Modeling DSC Annealing Peaks for Polyetherimide: Incorporation of Temperature Gradients
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during physical aging), DSC can be used to investigate this enthalpy recovery process.•'2's The results of enthalpy recovery experiments can be described using a mathematical model of structural recovery based on work done originally by Narayanaswamyx'5'6'7'8'9'1°, incorporating the 11 7 1012 13 ideas of Tool . However, there are several problems with the models ' ' ' , including an apparent dependence of model parameters on thermal history. The problems may result from 1) the presence of thermal gradients14'•s, 2) an inconsistent equation for describing the effects of temperature and structure on the relaxation time, or 3) incorrect representation of structural recovery using the Kohlrausch-William-Watts (KWW)16'17 function7. It is of interest in this work to examine the effects of the first two issues identified. The presence of thermal gradients in differential scanning calorimetric measurements have been suggested to be a factor in discrepancies between experimental data and model calculations. 14o15 Using Moynihan's formulation of the Tool-Narayanaswamy model, Hodge and Huvard14 found good agreement between calculated and observed values for the magnitude and placement of annealing peaks at low degrees of annealing (small overshoot peaks at Tg) using a single set of model parameters. However, for high degrees of annealing (large overshoots), calculations using the same model parameters overpredicted the magnitude of annealing peaks. The researchers postulated that these differences might be attributable to thermal lag effects. In later work, O'Reilly and HodgeIs found that model parameters varied with thermal history even for relatively slow heating rates (1.25*C/min) where thermal gradients were assumed to be negligible. Their results indicate that thermal gradients are not accountable for all of the shortcomings of the model. They do not address, however, the effects of thermal gradients on the shape of enthalpy recovery curves done in the DSC at typical rates (10°C/min) and the degree to which the gradients contribute to discrepancies between experimental data and model calculations. 177 Mat. Res. Soc. Symp. Proc. Vol. 455 ©1997 Materials Research Society
Model Calculations A full description of Moynihan's form of the Tool-Narayanaswamy model of structural relaxation is given elsewhere'; only a brief description follows. In Moynihan's formulation, the fictive temperature, Tf, originally defined by Tool ', is used as a measure of the structure of the glass. The relaxation process is both nonlinear and nonexponential and is represented by the generalized KWW function: drf dT= 1-expf- (t/-to) 1 dt (1) The nonexponentiality of the process is described by P; the nonlinearity is incorporated into the model by allowing the relaxation time -ro to be a function of both temperature and structure (Tf). Equation (1) is solved numerically for a given thermal history which begins at a temperature To above T for a specified thermal history consisting of cooling and heating ramps. To compare the calculatison to experimen
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