Phase equilibria critical concentration

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

Therefore, the physical meaning of the solubility curve of a surfactant is different from that of ordinary substances. Above the critical micelle concentration the thermodynamic functions, for example, the partial molar free energy, the activity, the enthalpy, remain more or less constant. For that reason, micelle formation can be considered as the formation of a new phase. Therefore, the Krafft Point depends on a complicated three phase equilibrium. 

The phase separation model follows exactly the description of a two-phase equilibrium, i.e., equating the respective chemical potentials of the particular surfactant in both phases (i.e., monomers in the nonpolar solvent and the micelles) at the critical concentration (CMC). Thus, (assuming ideal condition)... 

The multiple-equilibrium model which corresponds to the scheme of Eq. (3) does not provide any critical concentration and is considered in the frame of the present discussion to be the opposite limiting case with regard to those detergent systems which had to be assigned to the phase separation model. 

Fig. 6. Critical particle concentration, p Ip0, above which a dilute disordered phase and a concentrated ordered phase coexist in equilibrium for a sterically stabilized dispersion, as a function of the particle radius, a. System polyisobutene-stabilized silica particles in cyclohexane, 6=5 nm, T = 308 K, xi = 0.47, x2 0.10, A 4.54kT and v = 0.10.

For a specific polymer, critical concentrations and temperatures depend on the solvent. In Fig. 15.42b the concentration condition has already been illustrated on the basis of solution viscosity. Much work has been reported on PpPTA in sulphuric acid and of PpPBA in dimethylacetamide/lithium chloride. Besides, Boerstoel (1998), Boerstoel et al. (2001) and Northolt et al. (2001) studied liquid crystalline solutions of cellulose in phosphoric acid. In Fig. 16.27 a simple example of the phase behaviour of PpPTA in sulphuric acid (see also Chap. 19) is shown (Dobb, 1985). In this figure it is indicated that a direct transition from mesophase to isotropic liquid may exist. This is not necessarily true, however, as it has been found that in some solutions the nematic mesophase and isotropic phase coexist in equilibrium (Collyer, 1996). Such behaviour was found by Aharoni (1980) for a 50/50 copolymer of //-hexyl and n-propylisocyanate in toluene and shown in Fig. 16.28. Clearing temperatures for PpPTA (Twaron or Kevlar , PIPD (or M5), PABI and cellulose in their respective solvents are illustrated in Fig. 16.29. The rigidity of the polymer chains increases in the order of cellulose, PpPTA, PIPD. The very rigid PIPD has a LC phase already at very low concentrations. Even cellulose, which, in principle, is able to freely rotate around the ether bond, forms a LC phase at relatively low concentrations. 

Fitch-Roe approach. (At lower than critical emulsifier concentrations, micelles are not generated. With increasing amount of emulsifier, the physical properties of aqueous medium discontinuously change in the vicinity of its critical concentration. The critical emulsifier concentration is an important material constant.) Roe [140] proved that the two theories can be described by very similar quantitative relations. This latter theory stress the importance of dissolution equilibria for the equilibrium monomer concentration in the aqueous phase. 

The molecular weight dependence of the critical concentration for the establishment of uniformly anisotropic solutions of PBG is shown in Table I for various solvents that we have examined. Volume fractions ij>) of polymer quoted in this compilation correspond to the B-point in the nomenclature of Robinson (28-29). The B-point differs from the A-point, a lower concentration where the anisotropic phase just begins to form and is in equilibrium with isotropic polymer solution. 

Above the critical concentration v the solution becomes metastable and separates into two phases — isotropic and anisotropic. The condition of thermodynamic equilibrium of the two phases corresponds to the equality of the chemical potentials of each of the components in each of the coexisting phases. The concentration corresponding to a complete transition to the anisotropic state, v, is 1.56 times as high as vf (see also... 

In order to tmderstand the fundamentals of the retention mechanisms at finite concentration, it is necessary to study first the equilibrium of a single component in a simple chromatographic system i.e., using a piue mobile phase or a solution without strongly retained additives that would compete with the compoimd studied). This is the topic of this chapter. The important phenomenon of competition for interaction with the stationary phase at finite concentrations will be considered in detail in the next chapter. This phenomenon is critical because it has a major influence on the individual band profiles and, in the separation of mixtures, it eventually controls the throughput, the production rate, and the recovery yield of any high concentration chromatographic process. 

However, in nature, engineering and private life we deal with solutions. Addition of the second component considerably complicates the problem. In this connection, attention may be paid, firstly, to the transfer of the liquid-vapor critical curve and, hence, the binodals Ts p,c = const, c - the concentration in liquid phase) to the region of elevated pressures, see Fig. 1, and secondly to changes in the thermodynamic compatibility of components with temperature and pressure. These factors lead to considerable extent of the two-phase equilibrium region with respect to that of pure liquid and, consequently, to the principal increase in the requirements on the experimental methods and devices used to study this region. 

The high-energy input necessary for preparing multiphase polymer systems or other colloidal systems pushes these systems far from equilibrium at a critical concentration the energy input and the entropy export are so far above their critical values (i.e. they are supercritical) that a self-organisation process occurs in the form of a phase transition. This is the short, summarised main principle on which the new viewpoint [37] is based. 

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