Polydisperse systems spinodals

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... 

Introduction. After we have discussed examples of uncorrelated but polydisperse particle systems we now turn to materials in which there is more structure - discrete scattering indicates correlation among the domains. In order to establish such correlation, various structure evolution mechanisms are possible. They range from a stochastic volume-filling mechanism over spinodal decomposition, nucleation-and-growth mechanisms to more complex interplays that may become palpable as experimental and evaluation technique is advancing. 

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]. 

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

Brochard and de Gennes [67] discussed theoretically a flow-induced isotropic-mesophase transition in a polydisperse polymer system occurring through spinodal decomposition. Following Maier-Saupe s [50] theory of the nematic phase, the orientation-dependent interaction energy was taken as... 

These results indicate that spinodal decomposition can be induced by flow even in a homopolymer system, which results in an oriented nematic phase with long chains and an isotropic liquid with short chains. Though the above argument is based on a bimodal system, the same principle has also been applied to the polydisperse case [67]. However, this approach still does not take flow-induced conformational ordering into account, which may couple to the anisotropic interactions. 

For systems with polydisperse primary chains, the spinodal and critical conditions have to be determined from the appropriate Gibbs determinants. 

The hole theory offers an excellent basis to evaluate the phase behavior of polymer systems. The description of the spinodal conditions are almost quantitative without the introduction of empirical parameters. The cell free volume is very important for this quantitative success. The influence of polydispersity on the spinodal conditions in the Simha-Somcynsky theory is not restricted to the mass average molar mass. 

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. ... 

Beginning in the 1990s calculations of high-pressure phase equilibria of polydisperse polymer systems were performed. For example, Enders and de Loos calculated cloud-point and spinodal curves in the high-pressure range for methylcyclohexane + poly(ethenylbenzene) and compared their results with experimental data. Enders and de Loos ° used a Gibbs-energy model with pressure dependent parameters and models that include an equation of state, such as the lattice fluid model introduced by Hu et for the monodisperse and... 

The destruction of dense emulsions is a rich domain. A large variety of behavior is observed. At the macroscopic scale, the system may exhibit a demixtion, as represented by Figure 8.21, or an homogeneous destruction (the emulsion remains macroscopically homogeneous at any time). At the colloidal scale, the droplet size distribution may change from almost bimodal to polydisperse or very monodis-perse. In addition, when the continuous phase spinodally decomposes, one phase may cause severe destruction, as described in Section 8.4. 

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