Heterogeneous Nucleation of Colloidal Crystals on a Glass Substrate

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Heterogeneous nucleation of colloidal crystals on a glass substrate with depletion attraction Suxia Guo, Jun Nozawa, Sumeng Hu, Haruhiko Koizumi, Junpei Okada, and Satoshi Uda Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01906 • Publication Date (Web): 15 Sep 2017 Downloaded from http://pubs.acs.org on September 17, 2017

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Heterogeneous nucleation of colloidal crystals on a glass substrate with depletion attraction Suxia Guo, Jun Nozawa*, Sumeng Hu, Haruhiko Koizumi, Junpei Okada, Satoshi Uda Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan *Corresponding author Tel.: +81 22 215 2103; Fax: +81 22 215 2101 E-mail: nozawa@imr. tohoku.ac.jp (J. Nozawa)

ABSTRACT The heterogeneous nucleation of colloidal crystals with attractive interactions has been investigated via in situ observations. We have found two types of nucleation processes: a cluster that overcomes the critical size for nucleation with a mono-layer, and a method that occurs with two layers. The Gibbs free energy changes (∆G) for these two types of nucleation processes are evaluated by taking into account the effect of various interfacial energies. In contrast to homogeneous nucleation, the change in interfacial free energy, ∆σ, is generated for colloidal nucleation on a foreign substrate such as a cover glass in the present study. The ∆σ and step free energy of the 1st layer, γ1, are obtained experimentally based on the equation deduced from classical nucleation theory (CNT). It is concluded that the ∆G of q-2D nuclei is smaller than of mono-layer nuclei, provided that the same number of

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particles are used, which well explains the experimental result that the critical size in q-2D nuclei is smaller than that in mono-layer nuclei.

INTRODUCTION Two-dimensional (2D) colloidal crystals have attracted much interest owing to their important

potential

applications,

such

as

in

photonic

materials,1

sensors,2

masks

for

photolithography,3–6 and templates for the epitaxial growth of colloidal crystals.7 In addition to these applications, another important feature of colloidal crystals is that they can be used as a model system of phase transitions in atomic systems.8 Colloids show similar phase transitions as atomic and molecular systems, where typical length and time scales are easily accessible for optical microscopy. The colloidal model system has also been applied to investigate glass formation,9 nucleation,10 and epitaxial growth.11 Various methods have been employed to fabricate 2D colloidal crystals, for instance, solvent evaporation,12 spin coating,13 electric field-induced flow,14 and depletion attraction.15 2D colloidal crystals are generally grown on a foreign substrate such as glass in these studies. Though it is well known in the atomic system that nucleation and the growth process in heterogeneous crystallization are strongly affected by the substrate,16 to the best of our knowledge, no quantitative evaluation has been carried out for these effects in colloidal crystallization. An important parameter to investigate the substrate effect is the change in interfacial free energy associated with the formation of the 2D layer,


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∆σ, which includes the interfacial energies between substrate and solid, σsub-solid, the solid and liquid interface, σsolid-liquid, and the substrate and liquid interface, σsub-liquid. The nucleation rate and the growth mode are strongly dependent on ∆σ,17,18 especially in epitaxial growth. The shape and size of the nuclei are determined by the balance of the three above mentioned interfacial free energies. There have been studies on 2D nucleation processes of the mono-layer of colloidal crystals that is formed on a glass substrate. For instance, mono-layer 2D nucleation is driven by applying electric fields, in which the nucleation behavior is well explained by classical nucleation theory (CNT).14 Savage et al. reported that 2D nucleation on a cover glass can proceed through multiple distinct steps (non-CNT model) in the depletion attraction system.19 In these studies, however, ∆σ was not considered. The detailed 2D nucleation process on the terrace of colloidal crystals of an attractive system has recently been reported by Nozawa et al.,20 who also did not deal with ∆σ because 2D islands nucleated on the same material. In a hard-sphere system, interfacial free energy between solid and liquid, σsolid-liquid, for 3D crystals was obtained by computational methods,21 but the effect of ∆σ on the heterogeneous nucleation has not yet been measured in 2D colloidal crystals. In the present study, we have conducted in situ observations of heterogeneous nucleation of 2D colloidal crystals on a cover glass substrate. The number of particles in the critical size of nuclei, N*, at various area fractions, φarea, as the ratio of the total area of ad-particles to the area of substrate, which represents the supersaturation degree, was measured to evaluate ∆G for the nucleation process. The calculation of ∆G was conducted by taking ∆σ into account. Our study reveals the detailed


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heterogeneous nucleation mechanism on a glass substrate for colloidal crystals in an attractive system.

EXPERIMENTAL

Figure 1 (a) Illustration of the growth-cell for the nucleation of colloidal crystals. (b) Schematic of colloidal nucleation on glass. The number of ad-particles on the substrate (red line particles) is denoted by N1, and that on the 1st layer (blue line particles) is denoted by N2. A1 is the area chosen on the substrate (enclosed by purple dashed line), and A2 is the area of the 1st layer (enclosed by blue 1 2 dashed line). φarea is the area fraction of ad-particles on the substrate. φarea is the area fraction of

ad-particles on the 1st layer. Ω is the area of a particle. (c) Attractive potential between the particles. d is the center-to-center distance of the particles, a is the diameter of the particles. The interaction potential of van der Waals (VDW), UVDW, and depletion attraction, UAO, are show in blue and orange line, respectively. The sum of them is shown in black line.

The experiments were carried out with green fluorescent monodispersed polystyrene spheres 4 ACS Paragon Plus Environment

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(PS) (Thermo Fisher Scientific) with a particle diameter of 500 nm. The zeta potential was measured as −47.1 ± 5.9 mV. A charged sodium polyacrylate polymer (polymerization degree of 30000–40000) was mixed into the PS solution as the depletant to generate the attractive interaction. Next, 0.15 g/L of sodium polyacrylate was added. The volume fractions (Φ) of initial colloidal suspensions (Φ = 0.05, 0.1, 0.2, 0.25, 0.3, 0.4, 0.5, and 1%) were applied. A cover glass with a thickness of 0.12–0.17 mm was used as the substrate, then the mixed suspension was placed into a hole at the center of a silicone sheet (2 mm thickness) as shown in Figure 1. The principle of colloidal crystallization is based upon depletion attraction,22 in which interparticle interactions can be tuned by the concentration of a polymer. The attractive potential of particles is calculated in terms of depletion potentials and van der Waals (VDW) as follows. The strength of the depletion attractive potential, UAO(d), was derived by Asakura-Oosawa 23 and Vrij 24 as:

U AO ( d ) = − nb k BTV OV ( d ) =0

(a ≤ d ≤ 2Rd )

(1)

(d > 2Rd )

(2)

where nb is the polymer number density, Rd is the depletion radius which is obtained as the sum of the particle radius (a/2) and the radius of gyration of the polymer in water (Rg), VOV(d) is the overlap volume of spheres that have radii of Rd, d is the center-to-center distance of the particles, kB is the Boltzmann constant and T is the absolute temperature. VOV(d) is derived as:

VOV (d ) =

4π 3  3 d 1 d Rd 1 − +  3  4 Rd 16  Rd

  

3

  

(3)

Rg was determined by dynamic light scattering (DLS) to be 100 nm. The value of nb is obtained22 5 ACS Paragon Plus Environment

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nb ϕ = nb∗ ϕ ∗

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(4)

where nb* is the polymer number density at which the polymer coils overlap, ϕ is weight fraction of the polymer, and ϕ* is weight fraction of the polymer where the chains start to overlap. The concentration of polymer by weight, ϕ, is calculated to be 0.14 × 10 −3 wt%. The value of ϕ* is calculated by ϕ ∗ = 103 M W n∗b N A , where Mw is weight average molecular weight, which is obtained from the degree of polymerization (30000-40000) to be M w = 94 × 3.5 × 104 = 3.3 × 106 , in which 35000 was used as an average value of the degree of polymerization. Since n b∗ = nb =

ϕ ϕ∗

× nb∗ =

3 , equation 4 is transformed into 4πRg3 ϕN A 10 3 M W

.

The nb is calculated to be 0.26 × 1011 /g. The values of each parameter are substituted into equation 1 to obtain the potential of depletion attraction as a function of center-to-center distance, which is shown as blue line in Figure 1 (c). The VDW potential between two particles is given by U VDW (d ) = −

AH 6

2  a2  1 1  d 2 − (a )     + + ln  2  d 2 − (a )2 d 2  d 2    

(5)

where AH is the Hamaker constant. 25 UVDW and UAO+ UVDW are shown in orange and black line in Figure 1 (c), respectively. Crystallization occurs on the cover glass. In the polymer added solution, there are two depletion forces at work, one between particles and substrate and one between the particles themselves. Though the 500 nm particles exhibit strong Brownian motion, the depletion attraction between the particles and the substrate causes the particles to stay on the substrate. An oil


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immersion lens (magnification = 100 and N.A. = 1.3) was used with single-particle resolution of 500 nm particles. In this study, area fraction, φarea , is employed to represent the concentration of colloidal particles, which is defined as the area occupied by ad-particles divided by the corresponding area. We focus on the mono-layer and two-layer colloidal crystallizations. Therefore, two kinds of φarea are defined (Figure 1). The area fraction of ad-particles on the substrate,

1 φarea , defined as the area

occupied by ad-particles in the field of view (number of ad-particles on the substrate, N1, multiplied by the area of a particle, Ω), is divided by the corresponding area of a substrate (A1). The area fraction of ad-particles on the 1st layer,

2 φarea , is determined as the area of ad-particles on the 1st layer (number of

ad-particles on the1st layer, N2, multiplied by the area of a particle, Ω) divided by the area of the 1st layer as a terrace (A2). The number of particles in the critical nuclei on the substrate, N*, is investigated at various

1 φarea .

RESULTS AND DISCUSSION After mixing the colloidal suspension and the polymer, 2D nucleation takes place on the glass substrate at the bottom of the cell. First, with particles sinking due to gravity, the particle concentration in the vicinity of the glass substrate increases. As colloidal particles collide with each other, they form clusters. Some clusters shrink and disappear, while others continue to grow in the lateral direction to become nuclei, forming the 1st layer. The secondary layer occurs on the surface of


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the grown clusters. In the experimental time period, about 5 layers develop by the repetition of similar processes. The concentration of colloidal particles in the solution decreases over time because the particles are consumed for nucleation and growth. As the particle concentration decreases, only growth takes place, because a higher concentration is required for nucleation than for growth. When the particle concentration decreases to a certain value, growth then ceases. In this nucleation process, we discovered two strategies of nucleation: one occurs with a mono-layer and the other with two layers. In the former case, mono-layer nucleation, the cluster overcomes the critical size with a mono-layer, which is equivalent to 2D nucleation. In the latter case, quasi-two-dimensional nucleation (q-2D nucleation), the cluster overcomes the critical size by forming a two-layer structure. We call the two-layer structure a quasi-two-dimensional (q-2D) nucleation. Though the nucleus has two layers, it is not categorized as a three-dimensional (3D) nucleus because it does not have a 3D shape. To have a 3D shape, the ratio of the number of particles for each layer should be constant for any nucleus size; however, it is not constant for the q-2D nuclei observed in the present study. We classified this structure as q-2D. The q-2D structure is only observed for nucleation on a glass substrate, not for 2D nucleation on the terrace substrate of colloidal crystals. Therefore, this is characteristic of the nucleation process on a glass substrate, and further detailed observations were then carried out. Figure 2 shows the mono-layer nucleation process on the glass substrate by in situ observations. The images were taken immediately after solution preparation. The experiments were


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carried out at room temperature (25 ºC). The polymer concentration was 0.15 g/L. Some clusters with a mono-layer, enclosed in dashed white circles, shrink and eventually dissolve, whereas some others, enclosed in dashed red, keep growing. This suggests that a critical size for the nucleation process exists. In the nucleation process, a cluster overcomes the critical size by forming a mono-layer. We determined the critical size from the largest clusters that are not stable and still tend to dissolve. The number of particles in the critical nucleus, N*, in Fig. 2 is counted to be 38, where

1 φarea is around

11%.

Figure 2 Snapshots of the mono-layer nucleation process on the glass substrate. The clusters in dashed white circles disappear as shown in (a) and (b), while clusters in dashed red circles continue to grow as shown in (b), (c), and (d).


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We also observed q-2D nuclei formation where the 2nd layer forms on the 1st layer as shown in Figure 3. The images were taken several minutes after the solution was added into the growth cell. Two mono-layer embryos that appeared on the glass substrate are enclosed by a dashed white circle (I) and a dashed yellow circle (II) in Figure 3 (a). Several seconds later, a 2nd layer appears on the 1st layer in the enclosed dashed yellow embryo as shown in Figure 3 (b), where the inset shows a schematic of the q-2D structure. The embryos continues growing as seen in the dashed yellow circle (c–f). This nucleation process does not correspond to so-called two-step nucleation. Even though formation of the 2nd layer followed formation of the 1st layer, there is no intermediate phase that transforms into the final phase. Both 1st and 2nd layers constitute the building unit of the two-layer structure. On the other hand, the mono-layer structure shrinks and then disappears as seen in the dashed white circle (b–c). Though the number of particles in the q-2D structure is smaller than that of the mono-layer structure in Figure 3 (a), the former nucleates whereas the latter disappears. It should be noted that when subcritical embryos of the 2nd layer are growing at the edge, the nucleation process is probably affected by the boundary of the 1st layer because growth of the 2nd layer is inhibited by the boundary. The nucleation rate may be reduced due to this effect. However, we do not have enough data to quantitatively evaluate the effect on the nucleation rate. This will be investigated in future studies. The critical number of particles in the q-2D nuclei is obtained by measuring the minimum size of clusters that tended to grow. This is in contrast to the method for determining the number of


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particles in the critical mono-layer, in that the latter is obtained as the maximum size that tended to dissolve. This is due to the difficulties of observation. For q-2D islands, the frequency of disappearing embryos is significantly lower than the growing embryos. Thus, we rarely observe the disappearance of q-2D islands. In principle, the critical size is the largest size that will dissolve or the smallest size that will grow. The latter case is applied to the q-2D nucleation. The N* in the mono-layer nuclei and q-2D nuclei are counted to be 58 and 32, respectively, at

1 φarea ≈ 9% in Figure 3. The q-2D embryo

has a smaller N* than the mono-layer embryo.

Figure 3 q-2D nucleation. The mono-layer structure enclosed in dashed white disappears, while the q-2D structure in dashed yellow (a–f) keeps growing. Inset shows a schematic of the q-2D structure.

The number of particles for mono-layer and q-2D nuclei at various area fractions on the substrate,

1 φarea , is shown in Figure 4. For mono-layer nucleation, as discussed above, the number of


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particles in the nuclei is plotted as blue circles. The blue line based on these points corresponds to the N* at various

1 φarea . For q-2D nucleation, the number of particles in the nuclei is shown as green

diamonds. The smallest value among these points is the N* at each

1 φarea , which is shown as a green

line. It is clear that N* for the q-2D nuclei is less than that for mono-layer nuclei at a given

1 φarea . We

next investigated why N* for q-2D nuclei is smaller than that for mono-layer nuclei in terms of the driving force required for the nucleation process in these two cases.

Figure 4 Number of particles in nucleus as a function of

1 φarea for mono-layer nuclei (blue circles)

and for q-2D nuclei (green diamonds) on the substrate. The solid blue and green lines correspond to


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the number of particles in the critical size, N*, for mono-layer and q-2D nuclei, respectively.

Thus, the Gibbs free energy change, ∆G, for the two types of nucleation processes can be evaluated. According to the classical nucleation theory (CNT), ∆G for forming a 3D spherical nucleus of radius r is:26

∆G(r ) = −

4πr 3 ∆µ + 4πr 2 ⋅ σ 3v0

(6)

where v0 is atomic volume, σ is the interfacial energy, and ∆µ is the chemical potential difference between the bulk liquid and solid. There are two terms in Eq. 6: the decrease in volume free energy and the increase in surface free energy with the increase in r. The boundaries of all the islands are rough and the facet boundary is not observed. The rough boundary was reported for subcritical and supercritical nuclei,27-29 in which fractal dimension was calculated to obtain line tension. We have applied CNT to the rough islands assuming that islands have circular shapes with smooth boundaries. When mono-layer nuclei form on a foreign substrate, ∆G for the 2D heterogeneous nucleation, ∆G1, in terms of number of particles, n, is expressed as ∆G1 (n ) = − n∆µ +

2 3

π

n Ω ⋅ ∆σ +

2 3

π

πa n ⋅ γ 1

(7)

2 where a is the diameter of a colloid particle, Ω is the area per particle and is equal to πa 4 , ∆σ is the

change in interfacial free energy, and γ1 is the step free energy (line tension) of the mono-layer nuclei. Since there are voids between particles, the constant that is derived from the area fraction of a close packed circle,

π

, should be divided by the number of particles to express the area that is occupied

2 3


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by close packed particles and the square root of the value should be multiplied by the length of the area. Thus,

2 3

π

and

2 3

π

are multiplied by the number of particles in the second and third terms

of Eq.7, respectively. Here, ∆σ can be expressed as ∆σ = σ sub-solid + σsolid-liquid − σsub-liquid

(8)

where σsub-solid, σsolid-liquid, and σsub-liquid are the interfacial energies between the solid/substrate, solid/liquid, and substrate/liquid interfaces, respectively.

Figure 5 Schematic illustration of a q-2D embryo, showing parameters that are required to express the nucleation process. n1 and n2 are the numbers of particles of the 1st layer and the 2nd layer, respectively.

γ1 and γ2 are the step free energies of the 1st layer and 2nd layer, respectively. σsub-solid, σsolid-liquid, and σsub-liquid are the interfacial energies of the solid/substrate, solid/liquid, and substrate/liquid interfaces, respectively.

When q-2D nuclei form on a foreign substrate, parameters for ∆G2 are as shown in the schematic illustration of Figure 5. The equation for ∆G2 is written as ∆G 2 (n1 + n2 ) = −(n1 ∆µ1 + n2 ∆µ 2 ) +

2 3

π

n1Ω ⋅ ∆σ +

2 3

π

(

πa n1 γ 1 + n2 γ 2


)

(9)

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where n1 and n2 are the numbers of particles of the 1st layer and the 2nd layer, respectively. ∆µ1 and ∆µ2 are the chemical potential differences between the solution and the 1st layer, and the solution and the 2nd layer, respectively, at a certain supersaturation. γ1 and γ2 are the step free energies (line tensions) of the 1st layer and 2nd layer, respectively. For a given interparticle interaction, ∆µ can be expressed in terms of the supersaturation δ:30

∆µ = kBT ln (1 + δ )

(10)

where kB is Boltzmann’s constant and T is the absolute temperature. δ is related to the actual concentration (C) and equilibrium concentration (Ceq). Assuming C and Ceq correspond to φarea and equilibrium area fraction, φeq, δ is expressed by

δ=

φarea − φeq φeq

(11)

Substituting Eqs. 10 and 11 into Eq. 9, ∆G2 is expressed as

(

(

)

(

))

1 2 ∆G 2 (n1 + n 2 ) = − n 1 k BT ln φ area φ eq1 + n 2 k BT ln φ area φ eq2 +

2 3

π

n1Ω ⋅ ∆σ +

2 3

π

(

)

πa n1 γ 1 + n 2 γ 2 (12)

where φeq1 and φeq2 are the equilibrium area fractions of the colloidal particles on the 1st layer and 2nd layer, respectively. Among the parameters in Eq. 12, φeq2 and γ2 were obtained from Nozawa et al..20 To calculate ∆G2, the following parameters should be obtained: φeq1 , ∆σ, and γ1. 1 φeq1 is measured experimentally as φarea at which the growth rate of steps of the 2D island is

zero assuming that ∆µ is used only for growth kinetics. As nuclei grow, particles in the solution are consumed, leading to a decrease in supersaturation. Therefore, step velocity becomes small as time


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1 proceeds. The step velocity is close to 0 when φarea becomes close to φeq1 . The particles begin to

1 1 dissolve when φarea is smaller than φeq1 . The relationship between step velocity and φarea is obtained as

shown in Figure 6. The results show that φeq1 is determined to be about 0.21%.

1 . Figure 6 Step velocity versus φarea

∆σ and γ1 are also measured experimentally based on the CNT of mono-layer nucleation, in which these parameters determine the critical number of particles, N*. N* is obtained by differentiating Eq. 7 and setting it equal to zero as d∆G1 (n ) 1 =0 = − ∆µ + 1.1Ω∆σ + 1.05πaγ 1 dn 2 n

at n = N*

(13)

Then,

1 N∗

=

2k BT 1.1a∆σ 1 φeq1 − ln φarea 1.05πaγ 1 2 ×1.05γ 1

(

)

(14)


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2 by replacing Ω with πa 4 .

(

)

1 Figure 7 Reciprocal of the square root of the critical size, 1/√ ∗ , as a function of ln φarea φeq1 .

1 The number of particles for the critical mono-layer nuclei is obtained at various φarea (Fig. 4).

(

1 φeq1 The reciprocal of the square root of the critical size, 1/√ ∗ , as a function of ln φarea

)

is shown in

Figure 7. The fitting line is drawn without taking into account several unreliable data points that deviated significantly from the trend of the other data. The slope and intersection of the fitting line are 0.2 and -0.6, respectively, from which ∆σ and γ1 are derived as 3.5 kBT/a2 and 3.0 kBT/a, respectively. In a hard sphere system, the value of ∆σ on the hard-wall is obtained by computer simulation as almost zero.31 Since there are interactions between particles and substrate, ∆σ obtained from the experiments shows a non-zero value. The step free energy of the 1st layer, γ1, is five times larger than 17 ACS Paragon Plus Environment

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that of the hard-sphere system31 and almost the same as that of the 2nd layer.20 This increment is thought to be the sum of interactions between particles as well as between particles and substrate. In general, although ∆σ is an important parameter for nucleation such as epitaxial growth, it is very difficult to measure the critical nucleus in an atomic system. In contrast, observations at single-particle resolution of colloidal crystals are easy, which enables us to measure the critical size.

Figure 8 Gibbs free energy, ∆G, for mono-layer nucleation, ∆G1, and that for q-2D nucleation, ∆G2, at 1 various φarea . The solid blue and green lines correspond to the critical Gibbs free energy change, ∆G*,

for mono-layer and q-2D nucleation, respectively.

All the parameters, φeq1 , φeq2 , γ1, γ2, and ∆σ, are substituted into Eq. 7 and Eq. 12 for each point in Figure 4. The ∆G of mono-layer nuclei, ∆G1 , and two-layer nuclei, ∆G2 , at various area fractions are calculated as shown in blue circles and green triangles, respectively, in Figure 8. The blue 18 ACS Paragon Plus Environment

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and green lines based on these points are the minimum ∆G, which corresponds to the ∆G* at various 1 φarea .

These calculations suggest that ∆G* for q-2D nucleation is smaller than that for mono-layer 1 nuclei at the same φarea . This explains our experimental results shown in Figure 4 in which nucleation

occurs with a smaller number of particles for q-2D nuclei than for mono-layer ones. Although q-2D nucleation has a smaller ∆G, mono-layer nucleation is predominant when 1 1 φarea is larger than 12%. There is no q-2D nucleation observed in the φarea range over 12% in Figures 4

and 8. This is explained by the induction time of the second layer formation. For q-2D nucleation, a 1 , mono-layer nucleation certain time is required to form the 2nd layer. On the other hand, at high φarea

1 occurs very quickly and consumes particles for growth, which decreases φarea . Therefore, even though

q-2D nucleation has a smaller ∆G, it is prevented kinetically. In Figure 8, the parameter to express the quantity of the particles in the 2nd layer relative to those of the 1st layer is not included. Figure 9 shows the effect of the secondary layer on ∆G, in which the axis of n2/n1 is added, and the color contrast represents the value of ∆G. When n2/n1 equals zero, i.e., a mono-layer nucleus, blue circles in Fig. 9 correspond to those in Fig. 8. When n2/n1 is larger than zero, ∆ G 2 decreases with increasing n2. The appearance of the 2nd layer for q-2D nuclei reduces the interface between crystal and substrate, decreasing ∆G for the nucleation of particles. In other words, since formation of colloidal layers on the glass substrate is not as favorable as formation on the surface of colloidal crystals as a secondary layer (measured by ∆σ), nucleation with two layers is the


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lower path to overcome the energetic barrier for nucleation at a given number of particles.

1 Figure 9 Plots of ∆G versus φarea and n2/n1. Blue circles correspond to mono-layer nuclei, and green

diamonds represent q-2D nuclei. The colored surface shows ∆G of nuclei formation as a function of 1 φarea and n2/n1.

Summary Heterogeneous nucleation of colloidal crystals on a glass substrate has been investigated by in situ observations. Two types of nucleation processes were observed: mono-layer and q-2D nucleation. Interfacial free energy change, ∆σ, and step free energy of the 1st layer, γ1, were obtained from measurements of the critical size of nuclei and analysis based on the classical nucleation theory


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(CNT) equation. The ∆G calculations suggest that ∆G of q-2D nucleation is smaller than that for 1 , which explains the experimental results that N* for q-2D mono-layer nucleation at a given φarea

nuclei is smaller than that for mono-layer nuclei. We have succeeded in quantitatively examining the substrate effect as ∆σ via the nucleation process. Since γ and ∆σ are important parameters for controlling the size and shape of colloidal nuclei, our findings will contribute to wide fields of applications of colloidal crystals such as colloidal epitaxy and lithography.

ACKNOWLEDGEMENTS This work was supported in part by JSPS KAKENHI Grant Number 26870047 and the China Scholarship Council (CSC).

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Heterogeneous Nucleation of Colloidal Crystals on a Glass Substrate

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