Calculating concentrated polymer

The heat of dissociation in hexane solution of lithium polyisoprene, erroneously assumed to be dimeric, was reported in a 1984 review 71) to be 154.7 KJ/mole. This value, taken from the paperl05> published in 1964 by one of its authors, was based on a viscometric study. The reported viscometric data were shown i06) to yield greatly divergent AH values, depending on what value of a, the exponent relating the viscosity p of a concentrated polymer solution to DPW of the polymer (q DP ), is used in calculation. As shown by a recent compilation 1071 the experimental a values vary from 3.3 to 3.5, and another recent paper 108) reports its variation from 3.14 to 4. Even a minute variation of oe results in an enormous change of the computed AH, namely from 104.5 KJ/mole for oe = 3.38 to 209 KJ/mole for oe = 3.42. Hence, the AH = 154.7 KJ/mole, computed for a = 3.40, is meaningless. For the same reasons the value of 99.5 KJ/mole for the dissociation of the dimeric lithium polystyrene reported in the same review and obtained by the viscometric procedure is without foundation. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

Calculated concentrations of possible hydrolysis species as a function of the equilibrium pH are given in Figures 6 and 7 for americium ions and plutonium ions, respectively. In the americium solution, the predominant species at pH <7 are the Am3+ and Am(OH)2+ ions, whereas in the plutonium solution the Pu022+ ion is not the dominating species at near neutral media but polymers as postulated (Pu02)2(OH)22+ and(Pu02)3(0H)5+. 

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

Methods that calculate average polymer properties without the DRI are significant because they circumvent complications that arise from the measurement of the interdetector volume between the light-scattering and concentration detectors 10, 11). It is necessary, however, that each local signal,, be divided by the computed particle scattering function, P 6)i. 

The dependence of molecular weight of polymers on monomer concentration, temperature, Al/Co ratio and conversion are shown in Table 16. As molecular weights are lower than would be calculated from polymer and catalyst concentrations and polymer molecular weights, there is a transfer reaction but its nature is not yet established. Similar findings for the relationship between polymerization variables and molecular weight were found by Zgonnik et al. [170]. 

The usefulness of inverse gas chromatography for determining polymer-small molecule interactions is well established (1,2). This method provides a fast and convenient way of obtaining thermodynamic data for concentrated polymer systems. However, this technique can also be used to measure polymer-polymer interaction parameters via a ternary solution approach Q). Measurements of specific retention volumes of two binary (volatile probe-polymer) and one ternary (volatile probe-polymer blend) system are sufficient to calculate xp3 > the Flory-Huggins interaction parameter, which is a measure of the thermodynamic... 

Feng, W. et al.. Calculation of vapor-liquid equilibria of high concentrated polymer solution by a modified SAFT equation of state, presented at the 9th International Conference on Rroperties and Phase Equilibria for Product and Process Design, Kurashiki, Japan, May 20-25, 2001, 2001. 

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

The zero-shear viscosity of a concentrated polymer solution can be treated by a modified version of the method used to calculate the zero-shear viscosity of a polymer melt. The modifications take the two effects of the solvent (plasticization and true dilution of the polymer) into account. Approximations are involved, however, in determining the appropriate mixing rules for the plasticization effect and the magnitude of the true dilution effect. The zero-shear viscosity of concentrated polymer solutions will be discussed briefly in Section 13.G. 

Step 22. Calculate the viscosity of a polymer melt and/or the zero-shear viscosity of a concentrated polymer solution. The molecular weight dependence of the zero-shear viscosity is given by equations 13.2 and 13.3, where the critical molecular weight Mcr (Equation 13.6) is... 

Calculation of Concentrated Polymer Solution Viscosities A New Approach... 

A model for calculating viscosities of concentrated polymer solutions has been formulated and used successfully to predict viscosities of alkyd resin solutions in both pure aromatic solvents and in mixtures of hydrocarbons and oxygenated materials. It was also found to describe viscosity trends in polystyrene-diethylbenzene solutions accurately. The formulation explicitly accounts for the observation that concentrated solution viscosities increase markedly with decreasing compatibility between resin and diluent. The proposal of an empirical relationship which interprets the viscosity enhancement in poorer solvents in terms of increased chain-chain interactions is of interest. The model contains three constants which are fixed for a particular resin and are independent of diluent type. These are the Mark-HouuAnk constant, the parameter in the Martin viscosity equation, and the constant relating the postulated clustering to the solution thermodynamics of a particular solution. 

A new approach to calculating viscosities of concentrated polymer solutions has been presented. It consists of the derivation of a semi-empirical computational model containing three parameters characteristic of a particular polymer. Once these parameters have been established, the viscosity of any solution of the polymeric material in a solvent or solvent blend may be calculated. The method should be of particular interest to the coatings industry, where they often require a screening estimate of the potential viscosity-reducing power of a new solvent blend. 

Calculation of polymer concentrations at critical points in the process of polymerizing styrene in the presence of dissolved PBD is possible with use of... 

FIGURE 2.3 Comparison of observed and calculated concentration-time relationships for caprolactam (x), polymer chains (i ), and amino caproic acid (5i) fVo = 0.82 mol/kg, T = 259°C. Solid lines represent experimental data broken lines calculated using set-II parameters dotted lines calcnilated using set I parameters. (From Reimschuessel, H.K. and Nagasubramanian, K., Chem. Eng. ScL, 1972, 27, 1119. With permission.)... 

Table 2.10 summarizes the model equations for polymerization with multiple-site catalysts in batch, semibatch and continuous reactors. The moment equations are, as expected, the same as the ones developed for single-site types, but applied for each individual site type, as indicated by the subscript j. Notice that the molar balances for monomer and hydrogen use the concentration of all active site types and that the chain length averages are also calculated using polymer made on all active sites. The new variable n appearing in these equations is the number of active site types in the catalyst. Initial conditions were omitted because they are analogous to the ones shown in Table 2.9, now applied for each site type. 

Fig. 3.22 Calculated concentration of cations and the electric field stenth inside the polymer for an applied potenital of 2V. Reprinted with permission from [Pugal et at.

Many attempts have been made to find theoretical explanations for the non-ideal behavior of polymer solutions. There exist books and reviews on this topic, e.g., Refs. " " Therefore, only a short summary of some of the most important thermodynamic approaches and models will be given here. The following explanations are restricted to concentrated polymer solutions only because one has to describe mainly concentrated polymer solutions when solvent activities have to be calculated. For dilute polymer solutions, with the second virial coefficient region, Yamakawa s book provides a good survey. 

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