J. Phys. Chem. B 2007, 111, 14001-14005
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Multistep Crystal Nucleation: A Kinetic Study Based on Colloidal Crystallization Tian Hui Zhang and Xiang Yang Liu* Department of Physics, National UniVersity of Singapore, 2 Science DriVe 3, Singapore 117542 ReceiVed: June 22, 2007; In Final Form: September 24, 2007
Crystallization via an amorphous precursor, the so-called multistep crystallization (MSC), plays a key role in biomineralization and protein crystallization. MSC has attracted much attention in the past decade, but a quantitative understanding of it has so far not been available. The major challenge is that the kinetics governing the nucleation of crystals occurring in the metastable amorphous precursor remains unclear. In this study, the kinetics of MSC is addressed experimentally. Most importantly, a mathematical method is developed to calculate the local nucleation rate of the crystals in the amorphous precursor, which is not accessible to conventional methods. This local nucleation rate is critical to the understanding of MSC, but it has never been dealt with experimentally because of the difficulties of in situ observation. With the local crystal nucleation rates, the supersaturation for crystallization and the crystal-liquid interfacial free energy in the amorphous precursor are evaluated.
Introduction Control of crystallization, especially in its early stage, the so-called nucleation, is critical and strongly desirable in fields including protein crystallography,1 disease treatment,2 nanostructure fabrication,3 and so forth. However, it is now clear that precise control of nucleation is possible only after a complete quantitative understanding of nucleation kinetics is achieved.4 Currently, the widely employed quantitative understanding of nucleation is based on the classical nucleation theory (CNT).5-7 A basic assumption of CNT is that nuclei of a crystalline phase are created with spherical shape and are identical in structure to the bulk crystal. The relationship between nucleation rate and supersaturation is thus established on the basis of this assumption. This relationship makes it possible to predict the number and size distribution of crystal nuclei as a function of time under specified conditions. However, CNT loses its capacity to predict the nucleation rate of stepwise crystallization that occurs via metastable phases. A typical stepwise crystallization is the so-called two-step crystallization (TSC).8 According to TSC, dense amorphous droplets are first formed from the mother solution; crystalline nuclei are then created from the droplets. This mechanism has been confirmed by several other research groups.9-11 Moreover, it was found that TSC is a strategy widely adopted by biological systems in synthesizing functionalized crystalline structures. For instance, during the formation of calcite in sea urchin larvae, a transient amorphous phase is first formed before the crystalline phase emerge.12,13 Similarly, a transient amorphous phase is identified during the formation of aragonite controlled by mollusk bivalve larvae.13,14 It is widely believed that, in biological systems, the development of crystalline structures that are well-defined in shape and size is facilitated by the occurrence of transient amorphous phases.12-14 Furthermore, recent studies argued that TSC may be a mechanism underlying most crystallization processes occurring in typical atomic systems.15-17 * Corresponding author. Address: Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542. Tel: (65)6516-2812. Fax: (65)-6777-6126.
Because of its importance in science and technology, TSC has attracted much attention in the past decade. However, the understanding of TSC has, so far, remained poor. A key challenge in the approach is that the kinetics of creating the initial crystalline nuclei from the dense droplets remains unclear, making it impossible to predict the overall nucleation rate Jc of the crystals. To quantify the nucleation rate of TSC, Kashchiev et al. assumed that one dense droplet can produce only one crystal, the so-called mononuclear mechanism.18 On the basis of this assumption, the overall nucleation rate Jc of the crystals at the stationary state was found to be equal to the nucleation rate J of the dense droplets. J is larger than that of the crystals nucleated directly from the mother solution because of the lower nucleation barrier for the dense droplets. This means that the nucleation of crystals in TSC is enhanced because of the presence of metastable amorphous dense droplets, consistent with the previous theoretical predictions.8-11 One of the implications associated with the mononuclear mechanism is that all dense droplets can grow large enough to develop a crystal. However, a good deal of experimental evidence shows that, although many dense droplets were created initially, only a fraction of them could successfully produce a crystal.19,20 In this case, the overall nucleation rate Jc of the crystals would be, to a large extent, determined by the local nucleation rate of the crystals in the droplets, which is denoted by jc hereafter. However, in previous studies,19,20 jc has never been determined experimentally because of the difficulties in conducting an in situ observation of the crystal nucleation proceeding in the metastable amorphous droplets. Consequently, whether the occurrence of the metastable dense droplets can enhance the overall nucleation rate of the crystals remains questionable in practice. In the past decades, colloidal solutions have been studied extensively as a model of atoms. These studies have produced a good deal of insight into fundamental issues of condensed matter physics and materials sciences, including defects dynamics,21 nucleation22 and growth23 of crystals, melting,24 and glass transitions.25 In this study, we report an in situ investigation of crystallization preceded by an amorphous precursor, namely,
10.1021/jp074867w CCC: $37.00 © 2007 American Chemical Society Published on Web 11/21/2007
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Zhang and Liu
Figure 1. (a) Experimental setup. Colloidal suspension is sealed between two pieces of ITO-coated conducting glass plates separated by insulating spacers. The gap between the two glass plates is H ) 120 ( 5 µm. The dynamic process is recorded by a digital camera for analysis. (b) Phase diagram of the colloidal suspension: 2DC ) twodimensional crystals; 3DL ) three-dimensional liquid; 3DDA ) threedimensional disordered aggregation.
multistep crystallization (MSC), in a colloidal model system. In this system, the nucleation process can be observed at the single-particle level, and thus the kinetics of MSC can be investigated in detail. To deal with the local nucleation rate jc, a mathematical method is developed. Methods Figure 1a shows our experimental setup. Monodisperse colloidal particles (polystyrene spheres of diameter 0.99 µm and polydispersity e5%, Bangs Laboratories) are dispersed uniformly in deionized water. In our case, a volume fraction of 0.03% of the colloidal solid is chosen, and the surface potential of the colloidal spheres is adjusted to -72 mV by adding Na2SO4 (10-4 M). The pH of the suspension is measured at 6.35. The colloidal suspension is then sealed between two parallel horizontal conducting glass plates coated with indium tin oxide (ITO). Once an alternating electric field (AEF) is applied, the fluid flow induced by the AEF transports colloidal particles to the surface of the glass plates, where, under certain conditions, two-dimensional crystals (2DCs) are formed (Figure 1b). The growth processes are recorded for analysis by a digital camera (CoolSNAP cf, Photometrics), which is mounted on an Olympus BX51 microscope. To identify the crystalline nuclei formed in the amorphous dense droplets, a local two-dimensional bond-order parameter is defined by
ψ6(ri) ) M-1|
∑jei6θ | ij
(1)
where ri is the center of particle i, and θij is the angle subtended between the vector from particle i to its jth nearest neighbor and the arbitrarily chosen x-axis. M is the number of the nearest neighbors of particle i. The mean value 〈ψ6〉 of typical crystalline clusters obtained from our experiments ranges from 0.75 to 0.80. In perfect triangular lattice structures, 〈ψ6〉 should be 1. However, in real experiments,26 〈ψ6〉 was usually measured to
Figure 2. MSC observed at 800 Hz and 167 V/cm: (a) initial dilute liquid phase; (b) amorphous dense droplets are first created from the mother phase; (c) a few subcrystalline nuclei are created from the amorphous phase; (d) a stable crystalline nucleus is formed from the dense droplets.
be around 0.8, while, in simulations,27,28 〈ψ6〉 can be as high as 0.9. In our studies, particles with ψ6 g 0.8 and six nearest neighbors are considered as “crystal-like”. In our studies, choosing different ψ6 values within the range of 0.75-0.80 as the criterion for crystal-like particles has no significant impact on the related results. Results and Discussion A typical process of MSC, observed under conditions of E ) 167 V/cm and f ) 800 Hz, is presented in Figure 2. Colloidal particles in the initial mother solution distribute uniformly (Figure 2a). When an AEF is applied to the system, colloidal particles are transported onto the glass surfaces where they first form dense droplets (Figure 2b). Subsequently, a few subcrystalline nuclei are created from the droplets, as shown in Figure 2c. These subcrystalline nuclei are not stable and dissolve soon after they are created. Our investigations show that the crystal/ before line nuclei in the droplets need to reach a critical size Ncrys they can grow stably in the droplets, as shown in Figure 2d. In our experiments, a droplet can produce only one stable crystalline nucleus. Moreover, to form a stable crystalline nucleus with size beyond N/crys, the droplets have to first reach a critical size N*. More interestingly, although the sub crystalline nuclei in the droplets at the early stage are ubiquitously created by fluctuations, our investigations reveal that the stable crystalline nuclei in the dense droplets are formed by coalescence of the subcrystalline nuclei. This is distinct from the conventional picture. In our experiments, although, at an early stage, many small dense droplets are created, only three or four out of 20 droplets can reach the critical size N* and then successfully develop a stable crystalline nucleus. Most of the dense droplets will disappear gradually. This is consistent with previous observations concerning protein crystallization.19,20 In this case, the overall nucleation rate Jc of the crystals in the mother solution would be, to a large extent, determined by the local nucleation rate jc of the crystals in the dense droplets. This is different from the previous assumption of the mononuclear mechanism.18
Multistep Crystal Nucleation
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Conventionally, the nucleation rate J is defined by
J(t) )
n(t) tV
(2)
where n(t), a function of time t, is the number of supernuclei (nuclei larger than the critical size) created from the mother phase within a volume of V. At a stationary state, n(t) increases linearly with time, and nucleation rate J is time independent.4 In practice, the number of supernuclei is usually counted as a function of time, and J is thus evaluated from the slope of the linear part of the n-t plot. However, eq 2 loses its capacity to determine the local nucleation rate jc of MSC because a droplet can produce only one crystalline nucleus, and therefore the conventional method of counting the number of supernuclei as a function of time fails. It is now clear that a different approach has to be developed. Notice that, in the stationary state, eq 2 can be simplified as29
J)
1 τV
(3)
where τ is the average period needed by the mother phase with volume V to nucleate a new critical nucleus at the stationary state. We assume that, in MSC, eq 3 can be approximately applied to jc. Notice that the amorphous droplets in our case are two-dimensional; eq 3 has to be modified before it can be applied to calculate the two-dimensional jc. The result of the modification is
1 ) τA‚A jc
(4)
where τA is the time needed by an amorphous droplet with area A to create a critical crystalline nucleus with size N/crys. At a specified supersaturation, jc is independent of time and A. Therefore, in a homogeneous system, droplets with different A share a common feature: 1/jc. In our experiments, A increases with time as the droplets grow. Therefore, eq 4 has to be modified into an integration in terms of time before it can be applied to a growing subsystem. The modification gives
1 ) jc
∫0τ dt‚A(t)
(5)
where τ is the time for a growing droplet to reach the critical size N* and form a critical crystalline nucleus. To evaluate the nucleation rate jc from eq 5, τ and A(t) first have to be established. In practice, A(t) can be established from the droplet size N(t) through A(t) ) N(t)*π(d0/2)2. Here, N(t) is the number of colloidal particles contained by the growing droplets, d0 is the average center-to-center distance between two neighbor “liquid-like” colloidal particles in the amorphous droplets. Consequently, jc can be evaluated from N(t) by
1 ) jc
∫0τ dtA(t) ) ∫0τ dtN(t)‚π
() d0 2
2
(6)
N(t) can be established directly from the experimental investigations. Nevertheless, τ, the time needed for a droplet to create a critical crystalline nucleus, cannot be identified directly from / the experimental observations. To obtain τ, the critical size Ncrys has to be first identified. To achieve this purpose, the number of crystal-like particles Ncrys as a function of the droplet size N is established experimentally. Figure 3a presents a typical
Figure 3. (a) Number of crystal-like particles Ncrys as a function of the size N of the droplets. (b) d2Ncrys/dN2 reaches its maximum at the critical size N* ∼ 1440, where a critical crystalline nucleus is created. (c) The time at which N reaches the critical size N* is just the time τ needed by the growing droplets to create a critical crystalline nucleus.
Ncrys-N plot under conditions of E ) 167 V/cm and f ) 800 Hz. It can be seen that, at the early stage, the dependence of Ncrys on N is approximately linear. Subsequently, however, dNcrys/dN undergoes a gradual increase. The increase of dNcrys/ dN ends, and Ncrys becomes linearly dependent on N again when N grows beyond 1400. The detailed observation reveals that the second linear region (N > 1400) of the Ncrys-N plot is related to the existence of a stable crystalline nucleus in the droplets. Correspondingly, it is concluded that the creation of a critical crystalline nucleus should be responsible for the termination of the continuous rise of dNcrys/dN. On the basis of the above reasoning, d2Ncrys/dN2 should reach its maximum around the moment when a critical crystalline nucleus is produced by the amorphous dense droplets. Under conditions of E ) 167 V/cm and f ) 800 Hz, the size of the droplets for d2Ncrys/dN2 to reach the maximum is N* ∼ 1440 (Figure 3b). The corresponding critical size of the crystalline nuclei in the dense droplets is N/crys ∼ 161. The corresponding time τ is about 275 s. d0 is measured to be 1.24 ( 0.02 µm. With the experimentally established N(t) (Figure 3c), the numerical integration yields ∫275 0 dtN(t) ) 223399. Substitution of the above results into eq 6 yields jc ∼ 3.7 × 10-6 µm-2‚s-1. In our experiments, as can be seen in Figure 4a, jc increases with frequency. This result, according to CNT, indicates that increasing frequency leads to an increase of the supersaturation for crystallization in the dense droplets. This is supported by / and N*, as shown in Figure the decrease of the critical size Ncrys 4b. A more direct evidence of the increase of the supersaturation is the increase of the particle concentration in the dense droplets, which is reflected by the decrease of d0 (Figure 4c).
14004 J. Phys. Chem. B, Vol. 111, No. 50, 2007
Zhang and Liu crystalline nuclei cannot be created from the droplets. The particle concentration c in the droplets can be evaluated from the average distance d0 by c ) 4/πd02. Thus, eq 9 can be rewritten as
∆µ ) kBT ln
() dm2
(10)
d02
In our studies, d0 is measured experimentally, and the result is shown in Figure 4c, but it is difficult to measure dm directly. However, to determine ∆µ, the information of dm is critical. Notice that, if the value of jc at two different frequencies, jc1 and jc2, can be obtained, the combination of eqs 7 and 8 yields
(
)
2 sγ22 π sγ1 ln jc1 - ln jc2 ) kBT ∆µ1 ∆µ2
(11)
Assume that the critical crystalline nuclei are two-dimensional disks characterized by a radius rc; then, according to the twodimensional nucleation theory,29,30 we have
rc )
sγ ∆µ
(12)
Combination of eqs 10, 11, and 12 gives
F ) B ln dm Figure 4. (a) The local nucleation rate jc increases with frequency. (b) The critical sizes N* and N/crys as a function of frequency. (c) The average distance d0 between two neighbor liquid-like particles decreases with frequency, while the average distance dc between two neighbor crystal-like particles remains essentially constant in the crystalline core of the dense droplets.
The quantity of the supersaturation is critical for the estimate of the nucleation barrier and the crystal-liquid interfacial free energy in the dense droplets. According to CNT, nucleation rate is connected to nucleation energy barrier ∆G* by
(
)
∆G* J ) C exp kBT
(7)
∆G* is determined by thermodynamic driving force ∆µ and interfacial free energy γ.29 In the case of two-dimensional nucleation, ∆G* is given by
∆G* )
πγ2s ∆µ
(8)
where s is the average area occupied by a crystal-like particle. In our case, s ) πdc2/4, and dc is the average center-to-center distance between two neighbor particles in the crystalline regions, experimentally measured to be 1.07 ( 0.02 µm. Figure 4c shows that dc essentially remains constant at different frequencies. However, the increase of jc suggests that ∆G* is reduced by increasing frequency. According to eq 8, the reduction of ∆G* can be caused by either the decrease of γ or the increase of ∆µ. In our case, ∆µ can be defined in terms of the concentration c:4
∆µ ) kBT ln
() c0 cm
(9)
where c0 is the actual particle concentration in the droplets, and cm is the equilibrium concentration in the droplets below which
F ) ln
jc2 π + 2 (rc12 ln d01 - rc22 ln d02), jc1 s π B ) 2 (rc12 - rc22) (13) s
Here d01, d02 and rc1, rc2 are the average distances d0 and the critical sizes rc measured in conditions producing jc1 and jc2, respectively. Experimentally, rc can be evaluated from the critical size N/crys:
1 1 / / πrc2 ) Ncrys πdc2 rc ) dc xNcrys 2 4
(
)
(14)
Because N/crys, d0, dc, and jc have been established as shown in Figures 3 and 4, dm is the only unknown parameter in eq 13. Therefore, dm can be determined from the slope of the plot of F-B (Figure 5a). The result is dm ≈ 1.28 µm. Consequently, ∆µ can be determined from eq 10, and the supersaturation σ for crystallization in the droplets can be evaluated easily through ln(1 + σ) ) ∆µ/kBT. Since it is simple algebra, the results of ∆µ and σ are not presented. With dm ≈ 1.28 µm, γ can be calculated by combining eqs 10 and 12. The results of γ are shown in Figure 5b. It is found that γ increases slightly with frequency. However, from Figure 4c, we find that dc in the crystalline regions does not change within the frequency range we observe, indicating that the interaction between colloidal particles in the crystalline region does not change essentially. Given the interaction, γ should be constant at a fixed temperature. However, the results shown in Figure 5b suggest that this does not hold true. Notice that a key assumption contained in our calculation is that the shape of the critical crystalline nuclei in the amorphous droplets is disk-like, characterized by a radius of rc. This assumption leads to the minimum estimate of the length of the crystal-liquid interface. However, as can be seen in Figure 2d, the shape of the crystalline nuclei is normally irregular. Thus, the real lengths of the crystal-liquid interface are, in fact, much
Multistep Crystal Nucleation
J. Phys. Chem. B, Vol. 111, No. 50, 2007 14005 dense droplets are created by fluctuations. A critical crystalline nucleus is subsequently coalesced by the sub crystalline nuclei. To estimate the overall nucleation rate of the crystals in MSC, the local nucleation rate of the crystals in the droplets first has to be determined. A mathematical method is thus developed to calculate the local nucleation rates. Furthermore, the supersaturation and the interfacial free energy in the dense droplets are derived from the local nucleation rates. However, the estimates of the interfacial free energy are larger than their real values. Acknowledgment. This research is funded by ARF Project R-144-000-148-112. The authors are much indebted to Dr. C. Strom for her valuable suggestions and critical reading of the manuscript. References and Notes
Figure 5. (a) F-B plot. The slope of a linear fit gives the value of ln dm, which can be used to establish the supersaturation in the dense droplets. (b) The calculated line tension or the interfacial free energy γ increases with frequency. The assumption of disk-like twodimensional critical crystalline nuclei leads to a high estimate of γ.
larger than that used in our calculations. Given the supersaturation and the nucleation energy barrier, the lower estimate of the interface length would result in a higher estimate of γ. Therefore, the values of γ shown in Figure 5b should be larger than their real values. Moreover, it is clear in Figure 2d that the change of the particle concentration in the droplets, from the crystalline core to the amorphous fringe, is continuous. In fact, the crystalline region and its surrounding area are essentially identical in concentration. This makes it easy for the colloidal particles residing on the crystal-liquid interface to transform between crystal-like and liquid-like. As a consequence, the shape of the crystals in the droplets experiences strong fluctuation during the growth, giving rise to a large divergence from the assumed disk-like shape. Because of the shape fluctuation, the interface length estimated from the disk-like nuclei assumption contains a big error, leading to a big error in the estimate of γ. Figure 4c reveals that, as the frequency increases, the particle concentration in the amorphous region approaches that of the crystalline region. The reduced particle concentration difference between the crystalline region and its surrounding area enables the colloidal particles residing on the crystal-liquid interface to transform more easily between crystal-like and liquid-like, leading to a stronger shape fluctuation and thus a bigger error in the estimate of the interface length. Therefore, one possible reason for the increase of γ with frequency is that, at a higher frequency, a bigger error is contained in the calculation of γ. From the above discussion, it follows that, in reality, γ should be smaller than the values shown in Figure 5b. The experimental observations show that the length of the edge of the irregular crystals is at least two times the estimate based on the disk-like nuclei assumption. It follows that the values of γ shown in Figure 5b are at least two times their real values. Consequently, the real values of the interfacial free energy γ in the dense droplets may be less than 0.45-0.60 kBT/µm. Summary In this study, the kinetics of MSC is discussed. It is found that, at the early stage, the subcrystalline nuclei in the amorphous
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