Letter pubs.acs.org/JPCL
Kinetic Pathways and Mechanisms of Two-Step Nucleation in Crystallization Can Guo, Jincheng Wang,* Junjie Li, Zhijun Wang, and Sai Tang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China S Supporting Information *
ABSTRACT: Crystallizations often pass through multiple intermediate structures before reaching the final state, such as amorphous precursors, polymorphs, or denser liquid droplets. However, the atomistic pathways from these metastable phases to final crystals still remain unclear. Here, we investigated the structure evolution process from liquid to final crystals of homogeneous nucleation by atomic-scale simulations and analyzed the intrinsic mechanisms that influence the nucleation pathways. Three different pathways of two-step nucleation were found by visualizing the precursors’ evolutions, and some new micromechanisms of two-step nucleation are revealed. We suggest that the solid bond fluctuations can trigger the formation of intermediate precursors, while the precursors’ packing density dominates the structural transformation pathways from intermediate phases to crystals. These findings not only shed light on the mechanisms of nucleation but also provide guidance for future refinements of two-step nucleation theory. findings indicate that TS is fairly common and the intermediate phases play an important role in crystallization. To reveal the kinetic mechanism of TS, well-designed numerical and experimental research was carried out. Molecular dynamics (MD) simulations16 indicated that icosahedral configurations contributed to the formation of bcc nuclei. Phase-field crystal (PFC) simulations17 found amorphous precursors before the crystallization of bcc. Experimental investigations of colloidal18 and organic molecule19,20 crystallizations also reported that crystals are formed from amorphous precursors. However, understanding of the TS pathway is still limited, which is crucial for nucleation theory and industrial manufacturing. Recent colloid experiments21 showed that crystals are formed by direct nucleation inside of the amorphous precursors. Brownian dynamics simulations22 suggested that crystal nucleation may be “heterogeneous” as large amounts of middle-range ordered metastable precursors were found before the formation of nuclei. Tan23 found that the intermediate phases are the mixing of bcc-like, face-centered cubic (fcc)-like, and hexagonal close-packed (hcp)-like precursors and suggested two major pathways: from hcp-like or fcc-like precursors to fcc nuclei. These investigations bring us new insights on TS; however, the exact pathways about how the metastable precursor transforms into a crystal and the intrinsic mechanisms why the nucleation pathways are different for previous studies is far from revealing.
N
ucleation is a fundamental issue in biology, chemistry, condensed matter physics, and materials sciences. The morphology, size, and structure of nuclei as well as the nucleation pathways can strongly affect the performance of final products. Therefore, nucleation control has long been an important segment in industry fields, like casting, pharmaceuticals, chemical industry, and so forth. Nucleation, involving structure fluctuation, interface formation, and symmetry transformation, is a complex dynamic process that occurs at atomic length and diffusion time scales. Considering the initial liquid and the final solid as the only members in crystallization, Classical Nucleation Theory (CNT) was established and has successfully explained and predicted large amounts of experimental phenomena. However, CNT faces challenges as mounting experimental and simulation investigations indicated that crystallizations pass through intermediate states. This kind of crystallization, a so-called two-step nucleation (TS) has attracted much attention in the past decades and has been explained by Ostwald’s rule, which states that the first occurring phase in crystallization is normally the one closest in structure and free energy to the mother phase. Alexander and McTague1 argued that the body-center cubic (bcc) precursor was favored in all simple fluids in early studies. Numerical and colloid experiment studies2−4 for the Lennard-Jones (L-J) system also verified that precritical nuclei are bcc ordered predominantly. Further experiments showed that TS occurs not only in atomistic system but also in macromolecule and ionic solutions, such as amyloid fibril nucleation,5,6 proteins,7−9 biorelated10,11 and mineral12−14 crystallizations, and so forth. Moreover, Peng15 et al. found the existence of a liquid intermediate phase in the nucleation process of solid−solid transitions. All of these © XXXX American Chemical Society
Received: October 4, 2016 Accepted: November 22, 2016 Published: November 22, 2016 5008
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters Moreover, density fluctuation has long been considered as a key role in crystallization for one component system. It is believed that density fluctuation triggers the nucleation process as nucleation events occur inside of the dense regions for both classical nucleation and TS.6,8 However, recent simulation24 and experiment23 investigations suggested that spontaneous critical-like fluctuations of the solid bond numbers are more important. Tan23 even pointed out that the density fluctuation and symmetry development are entirely decoupling. Therefore, there is controversy about which is the key factor influencing crystallization. As fluctuations are fairly common and essential for crossing of the nucleation energy barrier, further investigations to clarify the effects of these two kinds of fluctuations can bring us new insights on understanding the nucleation mechanism. In this study, we addressed the nucleation issues using a simplified dynamic density function theory,25 the PFC model. The PFC model26 has shown its potential for nucleation issues, the energy relations of PFC and CNT have been built,27 and the appearance of amorphous precursors and TS17,28 have also been reported. Different from above studies, herein, we use the newly proposed multimode PFC,29 which can better describe the solid/liquid properties27,30 and investigate the complex structure transition process29,31 during nucleation compared with the classical PFC model. The free energy functional of PFC for pure material is written as
Figure 1. Phase diagram of the two-mode PFC model, where the dashed line below the peritectic line is the phase boundary of the metastable triangle phase. All parameters in our simulations were selected in the solid/liquid coexistence region. Here, the OS region is highlighted in yellow, the TS region is highlighted in gray, and the region where OS and TS occur in parallel is highlighted in blue.
0.1 is the phase boundary for the metastable triangle lattice. In our simulations, nucleation is induced by stochastic noises and all parameters are chosen in the solid/liquid coexistence region, as shown in Figure 1. The nucleation pathways in our simulations for various temperatures and supersaturations are also summarized in Figure 1. The one-step nucleation (OS) region is highlighted in yellow, THE TS region is highlighted in gray, while the transition region where both OS and TS occur in parallel is highlighted in blue. For TS, as suggested by Ostwald’s rule, we find that amorphous precursors appear first; this is because of the lower interfacial energy and nucleation barrier of the amorphous phase as compared with those of the crystals. Our simulations showed that crystallizations occur in one step only in low-supersaturation and low-temperature regions. The exact structural transformation pathways at the atomic length scale for five typical samples marked in Figure 1 are illustrated in Figure 2a−e. For sample A, as shown in Figure 2a1−a4, a small cluster with square symmetry emerges from the liquid first; then this cluster propagates rapidly and grows up into a square crystal. This is the typical OS described by CNT. Figure 2b1−b4 (for sample B) illustrated a TS process (type I). In the initial stage, Figure 2b1,b2, an amorphous precursor appears and propagates into the bulk liquid. The amorphous precursor is composed of liquid-like, square-like, and trianglelike atoms, where the square-like atoms are highlighted in blue (Figure 2b2). In the second stage, Figure 2b3, the square nucleus nucleates rapidly inside of the amorphous cluster and grows up into a square crystal. For sample C, as shown in Figure 2c1−c4, the nucleation is TS type (type II), and the square nucleus is also created from a dense amorphous precursor, but the kinetic mechanism is different from that in sample B. Sauter et al.33 found that crystallization of protein happened inside of the dense intermediate precursors, which has a similar pathway as Figure 2b. The square nucleus for sample C is produced through structure deformations; a shear parallel to the ⟨100⟩ direction of triangle-like precursors results in square symmetry, as shown in Figure 2c2−c4. This process happens throughout the amorphous cluster quickly. Figure 2d1−d4 shows another TS pathway for the square lattice (type III). For sample D, the amorphous precursor converts into a
⎡ ψ ( r ⃗)2 ψ ( r ⃗)3 ψ ( r ⃗)4 ⎤ − + ⎥ dr ⃗ ⎢ 6 12 ⎦ ⎣ 2 1 − ψ ( r ⃗) C2(| r ⃗ − r ′⃗ |)ψ ( r ′⃗ ) d r ⃗ d r ′⃗ (1) 2 where ψ is the time-averaged atom number density (supersaturation32). The Fourier form of the two-point correlation function for the multimode model is F=
∫
∫
C2(k) =
∫
∑ exp(−σ 2ki 2/2ρi βi ) exp(−(k − ki)2 /2αi 2) i
(2)
where i denotes the mode number, σ is equivalent temperature, and k, ρ, β, and α are crystallographic parameters (one can read ref 29 for details). The dynamic equation is given by ∂ψ δF ̃ = Γ∇2 +η ∂t δψ
(3)
where Γ is a mobility parameter and η is conserved stochastic noise. Here, η has the character of ⟨η(r,⃗ t)η(r′⃗ , t′)⟩ = ξσ∇2δ(r ⃗ − r′⃗ )δ(t − t′), where ξ is the noise strength and σ is an effective temperature parameter in the PFC model. In this work, noise with a wavelength shorter than the interatomic spacing is filtered, and ξ is set as a constant, ξ = 0.0001. Equation 3 is solved by using the semi-implicit Fourier spectral method with grid space Δx = a/8 and time step Δt = 0.01; here, a is the lattice constant. All of the simulations are conducted in a domain of L × M = 1024Δx × 1024Δx with periodic boundary conditions. Moreover, as we are only concerned with the nucleation process, the coarsening process and final equilibrium states are not shown in this work. The phase diagram for a 2D system is shown in Figure 1. The triangle phase is metastable for σ < 0.1.29 As all parameters are dimensionless, this phase diagram is for a general peritectic system instead of a specific material. The dashed line below σ = 5009
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters
Figure 2. (a−e) Snapshots of the structural transformation pathways during nucleation with different simulation parameters. For each row, four figures from left to right show the temporal evolution of crystal nuclei. Parameters selected for a−e are shown in Figure 1, corresponding to A−E, respectively. (a) σ = 0.05, corresponding to large undercooling, and the nucleation is OS type. (b−d) σ = 0.06−0.07, temperatures increase slightly, the nucleation is TS type, and the final crystal is square. (e) σ = 0.08, temperatures rise to the peritectic line, and there is a TS process for the metastable triangle crystal in the square stabilized region. (f−i) Temporal evolution of the atom fractions for square, triangle, and other configurations (mainly interface atoms and amorphous atoms) of the whole undercooled system for different temperatures: (f) σ = 0.05, (g) σ = 0.06, (h) σ = 0.07, and (i) σ = 0.08 (corresponding to points A, B, C, and E in Figure 1, respectively).
kind of TS has been widely investigated by colloidal experiments.18,21,34,35 The temporal evolution of the atom fractions for square, triangle, and other configurations (mainly interface atoms and amorphous atoms) of the whole simulation region (about the 15−20 trajectories in Figure 2a−e) with different temperatures is shown in Figure 2f−i. The atoms for different structures are distinguished by calculating the bond orientation order parameter q6(i),36 where i is the number of atoms, and the values of q6(i) for square and triangle lattices are 0.586 and 0.741, respectively. For large undercoolings (Figure 2f), only interface atoms and square atoms have been detected; the nucleation is OS type. As the temperature is raised, shown in Figure 2g,h, the number fractions and lifetime of triangle-like
metastable triangle nucleus instead of a square nucleus in the first stage. Then the square phase forms on the surface of the triangle phase through heterogeneous nucleation, where the faces contacting square and triangle crystals are (10)sq and (1− 10)tr. During crystallization, the triangle transforms into the square continually, and the motion of the phase boundary is shown in Figure 2d3,d4. Finally, all of the metastable triangle phase transforms into the square phase. For temperatures near the peritectic line, as shown in Figure 2e1−e4, the amorphous precursor transforms into the metastable triangle phase first, which is the same as in Figure 2d2, but further structural transformation into the thermodynamically stable square phase has not been observed. As for triangle stable systems, the nucleation pathway is the same as that of sample E, and this 5010
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters
nuclei would not be unified completely for a certain temperature (Figure 1). Density fluctuation has long been considered as the incentive of nucleation. Snapshots of spatial distributions for local densities (supersaturation) and precursors for TS are illustrated in Figure 4a,b. Comparing their distributions, we find that precursors always occurred in some relative dense regions but not the densest region. Moreover, the fluctuation of precursors is less frequent than the density fluctuation, and the nucleation velocity depends on the formation velocity of precursors. For this phenomenon, Tan23 and Russo24,40 concluded that it is the fluctuations in the solid bond order parameter (q6) rather than the density fluctuations that trigger the nucleation process, indicating that the polymorph selection also depends on Q6. Figure 4c shows the relation between the averaged q6 versus cluster size, Q6 vs N, where Q6 is defined as the average of q6 over all atoms within a cluster and N is the number of solid atoms in the cluster shown in Figure 2a−e. For sample A, the cluster has Q6 > 0.55, even when N < 8; however, for samples B−E, the Q6 of the target cluster just varies at around 0.4 even when N reaches 30. It is notable that the precursors are amorphous and cannot be distinguished by Q6 for N < 30. As for N > 30, the evolution of Q6 is different, which is due to the fact that their structural transformation mechanisms are different (see Figure 2a−e). For sample A, the nucleation is OS type; thus, Q6 increases to 0.59 (Q6 for the square phase) rapidly. For the TS process, the precursors are amorphous (Q6 < 0.4) in the initial stage, and thus, the value of Q6 increases slower than OS. It is also clearly shown that the value of Q6 for sample E is larger than that for samples B−D. This is due to the amorphous phase transform to the triangle (Q6 = 0.74) directly for sample E; the value of Q6 for the triangle is larger than that of other samples. For sample D, the nucleation passed through amorphous−triangle−square, and the cluster was a mixture with square, triangle, and interface phases; thus, the Q6 is less than that for sample E. As for samples B and C, the phase transition is from the amorphous phase to square, and their values of Q6 decrease again. The increase of Q6 for sample C is slower than that for sample B, which is due to the fact that the deformation-induced phase transition of sample C is more difficult and time-consuming. Thus, Q6 is a proper parameter to reflect the phase structure and nucleation pathway,22−24 and the fluctuation of Q6 is actually important for the building of precursors. However, due to the tiny difference of Q6 for these amorphous intermediate precursors before crystallization, the intrinsic factors that influence the structural transformation mechanism need to be further studied. Here, we find that the packing density ϑ of precursors has significant influence on the transformation pathway, where ϑ is defined as the crystal-like atom or molecular numbers per volume and normalized on the basis of the square nuclei. The critical packing densities for phase transformation from amorphous to crystal and liquid to crystal for samples A−E in Figure 2 are summarized in Figure 4d. For lower ϑ, there are more vacancies inside of the precursor and the free volume necessary for the mobility of atoms is larger. Thus, for ϑ = 0.4024, square crystals always nucleate inside of the amorphous precursor. As the packing density increases, for ϑ = 0.5835, there is not enough space for the formation of a new phase. In this case, the square nuclei are formed through deformation, where the underlying stresses may come from the surface tension. As for ϑ close to the density of the triangle crystal (0.866), the mobility of atoms decreases further, and very large
atoms during crystallization are increased. It is expected that the increase of atoms with triangle configurations can decrease the system’s nucleation barrier. As for temperatures close to the peritectic line, shown in Figure 2i, the metastable triangle phase is more favored than the square one, as described by Ostwald’s rule. At this time, if stable square phase nucleates on the triangle/liquid interface increase the system’s misfit energy significantly, then the nucleation of the square is suppressed. Until the driving force for crystallization of the triangle is exhausted, the phase transition from triangle to square could happen, and nucleation of the second phase starts at the grain boundaries. The kinetic pathways of the whole system shown in Figure 2f−i support the atomistic pathways shown in Figure 2a−e; thus, the atomistic pathways above represent the dominant pathway for each undercooling. In addition, the intermediate precursors are mixed with multiple phases (mainly triangle and square for our 2D system), which behave as the “seeds” of nuclei.23 The precursors’ structure diversity leads to the complexity of nucleation issues. Further investigations about the structure of the undercooled liquids can provide some useful information to understand nucleation mechanisms.37,38 The structure factor S(k)39 is calculated to characterize the liquid structure. The results for square nuclei, triangle nuclei, and the undercooled liquids are shown in Figure 3. Comparing the structure factors
Figure 3. Structure factors S(k) for different symmetries, (a) square nuclei (red line) and triangle nuclei (black line); (b) structure factors of undercooled liquids with different temperatures.
of square and triangle nuclei, we find a good match in the longwavelength peak (Figure 3a), which confirms their underlying structural connections. Figure 3b shows the S(k) for undercooled liquid with different temperatures, σ = 0.04, 0.06, and 0.08. According to Figure 3b, the difference between the first peaks of these samples is tiny, while the strength of the second peaks changed remarkably with temperature. For large undercoolings, the structure of undercooled liquids before crystallization is much closer to that of the square nuclei, and the square-like short-range order and square nuclei can be observed with high probabilities. However, for low undercoolings, the structure of undercooled liquids is close to that of the triangle nuclei, and the triangle-like short-range order and triangle nuclei can be observed with high probabilities. As for temperatures around σ = 0.06, the liquid structure will be a mixture of square-like and triangle-like short-range order; thus, competitive nucleation and growth between stable and metastable crystals occurred. In a word, the temperature can significantly influence the structure of an undercooled liquid and then affect the nucleation pathway. It also should be noted that, due to the existence of random fluctuations at the initial stage of crystallization, the atomistic pathways for all of the 5011
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters
Figure 4. Snapshots of spatial distributions for local densities (a) and precursors (b), where the data are taken in the early stage of crystallization. (c) Variations of the average bond order parameter Q6 of the target clusters in Figure 2. (d) Precursors’ packing densities (ϑ) for target clusters in Figure 2, where all of the data are normalized on the basis of square crystals. (e) The ϑ for the three different precursors for (ψ0, σ) = (−0.07, 0.06) (the blue region in Figure 1).
are compared and shown in Figure 1S in the Supporting Information. To illustrate the structural transformation mechanism of TS sufficiently, the free energy density evolutions versus N (the total number of atoms in the new nucleus) were calculated. We found that the distinctions between different stages of TS are very clear for the energy aspect, and the division of these stages is consistent with the atomistic snapshots in Figure 2 (for more details, one can refer to the Supporting Information, Figure 2S). Further, the nucleation energy barriers of proposed nucleation pathways are calculated quantitatively from Figure 2S, and the results are summarized in Table 1. After comparing
stress is needed to induce phase transition from triangle-like precursors to the square crystal by deformation. Then, in this case, the precursor tends to transform into a triangle nucleus by slight atom rearrangement; after that, a square nucleus is formed through “heterogeneous nucleation”. The formation of square nuclei is very hard for temperatures near the peritectic line, owning to the fact that forming a semicoherent interface ((atr − asq)/asq = 0.155) between the triangle and square is energy-disfavored for low undercoolings. To eliminate the influence of simulation parameters, we set (ψ0, σ) = (−0.07, 0.06), where multiple pathways happened in parallel, and three different nucleation pathways were observed, OS and two kinds of TS (types I and III); ϑ values are shown in Figure 4e. Compared with the results in Figure 4d, the values of ϑ for the same nucleation pathway are nearly equal. Therefore, ϑ is a key physical factor in determining the structural transformation pathway. Russo24,40 and Tan23 et al. reported that packing density is not crucial for nucleation. This is because the precursors’ packing density is very close to that of the solid for their investigation system, which just corresponds to “type III” of our work. While the precursors’ packing density values of our soft potential system can vary over a wide range, the influence of density fluctuations on nucleation pathways will be significant. To understand why the structure fluctuation is more important than the density fluctuation in the initial stage of nucleation, we calculated the local energy density distribution caused by both fluctuations. We found that the local energy density is lower in the regions with precursors, which indicates that structure fluctuation can decrease the system’s energy more effectively than density fluctuation does. The distributions of density, local structure, and energy density
Table 1. Energy Barrier of Each Stage of Homogeneous Nucleationa
stage I stage II a
type I (TS)
type II (TS)
type III (TS)
square (OS)
triangle (OS)
1.475 2.887
1.598 4.550
2.696
4.455
2.298
TS is for two-step nucleation, and OS is for OS nucleation.
these values, we found that the intermediate phases can significantly decrease the energy barrier of TS. For stage I, the formation of amorphous precursors is more energy favorable than that of the triangle, which is consistent with the structural analysis of intermediate phases in Figure 2. It should be noted that, although the energy barrier for type III (TS) is larger than that for the others, the nucleation of type III (TS) also passes through amorphous first (see the atomistic snapshots in Figure 2d). Because the amorphous phase is transformed into the triangle quickly in the initial stage of nucleation, the energy 5012
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters
(2) Shen, Y. C.; Oxtoby, D. W. bcc Symmetry in the Crystal-Melt Interface of Lennard-Jones Fluids Examined through Density Functional Theory. Phys. Rev. Lett. 1996, 77, 3585−3588. (3) Ten Wolde, P. R.; Ruiz-Montero, M. J.; Frenkel, D. Numerical Evidence for bcc Ordering at the Surface of a Critical fcc Nucleus. Phys. Rev. Lett. 1995, 75, 2714−2717. (4) Xu, S.; Zhou, H.; Sun, Z.; Xie, J. Formation of an fcc phase through a bcc metastable state in crystallization of charged colloidal particles. Phys. Rev. E 2010, 82, 010401. (5) Chong, S.-H.; Ham, S. Atomic-level investigations on the amyloid-β dimerization process and its driving forces in water. Phys. Chem. Chem. Phys. 2012, 14, 1573−1575. (6) Auer, S.; Ricchiuto, P.; Kashchiev, D. Two-Step Nucleation of Amyloid Fibrils: Omnipresent or Not? J. Mol. Biol. 2012, 422, 723− 730. (7) Di Profio, G.; Reijonen, M. T.; Caliandro, R.; Guagliardi, A.; Curcio, E.; Drioli, E. Insights into the polymorphism of glycine: membrane crystallization in an electric field. Phys. Chem. Chem. Phys. 2013, 15, 9271−9280. (8) Vivares, D.; Kaler, E. W.; Lenhoff, A. M. Quantitative imaging by confocal scanning fluorescence microscopy of protein crystallization via liquid-liquid phase separation. Acta Crystallogr., Sect. D: Biol. Crystallogr. 2005, 61, 819−825. (9) Wolde, P. R. t.; Frenkel, D. Enhancement of Protein Crystal Nucleation by Critical Density Fluctuations. Science 1997, 277, 1975− 1978. (10) Vatamanu, J.; Kusalik, P. G. Observation of two-step nucleation in methane hydrates. Phys. Chem. Chem. Phys. 2010, 12, 15065−15072. (11) Addadi, L.; Weiner, S. Control and Design Principles in Biological Mineralization. Angew. Chem., Int. Ed. Engl. 1992, 31, 153− 169. (12) Kovács, T.; Meldrum, F. C.; Christenson, H. K. Crystal Nucleation without Supersaturation. J. Phys. Chem. Lett. 2012, 3, 1602−1606. (13) Gebauer, D.; Cölfen, H. Prenucleation clusters and non-classical nucleation. Nano Today 2011, 6, 564−584. (14) Scheck, J.; Wu, B.; Drechsler, M.; Rosenberg, R.; Van Driessche, A. E. S.; Stawski, T. M.; Gebauer, D. The Molecular Mechanism of Iron(III) Oxide Nucleation. J. Phys. Chem. Lett. 2016, 7, 3123−3130. (15) Peng, Y.; Wang, F.; Wang, Z.; Alsayed, A. M.; Zhang, Z.; Yodh, A. G.; Han, Y. Two-step nucleation mechanism in solid−solid phase transitions. Nat. Mater. 2014, 14, 101−108. (16) Shibuta, Y.; Sakane, S.; Takaki, T.; Ohno, M. Submicrometerscale molecular dynamics simulation of nucleation and solidification from undercooled melt: Linkage between empirical interpretation and atomistic nature. Acta Mater. 2016, 105, 328−337. (17) Tóth, G. I.; Pusztai, T.; Tegze, G.; Tóth, G.; Gránásy, L. Amorphous Nucleation Precursor in Highly Nonequilibrium Fluids. Phys. Rev. Lett. 2011, 107, 175702. (18) Savage, J. R.; Dinsmore, A. D. Experimental Evidence for TwoStep Nucleation in Colloidal Crystallization. Phys. Rev. Lett. 2009, 102, 198302. (19) Ito, F.; Suzuki, Y.; Fujimori, J.-i.; Sagawa, T.; Hara, M.; Seki, T.; Yasukuni, R.; de la Chapelle, M. L. Direct Visualization of the Twostep Nucleation Model by Fluorescence Color Changes during Evaporative Crystallization from Solution. Sci. Rep. 2016, 6, 22918. (20) Harano, K.; Homma, T.; Niimi, Y.; Koshino, M.; Suenaga, K.; Leibler, L.; Nakamura, E. Heterogeneous nucleation of organic crystals mediated by single-molecule templates. Nat. Mater. 2012, 11, 877− 881. (21) Zhang, T. H.; Liu, X. Y. How Does a Transient Amorphous Precursor Template Crystallization. J. Am. Chem. Soc. 2007, 129, 13520−13526. (22) Kawasaki, T.; Tanaka, H. Formation of a crystal nucleus from liquid. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 14036−14041. (23) Tan, P.; Xu, N.; Xu, L. Visualizing kinetic pathways of homogeneous nucleation in colloidal crystallization. Nat. Phys. 2013, 10, 73−79.
barriers for each stage are hard to distinguish. As the crystallization driving force decreases with the temperature, the amorphous will transform into the triangle first (whose nucleation barrier is lower) rather than the square according to the step rule of Ostwald, as shown in Figure 2d,e. Further, we found that the energy barrier for type III is larger than the ideal value of nucleation of the triangle phase because the shape of the nucleus can rarely be circular in a real nucleation process, which leads to the increase of the interfacial free energy and the energy barrier. Moreover, the energy barrier of the second stage for type II is larger than the nucleation of OS. Therefore, this kind of nucleation is not an optimal energy pathway and scarcely appeared in our simulations. In conclusion, the structure of the precursors and the structural transformation mechanisms of homogeneous nucleation are investigated by atomic-scale simulations. By visualizing the precursors’ evolution process, we found three different phase transition mechanisms from amorphous into square crystals for TS: nucleation inside of the precursor, deformation, and heterogeneous nucleation. Furthermore, some new micromechanisms of TS were found by characterizing the precursors’ structure and density information during nucleation, and we suggest that the solid bond fluctuations can trigger the formation of intermediate precursors (stage I) while the precursors’ packing density dominates the structural transformation pathways (stage II). Herein, a 2D ideal system is studied, but further investigation of a three-dimension system will be more practical. Also, although the results by PFC simulations are more inclined to the nucleation behavior of soft matter systems, the conclusions obtained from PFC simulations are still significant and can provide new insights of nonclassical nucleation for more general systems.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b02276. Thermodynamic mechanisms of two-step nucleation (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Jincheng Wang: 0000-0003-3910-1020 Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work has been supported by the National Natural Science foundation of China (Grant Nos. 51571165, 51371151), the Natural Science Foundation of Shaanxi Province of China (Grant No. 2015JQ5151), and the Special Program for Applied Research on Super Computation of the NSFC−Guangdong Joint Fund (the second phase). We also thank the Center for High Performance Computing of Northwestern Polytechnical University, China for computer time and facilities.
■
REFERENCES
(1) Alexander, S.; McTague, J. Should All Crystals Be bcc? Landau Theory of Solidification and Crystal Nucleation. Phys. Rev. Lett. 1978, 41, 702−705. 5013
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
Letter
The Journal of Physical Chemistry Letters (24) Russo, J.; Tanaka, H. The microscopic pathway to crystallization in supercooled liquids. Sci. Rep. 2012, 2, 505. (25) Elder, K. R.; Provatas, N.; Berry, J.; Stefanovic, P.; Grant, M. Phase-field crystal modeling and classical density functional theory of freezing. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 75, 064107. (26) Elder, K. R.; Katakowski, M.; Haataja, M.; Grant, M. Modeling Elasticity in Crystal Growth. Phys. Rev. Lett. 2002, 88, 245701. (27) Guo, C.; Wang, J.; Wang, Z.; Li, J.; Guo, Y.; Huang, Y. Interfacial free energy adjustable phase field crystal model for homogeneous nucleation. Soft Matter 2016, 12, 4666−4673. (28) Backofen, R.; Voigt, A. A phase-field-crystal approach to critical nuclei. J. Phys.: Condens. Matter 2010, 22, 364104. (29) Greenwood, M.; Provatas, N.; Rottler, J. Free Energy Functionals for Efficient Phase Field Crystal Modeling of Structural Phase Transformations. Phys. Rev. Lett. 2010, 105, 045702. (30) Guo, C.; Wang, J.; Wang, Z.; Li, J.; Guo, Y.; Tang, S. Modified phase-field-crystal model for solid-liquid phase transitions. Phys. Rev. E 2015, 92, 013309. (31) Greenwood, M.; Ofori-Opoku, N.; Rottler, J.; Provatas, N. Modeling structural transformations in binary alloys with phase field crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 064104. (32) Tang, S.; Yu, Y.-M.; Wang, J.; Li, J.; Wang, Z.; Guo, Y.; Zhou, Y. Phase-field-crystal simulation of nonequilibrium crystal growth. Phys. Rev. E 2014, 89, 012405. (33) Sauter, A.; Roosen-Runge, F.; Zhang, F.; Lotze, G.; Jacobs, R. M. J.; Schreiber, F. Real-Time Observation of Nonclassical Protein Crystallization Kinetics. J. Am. Chem. Soc. 2015, 137, 1485−1491. (34) Sear, R. P. Nucleation: theory and applications to protein solutions and colloidal suspensions. J. Phys.: Condens. Matter 2007, 19, 033101. (35) Schö pe, H. J.; Bryant, G.; van Megen, W. Two-Step Crystallization Kinetics in Colloidal Hard-Sphere Systems. Phys. Rev. Lett. 2006, 96, 175701. (36) Lechner, W.; Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 2008, 129, 114707. (37) Kelton, K. F.; Lee, G. W.; Gangopadhyay, A. K.; Hyers, R. W.; Rathz, T. J.; Rogers, J. R.; Robinson, M. B.; Robinson, D. S. First XRay Scattering Studies on Electrostatically Levitated Metallic Liquids: Demonstrated Influence of Local Icosahedral Order on the Nucleation Barrier. Phys. Rev. Lett. 2003, 90, 195504. (38) Reichert, H.; Klein, O.; Dosch, H.; Denk, M.; Honkimaki, V.; Lippmann, T.; Reiter, G. Observation of five-fold local symmetry in liquid lead. Nature 2000, 408, 839−841. (39) Provatas, N.; Elder, K.: Phase-Field Methods in Materials Science and Engineering; Wiley: Hoboken, NJ, 2011. (40) Russo, J.; Tanaka, H. Crystal nucleation as the ordering of multiple order parameters. J. Chem. Phys. 2016, 145, 211801.
5014
DOI: 10.1021/acs.jpclett.6b02276 J. Phys. Chem. Lett. 2016, 7, 5008−5014
