The Effect of a Secondary Process on the Analysis of Isothermal Crystallisation Kinetics by Differential Scanning Calorimetry

Abstract

This paper demonstrates the application of a modified Avrami equation in the analysis of crystallisation curves obtained using differential scanning calorimetry (DSC). The model incorporates a square root of time dependence of the secondary process into the conventional Avrami equation and, although previously validated using laser flash analysis and infrared spectroscopy, is not currently transferable to DSC. Application of the model to calorimetric data required long-duration isotherms and a series of data treatments. Once implemented, the square root of time dependence of the secondary process was once again observed. After separation of the secondary process from the primary, a mechanistic n value of 3 was obtained for the primary process. Kinetic parameters obtained from the analysis were used in the model to regenerate the fractional crystallinity curves. Comparison of the model with experimental data generated R 2 values in excess of 0.995. Poly(3-hydroxybutyrate-co-3-hydroxyvalerate) was used as model polymer due to the prominent secondary crystallisation behaviour that this polymer is known to display.The crystallisation kinetics of a polymer are generally described by the following Avrami equation [15]:where X p,t is the fractional primary crystallinity; X p,∞ is the final primary crystallinity; Z p is the Avrami primary crystallisation rate constant; t is time and n an integer combining values attributed to both the geometry and mechanism of nucleation. Although this equation is widely used, it often generates non-integer n values. In addition, it does not take into account the secondary crystallisation process, which has been proven to occur during polymer crystallisation [16][17][18][19]. Numerous authors have proposed modifications to the Avrami equation to account for these issues, with the most notable being Hillier [20,21], Velisaris [17] and Hay [22].The method developed by Hillier [20] assumes an initial constant radial growth of spherulites, followed by a first-order increase in crystallinity at time θ. This leads to the standard Avrami equation for the primary process (Equation (1)) and a modified version for the secondary process (Equation (2)) in which the n exponent is taken as 1:where X s,t−θ is the fractional crystallinity formed after time θ; X s,∞ is the final fractional crystallinity; and z s is the secondary rate constant. These two equations can then be combined to give an expression for the total crystallinity at time t. X t = X p,t + t 0

The Effect of Powder Re-Use on the Coalescence Behaviour and Isothermal Crystallisation Kinetics of Polyamide 12 within Powder Bed Fusion

“…Therefore, for full phase transformation to occur, a greater contribution from secondary crystallisation is required. It is thought that secondary crystallisation is a thickening process that adheres to either a log [ 23 , 24 , 25 ] or a root time [ 18 , 47 ] dependence, so it occurs relatively slowly compared to primary crystallisation. For the secondary process to occur, rearrangement of relatively immobile, highly entangled inter-lamellar amorphous regions is required.…”

“…To enable visual comparison with the Hay model [ 22 , 51 ], cumulative crystallinity data were converted to true fractional crystallinity using Equation (2). For a detailed explanation of each model, refer to previous work from Kelly and Jenkins [ 15 , 47 ]. …”

“…These limitations have led various authors to propose alternative models that aim to resolve these problems by modifying the existing Avrami equation. These include, but are not limited to, the simplified Hillier [ 43 ], Tobin [ 44 ], Malkin [ 45 ], and Hay models [ 20 , 21 , 46 , 47 ].…”

Nonisothermal Crystallization Kinetics by DSC: Practical Overview

“…In turn, the crystallites can involve disordered or poorly ordered phases capable of further, i.e., secondary, crystallization. It is worth mentioning that there is a possibility to incorporate secondary crystallization into the Avrami model [38].…”

Determination of the activation energy under isothermal conditions: revisited

“…The isothermal kinetic analysis continues to be amply employed nowadays to study a wide range of processes. Thus, it is the most feasible option for reactions studied under realistic conditions of operation [19,20], long-term curing [21,22], oxidation and reduction processes [23,24], hydration and dehydration processes [25][26][27], reactions followed through X-ray diffraction [28] or those studies intended to discriminate the kinetic model under isothermal conditions [29][30][31][32].…”