Article pubs.acs.org/Langmuir
Crystallization Mechanisms in Convective Particle Assembly Philip Born,† Andres Munoz,† Christian Cavelius,‡ and Tobias Kraus*,† †
Structure Formation Group and ‡Nano-Cell Interaction, INM−Leibniz-Institute for New Materials, Saarbrücken, Germany S Supporting Information *
ABSTRACT: Colloidal particles are continuously assembled into crystalline particle coatings using convective fluid flows. Assembly takes place inside a meniscus on a wetting reservoir. The shape of the meniscus defines the profile of the convective flow and the motion of the particles. We use optical interference microscopy, particle image velocimetry, and particle tracking to analyze the particles’ trajectory from the liquid reservoir to the film growth front and inside the deposited film as a function of temperature. Our results indicate a transition from assembly at a static film growth front at high deposition temperatures to assembly in a precursor film with high particle mobility at low deposition temperatures. A simple model that compares the convective drag on the particles to the thermal agitation explains this behavior. Convective assembly mechanisms exhibit a pronounced temperature dependency and require a temperature that provides sufficient evaporation. Capillary mechanisms are nearly temperature independent and govern assembly at lower temperatures. The model fits the experimental data with temperature and particle size as variable parameters and allows prediction of the transition temperatures. While the two mechanisms are markedly different, dried particle films from both assembly regimes exhibit hexagonal particle packings. We show that films assembled by convective mechanisms exhibit greater regularity than those assembled by capillary mechanisms.
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INTRODUCTION Two-dimensional lattices of submicrometer particles are useful surface coatings: they induce well-defined roughnesses, curvatures, and periodicities to the interface. Surface roughness affects the interaction with biological systems like cells1,2 and arthropods3 and the contact angles on superhodrophobic4 and self-cleaning surfaces.5 Curvature is an important parameter for magnetic high density data storage6 or microlens arrays.7,8 Periodic surfaces modulate sound and light, for example, in planar waveguides,9 diffraction gratings,10 coatings with bandgaps for certain frequencies,11 and antireflective coatings.12 Metal particle layers provide plasmonic wave guides.13 Particle films can be used as sacrificial layer in particle lithography.14,15 Most of the applications exploit the tendency of the particles to form regular close-packed structures, often referred to as colloidal crystallization.16 A convenient bottom-up fabrication process of colloidal crystals is the convective particle assembly technique.10 Largearea films with a high degree of order can be deposited by convective assembly in simple experimental setups.17−24 The basic principle of convective assembly is the transport of particles from a reservoir into a meniscus that forms on a wetting substrate. Particles are confined at the contact line and deposited onto the substrate. When the substrate is slowly withdrawn, the meniscus moves and the accumulated particles are continuously deposited as a film. Crystallization occurs in a transition region between the meniscus and the dried particle film (see Figure 1). Microscopically, regular particle assembly requires guiding the particles to lattice positions. Two well-developed theories © 2012 American Chemical Society
Figure 1. (a) The three regions of convective particle assembly: In region I, the meniscus, free particles are convectively transported to the particle film. In region II, the particles are arranged to the particle film structure. In region III, the particles form a wet particle film with pronounced evaporation. A deposition in blade coating geometry as used in the experiments is shown. In this geometry a blade fixes the reservoir and straightens the meniscus. Panels (b) and (c) illustrate the crystallization mechanisms. (b) Scheme of convective steering. The particles are transported by the convective solvent flow Jc into niches that define a crystalline packing. (c) Scheme of capillary crystallization. The particles are transported into a thin solvent film. The deformation of the liquid surface induces attractive immersion forces Fi, which drag the particles into a crystalline packing.
describe the microscopic assembly mechanisms that arrange particles during convective particle assembly: convective steering25−27 and capillary crystallization.16,28,29 Received: December 9, 2011 Revised: February 14, 2012 Published: May 4, 2012 8300
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confinement and in external potentials.37,38 Such studies are typically performed on particles confined in slits with solid walls39,40 or on fluid interfaces using external magnetic fields.41,42 The convective assembly geometry we use here imposes little restrictions on particle materials, particle sizes, and substrates and provides easy access for observation. It is therefore suitable for the observation of crystallization and confinement effects on packing as well as particle−particle and particle−wall hydrodynamic interactions in confined geometries. To make quantitative studies possible, the basic assembly mechanisms of convective assembly have to be identified. In the next section, we develop a simple model to estimate the work required to move particles from their position back into the bulk suspension. We divide the convective particle assembly process into three regions (see Figure 1): Region III is the capillary pump, a wetted particle film. Region II is the rim of the liquid meniscus, the region where particles are confined and are attached to the film and where the structure formation takes place. Region I is the liquid meniscus in which the particles are transported from the reservoir to the rim. The action of the capillary pump has already been analyzed before.23 The shape of the meniscus in region I and its effects on film formation are also known.8,33 In this paper, we focus on the processes in region II. Our model describes how much work is required to remove particles from this region to the bulk suspension. Then we experimentally determine the liquid flow profile at different temperatures, analyze the temperature dependency, and identify the assembly mechanism. The different temperature dependencies of the two assembly mechanisms described above allow one to distinguish their roles. The exponential decay of the evaporation rate and the linear behavior of the thermal energy predict a regime where convective steering fails while capillary interactions still lead to regular assembly.
The capillary crystallization model was developed by Nagayama et al. to explain the assembly of particles in thinning liquid films16 (see Figure 1c). The authors identified an attractive capillary interaction, the “immersion force”, between particles that are deforming the liquid surface. It is much stronger than thermal agitation even for very small particles down to a few nanometers if a substrate supports the particles during deformation of the liquid surface.30 Under this strong attraction, small hexagonal packed nuclei form and grow into continuous polycrystalline, two-dimensional, close-packed particle films. Convective flow is neglected in this model of the assembly process; it merely ensures transport of particles from a reservoir to the assembly region of the liquid film. Capillary crystallization processes are temperature-dependent only insofar as the solvent’s surface tension depends on temperature. The surface tension of water drops almost linearly in the relevant temperature range, by ≈6% between 10 and 40 °C.31 The convective steering mechanism depends on the exact motion of the solvent (see Figure 1b). Norris et al. performed a network analysis of the fluid flow through a regular arrangement of particles to explain the striking yield of face-centered cubic particle packings that occur in the convective assembly of bulk crystals and far surpass the fraction of hexagonal closepacked or random hexagonal close-packed particle arrangements.27 They assumed that previously deposited particles form a scaffold for the placement of incoming particles. The solvent flows into the niches of this scaffold and drags the particles with them. The authors showed that niches without underlying particles have the greatest influx, which favors particle stacking in an ABCABC...-sequence, the structure of face-centered cubic crystals. Two types of convective steering are described. The first type occurs for macroscopic particles with large Reynolds numbers Re ≫ 1 and large Peclet numbers Pe ≫ 1, and the second for microscopic particles with Re ≪ 1 and Pe ≫ 1.26,27 Particles with Re ≫ 1 are not steered into specific niches and form a random hexagonal close-packing (rhcp), while particles with Re ≪ 1 are guided into niches to form a face-centered cubic (fcc) packing. The temperature dependence of convective steering stems from the evaporation, and its rate is proportional to the vapor pressure of the solvent, which exponentially depends on temperature. The vapor pressure of water drops by ≈84% between 40 and 10 °C.31 It is yet unknown which mechanism prevails in the convective assembly of two-dimensional crystalline films from a meniscus. A better understanding of the assembly mechanism will have practical implications for controlling the particle packing during convective particle assembly. Presently, only the thickness of the particle film17,23 and the areal fraction of submonolayer8 can be precisely controlled in pure convective assembly. Control over the crystalline domain size is a subject of ongoing research.24,32,33 Moreover, little control of layer structure is possible today. One would like to switch between amorphous and crystalline packings or between fine-grained and coarse-grained crystalline packings during assembly because amorphous films can tailor optical properties34 and have found applications as antireflective coatings5 and in Bragg mirrors.35 Adjustment of the dielectric constant of particle films36 or the structures formed by colloidal lithography14 is also desirable. Today, control over the particle packing requires combinations of different particles (e.g., particles with low and high size dispersities)17,34 or templates.20 Convective particle assembly mechanisms are also related to the phase transitions mechanisms of colloids widely studied in
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MODEL
Thermal agitation works against the confinement imposed by the convective flow. The ratio between convective work and thermal energy is expressed in the dimensionless Peclet number, Pe = vr/D, with a fluid velocity v, a particle radius r, the Stokes−Einstein diffusion coefficient D = kT/6πηr, the fluid viscosity η, and the thermal energy scale kT. The damping of the particle motion by the viscous fluid is expressed by the dimensionless Reynolds number, Re = ρvr/η, where ρ is the particle density. Particles with small Peclet numbers Pe ≪ 1 have so far been neglected in the investigation of the convective steering mechanism. However, the formation of ringlike stains (“coffee stain rings”) reported for evaporating sessile droplets containing nanoparticles and even solute molecules43 suggests that convective mechanisms can confine even objects with Pe ≪ 1. A general model that links the convective assembly mechanism, the convective steering mechanism, and the capillary crystallization is thus required. This model must consider the temperature and location dependence of the fluid velocity v. Here, we develop a model for the flow profile and calculate the drag force on the particles and the work required to move a particle quasi-statically against the flow. Other interactions are neglected. In steady state, the fluid flow force field is approximately conservative, and an equivalent potential energy is introduced. This representation is particularly convenient when analyzing the strength of the particles’ attachment to their lattice position in the growing film in the Crystallization section. A new parameter describing convective 8301
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particle assembly, Pe͠ , is introduced to connect the convective process to capillary crystallization at small Pe.
Volume conservation implies an inverse parabolic shape of the z-averaged velocity of the liquid in the meniscus:
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v (x , T ) = −
TRANSPORT We use Cartesian coordinates with the x-axis parallel to the substrate pointing toward the reservoir, the y-axis parallel to the growth front, and the z-axis perpendicular to the substrate (Figure 1). The deposition process is approximated by a quasistatic two-dimensional problem in the x−z-plane. The particle film is withdrawn from the meniscus at the same rate as the film grows so that the relative position of the growth front and le are constant. Figure 2 illustrates the modeling steps that lead from the meniscus geometry to the velocity profile, the drag force, and,
1 a(T ) + bx + cx 2
(2)
The actual flow profile is parabolic in the z-direction due to the nonslip boundary condition imposed by the substrate. However, the particle size in most of the experiments is comparable to the height of the meniscus at least up to a few particles diameter distance to the film growth front. The particles thus only experience an effective flow, in which the gradient in the z-direction is averaged out. We thus omit the z-component of the flow profile. Note that our choice of coordinates leads to negative velocities of liquid flowing into the film. Parameters b and c depend on the setup geometry, in particular on the position of the blade’s edge that pins the meniscus, and are expected to be constant.33 Parameter a is equal to the inverse fluid velocity at the film growth front at x = 0. The fluid’s flow rate is proportional to the evaporation rate from the particle film,10 which we assume to be proportional to the vapor pressure of the solvent here. The vapor pressure follows the law of Clausius−Clapeyron, which leads to an exponential dependency of the solvent velocity on the temperature for a fixed liquid geometry: v(0, T ) = −
1 ≈ d e −e / T a(T )
(3)
The particles transported by the convective flow have negligible inertia (Reynolds numbers are on the order of 10−4). We can therefore use their velocities (obtained from particle image velocimetry, see experimental section) to estimate the solvent velocity. A fit of the particle velocities to eq 2 yields the parameters a−c. Measurements of different temperatures provide a(T), which can be fit to eq 3 and yield the parameters d and e. A particle with radius r that is held at a constant position experiences Stokes drag exerted by the moving solvent: F(v , x , r , T ) = 6πηr v(x , T )
(4)
where η is the solvent’s viscosity. The work required to move a particle quasi-static from the film edge back into the reservoir therefore equals Φ(x , r , T ) = −
∞
∫0
F (x , r , T ) d x
∫0
∞
= 6πηr
1 dx a(T ) + bx + cx 2
(5)
The integral in eq 5 over the inverse parabolic function can be solved analytically44 to yield Figure 2. Modeling steps from the meniscus geometry to the work required to move a particle back into the reservoir. The parabolic shape of the meniscus h(x) impresses an inverse parabolic velocity profile v(x,T). Using Stokes drag, the force profile F(x,r,T) is derived. Finally, by integration over the force, the work Φ(x,r,T) is calculated. See text for details.
∫ a(T ) + 1bx + cx 2 dx =
⎞ ⎟ ⎟ ⎠
(6) x →∞
finally, the work required to drag a particle back through the convective stream into the reservoir. We start with the parabolic height profile h(x) of the meniscus’ liquid−air interface above the substrate at a distance x from the growth front:33 h(x) = a′ + b′x + c′x 2
⎛ 2cx + b arctan⎜⎜ 2 2 4ca(T ) − b ⎝ 4ca(T ) − b 2
The asymptotic behavior of the arc tangent, arctan(x) ⎯⎯⎯⎯⎯→ π /2 , and arctan(0) = 0 ensures a vanishing force far away from the assembly region in the reservoir and the normalization of the potential. The solvent flow causes the transport of particles from the reservoir to the assembly region. This convective transport
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surpasses diffusive transport if the work required to move particles away from the assembly region is larger than their thermal agitation: Φt (x , r , T ) = −
∫0
∞
F (x , r , T ) d x >
3 kT 2
(7)
If expressed in dimensionless form, the ratio ∞
∞
6πηr∫ v(x , T ) dx ∫ v (x , T ) d x Φ 0 = = 0 ≡ Pe͠ kT kT D
(8)
can be seen as a generalized Peclet number Pe͠ for convective assembly. For Pe͠ ≫ 3/2, particles will accumulate inside the meniscus, while an increase of concentration is impossible for Pe͠ ≪ 3/2. In the model above, the force F(x) created by the fluid flow is one-dimensional and conservative, and the work of removal Φt(x,r,T) defines a potential. The effects of the convective fluid flow are split into a potential that forces the particles into the particle film and a viscous medium that dissipates the particles’ kinetic energy and acts as a heat bath. We find this abstraction useful because it stresses the similarities between convective particle assembly and particle assembly in external potentials. The assembly of particles by sedimentation, for example, requires gravitation to transport and confine the particles and the solvent to provide dissipation and thermal agitation of the particles. We will use the terms “potential energy” or “potential” in the following to acknowledge the mathematical analogy and to emphasize the similarity between convective particle assembly and particle assembly by sedimentation or other external potentials.37,38
Figure 3. Schematic top view of convective particle assembly. In region I, the solvent stream advects free particles, in region II the particles arrange due to the force created by the convective stream and the restrictions by neighboring particles, and in region III the deposited particles form a scaffold for the addition of new particles. d = 2r is the distance particles must be able to diffuse against the solvent stream to become unrestricted in lateral motion.
Determining the shape of the potential from particle velocity measurements would enable predictions on the existence of a critical temperature Tc, defined as the temperature at which the binding energy of the particles decay below their thermal energy. At Tc, the arrangement of the particles would then change from a regime where convective steering ensures colloidal crystallization to a regime where capillary interactions solely control the arrangement process.
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CRYSTALLIZATION If the potential difference between reservoir and assembly region is smaller than 3/2kT, particles can diffuse back into the reservoir. However, even a much larger potential difference does not guarantee regular deposition of the particles. The force generated by the flow field in the above proposed model only acts in the x-direction and does not affect the lateral displacement of the particles. Without any further constraints, the particles would be deposited in a random manner. In the case of capillary crystallization, the attractions by neighboring particles form local potential energy minima that bind and arrange the particles.16 When convective steering governs the assembly, the required constraints originate from particles already attached to the growth front. They form “niches” into which the free particles are transported by the solvent flow.25 We can extend our model to account for this mechanism. The energy required to remove a particle from its niche must be larger than thermal agitation to ensure regular arrangement. If it was smaller, the film growth front would be constantly reconfigured (see Figure 3), and the convective steering must break down. We define the “binding potential” as the work needed to move a particle one diameter away from the film growth front: Φ b (x , r , T ) ≡ −
∫0
2r
F (x , r , T ) d x
EXPERIMENTAL PROCEDURES AND RESULTS
In the following, we experimentally assess the assumptions and predictions of the potential above. The used deposition setup has been previously described.33 The postulated parabolic decay of the fluid velocity with distance from the film growth front and the exponential dependency of the fluid velocity on substrate temperature were verified using particle image velocimetry (PIV) during depositions of 500 nm polystyrene (PS500) and 1 μm silica particles (SiO1000) at various substrate temperatures. The resulting velocity profiles were fit to eqs 2 and 3 to determine parameters a−e. We used these parameters to predict the prevailing crystallization mechanism for different substrate temperatures and particle sizes and tested the predictions via particle tracking. Barium titanate particles (BTO, 300 nm diameter) with high relative permittivities appeared as bright spots even in dense particle films so that we could observe their trajectories even within the deposited particle films. Electron micrographs of films produced from 100 nm polystyrene particles (PS100) under different conditions were analyzed by counting the number of next neighbors of each particle. Detailed descriptions of experimental procedures can be found in the Supporting Information. Transport and Binding Potentials. Figure 4 illustrates experimentally measured velocity profiles. Such profiles were obtained for SiO1000 particles at substrate temperatures of 25, 33, and 40 °C and for the PS500 particles at substrate temperatures of 15, 19, 25, 33, and 40 °C. In all cases, the particles were found to accelerate toward the assembly region, as expected. All particle velocity profiles fit eq 2 with regression coefficients better than 0.95 (Figure 5). Transport of particles to the assembly region in our experiments is described by eq 2 with b = 1.6 × 10−19 s/μm2 (with a standard deviation of 1.9 × 10−18 s/μm2) and c = 9.4 × 10−7 s/μm3 (with a standard deviation of 1.4 × 10−6 s/μm3) from averaging over all measurements. The variations in the linear and quadratic term of the flow profile indicate that the assumption of a static meniscus shape is critical over the full range of temperatures and withdrawal velocities tested in the experiments. The error in the binding potential calculated below caused by the uncertainties in b and c, however, stays below 0.9% for all evaluated particle sices.
(9)
Regular assembly can only take place if the binding potential Φb(x,r,T) exceeds 3/2kT. For a constant fluid velocity v independent of the position x, Φb/kT would equal two times the conventional Peclet number of the particles. 8303
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Figure 4. The process of particle image velocimetry: From two subsequent video frames (a) shifts in the correlation maximum for each interrogation area are calculated (b), which leads to the velocity profile after calibration to the magnification and the frame rate of the video (c). The example shows SiO1000 particles at 25 °C.
Figure 6. Plot of the depth of the binding potential in units of kT as calculated from eq 9. The plot is truncated at 100kT to maintain resolution at lower energies. A distinct dependency on the particle radius and the temperature can be found; below ≈250 nm and ≈20 °C the potential energy rapidly decays to kT.
Figure 5. Velocities v(0,T) of the convective fluid stream calculated for x = 0 at film growth front. The solid line is the result of the fit with eq 3. The trend follows the predictions by the model, and the fluid velocity increases with temperature. The labels indicate the particles used in the measurements.
where Φb takes values of several hundred kT. Below a particle radius of ≈250 nm and below a temperature of ≈20 °C, Φb rapidly decays to kT. This is insufficient to confine the particles at the film growth front and maintain order; the particles have enough thermal energy to escape from their lattice position. The effect of particle size is highlighted in Figure 7. As temperature increases, the depth of the potential well rapidly grows beyond 3/2kT for 500 nm diameter particles, by more than 50kT in the range from 5 to 18 °C. In contrast, the depth of the potential well only increases by 2.8kT between 5 and 20 °C for 100 nm diameter particles. Critical temperatures Tc at which the potential well reaches 3/2kT are shown in Figure 8. Note that only small particles (with radii below 100 nm) exhibit critical temperatures significantly differing from the dew point. For large particles, critical temperatures are around the dew point which is close to 5 °C for the ambient conditions in the laboratory. A further important result is the magnitude of the transporting potential (eq 7). Our model predicts that even small particles are transported and held in the meniscus. For example, a particle with a radius of 5 nm at a substrate temperature of 10 °C still encounters a potential difference of ≈50kT (or Pe͠ = 50) between the film growth
Maximal velocities v(0,T) = −1/a(T) from PIV are plotted in Figure 5. They fit the predictions of eq 3 with a regression coefficient of 0.8 and parameters d = 1.27 × 1010 μm/s (standard error of 5.73 × 109 μm/s) and e = 5376 K (standard error of 1453 K) (errors estimated from the goodness of the fit). As discussed below, there may be additional parameters not included in the model contributing to v(0), which limit the goodness of the fit. The force F(v,x,r,T) (eq 4) that would act on a stagnant particle in the fluid with velocity v(x,T) can now be recovered and the corresponding potential (eq 5) be calculated. Here, we are mainly interested in the magnitude of the binding potential, that is, the potential directly at the film growth front (eq 9). To ease applicability in the laboratory, we give the results of the calculations in units of °C instead of K. Figure 6 is a color-coded map of the binding potential Φb as a function of the particle radius and the substrate temperature. The binding potential shows a strong dependency on particle radius and temperature and diverges for large particles and high temperatures, 8304
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additional sequences can be found in the Supporting Information. For such particles, the binding potential well is too shallow to enforce crystallization below 10 °C; at 8 °C, the binding potential Φb becomes 0.6kT. The particle trajectories at 20 °C hardly deviate from the smooth stream lines of the solvent. They end at the growth front of the particle film, where all independent motion of tracer particles ceases and they merely follow the substrate. In contrast, at 8 °C, the transport of the particles is much slower. Trajectories are still directed toward the growth front but jagged by random detours. They do not end at the film growth front: particles diffuse along the front as predicted by the model. If particles can diffuse along the growth front, convective steering cannot be the prevailing mechanism of crystallization. We find, however, that ordered films still emerge. Capillary forces must be responsible for this crystallization. As Kralchevsky and Nagayama pointed out,30 the immersion forces even of very small particles are stronger than thermal agitation. When the liquid film is enclosing the particles’ film thins beyond the diameter of the particles at some distance from the growth edge, strong capillary interactions pull the particles into hexagonal close packed structures. We investigated the motion of particles inside the film by PIV to analyze this second assembly mechanism. A monolayer of PS100 particles with intermixed BTO particles was deposited. The critical temperature for 100 nm diameter particles is sufficiently above the dew point to avoid condensing water. For such particles, the binding potential Φb is expected to be 0.7kT at 13 °C and 2kT at 17 °C, the temperatures we chose for deposition. It is easy to observe mobility inside the film by observing the BTO particles’ displacements. Figure 10 indicates directions and magnitudes of the tracer displacement from one video frame to another with 0.6 s delay. Little mobility is seen at 17 °C. The particles merely follow the substrate. The particle film at 13 °C exhibits much greater mobility, in particular at the first 40 μm behind the film growth front. Random rearrangement of the particles occurs directly behind the front, while the particles further away from the front increasingly follow the withdrawn substrate. This indicates failing convective steering at the film growth front and a high mobility in the particle film which decays with thinning film. With interference microscopy, the range of the liquid meniscus being thicker than the particle diameter can be measured. This range corresponds to the length of high particle mobility in the deposited film. See the Supporting Information for interference microscopy measurement data. We find that the assembled films have characteristic features that indicate their assembly mechanism. Films of PS100 particles without the addition of BTO tracer particles were deposited at 13 and 17 °C. Figure 11 shows electron micrographs of the resulting particle arrangements. Both exhibit hexagonal dense packing of the polystyrene spheres. The surface coverage was 69.4% for the film deposited at 13 °C and 74.3% for the film deposited at 17 °C, both similar and far below the theoretical maximum of 90.7%. Pronounced differences in the particle packing between the two depositions are quantified by next-neighbors analysis (Figure 12). While most particles deposited at 17 °C have six next neighbors, most particles deposited at 13 °C have only four next neighbors. At 17 °C, most particles have a complete hexagonal neighborhood, while at 13 °C most particles sit at edges of hexagonal grains. The differences in assembly mechanism above and
Figure 7. Plot of the depth of the binding potential as a function of the substrate temperature for two particle diameters. In the case of 500 nm diameter particles, the depth of the binding potential well increases by more than 50kT in the given temperature range, while it increases only by 2.8kT in the case of 100 nm diameter particles.
Figure 8. Plot of the critical temperature Tc at which the potential well reaches 3/2kT as a function of the particle radius. For large particles, this temperature coincides with the dew point in the laboratory, while it diverges for particles below 100 nm. The inset gives an expanded plot of the sub-100 nm region. front and the reservoir. An accumulation of particles in the meniscus thus will happen at all practical temperatures. Crystallization Mechanisms. In the following, we analyze the trajectories of particles outside and inside the deposited film at various temperatures to find how they assume regular lattice positions. We used barium titanate (BTO) tracer particles to analyze particle trajectories above and below the critical temperature. Figure 9 shows two representative trajectories of BTO particles that were codeposited with PS500 particles at 20 and 8 °C. The original video sequences and
Figure 9. Trajectories of BTO particles codeposited with PS500 particles at 20 °C (left) and at 8 °C (right). The convective stream transported the particles from the right to the film growth edge. At 20 °C, the particle is build into the film and follows the withdrawn substrate without lateral displacement, while the particle at 8 °C stays mobile and diffuses along the film growth edge. 8305
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Figure 10. Displacements between two video frames with 0.6 s lag observed in 100 nm diameter polystyrene particle films with BTO tracer particles at 13 and at 17 °C. The film growth front is located at distance = 0 μm. The particles in the film deposited at 13 °C exhibit especially in the first 40 μm from the film growth front random displacements, whereas the particles in the film deposited at 17 °C merely follow the withdrawn substrate.
Together, Pe͠ , the Reynolds number Re, and Pe ≈ Φb/kT delineate four different assembly regimes of convective particle assembly (Table 1). The first and second regime in Table 1 Table 1. Four Regimes of Convective Particle Assembly
Figure 11. Electron micrographs of 100 nm particle films deposited at 13 °C (left) and 17 °C (right).
(1)
Re ≫ 1
Pe ≫ 1
Pe͠ ≫ 1
(2)
Re ≪ 1
Pe ≫ 1
(3)
Re ≪ 1
Re ≪ 1
Pe͠ ≫ 1 Pe͠ ≫ 1
(4)
Re ≪ 1
Pe ≪ 1
Pe͠ ≪ 1
random hexagonal close-packing of the particles fcc packing of the particles convectively assisted capillary assembly of the particles failing of convective assembly
involve crystallization by convective steering as described by Norris and co-workers.26,27 Depending on the Reynolds numbers, steering to specific niches is strong or weak, resulting either in random hexagonal close-packing or fcc packing of the particles. These two regimes cannot be distinguished in two-dimensional film deposition. The third regime describes convectively assisted crystallization by capillary interactions in two-dimensional particle films. It corresponds to the formation of the well-known “coffee stain rings” of solutes in sessile droplets. It involves a two-step
below Tc lead to different particle−particle packings in the deposited film. Below the critical temperature, small-grained particle films are formed.
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DISCUSSION We derived a potential energy model of convective particle assembly that considers the temperature- and position-dependent convective flow. Two new quantities are derived from the potential energy: the effective Peclet number for convective particle assembly Pe͠ and the critical temperature Tc.
Figure 12. Distribution of the numbers of next neighbors for 100 nm particle films deposited at 13 and 17 °C. Most particles have an incomplete neighborhood at 13 °C and thus sit at edges of crystallites, while at 17 °C most particles have a complete neighborhood and are situated within crystalline packings. 8306
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particle sizes and materials will clarify whether additional parameters are needed. Interactions beyond hard exclusion are entirely neglected in our model. The repulsive interactions of the charge-stabilized particles result in a finite distance between the particles in the wet deposited films even above Tc. They are overcome only in the last stages of drying, which causes the observed drying cracks. Possibly, the distance between the assembled particles can be derived from the magnitude of the binding potential and the cracking be predicted. The particles also may experience hydrodynamic resistance during horizontal motion close to the substrate. This effect itself is point of ongoing research.46,47 Convective assisted particle assembly would be an excellent experiment to investigate such hydrodynamic effects. The convectively transported particles match the speed of the solvent, resulting in little hydrodynamic coupling among the particles themselves. Comparing the convective flow profile predicted from interference measurements of the meniscus profile to high-resolution particle image velocimetry can resolve the increasing influence of the substrate with increasing confinement in the thinning meniscus. In summary, our model extends existing descriptions of the convective assembly process by explicitly taking into account the temperature dependence and the nonlinear variation of the fluid velocity with distance from the film growth front. The assembly process is thereby described over a broad range of particle sizes and temperatures, and particle packings are predicted depending on particle size and temperature.
capillary crystallization process that has not yet been reported for particle assembly. In the first and second regimes, crystallization occurs in a single step where advected particles are directly attached to a crystalline particle film. In the third regime, advected particles remain mobile even within the formed high density particle film. When the liquid meniscus thins below the particle diameter at some distance from the film growth edge, particles interact through capillary interactions and stop moving. This two-step crystallization mechanism differs from the capillary crystallization mechanisms described in literature, where particles are bound to a growing static particle film.16 The small-grained appearance of the final particle film suggests continuous nucleation of crystallites in the homogeneous precursor film. The mechanism is probably similar to that suggested by Aizenberg and co-workers:45 Clustering of equally distributed objects protruding through a lowering liquid surface by immersion forces introduces an effective repulsive interaction between neighbors. Small hexagonal particle clusters form, but repel their neighbors and cannot grow into larger grains. The last regime describes the failure of convective assembly when there is too little convective drag to hold solutes and particles in the meniscus. The temperature range of the regimes can be estimated from the evolution of the convective potential Φb with temperature for a certain particle size. The larger the particles, the steeper the increase in the potential with temperature (compare Figure 7). Thus, a micrometer-sized particle will switch from regime 4 to regime 2 around the dew point within a small temperature interval. A nanoparticle, on the other hand, will be assembled in regime 3 over a broad temperature range below Tc. A regime that would yield isotropic, amorphous particle coatings was not observed. One of the crystallization mechanisms is always active if a film is deposited at all. Amorphous particle films can only form with additional quenching, for example, by tailored particle−surface interactions, particle−particle interactions, or switchable external fields. The critical temperature Tc depends on particle size and indicates above which temperature the binding potential Φb surpasses thermal agitation. It delineates the convective steering from the capillary crystallization regime and thus provides a guide to choose the crystallization mechanism. The particle packings formed below and above Tc are qualitatively different. Adjusting temperature with respect to Tc thus aids fine-tuning of the particle packing, for example, to obtain equally spaced particle clusters at low surface coverage in the capillary crystallization regime or dense particle films without drying cracks in the convective steering regime. Appropriate temperature profiles also provide an unconventional option to anneal film defects: short detours to temperatures below the critical temperature Tc increase particle mobility at the film growth front and allow filling of voids and gaps. The simple model introduced here is easily extended. Presently, in the model the particle radius only is considered when calculating the Stokes’ drag. Experimentally we find sizedependent fluid velocities v(0) (Figure 5) particularly at high temperatures. Such deviations can originate from particle size effects on the meniscus geometry and on the evaporation rate of the drying particle film. The latter may also depend on the exact contact angle on the particles and thus the particle material, just as the pressure gradient generated in the wet particle film. Extensive measurements with a higher variety in
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CONCLUSIONS Convective particle assembly was described using a potential energy model. The model is based on the flow profile of the convective flow and the drag acting on the particles. A generalized Peclet number for convective assembly in parabolic menisci was introduced. By applying microscopic particle image velocimetry, a parabolic flow profile and an exponential temperature dependency of the fluid velocity were determined. From the temperature dependency, a transition temperature was predicted below which the convective flow does not confine the particles at the film growth front. The trajectories of tracer particles deposited below the predicted transition temperature indicated failing confinement and diffusion of particles along the film growth front as expected from the model. Hexagonal crystalline particle films formed in either case, albeit with qualitatively different particle packings. The transition temperature thus marks the transition between two assembly mechanisms, convective steering and capillary crystallization. Convective steering assembles free particles into a static particle film. Capillary crystallization assembles the particles after they have been deposited in a dynamic precursor film with high particle mobility. The results clarify the prevailing crystallization mechanism in convective assisted particle assembly. Their temperature dependence can be used to obtain specific structures of the deposited particle films and to anneal packing defects. We explain why hexagonal crystallization always occurs in films formed from well-stabilized, monodisperse particle suspensions in a stable convective process: Isotropic, amorphous particle coatings never form because one of the two assembly mechanisms ensures crystallization at every temperature that leads to film deposition at all. 8307
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ASSOCIATED CONTENT
S Supporting Information *
Detailed experimental procedures, results of interference microscopy measurements of the liquid meniscus shape during particle assembly, and additional videos of particle trajectories above and below the critical temperature to support the qualitatively different particle assembly mechanisms. This material is available free of charge via the Internet at http:// pubs.acs.org/.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Dr. Marcus Koch for scanning electron microscopy and Professor Dr. Eduard Arzt for his continuing support of the project. Funding from the German Science Foundation DFG is gratefully acknowledged.
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