Polydisperse systems thermodynamics

The conditions of stability of colloidal particles with respect to further dispersion down to molecular sizes can be found by analyzing the A (d) dependence at d - b. If the value of a does not change with the decrease in d down to the molecular dimensions, and if a can be used to describe the work of dispersing, further dispersion of particles down to molecular sizes is thermodynamically favorable. In a real polydisperse system the dispersed particles of colloidal range with some defined particle size distribution may, however, also fluctuationally form at a = const. 

The binodal curve is the boundary between thermodynamically stable and metastable solutions. The term binodal is used in truly binary systems while in actual polydisperse systems the correct denomination is cloud-point curve (CPC). Thus the experimental determination of this boundary always leads to a CPC. 

Continuous thermodynamics provides a simple way for the thermodynamic treatment of polydisperse systems. Such systems consist of a very large number of similar species whose composition is described not by the mole fractions of the individual components but by continuous distribution functions. For copolymers, multivariate distribution functions have to be used for describing the dependence of thermodynamic behavior on molar mass, chemical composition, sequence length, branching, etc. 

However, traditional chemical thermodynamics is based on mole fractions of discrete components. Thus, when it is applied to polydisperse systems it has been usual to spht the continuous distribution function into an arbitrary number of pseudo-components. In many cases dealing, for example, with a solution of a polydisperse homopolymer in a solvent (the pseudobinary mixture), only two pseudo-components were chosen (reproducing number and mass averages of molar mass of the polymer) which, indeed, are able to describe some main features of the liquid-liquid equilibrium in the polydisperse mixture [1-3]. In systems with random copolymers the mass average of the chemical distribution is usually chosen as an additional parameter for the description of the pseudo-components. However, the pseudo-component method is a crude and arbitrary procedure for polydisperse systems. 

Flory P J and Abe A (1978) Statistical thermodynamics of mixture of rodlike particles. 1. Theory for polydisperse systems, Macromolecules 11 1119-1122. 

Thermodynamics of Polydisperse Systems.— The simplest polydisperse system is perhaps that composed of hard spheres (or discs) with a continuous distribution f(a-) of particle diameter cr. For a multicomponent mixture of hard spheres in the Percus-Yevick approximation, the compressibility equation of state is ... 

The history and fundamentals of continuous thermodynamics will be briefly presented here and has been discussed in detail elsewhere. Before the 1980 s many authors applied continuous distribution functions to specific cases of non-equilibrium thermodynamics, statistical thermodynamics, the VLE of petroleum fractions and the LLE of polydisperse polymer systems. Starting in 1980 a consistent version of chemical thermodynamics directly based on continuous distribution functions was developed and called continuous thermodynamics. The work of Kehlen and Ratzsch," " Gualtieri et al., Salacuse and Stell, Briano and Glandt," are to be mentioned as sources of information. In the following years several groups applied continuous thermodynamics to nearly all important types of polydisperse systems." Cotterman and Prausnitz reviewed the literature up until about 1990. In the 1980 s continuous modelling of phase equilibria was mostly focused on polymer systems, petroleum fractions and natural gases. In the last ten years, this has been expanded to also include problems with asphaltene precipitation from crude oils and wax precipitation from hydrocarbon mixtures. In section 9.4 the more recent papers are discussed. 

Obtaining the spinodal and critical point of a polydisperse system with an equation of state has been illusive but the problem has been solved with continuous thermodynamics an analytical solution of the determinants of traditional thermodynamics has been shown to be possible. Phase equili-... 

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

Phospholipid (PL) vesicles are generally considered to be thermodynamically unstable but as the PL surfactant is relatively immobile and fusion is difficult they are kinetically stable for months (depending somewhat on the method of preparation), and the tendency to return to the lamellar state is very slow. It would appear that monodisperse vesicles tend to be more stable than polydisperse systems. 

The thermodynamic properties of silicate melts however, are complicated by the presence of a large number of different silicate anions. While silicate minerals are usually monor disperse, containing only a single type of anion (e.g. SiojJ in olivine), molten silicates contain a distribution of different polymeric silicate anions of different molecular weights and are thus polydisperse systems. The presence of a distribution of silicate anions in the melt can be inferred from the mixing properties of silicate melts (Richardson, 1956 Masson, I965). However, it has recently become possible to separate some of the... 

Now, when these two species are added to each other, in a given relative concentration, a new species appears with a much narrower size distribution. This is shown in Figure 10.18, where the P-index (a measure of the polydispersity) is plotted against the molar fraction of DDAB. The P-index drops from the initial value of 0.20 (a very broad distribution) to 0.04, a very narrow distribution (stable for months), at a relative percent of 0.4 DDAB to 0.6 oleate (Thomas and Luisi, 2004). Between DDAB molar fractions of 0.41 and 0.60, flocculation occurs, which indicates a thermodynamic instability, in agreement with other cationic systems (Kaler etal., 1989 Marques etal., 1998 Kondo etal, 1995). 

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