Continuous Random Network Theory

Attempts to describe the structures of amorphous solids date nearly 100 years back. In 1921, Lebedev developed the crystallite theory [2] according to which glasses are an agglomerate of microcrystallites, too small to be detected with (the then) conventional methods. Even though this theory did successfully explain some properties of glasses, it was strongly debated. 

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The first theory of the structure of glass to become widely accepted was that of Zachariasen (1932), called the random network theory [now commonly referred to as the continuous random network (CRN) theory]. This arose... 

Fig. 50. Continuous random network fit to the X-ray diffraction pattern of H20(as) (from Ref. 82>). Experimental data and X Theory---...

Clearly, any measurement that differentiates between the properties of high and low temperature forms of H20(as), and/or delineates the relationship between H20(as) and liquid H20, can be used to test the hypotheses advanced vis a vis their structures. These and the experimental tests suggested, together with the construction of continuous random network models more sophisticated than that for Ge(as), the increased use of computer simulation, and exploitation of the available experimental information to guide the choice of appproximations in a statistical mechanical theory should increase our understanding of H20(as) and, uitimately, liquid H20. 

While the rubber elasticity theory to be described below presumes a randomly cross-linked polymer, it must be noted that each method of network formation described above has distinctive nonuniformities, which can lead to significant deviation of experiment from theory. For example, chain polymerization leads first to microgel formation (9,10), where several chains bonded together remain dissolved in the monomer. On continued polymerization, the microgels grow in number and size, eventually forming a macroscopic gel. However, excluded volume effects, sUght differences in reactivity between the monomer and cross-linker, and so on lead to systematic variations in crosslink densities at the 100- to 500-A level. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

Percolation theory deals with the size and distribution of connected black and white domains and the effects on macroscopic observable properties, e.g., eleetrie conductivity of a random composite or diffusion coefficient of a porous roek. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely eonnected if is close to 1. Therefore at some well-defined intermediate value,... 

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