly for polymer systems, different molecular force fields have been proposed, including CHARMM (Chemistry at HARvard Macromolecular Mechanics) [28], and PCFF (Polymer Consistent Force Field) [29-30]. However, other general force fields such as DREIDING [31], COMPASS [32], and OPLS [33] have also been used for modeling polymeric systems [34-37].

The energy of the new configuration is then compared with the energy of the previous configuration (C/old) in order to accept or reject the new configuration. A very common acceptance criterion, known as the importance sampling scheme or the Metropolis condition [38] is based on the Boltzmann factor (/) (Eq. 3.5). If a uniform random number generated between 0 and 1 is lower than the Boltzmann factor, then the new configuration is accepted; otherwise, the new configuration is rejected, and the previous configuration is retained for the next iteration. This procedure is used as a strategy to overcome local energy minima that may lead to erroneous results.

For each configuration accepted during the previous procedure, the value of a certain thermodynamic property A is determined. Then, it is possible to determine the ensemble average as

where m is the total number of configurations considered. The total number of configurations required will depend on the thermodynamic property investigated, since some thermodynamic properties converge more rapidly than others. For example, heat capacities require in general a much larger ensemble sampling than internal energies [27].

MC methods are in general less efficient than molecular dynamic techniques. Therefore, MC methods have been applied only to systems where they are more effective, such as liquids or systems in solution. MC methods have been employed for the investigation of polymerization mechanisms and kinetics [39-45], the determination of thermodynamic properties of polymers [46-49], for the description of interfacial phenomena in polymers [50-53], their structural and morphological properties [54-57], and in heterophase polymerization in general [58-65].

Molecular Dynamics Simulation

Molecular dynamics (MD) simulation is a deterministic method used to follow the trajectories and velocities of an ensemble of atoms or molecules subjected to interatomic or intermolecular forces for a certain period of time. Although the atoms and molecules are composed of quantum subatomic particles, their motion can be satisfactorily described by classical Newton's equations of motion. From this information, it is possible to determine static and dynamic properties of the system, such as thermodynamic properties and transport coefficients.

The basic equations of motion employed are Newton's first law (Eq. 3.7] and the definition of velocity (Eq. 3.8):

where v,- is the velocity, mf- is the mass and x, is the position of the /-th molecule, Fy is the interaction force between the /-th and j-th molecules, and t is the time. The interaction force is calculated as the negative gradient of the interaction potential (Eq. 1.1]. Additional external or internal (mean field] forces can also be considered.

In conventional MD simulations, a system containing a finite number of particles N (atoms and/or molecules] is placed within a usually cubic cell of fixed volume. A set of velocities, usually drawn from the Maxwell-Boltzmann distribution (Eq. 3.9] at a given temperature T, is also selected and assigned in such a way that the net molecular momentum is equal to zero.

/мв is the probability of finding a particle of mass m with a velocity vk in the k-th direction.

The equations of motion are integrated numerically using very short time steps At (normally around 10'15 to 10'14 s). Larger time steps may lead to instability and erroneous results, whereas shorter time steps increase the computation time without improvements in accuracy. At each step, all the forces acting on each atom and/or molecule are calculated from the position-dependent interaction and external potentials and are used in Eq. 3.7 along with the current particle velocities to determine the new velocities one time step ahead. These new velocities are then used together with the current particle positions to determine the new positions one time step ahead (Eq. 3.8). These new positions and velocities are used again for determining the forces and iterating during the whole simulated time interval r (typically 10"11-10"1° s).

The interaction between atoms and/or molecules can be modeled using different interaction potentials, such as the Lennard- Jones potential, the Buckingham potential, the Morse potential, or even relatively simple interaction potentials such as the harddisk potential, which present no interaction between particles as long as they are not colliding and an infinite repulsion potential when they are in contact or superposing. The interaction potential is responsible for the attraction of atoms or molecules causing clustering or aggregation, and also for the repulsion resulting in elastic collisions, phase separation, and many other phenomena.

The trajectories obtained by MD simulation can be used to determine thermodynamic properties of the system, by averaging over the dynamic history of the system (by using Eq. 3.10). MD simulation, contrary to MC calculations, also allows the study of tim e-depende nt phenomena.

Numerous algorithms have been proposed to numerically integrate the MD equations of motion [66]. The most popular is the finite-difference method proposed by Verlet [67], which computes the position of the /-th particle (x,) and its velocity (v,), one time step ahead from the forces and positions at previous times:

The Verlet method (Eqs. 3.11 and 3.12) is easy to implement, is stable at sufficiently small time steps (10'1S s to 10"14 s), and is relatively fast. Other numerical integration methods of MD equations include the velocity Verlet [68], the Beeman [69], and the leapfrog

[70] algorithms, derived from the Verlet scheme; and predictor- corrector techniques such as the methods presented by Rahman

[71] and Gear [72].

During the integration of Newton's equations of motion, the energy is conserved. However, slow temperature drifts may occur as a result of the numerical integration and numerical truncations. The simplest method used to keep the system at a constant temperature Tref is rescaling the velocities at appropriate intervals by a factor of

yjTrc( / T{t), where T[t] is determined using Eq. 3.13.

MD simulations have rapidly emerged as a powerful tool to calculate structural and thermodynamic properties of complex liquids, molten salts, crystals, polymers, and proteins in solution [73]. Relevant applications of MD simulation in the field of polymers include

  • • Polymer thermodynamics [74-77]
  • • Phase transitions: Swelling [78], glass transition [79-80], crystallization [81], collapse [82], spinodal decomposition [83]
  • • Polymer fluid dynamics [84-89]
  • • Transport properties [90-93]
  • • Vibrational spectra of macromolecules [94-95]
  • • Mass transfer in heterophase polymerization. The molecular approach also appears to be a very good alternative for the prediction of non-equilibrium monomer concentration inside polymer particles in emulsion polymerization [96].

Some nice reviews of the application of MD simulation to polymers are currently available [97-99].

 
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