Modeling of Glass Transition

Several theories have been developed to describe glass transition, such as the thermodynamic theory [24, 25], free volume theory [26], or kinetic theory [27]. The former employs the lattice model to establish the partition function and the entropy of polymer can be calculated through this partition function. The latter takes the volume changes during the glass transition stage into account. 

Equation (2.12) is a differential equation with respect to time t that is able to take the effects of complex thermal loading (thermal loading at variable heating rates) into account. As any thermal loading procedure is also a function of time, and based on a finite difference method, the temperature at each finite time step can be approximated as a constant At a time step, J, with a constant heating rate, fip Eq. 2.12 can be converted to  

Arrhenius kinetics is used to calculate the reaction rate of a chemical process that typically is high. However, glass transition also becomes very fast, as far as, elevated temperatures and fire scenarios are involved. The modeling performance will be evaluated through a comparison to DMA glass transition results in Chapter 5. 

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Although glass transition is conventionally defined by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari-Goldstein (JG) secondary relaxation. The JG relaxation time, xjg, like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping xa constant in the equilibrium liquid state of a glass-former. For any fixed xa, the ratio, T/G/Ta, is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 — n, of the Kohlrausch function that fits the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. 

K. R. Sharma, Mathematical Modeling of Glass Transition Temperature of Partially Miscible Polymer Blends and Copolymers, 214th ACS National Meeting, Dallas, TX, 1998. 

H. Tanaka, Two-order-parameter description of liquids. I. A general model of glass transition covering its strong to fragile limit. J. Chem. Phys. Ill, 3163-3174 (1999). 

Lievonen, S.M. and Roos, Y.H. 2002a. Nonenzymatic browning in amorphous food models Effect of glass transition and water. J. Food Sci. 67, 2100-2106. 

Measurement of Glass Transition Temperature (Tg). obtained polymers were measured by DSC analysis (Du Pont Inc., 910 model) at a heating rate of 20°C/min. 

Finally, it is also worth mentioning that measurements of glass transition temperatures were carried out on model networks. It was shown that T% increases linearly with the reciprocal of the molecular weight of the elastic chains, for a series of homologous networks. The slope of the lines is the steeper, the higher the functionality of the crosslinks67. ... 

Figure 4.3 shows a plot of both characteristic times as a function of 1/T. When xc < xq, the polymer is able to reach, continuously, the equilibrium distribution of conformations. So it remains in the rubbery (or liquid) state. But when x > xq, the polymer cannot reach equilibrium in the time-scale of the experiment and it behaves as a glass. In the frame of this kinetic model, the glass transition may be defined as the temperature at which xc = xq (Fig. 4.3). 

The study of glass transition is an important subject in current research, and simulations may well be suited to help our understanding of the phenomenon. An example is the application of Monte Carlo techniques by Wittman, Kremer, and Binder.The authors employed a lattice method in two dimensions to model the system. The glass transition was determined by monitoring the free volume changes as well as isothermal compressibility. The glasslike behavior was determined by evaluating the bond autocorrelation function. The authors found that both the dynamic polymer structure factor and the orienta-... 

The -model possesses a line of glass transitions where the long time limit f = jumps discontinuously it obeys the equivalent equation to (21). The glass transition line is parameterized by (vJ, V2) = ((2A — with 0.5 < A < 1, and... 

Karmas, R. and Karel, M. The effect of glass transition on Maillard browning in food models, Maillard Reactions in Chemistry, Food, and Health, T.R Labuza, G.A. Reineccius, V.M. Monnier, J. O Brien and J.W. Baynes, eds.. The Royal Society of Chemistry, Cambridge, U.K., pp. 182-187, 1994. 

Khalloufi, S., El-Maslouhi, Y., and Ratti, C. Mathematical model for prediction of glass transition temperature of fruit powders, /. Food Sci., 65, 842, 2000. 

Provided care is exercised in the choice of models and questions asked, computer experiments will prove to be an increasingly valuable aid to the elucidation of glass transition phenomena and the study of amorphous structures, when carefully applied in conjunction with real experimental and theoretical studies. 

V. Truong, B.B. Bhandari, T. Howes and B. Adhikari, Analytical model for the prediction of glass transition temperature of food systems. In Amorphous Pood and Pharmaceutical Systems, H. Levine (ed.). Royal Society of Chemistry, Cambridge, 2002, pp. 31-47. 

Thermal Characterization. The DuPont 941 Thermal Mechanical Analyzer (TMA), attached to a model 900 thermal analyzer, was used for the measurement of glass transition temperature (Tg). The heating rate was set at 10 C/min. For wet samples, about 0.05 ml of distilled water was dropped into the TMA sample holder tube before running the experiments, to maintain the sample in a water saturated state during the measurement. 

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