Crystallisation Kinetics of PEEK Composites using Fractional Differential Equations
Topic(s) : Manufacturing
Co-authors :
Maria VEYRAT CRUZ-GUZMAN (UNITED KINGDOM), Jonathan P.-H. BELNOUE (UNITED KINGDOM), Stephen J. EICHHORN , James MYERS , Adam CHAPLIN (UNITED KINGDOM), John GRASMEDER (UNITED KINGDOM), Dmitry S. Ivanov (UNITED KINGDOM)Abstract :
Semicrystalline thermoplastic composites are experiencing a resurgence in interest and demand due to their attractive properties, such as their ability to be reshaped. This makes them appealing candidates for a range of applications requiring reusability, as well as emerging applications such as additive manufacturing. The crystallisation behaviour of thermoplastics is the most important phase transition to consider when processing thermoplastics. This is because crystallisation affects not only the microstructure of the polymer but also influences the bulk properties. It has been demonstrated that there is an important relationship between the degree of crystallinity and the interfacial strength of a composite, as well as other macroscale properties. [1] Hence, crystallisation modelling is essential for process optimisation in 3D printing, over-moulding, press-forming and other processes. The correct description of this complex phenomena yields benefits in mechanical properties, processing time, and energy consumption.
State-of-the-art crystallisation models, such as the differential forms of Avrami and Nakamura, tend to overpredict the degree of crystallinity and have difficulties when used for complex thermal history and when integrated with melting models. Additionally, they tend to exhibit a large number of material parameters and fitting constants, which are difficult to identify. [2,3]
DSC tests were conducted on PEEK to characterise its crystallisation behaviour at a range of temperatures and rates. Experimental crystallisation data, comprising quasi-isothermal and dynamic tests at different cooling rates, were mapped into a conventional kinetics space: crystallisation rate, temperature, and crystallisation. Fig1a. If the current models are appropriate for this material, then all the data should collapse on a single surface. However, there is a clear discrepancy between the two data sets and neither model can satisfy both sets.
This paper proposes to address the observed discrepancy by formulating kinetics equations in fractional space using the Caputo derivative of crystallisation instead of the crystallisation rate. By choosing the degree of differentiation, we demonstrate that the data sets can be reconciled towards a single surface. The discrete experimental data are then fitted with a simple double Weibull distribution, Eq.1., with a degree of accuracy. Fig.1b.
D^m X= k*W_1 (T)*W_2 (X)
Eq. 1) Double Weibull formula, W stands for the cumulative probability Weibull function, k is a scaling coefficient and T is the normalised temperature.
The validity of the fit was then verified by solving the fractional differential equation. Fig.2. The model is shown to reproduce the results of the tests with high precision up to the end of the tests, where those same tests conducted on different samples start to diverge. This demonstrates a promising and flexible tool for the analysis of crystallisation.
State-of-the-art crystallisation models, such as the differential forms of Avrami and Nakamura, tend to overpredict the degree of crystallinity and have difficulties when used for complex thermal history and when integrated with melting models. Additionally, they tend to exhibit a large number of material parameters and fitting constants, which are difficult to identify. [2,3]
DSC tests were conducted on PEEK to characterise its crystallisation behaviour at a range of temperatures and rates. Experimental crystallisation data, comprising quasi-isothermal and dynamic tests at different cooling rates, were mapped into a conventional kinetics space: crystallisation rate, temperature, and crystallisation. Fig1a. If the current models are appropriate for this material, then all the data should collapse on a single surface. However, there is a clear discrepancy between the two data sets and neither model can satisfy both sets.
This paper proposes to address the observed discrepancy by formulating kinetics equations in fractional space using the Caputo derivative of crystallisation instead of the crystallisation rate. By choosing the degree of differentiation, we demonstrate that the data sets can be reconciled towards a single surface. The discrete experimental data are then fitted with a simple double Weibull distribution, Eq.1., with a degree of accuracy. Fig.1b.
D^m X= k*W_1 (T)*W_2 (X)
Eq. 1) Double Weibull formula, W stands for the cumulative probability Weibull function, k is a scaling coefficient and T is the normalised temperature.
The validity of the fit was then verified by solving the fractional differential equation. Fig.2. The model is shown to reproduce the results of the tests with high precision up to the end of the tests, where those same tests conducted on different samples start to diverge. This demonstrates a promising and flexible tool for the analysis of crystallisation.