Article pubs.acs.org/Langmuir
Analytical Model for the Deformation of a Fluid−Fluid Interface Beneath an AFM Probe Daniel B. Quinn, Jie Feng, and Howard A. Stone* Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, United States ABSTRACT: We present an analytical solution for the shape of a fluid−fluid interface near a nanoscale solid sphere, which is a configuration motivated by common measurements with an atomic force microscope. The forces considered are surface tension, gravity, and the van der Waals attraction. The nonlinear governing equation has been solved previously using the method of matched asymptotic expansions, and this requires that the surface tension forces far exceed those of gravity, i.e., the Bond number is much less than one. We first present this method using a physically relevant scaling of the equations, then offer a new analytical solution valid for all Bond numbers. We show that one configuration with a large effective Bond number, and thus one requiring our new solution, is a nanothick liquid film spread over a solid substrate. The scaling implications of both analytical methods are considered, and both are compared with numerical solutions of the full equation.
1. INTRODUCTION Nanoscale interactions between fluids and solid bodies are commonplace in a host of industrial processes, and so are characterized by various laboratory techniques. Atomic force microscopy (AFM), for example, is a technique used to map the surface of solids and liquids at the nanoscale, and relies on slight changes in the interaction force between a fine tip and the surface. Tip radii in AFM setups range from atomic dimensions to upward of 1 μm,1 suggesting that van der Waals (vdW) intermolecular forces play a role in the liquid−solid interaction at the scale of the tip. The vdW force arising from the interaction of oscillating dipoles controls many aspects of the behavior of the materials,2 and vdW forces in general are known to affect wetting properties,3−5 surface tension,6−8 and colloidal interactions.9−11 In AFM experiments where the surface being measured is a liquid or liquid-film-covered substrate, the deformation of the fluid−fluid interface is due to a combination of surface tension, gravity, and vdW forces. This deformation indirectly determines the force−distance relationship between the probe and the substrate, an accurate model of which is necessary for effective use of the AFM and has thus been the subject of a number of investigations. The solid probe above the interface has largely been modeled in two ways: as a parabaloid, where theoretical studies have considered both the shape12 and the stability13 of the fluid− fluid interface, and as a sphere,14 where the simpler geometry has allowed both theoretical examination and experimental verification of scaling behaviors.15 In the latter case, the effects of varying Hamaker number, Bond number, and separation distance have been investigated numerically.16 Henceforth, we will consider only the case where the probe is modeled as a sphere. In addition to an accurate force−distance relationship, one characteristic of particular interest to AFM research is the “jump-to-contact” condition, when the sphere is suddenly © 2013 American Chemical Society
wetted by the liquid. This jump condition was the subject of a recent study where the minimum height before contact was predicted numerically and verified experimentally.17 AFM has also been used to measure the thickness of thin films, and such geometries have been studied experimentally18 and numerically,19 but only for one particular film depth. Despite continued progress, some fundamental issues of the modeling of liquid−solid interactions in the AFM environment are unresolved. The nonlinearity of the governing equation for the shape of the fluid−fluid interface has thus far limited full solutions to the realm of numerics. Fortunately, when modeling the probe as a sphere, some analytical solutions are possible in cases of small Hamaker number, /a = 4A /(3πγa 2), and/or Bond number, ) = (ρga 2 /γ )1/2 , where A is the Hamaker constant of the material system, γ is the surface tension, a is the radius of the sphere, ρ is the density of the lower fluid, and g is the acceleration due to gravity. Two of these solutions will be discussed below, one for /a ≪ 1 and ) ≪ 1, and one for /a ≪ 1 and arbitrary ) . All solutions to date have been the former type, since the Bond number is often small in AFM systems. As we will show in Section 6, however, this limit is not always the case. Table 1 summarizes the content and assumptions of the aforementioned literature. One feature that varies from study to study is the nondimensional scaling in AFM geometries. One option is the recent treatment by Ledesma-Alonso et al., where the radius of the sphere is used as a reference length.17 Here, we propose an alternate scaling based on a natural length scale that arises when balancing vdW and surface tension forces. With this Received: November 2, 2012 Revised: January 3, 2013 Published: January 7, 2013 1427
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Table 1. Summary of Previous Literaturea /a ≪ 1 and ) ≪ 1 refs Cortat & Miklavcic (2003) Cortat & Miklavcic (2004) Liu et al. (2005) Wang et al. (2007) Ledesma-Alonso et al. (2012a) Ledesma-Alonso et al. (2012b) Forcada et al. (1991) Lin et al. (2009) This paper
theory,1 where the disjoining pressure due to the sphere is of the form
/a ≪ 1 and ) = O(1)
2A π ⎛ x 3 − n[a(n − 4) − x] + (x + 2a)3 − n [a(n − 2) + x] ⎞ ×⎜ ⎟ (x + a)(n − 4)(n − 3)(n − 2) ⎝ ⎠
Πs(x , a) =
E
N
A
E
N
A
No
Yes
No
No
No
No
No
Yes
No
No
No
No
No Yes Yes
Yes Yes Yes
Yes Yes No
No No No
No No No
No No No
No
Yes
No
No
No
No
No No No
No No Yes
No No Yes
Yes No No
Yes Yes Yes
No No Yes
(2a) 3
=
for n = 6
(traditional vdW) (2b)
where A is the Hamaker constant. Substituting eq 2b into 1 reduces the problem to a second-order nonlinear ordinary differential equation. Finally, we write the curvature term in cylindrical coordinates and divide eq 1 through by γ to give a convenient form of the equation in terms of length scales alone
a
E: Experimental work; N: Numerical calculation; A: Analytical solution.
⎛ ⎞ dh ⎜ ⎟ r S2Aa3 1 d⎜ dr ⎟= h − 2 2 1/2 ⎟ 2⎞ r dr ⎜ ⎛ Sc ((S − h) + r 2 − a 2)3 ⎜ ⎜1 + d h ⎟ ⎟ dr ⎠ ⎠ ⎝⎝
scaling, all numerically determined quantities associated with the deformation are O(1). We will first reproduce the analytical solution using matched asymptotic expansions used in a couple of previous studies, then propose a new solution valid for all ) . Some implications of both models will be discussed, and they will both be compared to numerical solutions of the full nonlinear equation. Finally, we will examine the case of a thin nanofilm beneath an AFM probe and show that in this case the effective Bond number becomes large, which makes the solution via matched asymptotic expansions inappropriate.
( )
(3) 1/2
where S c = (γ/ρg) is the capillary length and S A = (4A/ (3πγ))1/2 is the natural length scale that arises when balancing vdW and surface tension forces.21 It is convenient to nondimensionalize using a as the typical radial length scale, but also to identify a length scale for interfacial displacement due to the vdW force. Thus, we let
2. GOVERNING EQUATION FOR INTERFACE DEFORMATION OVER A DEEP LIQUID LAYER As in previous studies of the equilibrium shape of a liquid surface influenced by the vdW interaction with a nearby spherical particle, we consider the modified Young−Laplace equation for the steady shape h(r) of a fluid−fluid interface 2γκ = Πs(x , a) − ρgh + p0
4Aa 3π (x 2 + 2ax)3
R=
r a
L=
H= S a
h 2 S A /a
/a =
⎛ ρga 2 ⎞1/2 a2 ⎟ = 2 )=⎜ Sc ⎝ γ ⎠
S2A a2
(4)
so that eq 3 can be written as ⎛ ⎞ dH ⎜ ⎟ R 1 d ⎜ dR ⎟ 1/2 ⎟ 2 R dR ⎜ ⎛ ⎞ ⎜ ⎜1 + / a2 dH ⎟ ⎟ dR ⎠ ⎝⎝ ⎠
(1)
where γ is the surface tension, κ is the mean curvature of the interface, Πs is the disjoining pressure due to the sphere, x is the distance to the sphere’s surface, a is the radius of the sphere, ρgh is the hydrostatic pressure due to gravity, and p0 is a reference pressure. The geometry of the sphere−interface system is shown in Figure 1. In the present case of a deep liquid bath, the reference pressure p0 may be set to zero without loss of generality, but we note its presence because it will become nonzero when we subsequently analyze a thin liquid film in Section 6. An expression for Πs can be found using Hamaker
( )
= ) 2H −
1 ((L − /aH )2 + R2 − 1)3
(5)
Note that for algebraic convenience we have defined the Bond number with a square root. Since the solutions we seek represent static equilibria, we further assume them to be smooth and symmetric about R = 0. Far from the origin, gravity will ensure the relaxation of the interface. These two conditions are encapsulated by the following boundary conditions: dH (0) = 0 dR
H (R → ∞ ) = 0
(6)
Equation 5 and boundary conditions 6 are the complete nondimensional problem statement. We will examine two analytical simplifications to eq 5, but will also solve it numerically to obtain H(R ; L , ), /a). The first three dimensionless parameters, /a , ) and L occur in the model of Ledesma-Alonso et al.,17 who chose a different scaling for h and obtained numerical solutions.
Figure 1. Sketch of the spherical probe deforming the surface of a semi-infinite liquid sample. 1428
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tempting then to neglect the third term of eq 7, we expect that the term’s presence is required for the interface to eventually become flat far from the spherical probe, as in the classic case of a static meniscus around a cylindrical needle.20 Instead, we retain the influence of ) despite its small magnitude by using the method of matched asymptotic expansions. We recognize that Liu et al.14 have previously demonstrated how asymptotic methods can be applied to this geometry. Nevertheless, Sections 4.2−4.4 offer new considerations of finite Bond number effects, scaling behavior, and predictions of the jump-to-contact condition. It will also be helpful to have an asymptotic solution that uses our scaling when drawing comparisons in Section 5. For these reasons, we begin by briefly reworking the method of matched asymptotic expansions to approximate a solution to 7. 4.1. Solution via Matched Asymptotic Expansion. First, we focus on large distances from the probe where gravitational effects make the interface flat. It is convenient to rescale the radial variable as R̃ = )R , for which R̃ = O(1). This rescaling means that radial dimensions are on the order of the capillary length S c, and eq 7 has an “outer” solution, H(o)(R̃ ) that satisfies
The nonlinearities of eq 5 preclude any straightforward analysis. Typical AFM geometries, however, suggest certain regimes of parameters that make the problem approachable via analytical means. To begin, we note that S A is nearly an atomic dimension, i.e., S A ≈ 1 nm, and a is typically about 20 nm, so /a ≈ 10−3. We are thus motivated to study eq 5 when /a ≪ 1. In this limit, eq 5 can be approximated as a linear differential equation of second order d2H 1 dH 1 + − ) 2H = − 2 ≡ f (R ) 2 R dR dR (L − 1 + R2)3
(7)
which is to be solved with the same boundary conditions as in the nonlinear case.
3. SCALING CONSIDERATIONS In order to complement the detailed calculation reported below, we begin with a series of scaling arguments that identify the length scales relevant to the interface deformation. The augmented Young−Laplace eq 1 includes surface tension and vdW interactions, where the latter introduces the separation distance, S ⊥, between the sphere and the interface (see Figure 2). Thus, the length scales we have at the outset are S ⊥, the
1 d ⎛ ̃ dH (o) ⎞ )4 ⎜R ⎟⎟ − H (o) = − 2 2 ⎜ R ̃ dR ̃ ⎝ dR ̃ ⎠ () (L − 1) + R̃ 2)3
(8)
Thus, in the case of small Bond number, ) ≪ 1, the right-hand side of eq 8 is negligible, and the outer solution for the shape is determined by the modified Bessel equation of order zero. The solution that is bounded away from the origin is H (o)(R̃ ) = c1K 0(R̃)
(9)
where K0 is the zeroth-order modified Bessel function of the second kind and c1 is a constant. Next, we return to eq 7 with ) ≪ 1, and find that the “inner” solution, H(i)(R), satisfies
Figure 2. Sketch enlarging the region of deformation and showing length scales relevant to the interface deformation.
sphere radius a, and the vdW−capillary length S A = (4A/ (3πγ))1/2. We also recognize the distance S ∥ along the interface over which the interface deformation changes significantly. We can now construct a scaling argument for the typical curvature κ and interface deflection h. From a balance of capillary forces (γκ) and vdW interactions (A/S 3⊥), we expect that in the case of low Bond numbers ) , the curvature κ = O(A/(γS 3⊥)). Since curvature scales with the second derivative of deformation, we also expect the curvature κ = O(h/S 2∥), and therefore h = O (S 2∥S 2A/S 3⊥). Finally, a geometric argument based on the gap between the sphere and interface leads to the common result that S 2∥ = O(aS ⊥). Using this result, we expect h = O(aS 2A/S 2⊥), so that the interfacial deformation varies as the inverse of the square of the separation distance. We then see that the relation of the curvature to the interface deflection is κ = O(S 2A/S 3⊥). Finally, combining these last two scaling arguments, we obtain κ = O(h3/2/(a3/2S A)). The variation of the curvature with the 3/2 power of the interface deflection was identified empirically by Ledesma-Alonso et al.17 We shall see below that the detailed calculations reflect the above scaling arguments.
1 d ⎛ dH (i) ⎞ 1 ⎜⎜R ⎟⎟ = − 2 R dR ⎝ dR ⎠ (L − 1 + R2)3
(10)
which upon integrating twice gives H (i)(R ) = c 2 + c3 ln R +
ln R 4(L2 − 1)2
ln(L2 − 1 + R2) 8(L2 − 1)2 1 + 8(L2 − 1)(L2 − 1 + R2)
−
(11)
where c2 and c3 are constants. We determine c3 by requiring that the solution remain finite as R → 0, and inspection of eq 11 shows that, for this to be the case, c3 = −1/(4(L2−1)2). Finally, we determine c2 in eq 11 and c1 in eq 9 by matching the inner and outer solutions; i.e., we demand that lim H (i)(R ) = ∼lim H (o)(R̃)
R →∞
4. LINEARIZED ANALYSIS FOR /a ≪ 1 AND ) ≪ 1 As the capillary length S c is usually millimeters, then the Bond number ) ≈ 10−5 in typical AFM geometries. While it is
R →0
(12)
Using the expansion of the modified Bessel function K0 at R = 0, we solve for c1 and c2 and arrive at the full inner solution for the shape of the interface 1429
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H (i)(R ) =
ln 2 − γE − ln ) 2
2
4(L − 1) +
−
ln(L2 − 1 + R2) 8(L2 − 1)2
1 8(L2 − 1)(L2 − 1 + R2)
(13)
where the Euler-Mascheroni constant γE ≈ 0.577. For a solution valid in both inner and outer regions, we add the inner and outer solutions and subtract their overlap, i.e., either side of eq 12, and obtain the composite solution H (R ) =
⎛ 1 1 − ln(L2 − 1 + R2) 2⎜ 4(L − 1) ⎝ 2 2
+
⎞ L2 − 1 + K 0()R ) + ln(R )⎟ 2 2(L − 1 + R ) ⎠ 2
(14)
The maximum interface deflection occurs at R = 0, where an expansion of K 0()R ) about the origin reveals that to leading order H(0) ≈
Figure 3. (a) Semilog plots show the height of the interface as a function of R for ) = {0.1, 1, 10} and dimensionless separation distance L = 1.2. Agreement between analytical and numerical results is seen to diminish with increasing ) . The case where ) = 10 is not shown, as it predicts only negative values for the height and is therefore unphysical. This is to be expected because the matched asymptotics assume ) ≪ 1. (b) The maximum deflection at R = 0 is plotted as a function of L for ) = {0.1, 1, 10}. The numerical data stop at the jump-to-contact condition, at which point lower values of L give no solution. The details of this no-solution region are discussed in the work of Ledesma-Alonso et al.16 The method of matched asymptotic expansions, which is based on ) ≪ 1, predicts no such stopping point but is shown cut off for cleanliness. All data shown are for /a = 10−5.
⎛ 1 1 1⎞ ⎜ln 2 − γ − ln ) − ln(L2 − 1) + ⎟ E 2 2⎠ 4(L2 − 1)2 ⎝ (15)
Our eqs 14 and 15 are similar to eqs 9 and 8 in Liu et al.,14 respectively, but use a scaling based on natural length scales rather than the probe radius. 4.2. Comparison with Numerics. To examine the validity of the asymptotic solution, we first compare it with numerical solutions of the full governing eq 5 calculated using MATLAB’s ODE45 function (see Figure 3). For a range of values for L (the dimensionless separation distance between the sphere and the interface), we examine the shape H(R) and maximum deflection H(0) of the interface using the analytical expressions in eqs 14 and 15. Whereas previous studies18,19 compared solutions only at small Bond number ) , here we examine three Bond numbers, two of which are O(1) or higher. From the results shown in Figure 2, we observe that for ) ≈ 0.1 the analytical and numerical results are in excellent agreement. Second, it is apparent that the solution breaks down for ) = O(1) and higher. Far enough from the origin, all solutions satisfy the boundary condition as R → ∞, but the inner region is inaccurate for ) = 1 and even predicts negative heights and is therefore unphysical for ) = 10. We see that, for ) = O(1) and higher, the assumptions behind the matched asymptotic expansion break down. 4.3. Scaling of the Curvature. The inner solution highlights the importance of the quantity L2 − 1, which we will denote by δ. This nondimensional quantity can be thought of as a representative distance between the sphere and the interface, one slightly larger than the separation distance S ⊥/a. According to eq 10, the dimensionless curvature κ0 at the tip (R = 0) satisfies
|κ0| ∝ δ −3
which was predicted in Section 3 using scaling arguments alone and was deduced empirically by Ledesma-Alonso et al.17 We note that the dependence on /a that they report is also consistent with the results shown here once the definitions of variables are taken into account. 4.4. Contact of the Tip with the Surface. When L is sufficiently small, the full equation for the shape (eq 5) has no solution with the given boundary conditions. We will denote this minimum separation distance as Lcontact. At this value of L, a jump-to-contact process initiates, and the interface wets the sphere. Leading up to this wetting process, the curvature may not be small, and the linearized eq 7 is no longer applicable. Nevertheless, we will assume small values of curvature, estimate Lcontact using the asymptotic solution, and then compare our estimate to one calculated numerically with the full equation for the shape, eq 5. For our analytical estimate, we will take contact to occur when the interface height is equal to the separation distance, i.e., when S − a = h(0), which in dimensionless terms is L − 1 = H(0)/a
(16)
We now seek the value of L that satisfies this equation. In terms of δ, which we assume to be small, i.e., L ≈ 1, we can write L − 1 ≈ δ/2. We can then use the estimate from the inner solution for the maximum interface deflection, H(i) 0 (0), in which case eq 19, after rearrangement, yields
Furthermore, according to the asymptotic solution 14 the maximum deflection scales as
H0(i)(0) ∝ δ −2
(17)
where the prefactor varies logarithmically with Bond number and δ. Combining these last two results yields |κ0| ∝ [H0(i)(0)]3/2
(19)
1/3 ⎡/ ⎛ 1 ⎞⎤ δ ≈ ⎢ a ⎜ln 2 − γE − ln ) + ⎟⎥ ⎣ 2 ⎝ 2 ⎠⎦
(18) 1430
(20)
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where we have excluded the logarithmic correction for δ since it is small compared to the other terms. Equation 20 provides an analytical characterization of contact as a function of the two dimensionless parameters /a and ) , according to 1/3 ⎡/ ⎛ 1 ⎞⎤ Lcontact ≈ 1 + ⎢ a ⎜ln 2 − γE − ln ) + ⎟⎥ ⎣ 16 ⎝ 2 ⎠⎦
W=
(21)
∫
=−
1 R
(24)
I0()ξ)ξ
R 2
∫0
(L − 1 + ξ 2)3 R K 0()ξ)ξ
dξ
(L2 − 1 + ξ 2)3
dξ
(25)
The solution to the original ODE 7 is then H(R) = Hh(R)+Hp(R), and the problem reduces to the determination of constants c1 and c2 such that the boundary conditions are satisfied. The first boundary condition involves the first derivative of H(R) dH (R ) = c1)I1()R ) − c 2 )K1()R ) dR R K 0()ξ)ξ dξ + )I1()R ) 2 2 3 0 (L − 1 + ξ ) R I0()ξ)ξ dξ − )K1()R ) 2 2 3 0 (L − 1 + ξ )
∫
∫
(26)
where L’Hôpital’s rule can be used to show that all terms vanish as R → 0 except for c 2 )K1()R ). Thus, we set c2 = 0 to satisfy the boundary condition at the origin. Applying the boundary condition at infinity is complicated by the singular behavior of eq 25 as R → ∞. Using L’Hôpital’s rule again, however, it is straightforward to show that ⎛ lim H(R ) = I0()R )⎜c1 − R →∞ ⎝
∫0
∞
⎞ d ξ ⎟ (L2 − 1 + ξ 2)3 ⎠ K 0()ξ)ξ
(27)
and so for H to vanish as R → ∞, we choose c1 =
∫0
K 0()ξ)ξ
∞ 2
(L − 1 + ξ 2)3
dξ
(28)
Having obtained the integration constants, we may now write the full expression for H(R) as
(22)
H(R ) = K 0()R )
∫0
+ I0()R )
I0()ξ)ξ
R 2
∫R
(L − 1 + ξ 2)3 ∞ K 0()ξ)ξ
dξ
(L2 − 1 + ξ 2)3
dξ
(29)
The expression involves irreducible integrals, but gives physical insight when considering the maximum deflection of the interface at the origin
R
1 u1(ξ)f (ξ)dξ W (ξ ) 0 R 1 − u1(R ) u 2(ξ)f (ξ)dξ r0 W (ξ)
∫0
− I0()R )
where c1 and c2 are arbitrary constants and I0 and K0 are zerothorder modified Bessel functions of the first and second kind, respectively. With a homogeneous solution in hand, we can solve the full nonhomogenous equation using the method of Variation of Parameters,24 which states that a second-order linear differential equation with homogeneous solutions u1(R) and u2(R) has a particular solution Hp(R) given by
∫r
)I1()R ) −)K1()R )
HP(R ) = K 0()R )
5. LINEARIZED ANALYSIS FOR /a ≪ 1 AND ARBITRARY ) The asymptotic method of Section 4 required that the Bond number ) ≪ 1. This assumption is traditionally valid in AFM geometries because of the nanoscale tip. Ultralow surface tensions in the range of O(10−7−10−5) N m−1 have been reported for oil−water mixtures with added surfactants.22,23 For a tip radius a = 30 nm, such low values of γ could bring ) as high as O(10−4). Even so, it seems reasonable that a low Bond number assumption would still be valid. In Section 6, however, we will discuss the case of a spherical particle over a nanoscale film of liquid resting on a solid substrate. In this case, we will show that the substrate’s first-order effect can be to increase the effective Bond number, and that the effective Bond number can be O(1) or higher. With this as motivation, we will now provide an exact solution to 7 that is valid for all ) . 5.1. Solution via Variation of Parameters. While nonhomogenous, eq 7 is linear and therefore tractable using classical methods. The pair of solutions to the homogeneous problem, i.e., f(R) = 0, is
Hp(R ) = u 2(R )
K 0()R )
By substituting eq 23 into eq 22, we can write the particular solution as
For example, Ledesma-Alonso et al.17 solved the full equation for the shape numerically, and calculated Lcontact = 1.1682 when ) = 10−5 and /a = 10−3. This value of Lcontact differs by about 7% from the estimate according to eq 21, which is Lcontact ≈ 1.0912. The discrepancy is reasonable considering our estimate came from a linearized model. It should be noted that the asymptotic solution demands small values of curvature, so eq 21 will always underpredict the minimum separation distance.
Hh(R ) = c1I0()R ) + c 2K 0()R )
I0()R )
H(0) = c1 = (23)
∫0
K 0()ξ)ξ
∞ 2
(L − 1 + ξ 2)3
dξ
(30)
This result shows, for example, that the interface deflection decreases with increasing ) and L, as one might expect on physical grounds. 5.2. Comparison with Numerics. Next, we compare our solution with numerical results for the full governing equation (see Figure 4). In this case, the analytical and numerical solutions show similar trends both near to and far from the origin. Unlike in the matched asymptotic expansion, the
where r0 is a constant and W denotes the Wronskian of the homogeneous solution set. Note that the value of r0 is immaterial, since the problem’s linearity implies that any change in r0 would simply cause the values of c1 and c2 to adjust accordingly. For simplicity, r0 will henceforth be set to zero. Using standard Bessel function identities, the Wronskian in our case evaluates to 1431
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Figure 5. Sketch of the spherical probe deforming the surface of a liquid nanofilm over a solid substrate.
on a different physical meaning. Specifically, we expect that the interface deflection h, curvature κ, and reduced pressure Πs all tend to zero as r → ∞, and so conclude that p0 = −Πb(0), i.e., it represents the disjoining pressure due to the substrate alone at h = 0. To obtain an expression for Πb(h), we can again recall the results of Hamaker theory,1 where the disjoining pressure near a half-space is given as. 2A Πb(h) = − π (n − 3)(n − 2)(hn − 3) (32a)
Figure 4. (a) Semilog plots show the height of the interface as a function of R for ) = {0.1, 1, 10} and dimensionless separation distance L = 1.2. Agreement between analytical and numerical results is unaffected by ) . (b) The maximum deflection at R = 0 is plotted as a function of L for ) = {0.1, 1, 10}. The numerical data stop at the wetting condition, at which point lower values of L give no solution. The method of variation of parameters predicts no such stopping point, but is shown cut off for cleanliness. All data shown are for /a = 10−5.
=−
for n = 6
(32b)
Substituting eq 32b into 31 gives the equation governing the interface of the thin film
maximum deflection eq 30 remains positive for all Bond numbers and, in fact, is more accurate as the Bond number increases. 5.3. Contact of the Tip with the Surface. As in Section 4.4, the analytical solution cannot predict jump-to-contact condition exactly, but it can be used to estimate the condition for direct contact. We again seek the value of L that satisfies L − 1 = H(0)/a, but this time use eq 30. With irreducible integrals, we are unable to solve for an exact expression for Lcontact, but it is straightforward to calculate the value numerically. Using the same sample values as before, ) = 10−5 and /a = 10−3, eq 30 gives Lcontact = 1.0912. This estimate differs from the numerical solution to the full equation for the shape 5 by about 7%, an accuracy comparable to the solution obtained by matched asymptotic expansions. That said, this estimate will remain valid for all ) , whereas the estimate from matched asymptotic expansion will become unphysical for ) ≥ 1.
⎛ ⎞ dh ⎜ ⎟ r 1 d⎜ dr ⎟ 1/2 ⎟ r dr ⎜ ⎛ dh 2 ⎞ ⎟ ⎜ ⎜1 + ⎟ dr ⎠ ⎠ ⎝⎝
( )
=
S2Aa3 h − (S − h)2 + r 2 − a 2)3 Sc2 S 2 (A / A ) ⎛ 1 1⎞ − A 2 1 ⎜ − 3⎟ 3 8 d ⎠ ⎝ (h + d )
(33)
where we have once again used S A and S c to write the equation in terms of length scales alone. Nondimensionalizing as in eq 5, we arrive at ⎛ ⎞ dH ⎜ ⎟ R dR 1 d ⎜ ⎟ 2 ⎞1/2 ⎟ R dR ⎜ ⎛ d H ⎟ ⎜ ⎜1 + / a2 ⎟ dR ⎠ ⎝⎝ ⎠
( )
6. LIQUID NANOFILM ON A SUBSTRATE One case where a solution to eq 7 via matched asymptotic expansion is inappropriate is when the liquid beneath the spherical particle is a liquid nanofilm resting on a solid substrate. In this case, the vdW attraction from the substrate can contribute significantly to the pull on the interface and affect its shape. The sphere−air−liquid−substrate system is shown in Figure 5, where d is the unperturbed depth of the nanofilm. In this new configuration, the modified Young− Laplace equation becomes 2γκ = Πs(x , a) + Πb(h) − ρgh + p0
A 6πh3
1 ((L − /aH )2 + R2 − 1)3 ⎛ ⎞ ⎟ 1 (⎜ 1 1 − − ⎜ ⎟ 3 8 + 3 ⎜ /aH ⎟ 1 + ⎝ D ⎠
= ) 2H −
(
)
(34)
where two new dimensionless groups have appeared: ( = A 2 /A1, the ratio of the Hamaker constants of the substrate and probe, and + = d /a, the unperturbed depth of the thin film scaled by the radius of the sphere. As mentioned above, we are particularly interested in the implications of small Hamaker numbers. If /a/+ ≪ 1 for
(31)
where we have added Πb(h), the disjoining pressure due to the substrate. In this case, the vdW attraction of the substrate persists for all values of r, and so the constant pressure p0 takes 1432
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instance, a Taylor series expansion of the final term in eq 34 gives ⎛⎛ ⎞ 3(/ H ⎞−3 1 a −3⎜ /aH − (+ ⎜⎜ + 1⎟ − 1⎟⎟ ≈ 4 ⎝ ⎠ 8 8+ ⎝ + ⎠
discussed at least one situation, that of nanofilms, where such an assumption breaks down. For ) ≥ 1, the method of Variation of Parameters gives more accurate predictions of the interface shape, and only introduces the numerical evaluation of two integrals. Neither method is able to capture the physics of the instability that lead to the wetting of the sphere, but both give estimates for contact that for sample conditions are within 10% of those calculated numerically. Finally, a liquid nanofilm was shown to be one example of a case with large effective Bond number, and it was discussed as a possibility for experimentally measuring the Hamaker constant of an unknown material.
(35)
We can identify two solution regimes based on the depth of the liquid compared to the radius of the spherical particle. If + ≫ 1, we observe that the right-hand side of eq 35 becomes small, and therefore conclude that the effect of the substrate is negligible. If, however, the film is thin and + < 1, it is possible that even though /a ≪ 1 and /a/+ ≪ 1, /a/+4 = O(1). For /a ≈ 4 × 10−4 , this limit corresponds to + ≈ 0.15, or d = 3 nm. In the experimental AFM studies of Forcada et al.,18 values of + ranged from 0 to 0.2, showing that the aforementioned assumptions are physically valid in certain AFM setups. Under the conditions where /a/+4 = O(1), eq 34 can be approximated as
■
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS We thank Harold Hatch and Ivan Christov for several helpful conversations about Hamaker theory. We also thank the NSF via grant CBET-1242574 for support of this research.
1 d ⎛⎜ dH ⎞⎟ ⎛ 2 3(/a ⎞ 1 R − ⎜) + ⎟H = − 2 R dR ⎝ dR ⎠ ⎝ 8+4 ⎠ (L − 1 + R2)3
■
(36)
The equation is the same as that for infinite depth, with the exception of the coefficient of H, so that we can think of
()
2
+
3(/a 8+ 4
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1/2
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as an effective Bond number. Since we expect
4
3(/a/(8+ ) = O(1), the effective Bond number may no longer be small. In fact, for low Bond numbers ) the coefficient of the H term is now dominated by the vdW interaction with the substrate. The method of matched asymptotic expansion is no longer applicable, and we must rely on the Variation of Parameters analysis described in the previous section. One reason the case of a liquid nanofilm is particularly interesting is that it suggests an experimental technique for determining the Hamaker constant of an unknown substance. With previous methods, experimental data for the force− distance relationship between the probe and the sample could be used to extract the Hamaker constant of the probe itself. But while many materials can be used for AFM probes, custom designing a probe with a specific material is costly. A cheap alternative would be to vary the material of the substrate beneath a nanofilm. Given a probe of known Hamaker constant, one could use eq 36 to derive a force−distance relation whose only unknown is the Hamaker constant of the substrate. This unknown Hamaker constant could be adjusted until the force−distance data are fitted to the predicted curve.
7. CONCLUSIONS We have considered two analytical solutions to the governing equation for van der Waals driven interface deflection when /a ≪ 1. Both approaches used a dimensionless formulation with a natural length scale that arises when balancing vdW and surface tension forces. The two analytical methods each have advantages and disadvantages. The method of matched asymptotic expansions gives explicit analytical formulas, and, for example, allows a prediction of the scaling of the curvature. A drawback of the method is that it requires ) , or more specifically the coefficient of H, in eq 7, to be small. This condition is usually the case in AFM geometries, but we 1433
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