Spinodal theoretical model

The liquid state is limited by two absolute limit spinodals, located well above the boiling temperature and below P < 0. They are the loci of the absolute stability limit of the homogeneous nucleation. The significance of this issue showed studies on unusual properties of water where the estimation of the spinodal within the negative pressures domain is the fundamental checkpoint for theoretical models. The significance of this issue taken into account in the novel pressure counterpart of the VFT equation recently proposed, namely " ... 

As an additional feature of these experiments we present an approach to estimate the interfacial tension. The standard methods suffer from large uncertainties when the interfacial tensions are very low (refer to the L77 case). We can apply the theoretical model of spinodal decomposition to back-calculate the interfacial tension from the characteristic length and time scales of the instability. 

The issue is more difficult when the composition of the quenched material lies close to the center of the miscibility gap. In this case, the final two-phase system exhibits a bicontinuous geometry, both phases occupying about the same volume fraction, and thus the structure cannot be modeled by a set ofisolated nano-objects. For the centtal part of the miscibihty gap, a theoretical model named spinodal decomposition was proposed (Calm, 1965). At advanced stages, even after nearly having reached the equilibrimn concentrations, both phases stiU evolve by a coarsening process. 

As theoretical models were proposed to give an account for the spinodal decomposition for unstable one-phase compositions (4-6) and for the nucleation and growth mechanism for metastable one-phase compositions (7), the necessity arose for information on the earliest stages of phase separation. It is the reason why additional techniques including neutron scattering (8, 9), N.M.R. spectroscopy (10, 11) and fluorescence emission were used with the aim to delect the phase separation on a molecular scale. 

The addition of water to solutions of PBT dissolved in a strong acid (MSA) causes phase separation in qualitative accord with that predicted by the lattice model of Flory (17). In particular, with the addition of a sufficient amount of water the phase separation produces a state that appears to be a mixture of a concentrated ordered phase and a dilute disordered phase. If the amount of water has not led to deprotonation (marked by a color change) then the birefringent ordered phase may be reversibly transformed to an isotropic disordered phase by increased temperature. This behavior is in accord with phase separation in the wide biphasic gap predicted theoretically (e.g., see Figure 8). The phase separation appears to occur spinodally, with the formation of an ordered, concentrated phase that would exist with a fibrillar morphology. This tendency may be related to the appearance of fibrillar morphology in fibers and films of such polymers prepared by solution processing. 

Various theoretical attempts have been made to describe the later stages of spinodal decomposition. The validity of the scaling laws given by Eqs. (35) and (36) results as a common feature of all these models. However, they differ in the prediction of the exponents a and p. Time-independent values, a = 0.212 and p = 0.81 or a = 1/3 and P = 1, were submitted in Refs. [78] and [79], respectively. More recently, it has been proposed that the scaling exponents themselves should be time-dependent [80-82],... 

It takes a finite time for the process to be completed the composition o slowly changes to the coexistent compositions (t>a and b- The process is illustrated by the horizontal arrow in Fig. 10.28. Kinetics of phase separation in a blend has recently gained considerable interest from both a theoretical and experimental point of view. Several experimental examinations were carried out to determine the concentration fluctuation during the phase-separation process taking PS/PVME as a model, which exhibits a LCST phase diagram. For example, Larbi et al. [168] observed fluorescence emission of anthracene-labeled PS in PS/PVME to investigate kinetics of both spinodal decomposition and nucleation growth. 

Theoretical miscibility windows for (a) VDAT-PS/T-PBMA and (b) A-PS/T-PBMA blends obtained from the Painter-Coleman association model PCAM ( ) spinodal curve and experimental data ( ) two-phase system (o) one-phase system. (Reprinted with permission from Kuo, S. W., and Hsu, C. H. 2010. Miscibility enhancement of supramolecular polymer blends through complementary multiple hydrogen bonding interactions. Polymer International 59 998-1005. Copyright 2010, Wiley-VCH, Germany.)... 

Finally, we have addressed only the phase behavior of solutions, and have not addressed eiflier the structural properties (e.g., as described by the various pair distribution functicms or the size of polymer coils) or the interfacial structure in phase-separated solutions (though we did pay attention to prediction of the interfacial tension between coexisting phases). Also, the kinetics of phase separation (via nucleation and growth or spinodal decomposition) has not been discussed. Thus, we emphasize that, although a few first and promising steps towards the computational modeling of polymer solutions via computer simulations have been taken, many further studies are still necessary to obtain a more complete theoretical understanding of polymer solutions and their properties. 

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