Adsorption, Desorption, Surface Diffusion, Lattice Defect Formation, and Kink Incorporation Processes of Particles on Growth Interfaces of Colloidal Crystals with Attractive Interactions

Good model systems are required in order to understand crystal growth processes because, in many cases, precise incorporation processes of atoms or molecules cannot be visualized easily at the atomic or molecular level. Using a transmission-type optical microscope, we have successfully observed in situ adsorption, desorption, surface diffusion, lattice defect formation, and kink incorporation of particles on growth interfaces of colloidal crystals of polystyrene particles in aqueous sodium polyacrylate solutions. Precise surface transportation and kink incorporation processes of the particles into the colloidal crystals with attractive interactions were observed in situ at the particle level. In particular, contrary to the conventional expectations, the diffusion of particles along steps around a two-dimensional island of the growth interface was not the main route for kink incorporation. This is probably due to the number of bonds between adsorbed particles and particles in a crystal; the number exceeds the limit at which a particle easily exchanges its position to the adjacent one along the step. We also found novel desorption processes of particles from steps to terraces, attributing them to the assistance of attractive forces from additionally adsorbing particles to the particles on the steps.

If attractive interactions between particles can be utilized for colloidal crystallization, we can achieve in situ observation of particle incorporation into crystals on the growing surface from a dilute dispersion of particles at the particle level.However, van der Waals attraction between large particles often causes irreversible coagulation, owing to a deep primary minimum of its potential curve of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [59]; van der Waals attraction is too strong to allow particles to reversibly attach to and detach from each other.Thus, we focus on depletion interaction between large particles that are suspended in a dilute polymer electrolyte solution as the second candidate [60].The depletion interaction is also an attractive interaction, whereas a reversible formation of crystalline aggregate has been reported by Kose and Hachisu [61].Although those authors showed clear microscopic images that indicate layer-by-layer stacking of two-dimensional regular arrays of polystyrene particles at the particle level, they did not focus on the surface dynamics of the particles.Although Toyotama et al. recently studied particle-level eutectic-formation processes from binary and ternery mixtures of polystyrene particles in depletion-induced attractive colloidal crystals [62], there are still a lot of unclarified elementary processes of crystal growth in this depletion-induced colloidal crystallization system.
In this study, we first use an optical microscope to examine adsorption, desorption, and diffusion of polystyrene particles on 2D regular arrays (crystals) of the particles in an aqueous sodium polyacrylate solution that induces depletion interactions between particles.We then examine lattice defect formation and incorporation processes of the particles into the crystals.

In Situ Observation of Growth Interface of Crystals
Growth surfaces of colloidal crystals were observed simply by using a 100ˆoil immersion objective (Figure 1).Not only the first layer of the crystals, but also the first three layers were easily resolved at the particle level.Particles on a layer moved around from one specific lattice position to another rather discretely, while those in bulk suspension moved continuously via normal bulk Brownian motion at the same time (Video S1).Such "hopping" behaviors of particles between adjacent lattice cites would provide more detailed information about activation processes of surface diffusion, if we conduct higher-frame-rate observation using high-speed cameras as the future works.Straight-shaped steps were also confirmed at the edges of a layer.From the viewpoint of periodic bond chain (PBC) theory [63], the straightness of the steps would reflect the strength of attractive interactions between particles along the steps.Furthermore, roughness of the steps is a direct measure of roughening transitions of steps.In the present setup, we can solve these problems by real-time and dynamical analyses of particle incorporation processes into the steps.We will also conduct these measurements in the near future.

Equilibrium Conditions
Equilibrium conditions (solubility, melting point, saturation pressure, and so on) are prerequisites for the quantitative analysis of crystallization kinetics.Thus, we first tried to find the equilibrium condition at which the growth and dissolution rates of crystals are equal.To determine the equilibrium conditions, we recorded the growth processes of crystals with time.At first, crystals nucleated, started to grow, and then grew layer-by-layer up to several thicknesses and sizes.Finally, the growth rates approached zero asymptotically; the system seemed to reach equilibrium.After that, however, the crystals started to dissolve, and in several days the crystals had disappeared completely and quite amazingly (Figure 2).Thus, we could not determine the equilibrium conditions in this system at the present stage.Although we considered Ostwald ripening and observed all parts of the growth cell repeatedly, we could not find any larger crystals, and all particles over the entire volume in the cell completely re-dispersed at the end of every experiment.In addition, no leaks of particles out of the growth cell were observed.Some other slow relaxation processes (interaction between water molecules and polyacrylic ions, for instance) would take several days.Although the mechanism of re-dispersion has not been clarified yet, crystallization processes in the early stages (at least on the first day under the conditions shown in Figure 2, for instance) seemed to be normal.Thus, in this study we focus on the dynamical behavior of particles on the growth interface of colloidal crystals with attractive interactions only in the early stages.Of course, we have to keep in mind that this "crystallization" process is examined under a dynamic and non-equilibrium environment at the present stage.

Equilibrium Conditions
Equilibrium conditions (solubility, melting point, saturation pressure, and so on) are prerequisites for the quantitative analysis of crystallization kinetics.Thus, we first tried to find the equilibrium condition at which the growth and dissolution rates of crystals are equal.To determine the equilibrium conditions, we recorded the growth processes of crystals with time.At first, crystals nucleated, started to grow, and then grew layer-by-layer up to several thicknesses and sizes.Finally, the growth rates approached zero asymptotically; the system seemed to reach equilibrium.After that, however, the crystals started to dissolve, and in several days the crystals had disappeared completely and quite amazingly (Figure 2).Thus, we could not determine the equilibrium conditions in this system at the present stage.Although we considered Ostwald ripening and observed all parts of the growth cell repeatedly, we could not find any larger crystals, and all particles over the entire volume in the cell completely re-dispersed at the end of every experiment.In addition, no leaks of particles out of the growth cell were observed.Some other slow relaxation processes (interaction between water molecules and polyacrylic ions, for instance) would take several days.Although the mechanism of re-dispersion has not been clarified yet, crystallization processes in the early stages (at least on the first day under the conditions shown in Figure 2, for instance) seemed to be normal.Thus, in this study we focus on the dynamical behavior of particles on the growth interface of colloidal crystals with attractive interactions only in the early stages.Of course, we have to keep in mind that this "crystallization" process is examined under a dynamic and non-equilibrium environment at the present stage.

Equilibrium Conditions
Equilibrium conditions (solubility, melting point, saturation pressure, and so on) are prerequisites for the quantitative analysis of crystallization kinetics.Thus, we first tried to find the equilibrium condition at which the growth and dissolution rates of crystals are equal.To determine the equilibrium conditions, we recorded the growth processes of crystals with time.At first, crystals nucleated, started to grow, and then grew layer-by-layer up to several thicknesses and sizes.Finally, the growth rates approached zero asymptotically; the system seemed to reach equilibrium.After that, however, the crystals started to dissolve, and in several days the crystals had disappeared completely and quite amazingly (Figure 2).Thus, we could not determine the equilibrium conditions in this system at the present stage.Although we considered Ostwald ripening and observed all parts of the growth cell repeatedly, we could not find any larger crystals, and all particles over the entire volume in the cell completely re-dispersed at the end of every experiment.In addition, no leaks of particles out of the growth cell were observed.Some other slow relaxation processes (interaction between water molecules and polyacrylic ions, for instance) would take several days.Although the mechanism of re-dispersion has not been clarified yet, crystallization processes in the early stages (at least on the first day under the conditions shown in Figure 2, for instance) seemed to be normal.Thus, in this study we focus on the dynamical behavior of particles on the growth interface of colloidal crystals with attractive interactions only in the early stages.Of course, we have to keep in mind that this "crystallization" process is examined under a dynamic and non-equilibrium environment at the present stage.

Surface Diffusion of Particles
The diffusion of particles on the growth interface of crystals was successfully observed as described in Section 2.1.The surface diffusion coefficient D s of particles was estimated using an equation expressed as, where x s is the mean displacement of an adsorbed particle and τ is the mean lifetime of an adsorbed particle before being dissolved again into the surrounding solution [64].Using 10 particles which adsorbed, diffused on the growth interfaces, and desorbed again into the surrounding solution, we directly measured x s and τ (Table 1).As a result, D s of particles was calculated to be (2 ˘1) ˆ10 ´13 m 2 ¨s´1 .This value is an order of magnitude smaller than the bulk diffusion coefficients of polystyrene particles in aqueous polyelectrolyte solutions [65].This is probably due to the activation energy for surface diffusion; the activation barrier for surface diffusion is probably higher than that for bulk diffusion in the solution.In addition, Video S1 shows that the speed of particle movements on a growth interface is clearly lower than that in the bulk solution, for instance.

Classification of Particle Incorporation Processes into Crystals
Particle incorporation processes were successfully observed at the particle level.Although solute incorporation into a crystal is well known to be an important elementary process of crystallization, to our knowledge there has been no direct and precise analysis of the process.First, we classified the routes of particles to be finally incorporated into a crystal at a kink site, as schematically shown in Figure 3.
Four routes are classified as follows.
‚ Kink incorporation processes via surface diffusion on lower layers ‚ Those via surface diffusion on upper layers ‚ Those via surface diffusion on lower layers and adsorption to the particle next to the kink site ‚ Those via direct incorporation from solution Although, in general, another classification-that via diffusion along the steps-would be added to this list, in the present study we did not observe kink incorporation via diffusion along the steps as far as we know; particles desorbed from the steps before arriving at the kink sites.Only in the "special" case of (c), particles incorporated into the kink site along the steps.Using 62 particles that were finally incorporated into kink sites, the number distribution of the incorporation routes was confirmed and are shown below as a bar graph (Figure 4).

Classification of Particle Incorporation Processes into Crystals
Particle incorporation processes were successfully observed at the particle level.Although solute incorporation into a crystal is well known to be an important elementary process of crystallization, to our knowledge there has been no direct and precise analysis of the process.First, we classified the routes of particles to be finally incorporated into a crystal at a kink site, as schematically shown in Figure 3.  Although, in general, another classification-that via diffusion along the steps-would be added to this list, in the present study we did not observe kink incorporation via diffusion along the steps as far as we know; particles desorbed from the steps before arriving at the kink sites.Only in the "special" case of (c), particles incorporated into the kink site along the steps.Using 62 particles that were finally incorporated into kink sites, the number distribution of the incorporation routes was confirmed and are shown below as a bar graph (Figure 4).These results indicate that surface diffusion processes (a) and (b) are major routes for kink incorporation processes, although we have not fully confirmed whether (a) is more frequent than (b) owing to Ehrlich-Schwoebel effects [66].The difference would depend on the difference in the area at this stage, since the order of magnitude of the size of the layers corresponds to that of xs in this study.A second major process is via adsorption to the particle at the edge of the kink site (c).Although a direct incorporation process from bulk solutions (d) is a minor one compared to the other processes shown in Figure 4, the possibility of finally incorporating particles into a kink site using this process is much greater than that of the diffusion process of particles along steps.These results are probably attributable to the difference in the number of recovered bonds of adsorbed particles, as shown in Figure 5. Dotted and solid circles represent the lower and upper layers, respectively.A, E, and S particles in Figure 5 are those that are adsorbed and form respectively three, four, and five bonds with particles in a crystal.In the (a) and (b) processes, adsorbed particles are classified as A particles.Therefore, the surface diffusion of A particles needs the activation energy to cut three bonds of A particles and move to adjacent lattice points.In the (c) process, adsorbed particles are classified as E particles.To move to the adjacent kink site, E particles should cut four bonds with three particles of the lower layer and the edge particle at the kink site; this higher activation energy for kink These results indicate that surface diffusion processes (a) and (b) are major routes for kink incorporation processes, although we have not fully confirmed whether (a) is more frequent than (b) owing to Ehrlich-Schwoebel effects [66].The difference would depend on the difference in the area at this stage, since the order of magnitude of the size of the layers corresponds to that of x s in this study.A second major process is via adsorption to the particle at the edge of the kink site (c).Although a direct incorporation process from bulk solutions (d) is a minor one compared to the other processes shown in Figure 4, the possibility of finally incorporating particles into a kink site using this process is much greater than that of the diffusion process of particles along steps.These results are probably attributable to the difference in the number of recovered bonds of adsorbed particles, as shown in Figure 5. Dotted and solid circles represent the lower and upper layers, respectively.A, E, and S particles in Figure 5 are those that are adsorbed and form respectively three, four, and five bonds with particles in a crystal.In the (a) and (b) processes, adsorbed particles are classified as A particles.Therefore, the surface diffusion of A particles needs the activation energy to cut three bonds of A particles and move to adjacent lattice points.In the (c) process, adsorbed particles are classified as E particles.To move to the adjacent kink site, E particles should cut four bonds with three particles of the lower layer and the edge particle at the kink site; this higher activation energy for kink incorporation results in a clearly reduced probability that the (c) process will achieve kink incorporation relative to the probability for the (a) or (b) processes.Furthermore, S particles are adsorbed on steps and form five bonds; the probability that these particles will arrive at the kink site via the diffusion along the steps probably becomes much lower than the others.

Formation of Stacking Disorders
Formation processes of lattice defects were observed in situ at the particle level.Figure 6 shows the formation of stacking disorders on the triangle lattice via two-dimensional nucleation and growth on different lattice sites.Similar images of stacking disorders had been already observed in our previous experiments, and we assumed that the stacking disorder was formed via two-dimensional nucleation and growth of colloidal crystals on the wall of the growth container [35], whereas the image was an ex-situ scanning-electron-microscope micrograph.Furthermore, the displacement of the boundary between different stacking layers was observed.The movement of the boundary is mainly promoted by the hopping of particles beyond the boundary between the different stacking layers.More detailed observation on the formation of lattice defects are planned as a future work.Furthermore, S particles are adsorbed on steps and form five bonds; the probability that these particles will arrive at the kink site via the diffusion along the steps probably becomes much lower than the others.

Formation of Stacking Disorders
Formation processes of lattice defects were observed in situ at the particle level.Figure 6 shows the formation of stacking disorders on the triangle lattice via two-dimensional nucleation and growth on different lattice sites.Similar images of stacking disorders had been already observed in our previous experiments, and we assumed that the stacking disorder was formed via two-dimensional nucleation and growth of colloidal crystals on the wall of the growth container [35], whereas the image was an ex-situ scanning-electron-microscope micrograph.Furthermore, the displacement of the boundary between different stacking layers was observed.The movement of the boundary is mainly promoted by the hopping of particles beyond the boundary between the different stacking layers.More detailed observation on the formation of lattice defects are planned as a future work.

Particle Desorption from Steps by Attractive Forces from the Other Particles on the Terrace
Novel desorption processes of a particle on a step of growth surfaces were found.Desorption occurred by the adsorption of other particles to the desorbing particle, as if the adsorbed particles peeled the desorbing particle off the step (Figure 7).Although no textbook on crystal growth has dealt with such a phenomenon, this occurrence seems to be naturally acceptable if the attractive interaction between particles is considered.Actually, not only this "peeling off" but also other cooperative rearrangements of multiple particles around steps were also observed.They also seemed to be due to the attractive interactions between particles and will be reported elsewhere.More detailed observation of these cooperative desorption and rearrangement of particles would be useful to clarify the elementary processes on dissolution processes of normal crystals deeply.
the formation of stacking disorders on the triangle lattice via two-dimensional nucleation and growth on different lattice sites.Similar images of stacking disorders had been already observed in our previous experiments, and we assumed that the stacking disorder was formed via two-dimensional nucleation and growth of colloidal crystals on the wall of the growth container [35], whereas the image was an ex-situ scanning-electron-microscope micrograph.Furthermore, the displacement of the boundary between different stacking layers was observed.The movement of the boundary is mainly promoted by the hopping of particles beyond the boundary between the different stacking layers.More detailed observation on the formation of lattice defects are planned as a future work.Novel desorption processes of a particle on a step of growth surfaces were found.Desorption occurred by the adsorption of other particles to the desorbing particle, as if the adsorbed particles peeled the desorbing particle off the step (Figure 7).Although no textbook on crystal growth has dealt with such a phenomenon, this occurrence seems to be naturally acceptable if the attractive interaction between particles is considered.Actually, not only this "peeling off" but also other cooperative rearrangements of multiple particles around steps were also observed.They also seemed to be due to the attractive interactions between particles and will be reported elsewhere.More detailed observation of these cooperative desorption and rearrangement of particles would be useful to clarify the elementary processes on dissolution processes of normal crystals deeply.

Methods
An inverted microscope (Nikon, Tokyo, Japan, TE-2000U) was used with a 100× oil immersion objective (Nikon, CFI Plan Achromat 100× oil).Crystal growth in particle dispersion was observed using a handmade observation cell (Figure 7).The dispersion of particles is sandwiched between a glass slide (Matsunami Glass Industries, Osaka, Japan, micro-slide glass S1126 76 × 26 mm) and cover glass (Matsunami Glass Industries, Osaka, Japan, micro-cover glass 18 × 18 mm), spaced with polystyrene strips (Evergreen Scale Models, 14″ dimensional strips; Item 102 0.28 × 1.0 × 350 mm), and sealed with a silicone adhesive (Konishi, Osaka, Japan, #46842) as schematically shown in Figure 8.To suppress vaporization of water from the dispersion during experiments, liquid paraffin was injected around the strips and filled into possible pores or hollows in thin solidified adhesive layers between spacers and glasses.

Methods
An inverted microscope (Nikon, Tokyo, Japan, TE-2000U) was used with a 100ˆoil immersion objective (Nikon, CFI Plan Achromat 100ˆoil).Crystal growth in particle dispersion was observed using a handmade observation cell (Figure 7).The dispersion of particles is sandwiched between a glass slide (Matsunami Glass Industries, Osaka, Japan, micro-slide glass S1126 76 ˆ26 mm) and cover glass (Matsunami Glass Industries, Osaka, Japan, micro-cover glass 18 ˆ18 mm), spaced with polystyrene strips (Evergreen Scale Models, 14" dimensional strips; Item 102 0.28 ˆ1.0 ˆ350 mm), and sealed with a silicone adhesive (Konishi, Osaka, Japan, #46842) as schematically shown in Figure 8.To suppress vaporization of water from the dispersion during experiments, liquid paraffin was injected around the strips and filled into possible pores or hollows in thin solidified adhesive layers between spacers and glasses.

Figure 1 .
Figure 1.An obtained colloidal crystal containing three layers of regular arrays of polystyrene particles.Individual particles are clearly resolved at the particle level.The focal plane in this picture is adjusted at the third layer.

Figure 2 .
Figure 2. Re-dispersion of particles from grown crystals.(a) 1 day after sample preparation; (b) 6 days after sample preparation.In a day, crystals grow up to several layers as shown in (a) at 25 °C.However, 3 days after sample preparation, the crystals start to dissolve, and 6 days after, the crystals have disappeared completely.

Figure 1 .
Figure 1.An obtained colloidal crystal containing three layers of regular arrays of polystyrene particles.Individual particles are clearly resolved at the particle level.The focal plane in this picture is adjusted at the third layer.

Figure 1 .
Figure 1.An obtained colloidal crystal containing three layers of regular arrays of polystyrene particles.Individual particles are clearly resolved at the particle level.The focal plane in this picture is adjusted at the third layer.

Figure 2 .
Figure 2. Re-dispersion of particles from grown crystals.(a) 1 day after sample preparation; (b) 6 days after sample preparation.In a day, crystals grow up to several layers as shown in (a) at 25 °C.However, 3 days after sample preparation, the crystals start to dissolve, and 6 days after, the crystals have disappeared completely.

Figure 2 .
Figure 2. Re-dispersion of particles from grown crystals.(a) 1 day after sample preparation; (b) 6 days after sample preparation.In a day, crystals grow up to several layers as shown in (a) at 25 ˝C.However, 3 days after sample preparation, the crystals start to dissolve, and 6 days after, the crystals have disappeared completely.

Figure 3 .
Figure 3. Schematic of the classification of particle incorporation processes into a kink site.Hexagons indicate crystals.Gray and indigo parts show the first (lower) and second (upper) layers, respectively.The borders between these layers are mono-particulate steps.Kinks on the steps indicate kink sites at which particles are finally incorporated into the crystal.Yellow circles represent the particles that are just incorporated into the crystal at the kink sites on the steps.(a) kink incorporation processes via surface diffusion on lower layers; (b) those via surface diffusion on upper layers; (c) those via surface diffusion on lower layers and adsorption to the particle at the edge of the kink site; (d) direct incorporation from the solution.Dashed and solid circles at both ends of arrows indicate particles at the previous and present positions, respectively.

Figure 3 .
Figure 3. Schematic of the classification of particle incorporation processes into a kink site.Hexagons indicate crystals.Gray and indigo parts show the first (lower) and second (upper) layers, respectively.The borders between these layers are mono-particulate steps.Kinks on the steps indicate kink sites at which particles are finally incorporated into the crystal.Yellow circles represent the particles that are just incorporated into the crystal at the kink sites on the steps.(a) kink incorporation processes via surface diffusion on lower layers; (b) those via surface diffusion on upper layers; (c) those via surface diffusion on lower layers and adsorption to the particle at the edge of the kink site; (d) direct incorporation from the solution.Dashed and solid circles at both ends of arrows indicate particles at the previous and present positions, respectively.

Figure 4 .
Figure 4. Number distribution of incorporation process routes.(a) to (d) indicate the same classifications described in Figure 3.At the top of bars, schematics used in Figure 3 are placed.

Figure 4 .
Figure 4. Number distribution of incorporation process routes.(a) to (d) indicate the same classifications described in Figure 3.At the top of bars, schematics used in Figure 3 are placed.

Crystals 2016, 6 , 80 6 of 11 Figure 5 .
Figure 5. Schematic of adsorption conditions.A, E, and S particles are those that form respectively three, four, and five bonds with particles in a crystal.Dotted and solid circles represent the lower and upper layers, respectively.

Figure 6 .
Figure 6.Formation processes of stacking disorders.(a) Two-dimensional nucleation on the first triangular layer (90 min.after the injection of a particle dispersion into a growth cell); (b) Growth of the nucleated layers (120 min); (c) Complete occupation of the second layer and formation of the boundary between two different stacks (200 min); (d) Displacement of the boundary accompanied by the growth of the third layers (280 min).Scale bar represents 10 µm.

Figure 5 .
Figure 5. Schematic of adsorption conditions.A, E, and S particles are those that form respectively three, four, and five bonds with particles in a crystal.Dotted and solid circles represent the lower and upper layers, respectively.

Figure 6 .
Figure 6.Formation processes of stacking disorders.(a) Two-dimensional nucleation on the first triangular layer (90 min.after the injection of a particle dispersion into a growth cell); (b) Growth of the nucleated layers (120 min); (c) Complete occupation of the second layer and formation of the boundary between two different stacks (200 min); (d) Displacement of the boundary accompanied by the growth of the third layers (280 min).Scale bar represents 10 µm.

Figure 6 . 11 2. 6 .
Figure 6.Formation processes of stacking disorders.(a) Two-dimensional nucleation on the first triangular layer (90 min.after the injection of a particle dispersion into a growth cell); (b) Growth of the nucleated layers (120 min); (c) Complete occupation of the second layer and formation of the boundary between two different stacks (200 min); (d) Displacement of the boundary accompanied by the growth of the third layers (280 min).Scale bar represents 10 µm.

Figure 7 .
Figure 7.The time course of a "peeling off" process caused by the additional adsorption of a particle from bulk solution onto the other particle on a step.(a) A particle approaches a top layer; (b) The particle adsorbs to the edge particle of the layer; (c) The edge particle is desorbing from the layer with the adsorbed particle; (d) The particle pair is completely peeled off the top layer of particles.

Figure 7 .
Figure 7.The time course of a "peeling off" process caused by the additional adsorption of a particle from bulk solution onto the other particle on a step.(a) A particle approaches a top layer; (b) The particle adsorbs to the edge particle of the layer; (c) The edge particle is desorbing from the layer with the adsorbed particle; (d) The particle pair is completely peeled off the top layer of particles.

Table 1 .
x s and τ of 10 particles on growth interfaces.