# Nonlinear determination of the equilibrium melting temperature from initial nonreorganized crystals of poly(3-hydroxybutyrate).

INTRODUCTIONThe microbially synthesized poly(3-hydroxybutyrate) (PHB) is a biodegradable and biocompatible polyester with a relatively high crystallinity, which makes it a quite brittle material (1). The intrinsic fragility has limited up to now the practical applications of plain PHB, even because it was found that both copolymerization with different hydroxyalkanoate units and the mixing with different polymers improve markedly the mechanical properties and processability of this polyester (2), (3). Notwithstanding, the knowledge of the thermodynamic and kinetic properties of pure PHB is crucial for the comprehension of PHB-based blends and copolymers properties.

For crystallizable polymers, like PHB, one of the most important thermodynamic parameter is the equilibrium melting temperature ([T.sub.m.sup.0]), that is the melting temperature of a crystal with infinite size and without defects (4), which can be used to determine, for example, the free energies of fold and lateral surfaces and the work required to fold a macromolecule (5). Since a direct measurement of [T.sub.m.sup.o] is impossible, due to the finite dimensions of the real crystals, extrapolation methods have been commonly used to determine the equilibrium melting point of PHB (6-19). One method is based on the Thomson-Gibbs (TG) equation, which, under the assumption that the lateral dimensions of the lamellae are much greater than their thickness, is given by (4):

[T.sub.m] = [T.sub.m.sup.0](1 - 2[[sigma].sub.e]/[DELTA][h.sub.m.sup.0][l.sub.c] (1)

where [T.sub.m], is the measured melting temperature, [[sigma].sub.c] the fold surface free energy, [l.sub.c] the lamellar thickness, and [DELTA][h.sub.m.sup.0] the heat of fusion of the fully crystalline state. According to Eq. 1, from the plot of the measured melting temperature against the inverse lamellar thickness, [T.sub.m.sup.o] can be calculated as the intercept value. The equilibrium melting temperature of PHB, obtained with this procedure, with [l.sub.c] values from small-angle X-ray scattering measurements, is around 473 K for samples with number-average molar mass included between 130,000 and 400,000 g [mol.sup.-1] (6-8).

Lower [T.sub.m.sup.o] values for PHB have been determined with the more frequently used Hoffman-Weeks (HW) procedure (20), with [T.sub.m.sup.o] ranging from 459 K to 471 K for samples with number-average molar mass included between 100,000 and 350,000 g [mol.sup.-1] (9-19). The Hoffman-Weeks method, which was derived by the TG equation in association with the Hoffman-Lauritzen (HL) theory (21) by introducing additional approximations regarding the lamellar thickness, determines the equilibrium melting temperature as the intersection between the linear extrapolated [T.sub.m] vs. [T.sub.c] experimental curve and the [T.sub.m] = [T.sub.c] line. Because of the sizeable approximations, this linear extrapolation is valid only at very small undercoolings (22).

The problem of the correct value of the lamellar thickness holds not only for the HW treatment, but also for the TG method, since the crystal lamellar thickness should be measured just before melting, and not at room temperature, as it is generally performed (22). If the crystals thicken at the solid state or melt and recrystallize during the heating from [T.sub.c], to [T.sub.m], it results that erroneous TG plots are generated as the [T.sub.m] values do not correspond to the measured lamellar thickness.

To overcome these problems, a new procedure for the determination of [T.sub.m.sup.0], of polymers that exhibit isothermal lamellar thickening and/or lamellar thickening at the solid state during heating has been proposed by Marand et al. (22), (23). Necessary, but not sufficient condition for the lamellar thickening is the existence of some segmental dynamics within the lamellar crystal, generally called [alpha]-relaxation process (24). For polymers that undergo [alpha]-relaxation, which include, among others, polyethylene, polypropylene, poly(oxymethylene), poly(oxyethylene) etc., the isothermal thickening has a logarithmic dependence on time (25), (26). Unfortunately, the method proposed by Marand cannot be directly applied to polymers that thicken not at the solid state, but after partial melting followed by recrystallization, like poly(ethylene terephthalate), poly(butylene terephthalate). isotactic polystyrene, etc. (27-30). In such a case, for a correct determination of the equilibrium melting point, it is essential to determine the conditions that prevent crystal reorganization through partial melting/recrystallization, in order to establish the real melting point of the initial primary crystals.

As detailed in a previous article of this series (31), for poly(3-hydroxybutyrate) crystals, which partially melt and recrystallize upon heating and for which the occurring of lamellar thickening in the solid state has been excluded (32), (33), a heating rate of 60 K [min.sup.-1] is able to prevent recrystallization of primary lamellae when the polymer is crystallized at a adequately high temperature and therefore when the crystals are sufficiently stable. On this basis, a procedure for the determination of the equilibrium melting temperature of 1311B is presented and discussed here. The procedure is completely based on data obtained by calorimetry and does not require measurements with nonroutine experimental techniques, as for example electron microscopy, atomic force microscopy, small angle X-ray scattering. In addition it may be extended to other polymers whose crystals improve their thermal stability during heating via partial melting/recrystallization process.

EXPERIMENTAL PART

Bacterial poly(3-hydroxybutyrate) (PHB), with number-average molar mass [M.sub.n] = 260,000 g [mol.sup.-1] and a narrow molar mass dispersity ([M.sub.w]/[M.sub.n] = 1.6), was purchased from Sigma-Aldrich Corp. The structure of natural PHB is perfectly isotactic, independently of the source of bacteria (34). The melting peak temperature of the as-received polymer, measured by DSC (first scan) at 10 K [min.sup.-1], was 446 K, in excellent agreement with values reported in the literature for pure PHB samples with very similar molar mass (35), (36). The glass transition, measured during a second scan at 10 K [min.sup.-1] after melting, was 278 K. The polymer had the consistency of fine powder and was used as received without further purification or thermal treatment.

DSC measurements were performed with a Perkin-Elmer Differential Scanning Calorimeter DSC7 equipped with an Intracooler II as refrigerating system. Dry nitrogen was used as purge gas at a rate of 30 mL [min.sup.-1]. The instrument was calibrated at 5 K [min.sup.-1] in temperature and energy with high purity standards (indium, naphthalene and cyclohexane) according to the procedure for standard DSC. For heating rates ranging from 1 to 90 K [min.sup.-1], the onset of indium melting was found to occur within a temperature interval of 0.15 K, which proves that, being superheating negligible for indium melting (37), the thermal inertia of the instrument (i.e. the delay in the heat transport from the heater to the sample) is insignificant (38).

In order to minimize and maintain constant the temperature gradients within the sample under analysis, the sample mass was kept as small as possible and approximately constant (3.5 [+ or -] 0.2 mg). This weight is the minimum quantity of PHB that permits to cover completely the bottom of the used DSC pan (diameter: 6 mm). A very small number of nuclei are generally produced during PHB nucleation process at high [T.sub.c]'s, since natural PHB is an extremely pure material, not containing catalyst residues and other impurities, that could promote heterogeneous nucleation at large extent (39). In this condition, as it was observed in some preliminary tests, if the thickness of the sample is not regular and uniform, crystals may grow only in a limited part of the sample under analysis, due to the extremely low nucleation density of PHB crystals, with consequent nonreproducibility of the crystallinity developed. A fresh PHB sample was employed for each analysis, in order to reduce the problems caused by thermal degradation during melting.

Before the analyses, each PHB sample was heated to 466 K at a rate of 30 K [min.sup.-1] and maintained at this temperature for 3 min with the aim of destroying all traces of previous crystalline order (40). These are the optimized experimental conditions that allow the complete fusion of previous crystal order, and at the same time ensure the minimum possible thermal degradation of PHB chains (41). Then the samples were cooled to the desired crystallization temperature at a nominal rate of 200 K [min.sup.-1]. At each [T.sub.c] the length of the isothermal step was varied in order to obtain several samples with different crystalline degree. After partial or complete crystallization, the melting behavior was recorded by heating the samples directly from [T.sub.c] up to 463 K at 60 K [min.sup.-1].

RESULTS AND DISCUSSION

The method for the determination of the equilibrium melting temperature of PHB, presented and discussed here, is based on the Thomson-Gibbs equation (Eq. 1), which can be rewritten as:

[T.sub.m] = [T.sub.m.sup.o]{1 - 2[[sigma].sub.e]/[DELTA][h.sub.m.sup.0][gamma][[C.sub.1]/[T.sub.m.sup.0] - [T.sub.c] + [C.sub.2]]} (2)

with [gamma] equal to the ratio [l.sub.c]/[l*.sub.c], where [l*.sub.c] is the initial lamellar thickness at [T.sub.c] (22), (42):

[l*.sub.c] = [C.sub.1]/[T.sub.m.sup.0] - [T.sub.c] + [C.sub.2] (3)

Equation 3 was derived experimentally for polyethylene (42) and is similar in form to the relationship proposed by Hoffman-Lauritzen (21). According to the HL theory, the term C1 is equal to 2[[sigma].sub.e][T.sub.m.sup.0]/[DELTA][h.sub.m.sup.0], whereas [C.sub.2] is a small negligible quantity approximately equal to 0.1 nm (21). As for polyethylene the [C.sub.2] parameter was found much larger than that predicted by the HL theory ([C.sub.2] = 6 nm) (42), criticisms have been leveled against the HL theory (22). The term [C.sub.2] is ignored in the classical HW treatment, which contributes to an underestimation of the equilibrium melting temperature (22).

In case of polymers exhibiting isothermal lamellar thickening, as the thickening rate changes with time and crystallization temperature, it is practically impossible to attain samples crystallized at different [T.sub.c] having the same coefficient [gamma], as required by the HW method. To overcome this problem, Marand and coworkers suggested to take into account only the melting temperatures of non-thickened crystals ([gamma] = 1), which were obtained by extrapolating the melting temperature of thickened lamellae to zero crystallinity (22), (23).

On the contrary, for polymers that do not thicken at the crystallization temperature and/or on heating, the thickening coefficient [gamma] remains equal to unity from [T.sub.c] up to [T.sub.m], unless the original crystals reorganize via partial melting and recrystallization [431. This process results in an increase of the lamellar thickness and a shift of the melting peak to higher temperatures. This type of crystal reorganization is more pronounced for low [T.sub.c]'s and heating rates, and progressively disappears with increasing crystallization temperature and heating rate (44). In case of polymer crystals that undergo partial melting and recrystallization, in order to obtain the melting temperature of original crystals with [gamma] = 1, it is necessary to prevent reorganization by operating at high crystallization temperatures and heating rates (27), (29). In a previous article of this series it was demonstrated that reorganization in PHB samples crystallized at [T.sub.c] [greater than or equal to] 388 K is hindered at heating rates equal or higher than 60 K [min.sup.-1] (31). The analysis was limited to samples crystallized at [T.sub.c] [greater than or equal to] 388 K, because at [T.sub.c] < 388 K it was impossible to stop crystallization before completion, as the process continued during the successive heating scans. Since the maximum of PHB crystallization rate is around 363 K (45), (46), [T.sub.c]'s higher than 388 K are temperatures sufficiently high to produce quite perfect crystals that do not undergo recrystallization upon heating at 60 K [min.sup.-1] (31).

The specific heat capacity of PHB in the melting region at 60 K [min.sup.-1] after complete isothermal crystallization at various temperatures is illustrated in Fig. 1. The melting endotherm is composed by a main peak, connected to fusion of dominant lamellae, that develop during primary crystallization, and a shoulder at about 1520 K below the main peak, which is caused by fusion of less stable crystals, which grow mostly in presence of more perfect crystals and melt earlier due to their lower stability (31). These less perfect crystals can melt and recrystallize, but only at heating rate lower than 50 K [min.sup.-1] (31). At the heating rate of 60 K [min.sup.-1] recrystallization of the secondary crystals is negligible. The enthalpy of fusion after complete isothermal crystallization (102 J [.g.sup.-1]) was found to be independent of [T.sub.c] in the whole analyzed temperature range.

An example of the dependence of the melting behavior on the crystallization time [t.sub.c] is shown in Fig. 2 for the crystallization at [T.sub.c] = 403 K. In the Figure the lines tangent to the steepest part of the main peak are also drawn for some curves. The intersection between the tangents and the baseline defines the onset temperature of the main peak ([T.sub.onset]).

The onset temperatures of the main endotherm, measured after different crystallization times at various [T.sub.c]'s, are reported in Fig. 3 as a function of the crystallization time in logarithmic scale (47). [T.sub.onset] initially decreases with [t.sub.c], then slightly raises again for 388 [less than or equal to] [T.sub.c] [less than or equal to] 403 K. These isothermal crystallizations were conducted up to completion. For [T.sub.c] = 408 K and 411 K only the decrease of [T.sub.onset], was detected, as only partial crystallizations were considered, because of the crystallization rate that progressively reduced.

Similar to the onset temperature, also the peak temperature ([T.sub.peak]) shows a discontinuity, as evidenced in Fig. 4, in which the peak temperatures are reported as a function of the crystallization time [t.sub.c]. An increase of the slope of [T.sub.peak] vs. log [t.sub.c] is observed at the same [t.sub.c] values at which [T.sub.onset] data reverse their trends. This crystallization time was found to correspond to the spherulite impingement point, so that the two different dependences were put in relation with primary and secondary crystallization respectively (31).

At the heating rate of 60 K [min.sub.-1] superheating and thermal lag effects cannot be neglected (47), (48). The term superheating indicates the increase of the melting temperature induced by the time dependence of the melting process, unlike the thermal lag, which is connected to the presence of thermal gradients within the sample (47). The onset temperature is influenced only by superheating, whereas the peak temperature depends on both superheating and thermal lag, as the latter causes a broadening of the transition endotherm. Before spherulite impingement, the [T.sub.onset] decrease that is observed with increasing [t.sub.c] in Fig. 3, can be explained considering that crystals that develop first are more perfect, successively the crystalline growth occurs through the formation of crystals with a higher defect concentration, which results in [T.sub.onset] reduction (31). In parallel, the rise of [T.sub.peak], which is monitored simultaneously with the [T.sub.onset] decrease, is due to thermal lag. After spherulite impingement, lamellae continue to grow in geometrically more restricted areas. Topological constraints can produce an increase of the energy barrier for the movement of the crystalline-amorphous interface, and thus raise the melting temperature (38), (47). As a consequence, the increase of both [T.sub.onset] and [T.sub.peak] after spherulite impingement is caused by crystal superheating (31).

A double dependence is exhibited also if the peak temperatures are plotted as a function of [(m[[DELTA]h.sub.m][beta]).sup.1/2] (Fig. 5). (For [T.sub.c] = 408 K and 411 K only one linear segment is observed, as the investigation was not conducted up to complete crystallization). The quantity [(m[[DELTA]h.sub.m][beta]).sup.1/2] is approximately the measurement of the [T.sub.peak] increase caused by the thermal lag, according to the relation (31), (49):

[T.sub.peak] - [T.sub.peak,o] = [(2Rm[[DELTA]h.sub.m][beta]).sup.1/2] (4)

where m is the sample mass, R the thermal resistance, [[DELTA]h.sub.m] is the enthalpy of fusion per gram, [beta] the heating rate and [T.sub.peak,o] the peak temperature with the instrumental effects that cause the broadening removed. Also the enthalpies of fusion at which the [T.sub.peak] plots change their trends in Fig. 5 were found to correspond to the spherulite impingement point (31).

The effects of superheating on both the onset and peak temperatures can be canceled by extrapolating the [T.sub.onset] and [T.sub.peak] values to [t.sub.c] [right arrow] 0, because fusion of very small crystalline areas, quite perfect and without constraints and restrictions, can occur with a reduced energy barrier for the crystalline-amorphous interface movement (31), (38), (47). On the contrary, thermal lag can be erased by extrapolating the [T.sub.peak] values to [[DELTA]h.sub.m] [right arrow] 0, according to Eq. 4. These extrapolations provide the true onset and peak temperatures of the initial crystals. The extrapolations of [T.sub.onset] and [T.sub.peak] in Figs. 3 and 4 were conducted linearly down to the values of zero crystallinity, i.e. down to the times at which the exothermal crystallization curves start to deviate from the baseline heat flow rate (50). On the contrary in Fig. 5 extrapolation was performed on the [T.sub.peak] data down to [(m[[DELTA]h.sub.m][beta]).sup.1/2] = 0. It is worth noting that the [T.sub.peak] values extrapolated in Fig. 5 are identical to those obtained in Fig. 4, which means that extrapolations to [[DELTA]h.sub.m] [right arrow] 0 or to [t.sub.c] [right arrow] 0 cancel both superheating and thermal lag.

For semicrystalline polymers, the peak point is often considered as the most representative melting temperature, because of the generally wide distribution of crystal size and perfection (51). The [T.sub.peak] value extrapolated to zero crystallization time corresponds to the melting temperature of the initial more perfect crystals that develop at a given [T.sub.c], as superheating and thermal lag are canceled by extrapolation. On the contrary, the extrapolated [T.sub.onset] values are related with the fusion of slightly less perfect initial lamellae. Thus the difference between the extrapolated [T.sub.peak] and [T.sub.onset] values represents the distribution of original crystal perfection at a particular [T.sub.c] and the fact that it decreases slightly with the crystallization temperature (from 4.1 K at [T.sub.c] = 388 K to 2.7 K at [T.sub.c] = 411 K) proves that the crystal perfection of the original dominant lamellae increases with the crystallization temperature. With the progress of crystallization, the less perfect crystals that develop successively bring to a reduction of the onset temperature, as illustrated in Fig. 3, and, as a consequence, to a further broadening of the peak. This event becomes progressively more marked with increasing the crystallization temperature, as also well visible in Fig. 1, where it can be observed that the width of the main peak increases with [T.sub.c].

The equilibrium melting temperature of PHB was calculated from the extrapolated peak temperatures through the Thomson-Gibbs equation (Eq. 2) by means of a nonlinear regression method, using the software OriginPro by OriginLab Corporation, with [T.sub.m.sup.0], [C.sub.1], and [C.sub.2] as fitting parameters, being [gamma] = 1 for initial original lamellae. The enthalpy of fusion [[DELTA]h.sub.m.sup.0] was 1.84 X [10.sub.8] J [m.sup.-3] (6), whereas the fold surface free energy [[delta].sub.e] was made to change in the range 0.040-0.060 J [m.sub.-2], on the basis of the available literature data (13), (45), (46), (52), (53). Variation of [[delta].sub.e], within this range does not affect the calculated [T.sub.m.sup.0], on the contrary both [C.sub.1] and [C.sub.2] were found to change. However, since the calculated [T.sub.m], vs. [T.sub.c] curves were all overlapping for the different [[delta].sub.e] initial values, the fold surface free energy that better reproduced through the parameters [C.sub.1] and [C.sub.2] the lamellar thickness data reported in the literature was chosen as [[delta].sub.e] fixed value. Considering that the error in the determination of the extrapolated [T.sub.peak] is ~0.5 K, and assuming [[delta].sub.e] = 0.060 J [m.sup.-2], the fitting procedure yielded the following values: [T.sub.m.sup.0] = (511 [+ or -] 3) K, [C.sub.1] = (3.33 [+ or -] 0.01) x [10.sup.2] nm K and [C.sub.2] = (1.4 [+ or -] 0.2) nm ([R.sup.2] = 0.984). Figure 6 illustrates the [T.sub.m] vs. [T.sub.c], plot for PHB. In this graph the equilibrium melting temperature corresponds to the intersection point between the fitted curve and the straight line [T.sub.m] = [T.sub.c]. According to Eq. 3, the estimation of PHB initial lamellar thickness is: [l*.sub.c] = (3.33 x [10.sup.2]/([T.sub.m.sup.0] - [T.sub.c]) + 1.4) nm, which corresponds to 4.1 [less than or equal to] [l*.sub.c] [less than or equal to] 4.7 nm for [T.sub.c] ranging from 388 to 411 K, in good agreement with available experimental data from small-angle X-ray scattering (5 [less than or equal to] l [less than or equal to] 7 nm) (8), (54). The quantity [C.sub.1] comes out approximately equal to 2[[delta].sub.e][T.sub.m.sup.0]/[[DELTA]h.sub.m.sup.0], as already found for polyethylene (42), whereas the parameter [C.sub.2], which corresponds to about 30% of the calculated [l*.sub.c] value at the investigated undercoolings, is confirmed to be a significant and non-negligible quantity. This finding substantiates the nonperfect correspondence between the Hoffman-Lauritzen predictions and the experimental evidences as regards the initial lamellar thickness. It is worth noting that the nonlinear fitting procedure applied to the extrapolated onset temperatures produced a final equilibrium melting temperature very close to that obtained from the peak temperatures ([T.sub.m.sup.0] = 512 K).

For comparison, the experimental [T.sub.peak data, treated on the basis of Marand's method, are reported in Fig. 7. According to this procedure, the true equilibrium melting temperature is the value for which the plot of [T.sub.m.sup.0]/([T.sub.m.sup.0] - [T.sub.m]) vs. [T.sub.m.sup.0]/([T.sub.m.sup.0] - [T.sub.m]) yields a straight line of unitary slope, as the melting temperatures refer to nonthickened crystals ([gamma] = 1):

[T.sub.m.sup.0]/([T.sub.m.sup.0] - [T.sub.m]) = [gamma]([T.sub.m.sup.0]/[T.sub.m.sup.0] - [T.sub.c] + a) (5)

where a = [C.sub.2][[DELTA]h.sub.m.sup.0]/2[[delta].sub.e] (22). Since Eq. 5 originates directly from the Eq. 2 by setting [C.sub.1] = 2[[delta].sub.e][T.sub.m.sup.0]/[[DELTA]h.sub.m.sup.0], according to the HL theory, the equilibrium melting temperature value that satisfies the condition of unitary slope, is equal to the one calculated by the nonlinear fitting procedure as for PHB the quantity [C.sub.1] has been found approximately equal to 2[[delta].sub.e][T.sub.m.sup.0]/[[DELTA]h.sub.m.sup.0].

The equilibrium melting temperature of PHB calculated by the method here presented is much higher than the values reported in the literature (6-19). It is ~50 K above the [T.sub.m.sup.0] determined with the Hoffman-Weeks procedure (9-19), and 40 K above the [T.sub.m.sup.0] obtained with the TG extrapolation (6-8). A difference of about 30 K between the linear and nonlinear extrapolations ([[DELTA]T.sub.m.sup.0]) was found by Marand et al. for isotactic polypropylene (23). Equilibrium melting temperatures higher than the values calculated by the HW procedure were found with the Marand's method for example for poly(L-lactide) ([[DELTA]T.sub.m.sup.0] = 32 K) (55), syndiotactic polypropylene ([[DELTA]T.sub.m.sup.0] = 42 K) (56), syndiotactic polystyrene ([[DELTA]T.sub.m.sup.0] = 25 K and 28 K for the [alpha] and [beta] forms, respectively) (57), poly(ethylene terephthalate) ([[DELTA]T.sub.m.sup.0] = 54 K) (58), poly(butylene terephthalate) ([[DELTA]T.sub.m.sup.0] = 27 K and 20 K) (58), (59). These large discrepancies originate from the contribution of the term [C.sub.2] to the lamellar thickness, neglected in the HW procedure, which leads to a nonlinear relationship between [T.sub.m] and [T.sub.c (20), (22). On the other hand, the smaller [T.sub.m.sup.0] values obtained for PHB from the [T.sub.m] vs. 1/l plots (6-8) can be ascribed to the fact that the reported [T.sub.m] values were experimentally obtained at too low heating rates (1020 K [min.sub.-1]), unable to suppress the recrystallization process. The consequent increase of the measured [T.sub.m] is more pronounced at higher undercooling, due to the smaller stability of the crystalline phase, which produces a lower slope of the [T.sub.m] vs. 1/l plot. Also for a correct use of the TG equation, it should be advisable to construct the [T.sub.m] vs. 1/l plots with melting temperature data referred to isolated, nonsuperheated and nonrecrystallized lamellae. If the crystals melt and recrystallize during the heating from [T.sub.c] to [T.sub.m], it results that erroneous TG plots are generated and lower [T.sub.m.sup.0] values are obtained, since the lamellar thickness measured at room temperature is always smaller than the true value at the melting temperature that refers to recrystallized lamellae.

CONCLUSIONS

The results presented in this study confirm that the equilibrium melting temperature of semicrystalline polymers is much higher than the one commonly calculated by the Hoffman-Weeks procedure. The underestimation of [T.sub.m.sup.0] is caused by the fact that significant contributions to the lamellar thickness are neglected in the HW procedure. Thus, for an accurate determination of [T.sub.m.sup.0] it is necessary to consider all the terms that contribute to the lamellar thickness of the crystals under analysis, which corresponds to a nonlinear extrapolation of the observed melting temperature data.

The need to perform nonlinear extrapolation of the [T.sub.m] - [T.sub.c] data had already been suggested by Marand (22). Marand's method is based on the Hoffman-Lauritzen theory: the equilibrium melting temperature is calculated from the Thomson-Gibbs equation, with the initial lamellar thickness defined by the Hoffman-Lauritzen equation plus an adjustable parameter.

Also in this study the equilibrium melting temperature is calculated from the Thomson-Gibbs equation, but the procedure is not connected to the Hoffman-Lauritzen theory: the initial lamellar thickness is defined by an empirical equation with two adjustable parameters (42), whose values are calculated by the nonlinear fitting procedure that allows to determine also the equilibrium melting tern perature.

The nonlinear extrapolation procedure here presented employs data obtained only by calorimetry and does not require measurements with nonroutine techniques. For polymers that do not thicken at the solid state, but melt and recrystallize upon heating, like poly(3-hydroxybuty-rate), the first step is to individuate the scanning rate which inhibits the recrystallization process at sufficiently high [T.sub.c]. Successively, the melting temperatures after various crystallization times at different [T.sub.c]'s are measured and extrapolated to zero crystallization time in order to cancel superheating and thermal lag effects. Thus [T.sub.m.sup.0] can be determined from the melting temperature of initial crystals, grown isolated and with the minimum content of defects, through a nonlinear fitting procedure.

Correspondence to: Maria Cristina Righetti; e-mail: righetti@ipcf.cnr.it

DOI 10.1002/pen.23199

Published online in Wiley Online Library (wileyonlinelibrary.com).

[c] 2012 Society of Plastics Engineers

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Maria Cristina Righetti, (1) Maria Laura Di Lorenzo (2)

(1.) Consiglio Nazionale delle Ricerche, Istituto per i Processi Chimico-Fisici, Via G. Moruzzi, 1, 56124 Pisa, Italy

(2.) Consiglio Nazionale delle Ricerche, Istituto di Chimica e Tecnologia dei Polimeri, c/o Compresorio Olivetti, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy

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Author: | Righetti, Maria Cristina; Lorenzo, Maria Laura Di |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Geographic Code: | 4EUIT |

Date: | Nov 1, 2012 |

Words: | 5754 |

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