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Langmuir 2006, 22, 2610-2619
Thin Film Drainage: Hydrodynamic and Disjoining Pressures Determined from Experimental Measurements of the Shape of a Fluid Drop Approaching a Solid Wall Roger G. Horn,* Mohammed Asadullah,† and Jason N. Connor‡ Ian Wark Research Institute, UniVersity of South Australia, Mawson Lakes, Adelaide, South Australia 5095, Australia ReceiVed August 24, 2005. In Final Form: January 16, 2006 Accurate measurements of the shape of a mercury drop separated from a smooth flat solid surface by a thin aqueous film reported recently by Connor and Horn (Faraday Discuss. 2003, 123, 193-206) have been analyzed to calculate the excess pressure in the film. The analysis is based on calculating the local curvature of the mercury/aqueous interface, and relating it via the Young-Laplace equation to the pressure drop across the interface, which is the difference between the aqueous film pressure and the known internal pressure of the mercury drop. For drop shapes measured under quiescent conditions, the only contribution to film pressure is the disjoining pressure arising from double-layer forces acting between the mercury and mica surfaces. Under dynamic conditions, hydrodynamic pressure is also present, and this is calculated by subtracting the disjoining pressure from the total film pressure. The data, which were measured to investigate the thin film drainage during approach of a fluid drop to a solid wall, show a classical dimpling of the mercury drop when it approaches the mica surface. Four data sets are available, corresponding to different magnitudes and signs of disjoining pressure, obtained by controlling the surface potential of the mercury. The analysis shows that total film pressure does not vary greatly during the evolution of the dimple formed during the thin film drainage process, nor between the different data sets. The hydrodynamic pressure appears to adjust to the different disjoining pressures in such a way that the total film pressure is maintained approximately constant within the dimpled region.
1. Introduction Colloidal interactions involving fluid objects (drops and bubbles) are more difficult to quantify than those involving only rigid solids. The reason is that fluid objects easily deform under the influence of surface and/or hydrodynamic forces, and deformation changes the drop geometry in a way that tends to amplify the interactions, whether they are attractive or repulsive. However, an understanding of fluid drop and/or bubble interactions is required if we are to develop a thorough knowledge of phenomena such as droplet coalescence affecting emulsion stability, droplet or bubble attachment and detachment from surfaces, the stability of froths and foams, bubble-particle interactions in flotation processes, biological interactions involving cells, drop impingement, dynamic wetting, and multiphase flow behavior in confined geometries such as porous media or microfluidic devices. From investigations of drop coalescence, foam stability, and drop or bubble attachment to surfaces, it is known that in many situations the rate-limiting step is thin film drainage, i.e., the time taken for the background fluid to be removed from the gap bounded by the surfaces of the drop (bubble) and the other material it is approaching. The viscosity of the background fluid means that energy is dissipated when it flows from the gap. Hydrodynamic pressure builds up in the gap, and this pressure can deform the shape of the drop. Calculating the pressure and drop shape is a complicated problem to unravel and solve in detail, since the hydrodynamic pressure itself depends on the changing drop shape, approach speed, and dimensions of the gap. * To whom correspondence should be addressed. E-mail: roger.horn@ unisa.edu.au. † Present address: Schlumberger Oilfield Australia P/L, Santos Petroleum Engineering Building, Frome Road, Adelaide, SA 5000, Australia. ‡ Present address: PELM Centre, Central Queensland University, Gladstone, Queensland 4680, Australia.
It has long been known that under some circumstances a “dimple” can form in the surface of a drop or bubble as it approaches another, or approaches a solid surface of low curvature, in a direction normal to the surfaces at their closest point.1,2 A dimple occurs when hydrodynamic pressure in the gap builds up sufficiently to invert the curvature of the drop (or bubble) surface and make it locally concave. The situation is illustrated in Figure 1, which also serves to define the coordinate system to be utilized in what follows. The system has cylindrical symmetry about the z axis. Once a dimple has formed, there is a circle (called the “barrier rim”) where the gap between the surfaces is narrowest, and this barrier inhibits the drainage of fluid from the dimple to the outer region. Rather than “gap”, we will henceforth refer to the film thickness of the intervening fluid and to the shape of the gap defined by the bounding surfaces as the film profile. In general, we will speak of a drop approaching a solid wall, but most of what we say would apply equally well to a bubble and to the mutual approach of two drops, two bubbles, or a drop and a bubble. Thin film drainage, hydrodynamic pressures, and the dimpling phenomenon have been investigated by many groups, both experimentally3-8 and theoretically,9-16 since the pioneering (1) Derjaguin, B. V.; Kussakov, M. M. Acta Physicochim. URSS 1939, 10, 25-30. (2) Platikanov, D. J. Phys. Chem. 1964, 68, 3619-3624. (3) Hartland, S. Chem. Eng. Sci. 1969, 65, 82-89. (4) Burrill, K. A.; Woods, D. R. J. Colloid Interface Sci. 1973, 42, 15-34. (5) Joye, J.-L.; Miller, C. A.; Hirasaki, G. J. Langmuir 1992, 8, 3083-3092. (6) Hewitt, D.; Fornasiero, D.; Ralston, J.; Fisher, L. R. J. Chem. Soc., Faraday Trans. 1993, 89, 817-822. (7) Goodall, D. G.; Stevens, G. W.; Beaglehole, D.; Gee, M. L. Langmuir 1999, 15, 4579-4583. (8) Hartland, S. J. Colloid Interface Sci. 1971, 35, 227-237. (9) Hartland, S.; Robinson, J. D. J. Colloid Interface Sci. 1977, 60, 72-81. (10) Dimitrov, D. S.; Ivanov, I. B. J. Colloid Interface Sci. 1978, 64, 97-106. (11) Lin, C.-Y.; Slattery, J. C. AIChE J. 1982, 28, 147-156. (12) Chen, J.-D. J. Colloid Interface Sci. 1984, 98, 329-341.
10.1021/la052314b CCC: $33.50 © 2006 American Chemical Society Published on Web 02/17/2006
Thin Film Drainage
Figure 1. Schematic illustration of the formation of a “dimple” in a fluid drop (bottom) as a flat solid surface presses down on it from above in the presence of a background fluid (patterned). The gray arrow represents the relative movement of the solid and the drop toward each other, and the white arrows show drainage of the background fluid from the film region under the dimple, which is bounded by a “barrier rim”, to the outer regions. Note that, in this and some of the following figures, the vertical scale is greatly exaggerated compared to the horizontal scale.
works of Derjaguin,1 Platikanov,2 and others.17 The equations describing the phenomenon are not difficult to write down once certain simplifying assumptions are made. They are based on simple hydrodynamics, with the Navier-Stokes equations for Newtonian liquids written in the lubrication approximation, neglecting inertial terms, and usually assuming either pure slip or no-slip boundary conditions at the bounding surfaces. This allows hydrodynamic pressure in the film to be related to the film dimensions and local rate of change in film thickness. At the same time, hydrodynamic pressure is coupled through the Young-Laplace equation to local curvature and interfacial tension of the fluid surface or surfaces that define the film thickness. This results in a nonlinear fourth-order partial differential equation describing the evolution of film thickness, h(r,t). Some authors13-15 have considered the situation when the fluids within the drop and the film have comparable viscosities, in which case the boundary condition is one of partial slip, and fluid flow within the drop cannot be neglected because it is coupled to flow within the film. Solving this more complicated case requires boundary integral methods. In general, it is a challenging numerical problem to solve the equations describing hydrodynamic dimple formation and evolution. The aim of the theoretical work is usually to find the evolution of the film shape, h(r,t), and to compare the model predictions with results of experiments that have also measured h(r,t) by various means, usually involving optical interferometry,2,4-6,14 reflectometry,4 or ellipsometry.7 Most previous numerical solutions are dependent on the initial conditions of drop shape and approach conditions (e.g., constant speed or constant total force constraints). The recent modeling work of Carnie et al.,16 however, is able to calculate h(r,t) for a prescribed approach velocity, and this could be adapted to more general cases. In this paper, we propose an alternative approach. The central idea is to calculate local interfacial curvature from experimental data and then use the curvature to calculate pressure in the film as a function of radial position and time. This approach was taken in an early paper by Hartland,8 and Lowry18 has used a similar method to calculate surface stresses due to axial flow past an axisymmetric liquid bridge with high accuracy. We are not aware of any other attempts to determine thin film pressures (13) Yiantsios, S. G.; Davis, R. H. J. Fluid Mech. 1990, 217, 547-573. (14) Abid, S.; Chesters, A. K. Int. J. Multiphase Flow 1994, 20, 613-629. (15) Klaseboer, E.; Chevaillier, J. P.; Gourdon, C.; Masbernat, O. J. Colloid Interface Sci. 2000, 229, 274-285. (16) Carnie, S. L.; Chan, D. Y. C.; Dagastine, R. R.; Lewis, C.; Manica, R. Langmuir 2005, 21, 2912-2922. (17) Frankel, S. P.; Mysels, K. J. J. Phys. Chem. 1962, 66, 190-191. (18) Lowry, B. J. J. Colloid Interface Sci. 1996, 176, 284-297.
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by this method since Hartland’s 1971 paper. Looking at film pressures rather than dimensions of draining films provides a different point of comparison between experimental and theoretical modeling.8,9 More importantly, it aids understanding of the processes occurring, since it is pressure that drives flow in the film. As we will see later, this approach gives new insight into what governs the behavior of a fluid drop approaching another surface. Deriving curvature from experimental data for h(r,t) is a risky business, since it requires first and second derivatives ∂h/∂r and ∂2h/∂r2 to be calculated. Even a single differentiation of data tends to amplify errors, so the procedure can only be contemplated if shape data are comprehensive and accurate. A suitable data set can be found in a recent paper by two of us,19 henceforth referred to as Paper I, in which measurements of the thickness of an aqueous electrolyte film between a flat mica surface and a millimeter-scale mercury drop are reported. The mica was driven toward the drop while the latter was held below it at the end of a fixed capillary. Aqueous film thickness measurements were made using optical interferometry recorded by a video camera in a variant of the FECO (fringes of equal chromatic order) method that has been used extensively in surface force apparatus measurements.20 This method provides a comprehensive set of measurements of h(r,t) (the notation D(r,t) was used in Paper I) with a resolution of 0.7 nm in h, 1 µm in r, and 0.02 s in t. (When the film is stationary, frame averaging allows an improved resolution of 0.3 nm in h). Typically ∼300 data points in an h(r) profile are obtained over a range of ∼250 µm in r at each time step, and the time steps of 0.02 s are set by the frame rate of a PAL video recording of the interference fringes. A particular feature of the mercury/aqueous/mica system is that the equilibrium surface force acting between the two surfaces bounding the film, i.e., the mercury and mica, can be controlled. At the film thicknesses investigated (tens to hundreds of nanometers), the surface force is dominated by electrical doublelayer repulsion.21 van der Waals forces are much smaller, never exceeding 2% of the total disjoining pressure, and no other contributions to the force are significant in this thickness range. By applying an electrical potential between the mercury phase and a reference electrode in the aqueous phase, the surface charge of the mercury is controlled, which in turn controls the doublelayer force acting between the mercury and mica surfaces. The force can be understood from the Poisson-Boltzmann equation describing the structure of the electrical double layers at the two surfaces.21 Static force measurements in the mercury/aqueous/ mica system reported in ref 21 were extended to dynamic ones to investigate how the drainage of the aqueous film was affected by different signs and magnitudes of double-layer force acting between the approaching surfaces (Paper I).19 Details of the dimple formation and evolution were reported for different values of mercury potential. Double-layer forces are determined by the distribution of ions in the aqueous film. With the assumption that the ion distribution can re-equilibrate on a time scale that is short compared to the time taken for the film dimensions to change (∼0.1 s), the surface force can be assumed to have its equilibrium value throughout the drainage process. A convenient way to describe the effect of surface forces in the present analysis is to consider the surface force per unit area of surface as an additional local pressure in the film, called disjoining pressure,22 that acts normal to the bounding surfaces. For gently curved surfaces (such as the flat (19) Connor, J. N.; Horn, R. G. Faraday Discussions 2003, 123, 193-206. (20) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975-1001. (21) Connor, J. N.; Horn, R. G. Langmuir 2001, 17, 7194-7197.
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mica and millimeter-scale mercury drop) and with the rapid equilibration mentioned above, disjoining pressure is a function only of film thickness. This allows a further step to be taken in the analysis that we propose. Disjoining pressure and hydrodynamic pressure both contribute to the film pressure. The film pressure is determined by finding the curvature of the fluid/fluid interface. Since the film thickness is also known, disjoining pressure can be calculated and subtracted from the total to find the hydrodynamic contribution to pressure. Thus, the method provides not only total pressure in the film but also the separate disjoining and hydrodynamic components for all radial positions, r, and times, t, throughout the drainage process. This is done for a variety of different double-layer forces ranging from strong repulsion to strong attraction between the drop and the solid surface. The results of this analysis, to be presented below, reveal a simple picture of how thin-film pressures develop and evolve during the approach of a fluid drop to another surface. For the entire time that a dimple is present, excess pressure in the film remains approximately constant throughout the film within the dimple region. The film pressure is about equal to the internal (Laplace) pressure in the drop. Furthermore, this result is not very sensitive to disjoining pressure. The total film pressure is essentially the same even when the disjoining pressure is varied between attraction and repulsion; the hydrodynamic pressure varies in such a way that it compensates for different disjoining pressures, keeping the total pressure quasi-constant.
2. Theoretical Background The basis of this analysis is simple and well known: the Young-Laplace equation which relates the pressure drop across a fluid/fluid interface to the interfacial tension and local curvature. Since the problem we are concerned with has axial symmetry about a vertical line passing through the top of the mercury drop and normal to the horizontal mica surface, it is appropriate to express this equation in cylindrical coordinates:
∆P ) γ
{
∂2z/∂r2 ∂z/∂r + 2 3/2 [1 + (∂z/∂r) ] r[1 + (∂z/∂r)2]1/2
}
(1)
Here z(r) is the location of an interface between two fluids, γ is the interfacial energy, and ∆P(r) is the pressure difference between the fluids. The terms in braces are the two principal components of local curvature of the interface. The YoungLaplace equation is valid in the absence of tangential stress differences across the interface, which is legitimate in the absence of fluid elasticity, and in situations where interfacial energy is not dependent on position (as it could be in the presence of temperature or surfactant concentration gradients, for example). For the present purposes, we use gauge pressures, that is, the excess pressures above the ambient, which is the hydrostatic pressure in the background fluid at the same height as, but remote from, the film. The pressure difference across the interface is
∆P ) Pint - Pfilm
(2)
where Pint is the pressure inside the drop and Pfilm is the pressure in the film. When the solid surface is far away from the drop, Pfilm ) 0, and the internal drop pressure is equal to the Laplace pressure given by
Pint ) 2γ/R0
(3)
where R0 is the radius of curvature of the drop when it is (22) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Consultants Bureau: New York, 1987.
undistorted by any interaction with the other surface. This radius is readily measured experimentally using the FECO method. In eqs 2 and 3, we have ignored the effects of gravity because the total variations in height for the data discussed below are in the range of a few hundred nanometers. Even with the high density of mercury, the maximum variation in hydrostatic pressure would be ∼0.1 Pa, compared to internal and film pressures which are ∼102 Pa. We assume in what follows that Pint is constant, given by eq 3, throughout the drainage experiments. There are two reasons why this may not be strictly true. First, the experiments were conducted with a constant Volume of mercury protruding from a capillary. However, the drop compression is only ∼10 µm compared with the drop dimensions of ∼1 mm, and numerical calculations23 show that under our experimental conditions the pressure of the drop remains constant to within 1%. The second reason is that hydrodynamic pressure within the drop is nonzero if internal fluid circulation occurs, which is expected to be the case when viscosities in the drop and the film are comparable. The present data is for mercury (whose viscosity at 20 °C is 1.55 mPa‚s) and water (1.00 mPa‚s), so fluid circulation within the mercury should not be ignored. However, according to estimates given by Yiantsios and Davis,13 the ratio of hydrodynamic pressures inside the drop and the film is of order xh0/R0, where h0 is the film thickness at which dimpling commences. In our experiments, h0 ≈ 270 nm and R0 ≈ 2 mm, so hydrodynamic pressure within the drop is only around 1% of the hydrodynamic pressure in the film. For the remainder of this paper, we will neglect the hydrodynamic pressure within the drop, even though we are conscious that flow within the mercury is coupled to flow in the aqueous film. For brevity, we will speak of hydrodynamic pressure in the film, whereas it would be more correct to speak of the difference in hydrodynamic pressures between the film and drop at a particular value of r. However, the above estimate justifies this economy of language in our discussion. From the coordinate system defined in Figure 1, h ) z - zs where the solid surface is located at zs ) constant, so the first and second derivatives of z with respect to r in eq 1 can be replaced by the respective derivatives ∂h/∂r and ∂2h/∂r2, which we denote by h′ and h′′. Including the radial dependence explicitly, eq 1 becomes
Pfilm(r) ) Pint - γ
{
h′′(r)
[1 + h′(r) ]
2 3/2
+
h′(r) r[1 + h′(r)2]1/2
}
(4)
This is the expression that we will use below to calculate pressure in the aqueous film from the first and second derivatives of the measured film profiles h(r) at each time step, t. Consistent with the lubrication approximation used for analyses of thin film hydrodynamics, it is assumed that pressure is uniform through the film thickness, i.e., Pfilm is a function of r and not of z. Finding derivatives by taking differences between successive data points results in large scatter, so the approach we take is to fit a smooth curve through the data and then differentiate it analytically. The high density of data in the experimental h(r) profiles allows this to be done reliably. A fifth-order smoothed spline function is adopted, using an algorithm due to Dierckx.24 Our fitting routine was based on Dierckx’s FITPACK software, downloaded from http://www.netlib.org/dierckx/. This routine allows the user to specify a smoothing factor, S, which balances the competing requirements of accuracy and smoothness of the (23) Connor, J. N.; Horn, R. G.; Zhang, Y. manuscript in preparation. (24) Dierckx, P. CurVe and surface fitting with splines; Oxford University Press: Oxford, 1993.
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fitted curve. The latter is specified by minimizing discontinuities in the kth derivative of the spline fit (k being the order of the spline), while the former is defined by the weighted sum of squares of differences between data and the fit at each data point. If there are N equally weighted points, the RMS error is xS/N. There are two contributions to the film pressure introduced in eq 2. One is the disjoining pressure, denoted Π, which arises from surface forces acting between the two bounding surfaces (mica and mercury in this case). As discussed in the previous section, this is a function only of film thickness. The other contribution is hydrodynamic pressure difference Phyd which is nonzero when there is flow of fluid in the film and/or the drop. Thus at any time, t
Pfilm(h′,h′′,r) ) Phyd(r) + Π(h)
(5)
in which we have included the explicit functional dependences of the various pressure components. After Pfilm is calculated from the local curvature of the drop, if Π(h) is known, then eq 5 can be used to find the hydrodynamic pressure. As discussed in the Introduction, disjoining pressure is dominated by double-layer interactions in the experiment under consideration. There are two possible approaches to finding Π(h). One is to use our theoretical knowledge of double-layer interactions between surfaces of known potential, together with the experimentally determined potentials of the mica and mercury surfaces. An alternative is to use the curvature analysis to establish the total film pressure in a situation where Phyd is negligible or zero, and from eq 5, set Π ) Pfilm. In practice, it is found to be possible to use the latter approach when the mercury/mica doublelayer force is repulsive. Under these conditions, the drop approaches a stable (flattened) equilibrium configuration at a finite distance from the solid and Phyd is negligible in the longtime data. However, in the presence of attractive forces, the film profile is never static, and in that case, we need to resort to a theoretical model for Π(h).
3. Results Paper I presents data showing the h(r,t) profiles at certain time steps during drainage measurements made under four different disjoining pressure conditions. The four conditions are labeled “strong repulsion”, “moderate repulsion”, “weak force”, and “strong attraction”. These terms describe qualitatively the doublelayer force acting between the mica (which has a negative charge when immersed in aqueous electrolyte solution) and mercury surfaces when the mercury potential is fixed at certain values ranging through large negative (-492 mV), moderate negative (-52 mV), small negative (-12 mV), and large positive (+408 mV) values compared to the potential at which the mercury surface is uncharged (the point of zero charge, PZC). As demonstrated previously,21 these values can be taken as the double-layer interaction potentials of the mercury. The measurements were made in KCl solution at a nominal concentration of 0.1 mmol/L; in this electrolyte, mica has a potential of -100 mV25 and the expected Debye length is 30 nm. Note that the term “weak force” refers to a slight double-layer repulsion when the surfaces are far apart, but this changes to an attractive force for aqueous film thicknesses less than about 57 nm. Calculations of corresponding disjoining pressure isotherms Π(h) from nonlinear Poisson-Boltzmann theory were presented in Paper I. Drainage measurements were made as the mica was driven vertically down toward the mercury drop at a nominal speed of 23 ( 1 µm/s, starting from an initial separation of 10.0 ( 0.2 µm and stopping (25) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531-546.
Figure 2. A typical fifth-order smoothed spline fit to experimental data19 for t ) 1.22 s in the strong repulsion case. The upper part of the figure shows the film profile h(r). Note the large difference between vertical and horizontal scales. The high density of data points and the small differences between data (diamonds) and the spline fit (solid line) make them virtually indistinguishable on this scale. The quality of fit is easier to discern from the lower part of the figure, which shows the difference, ∆h, between each experimental data point and the spline fit.
the drive after a travel of 30.0 µm. Time t is set to zero when the drop surface is visibly flattened just before the dimple forms, which occurs at a film thickness of about 270 nm. The mica drive stops at t ≈ 0.8 s. Each of the 76 draining film h(r,t) profiles published in Paper I has been fitted by a smoothed fifth-order spline. A typical result is shown in Figure 2. In this case, there are 323 data points, the smoothing factor is 30, the maximum error (the difference between the spline fit and each data point) is 1.2 nm, and the RMS error is 0.31 nm. Most of the fits are of comparable quality, although the smoothing factors had to be optimized for each profile. A compromise was required between the goodness of fit indicated by the RMS error, and the smoothness of the spline function indicated by lack of oscillation in the derivatives h′(r) and h′′(r). In practice, good results were obtained with smoothing factors between 20 and 80. The worst fit had an RMS error of 0.90 nm and a maximum error of 4.2 nm. As is usual with curve fitting routines, the fits are least reliable near the extremities (large |r|). This is manifest particularly in the derivatives h′(r) and h′′(r) derived from the spline fits, so pressures calculated at large |r| were unreliable and therefore discarded. From the measured curvature at the top of the undistorted drop and using the mercury/electrolyte interfacial tension of 426 mN/m,26 the Laplace pressure is calculated as Pint ) 443 Pa. Putting this value into eq 4, with h′(r) and h′′(r) calculated from the spline fits, enables the film pressures to be calculated for all experimental ranges of r and t. Figure 3a shows a typical calculation of film pressure at t ) 1.22 s in the strong repulsion case. Points near r ) 0 have been excised from the plot because the presence of r in the denominator of the last term in eq 4 creates a numerical instability in the calculation. Axial symmetry demands that the film pressure must have zero gradient at r ) 0, and it is fair to assume that the missing part of the pressure profile is flat and featureless in this region. It is not straightforward to quantify the uncertainty in pressure determined by this method. One estimate is obtained by comparing the left- and right-hand sides of the plots in Figure 3a. In principle, axial symmetry means that data on one side (either r > 0 or r < 0) is redundant. However, the full experimental profile covering both sides was used to produce the curve in Figure 3a. This means that pressures calculated for r < 0 and r > 0 generally (26) Vos, H.; Los, J. M. J. Colloid Interface Sci. 1980, 74, 360-369.
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Figure 3. (a) Film pressure at t ) 1.22s in the strong repulsion case calculated from interface curvature. The dashed line shows the internal pressure of the drop. (b) An expanded scale showing pressure calculated from four variations of the curve fitting procedure, as described in the text.
differ from each other, and these differences serve as an indicator of the typical uncertainty in the values obtained. Conversely, the similarity between the pressures calculated for positive and negative r can be taken as an indicator of the reliability of the calculations. The two sides are compared more directly by reflecting the r < 0 fit around r ) 0 and replotting it (blue curve) for comparison against the r > 0 half of the spline fit (black curve) in Figure 3b. This figure also includes results of calculating pressure from fits obtained by two other methods. In one case, the h(r) data are symmetrized by reflecting all data about r ) 0, i.e., setting h(r) ) h(-r), and combining this data set with the original data. Fitting a spline to the combined set results in a symmetrical curve (orange). The final curve (red) is the result of fitting a 10th-order polynomial function to the data. Only even orders are included in the polynomial, so this fit is also guaranteed to be symmetrical about r ) 0. We propose that the differences between these four variations give a reasonable estimate of the uncertainty in pressures using our method. The differences are generally less than 5 Pa, which is 1% of the internal (and typical film) pressure. Errors can be greater at large |r| where curvatures determined from the spline and polynomial fits are less reliable. The results presented in the remainder of this paper were all obtained by fitting a spline to the full data set (r < 0 and r > 0) without forcing the spline fit to be symmetric. This produces redundant curves insofar as the left half should be equivalent to the right half, but we will exploit this redundancy in presenting data in some of the later figures (7-9) below. Total film pressures calculated in this way are shown in Figure 4a-d for the strong repulsion, moderate repulsion, weak force, and strong attraction cases. Apart from the surface potentials on the mercury drop, all other details of the experiment, and in particular the drive function, are maintained the same between
Figure 4. Film pressures in pascals (upper curves in each plot) and film thicknesses in nanometers plotted on the same numerical scale (lower curves, with symbols giving the appearance of thicker lines) for (a) strong repulsion, (b) moderate repulsion, (c) weak force, and (d) strong attraction for various time steps shown in the legends.
these four cases. It is impractical to show all of the pressure profiles, so to avoid crowding, they are shown for selected time steps only. Film profiles h(r) at the same time steps are included in Figure 4 (to facilitate comparison, the same colors are used at the same time steps in all of these and subsequent plots). It can be seen that once a dimple is established the film pressures are fairly featureless, described to a first approximation by a flat-topped “top-hat” function which is about as wide as the
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barrier rim radius and as high as the initial Laplace pressure in the drop. This can be understood from the fact that drop interface curvature is small throughout the dimple region. Note that the vertical scale of Figure 2 is magnified 1000-fold compared to the horizontal scale and the dimple is almost flat (∼100 nm variation over ∼100 µm) on the millimeter scale of the whole drop. Such a low curvature means that the pressure difference across the interface is a small fraction of the drop’s internal pressure, so the film pressure must be close to the internal drop pressure. Closer inspection shows that the film pressure falls below the internal drop pressure at a radius slightly less than the barrier rim, and that there is a slight hump with the film pressure exceeding drop pressure near the symmetry axis. This hump is consistent with the concave curvature of the dimpled shape, associated with a negative pressure difference between drop and film (eq 2). It is noteworthy that the total film pressure does not vary much between the four different cases even though the surface forces are very different. This point will be taken up in the Discussion. The next step in the analysis is to find the disjoining pressure as a function of film thickness. As discussed in the previous section, there are two ways in which this could be done: from analyzing the film pressure when no hydrodynamic pressure is contributing or from a theoretical model. The former method can be used for the strong and moderate repulsion cases, but the latter is required for the strong attraction and weak force cases. The first method is applied as follows. At the end of a drainage measurement, the drop, solid surface, and intervening aqueous film all come to rest and hydrodynamic pressure in the film falls to zero. In the presence of a repulsive force, the drop adopts a flattened shape with a nearly parallel (uniform thickness) aqueous film separating them. The film thickness is that which corresponds to a disjoining pressure equal to the drop’s internal (Laplace) pressure, so the drop/film interface has zero curvature 23,27 (see the long-time profiles in Figures 3 and 4 of Paper I). Beyond the edges of the flattened region, the drop surface pulls away from the solid and reaches its normal shape (a spherical section if gravity is ignored) far from the solid. In the transition region just outside the flattened zone, h increases from its minimum value and Π(h) decays from its maximum (the drop pressure) to zero. Since hydrodynamic pressure is no longer present, Π is the only contribution to Pfilm. Hence, calculating Pfilm from the drop curvature in this transition region and plotting it against film thickness gives the disjoining pressure function Π(h). The results of doing this for the strong repulsion and moderate repulsion data sets are shown in Figure 5. Data from both sides of the profiles (r < 0 and r > 0) are included in this plot, and they overlay each other quite well. Clearly, the data are poor at large h and small Π because this corresponds to the experimental limits of the profiles at large |r| where the spline fits and their derivatives are unreliable. However, for a reasonable range of Π (about a factor of 2) and of h (about 20 nm, which is comparable to the Debye length), the data from different profiles and from the two sides of the profiles (r < 0 and r > 0) give consistent results. Figure 5 also includes theoretical calculations of the disjoining pressure due to electrical double-layer interactions, obtained by solving the nonlinear Poisson-Boltzmann equation using the Chan-Pashley-White algorithm.28 For strong repulsion (blue line in Figure 5), the parameters are -100 mV for the mica (27) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357-16364. (28) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283-285.
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Figure 5. Disjoining pressure isotherm obtained by calculating pressure in the shoulders of the drop profile at long t when hydrodynamic pressures are negligible for the strong repulsion (right) and moderate repulsion (left) cases. Note both sides of the profile data are included, r < 0 indicated by - symbols and r > 0 indicated by + symbols. Time steps are 9.62 (orange) and 13.62 s (lime green) for strong repulsion and 17.62 (violet), 25.62 (green), and 37.62 s (grey) for moderate repulsion. Data for Π j200 Pa come from the large |r| regions where the spline fits are least reliable, and the upturn in the violet curve is almost certainly an artifact caused by imperfect curve fitting in that region. The solid lines show theoretical calculations of electrical double-layer pressure as described in the text.
surface potential,25 -492 mV for the mercury surface potential,21 and 0.12 mmol/L for electrolyte concentration, corresponding to a Debye length of 28 nm. This concentration is slightly higher than the reported experimental concentration of 0.1 mmol/L;19 the difference is attributable to trace amounts of electrolyte present in the water at a level of a few times 10-5 mol/L, to which KCl was added at 1 × 10-4 mol/L. The theoretical curve for moderate repulsion (red line) is obtained with the same values for Debye length and mica surface potential and a mercury surface potential of -40 mV which differs slightly from the expected value of -52 mV based on the potential applied to the mercury. This difference is within the experimental uncertainty of relating mercury’s PZC to the potential applied with respect to a reference electrode.21 Clearly, the theoretical curves give accurate fits to the data, indicating that electrical double-layer forces are the predominant contribution to disjoining pressure in this range of film thickness. Pressure due to van der Waals forces (estimated in the nonretarded limit as 9 Pa at the minimum experimental thickness of 62 nm and decaying at least as rapidly as h-3) can be neglected at these film thicknesses. As expected for a double-layer pressure at a range of several Debye lengths, the data show an exponential decay of Π with film thickness. For the analysis using eq 5, it is convenient to use a simple exponential function
Π(h) ) Π0 exp(-h/λ)
(6)
Fitting the data of Figure 5 gives the decay length λ ) 28 nm, Π0 ) 12.7 kPa for the strong repulsion case and 4.06 kPa for the moderate repulsion case. Since film thicknesses never decrease below the flat-film thickness in the strong and moderate repulsion cases, the exponential form (eq 6) is an adequate description of disjoining pressure for the following analysis. For economy, this form rather than the full nonlinear solution is used in the calculations for the two repulsion cases. For the weak force and strong attraction cases, the drops are never stationary for long enough to justify ignoring hydrodynamic pressure, so this method of determining disjoining pressure cannot
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Figure 6. Profiles (from top to bottom, near r ) 0) of total film pressure, hydrodynamic pressure, film thickness, and disjoining pressure in the thin film between a dimpled drop and the solid surface at t ) 3.62 s in the strong repulsion case.
be applied. In those cases, we use a disjoining pressure, Π(h), calculated from the nonlinear solution of the Poisson-Boltzmann equation28 using the Debye length established above (28 nm) and the known surface potentials of mica and mercury. Having established total film pressure from the interfacial curvature and knowing the disjoining pressure as a function of film thickness, it is now a simple matter to calculate the hydrodynamic pressure in the film by subtracting disjoining pressure from total film pressure. An example of the results for one drop profile is shown in Figure 6. A selection of the pressure profiles thus obtained is plotted in Figure 7a-d for the strong repulsion, moderate repulsion, weak force, and strong attraction cases, respectively. In this figure, the symmetry of the data is exploited to plot disjoining pressures (taken from the r < 0 data) on one side and hydrodynamic pressures (from the r > 0 data) on the other. Figure 7 shows that the individual components, namely disjoining pressure and hydrodynamic pressure, are sensitive to the various force conditions (strong repulsion, etc.), whereas in Figure 4, we saw that the total film pressure is comparatively insensitive to this. This insensitivity shown in Figure 4 may have been anticipated from the fact that interfacial curvature is small within the dimple region, but Figure 7 reveals what is not at all obvious from the dimple shapes: that the hydrodynamic pressures are dramatically different between the four cases of different disjoining pressure.
4. Discussion The above analysis of fluid drop/fluid film interfacial curvature has successfully produced film pressure profiles from experimental data. Such an analysis is possible for a fluid/fluid system (provided the interface geometry can be measured with sufficient accuracy) because the local curvature depends only on the pressure difference across the interface at that point. In contrast, surface deformation of an elastic solid depends on stresses acting over all the surface, making it a more difficult proposition to determine a pressure distribution from measured deformation in that case. The only assumptions that have been required in this procedure to determine the distribution of pressures in the film are that (a) the interfacial tension is uniform, (b) there is no tangential stress difference across the drop/film interface, (c) the pressure is uniform through the thickness of the film, (d) pressure within the drop is not affected by drop deformation, and (e) hydrodynamic pressure within the drop is negligible. Assumption (a) is reasonable in the absence of temperature gradients and in the absence of surfactants whose nonuniform adsorption could cause Marangoni effects. The careful cleaning and drop renewal procedures used in the experiments whose data is analyzed here19
Figure 7. Disjoining pressures (left-hand side) and hydrodynamic pressures (right-hand side) for (a) strong repulsion, (b) moderate repulsion, (c) weak force, and (d) strong attraction for various time steps as shown in the legends.
were designed to remove surface active contaminants from the interface. Assumption (b) is implicit in other authors’ use of the Young-Laplace equation in many situations and requires the
Thin Film Drainage
absence of elasticity of the two fluids. Assumption (c) is inherent in the lubrication approximation that is routinely used to analyze thin film drainage. Part (d) is an approximation which we have shown by numerical calculations to be accurate within 1%. Assumption (e) does not require there to be no flow within the drop, only that there is negligible hydrodynamic pressure associated with that flow, i.e., that the fluid velocity gradients in the drop are much smaller than those in the film. This should be the case in the present system, as shown in Theoretical Background from the dimensional estimates given by Yiantsios and Davis.13 Note that no assumptions are required about the fluid/fluid flow boundary condition (slip, no slip, or partial slip). While it is not the main focus of this work, the data of Figure 5 show that the measurements of fluid interface curvature and film thickness can be used to determine disjoining pressure, at least over a limited range. Pompe and Herminghaus29 used a similar analysis to determine intersurface forces from accurate AFM measurements of sessile drop profiles. Once disjoining pressure is known (either from this method or from a theoretical model), it can be subtracted from the film pressure to calculate the other contribution, hydrodynamic pressure. The most important features of our results are as follows. (i) The total film pressure adopts a fairly simple profile when a dimple is formed. To a first approximation, this profile is a rather flat-topped top-hat function, having a plateau that is close to the internal pressure (Laplace pressure) inside the drop. The plateau extends nearly as far as the barrier rim radius then falls away to zero beyond the dimpled region. This pressure profile corresponds to the low curvature of the interface within the dimple region compared to the undistorted drop. (We are at pains to emphasize that this is only an approximate picture. The actual pressure profile is not as simple or as abrupt as a strict top-hat function, and the maximum in film pressure is not identical to the internal drop pressure.) Theoretical calculations of drainage between two fluid drops have previously noted this type of simple pressure profile.9,14,15 However, those calculations discussed hydrodynamic pressures only. Here we show it is the total film pressure, and not the hydrodynamic pressure, that has this simple form. As is evident from Figure 7, the hydrodynamic pressure is not as simple, and it does not adopt the same profile under different disjoining pressure conditions. (ii) The profile of total film pressure is insensitive to the value of disjoining pressure. Close examination of Figure 4 shows that the total pressure profiles at equivalent time steps are comparable in all four cases of different disjoining pressure. The design of the experiment was such that the same driving function was used for the approach of the mica surface to the mercury drop for all four cases, so the speed and distance of approach were similar at equivalent time steps. Once again, we note that the statement about comparable pressure profiles and their time evolution is made as a leading approximation, and we do not claim that they are identical in the four cases. They are, however, strikingly similar, while the disjoining pressure and hydrodynamic pressure profiles are distinctly different from each other (Figure 7). To illustrate this more clearly, we compare some of the pressure and film profiles for different disjoining pressure cases by plotting on the same graph the left-hand side of the data (r < 0) from one case and the right-hand side of the data (r > 0) from another. Figure 8 shows this comparison at equivalent time steps for the moderate repulsion case and the weak force case. In Figure 8a, it is seen that both film profiles initially show a dimple, but while the moderate repulsion case evolves to a flattened drop after about 13 s, the weak force dimple persists for 18.02 s, and then (29) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85, 1930-1933.
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Figure 8. Comparison between the moderate repulsion case (lefthand side) and weak force case (right-hand side) of (a) total film pressure (upper curves) and film profiles (lower curves), (b) disjoining pressures, and (c) hydrodynamic pressures at the time steps shown in the legends.
the film abruptly collapses. The total film pressures are quite similar to each other at the same times, the only significant difference being a slight hump at the shoulder (near the barrier rim) for the weak force case at 17.62 s. Figure 8b shows the corresponding disjoining pressures. Unsurprisingly, given that these cases correspond to different surface potentials on the mercury, the disjoining pressure profiles are quite different to each other. In Figure 8c, the hydrodynamic pressures are also distinctly different. At longer times, the pressure decays toward zero in the moderate repulsion case, whereas at t ) 17.62 s, there is a peak in hydrodynamic pressure near the barrier rim, associated with the rapid thinning and collapse of the film (at 18.02 s) in this region.19 In Figure 9, we show a similar comparison, this time between the extreme cases of strong repulsion and strong attraction. The comparison is only possible for short times because in the strong attraction case the film collapses after 0.64 s. Even in this short time, the differences in film profile and hydrodynamic pressure
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Figure 9. Comparison between the strong repulsion case (left-hand side) and strong attraction case (right-hand side) of (a) total film pressure (upper curves) and film profiles (lower curves), (b) disjoining pressures, and (c) hydrodynamic pressures at the time steps shown in the legends.
are evident, while the similarity in total film pressure (Figure 9a) is also apparent. Similar to what was noted in Figure 8a, there is a slight hump in total film pressure near the shoulder in the attractive force case. Why is it the total film pressure that appears quasi-invariant for the same driving function, rather than the hydrodynamic pressure? Our suggestion is that the behavior is ultimately driven by minimizing interfacial energy between the drop and surrounding fluid. When the drop is deformed by a combination of surface and hydrodynamic forces, it is the total pressure acting on its surface that determines its local curvature and hence deformed shape. The fact that the total film pressure tends toward a flat-topped profile with a value close to the internal drop pressure means that the pressure difference, and hence curvature of the interface within the dimple region, is small. Small curvature is associated with a low surface area in the dimple region. The observation of near-constant film pressure regardless of the
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disjoining pressure suggests that the drop is always maintaining the smallest surface area that it can when part of it is subjected to physical interaction with a nearby material. The experimental data reported in Paper 1 involve a dimple that is small (∼100 µm in radius) on the scale of the drop (∼1 mm in radius), and the above remarks may not apply for conditions that produce larger dimples. For small deformations, the macroscopic shape of the deformed drop is, to a first approximation, close to a truncated sphere. As estimated below, curvature in the dimple region never exceeds 10% of the drop’s curvature for the experiments analyzed here. The force exerted by the drop on the surface is ∼πr2bPint, and this is balanced by the force exerted by the surface (through the film) on the drop,8 given by 2π ∫R0 0 Pfilmr dr. Equality of these two forces is achieved if the film pressure has the top-hat function form indicated by our results, being equal to Pint up to the barrier rim radius, rb, and zero beyond that. The conclusion of this argument is that the details of dimple evolution are dictated by minimizing the drop’s total surface area throughout its collision with a solid wall. This statement would be self-evident in an equilibrium situation, but here it is observed to hold in a dynamic, nonequilibrium situation. This implies that the rate of adjustment of the overall drop shape is rapid compared to the drainage time of the film. Such adjustment requires movement of the fluid within the drop, including movement to or away from its surface whenever the interfacial area changes. A first correction to the zeroth-order argument given above would be to estimate the (concave) curvature under the dimple. If the dimple were a spherical indentation in the drop, of height δh ) h0 - hb where h0 is the film thickness at the center of the dimple and hb is the film thickness at the barrier rim, its radius of curvature would be approximately Rd ) r 2c /2δh, where rc is the radius of the barrier rim. When dimples are first formed in these experiments, their height is around δh ≈ 100 nm and rc ≈ 100 µm,19 giving Rd ≈ 0.05 m. This curvature corresponds to a film pressure near r ) 0 that exceeds the internal drop pressure by 2γ/Rd ≈ 20 Pa. It can be seen in Figure 4 that the film pressure at short times has a small peak at the center that exceeds the internal drop pressure by a few tens of pascals, comparable to this estimate. An obvious extension of this work is to integrate pressure over the drop surface to calculate the total force exerted on the mercury drop by the mica surface. The force is exerted by a combination of hydrodynamic and surface forces, which can be isolated by integrating the respective pressure components separately. This will be the subject of a future paper, but it is not difficult to anticipate the main result. Since the total pressure profiles differ little between the four cases of differing disjoining pressure, their integralsthe total force acting on the dropsis also very similar even when the surface forces change all the way from strong attraction to strong repulsion. The hydrodynamic and surface forces are distinctly different, and yet they compensate each other to maintain the same total force throughout the evolution of the dimple when a fluid drop collides with another surface.
5. Conclusion Calculation of the local curvature of a deformed drop from accurate experimental data has enabled pressure in a thin film between the drop and a solid wall to be determined with reasonable accuracy, estimated at 1% of the internal pressure of the drop that characterizes pressures in the system. Furthermore, knowledge of the disjoining pressure as a function of the film thickness
Thin Film Drainage
has allowed this component to be isolated from the hydrodynamic pressure and both to be established separately. The measurements provide a complete picture of the pressures as a function of position in the film and of time during the film drainage process involving a classic hydrodynamic dimple. Having an experimental determination of pressure in a thin draining film provides an alternative point of comparison for theoretical modeling. More importantly, knowledge of the pressure distribution allows a clearer insight into details of the fluid flow driven by hydrodynamic pressure gradients. The design of the experiment involved drainage runs that were repeated for various signs and strengths of surface forces while other parameters of drop size and approach speed of the solid surface were kept the same. Comparisons between different runs, with all parameters the same except for disjoining pressure, clearly illustrate that the total pressure in the film is quasi-invariant,
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while the film profile and hydrodynamic pressure are distinctly different between runs. The details of the drainage process depend on the different disjoining pressures, and the dimpled film shape evolves in such a way and at such a rate that hydrodynamic pressure compensates for different disjoining pressures. Theoretical calculations of hydrodynamic flow in draining films will only be accurate if they include a proper estimation of disjoining pressure. Acknowledgment. This work has been funded by the Australian Research Council through the Special Research Centre for Particle and Material Interfaces. We are grateful to Derek Chan, Steve Garoff, Vince Craig, Stan Miklavcic, Reinhard Lipowsky, and Olga Vinogradova for helpful discussions. LA052314B
