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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 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 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 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 ð Þ ð Þ p p 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 D ð Þ ¼ ; D ð Þ ¼ ð Þ 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 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. Denoting by pl(z) and pv(z) the liquid and vapor pressures at the two sides of the liquid– Since vapor interface z ¼ z(r), where 0 # r # r, the hydrostatic 0 ¼1; ¼ 1 ; ¼ r ; equations are tan f 0 sin f = cos f = r ð1 þ r02Þ1 2 ð1 þ r02Þ1 2

pl(z) ¼ pl(z)+gl[z(r) z] and pv(z) ¼ pv(z)+gv[z(r) z], (1) (5)

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 10288–10297 | 10289 where r0 ¼ dr/dz, and since dr0/dr ¼ r00/r0, we have where L is the length of the liquid–vapor interface in the middle 00 cross-section of the drop, and A is the area of that cross-section. df 1 r ¼ : (6) Since liquid pressure is constant along the line z ¼ const. dr 1 þ r02 r0 (Fig. 3b), the pressure difference pl(z) pv(z) along this line can The substitution of eqn (6) into eqn (4) yields be calculated as the pressure difference across the liquid–vapor 00 interface at z ¼ z, which is given by Young–Laplace’s expression 1 r Dp ¼ 2slvk; 2k ¼ ; (7) eqn (7), Dp ¼ 2s k. Consequently, by writing dA ¼ 2rdz, the area ð þ 02Þ1=2 ð þ 02Þ3=2 lv r 1 r 1 r integral in eqn (8) becomes ð ðh which is the axisymmetric form of Young–Laplace’s equation, expressing the equilibrium of an infinitesimal element of a drop’s plðzÞpvðzÞ dA ¼ 2slv ð2krÞdz; (9) surface locally. The derivation also delivers the geometric A 0 expression for the mean curvature k in terms of r and its deriv- atives, as given in eqn (7). with h as the height of the drop. Furthermore, it can be verified from eqn (7) that " # 0 3. Contact angle and its gravity independence = d rr 2kr ¼ð1 þ r02Þ1 2 : (10) 02 1=2 Fig. 3a shows a free-body diagram of one-half of the drop, dz ð1 þ r Þ together with a thin layer of the solid substrate on which it is When this is substituted into the integral on the right-hand side deposited. The forces with the components in the longitudinal of eqn (9), there follows direction are only shown, as only they participate in the equi- librium condition in that direction. For the general use of free- ðh L body diagrams in mechanics, ref. 42 and 43 can be consulted. The ð2krÞdz ¼ Rcos q; (11) 2 surface tensions within the solid–liquid and solid–vapor inter- 0 faces are denoted by s and s . The surface tension s acts in the sl sv lv 02 1/2 ¼ horizontal direction along the curved liquid–vapor interface of because (1 + r ) dz dL is the arc length along the liquid– 0 ¼ the vertical cut through the drop surface. The line tension along vapor interface in the (z, r) plane, r (h) 0 by the symmetry the triple contact line is denoted by s, and in the free-body around the z axis, and diagram of Fig. 3a it acts at the two points where the vertical cut 0ð Þ 0ð Þ¼ ; r 0 ¼ : r 0 cot q = cos q (12) intersects the triple contact line. The equilibrium condition in the 1 þ r02ð0Þ 1 2 longitudinal direction is obtained by making the sum of all force components in the longitudinal direction, acting on the free-body Consequently, eqn (9) gives ð shown in Fig. 3a, equal to zero. This gives ð plðzÞpvðzÞ dA ¼ slvðL 2Rcos qÞ: (13) ð Þ þ þ s ð Þ ð Þ ¼ ; ssl ssv 2R slvL 2 pl z pv z dA 0 (8) A A By introducing eqn (13) into eqn (8), we obtain the generalized Young’s equation, also referred to as the Boruvka–Neumann equation,44,45 s s s þ s cos q þ ¼ 0; (14) sl sv lv R i.e., s s s cos q ¼ sv sl : (15) slv Rslv If the line tension is ignored, eqn (15) reduces to Young’s equation. The presented derivation provides an explicit proof that the contact angle q between the drop and a substrate is independent of the gravity, or the specific weight of liquid, if the line tension is ignored, and if the surface tension is assumed to be independent of the curvature. An alternative proof available in the literature,31,33 based on the variational approach and energy minimization, is longer and more mathematically involved (Appendix). If the line tension is retained in the analysis, the gravity affects Fig. 3 (a) The free-body diagram of one-half of the drop atop a thin substrate layer. The forces with the components in the longitudinal the contact angle through the radius R, appearing in the second direction are only shown. The surface tensions of the solid–liquid, solid– term on the right-hand side of eqn (15), which depends on the vapor, and liquid–vapor interfaces are ssl, ssv, and slv. The line tension volume and the specific weight of the liquid. Since the typical 46 along the triple contact line is s. (b) The mid-section of the drop with value of the ratio s/slv is about 1 nm, and can be either positive indicated variables used in the integration procedure. or negative,47 the line tension effect cannot be neglected for

10290 | Soft Matter, 2012, 8, 10288–10297 This journal is ª The Royal Society of Chemistry 2012 droplets of initial radius below about 100 nanometers.48,49 The Thus, the adhesive force eqn (20) becomes positive line tension increases the equilibrium value of the 45 ¼ contact angle, while the negative line tension decreases it. Y slvsin q. (23)

2p p ¼ 4. Vertical component of the adhesive force In the absence of gravity eqn (17) reads (pl pv)R 2R Y 0. Since Dp ¼ pl pv ¼ 2(slv/R)sin q, eqn (23) follows. Fig. 4 shows a free-body diagram of the equilibrium configura- tion of the entire liquid drop. The liquid pressure at the solid– 0 5. Horizontal component of the adhesive force liquid interface is pl , and the constant vapor pressure at the 0 solid–vapor interface is pv. Denoting by pl(z) and pv(z) the liquid We next elaborate on the expression for the horizontal compo- and vapor pressures at the two sides of the liquid–vapor inter- nent of the adhesive force. Fig. 5a shows a free-body diagram of a face, z ¼ z(r), we can write thin surface layer of the solid substrate, with a removed drop. For the consideration of equilibrium in the horizontal direction, 0 ¼ 0 ¼ pl pl(z)+glz(r) and pv pv(z)+gvz(r). (16) only the forces in the horizontal plane of the substrate are shown. The equilibrium condition in the longitudinal direction is then

If H and Y are the horizontal and vertical components of the (ssv ssl)2R 2RH 2s ¼ 0. (24) adhesive force along the triple contact line, the equilibrium condition in the vertical direction for the entire drop is

ðR ðR The horizontal component of the adhesive force is, therefore, 0 ð Þ p p p ð Þ ¼ : s p1 pv z 2 rdr 2 RY gl 2 rz r dr 0 (17) H ¼ ssv ssl : (25) 0 0 R

Since, from eqn (16), If the longitudinal equilibrium is imposed to one half of the drop alone (free-body diagram without the solid substrate 0 underneath), as shown in Fig. 5b, the condition for the longitu- pl pv(z) ¼ pl(z) pv(z)+glz(r), (18) dinal equilibrium is ð the substitution of eqn (18) into eqn (17) gives slvL 2RH plðzÞpvðzÞ dA ¼ 0: (26) ðR 1 A Y ¼ Dprdr; Dp ¼ plðzÞpvðzÞ; (19) R 0 The integral in eqn (26) is equal to slv(L 2Rcos q), as shown in Section 3, so that eqn (26) yields an alternative expression for i.e., the horizontal component of the adhesive force, which is ðR slv ¼ Y ¼ 2krdr; (20) H slvcos q. (27) R 0 Furthermore, by reconciling eqn (25) and eqn (27), we recover because Dp ¼ 2slvk by eqn (7). the modified Young’s eqn (15). Further progress with eqn (20) can be made by observing that " # r 6. The shape of the deformed substrate surface 2krdr ¼ d : (21) ð þ 02Þ1=2 1 r In the absence of line tension s, the horizontal component of the adhesive force H ¼ s cos q is balanced by the surface tension When this is used to evaluate the integral in eqn (20), there lv difference s s , due to wetting of the solid surface below the follows sv sl drop. We assume that this self-equilibrating system of tangential ðR forces within the infinitesimally thin surface layer of the substrate ¼ ; ¼ 1 : 2krdr Rsin q sin q = (22) does not induce an appreciable deformation in the bulk of the 1 þ r02ð0Þ 1 2 0

Fig. 5 (a) The free-body diagram of a thin surface layer of the solid substrate, with a removed drop above it. For the consideration of equi- Fig. 4 The free-body diagram of the entire drop. The components of the librium in the horizontal direction, only the forces within the plane of the adhesive force along the triple contact line are H and Y. The liquid pressure substrate are shown. (b) The free-body diagram of one-half of the drop 0 at z ¼ 0ispl , and the vapor pressure over the surface of the drop is pv(z). above the substrate layer from part (a).

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 10288–10297 | 10291 substrate beneath it. This is consistent with a common practice of to first order in DR. By using eqn (29), and recalling that53 neglecting deformation of a thick substrate due to surface tension dEðkÞ 1 dKðkÞ 1 EðkÞ along the bounding free surfaces of the substrate. Consequently, ¼ EðkÞKðkÞ ; ¼ KðkÞ ; (32) dk k dk k 1 k2 we consider the substrate deformation caused by the vertical loading only. If there is no gravity, this loading consists of the it follows that 0 8 uniform vapor pressure pv all over the substrate, and the self- r > K ; r # R; equilibrated loading shown in Fig. 6, which consists of Laplace’s ð Þ < R Y 2 1 n Y pressure p ¼ 2kslv and distributed line force Y ¼ Rkslv. For a w ¼ (33) pG > R R sufficiently stiff substrate, the deformation caused by this loading : K ; r $ R: r r is negligibly small, but may be more observable or measurable for a softer substrate. The deformed surface shape of such soft The corresponding displacement at the center is wY(0) ¼ (1 n) substrates will be determined in this section by using the linear Y/G. The expression (33) is a novel expression, complementing an ¼ elasticity theory. Denoting by w w(r) the upward displacement earlier representation of wY given by (3.96a) of ref. 52, page 77. of the points on the surface of the substrate, we can write Since the line force (per unit length of the triple contact line) is

Y ¼ Rkslv ¼ slvsin q, while the pressure p ¼ 2kslv, with Rk ¼ sin w ¼ wp wY + . (28) q, the substitution of eqn (29) and eqn (33) into eqn (28) gives the expression for the vertical displacement 8 The expression for the displacement wp is well-known from the > r r ; # ; > K 2E r R classical linear elasticity,50–52 and is given by < R R 2w0 " # 8 w ¼ (34) r p > 2 > ; # ; > r R R R > E r R : 2 K 2E ; r $ R: < R R r2 r r 2ð1 nÞpR " # wp ¼ pG > r R R2 R :> E 1 K ; r $ R; In this expression, R r r2 r ð1 nÞs (29) w ¼ lv sin q (35) 0 G where G and n are the elastic shear modulus and the Poisson ratio is the magnitude of the displacement at the center, w0 ¼w(0). A of the substrate material (assumed to be isotropic), and simple representation of the expression for the vertical displace- pð=2 pð=2 ment w given by eqn (34) is novel. From eqn (35), we can identify dj 1=2 ð Þ¼ ; ð Þ¼ 2 2 the elasto-capillary length scale l ¼ (1 n)s /G, which represents K k = E k 1 k sin j dj 0 lv 2 2 1 2 1 k sin j the ratio of the surface tension slv and the elastic parameter G/(1 n). 0 0 For example, for a water drop on a Teflon substrate, l0 z 1.13 A, (30) while for a mercury drop on a Teflon substrate, l0 z 7.5 A. In are the complete elliptic integrals of the first and second kinds, the calculations, it was taken that the shear modulus and p respectively. The surface depression at the center is w (0) ¼ Poisson’s ratio of Teflon were G ¼ 0.35 GPa and n ¼ 0.46, (1 n)pR/G. respectively. The liquid–air surface tension slv of water is taken Y The expression for w can be conveniently obtained by to be 0.073 N m1 and of mercury 0.486 N m1.54 For very superimposing the solutions of two loadings and by performing deformable (silicon gel type) elastomer substrates, with the shear an appropriate limit. The two loadings are the pressure p over the modulus of 1 MPa and Poisson’s ratio close to 0.5, the length l0 circle of radius R, and the tension of magnitude p over the circle is about 350 A. For some gels and biological tissues, the shear D D of radius R + R ( R R). In the limit, the resulting modulus may be 50 kPa or less, and l0 can be of the order of displacement is micrometers.11,12,55 dwp For a given contact angle, determined from Young’s equation wY ðrÞ ¼ DR; pDR/Y; (31) ¼ ¼ dR cos q (ssv ssl)/slv, and for a given volume of the drop V0 3 4pR0 /3, the radius R of a spherical cap drop, appearing in eqn (34), is q 2R0cos ¼ 2 : R 1=3 (36) = q q sin1 3 1 þ 2cos2 2 2

Fig. 7a shows the plot of a nondimensional displacement

profile w(r)/w0. Fig. 8 shows the two displacement contributions p Y w and w (normalized by w0). The linear elasticity predicts a displacement singularity for wY, and thus w, along the triple contact line r ¼ R. The singularity can be eliminated by intro- Fig. 6 A self-equilibrated loading on the surface of the substrate, con- ducing a small but finite thickness of the liquid–vapor interface sisting of pressure p ¼ 2kslv and distributed line force Y ¼ Rkslv. layer and by considering Y to be distributed rather than

10292 | Soft Matter, 2012, 8, 10288–10297 This journal is ª The Royal Society of Chemistry 2012 6.1. Radial displacement

Fig. 7b shows the radial displacement of the points on the surface of the substrate, obtained from 8 < r r\R; ¼ R u u0: (37) 0; r . R;

where ð1 2nÞs sin q u ¼ lv : (38) Fig. 7 (a) The normalized vertical displacement (w ¼ wp + wY) of the 0 2G surface of the substrate due to combined (p, Y) loading shown in Fig. 6. The expression for the radial displacement eqn (37) is derived The normalizing displacement factor w0 is specified by eqn (35). (b) The 61 corresponding normalized radial displacement (u ¼ up + uY). The by the superposition of two contributions normalizing displacement factor u0 is specified by eqn (38). 8 < # ; pð1 2nÞ rrR up ¼ R2 (39) 4G : ; r $ R; r

and 8 < 0 r\R; Yð1 2nÞ uY ¼ R (40) 2G : ; r . R; r

with p ¼ 2Y/R and Y ¼ slvsin q. The plots of the displacement contributions up and uY are shown in Fig. 9. The displacement up is linear, while uY ¼ 0 within r < R. In addition, up ¼uY for Fig. 8 The normalized vertical displacement contributions wp and wY r > R, so that u ¼ 0 for r > R (Fig. 7b). The negative values of the from: (a) pressure p, and (b) line loading Y. Their sum gives the radial displacement u for r < R mean that the radial displacement displacement profile shown in Fig. 7a. The normalizing displacement there is toward the center. The linear elasticity also predicts that factor is specified by eqn (35). the radial displacement is discontinuous at r ¼ R. The radial displacement features shown in Fig. 7b were not previously noted or discussed in the literature. The displacement singularity and concentrated line force acting on the surface of the substrate. the displacement discontinuity will be eliminated in Section 6.2 Such an approach, previously used in ref. 11 and 12 will be dis- by distributing Y over a small, but finite thickness. The linear cussed further is Section 6.2. A nonlinear continuum model may elasticity predictions of infinite vertical displacement and also be used, although an atomistic/molecular dynamics discontinuous radial displacement can be interpreted as the approach is in essence needed to predict the uplifting of the indications of the tendency for large vertical displacement and substrate very close to a triple contact line. Analytical and large gradient of the radial displacement below the triple contact experimental determination of the deformation of a highly elastic line. If the substrate is incompressible (n ¼ 1/2), the radial silicone gel substrate due to a 10 mL water drop has recently been displacement vanishes at all points on the surface of the substrate reported by Jerison et al.,23 who measured surface and bulk (u ¼ up ¼ uY ¼ 0). deformation of a thin elastic film near a three-phase contact line using fluorescence confocal microscopy, but in their analytical 6.2. Finite thickness of the surface layer study they assumed that the contact angle is q ¼ p/2. Related work also includes the contributions.16–24,56–59 Earlier studies To eliminate the singularity in the vertical displacement and the considered drops which were deposited on circular plates with discontinuity in the radial displacement under the concentrated the bending stiffness,13 the analysis based on the plain strain line force, it is assumed that the line force Y is distributed over a approximation,55,60 and the analysis of a liquid droplet deposited small but finite thickness, related to the thickness of the interface in the middle of an elastic isotropic thin solid sheet,15 rather than of the liquid–vapor layer and the molecular interactions between a semi-infinite substrate as in this paper. The formation of the liquid drop and the solid substrate. This thickness may vary circular ridges by the action of capillary forces can have signifi- from 1 nm for harder substrates to millimicrons for softer rubber cant effects on the functioning of MEMS and other micro/nano or gel substrates.6,11,12,20 The vertical displacement along the devices, lubrication of magnetic hard disks, molten solder surface of the substrate due to the loading shown in Fig. 10 is spreading in electronic packaging, painting, coating, anti-stain or readily obtained from eqn (29) by superposing the displacements anti-frost treatments, etc. For example, the created wetting ridge due to tensile loading of magnitude q along r # R and the tension can significantly increase the spreading time of a deposited drop, loading of magnitude q along r < R0. This gives or can increase the roughness of the surface of a substrate upon the drop’s evaporation, both having important technological 2ð1 nÞqR wq ¼ W q; (41) consequences.16 pG

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 10288–10297 | 10293 Fig. 9 The normalized radial displacement contributions up and uY from: (a) pressure p, and (b) line loading Y. Their sum gives the displacement profile shown in Fig. 7b. The normalizing displacement factor is u0 from eqn (38), with p ¼ 2Y/R and Y ¼ slvsin q.

Fig. 10 Uniform tensile load q applied within an annular region R0 # r q # R over the surface of the half space. The radial displacement u of the Fig. 11 (a) The vertical displacement w along the surface of the points of this surface is measured along the r-direction, and the vertical substrate due to distributed loading shown in Fig. 10. The normalizing ¼ displacement w along the z-direction. displacement factor is w0 (1 n)q(R R0)/G. The solid curve is for R0 ¼ 0.9R, and the dotted curve for R0 ¼ 0.5R. (b) The corresponding q radial displacement u . The normalizing displacement factor is u0 ¼ where (1 2n)q(R R0)/(4G). 8 > r R0 r ; # ; > E E r R0 > R R R0 > > " # <> 2 r r R0 R0 R0 q E E K ; R # r # R; W ¼ 1 2 0 > R R r r r > > " # > 2 2 > r R R0 R0 R0 R R :> E E þ 1 K 1 K ; r $ R: R r r r2 r r2 r

The corresponding radial displacement is obtained in the same shows the corresponding radial displacement uq ¼ uq(r). The ¼ ¼ way by using eqn (39), with8 the result radial displacement gradient is discontinuous at r R0 and r ; # ; ¼ > 0 r R0 R. There is a sharp displacement gradient du/dr between r R0 > <> 2 and r ¼ R, which is sharper for smaller values of the difference ð Þ R0 q 1 2n q r ; R0 # r # R; u ¼ r (42) (R R0), giving rise to displacement discontinuity amounting to 4G > / / > 2 2 Y(1 2n)/(2G) in the limit R R0 and q(R R0) Y, as shown > R R0 : ; r $ R: earlier in Fig. 9b. It is noted that wq(0) ¼ wY(0), regardless of the r ratio R0/R. Both eqn (41) and eqn (42) are novel displacement expressions, Fig. 12 shows the total displacement components (w ¼ wp + wq not reported previously in the literature. and u ¼ up + uq), due to the tensile loading q within the range q ¼ q Fig. 11a shows the vertical displacement profile w w (r)in R0 # r # R, and the pressure p within 0 # r < R0, in the case of ¼ p p the case when R0 0.9R and 0.5R. The normalizing factor is R0 ¼ 0.9R (w and u are determined from eqn (29) and eqn (39), ¼ q w0 w (0). The formation of a blunted ridge under the load with in which R0 plays the role of R). The combined (p, q) loading is 2 2 2 an increase of the interface thickness R R0 is clear. Fig. 11b self-equilibrating, so that p ¼ q(R R0 )/R0 . The load q is

10294 | Soft Matter, 2012, 8, 10288–10297 This journal is ª The Royal Society of Chemistry 2012 Fig. 12 (a) The vertical displacement w ¼ w(r) along the surface of the substrate due to tensile loading q shown in Fig. 10 and the pressure 2 2 2 p ¼ q(R R0 )/R0 within r < R0, where R0 ¼ 0.9R. The normalizing displacement factor is w0 ¼ (1 n)p(R R0)/G. (b) The same for the radial displacement. The normalizing displacement is u0 ¼ (1 2v)p(R R0)/(4G).

on a solid substrate, and the Young equation for the contact angle between the two. This is accomplished by applying an integral, rather than a local form of the equilibrium condition for an appropriately selected finite portion of the drop, or the entire drop. It is demonstrated that, in the absence of line tension along a triple contact line, neither the contact angle nor the vertical adhesive force exerted on a drop by the substrate, or vice versa, depends on the gravity or specific weight of the drop. The simple analysis presented in this paper delivers all mechanics results related to the equilibrium drop’s shape and, as such, it represents Fig. 13 The virtual displacement dr(z) imposed in the r-direction takes the a valuable complement to the existing variational analysis. The ¼ surface of the drop r r(z) from its solid- to dashed-line configuration. Due uplifting of the surface of the substrate caused by the vertical to symmetry around the z-axis, only one-half of the surface profile is shown. adhesive force is determined by using a linear elasticity theory. The singularity in the vertical component and the discontinuity statically equivalent to the line tension Y ¼ s sin q in the sense sl in the radial component of the displacement below the triple that Y ¼ q(R R0). Since (p, q) is self-equilibrating, the q contact line are eliminated by distributing the capillary force over displacement w diminishes to zero at large r much faster than w a small but finite width around the contact line. It is shown that (cf. Fig. 11a and 12a). The vertical displacement at the center is the radial displacement vanishes on the surface of the substrate R ð1 nÞs Rsin q outside its contact with a deposited drop. A nonlinear continuum wð0Þ¼w0 ¼ : (43) R0 G R0 mechanics or an atomistic/molecular dynamics approach is needed to predict the uplifting of the substrate very close to the Furthermore, since the line force Y is distributed over the finite triple contact line. This is of importance for soft substrates thickness R R ¼ 0.1R, the discontinuity in the radial displace- 0 because the formation of circular ridges by the action of capillary ment at r ¼ R from Fig. 7b is eliminated, albeit the displacement forces can have adverse effects on the functioning of MEMS, gradient du/dr is still large in the range 0.9R < r < R (Fig. 12). The NEMS, microfluidic and medical devices. radial displacement vanishes for r $ R and, by symmetry, at r ¼ 0. The remarkable feature of vanishing radial displacement beyond the radius r ¼ R has not been observed in the literature before. Appendix: variational analysis of a drop’s shape and Rusanov12 studied the deformation of an elastic substrate caused contact angle by a deposited liquid drop, but did not impose the self-equili- We present in this appendix the derivation of Young–Laplace’s brating nature of the liquid pressure exerted by the drop and the and Young’s equations based on the variational approach and vertical component of adhesive force along the triple-contact line. energy minimization. The derivation is closely related to the Furthermore, his analysis did not proceed sufficiently far to reveal original derivations from ref. 30 and 31. Under constant ambient the vanishing radial displacement along the substrate outside its pressure and temperature, the free energy of the system, con- contact with the drop. The wetting angle, as determined by sisting of a liquid drop deposited in the vapor environment on a Young’s equation, is locally not affected by the formation of the stiff solid substrate, whose infinitesimal deformation can be blunted ridge due to substrate deformation, as noted in ref. 62. ignored, is

7. Conclusion ðh ¼ þð Þ 2p þ p s þ D : We have presented in this paper a new derivation of both the U slvS ssl ssv R 2 R gzdV (A1) Young–Laplace equation for the shape of a liquid drop deposited 0

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 10288–10297 | 10295 The last term on the right-hand side of eqn (A1) represents the because the slope at the apex of the drop r0(h) vanishes by gravitational potential energy, while symmetry, and the cosine of the angle between the tangent to the

ðh drop’s profile and the horizontal surface of the substrate is = S ¼ 2prðzÞds; ds ¼ 1 þ r02ðzÞ 1 2dz (A2) r0ð0Þ cos q ¼ : (A10) 0 1=2 0 1 þ r 2ð0Þ is the surface of the axisymmetric liquid–vapor interface, whose Consequently, by substituting eqn (A7) and eqn (A9) into eqn r ¼ r z profile ( ) is to be determined. If the liquid is assumed to be (A6), there follows incompressible and if there is no evaporation, the liquid volume V remains constant and the appropriate functional for the 2phrðzÞ 2slvkðzÞl þ zDg variational study is s þ 2pR slvcos q þ ssl ssv þ dðzÞ¼0: (A11) ðh R P ¼ U lV; V ¼ r2ðzÞpdz: (A3) For this to vanish, the two terms must vanish separately, which 0 gives The Lagrangian multiplier l has the dimension of pressure. In view of eqn (A1) and eqn (A2), the potential function P can be l 2slvk(z) zDg ¼ 0, (A12) written as ðh and ¼ ð ; ; 0Þ ; s P F z r r dz (A4) s s s cos q ¼ 0: (A13) sv sl lv R 0 The last expression is a modified Young’s equation, incorpo- where 44,45 rating the line tension s. Eqn (A12) yields the differential 02 1=2 2 F ¼ slv2prðzÞ 1 þ r ðzÞ r ðzÞpðl zDgÞ equation for the shape of the drop. To determine an expression 1 for the Lagrangian multiplier l, the fact that the surface of the þ ðs s Þr2ðzÞp þ 2prðzÞs dðzÞ: (A5) ¼ h sl sv drop near the apex z h is locally a sphere, of some radius b,so that eqn (A12) gives l ¼ hDg +2slv/b is used. Since Dp(h) ¼ 2slv/b, by the elementary consideration of a pressurized spherical shell, The Dirac delta function is d(z), and h is the height of the drop. the Lagrangian multiplier can also be interpreted as the pressure The variation of the functional dP, divided by an infinitesimal difference between the liquid and vapor phases at z ¼ 0, i.e., l ¼ variation dr(z) at an arbitrary z ¼ z (Fig. 13), must vanish at the Dp(0) ¼ Dp(h)+hDg. Thus, eqn (A12) can be rewritten in equilibrium state, the form of Young–Laplace’s equation 2slvk(z) ¼ Dp(z), where ¼ dP vF d vF vF z h Dp(z) ¼ Dp(0) zDg. ¼ h þ dðz zÞ ¼ 0: (A6) ð Þ 0 0 dr z vr dz vr z¼z vr z¼0

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