Equivalent thermal history reconstruction from a partially crystallized glass-ceramic sensor array

Abstract

The basic concept of a thermal history sensor is that it records the accumulated exposure to some unknown, typically varying temperature profile for a certain amount of time. Such a sensor is considered to be capable of measuring the duration of several (N) temperature intervals. For this purpose, the sensor deploys multiple (M) sensing elements, each with different temperature sensitivity. At the end of some thermal exposure for a known period of time, the sensor array is read-out and an estimate is made of the set of N durations of the different temperature ranges. A potential implementation of such a sensor was pioneered by Fair et al. [Sens. Actuators, A 141, 245 (2008)], based on glass-ceramic materials with different temperature-dependent crystallization dynamics. In their work, it was demonstrated that an array of sensor elements can be made sensitive to slight differences in temperature history. Further, a forward crystallization model was used to simulate the variations in sensor array response to differences in the temperature history. The current paper focusses on the inverse aspect of temperature history reconstruction from a hypothetical sensor array output. The goal of such a reconstruction is to find an equivalent thermal history that is the closest representation of the true thermal history, i.e., the durations of a set of temperature intervals that result in a set of fractional crystallization values which is closest to the one resulting from the true thermal history. One particular useful simplification in both the sensor model as well as in its practical implementation is the omission of nucleation effects. In that case, least squares models can be used to approximate the sensor response and make reconstruction estimates. Even with this simplification, sensor noise can have a destabilizing effect on possible reconstruction solutions, which is evaluated using simulations. Both regularization and non-negativity constrained least squares minimization approaches are shown to enable robust thermal history estimation. Based on the theoretical and simulation results, several general criteria can be established for the sensor performance and its practical implementation.