Mechanics of Particle Systems

With atoms and molecules taken to be single particles, earlier chapters have followed gas kinetic analysis of collisions, gas pressure, and transfer of energy as heat and work. However, the internal structure and mechanics of molecules— that they are not single point masses—can play a role in thermodynamic behavior and reaction energetics. This chapter focuses on the mechanics of vibration, an internal motion exhibited by all molecules. Though we start by using classical mechanics, it turns out to be an incomplete theory in that it fails to correctly describe very small, very low-mass particle systems. To go beyond classical pictures calls for us to invoke quantum mechanical ideas which are introduced here. The contrast and the correspondence between the classical and quantum pictures of the vibrational motion of molecules is a primary objective of this chapter. 

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A more general formulation of the mechanics of particle systems is based on Hamilton s principle, or the principle of least action. This principle states that the action S defined as... 

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics - the Schrodinger equation - lies beyond the potentialities of modem mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. 

We begin by describing the current understanding of the kinetics of polymerization of classical unsaturated monomers and macromonomers in the disperse systems. In particular, we note the importance of diffusion-controlled reactions of such monomers at high conversions, the nucleation mechanism of particle formation, and the kinetics and kinetic models for radical polymerization in disperse systems. 

First we need to define several quantities which enter into our discussion of the classical mechanics of the system. The symbols mk, Vk, Pk, Pk denote the mass, velocity, linear momentum and time rate of change of the linear momentum of the /cth particle in the laboratory-fixed frame. We remember that the momentum is defined by... 

The mechanical properties of rapidly polymerizing acrylic dispersions, in simulated bioconditions, were directly related to microstructural characteristics. The volume fraction of matrix, the crosslinker volume in the matrix, the particle size distribution of the dispersed phase, and polymeric additives in the matrix or dispersed phase were important microstructural factors. The mechanical properties were most sensitive to volume fraction of crosslinker. Ten percent (vol) of ethylene dimethacrylate produced a significant improvement in flexural strength and impact resistance. Qualitative dynamic impact studies provided some insight into the fracture mechanics of the system. A time scale for the elastic, plastic, and failure phenomena in Izod impact specimens was qualitatively established. The time scale and rate sensitivity of the phenomena were correlated with the fracture surface topography and fracture geometry in impact and flexural samples. 

Besides giving latices of narrow particle size distribution, mixed surfactant systems have shown several other interesting characteristics which lighten some aspects concerning the mechanism of particle nucleation in emulsion polymerization process. 

The mechanism of particle formation at submicellar surfactant concentrations was established several years ago. New insight was gained into how the structure of surfactants influences the outcome of the reaction. The gap between suspension and emulsion polymerization was bridged. The mode of popularly used redox catalysts was clarified, and completely novel catalyst systems were developed. For non-styrene-like monomers, such as vinyl chloride and vinyl acetate, the kinetic picture was elucidated. Advances were made in determining the mechanism of copolymerization, in particular the effects of water-soluble monomers and of difunctional monomers. The reaction mechanism in flow-through reactors became as well understood as in batch reactors. Computer techniques clarified complex mechanisms. The study of emulsion polymerization in nonaqueous media opened new vistas. 

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