Stable van der Waals-Induced Deformations of the Air−Water Interface

Langmuir 2006, 22, 6961-6968

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Stable van der Waals-Induced Deformations of the Air-Water Interface. Theoretical Predictions and a Suggestion for an Experiment S. J. Miklavcic*,† and L. R. White‡ Department of Science and Technology, UniVersity of Linko¨ping, S-601 74, Norrkoping, Sweden, and Department of Chemical Engineering, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania ReceiVed April 18, 2006. In Final Form: June 1, 2006

This article concerns the stability of the air-water interface subjected to a 2D attractive van der Waals stress. The physical problem models the setup of a Wilhelmy plate experiment prior to three-phase contact line formation. We present and employ an unambiguous condition to quantify the stability limit in terms of the distance of closest approach of a solid cylindrical plate of parabolic cross section to the fluid surface as a function of the strength of the van der Waals surface force and plate geometry. A numerical study spanning 4 orders of magnitude of the Hamaker constant and nearly 6 orders of magnitude of solid geometry characterizes the dependence of the stability limit on these physical parameters. Comparisons are also made with a previously published analytical condition guaranteeing a stable deformation of the fluid interface. A possible experiment for testing the theory is also described. Used together with the theory, the technique could be used as an independent means of determining system properties such as the surface tension or Hamaker constant.

I. Introduction A technique commonly used to investigate interfacial tension is the Wilhelmy plate method.1,2 In this technique, a rectangular plate of material, normally platinum, is immersed edge-on into a large-area trough of water exposed to air at atmospheric pressure and room temperature. When in proximity to the interface but still prior to three-phase contact, an attractive van der Waalstype surface force arises. Although short-ranged, this surface force can extend over a significant area of the fluid surface and cause it to deform in the direction of the solid. At some finite height above the base water level or some finite proximity, the deformed water surface will no longer maintain its continuous shape. Instead, the water interface will establish three-phase contact with the solid. Determining the point of instability of this van der Waals-induced deformation is the object of this article. Although the stability of this pre-immersion state may not have a direct bearing on the Wilhelmy plate method, the 2D model of this scenario serves as a simple but useful test case for the theoretical modeling of similar stability limits arising in noncontact systems involving colloidal interactions between solid objects and fluid drops.3-7 An equilibrium study of a related axisymmetric system of a point particle interacting with an infinite fluid surface was considered by Forcada et al.,8 who described mechanical stability limits in terms of solvability limits of the governing fluid interface equations. Such mathematical limits represent absolute bounds of existence and were extensively studied and quantified by Cortat * Corresponding author. E-mail: [email protected]. † University of Linko ¨ ping. ‡ Carnegie Mellon University. (1) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Wiley Science: New York, 1966. (2) MacRitchie, F. Chemistry at Interfaces; Academic Press: New York, 1990. (3) Butt, H.-J. J. Colloid Interface Sci. 1994, 166, 109. (4) Fielden, M. L.; Hayes, R. A.; Ralston, J. Langmuir 1996, 12, 3721. (5) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483. (6) Snyder, B. A.; Aston, D. E.; Berg, J. C. Langmuir 1997, 13, 590. (7) Preuss, M.; Butt, H.-J. Langmuir 1998, 14, 3164. (8) Forcada, M. L.; Arista, N. R.; Gras-Marti, A.; Urbassek, H. M.; GarciaMolina, R. Phys. ReV. B 1991, 44, 8226.

and Miklavcic9,10 for a system involving a vertical cylinder of parabolic cross section interacting through van der Waals forces with a fluid surface. These mathematical bounds describe limits under ideal conditions and should be distinguished from physical stability bounds that take into account a system’s sensitivity to external disturbances, which are always present in practice. Consequently, the physical limits are always those of most practical interest. An example of a physical and experimental application of equilibrium stability studies can be found in refs 11 and 12, which deal with systems of interacting thin films supported by solid substrates related to capillary condensation. Instabilities were also mentioned in the context of AFM experiments by Dagastine and White13 but in a manner that encompassed the entire AFM configuration rather than just the fluid interface deformation. Equilibrium interface stability, like the equilibrium fluid profile itself, is governed by the change in mechanical energy due to interface deformation induced by the applied van der Waals stress,

F ) γAΛ + G

∫∫∫∆V

z dV + Λ

∫ ∫ σ dS

(1)

Λ

In eq 1, G ) g∆F, with a fluid density difference ∆F and an acceleration due to gravity g; the intrinsic interfacial tension is γ. The terms in eq 1 are, respectively, the surface energy associated with the surface area change, the gravitational potential energy term, and finally the interaction term. Λ denotes the deformed interface. The equilibrium profile is determined by the first variation of eq 1 whereas the stability of this equilibrium profile is determined by the second variation.14,15 (9) Cortat, F. P. A.; Miklavcic, S. J. Phys. ReV. E 2003, 68, Art. No. 052601. (10) Cortat, F. P. A.; Miklavcic, S. J. Langmuir 2004, 20, 3208. (11) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16. (12) Forcada, M. L. J. Chem Phys. 1993, 98, 638. (13) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2002, 247, 310. (14) Courant, R.; Hilbert, D. Methods of Mathematical Physics; Wiley: New York, 1965; Vol. 1, Chapter 4. (15) Ewing, G. M. Calculus of Variations with Applications; Dover Publications: New York, 1985.

10.1021/la0610506 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/04/2006

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Some groundwork for the stability characterization of the deformed interface was laid in ref 16, which sought to provide a quantitative criterion for determining the onset of deformation instability. Focus was there placed on a function, Φ(x) ) Φ(x, z(x)), of the equilibrium fluid surface shape, z(x). Two conditions were established that respectfully guaranteed the stability or instability of the fluid interface. These conditions were based on direct comparisons between Φ(x) and explicit, closed-form functions (1 - E* exp(-(x/lG)2))/lG2 and (1 - E* exp(-(x/lG)2))/ lG2, with E* and E* being system-independent constants and lG being the capillary length of the system. These conditions were, however, only sufficient conditions, and because E* was not equal to E*, these left open the question of the state of stability if neither was satisfied, which turned out to occur in practice. In this article, we resolve this theoretical dilemma and present a criterion that unequivocally determines whether a fluid interface will be stable or unstable to an externally induced surface stress. In section 2, we quickly review background material on the equilibrium profile and its stability and reiterate the nature of the Φ function and show how it relates to fluid interface stability. The above-mentioned sufficient conditions are then restated in section 3. The new criterion representing both a necessary and sufficient condition for stability is derived in section 4 and applied in section 5 to a specific system involving a solid paraboloid cylinder interacting with the air-water interface. Numerical results are presented for extensive ranges of both van der Waals strength of interaction and solid geometry to gauge the validity of the sufficient conditions derived in ref 16 compared with the new criterion as well as to study the stability dependence on system properties. In this section, we also describe a possible experiment for comparing and testing our theoretical predictions of instability and otherwise for use in deducing system properties such as surface tension. Concluding remarks appear in section 6.

II. Equilibrium Shape and Stability Criterion of Fluid Interfaces An infinitely long plate, positioned edge-on over the fluid surface, forces a 2D symmetry to this system with x ) 0 as the symmetry plane. The equilibrium shape of the fluid interface and its state of stability are determined through consideration of the first and second variations of the 1D energy functional

F[zj] )

∫-∞∞ f(x, jz(x), jzx(x)) dx

(2)

with respect to functions jz belonging to the class, Ωx, of even, integrable functions that possess continuous derivatives up to second order, each of which is square integrable. The functional F is an energy per unit length, expressed as an (improper) integral of the multivariable function of (x, jz, p)

1 f(x, jz, p) ) W(p) h(x, jz) + Gzj2 2

(3)

where h ) γ + σ and W(p) ) (1 + We suppose that σ and therefore h are sufficiently differentiable so that f can be differentiated as needed. From symmetry, jzx(0) ) 0. We suppose that there exists a unique equilibrium profile, z ∈ Ωx, that gives an extreme value to the functional F. Any other function jz ∈ Ωx can then be represented as a sum jz(x) ) z(x) + η(x) for a suitably chosen η ∈ Ωx and real number . p2)1/2.

(16) Miklavcic, S. J. J. Phys. A: Math Gen. 2003, 36, 8829.

According to the fundamental theorem of variational calculus,14,15 the first variation of eq 2 leads to the Euler equation for z(x),17

(

)

h(x, z)zx W(zx)

) Gz + hz(x, z) W(zx)

(4)

x

with x ∈ (-∞, ∞). To arrive at this, we use the first partial derivatives of f (x, z, p) with respect to z and p() zx),

{

fz ) Gz + hz(x, z) W(p) h(x, z)p fp ) W(p)

(5)

Far from the plate, the effect of the van der Waals stress is negligible. Because the deformation here is expected to be small (i.e., |zx| , 1), eq 4 simplifies to the linear equation

γzxx ) Gz with exponential solution

z(x) ) Ce-|x|/lG

|x| . 1

(6)

where the characteristic decay length, lG ) xγ/G, is the capillary length of the system. The positive constant, C, is chosen to match this far-field solution with the near-field solution that must be determined numerically. The second variation14,15 of F takes the form16,18

δ2F )

2 2

∫-∞∞[fz z ηx2(x) + (fzz - (fzz )x)η2(x)] dx x x

x

(7)

where the second partial derivatives of f(x, z, p) with respect to x, z, and p() zx) are, respectively,

{

fpp )

h(x, z)

(1 + p2)3/2 fzz ) G + hzz(x, z) (1 + p2)1/2 phz(x, z) fzp ) (1 + p2)1/2

(8)

If eq 7 is positive for every possible function η that is inserted, then the equilibrium profile is said to be stable. If eq 7 is negative for any single perturbation, then the profile is said to be unstable, as it is unstable particularly to that perturbation. Consequently, a stability study is primarily concerned with the sign of eq 7, not its actual magnitude. If one can guarantee in general that fpp > 0, then it is possible to invoke the transformation V ) (fpp)1/2η. It can be shown16 that V itself belongs Ωx. Hence, the investigation of stability then reduces to a study of the integral,

I[V] :) δ2F )

∫-∞∞ [Vx2 + Φ(x)V2] dx

)

∫-∞∞ q(x, V(x), Vx(x)) dx

(9)

involving functions V over the space Ωx. Because we are interested in only the sign of the second variation, all positive constants (17) Miklavcic, S. J.; Attard, P. J. Phys. A: Math Gen. 2001, 34, 7849. (18) Miklavcic, S. J. ITN Res. Rep. Ser. 2002, (LiTH-ITN-R-2002-10, ISSN 1650-2612).

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that multiply this integral (e.g., the factor 2/2) can be dropped. In this expression, the function

Φ(x) )

( )

(fzz - (fzzx)x) [(fpp)x] 1 (fpp)x + + 2 2 fpp x fpp 4fpp

Φ(x) e

2

(10)

that we have introduced is the only function that need be considered for the stability state of the profile. The various terms appearing in eq 10 are evaluated using the explicit expressions in eq 8 with z(x) as the equilibrium profile. In terms of h, z, and p() zx), the terms in Φ are

{

(fzz - (fzp)x) 1 ) [GW3 - hzpx - (hzxp + hzz)W2] fpp h 2 [(fpp)z] 11 ) (h p + hx) - 3ppxW-2 4h z 4f 2

[

pp

]

and

( ) [

2 2 3(px2 + ppxx) (hzp + hx)2 1 (fpp)x 1 6p px ) + 2 fpp x 2 W4 W2 h2 (hxx + 2hxxp + hzzp2 + hzpx) h

]

The second and third derivatives of z, px() zxx), and pxx() zxxx) can be evaluated with help of the Euler-Lagrange equation (eq 4). Further details of the derivative contributions are unimportant to the following discussion and are omitted. (However, see ref 16 for more information.) It is not hard to see that when σ ) 0, Φ(x) equals the constant value, G/γ ) 1/lG2. Thus, if the underlying fluid is denser than the fluid above (which is the case here), then Φ and consequently the second variation (eq 2) will be positive for all V ∈ Ωx. The profile, z ) 0, will then be stable to all small perturbations. We particularly note that for a nonzero but locally confined surface stress, Φ asymptotes to G/γ in the limit |x| f ∞:

Φ(x) f

G 1 as |x| f ∞ ) γ l 2

However, if Φ(x) ) Φ(x, z(x)) satisfies the inequality

(11)

G

The question is then how negative can Φ(x) be in the vicinity of x ) 0, under typical conditions, for the functional I[V] to remain positive for all V and therefore guarantee stability.

1 - Ee-(x/lG) lG2

2

∀x ∈ R

(13)

for some E > E* ) 3/32[26x6 + x1944] ≈ 10.10, then the equilibrium profile will not be stable to arbitrary perturbations V ∈ Ωx. Detailed proofs of these statements can be found in ref 16. It was concluded that the first (stability) condition could be an effective tool for indicating the stability of solutions to the Euler-Lagrange equation. In the physical situation of a solid interacting with a fluid interface through van der Waals forces, it was found to be useful because the condition took advantage of Φ(x) decaying more rapidly than the Gaussian, ∼exp(-x2/ lG2), featured in the stability criterion. For the same reason, the second (instability) condition turned out to be less fruitful. Although still an unambiguous tool in terms of general solutions of Euler-Lagrange equations, it turned out to be impractical for deciding the instability state of a fluid interface deformed by van der Waals forces. The condition suits functions that decay more slowly than the Gaussian, which was not the case in practice.

IV. Necessary and Sufficient Condition for Interfacial Stability Our ambition is to establish conditions for Φ that are necessary and sufficient to ensure the stability of the interface. Simply stated, we are interested in knowing the minimum possible value of the integral in eq 9 for a specified Φ function. This is to be done with respect to members of Ωx. Alternatively expressed, our interest lies in identifying the function Vmin(x) that gives the minimum value of the functional, eq 9. Clearly, another application of the variational technique is the most natural and direct means of determining this function. The first variation of eq 9 leads to the Euler equation

( )

∂q d ∂q - )0 dx ∂Vx ∂V w

d2 min V (x) - Φ(x)Vmin(x) ) 0 (14) dx2

subject to the asymptotic boundary conditions min (x) f 0 as |x| f ∞ Vmin x (x), V

Because Φ(x) tends to the constant 1/lG2 as |x| f ∞, eq 14 tends to the equation

III. Established Conditions for Stability and Instability In ref 16, one of us proved the following two sufficient conditions for stability and instability, respectively, against infinitesimal disturbances. First, it was shown that if z ∈ Ωx is the solution of eq 4 for the equilibrium fluid interface under a local van der Waals stress and if Φ(x) ) Φ(x, z(x)) satisfies the inequality

Φ(x) g

1 - Ee-(x/lG)2 lG2

∀x∈ R

(12)

for some E e E* ) 5, then the equilibrium profile will be stable to arbitrary infinitesimal perturbations V ∈ Ωx.

Vmin xx -

Vmin )0 lG2

with asymptotic solution Vmin(x) ≈ exp(-|x|/lG). The above asymptotic boundary conditions can then be replaced by the single expression

Vmin x (x∞)

+

V

min

(x∞) )0 lG

(15)

and applied in practice at a conveniently large but finite x value, x∞. The minimum value of the functional is then identically

6964 Langmuir, Vol. 22, No. 16, 2006

Imin )

MiklaVcic and White

2 min (x)2] dx ∫-∞∞[Vmin x (x) + Φ(x)V

(16)

Although we used this particular approach earlier on functionals that we have used to compare with eq 9 and that involve closedform replacements for Φ(x),18 we have not previously applied it directly to the functional (eq 9). Although the present application necessitates further numerical calculations to solve eq 14 and then evaluate eq 16 and is not as aesthetically appealing as an explicit analytical condition, it does possess the unquestionable advantage of being both a sufficient and a necessary condition for the stability of the interface. First, a given physical situation gives rise to a unique function, Φ(x), which in turn generates a unique minimizing perturbation, Vmin(x). This perturbation in turn, together with its corresponding Φ, gives the least possible value of the functional (eq 9). Any other perturbation will, by definition, produce a greater numerical value of the integral. Consequently, if eq 16 is positive, then the equilibrium profile will unequivocally be stable to any infinitesimal perturbation. However, if it is negative, then the profile will be unstable to at least one perturbation, namely, that represented by the minimizing function, Vmin, itself. The positive value of Imin thus has the status of a sufficient and necessary condition for stability, which was our principal objective to obtain. With this we can now ascertain, albeit numerically, whether a calculated profile will likely appear in practice and therefore if its properties will be of any physical significance. As an aside, we mention the following detail with regard to the evaluation of the integral in eq 16. First, by symmetry one need consider only the half-interval [0, ∞). Second, multiplication of eq 14 by the solution Vmin and rewriting leads to the equivalent differential form

d min 2 (V ) - Φ(x)(Vmin)2 ) 0 dx The first integral (using the condition in eq 15) then gives the expression

Vmin(x)2 ) -

∫x∞ Φ(y)dyd Vmin(y)2 dy

) Φ(x)Vmin(x)2 +

∫x∞ Φy(y)Vmin(y)2 dy

Figure 1. Cross-sectional schematic of the system. The distance function, D(x), denotes the distance between the fluid interface point {x, z(x)} and the paraboloid surface point {xp(x), zp(x)}.

Derjaguin approximation19 and locally apply the expression for the triple-layer interaction between plane, parallel surfaces at the corresponding separation. This is justified for objects of low curvature at small separations. The expression we use for the interaction energy per unit area between two semi-infinite media separated by a distance, D, is19

σ(D) ) -

D(x, z) ) x(x - xp)2 + (z(x) - zp)2

∫0∞ Vmin(x)2 dx ) ∫0∞ Vmin (x)2[Φ (x) + xΦx(x)] dx

(19)

where the points (x, z(x)) and (xp, zp) are related through the equation

Hence, the minimal functional can be evaluated using the equivalent integral

∫0∞Vmin (x)2[2Φ(x) + xΦx(x)] dx

(18)

where A is the Hamaker constant, a parameter dependent on the dielectric properties of the three media. In the numerical work presented below, the solid is a parabolic cylinder positioned edge-on adjacent to the fluid interface. The cylinder’s cross-sectional profile is described by the equation zp(xp) ) zp0 + λxp2 (Figure 1), where (xp, zp) are the 2D coordinates of an arbitrary point on the cross section. zp0 is the minimum height of the solid at the apex, measured with respect to the undeformed fluid surface, and λ > 0 is a curvature parameter that we refer to as the splay constant. The distance function, D ) D(x, z), is defined as the distance from a point, (xp, zp), on the surface of the plate to the unique point, (x, z(x)), on the fluid interface, specified by the outward normal to the plate at (xp, zp).10,16 That is,

Consequently, after a little more algebra we find that

Imin )

A 12πD2

(17)

Either eq 16 or 17 will give the smallest value of the functional. However, in the results presented below, eq 16 was used because both Vxmin(x) and Vmin(x) were natural products of the numerical collocation solution procedure described below and because we conveniently avoid differentiating (numerically or analytically) the stability function Φ(x).

V. Specific Case Study of Fluid Interface Stability A. Mathematical Model. Exactly computing the van der Waals interaction between a deformed fluid interface and a solid across an immiscible fluid gap is a nontrivial task, even if the fluid interface shape is known and fixed. Instead, we invoke the

2λ2xp3 + xp(1 + 2λ(z0 - z (x))) - x ) 0 The physically relevant root of this cubic equation establishes xp ) xp(x)

xp(x) ) R -

1 + 2λ(z0 - z(x))

(20)

6λ2R

where

[ x

x + R) 4λ2

]

(1 + 2λ(z0 - z(x)))3 x2 + 16λ4 216λ6

1/3

(21)

(19) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992.

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The distance function, D(x, z(x)), depends both explicitly and implicitly on x through xp, zp, and the equilibrium profile, z(x). B. Numerical Procedure and Remarks on Scaling. As mentioned, from mirror symmetry we need only theoretically consider the half-interval [0, ∞). Furthermore, this half space can be split into a finite interval, [0, x∞], in which a full numerical study is required, and an infinite interval, [x∞, ∞), in which we can assume asymptotic behavior and utilize the analytic results. The nonlinear eq 4, with eqs 18 and 19, is solved in [0, x∞], together with boundary conditions applied at the ends of the interval:

{

zx (0) ) 0 z(x∞) zx(x∞) + )0 lG

(22)

The linear eq 14 is similarly solved numerically in [0, x∞] subject to the condition of eq 15 at the outer extreme and the symmetry condition at x ) 0. Numerical solutions of both of these equations were achieved using the general-purpose algorithm COLSYS20 that is used to solve ordinary differential boundary value problems by collocation. This technique creates a contiguous set of polynomial approximations, each valid within a subinterval of the stipulated x range. The set of subintervals makes up a mesh covering the main x interval. The method refines the mesh until a specified tolerance is satisfied. The polynomial spline approximations are then used to evaluate the solution (and/or its derivative) at any internal point in the given interval. The COLSYS algorithm was found to be extremely robust for solving both the nonlinear equilibrium problem, eq 4, and the linear stability problem, eq 14, despite the fact that both equations exhibit different behaviorial properties on different length scales. (See the discussion below.) In practice, the integral in eq 16 is the sum of two contributions: a numerical contribution over the interval [0, x∞] and an analytical contribution based on the asymptotic behavior for the remainder, [x∞, ∞). As a check on the numerical accuracy, the first contribution was evaluated in two ways. First, using a 40-point Gaussian integral procedure,21

I[Vmin] ≈

x∞ 40

∑wiq(xi)

2 i)1

(23)

where wi are the Gaussian weights and the argument of the integral, q, is evaluated at selected points xi ) x∞ζi/2 + x∞/2 specified by the Gaussian points ζi ∈ [-1, 1], i ) 1,.., 40.21 The second means was through use of the simple trapezoidal method

I[Vmin] ≈

x∞N - 1 q(xi) + q(xi+1) N

∑ i)0

2

(24)

where xi ) x∞/N, with N chosen to be sufficiently large, typically 5000 (although this value too was varied to establish sufficient accuracy). While for our purposes it is sufficient to state that these two methods gave the same sign for the integral and in particular predicted the same height at which the sign changed from positive to negative, they actually agreed quantitatively to at least three significant figures and often better in all of the cases studied. (20) Ascher, U.; Christiansen, J.; Russell, R. D. COLSYS, Netlib, http:// www.netlib.org (accessed 2004). (21) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications: New York, 1965.

Figure 2. Predictions of lowest stable minimum heights as functions of the Hamaker constant, A, and splay parameter, λ. Thick lines are predictions based on eq 18; matching thin lines are based on eq 12. The curves from top to bottom have been generated for λ ∈ {0.5, 5, 50, 500, 5000, 50 000, 100 000} m-1. Other system parameters are ∆F ) 1000 kg m-3 and γ ) 72.8 mN m-1.

An evaluation of the full improper integral, eq 16, is then completed by adding the contribution from the infinite part of the x range. In this infinite region we know that the function Φ(x) behaves as the constant 1/lG2 and the minimizing perturbation behaves as an exponential Vmin(x) ≈ exp(-x/lG). It is easy to show that the infinite contribution is then (Vmin(x∞))2/lG. We remark here that a large number of different length scales are present in this system. These are the capillary length, lG ) (γ/G)1/2, the height of the paraboloid, zp0, the lateral extent of the solid, λ-1, and a length scale measuring the relative strength of van der Waals stress to surface tension, lA ) (A/γ)1/2. These vary widely in magnitude and compete to different extents in different circumstances. This competition significantly complicates the numerics, which must accurately capture all of the important physical attributes of the system. Predominantly, these four length scales determine both the fineness of the x mesh used in the solution of the differential equations (eqs 4 and 14) and the evaluation of the numerical integral (eq 18) as well as the choice of the x limit, x∞. In our calculations, we have opted for the vertical scaling, zp0, and the horizontal scaling, (zp0lG)1/2. Although both the equilibrium profile and the minimizing perturbation, Vmin, decay on the length scale of lG, the practical range of influence of the van der Waals stress is much smaller. This necessitates a detailed description closer to the center of symmetry. At the same time, the longrange decay of deformation must also be captured. Consequently, we adopted the combination of the compromise scaling by the geometric mean (zp0lG)1/2 with a judicious choice of x∞, governed by the values of both λ and lA. In all cases shown here, we have confirmed that x∞ was chosen to be sufficiently large that the exponential decays of z(x) and Vmin(x) were reproduced to ensure the validity of eqs 15 and 22 without compromising the numerical accuracy of the integral evaluation in eq 23 or 24. Incidently, evidence for the relatively smaller range of influence of the van der Waals stress compared to lG is more clearly evident in the x dependence of the function Φ(x) in the Figures than in either z(x) or Vmin(x). This shorter range of influence is one reason for the lack of viability of the instability condition, (eq 13), derived in ref 16. C. Numerically Determined Stability Limits. In Figure 2, we summarize the principal results of our calculations. The Figure

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outlines, for a given A and λ, our predictions of the lowest heights to which a solid plate can be lowered and still ensure the stability of the fluid interface. The data covers 4 orders of magnitude of the Hamaker constant, which is the normal range encountered in practice, and nearly 6 orders of magnitude of the splay parameter, λ, that governs the lateral broadness of the solid. The reciprocal of 2λ is a measure of the solid’s radius of curvature at its lowest point, its apex. Thus, the values studied here cover curvature radii between 5 µm and 1 m, that is, between the experimental and the macroscopic scale. The significance of the results shown as thick lines in the Figure should be appreciated. These curves are predictions of the solid’s minimum limiting heights, based on the necessary and sufficient condition of the positive value of the integral in eq 16, as a function of the van der Waals strength, A, for given values of λ. These curves divide the zp0 - A half-space into two distinct regions. The region above a given curve represents all of those paraboloid positions (heights above the flat air-water interface) that give rise to stable deformations of the fluid surface. The region below represents those positions that result in unstable deformations. Consequently, equilibrium solutions of the profile equation, eq 4, found for solids positioned at some height under a given limiting curve will not likely arise in practice. Any physical characteristic associated with this hypothetical profile will then have no practical bearing. The Figure also contains predictions based on the direct comparison between the physical Φ values and the Gaussian, (1 - 5 exp(-(x[/lG)2))/lG2, in eq 12 (thin lines). Both conditions clearly predict the exact same qualitative dependence on both λ and A, as should be expected. Because the condition in eq 12 guarantees a positive value of the second variation, eq 9, and eq 16 focuses on zp0 points at the transition between positive and negative values in eq 9, eq 12 will always predict more conservative solid heights (i.e., larger zp0 values) than those deduced from eq 16. Figure 2 confirms this expectation, showing that the predictions of eq 12 always lie above the predicitions of eq 16. We observe in particular that for the very low values studied, λ ) 0.5 (and λ ) 1 not shown), the two conditions are actually in very good quantitative agreement. In fact, on the scale of the Figure the predictions are difficult to distinguish. (See the uppermost pair of lines in Figure 2.) Moreover, as one consequence of the increasing length scale, lA, the agreement gets better as the Hamaker constant, A, increases. (See, for example, the numbers quoted in Figure 3b) This implies that in practice eq 12 actually acts as both a sufficient and a necessary condition. As λ increases, however, the two predictions rapidly diverge, although the qualitative trends remain the same. Comparing the results in Figure 2, we see that the predictions of the critical heights of eq 12 rapidly converge to some form of asymptotic (in λ) prediction. One possible explanation for the low-λ agreement and high-λ disagreement is discussed below. Figure 3 shows profiles of the equilibrium solution, z(x), and corresponding Φ(x) function for the particular case of λ ) 0.5 m-1 and for Hamaker constants representative of the 4 orders of magnitude range that we have studied. Eight profiles are shown in each Figure (a and b). Four correspond to profiles evaluated at critical solid heights predicted by the sufficient condition of eq 12 (thin lines), and four correspond to profiles at the (lower) critical solid heights predicted by the condition of eq 16 (thick lines). As expected, the latter cases with the lower solid heights show higher z interfaces and deeper Φ profiles. The reader should

MiklaVcic and White

Figure 3. Profiles of (a) the equilibrium surface shape, z(x), and (b) the corresponding stability functions, Φ(x). The curves have been generated for the case of λ ) 0.5 m-1. Hamaker constants and paraboloid heights are given in part b. Other system parameters are the same as in Figure 2.

note the order of magnitude difference in the horizontal extent of the Φ dependence compared to z, which decays on the length scale of lG. For comparison, we show in Figures 4 and 5, analogous Φ -profiles for the two cases λ ) 50 m-1 and λ ) 5000 m-1. As mentioned earlier, λ-1 is one of the governing length scales of this system. The examples shown demonstrate that increasing λ by 2 orders of magnitude results in an order of magnitude decrease in the lateral extent of Φ: from 10-3 m in Figure 3 to 10-4 m in Figures 4 and 10-5 m in Figure 5. At the same time, the magnitudes of Φ at heights predicted by (18) also increase by a factor of 10. For this reason, the x -dependence of Φ at the critical heights predicted by (12), featured clearly in Figure 3(b), is lost on the vertical scales shown in Figures 4 and 5. From Figure 3b, we see the effect of decreasing the van der Waals length scale, lA, on the Φ profiles at the critical heights predicted by eq 16. At large lA, the range of influence of the van der Waals force is large, resulting in a broad but shallow Φ(x) profile. As lA decreases, the range of the surface force’s influence decreases, and because of instability, the Φ profiles must monotonically become narrower and deeper. This trend is also apparent in Figure 4. However, for reasons not completely clear the monotonic trend with decreasing Hamaker constant is not quite present in the case of λ ) 5000 m-1 in Figure 5. Presumably, this is a result of the competition or cooperation between the two length scales λ-1 and lA.

Van der Waals-Induced Deformations of an Interface

Figure 4. Profiles of the stability function, Φ(x). The curves have been generated for λ ) 50 m-1. Hamaker constants and paraboloid heights are given. Other system parameters are the same as in Figure 2.

Figure 5. Profiles of the stability function, Φ(x). The curves have been generated for λ ) 5000 m-1. Hamaker constants and paraboloid heights are given. Other system parameters are the same as in Figure 2.

The trend followed by the Φ functions as λ increases suggests one possible reason for the increasing disagreement between the two methods of predicting critical heights. For small λ values, the extent of influence of the van der Waals force is significant compared with the length scale, lG, over which the optimum perturbation decays. As λ increases by the orders of magnitude that we consider, the extent of influence rapidly diminishes and concentrates on a diminishing region around x ) 0. In terms of the integral, eq 9, the contribution of Φ(x) tends to the form of a singular function, centered at the origin. Such a function is not consistent with one of the assumptions made in the derivation of eq 12.16 In contrast, the form and x dependence of Φ for small λ is consistent with this derivation and results in a condition that is evidently both sufficient and necessary. D. Suggestion for an Experiment. It is possible to take advantage of the geometry of this particular example and the nature of the 2D deformation response of the fluid interface for quantitative testing by experiment. Viewed from along the x axis, the infinitely long plate and the corresponding deformation caused by the attractive interaction takes the form of a slit of width on the order of 100 to 1000 nm, that is, a width that is comparable to the wavelengths of visible light. As suggested in

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Figure 6. Schematic of the proposed experimental setup to determine the separation, D(0), between the fluid interface and paraboloid surface.

Figure 6, illuminating the slit from a normal direction should create a Fraunhofer diffraction pattern on the opposite side of the slit. Because the diffraction pattern of alternating light and dark patches is a function of the slit width,22 it is possible in principle to experimentally determine the separation at which the interface becomes unstable (as well as those heights preceding the instability point). This can be compared with the present predictions of lower limiting plate heights based on eq 16. Furthermore, if it is possible to experimentally establish the absolute height of the plate, then it will be possible to deduce from the diffraction data the maximum height of the fluid interface deformation. However, there may be some practical considerations. We imagine that the diffraction pattern will likely be apparent only above the water surface. Moreover, for an unobstructed view it is optimal for the container to have hydrophobic walls. An opaque material such as Teflon (as used in a commercial Langmuir trough2) or a hydrophobized glass surface could be used. Although a hydrophobic barrier is preferred, a material leading to a contact angle of 90° should suffice. A hydrophilic wall would give a wetting meniscus high enough to interfere with the projected pattern. A hydrophobic wall, however, would result in a meniscus that falls below the zero level of the water surface and thus would not present a problem for viewing the diffraction pattern. It might also be mentioned that greater optical contrast between light and dark diffraction fringes could be achieved using sharper plates (i.e., large λ), which results in a more localized fluid interface deformation and a thinner gap. Another issue is guaranteeing a horizontal plate. The diffraction pattern itself could be used to measure the extent to which the water surface and the plate are parallel. If the plate was not horizontal and therefore not parallel to the water surface, the fringe lines would diverge along their length. A rotation stage controlling the lateral tilt of the plate could be adjusted until the fringes are aligned parallel. Apart from the purpose of comparing experimental data with the present theoretical predictions of interface stability, the abovedescribed experiment might also be useful as an independent means of determining system properties. The splay parameter, λ, the strength of the van der Waals force, A, and the fluid surface tension, γ, are the three key physical parameters that determine the instability point. Alternately, by determining the separation at which instability sets in for a series of systems for which one (22) Born, M.; Wolf, E. Principles of Optics, 6th ed.; Pergamon Press: Oxford, England, 1991.

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of the interfacial tension in a given experiment. Figure 7 suggests that the best results would be obtained for solid material generating the greatest van der Waals interaction (Hamaker constant, A) between the solid and the water medium because the slopes of the curves increase with A and greater variation in γ occurs at the high-A end of the range.

VI. Concluding Remarks

Figure 7. Predictions of lowest stable minimum heights based on eq 18 as functions of the Hamaker constant, A, and surface tension, γ. The upper four curves are for λ ) 50 m-1, whereas the lower four curves are for λ ) 5000 m-1. From top to bottom, the curves in each set represent values γ ∈ {10, 30, 50, 72.8} mN m-1, resulting in capillary lengths lG ) 1.02, 1.75, 2.26, and 2.73 mm, respectively. Other system parameters are the same as in Figure 2.

or more of these is varied, it should be possible to deduce the remaining system variable(s). Figure 7 indicates the sensitivity to surface tension of the predicted critical height for a fixed splay parameter (λ ) 5000 and 50 m-1 are explicitly represented) as a function of the Hamaker constant. Because the spreading of surface-active agents (e.g., surfactants) on the water surface affects the surface tension but not the van der Waals interaction between the water and solid mediums (at least not on the scale of the separations involved here19), it could be possible to use data from experiments to quantitatively determine the absolute value

We have used an unambiguous means to quantify the physical conditions that lead to an unstable physical system. The sufficient and necessary condition (eq 16) has been applied to the particular case of a paraboloid cylinder interacting with an infinite planar air-water interface, and distances of closest approach have been calculated. These should be distinguished from the distances predicted by the mathematical limits of solvability of the governing profile equation, eq 4.9,10 Such closest approach distances would not be obtained in practice because the fluid interfacial system would likely become unstable to physical disturbances well before. Our future efforts involve developing an analogous stability criterion applicable to the experimental system of a fluid drop interacting with either a solid particle or a macroscopic surface. Because one object of these experimental studies is to extract system properties from measured surface forces and because these suffer from interpretational problems due to interface deformation, it might be possible to instead utilize stability criteria, such as those presented here, as an alternate means of obtaining such system information. We hope to elaborate on this in a future publication. Acknowledgment. S.J.M. thanks Professor Guy Berry for bringing to his attention the slit analogy central to the suggested experimental system. LA0610506