ISSN 10637745, Crystallography Reports, 2014, Vol. 59, No. 3, pp. 437–441. © Pleiades Publishing, Inc., 2014. Original Russian Text © E.K. Titaeva, V.B. Fedoseev, 2014, published in Kristallografiya, 2014, Vol. 59, No. 3, pp. 484–488.
CRYSTAL GROWTH
Specific Features of Crystallization of Supersaturated Solution in FemtoliterVolume Systems E. K. Titaeva and V. B. Fedoseev Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 603950 Russia email:
[email protected],
[email protected] Received July 1, 2013
Abstract—According to the thermodynamic model, a very high supersaturation of a solution (up to the com plete thermodynamic forbiddenness of crystallization) can be achieved in smallvolume systems with phase transitions of the solution–crystal type. Observations in favor of these regularities are described. In particular, “nonOstwald” behavior in an ensemble of droplets of femto and picoliter volume is demonstrated: the evaporation and crystallization times of small droplets significantly exceed the evaporation time of large droplets. It is shown experimentally that this behavior is characteristic of different crystals. The described reg ularities are proposed to have a general character. DOI: 10.1134/S1063774514030195
INTRODUCTION Phase transformations of a solution–crystal type in small volumes were described in [1, 2]. The great interest in the phase transitions in systems of femto and picoliter volumes is related to the intense develop ment of nanotechnology [3–5], as well as biotechnol ogy and medicine [6–8], where complex manipula tions with femtoliter volumes of materials must be per formed in some cases [9]. The effects that occur (when passing from pico to femtoliter volumes) in systems characterized by phase transitions are described in [10]. In particular, accord ing to the thermodynamic model, a very high supersat uration of solution (up to the complete thermody namic forbiddenness of the heterogeneous state) can be achieved in droplets of this size. When a system is diminished, the sizes and formation energy of critical nucleus increases. A model system was also proposed in [10] that made it possible to observe these effects experimentally. This study expands the range of mate rials for which such experimental observations were reproduced. This gives grounds to suggest that the reg ularities described in [10] have a general character. The first microscopic experiments [2], which were aimed at observing the evolution of crystal shape with time, were performed with droplets of saturated aque ous solution of NH4Cl in a cavity with a crosssec tional area of ∼30 µm (∼10–20 pl), formed in a hydro phobic organic material. Experiments on the crystalli zation of thiourea and NaOH aqueous solutions in a vaporizing droplet with a volume of about 100 pL are described in [3]. The influence of the solution concen tration and substrate properties on the size, structure, and properties of obtained crystalline formations was investigated, and droplet evaporation rate and the cor
responding solution supersaturation were theoretically estimated. The formation of the HgI2 crystalline phase in porous matrices was investigated in [11] by Raman spectroscopy, electron microscopy, and atomic force microscopy. In accordance with the results of this study, the formation of microcrystals of stable red modification is preceded by the formation of metasta ble yellow and orange HgI2 modifications. A similar size dependence for a sequence of phase transitions between stable and metastable TiO2 states (anatase– brookite–rutile) was observed experimentally in [12]. The relatively short duration of processes occurring in smallvolume objects makes it possible to investi gate the kinetic regularities of the diffusion and ther mal processes of nucleation, growth, crystal forma tion, as well as the kinetics of solvent evaporation. For example, the dependences of the sizes of K2SO4 crys tals in solution droplets on their growth time were obtained, and different velocity modes and stages of crystal growth were found in [13]. A linear dependence of the crystal dispersivity on the inverse time of their growth was found. A diffusion model of evaporation of a single spherical droplet with allowance for the decrease in the drop temperature and vapor pressure near the surface was developed in [14]. Some tech niques were proposed for measuring temperature dur ing droplet evaporation and crystallization [15, 16]. In particular, the mass and viscosity of relatively large drops can be measured using a quartz cavity [17]. The growth of K2SO4 crystals in a solution droplet with a volume of several microliters was investigated in [13]. More information can be obtained by observing an ensemble of droplets with different volumes [10, 18]. These observations allow one to compare (under similar conditions) the evaporation and crystallization
437
438
TITAEVA, FEDOSEEV
rates for drops belonging to different size fractions. An example of such observations is the “nonOstwald” sequence of evaporation of neighboring solution drop lets of femto and picoliter volumes; under these con ditions, the lifetimes (evaporation and crystallization times) of small droplets significantly exceeded the life time of large droplets. To check the general character of the size effects described in [10, 18] during phase transitions in sys tems of submicron size, we experimentally demon strated the existence, stability, and reproducibility of metastable states using the example of several systems of the solution–crystal type. THERMODYNAMIC DESCRIPTION OF CRYSTALLIZATION IN A SMALL VOLUME OF SOLUTION The most important feature of a solution as a mul ticomponent system is that phase transitions in a small volume are accompanied by a significant change in concentration of all solution components. A thermo dynamic model was proposed in [10] to explain the regularities of the behavior of supersaturated solution in smallvolume systems exhibiting phase transitions of the solution–crystal type. Dispersed systems with colloidal sizes belong to nonextensive systems [19]; currently, methods of non extensive thermodynamics are actively used for their description [20]. As was shown in [21], the classical and nonextensive thermodynamics leads in the gen eral case to consistent results; however, classical ther modynamics is preferred in considering multicompo nent systems, especially when equations of state of a real solution are used. Applying the main concepts of thermodynamic description [10], we will consider a closed twocom ponent system in which component 1 is a solvent and solid component 2 is limitedly dissolved in it. The con centration is set with a molar fraction x0 of component 2 in the system. The solvent is present only in the phase of the solution, while the dissolved component 2 is dis tributed between the solution and crystalline phase. Below the solution phase is denoted as sol and the crystalline phase is denoted as cr. The conditions of material conservation,
n1 = n1sol , n2 = n2sol + n2cr , n2sol n2 x0 = , x sol = , xcr = 1, n1 + n2 n1sol + n2sol
(1)
relate the number of mols of components in the sys tem, ni (i = 1, 2), and in the solution, n1sol and n2sol, to the concentrations of the mixture, x0, and solution phase, xsol. The numbers of components n1 and n2 are the parameters completely determining the initial composition and size of the system. Either the n2cr or xsol value is variable.
As an additional condition we will use the require ment of uniqueness of new phase inclusion. This lim itation is correct for only smallsize systems, because it excludes the case of formation of a crystalline phase as a dispersion. Let us consider a simple geometric configuration under the assumption that the system (droplet) has a spherical shape and the crystal is a cube. In this case, volume Vπ, radius Rπ, and externalboundary area Aπ of the system are completely determined by its compo sition n1, n2:
( )
1
3 Rπ = 3 Vπ , 4π
Vπ = n1V1 + n2V2,
(2) 2 3 3 Aπ = 4π Vπ , 4π where V1 and V2 are the molar volumes of the compo nents. Simplifying the problem, we can suggest that the molar volumes of the components are independent of the concentration and do not change when a com ponent passes from crystal to solution. This approxi mation makes it possible to exclude enthalpy compo nents related to a change in pressure or cavity forma tion from consideration. Let us analyze the case where the crystal is completely wetted by the solution and does not touch the external boundaries. Under these conditions, the volume Vcr and the radius Rcr of the crystalline phase and the interfacial area Asol/cr are completely determined by the n2cr value:
( )
Vcr = n2crV2,
1 Lcr = Vcr3 ,
Asol /cr =
2 3 6Vcr .
(3)
The Gibbs function of the solution–crystal system, with allowance for the surface energy of the external and internal interfaces, has the form
g = n1sol μ1sol + n2sol μ 2sol + n2cr μ 2cr + σ π Aπ + σ sol / cr Asol / cr , μ1sol = μ1osol + RT ln (1 − x sol ) , μ 2sol = μ 2osol + RT ln x sol ,
(4)
μ 2cr = μ 2ocr ,
where σπ and σsol/cr are the surface energies of the external boundary of the system and the interface, respectively; μ is the chemical potential; and µ ijo is the standard chemical potential of the components in the corresponding phases. The standard chemical poten tials of component 2 in the solution (µ o2sol ) and the crystalline phase (µ o2cr ) can be related based on the sol ubility data:
µ o2cr = µ o2sol + RT ln s ,
(5)
Here, s is the molar fraction of component 2 in a satu rated solution. With allowance for conditions (1)–(3), the Gibbs function g is a function of three variables: n1, n2, and
CRYSTALLOGRAPHY REPORTS
Vol. 59
No. 3
2014
SPECIFIC FEATURES OF CRYSTALLIZATION OF SUPERSATURATED SOLUTION GE, J/mol 1 2 3 4 0
Lсг, nm 120 110 5
–0.5
439
100
0.6
6
0.8
1.0 R, μm
Fig. 2. Dependence of the criticalnucleus size Lcr on the radius of a droplet of NaCl solution with a degree of super saturation of 1.1 at 20°C [10].
7 –1.0 0.05
0.10
0.15
ϕ
Fig. 1. Dependence GE(ϕ) for a NaCl solution with a degree of supersaturation of 1.1 at 20°C [10]. The curves correspond to spherical droplets with radii of (1) 0.45, (2) 0.5, (3) 0.55, (4) 0.6, (5) 0.65, (6) 1.5, and (7) 7 µm. The absence of a peak in GE(ϕ) at R ≥ 7 µm corresponds to the spinodal decomposition of the homogeneous state.
n2сr. The radius Rπ or volume Vπ of the system and the mixture concentration x0 can be used as parameters instead of n1 and n2. The third variable can also be cho sen in different ways. The use of values varying in the range from 0 to 1 is convenient when comparing char acteristics of systems significantly differing in sizes and/or concentrations. The dimensionless value n ϕ = 2cr , which is the fraction of component 2 in the n2 crystalline phase, was used in [18]; one can also use the ratio of the crystal volume to the volume of the system. We will present the Gibbs function in the coordi nates (n1, n2, ϕ) and consider only the excess (with respect to the homogeneous state) part of the function per mol of mixture for a system with a specified com position (n1, n2):
G E ( ϕ) =
g ( n1, n2, ϕ) − g ( n1, n2,0) . n1 + n2
(6)
In this form, the Gibbs function GE(ϕ) corresponds to the definition of excess thermodynamic quanti ties [22]. The characteristic form of GE(ϕ) is shown in Fig. 1. A more detailed study of the GE(ϕ) function makes it possible to obtain a size dependence for a critical nucleus of crystalline phase (Fig. 2) and the size dependences for the energy of nucleus formation, crit ical supersaturation, and saturated solution concen tration, which were described in [10]. CRYSTALLOGRAPHY REPORTS
Vol. 59
No. 3
EXPERIMENTAL Samples were prepared using distilled water and agents of pure and analytically pure grades. Experi ments were performed with solutions, the concentra tion of which ranged from 1.5 to 10 times below the saturatedsolution concentration. Dilution makes it possible to increase the time for searching and observ ing an appropriate ensemble of drops of different sizes (with radii in the range of 10–100 µm) with a decrease in the size of droplets (in which the solution reaches a critical supersaturation as a result of solvent evapora tion) to a femtoliter volume. Solutions were deposited onto a glass substrate using a button sprayer. Each image was fixed either as a video (1280 × 960, 15 frames/s) or as a series of indi vidual images, using DinoLite AM451 or AXIO LAB.A1 (Carl Zeiss) microscopes. The video filming time ranged from one to several tens of minutes, depending on the conditions. The experiments were performed in a room with out any special temperature or humidity control. RESULTS AND DISCUSSION One of the purposes of our study was to check the reproducibility of the “nonOstwald” behavior of an ensemble of microdroplets of solutions of different crystals. Previously, similar behavior was described for ensembles consisting of three [10] or more [18] drop lets of NaCl solution. Primarily we performed observations for the having a diagram of solubility similar to that for NaCl [23]. Figure 3 shows the sequence of states for ensembles of KCl, NaNO3, NH4Cl, and tartaric acid droplets. These figures demonstrate pronounced “nonOst wald” behavior of droplet ensembles of all materials under consideration. The effect is stably reproduced in experiments performed under different conditions, and there is no need for sample thermostating or maintaining air humidity in a certain range for its observation. Monitoring temperature and humidity is important when studying kinetic regularities.
2014
440
TITAEVA, FEDOSEEV (a)
(b)
(c)
(d)
50 μm Fig. 3. Sequence of states for an ensemble of aqueous solution droplets: (a) KCl, the time intervals from the video recording onset are (from left to right) 0, 418, 420, and 522 s; (b) NaNO3, the time intervals from the video recording onset are 0, 227, 449, and 1122 s; (c) NH4Cl, the time intervals from the video recording onset are 0, 338, 341, and 406 s; and (d) tartaric acid, the time intervals from the video recording onset are 0, 188, 200, and 252 s.
We took tartaric acid, which is used to produce tar trates [24, 25], as a representative of organic crystal line compounds. Figure 3d shows a sequence of states for an ensemble of tartaric acid droplets demonstrat ing the same regularities. The reproducibility of the behavior of the microdroplet ensemble suggests that the regularities observed are also characteristic of salts of organic acids. Solutions of salts forming crystallohydrates form a dispersed crystalline phase under the same conditions. Dispersed and fractal formations are observed when droplets of salts forming crystallohydrates (NaHCO3, CuSO4, etc.) are crystallized [2–4]. It was also shown in [26] that the fractal dimension of the structures aris ing in gel films is controlled by the formation condi tions of these films. According to [27], the fractal dimension can be considered a function of thermody namic conditions. In accordance with the thermody namic model [10], this can be explained by their high solubility or lower surface energy of the solution–crys tallohydrate interface. In this case, size effects should be observed in smaller droplets, which cannot be observed by optical microscopy. In the solutions used in this study, dispersed and fractal structures are formed in droplets having microliter volumes. CONCLUSIONS Size effects at phase transitions in femto and picoliter volumes are explained by two main factors: surface energy and conditions of material conserva
tion. First, the contribution of the surface energy of boundaries of the system and the solution–crystal interface becomes comparable with the Gibbs energy of the entire system with a decrease in its size. Second, the change in the concentration and chemical poten tial of solvent becomes significant even at a transition of a small amount of dissolved component from the solution to the crystalline phase and vice versa. With an increase in the size of the system, Eqs. (1)–(4) are smoothly transformed (without additional conditions) into equations of classical nucleation theory, in which the solution concentration barely changes during the formation of new phase nucleus; correspondingly, the chemical potentials of the solvent and dissolved com ponent are generally assumed to be constant. Size effects manifest themselves in enhanced crys tal solubility and in the achievement of high critical supersaturations of solution, up to complete thermo dynamic forbiddenness of supersaturated solution crystallization. The results of thermodynamic descrip tion are in agreement with the experimental observa tions. In particular, they make it possible to interpret the “nonOstwald” behavior of an ensemble of micrometersized solution droplets, at which the evaporation and crystallization times for small drop lets significantly exceed the evaporation time of large ones. The reproducibility of the effects for both inor ganic and organic materials suggests that the above described regularities have a general character.
CRYSTALLOGRAPHY REPORTS
Vol. 59
No. 3
2014
SPECIFIC FEATURES OF CRYSTALLIZATION OF SUPERSATURATED SOLUTION
ACKNOWLEDGMENTS We are grateful to V.N. Portnov for consultations on the history of the problem and for our fruitful discus sion of the results. This study was supported in part by the Russian Foundation for Basic Research, project no. 1303 12225ofim.
REFERENCES 1. G. G. Lemmlein, Morphology and Genesis of Crystals (Nauka, Moscow, 1973) [in Russian]. 2. M. O. Kliya, Kristallografiya 1 (5), 577 (1956). 3. L. V. Andreeva, A. S. Novoselova, P. V. Lebedev Stepanov, et al., Tech. Phys. 52, 164 (2007). 4. L. V. Andreeva, A. V. Koshkin, P. V. LebedevStepanov, et al., Colloid Surf. A 300 (3), 300 (2007). 5. L. A. Smirnova, T. A. Gracheva, A. E. Mochalova, et al., Ross. Nanotechnol. 5 (1–2), 79 (2010). 6. K. Provost and M.C. Robert, J. Cryst. Growth, 110, 258 (1991). 7. T. A. Yakhno, A. G. Sanin, O. A. Sanina, and V. G. Yakhno, Biophysics 57, 722 (2012). 8. Yu. Yu. Tarasevich and D. M. Pravoslavnova, Tech. Phys. 52, 159 (2007). 9. V. A. Nikitin, Tsitologiya 49 (8), 631 (2007). 10. V. B. Fedoseev and E. N. Fedoseeva, JETP Lett. 97, 408 (2013). 11. V. G. Dubrovskii, M. V. Nazarenko, and N. V. Sibirev, Tech. Phys. Lett. 35, 1117 (2009). 12. K.R. Zhu, M.S. Zhang, J.M. Hong, and Z. Yin, Mater. Sci. Eng. A 403, 87 (2005).
CRYSTALLOGRAPHY REPORTS
Vol. 59
No. 3
441
13. V. Yu. Fedorov, Tech. Phys. 55 (7), 972 (2009). 14. A. V. Kozyrev and A. G. Sitnikov, Phys. Usp. 44, 725 (2001). 15. T. A. Yakhno, O. A. Sanina, M. G. Volovik, et al., Tech. Phys. 57, 915 (2012). 16. M. Strub, O. Jabbour, F. Strub, and J. P. Bédécarrats, Int. J. Refrigeration 26, 59 (2003). 17. T. A. Yakhno, A. G. Sanin, C. V. Vacca, et al., Tech. Phys. 54, 1423 (2009). 18. V. B. Fedoseev and E. N. Fedoseeva, Prikl. Mekh. Tekhnol. Mashinostr., No. 1(20), 89 (2012). 19. K. A. Putilov, Thermodynamics (Nauka, Moscow, 1971) [in Russian]. 20. C. X. Wang and G. W. Yang, Mater. Sci. Eng. R. 49 (6), 157 (2005). 21. G. A. Abakumov and V. B. Fedoseev, J. Mater. Sci. Eng. A 2 (11), 747 (2012). 22. V. M. Glazov and L. M. Pavlova, Chemical Thermody namics and Phase Equilibria: TwoComponent Metal and Semiconductor Systems (Metallurgiya, Moscow, 1981) [in Russian]. 23. A. N. Kirgintsev, L. N. Trushnikova, and V. G. Lav rent’eva, Solubility of Inorganic Materials in Water: A Handbook (Khimiya, Leningrad, 1972) [in Russian]. 24. M. L. Labutina, M. O. Marychev, V. N. Portnov, et al., Crystallogr. Rep. 56 (1), 72 (2011). 25. A. E. Egorova, V. N. Portnov, D. A. Vorontsov, et al., Vestn. NNGU, No. 6, Part 1, 58 (2011). 26. E. N. Fedoseeva and V. B. Fedoseev, Polym. Sci. Ser. A 53, 1040 (2011). 27. V. B. Fedoseev, Pis’ma Mater. 2, 78 (2012).
2014
Translated by A. Grudtsov
