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Capillary against Adhesion Forces during Drying of Particle Submonolayers on a Solid Substrate Antoine Thill* and Olivier Spalla Service de Chimie Mole´ culaire, baˆ t. 125, C.E.A. Saclay, 91191 Gif-sur-Yvette Cedex, France Received December 18, 2001. In Final Form: March 18, 2002 This work deals with the common atomic force microscopy observation that small particles weakly adhering to a substrate tend to form aggregates at least when imaged under air. We show that it is possible to measure the coverage of mica surfaces with R-Al2O3 colloids in situ by means of static light scattering. The dynamic properties of the speckled pattern obtained from the layer are also used to probe the dynamic of the particles. It is then concluded that even the weakest adhesion of the particles does not allow in situ aggregation and surface mobility (at least, on the time scale of the laboratory experiment). The adhering particles aggregate only during the drying stage, due to the action of lateral capillary forces. A model for the aggregation mechanism is introduced taking into account the balance of forces acting on the particles during the drying stage.
I. Introduction Achieving a particular arrangement of adsorbed colloids on a substrate is a goal for many applications ranging from optical surface treatment to nanoelectronics. An inplane structured layer can be obtained with the help of chemical reactions at specific surface sites and/or by electrostatic interaction with patched surfaces.1,2 In general, the adsorption stage is therefore performed in the liquid phase followed by a drying stage in order to produce a substrate usable for further application. Each of the two stages raises important questions: (1) What is the structure of the adsorbed layer at the end of the adsorption? (2) Are the adsorbed particles laterally mobile while the solvent is still present? (3) Do the capillary forces acting between the particles modify the structure during drying? The present work mainly deals with the questions 2 and 3. The behavior of particles near a surface obviously depends on the strength of their adhesion to the substrate. For instance, in the case of monodisperse particles almost nonadhering to the substrate, it has been clearly shown that two-dimensional arrays were organized on a very large scale due to capillary forces acting during drying.3 In this case, several parameters altering the structure of the final deposited layer have been studied such as particle polydispersity4 or drying rate.5 On the other hand, when the particles adhere to the surface the situation is less controlled. To induce a surface ordering of the colloids, the lateral capillary forces have first to overcome the adhesion forces. Parameters acting in favor of the lateral capillary forces are a short separation between the adsorbed colloids (and thus a high surface coverage), the wettability and size of the colloids, and the surface tension of the liquid film.6 Nevertheless, the balance between adhesion and capillary forces is difficult to establish a priori and it is not clear, in general, whether (1) Tien, J.; Terfort, A.; Whitesides, G. M. Langmuir 1997, 13, 53495355. (2) Aizenberg, J.; Braun, P. V.; Wiltzius, P. Phys. Rev. Lett. 2000, 84, 2997-3000. (3) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183-3190. (4) Yamaki, M.; Higo, J.; Nagayama, K. Langmuir 1995, 11, 2975. (5) Rakers, S.; Chi, L. F.; Fuchs, H. Langmuir 1997, 13, 7121-7124. (6) Kralchevsky, P. A.; Nagayama, K. Langmuir 1994, 10, 23-36.
the capillary forces will affect the layer structure during drying or not. A good way to check this crucial point is to observe the layer before and after drying. Johnson and Lenhoff7 reported for instance that the structure of the adsorbed latex particle layer is different before and after drying especially when the surface coverage is high. They compared atomic force microscopy (AFM) images under liquid and under air of a dense latex layer on mica. Aggregates were observed under air, whereas in liquid, particles were well dispersed on the mica surface. Semmler8 has also observed that latex on mica forms aggregates at high salinity when the surface coverage is high. Although a role of capillary forces has to be taken into account, the occurrence of a 2D in situ aggregation cannot be excluded in some occasions. Such an in situ mechanism has been discussed for aggregation of gold colloids9 and glycoproteins10 on various substrates and for alumina colloids on mica.11 Nevertheless, it was not perfectly clear whether these aggregates were generated by an in situ aggregation involving lateral mobility of the particles or if they result from the action of capillary forces during drying. Indeed, the conclusion of an in situ mechanism was supported by the main observation that modifying the solvent properties and therefore the particle/surface and/or particle/particle interactions can control the degree of aggregation and the morphology of the aggregates. In the case of gold colloids, Derjaguin-Landau-VerweyOverbeek (DLVO) arguments were proposed to explain the aggregation. Nevertheless, in each of the mentioned examples, the lateral mobility of the small colloids at the solid interface was not observed in situ. Bohmer12 reported the observation by light microscopy of an in situ mobility and aggregation of electrodeposited 4 and 10 µm latex particles, but this mobility was however attributed to the electroosmotic flow at the solid interface. Finally, no direct (7) Johnson, C. A.; Lenhoff, A. M. J. Colloid Interface Sci. 1996, 179, 587-599. (8) Semmler, M.; Mann, E. K.; Ricka, J.; Borkovec, M. Langmuir 1998, 14, 5127-5132. (9) Thompson, D. K.; Collins, I. R. J. Colloid Interface Sci. 1994, 163, 347-354 (10) Lavalle, P.; DeVries, A. L.; Chen, C. C. C.; Scheuring, S.; Ransden, J. J. Langmuir 2000, 16, 5785-5789. (11) Spalla, O.; Desset, S. Langmuir 2000, 16, 2133-2140. (12) Bohmer, M. Langmuir 1996, 12, 5747-5750.
10.1021/la011824c CCC: $22.00 © 2002 American Chemical Society Published on Web 05/10/2002
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Figure 1. Schematic representation of the in situ light scattering sample cell: (a) cell inox body, (b) fixed piston, and (c) mobile piston.
observation of a real in-plane Brownian diffusion and aggregation of adsorbed colloids is yet available in the literature. The aim of the present study is therefore to probe directly the dynamic of adsorbed submicrometric particles in conditions corresponding to previously observed fast aggregation kinetics.11 Light scattering and AFM were performed on submonolayers of small alumina colloids adsorbed on mica. The light scattering setup was arranged in a way such to enable static light scattering measurements of wet or dry adsorbed layers. The speckles dynamic was also studied on a series of images obtained with a charge-coupled device (CCD) camera. The materials and methods are presented in section II. The standard protocols for adsorption and drying are defined in detail. The results are presented in section III and discussed in section IV on the basis of a new model involving capillary forces during drying. II. Materials and Methods 1. Materials. The R-Al2O3 particles (AKP50) were purchased from Sumitomo (Japan) and are pure R-alumina. Their average size is 200 nm in diameter with an irregular shape. Each particle is in fact composed of a few submonocrystallites sintered together during the synthesis process. Muscovite mica disks (15 and 8 mm diameter) with a low amount of defects were purchased from Metafix and cleaved just before use. All other chemicals were analytical grade and used as received. The imaging of the layer was performed under air with an AFM NIII from Digital Instruments using the tapping mode with high-frequency probes (200-300 kHz) from Nanoscience. Light scattering experiments were performed on a customized lowangle light scattering setup. The apparatus and the measurement procedure are described elsewhere.13 In brief, for each sample, several CCD images of the scattered light are recorded. The pixel intensities are averaged according to their distance from the central beam to obtain static light scattering intensities as a function of the scattering angle. To perform dynamical studies, CCD images acquired at successive time intervals are compared one with another. The positions of the speckles in these successive images are directly related to the dynamic of the scattering objects in the sample. A special sample cell was developed in order to perform measurements on adsorbed colloids. A schematic representation of the sample cell is shown in Figure 1. Two thin and freshly cleaved mica surfaces are glued on the fixed and mobile pistons of the cell. These two surfaces enclose a cell of variable optical length, where liquids can flow at a desired rate. The optical length can be adjusted from 0 to 2 mm. This length is controlled with an accuracy of (10 µm. On every experiment, we used an optical length of 1 mm. 2. Light Scattering from Adsorbed Layers in Water and AFM after Drying. Two circular mica substrates with an 8 mm diameter were cleaved on both sides and glued on the pistons of the sample cell. Ultrapure water (MilliQ) was first flowed into the cell to perform a background measurement of the light scattered by the mica sheets and water. A 1 g/L dispersion of R-Al2O3 was then allowed to flow in the sample cell for a time (13) Thill, A.; Desert, S.; Delsanti, M. EPJ 2001, 17, 201-208.
Figure 2. CCD image of light scattering by mica covered with alumina colloids and intensity of light scattered as a function of the scattering vector (q ) 4π sin(θ/2)/λ, θ is the scattering angle and λ is the wavelength in water) for the mica alone (- - -), the mica covered with alumina colloids (s), and the colloids (O). The grayed region corresponds to the range used to compute 〈I〉. ranging from 5 to 900 s. The light scattered by the adsorbed particles and mica substrates was measured after thoroughly rinsing the sample cell with MilliQ water until no further evolution with the rinsing time of the scattered light was noticeable. At the end of the experiments, the mica substrates were collected and air-dried for an AFM study of the adsorbed layers. 3. Light Scattering from Aged Layers in Water and AFM after Drying. The same procedure was followed to adsorb alumina colloids on the mica surface before a stage of in situ aging. The adsorption stage lasted 400 s. Then the sample cell was thoroughly rinsed to remove every nonadsorbed colloid. Finally, the layers were aged in a citric acid solution at the desired concentration and pH. The light scattered by the adsorbed colloids was measured both before and after filling the sample cell with a citric acid solution. A concentration of 1 mM citric acid at pH ) 7 was chosen. In these conditions, the alumina colloids are negatively charged and they have a weak adhesion with the mica surface.11 The water circulation was stopped, and the substrates were left at rest after the citric acid introduction. The light scattered by the adsorbed layers was recorded at several aging times. To study the dynamic of the layers, we have compared the positions of the speckles due to the adsorbed particles in different images acquired in a row with a fixed separation time. 4. Light Scattering and AFM after Drying. Freshly cleaved mica sheets are used for light scattering blank measurement in air. The sheets were then immersed in a 1 g/L dispersion of R-Al2O3 at pH ) 5 in order to adsorb colloids. The adsorption stage lasted 900 s. The mica sheets were then rinsed in three successive pots of pure water to eliminate any nonadsorbed particles. During the transfer from one pot to the other, care was taken to always keep a drop of water on top of the mica sheets to avoid drying. The mica sheets were finally put to age in a solution of citric acid at 1 mM and pH ) 7 for a given duration. After the desired aging time had elapsed, the mica sheets were either directly dried or further rinsed once again in ultrapure water before drying. The mica surfaces were pulled vertically at a controlled speed (from 4 to 100 µm/s) out of the solution. The drying was performed under ambient air (relative humidity of ∼60%). The light scattered by the dry mica sheets was measured, and the sheets were finally observed in AFM.
III. Results 1. Alumina Adsorption. The alumina particles were left to adsorb onto the mica surfaces from a 1 g/L suspension in the scattering cell for a time ranging from 5 to 900 s. An example of a CCD scattering image due to mica surfaces covered with alumina colloids after 600 s is presented in Figure 2 together with the corresponding
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Figure 4. 〈I〉 in cps measured with the in situ sample cell as a function of the surface coverage τ obtained by image analysis of the AFM scans on the same samples.
Figure 3. 15 × 15 µm height scans of the adsorbed R-Al2O3 colloids for contact times tc of (a) 5, (b) 15, (c) 180, and (d) 600 s. The image in (e) corresponds to a layer of adsorbed particles aged for 2 h in a citric acid solution at a concentration of 1 mM and pH ) 7 observed after drying.
annular averaged light scattering curve. The subtracted signal, corresponding to the colloids, is constant on the whole q range (from 2 × 10-5 to 3.5 × 10-4 Å-1). This flat scattering corresponds to a form factor of alumina particles having a radius smaller than about 500 nm. To compare the intensity of the light scattered by the adsorbed colloids from one experiment to another, the intensity is averaged in the q range from 2 × 10-4 to 3 × 10-4 Å-1 (represented by a gray area in Figure 2). This q range allows a good spatial averaging of the CCD light scattering measurement,13 and the results did not depend on its exact position as the light scattering signal is flat over the whole q range. The result of the averaging is noted hereafter as 〈I〉. After the light scattering measurements, the mica surfaces were recovered and dried for AFM observations. In Figure 3a-d is reported a series of 15 × 15 µm AFM height scans of the adsorbed layers obtained for increasing contact times. The surface coverage τ was calculated from the AFM height scans using the bearing function of the nanoscope image analysis software. In Figure 4, the average intensity 〈I〉 in counts per second (cps) is plotted versus the surface coverage τ. First, a very small surface coverage (about 1%) is detectable. The spatial beam radius being al ) 0.475 mm and the particle radius ap ) 100 nm, the smallest measured surface coverage corresponds to N ) 2τ(al/ap)2 ≈ 4 × 105 adsorbed particles in the laser beam area, yielding an intensity per particle of 〈I〉/N ) 4 × 10-3 cps.
Figure 5. Evolution of 〈I〉 in cps after injection of a 1 mM citric acid solution at pH 7 in the in situ sample cell.
2. In Situ Aging. The light scattered by the adsorbed alumina particle was measured just after injection in the cell of a 1 mM citric acid solution at pH 7. The average intensity 〈I〉 is plotted in Figure 5 as a function of the contact time. It remains constant with a value of about 2 × 104 cps even after more than 2 h of contact time. Moreover, a detailed inspection of the CCD images (see examples reported in Figure 6) reveals that the speckles, due to the adsorbed alumina, are immobile from one image to another during the whole experiment. For comparison, two images corresponding to a measurement of a 5 µm latex particle in water are also shown in Figure 6. In this later case, the speckles are completely decorrelated after 30 s. The fact that the speckles due to the adsorbed particles are immobile supports the conclusion that the adsorbed particles do not migrate on the surface. Finally, the 8 mm mica surfaces were recovered and dried after the in situ aging stage for AFM observation. The layer, shown in Figure 3e, contains aggregates as already reported in a former study.11 Therefore, a contradiction is obtained. On one hand, the particles are immobile under water as revealed by the constant speckle features, and on the other hand, the AFM analysis reveals aggregates previously attributed to a surface migration. 3. Drying Experiments. The mica sheets, after adsorption, rinsing, and aging, were dried under ambient air by pulling them vertically at a constant speed out of their aging solution. These sheets were observed by AFM and used for light scattering measurements. A size distribution was obtained by image analysis of the AFM height scans. A reference size distribution, shown in Figure 7, was obtained when the aging solution did not contain
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Figure 8. Light scattered by the dried colloidal layer after a contact time of 30 min with 1 mM citric acid solution at pH ) 7 as a function of the pulling speed (Vs) of the mica. The dotted line corresponds to the average light scattered by bare adsorbed R-Al2O3 colloids or by samples that have been rinsed in pure water for 30 min before drying.
Figure 6. Comparison of the speckle positions of two light scattering CCD images for the R-Al2O3 colloids in contact with a 1 mM citric acid solution at pH 7. Fifty minutes contact time has elapsed between the two pictures. As an example for comparison, the speckle position of a 5 µm latex particle suspension is shown. In this case, only 30 s has elapsed between the two pictures.
15 × 15 µm height scans clearly shifts toward larger sizes. The average scattered intensity also increases significantly from 7.5 × 104 cps to more than 15 × 104 cps. This confirms the aggregation process. Finally, the speed at which the mica surface is pulled out of water influences the aggregation process. For pulling speeds above about 10 µm/s, the adsorbed layer morphologies are comparable and the size distributions are similar to the 17 µm/s size distribution shown in Figure 7. The differences between a slow and a fast pulling speed are more pronounced for the fraction of remaining small particles than for the larger aggregates. Finally, the size distribution in Figure 7 corresponding to open circles corresponds to a sample prepared and aged in citric acid (C ) 1 mM, pH ) 7) but which has been re-rinsed in pure water for 30 min before drying. In this case, both the size distribution and average scattered light 〈I〉 correspond to the reference layer. It can be concluded that an aggregation occurs during drying, but only when the citric acid is present in solution. IV. Discussion
Figure 7. Size distribution computed from 15 × 15 µm AFM scans for adsorbed R-Al2O3 colloids. The time ta corresponds to the contact stage with a 1 mM solution of citric acid at pH 7, and the time td corresponds to the rinsing stage in pure water before drying.
citric acid. The distribution is centered on the average size of the particle S/Πap2 ) 1 meaning that they are not aggregated. The size distribution does not depend on the coverage. According to the size distribution, 90% of the particles have a radius smaller than 260 nm. This is consistent with the light scattering obtained from the in situ adsorbed alumina shown in Figure 2. Figure 8 presents the light scattering measurements performed on the dried layers. When no aggregates are present on the surface, the average light scattered at saturation of the mica surface is about 〈I〉 ) 7.5 × 104 cps whatever the drying speed, as shown by the dotted line. This value is higher than that in water due to the higher optical contrast in air. When the mica surfaces are pulled out of the citric acid solution after a contact time of 30 min, the layer contains 2D aggregates as observed by AFM. The distribution obtained from the image analysis of AFM
Our main results are as follows: (i) The adsorbed alumina particles are easily detected by light scattering. (ii) No in situ evolution of the average scattered light is observed when the adsorbed particles are aged in contact with a citric acid solution meaning that no aggregation occurs. (iii) The speckles due to the presence of the adsorbed particles are immobile meaning not only that the particles do not aggregate but also that they are immobile at their initial adsorption position. (iv) AFM observations reveal the presence of 2D aggregates only if the layer is dried in the presence of citric acid. These results are now used to discuss the mechanism of 2D aggregate formation and the influence of lateral capillary forces on the adsorbed layer structure. If an aggregation of the adsorbed colloids occurs at the interface, one would expect the scattered intensity to increase. When the adsorbed colloids are aged in a citric acid solution corresponding to a fast aggregation as previously observed,11 no increase of the scattered intensity, measured in situ, is observed even after 24 h. When the adsorbed colloid layers are dried out of the citric acid solution, a clear increase of the scattered intensity is obtained (see Figure 8). This increase corresponds to a 2D aggregation as observed by AFM. It can be concluded from these observations that no aggregation occurs in situ while
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Figure 9. Schematic representation of the geometry considered for the computation of the capillary force between two partially immersed colloids.
aggregates form upon drying when citric acid is present. Furthermore, the fact that the speckles measured in situ from the adsorbed colloids are static is strong evidence that the particles are immobile at their adsorption sites. Finally, in situ light scattering measurements allow one to conclude that no in situ aggregation or even mobility occurs. The aggregation of the adsorbed alumina colloids takes place during the drying stage. Hence, the mobility of the particles must be due to lateral capillary forces exerted during the drying stage of the sample preparation.3,6,14 Indeed, when a surface containing adsorbed colloids is dried, an attractive lateral capillary force is exerted between the colloids as soon as they are partially immersed. This force is mainly due to the deviation of the three-phase contact line from its horizontal position when two particles are close to one another (as drawn in Figure 9). When the colloids adhere to the surface, the lateral capillary force, Fc, has to overcome some resistance in order to promote a particle migration. The migration can occur by rolling or by sliding, but nevertheless, it is the out of balance capillary forces that drive aggregation. In the following paragraph, we perform a balance of the forces acting during drying in the case of rolling or sliding. When one considers the rolling hypothesis, the applied torque of moment Mc due to capillary force can be approximated by Mc ≈ Fcap considering that the lateral capillary force is applied to the center of the particle. On the other hand, the adhesion of the particle to the surface produces a torque of moment Ma that prevents rolling. Following the result of Sharma,15 the moment can be linked to the vertical adhesion force Fa by Ma ) Faac where ac is proportional to the radius of the contact area. The detailed influences such as the adhesion hysteresis arising from the difference between detachment at the rear of the contact area and the formation of an adhesive junction at the front of the contact are included in the effective value of ac. Finally, the rolling of a spherical particle on a plane surface occurs when the applied capillary torque exerted on the particle exceeds the adhesive resistance moment M a.
Mc > Ma or Fc > Faac/ap
(1)
When one considers that the mechanism of sliding is dominant, the balance between the two forces can simply be written as
Fc > µFa
(2)
where µ is the static friction coefficient. The adhesion force Fa is related to the adhesion energy per unit surface Ea through the Deryagin relation Fa ) (14) Budde, C.; Waltzel, P. Part. Sci. Technol. 2000, 18, 175-185. (15) Sharma, M. M.; Chamoun, H.; Sita Rama Sharma, D. S. H.; Schechter, R. S. J. Colloid Interface Sci. 1992, 149, 121-134.
Figure 10. Capillary forces divided by particle radius (Fc/a) as a function of the interparticle distance L. The calculations were performed with a contact angle of R ) 0, a surface tension of γ ) 72 mN/m, a density F ) 1000 g/L, and in the limit of complete drying (l0 ) 0). Different radii are considered: a ) 0.1, 0.2, 0.3, and 0.4 µm. The energy threshold (0.1 × 4πEa) (k ) 10 or µ ) 0.1) for colloidal mobility corresponding to the adhesion energy measured in a previous paper (ref 11), in the presence and absence of citric acid, is also shown.
4πapEa. If we suppose that ac is proportional to ap with ac ) ap/k (k > 1), then the mobility criterion for a rotation mechanism is Fc > 4πapEa/k and for a sliding mechanism is Fc > 4πapEaµ. In the following, the equations are based on the rotation notation. The same relations are obtained for sliding by replacing k by 1/µ. In the case of an alumina particle on mica, as no aggregation occurs in pure water, the exerted capillary force is less than the adhesion force Fc/ap < 4πEa/ k. In ref 11, the adhesion was found equal to Ea ) 1.6 × 10-3 J/m2. A first relation between Fc and k is thus obtained, Fc/ap < 2 × 10-2/k. When enough citric acid has adsorbed to the colloids, their surface charge is reversed and the adhesion was found to decrease to Ea ) 4.1 × 10-4 J/m2. In this case, the lateral capillary force induces a surface aggregation, that is, Fc/ap > 4πEa/k ) 5.2 × 10-3/ k. On the other hand, the lateral capillary forces between two spherical particles adhering to a surface can be calculated according to the procedure in Kralchevsky et al.6 The maximum force is obtained in the limit of complete drying (l0 f 0, see Figure 9). Figure 10 shows the capillary force Fc/a at a free water height of l0 ) 0 and a contact angle of R ) 0 as a function of the separation distance between two spheres of radius a ) 0.1 (ap), 0.2, 0.3, and 0.4 µm. For rough particles, a reasonable estimate of the radius of the contact area is ac/ap ) 0.1 (i.e., k ) 10) and µ ) 0.1 is a typical value for a static friction coefficient. In that case, the threshold of capillary force for triggering migration is Fc/ap ) 2 × 10-3 N/m in pure water and Fc/ap ) 5 × 10-4 N/m in the presence of citric acid for both a sliding and a rolling mechanism. Figure 10 shows that these criteria are in agreement with the theoretical evaluation of the capillary forces. The main factor controlling the force balance between two particles is their distance of separation. Therefore, the force criteria eq 1 or 2 is equivalent to a distance criterion L < Lc where Lc is a critical distance below which two colloids cannot resist the capillary forces. According to the adhesion measured in ref 11, the capillary force computed for spherical particles of radius 0.1 µm gives a critical distance for aggregation of Lc ) 0.42 µm for k ) 10. Such an aggregation criterion however fails to explain the kinetics of aggregation previously observed.11 One possible explanation for this aggregation kinetics could
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be the time required for the citric acid to adsorb on the alumina colloids.16 When the colloids are dried almost immediately after being contacted with the citric acid solution, the adhesion force is still strong enough to prevent the colloid surface migration. The fact that an adsorbed layer aged in citric acid does not present any aggregates if it is rinsed in pure water before drying can be explained by the reverse phenomena, that is, the desorption of the citric acid from the surfaces of the colloids increases their adhesion to the mica surface which can again balance the capillary forces. The last step is to construct an aggregation model that accounts for the size distribution of aggregates. Indeed, one important feature of the dried layers of alumina colloids is the presence of both isolated singlet colloids and large aggregates as shown in Figure 3e. This is different from a diffusion-limited 2D cluster-cluster or particle-cluster mechanism, as in these cases, monomers will barely coexist with large aggregates. Hence, a different process of aggregation has to be constructed, which reproduces the special distribution of sizes. This model is constructed as follows: on a rectangular surface of area S, one considers the random adsorption of n placed colloids of radius ap. The surface coverage is thus τ ) nπap2/S and the average distance d2 ) S/n. Then, following the criteria of out of balance capillary force, every couple of colloids for which the mutual separation distance is less than Lc will move to create aggregates. In practice, the particles are moved in the following manner: the first couple to be aggregated is the couple of colloids for which separation distance is the smallest below Lc. Then, the moves are generated following increasing distance of separation either from a singlet or an aggregate. The aggregation procedure runs until no isolated particle is closer than the threshold separation Lc from any other particle or aggregate. In such an aggregation procedure, it is possible to compute the number of remaining isolated colloids, ni, after the aggregation is completed. One just defines the probability for k particles to adsorb in a disk Lc around a given particle as
P(x ) k) ) Cnkpk(1 - p)n-k
(3)
with p ) πLc2/S. Then, the probability for a particle to remain isolated is
Q(x ) 0) ≈ P(x ) 0) + P(x ) 1)P(x ) 2) + ... (4) The first term, P(x ) 0), is the probability that a central colloid is initially isolated inside a disk of radius Lc. The second term is the probability that the same colloid has initially only one neighbor at a distance less than Lc which itself has two neighbors in its own disk and aggregates with one of the two. It is not worth taking into account further terms in Q(x ) 0) as they are negligible. Indeed, Figure 11 shows Q(x ) 0) as a function of Lc/d together with the simulation of 2D aggregated colloids for Lc/d ) 0.5, 1, and 1.5. The circles in Figure 11 correspond to the mean value of Q(x ) 0) ) ni/n obtained from 5 simulations starting from different random structures. The main feature of Figure 11 is that the fraction of isolated particles is high as soon as Lc < d. The final morphology of the aggregates obtained with this model is certainly not correct as capillary forces between aggregated particles induce some other local compaction that is not accounted for. (16) Desset, S.; Spalla, O.; Lixon, P.; Cabane, B. Langmuir 2001, 21, 6408-6418.
Figure 11. Fraction of remaining monomer particles at the end of the aggregation process as a function of the critical aggregation distance Lc/d computed using eqs 3 and 4. The points correspond to an average over 5 simulations. As an example, the morphologies of three different simulated layers are shown. The black point corresponds to Figure 3e. Note that only its vertical coordinate was measured (Q(x ) 0) ) 0.16) and the curve is used to deduce its horizontal coordinate Lc/d ) 0.85.
However, the knowledge of the number of remaining isolated monomers as a function of d is independent of such rearrangement of the aggregates. This result is important, and it allows an estimate of Lc from the counting of remaining monomers on a dried surface of a known rate of coverage. With spherical colloids, it would be in principle possible to deduce the maximum sustained capillary force directly from the measure of Lc. Finally, we can calculate Q(x ) 0) in the case of Figure 3e. It is possible to count the number, ni, of remaining isolated particles on the AFM image: ni ) 170. As the rate of coverage is τ ) 0.15, the total number of particles is n ) Sτ/πap2 ) 1074 and the average initial separation d ) 0.46 µm. Therefore, as Q(x ) 0) ) ni/n ∼ 0.16, the critical aggregation length must be 0.85d ) 0.39 µm according to Figure 11. The value for Lc determined from the morphology of the layer in Figure 3e is slightly below Lc ) 0.42 µm. This value was obtained from the force balance using the measured adhesion and capillary forces for spheres of identical radius (0.1 µm). Both values of Lc are of the same order of magnitude. Nevertheless, it is necessary to keep in mind the following points: (i) the determination of ni is not precise due to particle polydispersity; (ii) the determination of Lc from the adhesion forces depends on parameters k or µ that are arbitrarily chosen; (iii) the contact angle is not measured and may vary with the citric acid adsorption; (iv) the capillary force exerted on the alumina colloids can only be approximated by the value for spheres of radius 0.1 µm. Going further in the quantification would not be reasonable with such alumina colloids, and further quantitative validation of the proposed aggregation mechanism requires the use of well-defined particles. V. Conclusion The surface mobility of irreversibly adsorbed colloids was suspected from the observation of 2D aggregation kinetics of alumina onto mica11 and is often considered in the simulation of the deposition of colloids onto a solid
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surface.17,18 In this work, an original in situ light scattering experiment was used together with AFM microscopy to demonstrate that adsorbed alumina colloids do not aggregate in situ and are indeed immobile at the surface of mica. The 2D aggregates observed after drying for weakly adhering particles are due to the lateral capillary forces only. The capillary force is able to aggregate two particles (17) Oberholzer, M. R.; Wagner, N. J.; Lenhoff, A. M. J. Chem. Phys. 1997, 107, 9157-9167. (18) Gray, J.; Bonnecaze, T. J. Chem. Phys. 2001, 114, 13661381.
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separated by less than a threshold distance Lc. Furthermore, this threshold controls the number of remaining singlet particles after drying. A good agreement is obtained between the observed number of singlet particles and a calculation of Lc obtained by balancing the capillary force and the adhesion force. However, further quantitative validations of the proposed mechanism require a system in which the capillary interaction and adhesion forces can be fully calculated or measured. LA011824C
