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Interaction of Elastic Bodies via Surface Forces. 1. Power-Law Attraction Olga I. Vinogradova*,†.‡ and Franc¸ ois Feuillebois§ Max-Planck-Institute for Polymer Research, Postfach 3148, D-55021 Mainz, Germany, Laboratory of Physical Chemistry of Modified Surfaces, Institute of Physical Chemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 117915 Moscow, Russia, Laboratoire de Physique et Me´ canique des Millieux He´ te´ roge` nes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France Received November 28, 2001. In Final Form: March 20, 2002 We study theoretically the effect of finite elasticity on the attractive power-law interaction of two solids separated by a thin liquid (or gas) film. A new asymptotic technique is developed to determine the deformed shape of the surfaces and to calculate the elasticity contribution to the total force, i.e., an additional term present between the deformed bodies. Both the deformation and the elasticity contribution are found to be nonnegligible well before contact is reached, although they are of much shorter range than the surface force that caused them. This range can be characterized by a reference elasticity length, which depends on elastic constants and size of the solids, as well as on the attractive force that led to deformation. The total force vs separation profile for elastic surfaces is found to depend on how the measurements are made, namely, how the separation is detected: it can lead to either less or more attractive force compared with the case of rigid surfaces.
I. Introduction Surface forces, or the forces acting between microscopic or even macroscopic bodies when they are in close proximity, determine the stability of colloidal dispersions and thin films, as well as the properties of materials that have some granularity, or texture, or a structure on a scale much larger than atoms or molecules. Historically, most theoretical and experimental studies of surface forces have been conducted assuming that the surfaces are rigid.1 Theoretical description of surface forces is usually done for a flat geometry. For experimental reasons, during force measurements it is advantageous to have a configuration equivalent to a sphere-plane geometry.2-5 Theory and experiment can then be linked by invoking the Derjaguin approximation which relates the surface force (F ˜ ) normalyzed by the radius of curvature of the equivalent sphere (R) to the interaction free energy (E ˜ ) between two planar half spaces2,3
F ˜ (h) ) 2πRE ˜ (h)
(1.1)
where h is the separation at the point of closest approach. When the measurements deviate from theory, an extra attraction or repulsion is inferred to be present. For instance, the observation of an extra repulsion force in addition to that predicted by the Derjaguin-LandauVerwey-Overbeek (DLVO) theory6,7 has led to the discovery of hydration forces (see refs 8 and 9 and references * To whom correspondence should be addressed. E-mail:
[email protected]. † Max-Planck-Institute for Polymer Research. ‡ Laboratory of Physical Chemistry of Modified Surfaces. § Laboratoire de Physique et Me ´ canique des Millieux He´te´roge`nes. (1) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, 1985. (2) Derjaguin, B. V. Kolloid Z. 1934, 69, 155. (3) White, L. R. J. Colloid Interface Sci. 1983, 95, 286. (4) Vinogradova, O. I. Langmuir 1995, 11, 2213. (5) Vinogradova, O. I. Langmuir 1996, 12, 5963. (6) Derjaguin, B. V.; Landau, L. D. Acta Physicochim. URSS 1941, 14, 633. (7) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.
therein); an observation of extra attraction between hydrophobic surfaces led to the discovery of a hydrophobic attractive force,10 etc. Equation 1.1 ignores the effect of elasticity on the net force acting between the solids. However, solids are usually elastic (in some situations the interaction forces can even plastify them11), and the surface forces can cause their deformation well before contact is reached. This, in turn, can dramatically complicate the interaction of elastic bodies, and can be confused with an extra surface force acting between them.12,13 With this paper we start a series of publications devoted to the calculation of deformation caused by interaction forces. Early solutions of the elastic equations include the asymptotic results for repulsive interactions14,15 and numerical results16 for three fundamental types of surface forces, namely molecular attractive, electrostatic repulsive, and oscillatory (or solvation) forces. Therefore, we address the issues similar to those discussed in refs 1416. However, here we do not attempt to improve the accuracy of the existing computations. Instead, we adopt simplifying approximations, which we believe not to impair greatly the accuracy, and which lead to explicit asymptotic formulas that can be easily handled. The main point of this, the first paper, will be to calculate the deformation caused by power-law attractive forces, which represent both the attractive component of the DLVO potential and hydrophobic force, and are, therefore, important for understanding various phenomena, such as adhesion, (8) Rabinovich, Y. I.; Derjaguin, B. V.; Churaev, N. V. Adv. Colloid Interface Sci. 1982, 16, 63. (9) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (10) Israelachvili, J.; Pashley, R. Nature (London) 1982, 300, 341. (11) Butt, H.-J.; Do¨ppenschmidt, A.; Hu¨ttl, G.; Mu¨ller, E.; Vinogradova, O. I. J. Chem. Phys. 2000, 113, 1194. (12) Schmitt, F. J.; Ederth, T.; Wiedenhammer, P.; Claesson, P.; Jacobasch, H. J. J. Adhes. Sci. Technol. 1999, 13, 79. (13) Vinogradova, O. I.; Yakubov, G. E.; Butt, H.-J. J. Chem. Phys. 2001, 114, 8124. (14) Hughes, B. D.; White, L. R. Q. J. Mech. Appl. Math. 1979, 32, 445. (15) Hughes, B. D.; White, L. R. J. Chem. Soc., Faraday Trans. 1, 1980, 76, 445. (16) Parker, J. L.; Attard, P. J. Phys. Chem. 1992, 96, 10398.
10.1021/la011726r CCC: $22.00 © 2002 American Chemical Society Published on Web 06/01/2002
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cavitation, dewetting, and more. Our main result is the scaling expressions for the deformed shape of the bodies for small, intermediate, and large radial distance from the axis connecting their centers. These expressions are used to calculate analytically the interaction force of elastic bodies, and to demonstrate that it differs dramatically from the case of rigid surfaces. We also apply our analysis to compare the different techniques of the force measurements. Our paper is arranged as follows: In section II we briefly describe the principle of the force measurements and the main experimental techniques. Our system and approach are defined in section III, and the asymptotic results for deformation, pressure, and force are given in section IV. Here we also discuss the possible relevance of our results for the main force measurement techniques. II. Force Measurement Technique Recent years have seen an increase in the number of devices available for the direct measurements of surface forces. In the present paper we examine two main experimental techniques, the surface force apparatus (SFA), and the atomic force microscope (AFM). We also considers a third device, so-called the measurement and analysis of surface interaction and forces (MASIF) technique. The SFA17 uses mica surfaces, with a radius of curvature R ∼ 1-2 cm, glued to polished silica disks with an epoxy resin. One surface is mounted at the end of a double-cantilever force measuring spring. The surface separation is determined by multiple-beam interferometry, which also allows surface deformation to be monitored in situ. The zero of separation is measured directly by bringing the surfaces into contact in nitrogen or in liquid. The AFM18 uses so-called colloid probes of R ∼ 1.5-5 µm, glued or melted to the end of a single microfabricated cantilever. The absolute separation cannot be directly determined, but it is inferred from the deflection vs piezo position behavior. The contact is assumed to be at “constant compliance”, i.e., when the deflection of the cantilever becomes roughly linear with respect to sample displacement. In the MASIF19 spherical surfaces (R ∼ 1 mm) are mounted at the end of a piezoelectric bimorph, which permits the spring bending to be measured electronically. The surface separation is then calculated from this bending using a “constant compliance” region as in the AFM.
III. Analysis. In this section we define our system and summarize some earlier relationships which are pertinent to the present analysis. A. Model. We consider two smooth elastic spheres with radii R1 and R2, approaching along the line connecting the centers. Our first aim is to determine the shape of the deformed surfaces as a function of the radial distance from this line-of-centers of the spheres. We consider a situation in which the gap between the spheres is small compared to the smaller of their radii. In this approximation, the interaction of two spheres (MASIF, and some AFM experiments) is equivalent to the interaction of a sphere of an equivalent radius R ) R1R2/(R1 + R2) with a plane (AFM), which is, in turn, the same as for two crossed cylinders of equal radii (SFA).2-5 We require that the spheres are rigid enough so that the surfaces deform only within a small area near the axis of symmetry. This latter restriction was also made in14,20-22 and it lies at the heart of the Hertz contact theory of linear elasticity.23 For most (17) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (18) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (19) Parker, J. L. Prog. Surf. Sci. 1994, 47, 205.
Figure 1. Schematic of the deformation due to attractive force of two elastic spheres separated by a thin film. The solid curves “1” denote the undeformed surfaces, the dashed curves “2” denote actual deformed surfaces. The solid curves “3” are simply obtained by the shift of rigid surfaces “1” to get the real minimum separation d between deformed surfaces. The line “4” connects the centers of the interacting spheres.
solids, this restriction is very reasonable. The deformed and undeformed surfaces of the two spheres are sketched in Figure 1. The undeformed spherical surface H ˜ can be approximated by a paraboloid
r2 H ˜ (r) ) h + 2R
(3.1)
where h is the minimum separation between the undeformed spheres. The deformed gap profile can be locally given as
H(r) ) H ˜ (r) + w(r)
(3.2)
where w(r) ) w1(r) + w2(r) is the sum of deformations of the two surfaces from their original shape. A deformation toward the opposing surfaces is defined as negative. To determine the deformation we shall follow the ideas of the Hertz contact theory,23 and also the developments of authors in refs 14 and 20-22 who studied the deformation due to the action of surface and hydrodynamic forces. For completeness, we mention briefly some of the relationships. The essence of the theory is that an applied normal force distributed over the surface will cause it to deform. Provided that this deformation is small, it can be determined by integrating the surface stress distribution multiplied by a Green’s function over the area subjected to the stress, where the appropriate Green’s function is the fundamental solution of the linear-elasticity equation for an applied point force of unit magnitude. For our problem, the formulation of this integral is21,22
w(r) )
4θ π
∫∞0 Π(y)ψ(y/r) dy
(3.3)
where Π(r) is a disjoining pressure inside the gap. The parameter θ in 3.3 is defined as (20) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (21) Davis, R. H.; Serayssol, J.-M.; Hinch, E. J. J. Fluid Mech. 1986, 163, 479. (22) Vinogradova, O. I.; Feuillebois, F. J. Colloid Interface Sci. 2000, 221, 1. (23) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon Press: London, 1959.
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IV. Results and Discussion.
1 - ν12 1 - ν22 + E1 E2
(3.4)
where ν1,2 is Poisson’s ratio, and E1,2 is Young’s modulus of elasticity for spheres 1 and 2. The effective elastic constants of the mica-glue system in the SFA may be easily calculated from the measured radius of the contact zone and the pull-off force.24,25 The values of θ, recalculated from the measured elastic constants, are small, on the same order as that of silica, θ ) 1.9×10-11 m2/N, and only a factor 3 or 4 larger than that of mica, θ ) 6.7×10-12 m2/N. The average of all the measurements26 is (3.2(1.3)×10-11 m2/N. The MASIF and AFM experiments can be performed with rather soft materials, such as, for example, solid polymers.12,13,27,28 In this case θ can be as large as 5.5×10-10 m2/N The Green’s function kernel is given by
ψ(z) )
[
]
4z2 z K z + 1 (1 + z)2
(3.6)
where C is a constant. If C ) A/6, this equation gives the attractive van der Waals (the attractive component of the DLVO model) profile in the nonretarded approximation. In this paper we are not considering the repulsive Born part of the interaction, which is important for a contact problem.20,29 Typical measured values for the Hamaker constant (A) are 1.35 × 10-19 J for mica across air,30 2.2 × 10-20 J for mica across water,17 and 6 × 10-21 J for hydrocarbons across water. An attractive non-DLVO force of type (3.6) was also found between hydrophobic surfaces in an SFA experiment31 with C ∼ 1.0 × 10-19 J. Therefore, we have chosen values of C between 10-19 and 10-21 J. For rigid solids, application of the Derjaguin approximation eq 1.1 gives the connection of the interaction force with the force Π ˜ , applied to the unit area of the planar plates, by
∫∞h Π˜ p(H˜ ) dH˜ ) -RC h2
˜ p(h) ) 2πR F ˜ p(h) ) 2πRE
4θCR1/2 wp(r) ) - 2 5/2 Ip(ξ) πh
(3.7)
We stress that the essence of the Derjaguin approximation ˜ p(H ˜ (r)) and, is the possibility of expressing Π ˜ p(r) as Π therefore, to calculate a force with the simple expression 3.7. (24) Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 480. (25) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (26) Christenson, H. K. Langmuir 1996, 12, 1404. (27) Meagher, L.; Craig, V. S. J. Langmuir 1994, 10, 2736. (28) Considine, R. F.; Hayes, R. A.; Horn, R. G. Langmuir 1999, 15, 1657. (29) Attard, P.; Parker, J. L. Phys. Rev. A 1992, 46, 7959. (30) Israelachvili, J. N.; Tabor, D. Nature (London) 1972, 236, 106. (31) Claesson, P.; Christenson, H. K. J. Phys. Chem. 1988, 92, 1650.
(4.1)
with
Ip(ξ) )
(3.5)
where K is the complete elliptic integral of the first kind, and z ) y/r. To determine the deformed shape of the sphere surfaces we have to specify the disjoining pressure profile in the liquid layer between the surfaces. B. Interaction Forces. In the absence of deformation, the power law decay pressure Π ˜ p can usually be given by
C ˜))- 3 Π ˜ p(H πH ˜
A. Deformation Profile. The solution of the system of eqs 3.2 and 3.3, in general, requires a numerical method. However, for small deformations (w , h), these equations can be solved by an asymptotic method. In this case, in the first approximation the deformation can be determined via the pressure profile in the absence of deformation Π ˜ p(H ˜ ).21,22 Despite this initial restriction, in the case of attractive forces the results, as we show below, can be applied for any precontact situation. Putting eq 3.6 into 3.3, we obtain a deformation profile due to nonuniform power law disjoining pressure along the liquid film
ψ(ξ/η)
∫∞0 (1 + η2/2)3dη
(4.2)
Here we used the following scaling,
ξ)
y r , η) xRh xRh
In general, this integral must be performed numerically. However, analytic expressions may be obtained in the limits of small, intermediate, and large values of the radial position. Small Distances from the Axis (“Peak”). At small ξ, we have (see Appendix A)
Ip(0) )
3x2π2 5 1 - ξ2 + O(ξ4) 5 8 2
(
)
The deformation profile near the axis connecting the centers is, therefore, parabolic:
3x2θCR1/2 5 1 - ξ2 8 h5/2 8
wp(0) ∼ -
(
)
The maximum deformation wmax can be obtained from w(0) assuming ξ ) 0. The real separation d at the point of the closest approach can then be estimated as
(
d)h 1-
) (
3x2θCR1/2 3x2 Λ )h 18 h 8h7/2
(
)
7/2
[]
h 1 - 0.53
∼
)
7/2
[Λh ]
(4.3)
where the reference deformation length
Λ ) θ2/7C2/7R1/7
(4.4)
characterizes the relative importance of deformation for a given separation h. Deformation can be safely ignored if h is much greater than Λ, but it is crucial when h is of the order of Λ or smaller. An important point to note is that the dependence of the deformation length on the radius of surfaces is very weak. Estimates of Λ with the parameters of typical force measuring techniques (see Table 1) suggest that for the SFA experiment it is confined roughly between 0.5 and 2.5 nm; for a MASIF experiment, between 0.2 and 3.2 nm; and between 0.1 and 1.6 nm for the AFM force measurements.
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Table 1. Realistic Range of Values of Experimental Patameters in Three Force Measurement Techniques AFM θ, m2/N R, cm C, J Λ, nm
MASIF
SFA
min
max
min
max
min
max
6.7 × 10-12 1.5 × 10-6 10-21 0.1
5.5 × 10-10 5 × 10-6 10-19 1.6
6.7 × 10-12 1 × 10-3 10-21 0.2
5.5 × 10-10 1 × 10-3 10-19 3.2
1.9 × 10-11 1 × 10-2 10-21 0.5
4.5 × 10-11 2 × 10-2 10-19 2.5
One of the important consequences from this result is the possibility of calculating analytically the radius of curvature Reff of the “peak” region. This can be done if we recognize that in this region H ∼ d + r2/2Reff. If so,
R/Reff ∼ 1 +
15x2 Λ 32 h
[]
7/2
∼ 1 + 0.66
7/2
[Λh ]
This means that deformation causes not only a change in separation, but also elongation, i.e., the decrease in the local radius of curvature of the “peak”. However, the radius Reff describes only the local curvature at the “peak” and is not the radius to be used in the equations describing deformation, force, etc. Another consequence of our model is that the surfaces are coming into contact (d ) 0) at
h)
(
)
3x2CθR1/2 8
2/7
( )
)Λ
3x2 8
2/7
∼ 0.83Λ
Equation 4.3 demonstrates that at large surface separations the change in surface separation δd is equal to how far the surfaces have been moved δh. When surfaces are in close proximity this is no longer the case. Figure 2 illustrates this effect for the parameters typical for the SFA, MASIF, and AFM experiment. One can see that the surfaces are pulled toward each other, and d decreases more rapidly then h. Two main conclusions can be made from inspection of eq 4.3 and Figure 2. First, that there is, indeed, a substantial effect of surface deformation before the contact is reached. Second, that all the curves begin to deviate from the undeformed case (d ) h) only at separations which are much shorter than the range of surface forces that caused this deformation. This happens at h e 2-10 nm, depending on the parameters of the experiment, and is consistent with the concept of Λ. Intermediate Distances from the Axis (“Slope”). The expansion around ξ ) 1 is performed on the expression A3 of Ip. For this purpose, the function ψ is defined in term of function f, eq A5, such as that in Appendix A. The calculation of the series of I and J at ξ ) O(1) gives the following result for Ip ) I + J
I(Pad) (1) ∼ p 0.8189 + 0.3663ξ1 + 0.2520ξ12 + 0.0423ξ13 + 0.0025ξ14 1 + 1.1568ξ1 + 0.8976ξ12 + 0.3094ξ13 + 0.0768ξ14 with ξ1 ) ξ - 1. Then the first term of expansion around unity would be Ip(1) ) 0.82 - 0.58(ξ - 1) + O(ξ - 1)2 Therefore,
4θCR1/2 wp(1) ∼ - 2 5/2 (1.4 - 0.58ξ) πh i.e., in the first-order approximation deformation decays linearly with the radial distance. Large Distances from the Axis (“Tail”). For large distances r from the axis, we have η/ξ , 1, so that the integral Ip(ξ) behaves asymptotically as
Figure 2. A plot of the real separation d between solids at the point of closest approach against the undeformed separation h with the parameters corresponding to minimum (A) and maximum (B) values of deformation (see Table 1). In the absence of surface forces there is no deformation and as a consequence d ) h until contact (1). The power-law attraction causes a departure from linearity in the SFA (2), MASIF (3), and AFM (4) experiment.
Ip(∞) )
π + O(ξ-3) 4ξ
This leads to the hyperbolic decay of deformation
θCR1/2 wp(∞) ∼ πξh5/2 The formal boundaries between the regions of “peak”, “slope,” and “tail” should be defined by introducing the dimensionless parameters δ1 and δ2, so that the asymptotic
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expression for a “peak” can be used at ξ e 1 - δ1, for a “slope” at 1 - δ1 e ξ e 1 + δ2, and for a “tail” at ξ g 1 + δ2. The values of δ1 and δ2 can be determined from the intersection of scaling expression for Ip(ξ) and are found to be δ1 ) δ2 ) δ ∼ 0.5. The integral Ip(ξ) and the asymptotic results for different regions are plotted in Figure 3. The exact value of this integral has been calculated numerically. It is seen that with this value of δ the analytical formulas give an accurate description of Ip(ξ). We note that deformation depends on R, θ, and h, and the strength of the force. Taking into account that θ is of the same order in the AFM, MASIF, and SFA experiments, we conclude that differences in deformation are mostly because of the larger radius of curvature (by 3-4.5 orders of magnitude) of the surfaces in the SFA geometry. However, the dependence of deformation on radius is weak, so that this would give us only 1.5-2.2 orders of magnitude difference in deformation. We also stress that the film thickness is the main factor in determining the value of deformation. B. Extra Pressure Caused by Deformation. In ref 22 it has been shown that in the general case it is wrong to estimate the correction to pressure (force) by assuming that Π is a function of H(r). However, this approximation has been justified for the case of small deformation: w , H ˜ . If so, in the general case, one can write
Π(H) ) Π ˜ (H ˜ ) + ∆Π
(4.5)
The change in pressure due to deformation was shown22 to be ∆Π ) (dΠ ˜ /dH ˜ )w(r). We can therefore estimate this correction for power-law attraction as
3C ∆Πp ) 4wp(r) πH ˜
(4.6)
θC2R1/2 9x2 1 - 5ξ2/8 ‚ ‚ (4.7) 8π (1 + ξ2/2)4 h13/2
at ξ , 1, ∆Πp(0) ∼ -
θC2R1/2 3 1 ‚ 2‚ 13/2 h π ξ(1 + ξ2/2)4
at ξ . 1, ∆Πp(∞)∼ -
∫∞0 Π(r)r dr ) F˜ (h) + ∆F
(4.10)
The elasticity contribution is given by
∆F ) 2π
∫∞0 ∆Πr dr
(4.11)
The leading term for an elasticity contribution can be estimated assuming
∆F ) 2πRh(
1+δ ∫1-δ 0 ∆Π(0)ξ dξ + ∫1-δ∆Π(1)ξ dξ +
∞ ∆Π(∞)ξ dξ) ) ∆F(0) + ∆F(1) + ∆F(∞) ∫1+δ
(4.12)
1/2
12 1.4 - 0.58ξ θC R ‚ 3‚ (4.8) h13/2 π (1 + ξ2/2)4
at ξ ∼ 1, ∆Πp(1) ∼ -
pressure.32 The new pressure would then tend to make deformation smaller, so the first-order approximation will overestimate the deformation caused by repulsive forces and, at short separations, could lead to qualitatively wrong asymptotics. C. Total force between Deformed Surfaces. Now, we can calculate the force acting between elastic surfaces. In the general case, it can be presented as
F ) 2π
Direct substitution of the asymptotic expressions for deformation gives the following corrections for pressure for different distances from the axis connecting the centers of bodies:
2
Figure 3. Integrals Ip(ξ) calculated numerically (solid curve) and asymptotic expressions for “peak” (diamonds), “slope” (circles), and “tail” (squares).
Substituting into eq 4.12, the asymptotic expressions for ∆Πp(ξ), integrating, and keeping only terms of the order of δ2, we obtain contributions to the force due to elasticity from different areas
(4.9)
It follows from our results that Πp(H) is always less than Π ˜ p(H ˜ ), because both Π ˜ p(H ˜ ) and ∆Πp are negative ˜ p/dH ˜ is positive). This means (wp(r) is negative, while dΠ ˜ p(H ˜ ). The that the absolute value of Πp(H) is larger than Π new pressure tends to increase the absolute value of deformation, an effect we ignore in our first-order approach. Therefore, one can conclude that the first-order approximation we develop here, being formally applied for the situation of large deformation, will underestimate it. It is important to stress that qualitatively it will always give correct asymptotic behavior, and the next order corrections are coming into play only at very short separations (see Appendix B). Such a conclusion is, of course, applicable to any monotonically decaying attractive forces. Clearly, the opposite situation would happen in case of a repulsive interaction. While Π(H) will again be less than Π ˜ (H ˜ ), this will decrease the absolute value of
(
)
θC2R3/2 9x2 13 2δ 5δ2 ∼ 4 72 27 27 h11/2 θC2R3/2 - 11/2 (0.57 - 0.24δ - 0.59δ2) h
∆Fp(0) ∼ -
θC2R3/2 24 θC2R3/2 0.32δ ∼ - 11/2 0.78δ 11/2 2 h π h
∆Fp(1) ∼ -
θC2R3/26 ∆Fp(∞) ∼ - 11/2 π h
([
]
227 5arctan x2 + 648 8x2
)
16δ 64δ2 + 81 243
Hence, (32) Vinogradova, O. I.; Butt, H.-J.; Yakubov, G. E.; Feuillebois, F. Rev. Sci. Instrum. 2001, 72, 2330.
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θC2R3/2 (0.71 + 0.17δ - 0.09δ2) ∼ h11/2 Λ θC2R3/2 -0.8 11/2 ) 0.8F ˜ h h
∆Fp ∼ -
[]
7/2
(4.13)
Total force is, therefore,
Λ CR 1 + 0.8 2 h h
Fp ∼ -
(
7/2
[] )
(4.14)
Clearly, the elasticity contribution to the total force becomes comparable to the surface force between rigid bodies when h ∼ Λ, and dominates at shorter distances. The force curves and the elasticity contributions that correspond to the situation of the SFA deflection method and the AFM/MASIF method are shown in Figure 4. All the techniques measure the real force, i.e., the force distorted by deformation and described by eq 4.14. However, in the SFA experiment it is the distorted separation d which is measured, whereas in the AFM and MASIF experiment the undisturbed distance h is deduced. In general, the force curve expected for the SFA is slightly less attractive than it would be in case of rigid surfaces. Surprisingly, the SFA experiment with large θ gives reasonably accurate results. These small deviations from the interaction of the rigid surfaces are obviously due to partial cancellation of the correction for force and the correction for distance. In contrast to the SFA, the MASIF measurements give a more attractive interaction. The AFM situation is often nearly the same as for rigid surfaces, due to the small size of the colloidal probe in the AFM experiment. All the conclusions of the previous paragraph are made by assuming that the deflection technique can be used at any separation. In reality, this technique could be unsuitable for the measurement of strongly attractive elasticity contribution, because of the possibility of the jump instability, which could be either due to a force measuring spring17,33 or deformation itself29,34,35 (although that latter possibility represents only a small correction to the jump due to a spring instability if the spring constant is not infinite). We do not address the question of a jump separation in the current paper. However, one consequence for the force measurements by the jump method, made by gradually increasing the constant of the force measuring spring, has to be stressed. Our results mean that it would be wrong to extrapolate the force vs separation curve obtained by measuring the jump position at h . Λ, to the region of short separation h , Λ. In this region, the real force is more attractive then it follows from the jump method, applied at large separations, with dramatic consequences for adhesion and other phenomena influenced by the short-range interaction forces. To summarize, we have constructed the first-order analytical solution for a deformation profile caused by power-law attractive interaction, which captures the essential physics. Our results mean that in many real systems the elasticity contribution will dominate in the total interaction when separation is getting small enough and comparable with the deformation length. Our paper sets out the formal basis for studies of deformation and elasticity contribution in other situations. In particular, the significance of deformation in the presence of repulsive
Figure 4. The computed total forces and the elasticity contributions. The parameters are the same as in Figure 2 and correspond to the minimum (A) and maximum (B) deformation. The separation D ) d or h corresponds to the SFA method or AFM/MASIF measurements, respectively. The curves represent the interaction force that would act between rigid surfaces (1), the total force measured with the SFA (2), MASIF (3) and AFM (4) techniques, as well as the elasticity contribution to the SFA (5), MASIF (6), and AFM (7) force vs distance curves.
interactions, e.g., hydration and double-layer forces, will be the subject of a future investigation. Acknowledgment. We are grateful to R.G. Horn for helpful remarks on the manuscript. Appendix A: Expansion of the Integral Ip for Small ξ Let us find the expansion for small ξ of (3.5). Since ψ(z) has a logarithmic singularity in z ) 1, we write it in two parts, one from 0 to 1 and one from 1 to infinity. Let z ) η/ξ in the first part and z ) ξ/η in the second part. Then
Ip(ξ) ) ξ
ψ(z)
ψ(1/z)
∫01 (1 + ξ2z2/2)3dz + ξ∫01 (1 + ξ2/(2z2))3dz z2 (A1)
(33) Vinogradova, O. I.; Horn, R. G. Langmuir 2001, 17, 1604. (34) Pethica, J. B.; Sutton, A. P. J. Vac. Sci. Technol. A 1988, 6, 2490. (35) Smith, J. R.; Bozzolo, G.; Banerjea, A.; Ferrante, J. Phys. Rev. Lett. 1989, 63, 1269.
Now, from its definition, the function ψ satisfies for z * 0:
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Vinogradova and Feuillebois
1 ψ(1/z) ) ψ(z) z
(A2)
zˆ 3+m
3
Js )
1/ξ ∑ ξm-1(∫π/2 0 βm(R)dR)∫0 m)1
(zˆ 2 + 1/2)3
dzˆ (A11)
so that
Ip(ξ) ) ξ
∫0
1
ψ(z)
dz + ξ (1 + ξ2z2/2)3
∫0
1
3. Result for Ip. Analytical calculations give
z3ψ(z)
dz (z2 + ξ2/2)3 (A3)
Let the first integral be I and the second one be J. 1. Expansion of I. The expansion of I is
I)ξ
∫01ψ(z)dz + O(ξ3) ∫π/2 0
(A12)
3 Jr ) ξ -1 + π + O(ξ3) 8
(A13)
(
)
(A4) Js )
In the expression 3.5 of function ψ we will use the definition of K:
K(x) )
I ) ξ + O(ξ3)
15 3 3 x2π2 - πξ x2π2ξ2 + O(ξ3) (A14) 32 8 256
It is seen that, as a result of matching, which is performed by simply adding up the parts of the integral, Ip ) I + Jr + Js, some terms cancel out. The final result is
dR
x1 - x sin2R
so that
∫
ψ(z) )
π/2 0 f (z,R)
Ip ∼
dR
where
f(z,R) )
z
(A5)
x(z + 1) - 4z sin2R 2
Then the integral in eq A4 is calculated analytically: 1 ∫01ψ(z) dz ) ∫π/2 0 ∫0 f (z,R) dz dR ) 1
(A6)
2. Expansion of J. The calculation of J should be treated with care since the integral becomes singular for ξ f 0. This is a typical singular perturbation problem. The singularity is extracted by using an expansion of ψ for small z. We then need an expansion of f, eq A5, for small ξ, which is written symbolically as
∑
βm(R)zm + R(R,z)
(A7)
wmax ) 2θ
m)1
Then
J)ξ
2θ
∫01(z2 +zξ2/2)3∫π/2 0 R(z,R) dR dz ) 3 ∫01z13∫π/2 0 R(z,R) dR dz + O(ξ )
Js ) ξ
∑ (∫ m)1
π/2 0 βm(R)dR)
∫0
1
z3+m
(z2 + ξ2/2)3
1/2
+ 2θ
∫∞0 ∆Πp(r) dr
∫∞0 ∆Πp(r) dr ∼
[
(
)
This leads to
(
d ) h 1 - 0.53
dz (A10)
contains the singular part. The singularity is resolved in the classical way, using a stretched variable zˆ ) z/ξ, so that
]
3 40 ln 3 - 0.03δ 2 π2 81
(A9)
is regular because of the O(z4) behavior of R(z,R), and 3
∫∞0 Π(r) dr ) -3x2CθR 8h5/2
2θ2C2R -194 - 135 x2arctan(1/x2) + h6 256x2π
3
ξ
(A15)
(A8)
is written as Jr + Js, where
Jr ) ξ
)
The calculation of the integral has been performed similarly to a calculation of ∆Fp:
3
∫01(z2 +zξ2/2)3∫π/2 0 f (z,R) dR dz
(
A more detailed study shows that the O(ξ3) terms cancel out and the next term would be of order O(ξ4). Appendix B: Second-Order Approximation for Maximum Deformation We have calculated the deformation, assuming that pressure is not disturbed. However, as we have shown, the deformation causes a change in pressure. This new pressure tends to increase the absolute value of deformation. As a result, the real deformation will be larger than that predicted by the first-order solution. The calculation of the second-order solution can be obtained by using eq 4.5 instead of eq 3.6 in eq 3.3. This appendix only estimates the maximum deformation, which is the amount by which the central part of the two surfaces would be displaced elastically. This is given by36
3
f(z,R) )
3 5 x2π2 1 - ξ2 32 8
7/2
[Λh ]
7
[Λh ] )
- 0.29
LA011726R (36) Vinogradova, O. I. Langmuir 1998, 14, 2827.
