Molecular dynamics, simulations

MD simulation is a useful tool for the study of small-scale fluid flows, where numerical integration of Newton s law of motion for particles (atoms or molecules) is carried out using 

Researeh and development of the MD simulation method is still progressing. Currently, MD simulations are computationally restricted to very tiny time scales (about tens of nanoseeond) and very large shear rates ( 10 s ). The eost of eomputer time inereases dramatically with increasing number of atoms in the model of moleeular ehains. The growth of computational power and the development of parallel algorithms will help researchers to simulate erystallization for large systems. 

We will briefly discuss the molecular dynamics results obtained for two systems—protein-like and random-block copolymer melts— described by a Yukawa-type potential with (i) attractive A-A interactions (saa 0, bb = sab = 0) and with (ii) short-range repulsive interactions between unlike units (sab 0, aa = bb = 0). The mixtures contain a large number of different components, i.e., different chemical sequences. Each system is in a randomly mixing state at the athermal condition (eap = 0). As the attractive (repulsive) interactions increase, i.e., the temperature decreases, the systems relax to new equilibrium morphologies. 

To visualize the three-dimensional structures of these microphases more clearly, the density of A-segments was measured on a three-dimensional grid. In the A-rich domains the density of A-segments is high ( 1), and in the B-rich domains this density is low ( 0). The dividing surface between the A-rich domains and the B-rich domains is now represented by the isosurface where the density is midway between these values, i.e., 0.5. 

What is most important for our discussion is the fact that the spatial scale r of the segregated structure for protein-like copolymers is appreciably larger than that for random-block copolymers with the same composition and the same average block length. Also, MIST in the protein-like copolymer system occurs at a temperature higher than that of the random-block system, which is in agreement with the prediction of the polymer RISM theory [153]. 

Obviously, these features arise from the difference between the chemical sequences of the two copolymers. 

We may conclude from Section 4.3.1 that treatment of a chemical system using classical mechanics, based on potential calculated by quantum mechanics, should give quite accurate nuclear dynamics, except possibly for protons. The forces are calculated from the PES and included in Newton s equations, which are solved in an iterative way. This model is referred to as molecular dynamics simulation. 

The potential function of a particle corresponds to E IQ) in Equation 4.7, which we write as V(R). It consists of binding forces and forces from nonbonded nuclei groups and other nuclei. What is included in the molecular dynamics simulation depends on how detailed we want to be. Usually, it is sufficient to include CH bonds as a united atom and neglect the motion of the hydrogen atoms. In proteins, we may choose to ignore the bond distances and include only the torsion angles 

Potential functions may be calculated or approximated on the basis of experimental data or taken from calculated results. The bonding potential may be approximated using the harmonic approximation of Equation 4.26  

Anharmonicities may be ignored at normal temperature. R is the equilibrium distance between the atoms i and j. Equation 4.44 is part of the potential sum for the bonded atoms. 

The potential from bond angles and torsion angles is written in an analogous way  

Where r is the distance between the two interaction sites i and j. e and o are the LJ interaction parameters for the ij pair. All the parameters involved in the L J potential functions are listed in Table 8.5. 

The cross-parameters for the pair ij are calculated by the Lorentz-Bertherot rale. The cut-off distance for the intermolecular interaction was set at 3.5 

The simulation cell is divided into subcells. The density of the component k (either methane or ethane) in the Lth subcell is calculated by 

Where superscript L) denotes subcell number L, is the number of the molecules of the component k in the Lth subcell, is the volume of a subcell and is the distance in the x-direction of a subcell. The permeation flux (L ) of the component k in the Lth subcell is calculated by 

DP (A) has two different pore mouths one a large (pore a) and the other a small mouth (pore b). ZP (B) has zigzag shaped pores whose sizes (diameters) are all the same at the pore entry. SP (C) has straight pores which can be called slit-shaped pores. The minimum pore width (W ) is set equal to 0.5 nm for all three types of pores. The membrane thickness, given in Fig. 8.12 asXZ was set equal to 6.3,4.6 and 5.2 nm for DP, ZP and SP membranes. These values are based on the arrangement of micro-graphite crystallites (shown as a nodule in the figiue) whose unit size was fixed at 2.7 x 3.4 x 1.3 nm for the DP and ZP membranes. 

The pair potential employed was modified fi om that of the Born-Mayer-Huggins type used by Woodcock et al. without the dispersion terms [16], 

The result is a trajectory that specifies how the positions and velocities of the particles vary with time. In general the time increment is set to a few femtoseconds, in some cases even less. 

The potential energy of a single atom has to account for all the interaction energies with the atoms of the molecule, with those of the zeolite wall, and also for the internal bond bending, vibration, etc. Dual interactions are calculated 

The self diffusivity is obtained from the trajectory by means of the Einstein relation (3.5.4-1). 

Molecular Dynamics simulation of the diffusion of /rparaffins In the Silicalite (a zeolite). A zig-zag pores B straight pores. From Runnebaum and Maginn [1997]. 

---

Since the development of grazing incidence x-ray diffraction, much of the convincing evidence for long-range positional order in layers has come from this technique. Structural relaxations from distorted hexagonal structure toward a relaxed array have been seen in heneicosanol [215]. Rice and co-workers combine grazing incidence x-ray diffraction with molecular dynamics simulations to understand several ordering transitions [178,215-219]. 

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

Bartell and co-workers have made significant progress by combining electron diffraction studies from beams of molecular clusters with molecular dynamics simulations [14, 51, 52]. Due to their small volumes, deep supercoolings can be attained in cluster beams however, the temperature is not easily controlled. The rapid nucleation that ensues can produce new phases not observed in the bulk [14]. Despite the concern about the appropriateness of the classic model for small clusters, its application appears to be valid in several cases [51]. 

It is possible to use the quantum states to predict the electronic properties of the melt. A typical procedure is to implement molecular dynamics simulations for the liquid, which pemiit the wavefiinctions to be detemiined at each time step of the simulation. As an example, one can use the eigenpairs for a given atomic configuration to calculate the optical conductivity. The real part of tire conductivity can be expressed as... 

Figure Al.3.30. Theoretical frequency-dependent conductivity for GaAs and CdTe liquids from ab initio molecular dynamics simulations [42].

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Wang W, Nelson K A, Xiao L and Coker D F 1994 Molecular dynamics simulation studies of solvent cage effects on photodissociation in condensed phases J. Chem. Phys. 101 9663-71... 

Batista V S and Coker D F 1996 Nonadiabatic molecular dynamics simulation of photodissociation and geminate recombination of liquid xenon J. Chem. Phys. 105 4033-54... 

At any geometry g.], the gradient vector having components d EjJd Q. provides the forces (F. = -d Ej l d 2.) along each of the coordinates Q-. These forces are used in molecular dynamics simulations which solve the Newton F = ma equations and in molecular mechanics studies which are aimed at locating those geometries where the F vector vanishes (i.e. tire stable isomers and transition states discussed above). 

Wilson M R 1997 Molecular dynamics simulations of flexible liquid crystal molecules using a Gay-Berne/Lennard-Jones model J. Chem. Phys. 107 8654-63... 

Haile J M 1992 Molecular Dynamics Simulation Elementary Methods (New York Wiley)... 

Rapaport D C 1995 The Art of Molecular Dynamics Simulation (Cambridge Cambridge University Press)... 

SchlickT, Mandziuk M, Skeel R D and Srinivas K 1998 Nonlinear resonance artifacts in molecular dynamics simulations J. Comput. Phys. 140 1-29... 

Ciccotti G and Ryckaert J P 1986 Molecular dynamics simulation of rigid molecules Comput. Phys. Rep. 4 345-92... 

Procacci P, March M and Martyna G J 1998 Electrostatic calculations and multiple time scales in molecular dynamics simulation of flexible molecular systems J. Chem. Phys. 108 8799-803... 

Andersen H C 1980 Molecular dynamics simulations at constant pressure and/or temperature J. Chem. 

Tobias D J, Martyna G J and Klein M L 1993 Molecular dynamics simulations of a protein In the canonical ensemble J. Phys. Chem. 9712959-66... 

Alejandre J, Tildesley D J and Chapela G A 1995 Molecular dynamics simulation of the orthobaric densities and surface tension of water J. Chem. Phys. 102 4574-83... 

Holian B L and Lomdahl P S 1998 Plasticity induced by shockwaves in nonequilibrium molecular-dynamics simulations Soienoe 280 2085-8... 

Hilbers P A J and Esselink K 1993 Parallel computing and molecular dynamics simulations Computer Simulation In Chemloal Physios /o 397 NATO ASI Series Ced M P Allen and D J Tlldesley (Dordrecht Kluwer) pp 473-95... 

Niv M Y, Krylov A I and Gerber R B 1997 Photodissociation, electronic relaxation and recombination of HCI in Ar-n(HCI) clusters—non-adiabatic molecular dynamics simulations Faraday Discuss. Chem. Soc. 108 243-54... 

Sokal A D 1995 Monte Carlo and Molecular Dynamics Simulations in Polymer Science ed K Binder (New York Oxford University Press) oh 3... 

Kremer K and Grest G S 1990 Dynamics of entangled linear polymer melts a molecular-dynamics simulation J Chem. Phys. 92 5057... 

Figure C2.3.7. Snapshot of micelle of sodium octanoate obtained during molecular dynamics simulation. The darkest shading is for sodium counter-ions, the lightest shading is for oxygens and the medium shading is for carbon atoms. Reproduced by pennission from figure 2 of [36].

Weakliem P C and Carter E A 1993 Surface chemical reactions studied via ab /n/f/o-derived molecular dynamics simulations fluorine etching of Si(IOO) J. Chem Phys. 98 737-45... 

Barone M E and Graves D B 1995 Molecular dynamics simulations of direct reactive ion etching of silicon by fluorine and chlorine J. Appi. Phys. 78 6604-15... 

Helmer B A and Graves D B 1997 Molecular dynamics simulations of fluorosllyl Ions with silicon J. Vac. Sc/. Technol. A 15 2252-61... 

Hanson D E, Voter A F and Kress J D 1997 Molecular dynamics simulation of reactive Ion etching of SI by energetic Cl Ions J. Appl. Phys. 82 3552-9... 

---