Step Kinetics Dependent on the Kink Generation Mechanism in

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Step Kinetics Dependent on the Kink Generation Mechanism in Colloidal Crystal Growth Jun Nozawa, Satoshi Uda, Suxia Guo, Akiko Toyotama, Junpei Yamanaka, Junpei Okada, and Haruhiko Koizumi Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b00063 • Publication Date (Web): 04 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

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Crystal Growth & Design

Step Kinetics Dependent on the Kink Generation Mechanism in Colloidal Crystal Growth

Jun Nozawa,†,* Satoshi Uda,† Suxia Guo,† 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 Nucleation and growth of two-dimensional (2D) islands on a terrace are the dominant growth mechanisms of colloidal crystals whose particle interaction is attractive. The step velocity, vstep, of the 2D islands at various area fractions, φarea, and polymer concentrations, Cp, has been investigated. At low Cp (weak attractive interaction between particles), there is a nearly linear relationship between vstep and φarea, whereas it is parabolic for high Cp (strong attraction). Depending on the Cp, two manners of

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Crystal Growth & Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

kink generation at a step are observed: 1D nucleation under strong attraction and association of mound formation under weak attraction. As a result of mound formation, abundant kinks are created at steps, resulting in a linear relationship between vstep and φarea, whereas the relationship is parabolic for step propagation associated with 1D nucleation. Though the kink site is the most favored site for particles to be incorporated into crystals, weak attractive interaction makes the step front site an incorporation site as well, and this latter process is the main mechanism for mound formation. This study is the first to elucidate the relationship between step kinetics and the kink generation mechanism of colloidal crystals, and these new findings significantly contribute to better control of the growth of colloidal crystals.

1. Introduction Novel functional materials created with desired structures by colloidal particles are now garnering much attention in materials science. A vast number of methods have been explored to fabricate various structures of colloidal crystals, in which diverse kinds of materials as well as a variety of morphologies of colloidal particles are employed. Among numerous parameters that control the growth of colloidal crystals, the interparticle interaction is a critical factor. Essentially, three kinds of interactions are available: repulsive, attractive, and hard-sphere interaction. Among these, the repulsive system has widely been employed to prevent particle aggregation such as in the convective assembly method.1,2 Recently, the properly tuned attractive interaction has been used more frequently since it can fabricate a variety of complex structures that cannot be obtained by repulsive interactions. The typical available attractive interactions include magnetic,3–5 Coulombic,6–8 DNA mediated,9–11 and depletion forces.12–14 In particular, the depletion force is now being


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widely studied since it works on any type of particle without modification of the particle. The depletion attraction occurs in colloidal particles when a smaller particle or macromolecule (depletant) is introduced, in the region where the separation between the two particles is smaller than the size of the depletant.15–17 In contrast to the number of studies on nucleation employing the attractive system of colloidal crystals,18–21 few investigations have been reported on the growth process. One feature of growth in an attractive system is that it presents a smooth crystal face, where the particle density of the crystals and environment exhibit a huge difference. The growth mode is similar to vapor or solution growth in atomic systems. On a smooth surface, colloidal crystals grow by continuous two-dimensional (2D) nucleation and growth of 2D islands.22 The terrace-ledge-kink (TLK) model23 is, therefore, induced to operate on such a crystal surface. The surface pattern produced by mono-molecular steps of the layer-by-layer grown crystals offers us invaluable information on the growth mechanism and its kinetics. The recent availability of atomic force microscopy enables us to observe surfaces at the atomic scale, at which growth mode and evolution of detailed growth processes are revealed.24-26 Since step velocity determines the growth kinetics of the layer-by-layer growth, formation of kink sites is a critical factor for growth kinetics. There have been many studies focusing on the kink creation mechanism both by experiment27 and simulation.28 The step kinetics are important not only for growth but also for the dissolution process. 29,30

Surface observation is, therefore, the most suitable method to understand the

growth mechanism of colloidal crystals of an attractive system. Colloidal crystals have important applications. They have been employed as model systems of various phenomena in atomic systems, especially phase transitions. Processes such as solid-solid phase transition,31 superheating,32 and surface melting33



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have been modeled. Thus, the investigation of the growth mechanism of colloidal crystals provides new insights into growth mechanisms of atomic systems. We previously

applied

the

colloidal

system

to

impurity

partitioning

during

crystallization.34 An advantage for a colloidal crystal growth study is the availability of observation at single-particle resolution. In situ observation detects a number of particles on the crystal surface, which enables us to determine the precise thermodynamic driving forces such as supersaturation. Furthermore, real time observations are available to investigate step morphology, kink formation, and kink density of steps, which are directly related to the growth kinetics of crystals. Since the classical terrace-ledge-kink model assumes that the thermal fluctuation of steps is rapid enough to produce abundant kinks, it cannot interpret calcite crystal growth, for example, because of its low kink density and weak step edge fluctuation.35–38 Nonclassical growth behavior still remains to be revealed. The elucidation of the quantitative relationship between growth kinetics and nano-scale surface structures such as kink density or 2D clusters will significantly contribute to fundamental understanding of the crystal growth mechanism. This study focuses on the growth mechanism of two-dimensional islands on the terrace of colloidal crystals, in which added polymers generate a depletion attraction between particles. The growth rate of steps, vstep, is related to the area fraction, φarea, which is equivalent to the surface concentration of the particles, and polymer concentration, Cp, which yields a force roughly proportional to the strength of the attractive interaction. In atomic systems, the rate of step propagation greatly depends on the kink creation mechanism including thermal fluctuation and 1D nucleation. We discover the characteristic kink generation mechanism of colloidal


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crystals that is associated with mound formation at steps, which is attributed to a weak attractive interaction between particles.

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. The 500 nm green fluorescent monodispersed PS was used in the experiments without any further purification. The zeta potential of the particles was determined to be –47.1±5.9 mV. Sodium polyacrylate (polymerization degree of 30000–40000) was added as a depletant, which gave rise to an attractive force between particles. Since the adsorption of the depletant on the colloidal particles is negligible, modification of particles such as organic surface coating does not significantly affect the results of the study. Three concentrations of sodium polyacrylate, Cp (0.100, 0.125, and 0.150 g for 1 L of water), were applied. Since the depletion interaction depends on the concentration of the polymer, the attractive interactions become larger upon increasing Cp. From polymer concentrations and molecular weight of acrylate monomer residues [-CH2-CH(CO2Na)-] (= 94), the background electrolyte concentrations for polymer concentrations of 0.100, 0.125, and 0.150 g/L were estimated to be 1.0, 1.25, and 1.5 mM, respectively. At these high salt concentrations, the electrostatic interaction between the particles is significantly screened and negligibly small, though its magnitude appears to be strong enough at small separations to prevent random coagulation of the particles. The polymer solution and colloidal suspension were mixed and sealed in a



growth cell made up of cover glasses. An oil immersion lens (magnification is 100 and N.A. = 1.3) was used to achieve single-particle resolution of the 500 nm particles. The φarea is defined to represent the surface concentration of particles, in which the area occupied by particles is divided by the area of the terrace. The number of particles on the terrace is counted, multiplied by the area of one particle, and then divided by the area of the terrace. Since colloidal crystals grow by incorporating particles, the particle concentration in the solution gradually decreases. The φarea on the terrace also decreases, so the various φarea values depend on the elapsed time for each Cps. This study concerns the crystallization process of charged colloidal particles whose interaction is attractive. Although we used the charge and depletion attraction system in

this paper, we believe that the major conclusions from this study are applicable to attractive colloids in general. The results will also be applicable to particles of the general size of colloids (10-1000 nm). Outside of this particle range, it is difficult to obtain suitable Brownian motion for the crystallization.

3. Results and discussion At the initial stage of growth, two-dimensional (2D) nucleation occurs on the cover glass at the bottom of the cell when the particle concentration increases due to sedimentation by gravity. Colloidal crystals grow by lateral growth of the formed islands and further 2D nucleation on the terrace. We monitored the growth process at single-particle resolution by optical microscopy. Figure 1 shows a typical surface of growing colloidal crystals. Particles that are supplied from the solution diffuse on the terrace as ad-particles. While some ad-particles return to the solution, the ones that reach and attach to the steps of 2D islands are incorporated into the crystal, by which


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lateral growth of 2D islands occurs. The surface diffusion mechanism23 is found to work for colloidal crystals of attractive systems. As 2D islands on the same terrace grow, they coalesce with each other to form larger islands.

Figure 1. Nucleation and growth of 2D islands on the terrace of colloidal crystals. The 2D islands grow via incorporating ad-particles into steps.

All the steps observed in this study are steps of the 2D island that emerged on the {111} face. No spiral steps were recognized, because with the small number of layers, at most 5, in the formed colloidal crystals, it was difficult to generate spiral dislocation. Step velocities, vstep, of islands at various area fractions, φarea, were measured for three different polymer concentrations, Cp, of 0.10, 0.125, and 0.15 g/L. The step velocity was measured by tracking the displacement of a close packed row (indicated by white dashed line in Figure 1B) along the [211] direction. Since the growth rate was interrupted by adjacent steps, an isolated step, which is separated from adjacent steps by more than 10 µm, was selected for the measurement. The vstep was determined as the average value of several equivalent orientations of [211] of an island. Six-fold symmetry appeared on the {111} face, assuming that they had the same vstep. The equilibrium area fraction, φeq, where no growth of the step occurs was also determined. As experimental time elapsed, φarea decreased. This was associated with the consumption of particles in the solution by the growth of crystals. When φarea



reached a certain value, the steps no longer propagated on the surface. This value was determined as φeq. The φeq values for Cp of 0.10, 0.125, and 0.15 g/L were measured to be 0.008, 0.0048, and 0.001, respectively. As Cp increased, φeq decreased. This is understood in that a strong attractive interaction prevents particle detachment from crystals, which is driven by thermal fluctuation, leading to low solubility. Here, the repulsive interaction is constant among different Cps, whereas the attractive interaction increases with increasing Cp. Both detachment and attachment of particles from the kink sites occurred during both the crystal growth and dissolution processes. Therefore, the observed crystallization process is reversible. On φarea < φeq, detachments of particles from the kink takes place more frequently than attachments did, resulting in retreat of the steps. The first layer forms by nucleation on the cover glass. This is the nucleation on foreign substrate. The equilibrium concentration for the 1st layer is larger than that of the 2nd layer,39 because of the difference in changes of interfacial energy for the nucleation, which leads to larger growth rate of 2nd and larger number of layers than that of 1st layer under the same φarea. During experiment, any layers keep advancing as long as φarea is larger than φeq. Figure 2 shows vstep at various values of φarea subtracted by φeq for three Cps. Though Cps does not change during the experiment, φarea decrease due to a consumption of particles in the solution for growth of crystals. The step velocities at various values of φarea are obtained at different elapsed times of the experiments for each Cps. Since φarea–φeq is the net supply of particles to the step, which contributes to growth of the step, it is used to show the rate of step advancement. An almost linear relationship between step velocity and φarea–φeq is recognized for Cp of 0.10 and 0.125 g/L, whereas it is parabolic for the Cp of 0.15 g/L. It is also shown that a smaller Cp


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has a larger vstep at a given φarea–φeq value. The cause of these different relationships between vstep and φarea–φeq depending on Cp will be considered based on in situ observations.

Figure 2. Step velocity, vstep, as a function of φarea–φeq for three Cps. The φeq values for Cp of 0.10, 0.125, and 0.15 g/L were measured to be 0.008, 0.0048, and 0.001, respectively.

Figure 3 represents step morphologies at various φarea–φeq and Cp values. Steps in Figure 3 (A)–(C) are growing at different φarea–φeq under the same Cp, whereas (D)– (F) have different Cp under similar φarea–φeq. Among growth conditions of the same Cp, the shape of the step changes from straight to rounded as φarea–φeq increases, which corresponds to increasing kink density. This trend is also observed for



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decreasing Cp with similar φarea–φeq. Regarding the Cp change, since the strength of the attractive interaction is roughly proportional to the Cp, the variation in Cp corresponds to a change of bond energy, ψ. The step is likely to be straight when ψ is high so as to attain a minimum step surface energy. This is also understood from the kink density. In a Kossel crystal, there is a relationship between ψ and kink formation energy, ω, as ω = ψ/2.40 This relation is also valid for fcc lattice of colloidal crystals, because net energy change for kink formation in terms of bonding energy is the same.41 The number of molecules between kinks, nK, is described as23 nK =

 ω  1  +1 exp  2  k BT 

(1)

where kB is Boltzmann’s constant and T is the absolute temperature. This relationship clearly suggests that rough steps (smaller nK) appear at lower Cp (lower ψ). Note that eq. (1) is derived assuming all kinks are created by thermal fluctuations of the step, so the formula should be modified by the supersaturation term when it is applied to nonequilibrium steps that are growing. An increase of φarea–φeq corresponds to an increasing of the driving force, which causes smooth steps to turn into rough ones for kinetic reasons. The kink density, ρ, is formulated as a function of the saturation ratio (solute concentration divided by equilibrium concentration), S,42 ρ = 2 a −1 S 1 / 2 exp( −ω / k BT )

(2)

where a is the length of the growth unit. A large kink density is obtained at a high saturation ratio. The shapes of the steps at various conditions qualitatively follow the theories of atomic systems.


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Figure 3. Shape of steps at various Cp and φarea–φeq. The Cp of (A), (B), and (C) is 0.15 g/L with different (φarea–φeq) values from 0.001 to 0.015. (D), (E), and (F) have similar (φarea–φeq) of 0.006 with different Cp values from 0.10 to 0.15 g/L. The shape of the steps changes depending on Cp and φarea–φeq; straight steps change to rounded shapes upon either decreasing Cp or increasing φarea–φeq.

Figure 4 reveals the detailed process of step growth under two different conditions. The step in Figure 4A and B is growing at high Cp (0.15 g/L) and low

φarea–φeq (~0.001), with strong particle attraction and small driving force for growth. As mentioned above, a straight step appears under this condition. As shown in snapshots (Figure 4A), the creation of kinks is found to be caused by one-dimensional (1D) nucleation. The particle enclosed by a white dashed circle in Figure 4A initiates the 1D nucleation. Attachment of another particle next to the initial particle grows the row, creating two kinks at the step. Since associating 1D nucleation does not produce a further energy cost, it has no energetic barrier. The theory of 1D nucleation has been ACS Paragon Plus Environment


studied by many authors.43–45 Creation of kinks by thermal fluctuation of steps is also observed but it rarely occurs. In this growth condition, 1D nucleation is found to be the dominant kink generation mechanism. Figure 4B is a representation of the step dynamics. Images of the rectangular area enclosed by the dashed line in the left image of Figure 4B are extracted from the movie every minute, and these snapshots are arranged in order of time. The length and width of the rectangle are approximately 0.5 and 14 µm, respectively. The dashed lines are a guide to indicate the front position of the steps. Row-by-row growth of steps is clearly observed. Figure 4C shows snapshots of growing steps under low Cp (0.10 g/L) and medium φarea–φeq (~0.003). In this case, many kinks are available at the steps. The time evolution of steps indicates that mounds that have several particle depths are formed at a step (enclosed by dashed circle). Many kinks are generated as a result of mound formation. As in Figure 4B, images of the rectangular area from the movie are arranged sequentially in Figure 4D. As indicated by dashed circles, steps suddenly grow along the length corresponding to several particle sizes, where multiple kinks that are created by the mound advance steps simultaneously. In each Cp, when φarea is at equilibrium, kinks are generated only by the thermal fluctuation, which leads to straight shape of steps. As increasing φarea, kinks are generated by 1D nucleation and mound formation as well as thermal fluctuation, which kinetically rough the steps. Kink creation via mound formation is a transition state toward the kinetic roughening, which will occur at higher φarea. When kinetic roughening occurs, abundant kinks will be created at once without mound formation.


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Figure 4. Detailed growth process of steps under high and low Cp. (A) Snapshots of growing steps under high Cp (0.15 g/L) and low φarea–φeq (~0.001). Time is provided at the upper part of each image. 1D nucleation is the dominant mechanism for creating kinks at a step. (B) Rectangular area enclosed by dashed line in left image is shown every 1 minute and arranged sequentially, and hence the horizontal axis represents time. Row-by-row growth associated with 1D nucleation is observed. (C) Snapshots of growing steps under low Cp (0.10 g/L) and medium φarea–φeq (~0.003).



The time evolution of steps shows mound formation. The mounds generate many kinks. (D) The same process for images as in (B) is applied. The dashed circles indicate the step advances by several particles at once, which is caused by mounds formed at steps.

The kink formation mechanism depends on φarea–φeq as well as Cp. When φarea–

φeq is increased to ~0.016 (almost the highest value in the experiment) under high Cp (0.15 g/L), mound formation is also observed at steps similar to that of low Cp with medium φarea–φeq. When φarea–φeq is significantly decreased under low Cp (0.10 g/L), 1D nucleation occurs. Fig. 5 schematically represents how the kink creation mechanism depends on φarea–φeq and Cp. It should be noted that the transition of the kink creation mechanism occurs at lower φarea–φeq for low Cp. For low Cp of 0.10 and 0.125 g/L, the kink creation mechanism immediately converts from 1D nucleation to mound formation with increasing φarea–φeq, which makes the relationship between step velocity, vstep, and φarea–φeq almost linear as shown in Fig. 2. In contrast, at high Cp of 0.15 g/L, the kink creation mechanism is 1D nucleation up to high φarea–φeq, which makes the relationship between vstep and φarea–φeq parabolic. Step advancement associated with mound formation does not proceed at a constant rate. As is shown in Figure 4D, after several advancements occur as indicated by dashed circles, there are short pauses in growth. Once many kinks are created at the mounds, particles are incorporated into those kinks rather than used for new mound formation.


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Figure 5. Schematic illustrating relationship between φarea–φeq and Cp for kink formation.

Figure 6A shows snapshots depicting mound formation at a step. Attachment of the particles indicated by white arrows in the figure is found to be the key process for mound formation. Figure 6B illustrates the formation process of the mounds, in which a straight step that has a single kink site (i) is set as the initial state. Typically, a kink site is the most favored site for particles to be incorporated into a crystal, however, no mound formation takes place by incorporation only at kink sites. As indicated by white arrows in Figure 6A, particles attach at the front site of the step as well as the kink site, which corresponds to the site indicated by a white particle in Figure 6B(ii). This site is named the “step front site.” Based on this observation, a new particle more frequently attaches next to the first particles rather than at a kink site (iii). The new “step front site” is then occupied by subsequent particles (iv). As a result of this process, a mound forms at the step. We consider that the mound formation is attributed to a weak attractive



interaction between particles. When the Cp is high (strong attractive interaction), since the kink site has a larger number of neighboring particles compared to the step front site, it is significantly more favorable for new particles to be incorporated at the kink site. Under such a situation, the steps grow row-by-row by incorporation only at the kink site that is created by 1D nucleation. As the attractive interaction decreases due to decreasing Cp, the preference for kink site incorporation decreases. Particles often attach to the step front site as indicated in Fig. 6, which results in mound formation, where many kinks are created via mound formation. The step front site incorporation may also be attributed to geometric reasons. The kink sites are in recessed positions from the front most row, which decreases the frequency of particles reaching them compared to the step front site. Above discussion is microscopic picture based on observation of singleparticle level. On the other hand, this mounds formation process is understood as heterogeneous nucleation of 2D islands on steps, where substrate and island corresponds to steps and mounds, respectively. At low Cp, since smaller step free energy leads to smaller nucleation barrier and critical size, mounds formation easily occur. In contrast, large step free energy for low Cp has large nucleation barrier, which makes nucleation at low supersaturation difficult. We often observed formed mounds vanished at low Cp, due to disappearance of embryo that does not reach the critical size. Consequently, only 1D nucleation process works under large step free energy of higher Cp whereas mound formation is predominant for lower Cp.


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Figure 6. Formation process of mounds at the step. (A) Snapshots of formation of mounds. Attachment of particles indicated by white arrows is the key process to form mounds. (B) Schematic illustration to depict the mound formation process. By attaching particles at the white particle position, which is named the “step front site,” a straight step that has a single kink site (i) transforms into a mound (v).

Figure 7 represents the process by which a 2D island absorbs a 2D cluster. This does not occur frequently, however, it is likely observed under low Cp and high

φarea–φeq. The formation of 2D cluster appears to be attributable to homogeneous nucleation of mounds. It can occur when step free energy is sufficiently small at low Cp and at high supersaturation. This condition increases the lifetime of the 2D cluster on the terrace, which increases the probability for the cluster and 2D island to meet. This 2D cluster plays an important role in creating kinks for protein crystal growth at supersaturation.27,46 In protein crystal growth, mounds at steps are formed by incorporating the 2D cluster. A similar process is also observed in colloidal crystal



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growth, however, it is not the predominant mechanism for creating kinks under our experimental conditions.

Figure 7 Incorporation of a 2D cluster, indicated by white arrowhead, into the 2D island.

Results of in situ observations reveal the mechanism of kink generation by 1D nucleation at higher Cp and mound formation at lower Cp. Based on the theory of 1D nucleation, the rate of step advance by 1D nucleation, v1D,47 gives v1D = a 2 ρ 0 ω − σ (1 + σ

)1 / 2

(3)

where a is lattice constant, ρ0 is kink density, ω– is probability for detachment of an adatom from a kink position, and σ is supersaturation. The σ is expressed as (C– Ceq)/Ceq by concentration, C, and equilibrium concentration, Ceq. Assuming C and Ceq correspond to φarea and φeq, the relationship of v1D and φarea–φeq in eq. (3) is parabolic,


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which is consistent with that of high Cp in Figure 2. 1D nucleation occurs at low kink density. When kinks are sufficiently available, the growth rate of a step is expressed as,48

(

v = β Ω C − C eq

)

(4)

where β is the kinetic coefficient and Ω is volume per molecule in the solid. This equation leads to a linear dependence of v on C–Ceq. This is also consistent with the results for lower Cp in Figure 2. Under a strong attractive interaction (high Cp), the kink density is small and hence 1D nucleation operates for kink generation. Upon decreasing Cp, the attractive interaction is lowered, which increases the kink density. Additionally, frequent incorporation at the step front site leads to mound formation at steps, from which many kinks are created. A lower Cp gives rise to synergetic effects to increase the step velocity.

4. Conclusion The first observations revealing step kinetics of colloidal crystals have been conducted. Occurrence of 1D nucleation is revealed by in situ observations, which shows a parabolic relationship between vstep and φarea. On the other hand, a peculiar kink generation mechanism via mound formation for colloidal crystals has been discovered. Incorporation at the step front site is found to play an important role in the kinetics of step growth, in which the frequency is determined by the magnitude of the attractive interaction. These findings add new knowledge and practical guidance for the fabrication of colloidal crystals.



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


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For Table of Contents Use Only

Step Kinetics Dependent on the Kink Generation Mechanism in Colloidal Crystal Growth

Jun Nozawa, Satoshi Uda, Suxia Guo, Akiko Toyotama, Junpei Yamanaka, Junpei Okada, Haruhiko Koizumi

TOC graphic

Synopsys Depending on the polymer concentration, Cp, two manners of kink generation at a step are observed: 1D nucleation and association of mound formation. Though the kink site is the most favored site for particles to be incorporated into crystals, weak attractive interaction (low Cp) makes the step front site an incorporation site, which is the main mechanism for mound formation.


Step Kinetics Dependent on the Kink Generation Mechanism in

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