Determination Methods for Crystal Nucleation Kinetics in Solutions

Review Cite This: Cryst. Growth Des. 2018, 18, 540−551

pubs.acs.org/crystal

Determination Methods for Crystal Nucleation Kinetics in Solutions Yan Xiao,†,‡ Jingkang Wang,†,‡ Xin Huang,†,‡ Huanhuan Shi,†,‡ Yanan Zhou,†,‡ Shuyi Zong,†,‡ Hongxun Hao,*,†,‡ Ying Bao,†,‡ and Qiuxiang Yin†,‡ †

State Key Laboratory of Chemical Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, P. R. China ‡ Collaborative Innovation Center of Chemical Science and Chemical Engineering (Tianjin), Tianjin 300072, P.R. China ABSTRACT: The crystal nucleation process in solutions plays an important role in both the engineering processes and the physical science. To seek the truth of nucleation, researchers have worked for almost a century from either the kinetic viewpoint or the solution chemistry viewpoint. In this work, the nucleation rate measurement methods, including the deterministic method, droplet based method, double-pulse method, microfluidic method, and stirred small volume solution method, were reviewed, and their advantages and disadvantages were also discussed. Furthermore, problems in the nucleation kinetic investigation were also listed in the end.

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modeling16 and the non-steady state investigation17,18 through solving mathematical models based on the theory of the intermediate stage of crystal growth. Modeling results showed good agreements with the supersaturation decay curve and particle size distributions for several proteins19,20 whose data were easily found from the literature. However, more experimental data are needed to help verify and further modify the theoretical results. In classical nucleation theory, the nucleation process was simplified as a reversible chain reaction, molecules successively attach to and detach from the clusters of different sizes, and the frequency of clusters past the critical size is denoted as the nucleation rate.21 Therefore, molecular understanding of the nucleation process could be inferred from the accurate nucleation rate measurements. For decades, investigations on nucleation kinetics remain one of the most important ways to gain information on nucleation.22,23 Roger et al. and Peter et al.2,24−26 have reviewed the classical nucleation theory and two-step nucleation theory to interpret nucleation rate data for different supersaturations and solvents, or even different compounds, to distract molecular scale information on the nucleation process. The prevalent nucleation rate from solutions is defined as the number of crystals nucleated per unit volume (or area) during per unit time.27 Starting from the definition, the nucleation rate can be estimated directly by measuring the number or number density of crystals versus time profile. Alternatively, the time needed for the appearance of the “first crystal” can also be used to calculate the nucleation rate. Two broad classes of experimental methods are mainly used to characterize the

INTRODUCTION Nucleation phenomenon exists everywhere. It plays a significant role not only in the engineering processes, including condensation, boiling, crystallization, sublimation, etc., but also in the biology science, involved in the development of teeth, bone, kidney stones, gallstones, as well as the protein crystallization related diseases. Besides, it is also responsible for the formation of crystalline materials on Earth and certain aerosols which cause air pollution. Despite the importance of the nucleation process, it has not been fully understood.1−6 According to the origin of the nuclei, nucleation can be divided into primary nucleation, which means that nuclei are formed from an absolute pure supersaturated multicomponent system (or supercooled melt system for single component), known as homogeneous nucleation, or are catalyzed by the presence of existing foreign surfaces (such as dust, container walls, impellers), known as heterogeneous nucleation, and secondary nucleation, where nucleation is induced by the prior presence of seeds from the same material.7,8 Nucleation is the first step of the crystallization process, which may influence the crystal size and its distribution, polymorph, and purity of the crystalline products in many circumstances. Since many factors will affect the nucleation process, it is still quite difficult to design an ideal nucleation process to obtain products with desired physicochemical properties. Recently, although the development of spectroscopic techniques9−11 as well as the molecular simulation methodology12,13 enables us to investigate the molecules’ assembly state in solution, which may be related to the mature macroscopic crystal structures, there is still a long way to directly observe and characterize the nucleation process.14,15 Besides, many researchers have also worked a lot on the process © 2017 American Chemical Society

Received: August 30, 2017 Published: November 30, 2017 540

DOI: 10.1021/acs.cgd.7b01223 Cryst. Growth Des. 2018, 18, 540−551

Crystal Growth & Design

Review

constant artificially (solution concentration stayed the same and was monitored by using conductivity method), which could be related to the number of crystals, crystal morphology, and crystal growth rate. In pure solvents, the nucleation rate was a staircase-form function of time, while the rate was reduced and became more irregular with the presence of surfactants. Bhamidi et al.43,44 measured the crystal nucleation rates of hen egg-white lysozyme at different ionic strengths using a particle counter, which used the principle of near angle light scatter to detect particles and could directly export the number density of crystals in the crystallizing solution. The nucleation rates, determined from the slope of the crystal number density versus time plots, were correlated by the classical nucleation theory expression. The systematic deviation of model predictions toward the lower protein concentration region was illustrated by the authors that two nucleation mechanisms may exist: heterogeneous nucleation at lower protein concentration and homogeneous nucleation at higher protein concentration. Therefore, the data were split into two regions, and the corresponding kinetic parameters were estimated separately. Benefiting from the convenient colloid particle length scale (set by particle size, ∼ μm) and time scale (set by particle diffusion, ∼ ms to days), studies of colloids crystallization could be investigated using light scattering techniques.45 Cheng et al.33,46 used time solved Bragg light scattering to characterize crystal nucleation and crystal growth of colloidal hard spheres in a microgravity environment on a space shuttle. Also, they used UV−visible transmission spectroscopy to investigate the nucleation behavior of poly-Nisopropylacrylamide (PNIPAM) particles, which could be described by empirical power law relations. In-situ focused beam reflectance measurement (FBRM) was used to evaluate the nucleation rate of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) from γ-butyrolactone in unseeded batch cooling crystallization by Kim et al.36 The nucleation rates were estimated from the constant slope of total counts/s versus time plots recorded by FBRM, shown in Figure 1, before which a conversion from the counts/s to the actual number density of crystals was performed. By introducing a temperature term concerning the dependence of kinetics on saturation temperature, a modified classical nucleation rate model was proposed to correlate the nucleation rate data and showed a better prediction for the MSZW in the cooling crystallization.

elapsed time period, polythermal method, which measures the metastable zone width (MSZW), temperature (time) difference, at different constant cooling rates for the saturated solution to nucleate, and isothermal method, which determines the induction time needed for the commencing of nucleation at a constant temperature for supersaturated solutions with different supersaturation ratios. And the “first crystal” is usually treated as the point at which the number density or the volume fraction of crystals reaches some critical values. Therefore, both the MSZW and induction time are related to the nucleation and growth rate of crystallization process, which could provide the kinetics information indirectly.28−31 Many methods have been proposed to acquire massive nucleation rate data of solutions. As for the detection of nucleation process, because of the tiny size of critical nucleus and the limitation of analytical technique threshold, the detection could not be done directly and instantly. Therefore, indirect methods, such as monitoring the system property changes, seem to be better choices. In the publications, microscopes32 can be used to observe or count the number of crystals which already grow into a big size, while a laser,33 conductivity meter,34 thermometer,35 and even naked eyes can also be used to monitor the system property changes (refractivity, conductivity, temperature, etc.) which are caused by nucleation and the appearance of massive crystals. As to the devices used for measuring nucleation kinetics, jacketed vessels,36 quasi 2D cells,37 well plates,38 and an electrodynamic levitation trap apparatus39 are widely employed, and medium or high throughput devices, such as Crystal 16,40 crystalline parallel reactors, and microfluidic devices,41 are getting more popular for investigating the nucleation process because of easy operating and capability of indicating the stochastic nature of nucleation. The measurement of nucleation kinetics in solutions can also be divided into a deterministic (also called traditional) method and probability method, based on the multi-nuclei model and single nucleus model, respectively.42 Given the great importance of crystal nucleation kinetics in solutions, this work will summarize the methods used to obtain the crystal nucleation rate. Their corresponding pros and cons will be discussed. The future development of the crystal nucleation kinetics estimation will also be discussed at the end.

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DETERMINISTIC METHOD For large or industrial scale crystallizers, the crystal nucleation process tends to be a deterministic (repeatable) process, as a combination of primary nucleation, secondary nucleation, crystal growth, crystal break and agglomeration, and so on. Also, agitation is an essential process, which enhances nucleation in most of the crystallization engineering processes. The deterministic method is the most widely used one to estimate nucleation kinetics for the scale up or controlling of the crystallization process for many years as a result of its similar complexity with the real industrial process. According to the properties measured to characterize the nucleation process, it can be divided into three subcategories: the particle number based method, induction time or MSZW based method, and population balance equation (PBE), mass balance equation combined with the empirical kinetic equations based method. In 1978, Dugua et al.34 measured the nucleation rates of sodium perborate from aqueous solutions with and without the addition of the sodium salt of butyl ester oleic acid as the surfactant. Nucleation rates were estimated from the total precipitated crystal mass profile by keeping the supersaturation

Figure 1. Measurements of tonset, Tonset, and JFBRM from the operating profile in the unseeded batch cooling crystallization of RDX (Tsat,i = 343.15 K and q = 0.1 K/min). Reproduced with permission from ref 36. Copyright 2013 John Wiley and Sons. 541



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Liu et al.47 measured the nucleation kinetics of L-arginine trifluoroacetate (LATF) in water through induction time measurements, whose reciprocal was correlated to the nucleation rates. Results showed that nucleation is more probable for higher supersaturation. Similarly, nucleation rates of nesquehonite from water and water/dimethylformamide (DMF) solvent mixtures were determined by the reciprocal of induction time values to examine solvent effect on the nucleation kinetics by Zhao et al.48 (Figure 2), while the

indicated that the interfacial energy went up with the increase of solvent boiling point, and the nucleation constant showed a good linear relationship with the interfacial energy. PBE model and the mass balance equation, together with all kinds of in situ process analytical technologies (PAT), are widely used in the estimation of nucleation and growth kinetics for industrial crystallization process. Trifkovic et al.50 estimated the nucleation and growth parameters of paracetamol in an isopropanol−water antisolvent batch crystallization process by using the nonlinear regression method for the moments of the crystal population density. The moments were calculated from the measured chord length distribution (CLD) generated by FBRM, while the concentration profile was calculated through the ATR-FTIR spectroscopy measurements, as shown in the schematic experimental setup, Figure 3. Both the nucleation rate and growth rate expressions adopted the power law correlations,51 derived from the classical nucleation theory (CNT). Open-loop experimental results validated the crystallization model, using the estimated kinetic parameters. And the single- and multiobjective optimization on the antisolvent flow rate profiles by using the obtained model improved both the particle size and its distribution, as well as the yield. In the same year, Nagy et al.52 used the same method to establish the kinetic model for the cooling crystallization process of paracetamol from water by MSZW measurements. The combination of data obtained from RAMAN which was used to confirm the polymorph composition in the precipitated crystals, ATR-FTIR which was used to monitor the concentration, and FBRM which was used to gather the crystal size distribution information, was used by Trifkovic et al.53 to determine the solvent mediated polymorph transformation kinetics for buspirone hydrochloride (BUS-HCl) by separating the transformation experiment into three processes: the dissolution of metastable Form II, the nucleation of stable Form I, and the growth of stable Form I. As we can see, experiment results show no big dispersion and tend to be constant in the deterministic method. Therefore, several measurements will be enough to estimate the kinetic parameters and establish a process model. In terms of the setup, it is relatively simple, and the macroscopic size and stirring provide similar fluid dynamic conditions with the scaled-up crystallizers. Also, this method is crystallization method independent, and it can be used in cooling crystallization, antisolvent crystallization, and evaporation crystallization, either seeded or unseeded crystallization, batch or continuous crystallization, and so on. However, despite the advantages of time-saving and ease to scale up, the deterministic method is hard to operate at the desired temperature or supersaturation under exact uniform fluid dynamic conditions as a result of the large dimensions of the setup and transport phenomenon, rendering the system difficult to operate, control, and analyze. Also, many assumptions are applied in this kind of method,50 such as the crystallizer is well mixed, crystal agglomeration and breakage phenomenon are neglected or simplified in the PBE model, no growth dispersion happens, the crystals are born at size zero, etc. Usually, for simplicity, the crystal morphology or size is described by only one variable. Furthermore, the poor conversion from the chord length distribution (CLD) and total counts/s measured by FBRM to the crystal size distribution (CSD) and crystal number density restricted the wider use of FBRM and the PBE model method.54−56

Figure 2. Measured induction time of nesquehonite nucleation in pure H2O and H2O−DMF binary solvents at different saturation levels (A) before and (B) after solubility correction. The trend lines are drawn only for the purpose of guiding the eye. Reproduced with permission from ref 48. Copyright 2013 Elsevier.

induction time was calculated from the in situ conductivity detection of the solution. Results showed a positive correlation between nucleation rate and DMF concentration. Analysis based on the classical nucleation theory revealed an unexpected increase of surface energy with the increase of DMF content, implying that kinetic parameter, rather than the thermodynamic parameter, resulted in the enhancement of nucleation. Further analysis suggested that kinetic acceleration was possible with high surface energy if the nucleation process underwent a cluster aggregation mechanism instead of critical nucleus formation. Shiau et al.49 investigated the solvent influence on the pre-exponential factor and interfacial energy by measuring the MSZW data of salicylamide in different solvents. An integral model based on the classical nucleation theory and Stokes− Einstein equation was developed to determine the intrinsic nucleation rate constant and interfacial energy. Results 542



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Figure 3. Experimental setup. Reproduced with permission from ref 50. Copyright 2008 American Chemical Society.

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DROPLET BASED METHOD The crystal nucleation process usually happens quite fast, so that it is not very easy and suitable to testify for the prediction of nucleation theory, as a result of the unknown and inevitable catalytic surfaces or centers existing in the bulk system. Also, the inevitable existence of secondary nucleation affects the investigation of primary nucleation. Theoretically, if the system is subdivided into sufficiently small droplets, a large portion of the droplets should be free or at least having a substantially reduced amount of heterogeneous nucleant, which provides a method for investigating the homogeneous nucleation process in supersaturated systems. The small droplet based technique has been widely used to the gather nucleation rate data for different systems. Vonnegut57 was the first to recognize that miniaturization of a macroscopic sample into noncommunicating microsamples will suppress the effects of heterogeneous nucleation. The isothermal nucleation rates of supercooled liquid tin were determined using dilatometric measurements. The crystallization systems (Figure 4) were composed of many small droplets (1−10 μm diameter) dispersed by manual shaking and isolated by an oxide film. Results from different temperatures showed that the nucleation rates had a very large negative temperature coefficient corresponding to an activation energy on the order of ∼2 × 105 calories. In 1952, Turnbull58 investigated the solidification kinetics of supercooled liquid mercury using the same method with a droplet diameter of 2−8 μm. In addition to the volume change during the solidification of supercooled liquid, the heat released from crystallization process can also be measured to characterize the nucleation rate. Rasmussen et al.35 applied differential scanning calorimetry (DSC) to monitor the heat effect during the solidification of an alloy (90% tin−10% bismuth) emulsion droplet, stabilized in organic carried fluid. By rapid cooling to isothermal holding temperature or using a constant cooling rate, the temperature dependence of nucleation rate could be derived.

Figure 4. (a) Dilatometer. (b) Fraction of tin drops remaining unfrozen as a function of time from the dilatometer data. Reproduced with permission from ref 57. Copyright 1948 Elsevier.

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In 1959, White et al.59 extended this technique to the nucleation process from solutions. By using a droplet generating device, small droplets of potassium nitrate aqueous solution were produced continuously and isolated by mineral oil. The nucleation kinetics was recorded by observing the fraction of uncrystallized droplets with time by using a microscope. Results showed that with ordinary solutions, nucleation was catalyzed by impurities, which was a heterogeneous nucleation mechanism, while solutions without catalyzing centers which were removed by using a partial crystallization technique, turned to follow a homogeneous nucleation process. In 1978, Keller et al.60 applied this method to acquire the reaction crystallization nucleation rate of CaSO4· 2H2O aqueous solution by using an in situ generation of insoluble substance from a very soluble precursors protocol. The droplet based technique does diminish the effect of unknown catalytic centers in the system and testifies on the rationality of classical nucleation theory. It can also save materials used to measure nucleation kinetics, which is suitable for expensive products. The easy operation of dispersing droplets places less limits for the studied system, which is specifically suitable for high melting point materials.35,61 However, almost all the works ignored the crystal growth time which is defined as the elapsed time from the creation of nuclei to growing to detectable crystals. Besides, the generated droplets usually had a relatively large size variation which would influence the accuracy of the experimental results, as illustrated by Vonnegut and Filippone et al.57,61

Figure 5. (a) Schematic representation of the classical principle of separation in time of the nucleation and growth stages. Reproduced with permission from ref 62. Copyright 2010 American Chemical Society. (b) n [crystallite number] vs t [minutes] dependencies for surfaces rendered hydrophobic (by means of hexamethyl-disilazane). The corresponding concentration ratios σ = C/Csat are shown in the graphs; C is the actual concentration, and Csat is the equilibrium one. Reproduced with permission from ref 63. Copyright 1999 Elsevier.

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DOUBLE-PULSE METHOD One of the main reasons that nucleation is difficult to study is the nanosized critical nuclei, which makes the observation and locating of them almost impossible with the techniques to date.14,15 According to the idea proposed by Tammann38 in 1922, this question could be solved by separating the nucleation and growth processes of crystallization, which is called the double-pulse method and is schematically shown in Figure 5a.62 Its essence is that, during the first stage, crystals are only nucleated by creating high supersaturations for a relatively short time (minutes, sometimes hours), and crystal growth is not fast enough to exhaust the materials and decrease the supersaturation. Then, the supersaturation is rapidly lowered to the metastable zone, where the system is unable to produce further nuclei, and only growth of the existed nuclei could happen. For slow growing molecules, the growth time could be set as several hours or even days. By counting the number of macroscopic crystals formed during a certain nucleation time, the nucleation rate could be estimated, as shown in Figure 5b.63 This method eliminates the error resulting from ignoring growth times and has been successfully applied to many systems, especially proteins. In 1954, Dunning and Shipman64 used a counting cell which was a flat glass vessel, 13 cm diameter and 1 cm deep, to obtain the nucleation rate of a sucrose solution at different supersaturations and temperatures. The nucleation rate was obtained from the slope of crystal number vs time, while the intercept on the time axis was the induction time. On the basis of Volmer’s theory, the rate controlling step was considered to be the frequency of molecular clusters growing beyond the critical size and the corresponding activation energy was estimated. Even though they did not mention the growth time for the critical nuclei to be observed macroscopically, the time interval was considered to be included in the induction time

they denoted. Then in 1957, Dunning and Notley32 used a double-pulse method to investigate the nucleation rate of crystalline cyclonite from aqueous acetone in a jacketed vessel. The number of crystals were counted by direct microscopic examination or indirectly integration of the size distribution curves of the crystals. Results from different temperatures and binary solvent mixtures indicated the nature change of the critical nuclei. Galkin and Vekilov38,65 applied the double-pulse method to the direct measurement of nucleation rates of lysozyme. Teflon well plate containing 400 droplets of 0.7 μL was used to conduct the kinetics determination experiments. Similar to the droplet based method, nucleation events showed stochasticity in small volume solutions. Mean crystal number for a certain nucleation time was used to calculate the nucleation rates for different concentrations of protein and precipitant. Results showed that protein crystal nucleation may occur at or even beyond the boundary of applicability of classical, continuum nucleation models. Nanev et al. used a quasi-two-dimensional cell, shown in Figure 6,66 to study the effect of different surface templates on lysozyme nucleation kinetics based on the doublepulse methodology. Steady state nucleation rates were calculated from the number of nuclei vs time plots. It was found that hydrophobic surfaces (coated with hexamethyldisilazane)63 could promote nucleation rate, while templates of poly-L-lysine67 could suppress the nucleation rate, when compared with bare glass cells. Besides, they modified the experiment method by adding an ultrasonic field to the glass cell and found that the external ultrasonic field doubled the 544



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Figure 6. Quasi two-dimensional cell for crystallization investigations. It consists of two optically parallel glass plates (1) and two inlets (2). The protein solution is injected in the gap (δ) between the plates (1). This gap was changed stepwise. Reproduced with permission from ref 66. Copyright 2010 John Wiley and Sons.

nucleation rate of lysozyme, when compared to conditions without external field.68 In 2004, Nanev et al.69 modified the classical double-pulse method by using a supersaturation gradient (generated by a temperature gradient) along an insulin solution contained in a glass capillary tube, and successfully obtained nucleation rates at different supersaturations simultaneously. This method turned out to be time, labor, and material saving. Furthermore, using the same method, they also measured the crystal nucleation rate at different places of the cell respectively: bulk solution, glass support, air/solution interface, and solution/glass/air boundary for insulin and lysozyme. Interestingly, it was found that no true homogeneous nucleation but heterogeneous (surface) crystal nucleation happened even in the bulk solution, where biomaterial presented in the protein solutions could serve as foreign particles.62,66 The double-pulse method detaches the crystal nucleation and growth processes ingeniously, avoiding the observation of invisible nuclei and the systematic error caused by ignoring the growth time. However, the direct measurement of nucleation rate by counting crystals is a labor and time-consuming work and it also needs the growth rate to be slow enough to apply the separation strategy.38 Furthermore, the determination of experiment conditions is not as easy as the theory.62,66 The growth temperature needs to make sure that no new nuclei will form and the original nuclei can only grow, even though the Ostwald ripening may happen inevitably to dissolve some of the crystals, which underestimates the nucleation rate. Different solution concentrations may need different growth conditions. To count the crystal number clearly and accurately, the crystals should be separated from each other, which limits the supersaturation range we could study. All of these problems need to be solved through preliminary experiments, which add more work to researchers.

Figure 7. Design of the microdevice. The oil and aqueous phases are injected, respectively, in inlets 1 and 2. Oil can also be injected in inlet 3 to increase the velocity of the droplets. The top-right image shows the formation of the droplets. The two dashed areas are temperature controlled at T1 and T2. The dotted area in the middle of the serpentine corresponds to the observation zone through a stereo microscope. Reproduced with permission from ref 78. Copyright 2007 Elsevier.

(∼100 nL) and control their temperatures as they flow. With specific channel geometries, continuous droplets of potassium nitrate (KNO3) dispersed in silicone oil were formed and flowed along the channel. By using a stereo microscope and a CCD camera, movies of observation areas under crossed polarizers were made to estimate the probability of droplets with crystals at different nucleation times (corresponding to different chip area for the flowing droplets). Nucleation rates at different supersaturations and temperatures were derived from the probability temporal evolution curve. The results showed good agreement with the classical nucleation theory. Gong et al.77 applied microfluidic device to measure the nucleation rate of a thermoresponsive colloidal poly-Nisopropylacrylamide (PNIPAM) system. The colloidal PNIPAM suspension was injected into a microfluidic flow-focusing device to generate monodispersed droplets in oil. Temperature control can fine-tune the volume fraction of PNIPAM particles, while the flow rate can be changed to control the volume of droplets. An optical microscope with a pair of crossed polarizers was employed to detect the nucleation and the percentage of crystallized droplets (shown in Figure 8) within the investigated area over time. Therefore, the nucleation rate could be calculated from the linear line of crystallized droplets percentage versus time. Results showed that, in bigger volume droplets, multiple crystallites existed and interacted with each other, which affected the accuracy of the measurement of nucleation rate. Microfluidics could also be used to study the nucleation kinetics of the evaporation crystallization and precipitation process. Braatz et al.74,79 determined the nucleation kinetics of paracetamol and glycine crystals under conditions of timevarying supersaturation in a droplet-based microfluidics, named the Evaporation-based microwell crystallization platform. Vitry et al.80 developed a tailored microfluidic device which could

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MICROFLUIDIC METHOD The emerging of the high throughput microfluidic technique provides a new method for crystallization process investigation, such as polymorph or cocrystal screening, single crystal preparation, nanocrystal preparation, nucleation or crystallization kinetics, etc.70−75 It has been used to crystallize a variety of inorganic and organic compounds, including water, salts, amino acids, proteins, and active pharmaceutical ingredients. Apart from the good control of transport phenomena (enhanced mass and heat transfer), little gravity effect, confinement individual space and few impurities, the large quantities of nearly monodisperse droplets make microfluidic a convenient and high-efficient technique to conduct a statistical analysis of the stochastic nucleation process, with higher accuracy.41,76,77 Laval et al.78 developed a microfluidic device (shown in Figure 7) which allows us to produce monodisperse droplets 545



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synchronously generate drops of two different reagents and could ensure the total mixing process happened within a few milliseconds. By carefully designing experimental conditions and apparatus parameters, high nucleation rates of neodymium oxalate crystals formed from the reaction between neodymium nitrate and oxalic acid were measured successfully. Of course, the double-pulse method can be used together with the microfluidics device. Ildefonso et al.76 used a simply constructed microfluidics setup to measure the nucleation kinetics of lysozyme, shown in Figure 9. The nucleation time dependence of the mean number of crystals in 250 nL droplets was used to estimate the steady state nucleation rate. Compared to the work of Galkin and Vekilov,38 the diminished volume of solution widened the range of the measurable supersaturation or nucleation rate. Rossi et al.41 employed the double-pulse method to investigate the nucleation rate of adipic acid solutions by using a microfluidic device under both stagnant and flowing conditions. Instead of the number of crystals, the number of nucleated droplets was counted for different lengths of nucleation time to calculate the nucleation rate. It was found that nucleation was enhanced in moving droplets. It was considered to result from the internal circulation in the flowing droplets, which promoted the attachment frequencies of molecules to the nuclei. The microfluidic technique perfectly solves the size inconformity problem of the droplet method. It also has the advantages of large data quantity and material saving, which is helpful to accurately measure the nucleation rates. Therefore, it has been widely used to successfully depict the classical nucleation theory expression. However, this method also has some drawbacks,81 mainly associated with the design and manufacture of microfluidic devices,82−84 such as the device material needs to be transparent and compatible with the measured system, the temperature control and measurement strategy are hard for a temperature-varying device, the small diameter of the pipe containing the droplets allows clogs to happen easily, and the cleaning of the system needs to be done

Figure 8. Emulsion crystallization observed between a pair of crossed polarizers on an optical microscope. (a) PNIPAM droplets (500 μm size) at 23.6 °C. (b) PNIPAM droplets (100 μm size) at 23.6 °C. The multiple crystallites in the larger droplet system indicate interactions among nuclei at high nucleation volume. Arrows indicate the crystallized droplets in the smaller droplet system. Reproduced with permission from ref 77. Copyright 2007 American Chemical Society.

Figure 9. (a) Image of the device, (b) zoom of the inlets of the plug factory showing droplet formation, (c) image of the stored droplets, and (d−f) examples of lysozyme droplets observed after 20 h, at Tnucleation = 20 °C and Tgrowth = 40 °C (channel width 500 μm). Reproduced with permission from ref 76. Copyright 2011 American Chemical Society. 546



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Figure 10. (a) Two subsequent induction time measurements of the same sample of INA in ethanol at a supersaturation ratio of 1.44 (102 mg/mL) at 25 °C. The measured induction times are 2191 and 2826 s. (b) Two subsequent MSZW measurements of the same sample of INA in ethanol at a concentration of 102 mg/mL with a cooling rate of 0.4 °C/min. The measured MSZW are 12 and 10 °C. Reproduced with permission from ref 85. Copyright 2013 American Chemical Society.

Figure 11. (a) Induction time of 80 experimental data points for m-ABA solutions at the supersaturation ratio of S = 1.96. (b) Probability distribution of the induction times of these 80 points. Reproduced with permission from ref 40. Copyright 2010 American Chemical Society.

solutions. The experimental procedure is to disperse an equal amount (usually 1−1.6 mL) of bulk solutions into each HPLC vial and then set the hold temperature above the saturation point to completely dissolve the solute for an hour or so, then crash cool to a low temperature value to create the desired supersaturation and maintain for hours. In-line light transmission was used to detect the onset of crystallization for each vial. Induction time was taken as the difference between the time at which the transmission started to decrease and the time at which the low temperature was reached, which means the designed supersaturation is created. By repeating the thermal cycle several times, 80 or more induction times from the same conditions could be used to calculate its probability distribution (Figure 11 40 ). On the basis of the single nucleus mechanism,14,42,86 which has been observed in many stirred small volume solutions, and the assumption that the appearance of the nucleus (often referred as a critical nucleus) is a random process, independent of each other and follows Poisson Distribution,40 the probability curve could be fitted with a derived exponential expression, correlated to the nucleation rate and solution volume. It is worth noting that the time elapse from the occurrence of nucleation and the detection of crystals is defined as the growth time and estimated as the shortest induction time detected in the fitting equation. Using this method, the nucleation rates of m-aminobenzoic acid (m-ABA) and L-histidine (L-His) at different supersaturations were successfully determined, and it followed the trend expected from classical nucleation theory. Results indicated that heterogeneous nucleation happened, and the concentration of

carefully. Also, the massive data processing is time and labor consuming. Without combining with the double-pulse method, the microfluidic method alone fails to include the interference of crystal growth time.

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STIRRED SMALL VOLUME SOLUTIONS The droplet based method, double pulse method, and microfluidic methods mentioned above have been widely used to gather crystal nucleation kinetics, and could be related to the classical nucleation theory or two-step nucleation theory expressions to dig the nucleation mechanism from the molecule viewpoint. However, the kinetic parameters cannot be used in the industrial manufacturing process, due to the lack of stirring process. As we all know, stirring is necessary and inevitable in the crystallization process to achieve good mass and heat transfer and can cause secondary nucleation which will enhance the nucleation rate.40 Recently, two new methods have been developed to measure the nucleation kinetics in stirred small volume solutions, mainly for cooling crystallization. These two methods make use of the stochastic nature of crystal nucleation, which are reflected by the variations in induction time (Figure 10a) and metastable zone width (Figure 10b), respectively.85 With multiple repeats under the same experiment conditions, the nucleation rate can be characterized from the distribution of induction time or metastable zone width values. In 2011, Jiang et al.40 developed the induction time probability distribution method using medium throughput Crystal16 device (can hold 16 HPLC vials, Avantium) for the measurement of the nucleation rate in stirred small volume 547



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active foreign particles was relatively lower than prediction. Sullivan et al.23 used the same method to determine the nucleation rates of p-aminobenzoic acid and benzoic acid in polar and nonpolar solvents and linked the results to the solution chemistry investigated by FTIR and computational chemistry. For this system, solute dimerization and desolvation were found to be rate-determining processes in the overall nucleation pathway. The nucleation rates of racemic diprophylline (DPL) stable Form RI from dimethylformamide and metastable form RII from isopropanol were also determined from the induction time distributions by Brandel et al.14 The difference in nucleation behavior in the two solvents originates from the energy barrier for nucleation, and the longer induction time corresponds to a higher energy barrier. Mealey et al.87 applied the induction time method to acquire nucleation kinetics of risperidone in seven different solvents. FTIR spectroscopy and density functional theory (DFT) were employed to investigate the solvent−solute interaction, and it was found that the stronger the solvent bound to the risperidone molecule in solution, the slower the nucleation became. In addition to the induction time, metastable zone width (MSZW) is another widely used property to characterize nucleation rate, which is also called the polythermal method. Usually, MSZW is defined as the temperature difference between the supersaturation point and the nucleation point during a cooling crystallization process with a constant cooling rate. As a result of the stochastic nature of nucleation, MSZW is also not a determined parameter in small volume solutions and different MSZWs correspond to different temperatures. Therefore, a MSZW distribution inherently contains the nucleation rate−temperature dependency.85 Kadam et al.42 performed the MSZW measurements for paracetamol in water at 1 mL and 1 L scales to investigate the scale up of the stirred small volume nucleation kinetics. As we can see from Figure 12,42 large MSZW variations were displayed in 1 mL experiments, while almost no variations were found for 1 L scale ones. Interestingly, the measured MSZW values at 1 L scale corresponded to the onset of the MSZW distribution at the 1 mL scale, implying a simple scale up rule for laboratory MSZW data to larger scales. In their another work,88 MSZW data were measured for paracetamol−water and isonicotinamide−ethanol systems at four different scales from 500 mL to 1 L. Together with the results from their previous work mentioned above, it showed that the spread of MSZW increased roughly inversely proportional to the volume. The nonreproducible MSZW for smaller scale was explained by single nucleus mechanism, and the reproducible result for larger scale was explained by the deterministic population balance model. By using isonicotinamide−ethanol as the model system, both the induction time distribution (at constant temperature and different concentrations) and MSZW distribution (at different cooling rates and different concentrations) were measured and compared by Kulkarni et al.85 Mealey et al.15 also used both methods to investigate the crystal nucleation of salicylic acid in different solvents. Although the MSZW method is less labor intensive and could give the nucleation kinetic parameters from only one distribution, it failed to consider the temperature dependence of the parameters, and the changing temperature could cause a temperature difference between the vials and the heating block, which is difficult to account for. In comparison, the induction time method could provide more accurate data

Figure 12. MSZW (ΔT) for paracetamol concentration of 0.0150 g/ mL (a) and 0.0470 g/mL (b) at a cooling rate of 0.5 °C/min for different volumes. The solid line indicates the smallest MSZW detected with light obscuration at 1 mL, while the dashed line indicates the smallest MSZW detected by visual observation at volumes between 500 mL and 1 L. Reproduced with permission from ref 42. Copyright 2012 Elsevier.

and the data are much easier to analysize, although it is much more labor intensive. The induction time or MSZW distribution method to obtain crystal nucleation kinetics in small volume solutions saves material and reduces the experimental cost, especially for expensive pharmaceutical molecules or proteins. More importantly, it provides the potential to relate the lab scale data to the industrial scale crystallizer by introducing stirring in the measurements. However, to extract nucleation rate from such a large amount of data, a distribution curve, is kind of tricky and difficult. ter Horst et al.40,89 applied a Poisson distribution to describe the appearance of independent nuclei by assuming the supersaturation is constant during the nucleation process and used a single nucleus mechanism to illustrate the process from the onset of nucleation to the detection of crystals. Rasmuson et al.15 adopted log-normal cumulative probability function and a mean value of the induction time or MSZW value to qualitatively compare or calculate the nucleation rate. Mazzotti et al.90 presented two different models to describe a batch cooling crystallization process, where the interplay between the stochastic nucleation and the deterministic crystal growth was described differently. All of these data treatment methods need much work and different assumptions. Besides, in the stirred small volume solutions, sometimes one can only get one polymorph of the polymorphic compounds, which makes it impossible to evaluate the nucleation kinetics of other polymorphs.23 The “crowning” problem,23,89 solids form on the glass vial−solution boundary, 548



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distribution of experimental points (more points for low supersaturation or low nucleation rate),89 or might result from the intrinsic feature of nucleation, which means a tiny supersaturation change could cause huge difference in the nucleation rate for large supersaturation.64 This could also imply that the classical nucleation theory is so simplified that it cannot cover such a big nucleation rate or supersaturation range. Modified or new forms of nucleation kinetic expressions or nucleation mechanisms need to be developed to describe the nucleation process.94,95 Fourthly, in most of the crystallization processes, heterogeneous nucleation rather than homogeneous nucleation is proved to be dominant, as a result of a lower energy barrier. However, currently, we have little information about the catalytic centers in the system, and this clouds further investigation of the nucleation process. Nowadays, researchers have put a lot of effort into functional surface nucleation, which is an important step to understand the formation of critical nuclei.96−98 Finally, the kinetic data obtained from small volume or static methods is much more easily analyzed by nucleation theories and may provide a molecular mechanism of nucleation when combined with the solution chemistry or molecular simulation, while the deterministic method tends to be a good method to establish a kinetic crystallization model, which could be used for scaling up or optimizing the industrial manufacture processes. However, further connection between the small volume methods and the deterministic method is missing. With a better understanding of the nucleation process, the data from different methods and different scales should be analyzed together and give reliable kinetic results. In return, reliable nucleation rates can also help to further understand the nucleation puzzle.

may postpone the detection of crystals or even fail and the solution concentration could not remain constant. Also, the crystals can arouse secondary nucleation. To guarantee the effective detection of nucleation, the amount of crystals should fall in an appropriate range.

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CONCLUSIONS AND OUTLOOK Given the significance of crystal nucleation, its kinetics study has remained a hot subject for decades. Five measurement methods, including the deterministic method, droplet based method, double-pulse method, microfluidic method, and stirred small volume solution method were reviewed in this work. For the scaling up of lab scale crystallization experiments, the deterministic method, which covers the stochastic nucleation phenomenon using its large experimental setup, could be employed to estimate the nucleation and growth kinetics. The droplet based method diminishes the concentration of catalytic centers by subdividing the system into small droplets and works well for high melting point metals. But it fails to generate droplets with uniform size. The microfluidic technique helps to solve the droplet size nonuniformity problem, although the design of microfluidic device needs to be improved. The double-pulse method, which is widely used in protein crystallization research, applies the idea of separating the crystal nucleation and growth by controlling the supersaturation using the metastable zone concept to make the invisible critical nuclei visible, and nucleation rates can be obtained directly by counting the number of crystals in a certain time interval. As for the stirred small volume solution method, both the stochastic nature of nucleation, shown in the dispersion of measured induction time or MSZW values, and the effect of stirring on crystallization process are embodied. The latter four methods are all performed in relatively small systems or static conditions and the crystallization process is simplified to primary nucleation, (secondary nucleation, agglomeration, breakage processes are ignored) or single nucleus mechanism. The kinetics results can help to verify the classical nucleation theory or develop new nucleation theories, helping us to better understand nucleation process mechanisms.23,65,91 Therefore, to choose an appropriate method for estimating nucleation kinetics, the system properties, the device requirements and even the purpose to get the information should be taken into account. To further understand and uncover the mystery of the nucleation process, there is still a lot of work to do in the future. First, to improve the accuracy of nucleation kinetics, analytical techniques with higher accuracy and higher resolution should be developed.92,93 If the critical nuclei could be observed or characterized more directly, the nucleation rate could be obtained directly, instead of monitoring various system properties change which could cause extra uncertainty. Second, large amounts of experiment data need to be analyzed to estimate the nucleation kinetics, especially for the data with large dispersion. Middle value, average value, Poisson distribution assumption, log-normal cumulative probability function fitting, etc. have been used by different researchers. Qualitatively, they all can be used to compare the nucleation rate. But quantitatively, which method could provide the most reliable nucleation rate remains unknown. Third, for most of the nucleation rate data correlated by classical nucleation theory, the fitting result for high supersaturation points is always not as good as lower ones (model values are lower than experimental values).57 This might be caused by the uneven

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AUTHOR INFORMATION

Corresponding Author

*Tel: +86-022-27405754. Fax: +86-022-27374971. E-mail: [email protected]. ORCID

Hongxun Hao: 0000-0001-6445-7737 Ying Bao: 0000-0002-4461-8035 Qiuxiang Yin: 0000-0001-8812-0848 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is financially supported by National Natural Science Foundation of China (No. 21376165) and National Key Research and Development Program of China (No. 2016YFB0600504.)

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Determination Methods for Crystal Nucleation Kinetics in Solutions

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