Structure-property relationships discrete models

This method has been proposed for developing structure-property relationships of nano-structured materials and works as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with equivalent-continuum models. It has been shown that this substitution may be accomplished by equating the molecular potential energy of a nano-structured material with the strain energy of representative truss and continmrm models. 

FIGURE 1.5 Multiscale modeling in computational pharmaceutical solid-state chemistry. Here DEM and FEM are discrete and finite element methods MC, Monte Carlo simulation MD, molecular dynamics MM, molecular mechanics QM, quantum mechanics, respectively statistical approaches include knowledge-based models based on database analysis (e.g., Cambridge Structure Database [32]) and quantitative structure property relationships (e.g., group contributions models [33a]). 

The methods of discrete mathematics, introduced in this chapter, were sufficient for describing chemical compounds as discrete structures. However, once the relationships between properties and structures have to be modelled, non-discrete methods are required. Methods from supervised statistical learning theory and machine learning are particularly useful and thus some of these will be introduced in the next chapter. 

SAR work can be classified into two categories QSAR (quantitative structure-activity relationships) and qSAR (qualitative structure-activity relationships). In QSAR analysis, biological activity is quantitatively expressed as a function of physico-chemical properties of molecules. QSAR involves modeling a continuous activity for quantitative prediction of the activity of new compounds. qSAR aims to separate the compounds into a number of discrete types, such as active and inactive or good and bad. It involves modeling a discrete activity for qualitative prediction of the activity of new compounds. 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

The explanatory models are based on file basic concepts of electrical theory—potential, conductance, polarization, induction, etc. Knowledge about the physical mechanisms behind these phenomena is used to provide understanding of similar phenomena in biological materials, and the models are largely influenced by theories concerning the relationship between microanatomy and fundamental electrical properties. It is vital that these models only include discrete electric components for which the essential mode of operation is known. The models are explanatory because one believes that the components of the model represent isolated anatomical structures or physical processes, such that the dominating electrical property can be explained by means of the properties of the component. 

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