Drop Interacting

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Equilibrium Force Isotherms of a Deformable Bubble/Drop Interacting with a Solid Particle across a Thin Liquid Film Divesh Bhatt, John Newman, and C. J. Radke* Department of Chemical Engineering, University of California, Berkeley, California 94720-1462 Received July 10, 2000. In Final Form: October 25, 2000 Comparison of experimental force-distance isotherms obtained for a deformable drop/bubble and a solid particle interacting across a thin liquid film with colloidal theories, for example, the DLVO theory, currently relies on two incompatible and nonphysical assumptions. These are approximating the bubble/drop as a purely elastic solid and, contradictorily, as a nondeformable solid. To avoid these aphysical assumptions, we have developed a more rigorous method for interpreting the experimental force-distance isotherms. Equilibrium shapes of the bubbles/drops are calculated from the augmented Young-Laplace equation pertinent to the relevant geometry of the experiments. The overall bubble/drop-solid interaction force is then calculated using a macroscopic force balance. We find that the nondeformable and the elasticdeformation models for the bubble/drop are rather inaccurate. The developed methodology is of particular relevance to interparticle force measurements using an atomic force microscope (AFM). To this end, the theoretical method developed here is applied to several actual AFM experiments. It is seen that inaccurate deductions can be made regarding the range and form of thin-film forces if the analysis is done using the elastic and nondeformable assumptions. In particular, jump distances, that form the basis for estimating the range of thin-film forces, are grossly overestimated because of the above-mentioned assumptions. Finally, we provide an alternate method to compare experiments with theory that evaluates self-consistently the bubble/drop shapes and the macroscopic interaction force.

Introduction 1,2

Traditionally, the surface force apparatus (SFA) is used to measure interparticle forces between two nondeformable surfaces. Precise measurement of such forces when one or both of the particles is deformable has been more elusive. In the past few years, however, atomic force microscopy (AFM) has been extended from imaging surfaces to the direct measurement of forces between two surfaces,3 one or both of which may be deformable. The force resolution for the SFA is about 10 nN, whereas it is about 1-0.1 nN in the case of AFM. The distance resolution of the SFA is of the order of 0.1 nm, whereas that of the AFM is approximately 0.01 nm.4 Unfortunately, interpretation of AFM interparticle-force experiments is not straightforward when the surfaces deform during interaction. The macroscopic force between any two surfaces is a direct result of intermolecular forces between those surfaces attenuated by the surrounding liquid. When the liquid forms a thin film between the surfaces in question, the film influences the force enormously, leading to the term thin-film forces (i.e., disjoining-pressure isotherms). Numerous studies are available on interactions between two nondeformable solid surfaces;3-6 thin-film forces follow directly from these measurements.3 The case when one of the surfaces is deformable has, however, not been studied as extensively.7-13 The interest in systems where one of * To whom correspondence should be addressed. Phone: 510642-5204. Fax: 510-642-4778. E-mail: [email protected]. (1) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (2) Israelachvili, J. N. Acc. Chem. Res. 1987, 20, 415. (3) Ducker, W. A.; Senden, T. J. Langmuir 1992, 8, 1831. (4) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1991; Chapter 10. (5) Biggs, S.; Mulvaney, P. J. Chem. Phys. 1994, 11, 8501. (6) Larson, I.; Drummond, C. J.; Chan, D. Y. C.; Grieser, F. J. Am. Chem. Soc. 1993, 25, 11885.

the surfaces is deformable stems from the fact that many systems of technological importance have at least one deformable surface. These include, for example, the stability of foams and emulsions where the discontinuous phase is separated by thin liquid films and all interfaces are deformable.14,15 Another example involves the wettability of solids by either oil or water when the solid is immersed in the opposing phase.8,16 In this case, oil is separated from the solid surface by a thin film of water or vice versa. Yet another example is that of flotation where small mineral particles are separated from water by attachment to rising and deformable air bubbles. Because of the deformation of the interfaces, the exact separation distance profile between the surfaces is not known.12,13 Although there have been attempts at interpreting the experimental results by approximating the deformable interface as elastic,7-9 such treatments are not rigorous. A model based on linear elasticity assumes that a deformable object can be characterized simply by a stiffness parameter, the spring constant, which remains constant throughout the interaction between the deformable object and the solid across the thin liquid film. As we establish later, the assumption of linear elasticity for fluid bubbles or drops is not justified. The sole purpose of the linear elasticity model in AFM force measurements is to (7) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (8) Basu, S.; Sharma, M. M. J. Colloid Interface Sci. 1996, 181, 443. (9) Fielden, M. L.; Hayes, R. A.; Ralston, J. Langmuir 1996, 12, 3721. (10) Carambassis, A.; Jonker, L. C.; Attard, P.; Rutland, M. W. Phys. Rev. Lett. 1998, 80, 5357. (11) Mulvaney, P.; Perera, J. M.; Biggs, S.; Grieser, F.; Stevens, G. W. J. Colloid Interface Sci. 1996, 183, 614. (12) Snyder, B. A.; Aston, D. E.; Berg, J. C. Langmuir 1997, 13, 590. (13) Preuss, M.; Butt, H.-J. Int. J. Miner. Process. 1999, 56, 99. (14) Kruglyakov, P. M. Thin Liquid Films; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1998; Chapter 11, pp 767-827. (15) Ivanov, I. B. Colloids Surf., A 1997, 128, 155. (16) Jachowicz, J.; Berthiaume, M. D. J. Colloid Interface Sci. 1989, 133, 118.

10.1021/la0009691 CCC: $20.00 © 2001 American Chemical Society Published on Web 12/09/2000

Deformable Bubble/Drop Interaction with a Solid

estimate the separation distance between the two interfaces, or the thickness of the thin film, from the experimental force-distance result. To compare the experimental force-distance curve established using the model of linear elasticity with theories of colloidal interaction, for example, that developed by Derjaguin, Landau, Verwey, and Overbeek (DLVO),17 an assumption about the shape of the deformable object is required. An analytic expression of the force can be obtained, but only when the shapes of the objects are assumed to remain constant, that is, undeformed, during interaction. Thus, the current procedure to compare experiments with theory utilizes two mutually exclusive assumptions. The first is that the deformable bubble or drop behaves as an elastic solid, and the second is that the shape of the deformable object does not change during its interaction with the solid. Both assumptions have little physical significance. We anticipate that these aphysical assumptions perturb the shape of the force-distance curve deduced from the AFM experiments. Because the interpretation of the experimental force-distance curve forms the basis for deducing the behavior of thin-film forces, it is crucial to interpret the experiments in a more rigorous manner. The aim of this work is to provide a systematic approach to calculate quantitatively the macroscopic force-distance curve from the fundamental disjoining-pressure isotherms when the interfaces are fluid and, therefore, deformable. We consider only the case when a thin liquid film separates the two interacting interfaces. In other words, we do not consider the possible scenario of the rupture of the thin liquid film and the subsequent engulfment of the solid particle by the bubble/drop, a possibility that has been mentioned in the literature.12 A somewhat similar approach for an axisymmetric drop interacting with a flat solid has been suggested by Miklavcic et al.,18 later including two identical axisymmetric drops interacting with one another19 and the interaction between a drop and a flat solid in an SFA.20 However, we treat both the deformable bubble/drop and solid as curved objects, and we do not demand that the pressure inside the deformable object (a drop or a bubble) is a constant. Although we deal exclusively with the case when only one interface is deformable, the case where both the interfaces are deformable can be treated by extension (cf. Miklavcic19). Moreover, our procedure is of direct relevance to AFM experiments. Experimentally, the measured force-distance curve is translated into an interaction-potential curve. Here, however, we set first an interaction potential and then calculate the macroscopic force as a function of the separation between the bubble/ drop and the solid, instead of the inverse problem of specifying the force and calculating the thin-film interaction potential. With this systematic procedure, we check the validity of the nondeformable/elastically deformable approximations traditionally made when interpreting the experimental results, and we apply the proposed procedure specifically to AFM systems. Model Development The system we consider consists of two identical spherical solids approaching an axisymmetric bubble/drop (17) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (18) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357. (19) Miklavcic, S. J. Phys. Rev. E 1996, 54, 6551. (20) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483.

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Figure 1. Schematic of the system model with two spherical solids approaching an axisymmetric bubble/drop.

symmetrically, with the particles immersed in a liquid medium. Figure 1 shows the relevant geometry. Both of the solids have identical radii, Rs. The distance of closest approach between the bubble/drop and either solid is h0. The origin of the y axis is at the apex of the bottom solid, with the y distance of a point on the bubble/drop surface from the cylindrical r axis denoted as hˆ . h is the distance of that point normal from the solid surface. A noninteracting system boundary, which is much larger than either the solids or the drop/bubble, encloses the system. Hence, the liquid that forms a thin film between the solid and the drop/bubble is in equilibrium with a bulk liquid. In an actual AFM system, the bubble/drop is attached to a solid, and hence the system in Figure 1, with a “free” bubble/ drop, does not represent the real AFM system. However, we analyze this system first because it is useful for developing the fundamental aspects of the calculation and analysis. We deal with a more realistic model of the AFM system later. Augmented Young-Laplace Equation. Thermal equilibrium is imposed on the system, and the total mass and volume of the system are conserved. Thus, Yeh et al.21 establish that at equilibrium the appropriate energy to minimize is the Helmholtz free energy. However, the results of Yeh et al.21 apply only to the shape of a bubble/ drop over a flat solid. Following a similar analysis, Bhatt22 demonstrates that for a bubble/drop interacting with a curved particle, the pertinent free-energy minimum is stated by

∫0r

δ

max

[γx(1 + hˆ 2r ) + P(h) - pchˆ ]r dr ) 0

(1)

where rmax is the extent of the bubble/drop along the radial coordinate, γ is the equilibrium bulk interfacial tension of the bubble/drop against the liquid medium, P is the interaction potential between the (bubble/drop)/liquid and solid/liquid interfaces, and pc is the difference between the pressure inside the bubble/drop phase, pF, and the pressure in the liquid medium, pL, and is also called the (21) Yeh, E. K.; Newman, J.; Radke, C. J. Colloids Surf., A 1999, 156, 137. (22) Bhatt, D. M.S. Thesis, University of California, Berkeley, CA, 2000.

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capillary pressure. The subscript r in eq 1 denotes differentiation with respect to the radial coordinate, and the symbol δ denotes variational minimization. An important point in eq 1 is that the coordinate hˆ appears rather than the more usual distance parameter h. The reason is that the solid particle is curved, demanding a more suitable coordinate when r approaches Rs. hˆ describes the curvature correctly when P approaches zero. The interaction potential, P, in eq 1 depends only on h and not on the higher derivatives of h with respect to r. This assumption can be made if the two interfaces are locally almost flat and parallel.21 Near the point of closest approach of the two interfaces, that is, near r ) 0, the foregoing approximation is quite accurate. It becomes unreasonable at larger r, but then the two interfaces are sufficiently far apart that the thin-film forces can be neglected. Additionally, we chose to define P in terms of h instead of upon the y distance of a point on the (bubble/ drop)/liquid interface from the solid. This convention is adopted so that the disjoining pressure is determined by the distance of closest approach of a point on the (bubble/ drop)/liquid interface to the solid/liquid interface, consistent with that for flat solids. The interaction free energy (interaction potential) contribution to the total free energy is expressed as ∫P(h)2πr dr, which allows us to write eq 1. The term 2πr dr signifies a ring element which interacts with a segment of the (bubble/drop)/liquid interface that is at a distance h from the solid. In principle, P(h) should include the effect of the slope of the interfacial profile as well, but this is small when thin-film forces are significant. Thus, in practice, we accept this formulation of the interaction free energy as reasonably accurate. Use of 2πr dr for the area element when the solids are spherical is also questionable and can be made only when r , Rs.22 Physically, this implies that the thin-film forces must be important only in a small region near the point of closest approach of the bubble/drop and the solid. In other words, h must increase rapidly enough with r so that P(h) can be neglected in comparison to other terms in eq 1 as r becomes comparable to Rs. Application of the calculus of variations to eq 1, treating r, hˆ , and hˆ r as independent variables inside the integral, leads to an expression for the interfacial profile (i.e., the augmented Young-Laplace equation) for a bubble/drop interacting with a spherical solid particle22

hˆ rr (1 +

hˆ 2r )3/2

+

hˆ r r(1 +

hˆ 2r )1/2

1 ) (pc - Π(h)) γ

(2)

Figure 2. A schematic of the different kinds of (bubble/drop)/ liquid interfacial profiles possible over a solid sphere. Two separate bubble/drop profiles are labeled. Small dashed circles represent the regions where eq 2 is suspect.

Equation 2 is a second-order ordinary differential equation; hence, we need two boundary conditions to solve for the interfacial profile. If we consider an actual AFM system, we might adopt the distance between the center plane and the y ) 0 plane (i.e., ∆ in Figure 1) as a location for a boundary condition. However, for calculation purposes, it is more convenient to solve an initial value problem, and hence we set one boundary condition at the separation distance between the interfaces at r ) 0, that is, at the minimum separation distance h0, and the other along with the symmetry axis giving the two conditions

h ) h0 at r ) 0 hr ) 0 at r ) 0

Finally, for a liquid film of uniform thickness covering a flat solid, the film thickness is dictated by the intercept of pc and Π(h). In other words, pc ) Π denotes the occurrence of an equilibrium film of uniform thickness. Similarly, we expect a liquid film of uniform thickness covering a spherical solid for certain values of the film thickness. However, when the solid is spherical, that film thickness is no longer governed by the intercept of pc with Π. Let the thickness of the uniform, curved film be designated as hf. Because the solid is spherical, the liquid film of uniform thickness forms a spherical shell of thickness hf around the solid of radius Rs. Thus, the uniform film thickness is determined from eq 2 by

pc ) Π(hf) where Π is the disjoining pressure and is the negative of the derivative of the interaction potential, P, with respect to h (i.e., Π ) -dP/dh).21 It is important to note that hˆ ≡ h in Figure 1 if the solids are flat instead of spherical. As noted above, the contribution of thin-film forces to eq 2 is strictly valid only when r , Rs. Thus, for eq 2 to describe the interfacial profile accurately, it is necessary that the separation between the solid and the drop increases fast enough so that the thin-film forces can be neglected when r becomes comparable to Rs (i.e., Π(h) f 0 as r f ∞). Figure 2 illustrates the validity of eq 2 schematically, where two possible bubble/drop profiles are shown over a spherical solid. Profile 1, as shown, is expected to follow eq 2 quite accurately, whereas in the regions depicted by the dashed circles on profile 2, h is still close to h0 when r is comparable to Rs. Equation 2 is expected to be accurate only outside of the regions of the dashed circles in Figure 2.

(3)

2γ hf + Rs

(4)

Even though eq 2 is valid only when r , Rs and the film of uniform thickness extends beyond Rs while h is still hf, it is certainly valid near r ) 0; the criterion of eq 4 applies there. A similar expression, in a different context, for the thickness of a uniform film over a curved solid has appeared elsewhere.23 Such a film does not form a bubble/ drop profile. Force Balance. The governing equation for the (drop/ bubble)/liquid interfacial profile is given by eq 2. However, to arrive at a force-distance relationship, a macroscopic force balance is needed. Figure 3 illustrates the control volume required for the force balance. Because of symmetry, the system is cut along the center plane. Fluid forces on the side of the box, perpendicular to the r axis, do not contribute to the force balance in the y direction. (23) Kovscek, A. R.; Wong, H.; Radke, C. J. AIChE J. 1993, 39, 1072.


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Figure 3. A force balance on a control volume in Figure 1 obtained by cutting the system at the center plane.

Forces acting on the center plane in the y direction are portrayed with arrows. Interfacial tension applies a force directed along the positive y direction (i.e., the bubble/ drop profile intersects the center plane at 90°) of magnitude 2πrmaxγ, where, again, rmax is the maximum extent of the drop in the r direction. The y force on the center plane as a result of the surrounding liquid is pLπ(L2 - r2max), where L is the radial dimension of the control volume. Similarly, the y force on the lower surface of the control volume as a result of the surrounding liquid is pLπL2. On the center plane, the drop/bubble phase exerts a force of magnitude pFπr2max in the negative y direction. For the system to be at equilibrium, the applied force, F, on the solid surface must compensate all the other forces acting:

F ) πpcr2max - 2πγrmax

(5)

F can be thought of as the experimental force that the AFM cantilever exerts on the bubble/drop. Although thinfilm forces do not appear explicitly in eq 5, F arises solely from these forces because both pc and rmax depend on the disjoining-pressure isotherm in eq 2. In the absence of thin-film forces, the bubble/drop profile is a perfect sphere, and rmax is identically 2γ/pc leading to F being exactly zero. Solution Procedure First, we solve for the (bubble/drop)/liquid interfacial profile using the augmented Young-Laplace relation, eq 2, and the boundary conditions given by eq 3. In addition to the disjoining-pressure isotherm (or the interactionpotential curve), the solution requires three parameters, namely, pc, h0, and Rs. The chosen interaction-potential curve is constructed using the form and the parameter values given in Appendix A, with the resulting interactionpotential and disjoining-pressure isotherms shown in Figure 4. Figure 4a shows the interaction-potential isotherm, whereas Figure 4b illustrates the corresponding disjoining-pressure isotherm. Local maximum and minimum values are also defined on the disjoining-pressure isotherm as Πmax and Πmin, respectively. A value of Rs corresponding to a particular solid is chosen next. Once the disjoining-pressure isotherm and Rs are set, various

Figure 4. (a) Interaction potential as a function of h. (b) Disjoining pressure as a function of h. The local maximum and the minimum values of Π are shown as Πmax and Πmin, respectively.

(bubble/drop)/liquid interfacial profiles can be generated for different input values of pc and h0. Here, we employ a fourth-order Runge-Kutta algorithm with arc-length parametrization to integrate numerically the augmented Young-Laplace equation. Details are given in Appendix B. An interfacial profile corresponds to a physical bubble/ drop only when the angle that the profile makes with the r axis becomes 90° at some h. Attainment of this angle signifies that the interfacial profile intersects the center plane and hence denotes a bubble/drop. The volume of the drop/bubble, VF, can then be calculated from the bubble/ drop shape profile using simple quadrature. Up to now, we have not differentiated between a bubble and a drop. Each must satisfy a separate criterion. The pressure-volume relation for a bubble must obey an equation of state. For convenience, we choose the idealgas law. Conversely, a drop is taken as incompressible and, thus, must satisfy a constant-volume constraint. Hence, to identify a particular drop in our calculations, we scan the pc and h0 parameter space and choose only those sets of (pc,h0) values that lead to the same VF as that of the particular drop. More specifically, we set a value of h0 and identify the upper and lower limits on pc that approximately satisfy the constant-volume constraint using a sequential search algorithm. We then refine our estimate of pc using a binary search algorithm. This procedure is repeated for several values of h0. A similar argument is applicable to a bubble, with the ideal-gas law replacing the constant-volume constraint. Once the set of (pc,h0) values corresponding to a particular drop (or bubble) (i.e., to a particular VF relationship) are identified, we establish the force, F, on the drop (or bubble) as a function of h0 using eq 5. Results and Discussions Before presenting F-h0 curves for various drops/bubbles and solids of different sizes, it is instructive to consider the interfacial profiles for different values of pc, h0, and Rs. In all calculations, the value of the (bubble/drop)/liquid interfacial tension, γ, is chosen to be 72 mN/m, unless otherwise specified. Interfacial Profiles. Once the disjoining-pressure isotherm and the size of the solid particle are chosen, the parameters pc and h0 dictate both the bubble/drop volume

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Figure 5. Portions of the bubble/drop interfacial profiles for three different values of h0. The ordinate is hˆ - h0, and the solid is shown by the dashed line with hˆ as the ordinate. pc is 0.95 kPa, and Rs is 100 µm.

Figure 6. A portion of the (bubble/drop)/liquid interfacial profile for h0 ) 4 nm to highlight the formation of a nose near r ) 0. Rs is 100 µm, and pc is 0.95 kPa.

and the shape of the interfacial profile. From Figure 1, it is clear that the bubble/drop profile has a positive curvature near the center plane; for a macroscopic drop (i.e., one that, at least partially, lies outside the range of thin-film forces21), Π is zero near the center plane. This demands that pc must be positive for the interfacial profile to correspond to a physical bubble/drop. [A negative curvature indicates that the liquid and the bubble/drop phases are interchanged in Figure 1. Such a scenario implies that the liquid phase now forms a drop and interacts with the solid through a thin film of the original drop/bubble phase.] No experimental study of bubbles/drops with an initial undeformed radius, RD, of less than about 10 µm has yet been reported in the literature. Thus, the initial capillary pressure is of the order of 10 kPa (i.e., the capillary pressure when the bubble/drop is located far from the solid where Π ) 0). Accordingly, we confine ourselves to the shapes of those interfacial profiles that have a capillary pressure between zero and approximately 10 kPa. Because the value of Π at h ) 0 in Figure 4 is close to 165 kPa, there is always at least one value of the disjoining pressure that is coincident with the capillary pressure in the range we are interested in. Below Πmax, there are three coincident values of the disjoining pressure and the capillary pressure, each at a different separation distance h. In Figure 5, portions of the interfacial profiles are shown for three different values of h0 and for pc ) 0.95 kPa and Rs ) 100 µm. The ordinate is hˆ - h0 so that different profiles can be compared directly regardless of the respective h0 values. A part of the solid is also shown with a dashed line with the ordinate as hˆ , so that the apex of the solid lies at the origin. The h0 values chosen are 1000, 4, and 2.135 641 2 nm, respectively. When h0 ) 1000 nm, the bubble/drop is separated from the solid surface by a film thick enough so that Π is everywhere negligible compared to pc. Hence, the interfacial profile is unaffected by the presence of the solid and thus represents an undeformed bubble/drop of radius equal to 151.6 µm. As h0 is decreased to 4 nm, the liquid film separating the bubble/drop from the solid thickens much more rapidly in the radial direction. The reason is that at such an initial separation Π is negative (i.e., attractive), which results in a higher positive curvature at r ) 0 (cf. eq 2). A further decrease in h0 brings it closer to hf, obtained from eq 4, and the profile drapes increasingly over the solid forming a dimple in the bubble/drop. This dimpling is indicated by the interfacial profile with h0 ) 2.135 641 2 nm (hf ) 2.125 641 14 nm to 8 decimal places).

It must be emphasized that although these three interfacial profiles correspond to drops (or bubbles), they do not, in general, correspond to the same drop (or bubble) because nowhere is the constant-volume (or ideal-gas) criterion imposed. Experimentally, these three cases of different h0 can be thought of as individual experiments performed upon three different size drops/bubbles. There is an important aspect of the interfacial profile for h0 ) 4 nm that is not obvious from the range of the abscissa covered in Figure 5. An h0 value of 4 nm, as already pointed out, corresponds to the attractive part of the disjoining-pressure isotherm. Because of this initial attraction in the interfacial profile, a nose (or a pimple) is expected to form on the bubble/drop near r ) 0. To depict such an attractive nose, Figure 6 illustrates the interfacial profile for pc ) 0.95 kPa, h0 ) 4 nm, and Rs ) 100 µm for a larger range of the abscissa, r. Formation of a nose on the bubble/drop is now clearly evident. An important point emerges from the interfacial-profile calculations. When h0 corresponds to the attractive part of the disjoining-pressure isotherm, a nose is formed. Such a nose causes the interfacial profile to deviate away faster from the solid than the undeformed interfacial profile, resulting in a faster decay of the interaction force between the solid and the bubble/drop. Conversely, when h0 corresponds to a repulsive part of the disjoining-pressure isotherm a dimple is formed; the interfacial profile now drapes over the solid, and the total measured force increases. Thus, the contribution of the repulsive component of Π(h) to the measured force between the solid and the drop/bubble is more prominent than the attractive part. For a flat film, the positive sloping branch of the disjoining-pressure isotherm represents an unstable region of film thickness. However, because of the curvature of the bubble/drop profile, the film thickness changes continuously, and the criterion for instability of a flat film no longer applies. Deviation from flatness is emphasized even more when h0 for the drop/bubble corresponds to the positive sloping branch of the disjoining-pressure isotherm because of the formation of a nose. Thus, an h0 corresponding to a positively sloped branch of Π(h) does not imply that the bubble/drop is in an unstable regime. Force-versus-h0 Isotherms. For the same disjoiningpressure isotherm, the F-versus-h0 relationships for a compressible bubble and an incompressible drop are almost identical, although we do not show this result here (the thesis of Bhatt gives more details22). In other words, the F-versus-h0 isotherm for a bubble is coincident everywhere with that of a drop, provided, of course, that


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Figure 7. Comparison of the exact force calculations (solid and dashed-dotted lines) with the nondeformable-drop assumption in eq 6 (dashed line). The undeformed radius of the drop is 14.14 µm. The two different curves for the exact calculations correspond to two different solid radii, as labeled in the figure.

the undeformed radius, RD, is the same. This is to be expected because the maximum variation in the capillary pressure (and, hence, the variation of the bubble pressure) is negligible compared to the pressure of the surrounding liquid. Thus, the ideal-gas constraint approximates the constant-volume constraint. Hence, from this point onward the use of either a bubble or a drop implies the other as well. As emphasized in the Introduction, one of the current key assumptions when comparing the macroscopic experimental-force results to the underlying disjoiningpressure isotherm is that the drop remains undeformed. This permits the direct use of Derjaguin’s approximation for the interaction between two spheres.24 Under the undeformed approximation, F can be calculated directly from the knowledge of the interaction potential, P, at h ) h0, as

(

F(h0) ) 2π

)

RsRD P(h0) Rs + RD

(6)

This result suggests that we scale F by the geometric factor multiplying P(h0) when comparing the nondeformable result to the numerical solution of the augmented YoungLaplace equation and the exact force balance of eq 5. In Figure 7, a comparison of the appropriately normalized force as a function of h0 is made for a drop with an undeformed radius of 14.14 µm interacting with two different size solids of Rs values equal to 10 and 200 µm. The curve obtained using the nondeformable-drop assumption is also shown as a dashed line. If the nondeformable assumption is applicable, then the curves for both solids must coincide with that for the undeformed drop. As is obvious from the figure, the nondeformable assumption becomes poorer as the size of the solid increases. This is expected because a larger solid imparts more deformation to the drop. Also, as a result of deformation the attractive part of the curves in Figure 7 is altered more than the two repulsive parts. The reason behind this observation is the formation of a nose during attraction that leads to the drop profile deviating faster away from the solid than that for the nondeformable drop. Thus, the region of the drop that corresponds to the attractive part of Π(h) is diminished. (24) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989; Chapter 5.

Figure 8. Comparison of the exact force calculations (solid and dashed-dotted lines) with the nondeformable-drop assumption in eq 6 (dashed line). The solid radius is 200 µm. The two different curves for the exact calculations have different undeformed-drop radii, as labeled in the figure.

An increase in the size of the solid particle leads to a greater deviation from the nondeformable assumption. Similarly, we expect the nondeformable assumption to lead to a progressively worse match with the exact results as the drop size is increased, keeping the size of the solid fixed. Such a comparison is shown in Figure 8 for a solid with Rs ) 200 µm. The drop sizes considered have RD values of 14.14 and 151.6 µm, as labeled in the figure. Again, the appropriately normalized force is plotted against the minimum separation distance, h0, and the curve for the nondeformable assumption is shown as a dashed line. As RD is increased from 14.14 to 151.6 µm, the nondeformable assumption becomes worse. Only a very small attractive region is evident for the larger drop. Elastic Drop. We noted in the Introduction that a standard assumption for interpreting AFM force-distance curves, beyond that of nondeformability, is the perfect elasticity of the bubble/drop. This second approximation allows the calculation of h0 from the experimentally measured distances. If the elastic-drop assumption is made, then F must be related linearly to the deformation of the drop at its apex, δb, or

F ) Kbδb

(7)

where Kb is the “effective spring constant” of the drop. In current AFM force measurements, the procedure is to establish Kb and to then calculate δb from knowledge of the measured force.7-9 Fortunately, in our numerical analyses we calculate δb directly. Figure 9 illustrates the various deformations/deflections in the system, along with the relevant distance variables. Initially, an undeformed hemispherical drop, with an undeformed radius RD, sits atop a solid substrate. The initial position in the absence of interaction forces is shown with solid lines, and all deflections and deformations are positive in the upward direction. Initially, the cantilever is at a distance D from the substrate. Typically, a spherical solid is attached to the cantilever, but this is not depicted in Figure 9. [Because the spherical solid is rigid and undergoes no deformation, the presence or absence of a spherical solid attached to the cantilever is irrelevant to the development to follow.] A second feature of Figure 9 is that it corresponds more directly to an actual AFM system, as opposed to the system in Figure 1. In particular, the drop in Figure 1 is a free drop, rather than one pinned on a solid. Thus, the substrate in Figure 9 is analogous to the center plane in Figure 1, and the cantilever in Figure

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Figure 10. Force-drop deformation curve for an axisymmetric drop of undeformed radius 14.4 µm. The drop interacts with two identical flat solids. Arrows represent whether the drop forms a nose or a dimple. The dotted line is an extrapolation to zero drop deformation and zero force at a large separation distance. Figure 9. Description of the various deformations as the bubble/drop approaches the cantilever. Solid lines represent the initial positions, and dashed lines indicate the final positions. The dotted line depicts the fictitious final position of the bubble/ drop if it remained undeformed. Deformations are taken as positive in the upward direction.

9 denotes a spherical solid in Figure 1. Despite these obvious differences between Figures 1 and 9, the development to follow is applicable to both. When the substrate is moved upward by an amount δs, the cantilever deflects by an amount δc, and the resulting deformation of the drop at its apex is δb. New positions of the drop, the cantilever, and the substrate are depicted by the dashed lines in Figure 9. The new separation between the cantilever and the substrate is ∆, and the minimum separation distance is h0. The relationship between h0 and the various deflections/deformations is

h0 ) D - RD - δs + δc + δb

(8)

where both the cantilever deflection, δc, and the substrate movement, δs, can be measured accurately in an AFM apparatus because both these are relative quantities and not the absolute cantilever and substrate positions. The initial separation between the substrate and the cantilever, D, and the undeformed radius of the drop, RD, are constants. The separation distance ∆ is related to the experimentally measurable quantities, δc and δs, and to the constant D by

∆ ) D - δ s + δc

(9)

Because D is just a constant, it need not be known accurately as it drops out of the analysis with each differential change in ∆. Because changes in ∆ can be measured accurately, it is an appropriate experimental distance variable. In fact, Ducker et al.7 reported their “raw” experimental data as F as a function of ∆. Equation 9 can now be substituted into eq 8 to give

h 0 ) ∆ - R D + δb

(10)

which finally relates δb to the quantities that can be calculated conveniently in our numerical modeling procedure. Use of eq 10 simplifies the calculation of δb from the other parameters that either are input to our numerical procedure (i.e., h0) or can be calculated numerically (i.e.,

∆). This procedure to obtain δb, although exact for the numerical model solution, is not feasible for the AFM experiments because both h0 and δb are unknown. However, by calculating δb directly we can assess the validity of the assumption that the drop behaves as an elastic solid. Figure 10 graphs F against δb for a drop with RD ) 28.8 µm. The solid is flat in this calculation. A positive value of δb represents the formation of a nose, whereas a negative value of δb depicts a dimple. This definition of the sign of δb is consistent with the sign convention used in Figure 9. As the distance between the drop and the solids increases beyond the maximum in the force, both the force and the drop deformation decrease to zero. This is represented by the middle portion of the curve in Figure 10, although the curve stops at some point close to zero because the maximum h0 considered in the calculation is 100 nm. The dotted line depicts schematically the extrapolated force and δb for h0 larger than 100 nm. An intriguing aspect of Figure 10 is that in a certain region there are multiple values of F for a single δb and vice versa. The reason is that δb depends on the curvature of the drop at r ) 0 (i.e., the curvature at the point of the closest approach of the drop to the solid), which in turn is dictated by the difference between the capillary pressure and the disjoining pressure at the minimum separation distance. Due to the shape of the disjoining-pressure isotherm and the fact that the capillary pressure of the drop changes as the minimum separation is changed, there can be multiple possibilities of the minimum separation distance at which the difference between the capillary and the disjoining pressures remains the same. Clearly, from Figure 10, the force is not linearly related to the drop deformation. Hence, there is no basis for the assumption that a deformable drop behaves as a linearly elastic solid. Another point that emerges from Figure 10 is that the zero of drop deformation along the line of minimum approach distance to the solid does not occur at F ) 0, except as h0 f ∞ (dotted line). This is because the nose (or dimple) on the drop depends on the curvature of the drop profile at r ) 0, which in turn depends on the disjoining-pressure isotherm. As already mentioned previously, the maximum, the minimum, and the zero of the disjoining-pressure isotherm and the force-distance curve do not occur at exactly the same h0. Thus, the left, the middle, and the right branches of the F-versus-δb curve in Figure 10 correspond only roughly to the inner, the


Figure 11. Force-distance curve for an axisymmetric drop with an undeformed radius of 28.8 µm interacting with a flat solid. The solid curve is the F-versus-(∆ - RD) curve, whereas the dashed curve is the F-versus-h0 curve. The magnitude of the bubble deformation, δb, is also shown as the horizontal distance between the two curves: δb ) h0 - ∆ + RD.

middle, and the outer branches, respectively, of the disjoining-pressure isotherm of Figure 4. It might seem surprising that δb is positive (formation of a nose) while the force is still positive (cf. the nose region with positive force in Figure 10). This observation simply implies that the curvature at h ) h0 is larger than the curvature in the immediate vicinity of h0. Such a scenario is clearly possible even with a positive force, as the disjoining-pressure isotherm is positive and increasing between 20 and 33 nm (see Figure 4). Thus, at any h0 value between these two limits, the total force is positive (no attractive disjoining pressure at any subsequent h), and the curvature in the vicinity of h0 is less than that at h0. This curvature is determined by the difference between pc and Π and continuously decreases up to h ) 33 nm. Also, Figure 10 indicates that the drop deformation is almost linearly related to the force, but in distinct regions with different spring constants (different slopes of the F-δb curve). However, there are two serious objections to this line of reasoning. The first is that even if each branch of the F-δb curve may be approximated as a straight line with a different slope (i.e., the negative of the spring constant), the range of applicability of each spring constant cannot be established a priori. The second and more serious objection is the lack of any physical basis for this approach. Thus, we conclude that the imposition of a linear elastic solid as a model for a deformable bubble/drop is unwarranted. Recommended Distance Scale. Above, ∆ is identified as an appropriate experimental distance variable because it can be measured accurately. True, ∆ depends on D, but, as is evident shortly, the value of D is not of much importance. Numerically, it is also straightforward to calculate ∆ for given pc and h0 values. Thus, a plot of F against ∆ is an appropriate and recommended way to compare theory with experiment. In Figure 11, a plot of F against ∆ - RD is shown as a solid line for an axisymmetric drop with RD ) 28.8 µm interacting with a flat solid. The dashed curve represents the F-versus-h0 plot. The horizontal distance between the solid and the dashed curves is δb, the drop deformation. If the drop is nondeformable (i.e., δb is zero), then both the solid and the dashed curves in Figure 11 must coincide, which they clearly do not. The abscissa for the solid curve in Figure 11 is ∆ - RD, whereas we identified ∆ as the experimentally measurable distance. RD, like D, is a constant and leads merely to the

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Figure 12. Comparison of the exact F-h0 curve (solid line) with that obtained using the elastic-drop assumption (dasheddotted line). The curve obtained using the nondeformable assumption is also shown (dashed line). The effective drop stiffness is 0.11 N/m, RD is 28.8 µm, and the solid is flat.

translation of the abscissa of the F-versus-∆ curve. This translation does not affect either the decay behavior or the range of the forces. Moreover, translation of the abscissa is equivalent to setting the zero of distance in an AFM experiment. The implication of the sliding of the abscissa is that D and RD need not be known. From the numerical-calculation point of view, allowing an arbitrariness in the values of D and RD is not necessary. However, it is unreasonable to expect D and RD to be known accurately to a nanometer scale in the AFM experiments. To emphasize the utility of an F-versus-∆ (or ∆ - RD) plot, when the disjoining-pressure isotherm for a particular solution/drop/solid system is exactly known, then the numerically calculated F-versus-∆ curve must coincide with the experimentally measured F-versus-∆ curve after an appropriate translation of the abscissa. For this reason, we can deduce from the exact F-versus-∆ curve an approximate F-versus-h0 curve using the elastic-drop argument. Comparison of the so-deduced approximate F-versus-h0 curve and the exact F-versus-h0 curve sheds further light upon the validity of the elastic-drop assumption. Before such a comparison is made, the effective bubble stiffness parameter, Kb, needs to be established. The way this has been done experimentally is to measure the slope of the force-versus-substrate displacement curve in the region of constant bubble (or drop) compliance,7,9 which is essentially the linear region in the extreme left branch of the F-versus-δb curve of Figure 10. Thus, Kb is calculated to be 0.11 N/m from Figure 10. In Figure 12, the exact F-versus-h0 curve (solid line) is compared with the one calculated from the F-versus-∆ curve with the elastic-drop assumption (dashed-dotted line). Clearly, the elastic-drop assumption fails to give useful results, especially in the intermediate region between 0 and 60 nm. The shape of the F-versus-h0 curve is altered, and the decay behavior of the force is distinctly different. Moreover, the assumption of elasticity results in h0 being negative, which further highlights the aphysical nature of the elastic-drop assumption. The F-versus-h0 curve obtained using the nondeformable-drop assumption is also shown as a dashed line to point out the internal inconsistency of these two assumptions. Evidently, as discussed previously, the elastic-drop assumption is both inaccurate and inconsistent with the nondeformable-drop assumption. Comparison between the Rigorous and Approximate Models. In our more rigorous procedure, we start

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Figure 13. Comparison of the original disjoining-pressure isotherm with the one deduced by elastic-drop and nondeformable assumptions on the F-versus-∆ curve of Figure 11.

with the disjoining-pressure isotherm of Figure 4 and calculate the F-versus-∆ curve in Figure 11. However, in AFM experiments the problem is inverted, that is, F-versus-∆ is measured, and the Π(h) curve is to be deduced. To implement this inverse problem, as already mentioned, the elastic-drop assumption is first applied to convert the experimental F-versus-∆ curve to an F-versush0 curve, and then the nondeformable-drop assumption is applied to convert the approximate F-versus-h0 curve to the Π(h) curve (or the equivalent P(h) curve) using Derjaguin’s approximate integration scheme. Thus, we can further check the validity of these assumptions by starting with the F-versus-∆ curve obtained in Figure 11 and working backward with the elasticity/nondeformable assumptions to establish the approximate disjoining-pressure isotherm. The curve corresponding to the elastic-drop assumption in Figure 12 constitutes the first step. Next, we apply the nondeformable assumption to the elastic-drop curve in Figure 12 to obtain the interaction-potential isotherm (cf. eq 6). Finally, we convert the thus-obtained approximate P(h) into the corresponding Π(h) curve. The result for the same system as in Figures 10-12 is shown in Figure 13, where the solid line represents the original disjoining-pressure isotherm and the dashed line is the deduced Π(h) curve using the above-mentioned assumptions and the Fversus-∆ curve of Figure 11. Clearly, the currently used procedure in the literature significantly distorts the behavior of the underlying thinfilm forces. We see from the dashed curve that the decay lengths of each region in the original disjoining-pressure isotherm are substantially altered, as are the locations of the maximum and the minimum. Even negative separation distances arise. Such distortions introduced into the disjoining-pressure isotherm by the currently used interpretation method clearly can yield unusable results. Hysteresis. The shape of the F-versus-∆ curve in Figure 11 is intriguing in the sense that for certain ranges of ∆ - RD there are multiple values of F. Starting from large ∆, each point on the curve represents a smaller h0 than the previous one. This suggests the possibility of hysteresis in the system because of the fact that F is multiple valued. For an actual AFM system, ∆ depends on the cantilever deflection, δc, as well as on the substrate movement, represented by δs. Accordingly, the stiffness of the cantilever is expected to play an important role in any hysteresis. Moreover, the experimentally controlled variable is the substrate movement. Thus, if there is a jump in the force curve, it must occur at a constant substrate position. Although the substrate position can be inde-

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Figure 14. Hysteresis with a flexible cantilever on an F-versus(D - RD - δs) plot. The drop has RD ) 14.4 µm, the solid is flat, and the spring constant, Kc, for the cantilever is 0.08 N/m.

pendently monitored in an AFM apparatus, we need to deduce the substrate position from ∆ in our numerical model. To do this, we realize that the long, thin cantilever is expected to behave as a spring, with a spring constant denoted by Kc. Hence, we express the force as

F ) Kcδc

(11)

This relation, along with eq 9, allows the deconvolution of δs from ∆. To be specific, rearrangement of eq 9 gives

∆ - R D - δc ) D - R D - δs

(12)

and δc is given by eq 11. The expression on the right of eq 12 can be identified with the substrate position, because D and RD are constants. Thus, given the spring constant of the cantilever, we can plot F against the substrate position to identify any possible hysteresis. It must be emphasized that such a deconvolution of δs from ∆ is only needed for the numerical procedure because of the way we implement the calculation. In the AFM apparatus, δs can be monitored independently, and the cantilever spring constant is then used to calculate ∆ from δs. Figure 14 plots F against D - RD - δs for a drop with RD ) 14.4 µm on a flat solid, that is, Rs f ∞. The Kc value needed to arrive at this plot is chosen to be 0.08 N/m, which is in the range of usual cantilever springs for an AFM system.7 At point A, the substrate, and hence the drop, is far away from the cantilever; as the drop is brought closer to the solid by increasing δs, it traverses the path ABC. The force decreases beyond point C, and so does δs, if the force curve in Figure 14 is followed directly. However, because δs is continually increased in the experiment, the only possible alternative is that there is a jump from point C to point D. This jump is shown by a dashed arrow in the figure. Further decreases in D - RD - δs result in the force curve moving along from point D to point E. Thus, the path traversed as the drop is brought toward the cantilever tip is ABCDE. When the drop is retracted from the tip, however, a similar argument results in the path followed being EDFBA. These two different paths form the basis for hysteresis. Experimentally, the F-versus-(D - RD - δs) curve is expected to show such hysteresis. The implication is that the part of the F-versus-(D - RD - δs) curve lying between points C and F cannot be accessed experimentally. FB, FC, FD, and FF are the forces on the system at the points of interest. It must be emphasized here that such hysteresis is obtained only when Π is a nonmonotonic function of h.


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Figure 15. Hysteresis with a flexible cantilever on an F-versus(∆ - RD) plot. The drop has RD ) 14.4 µm, the solid is flat, and the spring constant, Kc, for the cantilever is 0.08 N/m.

On an F-versus-∆ plot, the jumps are not vertical because the abscissa depends on the cantilever deflection as well as upon δs. Because we know the forces at which the jumps occur as well as Kc, we can calculate δc from eq 11 at the relevant points for the jumps. Subsequently, we use eq 12 to calculate ∆ - RD for the relevant jump points to locate the points C, D, F, and B. Hence, we can locate the jumps in an F-versus-(∆ - RD) curve. This is done in Figure 15 for the same system as in Figure 14. Again, when the drop extends toward the solid the path is ABCDE, whereas the path is EDFBA upon drop retraction. Clearly, the jumps, from point C to point D and from point F to point B, are not vertical. An important result for the orientation of the jumps on an F-versus-(∆ - RD) plot can be obtained by considering eq 9 in more detail. Because the jumps take place at a constant δs, eqs 9 and 11 lead to

FD FC - ∆C ) - ∆D Kc Kc

(13)

where FC and FD are the forces at points C and D and ∆C and ∆D are the values of ∆ at those points. We can write a similar relationship for points B and F. Rearrangement of eq 13 gives

Kc )

FC - FD ∆C - ∆D

(14)

Hence, the slope of the lines representing the jumps in the F-versus-(∆ - RD) curve in Figure 15 is the spring constant of the cantilever. For a perfectly rigid cantilever, δc ) 0, and the jump from point C is vertical on an F-∆ plot as well. Similarly, for an infinitely flexible cantilever, the jump in ∆ is horizontal. Comparison to Experiment Here, we compare our modeling effort to the experimental work of Ducker et al.7 and that of Fielden et al.9 Both of these works study air bubbles interacting with silica particles in water with similar geometries. First, we redefine the system variables, as illustrated in Figure 16. The bubble is pinned at the hydrophobic solid surface, as is described in much greater detail in the original works,7,9 and has an undeformed radius RD above the pinning surface. The radius of the solid is again Rs, and rmax is the radius of the pinning surface. The positive direction of the y axis is now defined in the downward

Figure 16. Schematic of the AFM systems used by Ducker et al.7 and by Fielden et al.9

direction, as shown. The reason for the dashed line at the pinning point in Figure 16 is discussed shortly. The major difference between the system described in Figure 16 and that studied in Figure 1 is in the geometry. There are two distinct solid surfaces in Figure 16. In principle, we must account for the interaction of the bubble/ liquid interface with the lower solid in addition to the interaction with the spherical solid specifically studied. However, the interaction with the lower solid is not considered here because the bubble is in contact with the lower solid at the pinning point. Thus, the contact angle is not predefined, it is close to vertical because of the way the bubble is pinned, and the distance between the bubble/ liquid interface and the lower solid increases rapidly. Hence, any interaction of the bubble/drop with the lower solid is expected to be negligible compared to the interaction with the spherical solid particle. Boundary conditions required to solve this problem as an initial-value problem are the same as before; that is, hˆ ) h0 and hˆ r ) 0 at r ) 0. With these initial conditions, the augmented Young-Laplace equation for the axisymmetric geometry, eq 2, is solved as before. When the bubble/ liquid interface reaches the pinning point, integration is stopped. Hence, the condition for stopping the integration occurs when r ) rmax. An important point to bear in mind is that the condition for ceasing the numerical integration can occur at two values of hˆ , as is clear from Figure 16. Because of the initial state of the system, that is, that of no interaction with the solid sphere, we must make sure that the rmax value at which the integration actually stops corresponds to the larger of the two hˆ values. More particularly, we must ensure that the obtained bubble volume is constrained, and this requires iteration on pc for different h0 values. Force Balance. Although there is no symmetry plane for this geometry, the idea behind the force balance remains the same. Figure 17 displays a schematic of the requisite control volume for the new force balance. Instead of cutting the system at the center plane, as is done previously, we cut the system at hˆ ) hˆ c, which is chosen to be large enough that thin-film forces are negligible. At hˆ c, the bubble/liquid interfacial profile makes an angle θ with the r axis. r ) rc on the bubble/liquid interface at the cut. Application of the same argument for the force balance as that previously leads to the result

F ) πr2c pc - 2πrcγ sin θ

(15)

Numerically, it is quite easy to keep track of rc and θ at

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Figure 17. Force balance on the AFM experimental system.

a particular hˆ c. Thus, the force, F, can readily be calculated. It must be kept in mind that rc and hˆ c are used only to calculate the force. F is independent of rc and hˆ c as long as hˆ c lies outside the range of the thin-film forces. To calculate the bubble volume, VF, numerical profile integration must continue until r ) rmax. The criterion for the existence of a particular bubble is that it must satisfy the ideal-gas law. However, as is discussed above, there is practically no difference between a bubble and an incompressible drop. For this reason, we apply the incompressibility constraint, VF ) constant, to identify a particular bubble. This allows us to disregard the fraction of the bubble below the point of pinning (below the lower dashed line in Figure 16), the volume of which remains constant. Thus, given the volume of the bubble, we proceed as previously to calculate the force-distance relationship for the system. A minor point is that RD is not the appropriate variable to use in eq 8 for the system in Figure 16. Rather, RD must be replaced by the initial distance of the apex of the bubble from the pinning surface. This distance, HT, is related to RD by the geometric relation

HT ) RD + (R2D - r2max)1/2

(16)

Thus, for the given system with fixed values of RD and rmax, as is the case here, HT is a constant. Replacement of RD by HT in all the equations appearing in this section makes all the preceding analyses valid for the system in Figure 16 as well. Now, the disjoining-pressure isotherm, or the interaction potential, must be specified for both the systems before we can solve for the force, F. Also, Rs and RD have to be specified. Because these parameters for the systems of Ducker et al. and Fielden et al. are different, we treat each case separately below. In the experimental system of Ducker et al.,7 the spherical solid is silica with Rs ranging from 3 to 5 µm. The undeformed bubbles have radii ranging from 200 to 300 µm. Reported experimental results are for Rs ) 3 µm and RD ) 250 µm, and these parameters are adopted in our calculations below. rmax is taken to be 200 µm, as reported. The solid silica sphere and the air bubble interact through a thin film of water at a pH of 6. The DLVO theoretical interaction force, using Derjaguin’s approximation, is calculated by Ducker et al. for various air/water

Figure 18. (a) Interaction potential of Ducker et al.7 (b) Corresponding disjoining-pressure isotherm. The solid/liquid surface potential is -60 mV, and the bubble/liquid interface potential is -25 mV.

surface potentials. Use of Derjaguin’s approximation for two spheres, assuming that the bubble remains undeformed, yields the relation between the force, F, and the interaction potential, P, as given by eq 6. Although Ducker et al. plot F/Rs as a function of h0, this ratio is very close to 2πP because Rs , RD. Thus, we can determine the interaction potential and the disjoining-pressure isotherms from the calculated DLVO force curve. Ducker et al. calculated the DLVO force using the full retarded and screened van der Waals force and the solution of the full Poisson-Boltzmann equation for the double-layer force, using constant surface potentials on both of the interfaces in question. Here, we fit the parameters of our mathematical interaction potential to obtain exactly the same P(h) curve of Ducker et al. These parameters are given in Appendix A. The resulting interaction potential and disjoiningpressure isotherms are shown in parts a and b of Figure 18, respectively, for the interaction between a hydrophilic silica particle and an air bubble interacting in pH ) 6 water. The curve in Figure 18 corresponds to the F/Rs curves given for an air bubble with surface potential of -25 mV. The surface potential of the silica/water interface is -60 mV. In Figure 19, we show the comparison between the force reported in the AFM experiment (open squares), for a 3 µm silica sphere and a bubble with RD ) 250 µm in water, and the exact force calculation (solid line) outlined above for the interaction potential of Figure 18. F/Rs is plotted against ∆ - HT, where the reason for replacing RD by HT is already explained. The experimentally observed inward jump is shown in the figure with a solid arrow, whereas the dashed arrow represents the theoretical jump predicted by a flexible cantilever, as already discussed, with a spring constant of 0.04 N/m (i.e., determined by Ducker et al. in the AFM experiment). The outward jump was not shown explicitly by Ducker et al. for this system, although they mention it in passing. In Figure 19, we see that the experimental data fit the exact force curve at smaller ∆ - HT values. Thus, the form of the force (i.e., repulsive van der Waals force with retardation and screening) at small separations seems quite accurate. However, at larger separations the attraction given by the experiments is somewhat smaller than the electrostatic attraction ob-


Figure 19. Comparison of experimental F-versus-∆ curve with that numerically calculated using the interaction-potential curve in Figure 18.

Figure 20. ∆ - HT versus h0 for the force curve in Figure 19. The top figure, (a), shows the coordinates of the experimental jump. The bottom figure, (b), depicts the coordinates of the theoretical jump with the measured spring constant, Kc ) 0.04 N/m.

tained from the DLVO calculations. Apparently, the longer range electrostatic interaction force in the interactionpotential curve is not as precise. To arrive at the force curve, the zero of the distance ∆ - HT was chosen by Ducker et al. at a point where a large repulsive force was observed. Here, we shift the experimental data points to align them with the calculated curve. Thus, the point of commencement of the jump is also shifted. However, as previously mentioned, this shift in the ∆ scale is not particularly important. We can proceed further to transform the jump in ∆ - HT coordinates to a jump in the minimum separation distance, h0. In Figure 20a, we show a plot of h0 against ∆ - HT for the calculated force-distance curve in Figure 19. ∆1 and ∆2 represent the end points of the experimental jump, and h01 and h02 are the corresponding minimum separation distances. The resulting jump distance in the h0 coordinate is approximately 34 nm, as opposed to the 44 nm deduced by Ducker et al., using the elastic-bubble assumption. Figure 20b shows a similar plot for the theoretical jump distance, as indicated in Figure 19. Again, the subscripts 1 and 2 stand for the starting and the ending points of the jump, respectively. The magnitude of the theoretical jump distance, calculated by the method developed in this work,

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is 19 nm in the h0 coordinate. Ducker et al. report a theoretical jump distance, based upon the approximate force calculation using the nondeformable assumption, of 11 nm. They calculated this jump distance using the idea that the instability due to a flexible cantilever occurs when the slope of the theoretical F-versus-h0 curve equals the spring constant of the cantilever. Again, there is a considerable mismatch between the theoretical jump distance calculated using the exact analysis developed here and the approximate calculations of Ducker et al. The disagreement between the experimental and the theoretical jump as deduced by Ducker et al. is 33 nm, whereas from our more rigorous calculations, the disagreement is 15 nm. To explain the difference between the experimental and the predicted theoretical jump, Ducker et al. considered the possibility of instability in the system because of cavitation, or formation of bridging bubbles between the spherical solid and the bubble, and estimated the jump distance due to this instability as 15 nm. They thus estimated the maximum theoretically possible jump distance as 26 nm from a sum of the instability due to a flexible cantilever, 11 nm, and that due to cavitation. In our rigorous calculations, the effect of the flexibility of the cantilever results in a jump distance of 19 nm, instead of 11 nm, which suggests that the maximum theoretical jump distance is 34 nm, if we include the estimated jump due to cavitation. Because this value is exactly what we obtained earlier for the experimental jump, it appears that our detailed calculations explain the experimental jump exactly. However, we refrain from drawing such a conclusion, for several reasons. One is that the path followed by the F-versus-∆ curve during the theoretical jump is markedly different from that obtained for the experiment. From the starting and ending points of the experimental jump in Figure 19, we can calculate the expected spring constant of the cantilever from eq 14. This gives the value of Kc as approximately 0.02 N/m. However, the cantilever used by Ducker et al. has a higher spring constant of about 0.04 N/m. Moreover, we have neglected the dynamics of the cantilever that is intuitively expected to enhance the instability due to the flexibility of the cantilever. In spite of the differences in the actual experimental result and the predictions from our theoretical treatment, we conclude that the current approximate analysis overestimates the experimental and underestimates the theoretical jump distances in the h0 coordinate appreciably. The interest in correctly calculating the jump distance is heightened because the possibility of additional long-range attractive forces has been indicated in the literature,7 based partly upon the magnitude of the jump distance. Thus, if the deformation of the bubble is not taken into account, we may arrive at an erroneous conclusion about the origin and the range of thin-film forces. Moreover, our theoretical predictions from the disjoining-pressure isotherm of Figure 21b seem to explain the experimental results in Figure 19 much more closely than proposed by Ducker et al. using the same Π(h) curve and the nondeformable/elastic-drop assumptions. Fielden et al.9 also performed AFM experiments very similar to those done by Ducker et al.7 For the system of Fielden et al., RD, the undeformed bubble radius, is 275 µm. The solid silica sphere has an Rs value of 1.5 µm, and rmax is 175 µm. For the interaction of hydrophilic silica with the air bubble in water at a pH of 6.0, Fielden et al. measured a monotonically repulsive force for all scanner positions. This can be contrasted with the measurement of Ducker et al., who reported that the measured force is first attractive and then repulsive for the same system

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Figure 22. Force-versus-∆ curve calculated from the interaction-potential curve of Figure 21. RD is 275 µm, Rs ) 1.5 µm, and Kc ) 0.05 N/m in the figure. Dashed arrows show the predicted jumps as the cantilever tip is extended toward and retracted from the bubble. Figure 21. DLVO curves of Fielden et al:9 (a) interaction potential and (b) disjoining-pressure isotherm.

chemistry. It is intriguing that the same system gives completely different force behavior in the two different experiments. In another experiment on the same geometric system, Fielden et al. reported theoretical DLVO interactionpotential curves for an air bubble with a surface potential of -34 mV at infinity and a silica particle of -100 mV surface potential. The surface potential of the bubble is held constant, whereas the charge on the silica is held constant. A univalent electrolyte with a concentration of 10-4 M is also present in the surrounding aqueous solution. Because of the presence of the electrolyte, the solution chemistry is not quite the same as the one in which Fielden et al. reported the marked disagreement with Ducker et al., as mentioned above. The resulting interaction potential of Fielden et al. for the system with 10-4 M of univalent electrolyte is shown in Figure 21a. Figure 21b illustrates the disjoining-pressure isotherm. The specific parameters used to generate these curves are again given in Appendix A. For their experiments, Fielden et al. measured the value of γ as 72.8 mN/m. Kc is reported to be 0.05 N/m. With this and the preceding information, the expected forcedistance isotherms can be calculated. The value of the cantilever spring constant allows the prediction of jump distances on the F-versus-∆ curves. In Figure 22, a plot is shown of F as a function of ∆ - HT calculated using the disjoining-pressure isotherm given in Figure 21. Dashed arrows show the predicted jumps when the cantilever approaches the bubble or is retracted from it. It is clear from Figure 22 that there must be a distinct attractive jump when the cantilever is extended toward the solid if the interaction-potential curve in Figure 21 does indeed represent the thin-film forces in the system accurately. Because only a monotonically repulsive AFM force was observed, Fielden et al. mention the possibility that such a small attractive jump cannot be captured in the force window of the AFM. However, as is clear from Figure 22, the magnitude of the attractive jump is approximately 0.3 nN, which can be detected by the AFM apparatus. Thus, we conclude that the interaction potential given in Figure 21 does not represent the interaction in the system correctly. Fielden et al. report their experimental data in terms of the cantilever deflection measured in volts as a function

of the scanner position and then resolve these data into an F-versus-h0 curve using the argument that the bubble behaves as an elastic solid. To convert the cantilever deflection from volts to a distance unit (and then to a force, using F ) Kcδc), they argue that a change in the scanner position translates exactly into a change in cantilever deflection in the constant-compliance region. However, their argument is accurate only if the bubble is nondeformable. The precise relationship between δc and δs is given by eq 8, where it becomes clear that the presence of δb implies that a change in δs does not translate exactly into a change in δc. Because of finite bubble deformation, the force deduced by Fielden et al. from the plot of cantilever deflection in volts versus the scanner position is not accurate. This uncertainty in the “measured” force at a given value of the scanner position makes it impossible to compare their experimental data with our theoretical predictions. In view of the above discussion, we cannot compare the experimental data with the theory quantitatively. However, the qualitative comparison, where the applicability of the interaction-potential curve of Figure 21 to the present system is questioned, remains valid. Conclusions In this work, we discuss in detail the approach best used to calculate the thin-film force-distance relationship for a deformable object when it is brought near nondeformable objects. The procedure involves first solving the augmented Young-Laplace equation pertinent to the system geometry for the bubble/drop profile, using the given disjoining-pressure isotherm. Subsequently, we solve for the macroscopic force as a function of the minimum separation distance while satisfying the volume constraint for a bubble or a drop. We find that a drop, with a constant-volume constraint, and a bubble, obeying an ideal equation of state, follow essentially the same forcedistance curve. This is because the change in pressure of the deformed drop/bubble with respect to the atmospheric pressure is small as the drop/bubble approaches the solids. This observation allows us to use either a bubble or a drop interchangeably. This is true only when undulations in the disjoining-pressure isotherm are small compared to atmospheric pressure (or the base liquid pressure, to be exact), as is the case in our systems. The sizes of the drop and the solid (if curved) determine the exact force-distance isotherm followed by the drop. As the drop size or the solid size decreases, there is less


deformation of the drop/liquid interface, resulting in less force being observed at a given minimum separation distance. Because of the decrease in deformation, the nondeformable-drop assumption provides a progressively better result. However, such an assumption cannot be made a priori, because the deformation is appreciable over a wide range of drop sizes. The assumption of an elastic drop to calculate the minimum separation distance is not reasonable. This assumption has no physical basis and is inapplicable from the practical point of view as well, because the bubble deformation is not linearly related to the force. Although it is not practical to estimate h0 experimentally at this point, we still can compare experiments with theory. This is done by realizing that ∆ can be both measured experimentally and calculated numerically. Thus, a graph of F versus ∆ (or ∆ - HT) provides a convenient way to compare experiment and theory. Any method to calculate the bubble deformation elastically results in an error that changes the shape of the force curve. This change in shape, as already discussed, leads to erroneous conclusions about the interparticle force law. Unless a reliable way to measure h0 directly is developed, the alternate method of plotting the AFM force curve (using ∆ instead of h0) should be employed. Otherwise, as is clear from Figure 13, use of elastic/nondeformable-drop assumptions leads to a completely different disjoining-pressure isotherm than the correct one. Hysteresis is also predicted in the F-versus-∆ curve. The reasons behind hysteresis are the nonmonotonic shape of the disjoining-pressure isotherm and the drop deformation, which changes the shape of the force-distance curve. The onset of hysteresis can be estimated exactly by considering an F-versus-δs plot which can be extrapolated to the F-versus-∆ curve. The slope of a straight line depicting the hysteresis in the F-versus-∆ curve corresponds to the spring constant of the cantilever. Comparisons are made between the experimental data and the exact equilibrium force-distance relationships. Such a comparison for the data of Ducker et al.7 shows that if the deformation of the bubble is not accounted for, the experimental jump is overestimated and the theoretical jump is underestimated. Because these jump distances form the basis for assessing the range of the thin-film forces, it is crucial to estimate these accurately. Moreover, our more rigorous numerical procedure shows a much better agreement between the calculated DLVO thin-film force by Ducker et al. and their experimental AFM result than that obtained using the nondeformable/elastic-drop assumptions. Overall, we conclude that the F-versus-∆ curve is the appropriate way to compare experiments to theoretical predictions and that the nondeformable/elastic-drop assumptions are grossly inaccurate. Acknowledgment. This work was partially funded by the Assistant Secretary for Fossil Energy, Office of Oil, Gas, and Shale Technologies of the U.S. Department of Energy under Contract DE-AC03-76SF00098 to the Lawrence Berkeley National Laboratory of the University of California. Appendix A: Interaction Potential Before proceeding with the numerical solution of the augmented Young-Laplace equation, we need to specify the interaction potential or, equivalently, the disjoining pressure. This is constructed mathematically, using inverse power or exponential dependencies on the separa-

Langmuir, Vol. 17, No. 1, 2001 129 Table 1. Constants Used to Generate the P(h) Curve constant C1 C2 C3 C4 C5 C6 C7 d0

Figure 4 10-20

-4.0 × -1.8 × 10-4 3.3 × 10-8 3.0 × 10-68 8 0 1.0 1.0 × 10-8

Figure 18

Figure 21

0 -5.9708 × 10-4 2.0 × 10-9 4.7564 × 10-82 9 -1.0956 × 10-4 -1.55 × 10-10 -1.58 × 10-9

units

10-20

-4.0 × 0 1.0 3.0 × 10-68 8 0 1.0 1.0 × 10-8

J J/m2 m Jm(C5-2) J/m2 m m

tion distance, h, between the interfaces. The specific form we use is

P(h) )

C1 2

(h + d0)

( )

+ C2 exp -

h + C3

C4 (h + d0)

C5

( )

+ C6 exp -

h (A1) C7

where P(h) is the interaction potential and d0 is chosen so that P(h) is finite when h is zero. The values of the constants C1-C5 are chosen according to the particular interaction potential required. The first term in eq A1 can be identified with the London-van der Waals dispersion contribution to the interaction potential, and thus C1 is related to the Hamaker constant. The second term can be treated as equivalent to double layer forces, where C3 is an equivalent debye length. The third term represents a short-range repulsion; hence, the sign of C4 is chosen to be positive. The final term is similar to the second and is included for added flexibility to construct an interactionpotential curve. However, the correspondence of the terms in eq A1 to the actual interaction potentials of a system is arbitrary in the sense that eq A1 is a mathematical construct. This particular form is chosen primarily for its flexibility. Table 1 gives the values of the various constants. The resulting interaction potentials are shown in the text in Figures 4, 18, and 21. Appendix B: Numerical Procedure Upon rewriting eq 2 in terms of the arc length, s, and the angle of the bubble/drop profile with the r axis, θ, we obtain a system of three first-order ordinary differential equations

dr ) cos θ ds dhˆ ) sin θ ds 1 dθ 1 ) (p -Π(h)) - sin θ ds γ c γ

(B1) (B2) (B3)

where Π is evaluated as a function of h instead of hˆ . h and hˆ are related by the geometric relation

h ) {(hˆ + Rs)2 + r2}1/2 - Rs

(B4)

Equation B3 contains a singularity at r ) 0, and hence we need an asymptotic expression near r ) 0, or25

dθ 1 ≈ (p - Π(h)) near r ) 0 ds γ c

(B5)

(25) Yeh, E. K. M.S. Thesis, University of California, Berkeley, CA, 2000; Chapter 2.

130


This equation implies that both the curvature terms in eq 2 are approximately equal. Appendix C: List of Symbols Letters C1-C7 D d0 F HT h hˆ h0 hf K L P p pc r Rs RD

Parameters in the interaction-potential isotherm (see eq A1) Initial distance between the substrate and the cantilever (m) Parameter in the interaction-potential isotherm (m) Macroscopic force (N) Initial height of a pinned bubble above the substrate (m) Film thickness (m) Distance of the bubble/drop profile from the y ) 0 plane (m) Minimum liquid film thickness (m) Uniform film thickness (m) Effective spring constant (N/m) Radial dimension of the control volume (m) Interaction potential (N/m) Pressure (Pa) Capillary pressure (Pa) Radial axis of the cylindrical coordinate system (m) Radius of the solid (m) Undeformed radius of the bubble/drop (m)

Bhatt et al. rmax s V y

Maximum extent of the bubble/drop in the r direction (m) Arc length along the bubble/drop profile (m) Volume (m3) Axial coordinate of the cylindrical coordinate system (m)

Greek Letters γ ∆ δ θ Π

(Bubble/drop)/liquid interfacial tension (N/m) Distance between the center plane and the cantilever (m) Deformation/deflection (m) Angle between the tangent to the bubble/drop profile and the r axis Disjoining pressure (Pa)

Subscripts 1 2 b c s F L min max

Denotes a point value of ∆ Denotes a point value of ∆ Bubble/drop Cantilever Substrate Bubble/drop phase Liquid phase Minimum value Maximum value LA0009691

Drop Interacting

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