DOI: 10.1021/cg901058y
On the Metastable Zone Width of 1,3-Dipalmitoyl-2-oleoylglycerol, Tripalmitoylglycerol, and Their Mixtures in Acetone Solutions†
2010, Vol. 10 640–647
Keshra Sangwal*,‡ and Kevin W. Smith# ‡
Department of Applied Physics, Lublin University of Technology, ul. Nadbystrzcka 38, 20-618 Lublin, Poland and #Consultant, Loders Croklaan, Unilever Research Colworth, Shambrook, Bedfordshire, MK44 1LQ, U.K. Received August 31, 2009; Revised Manuscript Received November 12, 2009
ABSTRACT: The extensive experimental data reported by Smith et al. (Eur. J. Lipid Sci. Technol. 2005, 107, 583-593) on the metastable zone width of 1,3-dipalmitoyl-2-oleoylglycerol, (POP) tripalmitoylglycerol (PPP), and their mixtures in acetone are analyzed using a novel approach based on classical three-dimensional nucleation theory (Sangwal, K. Cryst. Growth Des. 2009, 9, 942-950) and a self-consistent N yvlt-like approach (Sangwal, K. Cryst. Res. Technol. 2009, 44, 231-247) to obtain quantitative information on the processes of nucleation of these systems. The relevant equations of the above approaches are (i) (T0/ΔTmax)2 = F(1 - Z ln R) and (ii) ln(ΔTmax)/T0) = Φ þ β ln R, where T0 is the saturation temperature and F and Φ are, respectively, the values of (T0/ΔTmax)2 and ln(ΔTmax)/T0) when ln R = 0, whereas Z is a parameter characteristic of the crystallizing system and the constant β is inverse of the apparent nucleation order m in the N yvlt approach. It was found that (1) the value of parameters β and Z is relatively insensitive to saturation temperature T0 but depends on the investigated compound and the technique used for the measurement of maximum supercooling ΔTmax and the presence of PPP impurity in POP solutions; (2) the values of β and Z for PPP and POP decrease with an increase in their solubility in acetone; and (3) the data of Φ and ln(F1/2) as a function of saturation temperature T0 follow the Arrhenius-type relation with an activation energy Esat. Analysis of the results revealed that (1) the values of β and Z are directly connected with the solid-liquid interfacial energy γ; (2) the value of Esat depends on the nature of ions/molecules participating in diffusion; (3) maximum supercooling ΔTmax measured by turbidimetry corresponds to a nucleation event alone, but that obtained from growth exotherm includes some contribution of growth of stable nuclei after nucleation; and (4) PPP impurity in POP solutions promotes crystallization by acting as growth centers for heterogeneous nucleation.
1. Introduction Wet fractionation involving the fractional crystallization of a solid phase and its separation from the liquid phase is an important process because it offers a natural means of controlling fat properties.1 Wet fractionation is particularly advantageous for isolating sharp-melting fractions for use in confectionary fats from raw materials such as palm oil. However, the efficiency of solid-liquid separation is crucial during a wet fractionation process and is determined by the nucleation of crystals and their development in the liquid phase. It is well-known from the literature on the crystallization of inorganic salts from aqueous solutions2,3 that the metastable zone width as determined by the maximum supercooling ΔTmax controls the solid-liquid separation process. The metastable zone width for inorganic compounds depends on a variety of factors such as saturation temperature, solvent used for preparation of supersaturated solutions, presence of impurities dissolved in the solution, presence of crystalline seeds in the solution, solution stirring, and cooling rate of solution from saturation temperature. However, there is not much systematic work on the metastable zone width of organic solvent systems. Recently, Smith et al.4 reported exhaustive experimental data on the metastable zone width of saturated solutions of †
PACS: 61.66.Hq; 64.60.Q-; 81.10.Aj; 81.10.Dn. *Corresponding author. Phone: þþ48 81 5384-504; fax: þþ48 81 5384731; e-mail:
[email protected]. pubs.acs.org/crystal
Published on Web 12/10/2009
1,3-dipalmitoyl-2-oleoylglycerol (POP), tripalmitoylglycerol (PPP), and POP-PPP mixtures in acetone using the conventional polythermal method at constant cooling rates R of solutions between 12 and 180 K 3 h-1. The nucleation event was determined4 by simultaneous recording of optical transmittance and temperature of an investigated solution as functions of time. In the former case, the nucleation event was defined by the detection of the cloud point Tc of the solution at which a sudden reduction in the optical transmittance of the solution was recorded due to the onset of solution turbidity. The recorded reduction in transmittance occurred before the detection of solution turbidity by the naked eye. In the latter case, the nucleation event was defined by the occurrence of major crystallization displayed on the temperature-time output recorder by the commencement of a growth exotherm at temperature Tg. The temperature at which the transmittance returned to its initial value when reheating the suspension was defined as the clear-point (i.e., equilibrium) temperature T0. The temperature difference (T0 - Tc) or (T0 Tg) was defined as the maximum supercooling ΔTmax, and the experimental data of ΔTmax so obtained are referred to as obtained by using turbidimetry and growth exotherms, respectively, in the present paper. Smith et al.4 also observed that the maximum supercooling ΔTmax for POP recorded by growth exotherms was always greater than that obtained by turbidimetry. This means that turbidimetry detects the nucleation event earlier than growth exotherms. Smith et al.4 analyzed their data of maximum supercooling ΔTmax obtained by using turbidimetry as a function of the r 2009 American Chemical Society
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Figure 1. Schematic illustration of determination of metastable zone width of solute-solvent system by the polythermal method.
cooling rate R using the traditional N yvlt’s equation.2,5 They found that the nucleation order m lies between 12 and 16 for the highly soluble POP, between 6 and 7.2 for the sparingly soluble PPP, but it is close to that of PPP in the case of POP containing up to 8 wt % additive PPP. However, the authors mentioned that the traditional equation “cannot provide fundamental quantitative rate data about the nucleation process, nor can they necessarily provide qualitative information that is uniquely related to nucleation (i.e. growth has to be considered as well)”. This is because N yvlt’s equation contains two parameters (i.e., nucleation order m and nucleation constant k), which have no physical significance. Recently, Sangwal6,7 advanced two new approaches to explain the dependence of metastable zone width on various factors. Both approaches assume that critically sized threedimensional (3D) nuclei are formed by the attachment of monomers to developing embryos during cooling of a solution below saturation temperature T0, but they differ in the dependence of nucleation rate J on the developed solution supersaturation ln S. The first approach is based on the classical 3D nucleation theory,6 while the second approach is based on power-law relationship7 between nucleation rate J and maximum supersaturation ln Smax. Sangwal8 also extended the approach based on the classical 3D nucleation theory to analyze the effect of impurities. The aim of this work is to examine the extensive experimental data reported by Smith et al.4 on the metastable zone width of POP, PPP, and POP-PPP mixtures in acetone using the recent approaches and to obtain quantitative information on the process of nucleation of these systems. An additional motivation for undertaking this work is that, in contrast to the behavior of aqueous solutions of inorganic salts, practically nothing is known about the nature of metastability of solutions containing organic solids dissolved in organic solvents, although organic systems are encountered inter alia in food and fuel industries. 2. Resume of Theoretical Approaches of Metastable Zone Width The polythermal method of metastable zone width is based on the determination of the maximum supercooling ΔTmax. This process is schematically shown in Figure 1. A solution of known saturation temperature T0 (point B) is cooled at a constant cooling rate R from unsaturated state (point A) to a temperature Tlim at which first crystals are detected in the
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solution (point C). Concentrations of solutions corresponding to temperatures T0 and Tlim are cmax and clim, respectively. The maximum supercooling ΔTmax is the difference between the saturation temperature T0 and the temperature Tlim; that is, ΔTmax = T0 - Tlim. However, the maximum concentration difference Δcmax = cmax - clim. Note that ΔTmax refers to the initial saturation temperature T0, while Δcmax refers to the temperature Tlim when the first nuclei are detected. There are two approaches to explain the dependence of metastable zone width on various factors. Both approaches assume the formation of critically sized 3D nuclei during the cooling of a solution below saturation temperature T0, but they differ in how the nucleation rate J depends on the developed solution supersaturation ln S. In the first approach, the relationship between nucleation rate J and maximum supersaturation ln Smax is described by a simple power law,7 that is, J µ (ln Smax)m, while in the second approach it is described by the classical 3D nucleation theory.6 2.1. Self-Consistent N yvlt-like Approach. According to the first approach called the self-consistent N yvlt-like approach, the relationship between the maximum supercooling ratio ΔTmax/T0 and the cooling rate R is given by7 1=m ΔTmax f ΔHs ð1 -mÞ=m 1=m ¼ R ð1Þ KT0 T0 RG Tlim where m is the apparent nucleation order, K is the nucleation constant different from that contained in the traditional N yvlt’s equation,2,5 ΔHs is the heat of dissolution, RG is the gas constant, and the constant f has units nuclei/volume. Both m and K depend on the processes of formation and growth of stable nuclei into visible crystals. A typical value of ΔHs/RGTlim is about 5 and ΔTmax/T0 ≈ 0.05. Taking logarithms on both sides of eq 1, one obtains the relation between ln(ΔTmax/T0) and ln R in the form lnðΔTmax =T0 Þ ¼ Φ þ β ln R ð2Þ where β = 1/m and 1 -m ΔHs 1 f Φ ¼ ln þ ln m m KT0 RG Tlim
ð3Þ
Equation 2 predicts linear dependence of ln(ΔTmax/T0) on ln R. This linear dependence enables one to calculate the values of nucleation order m from the slope and the nucleation constant K from the intercept Φ because (ΔHs/RGTlim) and the constant f can be calculated from solubility data of the investigated compound. Equation 2 is similar in form to the traditional N yvlt’s yvlt’s equation.2,5 However, it has three advantages over N equation: (1) there is a consistency in the units of the lefthand and right-hand sides; (2) the new nucleation constant K has the same units as the nucleation rate J (i.e., nuclei per unit volume per unit time); and (3) nucleation constant K and nucleation order m are related to constant B of the classical 3D nucleation theory by the relationships7 m ¼ 2B=ðln Seff Þ2 , K ¼ B=ðln Seff Þ0:565
ð4Þ
where ln Seff is the effective supersaturation for nucleation and B is given by eq 9. The relationship between K and B holds for B e 4. The effective supersaturation ln Seff < ln Smax. 2.2. Approach Based on Classical Three-Dimensional Nucleation Theory. According to the approach based on classical three-dimensional nucleation theory, the maximum
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Figure 2. Plots of ln(ΔTmax/T0) against ln R for (a) POP and (b) PPP solutions at various saturation temperatures T0 according to eq 2. Linear plots represent the best fit of data. See text for details.
Figure 3. Plots of (T0/ΔTmax)2 against ln R for (a) POP and (b) PPP solutions against ln R at various saturation temperatures T0 according to eq 5. Linear plots represent the best fit of data.
supercooling ratio ΔTmax/T0 and the cooling rate R are related by6 ðT0 =ΔTmax Þ2 ¼ F -F1 ln R ¼ Fð1 -Z ln RÞ where F ¼ F1 Z ¼ ¼ F
(
1 ΔHs 2 ZB RG Tlim ) -1 f ΔHs -ln AT0 RG Tlim
ð5Þ
ð6Þ
ð7Þ
The parameters A and B of eqs 6 and 7 are contained in the expression of the dependence of rate J of formation of stable 3D spherical nuclei on supersaturation ln Smax:2,3 J ¼ A exp½ -B=ðln Smax Þ2
ð8Þ
where the constant A is associated with the kinetics of formation of nuclei in the growth medium, and !3 16π γΩ2=3 ð9Þ B ¼ 3 kB Tlim In eq 9 γ is the effective solid-liquid interfacial energy, Ω is the molecular volume, and kB is the Boltzmann constant equal to RG/NA (NA is Avogadro’s number). When an impurity is present in the supersaturated solute-solvent system, the parameters F and Z may be given by8 F ¼ F0 ð1 þ b1 θÞ ð10Þ Z ¼ Z0 ð1 þ Reff θÞ
ð11Þ
where Reff is the effectiveness parameter for the impurity, θ is the surface coverage by the impurity and the constant ð12Þ b1 ¼ ð3b -Reff Þ
with the constant b ≈ 1. In the above equations, F0 and Z0 are the values of F and Z in the absence of impurity, respectively. The surface coverage is given by the usual adsorption isotherms. In the case of Langmuir adsorption isotherm, the surface coverage is given by ð13Þ θ ¼ KL ci =ð1 þ KL ci Þ where ci is the impurity concentration and KL is the Langmuir constant ð14Þ KL ¼ lnðQdiff =RG TÞ where Qdiff is the differential heat of adsorption of the impurity. As pointed out earlier,8 b1 of eq 12 can be positive or negative depending on whether 3b > Reff or 3b < Reff, respectively, depending on the chemical nature of impurity particles. When impurity particles inhibit the integration of growth units to the growing nuclei, 0 < Reff < 1. However, when impurity particles favor the integration of growth units to the growing nuclei, Reff < 1. Therefore, in principle, one expects that always b1 > 2. Equation 5 predicts that the quantity (T0/ΔTmax)2 decreases linearly with an increase in ln R, with slope Z and intercept F. As seen from eqs 6 and 7, the value of the parameter Z is related with the kinetic parameter A associated with the kinetics of formation of stable nuclei, while the value of F depends on the values of kinetic parameter A as well as the constant B determined by effective interfacial energy γ. The last two factors are strongly influenced by solute-solvent interactions and impurities. The main advantage of this approach is that the effects of different experimental parameters can be explained satisfactorily in terms of two parameters of the classical nucleation theory (see eq 8): (1) parameter A associated with the kinetics of formation of nuclei in growth medium, and (2) changes in the solid-liquid interfacial energy γ.
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Finally, it should be mentioned that the constant Z is not a dimensionless quantity. Instead, its value depends on the units of cooling rate R, but the term Z ln R is a constant quantity for a crystallizing system. In this work, the values of Z refer to cooling rate R taken as K 3 h-1. However, when R is taken as K 3 s-1, Z is increased by a factor of ln(3600) = 8.1887, since 1 h = 3600 s. Similarly, since the constant Z is approximately equal to the constant β = 1/m (see section 3.1), the so-called nucleation order m also depends on the choice of units used for cooling rate R. Incidently, the values of the nucleation order m, reported in most of the literature2 on the measurement of metastable zone width by the polythermal method, refer to R taken as K 3 h-1. 3. Analysis of Experimental Data 3.1. Dependence of Metastable Zone Width on Cooling Rate of PPP and PPP Solutions. Figure 2a,b presents the plots of ln(ΔTmax/T0) against ln R according to eq 2 for POP and PPP, respectively, while Figure 3a,b shows the above data for the two compounds as plots of (T0/ΔTmax)2 against ln R according to eq 5. The experimental data were obtained using turbidimetry and growth exotherms. These techniques are denoted in the figures by TBT and CRY, respectively. The best-fit values of the constants of eqs 2 and 5 and the Table 1. Best-fit Values of Constants of eq 2 TAG
T0 (K)
method
POP
292.65
TBT CRY TBT CRY TBT CRY TBT
296.35 297.95 PPP
295.95 299.45 303.25 304.35 308.05
-Φ
β
RC
3.146 ( 0.061 3.137 ( 0.073 3.254 ( 0.045 3.254 ( 0.023 3.265 ( 0.031 3.339 ( 0.032 3.443 ( 0.019 3.337 ( 0.073 3.696 ( 0.155 3.749 ( 0.119 3.774 ( 0.135
0.059 ( 0.015 0.060 ( 0.018 0.055 ( 0.012 0.060 ( 0.006 0.071 ( 0.008 0.088 ( 0.008 0.136 ( 0.005 0.116 ( 0.019 0.132 ( 0.039 0.116 ( 0.030 0.116 ( 0.034
0.885 0.850 0.922 0.981 0.976 0.983 0.997 0.951 0.859 0.887 0.860
Table 2. Best-fit Values of Constants of eq 5 TAG
T0 (K)
method
POP
292.65
TBT CRY TBT CRY TBT CRY TBT
296.35 297.95 PPP
295.95 299.45 303.25 304.35 308.05
F
Z
RC
508.9 ( 45.1 500.4 ( 52.8 634.3 ( 45.1 622.4 ( 23.9 617.2 ( 25.0 690.0 ( 33.5 717.5 ( 15.1 624.2 ( 38.4 1199 ( 192 1135 ( 175 1508 ( 177
0.083 ( 0.023 0.084 ( 0.027 0.079 ( 0.018 0.083 ( 0.010 0.091 ( 0.010 0.105 ( 0.012 0.131 ( 0.005 0.121 ( 0.016 0.127 ( 0.041 0.122 ( 0.031 0.121 ( 0.030
0.877 0.842 0.909 0.973 0.975 0.973 0.996 0.968 0.842 0.890 0.897
643
corresponding regression coefficients (RC) are given in Tables 1 and 2, respectively. It may be noted from Table 1 that the value of β is relatively insensitive to saturation temperature T0 but depends on the investigated compound and the technique used for the measurement of maximum supercooling; the value of β highly depends on the investigated compound. The value of β calculated from data obtained by turbidimetry is 0.062 ( 0.008 and 0.123 ( 0.010 for POP and PPP, respectively. However, analysis of the data obtained by growth exotherms for POP reveals that the value of β increases to 0.069 ( 0.016 from 0.062 ( 0.008 from data obtained by turbidimetry. The values of β calculated from data obtained using turbiditimetry gives nucleation order m = 1/β as 16.2 and 8.1 for POP and PPP, respectively. These are roughly the values reported by Smith et al.4 However, the value of β calculated from data obtained by growth exotherms gives m = 1/β as 14.5 for POP. Obviously, the different values of m for POP are associated with the sensitivity of the two techniques. The fact that the value of β obtained from the data obtained from growth exotherms is lower than that from turbidimetry implies that detection techniques which record the delayed onset of crystallization nucleation event developing nuclei lead to lower nucleation order m = 1/β. The value of Z shows trends similar to those of β (Table 2). The value of Z calculated from data obtained by turbidimetry is 0.084 ( 0.006 and 0.125 ( 0.005 for POP and PPP, respectively. However, the value of Z obtained from the data using growth exotherms is 0.091 ( 0.012 for POP. It is interesting to note that β ≈ Z for both POP and PPP. This equality means that ln Seff ≈ 2(ΔHs/RGTlim)2; cf. eqs 4 and 7. Since ln Seff < ln Smax (see section 2.2) and heat of dissolution ΔHs is related to solubility c0 of a solute in a solvent (see eq 16), the equality between β and Z implies that these parameters are connected with the solubility of POP and PPP in acetone. It is well-known2 that the solubility of different compounds increases with a decrease in the solid-liquid interfacial energy γ. Therefore, a decrease in γ also means a decrease in the factor B of eqs 4 and 7 and an increase in the values of β and Z. The higher the solubility of different compounds in a given solvent, the lower are the values of β and Z. Thus, the values of β and Z for a compound are intimately connected with its solubility in a solvent. However, it may be noted that the higher values of β and Z for a compound obtained from data using growth exotherms compared to those obtained from data using turbidimetry are caused by the growth of stable nuclei to detectable dimensions.
Figure 4. Plots of (a) ln(ΔTmax/T0) and (b) (T0/ΔTmax)2 against ln R for POP solutions containing different concentrations of PPP at saturation temperature T0 according to eqs 2 and 5, respectively. Linear plots represent the best fit of data.
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Table 3. Best-fit Values of Constants of eq 2 for 2.9% POP Solutions Containing PPP ci (w/w)
method
0
TBT CRY TBT CRY TBT CRY TBT CRY
2 4 8
-Φ
β
RC
3.146 ( 0.061 3.137 ( 0.073 3.440 ( 0.042 2.930 ( 0.040 3.663 ( 0.080 2.80 ( 0.046 3.544 ( 0.031 2.654 ( 0.011
0.059 ( 0.015 0.060 ( 0.018 0.123 ( 0.011 0.061 ( 0.010 0.135 ( 0.020 0.029 ( 0.012 0.104 ( 0.026 0.048 ( 0.003
0.885 0.850 0.985 0.948 0.957 0.778 0.897 0.993
Table 4. Best-fit Values of Constants of eq 5 for 2.9% POP Solutions Containing PPP ci (w/w)
method
0
TBT CRY TBT CRY TBT CRY TBT CRY
2 4 8
F
Z
RC
508.9 ( 45.1 500.4 ( 52.8 743.9 ( 23.3 321.2 ( 15.2 1130 ( 103 264.3 ( 20.1 992 ( 124 191.9 ( 3.6
0.083 ( 0.023 0.084 ( 0.027 0.124 ( 0.008 0.082 ( 0.012 0.131 ( 0.023 0.047 ( 0.019 0.116 ( 0.032 0.071 ( 0.048
0.877 0.843 0.991 0.959 0.943 0.771 0.876 0.991
In contrast to β and Z, the values of -Φ and F increase with an increase in saturation temperature T0 and a decrease in the solubility of the investigated compound. However, the detection method for ΔTmax has a relatively insignificant effect on their values. 3.2. Dependence of Metastable Zone Width on PPP Content in POP Solutions. Figure 4a,b illustrates the plots of ln(ΔTmax/T0) and (T0/ΔTmax)2 against ln R for 2.9% POP solutions containing different concentrations of PPP at saturation temperature T0 according to eqs 2 and 5, respectively. As in the case of additive-free POP and PPP, both types of data obtained from turbidimetry and growth exotherms follow the predicted linear dependences. The values of the constant of eqs 2 and 5 obtained from the best fit of the plots are given in Tables 3 and 4, respectively. The following feature may be noted from Tables 3 and 4: (1) With the addition of PPP, β and Z show a tendency to increase and decrease for the ΔTmax(R) data obtained by turbidimetry and growth exotherms, respectively. (2) In the presence of PPP impurity in POP solution, the values of β and Z from ΔTmax(R) data obtained by turbidimetry and growth exotherms attain values comparable with those for pure PPP. (3) With the addition of PPP, -Φ and F show a tendency to increase and decrease for the ΔTmax(R) data obtained by turbidimetry and growth exotherms, respectively. (4) In the presence of PPP impurity in POP solution, the values of -Φ and F calculated from data using turbidimetry exceed the values obtained for pure PPP. 3.3. Dependence of Constants Φ and ln(F1/2) on Saturation Temperature T0. The data of Φ and ln(F1/2) as a function of saturation temperature T0 may be described by an Arrhenius-type relation6,7 ln y ¼ ln y0 -Esat =RG T0 ð15Þ where ln y denotes -Φ and ln(F1/2), the factor ln y0 denotes the extrapolated values of -Φ0 and [ln(F1/2)]0, and Esat is the activation energy associated with the diffusion of solute molecules in the solution. Figure 5 shows the plots of -Φ and ln(F1/2) against 1/T0 for POP and PPP. The values of
Figure 5. Plots of -Φ and ln(F1/2) against 1/T0 for POP and PPP solutions according to eq 15. Linear plots represent the best-fit of data of Tables 1 and 2 obtained by eqs 2 and 5, respectively. Table 5. Values of ln y0, Esat/RG, and Esat for POP and PPP Solutions TAG
data
POP -Φ(T0)
method
TBT CRY [ln(F1/2)](T0) TBT CRY TBT PPP -Φ(T0) [ln(F1/2)](T0)
ln y0
Esat/RG Esat (kJ/mol) RC
10.14 ( 1.49 2.05 ( 0.44 14.15 ( 1.50 3.22 ( 0.44 9.15 ( 2.59 1.76 ( 0.76 12.09 ( 0.22 2.63 ( 0.07 14.55 ( 3.81 3.31 ( 1.15 15.19 ( 3.45 3.54 ( 1.04
17.0 ( 3.7 26.8 ( 3.7 14.7 ( 6.4 21.9 ( 0.5 27.5 ( 9.6 29.5 ( 8.6
0.977 0.990 0.917 0.999 0.856 0.891
Figure 6. Plots of ln c0 against 1/T0 for POP and PPP in acetone according to eq 16. Filled points denote the data reported, while circles with a plus sign denote data read from the plot of the temperature dependence of solubility c0 of POP in ref 4.
intercept ln y0, slope Esat/RG and, activation energy Esat are listed in Table 5. The following features may be noted from Table 5: (1) The values of intercept ln y0 and activation energy Esat calculated for a compound by using eqs 2 and 5 from the analysis of experimental data obtained by a particular technique are comparable. (2) The values of intercept ln y0 and activation energy Esat for a compound by using eqs 2 and 5 are higher in the case of growth exotherms than in the case of turbidimetry. (3) In the case of a particular detection technique, the values of intercept ln y0 and activation energy Esat obtained by using eq 2 or 5 are higher for PPP than that of POP. (4) The value of Esat for PPP is about twice its value for POP.
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Table 6. Values of ln y0, ΔHs/RG, and ΔHs for POP and PPP Solutions TAG
B1
POP PPP
55.9 ( 8.9 74.6 ( 2.3
10 3 ΔHs /RG 16.0 ( 2.6 23.4 ( 1.3 3
ΔHs (kJ/mol)
RC
133.0 ( 22.0 194.2 ( 10.7
0.938 0.995
molecules of POP and PPP solutes are indeed much larger than the molecules of solvent acetone. As seen from Table 5, the quantity ln y0 and the activation energy Esat are related. The higher the value of the activation energy Esat for a system, the lower is the value of the corresponding pre-exponential term ln y0. Figure 7 shows the dependence of ln y0 on Esat for the two compounds and may be represented by the empirical relation ln y0 ¼ A1 þ A2 Esat
Figure 7. Plots of -Φ0 and [ln(F1/2)]0 against activation energy Esat for nucleation from POP and PPP solutions. Best-fit linear plot is drawn according to eq 18.
The solubility c0, expressed as g of solute in 100 g of solution, of POP and PPP in acetone increases with increasing temperature following an Arrhenius-type relation (Figure 6) ln c0 ¼ B1 -ΔHs =RG T0
ð16Þ
where B1 is a constant and ΔHs is the activation energy for dissolution. These values are given in Table 6. Obviously, the activation energy ΔHs for POP and PPP is several times higher than the corresponding Esat, but the values of both ΔHs and Esat for PPP are higher than those for POP. Since the values of ΔHs reflect solute-solvent interactions, one can also attribute the values of Esat to such interactions. Sangwal6,7 suggested that eq 15 is a consequence of diffusion of solute ions/molecules in the solution. Therefore, it is natural to expect a relationship between the activation energy Esat and the nature of diffusing ions/molecules. According to the hole theory of liquids, for simple liquids in which the size of the holes is similar to that of ions/ molecules which jump into them (section 6.5.6, ref 9), the activation energy ED for self-diffusion in pure liquid electrolytes (e.g., Na in NaCl) is usually a constant, independent of temperature, and follows the relation (section 6.4.2, ref 9) ED =RG ¼ 3:7Tm
ð17Þ
where Tm is the melting point of the electrolyte. In the case of acetone as solvent with Tm = 178 K, from eq 17 one obtains ED = 5.5 kJ 3 mol-1. This value of ED is expected when the diffusing ions/molecules in solutions show ideal behavior. Such a behavior is possible at very low solute concentrations when solute ions/molecules do not associate with solvent molecules forming large-sized solvated entities and with themselves forming large-sized complexes. However, as seen from Table 5, the values of ΔHs for POP and PPP in acetone are much higher than the predicted value of ED. The fact that ED < Esat may be attributed to the participation of largesized entities in diffusion. The high solubility of POP in acetone indeed suggests strong solute-solvent interactions, leading to the formation of large clusters. Moreover, the
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ð18Þ
where the constants are A1 = 3.116 ( 0.107 and A2 = (0.412 ( 0.005) mol 3 kJ-1, with regression coefficient (RC) 0.9997. A similar relation was found earlier for aqueous solutions of different salts, but the values of constants were A1 = 4.75 and A2 = 0.367. No precise explanation of this difference is known, but one may speculate that the values of A1 and A2 are determined by the solvent used in the solution. In any case, these results suggest that the N yvlt-like relation 2 and relation 5 based on the classical 3D nucleation theory predict similar information on diffusion and nucleation processes involved during the occurrence of metastability in solutions. 3.4. Dependence of [(T0/ΔTmax)2]0, β and Z on PPP Concentration ci in POP Solutions. Figure 8a presents the plots of ratio [ΔTmax/T0]0 = exp Φ and F-1/2 against concentration ci of PPP in POP solutions. The values of ΔTmax/T0 corresponding to ln R = 0 for the determination of [ΔTmax/T0]0 were calculated from the plots of (1) ln(ΔTmax/T0) against ln R for temperature difference ΔTmax obtained by turbidimetry [i.e., ΔTmax = (T0 - Tc)], growth exotherms [i.e., ΔTmax = (T0 - Tg)], and appearance of turbidity and growth exotherm [i.e., ΔTmax = (Tc - Tg)], and (2) (T0/ΔTmax)2 against ln R for temperature difference ΔTmax obtained by turbidimetry and growth exotherms. The values of Φ and F1/2 for the calculation of [ΔTmax/T0]0 from data obtained by turbidimetry and growth exotherms are given in Tables 3 and 4, respectively. The following two features may be noted from the figure: (1) The calculated values of the ratio [ΔTmax/T0]0 from the values of Φ and F1/2 for a given concentration ci of PPP impurity in POP solution are comparable. (2) The value of [ΔTmax/T0]0 obtained from turbidimetry and growth exotherms decreases and increases, respectively, with an increase in the concentration ci of PPP impurity in POP solution. (3) In the case of temperature difference ΔTmax obtained from temperatures Tc and Tg of the appearance of turbidity and growth exotherm, the value of [ΔTmax/ T0]0 rapidly increases with an increase in the concentration ci of PPP impurity in POP solution. It may be seen from Figure 8a that, in the case of temperature difference ΔTmax = (Tc - Tg), the increase in [ΔTmax/T0]0 with an increase in ci of PPP impurity in POP solution is associated with the suppression of growth of POP by the impurity. This means that, in the case of data obtained from growth exotherms, the increase in [ΔTmax/T0]0 with an increase in ci is a result of the combined effect of the impurity on nucleation and growth of POP in the presence of the impurity. Obviously, the impurity PPP inhibits growth but promotes nucleation of POP. The above data of [T0/ΔTmax]0 as a function of ci of PPP in POP solutions may be described by the empirical relation ½ΔTmax =T0 0 ¼ ½ΔTmax =T 0 00 ð1 þ aθÞ ð19Þ
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Figure 8. Plots of (a) [ΔTmax/T0]0 and (b) [(T0/ΔTmax)2]0 against ci of PPP in POP solutions. See text for details. Table 7. Values of Constants of eq 19 for Data from Φ data
[ΔTmax/T0]00
a* (-)
KL (-)
TBT 0.0434 ( 0.0013 -1.411 ( 0.036 9.9 ( 0.5 CRY 0.0431 ( 0.0037 0.405 ( 0.150 120 ( 231 growth 0.0029 ( 0.0067 18.33 ( 45.25 53.6 ( 55.3
Qdiff (kJ/mol) 5.6 ( 0.1 11.6 ( 2.7 9.8 ( 1.6
Table 8. Values of Constants of eq 10 data technique [(T0/ΔTmax)2]0 F Φ
TBT CRY TBT CRY
508 500 541 530
b1 (-) 1.44 ( 0.76 -0.81 ( 0.04 1.92 ( 0.91 -0.89 ( 0.17
KL (-) Qdiff (kJ/mol) 46 ( 71 37 ( 5 86 ( 174 26 ( 12
9.3 ( 2.2 8.8 ( 0.3 10.8 ( 2.7 7.9 ( 0.9
where [ΔTmax/T0]00 is the value of [ΔTmax/T0]0 in the absence of additive PPP (i.e., θ = 0), a* is an empirical effectiveness parameter for the impurity, and θ is the fractional surface coverage for the impurity. Assuming that Langmuir isotherms hold, the best-fit plots were drawn for the data from Φ with the constants given in Table 7. In view of the empirical nature of eq 19 the data on [ΔTmax/ T0]0 as a function of ci of PPP in POP were analyzed according to the theoretical relation (10). Figure 8b shows the dependence of [(T0/ΔTmax)2]0 equal to F and exp(-2Φ) on ci, while the curves are drawn with the best-fit parameters given in Table 8. It may be noted that both eqs 19 and 10 describe the data on [ΔTmax/T0]0 as a function of ci of PPP in POP solutions. This suggests that the empirical impurity effectiveness parameter a* ≈ -b1/2 when the term b1θ , 1 in eq 10. As seen from Tables 7 and 8, the condition b1θ , 1 is satisfied in the case of growth exotherm data where a* ≈ -b1/2, but a* ≈ -b1 in the case of turbidity data. Since b1 = (3b - Reff) ≈ 2 and -1 for turbidity and growth exotherm data, respectively (see Table 8), Reff is equal to about 1 and 4, respectively. These values of Reff imply that PPP impurity in POP solutions does not inhibit the development of POP crystal nuclei by physically blocking the available growth sites, because 0 < Reff < 1 for this inhibition. Thus, it may be concluded that, because of its lower solubility, PPP impurity molecules act as centers for heterogeneous nucleation for the integration of POP molecules. This conclusion is corroborated by the observation that PPP impurity leads to the crystallization of a PPP-like POP phase.4 It is also consistent with the fact PPP-POP binary mixture exhibits the formation of eutectic phases for R, β0 , and β polymorphs induced by steric hindrance between the saturated and oleic acid moieties.10,11 An experimental technique for the measurement of metastable zone width records the temperature of formation of nuclei which have grown only to sizes detectable by the technique. Therefore, it can be assumed that turbidimetry measures maximum supercooling ΔTmax mainly for nucleation
Figure 9. Plots of β and Z against ci of POP in PPP solutions. Dashed curves 3 and 4 are drawn with |Reff| = 1, while solid curves 1 and 2 for |Reff| = 4 and 0.5 corresponding to data obtained from turbidimetry and growth exotherms, respectively. Table 9. Values of Constants of eq 11 data
curve
Z0 (-)
Reff (-)
KL (-)
Qdiff (kJ/mol)
TBT
1 (solid) 3 (dash) 2 (solid) 4 (dash)
0.0825 0.0825 0.0841 0.0841
4.0 1.0 -0.5 -1.0
1 5 500 30
0 3.9 15.0 8.2
CRY
with negligible contribution due to subsequent growth of stable nuclei. This means that the value of b1 obtained from turbidity data corresponds to the nucleation event alone, but that obtained from growth exotherm data also represents some contribution of growth of stable nuclei after nucleation. Figure 9 shows the dependence of β and Z on ci of POP in PPP solutions. The data of β and Z are taken from Tables 3 and 4, respectively, while the plots are drawn according to eq 11 of the theoretical dependence of Z on ci with two sets of constants given in Table 9. Figure 9 reveals that the nature of curves of the dependence of Z on ci for the data obtained from turbidity is very sensitive to the value of Reff (see Curves 1 and 3). In this case, the correct trend of the Z(ci) data can be predicted with different values of Reff lying between 1 and 4. However, the nature of the curves representing the data obtained from growth exotherms is relatively insensitive to the chosen value of Reff and may be represented with both Reff = -0.5 and -1 (see Curves 2 and 4). Obviously, the scatter in the calculated values of Z and the technique used for detection of nucleation event in the polythermal method play decisive role in the theoretical fit according to eq 11. It may be noted that the values of Langmuir constant KL and differential heat of adsorption Qdiff corresponding to
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Crystal Growth & Design, Vol. 10, No. 2, 2010
Z(ci) data from turbidimetry are very low in comparison with those obtained from data obtained from growth exotherms. This implies that, in the case of data obtained from growth exotherms, the value of Qdiff is essentially determined by the growth of stable nuclei to detectable dimensions. The fact that Reff g 1 for nucleation implies that the impurity particles favor the integration of growth units to the growing nuclei. 4. Conclusions From the present analysis of the metastable zone width of saturated solutions of 1,3-dipalmitoyl-2-oleoylglycerol (POP), tripalmitoylglycerol (PPP), and POP-PPP mixtures in acetone using the conventional polythermal method, the following conclusions can be drawn: (1) The values of parameters β and Z of eqs 2 and 5 of the self-consistent N yvlt-like approach7 and the novel approach based on classical three-dimensional nucleation theory,6 respectively, are relatively insensitive to saturation temperature T0, but depend on the investigated compound and the technique used for the measurement of maximum supercooling ΔTmax. The values of β and Z for a compound are intimately connected with its solubility in a solvent and is associated with the solid-liquid interfacial energy γ. The higher the solubility of different compounds in a given solvent, the lower are the values of β and Z. The higher values of β and Z for a compound obtained from data using growth exotherms compared to those obtained from data using turbidimetry are caused by the growth of stable nuclei to detectable dimensions. (2) The data of Φ and ln(F1/2) of eqs 2 and 5 of the self-consistent N yvlt-like approach7 and the novel approach based on classical three-dimensional nucleation theory,6 respectively, as a function of saturation temperature T0 follow the Arrhenius-type relation (15) with an activation energy Esat, which is associated with the diffusion of solute molecules in the solution. The value of Esat depends on the nature of ions/ molecules participating in diffusion. (3) The value of b1 ≈ 1 obtained from turbidity data corresponds to the nucleation event alone, but that obtained from growth exotherm data also represent
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some contribution of growth of stable nuclei after nucleation. Since b1 = (3b - Reff) and b ≈ 1, |Reff| ≈ 2, it implies that PPP impurity in POP solutions promotes the integration of POP molecules to the developing nuclei. This behavior of impurity particles is also reflected by the low values of the Langmuir constant KL and differential heat of adsorption Qdiff corresponding to Z(ci) data from turbidimetry in comparison with those obtained from data obtained from growth exotherms. The increasing value of Qdiff is associated with the growth of stable nuclei to increasingly larger dimensions. Acknowledgment. One of the authors (K.W.S.) expresses his gratitude to Loders Croklaan BV, Wormerveer, The Netherlands, for funding the experimental work. The authors are also grateful to the anonymous referees for their constructive comments on the first version of the manuscript.
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