Flory viscosity function

Different assumed values for the viscosity function F (or Flory constant) used to relate intrinsic viscosities to molecular dimensions in solution... 

One of the main assumptions which have been made in the study of polyesterifications is the concept of equal reactivity of functional groups. It was first postulated by Flory1 who, studying various polyesterifications and model esterifications, found the same orders of reaction and almost the same rate constants for the two systems. He concluded that the reaction rate is not reduced by an increase in the molecular weight of the reactants or an increase in the viscosity of the medium. The concept of equal reactivity of functional groups has been fully and carefully analyzed by Solomon3,135 so that we only discuss here its main characteristics. Flory clearly established the conditions under which the concept of equal reactivity can be applied these are the following ... 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

For each sample, the Flory Huggins constants (k, Eq. 5.25) were also determined (by viscosity measurements) as function of sonication time and are given in Tab. 5.17. 

As early as 1951, Flory and Osterheld [18] could show that partially ionized polyacrylic acid in aqueous solution of an inert salt shrinks in size if the concentration of the inert salt is increased. This shrinking process can be pushed towards the unperturbed dimensions of the NaPA chains. Known from neutral polymers as 0-state, it is reached for fully ionized NaPA [50] at T=15 °C and 1.5 M KBr. In several papers, the dependence of the intrinsic viscosity was investigated as a function of the molar mass [51]. Data were... 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

M and v are the molecular weight and the partial specific volume of the polymer, jjo p are the viscosity and the density of the solvent, respectively, and P and O are functions of relative chain length L/A and of the parameter of hydrodynamic interaction, d/A, respectively. These functions have been represented in an analytical form and tabulated over a wide range of changes in the L/A and d/A parameters At extremely high molecular weights (at IVA -> ), functions P and ap oach an asymptotic limit P— Po = 5.11 — 4>, = 2.862 x 10 (the Flory constant). This corresponds to the conformation of a hydrodynamically undrained Gaussian coil. 

Peak Shapes. In the case of the Wesslau MWD, the shapes of the peaks from the three detectors are always the same. For the Flory-Schulz distribution, the peak shapes are slightly different and the differences increase with increasing polydispersity. As the polydispersity increases, the LS and viscosity signals become narrower relative to the concentration detector signal and they also become less skewed. Figure 3 shows the peak variance of the viscosity and LS signals relative to the concentration detector peak variance as a function of polydispersity. The concentration detector peak variance increases from 0.25 mL when the polydispersity is 1.1 to 3.65 mL when the polydispersity is 3.3. The LS peak variance increases more slowly. The viscometer variance is in between the two but closer to the LS peak behavior. Figure 4 shows the relative skew of the peaks compared with the refractometer, where the skew is defined as... 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

While all relaxation times depend on temperature and pressure, only the global motions (viscosity, terminal relaxation time, steady-state recoverable compliance) are functions of Af , (and to a lesser extent MWD). The glass transition temperature of rubbers is independent of molecular weight because chain ends for high polymers are too sparse to affect this bulk property (Figure 3.14 Bogoslovov et al., 2010). The behavior can be described by the empirical Fox-Hory equation (Fox and Flory, 1954) ... 

The shear viscosity was found to be a decreasing function of shear rate (Fig. 3) but to have an asymptotic viscosity, rjo, at lower shear rates. Fox and Flory [F5] found the zero shear viscosity of rjo of linear polymer chains to depend on the 3.4 power of molecular weight. This was subsequently confirmed by various researchers [B39, F6, G20, P13, P14]. It was at first found that the zero shear viscosity for branched polymers was lower than that for linear polymers of the same molecular weight [S2] however, subsequently Kraus and Gruver [K20] showed that above a molecular weight, characteristic of the molecular topology, rjo for carefully prepared cruciform and Y-shaped polymers actually increases more rapidly perhaps with the sixth power. 

The right-hand side includes only the average over the equilibrium distribution function. It can be shown that eqn (4.187) gives a correct viscosity for two limiting cases, i.e., spherical particles and rigid rodlike particles. For flexible polymers in the 0 condition, however, the bound is rather weak in terms of the Flory-Fox parameter, one obtains (after some tedious calculation) ... 

It is known that for dilute polymer solutions and according to the Flory-Huggins equation, the reduced viscosity is a linear function of polymer concentration as follows ... 

Contrary to the accepted view that the rate of reaction of a polymer condensation would decrease as the reaction progressed and as the viscosity increased, Flory showed that this rate was independent of viscosity (6). He also showed that crosslinking would occur when reactants with more than two function groups were condensed. While at DuPont, Flory demonstrated his versatility by explaining chain propagatioh of free radical-initiated vinyl polymers (7). 

---