Critical polymer solutions

Wirtz D, Fuller GG (1993) Phase transitions induced by electric fields in near-critical polymer solutions. Phys Rev Lett 71(14) 2236-2239... 

D. Wirtz, D. E. Werner, and G. G. Fuller, Structure and optical anisotropies of critical polymer solutions in electric fields, J. Chem. Phys., 101, 1679 (1994). 

D. Wirtz, K. Berend and G.G. Fuller, Electric field induced structure in polymer solutions near the critical point, Macromolecules, 25, 7234-7246 (1992) D. Wirtz and G. G. Fuller, Phase transitions induced by electric fields in near-critical polymer solutions, Phys. Rev. Lett., 4,2236 (1993). 

If a fluid possesses two or more mesoscopic length scales, a competition between these scales may cause crossover between different behaviours, each associated with a particular meso-scale. In this section, we discuss only two examples of such competition near-critical polymer solutions and near-critical finite-size fluids. 

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

However, for more complex fluids such as high-polymer solutions and concentrated ionic solutions, where the range of intemiolecular forces is much longer than that for simple fluids and Nq is much smaller, mean-field behaviour is observed much closer to the critical point. Thus the crossover is sharper, and it can also be nonmonotonic. 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

S. Livne. A polydisperse polymer solution as a critical system. Macromolecules 27 5318-5328, 1994. 

As Eq. (3) sh vs, the critical composition (ticn can be controlled by the asymmetry of chain lengths. Particularly interesting is the limit Na = N, Nb = I (which physically is realized by polymer solutions, B representing a solvent of variable quality). Checking the deviations from the mean field predictions, Eq. (3), further contributes to the understanding of the statistical mechanics of mixtures. 

The formation mechanism of structure of the crosslinked copolymer in the presence of solvents described on the basis of the Flory-Huggins theory of polymer solutions has been considered by Dusek [1,2]. In accordance with the proposed thermodynamic model [3], the main factors affecting phase separation in the course of heterophase crosslinking polymerization are the thermodynamic quality of the solvent determined by Huggins constant x for the polymer-solvent system and the quantity of the crosslinking agent introduced (polyvinyl comonomers). The theory makes it possible to determine the critical degree of copolymerization at which phase separation takes place. The study of this phenomenon is complex also because the comonomers act as diluents. 

The polymer solubility can be estimated using solubility parameters (11) and the value of the critical oligomer molecular weight can be estimated from the Flory-Huggins theory of polymer solutions (12), but the optimum diluent is still usually chosen empirically. 

The term 6 is important it has the same units as temperature and at critical value (0 = T) causes the excess chemical potential to disappear. This point is known as the 6 temperature and at it the polymer solution behaves in a thermodynamically ideal way. 

Most pectin solutions behave like Newtonian liquids below a pectin concentration of about 1 % (w/w). Onogi (1966) derived the critical concentration of polymer solutions from plotting the double logarithmic curves of viscosity (ii) against concentration at constant shear rates. Each curve consists of two straight lines intersecting at the critical concentration. At higher... 

Reciprocals of the critical temperatures, i.e., the maxima in curves such as those in Fig. 121, are plotted in Fig. 122 against the function l/x +l/2x, which is very nearly 1/x when x is large. The upper line represents polystyrene in cyclohexane and the lower one polyisobutylene in diisobutyl ketone. Both are accurately linear within experimental error. This is typical of polymer-solvent systems exhibiting limited miscibility. The intercepts represent 0. Values obtained in this manner agree within experimental error (<1°) with those derived from osmotic measurements, taking 0 to be the temperature at which A2 is zero (see Chap. XII). Precipitation measurements carried out on a series of fractions offer a relatively simple method for accurate determination of this critical temperature, which occupies an important role in the treatment of various polymer solution properties. 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

For semi-dilute solutions, two regimes with different slopes are similarly obtained the powers of M, however, can be lower than 1.0 and 3.4. Furthermore, the transition region from the lower to the higher slope is broadened. The critical molar mass, Mc, for polymer solutions is found to be dependent on concentration (decreasing as c increases), although in some cases the variation appears to be very small [60,63]. 

Using this equation an attempt was made to find a critical Re-number which could be correlated to the onset of vortices observed with the naked eye, as has been done, for example, for Newtonian fluids [93], but no correlation could be found [88]. The reason is probably due to the fact that polymer solutions are viscoelastic fluids, also called second-order fluids. 

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

PI Freeman, JS Rowlinson. Lower critical points in polymer solution. Polymer 1 20-25, 1960. 

Figure 9 Qualitative phase diagram of a polymer solution showing phase separation both on heating (at the lower critical solution temperature) and on cooling (at the upper critical solution temperature). (From Ref. 31.)...

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)...

The above considerations apply to isolated polymer chains in solution, i.e. at very low polymer concentrations or volume fractions, < >p (i.e. in the limit < >p - 0). As p increases, interchain interactions becom important. Indeed, at a critical polymer concentration, interchain overlap begins, and beyond a second critical concentration, the chains are so overlapped... 

Figure 1. Schematic illustration of the two critical concentrations in a polymer solution

As another criterion of stability, a critical flocculation temperature(OFT) was measured. The measurement of CFT was carried out as follows the bare latex suspension was mixed with the polymer solution of various concentrations at 1+8 °C by the same procedure as in the adsorption experiments. Then, the mixture in a Pyrex tube(8 ml, U.0 wt %) was warmed slowly in a water bath and the critical temperature at which the dispersion becomes suddenly cloudy was measured with the naked eye. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

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