Non-isothermal crystallization

Modelling non-isothermal crystallization is the next important step in a quantitative description of reactive processing. This is particularly important, because crystallization determines the properties of the end product. Therefore, the development the spatial distribution of crystallinity, a, and temperature, T, with time throughout the volume of the reactive medium must be calculated. It is also noteworthy that crystallization and polymerization processes may occur simultaneously. This happens when polymerization proceeds at temperatures below the melting point of the newly formed polymer. A typical example of this phenomenon is anionic-activated polymerization of e-caprolactam, which takes place below the melting temperature of polycaproamide. 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and 

Isothermal curves derived from this equation are shown in Fig. 2.19. It is clear that this equation fits the experimental data. A comparison of the kinetic equation (2.48) and the Avrami equation shows93 that any experimental data described by the Avrami equation can be approximated by Eq. (2.48) for any arbitrary set of constants. The divergence of the curves does not exceed 1% in the range 0.2 a 1.0 and 8% intherangeO a 0.2. This means that the same experimental data (in the isothermal case) can be analyzed by both equations with practically the same reliability. Thus the choice of approximating equation depends on the goal of this procedure if we are interested in physi- 

To applying Eq. (2.47) to non-isothermal problems, it is necessary to generalize it by introducing temperature-dependent constants. The basic approach was proposed by Ziabicki94,95 who developed a quasi-static model of non-isothermal crystallization in the form of a kinetic rate equation  

The kinetic function f(T,a) was assumed to be a first-order equation. For a quasi-static approximation, we can write the following equation for the rate of crystallization  

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Verhoyen, 0., Dupret, F. and Legras, R., Isothermal and non-isothermal crystallization kinetics of polyethylene terephthalate mathematical modeling and experimental measurement, Polym. Eng. Sci., 38, 1592-1610 (1998). 

The non-isothermal crystallization dynamics were performed using DSC, employing cooling rates of 2.5, 5, 10, 20, 25, 30, 35 and 40°C/min. The isothermal crystallization dynamics were studied for each sample heated to 290 °C, with a 5 min hold time, and cooled to the isothermal crystallization temperature using a cooling rate of 200°C/min, and then holding for 40 min to obtain the crystallization exotherm. 

The Ozawa equation of isothermal crystallization dynamics applied to non-isothermal crystallization assumes that the crystallization proceeds under a constant cooling rate, from the valid mathematical derivation of Evans [47], In... 

F. J. Medellin-Rodriguez, C. Burguer, B. S. Flsiao, B. Chu, R. Vaia, S. Phillips, Time-resolved shear behavior of end tethered nylon 6-clay nanocomposites followed by non-isothermal crystallization, Polymer, vol. 42, pp. 9015-2023, 2001. 

L. Zhang, T. Tao, C. Li, Formation of polymer/carbon nanotubes nano-hybrid shish-kebab via non-isothermal crystallization, Polymer, vol. 50, pp. 3835-3840, 2009. 

In this case, the differential equations describing the process of non-isothermal crystallization in a batch-process reactor configured line a plane plate can be as ... 

K. Nakamura, K. Katayama, and T. Amano, Some Aspects of Non-isothermal Crystallization of Polymers. Part II. Consideration of Isokinetic Conditions, J. Appl. Polym. Sci., 17, 1031-1041 (1982). 

D. W. Henderson, Thermal Analysis of Non-Isothermal Crystallization Kinetics in Glass Forming Liquids , Journal of Non-Crystalline Solids, 30 301-315 (1979). 

The observations of Campos etal. (2002) showed that in non-isothermal crystallization, the slow reduction of temperature results in a lower crystal volume containing larger crystals and a more heterogeneous spatial distribution of the mass. This gives a softer fat compared to when milk fat is crystallized at a faster rate. In laboratory experiments using a Bohlin rheometer as a crystallizer, Breitschuh and Windhab (1998) demonstrated that compound crystals were formed during supercooling and that less compositionally differentiated fractions were produced. 

It is very important to understand non-isothermic kinetics to analyse the composition s technological properties, especially at very high sp s of cooling, which occur in thermoplasts and thermoplast-based compositions processing. The method for describing and forecasting non-isothermic crystallization kinetics is proposed in Ref [59]. Alternative methods for describing kinetics are in the studies cited in Ref [59] (see also Refs. [66, 67]). The essence of the method in Ref [59] is as follows. 

When the TTS principle is carried out, the non-isothermic crystallization kinetics can be described by replacing argument t in the dependence of non-crystallinity degree of time during isothermic crystallization. The replacement equation is... 

During polymer processing non-isothermal crystallization conditions, mechanical deformation, and shear forces may alter the morphology and orientation of polymers both at the surface and in the bulk. In addition, orientation effects of semicrystalline polymers that crystallize in contact with solids are considered. 

Rtx Rate of change of temperature during non-isothermal crystallization. 

A. Saleki-Gerhardt and G. Zografi, Non-isothermal crystallization of sucrose from the amorphous state, Pharm. Res. 11, 1166-1173(1994). 

PLA crystallizes usually between 83 and 150°C but its fastest rate of crystallization occurs between 95 and 115°C [83]. The value of the crystallization half time (t, j) varies according to author. In the temperature range 95-115 °C the tj 2 of PLLA for crystallization from the melt varies between 1.5 min to 5 min [45, 79, 84]. Nevertheless the optimum, 1.5 min, is obtained at around 110°C for isothermal crystallization from melt (Figure 8.6) [45]. Not only does the tj of PLA depend widely on the crystallization temperature, but it is also linked to the crystallization type (isothermal or non-isothermal, from cold or melted state). So upon isothermal crystallization from the cold state, t is below 2 min [79, 85, 86]. Eventually, upon non-isothermal crystallization, t also lies around 2 min [85,87,88]. The further the isothermal crystallization is from this optimum, the more tj increases. For isothermal crystallization below 90°C or above 130°C, tj can be beyond 10 min [45, 69]. 

Upon non-isothermal crystallization the Avrami exponent takes on values between 2.1 and 4.82, whereas the Avrami crystallization rate constant are found between 0.0104 and 0.685 [85,87]. 

Budrugeac et al. [123] examined the kinetics of the non-isothermal crystallization of (GeS2)o.3(Sb2S3)o.7 by employing the methods of Friedman and of invariant kinetic parameters and demonstrated that the process can be treated as a single step. A more complex kinetic situation has been encoimtered by Thomas and Simon in re-crystallization of nickel sulfide from the a- to P-form. Their analysis yielded evidence of at least two steps involved in the overall process [124]. 

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