Computer simulations -

What are the advantages to be gained from simulation Some are given in Section 1.4 and others are listed here  

Here again, no precise instructions can be given because each situation will demand a tailored approach. Note Numerical simulation in many ways resembles the what-if scenario technique available in spreadsheets programs. Several programs supplied with this book allow the reader to play with functions and noise levels.) 

Optimization of the experimental conditions, so that reliable data can be acquired, or that data acquisition is more economical. 

If a well-developed theory is available, the limits of the experimental system (chemistry, physics, instrumentation) can be estimated. 

Molecular and macroscopic models can be effectively tested by Monte-Carlo (MC) or molecular dynamics (MD) computer simulations. These techniques are a valuable source of data for the clarification of the properties of a system and the evaluation of a theory. For this reason they have been used extensively in studies at charged interfaces. 131-133,146,155 157 (jjg majority of this work concerns the properties 

The above result may raise questions, because the properties of adsorbed monolayers at charged interfaces should be governed by the long-range coulombic interactions. However, it can be easily explained if we take into account that, when we adopt an adsorption mechanism like that represented by Eq. (2), in fact, we model the adsorbed layer as a mixture of adsorbate A molecules and solvent clusters Sa with dimensions equivalent to A. That is, the adsorbed layer consists of species with dimensions greater than 0.25 nm and therefore the (hstance of the closest approach between two adsorbed dipoles cannot fall below 0.5 - 0.6 nm. Thus when we model the adsorbed layer as a mixture of adsorbate A molecules and solvent clusters Sa, the coulombic interactions stop to play the dominant role regarding the properties of this layer. This result is independent of whether we have polarizable or non-polarizable adsorbed molecules and, in fact, verifies the use of 

Due to its simplicity, this trend being developed for more than 20 years has reached its limits and therefore significant improvements in this area are not expected from now on. In contrast, we expect an essential development in the second trend. The second trend realizes the complexity of the charged interface and attempts to mimic it as close as possible. However, the consideration of local order and multistate solute and solvent molecules, as well as the treatment of the whole 

In any case there is still a long way until the development of a truly satisfactory molecular theory capable of predicting a priori and quantitatively the adsorption features of any solute. Until that occurs, the two trends mentioned above, together with computer simulations, will co-exist for different scopes The first trend for analyzing experimental data, and for applications to complicated adsorption phenomena as well as to interfacial phenomena affected by adsorption. The second trend along with computer simulations for a better understanding of the molecular nature of the adsorption on electrodes. 

Fowler and E. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, 1939. 

Currently, there are three different approaches to the problem one is theoretical and the other two are experimental. The three approaches are computer simulation (theoretical), homolog modeling, and de novo prediction (experimental). 

The underlying principles that characterize the theoretical approach are twofold  

The basic concepts which characterize the experimental approach are correct atomic positions, correct topology (connectivity of secondary structures), correct architecture (gross arrangement of secondary structure), and correct structural class 

Computer programs have been written for simulations from one stage to another in comparison with experimental data. 

Statistical Mechanics Approach In this approach, free energy is considered relatively as a constant. By using Monte Carlo simulation on model proteins, a random-number generator is employed to move molecules at random. Monte Carlo simulation aims at the calculation of the critical point at which the protein molecule collapses. This is basically the smdy of phase transition. 

Gas bubble formation and blistering effects have been widely observed in high-dose implantations of inert-gas ions. Backscattering measurements of depth distributions often show very low concentrations of implanted species in the nearsurface region. This indicates that the inert-gas atoms can escape from the material even without sputtering. In these cases, the simple model described in the previous sections does not apply. 

Sputtering has been modeled using both Monte Carlo and molecular dynamic computer simulations. A review of the simulation literature is given in Eckstein (1991). 

The SRIM program, a binary collision Monte Carlo approach, has been used to predict sputtering yields (Fig. 12.2). The incident ions and the recoil atoms are 

Andersen, H.H., Bay, H.L. Sputtering yield measurements. In Behrisch, R. (ed.) Sputtering by Particle Bombardment. I. Physical Sputtering of Single Element Solids, Topics in Applied Physics, vol. 47, pp. 145-218. Springer, Berlin (1981) 

Eckstein, W. Computer Simulations of Ion-Solid Interactions, chap. 12. Springer, Berlin, (1991) 

Hubler, G.K. Ion Beam Processing, NRL Memorandum Report 5928. Naval Research Laboratory, Washington, DC (1987) 

Therefore, one has to strike a balance between the degree of sophistication of the analysis and the practical usefulness of the analysis. This balance will depend on the particular interests of the individual. For industrial applications the degree of sophistication of the analysis of melting, as described in Section 7.3.1, is probably sufficient to analyze most practical extrusion problems. However, in some instances one may want to go into much more detail on certain aspects of the melting process. 

Analyses that require numerical techniques to arrive at solutions tend to be quite time consuming and require skilled personnel to develop the computer programs and to interpret the results of the computer simulations. Many people in the extrusion industry do not have the time or inclination to work through elaborate and complex analyses of melting. In this case, the preferred action is to use a less complicated analysis that yields analytical results. In most cases, actual predictions of melting performance can be made with a relatively simple programmable calculator. 

When using commercial simulation packages it is important that the theory behind it is clearly described with assumptions and simplifications. If this is not the case and the user does not really know what type of analysis they are actually using, the value of the results will be questionable. Early extruder simulation packages had 

Thermal degradation involves scission of covalent bonds which can be formally written as Eq. (36). 

With the increasing capabilities of computers and development of new numerical methods, it is now possible to predict polymer properties computationally. In addition to saving time, computer-aided chemistry can sometimes provide new insights into some decomposition mechanisms which are difficult to obtain by experimental techniques. Computer modeling has been used in an increasing number of ways to simulate thermal degradation. A few representative examples are described below. 

Ab-initio calculations were performed on a series of model structures to predict the effect of lateral alkyl substituents on the thermal stability of degradable polycarbonate. The optimized transition structures revealed that the Cc,-0 bond dissociates first, followed by abstraction of the j8-hydrogen atom, developing a carbocation character in the transition state on the atom. Substituents which stabilize the transition state will also accelerate the degradation rate [17]. 

Isotactic polypropylene (iPP) is a major commodity plastic material which cannot be utilized without thermal stabilizers. With a moderately complex structure, iPP is frequently used as a model system to test the different theoretical and experimental approaches to macromolecular degradation. 

It was shown from decomposition kinetics and by treatment with dimethyl sulfide that peroxides consist of two types a fast-decomposing one composed of peradds, and a slowly decomposing one consisting of hydroperoxides and hydro-peresters. During the induction period, the slowly decomposing hydroperoxides accumulate and the oxidation rate is controlled by the rate of decomposition, which may be finally catalyzed by metal ion residues. The autoacceleration stage is controlled by the fast-decomposing peracids [22]. 

---

The ultrasonic testing of anisotropic austenitic steel welds is a commonly used method in nondestructive testing. Nevertheless, it is often a problem to analyze the received signals in a satisfactory way. Computer simulation of ultrasonics has turned out to be a very helpful tool to gather information and to improve the physical understanding of complicated wave phenomena inside the samples. 

R. Rosseau, D. Degreve - Laborelec, Belgium. COMPUTER SIMULATION... 

The solution adopted by us is the use of computer simulations of mathematical models of the process and the mock-up situations. Eventually, simulation techniques will become so accurate, that the mock-up step can be discarded. For the time being it is reasonable to use such models to generate corrections for smaller differences between mock-up and process. 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

This is Camalian and Starling s (CS) equation of state for hard spheres it agrees well with the computer simulations of hard spheres in the fluid region. The excess Hehnholtz free energy... 

Figure A2.3.7 The radial distribution function g r) of a Lemiard-Jones fluid representing argon at T = 0.72 and p = 0.844 detennined by computer simulations using the Lemiard-Jones potential.

Figure A2.3.10 compares the virial and pressure equations for hard spheres with the pressure calculated fonu the CS equations and also with the pressures detemiined in computer simulations.

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

Perturbation theory is also used to calculate free energy differences between distinct systems by computer simulation. This computational alchemy is accomplished by the use of a switching parameter X, ranging from zero to one, that transfonns tire Hamiltonian of one system to the other. The linear relation... 

Koneshan S and Rasaiah J C 2000 Computer simulation studies of aqueous sodium chloride solutions at 298K and 683K J. Chem. Phys. 113 8125... 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Murrell J N, Stace A J and Dammel R 1978 Computer simulation of the cage effect in the photodissociation of iodine J. Chem. Soc. Faraday Trans. II 74 1532... 

From SCRP spectra one can always identify the sign of the exchange or dipolar interaction by direct exammation of the phase of the polarization. Often it is possible to quantify the absolute magnitude of D or J by computer simulation. The shape of SCRP spectra are very sensitive to dynamics, so temperature and viscosity dependencies are infonnative when knowledge of relaxation rates of competition between RPM and SCRP mechanisms is desired. Much use of SCRP theory has been made in the field of photosynthesis, where stnicture/fiinction relationships in reaction centres have been connected to their spin physics in considerable detail [, Mj. 

Interactions between macromolecules (protems, lipids, DNA,.. . ) or biological structures (e.g. membranes) are considerably more complex than the interactions described m the two preceding paragraphs. The sum of all biological mteractions at the molecular level is the basis of the complex mechanisms of life. In addition to computer simulations, direct force measurements [98], especially the surface forces apparatus, represent an invaluable tool to help understand the molecular interactions in biological systems. 

Classical ion trajectory computer simulations based on the BCA are a series of evaluations of two-body collisions. The parameters involved in each collision are tire type of atoms of the projectile and the target atom, the kinetic energy of the projectile and the impact parameter. The general procedure for implementation of such computer simulations is as follows. All of the parameters involved in tlie calculation are defined the surface structure in tenns of the types of the constituent atoms, their positions in the surface and their themial vibration amplitude the projectile in tenns of the type of ion to be used, the incident beam direction and the initial kinetic energy the detector in tenns of the position, size and detection efficiency the type of potential fiinctions for possible collision pairs. 

Ghrayeb R, Purushotham M, Hou M and Bauer E 1987 Estimate of repulsive interatomic pair potentials by low-energy alkalimetal-ion scattering and computer simulation Phys. Rev. B 36 7364-70... 

We carry out computer simulations in the hope of understanding bulk, macroscopic properties in temis of the microscopic details of molecular structure and interactions. This serves as a complement to conventional experiments, enabling us to leam something new something that cannot be found out in other ways. 

Computer simulations act as a bridge between microscopic length and time scales and tlie macroscopic world of the laboratory (see figure B3.3.1. We provide a guess at the interactions between molecules, and obtain exact predictions of bulk properties. The predictions are exact in the sense that they can be made as accurate as we like, subject to the limitations imposed by our computer budget. At the same time, the hidden detail behind bulk measurements can be revealed. Examples are the link between the diffiision coefficient and... 

Flere we consider various aspects of statistical mechanics (see also chapter A2.3 and [2, 3]) that have a direct bearing on computer simulation metiiodology. 

In this section we look briefly at the problem of including quantum mechanical effects in computer simulations. We shall only examine tire simplest technique, which exploits an isomorphism between a quantum system of atoms and a classical system of ring polymers, each of which represents a path integral of the kind discussed in [193]. For more details on work in this area, see [22, 194] and particularly [195, 196, 197]. 

Allen M P and Tildesley D J 1987 Computer Simulation of Liquids (Oxford Clarendon)... 

Sprik M 1993 Effective pair potentials and beyond Computer Simulation in Chemical Physics vol 397 NATO ASI Series C ed M P Allen and D J Tildesley (Dordrecht Kluwer) pp 211-59... 

Bates M A and Luckhurst G R 1996 Computer simulation studies of anisotropic systems. 26. Monte Carlo investigations of a Gay-Berne discotic at constant pressure J. Chem. Phys. 104 6696-709... 

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... 

Hockney R W and Eastwood J W 1988 Computer Simulations Using Particles (Bristol Adam Hilger)... 

---