Isoconversional Kinetics of Nonisothermal Crystallization of Salts from ...

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Isoconversional Kinetics of Nonisothermal Crystallization of Salts from Solutions Victoria L. Stanford, Calla M. McCulley, and Sergey Vyazovkin* Department of Chemistry, University of Alabama at Birmingham, 901 South 14th Street, Birmingham, Alabama 35294, United States ABSTRACT: In this study, differential scanning calorimetry (DSC) has been applied to measure the kinetics of nonisothermal crystallization of potassium nitrate and ammonium perchlorate from unsaturated and saturated aqueous solutions. DSC data have been analyzed by an advanced isoconversional method that demonstrates that the process is represented by negative values of the effective activation energy, which varies with the progress of crystallization. The classical nucleation model can be used to predict and understand the experimentally observed variation in the effective activation energy. The saturated and unsaturated solutions have demonstrated distinctly different crystallization kinetics. It is suggested that the unsaturated solutions undergo a change in crystallization mechanism from homogeneous to heterogeneous nucleation.

1. INTRODUCTION Crystallization of a solute from a liquid solution is one of the most common and important processes used for purification and production of crystalline compounds.1 Learning the crystallization kinetics aids in obtaining the mechanistic insights into the process as well as in its practical optimization. The kinetics of the solution crystallization is typically monitored by various techniques that measure directly either a decrease in the solution concentration or an increase in the amount of the crystalline solute. Process kinetics can also be followed calorimetrically (i.e., as the rate of the heat release that accompanies crystallization). Surprisingly, our literature search has not revealed any systematic studies of the liquid solution crystallization kinetics by calorimetric techniques such as differential scanning calorimetry (DSC). This is unexpected considering that DSC is used quite commonly to study the thermodynamics of crystallization, for example, in such systems as aqueous solutions of inorganic salts.2−8 The purpose of the present work is to initiate calorimetric kinetic studies that combine DSC with isoconversional kinetic analysis.9 Over the past decade, the isoconversional methodology has become a major tool for dealing with the kinetics of thermally stimulated processes. The key element of isoconversional kinetics is evaluation of the effective activation energy (Eα) as a function of substance conversion (α). The resulting dependence can be used for estimating other kinetic parameters, for making predictions, and for obtaining insights into the process mechanisms. The earliest and most common applications of the isoconversional kinetics have been in the area of thermally stimulated chemical reactions such as decomposition, oxidation, reduction, polymerization, and cross-linking. The applications of the isoconversional kinetics to physical processes (i.e., phase transitions) are relatively new and far less common. These new applications have been © 2016 American Chemical Society

focused primarily on the kinetics of nucleation driven processes such as polymer melt crystallization,10 gelation,11 morphological solid−solid transitions,12,13 coil-to-globule transition,14 and polymer melting.15,16 In this paper, we introduce some theoretical background to the process of nonisothermal crystallization of salts from solutions. We demonstrate that the supersaturation in the classical nucleation model can be replaced with the supercooling and that the resulting modified model can be used to predict a theoretical variation of the effective activation energy with temperature and progress of the process. The predictions are verified against the actual experimental dependencies obtained by applying an advanced isoconversional method17,18 to DSC data on crystallization of some inorganic salts from their aqueous solutions.

2. EXPERIMENTAL SECTION Potassium nitrate (Alfa Aesar) and ammonium perchlorate (Alfa Aesar) were purchased at 99.9% pure. Both salts were chosen due to their significant enthalpies of dissolution and strong temperature dependence of solubility as reported in the literature.1 Each salt compound was dissolved with 1 mL of deionized water produced from Purelab Ultra Genetic system (Siemens Water Technology, Germany) to obtain a saturated solution. The sample vials of each salt solution were placed in an ice/ water bath at 3 °C, in order to check whether crystallization occurs. Because in aqueous solutions water crystallizes below 0 °C, the occurrence of crystallization at a higher temperature is Received: April 15, 2016 Revised: May 22, 2016 Published: June 16, 2016 5703

DOI: 10.1021/acs.jpcb.6b03860 J. Phys. Chem. B 2016, 120, 5703−5709

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The Journal of Physical Chemistry B ⎛ −16πσ 3N 2 v 2 ⎞ ⎛ −E ⎞ w(T ) = w0exp⎜ 3 3 A 2 ⎟exp⎜ D ⎟ ⎝ 3R T (ln S) ⎠ ⎝ RT ⎠

indicative of the salt crystallization. Both salts demonstrated crystallization at this temperature. Approximately 20 mg of each sample were weighed into 100 μL Al pans that were sealed hermetically. The mass of salt per 20 mg solution was approximately 5−8 mg, which was recorded, and the samples were placed in the DSC. The samples were then heated to 45 °C at 10 °C min−1, held at 45 °C for 30 min to erase thermal history, and then cooled from 45 °C to −20 °C at the cooling rates: 16, 8, 4, 2, and 1 °C min−1. The saturated salt solutions were prepared in the same manner, but care was given to put some solid salt particles in the pan with the aqueous solution. Caution: it was found that KNO3 will corrode the Al pan within 24 h of being prepared. All calorimetric measurements were taken with a heat flux DSC (Mettler-Toledo, 823e). Indium and zinc standards were used to perform temperature, heat flow, and tau-lag calibrations. Melting of deionized water was used for calibration adjustment. The experiments were performed in the atmosphere of nitrogen flow (80 mL min−1).

(2)

where σ is the surface tension, v is the molecular volume, NA is the Avogadro number, R is the gas constant, T is the absolute temperature, and ED is the activation energy of diffusion. eq 2 can be rewritten in a simpler form ⎛ −A ⎞ ⎛ −ED ⎞ ⎟ w(T ) = w0 exp⎜ 3 ⎟exp⎜ ⎝ T (ln S)2 ⎠ ⎝ RT ⎠

(3)

where the constant A collects all parameters that are either independent or weakly dependent on temperature. As derived,1 eqs 2 and 3 do not include the supercooling (i.e., ΔT = T0 − T), where T0 is the equilibrium temperature, at which the solution of the concentration x would be at equilibrium with its solute (Figure 1). To introduce the supercooling, one needs to account explicitly for the temperature dependence of the supersaturation. The dependence is readily determined from the temperature dependence of the solubility. It can be expressed in many forms,1 the simplest of which is

3. THEORETICAL AND COMPUTATIONAL BACKGROUND The solubility of most salts decreases with decreasing temperature so that a solution unsaturated at higher temperature can become supersaturated at lower temperature. As a nonequilibrium system, the supersaturated solution would relax via crystallization (i.e., by pushing excess solute out of the solution to attain equilibrium). Figure 1 illustrates schematically

(4)

ln x = Z + BT

Note (Figure 1) that the supersaturation (i.e., excess of x relative to x0) is caused by dropping temperature from T0 to T. Then writing eq 4 for x and x0 and subtracting the latter from the former allows us to link the supersaturation to the supercooling as follows: ln S = B(T0 − T ) = BΔT

(5)

Substituting eq 5 into eq 3 yields ⎞ ⎛ −ED ⎞ ⎛ −A ⎟ w(T ) = w0 exp⎜ 3 ⎟exp⎜ ⎝ T (BΔT )2 ⎠ ⎝ RT ⎠

(6)

The derived eq 6 expresses the crystallization rate of a solution as a function of temperature alone. This equation can be used to predict a possible outcome of isoconversional calculations in the case of nonisothermal crystallization of the solutions. That is, to predict how the value of the effective activation energy can vary with the extent crystallization. The value is defined as the following derivative: E = −R Figure 1. Phase diagram of an aqueous solution of a salt.

dln w(T ) dT −1

(7)

Plugging eq 6 into 7 gives rise to the temperature-dependent activation energy:9 that dropping temperature below the equilibrium line between the one-phase (unsaturated solution) and the two-phase (saturated solution) region (Figure 1) forces solution to crystallize until it reaches the equilibrium concentration x0. The driving force of crystallization is the supersaturation, S, that is defined as the ratio of the nonequilibrium to equilibrium concentration:1

S=

x x0

E = ED +

⎤ AR ⎡ 3 2 − 2 ⎢ 2 2 3 ⎥ B ⎣ (ΔT ) T (ΔT ) T ⎦

(8)

It can be demonstrated that the expression in the brackets is negative at T > 0.6T0 (i.e., in a wide range of temperatures from T0 down to 0.6T0). It is also clear that at temperatures close to T0, the bracketed term approaches −∞. As temperature decreases from T0 to 0.6T0, the term increases from −∞ to 0. As a result, the effective activation energy of the solution crystallization should be a negative value that increases monotonically with decreasing temperature (i.e., with the crystallization progress) from −∞ to ED. Negative E values that increase with the extent of crystallization have been consistently

(1)

The rate of crystallization is proportional to the supersaturation in accord with the following equation:1 5704


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The Journal of Physical Chemistry B discovered10,19−26 by applying isoconversional kinetic analysis to crystallization of polymer melts. Note that the derived eq 8 represents accurately crystallization in a single component system such as melt (liquid) − crystal (solid). Per the Gibbs phase rule for isobaric conditions:

This information is helpful in understanding the results of the isoconversional treatment of the respective data. Considering crystallization on continuous cooling, the most general assumption would be that the process starts near one equilibrium state at higher temperature and finishes near another at lower temperature. Then the supersaturation would change from the value 1, pass through a maximum, and return back to 1. This behavior can be simulated by an inverted quadratic parabola. For illustration purposes, we will use the parabola of the form:

(9)

F=C−P+1

in a single component system (C = 1) equilibrium between two phases (P = 2) can be maintained by independently varying zero intensive variables (F = 0). This means that under isobaric conditions equilibrium in a single component system is defined by unique equilibrium temperature, T0. However, a solution of a salt is a system of two components (C = 2), solute and solvent. Per eq 9, equilibrium in such system can be maintained while varying one (F = 1) intensive variable (temperature or concentration). That is, changing temperature of a saturated solution would result in establishing a new equilibrium that is characterized by a new set of the equilibrium temperature (T0) and concentration (x0) that are linked to each other as presented in Figure 1. It means that eq 8 derived assuming T0 being constant would work as a reasonable approximation only when T0 does not change much. This would be the case when the solubility of a solute has a very strong temperature dependence (i.e., when the T versus x diagram (Figure 1) has a very shallow slope). It would also work for the initial stages of crystallization (i.e., not far beyond the induction period when x does not change much). Otherwise, one has to account for variation of T0 with the concentration. Let us now derive an equation for the temperature dependence of the effective activation energy that accounts for variability in T0. As shown above, the driving force of the solution crystallization can be expressed as supercooling (i.e., T0 − T). When a solution crystallizes on continuous cooling (i.e., nonisothermally), both T0 and T decrease. As long as a decrease in T causes the solute crystals to fall out, x is decreasing. Because x is linked uniquely to T0 (Figure 1), the latter is also decreasing. At any moment of time, x is determined by T, and T0 is determined by x so that T0 is a function of T, T0 = φ(T). Unless cooling is infinitely slow (i.e., crystallization occurs near equilibrium), the value of T0 would be lagging behind the T value. That is, the supercooling: ΔT = φ(T ) − T

S = 2 − 0.01(T − 273)2

The parameters of the parabola are chosen to have the equilibrium values of S at 263 and 283 K and some reasonable values of the supersaturation between (Figure 2). After taking

Figure 2. Temperature dependence of the supersaturation according to eq 12.

the logarithm of eq 12, we can equate the right-hand side of it and of eq 5 and solve the resulting equality for T0. This allows us to obtain an analytical expression for T0 as a function of T and its temperature derivative as follows:

(10)

T0 ≡ φ(T ) = T +

would remain positive. By substituting eq 10 into eq 6 and taking the derivative of eq 7, we arrive at eq 11:

ln[2 − 0.01(T − 273)2 ] B

dT0 0.02(T − 273) d φ (T ) ≡ =1− dT dT B[2 − 0.01(T − 273)2 ]

⎡ ⎤ dφ(T ) 2 dT − 1 ⎥ AR ⎢ 3 E = ED + 2 ⎢ − ⎥ B ⎢ (φ(T ) − T )2 T 2 (φ(T ) − T )3 T ⎥ ⎣ ⎦

(

(12)

)

(13)

(14)

In eq 5, B is typically a small positive value (e.g., 0.2). Substituting this value into eq 13 and looking at the difference between T0 and T allows us to visualize (Figure 3) how the supercooling changes as temperature decreases. As expected, T0 and T are similar near the initial and final equilibrium states. However, when the supersaturation reaches its maximum (Figure 2), the supercooling reaches its maximum as well, making T0 to lag a few degrees behind T. Finally, substitution of eqs 13 and 14 into eq 11 allows us to see (Figure 4) a general type of the E versus T dependence characteristic of nonisothermal crystallization of a solution. The dependence is more complex than the one that we can obtain from eq 8 that assumes T0 to be constant. Nonetheless, the initial portions of both dependencies are nearly identical. That is, both demonstrate that E has a negative value that tends to −∞ in proximity of the initial (high temperature) equilibrium

(11)

Unlike eq 8, this equation accounts for variation of T0 taking place during the nonisothermal crystallization. Practical application of eq 11 is rather complicated because the exact form of the φ(T) function is generally unknown, although it can be established. This would require measuring an exact phase diagram (Figure 1), that is, a dependence of T0 on x as well as measuring the process kinetics, that is, a dependence of x on T. However, this is beyond the scope of the present work. Here, we use eq 11 for the sole purpose of theoretically exploring possible types of the E vs T dependencies characteristic of nonisothermal crystallization of a solution. 5705


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crystallization rate passes through a maximum that separates the temperature regions of the nucleation and diffusion control. A similar phenomenon is known for crystallization of solutions. For instance, Mullin 1 demonstrates that the rate of crystallization of citric acid from an aqueous solution increases with decreasing temperature from 22 down to −5 °C, whereas from −5 down to −40 °C, the rate demonstrates a decreasing trend due to continuously increasing viscosity of the aqueous solution. Overall, our analysis of the nucleation model (eq 11) suggests that when a solute crystallizes out of a solution during continuous cooling, the effective activation energy of the process should take negative values as long as its rate is controlled by nucleation. It also suggests that as temperature decreases and crystallization advances, the negative E values should be increasing monotonically toward the positive ones. Because the extent of the solute conversion from the dissolved to crystalline state increases with decreasing temperature, we can expect an isoconversional method to yield the values of the effective activation energy that are negative and increasing toward positive values with increasing conversion. However, our analysis does not tell whether the positive E values can actually be accomplished during the continuous cooling runs. For example, multiple applications10,19−26 of isoconversional methods to the polymer melt crystallization kinetics yield negative E values that increase with conversion but never seem to cross into the area of positive values. The positive values are routinely obtained when isoconversional methods are applied to crystallization of polymer glasses that takes place on heating. Using the polymer melt crystallization as a prototype for the solution crystallization, we should not expect the respective effective activation energies to reach positive values in cooling experiments. Rather, positive E values should be observed for heating of supercooled solutions, which would be the case similar to crystallization of polymer glasses. In order to obtain actual dependencies of Eα on α for the solution crystallization, we have applied an advanced isoconversional method.18 The subscript α denotes a value related to a given value of conversion. The Eα-dependencies have been estimated from respective sets of DSC curves obtained at different cooling rates. The values of α are evaluated as partial areas of a DSC peak. According to this method, for a set of n runs conducted under different temperature programs, Ti(t), the effective activation energy is determined at any given α by finding Eα, which minimizes the function18

Figure 3. Temperature dependence of the supercooling derived from eq 13.

Figure 4. Temperature dependence of the effective activation energy obtained from eq 8 (dash line) and eq 11 (solid line).

state. As crystallization progresses with decreasing temperature, the E value increases monotonically. It is noteworthy that the shape of the E versus T dependence obtained from eq 11 is similar to the one we derived27 theoretically from the Hoffman−Lauritzen theory28,29 for crystallization of polymers. The characteristic feature of the dependence is that E continuously increases with decreasing T and changes its sign from negative to positive. The shape of the dependence was later confirmed experimentally by applying isoconversional analysis to calorimetric data on crystallization of various polymers.30−33 The sign change in crystallization of polymer melts is due to the crystallization regime switching from nucleation to diffusion control as temperature decreases. Initially the rate of crystallization increases with decreasing temperature because the free-energy barrier to nucleation decreases with increasing the supercooling. This is an antiArrhenian type of the temperature dependence that yields a negative value of the effective activation energy. However, as the melt becomes increasingly viscous, the rate starts to decrease with decreasing temperature because the migration rate of molecules toward nuclei slows down. This is an Arrhenian type of the temperature dependence that yields positive value of the effective activation energy. As a result, the

n

Ψ(Eα) =

n

J[Eα , Ti(tα)] J[Eα , Tj(tα)]

(15)

⎡ −E ⎤ α exp⎢ ⎥d t ⎣ RTi(t ) ⎦ α −Δα

(16)

∑∑ i=1 j≠i

where J[Eα , Ti(tα)] ≡

∫t

tα

In eq 16, α is varied from Δα to 1−Δα with a step Δα = m−1, where m is the number of intervals chosen for computation. The integral, J in eq 16, is estimated by using the trapezoid rule. Minimization is repeated for each value of α to establish the dependence Eα on α. Unlike simpler or, more precisely, rigid9 integral methods, this method can process data obtained under arbitrary temperature variation and can eliminate a systematic error found when Eα varies significantly with α. 5706


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4. RESULTS AND DISCUSSION Figure 5 demonstrates the DSC curves for crystallization of NH4ClO4 and KNO3 in unsaturated solutions. The heat flow is

discussed further, the maximum is related to the formation of the crystalline phase but not to any particular amount of it. Therefore, there is no obvious reason for it to appear at some particular value of conversion. After the maximum, the effective activation energy decreases with increasing the extent of conversion. The initial portion of the Eα dependencies is consistent with the theoretical prediction (Figure 4), but the subsequent decrease in Eα is not. The Eα dependencies of the salt solutions in Figure 6 correspond to the Arrhenius plot found in Figure 7 (solid line). The nonlinear anti-Arrhenian

Figure 5. DSC curves of the crystallization of unsaturated ammonium perchlorate and potassium nitrate on cooling at 2 °C min−1.

representative of the rate of the crystallization process throughout the supercooling of each salt. Therefore, it reflects the formation of the crystalline phase. The salts crystallize in different temperature ranges depending on the amount of supersaturation needed for the initial nucleus to form during supercooling. The DSC peaks demonstrate a shift to lower temperature with increasing the cooling rate, as was previously observed for crystallization of the polymer melts. Figure 6 shows the effective activation energy (Eα) versus the extent of conversion (i.e., crystallization) (α) for each

Figure 7. Schematic Arrhenius plots for nucleation process of salt solutions. The solid curve represents the unsaturated salt solutions, while the dash line represents the saturated salt solutions.

temperature dependence (Figure 7, dash line) is common for nucleation-driven processes taking place on cooling.9 The antiArrhenian behavior simply indicates that the rate of crystallization increases with decreasing temperature. However, the dependencies observed for the unsaturated solutions (Figure 6) correspond to the anti-Arrhenian dependence with a breakpoint (Figure 7, solid line). We hypothesize that the breakpoint is due to a change in crystallization mechanism from homogeneous to heterogeneous nucleation. Crystallization is a nucleation-driven process, where crystals precipitate from a solution initially beginning with a solid aggregate, called the nucleus. Nucleation can be either homogeneous or heterogeneous. Homogeneous nucleation is when, upon supercooling, a critical nucleus forms spontaneously being surrounded entirely by the original solution phase.1 Eq 11 and the predicted monotonic increase (Figure 4) in effective activation energy are based on the assumption of homogeneous nucleation. Thus, the initial increasing part the Eα dependence (Figure 6) is consistent with homogeneous nucleation. The decreasing part of the Eα versus α dependence in Figure 6 is likely due to a change in the mechanism to heterogeneous nucleation. Heterogeneous nucleation occurs when the nucleus forms on existing solid surfaces that may include the surface of already existing crystals. When nuclei form on the substrate surface, they are only partially surrounded by the parent phase. As a result, the surface area of the phase formed is significantly smaller than the one created in homogeneous nucleation, when the nucleus is surrounded entirely by the parent phase. Consequently, the free-energy barrier to heterogeneous nucleation, ΔG*het, is smaller than for

Figure 6. Effective activation energy versus conversion of crystallization of unsaturated solutions of potassium nitrate and ammonium perchlorate.

unsaturated solution. The Eα values are negative as has been described previously in the theoretical background section. It is seen that both unsaturated solutions demonstrate similar Eα dependencies that include a maximum around 30% conversion. The appearance of the maximum at approximately the same conversion for two different salts is likely accidental. As 5707


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The Journal of Physical Chemistry B the homogeneous, ΔG*, by a geometric factor f(Θ), which is smaller than 1 and depends on the contact angle, Θ,1 that is ΔG*het = ΔG*f (Θ)

(17)

A change in the crystallization mechanism from homogeneous to heterogeneous nucleation would cause a change in the free-energy barrier and, thus, in the temperature dependence of the crystallization rate. The change would reveal itself as a break point in the Arrhenius plot and, thus, in the effective activation energy, as shown in Figures 6 and 7. To test this hypothesis, we have studied crystallization of saturated solutions, that is, solutions that always contain the crystalline phase and, therefore, are likely to crystallize via heterogeneous nucleation. The respective DSC curves are shown in Figure 8. The Figure 9. Effective activation energy versus conversion of crystallization of saturated solutions of potassium nitrate and ammonium perchlorate.

negative numbers) than the respective values for the unsaturated solutions (Figure 6). The meaning of this change is readily understood from eq 11 and Figure 4. The term that affects the effective activation energy most is the supercooling, ΔT. The smaller this term is the larger the negative value of Eα. In other words, the Eα dependencies determined for the saturated solutions suggest that their crystallization takes place under significantly smaller supercoolings than in the case of the unsaturated solutions. Again, this result is consistent with the mechanism of heterogeneous nucleation that occurs easier and requires smaller supercoolings than homogeneous nucleation.

5. CONCLUSIONS The kinetics of the solution crystallization can be efficiently monitored by DSC. The classical nucleation model that describes the crystallization rate as a function of supersaturation has been extended to parametrize the rate as a function of supercooling. The extended model has been used to derive the temperature dependence of the effective activation energy of the process. This dependence is demonstrated to be helpful in understanding the Eα dependencies obtained when analyzing the crystallization data by means of an isoconversional method. The Eα dependencies have been obtained experimentally for crystallization of ammonium perchlorate and potassium nitrate solutions. It has been found that the extended nucleation model satisfactorily predicts the experimentally observed trends in Eα dependencies as long as there is no change in the crystallization mechanism. The saturated solutions have not revealed any remarkable changes in the mechanism throughout nearly entire process. However, in the unsaturated solutions the mechanism have changed markedly at early stages of crystallization. The change has been linked to the transition between homogeneous and heterogeneous nucleation. Overall, the results obtained indicate that DSC in combination with isoconversional analysis can be used to obtain important insights into the mechanism and kinetics of the solution crystallization.

Figure 8. DSC curves of the crystallization of saturated ammonium perchlorate and potassium nitrate on cooling at 2 °C min−1.

saturated solutions can be seen to crystallize at a higher temperature than the unsaturated solutions (Figure 5). For example, the onset of the unsaturated solution crystallization temperature for ammonium perchlorate is ∼5 °C, and for the saturated solution, it is ∼20 °C. Such a shift in crystallization temperature arises from a decrease in the energy barrier to nucleation (eq 17). Figure 9 is the Eα versus α graph of the saturated salt solutions. The Eα for NH4ClO4 is monotonically increasing throughout the whole process of crystallization. For KNO3, Eα increases monotonically up to about 65% conversion, after which it shows a minor decrease. Note that the Eα dependence for unsaturated KNO3 solution also shows some local minor peak around the same conversion (Figure 6). The coincidence of these two features appears to indicate that crystallization of a KNO3 solution is likely to undergo yet another mechanistic change in the final stages of the process. Most importantly, comparison of the dependencies obtained for saturated solutions (Figure 9) with those for the unsaturated solutions (Figure 6) reveals that the maximum in Eα observed around 30% conversion has disappeared. This is obviously consistent with our hypothesis that the breakpoint in the Eα and Arrhenius plots is associated with a change in the crystallization mechanism from homogeneous to heterogeneous nucleation. Another noticeable difference is that the Eα for the saturated solutions (Figure 9) have become markedly smaller (larger

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 205-975-9410. Notes

The authors declare no competing financial interest. 5708


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(22) Gesti, S.; Zanetti, M.; Lazzari, M.; Franco, L.; Puiggali, J. Study of Clay Nanocomposites of the Biodegradable Polyhexamethylene Succinate. Application of Isoconversional Analysis to Nonisothermal Crystallization. J. Polym. Sci., Part B: Polym. Phys. 2008, 46, 2234− 2248. (23) Papageorgiou, G. Z.; Achilias, D. S.; Bikiaris, D. N. Crystallization Kinetics and Melting Behaviour of the Novel Biodegradable Polyesters Poly(propylene azelate) and Poly(propylene sebacate). Macromol. Chem. Phys. 2009, 210, 90−107. (24) Papageorgiou, D. G.; Papageorgiou, G. Z.; Zhuravlev, E.; Bikiaris, D.; Schick, C.; Chrissafis, K. Competitive Crystallization of a Propylene/Ethylene Random Copolymer Filled with a β-Nucleating Agent and Multi-Walled Carbon Nanotubes. Conventional and Ultrafast DSC Study. J. Phys. Chem. B 2013, 117, 14875−14884. (25) Kalkar, A. K.; Deshpande, V. D.; Vatsaraj, B. S. Isoconversional Kinetic Analysis of DSC Data on Nonisothermal Crystallization: Estimation of Hoffman-Lauritzen Parameters and Thermal Transitions in PET/MMT Nanocomposites. Polymer 2014, 55, 6948−6959. (26) Tsanaktsis, V.; Bikiaris, D. N.; Guigo, N.; Exarhopoulos, S.; Papageorgiou, D. G.; Sbirrazzuoli, N.; Papageorgiou, G. Z. Synthesis, Properties and Thermal Behavior of Poly(decylene-2,5-furanoate): a Biobased Polyester from 2,5-furan Dicarboxylic Acid. RSC Adv. 2015, 5, 74592−74604. (27) Vyazovkin, S.; Sbirrazzuoli, N. Isoconversional Approach to Evaluating the Hoffman-Lauritzen Parameters (U* and Kg) from the Overall Rates of Nonisothermal Crystallization. Macromol. Rapid Commun. 2004, 25, 733−738. (28) Hoffman, J. D.; Lauritzen, J. I., Jr. Crystallization of Bulk Polymers with Chain Folding: Theory of Growth of Lamellar Spherulites. J. Res. Natl. Bur. Stand., Sect. A 1961, 65A, 297−336. (29) Hoffman, J. D.; Davis, G. T.; Lauritzen, J. I., Jr. In Treatise on Solid State Chemistry; Hannay, N. B., Ed.; Plenum: New York, 1976; Vol. 3, p 497. (30) Vyazovkin, S.; Dranca, I. Isoconversional Analysis of Combined Melt and Glass Crystallization Data. Macromol. Chem. Phys. 2006, 207, 20−25. (31) Bosq, N.; Guigo, N.; Zhuravlev, E.; Sbirrazzuoli, N. Nonisothermal Crystallization of Polytetrafluoroethylene in a Wide Range of Cooling Rates. J. Phys. Chem. B 2013, 117, 3407−3415. (32) Codou, A.; Guigo, N.; van Berkel, J.; de Jong, E.; Sbirrazzuoli, N. Non-Isothermal Crystallization Kinetics of Biobased Poly(ethylene 2,5-furandicarboxylate) Synthesized via the Direct Esterification Process. Macromol. Chem. Phys. 2014, 215, 2065−2074. (33) Bosq, N.; Guigo, N.; Persello, J.; Sbirrazzuoli, N. Melt and Glass Crystallization of PDMS and PDMS Silica Nanocomposites. Phys. Chem. Chem. Phys. 2014, 16, 7830−7840.

ACKNOWLEDGMENTS We gratefully acknowledge Mettler-Toledo for the donation of the DSC 823e instrument.

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