Two-Dimensional Nucleation on the Terrace of Colloidal Crystals with

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Two-dimensional nucleation on the terrace of colloidal crystals with added polymers Jun Nozawa, Satoshi Uda, Suxia Guo, Sumeng Hu, Akiko Toyotama, Junpei Yamanaka, Junpei Okada, and Haruhiko Koizumi Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b04532 • Publication Date (Web): 16 Mar 2017 Downloaded from http://pubs.acs.org on March 18, 2017

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Two-dimensional nucleation on the terrace of colloidal crystals with added polymers Jun Nozawa, † ,* Satoshi Uda, † Suxia Guo, † Sumeng Hu, † Akiko Toyotama, ‡ Junpei Yamanaka, ‡ Junpei Okada, † Haruhiko Koizumi, †

†

Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku,

Sendai 980-8577, Japan ‡

Graduate School of Pharmaceutical Sciences, Nagoya City University, 3-1 Tanabe,

Mizuho, Nagoya, Aichi 467-8603, Japan

*Corresponding author Tel.: +81 22 215 2103; Fax: +81 22 215 2101 E-mail: [email protected] (J. Nozawa)

Abstract Understanding nucleation dynamics is important both fundamentally as well as technologically in materials science and other scientific fields. Two-dimensional (2D) nucleation is the predominant growth mechanism in colloidal crystallization, in which the particle interaction is attractive, and has recently been regarded as a promising method to fabricate varieties of complex nano-structures possessing innovative functionality. Here, polymers are added to a colloidal suspension to generate a depletion attractive force, and the detailed 2D nucleation process on the terrace of the colloidal crystals is investigated. In the system, we first measured the nucleation rate at various area fractions of particles on the terrace, φ area . In situ observations at single-particle resolution revealed that nucleation behavior follows 1 ACS Paragon Plus Environment

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the framework of classical nucleation theory (CNT), such as single-step nucleation pathway and existence of critical size. Characteristic nucleation behavior is observed in that the nucleation and growth stage are clearly differentiated. When many nuclei form in a small area of the terrace, a high density of kink sites of once formed islands makes growth more likely to occur than further nucleation, because nucleation has a higher energy barrier than growth. The steady-state homogeneous 2D nucleation rate, J, and the critical size of nuclei, r*, are measured by in situ observations based on the CNT, which enable us to obtain the step free energy, γ , which is an important parameter for characterizing the nucleation process. The γ value is found to change according to the strength of attraction, which is tuned by the concentration of the polymer as a depletant.

1. INTRODUCTION Creation of novel functional materials with desired structures by colloidal particles is now opening a new field in materials science. Various materials as well as a variety of morphologies of colloidal particles have been examined for their potential applications. The particle interactions are also a significant key factor for fabricating these novel nanostructure materials. The attractive interactions, which includes magnetic, 1–3 Coulomb, 4– 6 DNA mediated, 7–9 and depletion forces, have recently been well utilized. 10–1 2 These attractive forces can be used to fabricate a variety of structures that are not obtained only by repulsive interactions between particles. In particular, the depletion force is now being quite widely studied since it works on any type of particle without modification of the particle. The depletion attraction occurs in colloidal suspensions when a smaller particle or macromolecule 2 ACS Paragon Plus Environment

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is introduced. 13–15 Colloidal crystals have also been used as a model system for crystallization kinetics since the colloid system shows analogous phase transitions to those of atomic or molecular systems. 16 The colloidal system has been applied to reveal kinetics of phase transitions such as glass transitions and nucleation processes. Recently, attractive colloids have been used as a model, since the attractive interaction mimics interactions of atoms or molecules in real systems. Processes such as superheating, 17 the Ehrich-Schwabel effect, 18 and surface melting 19 have been modeled by attractive colloids. This study focuses on two-dimensional (2D) nucleation on the terrace of colloidal crystals, in which added polymers generate a depletion attraction between colloidal particles. We found that 2D nucleation is the predominant growth mechanism in the system, where 2D islands nucleate repeatedly and spread on the surface. One important feature of the attractive system is that the density of the environment is quite lower than that of the crystallized solid phase, which is similar to the solution or vapor growth mode in atomic systems. Crystallization with attractive interaction does not proceed as typical hard-sphere crystallization nor as that of a charged sphere. Understanding the nucleation process is crucially important in engineering as well as materials science. Control of nucleation of a specific phase is critical for many cases of material design; however, the detailed nucleation mechanism is still under debate. Although analysis of most nucleation experiments relies on the framework of the classical nucleation theory (CNT), 2 0–25 it is difficult to predict quantitative nucleation rates with CNT. 26,27 Part of the reason for this is attributed to some 3 ACS Paragon Plus Environment

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assumptions on which the CNT is based, such as the so-called capillarity approximation that is contradicted by many experimental results. 28,29 One more main assumption is the one-step nucleation process. Recently, it has been reported that many substances show multi-step nucleation in which nucleation occurs via an intermediate metastable phase. 30 –40 Direct observation of nucleation is required to resolve the discrepancy between theory and experiment; however, this is quite difficult due to the short time and small spatial scales involved. Here, colloids pose huge advantages as a model system since the detailed process can be directly observed by optical microscopy at single-particle resolution. Many nucleation studies with colloidal systems, both experimental 4 1–45 and simulated, 2 7,4 6–49 have been conducted, especially for hard-sphere systems or those with repulsive interactions between particles. Most nucleation experiments of colloidal crystals so far have been conducted on three-dimensional systems. There have been several studies of 2D nucleation processes of the mono-layer of colloidal crystals that are formed on the surface of a glass slide. 50–52 In the two-dimensional case, we can obtain important information, such as the step morphology and the corresponding surface concentration. These can be measured from the from the number of ad-particles on the terrace, which cannot be accessed even by powerful in situ observation techniques such as TEM and AFM, where only the crystal phase is visualized. The study of the 2D nucleation of colloidal crystals well meets the requirement for fundamental understanding of the nucleation process. Furthermore, 2D colloidal crystals offer important applications that are different from 3D colloidal crystals, including colloidal nanolithography, 57–59 biosensors, 60 4 ACS Paragon Plus Environment

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and optical devices. 61 To grow high quality crystals with desired structures requires a fundamental understanding of the nucleation of colloidal layers. An understanding of 2D nucleation has greatly contributed to the ability to grow high-quality thin films of various semiconductors by epitaxial growth. In the present study, we investigate the nucleation process of 2D islands on the terrace of a colloidal crystal. Here, nucleation corresponds to growth of 2D islands at sub-critical size and growth corresponds to further growth of the islands. The nucleation rate, J, and the critical size of nuclei, r*, are related to the area fraction,

φ a rea , as the total area of ad-particles per area of terrace, which is equivalent to the surface concentration of the ad-particles. Our measurements reveal the detailed nucleation process of the attractive system of colloidal crystals.

2. EXPERIMENTAL The colloidal particles used in this study were charge-stabilized polystyrene (PS) spheres (Thermo Fisher Scientific) that were dispersed in Millipore-filtered water. Green fluorescent monodispersed PS, 500 nm in size, was used in the experiment without any further purification. The zeta potential of the particles was measured to be –47.1±5.9 mV. Sodium polyacrylate (polymerization degree of 30000–40000) was purchased from Wako Pure Chemical Industries, and was added as a depletant, which gave rise to an attractive force between particles. Three concentrations of sodium polyacrylate, C p (0.150, 0.125, and 0.100 g for 1 l of water), were examined. Since the depletion interaction depends on the concentration of the polymer, different polymer concentrations lead to different attractive interactions. 5 ACS Paragon Plus Environment

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Since

sodium

polyacrylate

is

a

polyelectrolyte,

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the

background

electrolyte

concentration of the solution is estimated from polymer concentrations and molecular weight of acrylate monomer residues [-CH 2 -CH(CO 2 Na)-] (= 94). At polymer concentrations of 0.150, 0.125, and 0.100 g/l, the electrolyte concentration was 1.5, 1.25, and 1.0 mM, respectively. Under these conditions, the electrostatic interaction between the particles is negligible. Therefore, we can safely assume that the driving force of the crystallization is the depletion attraction. We note, however, that the electrolyte concentrations inside the crystals are much lower than these values, because of the absence of sodium polyacrylate molecules. Various volume fractions ( Φ ) of colloidal suspensions ( Φ = 1.0, 0.5, 0.25, and 0.1%) were applied, and the suspension was sealed in a cell that was made of a cover glass and acrylic resin. The nucleation rate, J, and the particle area fraction of the terrace, φ area , were investigated in the present study. The φ area is defined as the area occupied by particles divided by area of the terrace, which corresponds to a kind of surface concentration. The area is limited to the terrace, which does not include any free areas. The number of particles on the terrace was counted, and then multiplied by the area of one particle and divided by the area of the terrace to determine φ area . Since the 2D nucleation on the surface of a colloidal crystal was investigated in the present study, we did not have to deal with the interfacial free energy between the nuclei and substrate, which should be taken into account for the case of nucleation of a monolayer on a foreign substrate.

3. RESULTS AND DISCUSSION 6 ACS Paragon Plus Environment

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In the initial stage of the crystallization, nucleation of monolayer colloidal crystals takes place on the bottom of the cell, where colloidal particles are concentrated due to the depletion interaction between the cover glass and the colloidal particles. 2D nucleation then occurs on the terrace of the colloidal crystals. The nucleation rate of the monolayer depends on the initial volume fraction of the colloidal particles, Φ . A high Φ corresponds to a large number of nucleated grains. Low-magnification images of typical nucleated colloidal crystals are presented in Fig. 1. The value of Φ for Figs. 1(a) and 1(c) is 0.25% and that for Figs. 1(b) and 1(d) is 0.50%, and the polymer concentration, C p , for all images is 0.15 g/l. All images in the figure were taken 50 minutes after the start of the experiment, when the colloidal suspension and polymer were mixed. Most of the grains in the figure have about 3 to 5 layers, and the stacking sequence is random. A larger number of grains were nucleated in Fig. 1(b) compared to in Fig. 1(a) (see Supporting Information Movie S1). The fluorescent images shown in Figs. 1(c) and 1(d) are magnified images of the images presented in Figs. 1(a) and 1(b), respectively. Nucleation and growth of 2D islands on the surface can be observed. The 2D nuclei formed repeatedly at random places on the terrace, then spread toward the edge of the terrace. The growth via 2D nucleation is characteristic for the attractive system of a colloid, suggesting that the growth mechanism is similar to that of vapor or solution growth. It should be noted that, at the time when the images in Figs. 1(a) and 1(c) were taken, the volume fraction was higher than that in Figs. 1(b) and 1(d), although the initial value of Φ for Figs. 1(a) and 1(c) was lower, which can be deduced from the higher 2D nucleation rate on the surface of the colloidal crystal in Fig. 1(c) than in Fig. 1(d). In addition, the shape of the grains in Fig. 1(d) is close to the equilibrium shape for a 7 ACS Paragon Plus Environment

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fcc structure, which in general is attained at lower concentration. As crystallization proceeds, Φ of the solution decreases because particles in the solution are consumed for crystal growing. Thus, the large number of nucleations in Figs. 1(b) and 1(d) reduces Φ of the crystal surface more than those of Figs. 1(a) and 1(c) (see Supporting Information Movie S2). Here, we introduce the area fraction, φ area , as the surface area fraction of colloidal particles. The relationship between the nucleation rate and the critical radius versus the φ area is investigated. In general, the solute concentration of the crystal surface and that of bulk solution is different for a crystal that is growing in solution. Precise measurements of surface concentration require a Mach-Zehnder interferometer, 62 which uses a sophisticated optical system and fringe analysis. With the interferometer, observation of the crystal surface while simultaneously measuring the concentration is very difficult. In the colloidal system, both are available simultaneously and they are interrelated.

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Figure 1 2D nucleation of colloidal crystals. The polymer concentration, C p , for all of the images is 0.15 g/l. A larger number of grains are nucleated on the cover glass for Φ = 0.5 (b) compared to Φ = 0.25 (a). More 2D nucleations occur on the surface of colloidal crystals for Φ = 0.25 (c) than for Φ = 0.5 (d), because formation of a larger number of grains decreases φ area for Φ = 0.5 (d) more than it does for Φ = 0.25 (c).

Fig. 2 is an image taken at single-particle resolution, which offers information on particles on the terrace (ad-particles). Particles are brought to the terrace from the solution, diffuse there over a certain length and time, and then return to solution. When a particle on the terrace reaches a step, it is incorporated into the crystals, 9 ACS Paragon Plus Environment

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which advances a step of 2D islands (see Supporting Information Movie S3).

Figure 2 Nucleation and growth of 2D islands observed with single-particle resolution. 2D islands nucleated at random places on the terrace, and then spread toward the edge by incorporating diffusing ad-particles.

Some particles occasionally form clusters, and a nucleus is formed when those clusters grow. Here, we recognized important features about the nucleation process (Fig. 3). Smaller embryos shrink and then disappear (white dashed circles in Fig. 3), 10 ACS Paragon Plus Environment

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whereas larger embryos keep growing (red dashed circles). This means that there is a critical size, which is in accordance with CNT.

Figure 3 A critical size exists for the nucleation process. Subcritical islands disappear (white dashed circles), whereas islands whose sizes overcome the critical size keep growing (red dashed circles).

Owing to these observations at single-particle resolution, the following problems that appear in the general nucleation experiment can be removed, i.e., distinguishing homogeneous nucleation from heterogeneous nucleation and separation of the 11 ACS Paragon Plus Environment

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nucleation process from subsequent growth, which is necessary to determine the exact concentration and nucleation rate. Heterogeneous nucleation can be distinguished from the homogeneous case. It occasionally occurs at impurities (dust in the solution) that are adsorbed on the terrace (see Supporting Information Movie S4). Nucleation occurs immediately after the dust reaches the terrace. Although the frequency at which this occurs is quite low, it should be distinguished to obtain a pure homogeneous nucleation.

Figure 4 Heterogeneous nucleation driven by dust in the solution.

The steady-state homogeneous 2D nucleation rate can be measured by in situ observation. The nucleation rate, J, was measured at various φ area . Here, J is the number of nuclei created per unit time, ∆t, per unit area, S, which is expressed as n = JS∆t

(1)

where n is the total number of nuclei. The value of n on the terrace was counted. The change in the area of the terrace was taken into account to obtain J. Figs. 5(a) through 5(d) show a typical nucleation sequence in the middle φ area range (J≈10 6 (1/m 2 s)). During nucleation, the terrace grows in the lateral direction, which increases its area. In this case, n can be written as 12 ACS Paragon Plus Environment

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∆t

n = J ∫ S (t )dt

(2)

0

Fig. 5(e) shows the area of the terrace as a function of time. We think the growth rate of the terrace can approximately be fitted with a linear function over the experimental time period of several hours, which makes it possible to derive a succinct relationship between the number of nuclei and nucleation rate. Given that the time variation of the area is S(t) = S 0 + α t, Eq. 2 can be written as 1  n = J  α∆t 2 + S 0 ∆t  2  

(3)

where α is the coefficient of change of area as a function of time, and S 0 is the initial size.

Figure 5 2D nucleation on the growing terrace (a)–(d). (e) Area of the terrace as a function of time.

Fig. 6 shows a typical nucleation process for high φ area (J≈10 8 (1/m 2 s)). The layer 13 ACS Paragon Plus Environment

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shown in Fig. 6(a) is the first layer, on which nucleation of the second layer is about to occur. Fig. 6(b) shows nucleation of the second layer on the first layer. After a certain number of nuclei form, further nucleation does not occur, and particles on the terrace are incorporated into the step of 2D islands rather than form new nuclei. When the terrace is fully covered by 2D islands, nucleation of a third layer begins (see Supporting Information Movie S5). This process is shown in Fig. 6(e), where the cumulative number of nuclei is shown as a function of time. Periodic nucleation ensures that the steady-state nucleation rate is being measured. To our knowledge, the stepwise nucleation and growth process have not been reported yet. This is due to the advantages of observing 2D islands as mentioned above in that it allows us to address the detailed process as compared to that of 3D islands.

Figure 6 (a) 1st layer of 2D islands, on which nucleation of the 2nd layer is about to occur. (b) Nucleation of 2nd layer on the 1st layer. (c) Only growth of 2nd layer 14 ACS Paragon Plus Environment

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takes place. (d) Nucleation of 3rd layer after 2nd layer is fully covered by 2D islands. (e) Cumulative number of nuclei versus time. Constant nucleation time of 2D islands leads to steady-state nucleation rate.

Single-particle observation can also distinguish coalescence of subcritical clusters from monomer-monomer nucleation (Fig. 7). The cluster on the terrace overcomes the critical size not by growing with incorporating a particle, but via coalescence of subcritical clusters whose size is smaller than the critical size. Although this phenomenon is rarely observed under experimental conditions in the present study, it should be distinguished from the homogeneous steady-state nucleation rate.

Figure 7 Subcritical clusters on the terrace overcome critical size via coalescence of clusters.

The critical size, r * , was also measured for the clusters from the in situ observation as the maximum size of clusters that still tend to dissolve. Figs. 8(a) and 8(b) show the variation of J and r * as a function of φ area , respectively. Since the experiment cannot be repeated under the same φ area , which depends on 15 ACS Paragon Plus Environment

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factors such as elapsed time and number of surrounding grains, each data point in the figure is obtained from a single observation. As φ area increases, J increases and r * decreases. Here, in order to apply CNT, the equilibrium volume fraction, φ eq , is measured experimentally as φ area at which the growth rate of steps of twodimensional islands is zero. The φ eq values for 1.5, 1.25, and 1.0 g/l of polymer concentration are 0.1%, 0.46%, and 0.8%, respectively. According to CNT, the Gibbs free energy change for the 2D nucleation is written as ∆G = −

πr 2 Ω

∆µ + 2πrγ

(4)

where r is a radius of a 2D island, ∆ µ is the chemical potential difference between solid and solution, Ω is the area per structural unit, and γ is step free energy (or line tension). Here, ∆ µ can be expressed as:

∆µ = kBT ln(1 + σ )

(5)

where k B is Boltzmann’s constant, T is the absolute temperature, and σ describes supersaturation that is related to the actual concentration (C) and equilibrium concentration (C eq ) by

σ = (C–C eq )/C eq

(6)

From Eq. (4), the nucleation barrier and the critical radius are: ∆G ∗ = r∗ =

πΩγ 2 ∆µ

(7)

Ωγ ∆µ

(8)

Substituting the concentration, C, for the surface area fraction, φ , and according to CNT, the steady-state 2D nucleation rate, J, and critical radius, r * , can be expressed assuming C/C eq = φ area / φ eq as follows (see reference 63 for detailed derivation of the 16 ACS Paragon Plus Environment

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following equations): ln J = ln A − r∗ =

πΩγ 2 ( k BT ) 2 ln(φ / φeq )

(9)

aγ k BT ln(φ / φeq )

(10)

where A is the pre-exponential factor related to attachment frequency of a particle to a critical nuclei, the Zeldvich factor, and the concentration of the crystal surface, and a is the diameter of the colloidal particles. The values in Figs. 8(a) and 8(b) were replotted based on Eqs. (9) and (10), respectively, assuming that the nucleation behavior follows the framework of CNT. The data are well fitted linearly with 1/ln( φ area / φ eq ) (Fig. 8 (c) and (d)). The step free energy, γ , is obtained from the gradient of the slopes in those figures, which results in 3.2k B T/a and 3.7 k B T/a for J and r*. The γ values agreed well although they were obtained from independent observations. These facts suggest that the nucleation process of colloidal crystals with added polymer agrees with the framework of CNT. Since a small number of particles in each nuclei generates a discrete nature of steps, different γ values may result. Since each data point in Fig. 8 (c) and (d) apparently is on a line, such an effect is not so critical under the present experimental conditions. Detailed investigations into how the configuration of particles comprising each nuclei changes γ will offer new insight into the capillary effect on the nucleation. The value of γ is 5 to 10 times larger than that of a hard-sphere system. 64 This is attributed to the attractive interaction. Since γ is free energy per unit length of the step, it includes both enthalpy and entropy terms. In the hard-sphere system, since there is no interaction, entropy is predominant. Here, there is a depletion attraction between particles, which gives rise to an enthalpy contribution to the gamma. These 17 ACS Paragon Plus Environment

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results suggest that more energy is required for the attractive system than in the hard-sphere to form steps on the terrace. At high φ area , the density of formed nuclei is high, which results in a short distance between neighboring islands. In this situation, depletion zones of formed islands may overlap if the electrolyte concentration is low enough. We deduce that this overlap affects the nucleation rate, especially the incorporation of particles into sub-critical islands. We consider that this effect is not significant under the high electrolyte concentration of our experimental conditions. How the interactions between formed nuclei affect the nucleation rate is important future work for us.


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Figure 8 (a) Nucleation rate, J, at various φ a rea . (b) Number of particles in the critical size of nuclei at various φ area . (c) lnJ as a function of 1/ln( φ area / φ eq ). (d) Radius of critical size divided by size of the colloidal particles, r * /a, as a function of 1/ln( φ area / φ eq ).

The nucleation rate for a polymer concentration, C p , of 1.5 g/l is compared to that for 1.25 and 1.0 g/l (Fig. 9). The depletion potential is roughly proportional to the number density of the depletant. When the polymer concentration is lowered, γ for 19 ACS Paragon Plus Environment

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1.25 and 1.0 g/l is 1.6k B T/a and 0.7k B T/a, respectively. This result is lowered correspondingly. The values are consistent in that a smaller bonding energy results in a smaller γ . We can address the interaction between colloidal particles based on CNT.

Figure 9 lnJ as a function of 1/ln( φ area / φ eq ) for different C p of 1.5, 1.25, and 1.0 g/l.

Extension of the lateral size of grown crystals depends on growth parameters such as 20 ACS Paragon Plus Environment

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initial volume fraction of particles, Φ , polymer concentration, C p , thickness of the cell, and substrate material. High Φ results in the production of a large number of tiny crystals, whereas no nucleation takes place under too low Φ . The C p affects the nucleation rate significantly. Thickness of the cell corresponds to how many particles are supplied to the growing crystals. The materials of the substrate change the thermodynamic barrier for nucleation. To obtain large crystals, we must explore the optimal conditions for the above mentioned parameters. In the hard-sphere system, gravity affects the nucleation.

65-66

Gravity changes the

density of particles in the melt in the vicinity of the crystal as well as in the crystal. However, we think particle diffusion on the terrace and incorporation into islands are dominant factors in the nucleation for the attractive system. Though gravity changes the particle supply to the terrace, that affects φ area , and the relationship between φ area and the nucleation rate will not change. However, the effect of gravity may not be negligible for the lower polymer concentration, where the process of diffusion and incorporation may be affected by gravity due to their weak particle interactions. The effect of gravity on the 2D nucleation requires additional investigation. The nucleation process in the attractive system of a colloidal suspension with added polymer is consistent with CNT, which is the same as that of a hard-sphere system. In a hard-sphere system, the densities of the environment and solid are very similar, suggesting that the growth mode is similar to the melt growth. In contrast, for the attractive system, the situation is close to those of solution or vapor growth, in which the density gap between the environment and the crystal is quite large. This allows us to obtain information on the surface of colloidal crystals, such as the nucleation of 2D islands, the growth of steps, and the behavior of ad-particles on the terrace. 21 ACS Paragon Plus Environment

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Further measurements of the above process will offer more a quantitative understanding of the nucleation process.

IV. CONCLUSION In situ observations reveal that the two-dimensional (2D) nucleation process on the surface of colloidal crystals agree with the framework of classical nucleation theory (CNT).

Observation

at

single-particle

resolution

was

shown

to

allow

for

determination of the surface concentration of ad-particles on the terrace as φ area , which is related to the nucleation rate, J. The steady-state homogeneous nucleation rate was successfully measured by distinguishing heterogeneous nucleation and coalescence of subcritical nuclei. The step free energy, γ , was obtained by J and critical size based on the nucleation theory, the values of which are in accordance with each other. A lower attractive interaction that is derived from the lower polymer concentration, C p , results in a lower γ , which makes sense since a lower γ is obtained with lower bonding energy. The colloid with an attractive system well describes the nucleation of real atomic or molecular systems, which is significant information to control the growth of colloidal crystals for producing novel nanostructured materials.

ACKNOWLEDGEMENTS This work was supported in part by JSPS KAKENHI Grant Number 26870047.

ASSOCIATED CONTENT Supporting

information

is

available

free

of

charge

http:/pubs.acs.org/. 22 ACS Paragon Plus Environment

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TOC

lnJ (1/m2・s)

25 20 15 10 5 0 0.24

0.29

0.34

0.39

1/ln(φarea/φeq)

0.44

3.5

radius (r/a)

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3 2.5 2 1.5 1

5 µm


0.22 0.24 0.26 0.28 0.3 0.32 0.34

1/ln(φarea/φeq)

Two-Dimensional Nucleation on the Terrace of Colloidal Crystals with

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