Short-Cut Method To Predict the Solubility of Organic Molecules in

Sep 9, 2005 - In this work, differential scanning calorimetry (DSC) was applied for estimating the solubility of organic molecules, in aqueous and non...
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Short-Cut Method To Predict the Solubility of Organic Molecules in Aqueous and Nonaqueous Solutions by Differential Scanning Calorimetry Rosana E. Tamagawa,*,† Wilson Martins,‡ Silas Derenzo,† Andre´ Bernardo,† Marlus P. Rolemberg,‡ Phillipe Carvan,‡ and Marco Giulietti†

CRYSTAL GROWTH & DESIGN 2006 VOL. 6, NO. 1 313-320

Instituto de Pesquisas Tecnolo´ gicas do Estado de Sa˜ o Paulo, Agrupamento de Processos Quı´micos, AV. Prof. Almeida Prado 532, Cidade UniVersita´ ria, 05508-901, Sa˜ o Paulo (SP), Brasil, and Rhodia Polyamide and Specialties, Fazenda Sa˜ o Francisco s/no, 0713140-000, Paulı´nia (SP), Brasil ReceiVed April 6, 2005

ABSTRACT: According to recent literature,1-4 differential scanning calorimetry (DSC) is being proposed as an alternative technique for the determination of solubility curves with the advantage of requiring a small quantity of sample (in the order of milligrams), besides being a fast method: solubility data can be accessed in a few hours. In this work, DSC was applied for estimating the solubility of organic molecules in aqueous and nonaqueous solutions. The method was carried out using four organic substances, referred to here as substances A, B, C, and D, A and B being solubilized in water, C in a mixture of methanol-water 20 wt %, and D in acetone. Introduction Determination of solid-liquid equilibrium is an essential step in the design and development of a crystallization process. It is, however, a time-consuming task, besides being restricted when samples are expensive and available in low quantities. The traditional methodology for determining solubility data consists of keeping the solution at a constant temperature under mixing and measuring its concentration periodically until the equilibrium is reached. A single equilibrium point may take a few hours to several days to be determined. The procedure is then repeated at other temperatures to determine the solubility curve. In this study, we used an alternative methodology to determine solubility curves with the advantages of being fast and consuming small quantities of sample (in the order of milligrams). The experimental procedure was the same as that used by Mohan et al.;2 however, different approaches were applied for the data analysis. In this methodology, the solidliquid mixture is heated in the DSC calorimeter until all of the solute is dissolved. Dissolution is confirmed by the presence of a peak in the DSC curve, with the end set temperature of the peak characterizing the end of the dissolution. The procedure is repeated at different heating rates causing an effect on the end set temperature, which usually increases as the heating rate is increased. This effect, known in the DSC theory as the thermal lag, is a delay on the instrumental response and can be assumed to be a linear function of the heating rate.5 Based on this, the approach used by Mohan et al.2 to determine the saturation temperature is the linear extrapolation of the end set temperatures for a heating rate equal to zero.2 The extrapolated temperature is assumed to be the saturation temperature for that solution, meaning that, if the solution is kept at this temperature with no heating process, all of the solute is dissolved at a certain instant. This procedure provides one point of the solubility curve. The procedure is then repeated for other concentrations to determine other points of the curve. * Corresponding author. Telephone: 55-11-3767-4682. Fax: 55-11-37674052. E-mail: [email protected]. † Instituto de Pesquisas Tecnolo ´ gicas. ‡ Rhodia Polyamide and Specialties.

The limitation of this method is that the dissolution rate must be fast enough to ensure that the solution is at equilibrium at every instant of the dissolution process. Otherwise, not only the thermal lag, but also a delay on the dissolution process, may affect the end set of the peak. With the thermal lag then not being the only parameter influencing the end set, the linear extrapolation for finding the solubility may not be valid. In this work, we propose a different approach, where the DSC curves are analyzed with regard to the energy and mass balances associated with them, allowing the application of the method for systems with slow dissolution. In this procedure, the dissolution peak enthalpy of each solute-solvent system is converted into concentration increments, which are then related to the respective solubility curves. The advantage of this strategy is that a whole range of the solubility curve can be estimated on the basis of the dissolution curves collected from just one sample, besides the suitability for slow dissolution systems. Experimental Section Instrumentation. The DSC data were collected with an 822e DSC calorimeter from Mettler Tolledo (Switzerland). This instrument was equipped with an automatic sampler robot programmable for automatically introducing and removing up to 34 samples into the furnace cell, and ensuring that the crucible was symmetrically placed over the heating disk. The measuring system was equipped with 56 thermocouples (28 per crucible position). The allowed range of temperature was from -65 to 700 °C, and the range of measurement was (350 mW, with a resolution of 0.04 mW (at room temperature). Data collection and analysis were performed with the Stare software (Mettler Toledo). Samples were weighted with an analytical balance (Mettler Toledo XS205) with a capacity of 220 mg and a readability of 0.1 mg. Sample Preparation and Pretreatment. Samples were prepared by weighing a certain amount of solute (1.0-5.0 mg) within the crucibles, followed by the addition of a certain amount of solvent (5100 µL). Water was used as the solvent for substances A and B, a 20.0 wt % mixture of methanol-water for substance C, and acetone for substance D. The aqueous solutions were prepared with deionized water, and other solutions were prepared with analytical grade solvents. Although crucibles were hermetically sealed, possible solvent loss during the heating intervals was checked by weighing the crucibles before and after each heating cycle. Two types of crucibles were used in this study: the 160 µL aluminum crucibles for the substance B solutions and the 40 µL aluminum

10.1021/cg050128y CCC: $33.50 © 2006 American Chemical Society Published on Web 09/09/2005

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Figure 1. Typical DSC curve recorded from the dissolution of substance A in water. Final solute concentration: 15.0 wt %. Heating rate: 1 °C/min. crucibles for the other solutions. The large volume crucibles were used in the case of substance B to allow the use of larger samples and to amplify the DSC signal up to a detectable level. The requirement of a certain amount of sample to provide a detectable signal depends on the dissolution enthalpy of each particular system. To define the proper amount of sample, a check through preliminary essays is necessary for most of the cases. During data acquisition, a heterogeneous distribution of the crystals inside the crucible may cause a not uniform heat flux, affecting the dissolution profile. To attenuate this variable, samples were submitted to a pretreatment, which consisted of heating the sample until complete dissolution of the mixture, holding the temperature for about 30 min, and finally cooling it to a temperature suitable for nucleation and recrystallization of the solute. A rate of 2 °C/min was used in the cooling process. Through this procedure, we intended to provide a more homogeneous distribution of the crystals inside the crucibles, improving the repeatability of subsequent dissolutions.

Results and Discussion Extrapolation of the End Set Temperature for Determining the Solubility. The DSC-based approach for determining solubility data by the linear extrapolation of the end set temperature to a heating rate equal to zero was evaluated in this study for four different substances. The results of each mixture were compared to the solubility data previously determined by the traditional isothermal method. The experimental conditions and the results of each system are presented below. (a) Substance A. Approximately 25.0 mg of the substance A-water solution at 15.0 wt % was initially heated for 30 min at 80 °C and cooled from 80 to 5 °C at a rate of 2 °C/min. After this pretreatment at which the solute had been completely dissolved and recrystallized, the mixture was maintained at 5 °C for 30 min to stabilize the heat flux, and heated from 5 to 80 °C. Once the dissolution peak was observed, sample was cooled again to 5 °C, prompting the sample for the next heating cycle at a different heating rate. The same procedure was carried out by applying the heating rates of 0.3-1.5 °C/min. In Figure 1, a typical DSC dissolution curve of substance A in water is presented. The end set temperature of the dissolution peak was 66.89 °C when the sample was heated at a rate of 1 °C/min. As expected, when the sample was heated at different heating rates, the end set temperature changed linearly with the heating rates, see Figure 2. By extrapolating the temperature to a hypothetical “zero” heating rate, we estimated the saturation temperature of substance A at the given concentration (C ) 15.0 wt %, Tsaturation ) 61.98 °C). In Figure 3, the estimated equilibrium point was plotted with the solubility curve determined by the traditional isothermal

Tamagawa et al.

Figure 2. The extrapolation of the end set temperature of the DSC dissolution peaks to a zero heating rate. Sample: substance A in water, 15.0 wt %.

Figure 3. The DSC solubility data for substance A in water and the equilibrium curve determined by the traditional isothermal method.

methodology. The DSC data matched reasonably well with the curve, indicating the suitability of the technique for this particular case. (b) Substance B. Approximately 90.0 mg of the substance B-water solutions at the concentrations of 3.0-5.0 wt % were prepared within aluminum crucibles and submitted to a pretreatment that consisted of keeping the sample for 30 min at 95 °C followed by cooling from 95 to -40 °C. After the pretreatment, each sample was heated at different heating rates (0.3-1.4 °C/min), providing a set of dissolution curves for each concentration. At the end of every dissolution segment, the temperature was held at 95 °C for 10 min and then lowered to -40 °C at a rate of 2 °C/min to recrystallize the solute. Such drastic cooling was necessary in this case, because we were not able to recrystallize the solute at higher temperatures. As expected, when samples were heated at different heating rates, the end set temperatures changed accordingly, following linear functions. However, the extrapolation of the end set temperatures for a zero heating rate did not agree with the saturation temperatures; values differed up to 15 °C from the expected temperatures (data not shown). The reason for the observed result was probably the slow dissolution rate, which, under the applied heating rates, did not allow the solution to reach the equilibrium in the course of the dissolution, contributing to the increase of the end set temperatures. Consequently, because the heating rate was not the only parameter influencing the end set temperature, the linear extrapolation of the end set to find the saturation temperature was not valid. To check the behavior of the end set temperatures for other heating rates, experiments were carried out at the rates of 0.05-3 °C/min. By doing so, we found that the relationship between the end set temperature and the heating rate assumed a nonlinear function (Figure 4) and the extrapolated temperatures approached a bit more to the expected saturation temperatures

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Figure 4. The extrapolation of the end set temperature of the DSC dissolution peaks to a zero heating rate. Sample: substance B in water. Concentrations: 5.0, 4.0, and 3.0 wt %.

Figure 6. The extrapolation of the end set temperature of the DSC dissolution peaks to a zero heating rate. Sample: substance C in methanol-water 20.0 wt %. Solute concentration: 7.8 and 10.8 wt %.

Figure 5. The DSC solubility data for substance B in water and the equilibrium curve determined by the isothermal method.

Figure 7. The DSC solubility data for substance C in methanol-water 20.0 wt % and the equilibrium curve previously determined by the isothermal method.

when lower heating rates were used (Figure 5). However, they still differed up to 5 °C from the theoretical curve We concluded then that this methodology was not suitable for substance B, because the end set temperature was not linear with the heating rate and the extrapolation for a zero heating rate did not converge to the saturation temperature. (c) Substance C. For substance C, we used a 20.0 wt % methanol-water mixture as solvent, solute concentrations of 7.8 and 10.8 wt %, and total sample masses of 21.70 mg. Samples were initially pretreated with an isothermal segment at 45 °C for 30 min, followed by cooling to -50 °C at a rate of 2 °C/min. After the pretreatment, they were heated from 25 to 45 °C, at the rates 0.05-0.5 °C/min. Every heating interval was followed by an isothermal segment at 45 °C followed by cooling to -50 °C to recrystallize the solute. As in the previous case, such drastic cooling was necessary, because when higher temperatures were applied no crystallization was observed. When the samples were heated at different heating rates, the end set temperatures changed accordingly (see Figure 6). However, as in the case of substance B, these functions were not linear. In Figure 7, the extrapolated saturation temperatures are compared to the solubility data previously determined by the traditional isothermal method. The estimated values deviated up to 2.4 °C from the equilibrium curve. (d) Substance D. To check the suitability of the DSC method for a nonaqueous system, we used acetone as solvent for the substance D. Samples of approximately 7.0 mg with 17.5 and 22.0 wt % of substance D in acetone were weighed inside the aluminum crucibles. Samples were initially submitted to a pretreatment, which consisted of holding the crucibles at 40 °C for 30 min followed by cooling to -60 °C to recrystallize the solute. After the pretreatment, samples were held at 10 °C for 60 min and heated from 10 to 40 °C at the rates of 0.02-0.5

Figure 8. The extrapolation of the end set temperature of the DSC dissolution peaks to a zero heating rate. Sample: substance D in acetone. Concentrations: 22.0 and 17.57 wt %.

°C/min. Each heating cycle was followed by an isothermal segment at 40 °C and a cooling segment from 40 to -60 °C. The end set temperatures of theses curves as function of the heating rates are presented in Figure 8. Again, theses values were better fitted with nonlinear functions. The extrapolated values were then compared to data previously determined by the traditional isothermal method (Figure 9). Although the extrapolation did not follow a linear function, the estimated saturation temperatures were in good agreement with the equilibrium curve. In this case, we suppose that at the lower heating rates equilibrium was reached and then the extrapolation coincided with the saturation temperature. In conclusion, the end set extrapolation approach worked reasonably well for two of the four studied compounds. For the other two, some disagreements were found as follows: (1) The end set temperature was not a linear function of the heating rate. (2) Even using a nonlinear relation instead, the extrapolation of the end set temperature failed. An explanation for such

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Figure 9. The DSC solubility data for substance D in acetone and the equilibrium curve previously determined by the traditional isothermal method.

Figure 10. The contribution of the solution heat capacity in the DSC dissolution curve (substance A in water 15.0 wt %). Heating rate: 0.5 °C/min. Total sample mass: 25.0 mg.

behavior is that the low dissolution rates of these systems affected the end set temperatures and then, because the thermal lag was not the only parameter affecting the end set of the peak, its dependence on the heating rate diverged from a linear function, at the applied heating rates. With these problems in the end set extrapolation method, other methodology for determining solubility based on DSC curves is being proposed in this study. This methodology consisted of coupling the enthalpy and the mass balances of dissolution and relating them to the equilibrium curves. To accomplish the enthalpy balance, the heat capacities of solvents and solutes were experimentally determined via DSC, allowing the simulation of the suspension heat capacities in the course of the dissolution and providing the estimative of the dissolution enthalpy profiles. The Enthalpy and Mass Balance of Dissolution and the Simulation of the Suspension Heat Capacity. Here, we briefly describe the enthalpy balances of the dissolution process recorded in the DSC curves, assuming that the measured enthalpy was comprised by the latent heat of the dissolution added to the heat flux provided by the suspension heat capacity (eq 1). The suspension heat capacity is given by the sum of the specific heat capacities of the solvent, solute, and nondissolved solid (Cpsolvent, Cpsolute, Cpsolid), multiplied by their respective masses (msolvent, msolute, msolid).

dissolution and allowing the solution to be at equilibrium; and (2) that dmsolute/dT was equal to zero, meaning that no dissolution occurred during the heating interval. We remind that dmsolute/ dT is equal to -dmsolid/dT, and m ) dm/dT*∆T. In practical terms, the first possibility would be the case of a solution with a dissolution kinetic that is fast enough to allow the solution to be at equilibrium at every temperature of the heating interval, whereas the second possibility would be the case of a solution with a dissolution kinetic that is so slow that the concentration change approaches to zero. These simulations were made on the basis of the fact that the real situation would be between these two possibilities. The simulation of the solution heat capacities based on the above assumptions gave an idea of the contribution of the heat capacity terms to the heat flux signal recorded in the DSC curves. In Figure 10, the simulated heat capacities for the substance A suspension with maximum and with no dissolution are plotted with the DSC dissolution curve. We observed that the difference between the heat capacities curves was relatively small, as compared to the magnitude of the dissolution peak, meaning that the suspension heat capacity does not change significantly as the dissolution takes place. Also, we noticed that the suspension heat capacity curves fluctuated around the baseline of the dissolution curve, demonstrating that the baseline referred to the suspension heat capacity. The same behavior was observed during the simulation of the heat capacities for the suspensions of substances B, C, and D. A certain disagreement between the baseline of the DSC curve and the simulated heat capacity was probably a consequence of a relatively low accuracy of the DSC signal due to the small amounts of sample. Considering that the baseline of the curves was then equivalent to the heat capacity of the mixtures, the peak area below the baseline was assigned to the dissolution enthalpy, according to eq 1. This demonstrates that, to isolate the dissolution enthalpy from the DSC signal, we should subtract the baseline enthalpy from the original curve. Determining the Enthalpy and the Dissolution Profiles. As demonstrated above, it is possible to distinguish the dissolution enthalpy from the enthalpy provided by the suspension heat capacity, by subtracting the baseline of the DSC curve. In this procedure, the experimental determination of an accurate baseline for each new suspension is not possible, because the masses of the suspension components at every instant of the dissolution process are unknown. However, as demonstrated above (Figure 10), the baseline does not change significantly as the dissolution takes place and can be approximated to a linear curve when compared to the magnitude of the peak. Actually, for the treatment of DSC data, a common and simple procedure for establishing the baseline of a peak is by

dHcurve dHdissolution ) + Cpsolvent‚msolvent + Cpsolid‚msolid + dT dT dmsolute dmsolid ‚Cpsolid‚∆T + Cpsolute‚msolute + ‚Cpsolute‚∆T dT dT (1) To validate this equation, the specific heat capacities, Cp, of solvents and solutes were determined experimentally by DSC and fitted to polynomial functions (Cp ) A + BT + CT2 + DT3 + ET4). The methodology consisted of heating each individual substance in the DSC cell, and measuring its enthalpy change during heating. For better accuracy, measurements were corrected with the sapphire (aluminum oxide) heat capacity. All substances were sampled with the 40 µL crucibles and heated at 5 °C/min. Sample masses were approximately 25 mg. Based on the Cp functions of solutes and solvents, the heat flux provided by the suspension heat capacity in the course of the dissolution processes was simulated according to eq 1. In this procedure, the exact masses of solute and nondissolved solid at every instant of the process were unknown; therefore, for the purpose of estimating the suspension heat capacities, we considered two possibilities: (1) that dmsolute/dT, the rate of the solute dissolution, was equal to that for providing the maximum

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Figure 11. The interpolated baseline of the DSC dissolution curve for the substance A-water sample. Concentration: 15.0 wt %. Heating rate: 0.5 °C/min.

Figure 12. Integral of ∆Hpeak along the heating interval applied in the dissolution of substance A in water. Concentration: 15.0 wt %. Heating rate: 0.5 °C/min.

interpolating a linear curve between the onset and the end set of the peak (Figure 11). For simplicity, we adopted this procedure, and then, after subtracting the baseline, we integrated the peak signal (∑dHpeak/ dT*∆T) to estimate the dissolution enthalpy profile (Figure 12). In this profile, ∆Hpeak increased gradually during the sample heating, and after a certain temperature it stabilized, indicating the end of the dissolution. This curve was similar to that expected for C versus T (concentration versus temperature) in the course of the dissolution. On the basis of this, we assumed that dC/dT, the rate of concentration change as function of temperature, was directly proportional to the peak enthalpy change (eq 2), meaning that ∆Hpeak (T),

dHpeak dC ) k‚ dT dT

(2)

the integrated value of the DSC peak at a given temperature, was a linear function of ∆C(T), the accumulated increment of concentration (eq 3):

∆C(T) ) k‚∆Hpeak(T)

(3)

At eq 2, dC/dT, the rate of concentration change as function of temperature, is dependent on the dissolution rate, dC/dt (wt %/min), and also on the heating rate, dT/dt (°C/min):

dC dC 1 ) ‚ dT dt dT/dt

(4)

The dissolution rate, dC/dt, specific of each system, is dependent on the crystal superficial area, concentration, and temperature. If the dissolution kinetic is fast enough to overcome the temperature change imposed by the heating rate, then the

Figure 13. Dissolution profiles C(T) of substance A in water at different heating rates. Solute concentration: 15.0 wt %.

concentration at every temperature of the heating interval, C(T), would match with Ceq(T), the equilibrium concentration. On the other hand, if the dissolution kinetic is not fast enough to overcome the rate of temperature change, the solution is not allowed to reach the equilibrium at every temperature of the interval. Consequently, the dissolution profile C(T) would be different from Ceq(T), and this difference would increase as the heating rate increases. For determining the proportionality factor “k” in eq 3, we assumed that ∆C, the concentration increment, was equal to zero when ∆Hpeak was zero, and equal to the concentration increment provided by the complete dissolution of the solute when ∆Hpeak reached its maximum. Once the factor “k” was determined, we were able to estimate the concentration increments ∆C(T) along the whole range of the heating interval, which allowed the determination of the dissolution profiles as a function of temperature, C(T), or as a function of time, C(t). For the estimation of such profiles, we established a value for the initial concentration, which was known to be somewhere between zero and the saturation. For mixtures with a fast dissolution, saturation may be attained in few minutes after the addition of the solvent. On the other hand, if dissolution is not so spontaneous, it can take hours to saturate the solution even under vigorous mixing. In the case of the samples within the crucibles without any mixing, it is difficult to ensure that equilibrium was reached. On the other hand, because crucibles had been kept at constant temperature from 30 to 60 min to stabilize the signal before the heating cycle started, and also considering that samples were prepared with considerable amounts of solid in excess, it is less probable that no dissolution occurred in this interval. Therefore, we assumed here that the solution was at equilibrium at the onset temperature of the peak, and then, by adding the estimated concentration increments to this initial concentration, we estimated the dissolution profiles C(T) from every DSC curve. In some cases, these dissolution profiles approached the equilibrium curve Ceq(T) (Figures 13 and 14), whereas in other cases they differed considerably from the equilibrium as the heating rate increased (Figures 15 and 16). The dissolution profiles of substances A and D (Figures 13 and 14) approached the saturation curves, indicating that their dissolution was relatively fast, allowing the solutions to reach the equilibrium at every instant of the heating interval, under the applied heating rates. Certain deviations were observed between the dissolution profiles and the equilibrium curve in Figure 13, probably because the baseline subtracted from the DSC curve (when calculating the peak enthalpy) was assumed to be a linear function of temperature, when in fact it was more likely a polynomial function. Even thought, these curves

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Figure 14. Dissolution profiles C(T) of substance D in acetone at different heating rates. Solute concentration: 17.6 wt %.

Figure 17. Concentrations of substance B during the heating interval determined from the DSC dissolution peaks at different heating rates.

Figure 15. Dissolution profiles C(T) of substance B in water at different heating rates. Solute concentration: 5.0 wt %.

Figure 18. Substance B solubility estimated from the extrapolation of the DSC dissolution profiles acquired at different heating rates.

Figure 16. Dissolution profiles C(T) of substance C in methanolwater 20 wt % at different heating rates. Solute concentration: 10.8 wt %.

provided a good estimative of the solid-liquid equilibrium of such system. The average errors in these estimations were 0.63 wt % and 0.06 wt % for substances A and D, respectively. In the end set extrapolation method, the average errors were 0.49 wt % and 1.23 wt % for substances A and D, respectively. With regard to substances B and C (Figures 15 and 16), the dissolution profiles C(T) clearly diverged from the equilibrium curves as the heating rate increased. This indicated that their dissolution was relatively slow and, consequently, affected by the heating rates (eq 4). For solutions with slow dissolution kinetic, better chances for C(T) approaching the equilibrium are in the low heating rates. The paradox of decreasing the heating rate and approaching to equilibrium (Ceq) is that kinetic, which is directly proportional to (Ceq - C), would also decrease. This would increase significantly the time for data collection and not attend the requirements of the methodology to provide the fast design of crystallization processes. Furthermore, decreasing the heating rate bellow certain values would decrease the sensitivity of the measurement.

Therefore, for those solutions, which did not converge to equilibrium even at low heating rates, we proposed a new strategy for estimating the solubility curve. Because the dissolution profiles were ruled by the heating rates, the strategy consisted of extrapolating the dissolution profiles to a “hypothetical” zero heating rate. Determination of Solubility by Extrapolating the Dissolution Profile. The dissolution profiles of substances B and C (Figures 15 and 16) deviated from the equilibrium curves as the heating rate increased. To estimate the equilibrium concentrations based on these data, we proposed the extrapolation of the dissolution profiles C(T) obtained at different heating rates, to a heating rate approaching to zero. First, we plotted the estimated concentration versus the heating rate for different temperatures, as seen in Figure 17 for the case of substance B. The extrapolation for a zero heating rate was then made by fitting polynomial functions to the concentrations estimated at different heating rates, and the extrapolated concentrations were assumed to be the equilibrium concentrations for the respective temperatures. In Figure 18, the extrapolated concentrations of substance B are plotted together with the equilibrium curve, indicating a reasonable agreement with the theoretical curve. The average error of the estimated curve (0.44 wt %) was lower than the average error observed in the end set extrapolation method (0.63 wt %). The same procedure was conducted with the dissolution curves of substance C. On the basis of the dissolution curves from Figure 16, we plotted the concentrations at different temperatures as a function of the heating rate and extrapolated the concentrations at each temperature to a heating rate approaching to zero (Figure 19). As in the previous case, the extrapolated concentrations were plotted with the equilibrium curve of substance C (Figure 20), indicating a good agreement with the expected values. The average error observed in these data was 0.13 wt %. The average error obtained in the end set extrapolation method was 2.46 wt %.

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Figure 19. Concentrations of substance C during the heating interval determined from the DSC dissolution peaks at different heating rates.

Figure 20. Substance C solubility estimated from the extrapolation of the DSC dissolution profiles acquired at different heating rates.

These results indicated that even if the equilibrium condition is not attained along the heating interval, it is possible to estimate the solubility curve through the DSC dissolution curves. This suggests that the DSC technique is suitable as a short-cut method

to predict solubility curves; however, the strategies and assumptions made in the data analysis should be used with caution. The consideration that the suspension is at equilibrium at every temperature of the heating interval cannot be stated only by applying a low heating rate. Consequently, the linear extrapolation of the end set temperature is better to be used with caution, unless the kinetic of the dissolution is well known, and the equilibrium condition within the heating interval is assured. In the new approach proposed here, the advantages are that the nonequilibrium condition can also be considered for determining the solubility curve, and a whole range of the solubility curve can be estimated on the basis of the DSC curves collected from only one sample. However, the extrapolation of the dissolution curves, as well as the whole methodology presented here, should still be checked for other types of solutions, before a better detailed procedure is proposed. The errors obtained in the proposed method were acceptable for most of the cases, when compared to the errors obtained in the isothermal method (Table 1). In comparison with the end set extrapolation method, the new method was more accurate for the studied solutions, except for the substance A solution. In this case, the first method was more accurate, because data followed a linear tendency. Errors were calculated with regard to the curves fitted on the solubility data determined by the traditional isothermal method. The errors obtained by the isothermal method for the substance B are not presented in the table, because, in this case, the solubility data were provided by the equation of an optimized curve. Conclusions In this study, we evaluated the applicability of DSC to determine the solubility of organic molecules in aqueous and

Table 1. Errors in the Solubility Determination through Different Methods substance A

substance B

error (wt %) T (°C) 5.0 10.0 15.0 20.0 22.0 25.0 27.0 30.0 30.0 32.0 35.0 35.0 37.3 40.0 43.3 45.0 50.0 54.4 55.2 58.6 59.5 62.0 64.5 68.0

σxa

M1

M2

0.17 0.02 0.13 0.01 0.54 0.09 0.01 0.37 0.20 0.40 0.06 0.12 0.52 0.10 0.28 1.03 0.69

substance C

error (wt %) M3

T (°C)

0.36 0.13 0.19 0.38 0.49 0.65 0.76 0.89 0.89 0.97 1.05 1.05 1.07 1.05 0.96 0.88 0.53 0.10 0.02 0.33 0.43

44.0 50.0 53.2 55.0 60.0 60.5 65.0 65.0 70.0

M2

T (°C)

M1

0.22 0.17

5.7 9.9 10.9 14.7 21.3 22.5 24.4 25.0 27.0 27.5 28.8 29.0 30.0 30.7 31.0 32.0 33.1 34.0 34.9 35.0 35.7 36.0 36.2 36.7 37.5 39.6 39.9

0.08 0.03 0.08 0.02 0.03 0.10 0.08

0.23 0.51 0.72 0.71 0.46 0.77

0.49 0.02 0.04

0.25

0.49

0.63

0.63

error (wt %)

M3

0.71

0.44

substance D

M2

error (wt %) M3

0.12 0.00 0.07 0.07

T (°C)

M1

10.4 15.5 20.3 25.0 30.9 31.9 35.9 36.0 40.6 45.9

0.39 0.50 0.21 0.14 0.10

M2

M3 0.07 0.01 0.05 0.08 0.09

0.95 0.33 1.51 0.11 0.09

0.04 0.04

0.01 0.01

0.15 0.28 0.37 0.28

0.02 0.03 0.04

0.01 2.93 0.05 0.05 1.99 0.02 0.00 0.00 0.04

2.46

0.13

0.23

1.23

0.06

σx ) [∑(Ci - Ccurve)2]1/2, M1) isothermal method, M2 ) DSC method based on extrapolation of the end set temperature, M3 ) DSC method based on determination of the dissolution profiles. a

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nonaqueous solutions. The conversion of the DSC curves into the solubility data was carried out through two different approaches. In the first one, the saturation temperatures were estimated through the extrapolation of the end set of the DSC peaks to a heating rate equal to zero, based on the assumption that the end set temperature was a linear function of the heating rate. The problems found in this approach were the nonlinearity between the heating rate and the end set temperature for most of the cases and the failure of the end set extrapolation for finding the saturation temperatures. The problem of this approach is that it is not suitable for solutions with low dissolution kinetics, at which the end set temperature is affected not only by the heating rate, but also by a delay in the dissolution process. In the second approach, we then coupled the enthalpy and the mass balances of the dissolution processes, converting the DSC enthalpy peaks into concentration profiles, which were then related to the respective solubility curves. In some cases, these concentration profiles were not affected by the heating rates (indicating a fast dissolution rate), and the solubility curves were directly provided by the concentration profiles, which matched with the equilibrium curves. In other cases, the concentration profiles were affected by the heating rates (indicating a low dissolution rate) and differed from the equilibrium curves as the heating rate increased. In these cases, considering that the concentration profiles were affected by the heating rates, we proposed the extrapolation of these concentrations to a heating rate equal to zero. The extrapolated concentration curves matched with the respective equilibrium curves, indicating the suitability of this procedure. The advantage of this second approach is that a whole range of the solubility curve can be determined using only one sample, whereas in the end set extrapolation method, only one point of the curve can be determined from each sample. Besides, the nonequilibrium condition is also considered in the second approach, allowing the estimation of solubility curves for slow dissolution systems. A source of error of this method is the simplification of the baseline to a linear curve, when it is in fact a polynomial function. The shape of the baseline affects

Tamagawa et al.

the calculated peak area as well as the enthalpy and the dissolution profiles. Even so, we were able to estimate the equilibrium curves of the studied solutions with reasonable accuracy. In conclusion, the determination of solubility data based on DSC is a promising methodology with the advantages of rapidity and low sample consumption. However, the development of a general protocol for routine applications should be made with some caution, with special attention to the particularities of each solute-solvent system, such as the dissolution kinetics and the dissolution enthalpy. These variables, specific for each particular system, combined with the DSC operational conditions such as heating rate, should fulfill the requirements for providing suitable dissolution curves. With regard to the operational conditions, we should use proper temperature intervals, heating rates, and sample masses. These parameters should be established experimentally, following the general DSC fundamentals. Acknowledgment. We thank Rhodia Polyamide and Specialties for sponsoring this project, the support from FAPESP (Fundac¸ a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo), and the Associated Laboratory of Micronal and IPT (LAMI) where the experimental work was carried out. References (1) Lorenz, H.; Seidel-Morgenstern, A. Binary and ternary phase diagrams of two enantiomers in solvent systems. Thermochim. Acta 2002, 382, 129-142. (2) Mohan, R.; Lorenz, H.; Myerson, A. S. Solubility measurement using differential scanning calorimetry. Ind. Chem. Eng. Res. 2002, 41, 4854-4862. (3) Park, K.; Evans, J. M. B.; Myerson, A. S. Determination of solubility of polymorphs using differential scanning calorimetry. Cryst. Growth Des. 2003, 3, 991-995. (4) Young, P. H.; Schall, C. A. Cycloalkane solubility determination through differential scanning calorimetry. Thermochim. Acta 2001, 387-392. (5) Ho¨hne, G.; Hemminger, W.; Flammersheim, H.-J. Differential Scanning Calorimetry - An Introduction for Practitioners; SpringerVerlag: Berlin, Heidelberg, 1996.

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