Kinetics crystallization

A core assumption when discussing the crystallization kinetics of polymers is that the theories must consider the effects of chain folding. In Chapter 2 the principles of crystallization were outlined and the importance of the differences 

Since is negative, the radius of the critical nucleus increases with decreasing degree of supercooling. Inserting eqn (6.27) into eqn (6.23) gives an expression for the free energy barrier (AG )  

Equation (6.29) predicts that nucleation occurs more readily at lower crystallization temperatures because of the lower critical nucleus size and the lower free energy barrier associated with the crystallization process. As with the case of simple crystal growth, nucleation can take a number of forms  

It has been proposed that the following equation can be used to describe the temperature dependence of both diffusive transport and nucleation. The overall crystallization rate (wc) at a general temperature is a combination of several factors. The crystallization process involves the diffusion (the first 

The stem adding to the surface is shown in white except in the tertiary case where the addition is to the trough. 

Chuah [54, 55] compared the isothermal crystallization kinetics of PET, PTT and PBT using DSC, and found PTT to have a crystallization rate in between those 

Using secondary nucleation analysis, Huang and Chang [51] found PTT to go through a transition in the multiple nucleation mechanisms from regimes II to III at around 194 °C. However, Lee et al. [50] found only regime II crystallization between 180 and 200 °C. 

At 71 °C, PTT crystallized and reached 80% of its equilibrium crystallinity in 1 min, while PET, which has a higher Tg, did not cold-crystallize at all. 

The well-know Avrami equation was used to describe the crystallization kinetics  

Cebe studied non-isothermal crystallization from 80 to 380°C, with rates from 1 to 50°C /min. The Avrami constant n was found 

This chapter presents first some fundamental aspects of nucleation, and second the general Avrami equation, which is frequently used to describe overall crystallization. The growth theories of Lauritzen and Hoffman and Sadler and Gilmer are discussed in sections 8.4.2 and 8.4.3. Molecular fractionation and orientation-induced crystallization are dealt with in sections 8.5 and 8.6. 

Let us, for the purpose of demonstration, select a particularly simple case a spherical crystal. The change in free energy on crystallization (AG) is then given by  

Crystal growth occurs by secondary and tertiary nucleation (Fig. 8.2). The initial step is the formation of a secondary nucleus, which is followed by a series of tertiary nucleation events. 

From a consideration of both diffusive transport and nucleation, the following general temperature (TJ dependence is obtained for the overall crystallization rate w )  

In this chapter, we shall focus on two relevant topics the effect of flow on crystallization kinetics, and the effect of crystallization on rheological and thermal properties. 

In the next chapter the rates of nucleation and ciystal growth are discussed. Both mechanisms are decisive for the quality of crystalline products. 

On combining these results with those of Baur [127,129], it is found that [124] 

This portion of the chapter can be summarized by noting that there is a substantial body of evidence demonstrating that formal phase-equilibrium thermodynamics can be successfully applied to the fusion of homopolymers, copolymers, and polymer-diluent mixtures. This conclusion has many far-reaching consequences. It has also been found that the same principles of phase equilibrium can be applied to the analysis of the influence of hydrostratic pressure and various types of deformation on the process of fusion [11], However, equilibrium conditions are rarely obtained in crystalline polymer systems. Usually, one is dealing with a metastable state, in which the crystallization is not complete and the crystallite sizes are restricted. Consequently, the actual molecular stmcture and related morphology that is involved determines properties. Information that leads to an understanding of the structure in the crystalline state comes from studying the kinetics and mechanism of crystallization. This is the subject matter of the next section. 

There are several methods by which the kinetics of crystallization of polymers from the pure melt, or from polymer-diluent mixtures, can be investigated. One procedure is to study the overall rate of crystallization using methods such as dilatometry, calorimetry, and various spectroscopies, for example. Another popular method by which to study the process of crystallization is to measure the rate of growth of sphemlites by direct hght microscopic examination. These two methods complement one another. Measurements of the rates of growth of specific crystal faces have also been employed in favorable cases for studying the kinetics of crystallization from dilute solution. 

The formal basis for analyzing the kinetics of crystallization from the pure melt has been developed substantially. With appropriate modifications, crystallization of polymers has been shown to follow the general mathematical theory that was developed many years ago for the crystallization of metals and other low molecular weight substances. The most elementary form, developed by von Goler and Sachs [132] postulated a process of nucleation and growth. However, in the original formulation there was no termination step, or demarcation for the end of the transformation. To remedy this problem, it was proposed independently by several different investigators that, when two crystallites collided, or made contact. 

Avrami found that the fraction transformed at a time t, namely 1 — A(t), can be 

On the other hand, heterogeneous nucleation is commonly originated by foreign nuclei, that is, any (low mass) particle with the correct size and surface, because it requires a much lower energy barrier than homogeneous nucleation. Commercial polymers typically nucleate in catalytic debris and any other impurities that are left over from their synthesis and/or their first processing before the material is commercialized. 

The crystallization of polymers has been smdied both in solution and from the melt. Crystallization also occurs during polymerization and during cooling from the melt with imposed orientation (e.g., injection molding). In this chapter, we focus particularly on the isothermal melt crystallization under quiescent conditions. 

The crystallization induced by orientation consists of stretching polymer chains to form fibrous crystals or fibers [25], The formation of such fiber-like morphology is accompanied by the formation of a typical shish-kebab or bottiebrush morphology [2,4-12,25-27]. A relevant reference for crystallization under orientation during different polymer processing operations has been recently published [28]. 

The crystallization of polymeric chains under quiescent conditions has been observed and studied for a very large number of synthetic and natural polymers from both dilute solution (leading to the preparation of single crystals and single-crystal mats) and the melt (usually yielding superstructural 3D structures like spherulites, although 2D stmctures like hedrites and axialites are also possible). 

The crystallization from dilute solutions has been used extensively to study the fundamental aspects of the structure and morphology at a molecular level [1-12,29]. On the other hand, the crystallization from the melt is more complex, because diffusion and kinetic effects can often dominate, but it is closer to the solidification conditions applied during processing operations. 

The tendency for a polymer to crystallize is enhanced by regularity and polarity. The molecules must fit together neatly, as well as attract one another, since most of the intermolecular energies vary inversely with the sixth power of the distance between molecules. Examples of the effect of regularity are given in Table 3.2. 

The isotactic form is not always the most crystalline. In poly(vinyI alcohol), for example, the syndiotactic form crystallizes more readily than the isotactic because there are strong repulsive forces between adjacent hydroxyl groups [37] in the isotactic form. 

Nonpolar Polypropylene (atactic), noncrystalline Polar Poly(vinyl alcohol) (atactic), somewhat crystalline Very polar Polyamide (nylon 6), very crystalline 

Although crystallization and melting are reciprocal of each other and occur at the same temperature for small molecules, this is rarely the case for polymers. The reason is that 

Most polymer crystallizations occur via a heterogeneous nucleation mechanism where extraneous nuclei such as dust particles, catalytic residues, or intentionally added nucleating agents initiate the crystallization. For either homogeneous or heterogeneous nucleation, the volume of a crystallite at time t will be V(t,x), where T represents the start of nucleus growth. Under isothermal conditions, one may assume that in isotropic growth, V t,x) obeys the relation  

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Samples can be concentrated beyond tire glass transition. If tliis is done quickly enough to prevent crystallization, tliis ultimately leads to a random close-packed stmcture, witli a volume fraction (j) 0.64. Close-packed stmctures, such as fee, have a maximum packing density of (]) p = 0.74. The crystallization kinetics are strongly concentration dependent. The nucleation rate is fastest near tire melting concentration. On increasing concentration, tire nucleation process is arrested. This has been found to occur at tire glass transition [82]. 

While there are several instances of redundancy among the Avrami exponents arising from different pictures of the crystallization process, there is also enough variety to make the experimental value of this exponent a valuable way of characterizing the crystallization process. In the next section we shall examine the experimental side of crystallization kinetics. 

Crystallization Kinetics. Crystallization of HDPE proceeds in two separate stages. During the first stage, HDPE crystaUizes rapidly. 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

Determination of Crystallization Kinetics. Under steady-state conditions, the total number production rate of crystals in a perfectly mixed crystallizer is identical to the nucleation rate, B. Accordingly,... 

Crystallization kinetics have been studied by differential thermal analysis (92,94,95). The heat of fusion of the crystalline phase is approximately 96 kj/kg (23 kcal/mol), and the activation energy for crystallization is 104 kj/mol (25 kcal/mol). The extent of crystallinity may be calculated from the density of amorphous polymer (d = 1.23), and the crystalline density (d = 1.35). Using this method, polymer prepared at —40° C melts at 73°C and is 38% crystalline. Polymer made at +40° C melts at 45°C and is about 12% crystalline. 

The crystallization kinetics defines the open time of the bond. For automated industrial processes, a fast crystallizing backbone, such as hexamethylene adipate, is often highly desirable. Once the bond line cools, crystallization can occur in less than 2 min. Thus, minimal time is needed to hold or clamp the substrates until fixturing strength is achieved. For specialty or non-automated processes, the PUD backbone might be based on a polyester polyol with slow crystallization kinetics. This gives the adhesive end user additional open time, after the adhesive has been activated, in which to make the bond. The crystallization kinetics for various waterborne dispersions were determined by Dormish and Witowski by following the Shore hardness. Open times of up to 40 min were measured [60]. 

Rielly and Marquis (2001) present a review of crystallizer fluid mechanics and draw attention to the inconsistency between the dependence of crystallization kinetic rates on local mean and turbulent velocity fields and the averaging assumptions of conventional well-mixed crystallizer models. 

The CSD from the continuous MSMPR may thus be predicted by a combination of crystallization kinetics and crystallizer residence time (see Figure 3.5). This fact has been widely used in reverse as a means to determine crystallization kinetics - by analysis of the CSD from a well-mixed vessel of known mean residence time. Whether used for performance prediction or kinetics determination, these three quantities, (CSD, kinetics and residence time), are linked by the population balance. 

Given expressions for the crystallization kinetics and solubility of the system, the population balance (equation 2.4) can, in principle, be solved to predict the performance of both batch and of continuous crystallizers, at either steady- or unsteady-state... 

In addition to induction time measurements, several other methods have been proposed for determination of bulk crystallization kinetics since they are often considered appropriate for design purposes, either growth and nucleation separately or simultaneously, from both batch and continuous crystallization. Additionally, Mullin (2001) also describes methods for single crystal growth rate determination. 

Evidence for secondary nucleation has came from the early continuous MSMPR studies. MSMPR crystallization kinetics are usually correlated with supersaturation using empirical expressions of the form... 

Plots of log population density versus crystal size of the type shown in Figure 5.14 enable the crystallization kinetics to be determined. Some early literature data reporting such analyses are summarized in Table 5.2. 

Crystallization kinetics erystal nueleation, growth, aggregation and disruption kineties (Chapters 5 and 6). 

Garside, J. and Shah, M.B., 1980. Crystallization kinetics from MSMPR crystallizers. Industrial and Engineering Chemistry Process Design and Development, 19, 509-514. 

Gutwald, T. and Mersmann, A., 1990. Determination of crystallization kinetics from batch experiments. In Industrial crystallization 90. Gamiiscli-Partenkirclien, September 1990. Ed. A. Mersmann, Dtisseldorf GVC-VDI, p. 331. 

Hostomsky, J. and Jones, A.G., 1991. Calcium carbonate crystallization kinetics, agglomeration and fomi during continuous precipitation from solution. Journal of Physics D Applied Physics, 24, 165-170. 

Hurley, M.A., Jones, A.G. and Drummond, J.N., 1995. Crystallization kinetics of cyanazine precipitated from aqueous ethanol solutions. Chemical Engineering Research and Design, 73B, 52-57. 

Jones, A.G. and Mullin, J.W., 1973. Crystallization kinetics of potassium sulphate in a draft-tube agitated vessel. Transactions of the Institution of Chemical Engineers, 51, 362-368. 

Nyvlt, J., 1989. Calculation of crystallization kinetics based on a single batch experiment. Collection of Czechoslovakian Chemical Communications, 54, 3187-3197. 

Qian, R., Chen, Z., Ni, H., Fan, Z. and Cai, F., 1987. Crystallization kinetics of potassium chloride from brine and scale-up criterion. American Institution of Chemical Engineers Journal, 33, 1690-1697. 

Qui, Yangeng and Rasmuson, A.C., 1994. Estimation of crystallization kinetics from batch cooling experiments. American Institute of Chemical Engineers Journal, 40, 799-812. 

Swinney, L.D., Stevens, J.D. and Peters, R.W., 1982. Calcium Carbonate Crystallization Kinetics. Industrial and Engineering Chemistry Fundamentals, 21, 31. 

Tavare, N.S., 1986. Crystallization kinetics from transients of an MSMPR crystallizer. The Canadian Journal of Chemical Engineering, 64, 752-758. 

Most of the high-pressure crystallization kinetics studies have shown that ECC grows one dimensionally at high pressures [113-116]. 

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