Excess polymer concentration

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. 

The excess polymer concentration is defined as the difference of the free-flowing polymer concentration and of the critical polymer concentration. 

The fraction of excess flowing polymer concentration retained in unit length is proportional to the excess polymer concentration ... 

In the following analysis, the objective is to find an equation which describes the distribution of excess polymer concentration in a linear system. 

In the distal region, z the excess polymer concentration decays exponentially to its bulk value ... 

Hoffman Degradation. Polyacrylamide reacts with alkaline sodium hypochlorite [7681-52-9], NaOCl, or calcium hypochlorite [7778-54-3], Ca(OCl)2, to form a polymer with primary amine groups (58). Optimum conditions for the reaction include a slight molar excess of sodium hypochlorite, a large excess of sodium hydroxide, and low temperature (59). Cross-linking sometimes occurs if the polymer concentration is high. High temperatures can result in chain scission. 

The viscosity of sodium pectinate was previously examined by Pals and Hermans [44]. In salt excess (NaCI 0.1 M), the viscosity of a pectate sodium form solution (ri) is related to the polymer concentration (c) and the molecular weight (or [rj] = KM ) by the relation ... 

An adsorption-desorption transition is illustrated schematically in Figure 1, where we plot a displacement isotherm, i.e. the adsorbed amount of a polymer as a function of the composition of a mixture of solvent and displacer. At the left in Figure 1, where the concentration of displacer is low, the polymer surface excess is positive. As we increase the proportion of displacer in the mixture, we observe a decrease in the adsorbed amount. At a certain composition the adsorbed amount of polymer becomes zero. The concentration at which the polymer surface excess just vanishes will be denoted as the critical displacer concentration cr. Beyond 4>cr, the surface excess of the polymer is negative (and very small if the polymer concentration is low). 

That the cleavage of nitrophenyl caproate by the polymer is truly hydrolytic was demonstrated in two ways. First, repeated additions were made of 1 x 10-4 M substrate, up to five times the concentration of polymer imidazole groups (6xl0-5 residue molar) nitrophenol was released completely each time. At lower polymer concentrations it was also possible to add a single injection of substrate, and in every case the molar amount of nitrophenol released was substantially in excess of the concentration of imidazole groups on the dodecylpolyethylenimine (Fig. 6). 

To compare the catalytic effectiveness of our polymer with that reported for other substances that accentuate nitrophenyl ester cleavage, we26 have carried out a series of experiments (at pH 7.3) in which the residue molar concentration of polymer imidazole groups was substantially in excess of the concentration of substrate, p-nitrophenol acylate. Pseudo-first-order rate constants k[ were determined at each of a number of polymer concentrations. Under these conditions k[ was found to be linear with [P-Im]0, the initial residue concentration of methylene-imidazole groups ... 

In this system k2 values were easily measured in solutions with large excesses (about 150 1) of catalytic sites over substrate concentrations. Under these conditions the observed first-order rate constant was unaffected by increasing polymer concentration. The variation of k2 over the pH range 3 to 7 is displayed in Fig. 20. It is bell shaped and exhibits a maximum at pH 4.5. This bell-shaped pH-rate profile is similar to that of other model primary amines57,58 63 as well as the enzyme acetoacetate decarboxylase.52... 

The scattered light intensity from a polymer solution arises from the fluctuations in both the solvent density and the polymer concentration. These fluctuations are considered as stable during the timescale of the measurement in the static mode of light scattering (for more details, see Evans (1972)). The light scattered from just the polymer (in excess of the light scattered from the pure solvent) is given by (Burchard, 1994)... 

Figures 1-3 show the observed reciprocal excess intensities of scattered light multiplied by sin 0 (to correct for the irradiated volume observed at each angle) plotted against sin2 (0/2) at constant polymer concentration for several temperatures above the phase-separation temperature. To give a clearer presentation the intensities are expressed in relation to intensities at the scattering angle 0 = 90°, as was also done by Eskin and Nesterow (11). In accord with the Debye theory, the plots give straight lines and can be represented by...

When the polymer concentration is larger than its solubility, the excess polymer is assumed to precipitate on the surface of the particles and/or to form micelles. This assumption allows us to write for the polymer surface density rj the expression... 

Our explanation implies that the excess polymer precipitates onto the surface of the particles when the polymer concentration becomes larger than its solubility, which decreases with increasing electrolyte concentration in the solution. 

Generally, one may establish that in some cases greatly enhanced concentration fluctuations occur under flow, in others, however, the size of concentration fluctuations is reduced and, obviously, flow promotes mutual miscibility of the polymers. Concentration fluctuations are accompanied by inhomogeneities of transport quantities as shear viscosity and diffusity. In a flow field the molecules are transferred into a non-equilibrium situation of extension. Two polymer molecules in a state of excess extension feel an additional repulsion due to the enhanced normal stress difference. Thus, the rate of dissipation by diffusion is low compared with the shear rate and the concentration fluctuations tend to grow. The opposite is true for a state of lower extension. In that case the dissipation of the concentration fluctuations is enhanced owing to an additional attraction between the chain molecules. 

In his classic paper, Flory predicted the phase behavior in solutions of rod-like particles (5). The resulting phase diagram related the solvent-solute interaction parameter %i ( -5) to the volume fraction, V2, for polymer rods with an axial ratio of 100. A positive Xi makes a positive or excess free energy contribution to mixing. Good solvents are characterized by small Xi values. Two of Flory s major predictions are that the minimum polymer concentration required for mesophase formation will increase as Xi decreases, sharply at first, then more gradually, and at certain Xi values two different anisotropic phases coexist. Our microscopical observations of conjugated phases may reflect the validity of the latter prediction. 

Oxide particles Above the stoichiometric line, excess polymer coexists with fully covered spheres. Below the stoichiometric line, excess spheres cause unlimited bridging and the separation of a concentrated gel from the pure solvent. Then at even lower polymer concentrations, below the optimum flocculation concentration (o.f.c.), the gel can no longer accommodate all the spheres, and some are rejected. 

Figure 5. Excess polymer can be used to bridge polymer-covered particles - but Its concentration must be high enough to allow It to penetrate the adsorbed layers.

In order to discuss a number of important quantities we consider the interfacial profile for the case of positive adsorption. Such a profile is sketched in fig. 5.6. It represents the polymer concentration dz) as a function of the distance z from the interface. The quantity c(z) is related to the volume fraction concentration profile of segments belonging to free molecules, having no contact with the surface. The excess adsorbed amount r (the amount of... 

The effect of 750 ppm Xanthan gum on the microemulsion phase behavior is shown in Figure 5(b). The observed phase behavior is similar except that the extent of the three-phase region is widened. Thus at both 0.8 and 1.0 gm/dl salt concentrations there exists a polymer-containing brine phase in equilibrium with the microemulsion phase. When no polymer is present, the microemulsion phase is in equilibirum with only excess oil. The volumes of the polymer phases are small and the interface between the polymer phase and microemulsion is diffipult to detect in Figure 5(b). However, phase separation is clearly visible in Figure 5(c), which illustrates the oil-equilibrated phase behavior at a higher polymer concentration of 1500 ppm. 

Polyelectrolyte Behavior. Figure 1 shows the characteristic Kc/R vs. c plot of lonomers in polar solvents the reciprocal reduced scattered intensity rises steeply from the intercept at zero polymer concentration, bends over, and becomes nearly horizontal at higher concentration. This type of behavior was reported for some salt-free polyelectrolytes in aqueous solution (23.26), although the reliability of these early measurements is rather poor because of very small scattered intensity from polyelectrolyte/aqueous solution systems. For example, the excess scattered intensity from salt-free sodium poly(methacrylate) in aqueous solution over that of water was only 10 to 100% (2i) - (0.1-1) x 10 °. in ionomer solutions,... 

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