Elsevier

Surface Science

Volume 517, Issues 1–3, 1 October 2002, Pages 157-176
Surface Science

Contact angle dependence of solid probe–liquid drop forces in AFM measurements

https://doi.org/10.1016/S0039-6028(02)02054-XGet rights and content

Abstract

In order to obtain the force between an atomic force microscopy probe particle and a liquid drop, we require the small change δR in the mean radius of curvature of the drop, which is proportional to the small change in liquid pressure inside the drop. These quantities depend on the total force F, the contact angle θ made by the unperturbed drop at the substrate, and the nature of the boundary conditions imposed at the contact line. Chan et al. [J. Colloid Interface Sci. 236 (2001) 141] developed a semi-analytical method for calculating the force in the fixed contact line case, while more recently Attard and Miklavcic [Langmuir 17 (2001) 8217] considered the fixed contact angle condition. Due to errors in perturbation theory calculations, the results presented by these authors for δR are incorrect, and their results [J. Colloid Interface Sci. 236 (2001) 141; Langmuir 17 (2001) 8217; J. Colloid Interface Sci. 247 (2002) 255] for the force are thus also in error. In contrast with the results of the above authors, we find that δR is singular at θ=0, and perturbation theory breaks down at small θ. Consequently we find that, for angles of practical interest, δR and thus the change in fluid pressure inside the drop can be orders of magnitude larger than previous calculations suggest. Modifying the method from [J. Colloid Interface Sci. 236 (2001) 141] accordingly, we present indicative numerical force F(X) curves (where X is the central probe-stage separation) for both boundary conditions. The general perturbation expansions are invalid near θ=π/2. We therefore extend the results into this experimentally significant region with a separate analysis at θ=π/2. Additionally, a nonperturbative matched expansion is developed to extend the results to small θ and to verify all analytical results.

Introduction

Following the early work of Ducker et al. [4], atomic force microscopy (AFM) has been applied to direct measurement of forces between gas bubbles [5], [6], [7] or liquid drops [8], [9], [10], [11], [12] and solid colloidal probe particles.

The theoretical challenge posed by these measurements is to allow, in the force calculation, for deformation of the interface, itself dependent on the induced pressure distribution. The problem has received attention along numerical [12], [13] and analytical [1], [2], [3], [14], [15] lines.

This paper addresses the interaction of a solid colloidal probe with an incompressible liquid drop, however as has been noted elsewhere [2], [13], the finite compressibility correction for gas bubbles is negligible in typical experimental situations. Thus the treatment here applies equally to bubbles.

Numerical solutions of the augmented Young–Laplace equation were used by Bhatt et al. [13] to obtain interface deformation and force–distance curves. An iteration on the central drop height and the pressure change in the drop was used to simultaneously implement fixed contact line and constant volume constraints. Similar numerical solutions were examined by Aston and Berg [12], in conjunction with experimental results.

Difficulties arise, however, in interpreting these numerical results, since the central drop height is sensitive to the small change in drop liquid pressure, requiring evaluation of the drop volume to very high numerical resolution [15].

Earlier numerical work focussed on bubble–bubble [16] and liquid drop–planar solid [17], [18] interactions. Bachmann and Mikavcic [18] assume the drop contact line is pinned to a capillary. Miklavcic et al. [17] make a similar assumption, but also discuss an example where the drop extends across a substrate and the contact angle θ is governed by Young’s equation,γcosθ=γSA−γSL,i.e. the constant θ condition, where γSL and γSA are respectively the surface free energy densities of the substrate–liquid drop interface and the substrate interface with the bathing aqueous solution, and γ is the droplet surface tension.

Young’s equation (1) will apply to real surfaces (i.e. in the presence of physical and chemical defects), provided the defects are sufficiently weak [19]. The presence of strong defects will lead to pinning [19] of the contact line, corresponding to contact angle hysteresis, and requires the use of fixed contact line boundary conditions.

Miklavcic [14] considered the interaction via electrical double-layer forces of a deformable droplet (pinned to a substrate) with a solid sphere, using perturbation theory to obtain the first order deformations away from sphericity. This approach was found to be appropriate for cases where the Debye length scale is at most a few orders of magnitude smaller than the unperturbed droplet radius of curvature, but it was noted that for cases where the droplet radius is many orders of magnitude larger, a matched asymptotic expansion would be more suitable.

This suggestion was implemented in Chan et al. [1] for purely repulsive potentials and fixed contact line, for the drop geometry and also for a spherical probe interacting with an (initially) planar deformable substrate. The matched expansion has the advantage that the Young–Laplace equation is solved analytically over the vast majority of the drop profile, alleviating the difficulties associated with purely numerical calculations. The method has been extended to potentials with attractive components by Dagastine and White [15].

Attard and Miklavcic [2] consider the problem under fixed contact angle assumptions in the weak force regime, also using a matched expansion approach.

In analyzing colloidal interactions between solid AFM probe particles and liquid drops, we wish to obtain the total force F(X) as a function of distance between the probe and the rigid substrate (i.e. the AFM stage). A crucial element in the calculation is the small change δR in the mean radius of curvature of the drop, which is proportional to the small change in liquid pressure inside the dropδPdrop=−2γδRR02.These perturbation quantities depend on the total force F, the contact angle θ made by the unperturbed drop at the substrate, and the nature of the boundary conditions imposed at the contact line.

In experiments such as those of Hartley et al. [11], where the drop is deposited onto a uniform substrate, the appropriate boundary conditions will depend on the surface defect structure at the contact line, which can of course be controlled [20]. We will see that the motion of a free contact line (implied by fixing θ) would typically occur on a submicron scale in AFM experiments.

The probe-AFM stage separation can be expressed [1] as a function of the central liquid drop–probe separation D0,X(D0)=z0+D0+H(D0)+G(D0)12lnaD04R02+P(θ),where a is the probe particle radius, H and G are certain integrals of the disjoining pressure over the (numerically calculated) drop profile in the inner (near to probe) region, z0 is the central height of the unperturbed drop and θ is the contact angle at the base of the unperturbed drop, which has curvature R0−1. The total force exerted by the probe is 2πγG, so that (3) provides the force–distance relationship. Typical length scales for AFM experiments [11], [12] are a∼5–10 μm and R0∼0.25–0.5 mm. From the maximum force values and surface tensions [11], [12], we see that Gmax∼0.01–0.4 μm, so that at the highest experimental loads the ratio G/R0 is typically of order 10−4 to 2×10−3, and is therefore a useful expansion parameter.

Under the fixed contact line assumption, where the radial extent r1 of the drop on the stage remains constant, the perturbation to R0, derived in [1], isδR=−G1−cosθ−2+13cos2θ+13cos3θ,which then leads [1] to a contact angle dependence in (3) ofP(θ)=1−cosθ2−13cos2θ−13cos3θ+12ln1+cosθ1−cosθ.We show here, however that the correct perturbation to R0 isδR=−G11−cosθ,resulting in the contact angle dependence,P(θ)=1+12ln1+cosθ1−cosθ.Note that in addition to the probe–drop calculation, a calculation was also carried out in [1] for a spherical probe interacting with a planar substrate. The planar results are unaffected by the erroneous δR calculation. In a follow-up paper, Dagastine and White [15] extend the treatment in [1] to include a general colloidal force, with both attractive and repulsive components. The expression (7) developed here is employed in [15] for P(θ).

The constant θ (free contact line) condition was treated in [2], with the perturbation resultδR=G1−cosθ2−cosθ−cos3θ,which is positive for G>0 (i.e. when the total force is repulsive), implying via (2) that the effect of a repulsive probe on the drop is to reduce the liquid pressure in the drop. Since this is physically implausible, (8) clearly cannot be correct. We find that the correct expression isδR=−G1(2+cosθ)(1−cosθ)and that the corresponding function P(θ) for constant θ is thenP(θ)=1+cosθ2+cosθ+12ln1+cosθ1−cosθ.

We show that the perturbation treatment breaks down for small θ (i.e. small drop volume), as would be expected physically. This breakdown is related to the singularity of δR and thus of the liquid pressure and surface curvature perturbations, evident in , , but absent in the relevant expressions (4) and (8) from [1] and [2] respectively.

A detailed analysis is given of the angular domain of validity of the perturbation theory results for δR.

The θ-dependent perturbation treatment presented here, in common with those of [1] and [2], is invalid near θ=π/2, due to the structure of the small G/R0 expansion. Since this region is obviously experimentally important, we conduct separate analyses for θ=π/2 for both constant θ and constant r1, leading in the latter case to a second order expansion. We show that the δR and P(θ) formulae, derived under the assumption that θ is not too close to π/2, do in fact give the correct θ=π/2 limit.

A nonperturbative matched asymptotic expansion is developed, avoiding inessential assumptions on θ. This method, while still invoking a small G/R expansion, does not assume that the change in curvature of the perturbed drop is small (i.e. of order G), in contrast with the perturbative treatment. We apply the resulting numerical method to determine the accuracy of the perturbative formulae for P(θ), confirming that the analytical formulae for P(θ) are useful for a large angular region, and, in particular, remain accurate in the region θπ/2.

Numerical results for F(X) are presented in some cases of interest, for both fixed r1 and fixed θ boundary conditions. We show that boundary condition effects on F(X) are greatest for obtuse drops with θ∼160°.

Section snippets

Formalism

The development in this section is based on that of [1], with some additional details provided for clarity. Throughout the paper, r1 denotes the contact line radius of the perturbed drop and θp is the perturbed contact angle.

We consider a drop or bubble of density ρ1, situated on an AFM stage, and immersed in a bathing medium of density ρ2. The unperturbed drop is taken to be a portion of a sphere of radius R0, with R0λ, where λ=(γg−1|ρ1ρ2|−1)1/2 is the capillary length (g is the

Small G/R0 expansion

The solutions to (30) can be expressed in terms of the incomplete elliptic integrals [21] E and F,z(r)=αR+E(K(r),q)−GRR+F(K(r),q)R+E(K(r1),q)−GRR+F(K(r1),q),withK(r)=arcsinR+2−r2R+2−G2R2R+−2,q2=1−G2R2R+4and the upper sign now refers to acute θp, and the lower to obtuse θp. The parameter α is used to specify the solutions corresponding to the upper and lower sections of the drop. For the acute case (where only the upper part is physically relevant) and for the upper part of the obtuse case, α

Fixed contact line

If we assume a fixed contact line, r1 remains constant, and is related to the unperturbed contact angle θ,cosθ=±1−r12R02,where the upper sign refers to acute, and the lower to obtuse unperturbed contact angles. Recasting (43) as an expansion in G/R0 and using the above relationship (45) between r1 and θ, we findVp−Vu=πR03δRR0−1+2cosθ−cos2θcosθ+GR0−1+cosθcosθ+OG2R02.This result differs drastically from the corresponding result (Eq. (A.51)) in [1]. Setting VpVu to zero yields, from (46), the

Fixed contact angle

When the contact angle θ remains fixed while the contact line is free to move, the perturbed contact radius r1 is a more complicated function of θ than in the fixed contact line case. A constant contact angle is equivalent to constant profile derivative at the contact line, i.e.tanθ=tanθp=∓G−r12Rr12G−r12R2,where we have used (30). The upper sign is for acute contact angles and the lower for obtuse. Solving for r12, we findr12=RG1+R2G+1tan2θ+R2Gtan2θ1+4GRsin2θ1+tan2θ.Expanding (49) in G/R0, we

Nonperturbative matched expansion in G/R

The nonsingular nature of both , at π/2 suggests that these formulae for P(θ) may be accurate beyond the angular range in which their derivation remains valid. To assess the true range of validity of these expressions, we pursue an approach which requires only that G/R≪1, for both fixed contact angle and fixed contact line cases. That is, we avoid expansions which introduce inessential conditions on the contact angle. This leads to a nonperturbative approach: we are still considering

The region θπ/2

Prior to applying the above numerical approach for θπ/2, we briefly digress to utilize , as the basis for extending the analytical (perturbation) treatment to special case θ=π/2, for both fixed r1 and fixed θ boundary conditions. We first expand the matching region terms in (56) in G/R,R0323cosθ+13cos3θ=R323+GR+OG2R2r13(R+2−r12)(r12−G2R2R+−2)±(RG)23R+F(K(r1),q)∓2R2+RGR+3E(K(r1),q),retaining in full the terms in r1 and using the estimate r0O(RG) for the matching coordinate.

The region θ∼0

We now examine the numerical solutions of , as θ is reduced to values where the perturbation analysis is expected to break down, but large enough that the nonperturbative treatment is valid.

The nonperturbative approach requires |sinθ|>GR/R0 in the small θ region (see Appendix A), which will be satisfied for |sinθ|>G/R0, since R<R0 (i.e. the perturbed curvature is greater). The G/R expansion in the matching region will lose accuracy for small θ, as R decreases with θ. This ratio must therefore

Numerical F(X) results

Having assembled analytical expressions for P(θ)|r1 and P(θ)|θ and determined their domains of validity, we are now in a position to calculate some indicative force curves F(X), using numerical solutions of the inner equation (20) to calculate the integrals G(D0)=(2πγ)−1F(D0) and H(D0). Then from (3), X(D0) is evaluated as a function of central separation D0, yielding F(X), parametrically in D0.

As in [1], a purely repulsive colloidal interaction is used for the disjoining pressure, calculated

Conclusion

We have investigated the contact angle dependence, P(θ), of the AFM observable, X, for the solid probe–liquid drop configuration, pursuing both an analytical perturbation theory approach and a nonperturbative numerical approach, for both fixed contact line (constant r1) and fixed contact angle (constant θ) conditions at the liquid boundary with the substrate.

The analytical results given here, which we confirm by extensive numerical investigation, differ radically from those given in Refs. [1],

Acknowledgements

The author was introduced to Ref. [1] by Derek Chan, who suggested the fixed θ calculation in order to examine the effect of boundary conditions on experimental results, and provided the Mathematica code used in [1] for numerical evaluation of F(X). This code was then modified for the present work.

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