Langmuir 2007, 23, 9933-9938
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Growth Dynamics of Self-Assembled Colloidal Crystal Thin Films G. Lozano and H. Mı´guez* Instituto de Ciencia de Materiales de SeVilla, Consejo Superior de InVestigaciones Cientı´ficas (CSIC), Ame´ rico Vespucio 49, 41092 SeVilla, Spain ReceiVed June 12, 2007. In Final Form: August 1, 2007 A theoretical and experimental analysis of the growth dynamics of colloidal crystal films deposited by evaporation induced self-assembly is herein presented. We derive an expression for the film growth velocity from which we obtain an equation that describes the evolution of the forming crystal thickness with time. Its validity is confirmed by comparison to the experimental profiles of a large number of films grown under different conditions. We find that, on top of the already reported linear increase in film width over long distances in the growth direction, periodic variations of the friction force at the meniscus give rise to short-range thickness fluctuations that are the main source of spatial inhomogeneities observed in these lattices. The key parameters that determine the period and the intensity of these fluctuations are identified.
Introduction Among all colloidal sphere crystallization techniques developed to date, evaporation-induced self-assembly (EISA)1 is probably the most widely employed. It offers the possibility to create large-scale face-centered-cubic crystalline films of controlled orientation and a low concentration of intrinsic defects. Although the quality of these lattices can still be greatly improved,2 it has been possible to study for the first time optical phenomena such as finite size effects,3 the full band gap opening,4 slow photon propagation at the band edges,5 fine spectral features at high energies,6 surface modes,7 the superprism effect,8 and light localization in planar defects,9 most of which would have been washed out by excess impurities. The EISA of 3D colloidal arrays onto flat substrates was essentially developed on the basis of the comprehensive work of Nagayama et al.10 on continuous convective assembly of fine particles into 2D arrays on horizontal solid surfaces, whose fundamental aspects have been thoroughly discussed.11 Briefly, substrates (typically silica glass slides or silicon wafers) are placed vertically or are tilted within a dilute suspension of monodisperse particles (typically silica or polystyrene microspheres). As the suspension evaporates, a crystalline film is deposited on the substrate at the contact line with the suspension meniscus. Although originally proposed as a method in which the colloidal crystal film thickness could be accurately controlled,1 later indepth studies revealed that thickness variations occur and are inherent to the growth mechanism.12,13 Braun et al. showed that film thickness increases linearly along the growth direction as * Corresponding author. E-mail:
[email protected]. (1) Jiang, P.; Bertone, J. F.; Hwang, K. S.; Colvin, V. L. Chem. Mater. 1999, 11, 2132. (2) Dorado, L.; Depine, R.; Mı´guez, H. Phys. ReV. B 2007, 75, 241101. (3) Bertone, J. F.; Jiang, P.; Hwang, K. S.; Mittleman, D. M.; Colvin, V. L. Phys. ReV. Lett. 1999, 83, 300. (4) Vlasov, Yu. A.; Bo, X. Z.; Sturm, J. C.; Norris, D. J. Nature 2001, 414, 289. (5) von Freymann, G.; John, S.; Wong, S.; Kitaev, V.; Ozin, G. A. Appl. Phys. Lett 2005, 86, 053108. (6) Mı´guez, H.; Kitaev, V.; Ozin, G. A. Appl. Phys. Lett 2004, 84, 1239. (7) Mihi, A.; Mı´guez, H.; Rodrı´guez, I.; Rubio, S.; Meseguer, F. Phys. ReV. B 2005, 71, 125131. (8) Prasad, T. D.; Mittleman, M.; Colvin, V. L. Opt. Mater. 2006, 29, 56. (9) Galisteo-Lo´pez, J. F.; Galli, M.; Andreani, L. C.; Mihi, A.; Pozas, R.; Ocan˜a, M.; Mı´guez, H. Appl. Phys. Lett. 2007, 90, 101113. (10) Dimitrov, A. S.; Nagayama, K. Langmuir 1996, 12, 1303. (11) Dimitrov, A. S.; Nagayama, K. Chem. Phys. Lett. 1995, 243, 462. (12) Schimmin, R. G.; DiMauro, A. J.; Braun, P. V. Langmuir 2006, 22, 6507. (13) Teh, L. K.; Tan, N. K.; Wong, C. C.; Li, S. Appl Phys. A 2005, 81, 1399.
a result of the increasing concentration of particles with time in the meniscus as the liquid dispersant evaporates. From a different perspective, Teh et al. qualitatively explained observed variations of thickness by the competition between surface tension forces due to the change in the contact angle, which results in a “stickslip” motion. Herein we propose an analytical model to describe the growth dynamics of colloidal particle films grown by EISA. We consider the array’s growing edge as a straight line advancing at a rate determined by the speed of the contact line between the meniscus and the substrate, which depends in turn on the colloidal particle flow in the meniscus. As a result, we obtain a growth equation in which we can identify a time-dependent friction force that gives rise to short-range thickness fluctuations. We prove that this equation can be used to fit the different types of experimentally observed film thickness profiles (oscillatory and steplike) obtained from the optical microspectroscopy analysis of colloidal crystals grown under different conditions.
Previous Analysis of Colloidal Crystal Thickness The thickness H of the colloidal films deposited by EISA is usually estimated using the expression originally derived by Nagayama et al.10 They applied the approach of the material flux balance at the array’s leading edge, from which they estimated the growth rate of the array. In their case, the substrate was withdrawn at that same speed to ensure the homogeneity of the deposited film. For the steady-state growth, the water evaporation Je must be exactly compensated by the water flow Jw from the bulk suspension into the array. These considerations yield the expression
H)
φβLVe (1 - )(1 - φ)Vg,tf∞
(1)
where φ is the particle volume fraction, β is the ratio of the mean velocity of the particles suspended in a moving fluid with respect to the velocity of that fluid, L is an evaporation length, is the porosity of the assembled spheres, Ve is the evaporation velocity, and Vg,tf∞ is the crystal growth velocity when steady state is reached. Recently, Braun et al.12 pointed out that when the evaporation velocity exceeds the sedimentation velocity Vs, particles will accumulate at the solvent-air interface, and they found experimental evidence that the thickness of colloidal films grown by EISA increases linearly with time (i.e., with distance
10.1021/la701737v CCC: $37.00 © 2007 American Chemical Society Published on Web 09/01/2007
9934 Langmuir, Vol. 23, No. 20, 2007
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Figure 1. (a) Pictures collected for 4 mm along the growth direction of a colloidal crystal grown at 35 °C, deposited from a 700-nmdiameter polystyrene latex suspension of 0.2% particle volume fraction. Areas with the same thickness have been shaded with the same color. The vertical arrow indicates the growth direction, and the scale bar is 0.3 mm. The number of layers corresponding to each colored region is also indicated. (b) Normal incidence specular reflectance spectra taken every 0.1 mm along the dashed line shown in part a, from 0.1-mm-side square spots. The arrow indicates the growth direction. These curves have been vertically shifted for clarity. (c) Corresponding thickness profile extracted from the micro-optical analysis.
measured along the growth direction). They provide a more detailed expression than eq 1 in which this effect is taken into account
H(t) )
[
( )]
φVe Vs βL +t 11 - (1 - φ)Vg,tf∞ Ve
(2)
where Vs is the sedimentation velocity and t is the elapsed time. In this previous work, thickness profiles of colloidal crystal films deposited from 1 µm polystyrene sphere suspensions of 0.8, 0.4, and 0.2% particle volume fraction at a constant temperature of 45 °C are fitted using eq 2. Each data set used in ref 12 is the aggregate of the thickness profiles of three replicate samples measured along a 15 mm line parallel to the growth direction using a 4× microscope objective. From these measurements, they obtained experimental thickness profiles that show a longrange linear increase from a few micrometers to several tens of micrometers.
Experimental Thickness Profiles However, when viewed under the optical microscope, colloidal crystal films show a series of thin terraces that are usually oriented perpendicular to the growth direction and whose width ranges
from 100 µm to 1 mm. An example of this is shown in Figure 1a, in which we present a series of photographs taken in reflection mode with a 4× microscope objective for a film made by EISA from a 700 nm polystyrene sphere suspension of 0.2% particle volume fraction and evaporated at a constant temperature of 35 °C. The arrow is parallel to the film growth direction and, for the sake of clarity, terraces have been artificially colored. These pictures indicate that abrupt fluctuations in thickness take place over short distances and help to establish the maximum spot area to be tested in order to attain an accurate profile of the colloidal crystal films. In our case, we have obtained thickness profiles taking advantage of detailed information on the film structure that can be extracted from the analysis of its photonic crystal properties using reflectance microspectroscopy. These measurements were performed using a Fourier transform infrared spectrophotometer (Bruker IFS-66) attached to a microscope. A 10× objective with a numerical aperture of 0.1 (light cone angle (5.7°) was used to irradiate the lattices and collect the reflected light at quasinormal incidence with respect to its surface. A spatial filter was used to detect light selectively from 100 × 100 µm2 square regions of the sample. Spectra were taken every 100 µm along directions parallel to the growth direction. In Figure 1b, we show the optical reflectance spectra measured from 100 × 100 µm2 spots at regularly spaced intervals of 100 µm along the dotted line drawn in Figure 1a. Reflectance results were fitted using a code based on the scalar wave approximation.14 From the optimum fittings, attained using squared minima, the crystal thickness was extracted at each measured spot, and we could build a detailed thickness profile along the growth direction, shown in Figure 1c in sphere monolayer units. Interestingly, in this example, an oscillatory behavior of the film thickness is clearly observed on top of a linearly increasing background. Very similar oscillatory profiles were attained along different parallel lines along the growth direction. Actually, we could confirm that the zones artificially shaded with the same color in Figure 1a were indeed terraces of similar thickness. The behavior reported in Figure 1c was confirmed for films prepared under different conditions on glass substrates following an EISA procedure described in detail elsewhere.15 Figure 2a shows thickness profiles of colloidal crystal films deposited from 700-nm-diameter polystyrene sphere suspensions (Ikerlat, average diameter of 700 nm, polydispersity below 3%, density F ) 1.1 g/cm3, refractive index n ) 1.59) of 0.2, 0.15, and 0.10% particle volume fractions evaporated at 35 °C (Figure 2a) and 40 °C (Figure 2b). In all cases, thickness increases linearly with distance in the growth direction at long range, with the respective slopes being directly proportional to the particle volume fraction, as previously reported by Braun et al.12 However, at short range, abrupt fluctuations of the thickness are observed, with varying period and amplitude of the fluctuations from one sample to another. As the particle volume fraction diminishes and temperature increases, short-range steplike fluctuations are more frequently observed than are oscillatory ones. The observed short-range fluctuations, either oscillatory or steplike, cannot be explained by using eq 2. To develop a colloidal (14) The scalar wave approximation has been repeatedly employed to successfully fit the optical properties of colloidal crystals and complex architectures built on the basis of them. Examples of its application to colloidal crystals can be found in the following references: (a) Mittleman, D. M.; Bertone, J. F.; Jiang, P.; Hwang, K. S.; Colvin, V. L. J. Chem. Phys. 1999, 111, 345. (b) Jiang, P.; Ostojic, G. N.; Narat, R.; Mittleman, D. M.; Colvin, V. L. AdV. Mater. 2001, 13, 389. (c) von Freymann, G.; John, S.; Wong, S.; Kitaev, V.; Ozin, G. A. Appl. Phys. Lett. 2005, 86, 053 108. (d) Mihi, A.; Mı´guez, H. J. Phys. Chem. B 2005, 109, 15 968. (15) Pozas, R.; Mihi, A.; Ocan˜a, M.; Mı´guez, H. AdV. Mater. 2006, 16, 1183.
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Langmuir, Vol. 23, No. 20, 2007 9935
Figure 3. Scheme of the forces acting on the contact line and particle and water fluxes in the vicinity of the forming lattice in the meniscus.
Figure 2. Thickness profiles of colloidal crystals vertically deposited from 700-nm-diameter polystyrene latex suspensions of 0.2% (squares), 0.15% (circles) and 0.1% (triangles) particle volume fraction at (a) 35 °C and (b) 40 °C.
crystal film growth model in which these thickness variations arise naturally, we make use of previous theoretical descriptions built up to understand the structure of thin films deposited onto horizontal solid surfaces as a result of the drying of droplets in which colloidal particles are suspended.16 As in our convective self-assembly process, as the liquid evaporates, particles will accumulate in the vicinity of the three-phase contact line forming a film. Although they are much more subtle in our case, the structural fluctuations that we observe resemble those reported for such films, except that in ref 16 no ordering of the particles occurs.
Growth Dynamics of Colloidal Crystals and Its Effect on Film Thickness Our main assumption is that the array’s growing edge is a straight line advancing at a rate determined by the speed of the contact line between the meniscus and the substrate. We use the model developed by Adachi et al.16 to calculate the evolution of the position of the contact line with time. A schematized drawing of the forming colloidal film is shown in Figure 3. The contact line is defined as the intersection of the extrapolated meniscus with the plane of the wetting-film surface. To establish the dynamics of this contact line, we need to clearly identify the forces acting on it. These are indicated in Figure 3. We assume that a liquid flow Jw, induced by its evaporation Je through the array surface, carries the particles from the bulk of the suspension to the boundary of the forming lattice in the meniscus. This force, coupled with the viscous drag on the particles, is responsible for particle motion in the meniscus, which can then be described as a net flow Jp. Thus, neglecting energy-transfer processes, we consider that the evaporative flow Je induces Jw and Jp. Because the suspension flow (a mixture of Jp and Jw) is viscous, a friction force σ acts on the wetting-film surface close to the contact line (16) Adachi, E.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057.
so as to prevent the meniscus from moving downward. The surface tension at the wetting-film-air interface γf also tends to prevent the meniscus from falling. In contrast, the surface tension at the liquid-air interface γl pulls down the contact line. In brief, γl competes with γf and σ at the meniscus contact line. When the suspension is treated as a continuous fluid, the flows Jp and Jw are defined as Jp ) φVp/Vp and Jw ) (1 - φ)Vw/Vw, where φ is the particle volume fraction in the wetting film; Vp (Vp) and Vw (Vw) are the volumes (average velocities) of a particle and a water molecule, respectively. The equation that governs the motion of the contact line and therefore the rate of film formation is
F*
d 2R ) γl cos(θ) - γf - σ(t) dt2
(3)
where F* is the effective mass of the contact line, R is the position of the contact line with respect to an arbitrary origin of coordinates that we locate at the top of the vertical substrate, and θ is the effective contact angle of the meniscus. For the sake of simplicity, the gravitational force is dismissed, which renders this model valid only for particles whose density is similar to the liquid phase of the dispersion, as is the case for polystyrene spheres suspended in water. The friction force is proportional to the particle flow at the contact line and can be defined as16
σ(t) )
2η βφ + R(1 - φ) V J (t) h βφ[φ + R(1 - φ)] p p
(4)
where η is the shear viscosity, h is the wetting-film thickness, and R is the ratio of the mass density of the particles suspended in the fluid to the density of that fluid. To find a solution for eq 3, another relation between Jp ) Jp(t) and R ) R(t) is necessary. As in ref 16, this relation is obtained by considering two conservation laws: one for the particle number and another for the total volume of the particle and water molecules during film formation, corresponding to eqs 5 and 6, respectively, as shown below
∫RR dr lhφp ) ∫0t dt lhVpJp
(5)
∫RR dr l(1 - φp)JeVw ) lh(VpJp + VwJw) + lhdR dt
(6)
0
0
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in which we have introduced R0 as the position of the contact line at t ) 0, φp is the volume fraction of the particles in the particle array and in the wetting film, and l is the horizontal length of the film perpendicular to the growth direction. Combining these two conservation laws then yields
[
JeVw dJp h d2R dR )+ VpJp dt hVp dt JeVw dt2
]
(7)
Equations 3 and 7 fully determine the motion of the contact line. By combining these two equations, one arrives at
d3R d2 R dR + 2λ + (λ2 + µ2) - Vs(λ2 + µ2) ) 0 3 2 dt dt dt
H(t) ) Hs(t) + Hl(t)
h βφ[φ + R(1 - φ)] (γl cos(θ) - γf) 2η βφ + R(1 - φ)
λ)
[
µ2 )
(9)
]
(10)
2ηJeVw βφ + R(1 - φ) - λ2 h2F* βφ[φ + R(1 - φ)]
(11)
2η βφ + R(1 - φ) 1 JV 2h e w F* βφ[φ + R(1 - φ)]
To achieve a solution, some assumptions are made. First, we take dR/dt|t)0 ) 0 and d2R/dt2|t)0 ) 0 as initial conditions (i.e., the velocity and acceleration of the contact line are 0 at t ) 0). Second, we consider a constant contact angle, θ. The particle volume fraction in the wetting film, φ, is supposed to be weakly dependent on time. Finally, a constant value of the wetting-film thickness, h, is considered, as it is implicitly done in ref 16. On the basis of these hypotheses, we obtain solutions of the master equation (eq 8) that explicitly show an oscillatory dependence of the position of the contact line with time:
1 R(t) ) R0 + Vs t - e-λt sin(µt) µ
[
]
(12)
Simultaneously, the average particle velocity Vg(t), and thus the particle flow and the friction force exerted on the contact line, oscillates as given by the expression extracted by substituting eq 12 into eq 3:
Vp(t) )
{
h β[φ + R(1 - φ)] (γl cos(θ) - γf) 2η βφ + R(1 - φ) Vs F* e-λt[(µ2 - λ2)sin(µt) + 2λµ cos(µt)] µ
{
}}
(13)
Assuming that the colloidal crystal film grows at the same speed, the contact lines shifts downward, and from R(t), we can obtain the crystal growth velocity Vg(t) as
Vg(t) )
{
dR(t) λ ) Vs 1 + e-λt sin(µt) - cos(µt) dt µ
[
(15)
(8)
When compared to the analogous result attained in ref 16 for cylindrical geometry, it can be readily seen that the master equation (eq 8) is much simpler because of the Cartesian geometry of our system, with nonlinear terms being found in our case. We have used similar notation to that employed by Adachi et al. for the constant coefficients of eq 8 because, as we will see later, it allows for better comprehension of their physical meaning. Their expressions as applied to our system are given by
Vs )
This expression explicitly shows that crystalline film growth occurs at a rate that depends on time through the product of an exponentially decreasing function with another function that is periodically dependent on time. The parameter Vs is the limiting crystal growth velocity attained when steady state is reached (t f ∞), µ is the period, and λ is a parameter linked to the extinction of those periodic fluctuations with time, which operates through the factor e-λt. The thickness H(t) of the colloidal films formed by EISA can now be obtained by substituting Vg(t) derived in eq 14 into a general expression of the form
]} (14)
where we have separated the functions describing the shortrange (Hs) and the long-range (Hl) behavior Hs(t) )
φVeβL (1 - )(1 - φ)Vs
[1 + e (µλ sin(µt) - cos(µt))]
Hl(t) )
-λt
φ (V - Vs)t (1 - ) e
-1
(16)
(17)
Please notice that the short-range time dependence is of the general form Hs(t) ) C/Vg and the expression for the constant C is derived by imposing the condition that for long times (Vg,tf∞ ) Vs) eqs 15 and 2 should be equal. In eqs 16 and 17, it is explicitly shown that there is short-range, periodic variation of the thickness of colloidal particle films, with the period µ given by eq 11 and long-range linear increases with the same slope as that derived in ref 12. It should be noted that we attain fluctuations of the deposited crystal thickness H(t) while keeping constant the wetting film thickness h within which the particles that will later deposit are confined. The short-range behavior is a direct consequence of the periodic dependence of the film deposition velocity Vg with time, which results in turn from the periodic fluctuation of the friction force originating from oscillatory particle flows at the meniscus. Thus, we arrive at an expression that qualitatively accounts for the presence of periodic fluctuations of the crystal thickness observed experimentally and reported in Figure 2.
Comparison between Theoretical and Experimental Profiles and Discussion We used eq 15 to fit the experimental thickness profiles shown in Figure 2. For fitting purposes, the constant wetting film thickness is taken to be at least like the maximum crystal thickness measured. Figure 4a,c shows the fittings of two thickness profiles corresponding to colloidal crystals deposited at 35 and 40 °C, from 700 nm polystyrene latex suspensions of 0.20 and 0.15% particle volume fractions, respectively. We have chosen these two examples because they well represent the kinds of profiles observed: oscillatory and steplike. Distance units in Figure 2 have been translated into time units in Figure 4 by measuring the time required to cover the total length of the crystal analyzed along the growth direction and assuming a reasonable value for the starting time of the process t0. Red squares in Figure 4a and green circles in Figure 4c correspond to the experimental data. (The connecting solid line is only a guide for the eye.) Superimposed dashed lines correspond to the best fit found using eq 15, and black squares and circles are those same fitting curves after considering the discrete nature of the substance being deposited (i.e., that the minimum increase in crystal thickness is on the order of one sphere diameter). Once this fact is taken into account, oscillatory and steplike profiles arise naturally as
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Langmuir, Vol. 23, No. 20, 2007 9937
friction force given by eq 4 also increases. Then, Vg is reduced, and the array’s growing edge position plateaus with time. Whereas the contact line position remains approximately constant, particles keep arriving at a fast rate to the meniscus, thus favoring the increase in film thickness. Consequently, crystal thickness reaches a maximum, according to eq 16. At some point, Vp starts decreasing and the friction force is gradually reduced, allowing for the meniscus to move downward more rapidly. Thus, the forming film gets thinner because particles arrive at the array more slowly (i.e., their flux is smaller) and they do not accumulate because of the fast shift of the contact line (i.e., Vg is large). Thus, the array’s edge undergoes a stick-slip motion that strongly affects the structure of the forming film. It is interesting to notice that in our approximation there is no need to take into account the crystallization process itself to explain the observed thickness variation qualitatively, except for the geometrical implications of the presence of stacks of close-packed layers considered in the discretization of the calculated profiles. Also, we assume no periodic variation of the contact angle with time, a hypothesis that had been proposed to account for the presence of stripes of different crystal thickness, although no detailed mathematical description of the process was provided.13 Considering the values of the adjustable parameters extracted from the fittings (t0, R0, Vs, λ, and µ) shown in Figure 4 (see caption), we can further approximate the growth velocity as
Vg = Vs[1 - e-λt cos(µt)]
(18)
and consequently Figure 4. (a, c) Experimental (black) and theoretical (colored) thickness profiles. The dashed line corresponds to the calculated thickness, assuming that the crystal grows continuously in each case. (b, d) Oscillatory motion of the contact line and evolution of the crystal growth velocity (dashed line) and mean velocity of particles arriving at the array (dotted line) with time for colloidal crystals deposited from 700 nm polystyrene latex suspension of φ ) 0.20% at 35 °C and φ ) 0.15% at 40 °C, respectively. Fitting curves shown in plots a and b were calculated using parameters t0 ) 0, R0 ) 0.2086 mm, Vs ) 1.670 × 10-8 m‚s-1, λ ) 5.134 × 10-7 s-1, and µ ) 6.452 × 10-5 s-1, and the fitting curves shown in plots c and d were attained using t0 ) 27 840 s, R0 ) -0.3317 mm, Vs ) 2.705 × 10-8 m‚s-1, λ ) 2.565 × 10-6 s-1, and µ ) 7.464 × 10-5 s-1.
a consequence of the different amplitudes of the thickness oscillations described in eq 16 for the different lattices under study. Only when the expected lattice thickness increment over a certain distance is larger than the interplanar distance along the [111] direction of the fcc lattice, which is the crystalline direction perpendicular to the substrate,1 does an actual increase in thickness occurs. Predicted oscillations will only occur in the real lattice if the difference between the thickness reached at the top of the oscillation and the thickness after the oscillation is completed is at least one lattice plane distance, as for the three oscillations observed in Figure 4a. Otherwise, it will give rise to a longer plateau, as can be seen in Figure 4c. Hence, both types of thickness profiles are explained within the same model and can be reproduced through the adequate choice of the fitting parameters. To provide some insight into the relation between the processes taking place at the meniscus and the resulting crystal thickness, we plot the curves of different relevant magnitudes versus time in Figure 4b,d for the specific cases under consideration. The evolution of the contact line position with time, the speed of the contact line, which we consider to be the film growth velocity Vg (eq 14), and the mean velocity of particles arriving at the array Vp (eq 13) are displayed. As Vp increases, the corresponding
Hs(t) )
φVeβL
1 (1 - )(1 - φ)Vs 1 - e-λt cos(µt)
(19)
These simplified expressions allow us to easily identify the parameters that determine the characteristics of the observed short-range fluctuations: parameter λ, which is the coefficient of extinction of the thickness fluctuations, and parameter µ, which represents the period of those fluctuations. With these two parameters, we understand the trends observed experimentally and displayed in Figure 2. After eq 10, the extinction of the oscillations is proportional to the evaporative flow and inversely proportional to the thickness of the wetting film. Therefore, the higher the temperature at which the EISA process takes place, the larger Je and the more attenuated the thickness fluctuations are expected to be. This is actually observed in Figure 2 when the profiles of crystals formed from suspensions of similar particle concentration but deposited at different temperatures are compared. At the same time, a thicker average wetting film h is expected for larger particle concentrations. This means that at a fixed temperature, oscillations will have higher amplitudes as we increase the particle concentration, as is actually the case in Figure 2. The fact that in the only precedent study of thickness profiles reported,12 no fluctuations were detected might be due to both to the averaging of the thickness of replicate samples and the coarser resolution employed when acquiring the optical reflectance spectra used for the analysis. It is interesting that in eqs 16 and 17 there is an implicit intrinsic competition between short- and long-range film uniformity: if we increase the evaporative flow (through, for instance, temperature), we can eliminate short-range thickness fluctuations, but we will increase the slope of the linear profile observed at long range.
Conclusions We have experimentally and theoretically analyzed the thickness profiles of colloidal crystal films. On the basis of our observations, we have proposed a model to describe the growth
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dynamics of colloidal particle films deposited by evaporationinduced self-assembly. We have shown that experimental observations can be explained by taking into account two basic concepts. First, for an evaporating colloidal suspension in which the evaporation velocity exceeds the sedimentation velocity, particles accumulate at the solvent-air interface, and this gives rise to a linear increase in thickness when analyzed over long distances, as proposed in ref 12. Second, during evaporation, the motion of the contact line resembles the stick-slip motion and shifts downward following an undulating dependence with time. This latter phenomenon causes short-range thickness fluctuations. With our approximation, the different types of thickness
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fluctuations with time observed experimentally (periodic oscillations and steplike) arise as particular cases of the same model and are fitted by the simulations. This analysis presents the possibility of finding the experimental conditions under which to prepare more uniform colloidal photonic crystals with higher optical quality. Acknowledgment. This work has been funded by the Spanish Ministry of Science and Education under grant MAT2004-03028. G.L. thanks CSIC for funding through a Ph.D. scholarship. LA701737V
