Polymer concentration, derivation equations

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

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... 

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... 

A laboratory scale, continuous process for the polymerization of acrylamide in aqueous solution is described. The reaction conditions can be held constant within narrow limits and the effect of small changes in individual variables, such as temperature, initiator concentration, and chain transfer agent concentration, can be quantitatively ascertained. Some experimental results are presented showing the effect of these factors on the molecular weight of the polymer. The data are examined vis-a-vis some theoretically derived equations. 

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. 

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. 

With a number of assumptions regarding the enthalpy of this reaction and the maximum chain-length of the polymer another mathematical equation was derived which is applicable at all temperatures but yields a polymer concentration of only 10 % at 120 C in contradiction to all analytical data available at the time. In a supplementary publication by Gee et al. [66] the enthalpy of the ring addition reaction was reduced to 13.3 kJ mor and the reaction entropy assumed as 31 J mol K but novel results were not obtained. The authors discussed however the possibility that the sulfur melt may contain rings larger than Ss and that the polymer So present at temperatures below 159 °C may consist of very large rings rather than chains. 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

Smith and Ewart [52] derived Equation (4.2) (to which reference if often made as the Smith-Ewart equation) in a more elaborate manner (see Appendix 4.1). They also derived an expression for the upper and lower limits of as a function of the concentrations of initiator and onulsifier and of the type of emukifier used. However, they did not consider the effect of the ionic strength of the aqueous phase or the valency of the counterions which also affect N [78,79]. They also assumed that as. the area occupied by an emukifier molecule at the polymer-water interface, would be the same as it is at the air-water interface and independent of the presence of monomer or other swelling agents in the particles. These assumptions have subsequently been found to be incorrect (see Tables 4.4-4.7 [80]). Nevertheless the predicted orders in initiator and emulsifler have been verified experimentally at least for styrene [66]... 

The diagram of the liquid-vapor phase equilibrium is eharacterized by a decrease in the derivative dp/dT with the polymer concentration (dp/dT —> 0 at k 0). This leads to increase in bofli die nucleation energy and the detachment size of a bubble (equation [7.2.59]) and, consequently, to reduction of the bubbles generation frequency. Note that in reality the critical work, W , for a polymeric liquid may exceed the value predicted by the formula [7.2.59] because of manifestation of the elasticity of macromolecules. 

This equation can be solved numerically using the Regula Falsi method, which does not need analytical derivatives in contrast to the Newton method. The interval of the search for the root should be specified. The polymer concentration of the polymer-rich phase must be above the critical concentration (Xg = 0.00756). The search interval is therefore between the critical concentration Xg = 0.00756 and a value at higher polymer concentration (xg 2 = 0.02). Using the Regula Falsi... 

Finally, the Equations (4.8) or (4.9), expressing the variation of the concentrations with time and space in the polymer and Equation (4.2) defining the so-called boundary condition, are the fundamental equations of diffusion through a sheet of an isotropic material, and thus of mass transfer between the polymer and the liquid. Equations (4.8) and (4.9) are partial derivative equations, in the sense that the concentration C depends on the two parameters of time and space. When the diffusivity is constant. Equations (4.9) and (4.2) should be considered, as well as the initial profile of concentration in the sheet. 

Domain Size Theory. Assuming spherical domains, Yeo and co-workers (38) derived equations for the domain size in sequential IPNs. The domain diameter of polymer II, Dn, was related to the interfacial tension y, the absolute temperature T times the gas constant R, and the concentration of effective network chains, ci and cn, occupying volume fractions v and Pn, respectively ... 

One review [24] considers the theoretical derivation of several such equations, and highlights the effect of polymer concentration and the nature of the solvent in the correlation of intrinsic viscosity values calculated using such equations compared with extrapolated value for [r]]. In summary it should be recognised that ... 

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