Thermal Analysis of Polymeric Materials

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Fig. 3.77

Fig. 3.78

3.6.3 Diffusion Control of Crystallization

Using the same polymer solution as in Fig. 3.78 for analysis of the crystal growth at lower temperature, the growth rate slows with time, as shown in Fig. 3.79. The crystal growth slows because the diffusion limits the number of crystallizable molecules in

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concentration and the build-up of solvent due to the slow diffusion. The distance (s s ) is a measure of the influence of the diffusion on crystal growth. The two sketches on the right suggest that diffusion control leads to faster crystal growth at corners of crystals and produces a dendritic morphology described in Sect. 5.2.5. In Fig. 3.81 an electron micrograph of polyethylene crystals with growth spirals in the center is given where the true crystallographic angle at the crystal tips is narrowed from its crystallographic angle of 67o due to diffusion control.

Fig. 3.81

3.6.4 Growth of Spherulites

Diffusion or thermal conductivity limitation may change the crystal morphology to such a degree that the dendritic morphology grows in all directions with approximately the same rate, namely that of the fastest growing apex. The electron micrograph of Fig. 3.82 shows the development such fast-growing tips in many directions. Additional discussion of dendritic crystal morphologies are given in Sect. 5.2.5 and an assessment of spherulites was given recently [32].

At sufficient supercooling, the spherulitic superstructure may be described with only one growth rate, va, as shown in Fig. 3.83. The increase in crystal volume, V, is given by the second equation and allows the computation of the enthalpy evolved by multiplication with the product of density and specific heat of fusion ( hf in J g 1). With nucleation data, the crystallization rate of the whole sample can be computed and linked to growth rates measured by dilatometry or calorimetry. Results for the LiPO3 crystallization, which is discussed in Sect. 3.1.6, are also listed in Fig. 5.83 [15]. The changes of the growth rate with temperature are given in the table. Note that the crystallization of the polymer is in this case coupled with the polymerization reaction [1], and increases with temperature.

260 3 Dynamics of Chemical and Phase Changes

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Fig. 3.82

Fig. 3.83

3.6.5 Avrami Equation

Knowing from the microscopic analysis the number of athermally growing nuclei and the linear crystal growth rate, it is easy to calculate the initial volume fraction, vc, crystallized at time t. This volume fraction is needed for the link of the crystallization

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data to dilatometry (see Sect. 4.1) or heat of crystallization (see Sects. 4.2–4.4). In Fig. 3.84 the total volume Vi of crystal i, and the sum over all crystals are computed for a crystal morphology as shown in Fig. 3.57. This calculation, using Eq. (2), is

Fig. 3.84

called the free growth approximation of crystallization. It applies to the initial time period before significant impingement of neighboring crystals occurs, as can be seen from a comparison of Figs. 3.55 and 3.56. The dependence of the volume on the third power of time given in Eq. (1) accounts for the fast increase in crystallization with time. In case the nucleation is thermal, more complicated equations must be derived. At present, one usually uses only linear increases of nuclei with time and, if needed, an additional induction time as seen in Fig. 3.58.

Next, the equation for the evaluation of the crystallized volume-fraction is derived in Eqs. (3 5) of Fig. 3.84. Equation (5) is also used to compute the volume fraction crystallinity of semicrystalline polymers from dilatometric experiments (see Sect. 4.1). The more common mass-fraction crystallinity is derived in Sect. 5.3.1. Simple density determinations as a function of time can, according to Eq. (5), establish the overall progress of crystallization.

The last step is the evaluation of the impingement of the crystals as observed in Figs. 3.56 and 3.57. It is based on the Poisson equation, written as Eq. (6) in Fig. 3.84, which gives the probability that a point in the melt is not overrun by a growing, spherical crystal, i.e., remains melted (P = 1 vc). The exponent x is the average chance, or the expectation value, simply the free growth approximation referred to the unit volume. Adding all together gives the Avrami equation, Eq. (8). Figure 3.85 shows the influence of n in the generalized Avrami equation. Experiments for the LiPO3 crystallization at different temperature are shown in Fig. 3.86 [15]. As long as the model used for nucleation and growth, and the assumptions that go into the

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Although the Avrami equation is derived for a very specific model, it is often generalized, as shown by the equation in Fig. 3.85. However, only when based on the Poisson equation can one expect a true link between experiment and crystallization model. In case the nucleation increases linearly with time, the Avrami exponent increases to four for a spherulitic crystallization. Often it is assumed that growth of lamellar and fibrillar morphologies can also be described with n changed to two and one (athermal nucleation) or three and two (thermal nucleation), respectively. In these cases, however, the condition that x is the free growth approximation in the unit volume is not fulfilled and geometric effects and the influence of the growth direction must be considered in evaluation of P in Fig. 3.84.

3.6.6 Nonisothermal Kinetics

An advantage for thermal analysis would be gained if one could evaluate crystal growth and chemical reactions with changing temperature in nonisothermal experiments. The simple kinetic models described in this chapter can be extended, making use of the change of reaction rates with temperature, as described by the Arrhenius equation described in Appendix 7, Fig. A.7.2. Figure 3.87 indicates the change of the rate constant with temperature, governed by an activation energy.

Fig. 3.87

Insertion of the Arrhenius equation into the rate expression leads to the an expression for the decrease in concentration d[A]/dt. Taking the logarithm of both sides of the equation removes the exponential, and additional differentiation leads to the second equation with the activation energy, pre-exponential factor, and order of the reaction expressed in terms of measurable quantities. This analysis is known under the name Freeman-Carroll method [33].

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experiments, the temperature is modulated with an amplitude, A, of ±0.5 K at a series of constant base temperatures. Data are taken after steady state is reached. At steady state, melting has been completed and practically no heats of transition are seen. Only the apparent heat capacity indicates the increasing fusion or crystallization as the temperature is changed. The higher heat capacity measured relative to the expected value calculated for 87.5% crystallinity marked in the figure may be largely due to beginning conformational motion within the crystals, as described in Sect. 2.3.7. Figure 3.89 shows a comparison of the quasi-isothermal TMDSC with a standard DSC trace of the same sample which records both, the reversible and irreversible effects. A large difference in scale is necessary to display the full melting peak.

Fig. 3.89

Figure 3.90 shows a plot of the modulated temperature versus the heat-flow rate for a low-molar-mass poly(oxytetramethylene), POTM650, molar mass 650. The comparison to an identical measurement in the melt which shows no latent heat effects allows an analysis of the range of crystallization and melting. The crystallization and melting ranges join at point A. To the left and right of temperature A, some reversible melting is indicated with symmetrical rates, as expected from Fig. 3.76. A supercooling, T, of about 2.4 K can be seen due to primary nucleation (see Sects. 3.5.3–4). The broad melting and crystallization range is probably due to slow diffusion of the crystallizable species to and from the proper crystal and the reorganization (annealing) which may occur immediately after initial crystal growth.

Data on the supercooling of aliphatic polyoxides in Fig. 3.91 can be compared to the paraffin and polyethylene data used for the illustration of the limits of molecular nucleation in Fig. 3.75. In the same range of chain length all three polymers, polyethylene, PE, poly(oxyethylene), POE, and POTM, decrease in amount of supercooling due to molecular nucleation. The polyoxides, however, show a second

266 3 Dynamics of Chemical and Phase Changes

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Fig. 3.90

Fig. 3.91

series of higher supercooling which is due to primary nucleation seen in experiments where all crystals melt before cooling as illustrated in Fig. 3.90. The polyethylenes and paraffins do not need primary nucleation in typical crystallization experiments by DSC, so that only the lower values of supercooling show.