Computer simulation methods

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

1 Time Averages, Ensemble Averages and Some Historical Background 

Suppose we wish to determine experimentally the value of a property of a system such as the pressure or the heat capacity. In general, such properties will depend upon the positions and 

, f) where pi is the momentum of particle 1 in the x direction and is its X coordinate). Over time, the instantaneous value of the property A fluctuates as a result of interactions between the particles. The value that we measure experimentally is an average of A over the time of the measurement and is therefore known as a time average As the time over which the measurement is made increases to mfinity, so the value of the following integral approaches the true average value of the property  

In Equation (6.3), E(p, r ) is the energy, Q is the partition function, kg is Boltzmann s constant and T is the temperature. The partition function is more generally written in terms of the Hamiltonian, Sf for a system of N identical particles the partition function 

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Kremer K 1996 Computer simulation methods for polymer physics Monte Carlo and Molecular Dynamics of Condensed Matter Systems vol 49, ed K Binder and G Ciccotti (Bologna Italian Physical Society) pp 669-723... 

Swope W C and Andersen H C 1995 A computer simulation method for the calculation of chemical potentials of liquids and solids using the bicanonical ensemble J. Chem. Phys. f02 2851-63... 

D. W Heermann, Computer Simulation Methods in Theoretical Physics, Springer, Berlin, 1986. 

The interface properties can usually be independently measured by a number of spectroscopic and surface analysis techniques such as secondary ion mass spectroscopy (SIMS), X-ray photoelectron spectroscopy (XPS), specular neutron reflection (SNR), forward recoil spectroscopy (FRES), scanning electron microscopy (SEM) and transmission electron microscopy (TEM), infrared (IR) and several other methods. Theoretical and computer simulation methods can also be used to evaluate H t). Thus, we assume for each interface that we have the ability to measure H t) at different times and that the function is well defined in terms of microscopic properties. 

Another special case of weak heterogeneity is found in the systems with stepped surfaces [97,142-145], shown schematically in Fig. 3. Assuming that each terrace has the lattice structure of the exposed crystal plane, the potential field experienced by the adsorbate atom changes periodically across the terrace but exhibits nonuniformities close to the terrace edges [146,147]. Thus, we have here another example of geometrically induced energetical heterogeneity. Adsorption on stepped surfaces has been studied experimentally [95,97,148] as well as with the help of both Monte Carlo [92-94,98,99,149-152] and molecular dynamics [153,154] computer simulation methods. 

Wetting phenomena on irregularly rough surfaces have not been studied so far. It seems quite reasonable to use computer simulation methods for this purpose. Of course, such computer simulation would be very expensive as the finite size of the simulation cells would require appropriate averaging over different spatial distributions of surface inhomogeneities. Nevertheless, with modern fast computers and using multispin coding techniques such calculations can be efficiently carried out for lattice gas systems. 

In addition to various analytic or semi-analytic methods, which are based on the theory of the liquid state and which are not the subject of this chapter, almost the entire toolbox of molecular computer simulation methods has been applied to the theoretical study of aqueous interfaces. They have usually been adapted and modified from schemes developed in a different context. 

H. Gould and J. Tobochnik, Computer Simulation Methods, Applications to Physical Systems, Addison-Wesley Publishing Co., Reading, MA, 1988. 

Perhaps the most common computer simulation method for nonequilibrium systems is the nonequilibrium molecular dynamics (NEMD) method [53, 88]. This typically consists of Hamilton s equations of motion augmented with an artificial force designed to mimic particular nonequilibrium fluxes, and a constraint force or thermostat designed to keep the kinetic energy or temperature constant. Here is given a brief derivation and critique of the main elements of that method. 

Shing, K. S. Chung, S. T., Computer-simulation methods for the calculation of solubility in supercritical extraction systems, J. Phys. Chem. 1987, 91, 1674-1681... 

So far, there have only been a few modeling studies to try to link local fluid flow to bed structure. Chu and Ng (1989) and later Bryant et al. (1993) and Thompson and Fogler (1997) used network models for flow in packed beds. Different beds were established using a computer simulation method for creating a random bed. The model beds were then reduced to a network of pores, and either flow/pressure drop relations or Stokes law was used to obtain a flow distribution. 

Several weaknesses and disadvantages of the computer simulation methods can also be mentioned. Foremost among these limitations is the fact that none of the commonly used models for water interactions... 

The roughening transition has also been studied by computer simulation methods . Figure 42 shows characteristic configurations of a f.c.c. (100) surface in the simple solid-on-solid (SOS) model, calculated by Gilmer . The roughening temperature in this model corresponds to a parameter k T/ = 0.6. 

Information about computer simulation methods is given in-Annex 4. 

The data on bamase and barstar may be analyzed by simple17 or computer-simulation methods.1819... 

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