Model-free isoconversional method applied to polymer crystallization governed by the Hoffman-Lauritzen kinetics

A new method is developed to obtain the kinetic parameters of polymer crystallization from measurements of the transformation rate usually done at (but not restricted to) constant heating or cooling rates. It can be considered a generalization of the Fiedman’s isoconversional method to deal with the Hoffman-Lauritzen temperature dependence. Apart from delivering the U* and Kg parameters as a function of the transformed fraction, this method allows to predict the crystallization course for an arbitrary thermal history. It has been applied to the crystallization of PET and PA6 samples, monitored by differential scanning calorimetry (DSC).

Second, an Arrhenian temperature dependence of the reaction rate is assumed: where and g() is characteristic of the "reaction model". In Eq.(3) E  is the activation energy, R, the gas constant and k 0 the pre-exponential constant.
For E and k 0 independent of , and constant heating rate Eq. (2) has been integrated for many particular reaction models leading to analytic functions describing the peak shape of the transformation rate, ) (T   [14,15]. However, many solid state reactions are no so simple and their description requires allowing for the dependence of E and k 0 on  [16]. The variation of E with  is usually associated with complex reactions where the ratecontrolling step switches from a low to a high activation energy step or vice versa (e.g. in crosslinking reactions [17] or in reactions that can be accomplished through two independent mechanisms acting in parallel [16]. Consequently, E  and g() (in fact, k 0α ·g()) must be considered as phenomenological functions of α that allow to predict the reaction course for arbitrary thermal histories [18]. Under these circumstances E  is referred as effective or apparent activation energy.
In general, description of the reaction rate by an Arrhenian dependence means that the process is limited by an energy barrier and, consequently, it goes faster at high temperature.
However, when the equilibrium temperature, T 0 , between the initial and final states is approached, an opposite temperature dependence appears because the reaction stops at T 0 .
Sometimes, like in solute precipitation from solid [19] or liquid [20] solutions or in solidification of pharmaceutical mixtures [21], an additional exponential term describes this dependence, e.g.: where f(T) is a function that varies slowly around T 0 . In Eq.(5), we have also considered the possibility of null molecular transport below T ∞ .
Polymer crystallization follows this kind of temperature dependence. The growth rate of spherulites, G, is usually described by the Hoffman-Lauritzen (H-L) equation [22]: glass temperature). In Eq.(6), U* is the activation energy for chain diffusion and Kg is called the "nucleation constant". Although the H-L theory describes the growth rate, many authors assume the same dependency for the crystallization rate, k cryst . However, in general, they will be different because k cryst also depends on  N . For 3D spherulitic growth [23]: and, thanks to the larger contribution of G on k cryst , the Kg and U* values obtained from k cryst will be usually close to those of G.
Despite the great difference between the Arrhenian and the H-L dependencies, some authors have taken advantage of the flexibility of the isoconversional methods to deduce U* and Kg from a fictitious dependence of the activation energy (Eq.(3)) on temperature [17,24].
Since its proposal in 2004 by Vyazovkin and Sbirrazzuoli [24], this particular method has been applied by many authors to study the crystallization of a broad range of polymer materials such as pure polymers (e.g. refs. [25,26]), polymer blends (e.g. refs. [27,28]) and nanocomposites (e.g. refs. [29,30]) (a more complete review can be found in ref. [31]). This method assumes that the kinetic constants (U* and Kg) are independent of α. Consequently, the great advantage of the isoconversional methods to give full description of non-ideal processes (i.e. those whose parameters depend on α) is lost and prediction of the crystallization course is not possible.
Finally, many authors (see for instance refs. [1,6,27,32,33]) fit to Eq.(6) the temperature dependence of the time elapsed to reach α = 0.5 after the incubation period under isothermal conditions. This procedure is much simpler than that of Vyazovkin but it also assumes that Kg and U* are independent of α.
The present paper describes the crystallization course of polymers assuming the dependence on α of Eq.(1) but without the need to suppose that the reaction rate constant has an Arrhenian dependence on temperature (Eq. (3)). After a brief description of the experiments (Section II), the proposed method of analysis will be explained (Section III) and tested against crystallization experiments on two polymers (PET and PA6) (section IV and VI). The paper will continue with a brief discussion and a concluding section.

II. Experimental
Pure PET and PA6 samples were used for this study.
where D is thermal diffusivity (1.8·10 -7 and 1.0·10 -7 m 2 /s for PET and PA6, respectively) and h is the sample thickness (0.4 and 1 mm, for the PET and PA6 samples), and from the heat of crystallization [35]: where A is the sample contact area with the pan (0.2 and 0.1 cm 2 for PET and PA6) [37], , thermal conductivity (0.29 W/(m·K) for PET and, also, for PA6) and Q MAX the heat evolved per unit time at the DSC peak maximum. Due to its higher thickness, h, and smaller section, A, these corrections were only significant for the experiments on PA6. At the highest cooling

III. Description of the isoconversional method
Notice that the H-L equation has two exponential factors whose dependencies on temperature are oposed. Whereas the exponential factor with U* continuously grows from zero at Obviously, the simplest experiments are those carried isothermally or at constant heating or cooling rates.
The crystallization kinetics will be fully described when the free parameters, U*, Kg, and k 0 together with the reaction model, g(), are determined. Let all of them vary with ; i.e. U  *, Kg  and k 0α ·g(). From Eq. (1) and assuming that the rate constant is given by Eq. (6), the crystallization rate can be written as: Let us consider that a given value of  is attained through several experiments performed at different constant heating or cooling rates,  i . Given that the sample will follow different thermal histories, i   will be different for each one. In any case, it will obey Eq.(10), now transformed into: Eq.(11) constitutes a system of non-linear equations on U  * an Kg  and k 0α g(). Thanks to the particular dependence on U  * and Kg  , the unknowns can be obtained after the iterative sequence of linear fittings described below.

Consider that you have done several experiments of melt and glass crystallization.
Since, on cooling, crystallization occurs nearer to 0 m T than to T  , the corresponding i   values are mainly governed by Kg  whereas they are weakly dependent on U  *. The contrary holds for crystallization during heating, because it occurs closer to T  than to 0 m T (Fig.1). Now, take a realistic guess value for U  *.This allows for Eq.(11) to be linearized: and to obtain k 0α ·g() and Kg  by linear fitting of (the "U*-plot"). Of course, since the guess value of U  * is not the solution, the experimental points will not be very well aligned and a second linear fitting (Kg-plot) will be done to obtain U  * from the equation: where Kg  is the value obtained in the former fitting. For the particular case of PET crystallization and α = 0.5 (Fig.2), convergence has been reached after 42 iterations. In this case, the final R-square correlation coefficients are very close to one (0.9979 and 0.9972) and the relative standard error of U  * and Kg  deduced from the last fitting is ±2%. However, the real uncertainty is higher. The ) ln( i   vs T i points have also been fitted by a non-linear procedure. The Kg and U* values coincide with those of the iterative method but with larger error bars of ±7%. This apparent discrepancy together with a hint of how to simplify the iterative procedure is explained in Appendix A.

IV. Crystallization of PET
Melt and glass crystallization experiments were done on thin (0.7 mm approx.) PET specimens of mass around 4 mg. The crystallization heat of the melt diminished steadily from the lowest (48.4 J/g at -1.25 K/min) to the fastest (46.0 J/g at -20 K/min) cooling experiments. During the heating experiments (2.5 < β < 20 K/min), Q cryst was smaller (42±1 J/g). These values should be compared to the heat of melting of 100% crystalline PET (ΔH f = 140 J/g [38]). The difference means that the final state reached with our experiments has a crystallinity below 50%. All the DSC peaks have a queue at long times that could correspond to a secondary crystallization process, similar to that reported by other authors [1,7]. Notice in Fig.3 (and in the upper part of Fig.1), that this process is more important during glass crystallization.
Application of the method described in the previous section, with T  = 312 K [17] and 0 m T = 553 K [1], has delivered the dependence on α of Kg, U* and k 0 ·g(α) shown by the solid curves of Fig.4. Whereas Kg is nearly constant from α = 0.1 to 0.8, U* increases steadily. The U* value at α = 0.9 almost triplicates the value at 0.1 (6000 J/mol). Since, in principle, one expects G to be independent of α (laser scattering [2] and transmission electron microscopy [39] have shown linear growth until impingement) this variation is surprising. A simple explanation arises if we consider that the long queue of the DSC peaks correspond to an independent crystallization process. In fact, it has been shown that, when crystallization occurs through two parallel processes, the activation energy determined by isoconversional methods exhibits a fictitious increment with α [16].
To analyze the main crystallization process (the one occurring first) we take into account that, at the peak temperature, α ≡ α MAX = 0.5 for melt crystallization. In contrast, for glass crystallization, α MAX is much smaller and depends on the particular experiment (α MAX = 0.33 -0.39). We thus make the hypothesis that, at the peak maximum, the degree of completion of the first crystallization process, α 1 , is always the same and equal to 0.5. So, we define: and we renormalize the crystallization rate, Now U* is nearly constant (7.3-8.2 kJ/mol) as expected, and Kg increases only slightly with α (4.7·10 5 to 5.9·10 5 K 2 ). An additional proof for the correctness of these values compared to those obtained above comes from error analysis; the error bars of U* shown in Fig.4c correspond to the solid line. This means that, before renormalization of α into α 1 , the points of the Kg-plot were not very well aligned (notably for α 1 > 0.5), indicating that the overall crystallization process does not follow the H-L kinetics.
Our Kg and U* values fall within the wide range reported in the literature (Table I).
Notice that, in Table I [2] where the density of crystallites grows steadily by 5 orders of magnitude from 220 to 130ºC.
We can determine the crystallization regime from the Kg value after applying the Lauritzen Z-test [41]. In regime I nuclei grow so fast that formation of any nucleus is followed by rapid completion of the substrate whereas, in regime II, nuclei grow slowly and many nuclei form before substrate completion. The lamellar width, L, is related to the socalled Z-parameter according to: where a 0 is the width of the molecular chain in the crystal (around 0.5 nm), and Z ≤ 0.01 or ≤ 1.0, and n = 4 or n = 2 for regimes I or II, respectively [41]. A realistic value for the lamella width (L < 34 nm) is only obtained for regime I that applies to the whole temperature range of our experiments (notice that all points are aligned in the U-and Kg plot of Fig.2). Once the crystallization regime is known, the product of the lateral energy times the fold surface energy, σσ e , can be calculated through [22]: where k B is the Bolzmann constant, Δh f , the melting enthalpy of the perfect crystal per unit volume (Δh f = ΔH f ·ρ c = 210 J/cm 3 for PET whose crystal density is ρ c = 1.5 g/cm 3 ) and b, the monolayer thickness (0.553 nm [24]). We obtain, σσ e = 12 10 -4 J 2 m -4 , that is similar to the values published by most authors [24].
Before leaving this section, we will check the goodness of our analysis by using the Kg α , U α * and k 0α ·g(α) values to calculate the overall crystallization rate (dα/dt) and compare it to the experimental DSC peaks. Eq.(2) has been numerically integrated following the method of Farjas developed in refs. [18,42], with the especial condition that crystallization begins after an incubation period. For crystallization experiments performed during cooling or heating ramps, this condition corresponds to dα/dt = 0 until T = T ONSET [2], where T ONSET is the experimental onset temperature of the DSC peak. In Fig.3, the calculated and experimental peaks are compared. There is good agreement despite that, as discussed above, the overall crystallization of our samples is a complex transformation that involves two individual processes. In fact, the discrepancies seen in Fig.3 mainly arise from the experiment (the DSC peaks do not have a steady evolution when the heating or cooling rate is changed).
Although the long queue of the DSC peaks makes that the isoconversional principle is not fully obeyed in this experiments of PET crystallization, this example constitutes an additional proof [10,11,18,42,43] of the flexibility of model-free isoconversional methods to predict the course of complex transformations.

V. Crystallization of PA6
The experiments on PA6 were done on single pellets (m  10 mg). Since quenching in liquid nitrogen didn't gave a 100% amorphous state, the reported results are those of melt crystallization (3.5 < β < 40 K/min ). As the cooling rate increased, Q cryst steadily decreased from 53 to 38 J/g indicating a higher difficulty to crystallize. Again, like with PET, Q cryst is much smaller than the enthalpy of full crystallization (ΔH f = 190 J/g [44]).
Once the DSC peaks have been shifted to correct for the sample thermal lag (Section II) and, in view of their long queue, we have subtracted this secondary crystallization process as shown in Fig.5. Then, the equilibrium melting temperature, 0 m T , has been determined by the Hoffmann-Weeks method [45]. We have obtained 0 m T = 222ºC. This value and T ∞ = 20ºC have been used as input parameters to apply the iterative method described in Section III to obtain the crystallization kinetic parameters of PA6. The result is shown in Fig.6. Kg is almost constant (1.02±0.05 10 5 K 2 for 0.05 < α < 0.95, whereas the U* dependence on α has a peak below α = 0.3. Above α = 0.3, U* = 3.1±0.2 kJ/mol. Probably, the peak of U* is related to the initial faster crystallization (high temperature side of DSC peaks) seen on the DSC peaks (Figs.5 and 7).
Our Kg value falls on the lower range of the reported values obtained from DSC experiments [46,47,32,33,26] or from direct measurement of G [48], and our U* value is closer to that extracted from G (4.0 kJ/mol) than that from non-isothermal DSC (3.3 kJ/mol [26]) (see Table II). However, without crystallization experiments at temperatures closer to T ∞ than to 0 m T , one cannot be very confident on the fitted value of U*. As with PET, most Kg values of PA6 obtained from DSC are higher than that obtained from G. For PA6, the Lauritzen Z-test tells us that crystallization occurs in regime II, and application of Eq.(16) (n = 2, b = 1.9 nm, ρ c = 1.24 g/cm 3 , Δh f = 236 J/cm 3 [46]) delivers σσ e = 17 10 -5 J 2 /m 4 .
The kinetic parameters of Fig.6 together with the T ONSET values (inset of Fig.5) allow us to calculate the melt crystallization rate during arbitrary thermal histories. Agreement with experiment is excellent when the experiments carried out at a constant cooling rate are simulated (Fig.7). Two further tests on the predictive capability have been done by cooling the melt at -20 K/min down to an isothermal stage (at 190ºC and 180ºC). The predicted transformation rates are compared with the DSC experiments in Fig.8. They show good agreement. Since at -20 K/min T ONSET = 83.7ºC, crystallization begins during the cooling period before the isotherm (inset of Fig.8). This short period has an influence on the crystallization course as illustrated by the comparison between the curves predicted with or without cooling ramp. As expected, the best agreement with experiments is achieved in the first case (Fig.8). Concerning the isotherm at 190ºC, crystallization begins after an initial incubation time that, from the T ONSET values (inset of Fig.5) has been estimated to be 30 s, following the method of ref. [49]. In view of the long crystallization time at this temperature ( Fig. 8), the incubation period is too short to be relevant.

VI. Conclusions and perspectives
In this paper, we have developed a new isoconversional method to obtain the kinetic parameters of Hoffman-Lauritzen model (U* and Kg) governing polymer crystallization. It involves an iterative procedure to optimize the linear fits of modified Friedman's plots [50] and can be applied to experiments carried out at any thermal conditions (constant or varying heating rates and isothermal). Compared with the current methods to obtain U* and Kg (fit to the isothermal crystallization half-time, and Vyazovkin's method for non-isothermal experiments [17]), these parameters are allowed to vary with the crystalline fraction, α. In case they are constant, the fitting procedure needs to be applied to a single value of α only and, as shown in Appendix A, becomes as simple as with the current methods.
It has been applied to analyse the DSC crystallization curves of PET and PA6 samples. Since PET crystallization has been done from the melt and from the glass states, accurate values of U* and Kg have been obtained as a function of α. In a preliminary analysis, U* remained constant around 7.5 kJ/mol up to a transformed fraction of 0.5 and, then, it increased steadily. It has been shown that this anomalous variation was fictitious because it disappeared when the secondary crystallization occurring at long times was subtracted from the DSC signal. Kg was much less sensitive to the secondary crystallization and remained almost constant around 5.3·10 5 K 2 .
In contrast with PET, PA6 crystallization could only be done from the melt and, consequently, the fitted value of U* was less reliable than that of Kg. Kg remained almost constant around 1.0·10 5 K 2 whereas U* showed a peak below α = 0.3 that has been attributed to a faster initial crystallization rate.
In addition to extracting the kinetic parameters, our method allows to predict the crystallization course for an arbitrary thermal history. This possibility has been tested by comparing the DSC curves measured at several heating and cooling rates with the predicted curves. Moreover, isothermal crystallization of PA6 after a cooling ramp has been successfully simulated.
To the best of our knowledge, this is the first time that an isoconversional method is able to obtain the dependence of the kinetic parameters on the transformed fraction and to make predictions of the crystallization course for a process that follows a temperature dependence as complex as that of Hoffman-Lauritzen (the easier case of crystallization just below the melting point was already done in ref. [51]). It opens the door to the analysis of a broader range of polymers and processes with temperature dependencies similar to that of Hoffman-Lauritzen, like those already commented on in the Introduction. Furthermore, even those processes not having well-established temperature dependence such as sol-gel transformation of gelatin [52] or collagen denaturation [53], but occurring near equilibrium, are also candidates for this kind of analysis.
Catalunya (contract num.SGR948) and from the University of Girona (program MPCUdG2016).