Langmuir 2006, 22, 2249-2257
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Formation and Drying of Colloidal Crystals Using Nanosized Silica Particles Fre´de´ric Juillerat, Paul Bowen, and Heinrich Hofmann* Laboratory of Powder Technology, Institute of Materials, Ecole Polytechnique Fe´ de´ rale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland ReceiVed August 24, 2005. In Final Form: December 8, 2005 Much interest has been generated in the fabrication of colloidal crystals from suspensions because of the promise of photonic band gap applications. However, since the case of small, nonsedimenting colloidal particles indeed remains rather rarely treated, spherical silica particles with diameters varying from 75 down to 20 nm have been used in the present work to fabricate colloidal crystals by drying the suspending liquid. Typical events that take place during the drying process of a particulate film, such as cracking, compaction and penetration of air into a porous network, have been evaluated using existing theories, and the maximum stress in the drying film could be approximated. Investigation on the dry film structure by scanning electron microscopy showed the arrangement of particles in a close-packed system. To interpret the formation of such crystals, the amplitudes of the interparticle and capillary forces have been estimated from existing models. The repulsive interparticle forces allow the particles to remain stable and thus rearrange up to fairly high particle concentration. These modeling results showed the dominance of the capillary contribution at the end of the drying process. Nitrogen adsorption/desorption measurements gave very coherent results regarding both pore volume and pore size of the dry particulate films when compared to the expected ordered packing arrangements.
1. Introduction Ordered assemblies of nanometer-sized particles represent an interesting class of nanomaterials that provide exceptional potential to achieve one-, two-, and three-dimensional structures for a wide variety of applications. The building of such patterns down to a nanoscopic level may be achieved by the use of a process such as the self-assembly, and, more generally, the selforganization, of colloidal particles.1 One important example is the photonic band gap crystals. These ordered three-dimensional particulate structures can be obtained by the use of such processes and are therefore also called “colloidal crystals”. These interesting materials possess a patterned periodic dielectric constant that creates an optical band gap2 that can be used to localize light to specific areas, inhibit spontaneous emission, or guide propagation of the light.3 With such aims, colloidal crystals have been built from a variety of materials, including polymers, ceramic materials, inorganic semiconductors, and metals.1,4 An area of potential application for such three-dimensional ordered structures that exploits the optical properties as well as others is sensors. Various gas sensors,5 biosensors6 or chemosensors7 have thus been studied using colloidal crystals with particles of different diameters and using specific aspects of their properties. Two features appear to be crucial for the use of colloidal crystals as efficient sensing devices. First, the ordering length needs to be optimized and the defect density reduced. Second, the crystal must have sufficient macroscopic dimensions with respect to the application and be free of cracks. Investigation of the stress that occurs during the crystal formation and drying stage thus appears to be essential. Applications such as photonic band gap crystals, but also tunable optical filters8 and other reversible photonic crystal * Corresponding author. E-mail:
[email protected]. (1) Dutta, J.; Hofmann, H. Encycl. Nanosci. Nanotechnol. 2003, 9, 617-640. (2) Park, W.; King, J. S.; Neff, C. W.; Lindell, C.; Summers, C. J. Phys. Status Solidi B 2002, 229, 949-960. (3) Park, S. H.; Gates, B.; Xia, Y. AdV. Mater. 1999, 11, 462-466. (4) Jiang, P.; Bertone, J. F.; Hwang, K. S.; Colvin, V. L. Chem. Mater. 1999, 11, 2132-2140. (5) Wohltjen, H.; Snow, A. W. Anal. Chem. 1998, 70, 2856-2859. (6) Haynes, C. L.; Van Duyne, R. P. J. Phys. Chem. B 2001, 105, 5599-5611. (7) Lee, K.; Asher, S. A. J. Am. Chem. Soc. 2000, 122, 9534-9537.
mirrors,9 act in the visible and thus require the crystal to be constituted of particles with diameters larger than 200 nm. For such sizes, sedimentation and drying from a colloidal suspension has most commonly been used10 to obtain colloidal crystals, but layer-by-layer deposition of binary particles,11 shear-flow selfassembly12,13 and convective self-assembly on patterned substrates14 have also been used as alternative methods. In the case of particles smaller than 100 nm, which are required to obtain materials that would exhibit specific luminescent2 or conductive15 properties for example, the sedimentation process can no longer be used. Murray et al. have widely reviewed the assembly process of particles of many different materials, smaller than 20 nm, into three-dimensional ordered arrays.16 They notably describe the use of slow destabilization by evaporation from a mixture of solvents to obtain ordered colloidal crystals. However, the use of complex solvents to obtain such ordered structures has rendered rigorous theoretical study and modeling of the formation mechanisms of such ordered three-dimensional structures difficult. There is indeed a lack of work dealing with fundamental contributions to the self-assembly process of nanoparticles such as interparticle, capillary or adhesion forces, and their use to describe the particulate crystal formation process. The analysis of such process should lead to a better understanding concerning the behavior of the investigated particles in their suspending medium, as well as the particulate film formation. The aim of the present work was therefore to first investigate a method to obtain colloidal crystals using aqueous suspensions with particles in the sub-100-nm range, where Brownian (8) Park, S. H.; Xia, Y. Langmuir 1999, 15, 266-273. (9) Xu, X.; Majetich, S. A.; Asher, S. J. Am. Chem. Soc. 2002, 124, 1386413868. (10) Xia, Y.; Gates, B.; Lin, Y.; Lu, Y. AdV. Mater. 2000, 12, 693-713. (11) Velikov, K. P.; Christova, C. G.; Dullens, R. P. A.; van Blaaderen, A. Science 2002, 296, 106-109. (12) Park, S. H.; Qin, D.; Xia, Y. X. AdV. Mater. 1998, 10, 1028-1032. (13) Gates, B.; Qin, D.; Xia, Y. AdV. Mater. 1999, 11, 466-469. (14) Hoogenboom, J. P.; Re´tif, C.; de Bres, E.; van de Boer, M.; van LangenSuurling, A. K.; van Blaaderen, A. Nano Lett. 2004, 4, 205-208. (15) Pileni, M. P. J. Phys. Chem. B 2001, 105, 3358-3371. (16) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Annu. ReV. Mater. Sci. 2000, 30, 545-610.
10.1021/la052304a CCC: $33.50 © 2006 American Chemical Society Published on Web 02/01/2006
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Table 1. Summary of the Properties of the Colloidal Suspensions Used in the Present Work
commercial name
concentration [g/mL] (vol fraction)
Klebosol 150H50 0.36 (∼0.16) Klebosol 1508-35 0.36 (∼0.16) Klebosol 20H12 0.20 (∼0.10)
density (pycnometry)a
mean volume diameter [nm]b
He
H2O
PCS
SEM
32.20 2.20 2.19
2.14 2.27
79 (5) 47 (8) 21 (2)
72 (7) 48 (6) 20 (2)
a Both He and water pycnometry were used to estimate the particle density. b The values in parentheses represent the standard deviation of the size distribution.
movement dominates gravity. The evaporation rate of the solvent, the pH, and the ionic strength have been investigated since they are all known to act on the various interparticle interactions involved in the process. The relatively simple system used in the present work, that is, particles suspended in water without additives (such as polymers), was chosen to allow the use of various theoretical models to interpret the macroscopic formation of ordered particulate films. In particular, the phenomenon of cracking and the general assembly process itself that leads to crystal formation were thoroughly invested to provide a solid basis for future applications where more specific geometries are of interest.
Figure 1. Schematic illustration of the Teflon-ring cell used for drying the colloidal suspension to obtain colloidal crystals.
2. Materials and Methods 2.1. Materials. Silica colloids, 75, 45, and 20 nm in diameter, dispersed in water, and supplied by Clariant, were used in the present experiments, and Table 1 summarizes their general characteristics. For practical reasons, this study has essentially focused on the 75 nm particles. To characterize these particles, the mean volume diameters have been determined by photon correlation spectroscopy (PCS) in dynamic light scattering (DLS) mode using a Brookhaven BI-9000AT, and by scanning electron microscopy (SEM) using a Philips XLS 30, measuring at least 300 particles. A Philips CM 20 transmission electron microscope was also used for further characterization. The pH of the 75 nm suspensions was initially 2.4 and rose to 3.5 after dialysis against ultrapure water. The ionic strength, Ic, of the original suspension was 2.1 × 10-2 M after conductance measurements (3850 µS), and decreased to 1.4 × 10-3 M (250 µS) after dialysis. Substrates used as supports for the preparation of colloidal crystals were generally glass microscope slides, but various others such as indium tin oxide (ITO) thin films, silicon wafers, and Au, TiO2, and Pt thin films on Si wafers were investigated as well. Successive washing of the support in acetone, 2-propanol, and ultrapure water was performed prior to use in experiments. 2.2. Method. The easiest way of achieving particulate films from a colloidal suspension is to evaporate a droplet of the colloidal suspension on a flat substrate. However, to gain more control over both the drying front and the drying rate, the use of a Teflon-ring cell in a RUMED climatic test cabinet was preferred. The Teflonring cell, suggested by the experiments of Dushkin et al.,17 consists of a Teflon ring with an inner diameter of 15 mm and a thickness of 5 mm fixed by a screw system onto the substrate, as illustrated in Figure 1. A volume of 100 µL of concentrated silica suspension (0.36 g/mL) is then added using a micropipet, and evaporation of the solvent can take place. The temperature and relative humidity (RH) in the climatic cabinet was varied between 10 and 30 °C and 10%-90% respectively. To investigate the effect of the suspending medium on the final particle structure and film, the pH was set to either 2 or 10 starting from dialyzed suspensions, which also set the ionic strength of the suspension to 10-2 M. However, since no significant differences in either the micro- or the macrostructure could be clearly established, the experiments presented here were (17) Dushkin, C. D.; Lazarov, G. S.; Kotsev, S. N.; Yoshimura, H.; Nagayama, K. Colloid Polym. Sci. 1999, 277, 914-930.
Figure 2. Optical micrographs of dry silica films after evaporation of a 0.36 g/mL suspension. (a) RH ) 10%, T ) 30 °C, L ) 100 µm; (b) RH ) 90%, T ) 10 °C, L ) 100 µm. (c) Profile of the dry film measured from SEM cross-section analysis. performed using suspensions “as received” (pH ) 2.4; Ic ) 2.1 × 10-2 M (3850 µS)). Investigations of the final nanostructures were performed by SEM, using a JEOL 6300F and a Park Scientific autoprobe CP atomic force microscope (AFM). Nitrogen adsorption/desorption measurements were performed on a Micromeritics ASAP2010 to determine the specific pore volume and pore size of the dry films. A Perkin-Elmer Lambda 900 UV/vis spectrophotometer via an optical fiber probe allowed the measurement of the absorbance characteristics of the liquid film during the evaporation step, and an optical microscope Leica MZ8 equipped with a JVC TK C1380 CCD camera was used to observe the film during drying. A Mettler TG50 thermobalance for very sensitive measurements in a noncontrolled atmosphere and a Mettler AE200 balance for measurements performed in the climatic test cabinet were used to determine the evaporation rate as a function of the atmospheric conditions.
3. Results and Discussion 3.1. Drying Process. Silica films obtained by drying the colloidal suspension in its original concentration (0.36 g/mL) in the Teflon-ring cell were cracked for all the experimental relative humidities and temperatures studied (see Figure 2a,b). The profile of the film was as shown in Figure 2c: its thickness varied from about 70 to 140 µm and is slightly higher at the edge, close to the Teflon ring. Transmission optical microscopy observations performed during the evaporation of the solvent indicated some important
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stages of the drying process, as illustrated in Figure 3a-e. The sudden appearance of cracks and the temporary loss of transparency are the most relevant observations. The latter effect was confirmed by UV/vis spectrometry measurements, where a strong increase in absorbance at 500 nm, close to the end of the evaporation process of the drying film, was observed (Figure 3f). Measurements were carried out to evaluate the evaporation rate of the particulate silica film during the evaporation of the suspending solvent by measuring the mass loss of a droplet of colloidal suspension at 30 °C over time. The results, giving the mass loss as well as the derivative of the mass loss versus time, which is proportional to the evaporation rate, are indicated in Figure 3g. The drying process of a particulate body can be separated into three different stages:18 the constant rate period, the first falling rate period, and the second falling rate period. First, the constant rate period, in which the decrease in volume of the particle network is equal to the volume of liquid lost by evaporation, that is, in which the rate of evaporation per unit area of the drying surface is independent of time. The stress in the drying film reaches its maximum value at the surface of the film at a critical point at the end of this constant rate period. At this point, shrinkage stops as a continuous “rigid” network of particles is formed. This is when cracking is most likely to occur because of the stress difference between the wet and dry parts of the body as the drying front moves toward the interior of the body. The likelihood of cracking is greater for a thin film that dries on a flat rigid substrate. Also, at this point, air starts entering the pores. The saturated body is translucent or transparent because of the similarity in refractive index of the liquid and the solid, but the lower index of air causes significant scattering of light. The drained regions may be big enough to scatter light, even if the pores themselves are too small to do so, explaining why the films turn opaque during the falling rate periods while it is transparent when fully saturated or fully dried. This effect was demonstrated and explained in a very elegant manner by Shaw,19 who investigated the immiscible displacement of liquid in porous media. The author used a film (bed) of packed silica spheres, initially filled with water and in an experimental configuration that only allowed the evaporation of water from one side, causing the propagation of a moving and visible drying front. In our experiments, spectrometry measurements and optical microscopy observations carried out in situ have demonstrated the appearance of cracks and the loss of transparency as air entered the pores at the critical point at the end of the constant rate drying period, in agreement with theoretical expectations.18 In the case of a film dried on a flat, rigid substrate, the network experiences a tensile stress that equals20
σx ) H
(
)( )[
]
1 - 2N V˙ E χ(L) cosh(χ(L)‚u) 1-N L sinh (χ(L))
(1)
where H ) K + (4/3)G is a term that depends on K, the bulk viscosity of the drained network, and G, the shear viscosity of the porous body drained of liquid; N is the Poisson’s ratio of the drained network; V˙ E is the rate of evaporation of the liquid; u ) z/L is the normalized coordinate along the vertical axis, z; L is the thickness of the film; χ(L) ) xL2ηL/D‚H; ηL is the viscosity of the liquid; and D is the permeability of the network. (18) Brinker, C. J.; Scherer, G. W. Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing; Academic Press, Inc.: San Diego, CA, 1990. (19) Shaw T. M. Phys. ReV. Lett. 1987, 59, 1671-1674. (20) Scherer G. W. J. Non-Cryst. Solids 1989, 109, 171-182.
Physically, such a stress finds its origin in the pressure gradient induced by the low permeability of the network that causes the tension in the liquid to be greater near the drying surface. Consequently, the contraction of the network is greater, and the difference in shrinkage rate between the inside and the outside of the body is the cause of a drying stress. Since our system is very similar to the one described by Brinker and Scherer,18 we can reasonably assume that the models they developed will be well adapted to our experimental configuration. After the constant rate period, the first falling rate period describes the process of liquid flow through partially empty pores, while the second falling rate period describes the final stage of drying, when liquid can escape only by diffusion via the vapor phase to the surface. The mass loss measurements (Figure 3g) show these three distinct periods and their duration. The calculation of the maximal stress in the film requires the knowledge of several parameters (eq 1). First, since the stress is maximum at the film surface, one has u ) 1. The Poisson’s ratio of the drained network, N, is known to be close to 0.2 for silica gel.21 To estimate H, one can compare the value of the capillary pressure, Pc, calculated using either H from previous studies20 or the hydraulic radius, rh.18 Since ∫10 [χ(L)cosh(χ(L)‚u)/sinh(χ(L))]du ) 1 and r ) rp ) 2rh at the critical point, with rp as the pore radius, these expressions lead to
2γLV H‚V˙ E ≈ rp L
(2)
where γ is the surface tension of water in air and is 0.072 J/m2, rp ) 2(1 - Fj)/SFjFs, Fj ) Fb/Fs is the relative density, Fb is the bulk density of the solid network (excluding the mass of the liquid), Fs is the density of the solid skeleton (density of silica, 2.2 g/cm3 from He pycnometry measurements), and S is the specific surface area. The relative density can be obtained by knowing Vp, the specific pore volume of the film using
Fj )
1 FsVp + 1
(3)
Nitrogen adsorption/desorption measurements gave a Vp of 2.16 × 10-1 cm3/g, thus one gets Fjd ) 0.678 for the relative density of the dry silica film. Finally, S can be calculated from the measured mean diameter and density, assuming spherical geometry, and one finds S ) 3/aFs ) 36.4 m2/g for 75 nm spherical particles. In reality, nitrogen adsorption gave for S a value close to 50 m2/g. This is attributed to a surface roughness effect, possibly linked to the surface properties of the silica as discussed later. H can now be evaluated by taking into account a mean value for different evaporation rates. As a result, one finds H ≈ 1.3 × 1010 Pa‚s. To check the validity of this approximation, the value of the viscosity η of a suspension with high volume concentration can be estimated from the Doolittle equation21 and compared to this value of H:
η ) ηLC exp
(
Aφ φm - φ
)
(4)
where ηL is the viscosity of the continuous phase (water), with the index L corresponding to “liquid”; C ) 1.2 and A ) 1.65 are two constants that have been determined elsewhere for 80 nm silica particles;22 φ is the actual particle volume fraction; and (21) Scherer G. W. J. Non-Cryst. Solids 1997, 215, 155-168. (22) Marshall, L.; Zukoski, C. F. J. Phys. Chem. 1990, 94, 1164-1171.
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Figure 3. In situ investigations on the experimental drying process: (a) initial deposition of 100 µL of the 75 nm silica suspension (0.36 g/mL) in the Teflon-ring cell, (b) final snapshot before cracking, (c) first cracks propagating in the film, (d) temporary loss of transparency, and (e) dry film. (f) Evolution of absorbance versus time at a wavelength of 500 nm observed by UV/vis spectrometry. (g) Mass loss measurements performed during evaporation of a droplet of highly concentrated 80 nm silica suspension (0.36 g/mL) at 30 °C. Changes in the drying speed close to the end of the drying process can be observed and separated into two different drying rate periods. Graphs f and g have been put together to show the correspondence between each event, as well as the location of images a-e.
φm ) 0.6377 is the volume fraction in the case of a random close packing of the spheres. Using this equation, one finds that η is on the order of magnitude of 1010 Pa‚s for φ ) 0.605. This value
is very similar to the one obtained when estimating the amount of water that remains in the film at the critical point and measured by mass loss measurements, which is φ ) 0.607 ( 0.005 (the
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Table 2. Values of the Maximal Stress in the Drying Film at the Critical Point, σmax, Estimated from Eq 1 for Different Atmospheric Conditionsa RH [%]
T [°C]
t [h]
σmax [MPa]
10 50 90
30 20 10
1.7 4.3 20.3
17.5 6.9 1.5
σx ≈
a The drying time necessary to totally evaporate the initial 100 µL of suspension is also given.
error on this value takes into account the error on the mass measurement and the error on the estimation of the precise location of the critical point). Furthermore, the permeability, D, can be calculated using the Carman-Kozeny model:18
D)
(1 - Fj)3 5(FjSFs)2
(5)
From the value of specific pore volume obtained by nitrogen adsorption/desorption measurements, one finds D ) 2.27 × 10-18 m2. Considering all of these values, one can deduce that χ(L) ≈ 0.018, which reduces σx to (3/4)(H‚V˙ E/L). The thickness of the film, L, was evaluated at 100 µm from SEM cross-section analysis (see Figure 2c), and ηL can be found in existing tables as a function of temperature. Finally, in the present experiments, V˙ E was varied in the climatic test cabinet from 1.5 × 10-8 to 1.8 × 10-7 m3/m2‚s (or 1.2 × 10-8 to 1.4 × 10-9 kg/s). These values were obtained after determination of the mass loss as a function of time for the conditions T ) 10 °C, RH ) 90% and T ) 30 °C, RH) 10%, respectively. The following equation then links the relative humidity and temperature to the rate of evaporation V˙ E:17,23
m ) m0 - FwVwAeV˙ E‚t
To estimate the effect of the substrate on the maximum stress, one can estimate the stress of a free film over its surface using following equation:20
(6)
In this equation, m0 is the initial dispersion mass, m is the mass at time t, Fw is the density of water, Vw ) 3 × 10-23 cm3 is the volume of a water molecule, and Ae ) 0.79 cm2 is the surface area of evaporation. The rate of evaporation during the constant rate period can then be obtained by measuring the slope of the curve m(t). Table 2 gives the maximum calculated stress in the film at the critical point and the drying time necessary to arrive at this point for three different atmospheric conditions. Since cracking was observed in all cases, one can deduce that the resistance of the network is lower than 1.5 MPa. Theoretically, decreasing the drying rate should allow the preparation of crackfree films. However, decreasing the stress down to 1 kPa, for instance, would require a drying time of more than 8 months! Further tests performed in a closed atmosphere, for which drying time lasted at least for 2 weeks (V˙ E ) 7.1 × 10-11 kg/s; σmax ) 0.4 MPa), did not avoid the propagation of cracks. In a previous study, Chiu and Cima24 used a substrate deflection method to measure the maximum stress in 40 µm thick particulate films. This value was typically close to 2 MPa when drying 400 nm alumina suspensions at drying rates of 1 × 10-8 to 1.2 × 10-8 kg/s. However, they did not observe cracking when slowing the drying rate down to 6 × 10-10 kg/s (σmax ) 0.4 MPa, from calculations using present models). These results lead to the conclusion that an additional interparticle force is needed to produce crack-free films using sub-100-nm particles, such as the use of a polymeric binder,25,26 by modifying the composition of the liquid16 or using supercritical drying.27 (23) Ro¨dner, S. C.; Wedin, P.; Bergstro¨m, L. Langmuir 2002, 18, 9327-9333. (24) Chiu, R. C.; Cima, M. J. Ceram. Trans. 1992, 26, 88-94.
(
)( )
1 - 2N LηLV˙ E 1-N 3D
(7)
In this case, the stress varies from 0.2 kPa (RH ) 90%, T ) 10 °C) to 1.6 kPa (RH ) 10%, T ) 30 °C). It appears from these values that, in the attempt to prepare crack-free films, the stress due to the substrate is a significant factor to consider. The use of a deformable substrate (liquid, gelatine) might be of practical interest.28 3.2. Microstructure Analysis and Interparticle Force Calculations. SEM and AFM investigations performed on the dry silica films showed arrays of particles in ordered closepacked configurations and throughout the whole film thickness (∼100 µm) (Figure 4). The typical size of these crystalline zones was about 1-10 microns. Regions of random arrangements of particles could also be observed, especially toward the center of the film, where the thickness is the lowest. To understand the mechanisms of formation of such colloidal crystals, we need to develop expressions for each force that can act on a particle initially in suspension as the system goes through the previously described drying steps. The Derjaguin-LandauVerwey-Overbeek (DLVO) theory assumes that the total potential interaction between two particles in suspension in a liquid is a sum of an attractive potential due to van der Waals forces and repulsive potentials that find their origin in mainly electrostatic or steric effects.29 In our case, for the particles still in suspension, only particle-particle interactions from electrostatic repulsive and van der Waals attractive potentials are expected to act. Particles in aqueous dispersions generally have a charged surface. This charge may have different origins, such as the ionization of surface groups (mainly hydroxyl groups), the adsorption of charged molecules, or simply the presence of ions. The amplitude of the surface charge is strongly linked to the pH and ionic strength of the solvent.30 According to Coulomb’s law, each charge generates an electrical potential that will be compensated by opposite charges. This means that around each negatively charged ion, an excess of positive charges will be present and vice-versa. This ion population of opposite charge is called the electric double layer, as it consists of a first layer of ions called the Stern layer at the surface of the particle itself, and a second, more diffuse layer called the Helmholtz layer. The thickness of this second layer also depends on the pH and the ionic strength.31 In the DLVO model, the repulsive forces result from the interaction of the electric double layers. Different expressions have been developed to allow the estimation of the repulsive potential as a function of the interparticle separation distance. One can express the repulsive interaction potentials between two particles (PP) by different expressions, notably, the constant potential or the Hogg-Healy-Fuerstenau approxima(25) Lee, K.; Asher, S. A. J. Am. Chem. Soc. 2000, 122, 9534-9537. (26) Weissman, J. M.; Sunkara, H. B.; Tse, A. S.; Asher, S. A. Science 1996, 274, 959-960. (27) Bellet, D.; Canham, L. AdV. Mater. 1998, 10, 487-490. (28) Belaroui, K.; Rapillard, G.; Bowen, P.; Hofmann, H.; Shklover, V. Key Eng. Mater. 2002, 206-213, 519-522. (29) Myers D. Surfaces, Interfaces and Colloids; VCH Publishers, Inc.: Weinheim, Germany, 1991. (30) Jolivet, J.-P.; Henry, M. De la solution a` l’oxyde: Condensation des cations en solution aqueuse, chimie de surface des oxydes; InterEditions CNRS: Paris, 1994. (31) Landolt, D. Corrosion et chimie de surfaces des me´ taux. Traite´ des mate´ riaux 12; PPUR: Lausanne, Switzerland, 1997.
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Figure 4. Micrographs of a colloidal crystal obtained by drying a highly concentrated (0.36 g/mL) silica 75 nm colloidal suspension: (a) ordered particle array at the film/air surface, (b) SEM cross-section micrograph, (c) detail of the film/substrate interface (the bright line indicates the ordering direction), (d) Fourier transform image of the area indicated in figure c, indicating an ordered close-packed structure, and (e) random arrangement of particles at the center of the film. PP 33,34 tion,32 the linear standard approximation (LSA), VLSA , and the superposition approximation.33 Since the two first expressions are the most common ones, and since they did not show significant difference in amplitude, we chose the LSA model for the present calculations. The following equation gives the variation of the PP particle-particle repulsive electrostatic potential, VLSA , with s, the surface-surface separation distance:30,33,34
PP VLSA ) 4π‚‚0‚a2(ζ exp(κ‚d))2
exp(-κ‚s) 2a + s
(8)
where is the relative permittivity of the suspending fluid (78.5 for water), 0 is the permittivity of vacuum (8.85 × 10-12 C2/ Nm2), a is the particle radius, ζ is the zeta potential of the particles in suspension, d is the thickness of the Stern layer, which can be estimated to be 0.5 nm,31 and κ is the Debye-Hu¨ckel parameter:
κ-1 )
(
0‚k‚T
2‚e ‚Ic‚NA 2
)
0.5
(9)
with NA being Avogadro’s number, k being the Boltzmann constant, e being the elementary charge, Ic ) 1/2 ∑icizi2 in mol/ dm3 being the ionic strength, ci being the ionic concentration, and zi being the number of charge. Note that, in eq 8, the surface potential was approximated to ζ exp(κ‚d) according to Jolivet.30 (32) Ohshima, H. D.; Chan, Y. C.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1983, 92, 232-242. (33) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (34) Gray, J. J.; Bonnecaze, R. T. J. Chem. Phys. 2000, 114, 1366-1381.
To calculate the van der Waals attractive forces between two particles, two assumptions are made. First, each atom of a particle interacts with every atom of the other particle. Second, the free energy resulting from these interactions is the sum of the contributions of every possible atom pair.35 A common model used to characterize the attractive potential between two spheres separated by a surface-surface distance s, in a specific medium, is the Hamaker nonretarded approximation. This potential, VPP H, is written as follows:36-38
( )[
VPP H ) -
(
)]
APP 2a2 2a2 4a2 + + ln 1 2 2 6 s + 4as (s + 2a) (s + 2a)2 (10)
where APP is the Hamaker constant for particles in suspension in a fluid (4.6 × 10-21 J for silica particles in suspension in water39). The forces resulting from such contributions may be obtained from the derivative of the potentials with respect to the separation distance. In the present case, numerical derivation was performed to calculate forces. In the case of the repulsive interactions, it appeared that the resulting forces were very close to those obtained using the following mathematical formula for the derivatives proposed elsewhere:18 (35) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain, 2nd ed.; WileyVCH: New York, 1999. (36) Bowen, W. R.; Jenner, F. AdV. Colloid Interface Sci. 1995, 56, 201-243. (37) Motte, L.; Lacaze, E.; Maillard, M.; Pileni, M. P. Langmuir 2000, 16, 3803-3812. (38) Hein, K.; Hucke, T.; Stintz, M.; Ripperger, S. Part. Part. Syst. Charact. 2002, 19, 269-276. (39) Bergstro¨m, L. AdV. Colloid Interface Sci. 1997, 70, 125-169.
Formation and Drying of Colloidal Crystals PP FLSA ) 2π‚‚0‚(ζ exp(κ‚d))2
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1 + κ(h + 2a) exp(-κh) (11) 2 h +1 2a
(
)
When the liquid starts being removed from the network, capillary forces start acting, generated by the radius of curvature of the liquid surface trapped between two solid surfaces. To evaluate the amplitude of the capillary potential, VPP C , and the , that pull particles toward one another, in the capillary force, FPP C specific case of two particles on a substrate separated by a liquid film (Figure 5), the following equations can be applied:40,41
VPP C ∼ FPP C
2πγLV‚rk2(sin2 ψk)a s
2πγLV‚rk2(sin2 ψk) ∼ s
(12)
Figure 5. Illustration of the capillary forces acting between two particles separated by a center-to-center distance L.
(13)
Table 3. Calculated Values of the Various Interaction Potentials (Bold Character) and Forces (Parentheses) Encountered during the Self-Assembly Process in the Case of Silica 75 nm Silica Particles Suspended in Water at pH 2 (Zeta Potential ) - 10 MV) and with an Ionic Strength of 10-2 Ma
where rk is the radius of the three-phase line at the particle surface, and rk ) [h(2a - h)]1/2; ψk is the meniscus slope angle with ψk ) arcsin(rk/a) - θ; γLV is the liquid-vapor interfacial energy (0.072 J/m2 for water/air); θ is the contact angle of the particle with the suspending medium (estimated experimentally to be 25°); and h is the height of emersion of the particle. The maximal capillary force is expected to occur when h ) a (i.e., rk ) a) and can be calculated by differentiating eq 12 with respect to the distance to obtain eq 13. As a result, one obtains the maximum capillary interactions as follows:
VPP C
2πγLV‚a3 cos2 θ ) s
(14)
FPP C
2πγLV‚a2 cos2 θ ) s
(15)
attractive Hamaker VPP H /kT (FPP H [N])
repulsive LSA VPP LSA/kT (FPP LSA [N])
attractive capillary VPP C /kT (FPP C [N])
1
-2.9 (-1.4 × 10-11)
3.9 (5.5 × 10-12)
-4.8 × 106 (-5.2 × 10-7)
10
-0.1 (-9.3 × 10-14)
0.2 (2.6 × 10-13)
-4.8 × 105 (-5.2 × 10-8)
100
-2.6 × 10-4 (-4.9 × 10-17)
-1.3 × 10-14 (1.8 × 10-26)
-4.8 × 104 (-5.2 × 10-9)
interparticle separation [nm]
a
Negative values are attractive terms, and positive values are repulsive.
In practice, Denkov et al.41 estimated the capillary force using s + 2a instead of simply s. However, it can be shown that relationships very similar to those in eqs 14 and 15 may be found using the value of the capillary pressure between two parallel plates:18
PC )
2γLV cos θ r
(16)
with r ) s/2 cos θ, as indicated in Figure 5, and h ) a. Multiplying PC by the surface that effectively takes part in the process gives the value of the force. This surface will, in any case, be smaller than a quarter of the sphere’s surface, π‚a2, which is immersed and “seen” by the other particle. Using these parameters, one obtains a force about twice as large as that from eq 15, showing that both approaches give the same order of magnitude for the capillary force. Table 3 lists the potentials and forces calculated for 75 nm silica particles at various interparticle separations: a pH of 2.4 and an ionic strength of 10-2 M. These results show the relative importance of capillary forces compared to interparticle forces in suspension. However, it has to be recalled that capillary forces are nonexistent as long as particles do not emerge from the liquid. This offers the possibility for the particles to rearrange until late in the drying process (40) Aizenberg, J.; Braun, P. V.; Wiltzius, P. Phys. ReV. Lett. 2000, 84, 29973000. (41) Denkov, N.; Velev, O.; Kralchevski, P.; Ivanov, I.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183-3190.
Figure 6. Total interaction potential (normalized with respect to kT) as a function of the interparticle separation calculated from the DLVO theory for 75 nm silica particles at pH 2.4 with an ionic strength of 10-2 M.
(probably only slightly before the critical point), as they can move in suspension. They are indeed expected to start agglomerating only for very small particle-particle separations (
