Universal and Scaling Behavior at the Proximity of the Solid to the

Aug 17, 2007 - Spherical solid model system: Exact evaluation of the van der Waals interaction between a microscopic or submacroscopic spherical solid...
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Langmuir 2007, 23, 12119-12124

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Universal and Scaling Behavior at the Proximity of the Solid to the Deformable Air-Water Interface Y. Z. Wang, D. Wu, X. M. Xiong, and J. X. Zhang* State Key Laboratory of Optoelectronic Materials and Technologies, and Department of Physics, Sun Yat-sen UniVersity, Guangzhou, 510275 China ReceiVed April 6, 2007. In Final Form: July 3, 2007 The scaling descriptions of the deformation of the air-water interface due to van der Waals attractive forces induced by a paraboloid shaped solid as well as of the force vs distance behavior are systematically discussed theoretically and experimentally. It is demonstrated that the force-distance curves at the proximity of the solid to the air-water interface without contact satisfy a simple and universal scaling law, which can be useful to help study various systems involved in the deformable interface. Moreover, an analytical solution to the E-L differential equation governing the deformation of the water surface profile is obtained from the scaling relation, and the two length scales that quantitatively evaluate the lateral and longitudinal deformation of the air-water interface respectively are hence determined.

I. Introduction The interactions of fluid entities with each other or with solid bodies in the controlled gaseous or liquid environments are of fundamental interest in technological and industrial areas, such as emulsions, froth flotation, and water treatment.1-3 To better understand these interactions, the interacting force and resulting interface deformation are the most essential elements. In recent years, great efforts have been made in both experimental measurements and theoretical works of these interactions. Atomic force microscopy (AFM), for example, has been used to investigate the properties of the liquid film on the substrate,4-5 the formation and rupture of water bridge between the tip and the glass substrate,6 the wetting and capillary bridging forces on various low/high-energy surfaces,7 as well as the surface forces and deformations at the oil-water interface or at the bubble surface in aqueous solutions.2,8 Meanwhile, researchers have also theoretically studied the mutual attractive interaction between the AFM tip and the liquid interface or between two solidsupported liquid films,5,9 the interaction between the emulsion droplets of different sizes10 and the colloidal interactions.11-20 However, few analytical solutions were given in these works * Corresponding author. E-mail: [email protected]. (1) Leja, J. Surface Chemistry of Froth Flotation; Plenum: New York, 1982. (2) Ducker, W. A.; Xu, Z. G.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (3) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (4) Mate, C. M.; Lorenz, M. R. F. F.; Novotny, V. J. J. Chem. Phys. 1989, 90, 7550. (5) Forcada, M. L.; Jakas, M. M.; Gras-Marti, A. J. Chem. Phys. 1991, 95, 706. (6) Sirghi, L.; Szoszkiewicz, R.; Riedo, E. Langmuir 2006, 22, 1093. (7) Malotky, D. L.; Chaudhury, M. K. Langmuir 2001, 17, 7823. (8) Hartley, P. G.; Grieser, F.; Mulvaney, P.; Stevens, G. W. Langmuir 1999, 15, 7282. (9) Forcada, M. L. J. Chem. Phys. 1993, 98, 638. (10) Denkov, N. D.; Petsev, N. D.; Danov, K. D. Phys. ReV. Lett. 1993, 71, 3326. (11) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357. (12) Bachmann, D. J.; Miklavcic, S. J. Langmuir 1996, 12, 4197. (13) Miklavcic, S. J. Phys. ReV. E. 1996, 54, 6551. (14) Miklavcic, S. J. Phys. ReV. E. 1998, 57, 561. (15) Miklavcic. S. J.; Attard, P. J. J. Phys. A: Math Gen. 2001, 34, 7849. (16) Bhatt, D.; Newman, J.; Radke, C. J. Langmuir 2001, 17, 116. (17) Attard, P.; Miklavcic, S. J. langmuir 2001, 17, 8217. (18) Attard, P.; Miklavcic, S. J. J. Colloid Interface, Sci. 2002, 247, 255. (19) Chan, D. Y. C.; Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2001, 236, 141. (20) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2002, 247, 310.

because the nonlinear equations corresponding to the practical interaction behavior can hardly be solved by general mathematical methods, and thus, some of the fundamental issues are still ambiguous.21 Following up Cortat and Miklavcic’s works,21-22 we employed the collocation-integration method23 to investigate the scaling behavior of the deformation of the air-water interface dependent on the solid geometry, the distance between the solid and the air-water interface, and the strength of the attractive van der Waals force in the solid-air-water system. As a result, the lateral and longitudinal deformation of the air-water interface as well as the interaction forces are quantitatively described by the universal scaling functions with the scaling parameters of the system (ω( λ,zp0,A),b,bF). It is demonstrated that the scaling behavior of the experimental force-distance curves is compatible with the theoretical prediction. At last a reliable analytical solution of the corresponding E-L differential equation is derived and thus well describes the relation between the deformation of the air-water interface and the system parameters λ,zp0 and A.

II. Theoretical Model In this paper, the issues are analyzed based on the theoretical model (see Figure 1) initially presented by Cortat and Miklavcic.21 Figure 1 gives the schematic of this solid-air-water system in cross-section. The solid is modeled as a paraboloid described by the function zp(r) ) zp0 + λrp2, where zp0 denotes the lowestmost point, and the splay parameter λ governs the shape of the tip. The water surface profile {r,z(r)} is determined in terms of minimization of the total free energy of the air-water interface, which can be written as21

F ) 2π

∫0∞ r[(γ + σ)W(r) + G2 z2] dr

(1)

where the first term is the surface energy resulting from the surface tension γ, the second term σ ) -A/2D2 corresponds to the attractive van der Waals energy, and the last one is the change in gravitational potential energy of the water lifted by the attractive force. Here, W(r) ) [1 + zr2]1/2 is an area scaling factor, G ) (21) Cortat, F. P. A.; Miklavcic, S. J. Langmuir 2004, 20, 3208. (22) Cortat, F. P. A.; Miklavcic, S. J. Phys. ReV. E 2003, 68, 052601. (23) Mathews, J. H.; Fink, K. D. Numerical Methods Using MATLAB, 3rd ed.; Electronics Industry Press: Beijing, 2002.

10.1021/la700980v CCC: $37.00 © 2007 American Chemical Society Published on Web 08/17/2007

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normalized profiles are quite superposed with each other (see in Figure 2, panels b* and c*). To further understand the properties of the interaction behavior, we perform the nonlinear fitting for the numerical data of the normalized profiles (Figure 2, panels a*-c*). By varying λ ∈ [0.1,1000] for fixed zp0 and A, varying A ∈ [5e - 21J,20e - 19J] for fixed splay parameter and zp0, or approaching the solid gradually without contact for fixed splay parameter and Hamaker constant respectively, the best fit of the form can be obtained

Figure 1. Schematic of the solid-air-water system in crosssection.21-22 D(r) is the distance vector establishing a one-to-one mapping between points on the profile{ r,z(r)} and those on the paraboloid zp(r) ) zp0 + λrp2, where zp0 denotes the lowest-most point, and the splay parameter λ governs the shape of the solid. The water profile is a solution to eq 2 for van der Waals interaction between the surface.

∆Fg is the notation by multiplying the density difference ∆F with the gravitational acceleration g, and D(r,z(r)) is the distance vector establishing a one-to-one mapping between points on the profile {r,z(r)} and those on the paraboloid { rp,zp(rp)}. In terms of the fundamental theorem that forces acting on the air-water surface are in balance and the static equilibrium can be achieved when the interfacial energy functional for a special interface profile is minimum or maximum, an Euler-Lagrange differential equation can then be obtained21

[

] [

]

∂σ(r,z) d rzr(r) (γ + σ(r,z)) ) r Gz + W dr W ∂z

(2)

Here we use the collocation-integration technique,23 quite different from the Runge-Kutta one21 but more powerful and convenient, to solve this E-L equation with two optimally refined boundary conditions {zr(0) ) 0,z(∞) ) 0} over the interval of integration r ∈ [0,1] cm, with 10-6 as absolute tolerance and 10-3 as relative tolerance (standard values in commercial solver packages).

III. Scaling Behavior as the Solid Approaches the Water Surface Without Contact A. Scaling Profile of the Deformable Air-Water Interface. Ignoring the unstable solution, which is corresponding to the maximum energy and is difficult to measure in the experiment,21 we focus on the stable solution and its dependence on the solid geometry, the distance between solid and water-interface, and the strength of the van der Waals force. Profiles of the air-water interface dependent on λ, zp0, and A are shown in Figure 2, panels a-c, respectively. After dividing the profile points by the apex value, we get the normalized profiles (Figure 2, panels a*-c*) which show the extent ratio of lateral deformation to longitudinal deformation. In Figure 2a, the change of deformation in both lateral and longitudinal directions induced by decreasing λ arises from the increased van der Waals force, which can be seen as a result of shortening the separations between the profile points of the solid [expect the apex (rp(0),zp(0))] and the air-water interface. Since the influence of decreasing λ on lateral deformation is larger than that on longitudinal deformation, the extent ratio increases and the corresponding curves become more sharper as λ decreases (see in Figure 2a*). However, this ratio exhibits much weaker dependence on zp0 and A, so that the

z(r) 1 ) z(0) 1 + a(r2/ω2)

(3)

ω ) 10β1λβ2zp0β3Aβ4

(4)

where ω is an empirical value and can be considered as the characteristic width of the normalized profile, β1 ≈ -0.7033, β2 ≈ -0.22652, β3 ≈ 0.34243, β4 ≈ -0.01171, and a ≈ 4.5. It is illustrated that despite the distinct differences among the water surface profiles in Figure 2, panels a-c, all of them can be scaled onto a single master profile (Figure 3) by scaling both z(r) and r of each data set with the corresponding peak value z(0) and characteristic width ω respectively. This scaling profile reflects the intrinsic deformation of the air-water interface induced by the van der Waals attractive interaction and has never been observed previously, despite data for r in the vicinity of (1 cm exhibits slight offsets, which mostly arises from the cutoff value rcutoff ) 1 cm set for the boundary condition z(rf∞) ) 0 during the numerical iterative procedure. Insets of Figure 3 show the relations between ω and the system parameters (λ, zp0, and A), respectively (note that the fitting curves are obtained from eq 4), and through comparing them with the profiles in Figure 2, panels a*-c*, it is demonstrated that ω( λ, zp0, and A) can be used to quantitatively describe the extent ratio of lateral deformation to longitudinal deformation. The larger the value of ω, the greater the extent ratio of lateral deformation to longitudinal deformation. B. Scaling Descriptions of the Deformation Peak versus the Solid Position and of the Force-vs-Distance Behavior. Figures 4a and 5a show the dependence of the deformation peak (z(0)) on the solid position (zp0) with different A and λ values, respectively. The curves in Figure 4a are analogous parallel lines, whereas in Figure 5a, they are not. It should be emphasized that in Figure 4a we exclude the leftmost points near the minimum allowed height of solid, zp0,min, because the slight discrepant behavior exists in the small confined distance as zp0 near zp0,min.21-22 Through scaling both z(0) and zp0 of each data set in Figure 4a by corresponding critical values of z(0)max and zp0,min respectively, all the points can be scaled onto a single master curve (Figure 4b). The similar scaling approach is also applied to the FSolid vs zp0 behavior for various A with the corresponding scale factors FSolid,max and zp0,min , where the force acting on the solid probe, FSolid, is obtained through the integration of the vertical (longitudinal) component of the van der Waals attractive force on the air-water interface, and FSolid,max is defined as the critical force at the lowest position of the solid zp0,min (for more details see ref 21). The resultant scaled force curve is shown in Figure 4c. Through fitting we find that

z˜(0) ) Cz˜(0) z˜p0-b

(5)

F ˜ Solid ) CF z˜p0-bF

(6)

where b and bF indicate the longitudinal deformation of the airwater interface, Cz˜(0) and CF˜ are constant coefficients. Here

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Figure 2. Profiles and normalized profiles for (a and a*) different shapes of the solid, (b and b*) different heights of the solid, and (c and c*) different strengths of the van der Waals interaction. The physical parameters are γ ) 0.072 mN/m, F ) 996.910 kg/m3 and constants A ) 20 × 10-21 J, zp0 ) 1424. 1 nm for (a) and (a*), constants A ) 20 × 10-21 J, λ ) 0.1 m-1 for (b) and (b*), and constants λ ) 0.1 m-1, zp0 ) 2460.2 nm for (c) and (c*).

Figure 3. Scaled profiles {r˜,z˜(r˜)} for different profiles given in Figure 2. The insets of Figure 3 show the relations between ω and the system parameters (λ, zp0, and A). The fitting curves are obtained from eq 4.

for λ ) 50 m-1, the values are b ) 2.16958, Cz˜(0) ) 0.47754, bF ) 2.0419, and CFˆ ) 0.63457. The reason for the slight difference between b and bF (or between Cz˜(0) and CF˜ ) is that the former one indicates the longitudinal deformation for the apex of air-water interface, whereas the latter one indicates the longitudinal deformation for the whole profile. Figure 5b shows the relation between the scaled deformation peak z˜(0) and the scaled solid position z˜p0 for various values of the splay parameter λ. Note that all of the scaled curves coincide with the scaling law described by eq 5, whereas the value of the exponent b varies with both λ and z˜p0. At large z˜p0 (approximately larger than 2), the scaled curves for different λ are separate and have different slopes, whereas at small z˜p0 (approximately smaller than 1.5), they are superimposed onto a master curve with a slope increasing as z˜p0 f 1 gradually. For example, b varies from

3.7 to 4.6 to 25.7 in Figure 5b when the value of z˜p0 decreases from 1.250 to 1.131 to 1.001. This is because the value of b is determined not only by the geometry of the solid but also by the profile of air-water interface. When the solid is far from the water surface, the van der Waals force is so small that the change of the water surface profile can be neglected. Then the value of b is merely determined by the geometric parameter λ of the solid probe. However, when the solid approaches very close to the air-water interface, the van der Waals force increases dramatically, leading to a great deformation of the water surface profile. Hence the value of b at small z˜p0 increases considerably. Likewise, decreasing λ would enlarge the deformation of the water interface, as stated above, and thus increase the value of b. Therefore, b and ω have a similar performance with respect to λ, as demonstrated by Figures 5c and 3a. By fitting the data, the

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Figure 4. (a) Logarithmic plots of the function z(0;zp0) for constant λ ) 50 m-1, with values of A ∈ {1,5,10,50,100,200,300,400,500} × 10-21 J; (b) Scaled behavior of z˜(0) for data presented in (a) as functions of the scaled value z˜p0, the inset shows the relationship between the scaled force F ˜ Solid acting on the solid and z˜p0. (Definitions of z˜(0), F ˜ Solid and z˜p0: z˜(0) ≡ z(0)/z(0)max, F ˜ Solid ≡ FSolid/FSolid,max and z˜p0 ≡ zp0/zp0,min ).

dependence of b on λ when the solid is far from the limiting position can be written as follows:

log10 b ) 0.3903 - 0.04191 log10 λ* + 0.00883(log10 λ*)2 (7) where λ* ) λ/λ0 and λ0 ) 1.0 m-1. It is important to note that eq 7 is valid for macroscopic solids only. For the accurate formula valid for the specialized applications involved in the experiments of the mesoscopic interaction between microscopic or submacroscopic solids and various fluid entities, the fitting of exact numerical calculations in the larger range of λ is required.21,22 From the above analysis, we can know that the interaction between the paraboloid solid and the air-water interface obeys a universal and simple scaling law described by eqs 5 and 6, similar to the power force law (F ) kd-n,valid for the interactions between two solid bodies with various geometries) obtained via integration using Derjaguin’s approximation.3,24-29 The scaling parameter b (or bF) is dependent on both the geometry of the solid and the profile of the air-water interface and, thus, can be used to characterize the longitudinal deformation and instability30 of the air-water interface for the solid with a fixed geometry, as illustrated in Figure 5b. In conclusion, the larger the value of b (or bF), the larger the longitudinal deformation (or the force gradient of the mutual interacting force) and the less stable the (24) Cappella, B.; Dietler, G. Surf. Sci. Rep. 1999, 34, 1. (25) Butt, H.-J.; Cappela, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1. (26) Burnham, N. A.; Colton, R. J.; Pollock, H. M. Nanotechnology 1993, 4, 64. (27) Hartmann, U. Phys. ReV. B 1991, 43, 2404. (28) Aston, D E.; Berg, J. C. J. Colloid Interface Sci. 2001, 235, 162. (29) Todd, B. A.; Eppell, S. J. Langmuir 2004, 20, 4892. (30) . Miklavcic, S. J.; White, L. R. Langmuir 2006, 22, 6961.

Wang et al.

Figure 5. (a) Logarithmic plots of the function z(0;zp0) for constant A ) 200 × 10-21 J, with values of λ ∈ { 0.1,0.2,0.5,1.0,5.0,10} m-1. The dash line depicts the critical values. zp0,min and z(0)max (the corresponding maximum deformation peak). (b) Scaled behavior of z˜(0;z˜p0) for data presented in (a) as functions of the scaled z˜p0, the inset shows the relation between b and λ for a fixed value of interaction strength A.

air-water interface. Moreover, the scaling strategy applied to the numerical calculations above can be useful to help study various systems with different kinds of interactions, such as the DLVO force between the silica particle and the air-bubble or oil droplet interface in the aqueous environment.2,8 It is also helpful for a better understanding of the force-distance curves measured by AFM (or SFA).31-33 C. Experiments by AFM. To examine the validity of the scaling laws so far discussed, in the following, we use a commercial AFM (SPA-300HV, Seiko Instrument Ltd.) equipped with a liquid cell to measure the interaction force between a solid probe and a planar air-water interface in air,34-35 and then the similar scaling approach presented above is applied to the forcedistance curves. The diameter of the liquid cell is about 17 mm, much larger than the dimensions of the tip (typically 15 µm in height), so that the water surface can be considered to be infinite and planar. To ensure a constant surface tension value, before measurements the probes were cleaned of organic contaminants by placing them for 10 min in pure acetone and then ethanol at 45 °C, and the liquid cell, which was filled with 0.5 mL of deionized water after being cleaned in acetone and ethanol by an ultrasonic cleaner6 and then dried with nitrogen gas, was placed peacefully on the sample stage. During the measurement, the AFM tip was moved toward the water surface by driving the step motor manually. The step interval of the motor was controlled to be about 50 nm, for which the movement of the liquid cell (31) Horn, R. G.; Bachmann, J. D.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483. (32) Christenson, H. K.; Fang, J.; Israelachvili, J. N. Phys. ReV. B 1989, 39, 11750. (33) Chen, N. H.; Kuhl, T.; Tadmor, R.; Lin, Q.; Israelachvili, J. Phys. ReV. Lett. 2004, 92, 024501. (34) Wu, D.; Wang, Y. Z.; Zhang, J. X. To be submitted. (35) Weisenhorn, A. L.; Hansma, P. K. Appl. Phys. Lett. 1989, 54, 2651.

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Figure 6. Approaching (+) and retracting (×) traces of force-distance scans taken with four different kinds of probes. (a) Tip of SN-AF01, k ) 0.02 N/m. (b-d) Tips of CSG11/Au and AF01-A, and k ) 0.03, 0.1, 0.15 N/m, respectively.

could be controlled smoothly enough to ensure the profile of the water surface stable. And the velocity of the piezodrive (i.e., approaching speed in force measurements) was maintained to be 0.28 µm/s so that force was measured under quasi-static conditions. Additionally, all of the equipment was placed on a vibration isolator to eliminate ambient disturbances. A detailed description of how the force-distance curves are measured and analyzed has been given elsewhere.34 Figure 6 shows a series of force-distance curves with four kinds of probes: the silicon nitride tip (SN-AF01) with a spring constant k of 0.02 N/m and three gold-coated tips (CSG11/Au/ A, CSG11/Au/B, and AF01-A) with spring constants k of 0.03, 0.1, and 0.15 N/m, respectively. It is demonstrated that, as the AFM probe approaches the air-water interface, the probe experiences a negligible attraction until it is very close to the water interface, where the attractive force increases dramatically and the unstable regime might emerge (Figure 6). It should be emphasized that Figure 6 only shows the force-distance curves measured at the stable-unstable transition point34 and that we only concerns the approaching process of noncontact in this manuscript. Here we assume the transition point (the onset point of the jump labeled in Figure 6) as the limiting position of the AFM tip, at which the instability of the water interface induced by the van der Waals interactions occurs.4-5,9,21,34 The zero distance of the force curves in Figure 6 is approximately estimated in terms of the strategy which is commonly applied to the infinitely hard materials with surface force.24,25,34 Through dividing the original data on the approaching curve by the values of the transition point, the approaching force curves measured with different kinds of probes can be scaled onto a single master curve (Figure 7). From the least-squares fitting, this curve can be also described by the scaling law of eq 6 with bF ≈ 1.082. It is much smaller than the value (bF ) 2.0419) obtained from Figure 4c because the splay parameter of the AFM probe used in the experiment is about 106 m-1, which is much larger than that we used in the theoretical analysis (λ ) 50 m-1). Additionally, the uncertainties induced by determining the zero distance or the transition point would also affect the determination of the minimum allowed height of the AFM tip and the corresponding critical force, and thus somewhat affect the value of bF, despite the fact that they would not mask this universal scaling behavior.

Figure 7. Normalized force-distance curves. Forces and distance are normalized with respect to the ones of the closest point respectively.

Although the similar scaling behavior for van der Waals interactions between sharp probes and flat surfaces of hard materials have already been discussed in ref 27, it is the first time that this scaling behavior is observed in experiments and in deformable surfaces. As expected, the experimental results are in good agreement with the theoretical analysis presented by us and by other scientists. However, there are still some problems not solved in our experiments. For example, it is hard to demonstrate the dramatic increase of bF as the probe approaches the vicinity of the limiting position, because the sampling rate of our equipment is too low to follow this variation, and the further validations of the dependences of the value of bF on λ and A require more precise experiments using various tips with different Hamaker constant or geometries. Moreover, some complicated factors existing in our experiments may influence the scaling behavior, such as the van der Waals repulsive interaction, the patch charge effect,26 or the inaccurately determination of zero distance.

IV. Analytical Solution for the Deformation of the Air-Water Interface Although the deformation of the air-water surface induced by van der Waals interactions can be well described by the numerical solutions and the scaling behavior presented above, to better understand the relations between the deformation and the parameters λ, zp0, and A, an accurate analytical solution of eq 2 which has hitherto not been known21-22 is necessary. Here

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an appropriate expression satisfying this objective can be deduced by substituting eqs 5 into eq 3

z(r, zp0, λ, A) )

(zp0)-b 1 + a(r /ω ) 2

2

(zbp0, min z(0)max)

(8)

where a ≈ 4.5, ω ) 10β1λβ2zβp03 Aβ4, b ≈ 100.3903 λ*-0.04191+0.00883 and zp0,min is written as

log10λ*,

zp0,min ≈ 10-6.4341 + 21µ1 λ-0.2379-0.0236log10λ Aµ1

(9)

in which µ1 ≈ 0.3057 + [0.1445 - 1.3252 × 10-2(log10 λ)2] log10 λ/(10 + 3 log10 λ). In terms of the similar mathematical and physical deduction of eq 9, which was first deduced by Cortat and Miklavcic,21-22 an analogous formula, accurately describing the dependence of z(0)max on λ and A, can be obtained

z(0)max ≈ 10

-6.89524+21µ2

-0.24162-0.02572log10λ

λ

A

µ2

(10)

in which µ2 ≈ 0.30733 + [0.1442 - 2.07229 × 10-2(log10 λ)2] log10 λ/(10 + 3.3079 log10 λ). Note that, in eqs 9 and 10, the value of a and the expressions of ω and b are only valid for macroscopic solids. For microscopic or submacroscopic solids with a large λ (such as λ ∈ [1000,1000000] m-1), other accurate fitting formulas and corresponding parameters are required.21,22,24 Equation 8, however, is unalterable and gives the essential dependence of the air-water interface on the parameters (λ, zp0, and A) with the van der Waals attractive force through substituting eq 9 into eq 10. Thus eq 8 can be considered as an analytical solution to the nonlinear E-L equation (eq 2). In addition, the two principal length scales which denote the lateral and longitudinal deformations of the air-water interface respectively are obtained from eq 8

lω(zp0, λ, A) ) lz(0)_V(zp0, λ, A) )

ω xa

( ) zp0,min zp0

(11)

b

z(0)max

(12)

It is noted that the above formulas (11 and 12) are only available in the solid-air-water system, and actually both lω and lz(0)_V are the complicated functions of λ, zp0, A, γ, and G, and quite analogous to lG ) [γ/G]1/2 and lV (zp0, λ, lA) presented in other

articles.21-22 From eqs 11 and 12 and previous works, we can know that the lateral deformation of the air-water interface depends not only on lG but also on the geometry of the solid, the solid/water distance, and the strength of the attractive force and that lz(0)_V (zp0, λ, A, γ, and G) actually amounts to lV (zp0, λ, lA), the exact formula of which has not yet known.22 Thus, eq 12 can be considered to be an accurate expression describing the relation between lV and the parameters (λ, zp0, and A) for the macroscopic solids in the solid-air-water system. Furthermore, by substituting eqs 11 and 12 into eq 8, a simplified formula can be given by z(r, lω, lz(0)_V) ) lz(0)_V/(1 + r2/lω2), which shows the simple relationship between the profile and the lateral and longitudinal deformation of the air-water interface.

V. Summary In this paper, we analyze the scaling behavior of the interaction between the solid probe and the air-water interface using the collocation-integration numerical technique. For the normalized water surface profile, the characteristic width ω(λ, zp0, A) is used to quantitatively describe the extent ratio of the lateral deformation to longitudinal deformation and the dependence of the lateral deformation on the system parameters. Moreover, we find that the deformation peak of the air-water interface and the force acting on the solid versus the solid position can be simply described by the scaling laws (eqs 5 and 6). The exponents (b and bF), apparently dependent on the solid geometry and the solid/water distance, can be used to well characterize the longitudinal deformation of the air-water interface. To further illustrate this scaling behavior, force measurements by AFM with four kinds of tips are performed. The results show that force curves measured with different AFM probes can be cast into a single master curve, as predicted by the scaling law. Besides, the analytical solution to the nonlinear E-L equation (eq 2) which describes the relation between the deformation of the air-water interface and the parameters (λ, zp0, and A) is obtained for the first time, and two length scales, lω(zp0, λ, A, γ, and G) and lz(0)_V(zp0, λ, A, γ, and G), are put forward to quantitatively describe the lateral and longitudinal deformation of the air-water interface, respectively. The strategy in this part also provides a suggestion for obtaining analytical solution of the complicated nonlinear differential equations which is hard to solve by common mathematical methods. LA700980V