Lamellar structure obtained by bringing the surface of a molten plaque into contact with a metal plate held at room temperature (microstructure just behind the surface) 

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... ®t of Eq. (11) to the data in Fig. 4 gave s o 1.5 ́ 10 ) 4 s and K N 110,000 K 2 (cf. Eq. 10). The form of Eq. (11) was chosen somewhat arbitrarily, but it was found to account well for the observed temperature dependence of s . (cf. the classical Turnbull±Fisher expression for primary nucleation, for example, in which the argument of the second exponential is raised to the power 2 [20]). The data obtained from nonisothermal crystallization at 40 Kmin ) 1 are compared in Fig. 9 with curves calculated using parameters derived from the isothermal data (Fig. 6, Eqs. 9, 11) and a simulation made after adjusting N o and T sd to optimize the ®t. Also shown is the optimized ®t plotted as a function of the sample temperature, T s , and the corresponding degree of conversion as a function of T s . This demonstrates that crystallization itself took place over a relatively narrow range of sample temperatures between 140 and 146 ° C, in spite of the fact that the oven temperature, T f , corresponding to the end of the crystallization exotherm was about 132 ° C. For nominal cooling rates above 40 Kmin ) 1 adequate control of the temperature ramp became dicult [9] and 146 ° C was therefore considered to be the lower limit of the temperature range which could be investigated in detail. DSC results for various cooling rates are shown in Fig. 10 along with the calculated curves. The adjusted values of N o ( T sd ) used to obtain these curves are compared with the results from isothermal experiments in Fig. 11 and the adjusted values of T sd are compared with the predictions of Eqs. (9) and (11) in Fig. 12. The dierences between the predicted and adjusted values of T sd were comparatively small. The dierences between the predicted N o ( T sd ) and adjusted values of N 0 were more signi®cant, as borne out by Fig. 9. However the predicted values relied on a relatively long extrapolation of data points con®ned to a narrow range of temperatures in which N o was changing relatively slowly with T . An exponential temperature dependence has been observed in other polymers [10, 11], but given the assumptions implicit in the use of Eq. (10) for G , corroborating evidence for the evolution in N o from optical microscopy was considered to be important. The results of direct estimates of the average spherulite radii are shown in Fig. 13, where they are compared with the radius of a sphere with volume N o À 1 . The agreement is satisfactory, although in the case of the sample cooled at 40 Kmin ) 1 it was dicult to characterize the morphol- ogy precisely, owing to overlap in the projections of individual spherulites and an ill-de®ned spherulitic texture. Time±temperature transformation diagrams are a con- venient way of representing either data or models for solidi®cation. The isothermal time±temperature trans- formation diagram for PA-12 inferred from the present model and extrapolated to higher undercooling using N o 9 ́ 10 41 exp( ) 0.153 T ) ( T in Kelvin) to interpolate and extrapolate N o ( T ) (the hatched curve in Fig. 11) is shown in Fig. 14. It should be emphasized that the induction time s is not expected to correspond to a true onset since it is derived from a ®t of a two-parameter Avrami model to data from a system in which the eective induction times may be distributed (and in which thermal nucleation may also be present). On the other hand, the curves for 50 and 99% conversion are realistic for the temperature range in which data were available. Figure 15a shows the non-isothermal time±temperature conversion diagram for a starting temperature of 220 ° C, chosen to be consistent with the DSC data (a dierent choice of starting temperature would simply result in a shift in the zero of the time axis). The curves were calculated from Eq. (6) and therefore do not include eects of latent heat evolution or thermal lag and will therefore, in general, not re ̄ect the apparent behaviour in practical situations. Indeed, as indicated in Fig. 15a, the predicted time for 99% conversion diverged from the experimental DSC data quite mark- edly at the highest cooling rates, even when these latter were corrected for thermal lag. Nevertheless certain situations, such as quenching against a cold metallic mould, will be approximately consistent with the ideal case. Also shown for completeness in Fig. 15b is a time±temperature degree of crystallinity diagram, tak- ing the enthalpy of crystallization of 100% crystalline sample to be 130 Jmol ) 1 [19] and using the data shown in Fig. 2. The question naturally arises as to the extent to which the extrapolation of the present model to lower crystallization temperatures and/or faster solidi®cation rates in Figs.14 and 15 is justi®ed. The extrapolation of G is arguably physically based [13, 21], and the D T dependence is widely observed, even if reservations have been expressed regarding the form of the transport term. N 0 , on the other hand, was interpolated/extrapolated as- suming a simple exponential dependence on T inferred empirically from the results. There is therefore no a priori basis for assumptions about N o in temperature regimes not accessible here. Also, even if it were possible to justify extrapolation of N o measured at high and intermediate crystallization temperatures, in practice these latter are often strongly dependent on the thermomechanical history (other parameters may also vary, particularly in the presence of a ̄ow ®eld). Therefore, although the application of the present approach to well-controlled DSC experiments has been demonstrat- ed, characterization of the microstructure after crystallization would still be necessary to verify any initial hypothesis regarding the nucleation density in other speci®c cases. In connection with this, the lamellar structures of various samples were investigated brie ̄y. The microstructures of samples crystallized isothermally at dierent temperatures and that of a sample taken from close to the surface of a molten plaque brought into contact with a metallic surface held at room temperature are shown in Figs. 16 and 17. In this latter case, the spherulite envelopes were ill de®ned, although lower-magni®cation images of heavily irradiated samples suggested a mean radius of about 0.5 l m. For comparison, extrapolation of N o 9 ́ 10 exp( ) 0.153 T ) predicted a radius of 5.4 l m at 130 ° C and 3.3 l m at 120 ° C. Estimates of the lamellar thicknesses from the samples shown in Fig. 16 gave values of 7.2, 6.7 and 6.3 nm for crystallization temperatures of 172, 166 and 158 ° C, respectively. If the lamellar thickness is assumed to scale as 1/ D T , the thicknesses expected at 130 and 120 ° C would be 5.7 and 5.65 nm, which is consistent with Fig. 17. Faster cooling by quenching a thin ®lm in ice±water from the melt, on the other hand, led to no discernible features in the TEM or in the optical microscope. There remains the question of the assumption of an induction time given by Eq. (11), which was necessary to account for both the isothermal and the nonisothermal data. However, if the dependence of N o on T is assumed to be a consequence of athermal nucleation, as suggested elsewhere, it is dicult to account for the existence of an induction time at high undercooling [10, 11] and it is at present not possible to justify the form of Eq. (11). It is therefore of interest to consider the consequences of zero induction time. The corresponding crystallization onset as de®ned by 1% conversion is given by the hatched line in Fig. 15a. In this case, the apparent cooling rate necessary for quenching to a predominantly amorphous state is shifted from about 800 Kmin ) 1 to about 1000 Kmin ) 1 . The crystallization kinetics of PA-12 has been success- fully modelled using a simple two-parameter Avrami model for isothermal crystallization, which has also been shown to be consistent with optical measurements of the spherulite growth rates and nucleation density. With small adjustments, parameters inferred from the isothermal DSC data also accounted for DSC data for nonisothermal crystallization, when the dynamic heat balance between the sample and the oven was taken into account. Further work is envisaged with the aim of investigating the balance between heterogeneous and homogeneous nucleation and the signi®cance of the observed incubation times, and the role of the prior thermomechanical history on N ...