Computer simulation even parts

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

Various issues in the development of a flow model and its numerical simulation have been already discussed in the previous section. It will be useful to make a few comments on the validation of the simulated results and their use in reactor engineering. More details are discussed in Part III and Part IV. Even before validation, it is necessary to carry out a systematic error analysis of the generated computer simulations. The influence of numerical issues on the predicted results and errors in integral balances must be checked to ensure that they are within the acceptable tolerances. The simulated results must be examined and analyzed using the available post-processing tools. The results must be checked to verify whether the model has captured the major qualitative features of the flow such as shear layers and trailing vortices. 

As we are going to show later in this chapter, computer simulations provide capabilities of solving problems which cannot be solved by theory or experiment, or even a combination of both. Let us start with asking how scientists and engineers approach physical systems. The question is fairly general, not even limited to materials science and engineering (MSE), let alone to polymer liquid crystals (PLCs). The answer is that one first describes the system, specifying the parts and their interactions, and then tries to make predictions what the system will do in the future under certain imposed conditions. 

In connection with the problem of pseudohomogeneity, it is clear that a reaction occurring inside a solid phase is not directly measurable. The internal flux of a metabolic reaction can thus only be checked by measuring the external fluxes in the liquid medium by means of computer simulation. Bar-ford and Hall (1979) found that an external overall flux does not reflect, even in an approximate manner, the internal fluxes. Moreover, in vitro examination of an isolated section of metabolism is inadequate for quantification of the coordinated and integrated biochemical control of an intact living system due to in vivo interactions between different parts of the metabolism. 

Combustion occurs with a large number of intermediate steps and even simple processes, such as the ones listed in Table 10.1, occur through dozens of coupled elementary reactions. With computer simulations it is possible to describe the interaction between the reactions, and concentration profiles can be calculated. In order to perform the computer calculations it is necessary to know the rate constants for the individual elementary reactions. Comparisons between theory and experiments are best made for a flat, premixed flame, which in its central part can be considered to have only onedimensional (vertical) variation, allowing computer calculations to be performed comparatively easily. The most important reactions are included in the computer description. In Fig. 10.1 experimental and theoretically calculated concentration curves are given for the case of low-pressure ethane/ oxygen combustion. As examples of important elementary processes we give the reactions... 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

In a similar way, computational chemistry simulates chemical structures and reactions numerically, based in full or in part on the fundamental laws of physics. It allows chemists to study chemical phenomena by running calculations on computers rather than by examining reactions and compounds experimentally. Some methods can be used to model not only stable molecules, but also short-lived, unstable intermediates and even transition states. In this way, they can provide information about molecules and reactions which is impossible to obtain through observation. Computational chemistry is therefore both an independent research area and a vital adjunct to experimental studies. 

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