Polydisperse systems points

A mathematical analysis of equilibrium behavior in a polydisperse system leads to the conclusion that from the slope at any point on curve C, we can obtain a weight average molecular weight at that local concentration of solute i. Several software programs are available for carrying out the necessary calculations [5]. 

Experimental data [94] were in good agreement with the theory at high volume fractions (0.99 < tj) < 1) at low to intermediate 4> values, empirical equations were fitted to the results. An interesting point is that extrapolation of the plot of (j) against osmotic pressure to zero 7t gives = 0.712 i.e. less than the volume fraction for monodisperse, undistorted spheres. This implies that droplet packing is less efficient in polydisperse systems. 

Fig. 3.19 Critical micelle temperature versus the polymer concentration calculated for an aqueous solution of Pluronic P105 (PEO37PPO56PEO37) (Linse 1994b). Results from a polydisperse model (MJMn = 1.2) are shown as solid lines and for the monodisperse polymer as a dashed line. The curves for the polydisperse system are labelled in terms of the number of components representing the polydisperse polymer. Points with constant micellar volume fractions (criterion of the cmc) are represented by dotted curves, the volume fraction being indicated. Experimental data from Alexandridis et al. (1994a) are also included as filled squares.

Ideally, it is desirable to solve equations 1 or 2 for the complete size distribution whenever polydisperse systems are being analysed. However, from the computational point of view, the direct application of the equations for monodisperse systems is more... 

Idealization of particulate fluidization provides the essential concept towards a basic understanding of fluid-particle systems in terms of the most significant factors, L/S in particular and G/S after appropriate corrections. The nature of fluid-particle motion cannot be properly understood, e.g., the preferential arrays of particles, L/S versus G/S, without supplementary studies of the basic mechanisms. Of immediate interest from a practical point of view is how to adapt the relatively simple relations derived for ideal fluidization to G/S systems, as already exemplified for fast fluidization. For polydisperse systems, the problem was oversimplified to the binary case, and... 

Using the results of [473] obtained for the flow field in the point force model, one can find the mean Sherwood number for a polydisperse system of particles [74] ... 

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. 

However, a major limitation of this model is the impossibility of fitting cloud-point curves for polydisperse systems. Moreover, it cannot deal with the fractionation effect accompanying phase separation, i.e. the dispersed phase will be enriched in the highest molar-mass fractions of modifier but in the lowest molar-mass fractions of the growing thermosetting polymer. This may produce variations in stoichiometry and conversion between both phases. These phenomena can be conveniently treated taking polydispersity of constituents into account. 

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... 

As pointed out by Klee et al., such a system can also be modeled using Eq. (6.33). This was accomplished by approximating the dilute phase as a polydisperse system of core-shell particles [37] and assuming a hard sphere structure factor however, the effect of polydispersity on the structure factor was not accounted for [62]. As well, the GIFT analysis has also been used to verify the presence of a core-shell structure [73]. Ultimately, there is more than one single model that can accurately describe the SAXS in terms of the peak intensity, average size, and Porod scattering law. For example, a eore-shell structure can correctly account for the shell of amphiphilic molecules at the surface of the microemulsion, which Eq. (6.36) cannot. 

This equation appears to have a number of names, of which the Mark-Houwink equation is the most widely used. In order to use it, the constants K and a must be known. They are independent of the value of M in most cases but they vary with solvent, polymer, and temperature of the system. They are also influenced by the detailed distribution of molecular masses, so that in principle the polydispersity of the unknown polymer should be the same as that of the specimens employed in the calibration step that was used to obtain the Mark-Houwink constants originally. In practice this point is rarely observed polydispersities are rarely evaluated for polymers assigned values of relative molar mass on the basis of viscosity measurements. Representative values of K and a are given in Table 6.4, from which it will be seen that values of K vary widely, while a usually falls in the range 0.6-0.8 in good solvents at the 0 temperature, a = 0.5. 

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