Polymer volume fraction concentration

Because of the rotation of the N—N bond, X-500 is considerably more flexible than the polyamides discussed above. A higher polymer volume fraction is required for an anisotropic phase to appear. In solution, the X-500 polymer is not anisotropic at rest but becomes so when sheared. The characteristic viscosity anomaly which occurs at the onset of Hquid crystal formation appears only at higher shear rates for X-500. The critical volume fraction ( ) shifts to lower polymer concentrations under conditions of greater shear (32). The mechanical orientation that is necessary for Hquid crystal formation must occur during the spinning process which enhances the alignment of the macromolecules. 

If the solvent concentration is very small, as in the case of gas sorption, the polymer volume fraction is near to unity and Eq. (1) becomes ... 

Figure 1 Is a flow sheet showing some significant aspects of the Iterative analysis. The first step In the program Is to Input data for about 50 physical, chemical and kinetic properties of the reactants. Each loop of this analysis Is conducted at a specified solution temperature T K. Some of the variables computed In each loop are the monomer conversion, polymer concentration, monomer and polymer volume fractions, effective polymer molecular weight, cumulative number average molecular weight, cumulative weight average molecular weight, solution viscosity, polymerization rate, ratio of polymerization rates between the current and previous steps, the total pressure and the partial pressures of the monomer, the solvent, and the nitrogen.

The theta (0) conditions for the homopolymers and the random copolymers were determined in binary mixtures of CCl and CyHw at 25°. The cloud-point titration technique of Elias (5) as moaified by Cornet and van Ballegooijen (6) was employed. The volume fraction of non-solvent at the cloud-point was plotted against the polymer concentration on a semilogarithmic basis and extrapolation to C2 = 1 made by least squares analysis of the straight line plot. Use of concentration rather than polymer volume fraction, as is required theoretically (6, 7 ), produces little error of the extrapolated value since the polymers have densities close to unity. 

If the polymer concentration increases so that the number of high order bead-bead interactions is significant, c>>c =p, (when c is expressed as the polymer volume fraction. Op), the fluctuations in the polymer density becomes small, the system can be treated by mean-field theory, and the ideal model is applicable at all distance ranges, independent of the solvent quaUty and concentration. These systems are denoted as concentrated solutions. A similar description appHes to a theta solvent, but in this case, the chains within the blobs remain pseudoideal so that =N (c/c ) and Rg=N, i.e., the global chain size is always in-... 

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

For comparison, a telechelic sulfonated polystyrene with a functionality f = 1.95 was prepared. In cyclohexane the material forms a gel independent of the concentration. At high concentrations the sample swells. When lower concentrations were prepared, separation to a gel and sol phase was observed. Thus, dilution in cyclohexane does not result in dissolution of the gel even at elevated temperatures. Given the high equilibrium constant determined for the association of the mono functional sample, the amount of polymer in the sol phase can be neglected. Hence, the volume fraction of polymer in the gel phase can be calculated from the volume ratio of the sol and gel phases and the total polymer concentration. The plot in Figure 9 shows that the polymer volume fraction in the gel is constant over a wide range of concentrations. 

Figure 9 Polymer volume fraction of the PS-(S03Li)2"40 gel at different concentrations in cyclohexane.

As established by Knoll et al. [49, 62], exposing thin films to well-controlled vapor pressure with a subsequent fast quench provides reproducible phase behavior under variation of the film thickness and of the polymer volume fraction (Fig. 15). At favored film thicknesses, cylinders orient parallel to the film plane, whereas a perpendicular orientation dominates at intermediate film thicknesses. In films thinner than 1.5 nm domain spacings and at high polymer concentration, the cylindrical... 

Figure 15.11 Slopes P=A/(X2-X 1) as a function of the polymer volume fraction at swelling equilibrium in cyclohexane < >e in a series of end-linked PDMS networks with various lengths between junctions (A, A Mn = 25000 g.mol 1 B, B Mn = 10500 g.mol1 C Mn = 3100 g.mol"1) and polymer concentration during crosslinking in toluene (A, B, C vc = 0.7 A , B vc =1)...

Consider a dilute polymer solution with Np polymer molecules of N monomers each. The polymer volume fraction p is often used as the concentration parameter it is given by < >p = NNpvp/(NNpvp + Nsvs), where vp and vs are the... 

A schematic T-c diagram is shown in Fig. 3 for fixed values of n, /, and w. Note that the excluded volume v is related to T through Eq. (4), and the polymer volume fraction

mass concentration c and chain number density p/n through... 

Pj concentration dependence coefficient of polymer volume fraction vp, in... 

The main aim was to determine the distribution of PEO molecules between the gel and the supernatant fluid at r = 0.1, c = 0.1 M, T = 5°C for M = 18,000 (bridging) and polymer volume fractions in the range between v = 0 and v = 0.12. The corresponding neutron diffraction traces are shown in Figure 12.6a. In comparing these structural analyses with an independent analysis of the concentration of the PEO in the supernatant fluid, we established the following protocol in preparing the samples. 

Let N cylindrical rods be situated in volume V, their concentration being c = N/V. The polymer volume fraction in the solution is then - jrpcd V4. Let us introduce the orientational distribution function for the rods f(u) cf(u)df2 is the number of rods per unit volume, which have the orientations within the small spatial angle dQ around the unit vector u. It is dear that in the isotropic state f(u) = const = l/4a. In the liquid-crystalline state the function f(u) has two maxima along the anisotropy axis. 

Let us assume that the solution of semiflexible macromolecules occupies the volume V. Let i be the polymer volume fraction in the solution. Then, the average concentration of segments is c = 4 fibrpd3, the total number of segments N = Vc, and, finally, the average concentration of macromolecules is c//L, where L denotes the contour length of one macromolecule. 

The Flory-Huggins parameter Xi was initially derived as an experimentally determined interaction enthalpy parameter which was supposed to be independent of poljnner concentration. However, experiments have shown that it is dependent on pol5mier concentration. Evans and Napper [40] have shown that the polymer volume fraction dependence of X can be given by... 

As already noted, the measured nonlinear shear relaxation modulus, for linear molecules with little polydispersity, is in excellent agreement with the Doi-Edwards model at long times. However, for melts or concentrated solutions of very high molecular weight (e.g., 10 for polystyrene, where 0 is the polymer volume fraction), the measuredfiamping function, h(y), is drastically lower than the Doi-Edwards prediction (Einaga et al. 1971 Vrentas and Graessley 1982 Larson etal. 1988 Morrison and Larson 1992). This anomalous... 

Consider a semidilute solution with polymer volume fraction (j>. The concentration dependence of the correlation length was discussed in Chapter 5 ... 

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