pubs.acs.org/Langmuir © 2009 American Chemical Society
Evidence of Critical Cooling Rates in the Nonisothermal Crystallization of Triacylglycerols: A Case for the Existence and Selection of Growth Modes of a Lipid Crystal Network Laziz Bouzidi and Suresh S. Narine* Trent University Biomaterials Research Program, Departments of Physics and Astronomy and Chemistry, Trent University, 1600 West Bank Drive, Peterborough, Ontario K9J 7B8, Canada Received September 10, 2009. Revised Manuscript Received October 11, 2009 The isoconversional method, a model-free analysis of the kinetics of liquid-solid transformations, was used to determine the effective activation energy of the nonisothermal crystallization of melts of pure and complex systems of triacylglycerols (TAGs). The method was applied to data from differential scanning calorimetry (DSC) measurements of the heat of crystallization of purified 1,3-dilauroyl-2-stearoyl-sn-glycerol (LSL) and commercially available cocoa butter melts. The method conclusively demonstrated the existence of specific growth modes and critical rates of cooling at specific degrees of conversion. The existence of critical rates suggests that the crystallization mechanism is composed of growth modes that can be effectively treated as mutually exclusive, each being predominant for one range of cooling rates and extent of conversion. Importantly, the data suggests that knowledge of the critical cooling rates at specific rates of conversion can be exploited to select preferred growth modes for lipid networks, with concomitant benefits of structural organization and resultant physical functionality. Differences in transport phenomena induced by different cooling rates suggest the existence of thresholds for particular growth mechanisms and help to explain the overall complexity of lipid crystallization. The results of this model-free analysis may be attributed to the relative importance of nucleation and growth at different stages of crystallization. A mechanistic explanation based on the competing effects of the thermodynamic driving force and limiting heat and transport phenomena is provided to explain the observed behavior. This work, furthermore, offers satisfactory explanations for the noted effect of cooling-rate-induced changes in the physical functionality of lipid networks.
Introduction Several theories and models with their underlying assumptions and limitations have been proposed to explain and examine the kinetics of liquid-solid transformations.1 The rate of phasechange reactions is typically given as a function of the temperature, T, and the extent of reactant conversion, R, as follows dR ¼ kðTÞ f ðRÞ dt
ð1Þ
where t represents time, k(T) is the temperature-dependent rate constant, and f (R) is a function called the reaction model. An implicit assumption in eq 1 is that the rate constant k(T) can be separated from the reaction model. The use and the physical interpretation of the Arrhenius equation (eq 2) for the description of the temperature dependence of the rate constant are supported by a good theoretical basis for most reactions.2,3 kðTÞ ¼ Ae -E=RT
ð2Þ
where A and E are the Arrhenius parameters (the pre-exponential factor and the activation energy, respectively) and R is the gas *Corresponding author. Tel: 1-705-748-1011. Fax: 1-705-748-1652. E-mail:
[email protected]. (1) Brown, M. E.; Dollimore, D.; Galwey, A. K. In Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Eds.; Elsevier: Amsterdam, 1980; Vol. 22, p 340. (2) Galwey, A. K.; Brown, M. E. Proc. R. Soc. London, Ser. A 1995, 450, 501– 512. (3) Galwey, A. K.; Brown, M. E. Thermochim. Acta 2002, 386, 91–98.
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constant. Although the temperature dependence of the overall rate can rarely be fit by a single Arrhenius equation, it holds well for a narrow temperature interval where one can estimate the effective value of the activation energy E. f(R), A, and E are the socalled kinetic “triplet”. The rate law for an elementary solid-state transformation reaction could depend on physical mechanisms such as the rate of nuclei formation, interface advance, diffusion, and/or the geometrical shape of solid particles. Mathematical forms (rate expressions) based on certain mechanistic assumptions have been proposed to model the transformations.1,4-6 The kinetics of crystallization and other transformations in lipidic materials are conventionally determined by fitting the kinetic curve R obtained at a single temperature or single processing rate7,8 (i.e., employing typical model-fitting approaches). The most popular rate functions used for the description of the crystallization of lipids from the melt are more or less sophisticated developments of the Avrami model and its early variants.9-13 (4) Jacobs, P. W. M.; Tompkins, F. C. In Chemistry of the Solid State; We, G., Ed.; Academic Press: New York, 1955; pp 184-212. (5) Galwey, A. K.; Brown, M. E. Thermal Decomposition of Ionic Solids: Chemical Properties and Reactivities of Ionic Crystalline Phases; Elsevier: Amsterdam, 1999. (6) Khawam, A.; Flanagan, D. R. J. Phys. Chem. B 2006, 110, 17315–17328. (7) Foubert, I.; Dewettinck, K.; Vanrolleghem, P. A. Trends Food Sci. Technol. 2003, 14, 79–92. (8) Himawan, C.; Starov, V. M.; Stapley, A. G. F. Adv. Colloid Interface Sci. 2006, 122, 3–33. (9) Kolmogoroff, A. N. Izv. Akad. Nauk SSSR, Ser. Math. 1937, 1, 335–359. (10) Johnson, W. A.; Mehl, R. F. Trans. Am. Inst. Min. Metall. Eng. 1939, 135, 416–442. (11) Avrami, M. J. Chem. Phys. 1939, 7, 1103–1112. (12) Avrami, M. J. Chem. Phys. 1940, 8, 212–224. (13) Avrami, M. J. Chem. Phys. 1941, 9, 177–184.
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They are principally based on the kinetics of nucleation and growth. In most practical situations, it is possible to assume an Arrhenius temperature dependence for both the nucleation rate N and the growth rate G.14,15 -EN N ¼ N0 exp kB T
ð3Þ
-EG G ¼ G0 exp kB T
ð4Þ
and
where EN and EG are the activation energies for nucleation and growth, respectively, and kB is the Boltzmann constant. However, despite their popularity, model-fitting procedures are very limited and do not take into account the complexity of the transformations. This method has been known to produce significantly differing kinetic triplets that are capable of satisfactorily reproducing the same experimental data, giving rise to different predictions.16-19 The results of the project initiated by the International Confederation for Thermal Analysis and Calorimetry, ICTAC,20,21 specially convened to evaluate the computational methods and forecast the tendencies for the future development of solid-state kinetics, have shown that single curve analysis methods are very limited and work poorly. ICTAC went as far as to recommend that single-rate methods not be used or published and urged the use of kinetic curves obtained via multiple heating/cooling programs for reliable kinetic evaluations. Narine et al.22 suggested the utilization of a modified Avrami treatment of the liquid-solid transformation of complex melts of TAGs that is based on the assumption that the kinetic triplet can change drastically over the course of the crystallization event. The mechanistic causes were rooted, the authors proposed, in constantly changing rates and the ease of nucleation and growth as a function of transport phenomena (both mass and heat). This treatment results in excellent fits to lipid crystallization kinetics data. Model-free approaches (also called isoconversional methods) (i.e., without explicitly assuming any reaction model) are popular and widely used; for example, see recent reviews of the thermal analysis literature.18 These methods that use multiple heating/ cooling programs have been particularly successfully used to study the kinetics of various processes that occur over a wide range of materials such as polymers,23,24 pharmaceuticals,25 and
(14) Frade, J. R. J. Am. Ceram. Soc. 1998, 81, 2654–2660. (15) Christian, J. W. The Theory of Transformations in Metals and Alloys: An Advanced Textbook in Physical Metallurgy, 3rd ed.; Pergamon Press: Oxford, U.K., 2002. (16) Vyazovkin, S.; Wight, C. A. Int. Rev. Phys. Chem. 1998, 17, 407–433. (17) Vyazovkin, S.; Wight, C. A. Thermochim. Acta 1999, 341, 53–68. (18) Vyazovkin, S. Anal. Chem. 2006, 78, 3875–3886. (19) Maciejewski, M. Thermochim. Acta 2000, 355, 145–154. (20) Brown, M. E.; Maciejewski, M.; Vyazovkin, S.; Nomen, R.; Sempere, J.; Burnham, A.; Opfermann, J.; Strey, R.; Anderson, H. L.; Kemmler, A.; Keuleers, R.; Janssens, J.; Desseyn, H. O.; Li, C. R.; Tang, T. B.; Roduit, B.; Malek, J.; Mitsuhashi, T. Thermochim. Acta 2000, 355, 125–143. (21) Burnham, A. K. Thermochim. Acta 2000, 355, 165–170. (22) Narine, S. S.; Humphrey, K. L.; Bouzidi, L. J. Am. Oil Chem. Soc. 2006, 83 (), 913–921. (23) Budrugeac, P. Polym. Degrad. Stab. 2005, 89, 265–273. (24) Vyazovkin, S.; Sbirrazzuoli, N. Macromol. Rapid Commun. 2006, 27, 1515– 1532. (25) Khawam, A.; Flanagan, D. R. J. Pharm. Sci. 2006, 95, 472–498. (26) Starink, M. J. Int. Mater. Rev. 2004, 49, 191–226. (27) Starink, M. J. J. Mater. Sci. 2007, 42, 483–489. (28) Smith, K. W.; Cain, F. W.; Talbot, G. J. Agric. Food Chem. 2005, 53, 3031– 3040.
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alloys.26,27 Only one study of a lipid system using isoconversional methods has been found in the literature.28 The basic assumption of the isoconversional methods is that a single equation (eq 1) is applicable only to a single extent of conversion and the (limited) temperature region related to this conversion. Isoconversional methods can detect any variation of activation energy as the reaction proceeds and provide information on possible changes in the reaction mechanism at different stages of conversion.29-31 The most popular are the methods devised for linear heating rates.32 -34 For processes that occur on cooling, such as the crystallization of lipid melts, these methods cannot be directly applied. In the method of Kissinger,34 for example, the activation energy E is determined from eq 5
β d ln T 2 M d T1 P
¼ -
E R
ð5Þ
where R is the gas constant and TP is the temperature corresponding to the maximum in a differential kinetic curve at a given heating rate dT ð6Þ β ¼ dt Equation 5 is then solved through linear regression using a number of heating rates, βi ln
!
βi TM, i
2
¼ const -
E RTP, i
ð7Þ
Kissinger’s method is a very popular procedure for kinetic evaluations of DSC and differential thermal analysis (DTA) data where the values of TP are peak temperatures. On cooling, the temperature decreases with time, giving rise to negative values of β in eq 6 that obviously cannot be substituted into eqs 5 and 7. Dropping the negative sign for β has been demonstrated35 to be a mathematically invalid procedure that generally makes these equations inapplicable to the processes that occur on cooling and may result in erroneous values of the activation energy. The differential method of Friedman36 and the more recent integral method of Vyazovkin37 are, however, alternative methods that can be used to obtain reliable values of the effective activation energy. A detailed analysis of the various isoconversional methods (i.e., the isoconversional differential and integral methods) for the determination of the activation energy has been reported in the literature.38-40 The development of a lipid crystal network can be traced as it develops from the melt (Narine and Marangoni, 2004).64 The structural hierarchy within a lipid is supported by microscopy techniques. The crystallization starts with the formation of nuclei followed by the growth, which is probably controlled by molecular (29) Vyazovkin, S.; Linert, W. Int. J. Chem. Kinet. 1995, 27, 597–604. (30) Vyazovkin, S. Int. J. Chem. Kinet. 1996, 28, 95–101. (31) Flammersheim, H. J.; Opfermann, J. Thermochim. Acta 1999, 337, 141–148. (32) Flynn, J. H.; Wall, L. A. J. Res. Natl. Bur. Stand. (U.S.) 1966, 70A, 487–523. (33) Ozawa, T. Bull. Chem. Soc. Jpn. 1965, 38, 1881–1886. (34) Kissinger, H. E. Anal. Chem. 1957, 29, 1702–1706. (35) Vyazovkin, S. Macromol. Rapid Commun. 2002, 23, 771–775. (36) Friedman, H. L. J. Polym. Sci. 1964, 6C, 183-195. (37) Vyazovkin, S. J. Therm. Anal. 1997, 49, 1493–1499. (38) Budrugeac, P.; Homentcovschi, D.; Segal, E. J. Therm. Anal. Cal. 2000, 63, 457–463. (39) Budrugeac, P.; Homentcovschi, D.; Segal, E. J. Therm. Anal. 2001, 66, 557– 565. (40) Budrugeac, P.; Segal, E. Int. J. Chem. Kinet. 2004, 36, 87–93.
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dynamics and kinetics, of the nuclei into nanoplatelets that aggregate via van der Waals interactions to form larger structures with several intertwined crystallites. These microstructural elements (∼1-5 μm) form the smallest repeating structure on a length scale visible under a light microscope. The microstructural elements eventually grow larger through further crystallization and continuing to aggregate, leading to intermediary-sized clusters that further aggregate to form larger clusters. The aggregation process is most probably controlled by mass- and heat-transfer limitations. The larger clusters, called microstructures, pack closely to form the fat crystal network. The manipulation of the rate of cooling during crystallization to achieve structural variations in crystallized lipids, which in turn delivers the desired physical functionality, has been proposed and demonstrated by a number of researchers.41-43 Indeed, our group has proposed that the crystallization of a particular lipid network can proceed along allowable growth modes defined by the rate of cooling and the extent of crystallization, these being kinetic effects limited by heat- and mass-transfer considerations (e.g. refs 22 and 43). We have also proposed that a knowledge of these growth modes and the limits to which cooling rates and extent of conversion is effective (or ranges within which they are ineffective) in promoting one mode of growth versus another can result in significant processing control over crystallized lipids so that their final physical functionality can be altered beneficially. It is well known that crystallized lipids can exist in local thermodynamic minima for extended periods of time, a situation without which the above arguments would not be valid. This study focuses on the crystallization kinetics under nonisothermal conditions of purified 1,3-dilauroyl-2-stearoyl-sn-glycerol (LSL) and a complex but extensively studied lipid sample, cocoa butter, and provides a description of the crystallization process. The kinetics of crystallization are examined using the isoconversional method so as to prove, without assuming a model-specific method of crystallization, that significantly different growth modes are possible for a crystallizing melt of TAGs, and indeed these growth modes are defined by clear differences in activation energy.
Materials and Methods Materials. The purified LLS TAG was synthesized in our laboratory according to known procedures.44 Its purity exceeded 98.0% as determined by gas chromatography. The cocoa butter sample was supplied by chocolaterie Bernard Callebaut (Calgary, AB Canada). Differential Scanning Calorimetry. DSC was used to carry out the nonisothermal crystallization experiments. A modulated DSC (model Q100, TA Instruments, New Castle, DE) with a refrigerated cooling system (RCS) was used in standard mode. The temperature and the cell constant of the DSC were calibrated with very pure indium (99.9999%, standard reference material 2232). The melting and crystallization behavior of samples of approximately 5 to 10 mg was monitored in hermetically sealed aluminum pans under 50 mL/min nitrogen flow. An empty aluminum pan was used as a reference. The data was analyzed using TA Universal Analysis software coupled with a method developed by our group.45 The sample was melted at 90 °C and held for 5 min to ensure complete destruction of all crystal memory and was then cooled at different rates (typically between (41) Narine, S. S.; Marangoni, A. G. Phys. Rev. E 1999, 59, 1908–1920. (42) Campos, R.; Narine, S. S.; Marangoni, A. G. Food Res. Int. 2002, 35, 971– 981. (43) Humphrey, K. L.; Narine, S. S. J. Am. Oil Chem. Soc. 2007, 84, 709–716. (44) Bentley, P. H.; McCrae, W. J. Org. Chem. 1970, 35, 2082–2083. (45) Bouzidi, L.; Boodhoo, M.; Humphrey, K. L.; Narine, S. S. Thermochim. Acta 2005, 439, 94–102.
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Figure 1. DSC cooling thermogram of a cocoa butter sample (cooling rate of 5 °C/min) with its calculated area-proportional baseline, B(T) = (1 - R) (a0 þ a1T) þ R[b0 þ b1T] where T is the temperature and R is the extent of conversion. Baseline 1: a0 = 1.2756, a1 = 0.001476, and R2 = 0.9977. Baseline 2: b0 = 1.3456, b1 = 0.004846, and R2 = 0.9899. 0.1 and 20 °C/min) down to a temperature of -50 °C where the crystallization was deemed completed. The number of cooling rates used in each case depended on the number of points required for linear fits. For example, 13 cooling rates (0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 1, 2, 3, 5, 7, 10, and 15 °C/min) have been used in the case of the LSL sample. Analysis Procedure. Using the heat-flow signal, S(t), at time t, the extent of conversion can be expressed as Rt ðSðtÞ -BðtÞÞ dt 0eRðtÞe1 ð8Þ R ¼ R ttsf ts ðSðtÞ -BðtÞÞ dt and the reaction rate can be expressed as dR ðSðtÞ -BðtÞÞ ¼ R tf dt ts ðSðtÞ -BðtÞÞ dt
ð9Þ
where ts is the time where the crystallization is just beginning and tf is the time where the crystallization is completed. B(t) is the socalled tangential area-proportional baseline that allows for the compensation of changes in the size of the heat capacity, Cp, of the reactant and product, and of changes in their temperature dependence.46 Special care has been taken in the determination of the baseline, which can significantly influence the determination of the kinetic parameters. It was calculated using appropriate tangents at the beginning and at the end of the measured DSC signal, as follows BðtÞ ¼ ð1 -RÞða0 þ a1 tÞ þ R½b0 þ b1 ðtf -tÞ
ð10Þ
where (a0 þ a1t) is the calculated tangent at the starting point and [b0 þ b1(tf - t)] is the calculated tangent at the ending point. B(t) was calculated iteratively using Universal Analysis software (Thermal Solutions TA Instruments, New Castle, DE) without user intervention. An example DSC signal with its calculated areaproportional baseline is presented in Figure 1 for the cocoa butter sample cooled at a rate of 5 °C/min. The Friedman approach has been used to evaluate the overall activation energy. The general kinetics equation, rewritten in logarithmic form, is given by dR ERi ln ¼ ln½Af ðRÞRi RTRi dt Ri
ð11Þ
(46) Hemminger, W. F.; Sarge, S. M. J. Therm. Anal. 1991, 37, 1455–1477.
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Figure 2. (a) Selected cooling DSC thermograms of the LSL sample; values of linear cooling rates (β, °C/min) are given on the right side above each curve. (b) Corresponding conversion, R, versus temperature, T, plots; the curves are shown with a decreasing cooling rate from 20 °C/min for the leftmost curve to 0.1 °C/min for the rightmost curve. (c) Corresponding plots of the logarithm of the reaction rate, ln(dR/dt), versus 1000/RT; cooling rates (°C/min) are given on the right side of the curves. The solid line is the isoconversional line at a conversion R of 0.99. where subscript R denotes the values related to a given extent of conversion, i is the ordinal number of the run carried out at cooling rate βi, and TRi the temperature, in Kelvin, at which the conversion was achieved. Therefore, ER/R is the slope of a straight line in the plot of the logarithm of conversion rate dR/dt over 1/T for a given R for the different cooling rates. The profile of the activation energy as a function of R is obtained by repeating the procedure at different values of R. In our case, the activation energy, ER, is directly obtained in kJ mol-1 by using the appropriate value of R. An averaged conversion of duplicates has been used, and the reported uncertainties in the activation energy are those directly obtained from the linear regression.
Results The reproducibility and hence the quality of the analysis was very good for all cooling rates. The calculated standard deviation of ln(dR/dt) for R between 0.01 and 0.99 was found to be less than 5% for all samples. All of the regression lines obtained by the application of the Friedman method were constructed by ensuring that they are statistically significantly different and nonparallel. Typical cooling DSC curves taken at different linear cooling rates and the corresponding plots of R versus T are illustrated in Figure 2a,b, respectively for the LSL sample. Note that all of the DSC cooling curves have a main exothermic asymmetrical peak. The dependence of the logarithm of the reaction rate, ln(dR/dt), versus 1/T for selected cooling rates is presented in Figure 2c. The R versus T plots demonstrate a monotonic increase with evident disruptions that mirror the differences in the thermograms and highlight the significant changes occurring in the crystallization kinetics. Table 1 lists the results of the linear regressions, and Figure 3a-c shows selected isoconversional ln(dR/dt)R versus 1000/RTR plots for the LSL sample. One can notice the obvious dramatic change in the slope and hence the very well defined critical rates where the change takes place (indicated by arrows in Figure 3a-c). The isoconversional plots yielded up to three 4314 DOI: 10.1021/la903420n
critical cooling rates. The largest critical rate in each plot, labeled βC3 (up triangles in Figure 3d), has been detected for all conversions, and it is the only one detected for conversions smaller than 3% and larger than 65%. The second and smallest critical rate in each plot, labeled βC1 (circles in Figure 3d), is detected for conversions of between 3 and 65%. The intermediary critical rate, labeled βC2, appeared for conversions of between 10 and 65%. Note that the values of the critical rates depend on the extent of conversion. βC1 varies steadily from 0.7 to 0.2 °C/min for R = 0.45 and then remains constant afterwards. βC3 varies from 0.7 to 3.0 °C/min for R = 0.05, remains constant in the interval R = 0.05-0.75, and then increases again. βC2 remains constant (0.7 °C/min) for the conversion interval where it appears. The critical rates delimit up to four intervals of cooling rates: (1) β > βC3, labeled range III. (2) β = βC2-βC3, labeled range II. (3) β = 0.1 °C/min-βC1, labeled range I. (4) β = βC1-βC2, labeled the intermediary range (Table 1). The activation energy as a function of R for the LSL sample is shown in Figure 4. Note that the data could not be exploited to obtain the activation energy when the conversion is greater than 95%. Range I (circles in Figure 4) displayed the highest activation energy, and the intermediary range of cooling rates (triangles in Figure 4) displayed the smallest. The variation of the activation energy versus R associated with range II (rectangles in Figure 4) is similar to that associated with range III of the cooling rates (lozenges in Figure 4) because both increase exponentially and plateau for R larger than 0.5, albeit the former has relatively higher values than the latter. Typical cooling DSC curves taken at different linear cooling rates and the corresponding plots of R versus T are illustrated in Figure 5a,b, respectively, for the cocoa butter sample. Note that all of the DSC cooling curves have a main exothermic asymmetrical peak and secondary well-resolved small peaks. The dependence of the logarithm of the reaction rate, ln(dR/dt), versus 1/T for selected cooling rates is presented in Figure 5c. The DSC cooling thermograms obtained with cooling rates smaller than Langmuir 2010, 26(6), 4311–4319
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Table 1. Results of the Linear Regressions of the Isoconversional ln(dr/dt)r versus 1000/RTr Segments in the Case of the LSL Samplea R
ER, range I
βC1
ER, intermediary range
βC2
ER, range II
0.01 -278 ( 84 0.90 0.02 -257 ( 36 0.90 0.03 -247 ( 39 0.80 0.04 -253 ( 53 0.80 0.05 -204 ( 25 0.70 0.07 -190 ( 37 0.60 0.10 -170 ( 39 0.60 1.0 -900 ( 15 0.15 -86 ( 36 0.40 0.7 -855 ( 28 0.20 -74 ( 11 0.40 -3761 ( 387 0.7 -769 ( 30 0.25 -71 ( 6 0.30 -3618 ( 268 0.7 -713 ( 46 0.30 -87 ( 31 0.30 -3072 ( 127 0.7 -680 ( 57 0.35 -40 ( 17 0.25 -2462 ( 114 0.7 -662 ( 53 0.40 -16 ( 25 0.25 -2322 ( 91 0.7 -627 ( 75 0.45 30 ( 13 0.25 -2074 ( 71 0.7 -606 ( 71 0.50 41 ( 8 0.25 -1794 ( 181 0.7 -526 ( 38 0.20 -1604 ( 177 0.7 -493 ( 50 0.55 22 ( 19 0.60 -1509 ( 61 0.7 -499 ( 65 0.65 -1270 ( 52 0.7 -474 ( 8 0.70 -560 ( 36 0.75 -504 ( 36 0.80 -478 ( 13 0.85 -475 ( 13 0.88 -445 ( 50 a R, conversion; ER, activation energy (kJ/mol); βC1, βC2, and βC3, critical cooling rates (°C/min).
5 °C/min present extra structure made of peaks and shoulders at the high-temperature end of the main peak. The R versus T plots demonstrate a monotonic increase with evident disruptions that mirror the differences in the thermograms and highlight the significant changes occurring in crystallization kinetics. Table 2 lists the results of the linear regressions, and Figure 6a-c shows selected isoconversional ln(dR/dt)R versus 1000/RTR plots in the case of the cocoa butter sample. Figure 6d shows the variation of the critical cooling rates with the extent of conversion. The isoconversional plots yielded two critical cooling rates (successively labeled βC1 and βC2), which defined three ranges of cooling rates. As can be noted, the span of the ranges also depends on the extent of conversion. The variation of the critical rates with the extent of conversion (Figure 6d) is particularly interesting because the β versus R curves almost mirror each other. Both were constant for conversions between 0.10 and 0.60. Note, however, that whereas βC1 increased monotonically for conversions larger than 0.80, βC2 jumped to 7 °C/min and remained constant. The activation energy associated with the different ranges of cooling rates as a function of R for the cocoa butter sample is shown in Figure 7. The variation of the activation energy with the progress of crystallization is complex. For cooling-rate ranges I and III (Figure 7a,b, respectively), ER grows exponentially. For range I and for conversion from 0.25 to 0.95, the small cooling rates appear to induce the same crystallization mechanism that led to a fairly small absolute value of ER, which can be related to what is usually calculated using basic assumptions for the nucleation mechanism.47,48 The behavior was quite different for smaller conversions for which the activation energy started with very small values when the conversion was less than 0.05 (approximately -1100 kJ mol-1) and then increased exponentially with increasing conversion. Similarly, the activation energy associated with range III (i.e., the large cooling rates) increased exponentially from a fairly small value of approximately -900 kJ mol-1 to a constant value of -280 kJ mol-1. (47) Cerdeira, M.; Candal, R. J.; Herrera, M. L. J. Food Sci. 2004, 69, R185– R191. (48) Marangoni, A. G.; Aurand, T. C.; Martini, S.; Ollivon, M. Cryst. Growth Des. 2006, 6, 1199–1205.
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βC3
ER, range III
2.0 2.5 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 5.0
-769 ( 23 -745 ( 70 -545 ( 36 -514 ( 7 -462 ( 30 -467 ( 11 -361 ( 17 -303 ( 15 -272 ( 19 -232 ( 15 -204 ( 18 -180 ( 15 -191 ( 17 -142 ( 11 -162 ( 6 -142 ( 11 -98 ( 9 -82 ( 11 -75 ( 9 -79 ( 8 -63 ( 6 -54 ( 6 -43 ( 32
Discussion The critical rates draw up the boundaries of ranges of cooling rates where the activation energies are different, indicating the presence of independent crystallization modes for the two successive ranges of rates. This is strong evidence that the cooling rate substantially affects the crystallization mechanisms and effects fundamental change in the growth mode of the resulting lipid network. This situation is not dissimilar to critical rates evidenced in the crystallization of metallic glasses.49 This is evidence in support of the theory that lipids crystallize and grow into networks via very specific growth modes and that the crystallization of a lipid network system could therefore be thought of as a succession of different crystallization events occurring in steps as suggested by Narine et al.22 It is worth noting that the variation of the activation energy of the pure LSL TAG is as complex as that of the cocoa butter sample that is composed of a complex mixture of TAGs, suggesting that even pure samples are subjected to similar kinetic constraints, probably because of the crystalline structure and polymorphism displayed by the lipid systems. As can be seen, the activation energy calculated for the LSL sample and therefore the crystallization mechanism associated with it depend strongly not only on the cooling rate but also on the amount of crystallized material (conversion, R). Crystallization proceeds via distinct mechanisms depending on the cooling rate. This is as yet the strongest evidence provided for the existence of different growth modes of the lipid crystal network. For small conversions (R < 5%), there are two mechanisms available: one with a relatively small absolute value activation energy “easy mechanism” associated with range I of the cooling rates (circles in Figure 4) and another with a larger absolute value activation energy associated with range III of the cooling rates (lozenges in Figure 4). For larger conversions (10% < R < 65%), range II (rectangles in Figure 4) evidenced a difficult mechanism, although with an absolute value activation energy much lower than that of the intermediary stage (triangles in Figure 4). For large conversions (R > 65%), only two mechanisms are available again. Range III represents the main crystallization mechanism as it is (49) Yuan, Z.-Z.; Chen, X.-D.; Wang, B.-X.; Wang, Y.-J. J. Alloys Comp. 2006, 407, 163–169.
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Figure 3. Selected isoconversional ln(dR/dt)Rversus 1000/RTR plots, where TR is the temperature at the given conversion. (a) R = 0.01, (b) R = 0.40, (c) R = 0.85, and (d) variation of the critical rates βC1 (b), βC2 (9), and βC3 (2) with the extent of conversion, R, for the LSL sample.
exp(-R/R0)). Note that the R0 value (a characteristic conversion) was 0.12 ( 0.01 for both ranges, indicating that the main crystallization mechanism took over in the early stages of crystallization; 85% of the increase has been achieved for R = 0.25. For conversions from 0.25 to 0.95, the small cooling rates appear to induce crystallization mechanisms characterized by fairly small absolute values of ER that can be related to what is usually calculated using basic assumptions for a nucleation mechanism.47,48 This can explain, as pointed out by Narine et al.,22 why a majority of authors publishing lipid crystallization studies utilizing a model-specific approach to explaining the kinetics of crystallization find it easier to work with the “main” segment of the crystallization event, usually spanning from 0.25 to 0.80. The dramatic effect of the cooling rate has been reported by Humphrey and Narine43 on a similar complex fat system (a cocoa butter alternative, CBA). They have reported significant differences in crystallization for cooling rates below and above 5 °C/min and have attributed it to differentiated effective mass-transfer limitations due to viscosity for the different fraction constituting the CBA. The continuous change in ER for both samples could be understood as a continuous balance change in the relative importance of nucleation and growth mechanisms, one continuously taking over the other. We define the overall activation energy as50 Figure 4. Activation energy (ER, kJ/mol) as a function of conversion, R, for the LSL sample (b, range I; 9, range II; (, range III; and 2, intermediary range of cooling rates). The lines are guides for the eye.
present during the whole crystallization process. The absolute value activation energy of the narrow intermediary range decreased almost linearly with increasing conversion, suggesting that the mechanism that it represents is taken over by the other mechanisms with which it coexists. For the cocoa butter sample, for ranges I and III of the cooling rates (Figure 7a,b, respectively), the activation energy started with very high negative values and then increased exponentially with increasing conversion following the function ER = E 0R(1 - A 4316 DOI: 10.1021/la903420n
E ¼
EN þ mEG mþ1
The overall rate constant is given by σN0 G0 1=ð1 þ mÞ -E k ¼ exp kB T mþ1 -E k ¼ k0 exp kB T
ð12Þ
ð13Þ ð14Þ
In the limit where EN . EG, nucleation takes place before growth, and the site-saturation approximation where nucleation is completed prior to crystal growth is the most appropriate Langmuir 2010, 26(6), 4311–4319
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Figure 5. (a) Selected cooling DSC thermograms of the cocoa butter sample; values of the linear cooling rates (β, °C/min) are given on the right side above each curve. (b) Corresponding conversion, R, versus temperature, T, plots; the curves are shown with a decreasing cooling rate from 20 °C/min for the leftmost curve to 0.2 °C/min for the rightmost curve. (c) Corresponding plots of the logarithm of the reaction rate, ln(dR/dt), versus 1000/RT; cooling rates (°C/min) are given on the right side of the curves. The solid line is the isoconversional line at a conversion R of 0.99. Table 2. Results of the Linear Regressions of the Isoconversional ln(dr/dt)r versus 1000/RTr Segments in the Case of the Cocoa Butter Samplea R
ER, range I
βC1
ER, range II
βC2
ER, range III
0.01 -969 ( 118 4.0 49 ( 102 9.0 -731 ( 30 0.02 -928 ( 63 4.0 1 ( 34 9.0 -940 ( 144 0.03 -1027 ( 46 4.0 -166 ( 23 9.0 -819 ( 109 0.04 -1053 ( 53 3.0 -292 ( 31 7.0 -648 ( 154 0.05 -1103 ( 45 3.0 -368 ( 14 7.0 -725 ( 54 0.07 -947 ( 23 3.0 7.0 -606 ( 91 0.10 -920 ( 190 3.0 7.0 -670 ( 66 0.13 -679 ( 82 2.0 -2103 ( 276 7.0 -512 ( 65 5.0 -584 ( 45 0.15 -497 ( 148 1.0 -2230 ( 114 0.20 -445 ( 76 0.8 -2420 ( 71 5.0 -418 ( 25 0.25 -344 ( 83 0.8 -2526 ( 166 5.0 -373 ( 15 0.30 -304 ( 77 0.7 -2340 ( 110 5.0 -334 ( 19 0.35 -278 ( 71 0.6 -2162 ( 123 5.0 -314 ( 10 0.40 -267 ( 48 0.6 -1935 ( 104 5.0 -303 ( 7 0.45 -225 ( 36 0.8 -1541 ( 114 5.0 -298 ( 8 0.50 -191 ( 27 0.8 -1501 ( 100 5.0 -274 ( 11 -1415 ( 141 5.0 -294 ( 22 0.55 -164 ( 21 0.8 0.60 -145 ( 18 0.9 -1469 ( 109 7.0 -292 ( 20 0.65 -134 ( 22 0.9 -1430 ( 84 7.0 -296 ( 19 0.70 -131 ( 15 1.5 -1406 ( 159 7.0 0.75 -130 ( 10 2.0 0.80 -121 ( 7 3.0 0.85 -129 ( 10 3.0 0.90 -131 ( 18 3.0 0.95 -124 ( 18 5.0 a R, conversion; ER, activation energy (kJ/mol); βC1 and βC1, critical cooling rates (°C/min).
description.51,52,14 Conversely, when EG . EN, either heterogeneous nucleation dominates and the site-saturation approximation is also the appropriate description or crystallization is driven by epitaxial growth. For the first and third sets of cooling rates in the LSL TAG case and for each crystallization rate, it is likely that the nucleationand-growth kinetics is manifested in the early stages of crystallization. The absolute value of the activation energy initially being Langmuir 2010, 26(6), 4311–4319
larger and continuously decreasing along with the development of crystallization suggests that the growth mechanism changes from initial interface-controlled growth with increasing nucleation rate to diffusion-controlled growth with zero nucleation rate. The normal grain growthlike mode dominates in the advanced stages of the transformation. The smaller absolute values of ER for small rates is understandable because at equal conversions the material has more time to crystallize for the decreasing cooling rate. It seems that in the intermediary stage, at given conversions, the cooling rate is large enough that the molecular mobility is hindered but not large enough to initiate thermally induced diffusion. The cooling-rate effect was found to be very sensitive to the amount of crystallized material, adding to the complexity of the crystallization process. The effect of solid material increase on the increase in the absolute value of the activation energy (more negative) could be explained by transport (energy and mass) arguments. The Fisher-Turnbull model53 can provide a mechanistic explanation of what was observed and justify the claim of selective growth modes being available for lipids. In this model, the nucleation rate, J, as described by the Fisher-Turnbull equation, is J ¼
NkT -ΔGc =kT -ΔGd =kT e e h
ð15Þ
where ΔGc is the activation free energy required to develop stable nuclei; ΔGd is the activation free energy for the diffusion of the crystallizing molecules with the correct conformation to the growing surface; k is the Boltzmann constant; N is Avogadro’s number; h is Planck’s constant; and T is the absolute temperature. (50) (51) (52) (53)
Farjas, J.; Roura, P. Acta Mater. 2006, 54, 5573–5579. Cahn, J. W. Acta Metal. 1956, 4, 572–575. Henderson, D. W. J. Therm. Anal. 1979, 15, 325–331. Turnbull, D.; Fisher, J. C. J. Chem. Phys. 1949, 17, 71–73.
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Figure 6. Selected isoconversional ln(dR/dt)R versus 1000/RTR plots, where TR is the temperature at the given conversion. (a) R = 0.02, (b) R = 0.50, (c) R = 0.85, and (d) is the variation of critical rates βC1 (b) and βC2 (9) with the extent of conversion, R, for the cocoa butter sample.
The activation free energy of nucleation, ΔGc, for a nucleus assumed to be spherical, is given by the Gibbs-Thompson equation ΔGc ¼
16 3 TM 2 πσ 2 3 ðΔHÞ ðTM -TÞ2
ð16Þ
where σ is the surface free energy of the crystal/melt interface; ΔT = (T - TM) is the crystallization driving force of the polymorph that forms and is defined as the degree of undercooling below the melting point TM; and ΔH is the heat of fusion. During undercooling at temperatures below TM, the system attempts to achieve thermodynamic equilibrium through nucleation. Undercooling provides the thermodynamic driving force that brings the molecules into a liquid lamellar structure. When liquid monomers aggregate to form a critical size, stable, solid nuclei are formed. The heat of fusion and the surface energy remain constant if one assumes that the polymorph that crystallizes is the same, leading to ΔGc µ [1/(TM - T)2]. Supercooling is therefore very low and ΔGc is very large when T is close to TM, and crystallization is driven by the nucleation rate. The ΔGd parameter is complex, with a number of dependencies on temperature. It is often taken as the activation free energy of viscous flow,54 in which case it is given by ηV ð17Þ ΔGd ¼ kT ln hN where η is the absolute viscosity and V is the molar volume. 4318 DOI: 10.1021/la903420n
Equation 17 is derived under the assumption that a fraction of molecules are in the right conformation for incorporation into a nucleus and the temperature T is such that the internal energy kT is sufficient for the molecules to diffuse over to the growing surface. With the kinematic viscosity of the liquid medium modeled roughly as, we may write μ ¼ AeB=ðD þ kTÞ
ð18Þ
Viscosity takes the following form: η ¼
μ A B=ðD þ kTÞ ¼ e F F
ð19Þ
Because viscosity is related to temperature in an exponential decay, the viscosity increases significantly as the temperature decreases. This is amplified for increasing cooling rates. With increasing cooling rate, the rapid drop in temperature, with its related viscosity effects during crystallization, induces significant barriers to nucleation. The competing effects of the provision of increased thermodynamic driving force due to increased effective undercooling as the rate is increased and limiting transport phenomena as the viscosity is increased rapidly as a result of the increases in cooling rate determine the crystallization behavior. As the solid network is formed, there is an increase in the surface area to volume ratio: there is more heat to transfer away from the growing centers, and the system becomes more limiting (54) Wood, G. R.; Walton, A. G. J. Appl. Phys. 1970, 41, 3027–3036.
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in ER is consistent with the decrease in the value of ΔGc that occurs on cooling. Note that if a different polymorph is formed or the molecular species is varied then both the surface energy and the heat of fusion will change. For TAGs, the main diffusional barrier to nucleation is the molecular structure. The TAG has to be in the right conformation before it can be incorporated into a nucleus.56 Nucleation and growth rates are decisive in determining the polymorphic occurrence in TAGs. The empirical Ostwald rule of stages, which states that the thermodynamically less stable phase is always formed first, is explained by the competition to nucleate among the polymorphic forms. The ordering dynamics as well as the interfacial free energy of the crystals, σ, will influence the activation energy for nucleation. σ is normally smaller for the less stable forms: σ(R) < σ(β0 ) < σ(β).57,58 At the occurrence of one polymorph from another, the induction time-temperature curve shows an abrupt discontinuity accompanied by an increase in the crystal/melt interfacial free energy.59 The growth rate from melts can be controlled by either the attachment rate of growing units at the crystal surface (surface kinetics) or by the transport of mass to or heat from the growing surface.60 As the solid content increases, mass transport can reduce the overall growth rate if, for instance, a significant increase in viscosity occurs, leading to a decrease in diffusivity,61 and heat transfer may increase the interfacial temperature by a significant amount.62 On a microscopic level, the transport of a considerable quantity of latent heat released during crystallization away from the crystal surface into the bulk liquid may be more important than mass transport, as recently shown from computer simulations of heat transport in molecular systems.63
Conclusions
Figure 7. Activation energy ER (kJ/mol) as a function of conversion, R, for the cocoa butter sample associated with the (a) first range, (b) third range, and (c) intermediary range of cooling rates. The solid lines are exponential fits.
to mass transfer, restricting the movement of the centers of growth and hence highlighting the significance of the degree of conversion. The nature of the initial polymorph formed will also dictate the shape and size of the crystallites that make up the microstructural elements55 and can also result in a restriction of the movement of the crystallites. Once ΔGc drops below a certain value, the nucleation rate becomes controlled by the ΔGd exponential term. Because of the opposing effects of the ΔGc and ΔGd exponential terms, their product demonstrates a maximum at some temperature, Tmax (isothermal crystallization performed at this temperature would accomplish a maximum rate). In the region TM-Tmax, crystallization demonstrates a behavior that is characterized by negative values of the effective activation energy. As T departs from TM, E remains negative but increases toward zero. Below Tmax, one should observe a behavior that is described by positive values of the experimental activation energy. In our experiments, we predominantly observe the negative values of the effective activation energy. The continuous increase (55) Narine, S. S.; Marangoni, A. G. Phys. Rev. E 1999, 60, 6991–7000.
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We have used a model-free method to demonstrate conclusively the existence of specific growth modes and critical rates of cooling at specific degrees of conversion that code for these growth modes. The existence of critical rates suggests that the crystallization mechanism is composed of growth modes strongly linked to the cooling rate and degree of crystallization that can be effectively treated as mutually exclusive, with each being predominant for one range of cooling rates and extent of conversion. Differences in transport phenomena induced by different cooling rates suggest the existence of thresholds for particular growth mechanisms and help to explain the overall behavior of lipid crystallization. The Turnbull-Fisher model has been used to elucidate the relative importance of the mechanisms involved in different stages of crystallization on the basis of the competing effects of the thermodynamic driving force and the limiting heat and transport phenomena. Acknowledgment. The financial support of the NSERC, the Ontario Soybean Growers, Elevance Renewable Sciences, and Trent University is gratefully acknowledged. (56) Sato, K. In Crystallization and Polymorphism of Fats and Fatty Acids; Garti, N., Sato, K., Eds.; Marcel Dekker: New York, 1988; pp 254-259. (57) Aquilano, D.; Sgualdino, G. In Sato, K., Garti, N., Eds. Marcel Dekker: New York, 2001; p 1. (58) Takeuchi, M.; Ueno, S.; Sato, K. Food Res. Int. 2002, 35, 919–926. (59) Ng, W. L. J. Am. Oil Chem. Soc. 1990, 67, 879–882. (60) Mullin, J. W. Crystallisation, 3rd ed.; Butterworth-Heinemann: Oxford, U.K., 1993. (61) Ghotra, B. S.; Dyal, S. D.; Narine, S. S. Food Res. Int. 2002, 35, 1015–1048. (62) Kloek, W.; Walstra, P.; van Vliet, T. J. Am. Oil Chem. Soc. 2000, 77, 389– 398. (63) Los, J. H.; Matovic, M. J. Phys. Chem. B 2005, 109, 14632–14641. (64) Narine, S. S.; Marangoni, A. G. In Microstructure. In Fat Crystal Networks; Marangoni, A. G., Ed.; Marcel Dekker: New York, 2004; pp 179-255.
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