THERMORHEOLOGICAL

Anandhan, S., De, P.P., De, S.K., Swayajith, S., and Bhowmick, A.K., Thermorheological properties of thermoplastic elastomeric blends of nitrile rubber and poly(styrene-co-acrylonitrile) containing waste nitrile rubber vulcanizate powder, Kautsch. Gummi Kunst., 11, 2004. 

Generally, the rheology of polymer melts depends strongly on the temperature at which the measurement is carried out. It is well known that for thermorheological simplicity, isotherms of storage modulus (G (co)), loss modulus (G"(complex viscosity (r (co)) can be superimposed by horizontal shifts along the frequency axis ... 

The rheological properties of insitu polymerized nanocomposites with end-tethered polymer chains were first described by Krisnamoorti and Giannelis [33]. The flow behavior of PCL- and Nylon 6-based nanocomposites differed extremely from that of the corresponding neat matrices, whereas the thermorheological properties of the nanocomposites were entirely determined by the behavior of the matrices [33]. The slope of G (co) and G"(co) versus flxco is much smaller than 2 and 1, respectively. Values of 2 and 1 are expected for linear mono-dispersed polymer melts, and the large deviation, especially in the presence of a very small amount of layered silicate loading, may be due to the formation of a network structure in the molten... 

For linear thermorheologically simple materials a single temperature-dependent shift factor, aT T), can be used to predict the transient thermal response [20]. The mechanical response is history dependent and involves the use of reduced times, ( ) and (t), which can be found from the shift factor as... 

Materials to which time-temperature superposition is applicable, are sometimes termed thermorheologically simple materials. The amount of shift, log aT, which is required to bring measurements at a given temperature, T, into superposition with measurements at a reference temperature, Tr, is described, usually within a range of temperatures Tg < T < (Tg + 100), by the WLF equation (6) ... 

Many amorphous homopolymers and random copolymers show thermorheologically simple behavior within the usual experimental accuracy. Plazek (23,24), however, found that the steady-state viscosity and steady-state compliance of polystyrene cannot be described by the same WLF equation. The effect of temperature on entanglement couplings can also result in thermorheologically complex behavior. This has been shown on certain polymethacrylate polymers and their solutions (22, 23, 26, 31). The time-temperature superposition of thermorheologically simple materials is clearly not applicable to polymers with multiple transitions. The classical study in this area is that by Ferry and co-workers (5, 8) on polymethacrylates with relatively long side chains. In these the complex compliance is the sum of two contributions with different sets of relaxation mechanisms the compliance of the chain backbone and that of the side chains, respectively. 

Since the relaxation mechanisms characteristic of the constituent blocks will be associated with separate distributions of relaxation times, the simple time-temperature (or frequency-temperature) superposition applicable to most amorphous homopolymers and random copolymers cannot apply to block copolymers, even if each block separately shows thermorheologically simple behavior. Block copolymers, in contrast to the polymethacrylates studied by Ferry and co-workers, are not singlephase systems. They form, however, felicitous models for studying materials with multiple transitions because their molecular architecture can be shaped with considerable freedom. We report here on a study of time—temperature superposition in a commercially available triblock copolymer rubber determined in tensile relaxation and creep. 

R is the gas constant and Ea the flow activation energy. The latter is a material-specific factor that is not dependent on the molar mass of the polymer and for thermorheological simple polymers also is not dependent on the shear stress. The activation energy Ea for polymer melts varies between 25 and 80 kj/mol. It can be determined from the slope of the line in the Arrhenius plot. 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

Few examples of the homogeneous diblock-incompatible homo-polymer behavior have been reported. One that has received considerable attention is the system polystyrene-poly-a-methylstyrene (2). Block copolymers of styrene and a-methylstyrene exhibit a single loss peak in dynamic experiments (2,3) and have been shown to be thermorheologi-cally simple (4) hence they are considered to be homogeneous. Mechanical properties data on these copolymers also has been used to validate interesting extensions of the molecular theories of polymer viscoelasticity (2,3,4). 

Calculation of Master Curves from Mechanical Models. The only way to obtain valid master curves for the thermorheologically complex systems (75/25 and 50/50 blends) is to calculate the moduli of the blends as a function of time, using an appropriate mechanical model. This method requires knowledge of the time and temperature dependence of the mechanical properties of the constituent phases. 

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

The time-temperature superposition method can be also applied to viscosity data (Ferry, 1980). For any viscoelastic parameter, exact matching of the adjacent curves is an important criterion for the applicability of the method. In addition, when possible, the same values of oy must superpose all the viscoelastic parameters and the temperature dependence of ar should have a reasonable form based on experience. One advantage of the method is that the range of frequencies are extended beyond those available experimentally. The time-temperature method has been also referred to as thermorheological simplicity (Plazek, 1996). 

Liao, H.-J. 1998. Simulation of continuous sterilization of fluid food products the role of thermorheological behavior of starch dispersion and process, Ph.D. thesis, Cornell University, Ithaca, NY. 

Plazek, D. J. 1996. 1995 Bingham medal address Oh, thermorheological simplicity, wherefore art thou J. Rheology 40 987-1014,... 

HEAT TRANSFER TO FLUID FOODS Thermorheological Models... 

In order to understand or study heat transfer phenomenon, the rheological behavior of a fluid food must be known as a function of both temperature and shear rate. For convenience in computations, the effect of shear and temperature may be combined in to a single thermorheological (TR) model. A TR model may be defined as one that has been derived from rheological data obtained as a function of both shear rate and temperature. Such models can be used to calculate the apparent viscosity at different shear rates and temperatures in computer simulation and food engineering applications. For a simple Newtonian fluid, because the viscosity, r), is independent of shear rate, one may consider only the influence of temperature on the viscosity. For many foods, the Arrhenius equation (Equation 2.42) is suitable for describing the effect of temperature on t] ... 

Figure 8-7 Thermorheological Data During Gelatinization of a Com Starch Dispersion (Yang and Rao, 1998b).

Figure 8-11 Thermorheological Data During Gelatinization of a Waxy Rice Starch Dispersion (liao, 1998).

Liao, H.-J., Rao, M. A., and Datta, A. K. 2000. Role of thermorheological behavior in simulation of continuous sterilization of a starch dispersion. iChemE Trans. Part C—Food and Bioproducts Process. 78(C1) 48-56. 

Lest one ignore the important role of rheological behavior and properties of fluid foods in handling and processing foods, they are covered in Chapter 8. Here, the topics covered include applications under isothermal conditions (pressure drop and mbcing) and under non-isothermal conditions (heat transfer pasteurization and sterilization). In particular, the isothermal rheological and nonisothermal thermorheological models discussed in Chapters 3 and 4 are applied in Chapter 8. 

Such residual stresses lead to warping, anisotropic shrinkage, and other product flaws. Thus, adequate thermorheological modeling remains a major engineering problem. 

A fundamental characteristic of the so-called thermorheologically simple systems is that consecutive isotherms have similar habits, so they overlie each other when they are shifted horizontally along the log t axis. In other words, the time-temperature correspondence principle holds. This property in creep experiments can be expressed by the relation (2,3)... 

For low molecular weight fractions, the variation in the values of the compliance function increases as either the chain length or the temperature increases. The changes observed in the compliance with temperature for very low molecular weight fractions are illustrated in Figure 8.17 (16). This lack of thermorheological simplicity was also observed for other amorphous polymers, specifically poly(ethyl methacrylate) (21), poly( -butyl methacrylate) (22), poly( -hexyl methacrylate) (23), and low molecular weight poly(methylphenyl siloxane) (24). 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

In thermorheological simple systems, the time-temperature correspondence principle holds. Chapter 8 gives examples of isotherms for compliance functions and relaxation moduli. The shift factors are expressed in terms of terminal viscoelastic parameters, and the temperature dependence of the shift factors is interpreted in terms of the free volume and the WLF equation. The chapter outlines methods for determining the molecular weight between entanglements, and analyzes the influence of diluents and plasticizers on the viscoelastic functions. 

Jones DS. Thermorheological (dynamic oscillatory) characterization of pharmaceutical and biomedical polymers. In Craig DQM, Reading M, eds. Thermal Analysis of Pharmaceuticals. Boca Raton CRC Press, 2007 311-58. 

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