Concentration-dependent shift factor

To obtain concentration-independent viscosity curves for PDMS/COj mixtures from Figure 13.12, Gerhardt et al. (1997) introduced a concentration-dependent shift factor a, defined by... 

In analyzing the viscosity data at various temperatures and shear rates, which were obtained via slit rheometry for molten PS, PMMA, polypropylene (PP), and LDPE with solubilized CO2 at varying concentrations, Royer et al. (2000, 2001) used the following expressions, analogues of Eq. (13.13), for the concentration-dependent shift factor at temperatures below T + 100... 

For concentrated polystyrene solutions (c>c ) in n-butylbenzene, Graessley and co-workers [36] observed that the shift factor hconc depends on the number of entanglements per macromolecule E. 

Figure 53. The dielectric loss of 2-picoline in mixtures with tri-styrene at different concentrations obtained at different temperatures but similar a-relaxation times of the 2-picoline component. For clarity, each spectrum is shifted by a concentration-dependent factor kc. Data from T. Blochowicz and E. A. Rossler, Phys. Rev. Lett. 92, 225701 (2004).

Figure 54. Dielectric loss spectra with the same maximum peak frequency for different concentrations of tert-butylpyridine (wt%) in tristyrene. For clarity, each spectrum is shifted vertically by a concentration dependent factor K. K = 1, 2, 1.3, 1, and 0.98 for 100%, 60%, 40%, 25%, and 16% TBP, respectively. The x axis is the real measurement frequency, except for the spectra of 100% and 16% TBP, where horizontal shifts of frequency by factors of 1.75 and 0.80, respectively, have been applied.

The explanation proposed by Ngai and Plazek [1990], was based on the postulate that the number of couplings between the macromolecules varies with concentration and temperature of the blend. The number of couplings, n, can be calculated from the shift factor, a. = [ (T)/ (T )] < ">, where ( g(T) is the Rouse friction coefficient. Thus, in miscible, single phase systems, as either the concentration or temperature changes, the chain mobility changes and relaxation spectra of polymeric components in the blends show different temperature dependence, i.e., the t-T principle cannot be obeyed. Similar conclusions were reached from a postulate that the deviation originates from different temperature dependence of the relaxation functions of the blend components [Booij and Palmen, 1992]. 

The added factor 1/(1 + s/X, s) in Equ. 5.88 represents the toxicity of the substrate at higher concentrations. Let us recall that the condition for calculation of the stationary state with nonvanishing biomass concentration is the relation fx(s) = D. This equation has only one solution if fi(s) is a monotonic function. But with characteristics as in Equ. 5.88, there are two solutions. Together with the washout state ( x, s) we have three stationary states. Two of them are stable ( x, and x, s), one of them is unstable ( x, s). Thus, we have a bistable system. The stationary values of the stable and the unstable stationary state are shown as a function of D in Fig. 6.11. Hysteresis may occur in shift experiments. Figure 6.12 shows how the final biomass concentration depends on the initial concentration. Figure 6.13 demonstrates that the phase plane is divided into two attraction domains. Both domains are touched by a separatrix in which the unstable stationary state lies. Note that, after an external disturbance, the system can cross over the separatrix and shift from one steady state to the other. This bistable behavior is a serious problem in, for example, waste treatment It takes place if substrates such as alcohols, phenols, or hydrocarbons occur in such high concentrations that the utilization of these substrates is inhibited. 

In addition to the free volume [36,37] and coupling [43] models, the Gibbs-Adams-DiMarzo [39-42], (GAD), entropy model and the Tool-Narayanaswamy-Moynihan [44—47], (TNM), model are used to analyze the history and time-dependent phenomena displayed by glassy supercooled liquids. Havlicek, Ilavsky, and Hrouz have successfully applied the GAD model to fit the concentration dependence of the viscoelastic response of amorphous polymers and the normal depression of Tg by dilution [100]. They have also used the model to describe the compositional variation of the viscoelastic shift factors and Tg of random Copolymers [101]. With Vojta they have calculated the model molecular parameters for 15 different polymers [102]. They furthermore fitted the effect of pressure on kinetic processes with this thermodynamic model [103]. Scherer has also applied the GAD model to the kinetics of structural relaxation of glasses [104], The GAD model is based on the decrease of the crHiformational entropy of polymeric chains with a decrease in temperature. How or why it applies to nonpolymeric systems remains a question. 

A viscoelastic shift factor, as, can be found from the ratio of the experimentally measured zero shear rate viscosity (at test conditions of P, T, and SCF cOTicentratiOTi), to the experimentally measured zero shear rate viscosity (at some reference conditions of Fref> Iref, and reference SCF concentration). Once as is determined experimentally, a master curve can be constmcted by plotting ij/as vs. as 7 where tj is the measured viscosity and y is the measured shear rate. If the fractional free volume, /, is estimated from an equation of state as/ = 1 — pjp, where p is the mixture density and p is the mixture close-packed density, then the shift factor due to the presence of the SCF can be calculated from Eq. (18.3), provided the constant B is known. Experimentally, B for SCF-swoUen polymers has been found to be near unity [130,131], in agreement the universal constants of the WLF equation [132] for the temperature dependence of pure polymers. 

Hence, description of the dynamics of the pol5mier-solvent system demands complete specification of both the complex Tg-concentration relationship and the tem-perature(s) of test. Importantly, unlike the case of time-temperature superposition, the shift factors for the relaxation times and the viscosity will not be the same. The latter will scale, as does the viscosity, as in equation 35. However, the relaxation times themselves will scale with a much weaker concentration dependence. 

At finite concentrations, the frequency dependence of G and G" — (aris can be examined directly without scaling the coordinates as in preceding figures. However, measurements at different temperatures may be combined by reduction to a standard temperature To G and G" — coris are multiplied by Toco/Tc and u> is multiplied by a shift factor aj which is given by rjo Vs)ocT/ t]o — Vs)coTq. Here r]o is the steady-flow (vanishing shear rate) viscosity and the subscript 0 otherwise refers to the reference temperature. This is the method of reduced variables which will be discussed fully in Chapter 11 with explanation of its rationale, and affords an extension of the effective frequency range. 

With increasing dilution, the temperature dependence of relaxation times referred to a fixed reference temperature Tq becomes less pronounced as Tg is depressed more and more below Tq and, in equation 21 of Chapter 11, c increases while c decreases. It is possible to analyze the temperature dependence of the shift factor ot at several different diluent concentrations and thus to obtain the dependence of relaxation times at different concentrations. More commonly, however, the magnitudes are compared through a shift factor and the latter is related directly to differences in free volume as explained in the following section. 

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