In Situ Observation and Measurement of Evaporation-Induced Self

Dec 30, 2010 - The growth processes of colloidal crystals in different cuvettes recorded by direct video observations revealed that solvent flow aroun...
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In Situ Observation and Measurement of Evaporation-Induced Self-Assembly under Controlled Pressure and Temperature Lei Yang, Kuiyi Gao, Yanhong Luo, Jianheng Luo, Dongmei Li, and Qingbo Meng* Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box, 603, Beijing 100190, China Received October 29, 2010. Revised Manuscript Received December 14, 2010 In situ observations of evaporation-induced colloidal self-assembly and in situ measurement of mass transfer process were carried out under a temperature and pressure controlling system. The growth processes of colloidal crystals in different cuvettes recorded by direct video observations revealed that solvent flow around the pore space of the crystal played a key role. By changing the circumstances (temperature and pressure) of the self-assembly system and properties of fluid (viscosity), different evaporation rate of solvent and growth rate of colloidal crystals were measured directly. It turned out that both evaporation rate and growth rate as functions of temperature and pressure fit Stefan’s law well. Furthermore, the transfer process of particles in the fluid flow was determined by the fluid-dynamic characteristics, which can be analyzed by the Reynolds number. The results obtained provide an insight into the growth mechanisms of self-assembly and theoretical basis for optimizing the experimental growth conditions of colloidal crystals.

1. Introduction Colloidal self-assembly, a complicated and intriguing process, has been studied by intensive efforts due to its important applications in fabricating structures with submicrometer dimensions. Over the past few decades, many typical methods have been developed to fabricate three-dimensional regular structures, such as electrostatic repulsion assembly,1,2 gravitational sedimentation,3,4 physical confinement,5,6 electrophoresis,7,8 spin or spray coating,9,10 and *Corresponding author: Tel þ86-10-82649242; Fax þ86-10-82649242; e-mail [email protected]. (1) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57–60. (2) Vos, W. L.; Megens, M.; vanKats, C. M.; Bosecke, P. Langmuir 1997, 13, 6004–6008. (3) Davis, K. E.; Russel, W. B.; Glantschnig, W. J. Science 1989, 245, 507–510. (4) vanBlaaderen, A.; Ruel, R.; Wiltzius, P. Nature 1997, 385, 321–324. (5) Park, S. H.; Xia, Y. N. Langmuir 1999, 15, 266–273. (6) Yin, Y. D.; Lu, Y.; Xia, Y. N. J. Am. Chem. Soc. 2001, 123, 771–772. (7) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706–709. (8) Holgado, M.; Garcia-Santamaria, F.; Blanco, A.; Ibisate, M.; Cintas, A.; Miguez, H.; Serna, C. J.; Molpeceres, C.; Requena, J.; Mifsud, A.; Meseguer, F.; Lopez, C. Langmuir 1999, 15, 4701–4704. (9) Deckman, H. W.; Dunsmuir, J. H. Appl. Phys. Lett. 1982, 41, 377–379. (10) Cui, L. Y.; Zhang, Y. Z.; Wang, J. X.; Ren, Y. B.; Song, Y. L.; Jiang, L. Macromol. Rapid Commun. 2009, 30, 598–603. (11) Zhou, Z. C.; Zhao, X. S. Langmuir 2004, 20, 1524–1526. (12) Meng, Q. B.; Gu, Z. Z.; Sato, O.; Fujishima, A. Appl. Phys. Lett. 2000, 77, 4313–4315. (13) McLachlan, M. A.; Johnson, N. P.; De La Rue, R. M.; McComb, D. W. J. Mater. Chem. 2004, 14, 144–150. (14) Jiang, P.; Bertone, J. F.; Hwang, K. S.; Colvin, V. L. Chem. Mater. 1999, 11, 2132–2140. (15) Im, S. H.; Kim, M. H.; Park, O. O. Chem. Mater. 2003, 15, 1797–1802. (16) Gu, Z. Z.; Fujishima, A.; Sato, O. Chem. Mater. 2002, 14, 760–765. (17) Dimitrov, A. S.; Nagayama, K. Langmuir 1996, 12, 1303–1311. (18) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Nature 1993, 361, 26–26. (19) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183–3190. (20) Zheng, Z. Y.; Gao, K. Y.; Luo, Y. H.; Li, D. M.; Meng, Q. B.; Wang, Y. R.; Zhang, D. Z. J. Am. Chem. Soc. 2008, 130, 9785–9789. (21) Zheng, Z. Y.; Liu, X. Z.; Luo, Y. H.; Cheng, B. Y.; Zhang, D. Z.; Meng, Q. B.; Wang, Y. R. Appl. Phys. Lett. 2007, 90, 051910. (22) Kitaev, V.; Ozin, G. A. Adv. Mater. 2003, 15, 75–78. (23) Wong, S.; Kitaev, V.; Ozin, G. A. J. Am. Chem. Soc. 2003, 125, 15589– 15598. (24) Joannopoulos, J. D. Nature 2001, 414, 257–258. (25) Vlasov, Y. A.; Bo, X. Z.; Sturm, J. C.; Norris, D. J. Nature 2001, 414, 289– 293.

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evaporation-induced self-assembly (EISA).11-26 It is worth noting that EISA has gained considerable attention for its facility and reproducibility. Yet, different ways for EISA have been utilized to fabricate high-quality colloidal crystal films. One main strategy is to change the environmental conditions of self-assembly, such as humidity, temperature, and pressure.20-26 However, these ways still cannot be harnessed to fabricate colloidal photonic crystals with near-perfect order. It is known that the quality of colloidal photonic crystals is mainly affected by the evaporation rate of solvent, the particle transfer in the fluid flow, and the crystallization process.27 In order to get higher ordered structure, the micromechanism of EISA in principle and its detailed mass transfer process during the evaporation of solvent to provide optimal and maneuverable experimental conditions are necessary. The growth process of EISA has been investigated experimentally and theoretically by many groups.17-19,28-39 As a powerful tool, different types of microscopes have been used to obtain the real-time information on EISA. Nagayama et al. observed the growing process of 2D particle arrays and revealed that capillary force induced by water evaporation is the driven force.17-19 Norris et al. investigated the process of thickness transitions in convective assembly.34 By using a confocal laser scanning microscope, (26) Chen, K.; Stoianov, S. V.; Bangerter, J.; Robinson, H. D. J. Colloid Interface Sci. 2010, 344, 315–320. (27) Ye, Y. H.; LeBlanc, F.; Hache, A.; Truong, V. V. Appl. Phys. Lett. 2001, 78, 52–54. (28) Perrin, J. Ann. Chim. Phys. 1909, 18, 5–114. (29) Pieranski, P.; Strzelecki, L.; Pansu, B. Phys. Rev. Lett. 1983, 50, 900–903. (30) Vanwinkle, D. H.; Murray, C. A. Phys. Rev. A 1986, 34, 562–573. (31) Haggerty, L.; Watson, B. A.; Barteau, M. A.; Lenhoff, A. M. J. Vac. Sci. Technol. B 1991, 9, 1219–1222. (32) Zhang, K. Q.; Liu, X. Y. Nature 2004, 429, 739–743. (33) Koh, Y. K.; Wong, C. C. Langmuir 2006, 22, 897–900. (34) Meng, L. L.; Wei, H.; Nagel, A.; Wiley, B. J.; Scriven, L. E.; Norris, D. J. Nano Lett 2006, 6, 2249–2253. (35) Ishii, M.; Harada, M.; Nakamura, H. Soft Matter 2007, 3, 872–876. (36) Yan, Q.; Gao, L.; Sharma, V.; Chiang, Y. M.; Wong, C. C. Langmuir 2008, 24, 11518–11522. (37) Jiang, Z.; Lin, X. M.; Sprung, M.; Narayanan, S.; Wang, J. Nano Lett 2010, 10, 799–803. (38) Brewer, D. D.; Allen, J.; Miller, M. R.; de Santos, J. M.; Kumar, S.; Norris, D. J.; Tsapatsis, M.; Scriven, L. E. Langmuir 2008, 24, 13683–13693. (39) Lozano, G.; Miguez, H. Langmuir 2007, 23, 9933–9938.

Published on Web 12/30/2010

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Ishii et al. showed that the crystallographic direction of the growth of the sphere array changes with the growth rate.35 Very recently, Wong et al. found that chemical conditions mediated by surface charge and surfactant additions play an important role in EISA.36 Since all the systems investigated in these papers are either a sessile drop or the solvent exposed to the external circumstance, the evaporation process, which is one of the most important factors in EISA, cannot be controlled precisely and continuously during in situ observations of EISA. In our previous study, we developed a controllable twoparameter (temperature and pressure) method for depositing colloidal crystals and obtained the optimal growth conditions for EISA of multilayer colloidal crystals.21 However, what happens when the growth condition of EISA is changed and how the growing condition affects the mass transfer process are still unknown. In this work, we applied the temperature and pressure controlling system to in situ observe the process of EISA in the vertical direction. Microscopically, the real-time movement of colloidal spheres under controllable temperature and pressure, which revealed the importance of solvent flow in EISA, can be recorded by our optical microscopy setup. Macroscopically, the growth rate of the colloidal crystal as a function of temperature and pressure was obtained by direct measurement. Unlike previous studies, this work was focused on the diffusion process of gaseous mixtures above the solvent. Effects of fluid-dynamic characteristics (velocity, viscosity, and geometrical shape) on the mass transfer process of EISA were also investigated systematically.

2. Experimental Section 2.1. Experimental Setup. Basically, the whole setup for in situ observations of colloidal self-assembly consisted of two parts and was built on a nonvibration optical table (see Figure 1a). The first part was the EISA system. Here, two kinds of cuvettes were used as the suspension container, and the polystyrene (PS) colloidal spheres were deposited on the walls of the cuvette. The cuvette was placed in a larger aluminum container, and the very small interspace was filled with silicone grease to ensure good thermal conduction. Temperature and pressure of the system can be adjusted simultaneously by self-designed controllers. The temperature controller, which contains temperature sensor, controlling circuit, thermal energy converter, and heat sink, is able to keep the temperature fluctuation less than 0.1 C. A diaphragm pump was used to maintain the low pressure in the cuvette, and the precision of the pressure controlling circuit is 0.1 kPa. The second part of the setup was a reflected light microscopy attached to a CCD camera (INFINITY 2-3C, Canada). In our case, the wall thickness of cuvettes was 1 mm and the front wall of the aluminum container was 2 mm. Therefore, an objective with a long working distance of 4 mm (Carl Zeiss LD EC 100x/0.75 HD DIC M27) mounted horizontally was used in the Kohler illumination light path. 2.2. Materials and Investigated Procedure. For most of our experiments we used a 1 vol % aqueous suspension of PS latex (China University of Petroleum, 1.09 ( 2.25% μm) dispersed in ultrapure water (Milli-Q Synthesis System, Millipore S.A., Molsheim, France). Different volume fractions (3%, 6%, 9%, and 12%) of ethylene glycol were added in the 1 vol % suspensions to study the influence of the liquid viscosity. The viscosity of the suspensions was measured at 40 C using a water bath by Brookfield LVDV-IIþPro (ULA). The front walls of two different types of cuvettes (Yixing jingke Optical Instrument Co., Ltd.) were performed as substrates (see Figure 1a). One was the standard cuvette (10 mm width) made of optical glass, and the other was the microcuvette (1 mm width slit) which was used to restrict the meniscus of the solution. Before assembly, the cuvettes were first immerged in chromosulfuric acid Langmuir 2011, 27(5), 1700–1706

Figure 1. (a) Scheme of the experimental apparatus. The light source is a tungsten bromine lamp. Both an aperture and a field iris diaphragm are employed to set up the illumination. There is a small window in the front wall of the container, so the uniform and bright light coming from the beamsplitter can illuminate the growth front at the walls of the cuvette. The cuvette can be sealed well by the cover which is connected to a pressure controller. The temperature sensor, which is attached to a temperature controller, is embedded in the container. So the whole growing process under different conditions can be recorded by the CCD camera. (b) Scheme of the particle and water fluxes in the vicinity of multilayer particle arrays growing on the front wall of the standard cuvette. (c) Scheme of the growth front of the particle arrays in the microcuvette. for more than 3 h, followed by washing with tap water and ultrapure water, and then dried in a stream of nitrogen. After the cuvette filled with PS suspension was placed in the aluminum container, the temperature and pressure controlling system was subsequently turned on. The whole environment of colloidal self-assembly can become steady within 2 min. Then, the objective was focused on the growth front, and real-time observations of the crystal growth could be obtained by the CCD camera. The growth rate of the sphere array was determined through the recordings. At the same time, the evaporation rate could be calculated by the mass loss of suspension in certain time. In our experiments, different growth and evaporation rate were obtained under different pressure and temperature.

3. Results and Discussion 3.1. Mass Transfer Processes in Different Kinds of Cuvettes. At first, the movement of PS particles in the standard cuvette at room temperature and normal pressure was observed (video S1 in Supporting Information). As can be seen in video S1, the thickness of water layer at the meniscus was much larger than the particle diameter (1.09 μm) and the water-air interface did not have a sufficiently high inclination toward the front wall of the cuvette. Therefore, the array growth cannot be initiated by the capillary force, and the particles were involved in Brownian motion. As we increased the temperature and depressed the DOI: 10.1021/la104338a

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Figure 2. Photographs and micrographs of colloidal crystals growing: (a, b) in the standard cuvette and (d, e) in the microcuvette. Simulative views of the growth front according to the video S2: (c) growing in the Æ112æ direction and (f) growing in the Æ110æ direction. The particle diameter is 1.09 μm.

pressure, the slope of water layer toward to the wall became smaller. Interference fringes near the border of ordered layers appeared because of the slowly varying thickness of water layer. Because of the increasing evaporation rate and more effective convection of solvent, the sequent more directional movement of spheres and higher growth rate of the sphere array were observed with the increasing temperature and decreasing pressure. Usually, the growth crystal face observed in the literature is (111) of fcc structure, which was also recorded when we used the standard cuvette (see Figure 2a,b). However, in situ observation of the growth process on crystal faces perpendicular to (111) has never been reported. By using the microcuvette, our original intention was to observe the growth front on the special crystal face (110) (see Figure 1b), which can characterize the thickness of colloidal crystals. Unfortunately, because of the restricted meniscus and influence of the inner corner, particles deposited on the edge of the slit (see Figure 2d), and what we observed was still the growth front of (111) as shown in Figure 1c. Nevertheless, the mass transfer processes in the two types of cuvettes were quite different. As shown in Figure 2 and video S2 in the Supporting Information, the particles in both two types of cuvettes started to form an ordered monolayer at the edge of the interference fringes, and the film gradually increased from monolayer to multilayer structures. In the standard cuvette, the array deposited on the front wall and grew in the Æ112æ direction, while on the slit of the microcuvette, the array grew in the Æ110æ direction. Furthermore, we can observe the whole process of the initial aggregation process of colloidal particles of the first line in the microcuvette (see Figure 1c and video S2). Near the three-phase contact line, particles were drawn up by the water flow to the growth front. Some of the particles were carried into the end of the first line, which was much longer than lines above it, while some of them lodged into the niches of the first line. The aggregation of the adjacent particles was mainly caused (40) Kralchevsky, P. A.; Denkov, N. D. Curr. Opin. Colloid Interface Sci. 2001, 6, 383–401.

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by the attractive lateral capillary force40 while the fluid flow into and through the packing array took the particles into the niches.38,41,42 One of the most interesting phenomena was that lodged particles in the niches can slip to the end of the first line, and particles incorporated to the end of the first line can also be suck into the above line due to the powerful flow into niches of the formed array. It is believed that this is the result of the competition between lateral capillary force and solvent flow. 3.2. Diffusion Process of Gaseous Mixtures during the Evaporation of Water. It is known that the convective transport of particles in EISA is mainly caused by water evaporation from the menisci.17 If we want to control the growth conditions by changing the evaporation process, it is crucial to understand the vapor transfer mechanism above the solvent during the evaporation of water. In M. Rohsenow’s book,43 problems of forced convection mass transfer in tubes and diffusion in binary gas mixtures are analyzed. According to the theory of this book, some assumptions have been made for our diffusion system: (a) In the binary mixture of vapor (gas a) and air (gas b), the Gibbs-Dalton laws hold: Pa þ Pb = P. (b) Both of them are ideal gases. (c) The diffusion coefficient D is constant at fixed temperature and pressure. (d) The whole process is steady and one-dimensional. Here, Pa and Pb are the partial pressure of vapor and air, respectively, and P is the total pressure. Theoretical expression for the diffusion coefficient D in gaseous mixtures is suggested by Gilliland:44 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 D ¼ 0:0069 þ 1=3 1=3 2 M Mb a PðVa þ Vb Þ T 3=2

ð1Þ

(41) Norris, D. J.; Arlinghaus, E. G.; Meng, L. L.; Heiny, R.; Scriven, L. E. Adv. Mater. 2004, 16, 1393–1399. (42) Gasperino, D.; Meng, L. L.; Norris, D. J.; Derby, J. J. J. Cryst. Growth 2008, 310, 131–139. (43) Rohsenow, W. M.; Choi, H. Y. Heat, Mass and Momentum Transfer; Prentice-Hall Press: Upper Saddle River, NJ, 1961; p 397. (44) Sherwood, T. K.; Gilliland, E. R. Ind. Eng. Chem. 1934, 26, 1093–1096.

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where (Na/A)abs is the absolute molal flux, A is the cross-sectional area of diffusion, R is the universal gas constant, and T is the absolute temperature. pa1 and pa2 are the boundary conditions: at x = 0, pa = pa1; at x = l, pa = pa2. Substituting D from eq 2 into eqs 3 and 4, following equations can be given:   Na kT pa1 - pa2 ¼ Rl A abs P

ðmodel IÞ

!   Na kT P - pa2 ðmodel IIÞ ¼ ln Rl A abs P - pa1

ð5Þ

ð6Þ

Since the diffusion of the new vapor phase (gas a) of our case is coming from the evaporation of water from menisci, the evaporation rate of the suspension Ve can be given as   Ma Na Ve ¼ F A abs

ð7Þ

Here, F is the density of the suspension. Likewise, the evaporation rate of the suspension Ve can be rewritten using eqs 5 and 6 as follows: Figure 3. (a) Model I: partial pressure profiles in equimolal counter diffusion. Model II: partial pressure profiles for diffusion of gas a in a stationary gas b. From ref 43. (b) Top: evaporation rate of pure water as a function of pressure at 40 C. Bottom: growth rate of array and evaporation rate as functions of pressure for 1 vol % of 1.09 μm PS suspension at 40 C. (c) Top: evaporation rate of pure water as a function of temperature under 10 kPa. Bottom: growth rate and evaporation rate as functions of temperature for 1 vol % of 1.09 μm PS suspension under 10 kPa. The curves and lines were simulated by model II (eq 10), and the constant k in the evaporation rate of suspension plot were calculated using parameters Ma = 18 g mol-1, F = 1 g cm-3, l = 1 m, R = 8.31 J mol-1 K-1 and (b) T = 313.2 K, (c) P = 10 kPa.

where T is temperature, P is pressure, Ma and Mb are molar masses, and Va and Vb are atomic volumes. Equation 1 shows D to be independent of the ratio of quantities of the gases present and a constant at fixed temperature and pressure. D is proportional to T3/2 theoretically, whereas actually in experiments, D ∼ T2 is usually found.45 In our case (temperature range of growth is from 30 to 45 C), it is also found that D is proportional to T2 and can be written as follows: D ¼ k

T2 P

ð2Þ

Here, k is a constant in the binary mixture of vapor and air. In terms of these assumptions, there are two different models in this binary mixture (see Figure 3a). One is equimolal counter diffusion of gas a and b (model I), and the other is diffusion of gas a through a stationary gas b (model II). The absolute flux of gas a in these two models are given respectively as43   Na D pa1 - pa2 ¼ RT A abs l 

Na A

 abs

DP P - pa2 ln ¼ RTl P - pa1

ðmodel IÞ

ð3Þ

! ðmodel II, Stefan0 s lawÞ ð4Þ

(45) Reid, R. C.; Sherwood, T. K. The Properties of Gases and Liquids; McGrawHill Press: New York, 1958.

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Ve ¼

kTMa Pa1 - Pa2 RlF P

kTMa P - pa2 Ve ¼ ln RlF P - pa1

ðmodel IÞ

ð8Þ

! ðmodel IIÞ

ð9Þ

In order to find out which model fits our case, we first fixed the temperature at 40 C (the reason for choosing this temperature is that the quality of the obtained film in our experiment was relatively high around 40 C) and then observed the growth process of the colloid particles maintained at different pressure. Different growth rates at different pressures were obtained (see Figure 3b, bottom). The evaporation rate of pure water and suspensions under different pressure at 40 C (see Figure 3b) were also measured. It is obvious that both evaporation rates decrease with the increasing pressure. Because of the interactions between particles and water molecules, the evaporation rate of suspensions, which is very close to the growth rate, is much less than that of pure water. Then, by applying these two models (eqs 8 and 9) to fit our experimental data (see Figure 3b), it is found that the whole evaporation diffusion process is basically diffusion of the new vapor phase (gas a) through the stationary air (gas b), which can be well described by Stefan’s law of model II. Consequently, the constant k in the diffusion coefficient D (eq 2) can be attained as k = 0.754 N K-2 S-1. Meanwhile, because of the close relationship between the particle growth rate and evaporation rate (we will discuss later), it appears that the growth rate as a function of pressure can also be simulated well by Stefan’s law (see Figure 3b, bottom). One more noticeable phenomenon is that the evaporation rate of pure water at 7 kPa (see Figure 3b, top) deviates badly from the simulative curve. The reason is that the pure water is under a near boiling and unsteady state at 7 kPa, which is below the saturated vapor pressure of pure water, 7.38 kPa, at 40 C.46 Besides, different growth rate and evaporation rate were also obtained by fixing pressure at 10 kPa and changing temperature (46) Dean, J. A. Lang’s Handbook of Chemistry, 15th ed.; McGraw-Hill Press: New York, 1999; Section 5.

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Figure 5. (a) Growth rate as a function of evaporation rate of suspension under the same condition. The red line is a linear fit of experimental data which shows the expecting proportional relation between the two rates.

Figure 4. Growth rate as a function of both temperature and pressure for 1 vol % of 1.09 μm PS suspension. Black squares and red crosses are experimental data from Figure 3b,c.

of our system as shown in Figure 3c. Obviously, both the evaporation rates of pure water and suspensions increase linearly with the increasing temperature, which is coincident with Stefan’s law, so does the growth rate. Although the evaporation rate and temperature in both of the two models have positive linear relationship, when the constant k is taken into account, only the simulative result of model II can have the coherent k (0.754 N K-2 S-1), and the Stefan’s law in our case should be modified as kðT - T0 ÞMa P - pa2 ln Ve ¼ RlF P - pa1

! ð10Þ

where T0 is the intercept of temperature axis. Therefore, we can conclude that the nonreactive, two-component diffusion system under different temperature and pressure can be analyzed by using model II, and the diffusion flux owing to a concentration gradient of the diffusing component (see Figure 3a) in EISA can be estimated using the modified Stefan’s law (eq 10). Most importantly, by in situ observations under controlled circumstances, we found that the growth rate of crystal arrays in suspensions also has the same rule with temperature and pressure (see Figure 4). Since holding the growth rate at an optimal value is crucial for the regular assembling of colloidal spheres,23 it is very convenient to obtain the certain growth rate by choosing the proper temperature and pressure according to this rule. 3.3. What Determines the Growth Rate?. The growth rate of film arrays deposited by EISA is usually presented using the following formula originally derived by Nagayama et al.17 Considering the material flux balance at the array’s leading edge (see Figure 1b), the water evaporation Je must be exactly compensated by the water flow Jw from the bulk suspension into the arrays for the steady-state growth. As a result, the rate of the array growth is Vg ¼

βl0 j Ve hð1 - εÞð1 - jÞ

ð11Þ

where β is the ratio of the mean velocity of the particles suspended in a moving fluid with respect to the velocity of that fluid, l0 is an evaporation length, j is the particle volume fraction in the suspension, 1704 DOI: 10.1021/la104338a

h is the thickness of the particle arrays, ε is the porosity of the arrays, Ve is the evaporation rate of the suspension, and Vg is the crystal growth rate when steady state is reached. In our experiments, different growth rates were achieved in quick succession under different conditions, and the thickness of the growing array did not change in a very short time. Hence, from eq 11, it is obvious that the growth rate (Vg) is proportional to the evaporation rate (Ve), which is in consistent agreement with our experiments (see Figure 5). As mentioned above, the growth of colloidal photonic crystals is mainly influenced by the evaporation rate of water (which has been discussed), the particle transfer in the fluid flow, and the crystallization process. Taking the particle transfer in the fluid flow into consideration, Norris made the hypothesis that the solvent flow may play a major role in evaporation-driven opal growth.41 What we are trying to answer is how the fluid flow influences the mass transfer process. In fluid mechanics,47 Reynolds’ similarity law describes the fluid-dynamic characteristics of a body exposed to a flow in one medium. If two kinds of flow in regions with boundaries have a same definite form, namely, bodies of the same shape are geometrically similar, the fluid-dynamic characteristics of two similar bodies are similar as long as their Reynolds numbers are identical. The Reynolds number is defined as47 Re ¼ Fvl=μ

ð12Þ

where F is the density of the medium, v is the velocity, l is the body dimension (the particle diameter), and μ is the viscosity of the medium. In terms of this law, we hypothesize that the growth process in the fluid flow can be determined by the parameter of hydrokinetics, namely, the Reynolds number. The same cuvettes and particles of the same size were used in our experiments to ensure that all kinds of fluid flow were geometrically similar. Here, the average velocity of the flow near the growth front, v, can be expressed as v ¼ jw

ð13Þ

17

For hfjw = l0je and Ve = je, the velocity of the flow, v, is then given by v ¼

l0 l0 j e ¼ Ve hf hf

ð14Þ

(47) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed.; Pergamon Press: London, 1987; p 56.

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Article Table 1. Viscosity (μ) of 1 vol % of 1.09 μm PS Suspensions with Different Volume Fractions of Ethylene Glycol at 40 C and Evaporation Rate (Ve) of These Suspensions under 9 and 12 kPa at 40 C volume fractions

0%

3%

6%

9%

12%

μ (cP) Ve at 9 kPa (μm/min) Ve at 12 kPa (μm/min)

0.79 103.6 83.9

0.84 73.9 63.6

0.91 65.6 51.1

0.99 58.3 46.9

1.07 53.6 43.9

ð15Þ

ethylene glycol at 40 C were measured, and the evaporation rates of these suspensions under 9.5 and 12 kPa at 40 C were also obtained (see Table 1). The evaporation rate decreases with the increasing viscosity of solvent as shown in Table 1. So, both the two factors influence the growth rate. According to eqs 12 and 14, the Reynolds number of these kinds of fluid was calculated to be 10-6 magnitude, which means that the particles are involved in creeping flows. In terms of Reynolds’ similarity law, the fluiddynamic characteristics of two similar bodies should be similar, and the growth rate should be the same as long as their Reynolds numbers are the same. As can be seen from Figure 6b, when the volume fraction of ethylene glycol is increased, the red circles and black squares explicitly show that the growth rate decreases with the decreasing Reynolds number monotonously, which confirms our hypothesis. However, some blue triangles from changing the pressure when there was no ethylene glycol did not agree well with the trend. It can be explained that the growth rate of colloidal arrays is determined by both the fluid-dynamic characteristics and the diffusion progress of vapor. When we fix the pressure or the two pressures are very close (9.5 and 12 kPa), the diffusion progresses of evaporation are nearly the same. And it is only the fluid-dynamic characteristic, namely, the Reynolds number, that decides the growth rate. However, the growth rate of some blue triangles at relatively higher pressure are much less than that of some red circles (see Figure 6b), though their Reynolds number are almost equal. This is because the diffusion progress of vapor also has a certain influence to the growth rate, when the pressure is far away from 12 kPa. Therefore, we can summarize that the behavior of particles in fluid flow do have a corresponding relation with the fluid-dynamic characteristics, and the growth rate is in good monotone linear increase dependence on the Reynolds number when the diffusion progresses of evaporation are much the same. This can also be interpreted from the definition of the Reynolds number which represents the ration of the dynamic forces (represented by the velocity v, the body dimension l, and the density of the medium F) to the friction forces (represented by the viscosity μ of the medium). The dynamic forces give particles momentum while the friction forces cause resistance; both of them determine the motion of particles.

which means that the velocity of the fluid flow determines the growth rate. Moreover, in order to further validate our hypothesis, different growth rate were obtained by changing the viscosity of the fluid. As shown in Figure 6a, the growth rate decreased by adding 9 vol % of ethylene glycol. While adding ethylene glycol changed the fluid-dynamic characteristic, the diffusion model in gaseous mixtures did not change and both the growth rates fit Stefan’s law well (see the simulative curves in Figure 6a). A similar result, wherein adding 9 vol % ethylene glycol induced a lower growth velocity, has been reported by Ishii et al.35 The question remains though;did the adding ethylene glycol change the fluid-dynamic characteristic or change the evaporation rate of suspension and then change the growth rate? Herein, the viscosities of 1 vol % of 1.09 μm PS suspensions with different volume fractions of

The mass transfer processes of EISA in different circumstances (cuvettes, temperature, pressure, and fluid viscosity) have been discussed in this study. By using an in situ monitoring set, the initial formation of the chain in early stages of growth indicates that solvent flows into and through the packing array play an important role in the assembly of colloidal crystals. According to the experimental results and theoretical fits, it is known to us that the diffusion process of gaseous mixtures above the solvent can be described well by the diffusion of vapor through the stationary air model (Stefan’s law). Most importantly, this model can be applied to control the growth conditions of EISA precisely (see Figure 4). In addition, it is confirmed from the view of hydrokinetics that the fluid-dynamic characteristic is a crucial factor for the mass transfer process. When the diffusion processes of evaporation are

Figure 6. (a) Growth rate as a function of pressure at 40 C for 1 vol % concentration of 1.09 μm PS suspensions without and with 9 vol % ethylene glycol. The curves are simulative results of Stefan’s law. (b) Growth rate as a function of the Reynolds number for 1 vol % of 1.09 μm PS suspensions at 40 C. The black squares and red circles correspond to different volume fractions of ethylene glycol at 9.5 and 12 kPa, respectively, and the blue triangles correspond to different pressures from 7 to 37 kPa (7, 10, 12, 17, 22, 27, 32, and 37 kPa) with no ethylene glycol. The Reynolds number, Re, was calculated using parameters F = 1 g cm-3 and l = 1 μm.

where jw is the water flux from the bulk suspension into the arrays, je is the water flux evaporated from the film, and hf is the thickness of the wetting film at the array’s leading edge. So substitution of eq 11 into eq 14 gives Vg ¼

βhf j v hð1 - εÞð1 - jÞ

Langmuir 2011, 27(5), 1700–1706

4. Conclusions

DOI: 10.1021/la104338a

1705

Article

identical, the growth rate is a function of the Reynolds number, and in our case their relation is a simple linear rule. Besides the better understanding of the mass transfer mechanisms in EISA, we hope our work is also of significance to provide theory guidance for optimizing the quality of three-dimensional photonic crystals. Acknowledgment. The authors appreciate the financial support from the Natural Science Foundation of China (No. 20725311, 20703063, 20873178, 21073231, 51072221 and 20721140647), the

1706 DOI: 10.1021/la104338a

Yang et al.

Ministry of Science and Technology of China (863 Project, No. 2009AA033101), Foundation of the Chinese Academy of Sciences (No. KJCX2-YW-W27, KGCX2-YW-386-1, KGCX2-YW-363). Supporting Information Available: Real-time video recordings showing the movement of particles in different circumstances (video S1) and different growth phenomena in different cuvettes (video S2). This material is available free of charge via the Internet at http://pubs.acs.org.

Langmuir 2011, 27(5), 1700–1706