Nucleation of Butyl Paraben in Different Solvents - Crystal Growth

Article pubs.acs.org/crystal

Nucleation of Butyl Paraben in Different Solvents Huaiyu Yang†,‡ and Åke C. Rasmuson*,†,§ †

Department of Chemical Engineering and Technology, School of Chemical Science and Engineering, KTH Royal Institute of Technology, SE 10044, Stockholm, Sweden ‡ Strathclyde Institute of Pharmacy and Pharmaceutical Sciences, University of Strathclyde, 27 Taylor Street, Glasgow, G4 0NR, UK § Department of Chemical and Environmental Science, Solid State Pharmaceutical Cluster, Materials and Surface Science Institute, University of Limerick, Limerick, Ireland ABSTRACT: The primary nucleation induction time of butyl paraben in pure solvents: acetone, ethyl acetate, methanol, ethanol, and propanol and in 70% and 90% ethanol aqueous mixtures has been determined. At each condition, about 100 experiments have been performed in 5 mL scale to capture the statistics of the nucleation process. The induction times at each condition show a wide variation. The data has been evaluated within the framework of the classical nucleation theory using several of the current approaches. Overall, the data obtained from the different methods of evaluation are surprisingly consistent. At comparable driving forces, nucleation is clearly fastest in acetone and slowest in propanol, with methanol, ethyl acetate, and ethanol in between. Adding water to the ethanol leads to a clear reduction in the nucleation rate. The solid−solution interfacial energy of butyl paraben in the different solvents decreases in the order: 70% ethanol > 90% ethanol > propanol > ethanol > ethyl acetate > methanol > acetone, which is surprisingly well-correlated to a decreasing solvent boiling point. It is shown that the same trend can be found for other systems in the literature. With the assumption that the stronger the bonding in the bulk phases, the higher the interfacial energy becomes, this observation is paralleled by the fact that a metastable polymorph has a lower interfacial energy than the stable form and that a solid compound with a higher melting point appears to have a higher solid-melt and solid-solution interfacial energy.

■

crystallizations,9−13 but the mechanisms are insufficiently understood and the outcome cannot be predicted. Becker and Döring14 and Zeldovich15 initiated the development of the theory today known as classical nucleation theory (CNT). The theory assumes that nuclei are formed by monomers in solution aggregating into clusters having the structure of the bulk crystalline material. The cluster16 will become thermodynamically stable and able to grow into a larger crystal if its size exceeds a critical size where the free energy gain of forming the bulk of the crystalline material overcomes the free energy cost of creating the phase boundary. Turnbull17 was perhaps the first to address the stochastic nature of the nucleation process and gave a simple equation. Recently, Jiang and ter Horst18 applied the steady-state stochastic formulation of the nucleation theory to crystallization of m-aminobenzoic acid and L-histidine. Toschev19 further developed this aspect of the nucleation theory, as well as the difference between steadystate and nonsteady-state nucleation. The nonsteady-state nucleation results in a time lag20,21 before establishment of steady-state nucleation.22 Little is known about this time lag, but theoretical studies have been made.23,24 In this paper, the nucleation of butyl paraben in 7 solvents has been investigated. The interfacial energy, critical nucleation

INTRODUCTION

Crystallization processes occur in nature (e.g., in the formation of bones and shells), have medical implications (e.g., formation of kidney stones), and are of significant importance in industry (e.g., in purification of pharmaceuticals).1 Nucleation denotes the initial step by which a new phase is formed and is of key importance for the control of product properties. At the same time, it is however the least understood mechanism of crystallization,2 which leads to significant problems in the design, operation, and control of industrial processes. For paracetamol nucleating in acetone−water mixtures,3 results reveal a clear influence of the solvent composition on the induction time. Influence of the solvent on the polymorphic outcome has been investigated4−6 and a link to solvent-induced self-assembly in solution has been established.5,7,8 By comparing FTIR spectra of solutions of benzoic and tetrolic acids with those of the solid phases, Davey et al.6 were able to find a link between solution structuring and the crystallizing solid phase. However, in the case of mandelic acid, this link could not be verified. Janik and et al.7 combined crystal structure data of the polymorphic forms with solvation models, and differences in desolvation may explain differences in the formation of different polymorphs. The results of Janbon et al.8 on p-anoxyanisole suggest that a certain preorganization in the liquid state may direct the nucleation process structurally but will not eliminate the nucleation barrier. The solvent can have a strong effect in © 2013 American Chemical Society

Received: January 29, 2013 Revised: August 9, 2013 Published: August 13, 2013 4226

dx.doi.org/10.1021/cg400177u | Cryst. Growth Des. 2013, 13, 4226−4238

Crystal Growth & Design

Article

free energy, pre-exponential factor, and critical radius of and number of molecules in the nucleus have been determined based on the classical nucleation theory. Five different methods of evaluation have been compared, and the influence of the solvent is evaluated. Butyl paraben is used for its antifungal25 and bactericidal properties in advanced cosmetic formulas, as a natural, nonirritating organically derived preservative to maintain the stability of the formulation, thus increasing the shelf life. The compound has been used for many years and is generally considered to be safe.26,27 As shown in Figure 1, the butyl paraben

Induction time experiments are usually evaluated by plotting ln tind versus T−3 (ln S)−2, for determination of the interfacial free energy σ from the slope, B, from which the critical free energy, ΔGc , of nucleation and the radius of the critical nucleus can be calculated. If the nucleation is assumed to be random and independent, the probability of finding m nuclei within a certain time frame is given by a Poisson distribution.18 The probability of finding any number of nuclei ≥ 1 at time, t, is given by Pt = 1 − exp( −N )

where N is the average number of nuclei formed within the same time frame. In a set of parallel experiments at equal conditions, the number of experiments that have nucleated will increase with time (i.e., Pt will increase with time because N will increase with time. Toschev18 suggested that N is proportional to the steady-state nucleation rate (JS) and Jiang and ter Horst17 used

Figure 1. Molecular structure of butyl paraben.

N = JS Vt

molecule includes an aromatic ring to which an alcohol group and ester group is attached in the para position. The solubility of butyl paraben from 10 to 50 °C in various solvents has been determined previously.36,37 There is only one polymorph reported for butyl paraben,48,49 and in the crystal structure, hydrogen bonding between the alcohol and carbonyl groups and pi-stacking are dominating features.

A similar equation was presented by Turnbull and for the two-step model, Knezic29 derived Pt = 1 − exp(−qta), where q and a are constants. By plotting the logarithm of the fraction of experiments not nucleated, ln(1 − Pt) versus time t, a straight line should be obtained from which the nucleation rate can be determined.33 An alternative evaluation of the nucleation rate from the same treatment1 is to use the fact that 1 J= Vte (9) where te is the time when 63.2% of the experiments have nucleated and can be directly read from a plot over cumulative fractional number of experiments that have nucleated versus time at Pt = 0.632. Both these methods rely on the cumulative distribution of nucleated tubes and can be described by eq 6. Behind the treatment leading to eq 8 is the assumption that the nucleation occurs at steady-state conditions with respect to the cluster distribution. Clusters constantly form and redissolve in each size class, but the distribution maintains a steady state in the solution. In the case of nonsteady state conditions, the nucleation distribution will show a curvature at short induction times, as shown by Toschev,18 and there is a time lag for the diffusional process to reach steady-state conditions.30,31 The nucleation rate of a nonsteady nucleation process is dependent on the time,21 and the average number of nuclei formed is given by

(1)

where σ is the solid−liquid interfacial energy, vm is the volume of one solute molecule, A is the pre-exponential factor, and Δμ is the difference in chemical potential between the solute in the supersaturated solution and in the crystalline bulk phase: x Δμ = kT ln S ≈ kT ln (2) x* where the activity coefficient ratio has been neglected, k is the Boltzmann constant, T is nucleation temperature, and S is the supersaturation, and x and x* are the actual and equilibrium solute mole fraction, respectively. The nucleus is defined as a crystalline particle of size sufficient for growth to be thermodynamically favorable. The critical size is given by 2v σ rc = m Δμ (3)

n=1 ⎡ ⎛ n 2 t ⎞⎤ ( −1)n π N = JS V ⎢t − τ − 2τ ∑ exp⎜ − ⎟⎥ 2 ⎢⎣ 6 ⎝ τ ⎠⎥⎦ n ∞

The induction time, tind, is the time period from the establishment of the supersaturated state to the first observation of crystals in the solution. Usually it is assumed28 that the relaxation time (tr) and the growth time (tr) are negligible and that the induction time is inversely proportional to the nucleation rate: B ln t ind V = −ln J = −ln A + 3 T (ln S)2 (4)

(10)

where τ is the time lag. When t > 5τ, eq 10 transforms into the simple linear relationship of eq 7,22 but at shorter time, the number of nuclei increases nonlinearly with time asymptotically approaching the straight line.28 In melt crystallization and nucleation of liquids or solids from vapor phase, the lag time23,32 is about 10−7s and is therefore negligible. However, Kantrowitz33 suggested that tr can be of importance, and Andres and Boudart26 reported several experimental cases in which the time lag cannot be neglected.19 Courtney34 showed that the transient period is dependent on the size of the cluster (number of molecules in one cluster) and the

16πσ 3vm 2 3k3

(8) 16

THEORY Under the assumption of a spherical shape of the nucleus, the classical nucleation theory leads to the basic equation:

B=

(7)

Pt = 1 − exp( −JS Vt )

■

⎛ ⎛ ΔG ⎞ 16πσ 3v 2 ⎞ J = A exp⎜ − c ⎟ = A exp⎜ − 3 3 m 2 ⎟ ⎝ kT ⎠ ⎝ 3k T (ln S) ⎠

(6)

(5) 4227



Article

temperature, and it is also dependent on the supersaturation.31 The time lag can be hours or even a day in viscous liquids.35 In the case where the relaxation time, tr, and the growth time, tg, cannot be neglected, eqs 6 and 7 should be replaced by −ln(1 − Pt) = JS Vtn = JS V (t ind − tr − tg)

mixtures of ethanol and water having 70% or 90% ethanol by weight (Table 1). Solutions of butyl paraben (100 mL) in each of the solvents were prepared in 300 mL glass bottles in a water bath held at a temperature of 30.0 °C. Each solution was stirred for 30 min to ensure that all of the butyl paraben was dissolved. By using a 10 mL syringe, each solution was quickly distributed into 10 tubes (about 5 mL per tube), each equipped with a small magnetic stir bar. The stir bar had a length × diameter of 1 (3/8) × (5/8) cm, and the tube dimensions were 12 (3/8) × 1 (7/8) cm. The tubes were carefully sealed by a screw cap and with Parafilm both inside as well as outside of the cap and agitated at 500 rpm for 30 min at 30.0 °C (well above the saturation temperature) in the water bath. Then, the tubes were transferred to another water bath (Figure 2) kept at a constant nucleation temperature (Table 1). The temperature of the bath was calibrated by a calibration thermometer with 0.01 degrees uncertainty before the nucleation experiments were performed, and the stability of the temperature is always within ±0.02 °C during the nucleation experiments. In the nucleation experiments, three different solvents were operated in parallel (i.e., all together 30 tubes). These tubes were fixed into a plastic transparent frame and placed on a multipole magnetic stirrer plate immersed in the water bath. The stirring rate used was 200 rpm. All tubes in the bath were simultaneously observed, and the nucleation recorded by a SONY camera (DCR-SR72) set at a declining angle, as illustrated in Figure 2. The solution was initially perfectly clear but became turbid as nucleation started. After all the tubes had nucleated, they were transferred back to the high temperature water bath for dissolution during 30 min at 30.0 °C and 500 rpm before a new nucleation experiment was performed. Afterward, all the recordings were played on a computer. The freeze-frames were replayed forward and backward around the nucleation point. Because of the high concentration of butyl paraben, the nucleation and crystallization occurred very rapidly once started. The time from the first appearance of crystals in a tube to the time when the tube was completely turbid (white) was always less than 10 s. Before nucleation, the white stirrer in each tube and black lines on the magnetic plate below were clearly visible (Figure 2), and the threshold for determination of nuclation was a reduced sharpness in the visibility in any of these two features. The induction time was determined as the time between the tubes were immersed into the nucleation water bath at the nucleation temperature and the time of the first observation of detection of this visibilty change. For each condition, more than 100 induction time data were determined.

(11)

Jiang and ter Horst17 assumed that the time of the first nucleation in a set of parallel experiments could be interpreted as tg and corrected eq 8 for that. Equation 11 illustrates that the time lag and the growth time will not change the slope of the plots but only induces a horizontal translation of the curve. Hence, the nucleation rate can be established from the straight line: −ln(1 − Pt)/V versus t that should appear after an initial time period.

■

EXPERIMENTAL WORK

The induction time of butyl paraben has been determined in seven different solvents and at three levels of supersaturation. At each condition, about 100 experiments have been performed to capture the stochastic variations. Butyl paraben (BP) > 99.0% mass purity was purchased from Aldrich and used without further purification. Methanol (ME) > 99.8% mass purity, ethanol (E) of 99.7% mass purity, 1-propanol (PR) > 99.5 mass purity, acetone (AC) > 99.0% mass purity, and ethyl acetate (EA) > 99.7% mass purity were purchased from VWR, and distilled water was used. The solubility of butyl paraben in these solvents from 10 to 50 °C has been determined previously.36,37 The induction time of butyl paraben has been determined in pure methanol, ethanol, propanol, acetone, and ethyl acetate and in

Table 1. Induction Time Experiments solvent

total number of data

nucleation temperature (°C)

70% E 90% E PR E EA ME AC

450 300 300 350 300 450 300

14.0, 13.0, 10.0 5.0, 3.0, 1.0 15.0, 13.0, 10.0 14.0, 13.0, 10.0 15.0, 13.0, 10.0 14.0, 13.0, 10.0 15.0, 13.0, 10.0

Figure 2. Schematic of nucleation experiment setup. 4228



■

Article

RESULTS AND EVALUATION Determination of Interfacial Energy and Pre-Exponential Factor. The induction times at each set of conditions expose a significant variation. Occasionally and especially at low supersaturation, there is a bias such that one tube nucleating slowly in one subset of experiments may nucleate slowly in another subset. However, in general, the nucleation in each tube appears to be essentially random. In Figure 3 is shown cumulative

significant decrease as the driving force increases, while in the other solvents the influence is weaker. Figure 5 shows the presentation of induction time data as a function of supersaturation, according to the classical method

Figure 3. Cumulative induction time distributions of butyl paraben in 90% ethanol at three different supersaturations. Each curve includes approximately 100 experiments.

Figure 5. Classical evaluation of induction time experiments based on eq 4. The ln tind of butyl paraben in 70% ethanol, 90% ethanol, propanol, ethanol, methanol, ethyl acetate, and acetone vs T−3(ln s)−2 with first-order correlation lines. Bars indicate the 95% confidence interval of ln tind.

induction time distributions of butyl paraben in 90% ethanol at S = 1.23, 1.26, and 1.28, as examples. The experimental induction time distribution is “S” shaped, however, usually tailing significantly toward longer times. With the use of Easyfit,38 the mathematical representation of this form has been explored. Among 55 different probability distributions, including, for example, the normal, exponential, Poisson, and log-normal distributions, the best fit is obtained by the Burr distribution. In Figure 4, it is shown how the width of the induction time distribution depends on the supersaturations and the solvents.

of evaluation, eq 4. The data at each supersaturation is represented by the median induction time value, and the bar represents the 95% confidence intervals of the natural logarithm of the median of tind. From the slope of each graph and from the intercept, the interfacial energy and the pre-exponential factor can be determined, respectively, and results are given in Table 2. From the interfacial energy, the free-energy barrier to nucleation, the size of the critical nucleus, and the number of molecules in the critical nucleus can be determined by eqs 3, 4, and 5. The higher the slope, the higher the activation energy for nucleation, and at equal driving force, the longer is the induction time. The data show that nucleation is much easier in acetone compared to the other solvents, and that nucleation is most difficult in propanol and in water−ethanol mixtures. The data also show that adding water to ethanol makes the nucleation more difficult. As shown in Table 2, the nucleation activation energy ranges from about just a small fraction of a kJ/mol to just above 10 kJ/mol. The interfacial energy is below 2 mJ/m2, the size of the critical nucleus is up to a few nanometers, and the number of molecules to make up a critical nucleus ranges down to a single molecule. Even though a very low number of molecules in the critical nucleus has been reported before,39−41 this does not appear to be realistic since in our experiments we always on average have an induction time much larger than zero. In acetone, the nucleation activation energy is the lowest, and the critical nuclei is the smallest, while in methanol at the lowest supersaturation, the critical radius is about 2 nm, and the number of molecules in the critical nucleus is above 100. The interfacial energy of butyl paraben in the different solvents decreases in the order: 70% E > 90% E > propanol > ethanol > ethyl acetate > methanol > acetone. The pre-exponential factor, A, is quite low: ranging from 2.5 × 103 to 9.6 × 103, (# s−1 m−3).

Figure 4. Width of induction time distributions. Coefficient of variation of the median tind vs driving force, RT ln S. Dashed lines are guides for the eye.

There is an overall tendency for the coefficient of variation to decrease with increasing driving force, which can also be observed in Figure 3. In addition, it is notable that the influence of the driving force on the distribution width depends on the solvent. For acetone, propanol and ethyl acetate there is a 4229



Article

Table 2. Nucleation Data for Butyl Paraben in Different Solvents As Determined by the Classical Nucleation Theory Approach, Eq 4, and Median Induction Time Valuesa solvent AC

ME

EA

E

PR

90% E

70% E

RT ln S (kJ/mol)

tind (s)

0.094 0.139 0.224 0.171 0.244 0.486 0.250 0.333 0.468 0.227 0.284 0.471 0.317 0.415 0.580 0.479 0.531 0.563 0.487 0.606 1.022

65 55 50 6861 624 148 587 178 111 1360 532 141 3172 444 181 170 113 99 854 329 144

σ (mJ/m2)

ΔGc (kJ/mol)

ln A [ln #/(s∇m3)]

0.74 0.34 0.13 10.51 5.14 1.29 5.72 3.24 1.63 7.10 4.52 1.65 10.3 6.24 3.30 4.84 3.94 3.50 5.50 3.55 1.25

0.30

1.07

1.13b

1.13c

1.62

1.64

1.73

rc (nm)

# molecules/nucleus

1.0 0.7 0.4 2.0 1.4 0.7 1.4 1.1 0.8 1.6 1.3 0.8 1.6 1.2 0.9 1.1 1.0 0.9 1.1 0.9 0.5

16 5 1 123 42 5 46 19 7 63 32 7 64 30 12 20 15 12 23 12 2

8.35

7.83

8.27

7.92

8.53

9.17

7.81

a Uncertainty (UT) = 0.01 K, Utime = 1 s, vm = M/ρNA = 2.62 × 10−28 m3, ln A expressed in accordance with eq 4. bσ = 1.125 (mJ/m2). cσ = 1.134 (mJ/m2).

Table 3. Interfacial Energy and Pre-Exponential Factor of Butyl Paraben in Different Solvents Determined by the Five Different Methods: (A) Median and (B) Average Values of Induction Time Distribution and (C) Eqs 8, (D) 9, and (E) 11 methods A

B

induction time median value

induction time average value

C

nucleation rate eq 8

D


E


ranges

in ethyl acetate

in acetone

in methanol

in ethanol

in propanol

in 90% ethanol

in 70% ethanol

σ (mJ m−2)

0.30

1.07

1.13

1.13

1.62

1.64

1.73

ln A (ln s−1 m−3) σ (mJ m−2)

8.35

7.83

8.27

7.92

8.53

9.17

7.81

0.46

1.06

1.18

1.37

1.74

1.74

1.87

ln A (ln s−1 m−3) σ (mJ m−2) ln A (ln s−1 m−3) σ (mJ m−2) ln A (ln s−1 m−3) σ (mJ m−2) ln A (ln s−1 m−3) σ (mJ m−2) ln A (ln s−1 m−3)

8.13

7.45

7.83

8.41

8.63

9.45

7.90

0.42 8.96

1.09 7.65

1.09 6.99

1.13 7.99

1.81 9.11

1.95 11.19

2.12 8.84

0.32 8.88

1.09 7.85

1.15 8.20

1.06 7.69

1.62 8.22

1.66 9.13

1.74 7.71

0.51 7.78

1.08 7.49

1.06 6.64

1.16 8.19

1.82 9.14

2.25 7.78

2.23 9.28

0.30−0.51 7.78−8.96

1.06−1.09 7.45−7.89

1.06 −1.18 6.64 −8.27

1.06−1.36 7.69−8.41

1.62−1.82 8.22−9.14

1.64−2.25 7.78−11.19

1.73−2.23 7.81−9.28

A comparison has been made with other methods of evaluating the experimental induction time data. The median value is commonly used in statistical analysis42 in medicine, social research, or other fields, where data is not normal distributed. However, it is not uncommon that the average value of the induction time is used and the outcome of this approach is shown as method B in Table 3. The average value of the induction time is always somewhat higher than the median value. However, the difference is not very large compared to the uncertainty caused by the spread in data. Figure 6 shows examples of the evaluation based on eqs 8 and 11, respectively. The dashed straight line is

used to evaluate the nucleation rate according to eq 11. One nucleation rate value is obtained for each supersaturation, and the result for each solvent is plotted in the standard format for evaluation of the interfacial energy and pre-exponential factor as exemplified in Figure 7. In Table 3, the result of using this method is labeled C. Of course, in consideration of the appearance of the data in Figure 6, it is quite obvious that data do not conform with the straight line depicted by eqs 8 and 9. This S-shape is, however, not uncommon.18,43−46 One possible explanation is that the assumption of steady-state nucleation on which eqs 8 and 9 are based is not fulfilled. In fact, the shape of 4230



Article

Figure 8. Nucleation rate determined by eq 9 in 90% ethanol.

Figure 6. Nucleation rate determination using eqs 8 and 11, respectively, in 90% ethanol. Equation 8 is a dashed line fitted to all data. Equation 11 is a solid line fitted to experiments marked by open symbols only.

Figure 8 illustrates determination of the nucleation rate using eq 9. The induction time data point most close to the Pt = 0.632 is taken as te. One nucleation rate value is obtained for each supersaturation, and the results for each solvent are plotted in the standard format for evaluation of the interfacial energy and pre-exponential factor, as is illustrated in Figure 7. For different solvents, the fit to a straight line in Figure 7 is more or less successful as will be further analyzed in the discussion. In Table 3, the result of this method is labeled D. Table 3 suggests that there is not a dramatic difference in the outcome of the five different methods. Figure 9 shows that with respect to the interfacial energy and the pre-exponential factor, the order between the solvents with few exceptions is the same

Figure 7. Determination of nucleation rate parameters in 90% ethanol from the nucleation rate by eqs 8, 9, and 11.

the data in Figure 6 agrees reasonably with the shape expected for nucleation under nonsteady-state conditions. In the experiments, the tubes are moved from a water bath keeping the solutions undersaturated to a water bath having the desired nucleation temperature kept constant. A short time is needed for the liquid in the tubes to receive the temperature of the second bath. In the nucleation theory, nonsteady state nucleation refers to nucleation under conditions where the cluster distribution has not adjusted to the supersaturation conditions in the solution. Usually, the time constant for cluster redistribution is assumed to be negligible, but the actual situation for organic molecules in organic solvents has not really been clarified. In relation to recent work on the history of solution effects,47 this redistribution time might be much longer than anticipated, and this might contribute to the curvature. The solid lines, in Figure 6, represent the straight line fit based on eq 11 and are drawn somewhat subjectively to capture the slope of the curve after the lag and growth times [i.e., neither the initial part (y value below about 1.0 × 105) of the curve nor the final part are included]. By plotting the nucleation rate thus obtained for each supersaturation versus the driving force function, T−3(ln S)−2, as in Figure 7, the interfacial energy and the pre-exponential factor can be determined. The corresponding plots for the other solvents look essentially the same, but the scatter around the straight line varies. In Table 3, the result of this method is labeled E.

Figure 9. Comparison of (a) interfacial energy and (b) pre-exponential factor for butyl paraben in seven different solvents evaluated by five different methods. 4231



Article

regardless of the methods. There is also not a very clear trend that one method consistently gives lower or higher values, even though there is a tendency for the methods based on eq 8 and 11 to give somewhat higher interfacial energies. All five methods are based on the classical nucleation theory and include one step in the evaluation where ln tind or ln J is plotted against T−3 ln S−2 for determination of the solid−liquid interfacial energy from the slope and the pre-exponential factor from the intercept, as is illustrated in Figures 5 and 7. In Figure 10, the goodness of fit (i.e., R2 − value) to a straight line is compared. Overall, it appears as if using the median value directly from the cumulative distribution provides for the best fit to a straight line. With concern to the three methods based on eqs 8, 9, and 11, respectively, the R2 − value is somewhat

Figure 10. Goodness-of-fit to a straight line in determination of interfacial energy and pre-exponential factor (see for example Figure 7).

Table 4. Solvent Properties and Nucleation Properties from Table 2a

AC ME EA E PR 90%E 70%E a

σ (mJ/m2)

ln A

dipole moment (D)

mole fraction solubility

solubility (mol/L)

surface tension σA (mN m)

ENT

Tb (°C)

viscosity (mPa s)

0.30 1.07 1.13 1.13 1.62 1.64 1.73

8.35 7.83 8.27 7.92 8.53 9.17 7.81

3.0 1.9 1.6 1.7 1.5 − −

0.31 0.22 0.25 0.26 0.29 0.24 0.16

3.07 3.23 2.12 2.95 2.88 2.95 2.53

23.3 22.6 24.0 22.3 23.7 23.2 25.5

0.36 0.76 0.23 0.65 0.62 − −

56.1 66.0 77.2 78.4 97.1 78.7 80.1

0.33 0.60 0.46 1.08 1.72 1.42 2.04

Solubility at 10.0 °C, σA55,56 at 20.0 °C, ENT ,57 Tb,58 and viscosity56 at 25.0 °C.

Figure 11. Crystal structure of butyl paraben: (a) primarily illustrating the hydrogen-bonded chains along the b axis, (b) primarily illustrating the π-stacking along the c axis; and (c) primarily illustrating, at greater magnification, the interaction of the aliphatic tails of two hydrogen-bonded chains. 4232



Article

better for the method based on eq 9 and is overall lowest for the method using eq 8. Influence of the Solvent. The experimental results show that nucleation of butyl paraben is quite easy in acetone. The interfacial energy is lower than in other solvents, and the induction time is short, even at low supersaturation. Among the pure solvents, the nucleation of butyl paraben appears to be most difficult in propanol where we also find the highest interfacial energy. In between, methanol, ethanol, and ethyl acetate are found with rather similar induction times at a comparable driving force and roughly the same interfacial energy. Adding water to ethanol has the effect of increasing the interfacial energy and reducing the nucleation rate. At very high supersaturation, the influence of the water content appears to be somewhat more complex as indicated by the small inset in Figure 5. With concern to the pre-exponential factor, overall, we have not found any clear correlation to the properties of the solvent. However, in the pure alcohols, the pre-exponential factor increases from methanol, through ethanol, to propanol, which correlates with the order found for most of the physical data given in Table 4. In comparing ethanol, ethyl acetate, and acetone, we note that the pre-exponential factor increases with decreasing viscosity. Important functionality of the butyl paraben molecule includes (Figure 1) an alcohol group, an ester group, an aromatic ring, and an aliphatic chain. The ether oxygen is well-embedded and has a very minor influence on the surface charge distribution. In the crystal structure of the only polymorph of butyl paraben48,49 (Figure 11), Figure 11b illustrates how the aromatic rings are arranged in planes approximately perpendicular to the c axis [parallel to (−2,0,5) plane] bonded together by π−π stacking, with a distance of about 3.825 Å. Within the plane, the alcohol group hydrogen of the paraben molecule is bonded to the carbonyl oxygen (2.758 Å) of another molecule, essentially along the b axis, as is shown in Figure 11a. In the a−b plane, these BP chains are held together by van der Waals interactions between the hydrocarbon tail groups alternating with essentially in-plane aromatic-to-aromatic bonding, illustrated from different angles and at different magnification in Figure 11, panels a and c. Methanol, ethanol, and propanol can hydrogen bond to the hydroxyl group as well as the carbonyl group of BP, while ethyl acetate and acetone can only hydrogen bond to the former. The relative difficulty of nucleation in propanol can be related to the length of the hydrocarbon chain. During the hydrogen bonding to the carbonyl oxygen of BP, the propanol molecule can also potentially interact with the hydrophobic parts of the BP molecule. The bonding becomes stronger, and the desolvation becomes slower. The relative ease of nucleation in acetone can be explained by the inability of the acetone molecule to form hydrogen bonding to the ester group of BP. Of course water tends to be strongest hydrogen bond donor and acceptor among the solvents, which would then explain why an increasing concentration of water in the ethanol−water mixtures leads to an increasingly more difficult nucleation. However, this simple analysis cannot explain why the nucleation in acetone is much easier than in ethyl acetate. In addition, we would have expected that the alcohols in general would hamper the nucleation more than the nonhydrogen bond donating solvents, since the alcohols can occupy both the carbonyl site and the alcohol site of BP. Hence, the molecular level analysis does contribute to an increased understanding of the influence of the solvent but is not able to provide a more detailed ranking. In Figure 12, the experimental interfacial energy values given in Table 2 are plotted versus the solubility of butyl paraben

Figure 12. Solid−liquid interfacial energy of butyl paraben in different solvents vs solubility. Dashed line is a guide for the eye.

Figure 13. Solid−liquid interfacial energy of butyl paraben in different solvents vs solvent dipole moment. Dashed line is a guide for the eye.

Figure 14. Solid−liquid interfacial energy vs solvent boiling point. Dashed line is a guide for the eye.

in the solvents at 283.15 K.36,37 Regardless of if the solubility is expressed as mole fraction or mol/L, there is a tendency that the interfacial energy decreases with increasing solubility, as suggested in the literature.37,50 However, the correlation is not very clear, and experimental values are about 3−4 times lower than those predicted by the Mersmann equation.51 In all five pure solvents, the activity coefficients52 are in the range of 0.61−0.77 at 283.15 K. Accordingly, there is a slight negative deviation from Raoults law (i.e., the actual solubility is higher than the ideal). Since the solution activity coefficient52 is 4233



Article

proportionality. The dipole moment of a molecule can be calculated within Materials Studio,59 and for the pure solvents the values obtained agree with the literature values. In Figure 14, the interfacial energy as determined by the experiments (Table 2) against the boiling point of the solvent is plotted. Obviously, there is a reasonably clear increase in the interfacial energy, as the boiling point of the solvent (solvent mixture) increases. As shown in Figure 15, this is actually a trend that can be found also for other compounds (e.g., paracetamol60,8 and the polymorph A of famotidine).61 Similarly, it is found that the metastable zone becomes wider, as the solvent boiling point increases for vanillin and ethylvanillin62 and for m-aminobenzoic acid63 and mandelic acid,64 as shown in Figure 16. The normal boiling point temperature of liquids is directly proportional to the enthalpy of vaporization (Trouton’s rule). Accordingly, it appears as if the nucleation becomes more difficult the stronger the solvent molecules bind together. For liquids where hydrogen bonding is significant, (e.g., water and the alcohols), the boiling temperature is lower than predicted by Trouton’s rule, which may explain why the corresponding points in Figure 14 are a bit higher. However, if we plot interfacial energy versus the enthalpy of vaporization of the solvent, the correlation is not improved, and the correlation against the cohesive energy density or the solubility parameter δ of the solvent is worse. The correlation between the interfacial energy of a compound in different solvents and the boiling point (as a representation of the bonding in the liquid state) of the solvents is paralleled by the fact that, in polymorphic systems, the interfacial energy is found to be lower for a metastable polymorph (Table 5) compared to the stable form, and the nucleation of the metastable polymorph tends to be favored. The metastable form has a higher Gibbs free energy than the stable form. Accordingly, in general we would expect the stable form to have a lower

inversely proportional to the solubility (at constant temperature), the nucleation behavior does not correlate to the activity coefficient, and there is no clear relation to the van’t Hoff enthalpy of solution.52 The empirical correlation found by Neumann53,54 relates the solid−liquid interfacial energy to the surface tension of the liquid, σL, and the surface energy of the solid. In fitting the equation to the experimental data, the best fit estimate of the solid surface energy becomes 13.7 mJ/m2, but there is up to a 60% deviation between the experimental interfacial energies and those given by the correlation. In Figure 13, the interfacial energy is plotted versus the inverse of the solvent dipole moment, showing a surprisingly clear direct

Figure 15. Solid−liquid interfacial energy of famotidine polymorph A and paracetamol in different solvents vs the solvent boiling point.

Figure 16. Metastable stable zone width vs solvent boiling point. 4234



Article

first diagram in Figure 17 shows the solid-melt interfacial energy of organic compounds plotted versus the melting point of the solid.58 Thirteen experimental interfacial energy values69,72−78 and 15 calculated interfacial energy values79,80 are included. Where experimental values are available, the calculated values80 are somewhat lower, but the order between compounds is essentially preserved. Experimental values have been determined by the shape of the grain-boundary-grooves, wedge-equilibrium method, capillary cone method, and from the depression of melting points of small crystals. The diagram shows an overall trend that an increasing melting point is associated with an increasing solid-melt interfacial energy, even though stearic and myristic acids show significant deviation. The second diagram in Figure 17 presents the solid-melt interfacial energy of 17 metals versus their melting points.81−88 The interfacial energy has been determined by methods like

Table 5. Solid−liquid Interfacial Energy of Polymorphs compound

solvent

eflucimibe indomethacin D-mannitol

ethanol ethanol ethanol aqueous −

1,4-transpolyisoprene

metastable polymorph

stable polymorph

reference

4.23 mJ/m2 17 mJ/m2 4.59 mJ/m2

5.17 mJ/m2 27 mJ/m2 5.04 mJ/m2

65 66 67

59.0 mJ/m2

91.5 mJ/m2

68

enthalpy and a higher melting point, reflecting a stronger bonding in the solid phase. This is in line with the expectation that the solid-melt interfacial energy would be proportional to the melting enthalpy of the solid,69,70 and that the solid−solution interfacial energy would be proportional to the enthalpy of dissolution.71 The

Figure 17. Organic compounds, metals, and inorganic compounds (from top to bottom) of the solid−liquid interfacial energy vs melting point. 4235



Article

entropy between a solute molecule at the solid surface and in the bulk solid. However, for the solvent, the entropy of molecules at the interface may very well be different from the entropy of solvent molecules in the bulk, which is a complicating factor. In addition, the situation is even more complex since the solution phase is not a pure solvent but a concentrated solution of the solute.

droplet homogeneous nucleation, maximum supercooling, and dihedral angles. Again a correlation of increasing interfacial energy with an increasing melting point can be observed. The last diagram in Figure 17 shows that the solid-solution interfacial energy, primarily determined by precipitation, of 40 inorganic solid salts in aqueous solution50,51,71,89,90 increases with increasing melting point of the solute,91 although not a very strong correlation and showing significant deviation with MgF2. The data presented in Figure 14 shows that the interfacial energy of butyl paraben crystals in various solvents is to some extent correlated to the boiling point of the solvent. We hypothesize that this stems from the fact that the boiling point fairly well characterizes the cohesion energy of the solvent and, hence, the bonding between the solvent molecules in the liquid state. It has previously been established69,70 that the solid-melt interfacial energy correlates with the melting enthalpy of the solid, a measure of the strength of the bonding in the solid phase. It has also been suggested71 that the solid−solution interfacial energy should correlate with the enthalpy of dissolution, being dependent on the strength of the bonding in the solid phase. In Figure 17, it is shown that these dependencies extend into correlations also to the melting temperature of the solid. In summary, there appears to be a general relation that the interfacial energy becomes higher the more bonding that is present in the solid phase and in the solvent, respectively. The nucleation process in solution involves desolvation of solute molecules, formation of the bulk solid phase, and establishment of the solid-solution interface. If the Gibbs free energy difference between the solute in the supersaturated solution and in the solid crystalline phase is given by the chemical potential difference, it includes also the desolvation and reorganization of the desolvated solvent molecules in the bulk solution. The interfacial energy is the Gibbs free energy increase per unit increment of interfacial surface area. It accounts for the fact that the solute molecules at the surface of the solid phase lack part of the solid phase bonding, and that solvent molecules at the interface lack part of the solvent−solvent bonding. At the interface, a certain degree of bonding can be established between solvent molecules and the solid surface, to some extent compensating for the part of the respective bulk phase bonding that is absent at the interface. A simple energy balance leads to E=

⎤ z⎡1 (WAA + WBB) − WAB⎥ ⎢ ⎦ a⎣2

■

CONCLUSIONS More than 2400 induction time experiments on a 5 mL scale, with magnetic stir bar agitation, are reported for butyl paraben in seven different solvents and at three different supersaturations. At each experimental condition, about 100 experiments have been performed, showing significant stochastic induction time variations at all conditions. The cumulative distributions are S-shaped but asymmetric with significant tailing at longer induction times. The distribution coefficient of variation increases with decreasing supersaturation. Within the framework of the classical nucleation theory, five methods of evaluation of the interfacial energy and pre-exponential factor have been compared. Overall, the data obtained from the different methods of evaluation are surprisingly consistent. However, using the median value of induction time distributions appears to provide for the best treatment of the data, under the assumption that the nucleation occurs under steady-state conditions. However, whether the nucleation can be treated as occurring under steady-state conditions remains to be clarified. With a comparable driving force, nucleation is clearly the fastest in acetone and the slowest in propanol, with methanol, ethyl acetate, and ethanol in between. Adding water to the ethanol leads to a clear reduction in the nucleation rate. The pre-exponential factor (ln A) is in the range from 7.81 to 9.17, and the nucleation activation energy is in the range from 1 to 10 kJ/mol. The critical nuclei radius ranges from 0.4 to 2 nm, and the number of molecules in the nucleus is below 125. The interfacial energy of butyl paraben increases in the order: acetone < methanol

Nucleation of Butyl Paraben in Different Solvents - Crystal Growth

Recommend Documents