Isothermal half-time

Fig. 9 Inverse of the crystallization half-time as a function of isothermal crystallization temperature for PCL11 homopolymer and for the PCL block of the indicated copolymers. All experiments were performed after the PPDX block had been previously crystallized until saturation. Solid lines are fits to the Lauritzen and Hoffman theory. (From [103]. Reproduced with permission of the Royal Society of Chemistry)...

Table 5.2 Half -times and induction times (in min) for different PET samples when crystallized isothermally from the melt [26]. From Wick, G., Characterization of PET polymer for bottle manufacturing, presentation given at the Society of Plastics Engineers Benelux Seminar, 20-21 May, 1980, Amsterdam, and reproduced with permission of KoSa GmbH Co. KG...

Applying the Avrami model to the analysis of the isothermal crystallization of interesterified and noninteresterified 20%SSS/80%000 at 30°C, 40 C and 50 C, many differences can be observed (Table 17.3). At 30°C and 40°C growth would be described as rodlike with instantaneous nucleation for both interesterified and noninteresterified samples. Also, for the noninteresterified system at 50°C spherulitic growth with instantaneous nucleation takes place. The half-time of nucleation... 

Duff et al. [27] reported a study made by means of DSC and WAXD on SPS/ PPE blends of various compositions, precipitated from ethylbenzene solutions, compression molded at 330 °C for 2 min and then slowly cooled to room temperature. In particular, the WAXD patterns show that in sPS-rich blends (>50 50 wt%) sPS is in a 0 or (3 form, while small amounts of a are present in the 50 50 wt% blend. The kinetics of crystallization and the mechanism of nucleation of sPS were investigated under isothermal and nonisothermal conditions as a function of blend composition and molecular weights of the components. The experimental curves show that the half-time to crystallization, t j2, increases with increasing content and molecular weight of PPE, but is not influenced by the molecular weight of sPS. The crystallization kinetics were... 

A sorption isotherm on excised human abdominal stratum corneum (female, age 68) is presented in Figure 9. The data were obtained on separate pieces of skin which were initially dried and then exposed to an air stream with a given RH uptake was followed until equilibrium was reached. Values of D calculated from half-time data for each humidity interval were 1.37-5.10 X 10 cm /sec. Scheuplein and Blank (16) reported a value of 5 X 10 cm /sec for human stratum corneum. The range of our values would indicate that D is concentration dependent. Full details will be reported in the future. 

Fig. 12. Half times t of the isothermal crystallization of PET PHB copolyesters as a function of crystallization temperature Tc. Parameter is the weight fraction of PET...

Isothermal crystallisation of IR rubber systems The crystallisation half-time value [t(0.5)-value] was first measured at temperatures between -3°C and -29°C to determine the optimum crystallisation temperature. These measurements were performed on three samples a natural rubber and the Ziegler polyisoprene systems Natsyn 2200 and Natsyn 400 (containing 2 %wt. stearic acid as crystallisation promotor). 

In Figure 3.12 typical crystallization isotherms were obtained by plotting a versus the crystallization time for the PEG/PEMA 80/20 blend at different crystallization temperatures. Erom such curves, the half time of crystallization, can be deduced. 

Shingankuli [1990] studied the crystallization behavior of PP in the presence of solidified PVDF domains. A higher crystallization temperature of the PP matrix phase was observed, indicating an enhanced nucleation in the blends. The degree of crystallinity of PP was found to increase by about 30 to 40% with increasing PVDF content. Isothermal crystallization studies also confirmed the acceleration of the overall crystallization rate in terms of shorter crystallization half-times for PP. 

Nadkami and Jog [1986] have reported on PPS/HDPE blends. The degree of crystallinity of HDPE was reduced when HDPE was the minor phase. Lurthermore, the T shifted to somewhat lower temperatures (by about 5°C) but only in those blends with a low HDPE content. Isothermal crystallization half-times for HDPE in its blends with PPS decreased as the HDPE content decreased, indicating an enhanced nucleation from the solidified PPS interfaces. 

The isotherms of crystallization were obtained by plotting the crystalline weight fraction at time t Xt) versus t. The half-time of crystallization, to.5 (defined as the time taken for half of the crystallinity to develop), is obtained from the curves and plotted against blend composition, for some T, in Fig. 6.2. 

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]. 

Figure 8.6 Half-time of crystallization (tl/2) of PLLA as a function of isothermal crystallization temperature reproduced with permission from [45]. Copyright Wiley-VCH Verlag GmbH Co. KGaA.

A quantitative measure of isothermal crystallization is the crystallization half-time, Crystallization times expressed as the reciprocal of (crystallization rate) are shown in Figure 9.4 as a function of crystallization temperature. These results indicate that the fastest crystallization for the PDS homopolymer can be obtained at approximately 47°C. For the sake of comparison, Figure 9.4 also includes spherulitic growth rates from HSOM. 

Logarithm of crystallisation half time vs. logarithm of molecular weight, for polyethylene crystallised isothermally at the temperatures indicated (from Mandelkern, L. J. Mater. ScL, 6,615, 1968). 

Overall rate constants j for isothermal crystallization is given by Equation (3). In analogous way, one may define a rate constant for non-isothermal crystallization on base of Equation (34). This is done by replacing half time under T = const by half time of crystallization under condition s = const, (ATIs It follows ... 

K " and n can be extracted from the intercept and the slope of Avrami plot, lg[-ln(l-.A0] versus lg(f-f ), respectively. The prime requirement of Avrami model is the ability of spherulites of a polymer to grow in a free space. Besides, Avrami equation is usually only valid at low degree of conversion, where impingement of polymer spherulites is yet to take place. The rate of crystallization of polymer can also be characterized by reciprocal half-time (/ 5). The use of Avrami model permits the understanding on the kinetics of isothermal crystallization as well as non-isothermal crystallizatioa However, in this chapter the discussion of the kinetics of crystallization is limited to isothermal conditions. 

FIGURE 6 (a) Half-time of crystallization from the crystallization peak for PHB12HHx crystallized isothermally at 110°C. (b) Plots of relative crystallinity versus crystallization time during isothermal crystallization of PHBSHHx ciystaUized at 112°C. 

The overall rate of isothermal crystallization of PTT (semicrystalline polymer) can be monitored by thermal analysis through the evolution of heat of crystallization by DSC as depicted in Figure 10. The sample is isothermally crystallized at preselected crystallization temperature (T) until complete crystallization. Half time of crystallization for the polymer is estimated from the area of the exotherm at r = const, where it is the time taken for 50% of the crystallinity of the crystallizable component to develop. The rate of crystallization of PTT can be easily characterized by the experimentally determined reciprocal half time, (tg j) . ... 

From the solutions for the fractional uptake in Table 9.2-4, the half time can be obtained by setting the fractional uptake F to one half, and they are tabulated in the third column of Table 9.2-4. For this case of rectangular isotherm, the half time is proportional to the square of particle radius, the maximum adsorptive capacity, and inversely proportional to the bulk concentration. The time it takes to reach equilibrium is half when the bulk concentration is doubled. This is because when the bulk concentration is doubled the driving force for mass transfer is doubled while the adsorptive capacity is remained constant (that is saturation concentration) hence the time to reach saturation will be half. Recall that when the isotherm is linear, the time scale for adsorption is independent of bulk concentration. Hence, for moderately nonlinear isotherm, the time scale would take the following form ... 

We demonstrate this with a spherical particle with no film resistance (that is Bi oo), and then present the results for slab and cylindrical particles. If the pore diffusion is the only diffusion mechanism and the adsorption isotherm is linear, the half time of adsorption is (from eq. 9.2-36b)... 

Solving eq. (9.2-45) numerically for the case of Langmuir isotherm using the programming code ADSORB 1 A, we obtain the following approximate correlation for the half time in terms of the parameter X = bCo, which is a measure of the... 

Thus, by combining the behaviour of the half time at various limits (eqs. 9.2-57 to 9.2-62), the following general equation for the half time for parallel pore and surface diffusion and any nonlinearity of the isotherm (Do, 1990) is obtained... 

The explicit form of eq. (9.2-63) is very useful for the determination of surface diffusivity. To do so we must rely on experiments conducted over the nonlinear region of the isotherm. Otherwise, the contribution of the surface diffusion can not be distinguished over the linear region. Half time is measured for each bulk concentration used. By rearranging the analytical half time equation (9.2-63a), we have... 

Half time is normally used to describe the time scale of the adsorption kinetics, and it is defined as the time to reach 50% of the equilibrium uptake. Thus, by setting F = 0.5 into eqs.(10.2-17), we get the necessary expressions for the half time written in terms of the size of the particle and the diffusivity. The following table shows the results of the half times for three shapes of the microparticle for the linear adsorption isotherm case. 

Table 10.2-5 Half times of microparticles with linear isotherm...

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