Critical solution curve, defined

Figure A2.5.17. The coefficient Aias a fimction of temperature T. The line IRT (shown as dashed line) defines the critical point and separates the two-phase region from the one-phase region, (a) A constant K as assumed in the simplest example (b) a slowly decreasing K, found frequently in experimental systems, and (c) a sharply curved K T) that produces two critical-solution temperatures with a two-phase region in between.

The critical point (Ij of the two-phase region encountered at reduced temperatures is called an upper critical solution temperature (UCST), and that of the two-phase region found at elevated temperatures is called, perversely, a lower critical solution temperature (LCST). Figure 2 is drawn assuming that the polymer in solution is monodisperse. However, if the polymer in solution is polydisperse, generally similar, but more vaguely defined, regions of phase separation occur. These are known as "cloud-point" curves. The term "cloud point" results from the visual observation of phase separation - a cloudiness in the mixture. 

Aniline point is the mixing temperature of equal volumes of pure aniline and the other liquid, usually a hydrocarbon. The aniline point may be as much as 1° C. lower than the CST because the curve of mixing is unsymmetrical (Figure 1). Terms analogous to aniline point can be defined for other solvents—for example, furfural points. No distinction is made in the tables between critical solution temperatures and aniline points (or their analogs), because of the small difference mentioned. 

The curve defined by (118) is a very well-known object it is the minimum energy path (MEP) which connects two minima of V(r) via a saddle point. How, why and by whom the MEP was first introduced in the context of molecular dynamics is not clear (to the author at least). The argument above indicates that it is the relevant object that concentrates most of the probability current of the reactive trajectories in its vicinity in the case of the overdamped dynamics when the temperature is small and the potential is sufficiently smooth (otherwise, if V r) has many critical points, (118) has many solutions, none of which taken alone is relevant). It is however not clear when such situations arise, and the MEP may often prove irrelevant. 

The critical solution temperature is an important quantity and can be accurately defined in terms of the chemical potential. It represents the point at which the inflexion points on the curve merge, and so it is the temperature where the first, second, and third derivatives of the Gibbs free energy with respect to mole fraction are zero. 

Figure 1. Demixing diagrams for PS in 0-solvents and poor solvents (schematic). The variable X might be pressure, Mw D/H ratio in solvent or solute, etc. See text for a further discussion, (a, top left) PS in a 0-solvent (monodisperse approximation). For X=Mw - the X=0 intercepts of the upper and lower heavy lines drawn through the minima or maxima in the demixing curves define 0Land0u, respectively, (b, top right) PS in a poor solvent (monodisperse approximation). The heavy dot at thecenterlocates the hypercritical (homogeneous double critical) point. (c, bottom right) The effect of polydispersity. BIN=binoda] curve, CP=cloud point curve, SP=spinodal, SHDW=shadow curve. See text Modified from ref. 6 and used with permission.

It is now established both theoretically and experimentally that many thermodynamic variables assume a simple power-law behaviour at or near critical points in both pure and mixed fluids. The actual functional dependence of one variable on another can be characterized by the so-called critical indices a, 5, etc. The critical index j8, for example, defines both the shape of the gas-liquid coexistence curve for a pure fluid and the liquid-liquid coexistence curve of a binary mixture in the vicinity of either an upper or a lower critical solution temperature. The correspondence between critical phenomena in one-, two-,... 

Any composition at a given temperature represented by points on the left of the curve AC or the right of the curve CB consists of only one layer. All compositions between pure water and point A yield a solution of phenol in water. Within the dome shaped area ACB, the system is heterogenous and two liquid phases exist, while in the area outside the dome only a single liquid layer, i.e., a homogeneous system exists. The upper critical solution temperature may, therefore be defined as the temperature above which the two partially miscible liquids become miscible in all proportions. For phenol-water system the temperature is 339 K. 

Figure 2.16 Analysis of phase behaviour of a binary blend of polymer and solvent or two polymers exhibiting an upper critical solution temperature, Top variation of Gibbs free energy with composition, 0(0 = 0i or 02) at four temperatures. The tie line CC defines the compositions on the binodal curve. The locus of points defined by the points of inflection (9 G/90 )t,p = 0 define the spinodal curve. At point A (inside the spinodal curve), the mixture will spontaneously phase separate (into domains with compositions 0 and 0") via spinodal decomposition. However, at point B (outside the spinodal curve) there is an energy barrier to phase separation, which then occurs by nucleation and growth...

At the critical solution temperature, the two sides of the spinodal curve meet a binodal point. The critical temperature is then defined by the conditions... 

A phase diagram relative to a polymer solution that phase separates on heating is shown in Figure 4. The solid line in this figure is called the binodal and it separates the stable from the metastable regions of the phase diagram. The dashed line is the spinodal curve, which separates the metastable and unstable regions. The spinodal touches the binodal at the critical point, Tc, which for a polsTner solution is defined by equations 17 and 18. 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

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... 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

The effect of solution concentration on nucleation rate is shown qualitatively in Fig. 9. At low levels of supersaturation, the rate is essentially zero but, as concentration is increased, a fairly well defined critical supersaturation is reached (point 1), beyond which nucleation rate rises steeply (curve 1-2). Point 1 may be regarded as the threshold of the labile region. Data from a series of such curves at different temperatures establish the locus of points at which nucleation starts, i.e., the Miers supersolubility curve discussed in Section II. 

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