A Probabilistic Approach to Model the Nonisothermal Nucleation of

Department of Food Science, University of Guelph, Guelph, Ontario, Canada N1G2W1, Department of Nutrition and Food ... Note. This paper contains enhan...
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A Probabilistic Approach to Model the Nonisothermal Nucleation of Triacylglycerol Melts Alejandro G.

Marangoni,*,†,§

Thomas C.

Aurand,†

Silvana

Martini,‡

and Michel

Ollivon§

Department of Food Science, UniVersity of Guelph, Guelph, Ontario, Canada N1G2W1, Department of Nutrition and Food Sciences, Utah State UniVersity, Logan, Utah 84322, and UniVersite´ Paris-Sud, Centre d’Etudes Pharmaceutiques, Laboratoire de Physico-Chimie des Syste´ mes Polyphases, UMR CNRS 8612, Chaˆ tenay-Malabry Cedex, France

CRYSTAL GROWTH & DESIGN 2006 VOL. 6, NO. 5 1199-1205

ReceiVed NoVember 28, 2005; ReVised Manuscript ReceiVed March 10, 2006

W This paper contains enhanced objects available on the Internet at http://pubs.acs.org/crystal. ABSTRACT: Crystallization studies are usually performed under isothermal conditions. Kinetic parameters characterizing the isothermal nucleation and growth processes can be obtained using classical nucleation and growth models. However, crystallization regimes found in nature, as well as those used in food and pharmaceutical processing, are rarely isothermal. Focusing on the nucleation stage, the approach followed in this work was to define a new parameter to characterize the driving force of nucleation, the supercoolingtime exposure (β), which not only depends on the difference between the melting temperature (Tm) and the onset temperature of nucleation (Tc) ∆Tc but also on the induction time of nucleation, tc, and therefore the cooling rate (φ), namely, β ) (1/2)∆Tctc ) ∆Tc2/2φ. An exponential probability density function of the values of β was utilized to model changes in nucleation rate as a x

function of β in the form J/Jmax ) ke-k β. From this parametrization procedure, the energy of activation for the nucleation process in palm oil, milkfat, and other palm oil based fats could be estimated. Introduction The crystallization of multicomponent triacylglycerol mixtures, e.g., edible fats, is of paramount importance in the manufacture of products such as chocolate, ice cream, and butter.1 In the industrial manufacture of these materials, crystallization takes place under nonisothermal conditions, where temperature is changing as the material crystallizes in time. Because fat is a multicomponent mixture of structurally diverse molecules, its crystallization behavior and structure are extremely sensitive to heat and mass transfer conditions during the crystallization process.2-6 Temperature, cooling rate, and shear profoundly affect the crystallization, final structure, and macroscopic properties of the material, such as mechanical strength, flow behavior, and sensory texture.7-16 W Polarized light microscopy movies of the crystallization of one of the fat samples used in this study (IHPO) crystallized from the melt at W 0.5, W 1.0, and W 5.0 °C/min are available in AVI format. Of particular interest is the nucleation behavior of these systems, since important structural features are a direct consequence of nucleation behavior, such as crystallite number, size, and morphology, as well as the spatial distribution of mass.17 To complicate matters even further, no theoretical tools exist to model the nucleation behavior of these complex organic mixtures under nonisothermal conditions. As a matter of fact, nonisothermal nucleation models for open systems, where the time dependence of the supersaturation is not known, do not exist. Crystallization is a first-order phase transition from the liquid to the solid state. When the temperature of a multicomponent * To whom correspondence should be addressed. Correspondence should be addressed to Universite´ Paris-Sud XI, Laboratoire de Physico-Chimie des Syste´mes Polyphases, UMR CNRS 8612, 5 rue Jean Baptiste Clement, 92296 Chaˆtenay-Malabry Cedex, France. Phone: (33) 1-4683-5623. Fax: (33) 1-4683-5312. E-mail: [email protected]. † University of Guelph. ‡ Utah State University. § Universite ´ Paris-Sud.

system in the liquid state is lowered below that of the melting point of its highest melting component, the system enters a metastable region. In this region, the concentration of some of the higher melting components in the melt exceeds their solubility limit, i.e., the melt becomes supersaturated in certain components. By necessity, the supersaturated components must crystallize and fall out of solution until a new equilibrium is established. The driving force for such a process is the difference in chemical potential between the solid and liquid components (i.e., the supersaturation), which in turn is a function of the difference between the melting temperature of the supersaturated components (Tm) and the temperature at which the crystallization process takes place (Tset), namely, ∆T ) Tm - Tset. This term is often referred to as the supercooling or the degree of supercooling.18 To model nonisothermal nucleation processes, Kashchiev19 assumed a quasi-stationary nucleation rate to model the timedependent nucleation rate. This approach only applies to closed systems where the time dependence of the supersaturation is an explicitly known function, and certain stringent conditions are satisfied. This approach does not yield a model that can be used to characterize and/or predict the nucleation behavior of materials under realistic conditions. Similar theoretical approaches have been used in the study of the nucleation of supersaturated vapor droplets by Rybin,20 and attempts to extend the nucleation theorem to nonisothermal situations have been made,21 but their results are difficult to implement. In addition, we are dealing with multicomponent mixtures of triacylglycerol (TAG) molecules. To model crystallization directly from molecular compositional data, one would have to know the activity of each TAG molecule in TAG mixed crystals at different temperatures, for each of the different polymorphic forms present in fats. Moreover, the activity of each TAG in the melt would also have to be known. This has been partially attempted by Wesdorp et al.22 for equilibrium isothermal conditions for certain polymorphic forms. The approach has not been generalized to different polymorphs and definitely not

10.1021/cg050630i CCC: $33.50 © 2006 American Chemical Society Published on Web 04/14/2006

1200 Crystal Growth & Design, Vol. 6, No. 5, 2006

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Table 1. Fatty Acid Composition of the Samples Used in This Study fatty acids

milkfat

PHPO

palm oil

IHPO

PH (PO/PS)

C4:0 1.5 ( 0.1 0 0 0 0 C6:0 1.4 ( 0.1 0 0 0 0 C8:0 1.0 ( 0.1 0 0 0 0 C10:0 2.5 ( 0.1 0 0 0 0 C12:0 3.0 ( 0.1 0 0 0.3 ( 0.1 0.2 ( 0.1 C14:0 9.8 ( 0.2 0.9 ( 0.1 0.9 ( 0.1 0.9 ( 0.1 1.0 ( 0.1 C14:1 1.6 ( 0.1 0 0 0 0 C16:0 28.4 ( 0.7 42.0 ( 0.6 40.3 ( 0.5 39.3 ( 0.2 45.3 ( 0.8 C16:1 3.0 ( 0.1 0 0 0 0 C17:0 0.8 ( 0.1 0 0 0 0 C18:0 12.4 ( 0.1 9.9 ( 0.4 7.8 ( 0.7 19.4 ( 0.2 11.4 ( 0.2 C18:1 27.2 ( 0.4 42.6 ( 0.2 39.1 ( 0.1 30.9 ( 0.2 35.3 ( 0.3 C18:2 4.5 ( 0.3 3.6 ( 0.3 10.4 ( 0.1 7.7 ( 0.1 5.7 ( 0.2 C18:3 1.1 ( 0.1 1.1 ( 0.1 1.5 ( 0.1 1.4 ( 0.1 1.1 ( 0.1 & C20:0 C20:1 1.1 ( 0.1 0 0 0 0 C22:0 0.6 ( 0.1 0 0 0 0 & C20:4 total

100

100

100

100

100

extended to nonisothermal conditions. What makes matters extremely difficult to model is the fact that for each point in time both the chemical composition of the solid and the melt are changing, and so is the supersaturation (chemical potential) of the different components. Moreover, shear and cooling rates will affect this entire dynamic profoundly in a very complex fashion. This makes this situation extremely complex and may represent the limit of what we can accurately predict from first principles. It is the view of the authors that it is questionable whether such an approach will ever allow us to model the complex crystallization behavior of mixtures of sometimes hundreds of different components. Thus, in this work we explore a different approach to the problem: we define a new quantity, or parameter, which accurately characterizes the dynamic of the system. In a nonisothermal crystallization process, the traditional concept of supercooling has no meaning. It is the time dependence of the supercooling that defines the supercooling history of the system. Armed with this new parameter, we can seek to define statistical laws obeyed by the systems and determine the energy of activation required for the formation of the nuclei. Even though this approach is less elegant than first principles approaches, it may represent a practical way to tackle the problem.

was injected into the chromatograph. The resulting peaks were integrated using a Shimadzu integrator (C-R3A Chromatopac). Three determinations of each of three separate samples were carried out. The average and standard deviation are reported. Differential Scanning Calorimetry (DSC). The thermal behavior of the samples was studied by means of a DSC2910 differential scanning calorimeter (DSC) (TA Instruments, Mississauga, Ontario). A 5 to 10 mg sample of melted fat was placed in an aluminum DSC pan and was heated from 20 to 60 °C, held at this temperature for 30 min, and then cooled at different cooling rates (0.5, 1, 2, 3, 4, and 5 °C/min) to the set temperature (Tset). This temperatures were set to 15, 25, 26, 30, and 20 °C for AMF, PH(PO/PS), PHPO, IHPO, and PO, respectively. The peak crystallization temperature (Tc) was determined from these profiles. Two different procedures were used to determine the melting temperatures by means of DSC. First, the sample was crystallized as described before and kept at Tset for 5 min, followed by heating from Tset to 60 °C at 5 °C/min. The peak melting temperatures were determined from these melting profiles. The second procedure used was to quench the samples from 60 to 20 °C and then they were stored at 5 °C for 1 month to ensure that the most stable polymorph was generated. Melting thermograms were obtained as before (heating from 10 to 60 °C at 5 °C/min) to obtain the peak melting temperatures of the samples. Three determinations of each of three separate samples were carried out. The average and standard deviations are reported. Induction Times of Crystallization (PTA). Induction times of crystallization were studied by means of a Phase Transition analyzer (PSA-70V-HT, Phase Technology, Richmond, BC, Canada). The analyzer is a light turbidimeter that detects the appearance of the first crystals during the crystallization process of a fat sample. Samples were melted for 30-45 min at 80 °C, and then 150 µL were placed in the crystallization cell. The temperature of the cell was set at 60 °C. Samples were held at this temperature for 15 min, and then cooled at 0.5, 1, 2, 3, 4, and 5 °C/min to the different Tset described in the DSC experiments. Both the crystallization temperatures and the induction times were calculated from these experiments as the time when the first crystals appear, which is evidenced by a deviation in the baseline of the laser signal. The induction time for nonisothermal nucleation (tc) was calculated as the time at which the first crystals were detected minus the time required to reach the melting point temperature. Three determinations on each of three separate samples were carried out. The averages and standard deviations are reported. This technique was also used to determine melting points. For this determination, melted samples (80 °C for 30 min) were placed in the PTA cell at 60 °C and then cooled to 10 °C at 40 °C/min. Upon reaching this temperature, samples were incubated at this temperature for 30 min and then heated at 1 and 5 °C/min until melted. The melting temperature was determined as the temperature at which the signal of the laser becomes constant and is equal to the baseline. As no significant differences were found between the different heating rates assayed using this method, the melting points reported in this study are the average of the two melting temperatures obtained at these heating rates and their standard deviations.

Materials and Methods Fat Samples. The fat samples used in this study included anhydrous milkfat (AMF), partially hydrogenated palm oil (PHPO), palm oil (PO), chemically interesterified and hydrogenated palm oil (IHPO), and a partially hydrogenated blend of palm oil and palm stearin (PH(PO/ PS)). The fatty acid profile of these samples is shown in Table 1. Chemical Composition. The fatty acid composition was determined using gas chromatography. A column of 5 mm o.d., 3 mm i.d. and 1.5 m long was filled with 10% silar 9CP on chromosorb W, AW 80/100 mesh. This column was placed in a GC equipment (Shimadzu GC-8A, Kyoto, Japan). The chromatograph oven was set at 60 °C, and then a temperature ramp was programmed from 60 to 210 °C at 8 °C/min. The detector and injector temperature was held at 230 °C. Nitrogen was used as the carrier gas, and both hydrogen and air were used to feed the FID detector. Before chromatographic analysis, the methyl esters of the TAG’s fatty acids were generated; therefore, 50 mg of sample was placed in a vial and dissolved in 2 mL of isooctane. A total of 200 µL of 2 N KOH 2 N in MeOH was added. The mixtures were vortexed for 1 min, and after resting for 5 min, 2 drops of methyl orange were added. Finally, the sample was titrated with 2 N HCl until a pink endpoint was observed. A total of 0.5 µL of the organic phase

Theory Nonisothermal Interpretation. Under nonisothermal conditions, the supersaturation, or supercooling, of the system is a dynamic quantity since ∆T is changing in time as the material crystallizes. Marangoni et al. (2006) defined a supercoolingtime parameter,23 which is the effective exposure of the system to the supercooling until the initiation of nucleation. This is calculated as the area under the supercooling-time trajectory from the time when the sample crosses Tm to the time where the first crystal nuclei appear (tc) at the crystallization temperature (Tc). Notice that Tc * Tset and therefore ∆T ) (Tm - Tset) * ∆Tc ) (Tm - Tc). This supercooling-time exposure (β) at nucleation can thus be defined as

β ) (1/2)∆Tctc

(1)

Since the cooling rate is defined as φ ) ∆T/∆t and at Tc t ) tc

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Crystal Growth & Design, Vol. 6, No. 5, 2006 1201

and t0 ) 0, then φ ) ∆Tc/tc. Substituting tc for ∆Tc/φ leads to an expression for the supercooling-time exposure at nucleation:

β)

1 (∆Tc) 2 φ

2

(2)

In our previous work,23 we demonstrated a relationship between the nucleation rate of fat systems and the supercooling-time exposure with the following form:

J x ) e-k β Jmax

(3)

Combining eqs 2 and 3, the nucleation rate is found to be an exponential function of the inverse square root of the cooling rate, namely

J x ) e-k(∆Tc/ 2φ) Jmax

(4)

Using this model, it also possible to determine the energy required to initiate the nucleation process, possibly the energy of activation for nucleation. From Tm to Tc, the onset of the phase transformation, no phase change has taken place. Thus, strictly specific heat has been removed from the system (Qm ) Cp∆T), where Qm is the heat removed from the system upon cooling per gram of material and Cp is the specific heat (heat capacity), which for vegetable oils is ∼2.0 J g-1 K-1 in this temperature range. Substituting Qm/Cp for ∆T in eq 4 leads to the expression:

J x x x ) e-k(Qm/Cp 2φ) ) e-(Qm/Z φ) ) e-(Χ/ φ) Jmax

(5)

where Qm is proposed here to possibly represent the energy of activation for nucleation per unit mass [J/g], and Z [J g-1 K-1/2 s1/2] is defined as

Z)

x2Cp k

(6)

Thus, from knowledge of k (eq 3), Z can be calculated using eq 6. Moreover, from the nonlinear fit of J/Jmax vs φ-0.5 data, the constant X can be obtained. It is thus possible to determine the energy of activation for nucleation as Qm ) Z × X [J/g]. This quantity can then be multiplied by the average molecular weight of the triacylglycerols to obtain the molar energy of activation for nucleation. Nonlinear Regression Procedure. Curve fitting of the model to the data was carried out using GraphPad Prism 4.0 (GraphPad Software, San Diego, CA). All equations were fitted to the data in the following form:

J ) (Jmax - Jlim)e-k

xβ

+ Jlim

(7)

where Jlim is the limiting nucleation rate as β f ∞. We found this was necessary with nucleation rate data obtained from the inverse of an induction time. In our previous work,23 where the nucleation rate was determined directly using polarized light microscopy, the introduction of Jlim was not necessary. This is due to the low sensitivity of the light scattering method at slow cooling rates. The introduction of the limiting nucleation rate (as opposed to a decay to zero nucleation rate) lead to a statistically significant improvement in the fit of the model to the data (P < 0.05).

Statistical Physics Considerations. In our previous work,23 we reported on the parametrization procedure and analytical methodology required for the determination of the energy of activation for nucleation under nonisothermal conditions. However, the justification for the use of the model, other than a phenomenological reason (i.e., excellent goodness of fit), was not provided. In this section, we will show how the form of the model is a direct consequence of the statistical nature of the nucleation process. The approach to the problem taken in this work is statistical in nature. First, it is imperative to realize that a parametrization of the data relative to temperature and time is required for a proper description of the phenomenon of nucleation under nonisothermal conditions. Both the degree of supercooling at nucleation and the nonisothermal nucleation induction time are included in the parameter β. The parameter β is distributed in an exponential fashion, with an exponential probability density function, p(x), of the form:

{

-λβ βg0 p(x;λ) ) λe 0 β 0. This probability density function (pdf) applies to values of the randomly distributed variable belonging to the set β ∈ [0;∞). The scale parameter (µ) is simply the inverse of the rate parameter and represents the mean, or expected value, of an exponentially distributed random variable, E[β] ) µ ) 1/λ. Thus, this pdf is quite appropriate to model our situation where our random variable has to always be greater than zero, and the mean is fixed. Exponential distributions are used to model memory-less Poisson, or stochastic, processes, which take place with constant probability per unit time (time series) or distance (random field). This is the reason exponential pdfs are extensively used to model Brownian motion and random walks, as well as diffusional processes. In our case, though, we assume that our nucleation phase transition initiation event takes place with a constant probability per unit supercooling-time exposure, β, possibly not unreasonable considering the constant cooling rates used. Another interesting property of an exponential pdf is that among all continuous pdfs, with support [0;∞), the exponential pdf with µ ) 1/λ has the highest entropy. Many physical systems tend to move toward maximal entropy configurations over time (Principle of Maximum Entropy). The rate of a reaction V is a function of the concentration of molecules with sufficient energy to overcome an energy barrier to the particular reaction (N*), nucleation in this case, V ) kp[N*], where kp is the rate constant for the reaction. In our case, this would correspond to the concentration of molecules in the metastable state, just prior to the nucleation event. The proportion of molecules in the appropriate state to undergo the nucleation reaction (from energetic and conformational considerations) will be given by (N*) ) p(x)(NT), where (NT) is the total concentration of reactive molecules. The rate of the reaction will thus be given by

V ) kpNTλe-λβ ) Vmaxλe-λβ

(9)

where Vmax is the maximum reaction rate. However, the dependence of the nucleation rate found in this work was to the square root of β. This suggests that our pdf is probably a special case of a Weibull pdf with k ) 0.5 or a “new” exponential probability density function. The Weibull

1202 Crystal Growth & Design, Vol. 6, No. 5, 2006

Marangoni et al.

Figure 2. Melting profiles of the fat samples used in this study immediately after reaching the crystallization temperature (A and C) and after 1 month of storage at 5 °C (B and D).

Figure 1. Temperature and crystallization profiles for materials crystallizing under isothermal (A), near isothermal (B), and nonisothermal (C) conditions. Reproduced with permission from ref 23. Copyright 2006 Elsevier.

model fit our data very well; however, its use would have prevented the extension of the model for the determination of the energy of activation for nucleation due to unit problems. Thus, in this work, we will modify the model to include the rate parameter of the distribution, λ (labeled k in this work). Other modifications include changes to eqs 3 and 5 to

J x x x ) ke-k β ) ke- (Qm/Z φ) ) ke- (Χ/ φ) Jmax

(10)

The determination of the energy of activation remains as before due to the eventual cancellation of the preexponential term. Results Classical nucleation theory mainly addresses isothermal processes18 in which the temperature drop from the melting temperature to a set crystallization temperature is instantaneous (Figure 1a). Under these conditions, the crystallization process can be characterized by an induction time (ti), which is the time required for the appearance of the first solid nuclei at Tset (Tset ) Tc in this case) under the influence of a driving force (∆T ) Tm - Tset). Experimental realities limit the speed at which a system can reach the set temperature, and therefore, most systems display a cooling profile similar to that shown in Figure 1b. Luckily, however, crystallization does not start until some time after the system has reached Tset. Hence, for this case, it is possible to determine the induction time of nucleation under isothermal conditions since crystallization still starts at Tset, and it is thus equal to Tc. In this case, it is important to remember that time zero corresponds to the time when the system crosses Tset. However, the situation described in Figure 1c (nonisother-

mal crystallization) is somewhat different. Here, crystallization starts before the system reaches Tset (and thus Tset * Tc), and both the induction time of nucleation (ti) and the supercooling for nucleation (∆T) have a different meaning that need further clarification.23 Determination of the Melting Temperature of a Complex Fat Mixture. The melting point is defined as the temperature at which a substance changes from solid to liquid state. A pure crystalline solid will melt at a specific temperature, and thus, the melting point is a characteristic of a substance that can be used to identify it. However, when dealing with natural fat systems, which are complex muticomponent mixtures of TAG molecules and other minor components, a melting point determination is not an easy task, especially if the fat tends to crystallize in different polymorphic forms as well. For this reason, it is very important to specify which techniques (capillary method, differential scanning calorimetry, mettler dropping point, etc.) and crystallization conditions were used to determine the “melting point”. For example, when using DSC, the peak temperature value obtained from the melting profile of a crystallized fat is usually a reasonable indicator of the average metling temperature of the sample. However, in most cases, the TAGs in the fat crystallize in different polymorphic forms (phases), and/or fractionation ocurs. In these cases, several peaks are obtained, and thus, choosing the correct peak that represents the melting temperature of the samples is not straightforward. Figure 2 shows an example of the melting profiles of all samples used in this work when crystallized at 1 °C/min. Figure 2A,B shows the melting behavior of samples PH(PO/PS), IHPO, and PHPO 5 min after reaching Tset (A) and after 1 month storage at 5 °C. (B) Figure 2C,D shows the same pattern but for AMF and PO. It is evident from this figure that not only the type of sample but also the time spent at the set temperature affect the thermal behavior of the sample and therefore the peak melting temperature value obtained. In the global analysis carried out in this work, we need an average melting temperature for the kinetic analysis. Thus, we have a challenge, since this temperature has to be related to the initial triacylglycerol fraction that nucleates. Moreover, the determination of this average/global melting temperature has to be reasonably reproducible. Figure 3 shows the melting profile of the five samples obtained using the PTA. From this figure, we can notice that the determination of a global or average value for the melting point of the fat is much more straightforward (however, not necessarily more correct) than when using the

Model of Nucleation of Triacylglycerol Melts

Crystal Growth & Design, Vol. 6, No. 5, 2006 1203

and becomes constant. The melting points obtained were 49.8 ( 0.9 °C, 35.5 ( 0.2 °C, 49.9 ( 0.2 °C, 51.3 ( 1.7 °C, and 45.6 ( 0.4 °C for IHPO, AMF, PH(PO/PS), PHPO, and PO, respectively. This method enables the determination of melting points as the temperature when the last crystals in the fat melt away. We believe that this method offers a convenient way to determine an average value for the melting point of a fat. Certainly this is not the true melting point of the fat, since the fat is composed of hundreds of different triacyglycerol molecules, each with a unique melting point. We resort to using this global melting temperature to be able to carry out the analysis. Otherwise, this analysis would not be possible. Table 2 shows the melting and crystallization temperatures obtained from DSC and PTA together with the induction times of crystallization determined by PTA. The DSC peak melting temperatures and the PTA end of melt temperatures reported

Figure 3. Phase transition analyzer (light scattering) profiles as a function of increasing temperature used to determine the melting point of the different fat systems.

DSC. Here, the melting temperature is determined as the temperature at which the signal intensity decreases to baseline

Table 2. Comparison between Melting (Tm) and Crystallization Temperatures (Tc) Calculated from the Melting Profile Determined by Using a Differential Scanning Calorimeter (DSC) and the Phase Transition Analyzer (PTA)a cooling rate (°C/min)

Tm (DSC) (°C)

Tc (DSC) (°C) AMF

0.5 1 2 3 4 5 10 15 20

33.6 ( 0.1 33.6 ( 0.2 34.0 ( 0.2 34.3 ( 0.1 34.2 ( 0.3 33.5 ( 0.1

17.8 ( 0.7 16.2 ( 0.2 15.1 ( 0.1 15.6 ( 1.4 14.7 ( 0.2 14.4 ( 0.1

0.5 1 2 3 4 5 10 15 20

45.7 ( 0.4 45.7 ( 0.1 46.0 ( 0.2 45.7 ( 0.1 45.6 ( 0.3 45.6 ( 0.1

27.3 ( 0.1 27.7 ( 0.2 27.4 ( 0.1 27.3 ( 0.1 27.2 ( 0.1 27.1 ( 0.1

0.5 1 2 3 4 5 10 15 20

43.7 ( 0.1 43.0 ( 0.1 43.3 ( 0.1 43.0 ( 0.3 43.0 ( 0.3 42.8 ( 0.5

20.3 ( 0.01 19.8 ( 0.1 19.5 ( 0.1 19.5 ( 0.1 19.4 ( 0.1 19.4 ( 0.1

0.5 1 2 3 4 5 10 15 20

47.2 ( 0.5 46.7 ( 0.1 47.0 ( 0.5 46.6 ( 0.4 32.7 ( 0.6 33.2 ( 0.2

31.9 ( 1.8 32.0 ( 0.1 31.3 ( 0.1 29.8 ( 0.1 30.1 ( 0.1 30.1 ( 0.2

0.5 1 2 3 4 5 10 15 20

45.7 ( 0.3 46.0 ( 0.1 46.2 ( 0.2 46.3 ( 0.1 46.3 ( 0.1 46.1 ( 0.2

PH (PO/PS) 30.0 ( 0.1 29.2 ( 0.5 28.7 ( 0.1 28.4 ( 0.3 28.3 ( 0.1 27.6 ( 0.3

PHPO

PO

IHPO

Tm (PTA) (°C)

Tc (PTA) (°C)

tc (s)

35.5 ( 0.2

21.4 ( 0.6 19.9 ( 0.4 18.9 ( 0.3 18.8 ( 0.2 18.3 ( 0.5 19.5 ( 0.1 16.0 ( 0.2 15.8 ( 0.9 15.9 ( 1.0

1734 ( 69 956 ( 25 508 ( 10 343 ( 14 264 ( 7 195 ( 22 119 ( 6 81 ( 7 60 ( 5

51.3 ( 1.7

34.6 ( 0.1 32.9 ( 1.7 30.6 ( 0.1 30.4 ( 0.3 30.1 ( 0.5 31.2 ( 0.7 29.0 ( 0.9 28.4 ( 0.6 28.7 ( 0.8

2050 ( 14 1128 ( 106 635 ( 8 428 ( 8 326 ( 8 247 ( 10 136 ( 7 94 ( 12 70 ( 3

45.6 ( 0.4

25.3 ( 1.0 24.0 ( 0.1 23.0 ( 0.3 22.9 ( 0.8 21.7 ( 0.8 23.0 ( 0.8 21.3 ( 0.3 21.1 ( 1.5 21.1 ( 1.2

2471 ( 169 1323 ( 18 691 ( 10 466 ( 16 359 ( 21 281 ( 13 149 ( 8 100 ( 7 75 ( 16

49.8 ( 0.9

37.4 ( 0.4 34.5 ( 0.2 34.1 ( 0.2 33.9 ( 0.1 33.8 ( 0.3 34.6 ( 1.1 33.0 ( 0.3 32.6 ( 0.7 32.4 ( 0.2

1527 ( 57 937 ( 10 483 ( 6 325 ( 4 230( 66 187 ( 16 103 ( 9 70 ( 12 54 ( 16

49.9 ( 0.2

35.1 ( 0.3 32.1 ( 0.1 31.4 ( 0.3 31.3 ( 0.4 30.8 ( 0.7 31.5 ( 0.9 30.0 ( 0 30.2 ( 0.5 29.1 ( 0.1

1806 ( 32 1089 ( 7 569 ( 11 380 ( 10 279 ( 54 226 ( 18 122 ( 30 80 ( 11 64 ( 11

a Induction times (t ) calculated from PTA runs are also included. All values represent averages and standard deviations of three separate determinations c on different samples.

1204 Crystal Growth & Design, Vol. 6, No. 5, 2006

in this table were determined from the melting profiles obtained after incubating the samples, 5 min (DSC) 30 min (PTA) after reaching Tset. Thus, they represent the melting temperature of the solids that are formed at the onset of the crystallization process. Nucleation Rate Determination. The nucleation rate was estimated from the inverse of the induction time of nucleation obtained from the PTA experiments (Table 2). This method of estimating the nucleation rate is convenient since it is experimentally accessible; however, it is only an approximation. A nucleation rate corresponds to the number of nuclei appearing per unit time, while this estimated nucleation rate is the inverse of an induction time. In our previous work,23 we determined the nucleation rate of fats using polarized light microscopy by counting the number of reflections appearing per unit time. The method used in this study is different. Even though we only had three cooling rates in our previous experiment, we plotted the nucleation rate obtained using the PTA vs the nucleation rate obtained by polarized light microscopy. The correlation coefficient (r2) obtained was 0.82 and the slope 1.1 ( 0.2. Thus, agreement was reasonable, thus validating the approximation J ∼ 1/τ. This, however, has to be validated further in other systems under different conditions. As shown in Table 2, crystallization temperatures obtained for the same sample at the same cooling rate were higher for PTA than for DSC, suggesting that the first technique was more sensitive than the second one at detecting the appearance of the first crystals (onset of crystallization). Because of the high sensitivity of the Phase Transition analyzer,24 we can consider that the induction times determined using this technique are reasonable estimates of the induction times of nucleation. For all the samples used in this study, the induction times were shorter when the samples were crystallized at higher cooling rates. This is an expected result since the higher the cooling rate, the shorter the time required to reach the crystallization temperature. Also, from Table 2 we can appreciate that when the cooling rate is high, samples crystallize at lower temperatures suggesting that the time of exposure to supercooling is of key importance. Considering that the induction times calculated during a nonisothermal crystallization include the time needed to reach a specific crystallization temperature, it is necessary to find some other parameter to describe the kinetics of crystallization for these situations. Therefore, a supercoolingtime exposure was proposed in our previous work.23 This supercooling-time exposure, β, includes both a thermodynamic (∆Tc) and a kinetic (φ) component. This parameter takes into consideration the amount of supercooling in time required for nucleation to start. Figure 4A shows the dependence of the relative nucleation rate to β. The smaller the β value, the less energy is required for the sample to nucleate, and therefore a higher J is obtained. It is for this reason that β can be considered as a supercoolingtime exposure required for the crystallization to occur. Conceivably, one could express the driving force or potential for nucleation as the inverse of the supercooling-time exposure. Energy of Activation for Nucleation. The exponential relationship between the relative nucleation rate and square root of β can be appreciated in Figure 4A. The fit of the model to the data was excellent and statistically better (P < 0.05) than for an exponential relationship between the relative nucleation rate and β. Moreover, the fit of the model to the data, when analyzing the systems separately, was significantly better than when analyzing all normalized nucleation rate data sets together as a single set (P < 0.0001). However, our model did fit the combined data set very well (Figure 4B), possibly suggesting

Marangoni et al.

Figure 4. (A) Variation of the normalized nucleation rate (J/Jmax) vs the supercooling-time exposure (β) for all systems studied. (B) Variation of the normalized nucleation rate (J/Jmax) vs the supercooling-time exposure (β) for all systems studied grouped together as a single data set. (C) Variation of the normalized nucleation rate as a function of the cooling rate for all systems grouped together.

the data parametrization procedure removed system-specific effects. It is interesting to notice, however, that when the normalized nucleation rate was plotted as a function of φ-0.5 (Figure 4C) all lines collapsed onto a single master curve and were not significantly different from each other (P > 0.05). This was also the case in our previous study using a different technique to determine nucleation rate.23 As described in Materials and Methods, it is possible to determine the energy of activation for nucleation using the approach developed in this study. Table 3 shows the values for k, Z, Jmax, and Qm determined for each sample. We can observe that the lower the Q value, the higher the J, as expected from kinetic theory, if Q represents the energy of activation for the nucleation process. We also reported values of k, Z, and Qm derived from the analysis of the combined data set. One interesting finding is that the Jmax values were not statistically different from each other (P > 0.05), with an average value of 1.25 and a standard deviation of 0.02. Thus, it thus possible predict the nucleation rate, or induction time of nucleation, of triacylglycerols at different cooling rates, under nonisothermal conditions, by using:

J [s-1] )

1 x ) ke- (1.25Qm/Z φ) tc

(11)

The parameters required are listed in Table 3; the energy of activation in the above expression should be used with units of kJ/mol.

Model of Nucleation of Triacylglycerol Melts

Crystal Growth & Design, Vol. 6, No. 5, 2006 1205

Table 3. Exponential Constants (k, Z), Energy of Activation (Q), and Maximum Nucleation Rate (Jmax) for the Systems Studied system AMF PHPO PO IHPO PH(PO/PS) average (n)5)

ka (K-1/2 s-1/2)

Z (J g-1 K-1/2 s1/2)

Qb (kJ/mol)

Jmaxa (s-1)

0.0775ab (0.00480) 0.0622bc (0.00339) 0.0558c (0.00329) 0.0848a (0.00510) 0.0701ac (0.00557) 0.070 (0.0050)

36.5

31.4

45.5

39.1

50.7

43.6

33.4

28.7

40.3

34.6

45.7

39.3

1.32a (0.079) 1.23a (0.056) 1.20a (0.056) 1.26a (0.070) 1.23a (0.084) 1.25 (0.020)

a Values reported are the average and standard error (n ) 9). Values with the same superscript letter within a column are not significantly different from each other (P > 0.05). b Molar energy of activation was calculated as Q ) Z*X*MW, where X ) 0.8598 K1/2 s-1/2, and the average molecular weight of a triacylglycerol used was MW ) 800 g/mol.

Moreover, a reasonable approximation of the nucleation rate of fats as a function of cooling rate can be obtained using:

J[s-1] )

1 x ) 0.0745e-(0.86/ φ) + 0.0008 tc

(12)

It is worth pointing out that the entire kinetic characterization was carried out using simple light scattering device. The PTA analyzer was used to characterize the melting as well as the crystallization behavior. This procedure could thus be completely automated and a standard method developed for the characterization of the nonisothermal nucleation behavior of fats. In summary, in this work we have developed a new way of characterizing nucleation kinetics under nonisothermal conditions by parametrization of the data, considering both time and supercooling effects. A supercooling-time exposure parameter was defined and found to be related to the nucleation rate in a simple exponential decay fashion. The parametrization procedure used in the analysis of the nucleation kinetics of the fats crystallized at different cooling rates allowed for the determination of a cooling-rate independent energy of activation for nucleation. Acknowledgment. The authors acknowledge the financial support of the Natural Sciences & Engineering Research Council of Canada (NSERC) and the Ontario Ministry of Food (OMAF). References (1) Hartel, R. W. Nucleation. In Crystallization in Foods; Hartel, R. W., Ed.; Aspen Publishers Inc.: New York, 2001; pp 145-188. (2) Chapman, D. Polymorphism of glycerides. Chem. ReV. 1969, 62, 433-456.

(3) Timms, R. E. Phase behavior of fats and their mixtures. Prog. Lipid Res. 1984, 23, 1-38. (4) Sato, K. Solidification and phase transformation behaviour of food fats -A review. Fett/Lipid 1999, 101, 467-474. (5) Sato, K.; Ueno, S.; Yano, J. Molecular interactions and kinetics properties of fats. Prog. Lipid Res. 1999, 38, 91-116. (6) Sato, K. Crystallization behavior of fats and lipids - A review. Chem. Eng. Sci. 2001, 7, 2255-2265. (7) Herrera, M. L.; Hartel, R. W. Effect of processing conditions on crystalization kinetics of a milk fat model system. J. Am. Oil Chem. Soc. 2000, 77, 1177-1187. (8) Herrera, M. L.; Hartel, R. W. Effect of processing conditions on physical properties of a milk fat model system: rheology, J. Am. Oil Chem. Soc. 2000, 77, 1189-1195. (9) Herrera, M. L.; Hartel, R. W. Effect of processing conditions on physical properties of a milk fat model system: microstructure. J. Am. Oil Chem. Soc. 2000, 77, 1197-1204. (10) Martini, S.; Herrera, M. L.; Hartel, R. W. Effect of cooling rate on crystallization behavior of milk fat fraction/sunflower oil blends. J. Am. Oil Chem. Soc. 2002, 79, 1055-1062. (11) Martini, S.; Herrera, M. L.; Hartel, R. W. Effect of processing conditions on microstructure of milk fat fraction/sunflower oil blends. J. Am. Oil Chem. Soc. 2002, 79, 1063-1068. (12) Campos, R.; Narine, S. S.; Marangoni, A. G. Effect of cooling rate on the structure and mechanical properties of milk fat and lard. Food Res. Int. 2002, 35, 971-981. (13) Mazzanti, G.; Marangoni, A. G.; Welch, S. E.; Sirota, E. B.; Idziak, S. H. J. Orientation and phase transitions of fat crystals under shear. Cryst. Growth Des. 2003, 3, 721-725. (14) Mazzanti, G.; Marangoni, A. G.; Welch, S. E.; Sirota, E. B.; Idziak, S. H. J. Effect of minor components and temperature profiles on polymorphism of milk fat. Cryst. Growth Des. 2004, 4, 1303-1309. (15) Dibildox-Alvarado, E.; Neves Rodrigues, J.; Gioelli, L. A.; ToroVazquez, J.; Marangoni, A. G. Effect of crystalline microstructure on oil migration in a semisolid fat matrix. Cryst. Growth Des. 2004, 4, 731-736. (16) Mazzanti, G.; Marangoni, A. G.; Idziak, S. H. J. Modeling phase transitions during the crystallization of a multicomponent fat under shear. Phys. ReV. E 2005, 71, 041607. (17) Marangoni, A. G.; McGauley, S. E. The relationship between crystallization behavior and structure in cocoa butter. Cryst. Growth Des. 2003, 3, 95-108. (18) Boistelle, R. Fundamentals of Nucleation and Crystal Growth. In Crystallization and Polymorphism of Fats and Fatty Acids; Garti, N., Sato, K., Eds.; Marcel Dekker: New York, 1988; pp 189-226. (19) Kashchiev, D. Nucleation at Variable Supersaturation. In Nucleation: Basic Theory with Applications; Kashchiev, D., Ed.; Butterworth Heinemann, Oxford, UK, 2000; pp 279-289. (20) Rybin, E. N. On the kinetics of non-isothermal nucleation. Colloid J. 2003, 65, 230-236. (21) McGraw, R.; Hu, D. T. Kinetic extensions of the nucleation theorem. J. Chem. Phys. 2003, 118, 9337-9347. (22) Wesrop, L. H.; van Meeteren, J. A.; de Jong, S.; v. d. Giessen, R.; Overbosch, P.; Grootscholten, P. A. M.; Struik, M.; Royers, E.; Don, A.; de Loos, Th.; Peters, C. Gandasasmita, Liquid-Multiple SolidPhase Equilibria in Fats: Theory and Experiments. In Fat Crystal Networks; Marangoni, A. G., Eds.; Marcel Dekker: New York, 2005; pp 481-709. (23) Marangoni, A. G.; Tang, D.; Singh, A. P. Nonisothermal nucleation of triacylglycerol melts. Chem. Phys. Lett. 2006, 419, 259-264. (24) Wright A. J.; Narine, S. S.; Marangoni, A. G. Comparison of experimental techniques used in lipid crystallization studies. J. Am. Oil Chem. Soc. 2000, 77, 1239-1242.

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