Nonisothermal Crystallization Of Polymers Research Articles

A note on the kinetics of non-isothermal crystallization of polymers

On the kinetics of non-isothermal crystallization of polymers, the reinterpreted Ozawa method proposed in a previous report is studied further to generate valuable information. The Ozawa method is used to investigate the crystallization kinetics with constant temperature scanning rates β and is concerned with the basic formula for apparent crystallinity, from which the true crystallinity is derived using the concept of an extended volume of the Kolmogorov–Johnson–Mehl–Avrami model. The formula is expressed by the temperature scanning rate β and a function of temperature T, βOz(T). This study investigates the temperature derivative of βOz(T), which represents the crystal growth rate in the presence of athermal nucleation from foreign heterogeneities including nucleating agents; this case is applicable to most polymer crystallization processes. The validity of this relationship was verified by comparing it directly to the crystal growth rate in numerical simulations and by comparing it to the crystallization peak and half times of isothermal crystallization for experimental results of poly(butylene terephthalate). This evaluation provides basic information on the crystal growth rate, which is necessary for understanding polymer crystallization. As is well known, the temperature derivative of βOz(T) also corresponds to the function required to derive Nakamura et al. model's suggested equation for polymer processing.

Analysis of non-isothermal polymer crystallization at constant scan rates based on the Avrami model

When polymer crystallization is initiated by an instantaneous athermal nucleation and controlled by two-dimensional secondary nucleation on the growth face, for crystallization from the melt upon cooling at a constant scan rate β, an analysis based on the Avrami model reveals that a plot of β/(ΔTpeak)2=β/(TM0−Tpeak)2, in which Tpeak represents the peak temperature of crystallization rate and TM0 the equilibrium melting point, is nearly equivalent to that of the inverse of crystallization half-time τ1/2 and peak time τpeak, which are obtained from isothermal crystallization and used as a measure of crystal growth rate. Therefore, a plot of β/(ΔTpeak)2 against Tpeak at different scan rates can be utilized to examine the temperature dependence of the crystal growth rate in the same way as that of τ1/2 and τpeak obtained under isothermal conditions at different temperatures. The applicability was confirmed via numerical calculations and experiments using poly(vinylidene fluoride) and poly(butylene terephthalate). The agreement of the temperature dependence estimated by the plot of β/(ΔTpeak)2 was much better than that of β/(Tpeak)2 used in the well-known Kissinger plot.

A fundamental rule: Determining the importance of flow prior to polymer crystallization

A continuum-level model for nonisothermal polymer crystallization following a complex flow is presented, along with a fundamental rule that may be employed to determine if the flow will influence the ensuing crystallization dynamics. This rule is based on two dimensionless parameters: the (Rouse) Weissenberg number and an inverse Deborah number defined by the ratio between the time taken to cool and the melting point vs the stretch relaxation time, which determines the time available for flow-enhanced crystallization. Moreover, we show how the time to reach the melting point can be derived semianalytically and expressed in terms of the processing conditions in the case of pipe flow—ubiquitous in polymer processing. While the full numerical model is required to quantitatively predict induction times and spherulite-size distributions, the proposed fundamental rule may be used practically to ensure, or eliminate, flow-enhanced structures by controlling the processing conditions or material properties. We discuss how flow-enhanced structures may be revealed only after postprocessing annealing and finally examine previous works that have successfully applied the model to extrusion-based three-dimensional printing.

Nonisothermal crystallization of polymers: Getting more out of kinetic analysis of differential scanning calorimetry data

Nonisothermal crystallization of polymers: Getting more out of kinetic analysis of differential scanning calorimetry data

A critical approach to the Kissinger analysis for studying non-isothermal crystallization of polymers

This work presents a new empirical method to analyse the non-isothermal crystallization kinetics of polypropylene. The Kissinger analysis, widely used in the literature to calculate an activation energy of the crystallization mechanism, was critically examined regarding its applicability to the crystallization process of semicrystalline polymers. The analogy between the systems modelled by Kissinger and polymer crystallization applies only in very restrictive cases, and the activation energy, as commonly used, is an ill-defined quantity. Nevertheless, it was shown that the Kissinger analysis yields a phenomenological parameter, here called REact, related to the crystallization kinetics at moderately high cooling rates (up to ~50 °C s−1). This parameter, created with data obtained on more than 20 poly(propylene) compounds based on four different base polymers of different chain structure and composition, correctly estimates the crystallization performance of different systems at higher cooling rates, demonstrated by a good correlation between REact and the morphology formed in cable extrusion and cast film extrusion processes.

Nonisothermal melt crystallization of PHB/babassu compounds

Poly(3-hydroxybutyrate) (PHB)/babassu compounds were prepared in a laboratory internal mixer with 10, 30, and 50 % by mass of fiber content. Nonisothermal melt crystallization behavior of PHB/babassu compounds was investigated using differential scanning calorimetry, and crystallization parameters were determined at cooling rates ranging between 2 and 32 °C min−1. Adding babassu fiber affected the melt crystallization behavior of PHB, and increasing filler content from 10 to 30 % has significant effects on the thermal characteristics of the system. Further increase in filler content from 30 to 50 % filler content has no effect on crystallization temperature and rate, but it has important positive consequences, once there is a considerably latitude in choosing the actual filler level in highly loaded PHB/babassu compounds without affecting processing characteristics. The melt crystallization kinetics of PHB/babassu compounds was analyzed by three empirical models widely used to represent nonisothermal polymer crystallization data: Pseudo-Avrami, Ozawa, and Mo. Kinetics analyses indicate that the Pseudo-Avrami model represented well the experimental data for both compounds in a wide interval of temperature, conversion, and cooling rates; the Ozawa model with two different sets of parameters, for low and high cooling rates, was found to correlate the data equally well, but over limited ranges of the variables, and the model proposed by Mo and collaborators did not adequately represent the experimental data for the systems and conditions tested.

Cooling rate dependence of the crystallinity at nonisothermal crystallization of polymers: A phenomenological model

ABSTRACTThe cooling rate dependence of the crystallinity of polymers is investigated via the example of different technical polypropylenes using fast scanning calorimetry (FSC) in a cooling rate range between 1 and 5000 K s−1. In the slower cooling rate range (below 100 K s−1) the crystallinity increases slightly with decreasing cooling rate. Above cooling at 100 K s−1 the crystallinity decreases substantially and vanishes at the critical cooling rate. We describe this behavior using a simplified model with two components: the generic crystallinity function and the retardation function. For a mathematical description, we use empirically fitted functions that describe the cooling rate dependence of the crystallinity and the critical cooling rate. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 42977.

Classical One-Phase Stefan Problems for Describing Polymer Crystallization Processes

A free boundary problem framework is proposed to approximate the solution of a deterministic nonisothermal polymer crystallization model in which crystallization fronts appear as the result of the combination of two heat transfer processes: the heat conduction due to the application of a cooling temperature below the polymer melting temperature threshold, and the latent heat production due to the phase change. When the latent heat is larger than the sensible heat of the crystallization process, a classical one-phase Stefan problem can be formulated which allows one to derive analytical approximations describing, for arbitrary applied cooling temperature profiles, the main features of the crystallization process: the relation between the latent heat and the specific heat capacity, the evolution of the temperature distribution, and the advance of the crystallization front. Analytical expressions of magnitudes of industrial interest such as the crystallization time are also derived, allowing the design of optimal cooling strategies for the applied temperature. The limits of the suitability of this framework are discussed, pointing out its applicability to other polymer crystallization models.

The influence of talc as a nucleation agent on the nonisothermal crystallization and morphology of isotactic polypropylene: The application of the Lauritzen–Hoffmann, Avrami, and Ozawa theories

AbstractA new method is proposed, which can be used to analyze the influence of different additives and fillers on the nonisothermal crystallization of polymers. The composites of talc in isotactic polypropylene (i‐PP) were prepared using a corotating twin‐screw extruder. The compounds were subsequently dried and injection molded. PP morphology and talc dispersion were visualized using optical microscopy and computed tomography. Wide‐angle X‐ray scattering and small‐angle X‐ray scattering measurements provided an insight into the crystal structure of PP. The data obtained from nonisothermal DSC measurements were fitted to the Avrami model for the nonisothermal case. The calculated Avrami's exponent (n), which takes into account the influence of talc on the nucleation and growth of the PP crystals, was used in the combination of Lauritzen–Hoffman and Ozawa models to calculate the nucleation parameter (Kg). A good agreement was found between the model predictions and literature values. The examination shows that the developed model extension gives an expected trend in the case of i‐PP filled with talcs from the same origin but with different particle sizes. Furthermore, it is shown that delaminated talc with a higher specific surface is more efficient in nucleation of i‐PP. Thus, the introduced model extension could be a useful tool for comparing of nucleation ability of different additives in the crystallization of polymers. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012

Comparison of the Ozawa and modified Avrami models of polymer crystallization under nonisothermal conditions using a computer simulation method

The Ozawa and the modified Avrami models were usually used to describe the nonisothermal crystallization of polymers. A computer simulation approach was adopted to verify the two models. The results show that the Ozawa model is suitable under constant heating and cooling rate conditions, but not the modified Avrami model. Therefore, the Ozawa model was extended to acquiring the linear growth rate of polymer entities. A nonlinear relationship between the logarithm of the rate function and temperature was found. The values of the Avrami exponent are approximate to the designed value and the linear growth rates of the entities obtained are consistent with the designed ones.

Crystal nucleation and growth in poly(ethylene terephthalate)/alumina‐nanoparticle composites

AbstractThe effect of nanoparticles on nonisothermal polymer crystallization was investigated using poly(ethylene terephthalate) (PET) nanocomposites with alumina (Al2O3) nanoparticles of average size 38 nm. The filler content in the nanocomposites was varied from 0 to 10 wt %. The interparticle spacing was observed to decrease (as expected) with an increase in loading of the nanoparticles. Contrary to previous reports in the literature on semicrystalline polymer‐based composites with micron‐size and macroscale particles, our differential scanning calorimetry, transmission electron microscopy, and X‐ray studies showed that the addition of the nanoparticles did not cause heterogeneous nucleation of PET crystals in nanocomposites containing up to 3 wt % Al2O3. This is attributed to the nanoparticle curvature being comparable to the radius of gyration of the polymer. The addition of the nanoparticles was found to disrupt the spherulitic morphology of the PET because of their physical presence and their proximity to one another. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007

Global weak solutions of non-isothermal front propagation problem

We show the global existence of weak solutions for a free-boundary problem arising in the non-isothermal crystallization of polymers. In particular, the free interface is shown to be of codimension one for every time t in two space dimensions; Holder continuity of the temperature u is proven.

Isothermal and nonisothermal crystallization of polymers: Analysis with a shear differential thermal analyzer

The working principle of an instrument developed for studying the effects of controlled shear pulses on the isothermal and nonisothermal solidification of polymers is presented. The device combines a capillary rheometer and a differential thermal analyzer (DTA). The capillary rheometer part of the system allows the production of shear pulses with controlled duration and intensity at any prescribed temperature up to 300°C, and the DTA records the thermal effects resulting from the solidification. Results obtained for quiescent isothermal and nonisothermal crystallizations compare well with those obtained from power compensation differential scanning calorimetry. The effect of controlled shear pulses on the overall isothermal crystallization kinetics enables the evaluation of a critical strain responsible for the saturation of crystallization in sheared polymer melts. Additional shear (increase of the shear rate or of the shearing time) does not accelerate the crystallization kinetics. The shear-induced nonisothermal crystallization of polyethylene filled with talc shows the acceleration of the crystallization kinetics with the shear rate increase, thus confirming this device as a valuable experimental tool for studying the effect of shear on the solidification of fast crystallizing materials.

Kinetics of Nonisothermal Polymer Crystallization

We adopt a cluster size distribution model to investigate the kinetics of nonisothermal polymer crystallization. The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avrami plots) are presented for different cooling rates. The incubation period is also investigated at different cooling rates and initial temperatures. The relationship between cooling rates and incubation time is presented graphically and compared with experimental measurements. The initial temperature (relative to melting point) has a significant effect on nonisothermal crystallization. A comparison of moment and numerical solutions of the population balance equations shows the influence of Ostwald ripening. Agreement between modeling results and experimental measurements at different cooling rates supports the application of the distribution kinetics model for nonisothermal crystallization.

A simplified model for nonisothermal crystallization of polymers

Recently, Burger and Capasso [1] derived a coupled system of partial differential equations to describe nonisothermal crystallization of polymers. The system is based on a spatial averaging of the underlying stochastic birth-and-growth process describing the nucleation and growth of single crystals. In the present work, we reconsider the scaling properties of the dimensional system as well as some special one-dimensional models. Moreover, using an appropriate scaling of the original system, we derive a simplified model which only consists of a reaction-diffusion equation with memory for the underlying temperature, such that the degree of crystallization can be explicitly given by a time integration of the temperature-dependent growth and nucleation rate. Numerical simulations indicate that the reduced model shows at least qualitatively the same behavior as the original model.

A comparison of some models for describing polymer crystallization at low deformation rates

There is now in the literature a number of mathematical models of the effects of flow on crystallization of polymers, and here we compare the performance of various models in slow simple shearing and sinusoidal strain (oscillatory) flows. We compare the model predictions to the experiments of Wassner and Maier [E. Wassner, R.-D. Maier, Shear-induced crystallization of polypropylene melts. in: D.M. Binding, et al. (Ed.), Proceedings of the XIII International Congress on Rheology, Cambridge, 2000, pp. 1–183] on isotactic polypropylene. Both the Kolmogorov [A.N. Kolmogorov, On the statistics of the crystallization process on metals. Bull. Akad. Sci. USSR, Class Sci., Math. Nat. 1 (1937) 355–359] and Nakamura et al. [K. Nakamura, T. Watanabe, K. Katayama, T. Amano, Some aspects of non-isothermal crystallization of polymers I, J. Appl. Polym. Sci. 16 (1972) 1077–1091] crystallinity formulations are discussed, and the advantages of the former are emphasized. For the cases considered, we see that strain-based crystallinity enhancement formulations appear to give results close to the experiments at the low flow rates which were studied, although they have some drawbacks. Several other ideas also lead to acceptable results; a new formulation based on both strain and strain-rate appears to give the best overall fit.

Crystal shapes and crystallization in continuum modeling

A crystallization model appropriate for application in continuum modeling of complex processes is presented. As an extension to the previously developed Schneider equations [W. Schneider, A. Köppel, and J. Berger, “Non-isothermal crystallization of polymers,” Int. Polym. Proc. 2, 151 (1988)], the model presented here allows one to account for the growth of crystals of various shapes and to distinguish between one-, two-, and three-dimensional growth, e.g., between rod-like, plate-like, and sphere-like growth. It is explained how a priori knowledge of the shape and growth processes is to be built into the model in a compact form and how experimental data can be used in conjunction with the dynamic model to determine its growth parameters. The model is capable of treating transient processing conditions and permits their straightforward implementation. By using thermodynamic methods, the intimate relation between the crystal shape and the driving forces for phase change is highlighted. All these capabilities and the versatility of the method are made possible by the consistent use of four structural variables to describe the crystal shape and number density, irrespective of the growth dimensionality.

Transient and athermal effects in the crystallization of polymers. II. Nonisothermal crystallization

AbstractNonisothermal crystallization of several polymers was investigated with differential scanning calorimetry and optical microscopy. The results indicated that as in the case of isothermal processes, crystallization starts with nucleation on noncompletely melted crystalline residues. It is assumed that if the crystalline residues are subcritical at melting temperatures, they can become stable by an athermal mechanism during cooling. There is also some contribution of nucleation on heterogeneities. The next mechanism of nucleation is a classical homogeneous process occurring by thermal fluctuations. The results showed the non‐steady‐state character of the nonisothermal crystallization of polymers. In the investigated range of cooling rates, the non‐steady‐state character of nonisothermal crystallization of polymers is dominated by the transient thermal effects. In the range of high temperatures, the transient homogeneous nucleation can be interpreted with the Ziabicki model, and the steady‐state rate determined from nonisothermal experiments coincides with the rate determined in isothermal crystallization. The athermal nucleation occurring at the beginning of crystallization from noncompletely melted aggregates seems to be independent of the applied cooling rate. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 68–79, 2003

MATHEMATICAL MODELLING AND SIMULATION OF NON-ISOTHERMAL CRYSTALLIZATION OF POLYMERS

This paper is devoted to the study of mathematical problems arising in the modelling of nonisothermal polymer crystallization under industrial conditions. A review of the hybrid stochastic-deterministic modelling procedure, which yields an averaged model,5,8 is given and supplied by the study of mathematical problems arising from it. Furthermore, multiple scale effects in the crystallization process are studied using different scalings, focussing either on the heat conduction or on the crystallization part. It is shown that in the limit of fast nucleation and slow growth, the model is equivalent to a direct extension of the classical Avrami–Kolmogorov model equations. In addition, numerical simulations are presented to illustrate the behavior during crystallization. Finally, open questions from mathematical and engineering view points are discussed.

Iterative Regularization of a Parameter Identification Problem Occurring in Polymer Crystallization

This paper is devoted to the mathematical analysis and regularization of an identification problem related to nonisothermal crystallization of polymers, which can be modeled by an initial-boundary value problem for a coupled system of parabolic and hyperbolic partial differential equations. The identification problem consists of estimating a material function of temperature, which appears as a nonlinearity in the equations. Existence and uniqueness of a solution of the direct problem is shown, as well as its stability with respect to the parameter. Furthermore, we develop algorithms for the application of various iterative regularization methods to this particular problem. Their use is justified by verifying the Fr{échet-differentiability of the parameter-to-output map, which is needed for their realization. The numerical performance of the iterative methods is compared with respect to speed of convergence, stability, and efficiency.