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Microfluidic Droplet Method for Nucleation Kinetics Measurements Philippe Laval, Aurore Crombez, and Jean-Baptiste Salmon* LOF, unite´ mixte Rhodia-CNRS-Bordeaux 1, 178 aVenue du Docteur Schweitzer, F-33608 Pessac cedex - FRANCE ReceiVed August 18, 2008. ReVised Manuscript ReceiVed October 17, 2008 In this work, we developed a microfluidic equivalent of the classical droplet method for investigating nucleation kinetics. Our microfluidic device allows us to store hundreds of droplets of small volume (∼100 nL) and to accurately control their temperature. We also monitor directly all the stored droplets, and thus perform statistical measurements on a large number of nucleation events. In the case of aqueous solutions of potassium nitrate, we manage to investigate nucleation kinetics and polymorphs and quantify the influence of impurities. The use of small droplets is crucial in such experiments, since it allows the sample to reach high supersaturations and to separate all the nucleation events. Moreover, we compare our results to the classical nucleation theory, and we demonstrate unambiguously using direct observations of the droplets that nucleation in aqueous solutions of potassium nitrate always occurs using heterogeneous mechanisms.
Introduction Understanding crystallization in solution is of great importance for both industrial and fundamental research. For instance, nucleation/growth mechanisms influence greatly the properties of crystals obtained in an industrial process.1 In a fundamental point of view, nucleation, polymorphism, and protein crystallization are examples of intense research topics, and their issues are far from being understood.2-4 Most of the difficulties come from the experimental investigations: in the case of nucleation, for instance, nuclei cannot be easily observed due to the small sizes involved, impurities are often governing nucleation, and statistical measurements have to be performed (stochastic phenomenon). Major recent breakthroughs concerning nucleation understanding were performed with numerical investigations on model systems5,6 and with experimental studies on crystallization of colloidal particles and proteins.7-9 Recently, it has also been shown that microfluidic tools may improve experiments of crystallization in solution, especially for the case of biological macromolecules.10-13 For instance, microfluidics allows perfect control of the kinetic pathways in the phase diagram,14-16 and * E-mail:
[email protected]. (1) Myerson, A. S. Handbook of industrial crystallization; ButterworthHeinemann: Boston, 2002. (2) Rodriguez-Hornedo, N.; Murphy, D. J. Pharm. Sci. 1999, 88, 651. (3) Bernstein, J.; Davey, R. J.; Henck, J.-O. Angew. Chem., Int. Ed. 1999, 38, 3440. (4) Zettlemoyer, A. C. Nucleation; Marcel Dekker: New-York, 1969. (5) ten Wolde, P. R.; Frenkel, D. Science 1997, 277, 1975. (6) Desgranges, C.; Delhommelle, J. Phys. ReV. Lett. 2007, 98, 235502. (7) Galkin, O.; Vekilov, P. G. J. Phys. Chem. B 1999, 103, 10965. (8) Gasser, U.; Weeks, E. R.; Schofield, A.; Pusey, P. N.; Weitz, D. A. Science 2001, 292, 258. (9) Gliko, O.; Neumaier, N.; Pan, W.; Haase, I.; Fischer, M.; Bacher, A.; Weinkauf, S.; Vekilov, P. G. J. Am. Chem. Soc. 2005, 127, 3433. (10) Hansen, C. L.; Skordalakes, E.; Berger, J. M.; Quake, S. R. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 16531. (11) Hansen, C. L.; Sommer, M. O.; Quake, S. R. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 14431. (12) Zheng, B.; Roach, L. S.; Ismagilov, R. F. J. Am. Chem. Soc. 2003, 125, 11170. (13) Zheng, B.; Tice, J. D.; Roach, L. S.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2004, 43, 2508. (14) Shim, J.; Cristobal, G.; Link, D. R.; Thorsen, T.; Jia, Y.; Piattelli, K.; Fraden, S. J. Am. Chem. Soc. 2007, 129, 8825. (15) Leng, J.; Lonetti, B.; Tabeling, P.; Joanicot, M.; Ajdari, A. Phys. ReV. Lett. 2006, 96, 084503. (16) Hansen, C. L.; Classen, S.; Berger, J. M.; Quake, S. R. J. Am. Chem. Soc. 2006, 128, 3142.
this control may be useful for protein crystallization.17,18 Moreover, microfluidic technologies permit the formation of droplets containing a given solution (typical volume 1-100 nL) and playing the role of chemical microreactors.19,20 The development of such droplet-based devices offers the possibility to perform high-throughput screening of protein crystallization conditions with small quantities of solutions.21,22 Gong et al. demonstrate recently that droplets generated by microfluidic tools can be used to investigate crystallization of thermoresponsive colloids.23 In recent work, we developed a microfluidic setup to store hundreds of droplets that contain a given aqueous solution and to control their temperature.24 We demonstrated that this device permits us to measure the solubility of metastable polymorphs and estimate the metastability extent of the solution. In the present work, we show that this device is also a microfluidic equivalent of the classical droplet method25-27 to measure nucleation kinetics. The principle of this method is to divide the volume of the studied solution in a large number of small, independent droplets in an inert oil. When the number of droplets is larger than the number of impurities initially present in the solution, one may observe homogeneous nucleation in some droplets. Moreover, when all the droplets have the same volume V and for small enough V, it is a priori possible to relate the probability that a droplet contains a crystal to the nucleation rate.28 Measurements of the temporal evolution of the fraction of droplets that contain crystals thus permit estimations of the nucleation kinetics. (17) Shim, J.; Cristobal, G.; Link, D. R.; Thorsen, T.; Fraden, S. Cryst. Growth Des. 2007, 7, 2192. (18) Talreja, S.; Kenis, P. J. A.; Zukoski, C. F. Langmuir 2007, 23, 4516. (19) Joanicot, M.; Ajdari, A. Science 2005, 309, 887. (20) Song, H.; Chen, D. L.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2006, 45, 7336. (21) Gerdts, C. J.; Tereshko, V.; Yadav, M. K.; Dementieva, I.; Collart, F.; Joachimiak, A.; Stevens, R. C.; Kuhn, P.; Kossiakoff, A.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2006, 45, 8156. (22) Lau, B. T.; Baitz, C. A.; Dong, X. P.; Hansen, C. L. J. Am. Chem. Soc. 2007, 129, 454. (23) Gong, T.; Shen, J.; Hu, Z.; Marquez, M.; Cheng, Z. Langmuir 2007, 23, 2919. (24) Laval, P.; Giroux, C.; Leng, J.; Salmon, J.-B. J. Cryst. Growth 2008, 310, 3121. (25) Vonnegut, B. J. Colloid Sci. 1948, 3, 563. (26) Turnbull, D. J. Chem. Phys. 1952, 20, 411. (27) Pound, G. M. Ind. Eng. Chem. 1952, 44, 1278. (28) Kashchiev, D.; Clausse, D.; Jolivet-Dalmazzone, C. J. Colloid Interface Sci. 1994, 165, 148.
10.1021/la802695r CCC: $40.75 2009 American Chemical Society Published on Web 12/23/2008
Nucleation Kinetics Measurements
Figure 1. (a) Picture of the microfluidic chip sealed on a glass slide (channel width and height: 500 µm). At the intersection between the oil and aqueous solution streams, monodisperse droplets are formed (here with a colored dye). Fluorinated oil may be injected after the droplet formation to insert one drop between each of solution. The droplets flow in the long microchannel when outlet 2 is opened and outlet 1 is closed (they are discarded for the opposite). (b) Schematic temperature profile to follow nucleation kinetics after a temperature quench below the solubility down to Tc. The PDMS device is sealed on a silicon wafer and placed on a Peltier module to control its temperature. Typical images (20 × 20 mm2) obtained under crossed polarizers of the storage area where birefringent crystals appear as bright spots (the symbols indicate the acquisition times).
However, the classical droplet method raises many experimental difficulties especially in the case of crystallization in solution. Indeed, droplets are often obtained using classical emulsification and are therefore never monodispersed, and they often contain surfactants. The polydispersity thus complicates the estimation of the nucleation kinetics,29 and the surfactants may interfere with the crystallization process. Moreover, nucleation events may not be independent in concentrated emulsions, as recently demonstrated.30 Finally, only indirect measurements of the fraction of the droplets containing crystals are often possible using this method, and it is difficult to rapidly induce a homogeneous supersaturation to all the droplets in the emulsion. We develop in our work an original microfluidic setup to overcome these drawbacks. Briefly, our setup allows us to store up to 300 monodisperse droplets (∼100 nL) containing an aqueous solution (see Figure 1a). These droplets are formed without surfactants at the intersection with a silicon oil stream, and only ∼100 µL of solution is required to perform an experiment. Fluorinated oil plugs are inserted between two solution droplets to avoid their coalescence when stored for a long period (typically a few hours).31 Temperature ramps can be applied rapidly to the droplets with perfect uniformity in the storage zone ( 40 g/100 g. Finally, it seems that the nucleation events of the stable form II are more frequent for small cooling temperature Tc. In the following, we focus on the nucleation kinetics for concentration C ) 40 g/ 100 g, and for cooling temperature down to 1 °C, we can thus neglect the occurrence of nucleation of form II. Kinetic Measurements. To investigate nucleation kinetics, ∼300 droplets of KNO3 at a given concentration are stored in the device as explained above. Temperature quench in the metastable zone is then applied at a rate of 0.1 °C s-1 down to a cooling temperature Tc (see Figure 1b for an example). The probability P that a droplet does not contain a crystal is simply estimated using custom-made analysis programs from images of the storage area at different times t (see Figure 1b for typical examples). Figure 7a displays the measured temporal evolution of P, for different cooling temperatures Tc and for a KNO3 solution with C ) 40 g/100 g of water. The probabilities P decrease as a function of time, at a rate that depends on the cooling temperature Tc. As expected, P(t) decreases more rapidly at low Tc than at higher Tc. If the nucleation process is identical for all the droplets, and characterized by a nucleation rate J, one expects that the probability P evolves as P(t) ) exp (-JVt), where V is the volume of the droplets.25-27,36 Figure 7b, where log (P) is plotted against t, demonstrates that there is not a unique nucleation rate in our experiments. This phenomenom has been observed in many other similar experiments using the droplet method and is explained by the presence of impurities that provoke nucleation in some droplets at small time scales. When all these specific droplets contain a crystal, the other droplets are expected to be governed by a (smaller) homogeneous nucleation rate. These nonlinearities thus suggest the presence of a distribution of impurities in the stored droplets. Evidence for Impurities. To evidence the presence of impurities-mediated nucleation events, we performed the following experiments. About 160 droplets of a KNO3 solution (C ) 30 g/100 g) are stored in the device above the solubility (35) Lide, D. R. Handbook of chemistry and physics; CRC Press: Boca Raton, 2004-2005. (36) Kashchiev, D.; Verdoes, D.; van Rosmalen, G. M. J. Cryst. Growth 1991, 110, 373.
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Figure 7. P(t) for different cooling temperatures Tc at C ) 40 g/100 g of KNO3 in water: (O) Tc ) 3.8, (0) 2.8, (]) 1.9, and (3) 0.9 °C. (b) Log (P) vs t for the same data set; the solid lines are fits of the data according to eq 3.
temperature of the forms II and III (18.3 and 14.9 °C at this concentration). Then, seven temperature cycles displayed in Figure 8a are applied on the droplets. During one cycle, temperature is first decreased at a rate of 0.1 °C/s from 23 to 0 °C, then maintained at 0 °C for 2 min to induce nucleation events in some droplets, and then increased again rapidly (in about 10 s) to 23 °C, in order to dissolve the crystals that have nucleated below the solubilities at 0 °C (the movie corresponding to Figure 8 is given in Supporting Information). We varied the temperatures at which we induce the dissolution of the crystals and also the duration of the cycles to induce the nucleation events, but it has no influence on the following results. Figure 8 shows images of the stored droplets taken at the end of the steps at 0 °C (the corresponding times are indicated on the temperature ramp by symbols). As above, crystals appear as bright pixels on these images taken under crossed-polarizers. At the end of each step at 0 °C, 31 ( 2 droplets contain a crystal; however, it appears that these events are not randomly distributed among all the droplets as suggested by the images of Figure 8. To quantify this result, we measured the probability pg that nc nucleation events occurred during the seven cycles in one specific drop. Figure 8b shows such estimated pg as a function of nc (bars). On the same figure is plotted the following expected probability
Figure 8. (a) Temperature profile to investigate the influence of impurities on the first nucleation events (C ) 30 g/100 g of water, solubilities of forms II and III: T2 ) 18.3, T3 ) 14.9 °C). The symbols indicate the acquisition times of the corresponding 14 × 25 mm2 images (only the first six images are shown for clarity). (b) Probability pg that a given drop contains a crystal nc times on the seven temperature cycles. Bars correspond to the experimental values and (9) to the theoretical values of pg in the case of random nucleation of crystals with p ) 0.19 ( 0.02 (see text and eq 2).
pg ) Cnncpnc(1 - p)n-nc
P(t) ) e-m(e-k0t - 1) + e-m exp(m e-k t)
(2)
if the nucleation events were randomly distributed. Equation 2 corresponds to the classical binomial distribution, and p ) (31 ( 2)/160 ) 0.19 ( 0.02 corresponds the estimated probability of nucleation at the end of each step at 0 °C. The large discrepancies between the experimental results and the estimated pg in the case of random events cannot be accounted for by temperature gradients across the storage zone, since variations of the temperature are below 0.05 °C (see the Microfabrication and Thermal Control section). We also rule out incomplete dissolution of the KNO3 crystals, since the kinetics of dissolution of this ionic crystal in the investigated range of temperatures is very rapid, and most importantly, because we check experi-
mentally that both the temperature and the duration of the dissolution step have no influence on the above results. Our experiments thus demonstrate unambiguously the presence of impurities in some droplets that provoke nucleation rapidely. Model of Distribution of Impurities. To get further information from the data P(t) of Figure 7, one can try to account for the presence of impurities randomly distributed among the droplets and get quantitative data on the supposed homogeneous nucleation rate at long time scales. Such attempts were first performed by Pound and La Mer,37 in the pionneering studies of nucleation kinetics of supercooled liquid droplets.25-27 If one assumes that impurities are randomly distributed among the droplets (Poisson distribution), and that m is the average number of active sites for heterogeneous nucleation in the droplets, then the probability P(t) that a droplet does not contain a crystal evolves as
(3)
where k0 is the homogeneous nucleation rate that one can observe in free-impurities droplets, and k′ is the heterogeneous nucleation rate for a unique impurity per drop.37 This model only takes into account a unique population of impurities, all characterized by the same activity according to the nucleation and randomly distributed among the droplets. This equation for P(t) accounts for the nonlinearities observed in the experimental data of P(t), as can be seen on the fits displayed in Figure 7b. Such a procedure thus enables us to extract estimations of m and of the expected homogeneous nucleation rate k0 from the data. (37) Pound, G. M.; LaMer, V. K. J. Am. Chem. Soc. 1952, 74, 2323.
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Figure 9. Evolutions of the parameters k0 and m as a function of the supersaturation S. These parameters are estimated from the Pound and La Mer model using the data of Figure 7.
Classical nucleation theory relates the nucleation rate to the frequency of monomer attachment to the critical nuclei and to the barrier energy to overcome surface tension effects (see ref 38 for a review). In this model, the nucleation rate is expected to follow the relation
(
k ) AVS exp -
B ln2 S
)
(4)
where A accounts for attachment frequency to the nucleus and depends linearly on the concentration of nucleation sites, S is the supersaturation of the solution, and B accounts for the surface tension between the nucleus and the saturated solution, and also depends on the nucleus shape. This relation holds for both homogeneous and heterogeneous nucleation; only the values of A and B differ in these two different mechanisms. In the classical nucleation theory, the prefactor A does not depend a priori on the concentration of salt molecules but on the concentration of nucleation sites,38 i.e., the concentration of impurities for heterogeneous nucleation, or the concentration of all the molecules of the solution in the case of homogeneous nucleation. Note that the nucleation rate k varies strongly with the applied supersaturation S over several orders of magnitude. Figure 9a displays the estimated k0 from the Pound and La Mer model applied to our data as a function of 1/ln2 S. The supersaturations S are estimated from the solubility curves displayed in Figure 5. More precisely, we approximate for our ionic crystal that S ≈ (C/Ceq)2 where Ceq is the solubility concentration of the different forms estimated from Figure 5.38 Despite intense literature researches, we have not found experimental data of the ions’ activity at the investigated metastable states. Moreover, we unfortunately have no experimental measurements of the solubility of form III for T < 5 °C. We therefore estimate the supersaturation S with the solubility of form II obtained from the literature data in this temperature range. Note that, for these low temperatures, the solubilities of the two forms seem to collapse as shown in Figure 5. A simple linear fit of the data of Figure 9a allows us to estimate A and B using eq 4: A ≈ 1.6 × 109 m-3 s-1 and B ≈ 37 (we assume that (38) Kashchiev, D.; van Rosmalen, G. M. Cryst. Res. Technol. 2003, 38, 555.
the evolutions of A and B with the temperature are negligible; Tc varies indeed from 1 to 4 °C in our case). In the case of homogeneous nucleation, one expects values for the prefactor A ranging from 1026 to 1030 m-3 s-1;38 our experimental estimation of A is very different (A ≈ 109 m-3 s-1) with more than 15 orders of magnitude off from the expected value. Such a huge discrepancy is often attributed to heterogeneous mechanisms, even with our analysis that enables us to dismiss impurities that act at small time scales in some droplets. However, no experimental values of A as large as the theoretical expected values have been measured for the crystallization of solutes, and the classical nucleation theory probably does not capture all the complexity of crystallization in solution. A more convincing argument for heterogeneous nucleation is the estimated evolution of m as a function of S, as shown in Figure 9b. As the applied supersaturation increases, the average number of impurities extracted from the data of Figure 7 increases significantly. This result cannot be understood with the Pound and La Mer model only and reveals the existence of different impurities. Indeed, all impurities are probably not equally effective as nucleating agents at a given temperature. It seems that foreign nucleation sites are not active at a given temperature but should be active at a lower temperature when supersaturation increases.37 It thus appears that, in the range of investigated supersaturations (up to S ≈ 7.5), and using the droplet method that isolate active impurities in droplets, nucleation rates at long time scales are probably always governed by heterogeneous mechanisms in the case of KNO3 aqueous solutions.
Conclusion In this work, we have developed an original microfluidic equivalent of the droplet method classically used to investigate nucleation kinetics.25-27 This setup allows us to store up to 300 small reactors (∼100 nL) and to control precisely their temperature. In such small volume, it is possible to measure solubilities of metastable polymorphs and quantify nucleation kinetics. On a specific system, potassium nitrate in water, and in conditions where nucleation of a metastable form is highly predominant, we have demonstrated with direct measurements that nucleation occurs always through heterogeneous mechanisms that involve impurities with different activities randomly distributed among the droplets up to supersaturations close to S ≈ 8. To possibly isolate impurities, one has to use smaller droplets, but the expected nucleation times may be too long to be easily reached with such a method. We believe that our device is wellsuited for investigating crystallization of many other compounds, since it allows perfect control of the crystallization conditions using small amounts of solutions. Acknowledgment. We gratefully thank J. Leng and M. Joanicot for fruitful discussions. We also acknowledge Re´gion Aquitaine for funding and support. Supporting Information Available: Movie of the nucleation/ dissolution cycles shown in Figure 8. Crystals appear as bright pixels in the stored droplets. This movie shows the presence of active impurities that provoke nucleation at small time scales in some droplets. This material is available free of charge via the Internet at http://pubs.acs.org. LA802695R
