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Transient Responses of a Wetting Film to Mechanical and Electrical Perturbations† Roge´rio Manica,‡ Jason N. Connor,§ Lucy Y. Clasohm,‡ Steven L. Carnie,‡ Roger G. Horn,| and Derek Y. C. Chan*,‡,⊥ Particulate Fluids Processing Centre, Department of Mathematics and Statistics, The UniVersity of Melbourne, ParkVille 3010, Australia, PELM Centre, Central Queensland UniVersity, Gladstone, Queensland 4680, Australia, Ian Wark Research Institute, UniVersity of South Australia, Mawson Lakes SA 5095, Australia, and Department of Mathematics, National UniVersity of Singapore, Singapore 117543 ReceiVed May 28, 2007. In Final Form: June 20, 2007 This article reports real-time observations and detailed modeling of the transient response of thin aqueous films bounded by a deformable surface to external mechanical and electrical perturbations. Such films, tens to hundreds of nanometers thick, are confined between a molecularly smooth mica plate and a deformable mercury/electrolyte interface on a protuberant drop at a sealed capillary tube. When the mercury is negatively charged, the water forms a wetting film on mica, stabilized by electrical double layer forces. Mechanical perturbations are produced by driving the mica plate toward or by retracting the mica plate from the mercury surface. Electrical perturbations are applied to change the electrical double layer interaction between the mica and the mercury by imposing a step change of the bias voltage between the mercury and the bulk electrolyte. A theoretical model has been developed that can account for these observations quantitatively. Comparison between experiments and theory indicates that a no-slip hydrodynamic boundary condition holds at the molecularly smooth mica/electrolyte surface and at the deformable mercury/electrolyte interface. An analysis of the transient response based on the model elucidates the complex interplay between disjoining pressure, hydrodynamic forces, and surface deformations. This study also provides insight into the mechanism and process of droplet coalescence and reveals a novel, counterintuitive mechanism that can lead to film instability and collapse when an attempt is made to thicken the film by pulling the bounding mercury and mica phases apart.
1. Introduction With the development of direct force measurements techniques based on the surface force apparatus (SFA) and the atomic force microscope (AFM), it is possible to study in some detail the interaction between fluid drops (bubble or liquid) in a continuous phase. In such soft systems, interfacial separations, surface forces, and surface deformations are all interlinked and can vary during interaction. Nonequilibrium effects can arise from the relative motion of the interacting interfaces due to hydrodynamic interactions. In addition, possible material transport between phases as well as within the interface can also be involved. Although such problems may be complex, they are central in applications such as the processing of droplets and emulsion dispersions, in solvent extraction technology, and in the capture of mineral particles by bubbles during flotation recovery. In the past decade, a number of studies, both experimental and theoretical and based on the SFA and the AFM, have been reported that focused on the effects of deformations on surface interaction1-13 and on dynamical effects due to the relative motion †
Part of the Molecular and Surface Forces special issue. * Corresponding author. E-mail:
[email protected]. The University of Melbourne. § Central Queensland University. | University of South Australia. ⊥ National University of Singapore. ‡
(1) Butt, H-J. J. Colloid Interface Sci. 1994, 166, 109. (2) Ducker, W. A.; Xu, Z.; Isrealachvili, J. N. Langmuir 1994, 10, 3279. (3) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357. (4) Fielden, M. L.; Hayes, R. A.; Ralston, J. Langmuir 1996, 12, 3721. (5) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483. (6) Aston, D. E.; Berg, J. C. J. Colloid Interface Sci. 2001, 235, 162. (7) Bhatt, D.; Newman, J.; Radke, C. J. Langmuir 2000, 17, 116.
of the interacting deformable drops or bubbles.14-22 Specifically, experimental results of interactions involving deformable mercury interfaces under static5,10 and dynamic15 conditions had been reported. Related experimental force measurements involving deforming oil/water interfaces using the atomic force microscope17,18 have been modeled in detail19-21 with excellent agreement between experiment and theory. Early attempts at quantitative modeling of the dimpling phenomenon observed on the deformable mercury/electrolyte interface22,23 have been (8) Chan, D. Y. C.; Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2001, 236, 141. (9) Nespolo, S. A.; Chan, D. Y. C.; Grieser, F.; Hartley, P. G.; Stevens, G. W. Langmuir 2003, 19, 2124. (10) Connor, J. N.; Horn, R. G. Langmuir 2001, 17, 7194. (11) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2002, 247, 310. (12) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2004, 269, 84. (13) Attard, P.; Miklavcic, S. J. Langmuir 2001, 17, 8217. (14) Aston, D. E.; Berg, J. C. Ind. Eng. Chem. Res. 2002, 41, 389. (15) Connor, J. N.; Horn, R. G. Faraday Discuss. 2003, 123, 193. (16) Gunning, A. P.; Mackie, A. R.; Wilde, P. J.; Morris, V. J. Langmuir 2004, 20, 116. (17) Dagastine, R. R.; Stevens, G. W.; Chan, D. Y. C.; Grieser, F. J. Colloid Interface Sci. 2004, 273, 339. (18) Dagastine, R. R.; Chau, T. T.; Chan, D. Y. C.; Stevens, G. W.; Grieser, F. Faraday Discuss. 2005, 129, 111. (19) Carnie, S. L.; Chan, D. Y. C.; Lewis, C.; Manica, R.; Dagastine, R. R. Langmuir 2005, 21, 2912. (20) Carnie, S.L.; Chan, D.Y.C.; Manica, R. ANZIAM J. 2005, 46E, C805. (21) Dagastine, R. R.; Manica, R.; Carnie, S. L.; Chan. D. Y. C.; Stevens, G. W.; Grieser, F. Science 2006, 313, 210. (22) Manica, R.: Connor, J. N.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C. Langmuir 2007, 23, 626. (23) Manica, R.: Connor, J. N.; Dagastine, R. R.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C. Phys. Fluids (submitted). (24) Clasholm, L. Y.; Connor, J. N.; Vinogradova, O. I.; Horn, R. G. Langmuir 2005, 21, 8243. (25) MacCormack, D.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 169, 177. (26) Connor, J. N. Ph.D. Thesis, University of South Australia, 2001. (27) Yiantsios, S. G.; Davis, R. H. J. Fluid Mech. 1990, 217, 547.
10.1021/la701562q CCC: $40.75 © 2008 American Chemical Society Published on Web 07/27/2007
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Figure 1. (a) Schematic diagram of the surface force apparatus with an aqueous electrolyte film between a mica plate and a protuberant mercury drop from a sealed capillary tube with radius r1 ) 0.5 mm or 1.5 mm. The film thickness h(r, t) is measured by fringes of equal chromatic order (FECO) detected by a spectrometer. (b) Disjoining pressure due to electrical double layer interactions that will stabilize an aqueous film when the repulsive maximum exceeds the Laplace pressure (2σ/R) of the mercury drop at a mica surface potential of -100 mV and a mercury surface potential of -35 mV at 0.1 mM 1:1 electrolyte concentration. The disjoining pressure for an unstable film with a mica surface potential of -100 mV and a mercury surface potential of -25 mV is also shown. The effect of van der Waals interaction is shown to be unimportant for the conditions under study.
equally successful. More complex deformation modes of the same surface have also been observed.24 In this article, we report more detailed experimental observations and modeling of transient responses of thin equilibrium aqueous films that arise from mechanical and electrical perturbations. Such films, with an initial thickness of less than 100 nm, are stabilized by electrical double layer repulsion between a molecularly smooth solid mica surface and a deformable mercury/electrolyte surface in the SFA (Figure 1). The surface potential of the mercury is maintained by an externally applied potential between the mercury and a calomel electrode in the bulk electrolyte. Mechanical perturbations to the film are generated by (28) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (29) Horn, R. G.; Asadullah, M.; Connor, J. N. Langmuir 2006, 22, 2610. (30) Teskov, R.; Vinogradova, O. I. Langmuir 2005, 21, 12090. (31) Ajaev, V. S.; Teskov, R.; Vinogradova, O. I. Phys. Fluids 2007, 19, 061702. (32) Yang, S.-M.; Leal, L. G.; Kim, Y.-S. J. Colloid Interface Sci. 2002, 250, 457. (33) Yoon, Y.; Borrell, M.; Park, C. C.; Leal, L. G. J. Fluid Mech. 2005, 525, 355.
Manica et al.
(a) pushing the mica surface toward the mercury/electrolyte interface for a fixed distance over a preset time interval or (b) pulling the mica surface away from the mercury/electrolyte interface. Electrical perturbations are applied by (c) making a step change in the applied voltage on the mercury, which results in a rapid change in the electrical double layer interaction between the mica and the mercury and can include switching from a stable film configuration to an unstable one (Figure 1b). By observing the response to mechanical perturbations, we gain insight into the deformation mechanism on the nanometer scale that thin films can undertake in order to make the transition from one equilibrium state to another under the constraints of surface forces, interfacial tension, and hydrodynamic interactions. The electrical perturbation studies yield information on the modes of collapse of unstable nanometer-thick aqueous films when the stabilizing effects of surface forces are suddenly removed. We also make quantitative comparisons between the observed time-dependent deformations of the mercury/electrolyte interface as the aqueous film accommodates such perturbations using a theoretical model that takes into account surface forces, hydrodynamic flow in the thin film, and deformations of the mercury/electrolyte interface. This comparison allows us to elucidate the respective contributions of surface forces and hydrodynamic pressure variations in determining interfacial deformations during the course of the transient response of the film. We can also use the theory to model situations that have not yet been accessed experimentally in order to provide insight into the phenomena occurring. Details about the modified SFA used to study such films have been presented previously.15 In brief (Figure 1), the applied potential on the mercury is adjusted to produce a repulsive electrical double layer disjoining pressure between the mica and the mercury drop (with an undeformed radius of ∼ 2 mm) that protrudes beyond the end of a sealed capillary. The thickness of the aqueous film between the mercury and mica is measured by an optical interference technique based on fringes of equal chromatic order (FECO). In the experiment, the mica surface is positioned close to or below the top of the mercury drop, and the repulsive disjoining pressure will cause the mercury/electrolyte interface to flatten and trap a circular equilibrium aqueous film that is typically between 20 and 100 µm in radius and less than 100 nm thick. From this equilibrium state, the mechanical and electrical perturbations described above are applied, and the resulting deformations of the mercury are recorded by video capture of the FECO. From these data, time variations in the film thickness can be resolved with subnanometer spatial resolution in the film thickness h and micrometer resolution in the radial position r. For theoretical comparisons, the Poisson-Boltzmann theory will be used to describe the dissimilar electrical double layer interaction between the mica and the mercury/electrolyte interface (e.g., ref 25). The accuracy of this theory for the present system has been verified experimentally.10 The deformation of the mercury/electrolyte interface is also shown experimentally to obey the Young-Laplace equation with a known constant interfacial tension.26 We assume that there is no material or charge transfer across the mercury/electrolyte interface. Because of the thin-film geometry indicated above, the Stokes flow Reynolds theory of film drainage will be used to describe hydrodynamic flow within the film with hydrodynamic boundary conditions to be specified at the mica/electrolyte and at the mercury/electrolyte interface. From previous work,22 it was observed that the
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stationary, no-slip boundary condition, hitherto referred as the “no-slip” boundary condition for brevity, at both the mica surface and at the mercury/electrolyte interface provides the most accurate description of hydrodynamic dimpling experiments in the same system and will therefore be used in the present analysis. A detailed description of the theoretical model has been given elsewhere.19,22 We note that the no-slip boundary condition at the mercury/electrolyte interface implies that flow within the mercury drop does not have to be considered. Further implications of the no-slip hydrodynamic boundary condition will be considered in the Discussion section. The thin film is assumed to be axisymmetric with constant shear viscosity µ, and its thickness, h(r, t) is a function of radial coordinate r and time t. The aspect ratio of the film permits the use of the lubrication approximation, and the thickness therefore obeys the form of the Stokes-Reynolds equation that applies when no-slip boundary conditions hold at the mica and mercury surfaces:
∂h 1 ∂ 3∂p ) rh ∂t 12µr ∂r ∂r
(
)
(1)
The Young-Laplace equation with constant interfacial tension σ is used to calculate deformations of the mercury/electrolyte interface consistent with the dynamic disjoining pressure, which is made up of the sum of the disjoining pressure, Π(h(r, t)) for the electrical double interactions between the mica surface and the mercury/electrolyte interface (van der Waals interactions are negligible at the film thicknesses investigated here as shown in Figure 1) and the hydrodynamic pressure, p(r, t) (measured relative to the bulk fluid),
2σ σ ∂ ∂h -p-Π r ) r ∂r ∂r R
( )
(2)
where the Laplace pressure is 2σ/R and the known constant interfacial tension of the mercury/electrolyte interface can vary parametrically with the applied potential on the mercury. The apex of the mercury has a measured radius of curvature R0 when it is far from the mica surface. The total instantaneous force between the mica and the mercury is given by
F(t) ) 2π
∫0∞ [p(r, t) + Π(h(r, t))] r dr
(3)
The numerical solution of eqs 1 and 2 in the spatial domain (0, rmax) requires boundary conditions. Note that rmax is chosen to be large enough to take the mercury surface beyond the range of disjoining pressure but is still much smaller than the capillary radius r1. Symmetry considerations require ∂p/∂r and ∂h/∂r to vanish at r ) 0. As r f ∞, it has been shown that h(r, t) has the asymptotic form19
r2 2R 1 1 + cos θ r F 1 + log + log 2πσ 2 1 - cos θ 2R0
h(r, t) ) X(t) - z0 +
[
(
)
( )]
(4)
where X, z, and θ are defined in Figure 1; z0 ) z(r ) 0, t ) 0) is the height of the apex of the undeformed mercury drop above the capillary. Therefore, by differentiating this result we have another boundary condition:
∂h dX(t) ) ∂t dt rmax 1 + cos θ 1 dF 1 1 + log + log 2πσ dt 2 1 - cos θ 2R0
[
(
)
( )]
at r ) rmax (5)
At large r, the pressure decays as r-4,27 and this condition is implemented as
∂p + 4p ) 0 at r ) rmax ∂r
r
(6)
For the initial condition, the mercury interface has a quadratic profile, so the initial film thickness is h(r, 0) ) hinit + r2/(2R0) when the mercury/electrolyte interface is far from the mica surface. In calculating the total force using eq 3, we divide the range of integration into two parts
F ) 2π
∫0r
max
[p + Π] r dr + 2π
∫r∞
pr dr
(7)
max
where the first integral is evaluated from the numerical solution and the second integral, in which the disjoining pressure Π is negligibly small, can be evaluated analytically from the asymptotic r-4 form of the pressure. A typical choice of the computational domain (rmax/R0)(σ/µV)1/4 ≈ 10-15 is sufficiently large to ensure that interfacial profiles, forces, and pressure are all independent of the precise value of rmax. In section 2, we present experimental results of deformations of the mercury/electrolyte interface starting from an equilibrium thin film as the mica plate is pushed toward the mercury. A detailed analysis of the physical basis of the observed “wimple”24 is provided. In section 3, we present results and analysis for the response of the film when the mica is pulled away from an equilibrium position. This study uncovers a novel mechanism for destabilizing thin films by a pull-off perturbation. In section 4, we analyze the collapse of a stable equilibrium film that follows a step change in the applied potential on the mercury that removes the stabilizing repulsive electrical double layer barrier. These novel observations suggest that there are different modes of coalescence considered in section 5 that depend on the initial shape of the film but capillary wave-induced instabilities appear not to be the collapse mechanism under the conditions of our experiments. The article closes with a discussion.
2. Compression of a Stable Film: The Wimple The experimental details for compressing an equilibrium film and observing the details of the resulting transition to another equilibrium state have been reported.24 Starting with a stable aqueous film of comparatively small radius (∼20 µm) stabilized by electrical double layer repulsion between the mica and the flattened mercury/electrolyte interface, the mica is driven toward the mercury drop for a distance of 10 µm in a time of less than 1 s. In response, the mercury/electrolyte interface deforms, develops a characteristic shape dubbed a “wimple”,24 and then subsequently evolves into the more familiar “dimple” shape of trapped fluid, which finally drains to form a new equilibrium film of larger radial dimension but with the same thickness as the initial film. The equilibrium film thickness is determined by the separation at which the disjoining pressure balances the internal Laplace pressure of the mercury drop. Because deformations of the mercury drop remain small on the scale of the drop, the Laplace pressure is essentially constant. The measured time sequence of the evolution of the mercury/ electrolyte (1 mM KCl) interface described above is shown in three stages in Figure 2. The response of the film during the interval when the mica is being pushed is shown in Figure 2a. The initial state (t ) 0) is a 50-nm-thick equilibrium aqueous film between the mica plate and the flattened mercury/electrolyte (34) Valkovska, D. S.; Danov, K. D.; Ivanov, I. B. Colloids Surf., A 1999, 156, 547.
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Figure 2. Response of an initially stable aqueous film between the mica plate and the flattened mercury/electrolyte interface when the mica is driven toward the mercury drop. Experimental results are taken from ref 24 at times (a) 0, 0.04, 0.08, 0.12, 0.2, and 0.28 s, (b) 0.28, 0.44, 0.6, 0.92, 1.24, 1.88, and 2.52 s, and (c) 2.52, 3.8, 6, 8, 10, 14, and 18 s. Theoretical results from the model are plotted at times (d) 0, 0.04, 0.08, 0.12, 0.2, 0.28, 0.44, 0.6, and 0.92 s, (e) 0.92, 1.24, 1.88, 2.52, 3.8, and 6 s, and (f) 6, 8, 10, 14, and 18 s. Arrows indicate the direction of motion with increasing time.
interface. When the mica is rapidly driven 10 µm toward the mercury, the central thickness h(0, t) of the film increases in thickness; that is, the central portion of the mercury/electrolyte interface backs away from the approaching mica surface while the thickness of the outer part of the film, beyond about r ≈ 50 µm, decreases. Shortly after the mica stops, at a time of 0.8 s, a second minimum in the film thickness develops at a large radial position at r ≈ 100 µm, in addition to the existing minimum at the axis of symmetry at r ) 0 (Figure 2b). These two minima give the characteristic wimple shape. After the formation of the wimple, the central thickness h(0, t) continues to increase while the thickness at the new film minimum at r ≈ 100 µm decreases
until the interfacial shape changes into a dimple profile with a barrier rim at r ≈ 100 µm at around 2 to 3 s. The final stage of film evolution is given in Figure 2c for t > 2 s, in which the dimple drains to an equilibrium film of the same thickness as the initial film. However, the final film radius is now several times larger at rfilm final ≈ 100 µm compared to an initial film radius of ∼20 µm. Throughout the entire process, which takes about 18 s in total, the film maintains axial symmetry. The corresponding stages of deformations of the mercury/ electrolyte interface as predicted by the Stokes-ReynoldsYoung-Laplace-Poisson-Boltzmann (SRYLPB) equations (eqs 1 and 2) are shown in Figure 2d-f. The experimental and
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Table 1. Nominal Experimental Parameters of the Mica-Mercury System and Corresponding Theoretical Values Used in the Calculation physical parameters
experimental
theoretical
interfacial tension σ (mN/m) viscosity of water µ (mPa s)
420 ( 6 0.89
420 0.89
Wimple mica surface potential (mV) [28] drop surface potential (mV) electrolyte concentration (mM KCl) drop radius R0 (mm) maximum mica travel ∆Xmax (µm) drive velocity V (µm/s)
-90 ( 10 -500 ( 20 1 ( 0.01 3.04 ( 0.1 10 ( 3 fast
-90 -500 1 3.04 7.2 8
Pull-Off mica surface potential (mV) -100 ( 10 drop surface potential (mV) -492 ( 20 electrolyte concentration (mM KCl) 0.1 ( 0.01 drop radius R0 (mm) 1.9 ( 0.1 drive velocity V (µm/s) 23 ( 1
-90 -492 0.11 1.9 24
Jump mica surface potential (mV) initial drop potential (mV) final drop potential (mV) electrolyte concentration (mM KCl) drop radius R0 (mm)
-90 ( 10 -47, -16 ( 20 -42, -11 ( 20 0.1, 1 ( 0.01 0.67, 0.9 ( 0.1
-90 -51, -17 -47, -11 0.11, 1.00 0.67, 0.74
theoretical parameters used to produce the results in Figure 2 are summarized in Table 1. Using parameter values within experimental uncertainties, the SRYLPB model gives semiquantitative agreement with experiment. The degree of agreement is not as precise as that found for earlier related SFA experiments on dimple formation when the mica surface is driven toward the mercury drop from a large separation.22 One reason for this is associated with the drive time being short (∼0.5 s) and comparable to the start-up and slow-down times of the motor used to drive the mica surface. As a consequence, the exact drive function of the mica, which is an input to the theoretical model, is not well characterized for the wimple experiment. Nonetheless, the main features of the experiment, namely, the initial increase in the central separation h(0, t), the development of the second minima at a large radial position, the formation of the characteristic wimple, the evolution from a wimple to a dimple, and the final drainage to the new equilibrium film with the same thickness as for the original film but with a larger flattened radius, are all predicted by the model. During the development of the wimple and subsequently the dimple shown in Figure 2a,b, the relative contributions to the total constant force from electrical double layer and hydrodynamic interaction change over time. An examination of this change provides insight into the physical processes that lead to the development of the wimple and subsequently the dimple during film drainage. The theoretical predictions for the disjoining pressure and the hydrodynamic pressure are given in Figure 3. For easy reference, the curves are labeled A-E as time progresses. At early times as the mica approaches the mercury, the film thickness at r ≈ 0 increases slightly (Figure 3a, curves A and B) as the mercury drop backs away from the approaching mica to reduce repulsive interactions at the expense of increasing the interfacial area. The increase in central film thickness causes the disjoining pressure to fall (Figure 3c, curves A and B), and the hydrodynamic pressure is small in magnitude and is slightly negative around r ≈ 0 (Figure 3d, curve A). However, the thinning of the film around r ≈ 30-50 µm (Figure 3a, curve B) causes the hydrodynamic pressure to increase locally (Figure 3d, curve B), which drives fluid toward r ) 0, causing the central portion of the film to continue thickening. At the formation of the wimple
Figure 3. Theoretical values of the film thickness h(r, t) in nanometers and the hydrodynamic p(r, t) and disjoining pressure Π(r, t) (in units of σ/R0) at various times corresponding to the wimple experiment in Figure 2. (A) 0 s, (B) 0.92 s, (C) 1.88 s, (D) 6 s, and (E) 18 s.
(Figure 3a, curve C), all parts of the mercury/electrolyte interface are more than 80 nm from the mica, so the disjoining pressure is negligible throughout the film (Figure 3c, curve C) while the hydrodynamic pressure dominates (Figure 3d, curve C). From this time onward, the rate of thinning of the draining film is controlled by hydrodynamic considerations. When the film thickness at the barrier rim in the film profile at r ≈ 100 µm reaches a thickness of ∼50 nm (Figure 3a, curve D), the repulsive disjoining pressure prevents further film thinning beyond this point. Now the inverted curvature of the dimple causes the fluid trapped within the dimple at r e 100 µm to drain slowly out of the barrier rim to form the final uniform film (Figure 3d, curve E). As the film thins everywhere in the r e100 µm region, the disjoining pressure in this region increases (Figure 3c, curve E) while the hydrodynamic pressure decreases as the film approaches equilibrium (Figure 3d, curve E), as we have previously demonstrated for dimple drainage.22,29 From this analysis, the combined roles of disjoining and hydrodynamic pressure in creating the complex wimple-dimple shape are evident. An earlier attempt to account for the wimple shape did not consider hydrodynamic effects,30 and from the above discussion, such an explanation would appear to be incomplete. The same group later added hydrodynamic pressure to its calculations and also disjoining pressure in the form of van der Waals repulsion,31 which is suitable for a gas bubble approaching a solid surface under water. Their treatment of far-field boundary conditions is not appropriate for the experiments discussed here. The time variation of the force between the mica and the mercury as predicted by the SRYLPB model is given in Figure 4. To a first approximation, the force is the internal pressure of the drop (which remains essentially constant because the extent of deformation is small compared to the undeformed drop radius) multiplied by the flattened surface area, so a 5-fold increase in contact radius corresponds to about a 25-fold increase in the total force from its initial value of 0.35 µN. This increase takes place rapidly in the short time interval (
