Critical dilution point

The ring-opening polymerization of is controlled by entropy, because thermodynamically all bonds in the monomer and polymer are approximately the same (21). The molar cychzation equihbrium constants of dimethylsiloxane rings have been predicted by the Jacobson-Stockmayer theory (85). The ring—chain equihbrium for siloxane polymers has been studied in detail and is the subject of several reviews (82,83,86—89). The equihbrium constant of the formation of each cychc is approximately equal to the equihbrium concentration of this cychc, [(SiR20) Thus the total concentration of cychc oligomers in the equihbrium is independent of the initial monomer concentration. As a consequence, the amount of linear polymer decreases until the critical dilution point is reached, at which point only cychc products are formed. 

By contrast, the corresponding values for the large cyclics decrease along the same series. The total weight fraction (experimental) of cyclics in these bulk equilibrates are listed in Table 4. The effect of dilution with cyclohexanone in the syntheses of these same polymers is illustrated in Figure 3. In all cases, a critical dilution point was reached as predicted but it was reached at a lower diluent volume than the theoretical value. Molar cyclization equilibrium constants for undiluted poly(methylphenyl)siloxane were later measured (19) and found to be similar to those previously obtained in the case of poly(ethylmethyl)siloxanes. 

Fig. 8. Experimental molar cyclization equilibrium constants Kx for (H(CH3)SiO)x at 273 K (denoted by and ) and for (CH3CH2(CH3)SiO)x at 383 K (denoted by o and ). Kx values in undiluted equilibrates are denoted by and o, and those measured in toluene solution close to the critical dilution points are denoted by and . Units of Kx are mol dm-3...

Critical dilution point %Volume polymer Toluene at 383 K 52 (1.5)... 

Detailed measurements of the solubility between the lower and upper critical end points have been made only for the solutions in ethylene of naphthalene,14 hexachlorethane,30 and />-iodochloro-benzene.21 Atack and Schneider2 have used dilute solutions of the last-named substance to study the formation of clusters near the gas-liquid critical point of ethane. 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

The solubility of a solid in a supercritical fluid has been described by Gitterman and Procaccia.(lO) The region of interest chromatographically will be for infinitely dilute solutions whose concentration is far removed from the lower critical end point (LCEP) of the solution. Therefore the solubility of the solute in a supercritical fluid at infinite dilution far from criticality can be approximated as,... 

The notion of the critical dilution is in harmony with our intuition, suppose a branching process molecular growth can take place only through the intermo-lecular linkages, whereas with increasing dilution the intermolecular reaction is suppressed. When the dilution reaches the well defined point, yc = 1, corresponding to Dc = 1, the probability of an infinite molecule emerging suddenly vanishes hence, the critical dilution. 

Solid lines (—) Eq. (102). (O) classical gel points (0) critical dilution. Experimental... 

Substituting these values into Eq. (102) (k = 1), we can now plot Dc as a function of y (Fig. 21). Agreement of the theory (solid line) with the experimental points ( ) by Ross-Murphy is remarkably good, giving a critical dilution yc = 0.96, greater than that in the adipic acid-pentaerythritol polymer system, which can be ascribed to the lower production of rings in this polymer system (Table 3). 

Fig. 23. Comparison of theory and experiment Dc vs. y [z / mol] curve The figure shown in the right side box is a magnification of the same plot. Solid lines (—) Eq. (102). (O) classical gel points ((x)) critical dilution. Experimental points by Muller and coworkers [67] (O k = 1)...

Table 12.5. Although the basis of the comparison presented therein is slightly different (per particle vs per doublet), the results predicted by both extreme models are not too widely disparate. The configurational free energy change associated with flocculation can obviously be as large as 10 A T in systems of usual interest. Dilute dispersions are clearly more stable on this basis than more concentrated systems. What this means in terms of the effect on the critical flocculation point depends critically upon the nature of the particular system concerned.

In phase equilibrium calculations, three unknowns, namely the polymer concentration in the dilute phase Xg, the polymer concentration in the concentrated phase Xg, and the temperature T, occur. Specifying one of the unknowns, the two other ones can be calculated by solving Eq. (10.15) simultaneously. In order to find suitable initial values, the spinodal and/or the critical solution point can be helpful. The spinodal (see Figure 10.2) separates the metastable from the unstable region in the phase diagram and can be calculated with the help of the stability theory, where the necessary condition reads... 

TAK Takada, M., Okano, K., and Kurita, K., Coexistence curve of dilute polymer solution in a mixed solvent having critical demixing point, Polym. J., 26, 113, 1994. 

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