Determination of Phase Behavior

In the following, we focus on a set of techniques commonly used to determine the phase behavior of oligomer mdts [47,48] and blends [81] to give an example of how 

MC techniques are applied in practice. The methodology is rather general and in principle can be applied to any molecular liquid [82] or spin system. It also has advantages over techniques like Gibbs ensemble Monte Carlo [83] because it can be combined with finite-size scaling in the vicinity of the critical point. In addition, the method yields interfadal properties. Our presentation follows Reference [47]. Simulations are typically performed in the grand canonical pVT ensemble with periodic boundary conditions, that is, we fix the chemical volume and temperature but allow for particle insertions or deletions. For a simple Lennard-Jones liquid, Eq. (1.13) becomes  

In a typical simulation run, a joint histogram of particle number and energy is accumulated. The system is at coexistence when an unweighted simulation spends an equal amount of time in the coexisting phases. If we plot the probability distribution as a function of particles in the melt, we obtain a double-peaked distribution at coexistence and the areas below the two peaks are equal [88, 90]. Coexistence densities can be calculated by determining the average particle numbers in the gas and the liquid peak and dividing the respective numbers by the volume of the simulation box. 

In practice, it is difficult to estimate the coexistence chemical potential ahead of time. However, if two distributions at p, T and p, V overlap sufficiently, it is possible to extrapolate data from p, T to p, and avoid a second simulation [91]. The probability of a certain configuration c at p, T is given by  

After a suitable normalization, the grand canonical partition sums Z and Z 

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Thewalt, J., Kitson, N., Araujo, C., MacKay, A. and Bloom, M. (1992). Models of stratum corneum intercellular membranes The sphingolipid headgroup is a determinant of phase behavior in mixed lipid dispersions. Biochem. Biophys. Res. Commun. 188 1247. 

Another aspect of lattice models concerns the determination of phase behavior. As far as continuous models were concerned we emphasized already that an investigation of phase transitions in such models usually requires a mechanical representation of the relevant thermod3marnic potential in terms of one or more elements of the micrascopic stre.ss teusor. The existence of sucli a mechanical representation was linked inevitably to symmetry considerations in Section 1.6, where it was also pointed out that such a mechanical expression may not exist at all. In this case a determination of the thermodynamic potential requires thermodynamic integration along some suitable path in thermodynamic state space, which may turn out to be computationally demanding. 

Chlorinated polymers/Copolyester-aniides Recent studies (5) of blends of chlorinated polyeAylenes with caprolactam(LA)-caprolactone(LO) copolymers have been able to establish a correlation between miscibiUty and chemical structure within the framework of a binary interaction model. In some of the blends, both components have the ability to crystallize. When one or both of the components can crystallize, the situation becomes rather more complicated. Miscible, cystallizable blends may also undergo segregation as a result of the crystallization with the formation of two separate amorphous phases. Accordingly, it is preferable to investigate thermal properties of vitrified blends. Subsequent thermal analysis also produces exothermic crystallization processes that can obscure transitions and interfere with determination of phase behavior. In these instances T-m.d.s.c has the ability to separate the individual processes and establish phase behavior. 

CR Yonker, JC Linehan, JL Fulton. The use of high pressure NMR for the determination of phase behavior for selected binary solvent systems. J Supercrit Fluids 14 9-16, 1998. 

Figure 20.1.10. A phase analyzer system for synthetic determination of phase behavior.

New solvents should be examined for compatibility with photochemistry. Laser light scattering is used with a variable-volume view cell in an almost fully automated setup for accurately determining the phase behavior of pure or mixed fluids. The automated light-scattering techniqne yields good data, is relatively quick, and is non-labor inten-... 

We determined the phase behavior of the HDPE/styrene/CC>2 using the method described by Berens et al. (1992), modeling mass uptake data as Fickian dilfusion into a planar sheet (Crank, 1975). Ethylbenzene was used as the penetrant to model styrene. 

Some current research areas using the methods described in the previous section are shown in Table 2. Many of these involve molecular simulations that exploit the limit of speed and storage of currently available supercomputers [6]. In this section I shall consider two examples from among these (1) the determination of phase equilibria by computer simulation and (2) the behavior of fluids in microporous materials. 

High viscous homogenous mixtures such as polymer blends of well defined components are often difficult to prepare and even more problematic is a production of the relative big sample quantities necessary for some conventional investigation methods. Hence there are demands for "mini method", especially for a method to determine the phase behavior as a function of temperature and pressure. This task is solved to one part by the recently developed "mini extruders" of DSM (1), which enable a very good premixing of small amounts of polymer samples. 

Figure 24.2 DSC traces for the N-methyl-N-ethylpyrrolidinium (P12) species. The y-axis has been adjusted to allow comparison between species. As observed for the P, series, the DCA and thiocyanate species show broad phase transitions in comparison to the very sharp TFSA salt. The significant differences observed between the species highlights the strong influence of the anion in determining the phase behavior.

In another study carried out in 2007, Anderson et al. [23] experimentally determined the phase behavior of the system H2-THF-H2O as a function of aqueous THF concentration within a temperature range of 260-290 K and at pressures up to 45 MPa. This is a wider range of temperatures and pressures than in Rovetto et al. s study on the same system. Similar to the findings of Rovetto etal. [22], they found that H2-THF clathrates showed maximum thermal stability at the stoichiometric value of 5.56mol% initial THF concentration. 

As we are again interested in determining the phase behavior of the binary mixture in confinement and near solid interfaces, we are essentially confronted with the same problem already discussed in Section 4.5, namely finding minima of the grand potential for a given set of thermodynamic (T, /x) and model parameters [see Eqs. (4.125)]. To obtain expressions for u> that are tractable, at least numerically, we resort again to a mean-field approximation. That... 

The conventional van der Waals approach where model parameters d and a are the constants cannot describe more than one first order phase transition and one critical point. Therefore a key question is a formulation of temperature -density dependency for EoS parameters generating more than one critical point in the mono-component matter. There are several approaches of the effective hard sphere determination from spherical interaction potential models that have a region of negative curvature in their repulsive core (the so-called core softened potentials). To avoid the sophistication of EoS and study a qualitative picture of phase behavior we adopt an approach Skibinsky et al. ° for one-dimensional system of particles interacting via pair potential... 

Davis summarized the concepts about HLB, PIT, and Windsor s ternary phase diagrams for the case of microemulsions and reported topologically ordered models connected with the Helfrich membrane bending energy. Because the curvature of surfactant lamellas plays a major role in determining the patterns of phase behavior in microemulsions, it is important to reveal how the optimal microemulsion state is affected by the surface forces determining the curvature... 

When determining the phase behavior of a ternary mixture, one of the pumps delivers the supercritical fluid of interest and the other delivers a binary mixture of fixed concentration. The overall loading of the solution delivered to the system is fixed by the flow rates of the pumps. The streams from each... 

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