Transition Temperature of Color Change in Thermochromic Systems and Its Description Using Sigmoidal Models

Symmetric sigmoidal models with four parameters based on an idealized S/Z-shaped curve are commonly used to analyze the optical parameters of thermochromic materials. However, our experimental findings show that this approach leads to systematic errors involving the incorrect estimation of the transition temperature or the possibility of a virtual indication of the hysteresis nature of a reversible thermochromic change. For this reason, we sought to find a five-parameter model that would appropriately avoid this problem.

Two commercial thermochromic pigments were used for the test and applied to a textile substrate at different concentrations. The optical properties were measured using reflectance spectrophotometry and then converted to Kubelka-Munk function values and colorimetric coordinates. The following statistics were used to assess the quality of the selected sigmoidal models: coefficient of determination, R²; adjusted coefficient of determination, AR²; root mean square error, RMSE; and Akaike Information Criterion, AIC.

The four-parameter models were compared with each other and with the five-parameter models using nested F-tests based on residual variance to obtain a statistical measure of superior performance. For all thermochromic color change data examined, the five-parameter models resulted in significantly better fitting. It could be shown that the five-parameter model showed significantly higher accuracy and precision in determining the transition temperature, like non-sigmoidal quantification methods.

We concluded that the asymmetric five-parameter model is a valuable extension of the symmetric model in the investigation of thermochromic color changes, providing better parameter estimates and a new approach to investigating the mechanisms contributing to the asymmetry of the thermochromic curve.