Evolution of Interparticle Capillary Forces during Drying of Colloidal

3692

Langmuir 2006, 22, 3692-3697

Evolution of Interparticle Capillary Forces during Drying of Colloidal Crystals Zuocheng Zhou,† Qin Li,*,† and X. S. Zhao‡ Department of Chemical Engineering, Curtin UniVersity of Technology, GPO Box U1987, Perth, WA 6845 Australia, and Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, Singapore ReceiVed NoVember 1, 2005. In Final Form: February 5, 2006 Photonic crystals are periodic structures that have the capability to manipulate the photons in the same way as semiconductors do for electrons. The self-assembly strategy that utilizes colloidal crystals as a template to form photonic crystals has received a great deal of recent research interest because it is simple and cost-effective. Experimental studies and theoretical analysis have speculated that capillary forces play a pivotal role in forming the colloidal crystals during the crystal growth process and that particularly during the drying stage the changing of the magnitude of capillary forces is critical to the resultant microstructure. This paper presents a computational analysis of the changing capillary forces, which may throw light on a refined strategy for controlling colloidal crystal growth.

Introduction It is widely accepted that self-assembly of monodisperse colloidal particles holds the most promising route for the costeffective production of three-dimensional (3D) photonic crystals due to the thermodynamic nature of this process.1,2 Over the past few years, many self-assembling techniques have been developed, such as gravitational sedimentation,3 confined cell method,4 vertical deposition (VD),5,6 and float packing method.7 Among them, the VD method attracts significant interest because the thickness and morphology of the colloidal crystals can be easily controlled.5 However, until now the fabrication of high quality colloidal crystals, in particular large-scale colloidal crystals, is an ever-presenting challenge, evidenced by the abundant grain boundaries, stack faults, dislocations, and cracks. Irrespective of the self-assembly techniques developed, a suspension of microspheres is the prerequisite to grow colloidal crystals. Therefore, the removal of the solvent is a necessary step to obtain solid colloidal crystals, and the quality of the colloidal crystal templates is of great importance to the resultant 3D photonic crystals. According to the different colloidal crystal growth processes, the self-assembly techniques available so far can be classified into two categories. The first one refers to the methods in which the ordered structures are formed in the bulk solvent and then separated from the system, such as the sedimentation3 and the confined cell methods.4 The second category covers all the evaporation-induced self-assembly methods, which have a common feature that the solvent removal and crystal growth are taking place simultaneously, such as the vertical deposition (VD) method.5 With regard to the second category, the solvent removal process is extremely important to the growth and the quality of the crystal * Corresponding author. E-mail: [email protected]. Tel: 61-08-9266 7704. Fax: 61-08-9266 2681. † Curtin University of Technology. ‡ National University of Singapore. (1) Stein, A. AdV. Mater. 2003, 15, 763. (2) Lo´pez, C. AdV. Mater. 2003, 15, 1679. (3) Mı´guez, H.; Meseguer, F.; Lo´pez, C.; Mifsud, A.; Moya, J. S.; Va´zquez, L. Langmuir 1997, 13, 6009. (4) Park, S. H.; Xia, Y. Chem. Mater. 1998, 10, 1745. (5) Jiang, P.; Bertone, J. F.; Hwang, K. S.; Colvin, V. L. Chem. Mater. 1999, 11, 2132. (6) Zhou, Z.; Zhao, X. S. Langmuir 2004, 20, 1524. (7) Im, S. H.; Lim, Y. T.; Suh, D. J.; Park, O O. AdV. Mater. 2002, 14, 1367.

because capillary forces are the main driver for self-assembly of colloidal spheres. It is also important to recognize that solvent removal in the VD method involves two stages. Stage 1 occurs during the formation of the colloidal films when a substrate is partially immersed in the bulk suspension of the colloidal system. Stage 2 takes place when the bulk residue liquid is removed and the colloidal film is dried in a chamber by heating. Mathematical models have been established to describe the evaporation-induced self-assembly processes.8-10 With these models, it has been concluded that the formation of cracks is mainly due to the capillary stresses at the drying fronts in stage 1. However, recently, it has been experimentally observed that stage 2 can also produce a significant amount of cracks.11,12 Zeng et al.11 observed that slow drying in stage 2 can prevent the formation of cracks. Kuai et al.12 showed that the colloidal crystals had a perfect crystal structure after stage 1 but became severely defective upon drying stage 2. These experimental results indicated that the capillary stresses caused by the drying fronts in stage 1 are not the only reason for crack formation. Considering the fact that there is still a small amount of solvent liquid between the colloidal spheres after vertical deposition,13 the drying stage 2 most likely also contribute to the formation of cracks. Drying stage 2 is a process used in both categories of selfassembly methods. By distinguishing the two drying stages in evaporation-induced self-assembly process, the problem of defect formation becomes clearer to understand. During the formation of the colloidal crystals, the colloidal spheres are transferred from the bulk solvent and self-organized into the ordered structures. The formation of amorphous structure, stack faults, and dislocations often occur in this step. When a colloidal crystal undergoes drying stage 2, the ordered structure has already formed despite the presence of interstitial liquid as shown in Figure 1a, (8) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088. (9) Lee, W. P.; Routh, A. F. Langmuir 2004, 20, 9885-9888. (10) Dufresne, E. R.; Corwin, E. I.; Greenblatt, N. A.; Ashmore, J.; Wang, D. Y.; Dinsmore, A. D.; Cheng, J. X.; Xie, X. S.; Hutchinson, J. W.; Weitz, D. A. Phys. ReV. Lett. 2003, 91, 224501. (11) Zeng, F.; Sun, Z.; Wang, C.; Ren, B.; Liu, X.; Tong Z. Langmuir 2002, 18, 9116. (12) Kuai, S.-L.; Hu, X.-F.; Hache´, A.; Truong, V.-V. J. Cryst. Growth 2004, 267, 317. (13) Gates, B.; Park, S. H.; Xia, Y. AdV. Mater. 2000, 12, 653.

10.1021/la052934c CCC: $33.50 © 2006 American Chemical Society Published on Web 03/14/2006

Drying of Colloidal Crystals

Langmuir, Vol. 22, No. 8, 2006 3693

Figure 1. (a) Schematic illustration of a monolayer 1D structure in drying stage 2; (b) schematic illustration of the drying process; (c) cracked colloidal structures due to heterogeneous initial conditions; and (d) a two-sphere system connected by a symmetric liquid bridge.

only cracking defects may be produced in the further drying process due to the capillary stress. In this work, we attempted to establish a dynamic model to describe the evolution of interparticle capillary forces during the drying stage 2. It is aimed to provide a detailed insight in the interparticle interactions during colloidal crystal drying, as well as a quantitative guidance to experimental data, thus assisting the design of a refined strategy for effective controls over colloidal crystal growth.

Computing Liquid Bridge Force To reduce the complexity of the problem, a monolayer 1D structure as sketched in Figure 1a represents the drying stage 2 in colloidal crystal formation. It is considered that the concave capillary bridge plays a significant role in pulling the particles together. Moreover, as the evaporation proceeds, the magnitudes of the capillary forces between particle pairs may vary significantly because of the change in the mean curvatures of the interstitial liquid surfaces. Minor heterogeneities in the initial configuration, which may be further pronounced with the dynamic changes in force magnitudes between particles during drying, could result in cracked structures as illustrated in Figure 1c. To obtain a quantitative description of the force evolution, we further simplify the model into a two-body system as shown in Figure 1d. Computing the Static Liquid Bridge Force. The force, F, acting between two equal radii spherical bodies, as shown in Figure 1d, is the liquid bridge force, which is a summation of the axial component of the surface tension exerted at the threephase contact line, and the hydrostatic pressure acting on the axially projected area of the liquid contact on either sphere.14 The followings are the major assumptions used in formulating the model: The effect of the gravity upon the liquid bridge is negligible. The liquid bridge between two rigid spheres is relative small and any effect of the buoyancy force is negligible. The liquid bridge force is given by:15

F ) 2πγa sin φ sin(φ + θ) + πa2∆p sin2 φ

(1)

where γ is the liquid surface tension, a is the radius of the spheres, φ is the half-filling angle, and θ is the contact angle. ∆p, the (14) Lian, G.; Thornton, C.; Adams, M. J. J. Colloid Interface Sci. 1993, 161, 138. (15) Princen, H. M. J. Colloid Interface Sci. 1968, 26, 249.

pressure difference across the curved air-liquid interface, can be described by the Laplace-Young equation

∆p ) γ

(

1 1 + r1 r2

)

(2)

Since one of the principal radii is a function of the radius of the meniscus’ cross section, which changes along the x axis, the liquid bridge force can only be calculated through analytical approximations or numerical methods. The numerical approach described by Lian et al.14 was adopted in this study. The liquid bridge force is calculated by

F* )

F ) 2π sin φ sin(φ + θ) + 2H* π sin2 φ γa

(3)

where F* is the dimensionless liquid bridge force, H* is the dimensionless mean curvature, which can be expressed as

2H* )

Y ¨ 1 ∆pa ) γ (1 + Y˙ 2)3/2 Y(1 + Y˙ 2)1/2

(4)

All of the variables used in the computation are converted into dimensionless form, the coordinates X, Y are defined as X ) x/a and Y ) y/a, respectively. Accordingly, the dimensionless halfseparation distance S* and the dimensionless liquid bridge volume V* can be expressed using eqs 5 and 6.14

S* ) S/a ) Xc - (1 - cos φ) V* ) V/a3 )

(5)

∫0Xc 2πY2 dX - 2π(1 - cos φ)2(2 + cos φ)/3

(6)

where Xc is the X value at position c in Figure 1d. Information on the dependence of liquid bridge force on the liquid bridge volume and separation distance can be found in the work of Lian et al.14 Evolution of Liquid Bridge Force. During the drying process, the volume of the interparticle liquid decreases with the evaporation of the liquid, which leads to the change of liquid bridge force between the two spheres. In the final structure formation of colloidal crystal, we postulate that the interplay between the liquid bridge force, viscous forces, and frictional forces is of vital importance.

3694 Langmuir, Vol. 22, No. 8, 2006

Zhou et al.

To establish a dynamic model to depict the temporal and spatial motion of the colloidal particles, further assumptions need to be made in addition to the assumptions for static liquid bridge analysis: The shrinkage process of a liquid bridge can be regarded as an integration of a series quasi-equilibrium steps; The spheres move without rotations; a constant translational velocity at each quasi-equilibrium step can be assumed as achieved instantaneously; The evaporation rate, jevap, does not vary with the change of the liquid bridge curvature; The viscosity and the liquid surface tension do not vary when evaporation proceeds. The dynamic model for the changing liquid bridge volume is given by

dV ) -jevapA dt

Figure 2. Evolution of liquid bridge profiles under evaporation rate of 0.0015 s-1. The initial liquid volume is 0.03 and the separation distance is 0.07. (The profiles were recorded every 5 s and the arrows indicate the changing direction of the curvatures.)

(7)

where the surface area A and its dimensionless form A* of the liquid bridge is given by

A* ) A/a2 ) 2π

∫x′x Yx1 + Y˙ 2 dX c

(8)

c

The dimensionless form of the evaporation rate, jevap (m‚s-1), can be defined as

j/evap ) jevap/a

(9)

please note that j/evap still has the unit of s-1. With eqs 7-9, the dimensionless form of liquid bridge volume at each quasi-equilibrium time step can be calculated as / ) V/i - j/evap A*∆t ) V/i - 2π j/evap ∆t Vi+1

∫x′x Yx1 + Y˙ 2 dX c

c

(10)

Therefore, the liquid bridge force at each time step can be calculated, and the force evolution as a function of time has become tractable. Input Parameters. The study of colloidal self-assembly generally uses colloidal systems of polystyrene (PS) or silica microspheres dispersed in water or ethanol solvent.2 In this work, colloidal systems of silica spheres dispersed in water solvent are used in computing. The contact angle between water and silica is taken as 27.5° and the surface tension of water is taken as 72.8 mN/m.16 In addition, we also calculated the effect of surface tension on the liquid bridge force by varying the concentrations of surfactant Tween-80. The data of the surface tensions and the contact angles as functions of the surfactant concentration are acquired from the report of Vogler et al.17 In the simulation, we use 3.99 × 10-6 mol/L as the initial Tween-80 concentration, which corresponds to a contact angle of 48.3°.17 The radius of the microspheres is set as 1 µm.

Results and Discussion Evolution of Liquid Bridge Force when Liquid Bridge Evaporates. Before the colloidal crystal dries completely, the colloidal spheres are not closely in contact because of the existence of the water.13,18 With the evaporation of the solvent, which progresses at all time, the colloidal crystal shrinks and the distance (16) Zhang, J.; Xue, L.; Han, Y. Langmuir 2005, 21, 5667. (17) Vogler, E. A. Langmuir 1992, 8, 2005. (18) Fustin, C.-A.; Glasser, G.; Spiess, H. W.; Jonas, U. Langmuir 2004, 20, 9114.

Figure 3. Evolution of liquid bridge forces under different evaporation rate. The dimensionless separation distance is set as constant 0.07 and initial dimensionless volume is 0.03.

between spheres decreases. However, in the cases when the selfassembly process is carried on a rough surface or a patterned substrate, the positioning of the colloidal spheres may be fixed, i.e., the distance between the spheres is kept constant. In this case, the complexity of the simulation is relatively low; however, it provides us a clear view of the dynamic evolution of the liquid bridge force. The changing of the liquid bridge force with time is calculated as described in section 2. During the computation, the evaporation rate is assumed as a constant, but the surface area of the liquid bridge decreases with the evaporation of the liquid. Therefore, the volume of evaporated liquid within constant time intervals decreases accordingly. The numerically computed profiles of the liquid curvature offer one a primary idea of the drying process, and as shown in Figure 2, the curvatures are recorded every 5 s under a dimensionless evaporation rate of 0.0015 s-1. It can be seen that, with the evaporation of the liquid, the meniscus shrinks (as indicated by the arrows) and both the half-filling angles φ and the neck radius Y0 decrease. The computation results of the evolution of liquid bridge force are shown in Figure 3. The dimensionless separation distance and initial volume are set as 0.07 and 0.03, respectively. It is seen that, with the evaporation of the liquid, the liquid bridge force decreases accordingly. Five evaporation rates, 0.0005, 0.001, 0.0015, 0.005, and 0.01 s-1, were used in our calculation. Figure 3 shows that, at a higher evaporation rate, the liquid bridge force decreases faster. In other words, high temperature and low humidity, which lead to a high evaporation rate, will cause a fast

Drying of Colloidal Crystals

Figure 4. Change of force magnitude differences vs time between two liquid bridges with initial volumes of 0.03 and 0.025 respectively, under various evaporation rates. The interparticle distance is fixed at 0.07.

descent in force magnitude between spheres when the interparticle distance is fixed. In our research, the effects of the initial liquid bridge volumes on the liquid bridge forces were also studied. Figure S1 shows the plots of the dynamic change of liquid bridge forces with initial volumes of 0.03 and 0.025, which have the same changing tendency under both conditions. To give a clearer view, we calculated the differences between the liquid bridge forces with the two initial volumes, i.e., (FV*)0.03 - FV*)0.025) and plotted them against time, as shown in Figure 4. It can be seen that under all of the five evaporation rates the force differences increased. It can be concluded that, under the same evaporation rate, the lower the initial volume, the faster the liquid bridge force decreases. In other words, during the evaporation process, the difference between the forces will be magnified if the initial liquid volumes are different. We can also find from Figure 4 that under low evaporation rates the force differences increase slower than that of the high evaporation rate. For example, at 10s the force differences are 12.4 × 10-9 and 47.3 × 10-9 N under evaporation rates of 0.0005 and 0.005 s-1, respectively. Particle Mobility when the Liquid Bridge Evaporates. During drying stage 2 of a real self-assembly system, the evaporation of the liquid and the motion of the particles usually take place simultaneously, so that a close-packed ordered structure is obtained when the colloidal crystal is completely dried. However, in the simulation, it is complex to determine the velocity of the spheres because it is a function of various forces, such as the liquid bridge force which changes with distance and liquid volume and the viscous force which is related to velocity.19 To simplify the problem, we applied additional assumptions as elaborated in “Evolution of Liquid Bridge Force” of section 2. A key point is that we assumed that the magnitude of resistance forces (friction force and the viscous forces) is always equal to the liquid bridge force, but in the opposite direction, thus the velocity of the spheres will keep constant once they move. Figure 5 illustrates the computational results for the liquid bridge force evolution under different evaporation rates. The spheres are assumed to move at a constant velocity of 0.0025 s-1. The simulation results show that under low evaporation (19) Pitois, O.; Moucheront, P.; Chateau, X. J. Colloid Interface Sci. 2000, 231, 26.

Langmuir, Vol. 22, No. 8, 2006 3695

Figure 5. Evolution of liquid bridge force under various evaporation rates. The initial liquid volume is 0.03, initial interparticle distance is 0.07, and spheres move at constant velocity of 0.0025 s-1.

Figure 6. Evolution of liquid bridge profiles under evaporation rates of 0.0005 and 0.01 s-1. The initial liquid volume is 0.03. The initial distance is 0.07 and the spheres velocity is 0.0025 s-1. The arrows indicate the changing directions of curvatures.

rates (0.0005, 0.001, and 0.0015 s-1) the liquid bridge forces increase with the increase of time and that under the high evaporation rate, 0.01 s-1, the liquid bridge force decreases as the evaporation proceeds. However, we can find that when the evaporation rate is 0.005 s-1, the liquid bridge force almost remains constant through the course. This is because, when the liquid volume is set as constant while two spheres move toward each other, the liquid bridge force increases with time,14 whereas when the distance is set as a constant, the liquid bridge force decreases with time due to evaporation as shown in Figure 3. Therefore, when these two conditions act simultaneously and satisfy a certain relationship, the two effects cancel off each other so that the force is kept constant. The evolution of the liquid bridge profiles under the two evaporation rates (0.0005 and 0.01 s-1) are shown in Figure 6, where the motion directions are indicated by the arrows. It can be found that, when the evaporation rate is 0.0005 s-1, the liquid bridge expands in the y direction, whereas under 0.01 s-1 it shrinks. The reason is that under both evaporation rates, the distance changing rates between two spheres are the same, however, the liquid volumes become diverse due to the different evaporation rates. As a result, the evolution of the liquid bridge force shows different tendencies under various evaporation rates. In addition, the evolution of liquid bridge force with initial volume of 0.025 was also computed. Shown in Figure S2A-E are the force plots with initial volumes 0.03 and 0.025 under

3696 Langmuir, Vol. 22, No. 8, 2006

Figure 7. Evolution of force magnitude differences between two liquid bridges with initial volumes of 0.03 and 0.025 respectively, under various evaporation rates. The dimensionless velocity of spheres is 0.0025 s-1.

evaporation rates of 0.0005, 0.001, 0.0015, 0.005, and 0.01 s-1. The translational velocities of the colloidal spheres are all assumed at 0.0025 s-1. It can be found that the forces in Figure S2A-C approach together with the increase of time, however, the forces in Figure S2D,E diverge further. To clarify these changing tendencies, the differences between the forces of different initial liquid volumes were plotted against time in Figure 7. It can be found that when the evaporation rates are low (0.0005, 0.001, and 0.0015 s-1), the force differences, which are caused by the different initial volume, decrease. However, the force differences increase with time when the liquid evaporates at high rates (0.005 and 0.01 s-1). These results can be used to explain the crack formation during the drying stage 2. As shown in Figure 1b, when the interstitial liquid volumes decrease between spheres due to evaporation, the liquid bridge force becomes the driving force to pull the spheres together. When the colloidal crystal proceeds to drying stage 2, the liquid volumes among the spheres may be different (as shown in Figure 1b, V2 < V1). As a result, the liquid bridge forces among the spheres are not of equal magnitude (as shown in Figure 1b, F1-2 and F3-4 are larger than F2-3, where the subscripts refer to the particle identity). Because at this moment the difference of the bridge forces is not large enough, the resistant forces may conveniently compensate the imbalance in liquid bridge forces without leading to defects. However, as the solvent further dries off, the volumes of the liquid bridges decrease with different rates and so do the interparticle liquid bridge forces. Hence, the magnitude difference between the two forces (i.e., F1-2 and F2-3 ) evolves accordingly. When the force differences between the two liquid bridges become large enough, they may lead to the breaking of the crystal. As shown in Figure 7, when the evaporation rates are high (0.005 and 0.01 s-1), the magnitude difference between the two bridge forces increases vs time, which may exceed the resistant forces. Consequently, sphere 2 will be dragged to the left side, similarly, sphere 3 will move to the right side. When the distance between spheres 2 and 3 reaches the liquid bridge rupture distance, cracking occurs between these two spheres. However, if the evaporation rate is low as shown in Figure 7 (0.0005, 0.001, and 0.0015 s-1), the magnitude difference in F1-2 and F2-3 decreases with the evaporation of the liquid; thus, the force balance can be well maintained, and no defect will be produced between spheres 2 and 3. This outcome can also be used to explain the cracks in multilayer colloidal

Zhou et al.

Figure 8. Evolution of liquid bridge force under various evaporation rates, when Tween-80 is used as the solvent with an initial concentration of 3.99 × 10-6 mol/L. Spheres move at a constant dimensionless velocity of 0.0025 s-1.

films. When a crack appears at the top layer, the gap will lead to the higher evaporation rate at the second layer. Thus, cracks will be formed at the exposed positions. As a result, the crack expands into the crystal and causes a deep crack defect. Zeng et al.11 have experimentally proven that keeping high humidity can produce high quality colloidal crystals; however, no explanation was provided upon the phenomenon. On the contrary, our computation results offer a clear rationalization. Effect of Surfactant. Recently, some experimental investigations have been carried out on studying effects of surfactants on the thickness and morphology of the colloidal crystals.20,21 Here, we numerically studied the effects of surfactant by using our liquid bridge force model. For most surfactant solution, both the contact angle and the surface tension will decrease during the evaporation of the solvent because of the augment in concentration. Both of these two factors have effects on the capillary force of the liquid bridge. Figure 8 plots the liquid bridge force vs time under a different evaporation rate with Tween-80 solution as the solvent with the initial concentration of 3.99 × 10-6 mol/L. It is found that the changing tendency of the force is the same as that with water as solvent (Figure 7). Figure 9 compares the liquid bridge force evolution between water and Tween-80. Due to the higher surface tension, the liquid bridge force of water is much larger than that of Tween-80 solution. The smaller liquid bridge force of Tween-80 has effects in 2-folds. On one hand, it could lead to less efficiency than water in the self-assembly process because of the smaller interparticle force. We conjecture that this is the reason for the formation of square arrays when the surfactant solution was used as solvent.20 On the other hand, from Figure 9, we can find that the force changes between 0 and 9 s of water and Tween-80 solution are 55.4 × 10-9 and 32.5 × 10-9 N, respectively. In other words, during the evaporation process, under the same evaporation rate, the change of the liquid bridge forces will be much smaller when the surfactant solution is used as solvent. Figure S3 shows the liquid bridge force evolution of surfactant solution when the initial liquid volumes differ (0.03 and 0.025) under different evaporation rates. The corresponding force magnitude differences vs time are plotted in Figure 10. It can be found that the changing tendencies are the same as that of water (Figure 7). Nevertheless, the magnitude differences are much smaller than that of water. For example, (20) Cong, H.; Cao, W. Langmuir 2003, 19, 8177. (21) Zhou, Z., Zhao, X. S. Langmuir 2005, 21, 4717.

Drying of Colloidal Crystals

Figure 9. Evolution of liquid bridge forces with water and Tween80 (initial concentration: 3.99 × 10-6 mol/L) as the solvent, respectively. The dimensionless evaporation rate is 0.0015 s-1 and the constant dimensionless velocity of spheres is 0.0025 s-1.

Langmuir, Vol. 22, No. 8, 2006 3697

and established a dynamic model and a numerical scheme to account for the evolution of interparticle liquid (capillary) bridge forces during drying stage 2. From the results, we found that during the drying process the change of the liquid bridge force is related to the evaporation rate and the approaching velocity of two neighboring spheres. Importantly, the heterogeneity in the initial conditions, e.g., the difference in the initial liquid bridge volumes, which may lead to the inequality of liquid bridge forces among spheres, could be the reasons for the crack formation. When the evaporation rate is low enough, this force imbalances may be minimized during drying, resulting in high quality colloidal crystals. Whereas, when the evaporation rate is high, the initial difference in liquid bridge force magnitudes may be enlarged significantly, causing the uneven motions among the spheres, rupturing the liquid bridge with the least strength, and starting crack formation. The derived knowledge from this work will be useful in designing a refined strategy to effectively control colloidal crystal growth. Nomenclature a ) radius of the spheres (m) A ) surface area of liquid bridge (m2) F ) liquid bridge force (N) H ) mean curvature jevp ) evaporation rate (m‚s-1)

Figure 10. Evolution of force differences between two liquid bridges with initial volumes of 0.03 and 0.025 under different evaporation rates. Tween-80 is used as the solvent (initial concentration: 3.99 × 10-6 mol/L) and the dimensionless evaporation rate is 0.0015 s-1.

with evaporation rate of 0.0015 s-1, at 8 s the force difference is 2.3 × 10-9 N, whereas for water the difference is 8.1 × 10-9 N as shown in Figure 7. This suggests that the surfactant can decrease the force difference induced by different initial volume and reduces the possibility of the formation of the cracks. To understand these results, particularly the influence of surfactant on the quality of the colloidal crystals, further experiments are needed, and these are currently in progress in our laboratories.

Conclusion In this paper, we differentiated the two drying stages involved in evaporation-induced self-assembled colloidal crystal formation

j/evap ) dimensionless evaporation rate (s-1) ∆p ) pressure difference across S ) half-separation distance (m) ∆t ) time (s) V ) volume of liquid bridge (m3) ∆V ) volume of evaporated liquid (m3) V ) velocity of sphere (m‚s-1) V* ) dimensionless velocity of sphere (s-1) X ) dimensionless horizontal coordinate Y ) dimensionless vertical coordinate Y0 ) neck radius at center of liquid bridge Y˙ ) first differentiation of Y Y¨ ) second differentiation of Y γ ) surface tension (mN/m) θ ) contact angle (°) φ ) half-filling angle (°) * ) dimensionless indicator of parameters

Acknowledgment. This work was supported by Australian Research Council (ARC) under DP 0558727. Supporting Information Available: The plots for evolution of liquid bridge forces with initial dimensionless liquid volumes of 0.03 and 0.025 under different evaporation rates. This material is available free of charge via the Internet at http://pubs.acs.org. LA052934C