Dimensionality, polydisperse systems

Numerical results for the some model polydisperse systems have been reported in Refs. 81-83. It has been shown that the effect of increasing polydispersity on the number-number distribution function is that the structure decreases with increasing polydispersity. This pattern is common for the behavior of two- and three-dimensional polydisperse fluids [81] and also for three-dimensional quenched-annealed systems [83]. 

Until now, the theoretical discussion has focused on monodisperse two-dimensional model systems. However, some studies have been performed on polydisperse systems, notably by Weaire et al. [68-72] The evolution of a soap froth of random cell sizes and shapes, known as a Voronoi network, was simulated by computer [68] (Fig. 7). The condition that three films must always meet at angles of 120° was again used. Cells with more than six sides were found... 

On deformation of the system, the bubbles are deformed, which increases their Laplace pressure p. Moreover, some films between particles are stretched and others are compressed, causing surface tension gradients to form, which also needs energy. Above a certain stress, yielding may occur, which means that bubbles (or drops) start to slip past each other, which generally occurs in planes about parallel to the direction of flow. Calculation of the shear modulus and the yield stress from first principles is virtually impossible because of the intricacy of the problem for a three-dimensional and polydisperse system, but trends can be predicted. One relation is that these parameters are proportional to the average apparent Laplace pressure... 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

Figure 28. Complex biomesogenic organizations modeling evolutionary developments (a) SFM visualization of polydisperse DNA [75] (b) nucleic acid textures (left to right) chicken-DNA, (U) (A) duplex (U) (A) (U) triplex, (G) (G) (G) (G) quadruplex, (U) (A) (L-Lys)5 complex [75] (c) building up of liquid-protein-nucleic acid multilayer systems as two-dimensional simulations of the grand evolutionary triad - at least in partial cooperation with a mediating water milieu [7 a, 17, 18, 33 a, c, p, q, 63a, 75].

We present the experimental results of the time evolution of the polydispersity, and the changes in the aggregation mechanism, and discuss the mathematical background of the observed phenomena. In the mathematical analysis we do not limit the dimensionality of the system thus, the results are readily applicable for 3D aggregating systems as well. 

Synthetic copolymers are always poly disperse, i.e., they consist of a large number of chemically similar species with different molar masses and different chemical compositions. Owing to this polydispersity, characterization of copolymers does usually not provide the number of individual molecules or their mole fraction, mass fraction, etc. but requires the use of continuous distribution functions or their averages. Continuous thermodynamics, developed by Ratzsch and Kehlen [1], can be directly applied to the calculation of thermodynamic properties, including phase equilibria, because this theoretical framework is based completely on continuous distribution functions, which include all the information about these functions and allow an exact mathematical treatment of all related thermodynamic properties. Continuous thermodynamics have been used for calculation of phase equilibria of systems containing two-dimensional distributed copolymers [1-8]. The purpose of this contribution is the application of continuous thermodynamics to copolymer fractionation according to the chemical composition and molecular weight. 

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