Ind. Eng. Chem. Res. 2003, 42, 4899-4909
4899
Primary Nucleation of Vanillin Explored by a Novel Multicell Device Osvaldo Pino-Garcı´a and A ° ke C. Rasmuson* Department of Chemical Engineering and Technology, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
A novel multicell apparatus is designed, constructed, and used to investigate crystal nucleation of vanillin in water/2-propanol solution (20 mass % of 2-propanol on a solute-free basis). The device contains 15 nucleation cells, with volumes of about 6 cm3 each, in which the induction times for nucleation are measured simultaneously. The nucleation in the cells is continuously video-recorded and analyzed offline. The induction time for nucleation of vanillin is determined at various levels of supersaturation and temperature, and by classical nucleation theory the solid-liquid interfacial energy is estimated to 7.3 ( 0.2 mJ m-2. A large variation in the experimental data is observed, and this variation is analyzed by statistical methods. 1. Introduction Crystallization is widely used for the purification and separation of many substances, including inorganic bulk chemicals and fine and specialty chemicals such as pharmaceuticals, biochemicals, and agrochemicals. Crystallization processes are usually dominated by two physical mechanisms: nucleation and crystal growth. Nucleation denotes the formation of a new threedimensional phase in the nanometer to low-micron size range and is classified as primary or secondary nucleation.1 Primary nucleation occurs by mechanisms that do not require the presence of crystalline matter in the solution. At high levels of supersaturation, the formation of primary nuclei takes place spontaneously at random sites in the pure bulk solution (homogeneous nucleation) or at solid surfaces (e.g., dust particles) acting as crystallization centers (heterogeneous nucleation). Homogeneous primary nucleation can be described by several nucleation theories,1 and heterogeneous nucleation is treated in a very similar way through the introduction of a coefficient that describes the reduction of the activation energy due to the catalyzing effect of the foreign surface. Primary nucleation is often studied by the measurement of induction times. The induction time is the time that elapses from the establishment of supersaturation to the moment when changes in solution properties that are related to the beginning of a phase transition can be observed.2 Because of the very small size range of molecular clusters and crystal nuclei, direct observation of the nucleation event is difficult, and the understanding of it is still highly insufficient. The determination of induction times is traditionally performed in jacketed agitated glass crystallizers, and the onset of nucleation is usually recorded as the formation of a new solid phase as observed by the naked eye. This method is not without problems,1 and more sensitive detection methods might be required.3 Advanced techniques that have been developed include crystalloluminiscence,4 dilatometry,5 laser diffraction,6 turbidimetry,7 differential scan* To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +46 8 790 82 27. Fax: +46 8 10 52 28.
Figure 1. Structural formula of vanillin.
ning calorimetry,8,9 interferometry,10 high-pressure nucleation,11 and the use of electrodynamic levitation with light scattering.10,12 Studies into primary nucleation also suffer from the stochastic nature of the process itself. The theory of homogeneous nucleation has been successfully applied to a large number of inorganic compounds, ranging from soluble salts (e.g., NaCl4,13 and KCl14) to sparingly soluble electrolytes (e.g., barium, calcium, lead, strontium, silver, and thallium salts2,15) Nucleation studies on organic substances and biochemicals include, for instance, those on urea,5 paracetamol,16,17 adipic acid,8 succinic acid,9 and amino acids and proteins.6,11,18,19 In the present work, nucleation of vanillin (C8H8O3, 4-hydroxy-3-methoxybenzaldehyde; CAS registry number 121-33-5) (Figure 1) is investigated. Vanillin has found a wide spectrum of applications,20-22 e.g., as a sweetener or flavor enhancer in foods and beverages and as a component in the production of balsamic fragrances, perfumes, and deodorants and in the manufacture of agrochemical products and pharmaceuticals. Vanillin is also used in the plastics and cosmetics (sun creams) industries because of its anti-UV properties, for the prevention of foaming in lubricating oils, as a brightener in zinc coating baths, as an activator in the electroplating of zinc, as an aid in the oxidation of linseed oil, and as an attractant in insecticides.21 Single crystals of vanillin are used in second harmonic generation (SHG) applications for nonlinear optics.23 Little published work is available on the fundamentals of vanillin crystallization. Lier24 studied the effect of pure organic solvents on the solubility of vanillin and on the width of its metastable zone. Hussain et al.25,26 measured the solubility and metastability of vanillin in water/alcohol solutions. The aim of this paper is to present a novel multicell device that has been developed to determine induction
10.1021/ie0210412 CCC: $25.00 © 2003 American Chemical Society Published on Web 09/09/2003
4900 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
times for the nucleation of vanillin. Data obtained from extensive induction time measurements are reported and used to evaluate the vanillin interfacial energy on the basis of classical nucleation theory. The performance and functionality of the novel experimental device, as well as the statistical significance of the results, are also explored. 2. Theory The rate of primary nucleation, J, is defined as the number of nuclei born in a unit volume and per unit time1,27,28 and is a function of the supersaturation, temperature, and interfacial energy
[
]
γ3ϑm2 J ) J0 exp -FN 3 3 k T (ln Sx)2
(1)
where J0 is the kinetic preexponential coefficient; FN is the geometrical shape factor of the nuclei, which is equal to 16π/3 for spheres; ϑm is the molecular volume; k is the Boltzmann constant; T is the absolute temperature; γ is the solid-liquid interfacial energy; and ln Sx represents the driving force for nucleation, i.e., the supersaturation. The thermodynamic driving force is the difference between the chemical potentials of the solute in the supersaturated solution, µs, and the solute in the equilibrium state of a pure solid or a saturated solution, µeq, and can be expressed in terms of activities or concentrations as
( ) ( )
as xs (µs - µeq) ) ln ≈ ln ) ln Sx RT aeq xeq
(2)
in which as and xs are the activity and mole fraction, respectively, of the solute in the actual supersaturated solution and aeq and xeq are the activity and mole fraction, respectively, of the solute in a solution that is in thermodynamic equilibrium with the crystalline solid state at the given temperature T. At low concentrations or when the activity coefficient ratio is close to unity, the activity ratio reduces to the mole fraction ratio. The most important physical chemical property in eq 1 is the interfacial energy (γ) between the solid nucleus and the actual supersaturated solution. The interfacial energy between a solid and a solution can be measured directly only with difficulty and not at all when the solid body is small. Indirectly, values can be obtained experimentally from the dependence of the nucleation rate on the supersaturation and temperature (eq 1). A weakness of this indirect approach is that the appearance of the quantity T-3(ln Sx)-2 in the argument of the exponential term in eq 1 gives the nucleation rate a strong nonlinear dependence on supersaturation and temperature. Moreover, the interfacial energy also appears in the preexponential coefficient as27,29-31
x
J0 ) 2ϑm
(
)
∆GD γ f0N3D exp kT kT
(3)
where f0 is an attachment (collision) frequency factor related to the number density of active nucleation centers, N3D, and ∆GD is the energy barrier for diffusion from the bulk solution to the cluster.27 The preexponential coefficient has a theoretical value ranging from 1025 to 1033 nuclei cm-3 s-1 for nucleation both from the melt and from solution.4,27 For the latter
case, the preexponential coefficient can be lowered to a value below 1010, which is probably a result of the need for desolvation of the solute before nucleation can take place.27 The classical theory of homogeneous nucleation27,32 assumes ideal stationary conditions and predicts immediate nucleation upon the creation of supersaturation in solution (e.g., by assessing a fast cooling). The induction time of nucleation, tind, is often assumed to be inversely proportional to the rate of nucleation, and hence, it is taken as a macroscopic measure of the nucleation event.1,27,33,34 However, in reality, the induction time is made up of three components: the transient period, i.e., a relaxation time needed to achieve a quasisteady-state distribution of molecular clusters; the period for the formation of stable nuclei, i.e., the nucleation time; and the period required for the critical nuclei to grow to detectable dimensions, i.e., the growth time. At any supersaturation level, the transient period is negligible.1,28,34 If the nucleation time is much greater than the growth time, then the nucleation step is ratecontrolling, and according to the classical theory of homogeneous nucleation, the induction time is inversely related to the nucleation rate (tind ∝ J -1). Then, from eq 1, the induction time can be expressed in logarithmic form as
ln tind ∝ ln J-1 ) ln J0-1 +
FNγ3ϑm2 k3T 3(ln Sx)2
)
ln J0-1 + βT-3(ln Sx)-2
(4)
which represents a linear dependence between ln tind and T-3(ln Sx)-2, the slope of which
β)
FNγ3ϑm2 k3
(5)
can be used to determine the interfacial energy γ. In addition to the assumption that the growth period is short compared to the nucleation time, the above analysis relies upon the following assumptions: (i) Primary nuclei are formed having a spherical shape, i.e., nuclei are isotropic, and the interfacial energy is constant and independent of supersaturation and temperature. Actually, the cause of anisotropy in grown crystals is the difference in interfacial energies for the different faces. However, whether a primary nucleus is faceted or spherical is a matter still under scientific discussion. (ii) The nucleation is primary and homogeneous, i.e., no impurities or crystalline materials are present in the supersaturated solution prior to the occurrence of nucleation. (iii) The bulk properties of a solution apply also at the molecular level.35 (iv) The preexponential factor J0 is relatively insensitive to changes in temperature and supersaturation.27,30 3. Experimental Section A novel multicell device has been developed for the determination of nucleation induction times. The main ideas behind the design and construction of this device are an increased experimental productivity through the operation of many nucleation cells in parallel, a rapid establishment of constant and homogeneous tempera-
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4901
ture and hence supersaturation, and a continuous supervision by video recording. Rapid establishment of the supersaturation level requires the use of material with a higher heat conductivity than glass and a thin wall over which the heat transfer can be carried out. Two additional important aspects are high surface area per unit volume and reasonably efficient convective heat transfer on both sides of the wall. A small volume increases the surface area-to-volume ratio and also reduces the consumption of chemicals, which is of importance in the development of processes for the production of expensive fine chemicals and pharmaceutical compounds. 3.1. Apparatus. The experimental device is called a multicell nucleation block (MCNB) and is designed and constructed for the determination of induction times for nucleation in 15 cells simultaneously. Figure 2 presents a schematic description of the MCNB. The MCNB consists of a set of 15 identical nucleation cells (labeled 1A, 1B, 1C through 5A, 5B, 5C), each with a volume of about 6 cm3. The cells are immersed and distributed in three flow channels (A, B, and C) through which either a cold or a hot fluid flows continuously. Each nucleation cell has a cylindrical shape with an internal diameter, Lint, of 14 mm and a total height of 36 mm. Each nucleation cell consists of a cylindrical wall, a flat base, and a flat cover. The cylindrical wall of each individual cell has a thickness, δ, of 1 mm and is made of stainless steel with a mirror-polished inner (solution-side) surface. At 21 °C, the thermal conductivity of stainless steel is about 17 times higher than that of glass.36 The covers and bases of the cells are made of chemically resistant and optically transparent plastic plates of a polycarbonate based on bisphenol A.37 The material is a window-clear thermoplastic. It is strong and rigid and has an exceptional impact resistance and a very high heat resistance (maximum-use temperature of 135 °C, brittle temperature of -135 °C). The stainless steel cylinder of each cell extends partly down into the base plate in drilled countersinks having the same diameter as the cylinder itself. Two thin silicon draining tubes connected to the cell cover are used for flushing in and out the solution during charging and discharging of the cells, respectively. Silicone sealing gaskets with vulcanized-on Teflon washer and O-rings are used to seal the cells hermetically. Teflon-coated magnetic spinbars are placed in the cells during assembly of the entire MCNB and are used to provide agitation to the solutions inside the cells during the experiments. The experimental setup is completed with a multipleposition magnetic stirrer, a light source, a cryostat, a thermostat, a video camera, a video recorder, and a color television set (Figure 3). The serial magnetic stirrer, which has 15 agitation units, is placed under the MCNB to give the same stirring speed to samples contained in all of the nucleation cells. A Variomag telemodul 40 S control unit (H+P Labortechnik GmbH, Germany) is used to control the agitation speed. The MCNB is connected to a Julabo FP 45 SP cryostat, with a temperature stability of (0.01 K, which is used to supply a constant cold fluid flow, thus controlling the temperature during experiments. The bath has a capacity of about 26 dm3 and is filled with a mixture of ethylene glycol and water (50% v/v). To reheat the solutions in the MCNB to 5 K above their saturation temperature, a Julabo FP 50 HP thermostat, filled with 8 dm3 of water at a fixed temperature, is used. A Pt-
Figure 2. Schematic of the multicell nucleation block (MCNB) and details of an individual nucleation cell.
resistance thermometer is used to measure the temperature of the cooling/heating flow. The cells are illuminated by optical fibers in four bundles (Fostec) connected to a remote light source (DDL quartz halogen lamp) that delivers a cold and extremely uniform light output. A solid-state rheostat is used to control the light intensity. The principle of illumination consists of supplying and distributing the light from four equivalent positions along the translucent base of the MCNB. The light reaches the bottom of each nucleation cell; it is partly dispersed by the turbid walls and bottom of the countersinks in the base plate up through the cell volume and transmitted through the polished transparent lid. A VHS video camera with an objective of appropriate focal length (zoom lens ) 8.7-70 mm) and a date/time function, a VHS videocassette recorder, and a color
4902 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 3. Experimental setup for nucleation experiments: (1) multicell nucleation block (MCNB), (2) multiple-position magnetic stirrer, (3) fiberoptic illumination system, (4) cooling bath, (5) heating bath, (6) video camera, (7) video recorder, (8) color television monitor, (T) temperature sensor.
television monitor are used to observe and record the sequences of the process in a high-resolution and highquality picture. 3.2. Chemicals. Food-grade white crystalline vanillin (99.9 mass % by GC; of USP, BP, and Eur. Ph. grades) supplied by Borregaard Synthesis (Norway) was used as received. Solvent mixtures (20 mass % of 2-propanol in water) were prepared from extra pure 2-propanol (99.5 mass % by GC; of USP, BP, and Eur. Ph. grades), supplied by Merck, and water (distilled, ion-exchanged, and filtered). 3.3. Solubility. The solubility of vanillin in 2-propanol/water was determined by the gravimetric method in the temperature range from 10 to 35 °C. The experimental setup consists of a thermostated water bath standing on a multiple-position magnetic stirrer. Glass flasks, each containing a Teflon-coated magnetic stirrer, are filled with excess solid vanillin and the solvent. The flasks are closed with plastic screw caps and are sealed with Parafilm to prevent evaporation losses. Each flask is immersed in the water bath, and the suspension is continuously stirred at the selected constant bath temperature (10, 15, 20, 25, 30, or 35 ( 0.05 °C). At least 96 h is allowed to ensure that equilibrium is reached. Afterward, undissolved residue is allowed to settle for at least 12 h at constant temperature without stirring. A sample of the clear saturated solution (approximately 5 cm3) is transferred with a preheated syringe through a 0.45-µm membrane filter into a previously weighed sample vial. The vial has a Teflon septum to prevent weight losses due to
Figure 4. Example of time flow diagram showing the main events during the determination of the induction time (e.g., tind ) 20 min) in a nucleation cell. In the temperature-time curve: TH ) 35.1 °C ) initial temperature of the hot flow, TS ) 30.1 °C ) saturation temperature, TC ) 20 °C ) supercooling temperature, point 0 ) moment of switching over from the hot fluid to the cold fluid, point A ) moment from which the induction time is measured (the actual experimental temperature has reached about 95% of TC), point B ) moment of detection of first changes in solution turbidity and following obscuration. In the obscuration curve, 0-1-2 ) clear solution, 2-3-4-5 ) solution causing increasing obscuration.
solvent evaporation during the weighing procedure. The mass of the sample vial with the saturated solution is measured. The septum is removed, and the solvent is allowed to evaporate in an air oven at 40 °C for approximately 12 days. The constant “dry residue” mass is then determined, and the solubility concentration, CS, expressed in grams of solute per kilogram of solvent, is calculated. 3.4. Induction Time Measurements. For induction time experiments, a mother solution of vanillin in 2-propanol/water is prepared by weighing predefined amounts of solid vanillin and the appropriate solvent mixture in a glass flask (250 cm3). The flask is closed with a screw cap and sealed with Parafilm. Complete dissolution of the crystalline vanillin is ensured by immersing the flask in a thermostated water bath under gentle warming and mixing the solution for at least 12 h. About 1 h before the start of each experiment, the water from the thermostat is allowed to circulate through the flow channels of the MCNB. The temperature is set and stabilized at a value, TH (Figure 4), that is approximately 5 K above the corresponding saturation temperature, TS, of the solution. This is done to ensure that no nuclei are formed during filling of the solution into the cells. A preheated syringe, provided with a 0.45µm membrane filter in its tip, is used to transfer the mother solution from the flask into the 15 thermostated nucleation cells of the MCNB. The solution is introduced into each individual nucleation cell through one of the draining tubes located in the cover of the cell. The cells are completely filled with solution, and the tubes are immediately closed. The clear solution in the nucleation cells is stirred at 500 rpm during the experiments. The light source, the
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4903 Table 1. Solubility of Vanillin in 2-Propanol/Watera
a
T (K)
CS ( SECS (g/kg of solvent)
283.15 288.15 293.15 298.15 303.15 308.15
24.20 ( 0.27 39.15 ( 0.39 61.80 ( 0.33 101.04 ( 2.08 184.54 ( 1.77 378.58 ( 11.61
20 mass % of 2-propanol on a solute-free basis.
video camera, the videocassette recorder and the television set are turned on (objects 3, 6, 7, and 8, respectively, in Figure 3). The time function is included in the recordings, and the starting temperature is noted. The temperature of the circulating water from the thermostat is lowered, so that the saturation temperature, TS, is slowly approached (Figure 4). To establish supersaturation, the desired supercooling
∆T ) TS - TC
(6)
Figure 5. Solubility of vanillin in 2-propanol/water (20 mass % of 2-propanol).
is rapidly generated by switching the circulating fluid over to the cryostat instead, which has been steadily kept at the desired experimental temperature (TC). The supercooling temperature TC is held constant during the process for as long as is necessary for crystals to form in all of the nucleation cells. All cells are continuously monitored by video recording. The onset of nucleation is easily observed as a very rapid change in solution turbidity, and the event finishes with complete obscuration of the light through the solution (Figure 4). The aim is to establish the temperature TC within the cells as rapidly as possible and to make the transient period of time required to do so negligible compared to the induction time. In separate determinations, we found that about 95% of the desired temperature change is established in the cells within 2 min. Because the rate of nucleation has a very strong dependence on the supercooling, it is reasonable to start the induction time measurement only when we are close to the final temperature. Hence, the induction time is calculated as
tind ) tAB ) t0B - t0A
(7)
with t0A taken to be 2 min. Only experiments where tind > 1 min are included in the evaluation. When the nucleation in all of the cells is completed and has been fully recorded, the temperature is raised again to dissolve the crystals and repeat the experiment or to facilitate cleaning of the apparatus. Each solution is used only twice and only during the same day because of the limited stability of vanillin.38,39 After that, the cells are cleaned, and fresh solution is used. During cleaning, the cells are drained, flushed thoroughly with alcohol, rinsed with distilled water, and dried with oilfree compressed air. 4. Results and Discussions 4.1. Solubility of Vanillin. The solubility values, CS, reported in Table 1 represent the arithmetic means of three samples from each solution, and the respective standard deviations, SECS, are also given. The solubility curve is illustrated in Figure 5. The solubility results obtained in this work are in good agreement with those reported by Hussain et al.25,26
Figure 6. Induction time data for nucleation of vanillin in 2-propanol/water.
4.2. Induction Time and Interfacial Energy. The induction times are determined at different molefraction supersaturation ratios Sx and absolute temperatures TC. Ten different solution concentrations, from 0.014 to 0.038 mole fraction of vanillin, and three different undercooling temperatures (283.15, 288.15, and 293.15 ( 0.05 K) were used to generate molefraction supersaturation ratios in the range 2.60-9.28. A total of 90 experiments with all 15 cells was performed, and hence, the study comprises 1350 data points. After exclusion of outliers, the logarithms of the induction times for the nucleation of vanillin were plotted vs [(ln Sx)-2T-3], as shown in Figure 6. For each value of [(ln Sx)-2T-3], the median (50th percentile) and the mean (arithmetic average) of the observed induction times were calculated and are also plotted in Figure 6. By linear regression, straight lines were fitted to the median values, to the mean values, and directly to all of the data. The corresponding standard errors were
4904 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 2. Interfacial Energies Calculated from Induction Time Data
based on
sample size N
slope (β ( SEβ ) × 10-7 (K3)
interfacial energy γ ( SEγ (mJ m-2)
medians means all data
15 15 414
8.99 ( 1.32 9.03 ( 1.29 8.59 ( 0.60
7.38 ( 0.36 7.39 ( 0.35 7.27 ( 0.17
calculated by the method of least squares.3,40 From the respective data the interfacial energy values and their standard errors were estimated and are presented in Table 2. The interfacial energy is 7.3 ( 0.2 mJ m-2. Inorganic salts of low solubility generally give quite high values (30-150 mJ m-2)7,15, whereas soluble inorganics generally give lower values (4-8 mJ m-2).34 Organic materials show even lower values of interfacial energy. Galkin and Velikov18 investigated the nucleation of lysozyme and obtained interfacial energy values between 0.51 and 0.64 mJ m-2. Interfacial energies of -caprolactam41 were found to range between 0.8 and 1.5 mJ m-2. Granberg et al.17 estimated the interfacial energy of paracetamol to be in the range 1.4-2.8 mJ m-2, whereas Hendriksen and Grant16 reported values between 1.7 and 3.5 mJ m-2 for the same substance. Interfacial energies of urea in alcohol/water solutions have been reported5 to be 3.8-8.9 mJ m-2. As far as we know, no interfacial energy data for vanillin have previously been published. 4.3. Statistical Analysis. If the nucleation process were deterministic, we would expect to observe the same induction time in all 15 cells of the MCNB if they were filled with the same mother solution and were operated under the same conditions. However, the molecular aggregation that takes place spontaneously at random sites in the bulk solution and the subsequent nucleation are intrinsically stochastic phenomena, as will be discussed in section 4.5. As a result, the nucleation data are scattered as illustrated in Figure 6. In the present study, the scatter is quite significant, and we need to verify that there is a reasonable statistical confidence in the interfacial energy that is determined. In determining the “best” fit of the straight line through the experimental points in Figure 6, the following assumptions were made: (i) Between the variables ln tind and (ln Sx)-2T-3, which, for convenience, will be called in this analysis output variable Y and input variable X, respectively, there is a linear relationship
Y ) R + βX
(8)
where R is the intercept and β is the slope of the straight line. (ii) Input variable X is determined with negligible error in comparison to the uncertainty in the determination of the output variable Y. (iii) The results of observations Y1j, Y2j, ..., Ynj, of output variable Y, measured for the same input variable Xj, represent independent random variables, i.e., the individual nucleation events are independent of each other. (j is a case number from 1 to 15 corresponding to the variable X or (ln Sx)-2T-3 ) 0.089, 0.106, 0.107, 0.122, 0.124, 0.134, 0.168, 0.201, 0.237, 0.240, 0.289, 0.319, 0.323, 0.352, and 0.457 × 10-7, respectively.) (iv) Systematic errors are negligible.
Figure 7. Histogram showing frequency distribution for induction time data at (ln Sx)-2T-3 × 107 ) 0.107, 0.201, and 0.323 × 10-7 (case numbers j ) 3, 8, and 13, respectively). First interval 0-0.50, second 0.51-1.00, and so on.
Is the Induction Time a Random Variable? Typical induction time frequency distributions are shown in Figure 7. Data are organized into equal-sized ln tind slots and include different experiments and different cells. Three case numbers are shown as examples: j ) 3, 8, and 13. The diagrams show that the data exhibit a fairly Gaussian distribution. Are the Cells Equivalent? A statistical test was carried out to determine whether a particular cell systematically nucleates early or late. The frequency analysis is presented below for one particular cell, i.e., 1A in Figure 2. The value Z denotes the order in which the cell nucleated compared to the other cells, i.e., Z ) 5 stands for that this cell was the fifth to nucleate in the particular experiment. The observed frequency value (ωobs) at, e.g., Z ) 5 in Table 3 states how many times this cell has been the fifth to nucleate. Assuming
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4905 Table 3. Frequency Table for χ2 Test of Significance for Nucleation Cell 1A order
observed
expected
difference
Z
ωobs
ωexpt
∆ω
(∆ω)2/ωexpt
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
9 5 8 2 7 5 4 11 2 4 10 3 9 6 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
3 -1 2 -4 1 -1 -2 5 -4 -2 4 -3 3 0 -1
1.50 0.17 0.67 2.67 0.17 0.17 0.67 4.17 2.67 0.67 2.67 1.50 1.50 0.00 0.17
Zmax ) 15
N ) 90
N ) 90
Σ)0
χ2 ) 19.33
that the group of observations is drawn randomly from a population that is uniformly distributed, then the expected frequency number ωexpt ) N/Zmax, equals 6 as Zmax ) 15 and N, the total number of experiments, is 90. The data cover all levels of supersaturation that have been studied. The observed and expected frequency numbers for each occurrence order are given in Table 3. The χ2 test statistic 2
χ )
Zmax (ωobs Z
∑
Z)1
2 - ωexpt Z )
ωexpt Z
(9)
with f ) Zmax - 1 degrees of freedom, is the quantity used here to evaluate the hypothesis that there is no evidence that the order of nucleation occurrence in cell 1A differs significantly in their frequency number (H0) compared to the alternative that the order of occurrence does differ significantly in their frequency number (H1). If sample χ2 > χ2crit, reject H0; otherwise, accept H0. The results are presented in Table 3. χ2 is 19.33 as compared to χ2crit ) 23.6842 for f ) 14 and a significance level R ) 0.05. Note that the sum of the ∆ω values is 0, as it should be. Because χ2 < χ2crit, the null hypothesis is accepted, i.e., the order in which nucleation occurs in 1A is a random variable and does not have a statistically significant influence on the results of the induction time measurements. A similar analysis was performed for the other cells, and in no case did we find a statistically significant deviation from the null hypothesis. Can a Straight Line Describe Data? To determine the interfacial energy, a straight line was fitted to the data according to eq 4. Residuals were calculated from
)Y kj - Y ˆ median Rmedian j j
(10)
)Y hj - Y ˆ mean Rmean j j
(11)
data data ) Yij - Y ˆ all Rall ij j
(12)
and the distributions of the residuals are plotted in Figure 8. Y kj is the median of the experimental values h j is the mean, Yij is the individual of ln tind for case j, Y value of experiment i for case j, and Y ˆ j is the corresponding value estimated by the linear regression for case j. Each case number j (j ) 1-15) corresponds to a
Figure 8. Distribution of residuals with respect to each supersaturation case number (j ) 1-15) corresponding to the variable (ln Sx)-2T -3 × 107 ) 0.089, 0.106, 0.107, 0.122, 0.124, 0.134, 0.168, 0.201, 0.237, 0.240, 0.289, 0.319, 0.323, 0.352, and 0.457, respectively.
value of the variable (ln Sx)-2T-3 ) 0.089, 0.106, 0.107, 0.122, 0.124, 0.134, 0.168, 0.201, 0.237, 0.240, 0.289, 0.319, 0.323, 0.352, and 0.457 × 10-7, respectively. If the fitting procedure is adequate, the residuals are expected to be normally distributed with a mean of zero and a constant variance. The error bars, shown in Figure 8a and b, represent the standard errors calculated as a propagation of uncertainty3 in estimating regressions and uncertainty in estimating the medians and the means and for each case number j. The standard error intervals about the residuals indicate that there are no outliers in the results as all of the error bars do cross the zero reference line. A clear trend would indicate systematic errors. No such trend is observed, although the residuals are not distributed perfectly evenly. Is the Slope Statistically Significant? The slopes of the straight lines fitted to N ) 15, 15, and 414 data values (median values, mean values, and all data, respectively) were used for the determination of the interfacial energy. This is only meaningful if the uncertainty in the slope is limited and especially does not include zero within the confidence range. For the analysis, it is assumed that the residuals are normally distributed. However, the residuals are correlated and have variances that depend on the location of the data
4906 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 9. Induction time for nucleation of vanillin showing homogeneous and heterogeneous regions.
points (i.e., on the case number or variable X in eq 8). It is a common practice to “studentize” the residuals so that all of the variances are homogenized. If the student t-distribution is applied, the slope deviates from 0 with 95% confidence if tstudent > tR/2 ) 2.145 (for medians and means) or tstudent > tR/2 ) 1.960 (for all data), where tR/2 is obtained from statistics tables42 for N - 1 degrees of freedom and R ) 0.05. The t-distribution42 is said to be robust in the sense that, even when the individual observations X are not normally distributed, sample averages of X have distributions that tend toward normality as the number of samples gets large. tstudent can be determined as40
tstudent ) β/SEβ
(13)
where β is the slope and SEβ is the standard error in the determination of the slope. Using the quantities reported in Table 2, we obtain tstudent ) 6.8, 7.0, and 14.3, respectively, for the slopes calculated on the basis of the medians, means, and all data. In all cases, tstudent exceeds tR/2; thus, the slopes β are statistically different from 0 and can be used to determine the interfacial energy values with confidence. The F-statistics42 can also be used to verify that the observed relationship between Y and X when all 414 data points are used explicitly did not occur by chance
F ) (SSreg/dfreg)/(SSresid/dfresid)
(14)
where SSreg is the sum of squared differences between the values of Y estimated by the regression and the average of Y, SSresid is the residual sum of squares, and dfreg and dfresid are the respective degrees of freedom. For R ) 0.05, dfreg ) 2, and dfresid ) N - 2 ) 412 (all data), the observed F-statistic (Fobs ) 204.56) is greater than the critical F-statistic value (FR,dfreg,dfresid ) 3.00),42 thus demonstrating that there is, in fact, a relationship between the variables that is statistically significant. Interfacial Energy Calculation. The slopes of the linear regressions calculated from the induction time plots (Figure 6) and based on the means, medians, and
all data were used to estimate the interfacial energy values, as defined in eq 5, and the results are reported in Table 2. The uncertainty in estimating the slopes (SEβ) is expressed as
SEβ ) SEY-X
x
1 SSXX
(15)
where SEY-X is the standard error in the prediction of Y values for each X in the regression and SSXX is the sum of squares of deviations from the mean for each X.3,40 Because the relationship between the interfacial energy γ and the slope β (eq 5) can be expressed as
γ ) (const)β1/3
(16)
the uncertainty (standard error) in the estimated interfacial energy values SEγ, as reported in Table 2, is calculated as a propagation of the uncertainty in the estimated slopes SEβ as3
SEγ 1 SEβ ) γ 3 β
(17)
4.4. Heterogeneous Nucleation. Measurements were also made at lower supersaturation. All data are shown in Figure 9. Quite clearly, two different regions can be observed. A change of slope occurs at [(ln Sx)-2T-3] ) 0.45 × 10-7, which corresponds to the supersaturation ratio Sx ) 2.6 at T ) 288.15 K. For [(ln Sx)-2T-3] > 0.45 × 10-7 the slope is much lower than that for [(ln Sx)-2T-3] < 0.45 × 10-7. In the determination of the interfacial energy above, the data in the latter region that correspond to the mole fraction supersaturation range 2.60-9.28 are used. For the region (ln Sx)-2T-3 > 0.45 × 10-7 corresponding to the supersaturation ratios Sx ) 1.6-2.6 and the temperature of 288.15 K, the slope is 17 times lower. In the literature, there are many more studies1,2,34 where the data exhibit the same behavior as here, i.e., the data cover two different regions. The usual interpretation is that, in these cases, data cover both heterogeneous and homogeneous nucle-
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4907
ation, the latter exhibiting a steeper slope. This interpretation is adopted in the present work even though we are aware that other explanations have been discussed in the literature. In heterogeneous nucleation, dust particles or the surface of the equipment catalyze the nucleation process by lowering the Gibbs free energy change for the formation of the critical nucleus, i.e., the activation energy for the nucleation. The catalyzing effect can also be described as a lowering of the interfacial energy. However, the interfacial energy of the total surface of the nuclei is in this case a combination of three different interfacial energies43
γ ) S21(γ21 - γ31) + S32γ32
(18)
where Sij is the surface area fraction of the ij interface; γij is the interfacial energy per unit area of the ij interface; and the subscripts 1, 2, and 3 denote the foreign surface (wall, dust), the solute nuclei (vanillin), and the solution, respectively. The total change in interfacial energy is lowered by favorable surface interactions between the nuclei and the foreign surface. For the heterogeneous nucleation region, the interfacial energy estimated from the slope in Figure 9 is 2.9 ( 0.5 mJ m-2, which is about 2.5 times lower than the value for the homogeneous nucleation region. 4.5. Stochastic Nature of Nucleation. There is significant scatter in the data of the present work, although the experiments were performed at the same conditions. Barlow and Haymet44 also reported a similarly large scatter for the heterogeneous nucleation of water on a silver iodide surface. Induction times between 100 and 10 000 s were obtained for more than 250 measurements under the same conditions. Induction times above a quench time of 112 s were found to follow an exponential frequency distribution, which could be related to the theory of nucleation as an activated process undergoing macroscopic first-order kinetics. Volmer and Weber32 established that the exponential term in the kinetic expression for nucleation (eq 1) is proportional to the frequency of formation of critical nuclei, and Einstein45 related the nucleation to spontaneous density fluctuations caused by the natural movement of the molecules. As the exponential term in eq 1 is so strongly dependent on supersaturation, any small perturbation leading to a local change in supersaturation might result in a significant change in the frequency of nuclei formation. Meyer46 concluded that nucleation might be the result of either isothermal and isobaric heterophase fluctuations or an adiabatic process induced by temperature fluctuations. The fluctuation mechanism in combination with mechanisms that control the face growth rate could be the reason for the large scatter observed in induction time measurements. Recent studies47 have shown that nucleation proceeds by fluctuations from a disordered phase into an ordered crystalline phase, passing through a sequence of slightly more ordered states, rather than by building up the crystalline state instantaneously. 5. Conclusions The new multicell nucleation device described in this work increases the experimental efficiency by allowing the performance of 15 measurements of induction time simultaneously and by reducing the volume of the
nucleation cells, thus reducing material costs and making it possible to study short induction periods. For each supersaturation value, a large spread in induction time data is observed among the different nucleation cells and in different experiments, thus confirming the stochastic nature of primary nucleation. Statistical analysis reveals that there is no systematic influence of the particular nucleation cells on the results. Statistical analysis also demonstrates that, despite the large spread in induction times, the solidliquid interfacial energy can be estimated with confidence. The interfacial energy for the nucleation of vanillin in water/2-propanol solutions (20 mass % of 2-propanol on a solute-free basis) is equal to 7.3 ( 0.2 mJ m-2. This value is in the range of values for other organic compounds found in the literature. Acknowledgment The authors gratefully acknowledge the Swedish Research Council (TFR/VR) and the Industrial Association for Crystallization Research and Technology (IKF) for financial support, and especially the Norwegian company Borregaard Synthesis for kindly supplying pure vanillin. Tomas O ¨ stberg, from our Department Workshop, is thanked for manufacturing the components of the multicell block. Nomenclature a ) activity CS ) solubility, g/kg of solvent df ) number of degrees of freedom for Fisher statistics F ) Fisher statistics FN ) geometrical shape factor of nuclei in eq 1, 16π/3 for spheres and 32 for cubes f ) number of degrees of freedom for χ2 test, Zmax-1 f0 ) attachment (collision) factor H0 ) null hypothesis H1 ) alternative hypothesis J ) primary nucleation rate, no. of nuclei m-3 s-1 J0 ) nucleation rate preexponential coefficient, no. of nuclei m-3 s-1 j ) case number for variable X k ) Boltzmann constant, R/NA, 1.381 × 10-23 J K-1 MW ) molar weight, kg kmol-1 N3D ) number density of active nucleation centers NA ) Avogadro’s number, 6.022 83 × 1023 mol-1 N ) number of experiments or sample size R ) universal gas constant, 8.314 J mol-1 K-1 Sij ) surface area fraction of interface ij, m2 Sx ) supersaturation ratio, xs/xeq SEβ ) standard error in estimation of the slope β, K3 SEγ ) standard error in estimation of interfacial energy γ, J m-2 SSreg ) sum of squared differences in a regression SSresid ) residual sum of squares T ) absolute temperature, K TC ) supercooling temperature, K TH ) heating temperature, K TS ) saturation temperature, K tind ) induction time for nucleation, min tstudent ) statistical t-distribution value X ) provisional variable designating the term (ln Sx)-2T-3 x ) mole fraction concentration Y ) provisional variable designating the term ln tind Z ) occurrence order Zmax ) maximum number of occurrence orders
4908 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Greek Symbols R ) significance level in statistical analysis or intercept of a straight line β ) slope of a linear regression, K3 γ ) interfacial energy, J m-2 γij ) average interfacial energy per unit area of interface ij, J m-2 ϑm ) molecular volume of vanillin, MW/(FcNA) ) 1.88 × 10-28 m3 or 188 Å3 µ ) chemical potential, J mol-1 ∆GD ) energy barrier for diffusion, J Fc ) crystal density of vanillin, 1347 kg m-3 χ2 ) chi-squared statistics ω ) frequency number Subscripts 1, 2, 3 ) denotes foreign surface, solute nuclei, and solution, respectively, in eq 18 3D ) three-dimensional eq ) equilibrium int ) internal i, j ) dummy variables N ) nuclei x ) mole fraction Superscripts all data ) related to all data values expt ) expected mean ) related to the mean values (arithmetic average) median ) related to the median values (50th percentile of a sample) obs ) observed Abbreviations BP ) British Pharmacopoeia CAS ) Chemical Abstracts Service const ) constant Eur. Ph. ) European Pharmacopoeia GC ) gas chromatography MCNB ) multicell nucleation block USP ) United States Pharmacopoeia
Literature Cited (1) Mullin, J. W. Crystallization, 4th ed.; Butterworth-Heinemann: Oxford, U.K., 2001. (2) So¨hnel, O.; Mullin, J. W. A Method for the Determination of Precipitation Induction Periods. J. Cryst. Growth 1978, 44, 377382. (3) Skoog, D. A.; Leary, J. J. Principles of Instrumental Analysis, 4th ed.; Saunders College Publishing: Orlando, FL, 1992. (4) Garten, V. A.; Head, R. B. Homogeneous Nucleation and the Phenomenon of Crystalloluminiscence. Philos. Mag. 1966, 14, 1243-1253. (5) Lee, F. M.; Stoops, C. E.; Lahti, L. E. An Investigation of Nucleation and Crystal Growth Mechanism of Urea from WaterAlcohol Solutions. J. Cryst. Growth 1976, 32, 363-370. (6) Biscans, B.; Laguerie, C. Determination of Induction Time of Lysozyme Crystals by Laser Diffraction. J. Phys. D: Appl. Phys. 1993, 26 (8B), B118-B122. (7) He, S.; Oddo, J. E.; Tomson, M. B. The Nucleation Kinetics of Barium Sulfate in NaCl Solutions up to 6 m and 90 °C. J. Colloid Interface Sci. 1995, 174, 319-326. (8) Myerson, A. S.; Jang, S. M. A Comparison of Binding Energy and Metastable Zone Width for Adipic Acid with Various Additives. J. Crystal Growth 1995, 156, 459-466. (9) Jang, S. M.; Myerson, A. S. A Comparison of Binding Energy, Metastable Zone Width and Nucleation Induction Time of Succinic Acid with Various Additives. In Crystal Growth of Organic Materials; Myerson, A. S.; Green, D. A.; Meenan, P., Eds.; American Chemical Society: Washington, DC, 1996; pp 53-58. (10) Mohan, R.; Kaytancioglu, O.; Myerson, A. S. Diffusion and Cluster Formation in Supersaturated Solutions of Ammonium Sulfate at 298 K. J. Cryst. Growth 2000, 217, 393-403.
(11) Waghmare, R. Y.; Pan, X. J.; Glatz, C. E. Pressure and Concentration Dependence of Nucleation Kinetics for Crystallization of Subtilisin. J. Cryst. Growth 2000, 210, 746-752. (12) Izmailov, A. F.; Myerson, A. S.; Arnold, S. A. Statistical Understanding of Nucleation. J. Cryst. Growth 1999, 196, 234242. (13) Hidalgo, A. F.; Orr, C. Homogeneous Nucleation of Sodium Chloride Solutions. Ind. Eng. Chem. Fundam. 1968, 7 (1), 79-83. (14) Polak, W.; Sangwal, K. Modelling the Formation of Solute Clusters in Aqueous Solutions of Ionic Salts. J. Cryst. Growth 1995, 152, 182-190. (15) Nielsen, A. E.; So¨hnel, O. Interfacial Tensions, Electrolyte Crystal-Aqueous Solution, from Nucleation Data. J. Cryst. Growth 1971, 11, 233-242. (16) Hendriksen, B. A.; Grant, D. J. W. The Effect of Structurally Related Substances on the Nucleation Kinetics of Paracetamol (Acetaminophen). J. Cryst. Growth 1995, 156, 252-260. (17) Granberg, R. A.; Ducreaux, C.; Gracin, S.; Rasmuson, Å. C. Primary Nucleation of Paracetamol in Acetone-Water Mixtures. Chem. Eng. Sci. 2001, 56 (7), 2305-2313. (18) Galkin, O.; Velikov, P. G. Are Nucleation Kinetics of Protein Crystals Similar to Those of Liquid Droplets? J. Am. Chem. Soc. 2000, 122, 156-163. (19) Mahajan, A. J.; Kirwan, D. J. Nucleation and Growth Kinetics of Biochemicals Measured at High Supersaturations. J. Cryst. Growth 1994, 144, 281-290. (20) Budavari, S., Ed. The Merck Index, An Encyclopedia of Chemicals, Drugs, and Biologicals, 12th ed.; Merck & Co., Inc.: Whitehouse Station, NJ, 1996; p 1693. (21) Kroschwitz, J.; Howe-Grant, M., Eds. Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed.; John Wiley & Sons: New York, 1997; Vol. 24, p 812. (22) Gerhartz, W., Ed., Ullmann’s Encyclopedia of Industrial Chemistry, 5th ed.; VCH: Weinheim, Germany, 1988; Vol. A11, p 199. (23) Singh, O. P.; Singh, Y. P.; Singh, N.; Singh, N. B. Growth of Vanillin Crystals for Second Harmonic Generation (SHG) Applications in the Near-IR Wavelength Region. J. Cryst. Growth 2001, 225, 470-473. (24) Lier, O. Solvent Effects in Crystallization Processes. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 1997. (25) Hussain, K. Nucleation and Metastability in Crystallization from Liquid Solutions. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 1998. (26) Hussain, K.; Thorsen, G.; Malthe-Sørenssen, D. Nucleation and Metastability in Crystallization of Vanillin and Ethyl Vanillin. Chem. Eng. Sci. 2001, 56 (7), 2295-2304. (27) Walton, A. G. Nucleation in Liquids and Solutions. In Nucleation; Zettlemoyer, A. C., Ed.; Marcel Dekker: New York, 1969; Chapter 5. (28) Myerson, A. S. Handbook of Industrial Crystallization, 2nd ed.; Butterworth-Heinemann: Boston, 2002. (29) Myerson, A. S.; Izmailov, A. F. The Structure of Supersaturated Solutions. In Handbook of Crystal Growth; Hurle, D. T. J., Ed.; North-Holland: Amsterdam, 1993; Vol. 1, Ch. 5. (30) Kashchiev, D. Nucleation. In Science and Technology of Crystal Growth; van der Eerden, J. P.; Bruinsma, O. S. L., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995; pp 53-66. (31) Mersmann, A. Supersaturation and Nucleation. Chem. Eng. Res. Des. 1996, 74 (A7), 812-820. (32) Volmer, M. Kinetik der Phasenbildung; Steinkopff: Dresden, Germany, 1939. Volmer, M.; Weber, A. Nucleus Formation in Supersaturated Systems. Z. Physik. Chem. 1926, 119, 277301. (33) Ny´vlt, J.; So¨hnel, O.; Matuchova´, M.; Broul, M. The Kinetics of Industrial Crystallization; Elsevier: Amsterdam, 1985. (34) So¨hnel, O.; Mullin, J. W. Interpretation of Crystallization Induction Periods. J. Colloid Interface Sci. 1988, 123, 43-50. (35) Liu, X. Y.; Bennema, P. The Relation between Macroscopic Quantities and the Solid-Fluid Interfacial Structure. J. Chem. Phys. 1993, 98 (7), 5863-5872. (36) Melcor Thermal Solutions Catalogue, 1999, p 45. (37) Kroschwitz, J.; Howe-Grant, M., Eds. Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed.; John Wiley & Sons: New York, 1996; Vol. 19, p 584. (38) Englis, D. T.; Manchester, M. Oxidation of Vanillin to Vanillic Acid. Anal. Chem. 1949, 21 (5), 591-593.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4909 (39) Jethwa, S. A.; Stanford, J. B.; Sugden, J. K. Light Stability of Vanillin Solutions in Ethanol. Drug Dev. Ind. Pharm. 1979, 5 (1), 79-85. (40) Johnson, R. A. Miller & Freund’s Probability and Statistics for Engineers, 6th ed.; Prentice Hall: Upper Saddle River, NJ, 2000. (41) Bouropoulos, N. C.; Kontoyannis, C. G.; Koutsoukos, P. G. Nucleation Kinetics of -Caprolactam Melts in the Presence of Water Impurity. J. Cryst. Growth 1997, 171, 538-542. (42) Perry, R. H.; Green, D. W.; Maloney, J. O., Eds. Perry’s Chemical Engineers' Handbook, 7th ed.; McGraw-Hill: New York, 1997. (43) Rodriguez-Hornedo, N.; Murphy, D. Significance of Controlling Crystallization Mechanisms and Kinetics in Pharmaceutical Systems. J. Pharm. Sci. 1999, 88 (7), 651-660. (44) Barlow, T. W.; Haymet, A. D. J. ALTA: An Automated Lag-Time Apparatus for Studying the Nucleation of Supercooled Liquids. Rev. Sci. Instrum. 1995, 66 (4), 2996-3007.
(45) Einstein, A. The Theory of the Opalescence of Homogeneous Fluids and Liquid Mixtures near the Critical State. Ann. Physik 1910, 33, 1275-1298. (46) Meyer, E. Adiabatic Nucleation in Superheated and Supercooled Liquids. Curr. Top. Cryst. Growth Res. 1995, 2 (1), 123133. (47) Terrill, N. J.; Fairclough, P. A.; Towns-Andrews, E.; Komanschek, B. U.; Young, R. J.; Ryan, A. J. Density Fluctuations: the Nucleation Event in Isotactic Polypropylene Crystallization. Polymer 1998, 39 (11), 2381-2385.
Received for review December 30, 2002 Revised manuscript received June 25, 2003 Accepted June 27, 2003 IE0210412
