Solid-Phase Nucleation of Bulk and Dispersed 3-Methoxy-4

Studies of the solid-phase nucleation of molten 3-methoxy-4-hydroxybenzaldehyde (vanillin) were performed either on bulk (a few cubic millimeters) or ...
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Ind. Eng. Chem. Res. 1997, 36, 874-880

Solid-Phase Nucleation of Bulk and Dispersed 3-Methoxy-4-hydroxybenzaldehyde Christine Jolivet-Dalmazzone, Pierre Guigon, Jean-Franc¸ ois Large, and Danie` le Clausse* URA CNRS, 1888 Ge´ nie des Proce´ de´ s, Universite´ de Technologie de Compie` gne, B.P. 529, 60205 Compie` gne Cedex, France

Studies of the solid-phase nucleation of molten 3-methoxy-4-hydroxybenzaldehyde (vanillin) were performed either on bulk (a few cubic millimeters) or on emulsified (cubic microns) samples. The results obtained from macroscopic observations and differential scanning calorimetry have pointed out undercooling phenomena characterized by roughly the same order of undercooling degrees ∆T ) Tm - Ts (Tm, melting temperature; Ts, the most probable solidification temperature) for steady cooled (-3 °C/min) bulk and emulsified samples, namely, ∆T ≈ 43 °C. At a fixed temperature, the solidification of the dispersed droplets can be described by an S-shaped curve that has been interpreted in terms of a polynuclear mechanism. Finally, the decrease of the undercooling degree exhibited after a solidification-melting process has been explained by the formation of nucleating sites at the liquid-emulsifying medium interface. Introduction It is generally thought that solidification is similar to melting, but experimental data show that it is a more complicated process which requires three necessary steps (Mullin, 1972): (1) The first one is undercooling: the liquid has to be cooled down to a temperature below the melting point and named the temperature of nucleation. The degree of undercooling ∆T is defined as the difference between the equilibrium temperature (or melting point) and the nucleation temperature. (2) The second one is nucleation. This step is characterized by the formation of critical nuclei in the undercooled liquid. Nucleation is a kinetic phenomenon for which only a most probable temperature of nucleation can be experimentally determined (Clausse, 1985). (3) The last one is the growth of the nuclei which depends on the thermodynamic driving force (i.e., undercooling) and which can be limited by the viscosity of the undercooled liquid. Undercooling is a parameter of prime importance in the industrial solidification processes involving the manufacture of powders or solid spheres such as atomization or prilling. The lack of data about undercooling phenomena can deeply affect the design and sizing of industrial solidification plants. Many authors studied the undercooling of a great number of compounds such as metals and alloys (Turnbull, 1950a; Turnbull and Cech, 1950; Turnbull, 1952; Rasmussen and Loper, 1975; Lemercier and Clausse, 1975), glasses (Zanotto, 1987), salts (Aguerd et al., 1984), and, of course, water (Bigg, 1953; Broto and Clausse, 1976; Langham and Mason, 1958), but few reports about organic compounds can be found in the literature (Dumas et al., 1987; Clausse et al., 1987; Cordiez et al., 1982; Tammann, 1925). 3-Methoxy-4hydroxybenzaldehyde (or vanillin) is of main interest from this standpoint because it is a solid organic compound at room temperature, displaying a very large undercooling, whatever the experimental conditions. Furthermore, an undercooled melt of vanillin can vitrify when it is submitted to high cooling rates, as was previously shown by Widmann (1987). * To whom all correspondence should be sent. S0888-5885(96)00348-X CCC: $14.00

The presumed complicated behavior of undercooled vanillin was therefore studied under various conditions from a more systematic standpoint. Differential scanning calorimetry (DSC) was used to determine the degree of undercooling of vanillin in the case of volumes of a few cubic millimeters and cubic microns. Our study was focused on the effect of several variables such as sample volume, temperature scanning rate, cooling/heating cycles, and time. Finally, a technique peculiar to our laboratory named the “copper plate technique” was used in order to analyze the behavior of small droplets undergoing a rapid cooling. Materials and Methods Materials. The chemical compound was 3-methoxy4-hydroxybenzaldehyde (also called vanillin) sold by PROLABO as a crystallized white powder with a purity of 99% and a melting point of about 81 °C. It is worth mentioning that the spontaneous solidification of bulk vanillin can give at least three polymorphic forms (McCrone, 1950). That is the reason why the melting point of solidified vanillin can vary from 79 to 83 °C, mainly 80 °C. In order to study samples of a few cubic microns by DSC, we used the “droplet experiment” first proposed by Vonnegut (1948) and intensively used to study emulsified metals (Turnbull, 1952; Rasmussen and Loper, 1975) and emulsified aqueous solutions (Aguerd et al., 1984; Broto and Clausse, 1976). The principle of the method is to disperse the material as an emulsion in a suitable medium, generally containing some emulsifier in order to ensure its stability. The main advantage of the droplet experiment is the suppression of several factors leading to the aleatory crystallization of a bulk melt, e.g., the walls of the container or impurities. The only way to obtain suitable emulsions was to slightly disperse the warm liquid into a silicone oil of high viscosity (100 000 mm2/s) at 130 °C. The typical composition of the emulsions was about 10% (by weight) of vanillin in silicone oil. It is noteworthy that no surfactant was used to stabilize the emulsions. The mean diameter of the droplets of vanillin in the emulsion was about 50 µm. Differential Scanning Calorimetry (DSC). The temperatures and enthalpies of melting or solidification © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 875

of vanillin were measured from the liberated heat of phase change by differential enthalpy analysis with a constant scanning rate (DSC). The method consists of monitoring the difference of heat flux between the sample and a reference cell. The fundamental equation of the differential enthalpy analysis is the following (Gray, 1968):

dTp dq d2q dh ) (Cs - Cr) - RCs 2 dt dt dt dt

(1)

It is composed of three terms: the first one is used to compensate the differences between the heat capacities of the sample (Cs) and of the reference (Cr); the second one represents the transitory phenomena due to the release (or absorption) of heat by the sample, R being the thermal resistance between the sample and the heater and RCs being the time constant of the cell containing the sample. The third one represents the production of heat by the sample (h). The DSC thermograms have been widely studied from a theoretical standpoint (Dumas et al., 1975). The apparatus used was a DSC 111 SETARAM. The crucibles were made either of aluminum or alumina. The “Plate Technique”. It is not always possible to achieve a suitable emulsion of a chemical compound in order to study its solidification characteristics from a statistical point of view, especially if this compound is solid at room temperature. The choice of the emulsifying medium and of the surfactant is mostly not easy. That is the reason why we decided to carry out a simple experiment to test the behavior of a population of small droplets undergoing a rapid cooling and to determine the most probable temperature of solidification. The principle of this technique, that we named the “plate technique”, was the following: A 2-mm-thick copper plate was put on a warming plate at a temperature higher than the melting point of the compound. Some liquid droplets were then deposited on that plate by means of a capillary tube. At time zero, the copper plate was quickly put into a thermoregulated bath, stabilized at a temperature lower than the melting point. The equilibrium temperature was reached in a few minutes, and the number of solidified droplets was recorded. The solidification of the droplets was visible to the naked eye because of their color change (from transparent to white). In the case of vanillin, the experiment was carried out on at least 100 droplets, in order to get a statistical representation of the phenomenon. Results DSC of Samples of a Few Cubic Millimeters. The effect of different variables on the degree of undercooling of vanillin was studied. The tests were performed on at least five samples in order to get information about the domain of variation of the solidification temperatures. The highest degrees of undercooling were obtained with the smallest samples (Table 1). This experimental result is often reported in the literature (Broto and Clausse, 1976). It is noteworthy that the higher the volume of the sample, the higher the degree of variation of the temperatures. This result is quite normal if we consider that the number of factors able to affect the crystallization is higher in a large sample than in a small one. Table 1 also shows the effect of the temperature scanning rate on the temperatures of solidification. The

Table 1. Effect of the Sample Weight and the Scanning Rate on the Degree of Undercooling (Degree of Overheating ∆T+ ) 6 °C) sample wt (mg)

scanning rate (°C‚min-1)

Ts interval (°C)

Ts (°C)

∆T (°C)

30 30 10 10 10

-1 -5 -1 -3 -5

40-42 38-40 41-43 35-41 34-40

41 39 42 38 37

39 41 38 42 43

Table 2. Effect of the Degree of Overheating ∆T+ on the Degree of Undercooling (Sample Weight ) 10 mg) ∆T+ (°C)

scanning rate (°C‚min-1)

Ts interval (°C)

Ts (°C)

∆T (°C)

6 6 16 16 46 46

-1 -5 -1 -5 -1 -5

41-43 34-40 38-44 35-41 38-47 34-38

42 37 41 39 42 36

38 43 39 41 38 44

Table 3. Effect of the Heating/Cooling Cycles on the Temperatures of Melting Tm and Solidification Ts (Sample Weight ) 10 mg) sample

temp (°C)

1st cycle

2nd cycle

3rd cycle

1 (∆T+ ) 6 °C)

Tm Ts Tm Ts Tm Ts Tm Ts

81.5 40.7 81.0 40.6 80.3 42.1 81.3 43.9

80.7 44.2 80.2 43.8 79.9 40.7 80.2 49.4

80.3 43.4 80.2 44.3 79.1 40.0 80.5 50.5

2 (∆T+ ) 6 °C) 3 (∆T+ ) 16 °C) 4 (∆T+ ) 16 °C)

mean degree of undercooling rises up with the temperature scanning rate. These results confirm the kinetic nature of the nucleation process. Table 2 illustrates the influence of the overheating of the molten compound before solidification. Some workers (Lemercier and Clausse, 1975) found an effect in the case of tin (Sn): the higher the degree of overheating, the higher the undercooling. In our case, there is no evidence of such an effect. To conclude this part of the study, the influence of successive cycles of heating and cooling was examined (Table 3). From an experimental point of view, the cycle effect depends on the compound under investigation (Clausse et al., 1987). In our case, the successive cycles allow a decrease of undercooling. DSC of Emulsions of Vanillin in Silicone Oil. 1. Constant-Temperature Scanning Rate. Open crucibles made of alumina were used to study emulsions of vanillin in silicone oil by DSC. These special crucibles allow preparation of samples of emulsions directly on a warming plate, avoiding any solidification of the sample before DSC. The temperature scanning rate was -3 °C/ min. Figure 1 shows typical thermograms of heating and cooling of such an emulsion prepared at 130 °C. The melting point of vanillin which solidified in emulsion depends on the different polymorphic forms of the spontaneously growing crystals in the melt (McCrone, 1950). In our case, it varies from 79 to 81 °C. The most probable solidification temperature is given by the actual temperature of the peak maximum: 37 °C[ ((1 °C), that means a degree of undercooling of about 43 °C. This is very close to the undercooling previously obtained with samples of a few cubic millimeters. Higher differences are generally found. In the case of emulsions prepared at 150 or 110 °C (instead of 130 °C), the results are quite similar. As

876 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

Figure 1. Heating and cooling thermograms of an emulsion of vanillin.

Figure 3. Solidification time of emulsions of vanillin versus temperature.

Figure 4. Solidification rate of vanillin droplets in the emulsion versus time (T ) 46 °C). Figure 2. Cooling thermograms obtained in the case of solidified emulsions of vanillin warmed up again to 87 (a) and 130 °C (b).

previously mentioned, we did not notice any significant influence of the initial overheating degree on undercooling. Some experiments were carried out by using an emulsion previously solidified in the calorimeter. The cooling thermograms drastically differed when the solidified emulsion was heated up to a temperature very close to the melting point (a few degrees above 80 °C). For instance, an emulsion solidified at room tempersture and warmed up again to a temperature of 87 °C sometimes exhibited a temperature of solidification Ts above the temperature observed in the case of freshly prepared emulsions (Ts ) 50 or 60 °C instead of 37 °C). This effect tends to disappear if the emulsion is warmed up to a temperature close to 130 °C (Figure 2). 2. Isothermal Studies. As reported before, the nucleation process can be affected by different variables such as volume, temperature scanning rate, or thermal treatments. Time is another important variable. A chemical compound can solidify at a temperature (T) above the most probable temperature of solidification after a time (t) which increases as the temperature gets closer to the melting point (Clausse, 1985). The influence of “time” on undercooling was therefore studied in the case of vanillin emulsions, in order to get a statisti-

cal representation of the phenomenon. The experimental procedure was the following: The sample was put into an alumina crucible which was heated on a warming plate at 130 °C. The sample was then inserted very quickly into the head of the calorimeter (t ) 0 s). The thermal signal was recorded as a function of time at a constant temperature. An exothermic peak was observed when solidification was occurring. Figure 3 shows the time corresponding to the maximum of the solidification peak as a function of temperature, in the case of five different emulsions. Up to 36 °C, the curve is almost parallel to the abscissa axis: solidification is probably taking place before reaching the equilibrium temperature. At about 40 °C, the slope is higher, and at 45 °C, it tends asymptotically to infinity. Figure 4 represents the solidification rate (from 0 to 100%) of droplets of vanillin in the emulsion versus time, at 46 °C (9 °C above the onset of solidification of the dispersed vanillin according to Figure 1). The S shape of this curve, already observed on water emulsions (Clausse et al., 1983), will be further explained. 3. Warming Up of Emulsions Quenched below Ts. We studied the effect of the cooling rate on the behavior of undercooled vanillin. The experimental procedure was the following: The sample of emulsion was deposited in an alumina crucible heated to 130 °C.

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Figure 6. Solidification rate of droplets versus temperature (plate technique).

Figure 5. Heating thermograms of emulsions of vanillin quenched to -20 (a) and -70 °C (b) and of pure vanillin quenched to -70 °C (c).

It was then rapidly put into the head of the calorimeter, previously regulated at a temperature T- below Ts (Ts ) 37 °C). The equilibrium was reached after about 2 or 3 min. The emulsion was then heated at a temperature scanning rate of 5 °C/min. Temperatures T- of -70, -40, -30, -20, and -10 °C were studied. An example of the thermograms obtained with this method is given in Figure 5. It clearly shows an exothermic peak of solidification at about 0 °C. This peak is sometimes followed by a smaller one at 20-30 °C (Figure 5a, T- ) -20 °C). The melting of the compound at 80 °C is then observed, as expected. In the case of Figure 5b (T- ) -70 °C), an endothermic peak at -50 °C corresponds to the melting of the silicone oil. It is obvious that the compound does not completely solidify for temperatures T- below -10 °C, because, at these low temperatures, the high viscosity of the undercooled liquid probably avoids the growth of crystal nuclei. But, when the emulsion is heated again, the viscosity decreases and allows the crystal growth at about 0 °C. We checked that vanillin was submitted to vitrification when cooling was performed down to -70 °C, by studying the behavior of a sample of pure vanillin (not emulsified) in the same conditions (Figure 5c). The thermogram clearly shows a baseline change at -30 °C, which is characteristic of a glass transition. This devitrification phenomenon was previously concealed by the melting peak of silicone oil in the case of emulsions. The Plate Technique. Figure 6 shows the rate of solidification as measured by the fraction of the droplets that are solid versus the temperature of the thermoregulated bath, at t ) 2 min and 2 min 30 s (see the description of the method in the section Materials and Methods). A typical S-shaped curve is observed: above

50 °C, no droplet solidifies, and below 30 °C, 100% of the population of droplets becomes solid. The most probable temperature of solidification is determined by reporting the temperature at which 50% of the droplets are solid. In our case, this temperature is about 35 °C. It is noteworthy that this value is very close to the temperature reported in the case of DSC experiments at -3 °C/min. The simple experiment of the plate technique therefore confirms the previously reported results. It is a very helpful method to study the solidification of organic compounds, because it rapidly gives interesting visual information on the behavior of compounds submitted to temperatures lower than the most probable temperature of solidification. Samples of a few grams of vanillin were rapidly quenched down to temperatures close to 0 °C: we observed that nucleation occurred during the quench (some white points appeared in the bulk melt), but the growth of crystals was very slow, because of the high viscosity of the melt. When the copper plate was put at room temperature again, a rapid acceleration of the solidification process was observed. This visual observation confirmed the assumption we made: at low temperatures, the high viscosity of the undercooled liquid avoids the growth of crystal nuclei. Discussion The main points to be discussed here are the following: (1) Do our experimental results agree with the previous ones? (2) Can our results be interpreted according to the classical nucleation theory? (3) How can we explain the effect of thermal treatments? Scanning and Isothermal Experiments. Our experimental study of vanillin by DSC clearly shows the occurrence of an interval of phase change temperatures. In the case of emulsions, the interval of temperatures is located between 25 and 42 °C. Of course, our emulsions are not actually monodispersed, but the reproducibility of the thermograms suggests that a solidification temperature distribution exists and that it is possible to determine a most probable temperature of solidification (Clausse, 1985). However, if it is possible to calculate the nucleation rate J (number of nuclei formed per time and volume

878 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

Figure 7. ln J versus 1/(T∆T2) (J ) nucleation rate, T ) temperature, and ∆T ) degree of undercooling).

unit), it is not possible to determine the place where the nuclei are formed into the volume. The appearance of a nucleus is essentially a random phenomenon for which only a probability P can be predicted (Clausse, 1985; Dufour and Defay, 1963). If we consider a number N0 of identical droplets whose volumes are V, Nt being the number of solidified droplets at time t, the probability of solidification is defined as:

P(V,t) )

Nt N0

(2)

Assuming that each solidified droplet observed corresponds to the appearance of one nucleus, the following relationship between J and P can be determined:

∫0tVJ(T) dt

Ln(1 - P) ) -

(3)

At a constant temperature T, J and V being not dependent on t, this equation can be expressed as:

Ln

(

)

N0 ) JVt N0 - Nt

(4)

The experimental results from the plate technique allowed one to determine the number of solidified droplets as a function of temperature at a given time. J was then calculated from eq 4 in the case of temperatures from 25 to 40 °C, at time t ) 2 min, considering that the average volume of the droplets is V ) 10-3 cm3. According to the classical nucleation theory, a linear relationship between ln J and 1/(T∆T2) should be observed (Turnbull, 1956; Dufour and Defay, 1963). Figure 7 shows the variations of ln J versus 1/(T∆T2) which are actually linear with a correlation coefficient of 0.99. Unlike many authors, we did not observe any significant effect of the sample volume on the nucleation temperature. Dumas (Dumas et al., 1987; Dumas, 1976), who studied several organic compounds by DSC, found the same behavior only in the case of aniline (C6H5NH2). The effect of the volume on the nucleation temperature is very controversial. Some authors think that the experimentally-observed influence of the volume is the result of the unavoidable presence of impuri-

ties in the bulk sample. But, if it is assumed that the total solidification is due to the appearance of the first nucleus, the probability of nucleation P is also volumedependent, since from eq 3, the larger the volume, the higher the probability of nucleation at a temperature T. In our case, it is now absolutely impossible to check this assumption from an experimental standpoint. It can only be suggested that the mechanisms of nucleation found in the case of emulsions or in the case of bulk samples of vanillin are identical, whatever they are (homogeneous or heterogeneous). If the experiment is performed at a constant cooling rate T˙ ) -dT/dt, then the expression of the probability of nucleation also depends on the temperature scanning rate (see eq 3): the larger the cooling rate, the smaller the probability of nucleation at any temperature T. In our case, the most probable temperature of nucleation does not significantly change with the scanning rate, but a modification of the interval of transformation can be observed. It becomes larger as the scanning rate increases. Dumas (1976) noticed the same behavior in the case of emulsions of benzene. Equation 4 can be expressed in a different way, that means:

Nt/N0 ) 1 - exp(-VJt)

(5)

In that case, the general shape of the curve showing the solidified droplets rate (Nt/N0) as a function of time t is an exponential one. If we consider the isothermal experiments (cf. section 2 of the section Results: DSC of Emulsions of Vanillin in Silicone Oil), it is clear that the curve obtained in this way has an S shape and not an exponential one, as expected from the theory. Clausse et al. (1983) have established a similar “S” curve in the case of the isothermal freezing of water. We have interpreted these “S” curves by using the analysis of the induction time and the metastability limit during a phase change applied to disperse systems (Kashchiev, 1989; Kashchiev et al., 1991, 1994). It appears that two approaches are generally used for the theoretical determination of the induction time, i.e., the time necessary to reach a given degree of crystallization. The mononuclear approach is based on the assumption that the appearance of the first nucleus is responsible for the nucleation process. In this case, the probability that at least one nucleus is formed at time t in a compact phase is given by the previously mentioned equation (5). The corresponding curve, therefore, exhibits an exponential shape. The polynuclear approach assumes that the formation and growth of a sufficient number of nuclei are responsible for the nucleation process. The probability of phase change is then given by the following relationship:

(

)

-Vex(t) V

P ) 1 - exp

(6)

with Vex ) extended volume, i.e., the volume of the solid phase if there was no overlap of the growing nuclei. In the particular case of a three-dimensional growth of spherical nuclei, the probability of phase change P is no more dependent on V (Kashchiev et al., 1991):

(

P ) 1 - exp

)

-πJG3t4 3

(7)

with G ) crystal growth rate. The corresponding curve exhibits an S shape.

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 879

According to this analysis, a polynuclear mechanism for which more than one nucleus is needed for the phase change has to be considered. That is likely the reason why no influence of the volume on the temperature of solidification was observed in our experimental results. Effect of Thermal Treatments. According to the classical nucleation theory, the nucleation rate J only depends on temperature, pressure, and nuclei size. The thermal history of the liquid (or metastable crystal) is surprisingly never taken into account straight. As previously described, we found an effect of thermal treatments on the temperature of solidification: (1) The successive cycles of heating/cooling show a decrease of undercooling (case of samples of a few cubic millimeters); (2) An emulsion of vanillin previously solidified at room temperature and warmed up again to a temperature slightly above the melting point sometimes shows a temperature of solidification above the most probable temperature determined during the first cycle. It seems that the first heating cycle modifies the sample and favors the nucleation process during the second cycle. This modification tends to disappear when the emulsion is warmed up to a temperature far above the melting point. The same type of behavior was previously observed in the case of water emulsions by Clausse et al. (1987). Different attempts of explanation have been made: (1) That effect could be due to a modification of the intensive properties of the phase, involving a transfer of the temperature of solidification, but no change in the melting point is observed and that assumption must be rejected. (2) A preservation of stable nuclei in the liquid phase can be proposed. However, it is generally admitted that the lifetime of that kind of nuclei is less than 10-10 s (Franck, 1958). (3) Some authors (Turnbull, 1950b; Fukuta, 1966) think that very small particles of crystal are confined in pores on the walls of the container after melting. A thermodynamic study shows that the melting point of nuclei confined in pores ( TD. The authors determined the values of TD for different crystalline substrates. To explain their experimental results, they assumed the existence of an adsorbed monolayer on the substrate after the first crystallization and that this layer can keep the structure of the crystal. This ordered layer would not disappear at Tm but at the higher temperature TD. At the following cooling, if an aggregate appears on an ordered layer, the potential barrier is weaker because of the similarity of structure and the crystallization is very

probable. If the highest temperature reached above melting is higher than TD, the ordered layer disappears and nucleation occurs on a disordered layer or in the volume, which gives a lower probability of crystallization. Clausse et al. (1987) made similar assumptions to explain their experimental results in the case of water emulsions, considering that the freezing of water in emulsion around a temperature T above the most probable temperature of nucleation is possible by the existence of nucleating surfaces at the liquid-emulsifying medium interface (heterogeneous nucleation), whose areas depend on the number of thermal cycles or on the value of the temperature reached above Tm. This explanation can be used to interpret our experimental results in the case of emulsions or samples of a few cubic millimeters. It can be assumed that an adsorbed and ordered liquid layer is formed at the interface during thermal treatments. If the nucleation occurs on this layer, a smaller undercooling is observed during the next cycle. This effect disappears if the sample is warmed up to a temperature T close to 130 °C (.Tm). Conclusion The main conclusions of the present work are as follows: (1) Differential scanning calorimetry (DSC) is a very powerful technique to study the nucleation and solidification of liquid samples. It allows an easy determination of undercooling as well as a precise characterization of some important phenomena more or less linked to undercooling. But when DSC is not available, the plate technique can be a helpful “replacement technique”. The degrees of undercooling can be easily determined by recording the fraction of the droplets that are solid versus temperature. Furthermore, it rapidly gives an idea of the behavior of a sample submitted to a more or less rapid cooling. For instance, it allows the determination of a domain of temperatures where the growth rate of crystal nuclei drastically decreases. (2) The experimental results obtained from the DSC of samples of a few cubic millimeters and emulsions can be interpreted according to a mechanism of polynucleation for which several nuclei are responsible for the nucleation process. By this approach and in the particular case of a three-dimensional growth of spherical nuclei, the probability of phase change is given by an equation describing an S-shaped curve, as was experimentally observed during the isothermal crystallization of vanillin. Furthermore, this probability is no more dependent on the sample volume. It can therefore explain why we did not observe any significant difference between the most probable temperature of solidification in large and small volumes. (3) The effect of thermal treatments is not easily explainable. Our experimental observations are very similar to those obtained by Clausse et al. (1987) on water emulsions, and we agree with their assumption of the formation of nucleating surfaces at the liquidemulsifying medium interface. These nucleating surfaces could be ordered layers whose areas depend on the number of thermal cycles or on the value of the temperature reached above the melting point during the warming. If they remain after the heating cycle, a smaller undercooling is observed during the next cooling cycle. Nomenclature Cs ) heat capacity of the sample Cr ) heat capacity of the reference

880 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 dh/dt ) power due to a thermal phenomenon dq/dt ) power registered by the calorimeter G ) crystal growth rate J ) rate of nucleation N0 ) number of droplets Nt ) number of solidified droplets at time t P ) probability of nucleation R ) thermal resistance between the sample and its heater t ) time T˙ ) cooling rate T ) temperature T- ) temperature of quenching Tm ) melting temperature Ts ) temperature of solidification Ts ) mean temperature of solidification ∆T ) undercooling degree ∆T ) mean undercooling degree Tp ) temperature of the thermostatic block V ) volume

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Received for review June 17, 1996 Revised manuscript received October 23, 1996 Accepted November 5, 1996X IE960348W

X Abstract published in Advance ACS Abstracts, January 15, 1997.