Langmuir 2001, 17, 4867-4875
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Structural Characterization of Dense Colloidal Films Using a Modified Pair Distribution Function and Delaunay Triangulation Peter H. F. Hansen, Sandra Ro¨dner, and Lennart Bergstro¨m* Institute for Surface Chemistry, P.O. Box 5607, SE-11486 Stockholm, Sweden Received December 1, 2000. In Final Form: May 16, 2001 Methods for structural characterization of particle films have been developed. Monodisperse silica particles with alkoxy chains grafted on the surface formed dense colloidal films at the air-liquid interface. The positions of the partly immersed particles were determined by light microscopy and image analysis. The average size of the ordered domains could be estimated from the pair distribution function. We were also able to determine the distribution of pores and defects in the colloidal film using Delaunay triangulation. The two methods were used to study the effect of interparticle bond strength on the film structure. We found that the size of the ordered domains decreased exponentially when the bond strength increased, while the pore density increased. The effect of drying on structural changes of the colloidal film was also discussed.
Introduction Controlled self-assembly of colloidal suspensions is one of several routes that currently attracts much interest for tailoring the properties and the microstructure of new materials.1-5 Photonic crystals are very interesting materials that can be constructed from ordered arrays of micron- and submicron-sized particles.1,2 Band gap properties of photonic crystals, embedded in liquid crystals, can be controlled by electrooptic tuning.3 Two-dimensional assembly of colloidal particles can, for example, be used for very small lasers.4 Porous films and particle gels are also of technical interest; the surface conductivity on composite substrates may, for example, be tailored by the incorporation of a porous network of nanoparticles,5 useful for sensor applications. Different processes have been proposed to produce 2D colloidal films with varying degrees of homogeneity. Picard used dynamic thin laminar flow to form 2D colloidal films at the air-water interface.6 Both large, highly ordered crystals and porous networks could be deposited on substrates by manipulating the particle interactions and the compression rate. Nagayama et al. investigated a method to deposit colloids on a substrate.7,8 They found that the homogeneity of the formed colloidal film was controlled by the evaporation rate and by the rate with which the substrate was pulled out of the suspension. A third method to create a 2D colloidal film is based on a two-step process.9 First, a colloidal film is formed by the self-assembly of floating particles with an attractive interaction; second, the film is transferred to a substrate where it is dried and consolidated. * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Asher, S. A.; Holtz, J.; Weissman, J.; Pan, G. MRS Bull. 1998, 23, 44-50. (2) Blaaderen van, A. MRS Bull. 1998, 23, 39-43. (3) Yablonovitch, E. Nature 1999, 401, 539-541. (4) Painter, O.; Lee, R. K.; Scherer, A.; Yariv, A.; O’Brien, J. D.; Dapkus, P. D.; Kim, I. Science 1999, 284, 1819-1821. (5) Wang, Y.; Anderson, C. Macromolecules 1999, 32, 6172-6179. (6) Picard, G. Langmuir 1997, 13, 3226-3234. (7) Nagayama, K. Colloids Surf., A 1996, 109, 363-374. (8) Dimitrov, A. S.; Nagayama, K. Langmuir 1996, 12, 1303-1311. (9) Kondo, M.; Shinozaki, K.; Bergstro¨m, L.; Mizutani, N. Langmuir 1995, 11, 394-397.
Despite the work done on the formation of colloidal films,6,7,9-14 surprisingly little has been done to characterize the structure in detail. Colloidal films, formed at the air-liquid interface due to compression in a LangmuirBlodgett trough, are frequently characterized by visual appearance (i.e., microscopy) and by the average packing density given by the surface pressure-surface area isotherm,10,11,14 occasionally complemented with information about the lattice dimensions.13 The crystallinity of highly ordered 2D arrays of colloids, deposited on elevated substrates, has also been examined by displacement field analysis of the Fourier spectrum.7 This study proposes the use of two complementary methods to better characterize the structure of colloidal films. Colloidal films with varying degrees of order and porosities were formed by controlling the bond strength between grafted silica particles, partly immersed in different organic liquids.9 The position of the particles was determined using ordinary light microscopy and image analysis. We will show that the size of ordered domains may be estimated from the experimentally determined pair distribution function, g(r). This approach builds on the work by Grier and Murray,15 who used the pair distribution function to produce a measure of how well a colloidal film resembles a triangular lattice. The pore size distribution was characterized using Delaunay triangulation. The combination of Delaunay triangulation together with its conjugate, the Voronoi cell, has recently been used as a tool to determine the area, the volume, and the connectivity of any pores in both 3D and 2D.16 (10) Ho´rvo¨lgyi, Z.; Ne´meth, S.; Fendler, J. H. Colloids Surf., A 1993, 71, 327-335. (11) Fulda, K.-U.; Tieke, B. Supramol. Sci. 1997, 4, 265-273. (12) Matsushita, S.; Miwa, T.; Fujishima, A. Langmuir 1997, 13, 2582-2584. (13) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969-1976. (14) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 8820-8828. (15) Grier, D. G.; Murray, C. A. Direct Imaging of the Local Dynamics of Colloidal Phase Transitions; Arora, A. K., Tata, B. V. R., Eds.; VCH Publishers: New York, 1996; pp 69-100. (16) Sastry, S.; Corti, D. S.; Debenedetti, P. G.; Stillinger, F. H. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 5524-5532.
10.1021/la001683z CCC: $20.00 © 2001 American Chemical Society Published on Web 07/12/2001
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Materials and Methods Preparation of Alkoxylated Particles. Silica particles were chemically grafted with silanes in toluene by the method proposed by Brandriss and Margel.17 The silica (Tokuyama Soda Co., Japan) particles were spherical and highly monodisperse with a mean particle diameter of 1.97 µm and a standard deviation of 0.07 µm, as determined by a scanning electron microscope. Octyltrichlorosilane (OcTS) and octadecyltrichlorosilane (OTS) at 95% purity were purchased from Aldrich. Although a small amount of bound water is needed as a catalyst to form a smooth monolayer of the silane on the surface,18,19 an excess of water results in undesired polymerization of the silane.19 Consequently, the glassware was thoroughly cleaned and vacuum-dried before use. The silica was dried in a vacuum oven (60 °C, 45 mbar, 24 h) and dispersed in toluene (Baker, Merck), which was distilled prior to use. The silane was added to the silica dispersion in toluene, and the reaction proceeded, while being stirred, at room temperature in a nitrogen atmosphere for 20 h. After the grafting reaction, the dispersion was cleaned by centrifugation, and the particles were redispersed in fresh liquid. This rinsing procedure was repeated three times with toluene followed by a final rinse in ethanol (99.5%, Kemetyl). Finally, the particles were dried and stored in closed containers. Preparation of Interfacial Colloidal Film. An ordinary glass Petri dish served as the experimental cell containing the organic liquid. The bottom half and the lid were sealed by a rubber fitting in order to minimize air convection. The colloidal film was formed from a silica suspension with a dispersed silica content of 61 mg/cm3 in ethanol. A series of dispersion droplets (ca. 7 µL each) was gently added to the surface of the organic liquid. The ethanol drops spread quickly on the surface of the liquid, forming a thin liquid film containing the alkoxylated silica particles. The particles became trapped at the air-liquid interface when the ethanol evaporated. The immersion depth is controlled by the three-phase contact angle between the grafted silica particles and the organic liquid. The particle film formed at the air-liquid interface was also transferred to a precleaned glass substrate. The substrate was originally located at the bottom of the Petri dish and then elevated horizontally through the air-liquid interface, collecting part of the colloidal film. The drying process was studied in situ using the microscope at ambient conditions. Instrumentation and Data Acquisition. The colloidal film at the air-liquid interface was observed using an inverted microscope (Zeiss Axiovert 100) equipped with long workingdistance optics placed on a vibration-reducing stone table. This setup provided a minimal disturbance from convection or interference by condensed liquid on the optics. The individual particles were resolved with an objective of 32× magnification, resulting in a maximum field-of-view of 160 × 125 µm. Each particle (2 µm φ) occupied the equivalent area of 8 × 8 pixels, which enabled the individual particles to be well-resolved. Ordinary light field microscopy or phase-contrast microscopy was used. The latter resulted in the best image quality at a large immersion depth of the particles. Images of the film were captured by a 1/3-in. CCD camera (Hamamatsu C5405-01) mounted to the microscope. The video signal was contrast enhanced by a preprocessor (Hamamatsu Argus-20) before the images were digitized using an SLIC-frame grabber (MultiMedia Access Corporation) and stored on a Sun Ultra UNIX work station. The images were video recorded (Sony SVO 9500 MDP) as a backup reference. The images were filtered to enhance the contrast of the particles, and a texture operation was used to produce a binary projection of the particle centers. A set of customized masks and object classifications distinguished small pores from particles; hence, the number of artifacts was minimized. The particle positions were acquired with subpixel resolution from the center of mass of the particle projections. These data were further processed in order to characterize the structure. (17) Brandriss, S.; Margel, S. Langmuir 1993, 9, 1232-1240. (18) Brzoska, J. B.; Ben Azouz, I.; Rondelez, F. Langmuir 1994, 10, 4367-4373. (19) Wasserman, S. R.; Tao, Y.-T.; Whitesides, G. M. Langmuir 1989, 5, 1074-1087.
Hansen et al. Calculation of the Pair Distribution Function, g(r). The positional data allows the calculation of the separation distance between all the particles in the colloidal film. These data were represented in the form of a histogram, which was used to obtain the pair distribution function in a discrete form. Let ni(r) denote the number of particle pairs assigned to a histogram bin corresponding to a separation distance within the interval {r, r + δr}. This is equivalent to the number of particle centers contained in a ring of radius r and thickness δr, centered at particle i. The distribution of particle centers around particle i is given by the normalization of ni(r) with respect to ngas(r) ) Fπ((r + δr)2 - r2). The property ngas(r) equals the number of homogeneously distributed particle centers contained in the ring at a particle density of F. The pair distribution function, g(r), is the average over N particles and may be written as
g(r) )
1
N
Aoni(r)
∑ π(δr N i)1
2
+ 2rδr)
(1)
where Ao is the average area per particle in the image (i.e., 1/F). In the calculations, the radial distance r was represented in a discrete form of r ) mδr (m ) 1, 2, ...) with a chosen maximum value of rmax ) 16 particle diameters. Edge effects were absent because only particles located at a sufficient distance from the edge of the image where chosen as center particles. Particles from approximately 60 images from each system produced the final pair distribution function.
Results and Discussion Aggregation and Film Formation. Previous work showed that hydrophobic silica particles, trapped at the air-liquid interface, form ramified networks by clustercluster aggregation at low area fractions of particles.20 Compact, mesoporous films, however, could be obtained directly at the spreading of one large dispersion drop.9 In this work, we create a continuous colloidal film by spreading a series of relatively small dispersion drops on the liquid surface. The ethanol dispersion spread quickly at the air-liquid interface; addition of sequential dispersion drops at the middle of the liquid surface resulted in a transport of already trapped particles toward the dish wall. Consequently, the film was built up from the container walls inward to the position of drop deposition under the influences of rearrangement and compression. When approximately 15 drops (100 µL) had spread at the interface, the entire liquid surface in the container was covered by a particle film. The Williams and Berg model21 can be used to estimate the interparticle energies between the alkoxylated silica particles, partly immersed at the air-liquid interface. This model uses the linear fractional immersion to obtain an effective Hamaker constant, Aeff, determining the magnitude of the attractive van der Waals interaction. The length of the grafted hydrocarbon chain relates to the range of the steric barrier. The Hamaker constant across air, Aair ) 15.8 kT, was taken from Bergstro¨m,22 whereas the Hamaker constants across different organic liquids were calculated using the Lifshitz theory.22,23 The degree of immersion is related to the contact angle between the particle and the liquid; a high contact angle results in a low degree of immersion, hence, the van der Waals attraction becomes strong, as in Figure 1. The contact angles were estimated by the sessile drop method (20) Hansen, P. H. F.; Bergstro¨m, L. J. Colloid Interface Sci. 1999, 218, 77-87. (21) Williams, D. F.; Berg, J. C. J. Colloid Interface Sci. 1992, 152, 218-229. (22) Bergstro¨m, L. Adv. Colloid Interface Sci. 1997, 70, 125-169. (23) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3-41.
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Figure 1. Schematic picture of particles trapped at the airliquid interface. The immersion depth of the particles is controlled by the contact angle, θ, between the coated particles and the liquids. Table 1. Contact Angles and Effective Hamaker Constants of Alkoxylated Silica Particles Trapped at the Air-Liquid Interface liquid
θ OcTS (°)
benzene
5(4
toluene
9(3
1,4-dioxane
26 ( 4
2-methoxyethanol
42 ( 2
1,2-dichlorobenzene
30 ( 2
water
104 ( 2
θ OTS (°) 7(3 20 ( 2 40 ( 3 53 ( 4 49 ( 2 120 ( 4
Aeff (kT)
Vmin (kT)a
0.62 0.62 0.81 0.85 1.42 1.87 2.11 2.95 0.71 1.73 11.25 13.7
-20 -10 -26 -13 -46 -30 -68 -47 -23 -27 -364 -218
a
Values for the minimum interparticle energy, Vmin, are evaluated at a separation distance of 2.5 nm for OcTS coatings and 5.0 nm for OTS coatings.
using flat silica substrates grafted with the two silanes, OTS and OcTS. The result is presented together with the estimated effective Hamaker constants in Table 1. The silane coating acts as a repulsive steric interaction that prevents the particles from approaching closer than the smallest distance given by the coating thickness. We estimate a minimum separation distance of 2.5 and 5 nm for the OcTS and OTS coatings, respectively, assuming fully stretched and densely packed alkoxy chains at the particle surfaces.19 Particles, trapped at an air-liquid interface, are subject to capillary interactions only when the particles induce a curvature in the air-liquid interface. Applying the model by Chan et al.24 shows that the curvature of the interface is negligible. Furthermore, the interaction energy due to capillary forces at particle contact was calculated to be on the order of 10-5 kT for the systems in this paper. Hence, capillary interactions could be safely neglected in this study. Similar to previous studies,6,9-11 we find that a lower bond strength between particles results in a more ordered structure with a higher packing density. Figure 2 shows the normalized packing density φ* ) φ/φmax as a function of the estimated contact energy, |Vmin|. The normalization constant, φmax ) π/2x3, is the area fraction of particles in a perfect triangular lattice in 2D. We observe an exponential relation between the packing density and the bond strength between particles. Assuming that this relation can be extrapolated up to the limit of a perfect triangular lattice, φ* ) 1, we obtain the limiting bond strengths of formation of a perfectly ordered colloidal film at -5 and -0.5 kT for the OcTS and OTS coatings, respectively. (24) Chan, D. Y. C.; Henry, J. D. J.; White, L. R. J. Colloid Interface Sci. 1981, 79, 410-418.
Figure 2. Normalized packing density, φ*, plotted against |Vmin|, the minimum interparticle energy, for (×) OTS coated silica particles and (O) OcTS coated silica particles.
Hence, it appears that the formation of large-scale ordered colloidal films requires a lower contact energy using OTS as compared to OcTS coated particles. Figure 2 suggests that the combined effect of coating thickness and contact angle has to be further optimized (i.e., thicker coating and lower contact angle) in order to reach down to the limiting values of contact energy where large-scale ordered colloidal films appear. Alternatively, the use of a smaller particle size could improve the ordering if the disorder induced by Brownian motion does not become too large. For instance, an OcTS coated silica particle with a radius of 0.25 µm trapped at the airbenzene interface should have a contact energy of -4.7 kT according to our estimates. Structural Characteristics of Colloidal Films. Size of Ordered Domains. All of the colloidal films displayed regions of mostly hexagonally ordered particles. The size of these regions and the degree of order are important characteristics for a number of potential applications of colloidal films, e.g., photonic band gap materials. We utilized the experimentally determined information on the particle positions to obtain the pair distribution function. This experimental pair distribution function, g(r)exp, was fit with an empirically modified pair distribution function based on a structure of triangular (hexagonally packed) order, g(r)triang. The peaks in g(r)triang were broadened by a normal distribution, with a standard deviation σ(r) to account for the statistical fluctuations of the particle positions around the triangular lattice points. We also enveloped g(r)triang with an exponentially decaying function to account for the limited size of ordered domains. The final expression for the calculated pair distribution function is15
g(r) )
[∫
δr × gtriang(r - x) (2π)1/2σ(r) r x2 exp dx - 1 exp - + 1 (2) ζ 2σ(r)2
(
) ] ( )
where x is the distance from the position a particle would have in a lattice of perfect triangular order. The parameter ζ is a measure of the correlation length. In the experiments, gexp(r) was measured in a discrete form in steps of δr (the particle diameter equaled 26 steps in r). Consequently, the integral in eq 2 was transformed into a summation to facilitate fitting to the experimental data. We found that the standard deviation, σ(r), had to
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Figure 3. Representative images of colloidal films with corresponding g(r)exp for five different combinations of particle coating and liquid: (a) OcTS and toluene, (b) OTS and toluene, (c) OcTS and 1,4-dioxane, (d) OTS and 2-methoxyethanol, and (e) OcTS and water.
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Table 2. Structural Characteristics of Particle Films Formed at the Air-Liquid Interface of Various Liquidsa OTS liquid
ζ/db
φ*
benzene toluene 1,4-dioxane 2-methoxyethanol 1,2-dichlorobenzene water
12.7 10.2 5.3 3.3
0.918 0.902 0.893 0.879
2.5
0.827
OcTS Vmin (kT)
ζ/db
φ*
Vmin (kT)
-10 -13 -30 -47 -27 -218
14.0 15.3 6.4 3.2 13.4 3.2
0.940 0.936 0.902 0.887 0.935 0.822
-20 -26 -46 -68 -23 -364
a
The silica particles were coated with OTS or OcTS. b The correlation length, ζ, was normalized with respect to the particle diameter, d.
be represented as a function of r to enable a good fit to gexp(r). It is expected that fluctuations in ordered colloidal films propagate quite far from the origin. We expressed the observed behavior of σ(r) ad hoc with
( (br) )
σ(r) ) σo 1 +
2/3
(3)
where σo relates to the standard deviation at short distances and b determines how strongly the standard deviation increases with r. The empirical fit to g(r)exp was optimized using the parameters σo, b, and ζ with respect to the positions and amplitudes of the peaks. Figure 3 shows images of the particle films together with the obtained g(r)exp of five different systems. This figure exemplifies the relationship between the structure of the colloidal film and the features of the pair distribution function. The sequence a-e in Figure 3 is related to colloidal systems with increasing bond strength; Figure 3a represents a system with low bond strength, and Figure 3e represents a system with high bond strength. The systems are described in more detail in Table 2. The octyl system with the lowest interparticle bond strength yields a colloidal film with large, well-ordered domains (Figure 3a). This is reflected in the pair distribution function, which displays a slow decay, corresponding to a large correlation length. We observed that the ordered domains occasionally displayed some stacking errors (quadrilateral packing) and some pores, associated with missing particles. Changing the particle interactions by using particles with octyldecyl- instead of octylalkoxy chains grafted on the particle surfaces resulted in smaller domains of ordered particles and larger pores in the grain boundaries (Figure 3b). The increase in the number of pores lowered the average packing and the smaller domain size in Figure 3b is reflected in a stronger damping of g(r)exp as compared to the OcTS system (Figure 3a). Figure 3c represents OcTS coated particles floating at a dioxane interface. This system has a similar packing density as the system in Figure 3b but displays a different structure. The ordered domains are smaller, and the boundaries contain mostly small pores. Furthermore, the number of stacking faults are further increased. Stacking faults are local particle arrangements, and any area of quadratic ordered particles is small. Hence, quadrilateral structures affect g(r)exp at small length scales only. The correlation length is selected to enable a fit to g(r)exp at all length scales; a lower value of ζ is associated with the smaller size of ordered domains in the system shown in Figure 3c. The last two examples of interfacial films in Figure 3 are OTS coated particles at 2-methoxyethanol (Figure 3d) and OcTS coated particles floating at a water interface (Figure 3e). The correlation lengths of these two systems
Figure 4. Normalized correlation length, ζ/d, plotted against |Vmin| for (×) OTS coated silica particles and (O) OcTS coated silica particles.
are similar, but the aqueous based system has a lower average packing density due to the presence of large pores. This system does not display a long-range order; the pair distribution function decays within a couple of particle diameters to an average value. In the 2-methoxyethanol system, a substantial amount of quadrilateral packing affects the relative amplitudes of the double peak around 2d. This suggests that this system would perhaps be better described by a linear combination of triangular and quadratic order. The standard deviation at each position in g(r)exp was 10-15% of the amplitude, with a higher standard deviation for the more ordered systems. Hence, interpretation of the local structure from the peak positions and amplitudes should be made with care; however, the decay of g(r)exp is less affected by the scatter, which makes the estimate of ζ more accurate. The combination of σo and b in eq 3 determines the width of the normal distribution of particles at a distance, r, from the origin of the central particle. The first peak in g(r) fits the radial distribution of particles in the first layer around the central particle. The standard deviation of the normal distribution of this peak is essentially expressed by the parameter σo, which was set to a value of ≈0.03 particle diameter for all systems. The parameter b expresses the radial dependence of σ(r) and indicates to what degree the distribution of particles at distance r is disturbed by the distribution of particles closer to the central particle. Generally, we find that a decrease in the domain size as described by a lower value of ζ was accompanied with a lower value of b as well. The parameter b was about 3-4 times the value of ζ. For the least ordered systems (methoxyethanol and water), the local disorder within the domains was so large (b ≈ two particle diameters) that σo had to be adjusted in order to fit the first peak in combination with b as expressed by eq 3. We find that the correlation length displays an exponential dependence with the estimated bond strength, |Vmin|, (Figure 4). However, the strongly aggregated systems of alkoxylated particles floating at the air-water interface deviate from this relation. This is not so surprising since the aqueous systems have a contact angle larger than 90°, and the point of contact between particles is located above the air-liquid interface. Hence, films of the aqueous systems are probably formed by a fundamentally different process as compared to particles partially immersed in the organic liquids.
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Figure 5. Normalized correlation length, ζ/d, plotted against the normalized packing density, φ*, for (×) OTS coated silica particles and (O) OcTS coated silica particles.
Plotting the correlation length against the normalized packing density, φ*, results in a linear relation (Figure 5). There is a subtle difference between the OTS and the OcTS systems; the former systems produce a slightly larger correlation length at equal packing density. This effect is associated with differences in the pore structure. Colloidal films with relatively large pores between large ordered domains may result in the same average packing density as films with smaller ordered domains having a disordered particle arrangement with no large pores at the domain boundaries. Hence, the results imply that the OTS coated systems contain fewer but larger pores as compared to the OcTS coated system. Pore Structure Characterized by Delaunay Triangulation. The pair distribution function analysis of the colloidal films provides much information on the size of the ordered domains and the degree of order. However, we also need a detailed analysis of the pore distribution, including dislocations and stacking faults, for a fuller description of these complex systems. A tessellation (Delaunay triangulation) of the image was applied for this task. The limited number of pore sizes in the dense colloidal films facilitated the analysis. Tessellation of space among particles (considered as a point set) has mainly been used to investigate phase transitions of one component liquid in two dimensions.15,25,26 Sastry et al. proved that dual tessellation can be used to obtain the volume, the area, and the connectivity of pores in a media16 and used this to investigate void fluctuations in a Lennard-Jones liquid.27 We characterized the colloidal films by the set of Delaunay triangles obtained for each image. A Delaunay triangulation is a triangulation of a point set with the property that no point in the point set falls in the circumcircle (circle that passes through all three vertexes) of any triangle in the triangulation.28 In our images, the point set is given by the position of the particle centers. Hence, the length of the sides and the area of the Delaunay triangles are obtained from the positions of the particles. (25) Marcus, A. H.; Rice, S. A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 55, 637-656. (26) Somer, F. L.; Canright, G. S.; Kaplan, T. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 58, 5748-5756. (27) Sastry, S.; Debenedetti, P. G.; Stillinger, F. H. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 55335543. (28) Tanemura, M.; Ogawa, T.; Ogita, N. J. Comput. Phys. 1983, 51, 191-207.
Figure 6. Examples of the geometry of different particle arrangements together with the associated number of Delaunay triangles. The pores are shaded gray, and the normalized pore area, Ap, is included.
A perfect triangular lattice is defined by equilateral triangles with a side length of 2ao. An opening between two neighboring particles is thus identified by a side length exceeding 2ao + ∆, where ∆ is an error tolerance of the particle positions. Different kinds of pores (particle arrangements) may be identified by the number of Delaunay triangles, nt, that are connected in order to envelop a pore. The relations between the number of triangles and types of pores are sketched in Figure 6. The void between three particles in an equilateral triangle is enclosed by one triangle, which is characteristic for the triangular geometry of hexagonally ordered domains. A pore enclosed by two triangles is the result of four particles bound in a quadrilateral configuration. The pore has a different shape and area depending on the angles of the quadrilateral configuration. Three triangles enclosing a pore with five particles forming the enclosure are associated with a stacking fault in the film. Four triangles are associated with a larger stacking fault or a missing particle, i.e., a hole surrounded by six particles in a hexagonal arrangement.
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Figure 7. Example on pore classification by Delaunay triangulation. Circles mark the center position of particles and the triangular mesh illustrates the Delaunay triangulation. Different degrees of gray indicate the different pore classes. Most numerous are the hexagonally arranged particles (white) followed by the quadrilaterally arranged particles (light gray).
There are some geometrical relations that follow from the Delaunay representation. If nt triangles are connected to enclose a pore, nt + 2 touching particles confine this pore. Furthermore, the normalized area of one pore, Ap, is given by
Ap )
1 πa2o
nt
∑ j)1
|r21(j) × r31(j)| 2
-
nt 2
(4)
where j is the index of the nt triangles (the numerical indexes refer to the three particles associated to the triangle) and r is the positional vector between a pair of particles. The sum in eq 4 is the area of the triangles, and the second term corresponds to the area of the triangles occupied by particles. The maximum value of Ap is obtained when the angles between particle pairs along the pore enclosure are equal. Examples of Ap, normalized with the particle cross-area πa2o , are included in Figure 6. The colloidal film may be characterized by counting the number of pores with a certain nt, m(nt), and by measuring the distribution of Ap for all pores of a certain nt, Ap(nt). The Delaunay triangulation data was represented in a matrix where m(nt) and Ap(nt) were obtained by matrix multiplication. The pore class distribution, m(nt), was corrected for pores removed along the image edge according to a modified version of an unbiased counting frame,29 obtained by counting the pores that were traversed by a test line. Figure 7 illustrates a triangulated subregion in an image of a colloidal film. Note that quadrilateral arrangements often are gathered in small regions and that stacking faults often are located around these regions. Figure 8 shows the distribution of pores, m(nt), normalized with respect to the total number of particles, N. We find that there is an exponential decrease of the number of pores with an increase in nt. The OcTS coated systems (Figure 8a) all display a similar slope in the semiloga(29) Gundersen, H. J. G. J. Microsc. (Oxford) 1977, 111, 219-223.
Figure 8. Natural logarithm of the normalized pore size distribution, m(nt)/N, of colloidal films floating at the air-liquid interface for (a) OcTS coated silica particles and (b) OTS coated silica particles.
rithmic plots, but the amplitude of the curves shift upward with increasing bond strength. It is interesting to note that the behavior is different for the OTS coated particles, as in Figure 8b. Here, the fraction of quadrilateral packing m(nt ) 2) increases, while the frequency of larger pores m(nt > 5) decreases when the contact angle and |Vmin| increases. The reason for this behavior is not clear at the moment, and more experiments are planned to clarify this issue. Effects of Drying. We found that the structural changes induced by the transfer of the floating films to the substrate and subsequent drying varied depending on the cohesion of the films. Colloidal films characterized by a low interparticle bond strength (i.e., colloidal films formed at the benzene or toluene surface) showed a reduction in the normalized packing fraction, φ*, and size of the ordered domains, ζ, after transfer to the substrate and subsequent drying. The structural characteristics of the stronger films (i.e., films characterized by high interparticle bond strength) remain essentially unchanged by this treatment. It is clear that the transfer of the floating colloidal film to the substrate will strain the film and may induce local rupture and stretching. Drying of the film, however, induces a compressive force due to the capillary forces that tend to shrink the film. Our results suggest that the defects induced by the transfer process do not fully heal during the drying of the low cohesive films. The stronger films are apparently sufficiently strong enough to be less affected by this treatment.
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Figure 10. Normalized pore area distribution, Ap(nt)/m(nt), for large stacking faults (i.e., nt ) 4) for colloidal films of OTS coated silica; O represents the film floating at the air-liquid interface of toluene, and * represents the film after transfer and subsequent drying.
Figure 9. Natural logarithm of the normalized pore class distribution, m(nt)/N, of dried colloidal films for (a) OcTS coated silica particles and (b) OTS coated silica particles.
The pore distribution was analyzed in more detail using Delaunay triangulation. Figure 9 shows the normalized distribution of pores, m(nt)/N, in the colloidal films after being transferred and subsequently dried. We find that the dried films contained fewer pores in the range of 2 e nt e 4, whereas larger pores are more frequent as compared to the floating colloidal films (see Figure 8). This effect was observed for all systems and also for the strong films that did not show any significant reduction in φ* after being transferred to the substrate and dried. Hence, the creation of larger pores occurs together with a disappearance of some of the small pores. This process is probably driven by the capillary forces that are sufficiently strong enough to transform small pores into triangular order but not sufficient enough to heal the large voids and cracks. The Delaunay description allows a characterization of the area of the pores, Ap, which was used for a statistical analysis of small changes in the geometry of a certain class of pores. The pore size distribution of large stacking faults, Ap(nt ) 4), (Figure 10), is bimodal, where one fraction relates to missing particles (Ap ≈ 1.3) and the other fraction of a smaller pore area belongs to large stacking faults. The change in the ratio between the two fractions when the films are dried suggests that the particle positions are adjusted such that the stacking faults are either transformed to a more symmetrical geometry of a missing particle or recombined into another class of pore (i.e., nt * 4).
An interesting observation is the systematic shift of the bimodal distribution toward larger areas of a large stacking fault for the dried system, as in Figure 10. The criterion for “touching” particles was set to 2ao + ∆ in the analysis of the positional data to account for errors in the measurement of the particle positions. For stretched films (e.g., induced by the transfer process) more particle pairs will have a separation distance close to 2ao + ∆ rather than the normal 2ao. Hence, some sides in the triangles may attain a larger value, and the distribution of the triangle area as well as the pore area becomes shifted toward a larger value. We speculate that this is an indication that the transfer process stretches the disordered domain boundaries. This in turn would generate more degrees of freedom for the capillary forces to rearrange the particles in the boundaries between the ordered domains. Conclusions Dense films of alkoxylated silica particles were formed at the surface of various organic liquids and thereafter transferred and dried on glass substrates. The positions of all particles in the colloidal films were obtained by microscopy and image analysis and further analyzed in two ways. First, the average size of hexagonally ordered domains was estimated using a theoretical fit to a modified pair distribution function based on triangular order. Two fitting parameters were used; the standard deviation σ(r) accounted for the statistical fluctuations of the particle positions around the triangular lattice points, and ζ determined the exponential decay of the pair distribution function due to the average size of the ordered domains. Second, the distribution of pores and defects in the colloidal films was determined using Delaunay triangulation. The pores and voids between all particles were classified by the number of Delaunay triangles, nt, associated with each pore. The pores were analyzed both by their number distribution among pore classes and by the distribution of pore areas within the individual pore classes. The average size of the hexagonally ordered domains, estimated from ζ, decreased exponentially as the interparticle bond strength increased. We also found that the distribution of different pore classes, m(nt), decayed
Dense Colloidal Films
exponentially as size increased, characterized by nt. The transfer and subsequent drying process facilitate the rearrangement of particles driven by the capillary forces to transform small pores into triangular order and simultaneously make some of the voids and cracks larger.
Langmuir, Vol. 17, No. 16, 2001 4875
Acknowledgment. Support for this project was provided by the Swedish Research Council for Engineering Sciences, TFR. LA001683Z
