Simple Force Functions

The simple force functions described in Secs. 8-2 and 8-3 can be modified in several ways in order to give a more accurate description of the vibrational frequencies. Perhaps the most obvious procedure is the introduction of a few judiciously chosen interaction constants. Physically, this amounts to taking into account the change in the stiffness of a bond resulting from the distortion of other bonds. 

Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 

The rapid rise in computer power over the last ten years has opened up new possibilities for modelling complex chemical systems. One of the most important areas of chemical modelling has involved the use of classical force fields which represent molecules by atomistic potentials. Typically, a molecule is represented by a series of simple potential functions situated on each atom that can describe the non-bonded interaction energy between separate atomic sites. A further set of atom-based potentials can then be used to describe the intramolecular interactions within the molecule. Together, the potential functions comprise a force field for the molecule of interest. 

As a simple example of the general method outlined above, consider the vibrations of the harmonic oscillator under the forcing function

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Some of the simple linear equations that we will simulate can, of course, be solved analytically by the methods covered in Part III to obtain general solutions. The nonlinear equations cannot, in general, be solved analytically, and computer simulation is usually required to get a solution. Keep in mind, however, that you must give the computer specific numerical values for parameters, initial conditions, and forcing functions. And you will get out of the computer specific numerical values for the solution. You cannot get a general solution in terms of arbitrary, unspecified inputs, parameters, etc., as you can with an analytic solution... 

To begin, it is essential to rationalize the equilibration of water within the membrane at AP = 0, APs = 0, j = 0, and = 0. The suggested scenario of membrane swelling is based on the interplay of capillary forces and polymer elasticity. In order to justify a scenario based on capillary condensation, isopiestic vapor sorption isotherms for Nafioni in Figure 6.9(a) are compared with data on pore size distributions in Figure 6.9(b) obtained by standard porosimetry.i In Figure 6.9(a), a simple fit function. 

The formation of complexes is not restricted to mixtures of polyectrolytes and surfactants of opposite charge. Neutral polymers and ionic surfactants can also form bulk and/or surface complexes. Philip et al. [74] have studied the colloidal forces in presence of neutral polymer/ionic surfactant mixtures in the case where both species can adsorb at the interface of oil droplets dispersed in an aqueous phase. The molecules used in their studies are a neutral PVA-Vac copolymer (vinyl alcohol [88%] and vinyl acetate [12%]), with average molecular weight M = 155000 g/mol, and ionic surfactants such as SDS. The force measurements were performed using MCT. The force profiles were always roughly linear in semilogarithmic scale and were fitted by a simple exponential function ... 

Molecular mechanics models differ both in the number and specific nature of the terms which they incorporate, as well as in the details of their parameterization. Taken together, functional form and parameterization, constitute what is termed a force field. Very simple force fields such as SYBYL, developed by Tripos, Inc., may easily be extended to diverse systems but would not be expected to yield quantitatively accurate results. On the other hand, a more complex force field such as MMFF94 (or more simply MMFF), developed at Merck Pharmaceuticals, while limited in scope to common organic systems and biopolymers, is better able to provide quantitative accounts of molecular geometry and conformation. Both SYBYL and MMFF are incorporated into Spartan. 

It is not possible to discuss real-time control without a brief discussion of sensors and the measurements they represent. In traditional process control, the measurements and the properties to be controlled are identical. For instance, one controls the temperature of a fluid using feedback from a thermocouple. There is also generally a fairly predictable relationship between the measurement and the forcing function necessary to change that measurement. Except for unusually simple cases, that is not true of polymer processing. The multiple, complex properties to be controlled cannot be measured and are not always... 

Forcing function is a term given to any disturbance which is externally applied to a system. A number of simple functions are of considerable use in both the theoretical and experimental analysis of control systems and their components. Note that the response to a forcing function of a system or component without feedback is called the open-loop response. This should not be confused with the term open-loop control which is frequently used to describe feed-forward control. The response of a system incorporating feedback is referred to as the closed-loop response. Only three of the more useful forcing functions will be described here. 

The response of the controlled variable to different types of perturbation (forcing function) in set point or load can be determined by inverting the appropriate transform (e.g. equation 7.112). This is possible only for simple loops containing low order systems. More complex control systems involving higher order elements require a suitable numerical analysis in order to obtain the time domain response. 

An entirely different approach, based upon classical mechanics, is the molecular mechanics or empirical force field method (82MI5 83AG(E)1 86MI2). It is assumed that the steric energy ( s) of a molecule can be expressed as a sum of energy contributions [Eq. (6)], where each term is obtained from a simple potential function, such as the one given by Hooke s law. 

In conventional MM the potential energy function (PEF) is parameterized and these optimized parameters are called force-field parameters. Such methods are widely applied in the studies of nucleic acids, proteins, their complexes and other biomolecular systems. A typical, simple force field of a molecular system is defined by the following equation ... 

Since the intermolecular force constants are mainly due to the repulsive part of the potential, simple exponential functions are often sufficient to describe the intermolecular force constants (Fig. 2.5-7 Harada and Shimanouchi, 1967 Schrader, 1978). 

As the application of this treatment depends on a knowledge of the function E r, a), which depends on the nature of the molecular force field, it is relevant next to enquire whether any simple force field leads to a simplification of the results. A good first approximation to the molecular force field is the simple valence force field (see Linnett toj, This approximation implies that dEjdrk for all... 

These enthalpies and entropies are of course a function of the structure of the compound. The thermodynamic quantities for vaporization were discussed previously (page 14), and it was seen that this factor varied in a reasonably predictable way with a change in structure. The thermodynamic quantities for sublimation are the sum of those for vaporization and for fusion, and it is now necessary to consider the latter. These are not as simple a function of the structure as is the boiling point, because they depend on the crystal structure which is possible with the compound and on short-range attractive forces which operate in the crystal. Certain generalizations may, however, be made. 

Figure 28-7 In force field calculations, different levels of approximations are used to reproduce the stretching and compression of chemical bonds The plot shows a Morse potential energy function siipenmposed with various power series approximations (quadratic, cubic, and quartic functions) Note that the bottoms of the curves, representing the bond length for most chemical bonds of interest to medicinal chemists, almost overlap exactly. This nearly perfect fit in the bonding region is the reason simple harmonic functions can be used to calculate tx>nd lengths for unstrained molecular structures in the force lield method.

In the preceding sections we have studied diatomic interactions via U(R). However, the study of diatomic interactions can also be carried out in terms of the force F(R) instead of the energy U(R), where R denotes the internuclear separation. Though there are several methods for the calculation of the force, the electrostatic theorem of Hellmann (1937) and Feynman (1939) is of particular interest in this section, since the theorem provides a simple and pictorial method for the analysis and interpretation of interatomic interactions based on the three-dimensional distribution of the electron density p(r). An important property of the Hellmann-Feyn-man (HF) theorem is that underlying concepts are common to both the exact and approximate electron densities (Epstein et al., 1967, and references therein). The force analysis of diatomic interactions is a useful semiclassical and therefore intuitively clear approach. And this results in the analysis of diatomic interactions via force functions instead of potential ones (Clinton and Hamilton, 1960 Goodisman, 1963). At the same time, in the authors opinion, it serves as a powerful additional instrument to reexamine model diatomic potential functions. 

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