The Logistic Function in Glass Transition Models of Amorphous Polymers: A Theoretical Framework for Isobaric Cooling Processes

Studying the macroscopic-phenomenological behavior of amorphous polymers at glass transition is often subject to limitations because the ordinary differential equations that describe the material behavior require numerical solution. To avoid these limitations, ad-hoc-formulated models of the glass transition have been proposed. However, their scope of application is expected to be limited due to insufficient theoretical foundation. This work establishes a theoretical framework for models that use the logistic function to approximate the macroscopic-phenomenological behavior of amorphous polymers at glass transition. For this purpose, an exactly-solvable Riccati equation is derived within thermodynamics with internal state variables. A closed-form expression in terms of mathematical functions for the temperature derivative of a single internal state variable is the result. This closed-form expression contains the logistic function thus featuring a continuous transition region centered around a pressure and cooling-rate dependent transition temperature. Based on comparison of existing models with the exact solution derived from the Riccati equation, generalized models that approximate the thermal expansion coefficient and heat capacity at glass transition are proposed. This work thus demonstrates the validity of the logistic function in glass transition models of amorphous polymers and provides suggestions as to how existing models can be extended in their applicability.