Using Stable and Unstable Profiles To Deduce Deformation Limits of

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Langmuir 2004, 20, 3208-3220

Using Stable and Unstable Profiles To Deduce Deformation Limits of the Air-Water Interface F. P.-A. Cortat and S. J. Miklavcic* Department of Science and Technology, Linko¨ ping University, Campus Norrko¨ ping, S-601 74, Norrko¨ ping, Sweden Received September 5, 2003. In Final Form: January 7, 2004 The shape of the air-water interface deformed by a van der Waals stress induced by a paraboloid shaped solid body is addressed and discussed. Emphasis is placed on the existence limit of solutions to the governing Euler-Lagrange equation for the equilibrium shape. Two legitimate solutions, one stable and one unstable, are found to converge at the existence limit, giving a numerical criterion for establishing critical physical conditions guaranteeing absolute stability. Insight is aided by a study of an analogous mechanical problem that exhibits very similar properties. Among numerical data produced are critical lower height limits of the paraboloid to the air-water surface and associated peak deformation heights and their dependencies on physical parameters. Of further interest to experimentalists in the surface force field are the variations in peak deformation height and total surface force on the solid as a function of position of the paraboloid, paraboloid geometry, and strength of the van der Waals stress.

I. Introduction A number of different experimental techniques directly involve the immersion or near immersion of a solid through a fluid interface. One principal application is the fundamental study of interactions between fluid entities and solid bodies. The atomic force microscope (AFM), for example, is now being employed to quantify the noncontact interaction between a colloidal probe particle and a finite volume fluid drop. In particular, investigations have been conducted with the fluid drop being a gas bubble hanging from a horizontal substrate in a bulk aqueous liquid.1-3 In other cases, the fluid drop was an immiscible oil-based liquid similarly pinned to a substrate and immersed in an aqueous environment.4-6 Another device, a modification of the surface forces apparatus (SFA),7,8 has been used to measure the interaction between a flat surface (mica) and a purified mercury drop across water (under natural as well as applied potential conditions).9,10 Connor and Horn11 found that the limiting separation between the mica and mercury surfaces could be controlled by an applied potential. This allowed them to experimentally reproduce the electrical double layer potential profile. In a more recent study, the mercury drop was replaced with an air bubble and the resulting interaction was measured in the same manner.12 These systems have a number of common features. When the solid and fluid are close to contact, measured on a scale much less than the dimensions of the bodies (1) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (2) Butt, H.-J. J. Colloid Interface Sci. 1994, 166, 109. (3) Fielden, M. L.; Hayes, R. A.; Ralston, J. Langmuir 1996, 12, 3721. (4) Basu, S.; Sharma, M. M. J. Colloid Interface Sci. 1996, 181, 443. (5) Hartley, P. G.; Grieser, F.; Mulvaney, P.; Stevens, G. W. Langmuir 1999, 15, 7282. (6) Aston, D. E.; Berg, J. C. J. Colloid Interface Sci. 2001, 235, 162. (7) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (8) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (9) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483. (10) Connor, J. N.; Horn, R. G.; Miklavcic, S. J. Uzb. J. Phys. 1999, 1, 99. (11) Connor, J. N.; Horn, R. G. Langmuir 2001, 17, 7194. (12) Connor, J. N. Measurement of Interactions Between Solid and Fluid Surfaces. Doctoral Thesis, University of South Australia, Adelaide, South Australia, Australia, 2001, ISBN 0-868039152.

concerned, an effectively short-ranged equilibrium surface force arises between their two surfaces. This surface force or stress results in the free surface becoming deformed. On this point, use of the modified SFA is particularly advantageous since it possesses the ability to actually monitor the shape of the fluid interface under interaction conditions. One can thereby follow the development of fluid interface deformation as a function of the distancedependent interaction, whether it be an equilibrium interaction or a dynamic one.9,10,12 Especially significant is the fact that one can follow the transition from stable interface deformation to unstable deformation. With the AFM, on the other hand, one is necessarily working blind, which can lead to serious interpretation problems. To illustrate, AFM measurements of thicknesses of condensed liquid films on solid substrates were performed by Mate et al.13 These were based on determining the difference in vertical positions of the AFM tip at two points, one corresponding to the onset of a strong adhesion (formation of three-phase contact) and one corresponding to a strong hard wall repulsion (tip-substrate contact). The measurements, however, were not corroborated by ellipsometric measurements, the latter being consistently smaller. The discrepancy was subsequently attributed to the neglect of the gradual deformation of the film in the AFM case, due to surface forces.14 Theory too has made significant advances, some of which have been subsequently confirmed by experiment. Fluid surface deformation by colloidal surface forces has been investigated in related contexts by a number of theoretical research groups including ourselves and our colleagues.14-28 While details may depend somewhat on (13) Mate, C. M.; Lorenz, M. R.; Novotny, V. J. J. Chem. Phys. 1989, 90, 7550. (14) Forcada, M. L.; Jakas, M. M.; Gras-Marti, A. J. Chem. Phys. 1991, 95, 706. (15) Forcada, M. L.; Arista, N. R.; Gras-Marti, A.; Urbassek, H. M.; Garcia-Molina, R. Phys. Rev. B 1991, 44, 8226. (16) Forcada, M. L. J. Chem. Phys. 1993, 98, 638. (17) Denkov, N. D.; Petsev, N. D.; Danov, K. D. Phys. Rev. Lett. 1993, 71, 3326. (18) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357-16364. (19) Bachmann, D. J.; Miklavcic, S. J. Langmuir 1996, 12, 41974209. (20) Miklavcic, S. J. Phys. Rev. E 1996, 54, 6551. (21) Miklavcic, S. J. Phys. Rev. E 1998, 57, 561.

10.1021/la035651y CCC: $27.50 © 2004 American Chemical Society Published on Web 03/13/2004

Deformation Limits of the Air-Water Interface

geometry, the findings possess a general relevance that has proved useful in the development of new experimental approaches as well as for the analysis of experiments. For example, a repulsion between a macroscopic (flat) solid and a finite drop was shown to give rise to a stable limiting separation due to liquid interface flattening.18 A similar limiting event appears in the case of a drop interacting with an object of finite curvature, with the modification that the drop surface curvature inverted to mimic the curvature of the solid (modulo a correction from the separation distance).6,24,25,28 More recently, we have addressed the issue of the actual physical manifestation of predicted equilibrium solutions.29-31 Under extreme conditions, the surface forces that cause the fluid surface to deform may also be responsible for its loss of stability. However, even without such explicit stability analyses, it should be apparent that the existence of a solution to the profile equation (the Euler-Lagrange equation) will not always be guaranteed. As with all differential equations, certain conditions must be met for a solution to exist (and possibly others besides for it to be unique). These give rise to mathematical stability criteria; if a mathematical solution does not exist, then neither will a physical solution. Thus, circumstances can arise in which the system parameters are no longer consistent with the existence conditions. This fact was known to Forcada et al.14 and again by Forcada et al.15 in their study of a point particle interacting with a dielectric liquid half space. We continue and extend the discussion in this paper in which an unambiguous criterion of existence is employed. Much theoretical effort has been predisposed to the system typically found in surface force measurements: finite volume fluid drops interacting either with a colloidal particle1,2 or with a macroscopic surface.9 The problem we address in this paper is similar in philosophy yet is representative of a different series of problems involving deformable fluid surfaces. The significant difference is that the undeformed fluid surface is planar and of infinite extent. It interacts with a solid body of convex curvature via van der Waals surface forces. The motivation for this study is to investigate the behavior of systems such as that of the Wilhelmy plate experiment32,33 or any system involving the prodding of a body toward and into an infinite body of liquid having a flat interface. Examples can be as commonplace as a finger dabbing into a pool of water. We present and discuss results of a numerical study of the shape and stability dependence of a planar fluid surface on the strength of the van der Waals force and proximity to the solid body. The solid is modeled as a rigid paraboloid, while the free surface profile is determined self-consistently by numerically solving an Euler-Lagrange equa(22) Bhatt, D.; Newman, J.; Radke, C. J. Langmuir 2001, 17, 116. (23) Miklavcic, S. J.; Attard, P. J. Phys. A: Math Gen. 2001, 34, 7849-7866. (24) Attard, P.; Miklavcic, S. J. Langmuir 2001, 17, 8217-8223. (25) Chan, D. Y. C.; Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2001, 236, 141-154. (26) Attard, P.; Miklavcic, S. J. J. Colloid Interface Sci. 2002, 247, 255. (27) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2002, 247, 310. (28) Bardos, D. C. Surf. Sci. 2002, 517, 157. (29) Miklavcic, S. J.; Attard, P. J. Phys. A: Math Gen. 2002, 35, 4335-4347. (30) Miklavcic, S. J. EMAC 2003 Proceedings; May, R. L., Blyth, W. F., Eds.; University of Technology: Sydney, 2003; p 153. (31) Miklavcic, S. J. J. Phys. A: Math. Gen. 2003, 36, 8829. (32) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Wiley Science: New York, 1966. (33) MacRitchie, F. Chemistry at Interfaces; Academic Press: New York, 1990.

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Figure 1. A schematic of the system in cross-section. The distance vector D(r) establishes a one-to-one mapping between points on the profile {r, z(r)} (lowest solid curve) and those on the surface of the paraboloid {rp(r), zp(r)} (upper solid curve). The closest distance is always at the apex. The dotted-dashed line corresponds to the far-field approximation, eq 5. The solid horizontal line z ) 0 represents the undisturbed fluid interface.

tion. A special focus of this work is on the limits of solvability of this profile equation. The structure of the paper is as follows. In the next section, we introduce the physical and mathematical problem. Moreover, we present a simple but enlightening mechanical model which is analogous to our system and which exhibits similar qualitative behavior. In section III, we describe the method of solution. In particular, we outline the numerical strategy for determining the solvability limits of the governing equation as a function of system parameters. In section IV, we present and discuss our numerical results demonstrating typical fluid interface profiles and their dependence on input parameters. We devote some space to describing the critical (minimum) heights to which the solid body can be lowered toward the fluid surface and their dependence on interaction strength and solid geometry. II. Equation for the Equilibrium Profile A. Model System and Equilibrium Equation. In a system involving an attractive van der Waals interaction between a fluid interface and a solid, three effects compete: the van der Waals interaction tends to raise the water interface, creating a bulge centered under the base of the solid (Figure 1); the surface tension, denoted γ, and the gravitational force (weight of water displaced) act to oppose the deformation of the interface. The energy of the deformed air-water interface is thus given by a sum of these three contributions written as integrals over, or otherwise involving, the deformed surface. Since the water molecules in the bulk liquid reservoir are not in chemical potential equilibrium with those in the gas phase (not a saturated vapor), the problem thus formulated is one of mechanical rather than thermodynamical equilibrium. Similarly, the instability we shall shortly refer to is a mechanical instability. For our purposes, it suffices to take a simple model of an object possessing axial symmetry. Specifically, we feature an infinite cylindrical paraboloid whose vertical cross-section is defined by the relation

zp ) zp0 + λrp2 where λ g 0 is termed the splay constant or parabolicity coefficient and describes the broadness or bluntness of the solid (see also ref 34). If the solid is also homogeneous in its material properties, then the entire system will have cylindrical symmetry. Consequently, in this geometry both

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Cortat et al.

surfaces can be specified in terms of functions of the radial coordinate, r. Points on the liquid surface are described by a set of coordinate pairs {(r, z(r))}, under the proviso that the mapping r f z(r) is injective. The solid surface can be expressed by the set of function pairs {(rp(r), zp(r))} determined uniquely by geometry. In cylindrical coordinates, the interfacial energy of deformation can be written as a radial integral,

F ) 2π

∫0

∞

G r (γ + σ)W(zr) + z2 dr 2

[

]

(1)

where we have used the notation G ) g∆F, where ∆F is the density difference across the interface, g is the gravitational acceleration, and W(zr) ) (1 + zr2(r))1/2 is a surface area scaling factor associated with the deformation. The first term of eq 1 corresponds to the bare surface energy due to deformation. The last term is the change in gravitational potential energy of the water lifted above the reference level, z ) 0. The second term is a surface energy resulting from the interaction of the air-water interface with the solid paraboloid. An approximate closed form expression for the attractive van der Waals energy exists in the case of two infinite parallel planar surfaces separated by a distance, D,8

σ)-

A 2D2

(2)

where A is the Hamaker constant specific to the three given media; A is a measure of the strength of interaction between the two surfaces. We adopt the further approximation that eq 2 holds locally, that is, σ ) σ(r, z), by replacing D with the locally varying separation, D(r; z(r)), between the fluid surface point, (r, z(r)), and its unique partner point on the solid paraboloid, (rp(r), zp(r)) (Figure 1):

D(r; z(r)) reflects the geometry of the system and is both an explicit and implicit function of r. F is a real-valued functional of the possible profiles z ) z(r): F ) F[z], that is, a mapping from a given function space to the real numbers. Neglecting dynamic effects, static equilibrium is achieved when for a specific interface profile the energy functional is a minimum and the three competing effects of surface tension, imposed stress, and gravity are in balance. By the fundamental theorem of variational calculus,35 this specific equilibrium profile, for a given paraboloid geometry and a given minimum height, zp0, is determined by the Euler-Lagrange differential equation for the functional 1. Assuming that this equation has at least one solution (and the functional at least one extremum), denoted by z(r), the Euler-Lagrange equation takes the form23

[x

r zr(r)

1 + zr(r)

2

]

(γ + σ(r, z(r))) )

[

r G z(r) + x1 + zr(r)2

d dr

[x

]

∂σ(r, z(r)) (3) ∂z

No analytical solution to this nonlinear equation is yet known. Thus, a numerical treatment is unavoidable. Note (34) Cortat, F. P.-A.; Miklavcic, S. J. Phys. Rev. E 2003, 68, 052601(1-4). (35) Courant, R.; Hilbert, D. Methods of Mathematical Physics; Wiley: New York, 1965; Vol 1.

]

r zr(r)

)

1 + zr(r)2

(

)

σz - σrzr(r) r G z(r) + γ+σ x1 + zr(r)2

(4)

where on the right-hand side there appears a scalar product between the gradient derivative of σ and (-zr, 1)T/W, the unit outward normal vector to the liquid surface at the point (r, z(r)). The last term in eq 4 is then interpreted as the directional derivative of σ normal to the surface at (r, z(r)). This shows that the dominant contribution from the interaction to the deformation process comes from the field component normal to the equilibrium profile.23 γ + σ(r, z) and its derivatives involve the distance function, D(r; z(r)), and derivatives thereof. The distance function itself is fully determined once the function rp(r), relating a point (r, z(r)) on the profile to the closest point (rp, zp) on the surface of the solid, is specified. Using vector geometry, it can be shown that

rp(r) )

]

[

r + x‚ 4λ2

1/3

1 + 2λ(zp0 - z(r)) 2

6λ

[

]

r + x‚ 4λ2

-1/3

with

x‚ ≡

D(r; z(r)) ) x(r - rp(r))2 + (z(r) - zp(r))2

d dr

that eq 3 is similar to the differential equation governing the deformation of a sessile drop,23 except for the absence of a constant Lagrange multiplier term that would be associated with a constant volume constraint. By performing the derivative of the product on the lefthand side and collecting derivatives of σ, eq 3 can be rewritten as

x(

) ( )

1 + 2λ(zp0 - z(r)) 6λ2

3

+ -

r 4λ2

2

B. Far-Field Approximation. For points far from the paraboloid, that is, points (r, z(r)) with r very large, the interaction term, σ(r, z), and all its derivatives are negligible. Furthermore, far from the influence of the solid, the profile and its slope deviate only marginally from their neutral values, that is, |z|, |zr| , 1. Invoking these approximate properties in eq 3 results in a simplified differential equation for the profile, valid asymptotically. To first order in z and its derivatives, eq 3 becomes

r2 zrr(r) + r zr(r) -

G 2 r z(r) ) 0 γ

for large r. This is a modified Bessel equation whose general solution is given by

z(r) ) C1 I0(η r) + C2 K0(η r) where η2 :) G/γ and I0 and K0 are the zeroth-order modified Bessel functions of the first and second kinds, respectively. The far-field asymptotic condition

lim z(r) ) 0 rf∞

forces C1 ) 0. Therefore, the equilibrium profile far from the object is prescribed by the radial function

z(r) ) C K0(η r)

C>0

(5)

up to the unknown constant C. As (d/dx)(K0(x)) ) -K1(x)


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is strictly negative for all x > 0 and since zr must be a strictly negative quantity for an attractive van der Waals stress, C must be a positive constant. C. Total Interaction. The van der Waals force exerted by the paraboloid on the profile can be determined by integration of the stress over the deformed profile. Since the radial component of this force is zero by axial symmetry and since most practical interest lies with the vertical (z-)component of the stress integral anyway, we shall focus attention only on the z-component of the force expressed by the integral

Fvert )

∫02π dθ ∫0∞ W(zr(r)) 3σ‚kB r dr

) 2π

∫0∞ W(zr(r)) σz(z(r)) r dr

= 2π

∫0r

∞

(6)

W(zr(r)) σz(z(r)) r dr

Figure 2. Schematic of the mechanical analogy.

where B k ) (0, 0, 1) and in the last expression we have introduced an effective radius, r∞, beyond which σz,

∂D(r; z) A σz(r) ) 3 ∂z D(r; z) is negligible by assumption (section B). Equivalently, the van der Waals force on the profile is equal and opposite to the force exerted on the paraboloid cylinder (the actual quantity measured in experiments). Thus, the vertical component of the force on the profile can equally well be expressed as an integral over the surface of the paraboloid,

Fvert ) 2π = 2π

∫0∞ W(rp) σz(rp) rp drp ∫0r

p,∞

(7)

W(rp) σz(rp) rp drp

If r∞ is chosen properly, the choice of the upper limit, rp,∞, in this last integral is not crucial in order to get exact numerical correspondence with eq 6c, provided rp,∞ g rp(r∞), the latter determined self-consistently. Neither eq 6 nor eq 7 can be explicitly written in closed form since the profile, z(r), itself can only be given as numerical data. In section IV, the results of numerically evaluating these integrals are presented under various parameter conditions. To better exhibit the effect of deformation on the force and where it sets in, we compare our numerical calculations with the approximation to eq 7 obtained by setting z ) zr ≡ 0. This corresponds to the usual situation of two interacting solids commonly encountered in an atomic force microscope experiment. D. A Mechanical Analogy. The physical system just introduced can be likened to a much simpler mechanical problem. Hence our results can be better understood if presented with reference to the expectations of this simpler mechanical system. Consider two point objects, for convenience referred to as a sphere and a box, interacting via an attractive electrostatic force, with strength Q > 0. The sphere is fixed in space, while the box, positioned beneath the sphere, is relatively free to move unilaterally in the vertical direction. The box is connected to one end of a spring of strength k, whose other end is attached to a fixed point. The forces that act on the box are thus a restoring spring force proportional to the box displacement from neutral equilibrium; a gravitational force proportional to the box’s mass, m, directed down; and finally the attractive electrostatic force due to the sphere.

Clearly the forces that arise in the actual system of interest possess a direct association with the forces that act in this model. The force of surface tension, measured by γ, corresponds to the spring force of strength k; the van der Waals force with Hamaker constant, A, corresponds to the electrostatic force of strength Q; the gravitational force is obviously associated with the mass of fluid raised above neutral equilibrium and corresponds to the box mass, m. Note, however, that in the mechanical analogy the box mass is constant (irrespective of its position) while in the real physical system the mass of water being displaced increases as the van der Waals stress increases. Since the systematic mass dependence is unknown and an exact analysis is possible assuming a constant mass, we restrict ourselves to the latter case in the analogy. Both the box displacement, z, and the sphere position, zs, are measured with respect to an origin centered at the box end of the unstretched spring (Figure 2). At equilibrium, we require z < zs. Equilibrium itself is established when the total energy is a minimum or equivalently when the forces acting on the box are in balance:

-kz - mg +

Q )0 (zs - z)2

This is a third-order algebraic equation for the equilibrium position of the box, z. Defining K :) Q/k and M :) mg/k, this equation can be rewritten as

z3 + (M - 2zs)z2 + zs(zs - 2M)z + Mzs2 - K ) 0 (8) There are three solutions to this equation which are all real provided the discriminant

D ) -K

[(

) ]

zs + M 3

3

-

K ): -K[X3 - Ω3] 4

is negative, which it will be provided zs > 3Ω - M. The (real) solutions are

{

z1 ) 2X(1 + cos φ) - M z2.3 ) X(2 - cos φ ( x3 sin φ) - M

(9)

where

φ :)

(

2Ω3 1 cos-1 3 - 1 3 X

)

A simple examination of these solutions shows that in the

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Cortat et al.

Figure 3. The three real solutions of eq 8 as a function of the sphere height, zs. The two dotted lines emphasize the asymptotic behavior of the solutions. The following arbitrary parameter values were used: Q ) 150, m ) 0.25, g ) 9.81, and k ) 0.2.

limit zs f ∞, that is, in the absence of the electrostatic interaction, φ f π/3 and the solutions take the following limiting values: z1(zs f ∞) ) z2(zs f ∞) ) zs, while z3(zs f ∞) ) -M. Since the third result is the only physically realistic one, z3 is the solution of physical relevance at finite sphere heights. Nevertheless, all three solutions are plotted in Figure 3 for comparison as a function of sphere height. Solution z1(zs) > zs places the box above the sphere (upper curve in Figure 3) which is unphysical. Of the remaining two solutions, z2(zs) < zs corresponds to a local maximum of the energy while z3(zs) < zs corresponds to a local minimum. Thus, although both z2 and z3 fulfill the physical requirement that they place the box below the sphere, z2 is an unstable position while z3 is a stable position. Note from Figure 3 that z2 decreases with zs while z3 increases with zs, the latter exhibiting the physically expected behavior. As zs decreases, the two physical solutions converge monotonically to form a double real solution. This occurs when the discriminant, D, vanishes, that is, when

zs,min ) 3Ω - M

(10)

At this critical height, Xmin ) Ω w φ ) 0 and the unphysical solution becomes

z1(zs,min) ) 4Ω - M ≡

4zs,min + M 3

while the stable and unstable ones become

z2(zs,min) ) z3(zs,min) ) Ω - M ≡

zs,min - 2M (11) 3

This is a particularly significant fact given the numerical results to come. The ratio of zs,min to the above unique solution is

{

3Ω - M g3 for 0 e M < Ω R) M>Ω Ω - M e1 For a negligible mass, R ) 3. As the mass increases, so too does R diverging for M ) (K/4)1/3. From this relation and eq 11, we infer that for negligible mass there is a simple factor of 3 between the height of the sphere and the equilibrium position of the box. Even with a finite box mass, eq 11 indicates that there is still a linear relation

Figure 4. The stable (full lines) and unstable (dashed lines) solutions of eq 8 as a function of sphere height, zs, for different masses ranging between 0 and 0.33. The solutions are rescaled with respect to z3(∞) ) -M. The parameters are as in Figure 3.

between the two with a proportionality factor of 3 independent of any system parameters but with an offset proportional to the amount of mass. In Figure 4, we plot as a function of zs the stable (and unstable) solution relative to the solution value at infinite sphere height, that is, z3(zs) - z3(∞) ) z3(zs) + M, for various increasing masses. Note first that

z3(zs,min) - z3(∞) )

(K4 )

1/3

≡Ω

a constant (i.e., the apex of the curves lie on a horizontal line). Second, note that as the mass of the box increases the stable solution takes on a more curved dependence on z s. Finally, we point out, from both Figure 3 and Figure 4, the most important fact: for sphere positions lower than zs,min (the point where the stable and unstable solutions are in coincidence), there exists no physically possible solution. This point thus earmarks an absolute (mathematical) stability limit to the system. From eq 10 (and Figure 4), this stability limit can be seen to decrease as the mass of the box increases. III. Numerical Method For a number of computational reasons, it is useful to rewrite eq 3 in the form zrr ) f(r, z, zr). However, some care is required because of the possibility of division by zero at the regular singular point r ) 0. To obtain the r ) 0 value, one must employ a limit process. Following Miklavcic and Attard,23 we find

{

zrr ) Gz0 + σz(0, z0) 2(γ + σ(0, z0)) 2

1 + zr

γ + σ(r, z)

[

Gz W(zr) - zr σr(r, z) + σz(r, z) -

r)0

]

zr(γ + σ(r, z)) r

r>0

(12)

In theory, the function γ + σ(r, z) can also introduce a singularity, which is likely to be nonintegrable (although this has not been verified). However, this complication can be avoided by working under conditions such that this function remains positive definite. This in turn restricts the closest approach distance between the solid and the fluid interface to be Dmin > (A/2γ)1/2. In our calculations, this quantity proved to be at least 3 orders


Langmuir, Vol. 20, No. 8, 2004 3213

of magnitude smaller than the minimum separation actually attained under any set of conditions. Iterative Procedure. For practical purposes, numerical integration of eq 12 extends up to a finite radius, r∞, large enough so that the far-field approximation 5 holds there. Rotational symmetry demands zr(0) ) 0, while the asymptotic condition, z f 0 as r f ∞, must be replaced by the finite condition + z(r∞ ) ) z(r∞ ) ≡ CK0(η r∞)

(13a)

to be satisfied by the numerical (-) and asymptotic (+) solutions. Equation 13a is complemented by + zr(r∞ ) ) zr(r∞ ) ≡ -Cη K1(η r∞)

(13b)

equating numerical (-) and asymptotic (+) derivatives of the profile at r ) r∞. Two unknowns, the constant C and the apex height, z(0) ) z0, are determined by solving the two nonlinear algebraic eqs 13. Consequently, more than one real solution pair (C, z0) is possible. The final algorithm employs a so-called shooting method, consisting of the following steps: (1) Provide a set {zi0} of guesses for the unknown z(0) (can involve only one guess). (2) For each zi0, determine the profile {r, z(r)}i by integrating eq 12 using a Runge-Kutta technique. (3) For each i, determine the integration constant Ci, using eq 13b. (4) Determine the asymptotic value, z(r+ ∞ ), using eq 5. ) z(r(5) Form the difference δi :) z(r+ ∞ ∞ ) between asymptotic and numerically determined heights at r∞. This generates a set of points {δi(zi0)} of the function δ(z0). (6) Zero the function δ(z0) using a Newton-Raphson method. This is the refinement step. (7) Perform final numerical integration of eq 12 using the optimally refined boundary conditions {z(0) ) z0, zr(0) ) 0}. At the Runge-Kutta stage, a very accurate embedded variable step length technique was used, referred to as Fehlberg78 (see, e.g., ref 36). Solutions were calculated over the interval of integration r ∈ [0, 1] cm, with 10-6 as absolute tolerance and 10-3 as relative tolerance (standard values in commercial solver packages). The physical parameters used are consistent with an air-water system: γ ) 72 mN/m, ∆F ) 996.910 kg/m 3, and g ) 9.81 m/s2. The Hamaker constant was chosen to cover more than 2 orders of magnitude, 1 × 10-21 J e A e 5 × 10-19 J, while the splay parameter range covered 5 orders of magnitude, 0.05 m-1 e λ e 10 000 m-1. The paraboloid position zp0 was chosen never to exceed 1 × 10-4 m. IV. Results and Discussion A. On the Nature of Solutions. It is often pointed out that being a solution to eq 3 is only a necessary condition for a stable equilibrium profile. Solutions to eq 3 are extrema of the energy functional and can either be local maximum energy solutions or local minimum energy solutions. Previously we and others declared (but did not confirm) solutions to equations like eq 3 as being stable, local minimum energy solutions. The possible existence of other solutions, albeit unstable ones, was not considered. The present numerical method explicitly brings to light the existence of two possible numerical solutions for any (36) Hairer, E.; Nørsett, S. P.; Wanner, G. Solving Differential Equations 1, Nonstiff Problems, 2nd ed.; Springer: Berlin, 2000.

Figure 5. Stable (a) and unstable (b) solutions of eq 3 and their convergence as a function of the paraboloid heights. The full line corresponds to the critical solution, at the minimal height zp0min ) 1408.858 nm. The physical parameters are A ) 50 × 10-21 J, λ ) 0.5 m-1, γ ) 0.072 N/m, and ∆F ) 996.910 kg/m3.

one set of system circumstances. Considered individually these appear as perfectly reasonable air-water interface profiles. However, one of these is stable and one is unstable. This is supported by the fact that they are both energy extrema and the energy of one is less than the energy of the other. There is also a clear similarity between the physical and the analogous mechanical systems. The existence of a second, unstable solution was noted by Forcada et al.,16 who investigated the deformation of a planar fluid interface arising from the interaction with a point particle. In Figure 5, we show both stable (Figure 5a) and unstable (Figure 5b) profiles obtained by solving eq 3 for a sequence of paraboloid heights, zp0. While the stable profiles show increasing deformation as the solid is lowered (and stress increases), the unstable profiles show the opposite and unphysical tendency of diminishing their degree of deformation. The sequences of curves converge in both figures to the same profile (solid curve) which represents the limiting profile found at the critical solid height of zp0,min = 1408.86 nm; the stable profiles merge to this curve from below, while the unstable profiles merge from above. For paraboloid positions below this critical height, there are no solutions to eq 3. The overall behavior in Figure 5 is precisely that found in the mechanical analogy problem (Figure 3 and Figure 4). Clearly, only the stable solution will be selected by nature and is thus the only one of physical interest. However, the existence of a second, unstable solution and the fact that the two merge to the same critical profile

3214


Figure 6. Profiles for (a) different heights of the solid and (b) different strengths of the van der Waals interaction. The physical parameters are λ ) 0.1 m-1, γ ) 0.072 N/m, ∆F ) 996.910 kg/m3, and a constant A ) 10 × 10-21 J for (a) and a constant zp0 ) 3555 nm for (b).

present us with a well-defined mathematical limit to the existence of a physical solution and a numerical criterion with which to uniquely determine important factors associated with it. Note that implementation of chemical potential equilibrium between the water molecules in the two phases will modify this existence limit: it will appear at greater heights, zp0, than those we find here. B. Profile Dependence on System Parameters. Forthwith we focus attention only on the stable solution to eq 3, in particular its dependence on system parameters. When the solid is sufficiently far above the liquid interface, the deformation induced on the interface is negligible. Although the deformation grows monotonically as the paraboloid approaches, the rate of increase of deformation is not constant. Even a cursory examination of Figure 6a is sufficient to infer that the rate of deformation with respect to paraboloid position must undergo a dramatic change. For a zp0 height variation of only 15%, a factor of 3.5 increase is evident from the curves shown in the figure. Compare this with the slower deformation growth rate that must be involved with the change from the neutral state, z ) 0, when the solid is at infinite separation, up to the least deformed profile resulting with the paraboloid at its highest documented position (zp0 ) 1500 nm). As we shall see, there is a change in growth rate from power law behavior to near singular behavior. This transition is associated with the convergence of the stable and unstable profiles and the approach to the limit of solution existence, here found when zp0 = 1167.74 nm.

Cortat et al.

Figure 7. Normalized profiles for (a) different shapes of the solid and (b) different interaction strengths. The physical parameters are zp0 ) 1170 nm, γ ) 0.072 N/m, ∆F ) 996.910 kg/m3, and a constant A ) 10 × 10-21 J for (a) and a constant λ ) 0.1 m-1 for (b).

Despite an increase in vertical deformation with decreasing zp0 (Figure 6a), it remains concentrated to a region within 1 cm of the central vertical axis of symmetry. In these cases, the far-field approximation continues to remain valid beyond this point with an appropriate value of the amplitude coefficient, C. Apart from the height, zp0, there are two other parameters that characterize the solid: its geometry, summarized in the splay parameter λ, and the strength of the van der Waals interaction between the solid and the airwater interface, expressed by the Hamaker constant, A > 0, reflecting the nature of the three media. Figure 6b demonstrates the effect of varying the interaction strength. Although the expected response to an increase in interaction strength is an increase in deformation, it may not be expected that the interaction strength plays no significant role in the lateral extent of the deformation. In Figure 7b, we have replotted the results of Figure 6b after dividing the profile points by the peak value of the deformation at the apex. Plotted on this scale, all the curves superimpose. Incidentally, plotting the curves of Figure 6a on such a normalized scale also results in a complete superposition of all curves. This is in marked contrast to the case of plotting the numerical curves for various splay parameters in this way (Figure 7a). Here, there is a clear lateral dependence on λ. The sharper the paraboloid, the more concentrated the deformation. Since the van der Waals interaction has a natural length scale associated with it (given by the


Figure 8. Plots of apex heights z(0) versus height of the solid zp0. The function z(0; zp0) for (a) λ ) 0.5 m-1, for values of A ∈ {1, 2, 5, 10, 20, 50, 100, 200, 500} × 10-21 J, and (b) A ) 100 × 10-21 J, for values of λ ∈ {0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000} m-1. Further, γ ) 0.072 N/m and ∆F ) 996.910 kg/m3.

relative strength of the van der Waals stress to surface tension, lA ) (A/γ)1/2) and since the interaction acts pointwise, it is reasonable that a decreasing splay parameter allows more of the solid to come within effective van der Waals range of the fluid interface. This results in the observed increase in lateral extent of deformation. Plotting these curves in the standard manner gives rise to a series of curves qualitatively similar to those shown in Figure 6a. In conclusion, of the three parameters that characterize the pin λ, A, and zp0, it is only the geometry parameter that governs the spreading effect of deformation. C. Dependence of Profile Peak on Height of Solid. For a quantitative study of deformation, one needs to focus on some specific function of the factors influencing deformation, f(zp0, λ, A). One function-factor relation that may be of particular interest to experimentalists working in the surface force area1,2,37 is the dependence of the deformation peak, z(0), on the paraboloid position, zp0. We implicitly considered this dependence in the qualitative discussion of Figure 6a. Figure 8 illustrates the general qualitative and specific quantitative behavior of the function z(0; zp0) for various values of interaction strength (Figure 8a) and splay parameter (Figure 8b). In light of (37) Snyder, B. A.; Aston, D. E.; Berg, J. C. Langmuir 1997, 13, 590-593.

Langmuir, Vol. 20, No. 8, 2004 3215

Figure 9. Logarithmic plots of the function z(0; zp0) for (a) constant λ ) 10 m-1, with values of A ∈ {1, 2, 5, 10, 20, 50, 100, 200, 500} × 10-21 J, and (b) constant A ) 5 × 10-21 J, for values of λ ∈ {0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000} m-1.

previous discussions, the results are in line with expectations: deformation increases as the solid is lowered toward the interface with the extent of deformation increasing with increasing A and decreasing λ. From these curves, it is perhaps clearer that a change in functional dependence on zp0 takes place. The leftmost extreme point of each of the curves earmarks the mergence of the stable and unstable solutions. As already stated, no physical and stable solution to eq 3 exists for paraboloid positions, zp0, to the left of (lower than) these critical values, zp0,min. These points thus indicate the absolute stability limit of physically meaningful fluid interface profiles. For smaller zp0 values, a completely different physical scenario is anticipated. Featured in a followup paper38 is the hypothesis that below zp0,min the most favorable configuration is the one in which the fluid interface has engulfed the solid and formed a stable three-phase line of contact. The particularly interesting feature exhibited in Figure 8 is the fact that the sets of critical limiting points, z(0; zp0,min) (dashed lines in Figure 8), lie to a very good approximation along straight lines. In Figure 8a, the line is of constant λ, while in Figure 8b it is of constant A. These lines indicate a nearly constant ratio between z(0; zp0,min) and zp0,min; note that the intercept is zero in both cases. This ratio is not only constant for a constant A or λ, but it is nearly constant for the entire Aλ-parameter space investigated. The slopes of the two lines shown in (38) Cortat, F. P.-A.; Miklavcic, S. J. To be published.

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Table 1. Values for the Coefficients r (Upper Table) and log10(ζ) (Lower Table)a R A λ

1

2

5

10

20

50

100

200

500

0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000

-2.908 66 -2.845 65 -2.734 45 -2.642 39 -2.553 67 -2.448 84 -2.381 18 -2.323 72 -2.261 23 -2.221 07 -2.183 08 -2.127 53 -2.073 15

-2.910 77 -2.848 64 -2.738 07 -2.646 00 -2.557 03 -2.451 69 -2.383 60 -2.325 71 -2.262 70 -2.222 15 -2.183 72 -2.127 42 -2.072 24

-2.913 48 -2.852 51 -2.742 83 -2.650 77 -2.561 48 -2.455 49 -2.386 83 -2.328 37 -2.264 66 -2.223 58 -2.184 58 -2.127 26 -2.071 01

-2.915 47 -2.855 39 -2.746 41 -2.654 38 -2.564 85 -2.458 38 -2.389 29 -2.330 41 -2.266 15 -2.224 67 -2.185 22 -2.127 13 -2.070 05

-2.917 42 -2.858 23 -2.749 98 -2.657 98 -2.568 24 -2.461 28 -2.391 77 -2.332 46 -2.267 66 -2.225 77 -2.185 87 -2.126 98 -2.069 06

-2.919 91 -2.861 90 -2.754 65 -2.662 74 -2.572 72 -2.465 15 -2.395 08 -2.335 19 -2.269 67 -2.227 24 -2.186 72 -2.126 76 -2.067 71

-2.921 75 -2.864 62 -2.758 16 -2.666 34 -2.576 12 -2.468 09 -2.397 60 -2.337 28 -2.271 20 -2.228 35 -2.187 37 -2.126 57 -2.066 66

-2.923 53 -2.867 29 -2.761 65 -2.669 93 -2.579 53 -2.471 04 -2.400 13 -2.339 38 -2.272 75 -2.229 48 -2.188 02 -2.126 38 -2.065 59

-2.925 84 -2.870 76 -2.766 23 -2.674 67 -2.584 04 -2.474 97 -2.403 51 -2.342 19 -2.274 82 -2.230 98 -2.188 87 -2.126 09 -2.064 12

log10(ζ) A λ

1

2

5

10

20

50

100

200

500

0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000

-24.673 70 -24.440 38 -24.045 67 -23.743 00 -23.481 07 -23.226 00 -23.108 20 -23.052 44 -23.059 16 -23.107 50 -23.172 09 -23.237 38 -23.227 82

-24.381 50 -24.151 86 -23.759 84 -23.457 12 -23.194 16 -22.937 00 -22.817 39 -22.759 85 -22.764 34 -22.811 03 -22.873 80 -22.935 90 -22.922 94

-23.994 88 -23.770 12 -23.381 83 -23.079 17 -22.814 89 -22.555 02 -22.433 05 -22.373 14 -22.374 67 -22.419 16 -22.479 48 -22.537 27 -22.519 75

-23.702 15 -23.481 09 -23.095 76 -22.793 23 -22.528 00 -22.266 12 -22.142 38 -22.080 67 -22.079 94 -22.122 75 -22.181 18 -22.235 66 -22.214 63

-23.409 20 -23.191 85 -22.809 58 -22.507 24 -22.241 12 -21.977 26 -21.851 75 -21.788 26 -21.785 26 -21.826 36 -21.882 89 -21.933 99 -21.909 40

-23.021 62 -22.809 15 -22.431 09 -22.129 13 -21.861 88 -21.595 46 -21.467 64 -21.401 78 -21.395 77 -21.434 59 -21.488 55 -21.535 12 -21.505 76

-22.728 18 -22.519 40 -22.144 63 -21.843 04 -21.575 00 -21.306 69 -21.177 13 -21.109 49 -21.101 18 -21.138 25 -21.190 24 -21.233 32 -21.200 30

-22.434 55 -22.229 44 -21.858 06 -21.556 91 -21.288 12 -21.017 95 -20.886 67 -20.817 24 -20.806 64 -20.841 94 -20.891 93 -20.931 46 -20.894 73

-22.046 10 -21.845 81 -21.479 03 -21.178 59 -20.908 88 -20.636 32 -20.502 79 -20.431 00 -20.417 33 -20.450 27 -20.497 57 -20.532 34 -20.490 63

a

λ is given in units of m-1, while A is in 10-21 J. Calculations were performed with γ ) 72.8 mN/m and ∆F ) 996.91 kg/m3.

Figure 8 are 0.350 067 and 0.350 529, which can be compared with the slope of 1/3 of the line relating the dependence of the critically stable position of the box on the sphere position in our mechanical analogy problem (in the absence of a box mass this line also goes through the origin). Figure 9 shows curves of z(0) versus zp0 plotted on loglog scale. Here, the two regimes of zp0 dependence are more transparent than on the linear scale and are that of a power-law dependence for large zp0 and a marked singular dependence on the approach to the critical limit. The curve sets show variation with the Hamaker constant (Figure 9a) and variation with the geometric splay constant (Figure 9b). To leading order, the far-field regime can be represented by the expression

log z(0) ) R log zp0 + log ζ where R and ζ are constants which depend on A and λ (and γ and G). Except for small λ-values (Figure 9b), the lines are approximately parallel, and thus R appears largely independent of A (1% variation for fixed λ) and even λ (30% variation for a change in λ of 4 orders of magnitude for fixed A). The constant ζ is strongly dependent on all four parameters. The full extent of variation is given in Table 1. The power-law dependence is presumably characteristic of a balance between the applied surface stress and the opposing surface tension. Approaching the critical solid height, the amount of water raised above neutral level increases significantly. This gives rise to a more complicated, three-term balance. As is the case in Figure 4, increasing the mass increases the curvature of a given line. However, since the mass is not constant but is here dependent on zp0, the effect is more subtle and gradual than in the mechanical analogy.

Figure 10. Behavior of both the stable (solid line) and unstable (dashed line) deformation peaks of the air-water surface, for λ ) 100 m-1 and A ) 5 × 10-21 J. The dotted line indicates the height of the paraboloid. Compare this with Figures 3 and 4.

Even though the van der Waals stress acts pointwise, the extended lateral influence of the solid found with small λ (Figure 7a) is felt over all the profile. As the latter is determined via differential eq 3, the extended range of deformation requires a greater z(0) value for consistency. This accounts for the steeper rise of the curves in Figure 9b occurring for small λ. In Figure 10 appears this system’s equivalent to Figures 3 and 4: apex peaks of stable and unstable profiles plotted as a function of solid height, zp0, down to the critical height where the two converge. The asymptote (dotted line) to the unstable profile peak shows the position of the paraboloid.


Langmuir, Vol. 20, No. 8, 2004 3217

Figure 11. Critical heights, zp0,min, as a function of A and λ. (a) The dependence of zp0,min on the splay parameter λ, for values of A ∈ {1, 10, 100, 500} × 10-21 J (diamonds), and the corresponding maximal deformation height z(0) for A ) 1 × 10-21 J (solid triangles). (b) The dependence on A, for values of λ ∈ {0.1, 1, 10, 100, 1000, 10000} m-1 (diamonds), and the maximal deformation z(0) for λ ) 10 000 m-1 (solid triangles). Other physical parameters are γ ) 0.072 N/m and ∆F ) 996.910 kg/m3.

D. Minimum Allowed Height of Solid. In a previous communication,34 we presented and discussed advantages of using a simple formula that summarizes numerically determined limits to approaching the air-water interface with a solid object. The reader may by now appreciate that the convergence of the stable and unstable solutions provides a convenient means of establishing the mathematical solution limit of the profile equation. Reiterating, this solution limit defines an absolute minimum height to which an object can be lowered toward the fluid surface. Since this clearly depends on the interaction strength and solid geometry (as well as interfacial tension and fluid density difference), one finds a variation in approach distances which we have captured in the formula34

zp0,min ) 10R(λ)Aβ(λ)

(14)

with

{

R(λ) ≈ -6.4341 - 0.2379 log10 λ - 0.0236(log10 λ)2 β(λ) ≈ 0.3057 + (0.1445 - 1.3252 × 10-2(log10 λ)2) log10 λ (15) 10 + 3 log10 λ

Figure 12. The ratio zp0,min/z(0; zp0,min). The parameters are as in Figure 11.

These expressions are valid for [log λ] < 10/3 and provide an accurate representation of numerical data to within 0.5% relative error for macroscopic objects with effective splay parameters λ ∈ [0.05, 10]. Some representative curves are shown in Figure 11 for various parameter values. Explicit numerical data upon which formula 14 is based are given in Table 2. Included in both Figure 11 and Table 2 are limiting peak heights of the air-water interface, z(0; zp0,min), corresponding to these minimum allowed paraboloid heights. One clear distinctive feature is again the near consistent factor of 3 between the respective numbers. In fact, the ratio zp0,min/z(0; zp0,min), plotted in Figure 12, has values varying between 2.85 and 2.99 (see also Table 2). In general terms, the trend is as follows: the greater the strength of the van der Waals stress (increasing Hamaker constant, A), the smaller the ratio; the broader the solid (decreasing splay constant, λ), the smaller the ratio. It is no coincidence that a correlation exists between the extent to which this ratio deviates from the value of 3 and the amount of fluid mass raised above the neutral level. It has now been established that a broader (small λ) and more attractive (large A) solid leads to a greater deformation in both the vertical and horizontal extents. This results in a larger mass of water raised. In the mechanical analogy problem, we had seen that for zero box mass an exact factor of 3 features in the linear relation between the critical sphere height and the corresponding box position. Furthermore, as the box mass is gradually increased there is a lowering of the box position compared to the zero mass case. Clearly, these same features appear in the physical system.

3218


Cortat et al.

Table 2. Values for zp0,min and Corresponding z(0) for the Air-Water Systema A λ 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 20 50 100 200 500 1000 2000 5000 10000 a

1

2

5

10

20

50

100

200

500

668.688 234.1750 596.494 208.6687 523.353 182.2264 481.442 167.3293 452.483 156.9919 430.584 149.0888 413.104 143.0221 398.639 137.8004 386.353 133.4540 375.711 129.8056 366.351 126.5689 308.707 106.2195 278.252 95.4491 258.112 88.3828 243.317 83.3456 231.756 79.3528 222.345 75.9628 214.456 73.3789 207.698 71.0943 201.810 68.8589 166.589 56.6930 128.525 43.5936 105.244 35.5961 85.959 29.0832 65.570 22.0744 53.310 17.9376 43.273 14.5254 32.776 10.9775 26.527 8.8688

816.003 285.8091 731.087 255.7163 644.004 224.5630 593.676 206.6098 558.733 194.2371 532.223 184.6162 511.012 177.1472 493.426 170.8533 478.467 165.4321 465.493 160.9316 454.070 156.7936 383.476 132.1345 346.024 118.8818 321.203 110.0934 302.943 103.9040 288.660 98.8545 277.023 94.8067 267.263 91.4066 258.897 88.5727 251.604 86.0364 207.921 70.9962 160.600 54.5412 131.602 44.6782 107.553 36.3533 82.092 27.7385 66.772 22.5210 54.221 18.2632 41.087 13.7972 33.265 11.1313

1059.853 371.0634 955.143 334.3420 845.959 295.4645 782.109 272.5386 737.479 256.7501 703.465 244.5008 676.158 235.0448 653.458 226.7712 634.107 219.7832 617.294 213.9602 602.468 208.5298 510.399 175.8376 461.262 158.7935 428.595 147.4787 404.516 138.7614 385.652 132.1628 370.265 126.8760 357.348 122.4330 346.268 118.5647 336.603 115.0717 278.592 95.0831 215.536 73.3376 176.795 59.9445 144.607 49.0248 110.473 37.3819 89.915 30.3345 73.050 25.0307 55.390 18.6087 44.862 15.0500

1289.952 451.8417 1167.739 409.0866 1038.634 363.0969 962.429 335.6660 908.878 316.9939 867.917 301.8886 834.943 290.5381 807.474 280.4012 784.016 272.4323 763.607 264.5914 745.586 258.2138 633.235 218.6890 572.986 197.1908 532.828 183.3299 503.178 172.7450 479.921 164.8076 460.934 158.1186 444.983 152.4640 431.291 147.8548 419.341 143.7333 347.502 118.5865 269.198 91.7505 220.986 75.0577 180.873 61.3591 138.278 46.8058 112.596 37.9799 91.517 30.9047 69.425 23.3637 56.250 18.9040

1568.222 548.5141 1426.077 499.8083 1273.891 445.7973 1183.195 413.8328 1119.109 390.2384 1069.901 372.5198 1030.176 358.6222 997.008 346.8410 968.633 336.3921 943.906 327.5820 922.045 319.6913 785.179 271.2182 711.405 245.2653 662.100 227.8684 625.630 215.1312 596.988 205.0478 573.580 197.0599 553.900 189.9179 536.996 184.0611 522.234 178.9255 433.332 148.1336 336.147 114.4865 276.173 93.8071 226.199 76.8860 173.061 58.5708 140.990 47.6863 114.641 38.7320 87.011 29.3738 70.524 23.7009

2026.715 708.3561 1854.113 649.7063 1665.896 583.9791 1552.246 542.7822 1471.323 513.6751 1408.858 491.4876 1358.232 473.2980 1315.829 458.1888 1279.460 444.9987 1247.698 434.1615 1219.566 423.8938 1042.392 360.7357 946.194 326.8200 881.653 304.0277 833.792 287.2319 796.134 274.2286 765.315 263.1360 739.373 253.9591 717.070 246.1214 697.577 239.3878 579.888 198.6983 450.695 153.8201 370.711 126.3449 303.925 103.2275 232.772 78.9489 189.755 64.3860 154.394 52.2728 117.265 39.6896 95.092 32.0947

2457.422 857.2753 2258.389 791.6312 2038.202 714.3351 1903.875 666.2114 1807.652 632.4557 1733.068 605.2323 1672.429 583.2311 1621.512 565.1430 1577.751 549.2922 1539.468 535.7301 1505.509 523.6899 1290.613 446.8759 1173.244 405.8610 1094.251 377.9764 1035.552 356.9845 989.297 340.5961 951.397 327.4268 919.465 316.2128 891.991 306.5728 867.963 298.1683 722.594 247.5417 562.470 191.8802 463.077 157.6232 379.942 129.2566 291.237 98.8458 237.539 80.4444 193.365 65.5373 146.948 49.7595 119.211 40.1601

2976.293 1036.4425 2747.691 962.6162 2490.984 873.0301 2332.716 816.8736 2218.639 776.8477 2129.833 744.4110 2057.394 718.3742 1996.411 696.5067 1943.885 678.1231 1897.852 661.9401 1856.955 647.0005 1596.841 553.7204 1453.889 503.0086 1357.354 469.1774 1285.461 443.7883 1228.718 423.7513 1182.167 407.5377 1142.907 393.5098 1109.100 381.5116 1079.512 371.3351 900.113 308.5130 701.783 239.8401 578.332 197.3304 474.889 161.6347 364.337 123.5552 297.324 100.6899 242.159 81.9732 184.137 62.3479 149.444 50.4884

3827.423 1328.9421 3554.646 1243.9683 3241.910 1137.3577 3046.300 1068.5139 2904.109 1017.8859 2792.757 977.6516 2701.517 944.9760 2624.430 916.7582 2557.835 893.5124 2499.326 871.9428 2447.231 853.4013 2113.539 734.5577 1928.536 668.6400 1803.011 623.5406 1709.233 590.5139 1635.047 564.5299 1574.079 542.7574 1522.585 525.1454 1478.192 508.9519 1439.300 495.2846 1202.742 412.7444 939.861 321.2976 775.581 264.5713 637.574 217.1368 489.748 166.3447 399.977 135.7228 325.988 110.3237 248.090 84.1253 201.473 68.0549

λ has the units m-1, and A is in 10-21 J. Parameter values are γ ) 72.8 mN/m and ∆F ) 996.91 kg/m3.

To further demonstrate the influence of mass, we have performed a similar series of calculations with a reduced gravity contribution obtained by reducing the fluid density difference by 3 orders of magnitude from ∆F ) 996.91 kg/m3 to ∆F ) 1.0 kg/m3. The results for the critical height are tabulated in Table 3 and have been least-squares fitted to the function 14 with

{

R(λ) ≈ -6.3029 - 0.2923 log-10 λ 9.705 × 10-3 log10 λ β(λ) ≈ 0.3223 + 3 + log10 λ

In comparison with eq 15, this shows a simpler dependence on the splay parameter, perhaps indicating the trend toward a two-term force balance. Supporting this conjecture is the fact that the ratio zp0,min/z(0; zp0,min) is now in general much closer to 3 with values ranging from 2.9001 in the most attractive case (λ ) 0.05 m-1 and A ) 500 × 10-21 J) to 2.9604 in the least attractive case (λ ) 10.0 m-1 and A ) 1 × 10-21 J). Compare these with the variation 2.8801-2.9308 found in the true air-water system. E. Force of Interaction. Another quantity of importance to the surface force community is the force acting


Langmuir, Vol. 20, No. 8, 2004 3219

Table 3. Values for zp0,min and Corresponding z(0) for a “Fake” Air-Water Systema A λ

1

2

5

10

20

50

100

200

500

0.05

1171.673 401.1886 967.915 330.5004 796.375 271.2728 709.369 241.3864 653.056 221.9282 612.27 207.9686 580.725 197.396 555.25 188.7885 534.036 181.3611 515.96 175.1718 500.282 169.6147 407.859 138.3037 361.597 122.626 331.87 112.2085 310.443 104.9986 293.931 99.4029 280.634 94.9859 269.588 94.9859 260.194 87.8446 252.06 85.144

1460.459 500.0662 1207.855 412.5573 994.721 339.0328 886.463 301.8 816.341 277.5733 765.527 260.3913 726.213 246.6554 694.453 235.9996 667.999 226.8923 645.454 219.0637 625.895 212.5535 510.537 173.152 452.754 153.2182 415.607 140.7354 388.826 131.5783 368.184 124.4292 351.558 118.8188 351.558 118.8188 325.994 110.2555 315.82 106.7402

1953.253 669.3294 1617.997 553.643 1334.235 455.0717 1189.808 405.7148 1096.157 373.1197 1028.244 349.9827 975.672 331.7382 933.185 317.151 897.783 305.1198 867.603 295.2111 841.416 285.7493 686.838 233.0138 609.331 206.8775 559.479 189.3803 523.523 177.2726 495.801 167.8752 473.468 160.3767 473.468 160.3767 439.121 148.4121 425.447 143.8935

2432.784 834.5147 2017.811 690.3111 1665.673 568.449 1486.149 506.7383 1369.637 466.4255 1285.097 437.5024 1219.625 415.1047 1166.695 397.2957 1122.581 381.5634 1084.965 368.7109 1052.318 357.5531 859.493 291.6506 762.732 258.4747 700.466 237.6712 655.545 222.2582 620.903 210.3627 592.99 201.0709 592.99 201.0709 550.052 186.0834 532.955 180.4347

3028.935 1039.8081 2515.674 861.4023 2078.939 709.8723 1855.895 633.0771 1711.003 582.991 1605.806 546.9298 1524.299 519.1206 1458.383 496.4531 1403.429 477.3485 1356.559 461.341 1315.872 447.4634 1075.402 365.5311 954.631 324.0115 876.88 297.393 820.769 278.2731 777.488 263.2532 742.607 251.5771 742.607 251.5771 688.939 233.0743 667.566 225.858

4044.464 1390.013 3365.514 1153.8927 2785.538 952.124 2488.599 850.3203 2295.449 783.1057 2155.093 734.5121 2046.275 696.997 1958.227 667.1261 1884.792 641.7329 1822.139 620.1569 1767.735 601.6122 1445.898 491.3918 1284.072 435.7695 1179.823 400.1171 1104.556 374.6023 1046.48 354.6155 999.664 338.8889 999.664 338.8889 927.611 314.0594 898.909 304.4281

5030.882 1730.6427 4192.755 1438.3661 3474.524 1188.6112 3106.054 1060.9652 2866.116 978.4773 2691.638 917.8552 2556.294 871.8031 2446.741 833.5689 2355.338 802.5813 2277.335 775.3559 2209.586 752.0862 1808.505 614.6374 1606.643 545.4366 1476.535 501.4955 1382.567 468.9832 1310.041 444.397 1251.566 424.374 1251.566 424.374 1161.546 393.7699 1125.681 381.3219

6255.145 2153.3279 5221.49 1792.9652 4332.686 1483.7443 3875.719 1325.3548 3577.815 1222.0452 3361.021 1147.0845 3192.761 1089.0654 3056.504 1042.6189 2942.784 1003.0449 2845.706 969.5549 2761.369 940.7057 2261.695 769.3311 2009.962 682.7681 1847.623 627.3898 1730.333 587.2111 1639.783 556.2504 1566.76 531.2384 1566.76 531.2384 1454.316 493.2957 1409.508 477.6366

8336.295 2874.5213 6974.584 2396.6745 5797.996 1986.9371 5191.198 1776.3946 4794.973 1639.7987 4506.32 1539.6306 4282.114 1461.538 4100.441 1399.9216 3948.742 1347.7268 3819.191 1302.0637 3706.603 1263.5215 3038.812 1034.0935 2701.912 918.5675 2484.486 844.144 2327.315 790.476 2205.93 749.5626 2108.012 715.3668 2108.012 715.3668 1957.183 663.7593 1897.06 643.4461

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 a

λ is given in units of m-1, and A is in 10-21 J. Parameter values are γ ) 72.8 mN/m (actual value for water) and ∆F ) 1.0 kg/m3.

on the solid and its dependence on position relative to the air-water interface. The force function, Fvert(zp0 - z(0)), is shown in Figure 13 (on a logarithmic scale, -log|Fvert| versus log(zp0 - z(0))) for various values of interaction strength, A, and splay constant, λ. This numerical quantity is compared with the approximation obtained assuming that the fluid interface remains undeformed. The most immediate and significant conclusion that can be drawn from this comparison is that when plotted on a relative surface separation scale the two data sets all but superimpose. There is some slight discrepancy appearing at separations approaching the critical height limit of the paraboloid, but this is noticeable only for the smallest λ values. Deformation results in two effects: a reduction in separation and a corresponding increase in attractive force, both of which are discernible when plotted against absolute paraboloid height. These two effects, however, appear to cancel to leading order if the results are plotted on a relative scale. The overall impression one gets is that, in this system at least, one can utilize the zero-deformation approximation to give quantitative information on the force versus separation relation. The only piece of information lacking is the minimum separation to which the approximate force should be calculated. However, even this is possible. Equation 14 gives an accurate quantitative estimate of the lowest height to which the object can be lowered. Figure 12 shows that at this distance the height of the profile at

its apex is approximately 1/3 that of the critical height of the solid (as discussed in the preceding section). Consequently, the zero-deformation approximation to the force, if plotted down to a distance of 2/3 of zp0,min, will give a good representation of the true force versus separation data, which actually extends to this much smaller separation. A more accurate estimate of the limiting separation can of course be obtained by a least-squares fitting of the data in Figure 12, fully but implicitly presented in Table 2. On a more fundamental level, we see that both the true and the approximate force curves lie more or less parallel, except for the broadest paraboloid case (λ ) 0.1 m-1). This latter result is again a trait particular to the very broad solid geometries. The force, being an integral quantity, increases more rapidly with separation (i.e., has a larger slope) when the lateral extent of influence of the van der Waals stress and resulting deformation is great. V. Summary and Final Remarks This paper concerns a numerical investigation of deformation of an air-water interface in response to a van der Waals surface stress induced by a nearby solid. The principal aim has been to document the overall deformation response as a function of parameters that describe the interacting solid: its geometry and the strength of the interaction. Two real-valued solutions of

3220


Figure 13. The force acting on the solid (negative of the force acting on the profile), in terms of the separation, zp0 - z(0). (a) Constant λ ) 0.5 m-1, for values of A ∈ {1, 2, 5, 10, 20, 50, 100, 200, 500} × 10-21 J. (b) Constant A ) 50 × 10-21 J, for values of λ ∈ {0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000} m-1. The dashed lines (barely visible) correspond to the z ) 0 approximation.

the Euler-Lagrange (E-L) equation, one representing a stable (local minimum energy) physical solution and one an unstable (local maximum energy) unphysical solution, have been monitored in the process. We have quantified critical factors describing the limits of deformation of the

Cortat et al.

air-water interface. Insight into the behavior of the system has been aided by studying a simple analogous mechanical system exhibiting very similar properties and tendencies. For example, the mechanical system possesses two physically relevant states of local energy extrema, one being stable and the other unstable. These two solutions, like the E-L solutions in the actual problem, converge with diminishing distance between the interacting objects to a single solution that earmarks the existence limit of a physical equilibrium state (profile). Extensive numerical data are tabulated and in some cases fitted to effective formulas for ease of quantitative reproduction. Some points of particular importance are as follows: (1) There is a close association between the fluid-solid surface interaction and an effective two-body problem assuming one is attached to a spring. What has been done in the past is to liken a fluid interface suffering a colloidal stress to a force-spring system.1 One conclusion that can be drawn from the present study is that such a model is optimally suited to fluids of small density difference. For large density differences, it is necessary to bring in a finite mass-gravity contribution in order to get correct qualitative representation. (2) The lateral extent of deformation is primarily dependent on the breadth of the encroaching solid, while geometry, height (separation), and interaction strength govern the vertical extent of deformation. (3) Convergence of stable and unstable solutions provides an unambiguous means of numerically determining mathematical existence limits and thereby absolute limits to physical deformation. At the deformation extremes, there is a difference of a consistent factor of approximately 3 between the height of the deformation peak and the critical height of the solid. This appears also in the simpler mechanical analogy. (4) Force curves, when plotted against relative separation, coincide to a good approximation with force predictions made assuming no deformation. If use of the latter is combined with formulas determining the limiting height to which a solid can be lowered to the fluid interface, one can effectively reproduce force data over the entire allowed separation regime. LA035651Y

Using Stable and Unstable Profiles To Deduce Deformation Limits of

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