Polydisperse systems critical points

Polymer networks are formed from functional precursors by covalent bond formation [1], As a result, molecular weights and polydispersity increase and the system passes through a critical point, the gel point. At this point, an infinite structure (molecule) is formed for the first time. Beyond the gel point, the fraction of the infinite structure (the gel) increases at the expense of finite (soluble) molecules (the sol). The sol molecules become gradually bound to the gel and eventually all precursor molecules can become a part of the gel - the network. This is not always the case for different reasons sometimes sol is still present after all functional groups have reacted. In passing from the gel point to the final network not only the gel fraction increases, but also the network becomes denser containing increasing amounts of crosslinks and strands between them called elastically active network chains. 

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.

The maximum of the dissymmetry lies at 5-6 wt % of the polymer near the quasi-binary spinodial. All maxima are indicated by arrows in Figure 6. In our opinion polydispersity is the main reason that the maximum of critical opalescence is not found at the critical point. In a system consisting of a polydisperse polymer and a solvent the shape of the spinodial surface may be such that highly unsymmetrical fluctuations may occur in the critical region and give rise to the above mentioned... 

The simple thermodynamic model derived in Sect. 2.1 has been useful to get a qualitative insight into the phase separation process. When one intends to apply it to an actual system, the significant influence of polydispersity is clearly evidenced. For example, Fig. 13 shows the experimental cloud-point curve for a DGEBA-CTBN binary mixture, together with binodal and spinodal curves calculated by assuming monodisperse components [66] (curves are arbitrarily fitted to the critical point). The shape of the CPC and precipitation threshold temperature (maximum of the CPC) appearing at low modifier concentrations are a clear manifestation of the rubber polydispersity [77]. 

Continuous thermodynamics has also been applied to derive equations for spinodal, critical point and multiple critical points. To do so with continuous thermodynamics is much easier than in usual thermodynamics. Spinodal and critical points may be calculated for very complex systems or for cases in which the segment-molar excess Gibbs energy and depends on some moments of the distribution function. In simple cases (for example, a solution of a polymer in a solvent, where the segment-molar excess Gibbs energy is independent of the distribution function) the equations of the spinodal and the critical point are known from the usual thermodynamic treatment. However, for more complex systems continuous thermodynamics has achieved real progress, for example, for polydisperse copolymer blends, the polydispersity is described by bivariant distribution functions. ... 

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... 

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