Critical Polymer Concentration

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... 

Blend solutions. Solutions of blends comprising immiscible polymers Pj and P2 in a nonselective solvent have miscibility gaps as shown schematically in Fig. 14. When the polymer concentration increases by solvent evaporation the polymer coils start to interpenetrate above a certain concentration. As a consequence, interactions between the polymers become operative and phase separation must start above a critical polymer concentration p. The composition of the new phases will be situated on the branches of the coexistence curve. Finally, the unmixing process is arrested owing to enhanced viscosity. This simple scheme reveals the factors directing morphology evolution in blend solutions ... 

Pc mean degree of polymerization at critical polymer concentration... 

Turner et al. [41-43] do not consider reduced coil mobility with increasing polymer concentration as decisive. Under these conditions, macromolecules behave as rigid balls, mutually hardly penetrable the probability of radical encounter decreases. The onset of autoacceleration is also ascribed to mutual coil entanglement which only occurs at and above the critical polymer concentration Cc and a certain minimum mean molecular mass A (0.5 < a < 1). The product of these critical quantities is constant... 

The above-described behaviour is obtained above a critical polymer concentration (C ), which can be located from plots of log tj versus log C, as is illustrated in Figure 10.26. Below C, the log //-log C curve has a slope in the region of 1, whereas above C the slope of the line exceeds 3. In most cases a good correlation between the rate of creaming or sedimentation and //(O) is obtained. 

It is obvious that, as with nonaqueous dispersions, the critical polymer concentration required for flocculation decreases with increasing molecular weight. Increasing the temperature rendered the latices more susceptible to flocculation (Fig. 16.6). 

The molecular weight dependence of the critical polymer concentration for depletion stabilization... 

What is understood by the term "traditional thickener" in general is a thickening agent which, when present in the coating, increases the viscosity of the aqueous phase and which do not significantly associate with the other ingredients. The polymer concentration at which polymer coils start to overlap is indicated by the critical polymer concentration (c ). Above this concentration, the viscosity is increased by a traditional thickener polymer dissolved in the aqueous phase, because of the entanglement of the polymer chains. (Fig. 1). 

The following two parameters are most important for the formation reactions of intermolecular metal complexes the critical polymer concentration c and the temperature Tt of the liquid <=> gel transition (Eq. 3-23). 

Flory s analysis focuses on the thermodynamic interactions between polymers, and defines the theta point at the critical polymer concentration for phase separation (equal to the critical concentration of chain units within a single chain upon collapse transition), similar to the Boyle point of the non-ideal gas. We can perform Virial expansion on the osmotic pressure of dilute polymer solutions, as... 

The triblock copolymer EO PO EOn is soluble in water and forms micelles above a critical polymer concentration and above a characteristic temperature. This was shown by Zhou and Chu [352,375,376]. These authors afiBmed that micelle formation is temperature dependent because the hydrophilic/hydropho-bic characteristics of these polymers can be easily modified by changing the temperature. They mentioned the presence of three temperature regions a uni-mer region, a micelle region and a transition region where unimers and micelles coexist. The micelles seem to be monodisperse in size, even if the polymer itself is polydisperse, their radii being independent of concentration but dependent on temperature [377],... 

Dependence of the adsorbed amount on the polymer concentration is given for a typical unhydrolyzed and a partially hydrolyzed polyacrylamide in Figures 5 and 6. Apart from the fact that the absolute values indicate considerable differences, it is characteristic that n increases with the pol3nner concentration. Within this general tendency, as it appears from these curves, the steep increase occurs after a definite pol3nner concentration in all cases. Before this critical concentration—similar to the equilibrium section of static adsorption isotherms—the adsorbed amount does not depend significantly on polymer concentration. Incidentally, the critical polymer concentration in 2 gdm" NaCl is increased twofold over values measured in distilled water for both polymers. 

Willhite and Dominguez (15,23), referring to unpublished data, reported that molecular interactions and aggregation can take place long before the critical polymer concentration. Consequently a real solution does not exist in the vicinity of the critical con-... 

The assumption could explain the role of polymer concentration and flow velocity. Over a critical polymer concentration and flow velocity, the number of arriving molecules at a given location of the outer polymer layer exceeds the number of disentangled molecules. 

As we have seen above, the development of unsteady-state flow is concentration dependent. The number of molecules expelled into the dynamic trap in unit time is proportional to the difference between the actual flowing concentration and the critical polymer concentration. This concentration difference, called excess polymer concentration is large at the inlet face, therefore, the rate of polymer buildup is also large. Far from the inlet face, this excess polymer concentration is greatly reduced since the rock has already stripped out a portion of this excess polymer concentration. The rate of polymer buildup will consequently be reduced far from the inlet surface. The experimental verification of this fact will be given in the Discussion. 

Using a certain flow velocity at which the injected polymer concentration exceeds the critical polymer concentrations, the porous material will strip out all the excess polymer concentration above the critical concentration value. The accumulations of this excess polymer concentration take place not only at the inlet surface, but also in depth. If the injected concentration is equal to or less than the critical polymer concentration, no continuous polymer buildup will occur anywhere in the porous medium. The number of entrapped polymer molecules should be proportional to the polymer concentration above the critical polymer concentration. Since the excess polymer concentration is greatest at the injection face, the rate of polymer entrapment is also highest at this location. 

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