Polymer concentration, derivation

For polymer chains in a -solvent the scaling exponent y takes its mean-field value y — 1. The polymer concentration derivative of the reduced osmotic pressure follows from (4.29) as... 

Appendix A Derivation of Equations for Polymer Concentration This Appendix shows the derivation of Equations (1) and (3) in the text. 

The amounts oi adsorption of the polymer on latex and silica particles were measured as follows. Three milliliters of the polymer solution containing a known concentration was introduced into an adsorption tube(lO ml volume) which contained 2 ml of latex (C = l+.O wt %) and silica(C = 2.0 wt %) suspensions. After being rotated(l0 rpm) end-over-end for 1 hr in a water bath at a constant temperature, the colloid particles were separated from the solution by centrifugation(25000 G, 30 min.) under a controlled temperature. The polymer concentration that remained in the supernatant was measured colorimetrically, using sulfuric acid and phenol for the cellulose derivatives(12), and potassium iodide, iodine and boric acid for PVA(13). From these measurements, the number of milligrams of adsorbed polymer per square meter of the adsorbent surface was calculated using a calibration curve. 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The treatment of batch polymerizations is in principle more difficult than that of continuous polymerizations in the latter, it may be permissible to assume steady-state conditions, though this has been questioned as indicated above but in the former it can never be permissible. It may, however, be permissible to make the assumption usual in treatments of the kinetics of radical polymerizations that the time-derivatives of radical concentrations may be neglected, though Kuchanov and Pismen (94) have questioned this assumption but the time-dependence of the polymer concentration must always be taken into account. 

We have synthesized and tested an example boron-chelating polymer based on a commercially available dendrimeric poly(amido amine) (PAMAM). Dendrimeric chelants offer several advantages over polymers typically used in PAUF. Foremost among these is the reduced viscosity of dendrimer solutions as compared to solutions of linear polymers[6]. This allows the use of higher polymer concentrations than previously feasible (though in the present study we worked at polymer concentrations of less than 5% due to the expense of the starting dendrimer). In this study, the dendrimeric chelant also serves as a convenient, monodisperse polymer with which to test the mathematical model for boron speciation which is derived from the work of Wise and Weber[l],... 

The concept of a unique hydrodynamic volume for all rodlike polymers was derived from examination of the Mark-Houwink constants, K and a, of the equation [rj ] = KMa. Macromolecules with values of a greater than unity are commonly accepted to be stiff or rigid rods. However, it was also found that such molecules (even for values of a less than unity) obey a relation illustrated by close concordance with the curve in Fig. lb (13) flexible, branched or otherwise irregular polymers, on the other hand, show dispersion around the upper part of the curve. The straight line curve in Fig. lb implies that the constants K and a are not independent parameters for the regular macromolecules to which they apply. Poly (a- and polyQJ-phenylethyl isocyanide) fall on this line the former has a value of a > 1 while the latter has a value a < 1 (14) both polymers give linear concentration dependence of reduced specific viscosity for fractionated samples... 

The effect of polymer concentration on [M]e has also been shown in the study of Tobolsky et al. (35). Unfortunately, their experimental technique was not sufficiently refined, and the wide scatter of experimental points prevented quantitative deductions from their data. Comment is necessary with respect to one statement made in their paper. The authors assumed that the "true value of [M]e could be derived by linear extrapolation from the experimental [M]e s, obtained at high polymer concentrations, up to the point of intersection with the line [M]e= [M]0. However, this extrapolation is not valid because, as has been shown by Vrancken et al. (13), the results depend on the concentration of living ends. The proper determination of the true [M]e requires studies at low concentrations of living ends, as was done by Worsfold and Bywater (12) and by McCormick (11). 

G. G. Fuller and L.G. Leal, Network models of concentrated polymer solutions derived from the Yamamoto network theory, J. Polym. Sci. Phys. Ed., 19,531 (1981). 

The mechanism of separation with linear polymers is as follows. At a certain polymer concentration known as the entanglement threshold, the individual polymer strands begin to interact with each other, leading to a meshlike structure within the capillary. This allows DNA separation to take place. Many of the common polymers are cellulose derivatives, such as hydroxyethyl cellulose, hydroxypropyl cellulose, hydroxypropylmethyl cellulose, and methylcellulose. Other applicable polymers include linear polyacrylamide, polyethylene oxide, agarose, polyvinyl pyrrolidone, and poly-N. Ar-dimethylacrylamide. High-resolution separation up to 12,000 bp has been reported using entangled polymer solutions. 

It should be pointed out that since I.G.C. measures the total free energy of the interaction, any value of the Flory-Huggins interaction parameter which is derived will be a total value including combinatorial and residual interaction parameters as well as any residual entropy contributions. Similarly when using Equation-of-state theory one will obtain Xj2 rather than Xj. The interactions are measured at high polymer concentration and are therefore of more direct relevance to interactions in the bulk state but this does not remove problems associated with the disruption of intereactions in a blend by a third component. 

Formulas for the Leslie viscosities, in turn, were derived from the Smoluchowski equation for hard rods by Kuzuu and Doi (1983,1984 Semenov 1987), and are given in Eqs. (10-20) with ao = 0- These formulas require as inputs values of 2,54, k, and Dr, which are functions of polymer concentration C. Reasonably reliable analytic functions for these dependencies were obtained by Kuzuu and Doi using a perturbation expansion for large order parameter, yielding... 

The oxidative coupling reaction of terminal alkynes is critically dependent on the water concentration in the reaction mixture (see Section 2.5.2). Since water is produced during the reaction, careful elimination of it may be required. Challa and Meinders have demonstrated that the polymer catalyst derived from copper(II) chloride and either N,/V-dimethylbenzylamine or N,/V-dimethylaminomethylated atactic polystyrene (37) provides an extra protection of the catalytic copper complexes against water in the coupling reaction of phenylacetylene (equation 23), resulting in a higher reaction rate than the low molecular weight catalyst. 

Wesslau [3] and Tung [4]). Often these simple models do not describe adequately the measured distribution of Commerical polymers. Theoretical derivations for the blending of two components with known distributions have been reported 15]. These studies concentrated on the effect of the parameters on the MWD of the blend. 

Three situations are derived from this equation. First, if Ah is negative (exothermic reaction), the polymer concentration, [CR], decreases as temperature increases. If AS is also negative, no polymer can exist above a ceiling temperature. 

Since their discovery in 1991, carbon nanotubes (CNTs) have been a constant source of scientific inspiration [307]. Among the most intriguing properties of CNTs is the electric field enhanced electron emission from the nanotubes [308]. Initial studies combining CNTs and conjugated polymers concentrated on the diode properties in a CNT-polymer heterojimction [309]. Romero et al. demonstrated fight-sensitive photodetectors in combination with a PPV derivative. The authors showed that hole injection from the CNT electrode proved to be much more efficient than using an ITO electrode, and related this phenomenon to the enhancement of the local electric field at the tip of the nanotubes [309]. 

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