Osmotically Driven Deformation of a Stable Water ... - ACS Publications

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Osmotically Driven Deformation of a Stable Water Film Sue A. Chen,*,†,‡ Lucy Y. Clasohm,§ Roger G. Horn,∥ and Steven L. Carnie† †

Department of Mathematics and Statistics, The University of Melbourne, Parkville 3010, Australia IBM Research - Australia, Carlton 3053, Australia § Ian Wark Research Institute, Mawson Lakes 5095, Australia ∥ Institute of Frontier Materials, Deakin University, Burwood Highway, Burwood 3125, Australia ‡

S Supporting Information *

ABSTRACT: An aspect of dynamic colloidal interactions that has received little attention is the osmotic stress associated with nonequilibrium distribution of solutes. Recent experiments on a mercury drop near a mica surface show a dimple forming on the mercury/water interface when there is a sudden change in the electric potential of the mercury drop coated with a selfassembled monolayer (SAM) of 11-mercapto-1-undecanoic acid thiol molecules. A reasonable hypothesis is that the dimple formation is due to the desorption of a fraction of the SAM from the mercury drop surface when the surface potential is changed. The osmotic pressure in the thin film region increases as a result of the presence of the thiol molecules in the region, giving rise to the observed dimple. A model including the effects of osmotic flow, disjoining pressure, interfacial tension and hydrodynamic pressure is developed to test the hypothesis. The simplest version of the model, in which desorption is uniform and instantaneous, can produce a dimple whose growth is significantly more rapid than its decay, in qualitative agreement with the data. However, quantitative agreement is lacking. Several refinements to the model, including effects such as the change in interfacial tension as thiols are desorbed, gradual thiol desorption, a change in disjoining pressure as charged thiols are desorbed and nonuniform desorption do not change the qualitative picture. The qualitative success of the model suggests the osmotic pressure mechanism is correct, but the detailed picture of the SAM desorption at positive mercury surface potentials is not sufficiently well understood. The model reveals that the osmotic dimple is not the time-reverse equivalent of the usual hydrodynamic dimple phenomenon. We suggest that transient deformation of thin films by osmotic flow is a new and littlestudied mechanism influencing the structure of stable thin films and the interaction of deformable drops. This has implications for colloidal interactions in a broader range of systems where solute concentration may not be homogeneous, for example in solute transfer processes.

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INTRODUCTION The area of microfluidics research has experienced phenomenal growth in the past two decades due to extensive applications in fields ranging from physics to engineering to biomedicine. Among the applications that merge the field of nanotechnology with microfluidics are lab-on-a-chip systems for drug delivery, integration of microfluidics with nanoneuroscience in studies of the nervous system and nanomaterial synthesis. The development of the Surface Force Apparatus (SFA) has facilitated the direct measurement of surface forces1,2 acting at the nanoscale between molecularly smooth surfaces. Another measurement tool, the Atomic Force Microscope (AFM), has enabled the measurement of in situ interaction forces between many kinds of surfaces. The mechanism behind the SFA includes an optical technique using fringes of equal chromatic order (FECO) to measure the evolution of surface profiles of drop deformation during interaction. In the SFA, white light is shone through two thin mica sheets, each coated with a layer of highly reflective silver on the outer surfaces, and aligned orthogonally to form a © 2015 American Chemical Society

crossed cylinders geometry. To extend the capabilities of the SFA, Connor and Horn3 modified the SFA to include a fluid surface (mercury). This modification expanded the range of deformable surfaces on which SFA techniques can now be used to include bubbles and drops. One application of this experimental system is the investigation of the shape of a mercury drop deformed by hydrodynamic forces when it is in motion normal to the mica surface through analysis by video recording of FECO, detected in the light reflected from mercury.4 The modified SFA also provides real time measurements of the hydrodynamic drainage process of the aqueous film for thicknesses down to around 50 nm with subnanometer precision. On the other hand, the AFM is set up such that a cantilever is pressed against a surface and the distance between the cantilever and surface is controlled by a piezoelectrical tube. Received: June 18, 2015 Revised: August 7, 2015 Published: August 11, 2015 9582

DOI: 10.1021/acs.langmuir.5b02220 Langmuir 2015, 31, 9582−9596

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at the mercury/water interface, reminiscent of the dimple seen when a drop approaches a solid surface. The dimple forms quite quickly and then gradually drains, sometimes over a period of 30 min. This is much slower than conventional thin film drainage which occurs within tens of seconds when a bare mercury drop approaches mica in a similar experimental setup.3 Drainage is known to depend on film viscosity, interfacial tension and hydrodynamic boundary conditions,23 none of which would be sufficiently affected by the adsorbed thiol monolayer to account for substantially longer drainage times observed here. One suggestion37 is that the formation of the dimple is due to the desorption of a fraction of the SAM from the surface of the mercury drop when the surface potential is changed. The osmotic pressure in the thin film region increases as a result of the presence of the thiol molecules in the film and this in turn causes the observed dimple. In this paper, we attempt to verify this explanation by constructing a detailed model incorporating osmotic flow into the thin film model.

The cantilever deflection measurements are subsequently converted into force. The range of velocities involved in the motion of drops of around 50 μm radius encompasses the range of thermal velocities for drops of this size. The AFM is also less sensitive to contamination due to the small surface areas used. Dagastine et al.5 made the first AFM measurements of the force between two approaching drops of decane (40 μm radius) coated with sodium dodecyl sulfate in an aqueous solution. The work done by Nespolo et al.6 and Dagastine et al.7 describe the difficulty in the measurement of equilibrium surface forces involving liquid particles due to the interfacial deformability. Extensive research done in the past decade, both experimental, based on the SFA and AFM, and theoretical probed the effects of deformation on equilibrium surface interaction3,6,8−17 and on dynamical effects as a result of the relative motion of deformation drops and bubbles.4,5,18−23 Experimental results obtained using the SFA on the interaction of a smooth mica surface and a deformable mercury drop for both repulsive and attractive interactions were compared to the predictions of a theoretical model derived using a combination of both lubrication theory and the normal force balance at the mercury/water interface. It found excellent agreement between the experimental and theoretical approaches.23 The same model was applied to the analysis of force curves between drops of around 50 μm in radius measured with the AFM19 and its results gave good agreement with the experiments.5 The good fit between theory and experiment has greatly increased our understanding of the dynamics of thin liquid films of thicknesses 20−100 nm. When a drop or bubble approaches a solid surface immersed in liquid, the liquid film between them can, under the right circumstances, generate sufficient pressure to reverse the curvature of the fluid/fluid interface, producing the so-called hydrodynamic dimple. There have been extensive experimental measurements of hydrodynamic dimpling.4,24−31 Attempts to quantitatively model the phenomenon date back to work by Hartland,27 which was then refined by Slattery.32 Among the most detailed attempts to fit a model to the dimple phenomenon were those by Manica et al.23 A more complex interface shape, dubbed the wimple, was observed in 200533 and first fitted in a theoretical model by Tsekov and Vinogradova.34 Later models incorporating hydrodynamic effects successfully reproduced this phenomenon.35,36 Central to such systems is the interplay between thin film hydrodynamics, interfacial tension forces and surface forces included through the disjoining pressure. A key component in the modeling of these SFA experiments was the development of a novel asymptotic boundary condition that correctly describes the far-field deformation at the outer edge of the computational domain, incorporating assumptions about the three-phase contact line where the mercury drop protrudes from a capillary held at constant pressure.19 Clasohm37 reported a different and new type of film deformation which has not yet been explored theoretically. A mercury drop is first coated with a layer of thiol molecules (11mercapto-1-undecanoic acid) that spontaneously adsorb to form a close-packed monolayer at the mercury-water interface, thus modifying the surface properties of the drop. The structure of such a layer is aptly called the self-assembled monolayer, otherwise known as SAM.38 A sudden change in the surface potential of the mercury drop from −150 to −200 mV vs a saturated calomel electrode (SCE) results in a dimple forming

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EXPERIMENTAL OBSERVATIONS

Details of the experimental system and methods can be found in a previous publication.39 In brief, a mercury drop is immersed in a chamber containing aqueous electrolyte (1 mM KCl), and forms the working electrode in a 3-electrode system with a Pt counter-electrode and saturated calomel electrode (SCE) as the reference. The mercury drop is held captive at the open top of a vertical capillary of internal diameter 3 mm while a horizontal surface of mica is mounted in the upper part of the chamber and can be moved vertically down to approach the drop with nanometric position control. The mica is a thin uniform sheet whose upper side is part-silvered; white light passing down through a window above the mica and reflected back undergoes interference between the silver layer and the mercury surface. Analysis of the reflected spectrum in a spectrograph allows the thickness of the aqueous film separating the mica and mercury to be measured with an accuracy of ±0.5 nm in the vertical direction and a resolution of ∼1 μm along the horizontal. Since the mica reference surface is flat, this gives a cross-section of the profile of the mercury drop measured through its apex.4 A droplet of 10 mM/L solution of 11-mercapto-1-undecanoic acid in ethanol is injected into the aqueous medium close to the mercury drop. Within 10 min a self-assembled monolayer of the thiol compound forms on the mercury surface, with a density of (95 ± 10) μC/cm2 (corresponding to (17 ± 2) Å2 /molecule) estimated from separate cyclic voltammetry desorption experiments.39 Desorption occurs over a range of applied potential from −200 to −800 mV SCE. To start an experiment the mica surface is moved down toward the mercury. As a result of double-layer repulsion between the mica and mercury surfaces, the top of the mercury drop is flattened and a uniform aqueous film forms with a thickness at which double-layer pressure equals the initial Laplace pressure in the drop.3 The presence of repulsion shows that the acid thiol-decorated mercury has the same sign of surface charge as the mica, which is well-known to be negative. The decorated mercury remains negative over the entire experimental range of applied potential from 0 to −1500 mV SCE, in contrast to bare mercury which is negative only for applied potentials less than about −400 mV SCE.3 From an initial configuration of a flattened drop (uniform film thickness) at a potential of −150 mV, the applied potential is changed abruptly to −200 mV SCE. The cyclic voltammogram shows that some desorption of the undecanoic acid occurs in this range.39 Figure 1a and b show a time series of aqueous film profile measurements following the step in applied potential that causes partial desorption. It is seen that a “dimple” shape forms, reaching a maximum amplitude after 225 s, after which it decays over a further 30 min until the uniform film is restored at a similar thickness to the original film. However, in the present case the mica has not been moved with 9583

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terms of an osmotic pressure, which is another way to formulate the solvent chemical potential. Since osmotic pressure is proportional to solute concentration for dilute solutions, the solute concentration c(r,t) is introduced as a new dependent variable in the system and the transport of the solute is described by a convection-diffusion equation. The thin film equation and the convection-diffusion equation combine to give a system of coupled partial differential equations. The work done here bears some similarities with the work of Danov et al.,40 who observed a cyclic dimple formation in a thin film system where a reservoir of solute in the (oil) drop enters the thin film by mass transfer across the oil/water interface. Their numerical treatment of the problem was based on the lubrication approximation for the thin film hydrodynamics and took into consideration the solute fluxes due to convection and diffusion. What sets our system apart from that of Danov’s is that the mercury/SAM system here does not involve mass transfer across the interface and no renewal of solute takes place. The dimpling phenomenon is a transient one since the film returns to its original thickness after the drainage process has ended. The model requires equations describing the hydrodynamics in the thin film, the motion of solute in the film and the deformation of the mercury/water interface. • The application of the Stokes-Reynolds theory of film drainage to describe the hydrodynamic flow within the film is appropriate, given the thin film geometry of the system, represented by film thickness h(r,t). The incorporation of solute concentration c(r,t) as a driving force in the hydrodynamic flow is new. • The motion of the solute (averaged over the thickness of the film) is described by convection and diffusion in a film with time-varying thickness. • The deformation of the interface is described by the normal force balance at the mercury/water interface, known as the Young−Laplace equation, including a disjoining pressure term Π(h) to account for surface forces. To describe the dissimilar electrical double layer interaction between the mica and mercury electrolyte interface, we apply the Poisson−Boltzmann theory on which the Gouy−Chapman model is based. Experimental work by Connor3,41 proved the accuracy of the theory for this system. Given that κh > 2 in the cases studied here, where κ−1 is the Debye length and h the film thickness, a simple superposition approximation for given values of electrolyte concentration ce and surface potentials on the two surfaces Ψd and Ψm is used to represent the disjoining pressure

Figure 1. (a) Formation of a dimple on sudden change of applied potential from −150 to −200 mV (SCE). The legend shows the times in minutes and seconds at which measurements were made. (b) Relaxation of the dimple over the next 7 min. This dimple takes 30 min to return to the original flattened state. respect to the fixed capillary holding the mercury, and the height of the drop above the capillary would not be changed by a change in interfacial tension between mercury and water because the latter is sensibly constant over this range of potential.39 In other cases, a shorter-lived dimple is formed, such as when the applied potential is changed abruptly from −340 to −360 mV SCE.37 This is shown in Figure 2.

Figure 2. Formation of a dimple on sudden change of applied potential from −340 to −360 mV (SCE). The dimple has formed before the first trace was recorded at 25 s, and then relaxes over 5 min.

⎛ Ψ ⎞ ⎛Ψ ⎞ Π(h) = 64kTce tanh⎜ d ⎟tanh⎜ m ⎟e−κh ⎝ 4kT ⎠ ⎝ 4kT ⎠

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(1)

where k is the Boltzmann constant and T is the temperature. Governing Equations. The governing equations are presented here. A more complete description of their derivation and the computation method is provided in the Supporting Information. The flux of the solvent can be decomposed into two components, one being the flux due to pressure gradients, and the other the flux due to solute concentration gradient. The pressure that would have to be applied to a pure solvent to inhibit the inward flow into a given solution by osmosis is

THEORY: A FIRST MODEL We develop a model for this experimental system by assuming that a certain fraction α of the adsorbed thiol molecules desorbs uniformly and enters the aqueous solution instantaneously. We treat the thiol as a simple uncharged solute, that produces a gradient of solvent chemical potential and hence a flow of solvent into the thin film region, such a flow is called an osmotic flow. This extra flow has to be included in the usual film thinning equation. Osmotic flows are often described in 9584

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where both c and h are functions of r and t. This equation is the required convection-diffusion equation for a channel with timevarying height h(r, t) . The Young−Laplace equation, describing the normal force balance across the mercury/water interface, is unchanged from the usual expression

(2)

where c is the solute concentration in the experimental system. This extra pressure term features in both the convectiondiffusion equation and the thinning equation. The driving force for solvent flow is the solvent partial pressure42 ppart = p − kTc

p+Π=

⎤ ∂h 1 ∂⎡ 3∂ = ⎢rh ppart ⎦⎥ ⎣ ∂t ∂r 12μr ∂r ∂ ∂c h3 ∂ Dh ∂ ⎛⎜ ∂c ⎞⎟ [c(r , t )h(r , t )] = ppart + r ∂t ∂r 12μ ∂r r ∂r ⎝ ∂r ⎠ ppart + kTc + Π =

(4)

(5)

Applying these boundary conditions and the usual lubrication assumptions yields the following thinning equation ⎤ 1 ∂⎡ 3∂ ∂h = ⎢rh (p − kTc)⎥⎦ 12μr ∂r ⎣ ∂r ∂t

Γ = α ΓSAM (6)

(10)

Assuming the solute concentration rapidly becomes uniform across the film thickness, we obtain the following initial condition

which differs from the one used previously only by the contribution of solute concentration to the driving force. Here μ the viscosity of the continuous phase which is assumed to be constant even at very small separations. The mass flux for the solute, Jsolute, must be written in terms of a convective flux and a diffusive flux Jsolute = c u − D∇c

σ ∂ ⎛ ∂h ⎞ 2σ ⎜r ⎟ − R r ∂r ⎝ ∂r ⎠

These equations are equivalent to eqs 6, 8, and 9 that we prefer to use, since they make the role of solvent motion under an osmotic gradient clearer, as discussed in Hunter, Appendix A6.43 Initial and Boundary Conditions. For an initial profile, we assume h(r,0) to be the flattened drop from a previous push toward the mica surface which also gives the initial disjoining pressure profile, the hydrodynamic pressure being zero. The initial solute concentration profile is derived using the conditions described in Clasohm.37 We assume a uniform instantaneous desorption of thiol molecules across the interface. The surface excess Γ is approximated by the surface density (molecules/area), which in turn is a product of the close-packed surface density of SAM molecules ΓSAM and α, describing the fraction of SAM desorbed into the thin film region

At the wall, the no-slip and no-penetration boundary conditions are uz = ur = 0 at z = 0

(9)

where σ is the interfacial tension and R is the radius of the mercury drop. An alternative formulation to the problem is to choose the partial solvent pressure ppart to be the driving force. This results in the conventional thinning equation, a convection-diffusion equation and a Young−Laplace equation that has an extra term due to the contribution from osmotic pressure Πosm = kTc,

(3)

where p is the hydrodynamic pressure in the system. This expression ensures that solvent flows up gradients of solute concentration as well as down pressure gradients. Since the drop is assumed to be part of a flattened sphere initially and the solute is desorbed uniformly, the problem is written in cylindrical coordinates (r,θ,z) with z = 0 representing the mica surface. The velocity vector u is defined by ur, uz since the nature of the problem implies an axisymmetric flow. In addition, the large difference in the length scales implies a weak variation of the film thickness over a typical radial scale so that a lubrication treatment is appropriate. The experiment modeled here has low velocities and small length scales, implying that Ca ≪ 1 and Re ≪ 1 (Stokes flow). The no-slip boundary condition is applied at the solid/liquid interface. Previous experimental results on the same SFA system have validated the accuracy of the no-slip boundary condition rather than the full slip condition associated with the continuity of tangential stress across the mercury/water interface,35 that is, the interface is immobile. Consequently, the tangential component of velocity at the drop/liquid interface must be zero ur = 0 at z = h(r , t )

σ ∂ ⎛ ∂h ⎞ 2σ ⎜r ⎟ − R r ∂r ⎝ ∂r ⎠

c(r , 0) =

α ΓSAM h(r , 0)

(11)

where c(r,0) and h(r,0) are the initial solute concentration and initial film thickness, respectively, and α represents the fraction of molecules that have desorbed by the potential change at t = 0. The desorption fraction α is the only unknown parameter in this model. Due to the axisymmetric nature of the problem, the following conditions must hold at r = 0

(7)

where D is the diffusivity constant. The velocity in eq 7 is the same as that used to derive eq 6, that is, a radial flow with a parabolic profile. Using a balance equation for the number of solute particles in a control volume of varying height h(r,t) due to the solute flux in eq 7, we get ∂ ∂c h3 ∂ Dh ∂ ⎛⎜ ∂c ⎞⎟ [c(r , t )h(r , t )] = (p − kTc) + r ∂t ∂r 12μ ∂r r ∂r ⎝ ∂r ⎠ (8) 9585

∂p =0 ∂r

(12)

∂h =0 ∂r

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∂c =0 (14) ∂r Assuming that (to first order), the film profile at large r well beyond the barrier rim is parabolic and applying appropriate substitutions to the thinning equation yields an expression involving the derivative of the solvent partial pressure r

∂ (p − kTc) + 4(p − kTc) = 0 ∂r

From eq 2, it is necessary that the concentration scale cc is consistent with the pressure scale, and this results in the form of cc as given above. If we choose

(15)

(17)

∞

(19)

where G represents the force within the film, scaled by the interfacial tension to give dimensions of length. The contact angle θ is related to the drop radius and capillary radius by sin θ = rcap/R. This boundary condition was used before19,23,35 but V = 0 here, since the mica is kept stationary during the experiment. A low-deformation condition G/R ≪ 1 is required to derive this boundary condition. Scaled Equations. The problem is nondimensionalised with the following set of scales, rc =

hcR

R3μ σD

(26)

σ RkT

(27)

⎤ ∂h 1 ∂⎡ 3∂ = ⎢rh (p − c)⎥⎦ ⎣ ∂t 12r ∂r ∂r

(28)

∂ ∂c h3 ∂ h ∂ ⎛⎜ ∂c ⎞⎟ [ch] = (p − c ) + r ∂t ∂r 12 ∂r r ∂r ⎝ ∂r ⎠

(29)

1 ∂ ⎛ ∂h ⎞ ⎜r ⎟ r ∂r ⎝ ∂r ⎠

(30)

where the variables have now been made dimensionless by the scales given above, subject to the boundary conditions at r = 0,

with r[p(r , t ) + Π(h)]dr

tc =

p+Π=2−

(18)

∫0

(25)

The three nondimensional governing equations are the thinning eq 28, the convection-diffusion eq 29 and the normal stress balance eq 30

2 ⎡ 1 ⎛ r ⎞ 1 ⎛ 1 + cos θ ⎟⎞⎤ ̇ ⎥ = −V hrmax + Ġ ⎢1 + ln⎜ max2 ⎟ + ln⎜ ⎢⎣ 2 ⎝ 4R ⎠ 2 ⎝ 1 − cos θ ⎠⎥⎦

1 σ

R3μD σ

cc =

This boundary condition has no effect until very long times compared to the times probed in the experiments. The final boundary condition was derived for a drop placed on a solid stage, having a pinned three-phase contact line, with the drop making an angle with the solid surface called the contact angle.19 For the case of a drop with initial contact angle θ interacting with a solid plane moving toward the drop with velocity V in the SFA, the boundary condition we use is

G=

rc2 =

(16)

at r = rmax, the outer boundary of the computational domain. For simplicity, we implement a no-flux boundary condition on the convective-diffusive motion of the solute concentration c(r,t) at r = rmax ∂c ∂h h+ c=0 ∂t ∂t

(24)

to simplify eq 6, the remaining scales become

Following previous work,19,35 we implement an asymptotic boundary condition for p to reflect p ∼ r−4 at large separations

∂p r + 4p = 0 ∂r

DμR σ

hc =

∂p =0 ∂r

(31)

∂h =0 ∂r

(32)

∂c =0 ∂r

(33)

the boundary conditions at r = rmax, ∂c ∂h h+ c=0 ∂t ∂t

(34)

dp + 4p = 0 dr

(35)

dh dG +β =0 dt dt

(36)

r

(20)

r2 tc = c (21) D p cc = c (22) kT σ pc = (23) R where the length scale rc is the natural length scale used in thin films, and the time scale tc the diffusive time scale. Diffusion is the slowest process here and thus, the diffusive time scale controls the time dependence of other processes. The pressure scale pc is half the Laplace pressure needed to flatten a drop.

where G=

∫0

∞

β=1+

r[p(r , t ) + Π]dr

⎛ 2 1 ⎜ rmax ln⎜ 2 2 ⎝ 4R

R3μD σ

(37)

⎞ ⎞ ⎛ ⎟ + 1 ln⎜ 1 + cos θ ⎟ ⎟ ⎠ ⎝ 2 1 cos − θ ⎠

(38)

and the initial condition c(r , 0) = 9586

R 1 (kT ΓSAM) σμD h(r , 0)

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previously37,39 with no definitive explanation given; the thickness is not very sensitive to the choice of Ψm, Ψd or σ within credible bounds. This difference between experiment16 and the model system should be borne in mind in what follows. The value of desorption fraction α is not known directly, so it is treated here as an adjustable parameter. However, an approximate calculation is made in Clasohm37 based on an estimate of the osmotic pressure, which is related to the total desorbed thiol concentration in the aqueous film. That estimate gave a value of α = 0.006, and the range we have chosen for the calculations (0.0001−0.045) brackets that value. For very small values of α, the amount of solute desorbed into the thin film region will be insufficient to have any impact on the film thickness. Only at a certain value of α does the film deform. We call this the critical desorption value αc; it is calculated by balancing the osmotic pressure due to the solute with the Laplace pressure required to flatten the film

where h(r,0) is found numerically from a previous run. To calculate G numerically, the range of integration is divided into two parts where the first integral is evaluated numerically and the second, where the effect of the disjoining pressure Π is negligible, is evaluated analytically from the asymptotic r−4 form of the pressure. G=

∫0

rmax

r[p(r , t ) + Π], dr +

∫r

∞

rp(r , t )dr

max

(40)

Central differences are used in the approximation of the first and second derivative terms in the equations of the system. This gives rise to a system of 2n + 2 differential equations plus an algebraic equation coming from the functional G by evaluating the following integral using Simpson’s rule, G=

∫0

rmax

r[p(r , t ) + Π]dr

This expression relates G to all the variables hj as an algebraic constraint. We note that as r approaches zero, eq 28 has a singular behavior. Taking the limit of 28 and applying boundary conditions at r = 0 provides us the equation for ḣ0. Similarly, taking the limit of eq 29 as r goes to 0 and applying the boundary conditions at r = 0 gives us the form of ċ0 ḣ0. The boundary condition at r = rmax, eq 36 gives the equation for ḣn and likewise, the boundary condition eq 34 provides the behavior of cṅ. Since ck is one of the unknowns in our system, we implement a no-flux boundary condition at r = rmax for the equation ċn ḣn. The resulting system of equations has a singular state dependent mass matrix and is a differential-algebraic equation of index 1. We solve this system using a built-in ODE solver ode15s in MATLAB for the values cj, hj, G for j = 0,1,..., n as a function of time.

kBTα ΓSAM 2σ = h(0, 0) R

with the parameter values given in Table 1 and initial film thickness of 51 nm, giving αc ≈ 0.0011 (42) which sets the boundary between the solute transport in a fixed geometry film and solute-driven film deformation. Figure 3 shows the expected purely diffusive solute transport in a film of fixed geometry for a desorption fraction α much smaller than αc. There is no detectable film deformation at this value of α. The magnitude of solute concentration in the film region for a desorption fraction of 0.0001 is approximately 10−5 M. The steep concentration gradients observed initially at approximately 130 μm, are due to the edge of the thin film region. The concentration of thiol molecules is much higher for 0 ≤ r ≤ rrim than for r > rrim with a very steep gradient in the solute concentration at r = rrim, corresponding to the end of the narrow channel. Initially, diffusion occurs only at the edge of the thin film driven by the gradient from higher concentration in the thin film compared to much lower concentrations outside. At this value of α, there is little film pressure generated so little solute convection occurs and the process is mainly diffusive. Once the concentration at the film center r = 0 has reduced appreciably, subsequent diffusion occurs over the whole region. For clarity, we have separated these two regimes into the initial traces where diffusion occurs only at the film edge (Figure 3a) and the subsequent diffusion over the whole domain (Figure 3b). In order to produce some film deformation, we next consider a desorption fraction α = 0.002, a bit more than αc. Figure 4a and b show the film profile evolution h(r,t) for this value of α. Notice that the film is plotted to show the mercury/water interface below the mica surface at h = 0, in order to mimic the experimental situation seen in Figures 1 and 2. The magnitude of solute concentration in the film region for a desorption fraction of 0.002 is approximately 3 × 10−4 M. The concentration gradient at the edge of the film is now large enough to drive significant osmotic flow of solvent into the film. This flow increases the hydrodynamic pressure in the film and so increases the total film force G. This stage of dimple formation coincides with the interval over which G increases, which last about 10 s, and is shown in Figure 4a. A deformation of this magnitude would be detectable in an SFA experiment, given that the changes of the film thickness are around 10−20

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RESULTS: THE FIRST MODEL With the exception of the desorption fraction α, which is discussed below, we use parameter values determined in the experiments37 (R, rcap and σ) or taken from literature.44−46 These are listed in Table 1. The mica surface potential is known Table 1. Parameter Values Used in the Model parameter

value

source

undeformed radius of drop at apex, R radius of capillary, rcap interfacial tension, σ film viscosity, μ thiol diffusion constant, D SAM packing density, ΓSAM surface potential on mica, Ψm surface potential on drop, Ψd 1:1 electrolyte concentration, ce temperature, T

1.80 ± 0.05 mm 1.50 ± 0.01 mm 420 mN/m 0.89 × 10−3 Pa s 4.5 × 10−10 m2/s 5 × 1018 molecules/m2 −80 mV −150 mV 1.0 mmol/L 298 K

ref 37. ref 37. ref 37, 39. ref 45. ref 44. ref 46. ref 3. see text ref 37. ref 37.

(41)

from previous experiments3 and the stable aqueous film observed between mica and the SAM-decorated mercury drop39 indicates a large negative surface potential on the drop; here we use −150 mV. The initial flattened film profile is generated using code previously developed for the approach of a mercury drop toward a mica surface.23 These surface potentials produce an equilibrium film thickness of 51 nm, to be compared with the experimental thickness of 63 nm. Possible reasons for this discrepancy have been discussed 9587

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Figure 4. (a) Film profile evolution for critical desorption (α = 0.002) when the film force G is increasing: t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E), 12.8 (F) s. The colored characters correspond to their respective traces. (b) Film profile evolution for critical desorption when the film force G is decreasing: t = 14.08 (G), 35 (H), 63.51 (I), 114.36 (J), 211 (K), 500 (L) s. (c) Disjoining pressure profile for critical desorption (α = 0.002) when the film force G is increasing: t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E), 12.8 (F) s. Same-colored traces correspond to their respective film profiles in (a). (d) Disjoining pressure profile for critical desorption when the film force G is decreasing. Same-colored traces correspond to their respective film profiles in (b).

Figure 3. (a) Solute concentration profile evolution for weak desorption (α = 0.0001) at early times: t = 0 (A), 0.06 (B), 0.4 (C), 1.55 (D), 5.22 (E) s. The colored characters correspond to their respective traces. (b) Solute concentration profile evolution for weak desorption (α = 0.0001) at later times: t = 8.3 (F), 14.27 (G), 24.08 (H), 39.35 (I), 62.94 (J), 470.34 (K) s. This figure shows radial solute diffusion in a film of fixed geometry since α ≪ αc.

As the drop deforms and recedes from the mica surface, the disjoining pressure gradient at r = rrim moves inward since the part of the drop at the minimum film thickness is shifting radially inward. The disjoining pressure profile appears to retreat until the start of the dimple formation. When this happens, the maximum in disjoining pressure at the barrier rim becomes more pronounced. The region of significant disjoining pressure then starts moving outward, barrier rim first, as the dimple begins to drain and settles at its initial position. Notice that initially and finally the total load is taken by the disjoining pressure in the absence of hydrodynamic pressure. The formation of the barrier rim shows up very clearly in the first trace of Figure 4d. Figures 5a and b present the solute concentration profiles for the critical desorption fraction case. The solute concentration produced by the desorbed thiol is about 3 × 10−4 M. Although the solute concentration profiles for the critical desorption case look similar to those for the weak desorption case, the former indicates the presence of convection by the osmotic flow in addition to diffusion, as the concentration gradients beyond r = rrim are less steep than those in Figure 3, whereas the latter is a purely diffusive process, restricted by the smaller channel width. Finally, in Figure 5c and d we show hydrodynamic pressure profiles which act to drive solvent flow in and out of the film. The initial hydrodynamic pressure profile is almost flat since there is no flow at the start of the desorption. Initially osmosis drives solvent into the narrow channel thereby forcing open the channel. This produces a positive pressure zone just outside the film edge, and a negative pressure zone where the film is receding from the mica surface. This same pattern moves

nm. The barrier rim, the annular region of minimum film thickness separating the dimple from the region external to the thin film, does not form immediately after the potential step at t = 0 but only occurs after the drop recedes from the surface and the thin film region shrinks radially. We note that while the experimental barrier rim widens, the barrier rim in the simulations decreases in width. Once the dimple is formed, concentration gradients have been greatly reduced and osmotic flow largely ceases. Then the dimple drains under hydrodynamic pressure gradients as the film force G decreases, as shown in Figure 4b. The dimple produced for this value of α is quite short-lived, as it drains within 5 min of the simulation, reminiscent of the dimple seen in Figure 2, but with a much smaller magnitude. Another view of the dimple formation and drainage process is afforded by examining the disjoining pressure profiles over the same time interval. At separations larger than about 80 nm, the disjoining pressure is negligible. A positive disjoining pressure in the aqueous film corresponds to a repulsive force between the negatively charged mica and mercury surfaces. When the film pressure is equal to the internal pressure of the mercury drop, there is no pressure difference across the interface and this implies that the top of the drop is flat. This is consistent with the initial disjoining pressure profile given in Figure 4c. 9588

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Figure 5. (a) Solute concentration profile evolution for critical desorption (α = 0.002) when the film force G is increasing: t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E), 12.8 (F) s. The colored characters correspond to their respective traces. (b) Solute concentration profile evolution for critical desorption when the film force G is decreasing: t = 14.08 (G), 35 (H), 63.51 (I), 114.36 (J), 211.35 (K), 500 (L) s. (c) Hydrodynamic pressure profile for critical desorption (α = 0.002) when the film force G is increasing: t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E), 12.8 (F) s. Same-colored traces correspond to their respective film profiles in (a). (d) Hydrodynamic pressure profile for critical desorption when the film force G is decreasing. Same-colored traces correspond to their respective film profiles in (b).

Figure 6. (a) Film profile evolution for moderate desorption (α = 0.009) when the film force G is increasing: t = 0 (A), 0.01 (B), 0.08 (C), 0.39 (D), 0.83 (E), 1.68 (F), 3.59 (G), 5.59 (H), 9.15 (I), 19.37 (J) s. The colored characters correspond to their respective traces. (b) Film profile evolution for moderate desorption (α = 0.009) when the film force G is decreasing: t = 20.09 (K), 34.47 (L), 35.6 (M), 68.32 (N), 108.28 (O), 191.90 (P), 500 (Q) s.

inward radially as the film edge does, until there is a single maximum in the pressure profile at the film edge, decaying to zero near r = 0 and for r > rrim. The pressure gradient from the rim to the film center drives solvent inward, augmenting the osmotic flow; that from the rim to the external fluid drives solvent outward, opposing the osmotic flow. The film force G is continually increasing as the pressure becomes more positive. Once the pressure maximum has reached r = 0 (last trace of Figure 5c), G reaches its maximum and drainage commences, with the hydrodynamic pressure gradient driving solvent out of the dimple. Osmosis is a minor factor at this stage, with drainage primarily controlled by the film thickness at the barrier rim. As the desorption fraction α is increased, the general picture described above still holds. The magnitude of the film deformation increases but the qualitative features remain. Figure 6 shows that for α = 0.009, the resulting dimple has a depth of approximately 100 nm. As in preceding cases, the film initially shrinks radially before a barrier rim forms. However, the barrier rim in this instance moves much further away than seen in Figure 1a. The larger gap between the barrier rim and mica surface facilitates the dimple drainage, resulting in a shorter lived dimple. Although closer to the time scale seen in Figure 2, the barrier rim moves radially outward during drainage in the model, but the barrier rim barely moves radially in Figure 2. For a larger value of α = 0.09, a dimple with a depth of about 600 nm is formed over about 10 s but the barrier rim recedes by about 300 nm, and the whole drainage process last about 5 min.

It is instructive to compare the simulation results for a range of desorption fractions. The profiles of the force G over time for α = 0.0001, 0.002, 0.009, and 0.045 are compared in Figure 7. The exact value of α in the experiments is unknown and for the purposes of comparison, we include the simulation for α = 0.045 as this results in a dimple of a much greater magnitude than that observed in Figures 1b and 2. For all values of α considered, the immediate increase in film force G is fast, ranging from 10 to 50 s into the simulation. The increase in film force is due to the instantaneous and uniform desorption process leading to osmotic flow into the film region, as evidenced by the formation of the dimple and barrier rim. Subsequently the film force decreases gradually due to hydrodynamic drainage with a rate determined primarily by the film thickness at the barrier rim. The greater the degree of desorption, the larger the dimple, the greater the transient increase in film force and the longer the drainage process, even though the barrier rim is forced further away from the surface. These evince the transient nature of the scaled force G, evolving under zero velocity. More details of the dimple formation process can be gained by plotting the barrier rim height over time for α = 0.0001, 0.002, 0.009, and 0.045 as in Figure 7b. Here we can see that the dimple formation and barrier rim motion away from the surface are very fast, lasting only a few seconds. Since the barrier rim is now so far away from the surface, there is a fast 9589

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Article

THEORY: REFINEMENTS TO THE MODEL

The simplest mathematical model in which the desorption of the thiol molecules on the mercury drop occurs uniformly and instantaneously across the interface results in a short-lived dimple, due to the retreating barrier rim of the mercury drop in the simulations. It is thus natural to explore a model that endeavors to keep the barrier rim fixed at its initial position and will subsequently result in a longer-lived dimple. A Model with Time-Dependent Interfacial Tension during Desorption. Although we have treated the mercury/ water interface so far as having constant (and uniform) interfacial tension, electrocapillarity measurements39 show that the interfacial tension rises somewhat as the applied potential is decreased (and thiol is desorbed), by up to 30 mN/m from around 390 mN/m, depending on the starting potential and the change in potential. As the interfacial tension rises, the mercury drop will deform less from a spherical shape (be less flattened), and so push up toward the mica, that is, the higher interfacial tension acts as a kind of global “push” that keeps the barrier rim in place. For this reason, we explore a model where the interfacial tension of the drop changes as the thiol desorbs, with the change of interfacial tension being proportional to the amount of desorption. Since it is not feasible to solve the equations with a jump change in interfacial tension, we let both interfacial tension and surface density of thiols vary linearly with time over a short time interval, that is, a brief but gradual desorption rather than the instantaneous desorption considered so far. This calculation also provides a simplistic model to explore the possibility that a new equilibrium following the step change in mercury potential is not reached immediately, but only after a finite time for desorption and readsorption to rebalance and establish the new value of interfacial tension. Since all of our scalings so far use the interfacial tension, which is no longer constant, we need to scale using the initial interfacial tension σ0 and introduce a new parameter η(t), being the ratio of the changing interfacial tension to the initial interfacial tension, to the Young−Laplace equation. The evolution equation for the film thickness h(r,t) remains unchanged and thus, we apply the dimensional eq 6 as before. The other new feature of the model is the gradual desorption of the thiol molecules adsorbed on the mercury drop. A linear desorption profile is integrated into the existing model and desorption occurs in the same time frame as the increase in interfacial tension. Since one of the quantities affected by the newly introduced gradual desorption feature of the model is the solute concentration c(r,t), the convection-diffusion equation differs from the one in the previous model, eq 8, in that a source term that describes the gradual desorption process has to be incorporated. We still assume that the solute concentration is uniform across the film. The revised following convection-diffusion equation is thus

Figure 7. (a) Comparison of the film force (scaled with interfacial tension) G over time for values of desorption fraction α = 0.0001, 0.002, 0.009, and 0.045 (from bottom to top). (b) Comparison of the barrier rim height over time for values of desorption fraction α = 0.0001, 0.002, 0.009, and 0.045. All of the curves for the simulations being compared here start at 51 nm. For intermediate α values, the barrier rim forms some time into the simulation whereas for large α, the barrier rim forms very early in the simulation, rising very sharply at small t such that the first calculated point has a much larger value of barrier rim height. This is subsequently replaced by a complicated deformation between 15 and 40 s, after which the barrier rim is seen again.

initial phase of drainage in which the barrier rim relaxes back toward the surface over the next 20−40 s. By this stage the dimple is also much smaller. This stage corresponds roughly to the maximum in the film force. We have successfully formulated a model to incorporate osmotic flow to produce a transient dimple in a stable water film with qualitatively correct time scale and film profile, and a rise time significantly faster than the drainage time, as observed experimentally. However, the results show several characteristic features different to those observed in the experiments of Clasohm.37 This is because the barrier rim of the simulated dimple backs away from the mica surface even as the dimple is being formed. Clasohm’s experiments37 show that the barrier rim of the mercury drop remains approximately at its initial position during dimple formation and drainage, with slight fluctuations. These discrepancies between the theoretical and experimental results motivate the investigation of more complex models of the experimental system in the next section.

dα Dh ∂ ⎛⎜ ∂c ⎞⎟ ∂ ∂c h3 ∂ [ch] = ΓSAM (p − kTc) + r + dt r ∂r ⎝ ∂r ⎠ ∂t ∂r 12μ ∂r (43)

where the solute concentration c and film thickness h are functions of r and t and dα is the rate at which the desorption dt

dα

fraction changes. We have a new source term, ΓSAM dt , compared to eq 8, representing the gradual desorption of thiols. We take the desorption to be linear in time, so that 9590

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Langmuir ⎧ αf ⎪ t, t < T α (t ) = ⎨ T ⎪ α , t > T. ⎩ f

G0 = hc (44)

G=

and the final desorption fraction αf plays the same role as α in our first model. It is assumed that at the start of the desorption process, the initial interfacial tension is σ = σ0. In addition to this, the pressure at the start of the desorption process is at equilibrium and thus in the absence of dynamic pressure at equilibrium, the Young−Laplace equation can be expressed as 2σ σ ∂ ⎛ ∂heq ⎞ Π(heq (r )) = 0 − 0 ⎜r ⎟ R0 r ∂r ⎝ ∂r ⎠

(45)

(46)

(47)

(48)

G0 η(t )

∫0

(51)

G0 dη η 2 (t ) d t

(52)

(53)

where V represents the applied velocity. The magnitude of the right-hand side of the boundary condition eq 52 is approximately −44 nm/s and this is about 3 orders of magnitude smaller than the right-hand side of eq 53 which was typically about −10 μm/s in previous SFA experiments.23,35 This already tells us that, although the changing interfacial tension has the same effect as an external push, the effect is small and only lasts during the gradual desorption. Any other effects of an increased interfacial tension must show up in the Young−Laplace equation. We test these ideas quantitatively with a simulation for a moderate desorption fraction. The parameters used in the simulations are given in Table 1 with the only differences being: σ0 is 390 mN/m and σf is equivalent to σ (420 mN/m). The time T1 over which desorption takes place must be smaller than the observed time of dimple formation, for example, 25 s in Figure 3. It cannot be too small or the sudden change in η causes problems in the numerical solution. Here we have chosen one plausible value that is T1 = 20 s. We consider the case where the desorption fraction is sufficiently high to cause a dimple while simultaneously increasing the interfacial tension σ on the mercury drop. In the absence of desorption, the act of increasing the interfacial tension results in the barrier rim approaching the mica surface and subsequently yields a final film thickness that is less than the initial film thickness. It is expected that restricting the width of the channel will limit the amount of the desorbed thiol molecules diffusing out of the thin film region, consequently lengthening the time taken for the dimple to drain. One might hope that by incorporating the increase in interfacial tension, a long-lived dimple as seen in Figures 1a and b might be achieved. The model is run with the final desorption fraction set to 0.002. The initial film thickness is approximately 51 nm. Upon running the simulation, the development of an area of minimum film thickness is once again observed in the simulation. The mercury drop continues to back away, while simultaneously forming a dimple of 13 nm depth, over a period of 30 s. The unique behavior of the mercury drop where it first backs away from the mica, forming a dimple along the way before approaching the mica again and settling at a position

As the interfacial tension σ increases, it is more convenient to use the following expression to describe the force G

1 η(t )σ0

r(̂ p ̂ + Π̂)dr ̂

dh dG +β = −V dt dt

2 ⎡ 1 ⎛ r ⎞ 1 ⎛ 1 + cos θ ⎞⎟⎤ ̇ ⎥=0 + Ġ ⎢1 + ln⎜ max2 ⎟ + ln⎜ hrmax ⎢⎣ 2 ⎝ 4R ⎠ 2 ⎝ 1 − cos θ ⎠⎥⎦

=

∞

(50)

where β has been previously defined in eq 38. The above boundary condition is fundamentally different from the one we had previously, eq 36, as there is a nonzero term on the righthand side for t ̂ 0 for t ̂ < T̂ and β < 0. We compare this to the boundary condition implemented in the hydrodynamic dimple case,23

assuming that for small changes, the change in interfacial tension is proportional to the change in the surface packing density on the mercury/water interface. The problem at hand is an axisymmetric one, and therefore, the boundary conditions eq 12, eq 13, and eq 14 at r = 0 must hold. Similarly, we apply the boundary condition specified in eq 16. The no-flux boundary condition on the convective-diffusive motion of the solute concentration c(r,t) at r = rmax, eq 17, is implemented as before. For the case of a drop interacting with a stationary solid plane in the SFA, the boundary condition is

G =

∫0

r(̂ p ̂ + Π̂)dr ̂

̇ hrmax + Ġ0β = β

where η(t) has the following time-dependence ⎧ σ − σ0 t + 1, t < T ⎪ ⎪ σ0T η( t ) = ⎨ σ ⎪ t > T. , ⎪ σ ⎩ 0

hc η(t )

∞

where r̂ = r/rc,p̂ = p/pc and Π̂ = Π/pc. Using eq 49 in eq 48 yields

where the terms σ0 and R0 represent the interfacial tension at the start of the desorption and the undeformed drop radius, respectively. As thiol molecules desorb from the mercury surface, the interfacial tension increases and thus, the dynamic pressure, p is no longer negligible. To account for the evolving interfacial tension behavior, the new parameter η is introduced. We replace σ by σ0 η where η ≥ 1. With the Young−Laplace equation no longer in equilibrium, the pressure approaches equilibrium again at a different (higher) interfacial tension, described by the following equation ⎡ 2σ σ ∂ ⎛ ∂h ⎞⎤ p + Π(h(r )) = η(t )⎢ 0 − 0 ⎜r ⎟⎥ r ∂r ⎝ ∂r ⎠⎦ ⎣ R0

∫0

∞

r[p + Π]dr, (49)

with η(t) given above. Scaling with the appropriate scales yields 9591

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increasing interfacial tension providing a mechanism to keep the barrier rim in place has not proved fruitful, providing motivation for further modeling. Several simulations with higher desorption fraction α were performed where we found that the cutoff time for the desorption and change in interfacial tension processes affects the behavior of the solution. These simulations show the intricate relationship between the cutoff time T1 and the desorption α on the drop deformation. For higher α, a longer time is needed to counter-act the effects of increasing the drop interfacial tension. That is, if T1 is too short, we are only able to observe the effects of increasing the interfacial tension and no dimple is recorded. On the other hand, for smaller α, the right balance for T1 needs to be found in order to produce a dimple. If we allow a longer time for the desorption and increase in interfacial tension processes, we see that the effect of increasing the drop interfacial tension dominates the effects of desorption while for too short a time interval T1, the drop recedes away from the wall. Although the model does not reproduce the experimental observations, it has increased our understanding of the influence of interfacial tension on the final equilibrium film thickness. Additionally, we were able to gain an understanding on how the desorption fraction profile affects the deformation of the film. The results suggest that a slow approach to a new adsorption/desorption equilibrium is not the explanation for the slow drainage times observed in the experiment. The Effect of Charged solute. The previous models did not take into account the fact that the thiol molecules are weak acids, so can themselves be charged. The experimental observations39 are consistent with the following picture. • For sufficiently negative applied potentials (say less than −500 mV vs SCE) the mercury is negative, the undecanoic acid is undissociated and adsorbs to the mercury via its terminal carbon in a similar manner to uncharged thiols. The stable film is due to repulsion between the mercury and the mica. • For less negative potentials (say more than −400 mV vs SCE) the mercury becomes positive, the undecanoic acid is dissociated and adsorbs electrostatically, overcompensating the charge on the mercury to produce a negative dressed surface. The stable film is due to repulsion between the dissociated SAM and the mica. The dimples are only observed in this regime. This suggests that, as the thiols desorb, the charge on the dressed mercury surface will become less negative, reducing the repulsive disjoining pressure. It is thus convenient to characterize the disjoining pressure by the surface charge density on the drop and relate this to the desorption fraction α. Should a fraction of these thiol molecules desorb from the mercury drop, the charge on the mercury drop and the resulting disjoining pressure will change. To incorporate the charge on the thiol molecules, it is necessary to calculate the amount of charge generated by the surface excess packing density on the mercury drop. At separations more than a few Debye lengths, as is the case here, where the separation is the distance between the mica surface and the mercury drop, there is little distinction between the disjoining pressure from using constant charge or constant potential, and the nonlinear superposition approximation used in eq 1 is accurate.47 The surface charge q, a function of the

closer than its initial configuration suggests that although we have set the model to desorb and increase its interfacial tension in the same time frame, the drop appears to only feel the effects of one action at any given time. Despite implementing a change in interfacial tension during the desorption process, the barrier rim of the drop does not stay fixed at its position and it moves together with the dimple. Consequently, the dimple in this model is also a short-lived one, draining within 30 s of the simulation. As the osmotic dimple drains and approaches its initial flattened shape, it is observed that the drop settles at a distance less than what it started off at and this is due to the higher final interfacial tension of the drop, producing in turn a higher Laplace pressure. Figure 8 presents the solute concentration profiles for the uniform interfacial tension case and for the current model that

Figure 8. (a) Solute concentration profile when film force G is increasing for critical desorption with constant interfacial tension σ = 0.42 N/m: t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E) and 12.8 (F) s. The colored characters correspond to their respective traces. (b) Solute concentration profile when film force G is decreasing for critical desorption with constant interfacial tension σ = 0.42 N/m: t = 14.08 (G), 35 (H), 63.51 (I), 114.36 (J), 211.35 (K) and 500 (L) s. (c), (d) Similar to (a), (b) except with varying interfacial tension with σ0 = 0.39 N/m. The concentration profiles in (c) have been labeled according to chronological order for easier viewing: t = 0 (A), 4.01 (B), 9.03 (C), 14.05 (D), 19.06 (E) s. The comparatively lower cmax for the refined model is due to the competition between α(t), diffusion and osmosis processes.

assumes a gradual increase in interfacial tension. The initial solute concentration profile for the current model is just zero everywhere, unlike the initial solute concentration profile for the simplest model. The solute concentration profile takes on a bell-like curve of approximately 60 μm width with a maximum at r = 0 as the thiol molecules gradually desorb. The heights of the concentration curves increase as α(t) increases and, once t > T, the thiol molecules begin to diffuse out of the thin film region, as evidenced by the increasing widths and decreasing heights of the subsequent curves. The barrier rim in this refined model still moves away by a distance similar to that observed in the first model simulated. This is somewhat surprising given that the concentration of desorbed solute only reaches about half the value in the preceding model. In any case, the idea of 9592

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Langmuir desorption fraction α and the degree of dissociation ξ, is given by the following expression q = e ΓSAMξ(1 − α)

mean that it is unnecessary to solve the full governing equations as in the previous two models.

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DISCUSSION We have devised some models that incorporate the effect of osmotic flow caused by concentration gradients on a stable thin film, showing a dimple forming much faster than the subsequent drainage, as observed in experiments. This suggests that the basic mechanism is reasonable. The first model assumed an instantaneous and uniform thiol desorption across the interface and we found that this results in a short-lived dimple that forms and subsequently drains in approximately a minute. This is due to the increased thickness of the channel from the movement of the barrier rim. A refinement to the first model then tries to deal with this shortcoming by incorporating a gradual increase in interfacial tension and desorption fraction with hopes that this will result in a fixed barrier rim. This, however, is in vain as the effects of increasing the interfacial tension and the desorption fraction are not simultaneous and the barrier rim still moves a considerable distance away. The third model asks whether the change in surface charge density due to desorption of charged thiol molecules could significantly reduce the repulsive disjoining pressure within the film, which we have answered in the negative. This model, as well, could not produce the appropriate initial film thickness and flattened region widening behavior as recorded in the experiments.37 Figure 1a indicates that not only does the barrier rim stay at its initial separation from the mica surface, but the width of the flattened region of the drop actually increases somewhat as the dimple is formed. The similarities of the characteristics seen in the osmotic dimple phenomenon to those of the hydrodynamic dimple that forms when a normal push motion is applied to the mercury drop23,27,32 motivate us to implement a push-like action in the model assuming an instantaneous and uniform desorption.Thus, we consider implementing the simplest model with an added term in the boundary condition at r = rmax, that is, eq 53 to see whether a global push on the drop can keep the barrier rim fixed in the presence of osmotic flows producing a dimple. We use a small velocity V = 0.1 μm/s to mimic a possible slow drift of the piezo stage, which can happen due to thermal effects. We compared the simulation results for a mercury drop with an instantaneous and uniform desorption with near-critical desorption fraction α = 0.001 and a mercury drop that is pushed toward the mica surface with velocity V (0.1 μm/s) for the first 15 s while simultaneously desorbing instantaneously and uniformly across the mercury/water interface with desorption fraction α = 0.001. The initial film thickness and width of the flattened region for both cases remain unchanged from before. While the film profile evolutions corresponding to the maximum film force for the two cases appear to be identical, the intermediate traces in the case with an applied push velocity indicate that the drop experiences a higher degree of deformation than its counterpart. In addition, the barrier rims of the profiles in the latter recede much further away than in the profiles computed using simplifying assumptions. The drop with an added push, however, does settle at its initial position with a slightly wider flattened region. It is apparent that the combined effects of pushing the drop while desorbing do not produce a long-lived dimple we had hoped for, at least for a slow push chosen to mimic inadvertent drift of the piezo stage. We have not explored higher push

(54)

where e is the fundamental charge and ΓSAM the surface packing density of a close-packed monolayer. To express the disjoining pressure in terms of the surface charge density, we use the relation between the surface charge density, scaled by εε0 kTκ/ e, and the surface potential for an isolated flat surface ⎛ eΨ ⎞ s = q/(εε0kTκ /e) = 2sinh⎜ d ⎟ ⎝ 2kT ⎠ Ψd 4kT

( ) in eq 1 can be written in terms of

Then the factor tanh surface charge as

⎛ Ψ ⎞ s tanh⎜ d ⎟ = ⎝ 4kT ⎠ 2 + 4 + s2

showing again the saturation with either large potential or large surface charge. Assuming a value for ξ, say ξ = 1 for fully dissociated thiol acids, we then have the disjoining pressure as a function of the desorption fraction α. The governing equations used in this model are the same as the ones used in the first model. The only difference in the third model from the first is the use of the scaled surface charge s in the computation of disjoining pressure. Experimenting with possible dependence of disjoining pressure on the desorption fraction revealed, however, that the full dynamical model was not worth pursuing. We were originally content to use a film thickness of 51 nm in our first model but the measured film thickness, after the electrolyte solution is injected into the chamber, is 63 nm. This discrepancy in the initial film thickness could be the reason why the amount of spread of the flattened region observed in the experiments is not observed in any of the simulations. Various reasons for the observed film thickness were suggested, largely without success.39 Even at zero desorption (α = 0) and assuming full dissociation, the amount of repulsion on the mercury drop generated by the presence of the negatively charged thiol molecules is insufficient to result in the observed wetting film thickness of 63 nm. This case produces a surface potential on the drop of −312 mV, corresponding to scaled surface charge density s = −434, and results in a film thickness of approximately 52 nm, which is not significantly different from the film thickness in the first model. At any value of α more than the densely packed surface case, the repulsion between the mica surface and the thiol molecules will be less and cannot produce the film thickness observed.39 Since the rate of dimple drainage is sensitive to the film thickness at the barrier rim, we cannot expect to obtain quantitative agreement without the correct film thickness. The very large value of s shows that the disjoining pressure is very insensitive to surface charge in this regime. A fall in s by 10% (corresponding to α = 0.1, larger than any value used in our first model), produces just a 0.05% change in disjoining s pressure through the factor . We doubt that such a 2 2+ 4+s

small change in disjoining pressure could have any material effect. This, and the inability to produce an initial profile with a film thickness of 63 nm and thus, a wider flattened region, 9593

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radially. Once the zone of high pressure has spread to the edge of the film, the hydrodynamic dimple begins to drain. Due to the outward movement of water into the region beyond the channel, the hydrodynamic pressure profiles in Figure 9 start to flatten until it ultimately reaches zero. By then, the hydrodynamic dimple drainage has ended. Note that the dynamic pressure is never high beneath the barrier rim. The hydrodynamic pressure profiles corresponding to the osmotic dimple have been analyzed above. The first few hydrodynamic pressure profiles in Figure 9 have maxima at different location, explaining the fact that the barrier rim of the osmotic dimple does not stay fixed. The maxima in dynamic pressure lift up the barrier rim and force it away from the surface. In addition, the development of negative pressure zones in the osmotic dimple causes suction of water into the thin film. This analysis allows us to understand a barrier rim that recedes away from the mica surface in the osmotic dimple but not in the hydrodynamic dimple. The osmotic flow in the osmotic dimple is caused by the difference in solute concentration between the thin film region and the area beyond the thin film. The high solute concentration gradients drive water into the thin film region, producing an increase in hydrodynamic pressure that pushes the drop away from the mica. The flow involved in the formation of the osmotic dimple is thus inward, evidenced by the inward gradients of the pressure profiles in Figure 9a. Something similar is seen in the modeling of the so-called “wimple”.35 In that case, a drop with a stable film is pushed again toward the surface. The outer part of the drop approaching the surface sends water flowing inward toward the thin film region, which is, as here, thrown away from the surface to form the rippled film profile characteristic of the wimple. On the other hand, the formation of the hydrodynamic dimple produces motion of water being pushed out of the thin film region into the chamber, a much larger region. The slopes of the pressure profiles corresponding to the hydrodynamic dimple are outward, as seen in Figure 9c. Since the outward flow of water is not restricted by the width of the channel, as in the osmotic dimple, there is no increase in pressure at the barrier rim and so no reason for the hydrodynamic dimple to move away from the wall. The differences in the hydrodynamic pressure profiles provide evidence that the osmotic dimple is not the time-reverse equivalent of the hydrodynamic dimple, but rather a new phenomenon. The reason that the barrier rim recedes from the surface is that water is being drawn into a confined region and so a high pressure zone under the barrier rim forms, and pushes the barrier rim away. This led us to consider the possibility of an osmotic flow being confined more to the center of the film, drawing water initially from the outer part of the film, not directly from the region outside the film. For this reason, we briefly explored the situation where a higher density of adsorbed thiol molecules desorb at the center of the mercury/water film, for some unknown reason. The initial solute concentration profile is modeled with a Gaussian curve, centered at r = 0. For simplicity, we set the width of the Gaussian curve to 100 μm, comparable to the radial extent of the film, that is, there is little concentration gradient initially at the film entrance. The magnitude of the solute concentration at t = 0 corresponding to a desorption fraction α = 0.001 is approximately 1.6 × 10−4 M. Although the barrier rim did stay at its initial location for a short time, eventually water flows into

speeds, which may well be able to keep the barrier rim in place, simply because we have no rationale for such a push in the experimental protocol. It was suggested by one of us37 that the processes involved in the formation of the osmotic dimple are the time-reverse equivalent of the hydrodynamic dimple described in many studies.23,27,32 In the case of draining of the hydrodynamic dimple, the flow is outward since the water is being pushed out of the thin film region, meaning that dynamic pressure effects dominate those of the disjoining pressure, resulting in the barrier rim being secured at its initial position. We expect the reverse to be true in the osmotic dimple case, with the flow now inward due to movement of water into the flattened region. The question is why should the barrier rim stay fixed near the surface in one case, but recede from the surface in the osmotic case? To answer this question, we need to understand the role of hydrodynamic pressure in the osmotic dimple and hydrodynamic dimple so we compare the hydrodynamic pressure profiles for these two phenomena in Figure 9. The hydro-

Figure 9. (a) Hydrodynamic pressure when film force G is increasing for critical desorption (α = 0.002) for osmotic dimple with t = 0 (A), 0.02 (B), 0.13 (C), 0.57 (D), 2.73 (E), 12.8 (F) s. The colored characters correspond to their respective traces. (b) Hydrodynamic pressure when film force G is decreasing for critical desorption (α = 0.002) for osmotic dimple with t = 14.08 (G), 35 (H), 63.51 (I), 114.36 (J), 211.35 (K), 500 (L) s. (c), (d) As for (a), (b) but for a hydrodynamic dimple and at different times.

dynamic pressure profiles corresponding to the osmotic dimple, assuming a uniform thiol desorption across the surface and instantaneous entry into the thin film region, are given in Figure 9a and b. As before, we have separated the profiles to the regimes where the film force is increasing and where the film force is decreasing. Figure 9c and d present the hydrodynamic pressure profiles for the hydrodynamic dimple. We note that in Figure 9c, applying a push to the drop at the beginning of the simulation causes the hydrodynamic pressure to increase immediately at r = 0 to approximately the Laplace pressure (scaled pressure =2). During the time interval over which the film force G increases, the hydrodynamic pressure profile for the hydrodynamic dimple does not undergo much change, indicating that the pressure at r = 0 is still quite high. The zone of high pressure does, however, move outward 9594

DOI: 10.1021/acs.langmuir.5b02220 Langmuir 2015, 31, 9582−9596

Article

Langmuir the central region from outside and the barrier rim still moves away. So nonuniform desorption also does not improve quantitative agreement with experiment, at least within the model used here. The present model does not take account of interfacial tension gradients that could accompany nonuniform desorption and drive Marangoni flow. However, in the absence of any information on nonuniform desorption (experimentally, the potential jump occurs uniformly throughout the mercury drop, so uniform desorption is expected), further embellishment of the model to include interfacial tension gradients has not been attempted.

thiol monolayer adsorbed on mercury still remaining to be explained. The modeling shows that osmotic effects can significantly deform stable thin films in a transient way, joining mechanical and electrical perturbations as ways to influence thin film behavior.35 These results have implications for colloidal interactions in a broader range of systems where solute concentration may not be homogeneous, for example in solute transfer processes. We have shown that nonequilibrium osmotic pressure can be comparable to the strength of other colloidal forces such as double-layer forces.

CONCLUSIONS Dynamic colloidal interactions could include not only hydrodynamic forces, but also effects arising from osmotic stresses associated with nonuniform solute concentrations. We have conducted experimental and theoretical investigations of one example of this class of dynamic colloidal interaction. We have reported observations of novel long-lived deformations of a stable water film between a mercury drop coated with a self-assembled monolayer of charged thiol molecules and a mica surface, produced when the potential on the mercury drop is reduced (the underlying mercury potential made less positive). Previous experiments39 showed that the thiol molecules are likely desorbing in this circumstance, suggesting that the dimple is formed by osmotic effects. In many ways, the formation of the dimple appears like the timereverse of the drainage of the well-known hydrodynamic dimple formed when a drop is pushed toward a surface. Detailed modeling including osmotic flow caused by concentration gradients arising from thiol desorption reveals several surprising features. The simplest model, which treats the thiol as an uncharged solute desorbing uniformly and instantaneously, can produce dimples of the magnitude observed but with a different time-dependence. The experimental dimples are formed over a period from 20 s to 5 min, depending on the applied potential and potential step on the mercury. Having formed they drain slowly, over from 5 to 30 min, much like the hydrodynamic dimple formed when a drop is pushed toward a solid surface. The model shows the same fast dimple formation and a slow drainage, but with each process faster than experimental observations. The formation of a dimple occurs over a time frame of a few seconds with the simultaneous movement of the barrier rim away from the surface, leading to the drainage of the dimple, slower than its formation but again faster than observed. Examination of hydrodynamic pressure profiles in the two cases show that the osmotic dimple formation is not the time-reverse of the hydrodynamic dimple but a new phenomenon, with some similarities to the formation of the “wimple” in the same system. More refined models incorporating effects such as a gradual desorption with concomitant increase in interfacial tension of the mercury, the effect of desorbing charged thiols on the repulsive disjoining pressure or an extra push mimicking a slow steady drift of the piezo stage of the SFA were unable to improve the quantitative agreement with observations. Only a putative nonuniform desorption process was able to capture some of the missing features. The model predictions are physically sensible. The lack of qualitative agreement with experiments, together with previous failure to quantitatively explain the observed film thickness at small applied potentials, suggest that there are features of the

* Supporting Information

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ASSOCIATED CONTENT

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02220. A fuller description of the first model and the computational methods used (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.A.C. acknowledges the University of Melbourne for support through a Melbourne International Fee Remission Scholarship and Melbourne International Research Scholarship. S.L.C. is a member of the Particulate Fluids Processing Center, a Special Research Centre funded by the Australian Research Council.

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REFERENCES

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DOI: 10.1021/acs.langmuir.5b02220 Langmuir 2015, 31, 9582−9596