Simple empirical models

In this section we focus on simple molecular-scale empirical models which aim at describing the properties of classes of systems characterized by some common physical features. These models target the qualitative prediction of general trends 

The original form of the GB potential applies to systems consisting of identical uniaxial particles. Some extensions of the Gay-Beme model have been proposed to overcome this Hmitaticai by generalizing the potential to dissimilar biaxial molecules [69, 70]. Moreover, the GB model can easily be combined with other potentials to add a few chemical details and establish a closer link with the structure of real molecules [71]. TTie effects of adding electric multipoles to the GB potential have been studied and important modifications have been observed in the overall molecular organization [68, 72]. 

GB particles are quite realistic models for small molecules, but once linked together with specific bonded interactions similar to those in (3), they can also be used as building blocks of more complex systems, such as polymers [73-77], LC dendrimers [78], and elastomers [79], as well as covalently bonded fullerene-mesogen systems [41, 80] or end-capped oligomers [81]. Among these studies, a few attempts have also been made to use the GB potential as a basis for CG-like parameterization, using atomistic simulations as the reference potential [68,76,81]. 

A computationally more efficient approach to the modeling of polymer systems is the Dissipative Particle Dynamics (DPD) method, introduced by Hoogerbmgge and Koelman [84] to describe the dynamics and rheological properties of complex fluids, 

Espanol and Warren [90] showed that, unlike the conservative force, the dissipative and stochastic forces cannot be independent and must be coupled together through a fluctuation-dissipation relation  

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Fissore, A. A., and G.. A. Lieheck. 1991. A simple empirical model for predicting velocity distri butions and comfort in a large slot ventilated space. ASHRAE Transactions, vol. 97, no. 2. 

Trends in air pollutant concentrations can be predicted with simple empirical models based on atmospheric and laboratoiy data. Concentrations of nonreactive pollutants from point sources can be predicted vfith accuracy well within a factor of 2 predictions are more likely to be too high than too low, especially predictions of concentration peaks. Concentrations of reactive pollutants, such as ozone and other photochemical oxidants, can be predicted reasonably well with photochemical-diffusion models when detailed emission, air quality, and meteorolc c measurements are available most such predictions of air pollution in Los Angeles, California, have been accurate to within approximately 50% for ozone. Detailed performance analyses are found elsewhere in this chapter. 

Many mixtures exhibit edge effects such that the behavior of the formulation shows drastic changes when one or more of the components is omitted from the mixture [Anderson and McLean (1974)]. Thus, if simple empirical models such as Equations 12.90 and 12.91 are to be used to model the system, it is often best to work in regions that have all components present. Such systems can be prepared with so-called pseudo-components [Cornell (1990)] as shown in the lower two panels of Figure 12.33. The pseudo-components correspond to the vertexes in these designs and are seen to be mixtures that are relatively rich in one of the components. In practice, the pseudo-components can be prepared first, and then the other mixtures in the design can be prepared from these pseudo-components. 

This simple empirical model for predicting the I.S. shift of a Mossbauer nucleus placed in a metallic system (alloys, as well as intermetallic compounds), uses differences in the tabulated macroscopic work functions and bulk moduli to model differences in the microscopic electronegativities and electron densities at... 

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

Advances in NMR instrumentation and methodology have now made it possible to determine site-specific proton chemical shift assignments for a large number of proteins and nucleic acids (1,2). It has been known for some time that in proteins the "structural" chemical shifts (the differences between the resonance positions in a protein and in a "random coil" polypeptide (3-5),) carry useful structural information. We have previously used a database of protein structures to compare shifts calculated from simple empirical models to those observed in solution (6). Here we demonstrate that a similar analysis appears promising for nucleic acids as well. Our conclusions are similar to those recently reported by Wijmenga et al (7),... 

There are various approaches to parameterizing the process of formation and destruction of the ozone layer. The difficulty of deriving dynamic models of the ozone cycle in the atmosphere has to do with the participation in the cycle of more than 75 chemical reactions, a qualitative and quantitative description of which is impossible without deriving detailed models of the many minor gas components of the atmosphere. Nevertheless, there are empirical models of the ozone layer, which make it possible, under the present climatic situation, to obtain adequate spatial distributions of ozone. For instance, Bekoryukov and Fedorov (1987) derived a simple empirical model of total ozone content confirmed by observational data for the Southern Hemisphere ... 

Marcus, Y. (1994). A simple empirical model describing the thermodynamics of hydration of ions of widely varying charges, sizes, and shapes. Biophys. Chem. 51, 111—127. 

The present system is sufficiently general to incorporate any type of reaction network, from simple empirical models to detailed mechanistic ones. However, in the following discussion, pyrolysis models reported in our previous publications (12,13,14,16) will be employed. 

Isothermal flow curves are often summarized by simple empirical models to understand fabrication performance. The Cross model [39,40], given by Equation (1), is well-suited for fitting SAN copolymer data as seen in Figure 13.4. 

A practical approach we have used in the past is to define a simple empirical model for the decomposition of feedstock (single or mixed feeds). Then yields, product gas composition, expansion on cracking, partial pressure, and so forth, are all calculated by using a back-up program which relates yields to decomposition or other internal measures of severity and to the other conditions of cracking (reaction time, average partial pressure). 

As explained in Section 6.2 simple empirical models such as those of Eq. (6.1) and Eq. (6.2) are usually applied. They can be easily generalized to more than two variables. Usually not all possible terms are included. For instance, when including three variables one could include a ternary interaction (i.e. a term in. vi.vi.vy) in Eq. (6.1) or terms with different exponents in Eq. (6.2). such as. vi.v , but in practice this is very unusual. The models are nearly always restricted to the terms in the individual variables and binary interactions for the linear models of Eq. (6.1), and additionally include quadratic terms for individual variables for the quadratic models of Eq. (6.2). To obtain the actual model, the coefficients must be computed. In the case of the full factorial design, this can be done by using Eq. (6.5) and dividing by 2 (see Section 6.4.1). In many other applications such as those of Section 6.4.3 there are more experiments than coefficients in the model. For instance, for a three-variable central composite design, the model of Eq. (6.2)... 

Three types of surface are in use for water simulations. The first consists of simple empirical models based on the LJ-C potential. There seems to be no purpose in continuing to develop and use such models as they give little, if any, new information. A second group attempts to improve the accuracy of the potential using semiempirical methods based on a comprehensive set of experimental data. These models allow for physical phenomena such as intramolecular relaxation, electrostatic induced terms, and many-body interactions, all of which are difficult to incorporate correctly in liquid water theories. There is room for much more work in these areas. The third group makes use of the most advanced ab initio methods to develop accurate potentials from first principles. Such calculations are now converging with parameterized surfaces based on accurate semiempirical models. Over the next few years it seems very likely that the continued application of the second and third approaches will result in a potential energy surface that achieves quantitative accuracy for water in the condensed phase. 

It is likely, in the interim, while we await models from the molecular modeling perspective for the more difficult complex fluids, that the most success in predicting fluid mechanics results for non-Newtonian fluids will come from a hybrid approach combining some elements of both continuum mechanics and molecular modeling to produce relatively simple empirical models. There is a great deal of current research focused on all aspects of constitutive model development on numerical analysis of flow solutions based on these models and on experimental studies of many flows. There are a number of books and references available, but this is a complicated field that really requires a textbook/class of its own. At this point, it is time to return from our little sojourn into the land of complex fluids and come back to the principle subject of Newtonian fluids. 

The angular overlap model is a relatively crude method which appears to yield results at least as good as those afforded by the crystal field model. As with all simple empirical models, the AOM depends on many approximations and assumptions which cannot be expected to be even approximately correct. Thus, for example, the parameter a is assumed to depend only on the identity of the metal and the ligand, and on the internuclear distance it is independent of the stoichiometry or stereochemistry. The theoretical basis for assuming the proportionality of the AOM matrix elements to overlap integrals is closely related to the Wolfs-berg-Helmholz approximation for the off-diagonal matrix elements of the one-electron operator ... 

Antolini and Gonzalez recently proposed a simple empirical model to evaluate the contribution of alloyed and non-aUoyed platinum and tin to the ethanol oxidation reaction on Pt-Sn/C catalysts for DEFC [194]. On the basis of the model, the ethanol oxidation on partially alloyed catalysts occurs through a dual pathway mechanism, separately involving the PtsSn and Pt-SnOx phases. The model, validated by experimental data, can predict the performance of a DEFC by varying the Sn content and/or the degree of alloying of Pt-Sn/C catalysts used as the anode material (Fig. 8.20). 

An alternative approach is to start from the onset with a low resolution model, which is set up essentially by physical intuition, where a whole molecule or monomer is replaced by one or few beads. The typical aim of those simple empirical models is to examine the minimum set of molecular features needed to obtain a given molecular organization, for instance a certain anisotropy of shape or the presence of electric multipole moments, and the qualitative relatirm between variations in the microscopic model and macroscopic properties. This type of modeling does not necessitate a preliminary weU-defined and known chemical structure, but is more akin to a reverse molecular engineering process, where one guesses what key features are needed to achieve the desired macroscopic behavior before actually trying to write down and possibly synthesize a certain molecule. 

The quantity, would then be used to calculate the apparent viscosity of the polymer solution within the porous medium, /(C,y ), which will have been established experiinentally or by using a simple empirical model. 

Ochsenfeld and coworkers have compared the sensitivity of ab initio methods against empirical methods in calculations of NMR chemical shifts from molecular structures of several prototypical peptides. They find that although simple empirical models can give rather good agreement with experimental results they are highly insensitive for structural (and electronic) changes. On the other hand the cheap empirical methods are continuously improved while ab initio are not feasible at all to use. Ab initio methods can also be used to improve the empirical methods same way as it is done for the empirical force fields. 

The statistical adiabatic channel model was originally introduced as a simple empirical model to describe kinetic experiments with as few parameters as possible, providing nevertheless a connection to a more fundamental theory. While it was clear from the start that the model could also be used as an ab initio theory, the numerical computations necessary in this context seemed impractical at the time of the invention of the model. Indeed, even the simple empirical approach was computationally demanding by the standard of the resources available then. During the past decades the computational... 

Software packages for data interpretation that are provided by rheometer manufacturers sometimes include programs to calculate the parameters of empirical viscosity equations based on experimental data. It is important to keep in mind, however, that these are simple empirical models and that they do not provide an accurate fit of the entire viscosity curves of real materials. Also, there is no unique procedure for inferring parameter values from data. When such equations are fitted to experimental data, information is lost. For example, it is not possible to use such an equation to infer the molecular weight distribution using the methods described in Chapter 8. 

Barnes, H. A. Roberts, G. P. A simple empirical model describing the steady-state shear and extensional viscosities of polymer melts, J. Non-Newtonial Huid Mech., 44, pp. 113-126 (1992). 

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