Kinetics of Liquid Annulus Formation and Capillary Forces - Langmuir

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Kinetics of Liquid Annulus Formation and Capillary Forces Y. I. Rabinovich, A. Singh, M. Hahn, S. Brown, and B. Moudgil* Particle Engineering Research Center, University of Florida, Gainesville, Florida 32611, United States ABSTRACT: The dependence of the capillary adhesion force Fcap between a silica microsphere and a flat silica surface versus a time period t of the samples’ contact (i.e., dwell-in time) is experimentally investigated using atomic force microscopy (AFM). Fcap was found to be dependent on t if the humidity was >30
35%. This dependence is exponential, with decay (characteristic) times of ∼10 s. It is suggested that the kinetics of the adhesion process are related to the growth of the water annulus between surfaces. Furthermore, we propose that the growth kinetics has two components: (1) water vapor diffusion from the surrounding humid media into the gap between samples and (2) water drainage from the gap. The theory of diffusion through thin pores (i.e., gaps) is developed, and analytical formulas are obtained for the dependence of the meniscus radius r versus time t. However, the experimental dependence of Fcap versus t and, consequently, r versus t obtained in this article disagrees with the theoretical prediction by several orders of magnitude. Similar results were obtained from the literature data for capillary forces between an AFM cantilever tip and a flat surface. Possible reasons for the deviation from diffusion theory are suggested: surface and Knudsen regimes of vapor diffusion, nonsteady state vapor flow, and tortuosity. Taking into account the viscous drainage of water from the multilayer gap can explain the experimental kinetics of bridge formation, but only if the viscosity of the adjacent multilayer of water is several orders of magnitude larger than the bulk viscosity.

I. INTRODUCTION Capillary forces play an important role in many industrial processes, such as the pharmaceutics, paint, and food industries, and impact the flow behavior of powder/liquid mixtures in silos and hoppers. The origin of capillary forces is also an important fundamental problem related to the adhesion of surfaces. They have been investigated experimentally and theoretically for a long time by many researchers. In particular, a classic formula for ideally smooth surfaces suggesting a spherical meniscus was obtained many years ago on the basis of the Young
Laplace equation.1 For nonspherical but smooth particles, the role of shape was evaluated numerically.2,3 There have been many theoretical and experimental investigations of capillary forces.1,4,5 Over the past few years, model measurements of the capillary force were performed using atomic force microscopy (AFM) on a flat surface using either the cantilever tip or an individual particle mounted on the AFM cantilever.6
10 Such experiments are very precise, and many features of capillary forces were investigated. The effect of sample roughness on the capillary force was evaluated by Rabinovich et al.,8 Farshchi et al.,6 and Butt.7 Rabinovich et al.8
10 measured the detachment force between a silica tip and a flat silica surface in a humid atmosphere8 or using oil as a liquid interface.9,10 In ref 8, the effect of roughness was evaluated experimentally and theoretically, and a simple analytical formula was developed for capillary forces between rough solids. This theory suggests an interaction between the sphere (or tip) and the flat plate that is separated by a distance equal to the peak height. Later, Drelich11 suggested formulas for various models of roughness and surface deformation. A more precise r 2011 American Chemical Society

theory was created by Butt et al.,7,12 where the authors suggested calculating capillary forces between rough solids by taking into account the number of contacts between the rough peaks and a shape (either spherical or nonspherical). The authors also considered the contribution of the dispersion force to the total adhesion force between solids separated by liquid interfaces of various thicknesses.12 One disadvantage of the capillary force measurements using an AFM cantilever tip is the wearing of the tip shape during the measurements. Butt et al.12 have shown that the significant deviations in the experimental data for the capillary force obtained by various authors can be explained by tip wearing. The authors suggested a numerical method to evaluate this effect by using a two-sphere model taking the tip deformation into account, which explained the change in capillary force (increase, decrease, or constant) for high humidity. The problem remains, however, of how to control the tip shape during the experimental measurements rather than before and after it. Experiments for dynamic measurements of adhesion forces13 investigated the dependence of the adhesion force on the rate of the loading force between a sphere on an AFM cantilever and a flat surface coated with a monolayer of thiol. The adhesion force increased with the increasing rate of the loading force. The kinetics of the capillary forces were also investigated by Ally et al.,14 who measured the capillary force as a function of the approach/retraction velocity between a polystyrene microsphere Received: June 10, 2011 Revised: September 20, 2011 Published: September 23, 2011 13514

dx.doi.org/10.1021/la202191c | Langmuir 2011, 27, 13514–13523

Langmuir and glass coated with a film of silicone oil and water with thicknesses of 0.2
1.8 μm. In the case of wetting films of various thicknesses, the kinetics of capillary forces are determined by the velocities of sample wetting and film drainage. However, the dependence of capillary adhesion forces on the sample’s contact time remains almost uninvestigated. In particular, the kinetics of the development of the capillary force immediately after sample contact was investigated in only two papers;15,16 the authors measured the capillary force Fcap for water between an AFM cantilever tip (silicon nitride) and a plane of either mica15 or silicon16 as a function of the sample’s contact time before detachment. When the humidity was >25%, the capillary force increased with contact time t. However, the authors of ref 15 did not explain how the capillary adhesion force is related to the annulus size. Moreover, the authors investigated the dependence of the adhesion force on the contact time at only one level of humidity (42%). Experimental data for the kinetics of the capillary force were not compared with theory; instead, only a qualitative explanation of the kinetics was suggested on the basis of vapor diffusion. In ref 16, capillary forces were measured versus the contact time for various humidity levels, with increasing contact times resulting in increased forces. At 65% humidity, the experimental value of the characteristic time was ∼2 s, whereas at lower humidity, the characteristic time was shorter. Wei and Zhao16 repeated the theoretical development of Kohonen et al.17 to obtain the kinetics of vapor diffusion in the sample gap, considering Knudsen’s regime of diffusion instead of bulk diffusion. Then, for 65% humidity, this formula predicts a characteristic time of ∼1 ms (i.e., much shorter than the experimental result reported in ref 16). They also evaluated the characteristic time for liquid transport into the meniscus through the equilibrium liquid film, supported by van der Waals forces; this predicted value was similar to the experimental value. The problem remains, however, that both processes (the vapor diffusion and the liquid transport through the liquid film) occur simultaneously; consequently, the characteristic time should be dominated by the faster process. Another approach to capillary annulus growth has been made by Kohonen, Maeda, and Christenson.17 The authors did not consider the possible relationship between the capillary force and the annulus growth in the formation process. Instead, in this study the surface force apparatus (SFA)18 was used for optical measurements of the diameter of the annulus versus the time for water and various organic liquids between mica cylinders. The vapor pressure was near saturation (>99%), the radius of the mica samples was ∼2 cm, and the radius of the meniscus calculated from the annulus size was between 20 and 300 nm during the annulus-formation process. The radius versus time curve was exponential, as theory predicted; however, the experimental characteristic time was found to be near 100
2000 s (for various liquids and vapor pressures), which was several times larger than the theoretical value. Butt and Kappl reveal even more data about the kinetics of annulus formation.4 They plotted the theoretical dependence of the meniscus radius versus time for particles of 5 μm radius under humidities ranging from 50 to 90%. (See Figure 12 in ref 4.) Using the modernized equation of Kohonen et al.,17 they have shown that the characteristic time is between 13 and 50 μs for humidities of 50
90%. However, this figure is theoretical rather than experimental. The kinetics of capillary adhesion forces are important for processes where short-time contacts between particles occur in a

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Figure 1. AFM image of a flat silica surface. The rms of the roughness is 0.23 nm.

humid atmosphere. For example, the contact time of particles in dispersions and flowing through hoppers19
21 can be on the order of milliseconds. If the capillary force changes during the course of annulus formation, then the theory must be changed to take into account these mechanical properties of liquid powder mixtures, where such a fast relative movement of particles takes place. In this article, AFM was used to measure the capillary adhesion force between a silica microsphere and a flat silica surface at various humidity levels for different dwell-in times before sample detachment (and in some cases also for the dwell-out time after detachment). For humidities in the range of 30 to 60%, the capillary force depends on the dwell-in time and is related to the growth of the annulus. A theory of vapor diffusion is developed, and an analytical formula for the meniscus radius versus time is obtained. The characteristic time obtained from diffusion and drainage theories is compared with experimental results.

II. EXPERIMENT: METHODS AND MATERIALS The capillary force was measured using an Asylum Research MFP3D AFM (Santa Barbara, CA). The method of force measurements with AFM suggested by Ducker, Senden, and Pashley22 is known as a colloid probe method and was used to measure the capillary force in many studies. In this article, the capillary force was measured as the detachment force between silica microspheres with radii near 3 μm (Bangs Laboratories, Inc., Fishers, IN) and either a flat silica surface (microscope slide) or a silicon wafer coated with silica (Silicon Quest International, San Jose, CA, U.S.). The microsphere was mounted on the cantilever tip with epoxy glue. Before measurements, the spheres were cleaned with a plasma discharge and the flat silica was cleaned with acetone, 200 proof ethanol, and nanopure water, followed by plasma discharge. An AFM image of the typical flat silica surface is shown in Figure 1. The root-mean-square (rms) value of roughness measured by AFM was ∼0.2
0.3 nm for flat silica and for the sphere. Measurements were made in standard AFM humidity-controlled cells in a nitrogen or air atmosphere with controlled humidity. A schematic of the humidity monitoring is shown in Figure 2. Cell 1 with a volume of ∼1 cm3 (coated with a rubber membrane, not shown) was connected to relatively large vessel 2 with a volume of ∼2 dm3, which contained water. Dry nitrogen flowed into cell 1 from gas tank 3 with control provided by flow meter 4 and valve 5. Humid air flowed from vessel 2 through flow 13515

dx.doi.org/10.1021/la202191c |Langmuir 2011, 27, 13514–13523

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Figure 3. Statistical distribution of the capillary force Fcap: number of cases vs the force value. R = 3.2 μm, dwell-in time = 10 s, and humidity ψ = 46%.

Figure 2. Schematic of humidity monitoring. (1) The cell, (2) a large vessel, (3) a gas tank, (4) flow meters, (5, 6) valves, and (7) a humidity meter. meter 4 and valve 6. The humidity, ψ, was monitored by controlling valve 6 near flow meter 4 and measured with humidity meter 7 within the cell 1. Measurements were made in the humidity range of 8 to 70%. The experimental cycle has been carried out as follows: the piezocrystal moves samples closer to each other (at a rate of 2 μm/s) until the repulsive force set point is achieved (usually 10 nN). Then the piezocrystal stops for the dwell-in time (between 0 and 30 s). The interaction force remains constant during the dwell-in time, after which the piezocrystal moves samples away from each other. The maximal attractive force is the detachment force, which is equal to the capillary force (plus ion
electrostatic and dispersion forces, which, however, are negligible at moderate and high humidity). The samples remain in the separated position during the dwell-out time (between 0 and 10 s), after which the experimental cycle is repeated. The movement of the piezocrystal for the whole cycle was controlled automatically by the AFM program. Another method that was applied was force
volume measurements. In this case, the above-mentioned experimental cycle took place at one contact point, after which the piezocrystal (together with one sample) moved automatically in the xy direction to the next point and the cycle repeated at this point. A typical experiment would include 8  8 points, with a distance of 2.5 μm between each point. This data provides an idea of the detachment force variability due to inherent sample surface heterogeneity (e.g., roughness). Twenty consecutive force
distance measurements were made at each humidity level and dwell-in and dwell-out times. The dwell-in time was changed, and the measurement cycle was repeated at the same sample spot under the same humidity. Then the humidity was changed by valves near the flow meters and the next measurement’s cycle (for various humidity values and dwell times) was conducted at the same contact place. Dwell-in and dwell-out times (in the contact sample’s position and after the sample’s detachment, respectively) were varied from 0 to 10 s. Because of the restricted velocity of the piezocrystal near 2 μm/s, the real minimal time of sample contact was ∼0.1 s. Comparing the results for various dwell-in times with the results for the same dwell-in time and an additional dwell-out time allowed us to draw conclusions about the equilibrium (or nonequilibrium) state of the wetting film between measurements. The vertical scan distance was 1 μm, and the time of one measurement (excluding the dwell time) was 1 s. We attached a silica microsphere to a silicon tipless cantilever (MikroMasch NSC12/tipless/ AlBS). The stiffness of each cantilever was measured by a thermofluctuation method.23 The average cantilever spring constant was ∼20
30 N/m. The loading force was 10
20 nN, which is significantly less than the capillary force measured.

Figure 4. Schematic of the interacting sphere and plane. θ is the contact angle; j is the half angle surrounding the deformed zone and liquid annulus; x is the coordinate of the liquid meniscus; d is the coordinate x corresponding to the equilibrium meniscus; and R and r are the radii of the sphere and meniscus, respectively. The capillary force was calculated as the average value of 20 measurements of the pull-off force at each humidity and dwell time. The statistical distribution of the capillary force measurements (number of cases vs capillary force) is shown in Figure 3. The variance of the resulting experimental force was