SESSILE DROPS ON SLIGHTLY UNEVEN HYDROPHILIC SURFACES

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 7, Number 3, Fall 1999 SESSILE DROPS ON SLIGHTLY UNEVEN HYDROPHILIC SURFACES S. PENNELL AND J. GRAHAM-EAGLE ABSTRACT. The problem of determining the shape of a sessile drop on a hydrophilic surface that is nearly, but not perfectly, horizontal is considered. The drop is assumed to be nearly axisymmetric. A partial differential equation determining the shape of a nonaxisymmetric drop is derived, and a procedure is described for generating an approximate solution of this equation based on linearization about the axisymmetric solution. The results of two sample computations are presented. 1. Introduction. The problem of determining the equilibrium shape of a drop of liquid in contact with a solid surface has interested researchers for many years. One reason for this interest is that measurements of drop shapes can be used to determine important material characteristics such as surface tension coefficients. It is well known that in the absence of gravity the drop assumes a spherical shape because of surface tension. If both gravity and surface tension play a role, it is not possible to give an exact analytical formula for the shape of the drop, and numerical or asymptotic methods must be used. See, for example, the numerical solutions of Padday [ll]and Hartland and Hartley [5] and the asymptotic solutions of Chesters [2], Rienstra [12] and O'Brien [9, lo]. Finn [4] provides an extensive compilation of mathematical results regarding equilibrium drop shapes. Studies of equilibrium drop shapes have focused on axisymmetric drops resting on or hanging from a horizontal surface. When viewed from above, an axisymmetric sessile drop appears circular. In practice, however, sessile drops often appear noncircular when viewed from above. Such asymmetry can be caused by a supporting surface that is not perfectly flat. Cox [3] considered the problem of determining equilibrium shapes of drops resting on a rough surface with a view toward investigating contact angle hysteresis. (The contact angle is the angle between the drop surface and the supporting surface.) To Accepted for publication on August 12, Copyright Rocky Mountain Mathematics Consortium

2 284 S. PENNELL AND J. GRAHAM-EAGLE that end he assumed that the surface slope was small and the drop volume was large, and he neglected the effects of gravity. In this paper we also consider the problem of determining equilibrium shapes of drops resting on a rough surface with small slope, but we include gravitational effects and we consider only drops whose volume is O(1). We further assume that the supporting surface is hydrophilic and nearly, but not perfectly, horizontal, and we assume that the drop is nearly axisymmetric. Because we do not deal with large-volume drops we do not observe the hysteresis effects studied by Cox. Li et al. [8] also studied the effect of surface roughness on the contact angle, but they did not compute drop shapes. The shape of an axisymmetric drop can be determined by solving an ordinary differential equation, but the determination of the shape of a nonaxisymmetric drop requires the solution of a partial differential equation. In the next section we derive this partial differential equation, and in the third section we describe a procedure for generating an approximate solution based on linearization about the axisymmetric solution. In the fourth section we present the results of two sample computations. 2. Formulation of the problem. Consider a liquid drop of volume V at rest on an impermeable surface S. We describe the drop's surface by z = f (r, 8), and we describe the base surface by z = s(r, 8), where r, 8, and z are the usual cylindrical coordinates. The function s is assumed known, up to an additive constant; we seek to determine the function f. In order to ensure that f is single-valued, we restrict our attention to hydrophilic surfaces, for which the contact angle, i.e., the angle between the drop's surface and S, is less than 90". We note that the condition f (T, 8) = S(T, 8) implicitly defines a function r = R(8) describing the contact line, where the drop meets the supporting surface. We first derive a partial differential equation describing the shape of the drop. It is well known that the shape is determined by the Laplace equation of capillarity: where Ap denotes the pressure jump across the droplair interface, y denotes the liquidlair surface tension coefficient, and KI and I E ~ are the

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