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Langmuir 2008, 24, 160-169
Rupture Work of Pendular Bridges P. C. T. de Boer*,† and M. P. de Boer‡,§ Sibley School of Mechanical and Aerospace Engineering, Cornell UniVersity, Ithaca, New York 14853, and MEMS Design and Reliability Department, Sandia National Laboratories, Albuquerque, New Mexico 87185 ReceiVed April 29, 2007. In Final Form: September 6, 2007 Capillary bridging can generate substantial forces between solid surfaces. Impacted technologies and sciences include micro- and nanomachining, disk drive interfaces, scanning probe microscopy, biology, and granular mechanics. Existing calculations of the rupture work of capillary bridges do not consider the thermodynamics relating to the evaporation that can occur in the case of volatile liquids. Here, we show that the occurrence of evaporation decreases the rupture work by a factor of about 2. The decrease arises from heat taken from the surroundings that is converted into work. The treatment is based on a thermodynamic control-volume analysis of the pendular bridge geometry. We extend the mathematical formulation of Orr et al., solving the meniscus problem exactly for non-wetting surfaces. The extension provides analytical results for conditions at the rupture point and at a possible inflection point and for the rupture work. A simple equation (eq 32) is shown to fit the rupture work for the two cases over a meniscus curvature range of 3 orders of magnitude. Coefficients for the equation are given in tabular form for different contact angle pairs.
1. Introduction
* To whom correspondence should be addressed. † Cornell University. ‡ Sandia National Laboratories. § Present address: Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213.
the first and second kind, for the case of a spherical asperity touching a plane substrate. His treatment is limited to cases without an inflection point. Erle et al.,7 Hotta et al.,8 and de Bisschop and Rigole9 developed numerical procedures to solve the YoungLaplace equation. A very extensive treatment of the problem is that of Orr et al.,10 who paid special attention to cases with inflection point. Orr et al. listed detailed tables with formulas in terms of elliptic integrals, as well as numerical results. Lian et al.11 presented results for the capillary force as a function of distance between two spheres. They integrated the Young-Laplace equation numerically, after representing the meridian profile by a truncated Taylor series to obtain a recurrence equation (the modified Euler method). Gao et al.12,13 considered the capillary force between a spherical asperity and a flat plate, as well as that between a conical asperity and a plate. They also considered more complicated geometries, such as craters. The force was obtained as the derivative of the free energy with respect to displacement. Gao et al. accounted for the disjoining pressure due to surface interactions, which is of importance when a thin liquid film wets the surface. They integrated the Young-Laplace equation numerically, using boundary conditions appropriate for dry, as well as for wet, solid surfaces (a wet surface is one covered by a thin condensation layer). They pointed out that, for volatile liquids, the volume of the liquid bridge may change as the asperity is withdrawn. They presented results for the capillary force as a function of distance for assemblies of spherical and conical bumps. Matthewson14 pointed out that at high rates of deformation, the capillary force is increased significantly due to viscous effects. He derived an expression for the viscous contribution to the capillary force as a function of rate of deformation. The predictions of his analysis were confirmed by experiments using a force
(1) Delauney Sur la surface de re´volution dont la courbure moyenne est constante. Journal de M. LiouVille 1841, 309 (as cited in ref 4). (2) Lamarle The´orie ge´ome´trique des rayons et centres de courbure. Bulletin de l’Acad. 1857, 2nd series (as cited in ref 4). (3) Beer Tractatus de theoria mathematica phaenomenorum in liquidis actioni gravitatis detractus observatorium. Bonn 1857 (as cited in ref 4). (4) Plateau, J. The Annual Report of the Smithsonian Institution, 4th Series; The Smithsonian Institution: Washington, DC, 1864; pp 338-369. (5) Plateau, J. Statique expe´ rimentale et the´ orique des liquides soumis aux seules forces mole´ culaires; Gautiers-Villars: Paris, 1873. (6) Melrose, J. C. AIChE J. 1966, 12, 986.
(7) Erle, M. A.; Dyson, D. C.; Morrow, N. R. AIChE J. 1971, 17, 115. (8) Hotta, K.; Takeda, K.; IInoya, K. Powder Technol. 1974, 10, 231. (9) de Bisschop, F. R. E.; Rigole, W. J. L. J. Colloid Interface Sci. 1982, 88, 117. (10) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J.F.M. 1975, 67, 723. (11) Lian, G. P.; Thornton, C.; Adams, M. J. Colloid Interface Sci. 1993, 161, 138. (12) Gao, C. Appl. Phys. Lett. 1997, 71, 1801. (13) Gao, C.; Dai, P.; Homola, A.; Weiss, J. J. Tribology 1998, 120, 358. (14) Matthewson, M. J. Phil. Mag. A 1988, 57, 207.
Liquid bridges are of interest in many different fields. They are the source of capillary adhesion, which is of importance in granular materials and powders at the macroscale. This includes granular flows, powder granulation, water saturation in soils, condensation and evaporation in porous media, adhesion of dust and powder to surfaces, etc. Recent areas in which liquid bridges have become of interest include the formation of water films in biological processes, scanning probe microscopy (SPM), surface force apparatus (SFA), magnetic head-disk interface (HDI), and adhesion of microelectromechanical (MEMS) surfaces. The study of liquid bridges has a long history. The earliest papers on the subject appear to be those of Delauney,1 Lamarle,2 and Beer,3 which date back to 1841, 1857, and 1857, respectively. A crucial aspect of the study is the shape of the meniscus at constant mean curvature, which is governed by the YoungLaplace equation. The corresponding meniscus profiles were classified in 1864 by Plateau.4 Plateau showed that the profiles can be categorized as either “nodoids,” “catenoids”, or “unduloids.” It was known at the time that the differential form describing surfaces of revolution with constant mean curvature is completely integrable by elliptical functions.3-5 Nevertheless, Plateau carried out his classification without recourse to such functions. Instead, he combined experimental results with the use of a property discovered by Delauney1 and demonstrated geometrically by Lamarle.2 Melrose6 presented the meniscus profile for nodoids in terms of incomplete elliptic integrals of
10.1021/la701253u CCC: $40.75 © 2008 American Chemical Society Published on Web 11/28/2007
Rupture Work of Pendular Bridges
pendulum. Detailed measurements of the viscous contribution to the force were also reported by Ennis et al.15 and by Pitois et al.16,17 Good agreement again was obtained with the expression derived by Matthewson. Pepin et al.18,19 and Rossetti et al.20,21 showed the influence of wettability by comparing results for silanized and untreated particles. Mehrotra and Sastry22 studied the pendular bond strength between spherical particles with different radii. Willett et al.23 developed closed-form approximations to calculate the capillary forces between equal and unequal spheres, as well as for the rupture distance. In another paper, Willett et al.24 considered the effects of wetting hysteresis. Approximate expressions for the capillary force in nanoparticlesurface interactions were developed by Rabinovich et al.25 and by Pakarinen et al.26 A detailed comparison between toroidal and nodoidal profiles was recently presented by Mayer and Stowe.27 In another recent paper, Sirghi et al.28 considered the contant volume rupture process of a nanoscale water bridge. Other recent work studied the pull-off force by using Monte Carlo calculations.33,34 Because the exact solution of the Young-Laplace equation is rather involved, many authors instead have used the so-called circle or toroidal approximation. It is based on taking the meniscus profile to be part of a circle. The idea was introduced by Haines.29 Certain errors in Haines’s work were corrected by Fisher.30 According to the Young-Laplace equation, the effective curvature of an axisymmetric meniscus is the average of the meridian curvature and the azimuthal curvature. For given effective curvature, both of the latter vary with distance from the plate. Ignoring this variation by choosing the meridian curvature to be constant makes the azimuthal curvature constant too. This is contrary to the physical three-dimensional situation. Another deviation from the actual situation is that the expression for the capillary force at fixed separation becomes a function of distance from the plate. Nevertheless, it has been found that results based on the circle approximation often are close to the exact results.11,15,22,26,27 Common choices for the azimuthal curvature are the value corresponding to the intersection of the meniscus and one of the solid objects (one of the three-phase points), or the value corresponding to the narrowest part of the liquid bridge (“neck” or “gorge” method).8 The general pendular bridge problem has many aspects. A chronological summary of papers appearing up until 1977 is given in ref 22. Items listed are the two-particle solid system, contact angle, method of finding the meniscus profile, and whether expressions were obtained for the volume of the bridge and for (15) Ennis, B. J.; Li, J.; Tardos, G. I.; Pfeffer, R. Chem. Eng. Sci. 1990, 45, 3071. (16) Pitois, O.; Moucheront, P.; Chateau, X. J. Colloid Interface Sci. 2000, 231, 26. (17) Pitois, O.; Moucheront, P.; Chateau, X. Eur. Phys. J. 2001, B23, 79. (18) Pepin, X.; Rossetti, D.; Iveson, S. M.; Simons, S. J. R. J. Colloid Interface Sci. 2000, 232, 289. (19) Pepin, X.; Rossetti, D.; Simons, S. J. R. J. Colloid Interface Sci. 2000, 232, 298. (20) Rossetti, D.; Pepin, X.; Simons, S. J. R. J. Colloid Interface Sci. 2003, 261, 161. (21) Rossetti, D.; Simons, S. J. R. Powder Technol. 2003, 130, 49. (22) Mehrotra, V. P.; Sastry, K. V. S. Powder Technol. 1980, 25, 203. (23) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396. (24) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Powder Technol. 2003, 130, 63. (25) Rabinovich, Y. I.; Esayanur, M. S.; Moudgil, B. M. Langmuir 2005, 21, 10992. (26) Pakarinen, O. H.; Foster, A. S.; Paajanen, M.; Kalinainen, T.; Katainen, J.; Makkonen, I.; Lahtinen, J.; Niemimem, R. M. Model. Simul. Mater. Sci. 2005, 13, 1175. (27) Mayer, R. P.; Stowe, R. A. J. Colloid Interface Sci. 2005, 285, 781. (28) Sirghi, L.; Szoszkiewicz, R.; Riedo, E. Langmuir 2006, 22, 1093. (29) Haines, W. B. J. Agric. Sci. 1925, 15, 529. (30) Fisher, R. A. J. Agric. Sci. 1926, 16, 492.
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the capillary force. A related aspect is whether the solid surfaces are wetted or dry. Other aspects are whether the capillary force was obtained as function of distance between the two solid systems and whether expressions were obtained for the work of rupture. Still other aspects are whether the volume of the liquid bridge was assumed to be constant and whether evaporation and condensation governed by the ambient humidity was assumed to occur. The motivation for the present paper is two-fold. The first purpose is to elucidate the differences between the case of constant bridge volume and the case of constant bridge pressure (i.e., constant meniscus curvature). The case of constant volume applies to nonvolatile liquids. It also applies to volatile liquids at sufficiently fast rupture processes to allow neglecting evaporation and condensation processes but not fast enough to cause viscous effects to be of importance. The case of constant pressure applies when evaporative equilibrium is maintained throughout the rupture process. The two cases are analyzed using the first law of thermodynamics for a control volume.31 The second purpose is to derive exact results for the work required to rupture the pendular bridge. This work is found as a function of the nondimensional meniscus curvature. The initial situation is supposed to be a pendular ring in thermal equilibrium with the partial pressure of the surrounding atmosphere. This differs from the initial situation considered in many of the papers referenced, where the pendular bridge resulted from injection of a known volume by syringe. In the latter case, the liquid bridge volume is an independent parameter which can be used to find an effective meniscus radius. The present paper complements work described in a concurrent paper by the same authors.32 In the latter paper, the pendular bridge is analyzed using a simplified asymptotic analysis, as well as using the elliptic integral formulation of Orr et al.10 Various geometries are considered, and the results are applied to the determination of thin-film adhesion energy in MEMS.32 In the present work, exact analytical results are derived for conditions at the rupture point and at an inflection point. In the case of the rupture point, the condition used is that the derivative of the distance between asperity and plate with respect to the filling angle equals zero. In the case of an inflection point, the condition used is that the second derivative of the meniscus profile equals zero. The analytical results obtained are in a form suitable for numerical evaluation. Convenient, accurate curve-fitting results are provided for the rupture work at constant volume and at constant pressure over a wide range of parameters. This range includes conditions of importance in MEMS adhesion processes.32 Section 2 contains the thermodynamic analysis of the problem. The geometry considered is that of a liquid bridge between a spherical asperity and a flat plate. The influence of gravity is neglected, as is the possible influence of disjoining pressure. Energy balances are presented for both the constant volume and the constant pressure case. Detailed results for the constant pressure case for a contact angle of 40° are derived in Section 3. They include the capillary force as function of withdrawal distance, the conditions at rupture, and the work of rupture as a function of the dimensionless Kelvin radius. In this case, inflection points do not arise. This contrasts with the case of constant volume, which is treated in Section 4. Equations are derived for the conditions at which the inflection point occurs. Determination of the conditions at breakdown in the constant volume case is rather involved, and is described separately in (31) Moran, M. J.; Shapiro, H. N. Fundamentals of engineering thermodynamics, 3rd ed.; John Wiley and Sons, Inc.: New York, 1995. (32) de Boer, M. P.; de Boer, P. C. T. J. Colloid Interface Sci. 2007, 311, 171.
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Figure 2. Surface tensions at foot of meniscus.
Figure 3. Control volume just surrounding meniscus
Figure 1. Model used for pendular bridge (∆p < 0 here).
Appendix A. In section 5, analogous results are presented for various other contact angles. Conclusions that can be drawn from the results presented are described in Section 6.
2. Thermodynamic Analysis We consider the case of a pendular bridge between a spherical asperity and a substrate consisting of a flat plate (see Figure 1). The asperity initially is just touching the plate and then is gradually withdrawn. The pendular bridge assumes different shapes and eventually ruptures. The work required for the rupture process follows from applying the first law of thermodynamics
dE ) hidmi - hedme + δQ - δW ˆ + γadSˆ a + γpdSˆ p (1) to the control volume just enclosing the bridge and its surfaces. Here, dE is the change in total energy of the system, hidmi is the enthalpy added to the system by flow across the boundary, hedme is the enthalpy leaving the system by flow across the boundary, δQ is the heat is added to the system, δW ˆ is the work done by the system on its surroundings, γa is the solid-vapor surface tension of the asperity, dSˆ a is the change in interfacial area between the liquid bridge and the asperity, γp is the solid-vapor surface tension of the plate, and dSˆ p is the change in interfacial area between the liquid bridge and the plate. The circumflex denotes dimensional quantities that later appear in nondimensional form. By definition, the change in total energy equals
dE ) γldSˆ m + γladSˆ a + γlpdSˆ p + FudVˆ
(2)
where γl is the surface tension of the meniscus, Sˆ m the surface area of the meniscus, γla the surface tension between the liquid and the asperity, γlp the surface tension between the liquid and the plate, Vˆ the volume and Fu the internal energy per unit volume of the bridge. The pressure difference ∆p across the meniscus is given by
∆p ) 2Hγl
(3)
where H is the mean curvature of the meniscus (i.e., the inverse of the mean radius of curvature of the meniscus). For a liquid bridge in evaporative equilibrium with the surrounding atmosphere, ∆p is given by the Kelvin equation
∆p ≡ p - patm ) (RT/VoL) ln(Peq/Po)
(4)
where p is the pressure of the liquid, patm is the pressure of the sourrounding atmosphere, R is the gas constant, T is the absolute temperature, VoL is the molar volume of the liquid, Peq is the pressure of the vapor in the surrounding atmosphere, and Po is
saturation vapor pressure (i.e., the pressure of vapor in equilibrium with bulk liquid at temperature T with a flat meniscus; Peq/Po is the relative vapor pressure). The present work concerns cases in which the liquid bridge initially is in evaporative equilibrium with the surrounding atmosphere. The vapor in the latter is assumed to be unsaturated, and the initial value of H is negative. Hence, the initial pressure inside bridge less than atmospheric pressure. The pressure difference ∆p and the mean curvature H remain constant in the case of continuing evaporative equilibrium (constant pressure) but vary in the case of constant volume. 2.1. Constant Volume. In the constant-volume case, there is no flow across the boundaries of the system, and hence, hidmi ) hedme ) 0 , δVˆ ) 0. Furthermore, since no heat is added or withdrawn from the system, δQ ) 0. It follows from integrating eqs 1 and 2 that the rupture work is given by
-W ˆ V ) (Ef - Ei) - γa(Sˆ af - Sˆ ai) - γp(Sˆ pf - Sˆ pi) ) γl(Sˆ mf - Sˆ mi) + (γla - γa)(Sˆ af - Sˆ ai) + (γlp - γp)(Sˆ pf - Sˆ pi) (5) where subscripts i and f indicate initial and final states, respectively. The minus sign in front of W ˆ V results from the convention that the work is counted negative when it is done on (rather than by) the system. The surface tensions are related by
γl cos θa + γla ) γa, γl cos θp + γlp ) γp
(6)
where θa ≡ θ1 and θp ≡ θ2 are the contact angles for asperity and plate, respectively (see Figure 2 for the second of eqs 6). The nondimensional form of eq 5 thus becomes
ˆ V/(γlR2) ) -WV ≡ -W Smf - Smi - (Saf - Sai) cos θa - (Spf - Spi) cos θp (7) where R is the radius of the spherical asperity and the surfaces have been nondimensionalized by setting S ) Sˆ /R2. 2.2. Constant Pressure. The condition of constant pressure ∆p results in changes of the bridge volume Vˆ during the rupture process. In most cases of present interest, the volume decreases due to evaporation. The evaporation process is taken to be slow enough to ensure that the liquid remains at constant temperature. The heat of evaporation Q then is taken from the surroundings, i.e., from the asperity and the plate. The heat capacity of the surroundings is assumed to be large enough to neglect variations in temperature. As follows from considering a control surface just surrounding the meniscus (see Figure 3), Q is given by
Q ) (uV + p/FV)∆m - [u + (p + ∆p)/F]∆m
(8)
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written in Orr et al.10 but is required because the meridional radius of curvature must be taken negative when the meridional surface is concave. This occurs when d2y/dx2 and dy/dx have opposite signs (see Figure 4). Conversely, the meridional curvature is positive when the meridional surface is convex, i.e., when d2y/dx2 and dy/dx have the same sign. Melrose6 showed that the Young-Laplace equation can be solved by setting z ) -sin , where is the angle made by the normal to the meniscus with the negative vertical axis (see Figure 1). It is noted that 0 < < π and, hence, that z < 0. Using Figure 4. Sign of meridional radius rm.
where subscript V denotes vapor, and ∆m is the evaporated mass. Because evaporation results in a decrease of mass m of the system, ∆m ) -F(Vˆ f - Vˆ i). The term (uV + p/FV)∆m represents the enthalpy outflow of this control surface, the term [u + (p + ∆p)/F]∆m the enthalpy inflow. Equation 8 also covers the case of condensation, where Q is heat supplied to the surroundings. It follows from combining these results with the integrated forms of eqs 1 and 2 that the rupture work now is given by
-W ˆ P ) (Ef - Ei) + (uV + p/FV)∆m - Q - γa(Sˆ af - Sˆ ai) γp(Sˆ pf - Sˆ pi) ) γl(Sˆ mf - Sˆ mi) - γl cos θa(Sˆ af - Sˆ ai) γl cos θp(Sˆ pf - Sˆ pi) - (p + ∆p)(Vˆ f - Vˆ i) (9) The term -p(Vˆ f - Vˆ i) must be deleted. It represents the work obtained by decreasing the volume of the pendular bridge in the surrounding atmosphere at pressure p. This work is transferred to the asperity through its wetted surface. Because the rest of the asperity is surrounded by air at the same pressure p, the net work provided to the asperity for this effect is zero. This is equivalent to neglecting the pressure p in finding the downward force on the sphere (see eq 26). Comparison of eqs 9 and 5 shows that the rupture work in the constant pressure case is less than that in the constant volume case by the term ∆p(Vˆ f - Vˆ i). This term represents the work derived from the heat of evaporation Q taken from the surroundings. Using eq 3, the dimensionless form of eq 9 becomes
ˆ P/(γlR2) ) Smf - Smi - (Saf - Sai) cos θa -WP ≡ -W (Spf - Spi) cos θp - 2HR(Vf - Vi) (10) where V ) Vˆ /R3.
3. Basic Equations for the Pendular Bridge The equations describing pendular bridges and their solutions have been presented by Orr et al.10 They are reproduced in this section, albeit in slightly different form. The mean curvature is given by the dimensionless Young-Laplace equation for an axisymmetric meniscus
2HR )
R R + ) rm ra
d2y/dx2 |dy/dx| sgn(dy/dx) + (11) 2 3/2 [1 + (dy/dx) ] x[1 + (dy/dx)2]1/2 Here, H is the mean curvature of the meniscus, R the radius of the sphere, rm the meridional radius of curvature, ra the azimuthal radius of curvature, x ≡ r/R the dimensionless radius, and y ≡ z/R the dimensionless height (see Figure 1). The excess pressure inside the pendular bridge over that in the atmosphere is given by ∆p ) 2Hγl. This means that negative values of H correspond to negative ∆p. The absolute value signs in the last term of eq 11 are required because the azimuthal curvature always tends to increase ∆p. The factor sgn(dy/dx) in the first term is not explicitly
dy/dx ) tan
(12)
2HR ) -dz/dx - z/x
(13)
z ) C/x - HRx
(14)
eq 11 becomes
with the solution
where C is the constant of integration. Solving this equation for x under the boundary condition x1 ) sin ψ yields
x)
1 sin ( xsin 2 + c 2HR
(
)
(15)
where
c ) 4HRC ) 4(HR)2 sin2 ψ - 4HR sin ψ sin (θ1 + ψ) (16) As detailed by Orr et al.,10 three different cases arise in determining the choice of upper or lower sign in eq 15. (i) The first case is HR < 0. It is noted that sin > 0. Furthermore, sin ψ and sin(θ1 + ψ) are always positive for the situations of present interest. It follows that in this case c > 0. Since x must be positive (cf. Figure 1), the minus (lower) sign in eq 15 must be chosen. (ii) The second case is HR > 0, c > 0, which requires the plus (upper) sign in eq 15. For this case, HR > sin(ψ + θ1)/sin ψ. (iii) There remains the case HR > 0 , c < 0. Here, x is positive for both choices of the sign. The two choices now correspond to positive and negative meridional curvature, respectively.10 This follows from comparing the sign of dy/dx with the sign of
2HR 1 d2y ) dx2 cos3 1 ( sin /xsin2 + c
(17)
The former equals sgn(cos ), while the latter equals
sgn(cos ) × sgn ( sin /xsin2 + c
(
)
(18)
since sin /xsin2+c > 1 in this case (sin > 0, c < 0). The two signs are the same with the upper (+) sign in eqs 15 and 18, and opposite with the lower (-) sign. Hence the upper sign goes with parts of the meniscus surface that have positive meridional curvature, the lower sign with parts that have negative meridional curvature.38 For this case, 0 < HR < sin(ψ + θ1)/sin ψ. The same test shows that the meridional curvature is negative when HR < 0 (Case (i)) and positive when HR > 0 and c > 0 (Case (ii)). The values HR ) 0 and HR ) sin(ψ + θ1)/sin ψ are special cases corresponding to a catenoid and a sphere, respectively. The meridian profile y[x()] is obtained by integrating eq 12. The distance between sphere and plate is given by
δ ) -1 + cos ψ + y1
(19)
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The downward force on the sphere is given by
F ˜ s ) -∆pπr12 + 2πr1γl sin(θ1 + ψ)
(20)
where account is taken of the circumstance that the apparatus is surrounded by air at atmospheric pressure. Its dimensionless form is
Fs ) -2HRπx12 + 2πx1 sin(θ1 + ψ) ) -2πC
(21)
Similarly, the dimensionless form of the upward force on the plate equals
Fp ) -2HRπx22 + 2πx2 sin θ2 ) -2πC
(22)
4. Results for the Case of Constant Pressure As mentioned in Section 2, we consider situations where initially HR < 0 and the asperity contacts the substrate. This corresponds to Case (i) of the preceding section, and the lower sign in eq 15 applies. In the case of constant pressure, HR and ∆p both remain constant during the rupture process. Using analytic geometry, the following results obtain
y)
∫
2
dx 1 tan d ) d 2HR
δ ) -1 + cos ψ +
1 2HR
x1 ) sin ψ, x2 )
( ∫ ( ∫
sin 1 -
2
1
sin 1 -
2
(
sin
xsin2 + c sin
xsin2 + c
) )
1 sin 2 - xsin2 2 + c 2HR
d (23)
d (24)
)
Figure 5. Filling angle ψ as function of distance δ. The + symbol denotes the point where the inflection first occurs; the * symbol denotes the cutoff point.
Figure 7 shows a plot of the initial angle ψi and of ψ* (denoted by ψ/P) as function of HR over the domain -1 e HR e 0 , for the case θ1 ) θ2 ) 40°. The figure was obtained by specifying successive values of HR and by finding the corresponding initial values ψi from eq 24 with δ ) 0. For each HR , eq 27 then yields ψ/P. The various quantities appearing in eq 10 are given by
(25)
F() Fp ) Fa) ) πc/(2HR) )
Sp ) πx22
(28)
Sa ) 2π(1 - cos ψ)
(29)
2π[HRsin ψ - sin ψ sin (θ1 + ψ)] (26) 2
In these expressions, 1 ) θ1 + ψ, 2 ) π - θ2 with θ1 ≡ θa, θ2 ≡ θp (see Figure 1). The quantities ψ and F are plotted as function of δ in Figures 5 and 6, respectively, for the case θ1 ) θ2 ) 40°, ψi ) 60° ) 1.0472 radian. It follows from eq 24 with δ ) 0 that these values correspond to HR ) -0.1530. The results were obtained by finding δ from eq 24 and F from eq 26 at successively decreasing values of ψ. It is seen that for each value of δ there are two solutions for ψ and F. The lower branches of these have been dotted because they are unstable with respect to condensation. This can be seen by noting that the area enclosed by the ψ vs δ curve corresponds to values of HR that are smaller than the value belonging to the curve. Any point in this enclosed area thus corresponds to an absolute value of HR larger than the equilibrium absolute value; i.e., to an increased pressure difference between the liquid and the atmosphere. In turn, this leads to condensation, which continues until the upper branch is reached. When the asperity is gradually withdrawn, the filling angle ψ decreases until δ attains it maximum value δ*, at which point the pendular bridge ruptures. Consequently, the area under the upper branch of the F vs δ curve equals the rupture work WP given by eq 10. The angle ψ* at which the column breaks follows from differentiating eq 24 with respect to ψ, and setting dδ/dψ equal to zero. The resulting equation is
-sin ψ +
(
)
sin 1 + [2HR sin(2ψ) sin 1 1 2HR xsin2 + c sin(θ1 + 2ψ)]
∫
1
sin2 d ) 0 (27) 2 (sin2 + c)3/2
where ψ ) ψ*, /1 ) θ1 + ψ*, c ) c(ψ*).
π Sm ) 2(HR)2 π V)8(HR)3
∫
1 2
∫
1 2
(sin - xsin + c) d 2
2
(30)
xsin2 + c
sin sin - xsin2 + c
(
xsin2 + c
3
) d -
1 2 π -cos ψ + cos3 ψ + (31) 3 3
(
)
The initial values of these quantities are obtained by using the initial value ψi, the final values by using the value ψ/P. Since HR < 0 in this case, no inflection points occur in the meniscus profiles. The resulting plot of WP given by eq 10 over the domain -1 e HR e 0 for the case θ1 ) θ2 ) 40° is shown in Figure 8. Over this domain, the value of WP varies from about 1 to about 2.5. Figure 9 shows a similar plot over the domain -100 e HR e -0.1 using logarithmic scales. It can be seen that WP is approximately inversely proportional to -HR at large -HR. The asymptotic result at large -HR is derived in ref 32 and is given by WP ) 2π(cos2 θ)/(-HR) (eq 41 of ref 32). As a check on the correctness of the results obtained for WP (eq 10) and FP (eq 26), it was verified analytically as well as numerically that FP ) dWP/dδ. For practical purposes, the result for WP can be approximated by the quadratic equation
Y ) p2X2 + p1X + p0
(32)
where Y represents an approximation to log10 WP, X ) log10(-HR), p2 ) -0.1470, p1 ) -0.4877, p0 ) 0.0383. This
Rupture Work of Pendular Bridges
Figure 6. Force F as function of distance δ. The + symbol denotes the point where the inflection first occurs; the * symbol denotes the cutoff point.
Langmuir, Vol. 24, No. 1, 2008 165
Figure 9. Rupture work WV and WP as function of mean curvature HR (θ1 ) θ2 ) 40°, logarithmic axes). Dashed lines represent quadratic curve fits; dotted lines represent asymptotic results of ref 32.
5. Results for the Case of Constant Volume
Figure 7. Initial angle ψi, rupture angles ψ/P and ψ/V and inflection angle ψ ˜ V as function of mean curvature HR.
The procedure to find results for the case of constant volume is similar to that of constant pressure, albeit more complicated. One of the complications is that the mean curvature HR of the meniscus varies during the rupture process. As a consequence, the pressure ∆p inside the pendular bridge varies with separation distance δ. It is assumed that this pressure remains uniform and that the meniscus is isobaric for given δ. A second complication is that an inflection point may originate in the meniscus during pull-off. The cases without and with inflection point are governed by different equations. These cases are treated separately in the following subsections. 5.1. Meniscus without Inflection Point. The initial value ψi of ψ is calculated by solving eq 24 with δ ) 0 for a specified initial value (HR)i of HR. Next the volume V is determined from eq 31. The initial values Spi, Sai, and Smi of Sp, Saand Sm follow from eqs 28-30. Successively decreasing values of ψ < ψi then are specified. Each corresponding value of HR is determined by solving eq 31 with the given value of V and the new value of ψ. With ψ and HR known, the value of δ follows from eq 24, that of F from eq 26. Results for ψ and F as function of δ for the case of constant volume are included in Figures 5 and 6, respectively. When the inflection point is close to coming in, the value of sin2 + c approaches zero. The integrands of eqs 31, 24, and 30 then become very large near the lower limit. For purposes of numerical evaluation, it thus is useful to split the range of integration in these equations into two parts. In the part near the lower limit, the variable of integration is changed to p )
Figure 8. Rupture work WV and WP as function of mean curvature HR (θ1 ) θ2 ) 40°, linear axes).
approximate result is represented by the dashed line nearly coinciding with WP in Figure 9.
xsin2+c. Because this would lead to a singularity at ) π/2, the variable is retained in the remaining part. The division between the parts is denoted by m. In principle, any value between 2 and π/2 may be chosen for m. A reasonable choice is m ) 0.5(2 + π/2). Upon noting that near the lower limit cos < 0
166 Langmuir, Vol. 24, No. 1, 2008
and hence cos ) respectively,
[∫
x1-p2+c, eqs 31, 24, and 30 become,
xsin2m+c π V)8(HR)3 xsin22+c
∫
1
(xp - c - p) dp +
2
xsin2 + c
m
3
2
sin sin - xsin + c
(
de Boer and de Boer
x1 - p2 + c 3
) d
]
Figure 10. Angle as function of y, with inflection point at 3.
1 2 -π -cos ψ + cos3 ψ + ) 0 3 3
(
δ ) -1 + cos ψ -
[∫
π 2(HR)2
(33)
1
sin sin - xsin2 + c
(
xsin2 + c
m
( xp - c - p) 2
xsin2m+c
)d
∫
m
]
(34)
2
dp +
xsin22+c xp2 - cx1 - p2 + c 1
( sin - xsin + c) d 2
2
xsin2 + c
]
(35)
The angle ψ/V and the value of HR* at which the column ruptures follow from eq 33 with the given value of V, together with the condition dδ/dψ ) 0. For the latter condition, δ is given by eq 34. The differentiation with respect to ψ is involved, because HR is a function of ψ, and c is a function of both HR and ψ. Equation 34 thus is of the form δ ) δ{ψ, HR(ψ), c[HR(ψ),ψ]}. The details of the derivation are shown in Appendix A.1. The condition dδ/dψ ) 0 is represented by eq A1. Substituting ψ* and HR* in eqs 28, 29, and 30 provides Spf, Saf, and Smf, respectively. Using the resulting values in eq 7 yields WV. 5.2. Meniscus with Inflection Point. At some point during pull-off at constant volume, an inflection point may originate at the plate. The inflection point is characterized by the condition d2y/dx2 ) 0 , which occurs when c ) -sin2 2 ≡ c˜ (cf. eq 17). The value c˜ can be used to find conditions at origination of the inflection point (these conditions are denoted by a tilde). Substituting c˜ in eq 33 relates the value ψ ˜ to the value H ˜ R. The second equation between ψ ˜ and H ˜ R is obtained by solving eq 16 for HR, again using c ) c˜ ) -sin2 2
˜ ) - xsin (θ1 + ψ ˜ ) - sin 2 sin(θ1 + ψ H ˜R ) 2 sin ψ ˜ 2
2
[
∫
∫
xp2 - c˜ - p sin2m+c˜ x dp + 0 x1 - p2 + c˜ 2 ˜ 1 sin sin - xsin + c d (37) m xsin2 + c˜
(
)
]
(38)
The locations where the inflection point first occurs for the case θ1 ) θ2 ) 40° are marked with a + symbol in Figures 5 and 6. Figure 7 includes the curve for the angle ψ ˜ V as function HR, which in this case represents the initial mean curvature (HR)i. As ψ decreases below ψ ˜ , the inflection point moves away from the plate. The meridian curvature then is negative in the region with small y. This requires the upper (positive) sign in eq 15. As y increases from 0, now increases beyond 2 (see Figure 10). It reaches its maximum value 3 at the inflection point, where d2y/dx2 ) 0. Equation 17 shows that 3 is given by sin2 3 + c ) 0. The equations for y , δ, Sm, and V all have an integrable singularity at the inflection point. The singularity again is transformed away by changing the variable of integration to p ) xsin2+c in the region of small y. Each integral now is split into four parts. The first part covers the meniscus surface with positive meridional curvature, and extends from 2 to 3. The second part extends from 3 to some arbitrary value 4 between 3 and 1. It is reasonable to take 4 ) 2. The first and second parts then have identical limits and can be combined. The third part is over the domain from 4 to m, with p used as the variable of integration. The fourth part is from m to 1. The second, third and fourth parts cover the meniscus surface with negative meridional curvature. Equation 31 is replaced by
-V -
[
π -2 8(HR)3
∫
xsin22+c
0
∫x
xsin2m+c
∫
1
xp2 - c(4p2 - c) dp x1 - p2 + c
(xp - c - p) dp + 3
2
x1 - p2 + c 3 sin ( sin - xsin2 + c) sin22+c
]
d -
xsin2 + c
m
1 2 π - cos ψ + cos3 ψ + ) 0 (39) 3 3
(
(36)
The minus sign is needed in view of the inequalities c˜ < 0 , H ˜R > 0. After solving eqs 36 and 33 with c ) c˜ numerically for ψ ˜ and H ˜ R, the value of the distance δ at origination of the inflection point follows from
˜ - 1 δ˜ ) -1 + cos ψ 2H ˜R
The force at this point is
F ˜ ) -πc˜ /(2H ˜ R) ) π sin2 2/(2H ˜ R)
xsin2m+c xp2 - c - p 1 dp + 2HR xsin22+c x1 - p2 + c
∫
Sm ) -
[∫
)
)
where c(ψ, HR) is given by eq 16. This equation again is solved for HR at successively decreasing values of ψ. The force still follows from eq 26, while the displacement is given by
δ ) -1 + cos ψ -
∫ xx
[
1 -2 2HR
∫
xsin22+c 0
xp2 - c
dp -
x1 - p2 + c
xp2 - c - p dp +c x1 - p2 + c
sin2m+c
sin2
2
∫
1 m
(
sin 1 -
sin
xsin2 + c
)
]
d (40)
Rupture Work of Pendular Bridges
Langmuir, Vol. 24, No. 1, 2008 167
The result for the filling angle ψV as function of δ for the case V ) constant, θ1 ) θ2 ) 40°, ψi ) 60° is included in Figure 5. The situation where the inflection point first occurs is indicated by a + symbol. It is seen that this occurs just before rupture. The value of δ at rupture is significantly larger than that for the case of constant pressure. In the latter case, evaporation reduces the volume of the liquid, which leads to an earlier rupture. The result for F in the case θ1 ) θ2 ) 40°, ψi ) 60° is included in Figure 6. It is seen that the area under the curve for constant volume is significantly larger than that under the curve for constant pressure and thus represents a significantly larger rupture energy. The angle ψ* and the value of HR* at which the column ruptures in this case follow from solving eq 39 with the given value of V, together with the condition dδ/dψ ) 0. In the latter condition, δ now is given by eq 40. The details of working out the condition are shown in Appendix A.2. Figure 7 includes the curve for the rupture angle ψ/V as function of the initial value of HR. Substituting ψ* and HR* in eqs 28 and 29 provides Spf and Saf, respectively. The area of the meniscus surface in this case is given by
S/m )
[∫
Figure 11. Rupture work WP as function of mean curvature HR for various values of θ1 ) θ2. Unmarked curves are at θ1 ) θ2 ) 10°, 20°, 30°, and 40°, respectively.
xsin2 2 + c π 2p2 - c dp + 2 2 0 2(HR) xp2 - cx1 - p2 + c
∫x
xsin2 m + c sin2 2 + c
(xp - c - p) 2
2
dp -
xp2 - cx1 - p2 + c 2 2 ( sin - xsin + c)
∫
1
m
xsin2 + c
]
d (41)
where ψ ) ψ* and HR ) HR*. Using the resulting values in eq 7 yields WV. ˜ in the For the case θ1 ) θ2 ) 40° it was found that ψ* < ψ entire domain of (HR)i considered and, hence, that there always is an inflection point in the meniscus at breakdown. Results for this case are included in Figures 8 and 9. It is seen that WV is about twice as large as WP over the domain shown. The asymptotic result at large - HR is given by WV ) 4π(cos2θ)/(-HR), which is exactly twice the corresponding expression for WP.32 As a check on the correctness of the results obtained for WV (eq 7) and FV (eq 26) in the case of constant volume, it was verified numerically that FV ) dWV/dδ. The dotted line in Figure 9 nearly coinciding with WV represents the curvefit of eq 32 for WV with p2 ) -0.1604, p1 ) -0.4433, p0 ) 0.3258.
Figure 12. Rupture work WV as function of initial mean curvature (HR)i for various values of θ1 ) θ2. Unmarked curves are at θ1 ) θ2 ) 10°, 20°, 30°, and 40°, respectively. Table 1. Curve-Fitting Coefficients of Equation 32
6. Results at Other Values of θ1 ) θ2 In addition to the results for the case θ1 ) θ2 ) 40° listed in the previous sections, results for the rupture work of a pendular bridge were obtained for the cases θ1 ) θ2 ) 0°, 10°, 20°, 30°, 50°, and 60°. They are shown in Figure 11 for the case of constant pressure and in Figure 12 for the case of constant volume. The corresponding curve-fitting coefficients are shown in Table 1. The rupture work for the case θ1 ) θ2 ) 10° is very close to that of 0°. The separation between successive curves gradually increases as θ1 and θ2 increase. No inflection points occur for the case of constant HR < 0. For the constant volume case, it was found that there is no inflection point at rupture when θ1 ) θ2 ) 0°, 10°, or 20°. For θ1 ) θ2 ) 30°, there is an inflection point at rupture in the domain HR < -0.5443 and no inflection point in the remainder of the HR domain shown in Figure 12. For θ1 ) θ2 ) 40°, 50°, and 60° there always is an inflection point at rupture. For θ1 ) θ2 ) 40°, this inflection point becomes very close to the rupture point as -HR becomes large (i.e., close to 100).
constant pressure θ
p2
60° 50° 40° 30° 20° 10° 0°
-0.1636 -0.1554 -0.1470 -0.1387 -0.1307 -0.1240 -0.1209
p1
constant volume p0
p2
p1
p0
-0.3973 -0.4615 -0.1727 -0.3904 -0.0889 -0.4470 -0.1687 -0.1660 -0.4215 0.1572 -0.4877 0.0383 -0.1604 -0.4433 0.3258 -0.5221 0.1866 -0.1557 -0.4592 0.4412 -0.5509 0.2889 -0.1515 -0.4706 0.5153 -0.5724 0.3508 -0.1484 -0.4772 0.5542 -0.5817 0.3723 -0.1475 -0.4776 0.5631
7. Concluding Remarks The situation considered is that of a spherical asperity initially touching a substrate in the form of a flat plate. A pendular bridge in evaporative equilibrium with the surrounding atmosphere has formed in between them. The asperity is gradually withdrawn. At a certain withdrawal distance, a solution for the meniscus profile no longer exists and the bridge ruptures. Equations are (33) Jang, J.; Ratner, M. A.; Schatz, G. C. J. Phys. Chem. B 2006, 110, 659.
168 Langmuir, Vol. 24, No. 1, 2008
presented for the work of rupture in two limiting thermodynamic cases: constant volume and constant pressure. Detailed results are presented for the case of contact angle θ1 ) θ2 ) 40° (Figures 5-9). Graphs showing the rupture work for the cases θ1 ) θ2 ) 0°, 10°, 20°, 30°, 40°, 50°, and 60° are shown in Figures 11 and 12. Simple quadratic curve fits are given for these results (eq 32 and Table 1). The rupture work for the constant-pressure case is about two times smaller than that for the constant-volume case over much of the range of HR. The ratio is exactly 2 in the limit of large negative values of the mean curvature HR.32 The difference arises because, in the constant pressure case, heat is taken from the surroundings during the evaporation process. This heat equals the heat of evaporation and in the large HR limit reduces the work of adhesion to one-half the surface energy created.32 Meanwhile, in the constant volume case, the work of adhesion equals the surface energy created.32 As in ref 32, various secondary effects have been neglected. These include the possible occurrence of wetting hysteresis. This effect occurs on a majority of real surfaces.18,24 When the liquid is receding during the rupture process, the hysteresis causes the contact angle to decrease. The decrease is gradual, until a constant value called the receding contact angle is reached. If the values of the contact angle during the rupture process were known, the effect on rupture work could be estimated by interpolating the results shown in Figure 11 for constant pressure, or in Figure 12 for constant volume. As a first approximation, the contact angle could be taken to be equal to the receding contact angle during the entire rupture process. Another circumstance that has been neglected is that transfer of heat can occur only over a finite temperature difference. In reality, not all of the heat of evaporation can be taken from the surroundings. In this sense, the constantpressure results are lower limits to the work of rupture. The constant-volume case is an upper limit, corresponding to the absence of evaporation. For intermediate cases with nonequilibrium evaporation, the rupture work can be estimated by interpolating the results shown in Figures 11 and 12. Also neglected is the disjoining pressure, which is of importance as the thickness of a liquid film becomes comparable to the Kelvin radius.12,13 The consequences of this are discussed in Section 8 of ref 32. Still another neglected effect is the influence of viscosity, which can become of importance for highly viscous fluids.14-17 Many experimental results have been reported for the constantvolume case. In general, good agreement was found with the theoretical results of Orr et al.,10 on which the present constant volume results are based. Experimental data for the constant pressure case are reported in refs 36 and 37. These experiments were conducted under conditons with large disjoining pressure, where the present results are not applicable. As mentioned in the Introduction, liquid evaporation during the rupture process was previously considered by Gao et al.12,13 They presented equations for the bridge pressure in the case of thermodynamic equilibrium between the liquid film and the surrounding vapor. They developed a corresponding theory for the meniscus forces and profiles but did not study the effect of evaporation experimentally. The present results are in a form suitable for numerical evaluation involving proper integrals. They could easily be extended to the case of two equal spheres. In this case, the plate must be replaced by a plane of symmetry, which means that the contact angle there must be taken equal to 90°. In any case, the (34) Jang, J.; Sung, J.; Schatz, G. C. J. Phys. Chem. C 2007, 111, 4648. (35) de Boer, M. P. Expl. Mechanics 2007, 147, 171. (36) Mate, C. M.; Lorenz, M. R.; Novotny, V. J. J. Chem. Phys. 1989, 90, 7550. (37) Bowles, A. P.; Hsia, Y-T.; Jones, P. M.; Schneider, J. W.; White, L. R. Langmuir 2006, 22, 11436. (38) This was concluded by Orr et al.10 by considering the quantity sin K) -sin - (sin2 + c)1/2
de Boer and de Boer
results obtained can serve as a basis for statistical treatment of the energy of adhesion of thin films in MEMS.32,35
Appendix A. Determination of ψ* and HR* in the Constant-Volume Case A.1. No Inflection Point at Rupture. Application of the chain rule to eq 34 yields
|
|
∂δ dδ ∂δ dHR + ) + dψ ∂ψ HR,c ∂HR ψ,c dψ ∂c dHR ∂c ∂δ ) 0 (A1) + ∂c HR,ψ ∂HR ψ dψ ∂ψ HR
| [ |
|]
The quantity dHR/dψ is obtained by differentiating eq 33 with respect to ψ. This equation is of the form f ) f{ψ, HR(ψ), c[HR(ψ), ψ]} ) 0. Using the chain rule and solving for dHR/dψ yields
dHR ) dψ ∂f ∂f ∂f ∂c ∂f ∂c + + (A2) ∂ψ HR,c ∂c HR,ψ∂ψ HR ∂HR ψ,c ∂c HR,ψ∂HR ψ
( |
|
| )/( |
|
|)
The various partial derivatives are given by
|
∂δ ∂ψ
∂δ HR
|
) -sinψ +
HR,c
) ψ,c
[
∫
(
1 sin 1 1 2HR
|
xsin2 1 + c
xsin 2+c xp2 - c - p 1 dp 2(HR)2 xsin2m+c x1 - p2 + c
)-
HR,ψ
)
(A3)
2
∫ ( 1
m
∂δ ∂c
sin 1
sin 1 -
[ (
sin
xsin2 + c
)
]
d (A4)
)
sin m 1 1 1+ 4HR cos m xsin2 + c sin 2 1 1+ cos 2 2 sin + c x 2
∫ xx
(
)
sin m+c
1 - pxp2 - c
sin2 2+c
(1 - p2 + c)3/2xp2 - c
2
∫
dp -
]
1
sin2 d (A5) m (sin2 + c)3/2
|
∂c ) 8HR sin2 ψ - 4 sin ψ sin(θ1 + ψ) ∂HR ψ
(A6)
|
∂c ) 4(HR)2 sin(2ψ) - 4 HR sin ψ sin(θ1 + 2ψ) (A7) ∂ψ HR
|
∂f ) ∂ψ HR,c π 8(HR)3
sin 1 -sin 1 + xsin2 1 + c
(
xsin
2
1 + c
3
) - π sin ψ (A8) 3
Rupture Work of Pendular Bridges
|
π ∂f ) × ∂c HR,ψ 16(HR)3
[
(- sin
m
Langmuir, Vol. 24, No. 1, 2008 169
+ xsin2 m + c
cos mxsin2 m + c
A.2. Inflection Point Present at Rupture. If an inflection point is present at rupture, eqs A1-A3 and A6-A8 remain unchanged. Equations A4, A5, A9, and A10 become, respectively,
3
)-
(- sin + xsin + c) + 3
2
2
cos 2xsin2 2 + c
∫x
2
2
2
(
)(
(sin2 + c)3/2
|
∂f 3π )× ∂HR ψ,c 8(HR)4
∫
m
|
∫ xx
[
∫x
∫
2
2
-
HR,ψ
[∫
tan 2
xsin2 2 + c
xsin22+c
0
1
m
sin 1 -
sin
xsin2 + c
)
d
]
+ 1
dp +
(1 - p - c)3/2 xp2 - c
cos m xsin2 m + c
-
sin 2 - xsin2 2 + c
(- sin
m
cos 2 xsin2 2 + c
sin2m+c
1 - pxp2 - c
sin22+c
(1 - p2 + c)3/2 xp2 - c
∫
[
dp -
]
sin2 d (A12) m (sin + c)3/2 1
2
∫x
2
+ xsin2 m + c
cos mxsin2 m + c
) - (- sin + xsin + c) + 3
3
2
2
2
cos 2xsin2 2 + c
2 2 2 xsin2m+c xp - c - p 2p - 2c - 3 + pxp - c
(
sin22+c
∫
1 m
xp2 - c(4p2 - c) dp + x1 - p2 + c
(xp - c - p) dp + 2
3
x1 - p2 + c 3 sin ( - sin + xsin2 + c)
1
xsin2 + c
m
]
d (A14)
Nomenclature C c E F H h m p Q R r S u V W x y z
see eq 16 see eq 16 total energy force mean curvature of meniscus specific enthalpy mass pressure heat radius of asperity radius of curvature surface area specific internal energy volume rupture work see Figure 1 see Figure 1 ) -sin
Greek symbols
+
tan 2(4 sin 2 + 3c) π ∂f + ) × -2 3 ∂c HR,ψ 16(HR) 2 sin + c x 2 4 2 2p 6p 4p2c + 2c2 + 3c 2 dp + 2 0xsin 2+c (1 - p2 + c)3/2 xp2 - c
∫
xsin22+c 0
2
sin m - xsin2 m + c
∫ xx
∫ (
(A11)
1 2 4HR 2
|
d (A10)
xsin2 + c
1
xp2 - c - p dp sin +c x1 - p2 + c
|
]
sin22+c
sin2m+c
∂δ ∂c
3
2
x1 - p2 + c 3 sin ( - sin + xsin2 + c)
[
ψ,c
]
(xp - c - p) dp +
xp2 - c 1 sin22+c x dp -2 0 2(HR)2 x1 - p2 + c
)
)d
∫
sin22+c
(A9)
xsin2m+c
[
xsin2m+c
2
sin -sin + xsin2 + c sin + 2xsin2 + c
m
∂δ ∂HR
2
(1 - p2 + c)3/2xp2 - c
sin22+c
1
∫x ∫
( xp - c - p) (2p - 2c - 3 + pxp - c)dp +
xsin2m+c
∫
|
3π ∂f )× 2 ∂HR ψ,c 8(HR)4
2
2
)(
(1 - p2 + c)3/2xp2 - c
sin - sin + xsin2 + c
(
2
)(
(sin + c) 2
)dp +
sin + 2xsin2 + c
3/2
)d
]
(A13)
δ γ θ F ψ
separation distance see Figure 1 surface tension contact angle density see Figure 1
Subscripts 1,2 a f i l m P p V
see Figure 1 asperity final initial liquid meridian constant pressure plate constant volume
Superscripts ∧
∼ *
denotes dimensional quantity denotes inflection point denotes quantities pertaining to rupture point
LA701253U
