The methods of computer simulation

This gap, between theories that can be applied only to idealized systems, and experiments that are restricted to liquids or to circumstances that are not readily treated theoretically, can be filled by the method of computer simulation. Here %(r ) is specified and, within reason, can have as realistic a form as we wish. Computer simulation is then a technique which tells us the macroscopic physical properties such a realistic model system would possess. It has two aspects first, with respect to statistical theories, it is an experiment which tells us the properties of a system of given configurational energy. This has been its primary use and the one we exploit in this Chapter. A second aspect, with which we are less concerned, is its use in the determination of intermolecular forces. By comparing the properties of the simulated system with those of a real system we can judge how closely the chosen form and strength of V match those of the real molecules. 

The two methods of computer simulation are known by the labels of Molecular Dynamic (MD) and Monte Carlo (MQ simulation. In the fimt the evolution of an assembly of N molecules is followed by numerical solution of Newton s equations of motion. The system is one of fixed N, V, and U and so is the simulation of a micro-canonical ensemble, but since the sequence of states is that of real time both equilibrium and dynamic information can be obtained. In the second method a sequence of states is generated such that each state occurs with a probability proportional to its Boltzmann factor, exp(-%(i )/fcT0. The sequence is (usually) specified by fixed values of N, V, and T, and so the ensemble represented is canonical. The ordering of the steps of the sequence is arbitrary (that is, it contains no information) and so only thermodynamic properties can be calculated. The principles and practice of these techniques are described elsewhere both have been used to study the liquid-gas surface and here we describe only the special problems which these studies involve. 

Unless otherwise stated, all simulations discussed in this chapter have used the same form of namely a Lennard-Jones (12,6) potential function between each pair of molecules, 

Even the largest computer cannot handle a system of /V 10 molecules, but is restricted to It is this restriction that causes 

If the system is not uniform but separated into two fluid phases then the convention of repeating coordinates is adequate for the calculation of bulk thermodynamic properties but not for the study of the interface which would be of irregular, ever-changing, and unidentifiable shape. A simple change of the boundary conditions leads to a much more useful configuration repetition of the coordinates is retained in the x and y directions, but the cell is bounded by reflecting walls at z = 0 and z = L. If now the simulation is started with a flat liquid surface in the x, y (or horizontal) plane, then the repetition of these coordinates tends to maintain its position. Other constraints can be added to enhance the stability. 

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The book explains how to solve coupled systems of ordinary differential equations of the kind that commonly arise in the quantitative description of the evolution of environmental properties. All of the computations that I shall describe can be performed on a personal computer, and all of the programs can be written in such familiar languages as BASIC, PASCAL, or FORTRAN. My goal is to teach the methods of computational simulation of environmental change, and so I do not favor the use of professionally developed black-box programs. 

The method of computer simulation via molecular dynamics has been applied to classical atomic fluids for some twenty years a wealth of... 

The methods of computer simulation of adsorption (and diffusion) in micro-porous solids were described in Chapter 4 a summary is given in Table 4.1. These techniques are now sufficiently well developed for physisorption that thermodynamic properties can be predicted routinely for relatively simple adsorbates, once the structural details of the host are known. Molecular mechanics using standard forcelields are very successful for zeolitic systems, which take into account dispersive interactions satisfactorily, but it is also possible to use higher level calculations. 

The results discussed in the last two sections are the main part of the work so far done by the method of computer simulation, but other problems have been tackled and there are still further ones that are within the scope of modern computers. Here we note some of this work, and suggest how it might be extended. 

The method of computer simulation of protein dynamics is based on the solution of the classical equations of motion for all N atoms (i) of the protein... 

K is a temperature, where most of the natural proteins denaturate. Nevertheless one can do computer simulations of protein dynamics also at such temperatures. The advantage is that one can observe also processes which at lower temperatures are too slow to appear in the time regime accessible for the method of computer simulation. Pig. 4 displays the time decay of the non-bonded energy autocorrelation function at 500K. There is now an additional slow decay component with a time constant of about 5ps. This component is more pronounced for the electrostatic contribution to the energy autocorrelation function. 

The most popular approach to solve these problems is to use experience and good engineering judgment. A quick experiment may be another solution. A computer simulation is a third option. All those approaches may eventually lead to success. This chapter presents various methods of computer simulation for industrial ventilation design. 

The use of computer simulations to study internal motions and thermodynamic properties is receiving increased attention. One important use of the method is to provide a more fundamental understanding of the molecular information contained in various kinds of experiments on these complex systems. In the first part of this paper we review recent work in our laboratory concerned with the use of computer simulations for the interpretation of experimental probes of molecular structure and dynamics of proteins and nucleic acids. The interplay between computer simulations and three experimental techniques is emphasized (1) nuclear magnetic resonance relaxation spectroscopy, (2) refinement of macro-molecular x-ray structures, and (3) vibrational spectroscopy. The treatment of solvent effects in biopolymer simulations is a difficult problem. It is not possible to study systematically the effect of solvent conditions, e.g. added salt concentration, on biopolymer properties by means of simulations alone. In the last part of the paper we review a more analytical approach we have developed to study polyelectrolyte properties of solvated biopolymers. The results are compared with computer simulations. 

The method of mathematical simulation has many advantages, and is very close to the physical experiment. However the further development of this approach /a consideration of volume effects, reversible reactions and so onj can be rather difficult because it will reau.ire too much computer time, therefore it is expedient to search some simple analytical or semianalytical approximate approaches to the calculation of cross-linking kinetics and conformational properties of cross-linked macromolecules. The results obtained bv the Fonte Carlo calculation can serve as criteria of the accuracy of such approximation. 

The flow resistance behavior of the reconstructed medium can now be examined by performing 3D flow simulations with the Lattice Boltzmann method (Chen and Doolen, 1998), and obtaining the permeability of the material (Konstandopoulos, 2003). Figure 8(a) depicts a visualization of 3D flow tubes and flow velocity distributions at different cross sections in a reconstructed filter material. Figure 8(b) shows the comparison of computer simulated and experimental permeabilities obtained with the experimental protocol described in Konstandopoulos (2003). 

It is difficult to attribute quantitatively by experiment the rate enhancements of the different factors contributing to catalysis. Protein engineering can get close to accurate answers when dealing with nonpolar interactions, especially in subsites. But analysis of mutation is at its weakest when altering residues that interact with charges (Chapter 15). The next development must be in improved methods of computer simulation. Controversies arise when there are no intermediates in the reaction because the kinetics can fit more than one mechanism. Again, computer simulation will provide the ultimate answers. 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

The first [5] super-cyclodextrin whose nano-sized cyclo-pentameric array is held only by a mechanical bond was synthesized by the pentakis-azo coupling of a new hermaphrodite monomer with 2-naphthol as a stopper, isolated by chromatography, and characterized by MS, 2D NMR, and visible spectral methods with the help of computer simulation [60], Cyclic pentamer (as red film, 15% yield) accompanied with the corresponding monomer (20%) (Figure 22) was obtained. 

A prime application of graphical methods in modern distillation technology is for analyzing the results of computer simulations. Several of the graphical construction rules can be bent in order to benefit from computer accuracy and to reduce effort. Johnson and Morgan (28) described several key considerations their work is expanded here using the author s experience. 

As announced above these findings are in astonishing agreement with the heuristic pictures of the diffusion mechanism discussed in the framework of some microscopic diffusion models. But, besides being free of the conceptual drawbacks (the ad hoc assumptions) of the classical diffusion models, the MD method of computer simulation of diffusion in polymers makes it possible to get an even closer look at the diffusion mechanism and explain from a true atomistic level well known experimental findings. For example the results reported in (119,120) on the hopping mechanism reveal the following additional features. 

As most chemical and virtually all biochemical processes occur in liquid state, solvation of the reaction partners is one of the most prominent topics for the determination of chemical reactivity and reaction mechanisms and for the control of reaction conditions and resulting materials. Besides an exhaustive investigation by various experimental methods [1,2,3], theoretical approaches have gained an increasing importance in the treatment of solvation effects [4,5,6,7,8], The reason for this is not only the need for sufficiently accurate models for a physically correct interpretation of the experimental data (Theory determines, what we observe ), but also the limitation of experimental methods in dealing with ultrafast reaction dynamics in the pico- or even subpicosecond regime. These processes have become more and more the domain of computational simulations and a critical evaluation of the accuracy of simulation methods covering experimentally inaccessible systems is of utmost importance, therefore. 

The problems being addressed in recent work carried out in various laboratories include the fundamental nature of the solute-water intermolecular forces, the aqueous hydration of biological molecules, the effect of solvent on biomolecular conformational equilibria, the effect of biomolecule - water interactions on the dynamics of the waters of hydration, and the effect of desolvation on biomolecular association 17]. The advent of present generation computers have allowed the study of the structure and statistical thermodynamics of the solute in these systems at new levels of rigor. Two methods of computer simulation have been used to achieve this fundamental level of inquiry, the Monte Carlo and the molecular dynamics methods. 

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