Noncontact friction via capillary shear interaction at nanoscale

Friction in an ambient condition involves highly nonlinear interactions of capillary force, induced by the capillary-condensed water nanobridges between contact or noncontact asperities of two sliding surfaces. Since the real contact area of sliding solids is much smaller than the apparent contact area, the nanobridges formed on the distant asperities can contribute significantly to the overall friction. Therefore, it is essential to understand how the water nanobridges mediate the ‘noncontact' friction, which helps narrow the gap between our knowledge of friction on the microscopic and macroscopic scales. Here we show, by using noncontact dynamic force spectroscopy, the single capillary bridge generates noncontact friction via its shear interaction. The pinning–depinning dynamics of the nanobridge's contact line produces nonviscous damping, which occurs even without normal load and dominates the capillary-induced hydrodynamic damping. The novel nanofriction mechanism may provide a deeper microscopic view of macroscopic friction in air where numerous asperities exist.

(main text), one should be able to vary R ψ by small amounts. However, it is experimentally difficult to achieve fine variations of R ψ only by changing RH; we actually realized slight increase of R ψ by repeating the approach-retraction cycles at a given position on the substrate as well as at a given RH. In other words, as the water-bridge formation is performed between the tip and sample, some water molecules are left on the surfaces of tip and sample, which makes the water bridge that is obtained at the next approach-retraction cycle slightly bigger than the previous one due to the presence of the pre-existing water layer on the surfaces. As shown in the figure, while R ψ shows an overall increase with RH, several closely-spaced values of R ψ are obtained at a given RH. This feature was observed in all the experiments on RH dependence, especially prominent at high RH. Error bars indicate the maximum variation (±1%) of RH. It is important to notice that the same value of k eff or b eff w is observed for the identical value of R ψ (Fig. 4, text) even if R ψ can be obtained at different values of RH, which validates our experimental method. Notice also that measurements at different values of RH are made at different positions on the sample surface.

Supplementary Note 1
In ambient condition, water bridges are typically formed below ~10 nm separation. Therefore, between two macroscopic contacting surfaces with roughness of ~10 nm, most of the noncontact asperities will form the capillary-condensed bridges. On the other hand, for contact sliding between macroscopic surfaces with very large roughness, water bridges may not be easily formed between noncontact asperities. The actual distance between noncontact asperities depends on the weight of the macroscopic bodies themselves as well as the externally exerted normal force.

Determination of the tip-sample contact point
In our shear-mode AFM, we carefully define the tip-sample zero distance, based on the full approach-retraction curves of the elastic and damping coefficients, k eff and b eff w, as well as the energy dissipation E dissi [1]. Supplementary Figure 1(a) shows a full cycle of approachretraction of k eff and b eff w. As the tip approaches the sample surface, the capillary condensation first occurs at z=0 with the subsequent increase of k eff and b eff w, the values of k eff and b eff w show abrupt increase at about z=-4.6 nm, and then the tip retracts from z=-5 nm to the original position. Interestingly, both k eff and b eff w increase rapidly at z=-4.6 nm during approach, whereas the energy dissipation E dissi first increases, but then decreases suddenly near the same position of z=-4.6 nm, as shown in Supplementary Figure 1 This abrupt decrease of E dissi indicates that the tip-substrate interaction becomes increasingly elastic and thus energy dissipation becomes negligible, which occurs when the tip-sample contact is made. In our analysis of experimental results, therefore, we have set the contact point at z=-4.6nm, where k eff and b eff w abruptly increase and E dissi rapidly decreases. We find that this definition of the contact point is proper enough to describe the geometry of the capillary bridge formed in the tip-sample gap and the associated interaction forces. Moreover, we emphasize that the experimental results vary only within less than 20% variation (z=-4.6±1 nm) in the contact point and thus the main conclusions made are not affected. Notice that there have been reports on the capillary condensation of the water nanobridges at the distance far beyond the Kelvin radius, similar to our case (see for example, [2]).

Determination of the geometrical quantity R ψ of the capillary liquid-bridge
To obtain the contact-line radius R ψ (Supplementary Figure 2(a)) at a specific tip-sample distance (for example, z=1 nm or h ψ ≈ 5.6 nm as in the case of Fig. 4 of the main text), which is an important parameter as shown in Supplementary Equations (1-3), we use the theoretical formalism described in ref. [3], based on the Young-Laplace equation. This method allows to determine the detailed geometrical parameter of a liquid bridge such as R ψ using the boundary condition of constant volume, for the given tip's radius of curvature R, tip-bridge contact angle θ, substrate-bridge contact angle θ', and volume of the liquid bridge V 0 . Here the contact angle θ is obtained as where φ 0 is the equilibrium angle at x=0 (see Supplementary Figure 2(a)). In addition, we use Eq. (14) in ref. [4] to obtain V 0 from the experimentally measured rupture distance d r (measured with respect to the contact point; see Supplementary Note 2) under the assumption θ'=θ. Therefore, once R, θ and d r are known, R ψ at a specific distance h ψ can be uniquely determined by this formalism.
Supplementary Figure 3 presents the flow diagram that details the iterative derivation of the liquid-tip contact radius R ψ . Although d r and h ψ can be directly measured by experiment, it is difficult to know a priori the values of R and θ experimentally. Therefore, to calculate R ψ with unknown R and θ, we use the following iterative procedures: we (i) first pre-assume the values of R and θ, (ii) calculate R ψ by using experimentally measured d r and h ψ , (iii) extract θ from the slope of the linear fit for k eff (R ψ ), which is given by k eff h ψ /(γπ sinφ 0 ) (using Supplementary Equation (2) and κ k ≈1/h ψ as shown in Supplementary Note 5), and then (iv) compare the extracted θ to the pre-assumed θ to find out the self-consistent values of R and θ, which allows the desired value of R ψ . By this method, we obtained R=400 nm and θ =33˚. As a result, once R ψ is known, we can plot k eff as a function of R ψ , as shown in Fig. 4 of the main text. Notice that we also confirmed the linearity of k eff and b eff w (in Fig. 4) is unaffected for 12 different combinations of parameters; R=300, 400, 500 nm, infinity (flat) and θ=23, 33, 43°, while only about 10% variation occurs in the calculated absolute values of R ψ .

Supplementary Note 4 Estimation of the hydrodynamic interaction on the tip-liquid contact interface
Hydrodynamic interaction between the tip surface and the liquid bridge is associated with the fluid dynamics within the confined liquid and the corresponding shear stress exerted on the tip surface. The resulting interaction is estimated by the sum of the damping forces arising from (i) the Couette flow in the most area of the tip-liquid interface and (ii) the viscous flow within the wedge of the liquid near the moving contact-line. The hydrodynamic damping of (i) is well described by the formula, wμσ/h ψ , presented in the main text, for the fluid system of low Reynolds number (≡ρwAh ψ /μ ~10 -6 ). This interaction results in a damping coefficient of about 10 -5 N/m, much smaller than the observed value of 1 N/m, as discussed in the main text. On the other hand, the viscous interaction of (ii) occurring in the proximity to the contact-line originates from the bending of the liquid/vapor interface, resulting in the change in the contact angle. The contact-angle change is estimated to be φ 3 -φ eq 3 =9 Ca l [5], as shown in the main text. Here l=log(l m /l 0 ) is the ratio of macroscopic to microscopic length scales, which typically falls into the range of 1~100. And Ca is the capillary number defined by μwA/γ, where μ is the viscosity of liquid, w the tip's oscillation frequency, A the oscillation amplitude and γ the surface tension. Then, we obtain a relation between the angle change and the velocity, given by Δφ/A=μwl/(3γφ eq 2 ). By using Supplementary Equation (3), and also employing the numerically obtained parameters for our tip-fluid system described in Supplementary Note 3, we can finally estimate the damping coefficient as ~10 -4 N/m for R ψ =15 nm and l=100, which is 10 4 times smaller than the observed damping ~1 N/m. Therefore, we conclude that the hydrodynamic interaction cannot address the observed enhanced damping as well as the elastic forces.

Theory of the contact-line-induced shear interaction
To derive the capillary 'shear' interaction originating from the three-phase contact-line on the tip-bridge interface, we discuss how the tip-liquid angle φ is changed as the tip (radius of curvature R) moves laterally (Supplementary Figure 2(a)). When the tip moves to the right in the x direction, the shape of the liquid bridge capillary-condensed in the tip-substrate nano gap is slanted, changing the tip-liquid angle φ(Ω). We define φ(Ω=0)≡φ r (i.e., the angle at the very right end in the side view) and φ(Ω=π)≡φ l (i.e., the angle at the very left end). In general, φ(Ω) depends on the tip surface-liquid wetting property as well as the tip's motion. For example, if the tip is super-hydrophilic (super-hydrophobic), then φ(Ω)=ψ (φ(Ω)=ψ+π) by definition. Although one can derive φ(Ω) by considering the three-dimensional shape of the slanted liquid bridge for a complete description, we assume a circular distribution of φ(Ω) for small displacement x to obtain the analytical form of the interaction, as described in Supplementary Figure 2(a).
Here the inner circle with a radius r in the ξ-η plane (the blue-colored coordinate frame in Supplementary Figure 2(a)) determines the magnitude of the horizontal component of the tangential force f in such a way that the force is proportional to the physical length of the red arrow l; f(Ω)= τ l(Ω), where τ is a constant. Then, the two unknowns r and τ can be exactly determined by the two known relations, f(0)=γcosφ r and f(π)=γcosφ l , where γ is the surface tension of the liquid. By integrating f(Ω) over 0 ≤ Ω < 2π, we obtain the contact-line-induced force exerted by the capillary liquid-bridge,

(Supplementary Equation 1)
Supplementary Equation (1) describes the instantaneous value of the capillary shear force in terms of the two contact angles φ r ≡φ(Ω=0) and φ l ≡φ(Ω=π), which generally depend on the tip displacement x and velocity v, so that φ r =φ r (x,v), as schematically presented in Supplementary Figure 2(b). In order to investigate the quantitative characteristics of the shear force, we adopt a simple form of φ r =φ r (x,v) as an ellipse that connects smoothly the four fixed points φ -A , φ A , φ max , and φ min . As a result, a functional form of F line can be obtained, from which we can derive the elastic and damping constants of the shear interaction using the relations, k eff =F k /A and b eff w=F b /A, Here κ k ≡(φ -A -φ A )/(2A) is the slope of the fitted ellipse, κ b ≡ (φ max -φ min )/(2A) characterizes the hysteresis at x=0, φ 0 is the equilibrium angle at x=0, and R ψ is the contact-line radius (Supplementary Figure 2(a)).
Although Supplementary Equations (2,3) formally describe k eff and b eff in terms of the angle changes κ k and κ b , respectively, this mechanistic description alone does not predict the detailed values of κ k and κ b for a given shear motion of the tip. First, for quantitative investigation of κ k , we consider simply an ideal cylindrical shape of the liquid bridge. When shearing the liquid column (φ 0 =π/2), we can obtain analytically the angle change under small oscillation as follows, Using this approximation, we find k eff =πγ (1/h ψ )R ψ and this is exactly the leading term of k eff derived from Eq. (1) in the main text, which is based on the explicit calculation of the interfacial energy change for the cylindrical bridge (see the text below Eq. (1) in the main text).
For the damping coefficient, on the other hand, we consider the 'hopping' dynamics of the capillary contact-line, as discussed in the main text. For detailed derivation of the damping constant, one has to sum over the externally imposed forces per unit length on the contactline, f l =γ(cosφ eq -cosφ d ), and the explicit summation results in the analytic expression that is proportional to the term, (cosφ r -cosφ l ). Notice that Supplementary Equation (1) connects this term, (cosφ r -cosφ l ), to the total shear force F line , which is reduced to F b for x=0 (or v=wA). Furthermore, the 'hopping' model relates the damping force F b (x=0, v=wA) with the contactline velocity V, which is proportional to the tip's shear velocity wA (refer to the main text for details). Following these steps, we finally obtain Eq. (2) in the main text.

Supplementary Note 6
Derivation of the surface-energy change U l for a sheared cylindrical column Here we present detailed derivation of the surface-energy change U l for the simple cylindrical column of water (see inset of Fig. 2(a) in the main text). Consider the top circular surface of the cylindrical column whose radius is R ψ . Under the shear displacement of x along the x-axis, the incremental surface area dS of the side wall of the cylinder is given by, where is the polar angle with respect to the x-axis and is the height of the sheared cylinder. The corresponding increase of the surface energy for a given x, dU l , is then written as where γ is the surface tension of water. Therefore, the resulting total increase of the surface energy U l is given by, which can be rewritten as,

(Supplementary Equation 5)
Here the function E is the complete elliptic integral of the second kind. Notice that the total increase of the surface area S for a given x is S=4 dS=

Supplementary Note 7
Shear-amplitude dependence of k eff and b eff w for a liquid bridge with decreasing R ψ In the main text, we have presented the shear-amplitude dependence of k eff and b eff w for a liquid bridge with the constant volume or constant R ψ (Fig. 3 in the main text). We emphasize this condition of constant volume is confirmed by the fact that the interaction force measured after the amplitude-dependence experiment done at a given RH is fully recovered to the initial value of the force (Supplementary Figure 4(a)). Under this condition of constant bridge volume, we have observed the logarithmically increasing shear frictional force with the increase the shear amplitude (or velocity), as shown in the inset of Fig. 3(b) and also predicted by Eq. (2) in the main text. Now let us consider the noncontact sliding friction mediated by a liquid bridge whose volume decreases in time during the shear motion. This diminishing bridge volume leads to the decreasing R ψ during the friction measurement, as typically observed in the contact friction experiments that employ a linearly moving cantilever, which is due to the finite nucleation or growth time of the bridge [6]. Interestingly, in our experiment, the similar decrease of R ψ is observed in shear measurement for the liquid bridge elongated up to z=5.2 nm, much longer than that in Fig. 3 of the main text. For this longer bridge, the evaporation process of the bridge is more favorable and thus R ψ decreases rapidly with the shear motion.
Supplementary Figure 4(c,d) shows the experimentally measured k eff and b eff w for a capillary bridge whose R ψ decreases with the increase of A, i.e., for the bridge sheared at z=5.2 nm. As expected, we find that both the elastic and damping forces start increasing following the curves similar to those for the bridge sheared at z=1 nm, but then both decrease rapidly due to the diminishing R ψ despite the increase of A. In fact, we observed no remaining shear forces when measured after the amplitude-dependence experiments were performed, unlike the z=1 nm case, which indicates that the liquid bridge indeed vanished during shear motions ( Supplementary Figure 4(b)). In other words, the friction initially increases due to the dominant pinning-depinning dynamics of the contact-line, but then diminishes because of the reducing volume or R ψ (see Eq. (2) in the main text).