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Soft Matter PAPER Soft Matter Dynamic Article LinksC< Cite this: Soft Matter, 2012, 8, 10288 www.rsc.org/softmatter PAPER Mechanics of a liquid drop deposited on a solid substrate Vlado A. Lubarda*ab Received 30th March 2012, Accepted 16th July 2012 DOI: 10.1039/c2sm25740h New derivations of Young–Laplace’s and Young’s equations for a liquid drop deposited on a smooth solid substrate are presented, based on an integral form of equilibrium conditions applied to appropriately selected finite portions of the drop. A simple proof for the gravity independence of the contact angle is constructed and compared with a commonly utilized but more involved proof based on the energy minimization. It is demonstrated that the vertical component of the adhesive force along a triple contact line does not depend on the specific weight of the liquid, provided that the line tension along a triple contact is ignored. The uplifting of the surface of the substrate due to vertical force is calculated by using a linear elasticity theory. The resulting singularity in the vertical displacement and the discontinuity in the radial displacement along a triple contact line are eliminated by incorporating in the analysis the effective width of the triple contact region. It is shown that the radial displacement vanishes on the surface of the substrate outside its contact with a deposited drop. 1. Introduction Furthermore, a significant amount of research has been devoted to the determination of the shape of the deformed surface of a The study of a drop’s spreading over solid substrates is a 1–3 soft substrate, particularly the uplifting of the surface caused by fundamental problem in the mechanics of wetting. Two central the vertical component of the capillary force.11–24 Yet, in many results in this field are Young–Laplace’s equation, relating the treatises of the subject, the vertical force is not shown on the pressure difference across the liquid–vapor interface of a drop to graph and the consideration of the balance of forces in the D ¼ its curvature and surface tension, p 2slvk, and Young’s vertical direction is omitted altogether.25 This is in contrast with equation, relating the contact angle between a drop and a solid substrate to the surface tensions of the problem, cos q ¼ (ssv À ssl)/slv. The indices (sv, sl, lv) designate the solid–vapor, solid–liquid, and liquid–vapor interfaces, and s stands for the surface tension or energy. Different derivations of Young–Laplace’s and Young’s equations are possible. The original derivation by Young4 and Laplace5 is based on the consideration of the equilibrium of an infinitesimal element of a pressurized thin membrane. Young’s derivation of the expression for the contact angle is based on the consideration of the equilibrium of a material element around a triple contact line, which consists of an infinitesimal segment of a liquid–vapor interface and a thin layer from the surface of a solid substrate (Fig. 1a). The equilibrium condition in the horizontal direction then gives slvcos q ¼ ssv À ssl. Although Young did not refer to it, the vertical component of the surface tension slvsin q is balanced by the reaction force Y exerted along the contact line by the substrate. The existence and the origin of such adhesive force Fig. 1 (a) An infinitesimal element around a triple contact line con- required for the vertical equilibrium has, however, been dis- sisting of a thin surface layer of a solid substrate under surface tensions 2,6 7–10 s s cussed in the literature, sometimes with the opposing views. sv and sl, and an infinitesimal portion of a liquid–vapor interface under surface tension slv. The tangential component slvcos q is balanced by ssv À ssl, and the vertical component slvsin q by the adhesive force Y. (b) The free-body diagram of the extracted edge ring of the drop which was aDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA. E-mail: in contact with the substrate in part (a) along its horizontal side. (c) The [email protected]; Fax: +1 858-534-5698; Tel: +1 858-534-3169 free-body diagram of the isolated thin surface layer of the substrate. bMontenegrin Academy of Sciences and Arts, Rista Stijovica 5, 81000 The free-body diagram from part (a) is recovered by superposing the free- Podgorica, Montenegro body diagrams from parts (b) and (c). 10288 | Soft Matter, 2012, 8, 10288–10297 This journal is ª The Royal Society of Chemistry 2012 the well-known equilibrium considerations of the floating lenses, or drops deposited on liquid substrates, which are based on Neumann’s triangle construction.26–28 The variational derivation of Young–Laplace’s and Young’s equations is based on the energy minimization, in the spirit of the Gibbs thermodynamics.29 The Young–Laplace’s equation follows as the Euler–Lagrange equation of the variational prin- ciple, while Young’s equation is obtained as the transversality condition accounting for the boundary conditions.30–35 This approach is particularly appealing because it incorporates in the analysis the gravity, demonstrating explicitly that the specific weight of the liquid does not affect the equilibrium contact angle, provided that the line tension along the triple contact line is omitted from the analysis, which has previously been a topic of an active debate.36–41 The variational approach is powerful but mathematically involved, having some physical aspects of the problem embedded in the analysis only implicitly. It is, therefore, desirable to provide a more direct derivation of Young–Laplace’s and Fig. 2 (a) A liquid drop deposited on a solid substrate in a vapor Young’s equations, with an additional insight into the physical atmosphere. The gravity field is g, the radius of the contact circle between the drop and the substrate is R, and the contact angle is q. (b) The free- origin of the gravity independence of the contact angle. This is body diagram of the portion of the drop above the level z ¼ z(r). The accomplished in this paper by using equilibrium conditions from liquid pressure at that level is pl(z). The vapor pressure along the liquid– Newtonian mechanics in an integral form, applied to appropri- vapor interface is pv(z) and the liquid–vapor surface tension along the ately selected finite portions of the drop, or the entire drop. The perimeter of radius r is slv. presented analysis is remarkably simple and is able to deliver all mechanics results related to the equilibrium drop’s shape. As such, it represents a valuable complement to the existing varia- where g ¼ r g and g ¼ r g are the specific weights of liquid and tional analysis, reviewed in the Appendix of this paper. The l l v v vapor. In equilibrium the net force should be zero. Thus, making integral equilibrium conditions furthermore deliver the expres- the sum of all force components in the z-direction for the portion sions for the horizontal and vertical components of the adhesive of the drop shown in Fig. 2b equal to zero gives force between a drop and a solid substrate along a triple contact ðr ðr line, demonstrating explicitly their independence from the  à ð Þ ð Þ p À p À p ½ ð Þ ¼ : specific weight of the drop, if the line tension is ignored. In the pl z pv z 2 rdr 2 rslvsin f gl 2 r z r z dr 0 last section of the paper we shed new light on the determination 0 0 of the deformed shape of the surface of the substrate by using a (2) linear theory of elasticity, and discuss the singularity of the vertical component and the discontinuity of the radial compo- The total force in the z-direction from the pressure pv, acting nent of displacement below the triple contact line. The singu- over the curved surface of the drop, is equal to the integral of pv larities are eliminated by incorporating in the analysis the over the projected area orthogonal to the z-direction, which effective width of the triple contact region. It is shown that the yields the corresponding contribution to the first integral on the radial displacement vanishes on the surface of the substrate left-hand side of eqn (2). The radius of the circular base at the outside its contact with a deposited drop. level z is r, and f is the indicated angle between the drop’s profile z ¼ z(r) and the base z ¼ z. By using eqn (1), eqn (2) becomes ðr À Á À Á À Á À Á D ð Þ ¼ ; D ð Þ ¼ ð Þ À ð Þ : 2. Derivation of Young–Laplace’s equation r p z r dr slvrsin f p z r pl z r pv z r 0 Consider the equilibrium configuration of a liquid drop, sur- (3) rounded by its equilibrium vapor (or another, less dense fluid), deposited on a solid flat substrate (Fig. 2a). The mass density of The notation pl(z(r)) means pl evaluated at the level z(r). The application of the derivative d/dr to both sides of eqn (3) then the liquid is rl, the mass density of the surrounding vapor is rv, and the gravity field is g. Fig. 2b shows a free-body diagram of gives the portion of the drop above the level z ¼ z(r). The surface À Á df rDp zðrÞ ¼ slv sin f þ rcos f : (4) tension slv acts along the perimeter of radius r at height z in the dr direction tangential to the drop surface.
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