THE MODIFIED YOUNG S EQUATION FOR THE CONTACT ANGLE OF A SMALL SESSILE DROP FROM AN INTERFACE DISPLACEMENT MODEL

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1 International Journal of Modern Physics B, Vol. 13, No. 7 (1999) c World Scientific Publishing Company THE MODIFIED YOUNG S EQUATION FOR THE CONTACT ANGLE OF A SMALL SESSILE DROP FROM AN INTERFACE DISPLACEMENT MODEL HARVEY DOBBS Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium Received 30 August 1999 We derive the modified Young s equation for the contact angle of a fluid droplet on a rigid substrate using an interface displacement model and identify the line tension with the excess free energy per unit length calculated previously for a straight three-phase contact line. A large fluid droplet sitting on a rigid substrate exhibits a contact angle θ, given by Young s equation cos θ =1+S/γ, whereγis the surface tension of the fluid and S(< 0) is the spreading coefficient of the fluid on the substrate. 1 For a small droplet of finite lateral radius r, the contact angle θ r is given by a modified equation cos θ r =cosθ τ γr (1) where the line tension τ is the excess free energy per unit length of the contact line at the foot of the droplet.,3 This relation is useful, because it shows how measurement of the contact angles of droplets of various radii can reveal the value of τ.,4,5 Equation (1) is usually derived using thermodynamic arguments, 6 by which the line tension term arises in a manner analogous to the Laplace pressure γ/r across a curved surface. In this note we show that (1) can also be derived from an interface displacement model 7 and that τ corresponds to the quantity previously identified as the excess free energy of a straight contact line. 8 We begin with an expression for the free energy of a fluid film of varying thickness l(x), relative to that of a uniform film Ω[l] = [γ({1+( l) } 1/ 1) + V (l) µl]d x. () The first term is the increased area of the film multiplied by the surface tension γ. We keep the gradient term l to all orders, 9 rather than making a gradient-squared harvey.dobbs@fys.kuleuven.ac.be; Tel: , Fax:

2 356 H. Dobbs approximation which would only yield Young s equation for small contact angles. The net interaction between film and substrate is given by the interface potential V (l). 10 When the contact angle θ is nonzero (partial wetting 1 ), this has a minimum at a finite distance corresponding to the equilibrium thickness of a uniform film, with a normalised value V ( ) = 0. For large l, V tends to the value E = S. The final term µ can be interpreted either as a Lagrange multiplier which determines the thickness of the fluid film in the case of a non-volatile fluid, or as a measure of the distance from two-phase coexistence in the case of a volatile fluid. 11 The profile of a droplet is determined by minimizing the free energy () with respect to l. Assuming cylindrical symmetry, this gives the Euler Lagrange equation ( d dx + 1 x )( l x (1 + l x) 1/ ) = dṽ dl µ (3) where Ṽ = V/γ, µ=µ/γ and x is the radial coordinate measured from the axis of the droplet. A subscript x indicates differentiation. The left hand side of (3) is the familiar expression for the local curvature of a surface with cylindrical symmetry. The Laplace pressure resulting from this curvature is balanced by the two terms on the right hand side, which represent the disjoining pressure due to the interaction with the substrate 7 and an externally applied pressure that controls the volume of the fluid film and the droplet. We look for solutions to (3) with a given height h above the substrate at x =0, where the derivative vanishes l x (0) = 0 and which tend to a uniform thin film of thickness f for large x. For large droplets, the difference (f )=O( µ)is small. An example of a numerically determined solution for a model potential V (l) is shown in Fig. 1. Except in regions where the interface is close to the substrate, µ is the dominant term on the right hand side of (3) and the solutions form a family of spherical Fig. 1. The full line is a numerical solution of Eq. (3) for the model potential Ṽ (l) =Ae (l 1) + Be (l 1) + Ce 3(l 1) + Ẽ with A =3.3, B = 7.0, C =3.5, and the normalization constant Ẽ =0.03. The droplet height above the substrate is h =8.7. The dotted lines are a spherical dividing surface touching the top of the droplet and a flat dividing surface for the fluid film. The dividing surfaces intersect with a contact angle θ r along a circular dividing line of radius r.

3 The Modified Young s Equation for the surfaces with radii set by µ. As shown in Fig. 1, we take that sphere which touches the droplet at x = 0 as a dividing surface 1 marking the edge of the fluid phase. The plane of the thin film l = f acts as the dividing surface for the substrate 8 and both dividing surfaces intersect along a circle of radius r the dividing line. 8 The angle at which the dividing surfaces meet is the contact angle θ r. This definition of r and θ r corresponds to an experimental procedure by which the shape of a droplet, measured by interferometry for example, is fitted to a spherical surface. The parameter µ is related to the radius and the contact angle by and to the height of the droplet by µr =sinθ r (4) µ(h f) =(1 cos θ r ). (5) We now consider an expansion of the droplet shape in powers of 1/r (more formally ξ/r, whereξis the range of the interface potential). In the vicinity of the droplet edge, the derivative of the interface potential dṽ/dldominates µ on the right hand side of (3). To zeroth order in 1/r, both µand the term 1/x 1/r on the left hand side can be ignored, so that (3) is approximately the same equation as for a straight contact line, 9 which has a useful first integral l x (1 + lx ) 1/ (Ṽ Ṽ ) 1/. The full form of Eq. (3) can also be integrated once by writing the derivative with respect to x as (dl/dx)d/dl. 13 With the boundary conditions at x = 0 and for large x µ(h f) h ( l x Ṽ (h) Ṽ (f) = + x(1 + lx) + µ ) dl. (6) 1/ f For large l where the solution is given approximately by the spherical solution, the integrand in (6) tends to zero. The only region over which the integrand is significant is in the vicinity of the thin film l f where the first term under the integral can be approximated by the one-dimensional solution with x r. Since (f )=O(1/r) andṽ(f)=o(1/r ), then to first order in 1/r Ṽ (h) = µ(h f) r 1 h [ (Ṽ Ṽ ) 1/ µr ] dl. (7) Provided the intermolecular forces decay sufficiently quickly Ṽ (h) E/γ = Ẽ and the integral in (7) is convergent when the upper limit is taken to infinity. Using Eqs. (4), (5) and Ẽ =1 cos θ yields the modified Young equation with the line tension ( ) 1/ ) 1/ τ = 1/ γ Ṽ Ṽ ( Ẽ Ẽ dl. (8)

4 358 H. Dobbs Fig.. The full line and symbols show the variation of cos θ r with droplet radius r determined numerically for same interface potential as in Fig. 1. The dashed line is the modified Young s Eq. (1) with the line tension τ given by Eq. (8). Inset: The next-to-leading order correction =cosθ τ/(γr) cos θ r. This is the expression for the free energy of a contact line, when the gradient term in Eq. () l is kept to all orders. 9 It reduces to a form more commonly used in the gradient-squared approximation, 8 valid for weak interface potentials τ = 1/ γ [Ṽ(l)1/ Ẽ1/ ]dl. (9) Higher order corrections of order 1/r can also be calculated, although we can find no simple expression for the amplitude in terms of the interface potential. In Fig. we show numerical results for θ r with the model interface potential used for Fig. 1. It is seen that Eq. (1) holds as r with τ given by (8) and that the next-to-leading term is O(1/r ). In some systems with long range intermolecular forces, more caution is needed in taking the limit h to infinity in Eq. (7). Specifically we consider a system in d spatial dimensions, with intermolecular potentials which decay with separation r as r (d+σ). The interface potential V (l) then approaches its value E for large l with a power law l (σ 1). 8 For cases σ 3, there is no problem with the limit and both leading (1/r) and next-to-leading (1/r ) terms appear in the expansion of cos θ r. The boundary case σ = 3 corresponds to a system with (non-retarded) van der Waals forces in 3 spatial dimensions. For longer range potentials with <σ<3, the line tension τ is convergent 8 for partial wetting and the modified Young s equation gives the correct leading order (1/r) term. However, the next-to-leading term is of order 1/r (σ 1). For even longer range potentials 1 <σ, the finite value of h must be retained in (7). In such cases the line tension is divergent 8 and finite size effects have to be considered. 14 The leading order term in the expansion of cos θ r is

5 The Modified Young s Equation for the then of order 1/r (σ 1), although there is a logarithmic correction for σ =. Cases of extreme long range forces σ 1 can not be studied with the interface displacement model (), as the surface tension γ is divergent in such systems. Finally we remark that the modified Young equation (1) takes no account of the dependence of the surface tension γ on the curvature of the fluid interface. 1 This effect is also not included in the interface displacement model. However, it can be included in simple balance-of-forces arguments which suggest that the line tension τ in (1) be replaced by a term (τ γδ sin θ), where δ is Tolman s length for the fluid interface. 1 Both the ratio of the line tension to the surface tension τ/γ and δ are of the order of a molecular diameter, so that it would appear that the curvature dependence is significant. However for small contact angles θ 0, the correction vanishes, while in the most frequently encountered systems with a first-order wetting transition the line tension tends towards a finite positive value. 5,8 Acknowledgments It is a pleasure to thank Dmitry Tatianenko for suggesting this problem, and Ralf Blossey and Joseph Indekeu for helpful comments. This work has been supported by the EC-TMR Research Network ERB 4061 PL97, the VIS/97/01 project of the Flemish Government and the Belgian IUAP programme. References 1. P. G. de Gennes, Rev. Mod. Phys. 57, 87 (1985).. B. V. Toshev, D. Platikanov and A. Scheludko, Langmuir 4, 489 (1988) (and references therein). 3. J. O. Indekeu, Int. J. Mod. Phys. B8, 309 (1994). 4. A. Dussaud and M. Vignes-Adler, Langmuir 13, 581 (1997). 5. J. Y. Wang, S. Betelu and B. M. Law (submitted to Phys. Rev. Lett.). 6. B. Widom, J. Phys. Chem. 99, 803 (1995). 7. N. V. Churaev, V. M. Starov and B. V. Derjaguin, J. Colloid Interface Sci. 89, 16 (198). 8. J. O. Indekeu, Physica A183, 439 (199). 9. H. T. Dobbs and J. O. Indekeu, Physica A01, 457 (1993). 10. R. Lipowsky, D. M. Kroll and R. K. P. Zia, Phys. Rev. B7, 4499 (1983). 11. R. Blossey, Int. J. Mod. Phys. B9, 3489 (1995). 1. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 198). 13. R. Bausch and R. Blossey, Phys. Rev. E48, 1131 (1993). 14. J. O. Indekeu and H. T. Dobbs, J. Phys. I France 4, 77 (1994).