Quiescent and shear-induced crystallization of polyprophylenes

Abstract

In this paper, the effect of shear on the flow-induced crystallization (FIC) of several polypropylenes of various macrostructures was studied using rheometry combined with polarized microscopy. Generally, an increase in strain and strain rate or decrease of temperature is found to decrease the thermodynamic barrier for crystal formation and thus enhancing crystallization kinetics at temperatures between the melting and crystallization points. Secondly, popular models based on suspension theory which are used to relate the degree of crystallinity to normalized rheological functions (such as viscosity) are validated experimentally. For this purpose, the space filling of crystals in the polarized micrographs determined from image processing was plotted as a function of normalized viscosity under various shear rates. It is found that the constant(s) of various suspension models should be dependent on the flow parameters in order for the suspension models to describe the effect of shear on FIC, particularly at higher shear rates.

Appendices

Appendix A

The coupled energy equations for the system shown in Fig. 18 where the polymer in the space between two quartz plates is crystallized are developed. This is necessary to calculate the increase of temperature in the optical microscopy results due to the heat of crystallization which is accumulated in the sample. This has an effect on the degree of crystallinity which needs to be quantified in order to explain the differences between the POM and DSC results with reference to Fig. 16a.

Fig. 18

The schematic of system used in this study

Crystal formation releases heat within the polymer at a rate of

$$ \dot{q} =\Delta H_{f\left({100\% \;\text{crystalline}\;\text{polymer}} \right)} \times \rho \times \dot{\emptyset} $$

(7)

where $\Delta H_{f\left ({100\% \;\text {crystalline}\;\text {polymer}} \right )}$ is the heat of melting if the polymer was 100 % crystalline, ρ is the density of crystals, and $\dot {\emptyset }$ is the rate of volume fraction increase of crystals. The heat transfer equations for the polymer and the quartz plate located on top of the polymer (Fig. 18) are as follows:

$$ \left\{ {{\begin{array}{*{20}c} {\rho_{\text{polymer}} C_{p,\text{polymer}} \frac{\partial T}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left({k_{\text{polymer}} r\frac{\partial T}{\partial r}} \right)+\frac{\partial }{\partial z}\left({k_{\text{polymer}} \frac{\partial T}{\partial z}} \right)+\dot{q} } \\ {\quad \;\;\rho_{\text{quartz}} C_{p,\text{quartz}} \frac{\partial T}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left({k_{\text{quartz}} r\frac{\partial T}{\partial r}} \right)+\frac{\partial }{\partial z}\left({k_{\text{quartz}} \frac{\partial T}{\partial z}} \right)} \\ \end{array} }} \right. $$

(8)

The equations are solved using POLYFLOW with the following boundary conditions. A convection boundary condition is imposed on the plate in contact with nitrogen gas which is purged into the oven chamber of the rheometer as the heating/cooling gas. The heat transfer coefficient used in these calculations was h = 100 W/(m ²⋅K). The quartz plate underneath the polymer imposes an adiabatic boundary condition at the lower side of the polymer. Continuity of T and heat rate at the polymer/quartz interface completes the required BC.s.

After solving the system of Eq. 8, the temperature elevation within the polymer sample can be obtained as a function of T. The temperature rise as a function of time at r = 0.75R and z = 0.25 mm is shown in Fig. 19. The temperature rise in the polymer matrix decreased as the temperature is increased. At higher temperatures, the kinetics of crystallization slows down, and thus less heat is released per unit of time. This essentially explains the larger differences between OM and DSC results depicted in Fig. 16a at the lower temperatures.

Fig. 19

The temperature rise at r/ R = 0.75 and z/ h = 0.25 in a polymer sample that crystallizes at different temperatures. The temperature increases of 0.2 to 1.9 °C at the crystallization temperatures of 131.4 to 121.7 °C, respectively, explain adequately the differences in Fig. 16a

Appendix B

The Avrami equation is often used to predict the isothermal crystallization behavior under quiescent condition. This equation can be written as follows:

$$ X \mathord{\left/ {\vphantom {X {X_{f} }}} \right. } {X_{f} }=1-e^{-kt^{n}} $$

(9)

where X is the degree of crystallinity, X _f is the total crystallinity at the end of primary crystallization process, n is the Avrami index with a value of 2 for the resin studied, and k is the Avrami rate parameter. The Avrami parameter k contains the temperature dependence of the nucleation and crystal growth processes, while the exponent n depends on the geometry and dimensionality of the growth as well as on the nature of the nucleation process. The Avrami equation best predicts the behavior in primary growth region of crystals which corresponds to a relative crystallinity in between 30 and 70 %. The Avrami index determined experimentally varied in the range of 1.9–2.5, and therefore, the theoretical value of 2 (1.9–2.7) was chosen for the investigated temperatures (120–135 °C). The Arrhenius Eq. 10 is used in this study to represent the dependency of the Avrami rate parameter on temperature. The Arrhenius equation is then used in combination with the Avrami equation to predict the behavior of the sample in the microscopy setup.

$$k\left(T\right)/k\left({T_{o}}\right)=\exp \left[{E_{a}/R}\left({1/T}-1/{T_{o}}\right)\right] $$

(10)

The value of the Avrami rate parameter obtained using the temperature history in Fig. 19 along with the Arrhenius equation is used in the Avrami equation (Fig. 20). The prediction of the Avrami model is shown in Fig. 16b and compared with the experimental data from the microscopy setup. The agreement was found to be excellent which shows that the difference in Fig. 16a is related to different thermal histories during crystallization.

Fig. 20

Fit of the Arrhenius equation to the Avrami rate parameter k using n = 2