J. Chem. SOC., Faraday Trans. I , 1984,80, 37-45 Profile and Contact Angle of Small Sessile Drops A More General Approximate Solution BY MARTIN E. R. SHANAHAN Centre de Recherches sur la Physico-chimie des Surfaces Solides, 24 avenue du President Kennedy, 68200 Mulhouse, France and Laboratoire de Recherches sur la Physico-chimie des Interfaces de 1'Ecole Nationale Superieure de Chimie de Mulhouse, 3 rue Alfred Werner, 68093 Mulhouse-Cedex, France Received 9th February, 1983 The second-order differential equation describing the profile of a sessile drop has been derived in polar coordinates using the criterion of minimum free energy and applying the calculus of variations. Application of a form of perturbation theory leads to an approximate solution valid for drops of sufficiently small maximum diameter.In the case of drops of contact angle > 90°, this solution can be exploited directly to obtain the contact angle from a knowledge of drop height, maximum diameter and diameter at the plane of contact with the solid. If this last datum is lacking, the contact angle can still be obtained by a reiterative method or graphically. For contact angles c 90" this last procedure must be used and thus little advantage is gained over a solution previously obtained in cartesian coordinates. Although the solution is less accurate than data obtained from numerical integration, its relative simplicity should prove useful for the objective determination of contact angles. The well known analytically insoluble second-order differential equation describing the profile of an axisymmetric drop (or related meniscus) can be derived either from the Laplace equation1 for the equilibrium of forces at a liquid/fluid interface [e.g.ref. (2)-(7)] or by exploiting the calculus of variations to minimise the free energy of the system in question [e.g. ref. (8)-(lo)]. The most frequently employed methods of approximate solution of this equation involve numerical integration by computer, and some very accurate data have been obtained and t a b ~ l a t e d . ~ ? ~ Another approach for obtaining approximate solutions is to employ perturbation theory. This is generally less accurate (except for small drops) than the numerical techniques but has the advantage that continuous functions are obtained to describe drop profiles rather than tabulated point values.The first reference to the use of perturbation theory in this context that the author has encountered is that of Ehrlich,ll in which contact angles Bo > 90' can be evaluated from sessile drops from a knowledge of maximum drop diameter and diameter at the plane of contact with the solid. Chesters12 employed perturbation theory to obtain the approximate profile of a pendant (and by extension, sessile) drop. Two solutions were found necessary, one describing drop profile in the region near the maximum diameter and the other representing (most of) the rest of the profile. In recent work by the author13 an approximate solution was obtained by using a novel form of perturbation theory originally suggested by Roth.l* This solution bears a marked resemblance to one of Chesters' equations,12 although both its derivation and method of application are quite different.The resulting equation is useful in the evaluation of contact-angle data, but, being based on a cartesian- coordinate system, is only valid for sessile (or pendant) drops of 8, < 90' in the form 3738 PROFILE AND CONTACT ANGLE OF SMALL SESSILE DROPS presented. A solution for drops of 8, >/ 90° can be obtained but is rather too involved in its application to be of any practical use.15 The present study involves the application of the same basic perturbation method as that of ref. (13), but by using a polar-coordinate system drops of 9, > 90" may be treated. Although the resulting equations to be solved are slightly more complicated than in the previous study, the final approximate solution for the drop profile is in fact simpler.With a knowledge of drop height, maximum diameter and diameter at the plane of contact with the solid, the solution can easily be used to obtain 8,. Even if the contact diameter is not available, the contact angle can be evaluated by a reiterative met hod. THEORY The usual implicit assumptions in axisymmetric meniscus calculations are made here (solid and fluids homogeneous energetically, immiscibility of phases etc.). Consider fig. 1, which represents an axisymmetric drop of liquid 1 resting on a solid surface S of area A in the presence of a less dense fluid phase 2. The contact angle, 8,, is assumed > 90°, and the free energy of the system stems from the three free S Fig. 1.Liquid drop (phase 1) lying on flat solid surface (phase S) in presence of fluid (phase 2) and coordinate system adopted. interfacial energies, yI2, ysl and ysz, and the potential, gravitational energy; the latter is calculated in the present case, for the sake of convenience, with respect to the height of the origin, 0, which corresponds to the height of the maximum drop diameter. (In the previous paper13 the gravitational energy was evaluated with respect to the solid surface, but in fact this difference is of no consequence, introducing only an additive constant which disappears later in the calculation.) A polar-coordinate system [r(8), 81 is applied to the drop as shown. (Unfortunately, convention has it that 8 is used both for the polar angle and the contact angle, but confusion should not arise since in the present context 8, is adopted as the symbol for contact angle.)M.E. R. SHANAHAN The free energy of the drop to be minimised for equilibrium is given by 39 = loec Ee d8 + constant where r =f(8) is the function describing drop profile (0 < 8 < O,), re = dr/dO, p is the density difference of the fluids (pl -p2) and g is gravitational acceleration. The volume of the drop is a constant, V : nr3(8~) sin2 8, cos e, V = -jOecr3 sin Ode-- 2n 3 3 = s,” VedO + constant. (2) The Euler equation from the calculus of variations which constitutes a necessary condition for E to be stationary with the given volume constraint is16* where A is a Lagrange multiplier. Application of eqn (3) to eqn (1) and (2) leads to Ar2 Y12 sin e+c9 sin ocos o+- sin 8 (4) where c = p g / ~ , , .~ , 1 3 9 la* l9 Eqn (4) is an analytically insoluble second-order differential equation whose solution describes the profile of a sessile drop for a given value of c and a certain fixed, although undetermined, value of A. A perturbation method will be used to solve eqn (4) considering that c is sufficiently small. Radius r then becomes a function of both 8 and C; r(8, c). We consider eqn (4) from a purely mathematical standpoint for the moment and wish (hypothetically) to vary c from zero to its ‘real’ value. For each value of c in this range, we can choose a value of A such that we have a solution r(8, c) of eqn (4) satisfying the imposed boundary conditions r ( x / 2 , c ) = R and (Ck/aO)l(n,2, c ) = 0.These boundary conditions are thus compatible with eqn (4) provided that A is a function of c. The constant R represents the radius of the unperturbed circular profile as shown by the dotted line in fig. 1. Consideration of eqn (4) for the unperturbed case when c = 0 leads to the relation r(8, 0) = R = -2y12/A(0). ( 5 ) The significance of imposing the above boundary conditions is to ensure that the family of solutions to be considered, of which our final result will be a member, all pass through the point on the profile corresponding to the maximum horizontal40 PROFILE AND CONTACT ANGLE OF SMALL SESSILE DROPS diameter of the unperturbed drop (point P in fig. 1) and have a vertical tangent there. (Other boundary conditions could have been chosen of course, but as shown below those adopted lead to a tractable final solution.) Having established the desired boundary conditions, we assume that the profile of the drop, when c is non-zero, may be expressed as a Maclaurin expansion.Thus Terms O(c2) will be ignored in this first-order perturbation approximation. We define z(e) = r/ ac ( e , o ) whence Differentiation of eqn (4) with respect to c followed by evaluation at c = 0 (where r = R and re = 0) and substitution of the terms z and zo leads to "(. sin 8) = R3 sin 8 cos 8-22 sin d8 has been made of the fact that A = -2y,,/R when c = 0 [eqn (5)]. can be rewritten zeo+ze cot 8+22 = R3 cos 8+-- (7) where 208 = d2z/d02. the complementary function is thus readily found [AP,(cos 8) + BQ,(cos O)].With the right-hand side of eqn (8) equal to zero, Legendre's equation results and By making the substitutions x = cos 8 and eqn (8) can be solved by the method of variation of parameters20 and the particular integral obtained is - R3 z = -- - - cos 8 In (1 - C O S ~ 8). (9) The final general solution of eqn (8) is then (10) where the three constants A , B and (R2/2y12) (dA/dc)(,-, remain to be found. Since z(0) must be finite, terms in In (1 - cos 8) must be removed. This leads to B = - R3/3. The principal term in eqn (6), i.e. r(8,0), is equal to R. Bearing in mind the boundary conditions imposed above, we can thus deduce that both z(n/2) and ~e(n/2) are zero. These conditions imply respectively that both the total additive constant and A in eqn (10) are zero.The final first-order perturbation term is thus R3 3 z(8) = -- cos 8 In (1 +cos 8)for 0 d 8 -= Z, approximation M. E. R. SHANAHAN and the radius, r, of the sessile drop is thus by 41 given to a first provided the drop is sufficiently small. Note that in the above solution the unperturbed circular profile of radius R shown in fig. 1 does not correspond to the same physical drop where c is zero, since both volume and contact angle are incorrect. This profile is only an artefact leading to the obtention of eqn (12). [A similar procedure was used in ref. (1 3).] We now consider an approximate upper limit to the validity of eqn (12). At 8 = 0, we have This corresponds to the height of the liquid drop above 0. Now this height should increase with drop size such that a maximum drop height is attained.According to Padday and Pitt,21 the maximum (overall) drop height is attained at a value of the drop parameter B( = pgb2/y12, where b is the radius of curvature at the drop apex) between ca. lo3 and lo4 for drops possessing contact angles in the range 90-180°. Only after this maximum does a continuous increase in drop volume lead to a slow reduction in drop height reaching an asymptotic value towards values of /3 of ca. 10l2. However, it is clear from eqn (13) that r(O,pg/y,,) starts to decrease for drops of R > (y,,/pg In 2);. Now the radius of curvature, b, in the polar coordinate system Using eqn (12) and (13) and the limiting value, R = (y12/pg In 2)4 in eqn (14), it is easy to show that r(O,pg/yl2) starts to decrease at a value of of ca.33, well below the values obtained by Padday and Pitt. It is therefore reasonable to consider the perturbation solution to be valid for the range of maximum drop radii of Conslder fig. 2, which represents schematically the contact area of the drop on the solid surface. In the limit that the increment of angle, 68, tends to zero the contact angle of the drop, e,, is given by R 5 (Yl?/Pg In 214. e, = e,+v where 0 Fig. 2. Schematic representation of contact area of drop.42 PROFILE AND CONTACT ANGLE OF SMALL SESSILE DROPS From eqn (1 2) we have pgR2 sin 8,[( 1 +cos 8,) In (1 + cos 8,) + cos 8,] (1 +cos 8,) [3y12-pgR2 cos 8, In (1 +COS 6,)J Thus combining eqn (1 5)-( 17) leads to the final expression for contact angle, 8,. the shaded cone, K, at the base of the drop is given by The volume of the drop, V , can be calculated as follows.In fig. 1 the volume of cos 8, In (1 + cos 8,) --" ( 8, -nR3 3 3 sin2 8, cos g c z ____ K = The volume of the rest of the drop, &, is & = $Cc r3 sin 8 d 8 x 2"3R3 - Ioe' (1 -? cos 8 In (1 +cos 0) However, the perturbation approach used to obtain eqn (1 2) ignored terms of O(c2) and thus the volume, V = V = - 2 + V2, should be written as [l - cR2 cos 8 In (1 +cos O)] sin 8d8 - sin2 8, cos 8,[ 1 - cR2 cos 8, In (1 +COS O,)] + O(c2). (20) 1 nR3( 3 Joec Reintroducing the notation c = pg/y12 and introducing q = (1 +cos O,), we have (21) nR3 3 V = - (q - 2)2 [q + 1 - cR2(q2 In q + $)I + O(c2). APPLICATION OF THE THEORY The application of the above theory is basically straightforward and it can be seen that for a sessile drop of 8, > 90" the profile can be immediately assessed from a knowledge of p, g, y12 and the maximum radius, R.Although the method could be applied to drops of 8, < 90°, a reiterative or graphical technique would need to be employed in order to obtain the parameter R (which would no longer have any physical meaning). Thus for 8, < 90" the method presents little advantage over the previously developed theory in Cartesian coordinates. l3 However, assuming that 8, > 90°, the method can be used successfully to obtain contact-angle data in an objective manner. Let us first assume that we are in possession of the overall drop height, H, the maximum radius, R, and the radius of the circle of contact with the solid surface, R, (see fig.1). The position of the origin, 0, can be immediately obtained from eqn (13). The value of 8, can then be found since Ro tan$, = ____ r(0) - H ' With p, g, yI2, R and 0, known, eqn (1 5)-( 17) can be used directly to evaluate the contact angle, 0,. If, however, as is sometimes the case experimentally, R, cannot be easily measured, all is not lost (by contrast with Ehrlich's method1'). As before, the position of the origin, 0, can be determined from eqn (1 3 ) . From fig. 1 (23) H - ~ ( o ) = r(e,) cos (n-e,) = -r(oc) cos 8,.M. E. R. SHANAHAN 43 Therefore, using eqn (1 2) H- r(0) = pE cos2 8, In (1 + cos 6,) - R cos 6,. (24) 3Y12 Now eqn (24) can be used either reiteratively or graphically to obtain 6,. As before, the contact angle, O,, is then calculated from eqn (15)-(17).This method is clearly more tedious, but may be of practical use. RESULTS AND DISCUSSION The above theory was tested experimentally using a system known to give contact , angles > 90’. Photographs in profile were taken of drops of freshly distilled mercury on glass and these were enlarged to x 30 for analysis. Fig. 3 represents a tracing of one of the larger drops studied together with two half-profiles corresponding to the perturbation solution from this study and that from the ‘best’ Bashforth and Adams solution available in the tables.2 The circular profile corresponds to a sessile drop of Fig. 3. Drop of mercury on glass (drop 6); 0, calculated profile; 0, profile of Bashforth and Adams for p = 1.5.Circular profile represents drop of equal V and 0, in absence of gravity. the same volume and contact angle but in the absence of gravity as calculated from the well known equation where R, represents the radius (R, # R). (This circular profile is not to be confused with that of fig. 1, which does not correspond to a ‘real’ drop.) As can be seen from fig. 3, there is good agreement between the experimental profile and the perturbation solution. The agreement shown by the tabular results of Bashforth and Adams is perhaps less convincing (near the contact region), but this is hardly surprising since clearly the value of /? for this drop does not correspond exactly to the nearest value for which profile data have been published (B = 1.5). It was found experimentally that the limiting value of R for the validity of the theory, as calculated above [ s(y,,/pg In 2):], was a little optimistic. The theoretical value in the present case corresponds to ca.2.3 mm whereas in practice the perturbation technique is valid up to ca. 90% of this value. Of course, this criterion is fairly arbitrary since the judgement of acceptability of approximate solutions is a subjective matter.44 PROFILE AND CONTACT ANGLE OF SMALL SESSILE DROPS Owing to the difficulty of estimating volumes of drops of mercury from syringe measurements, another technique was employed to investigate the validity of eqn (21). Small drops of arbitrary volume were formed and individually photographed for analysis. These were then brought together to coalesce and photographs of the final relatively large drops were taken.Fig. 3 represents such a final drop, and data concerning it (drop 6) and its constitutive smaller drops (1-5) are given in table 1. Table 1. Data for mercury drops on glassa measured values contact angle drop no. H/mm R/mm R,/mm 9Jo V/mm3 B O O / " wo 1 1.010 0.627 0.400 135 0.94 0.11 141 139 2 1.103 0.700 0.493 131 1.27 0.14 137 136 3 1.430 0.965 0.705 127 3.16 0.29 137 138 4 1.677 1.130 0.890 124 4.85 0.42 136 136 5 1.950 1.435 1.157 118 9.00 0.79 134 134 - total 19.22 6 2.277 1.917 1.550 118 19.17 2.68 144 141 a yI2 = yT, = 480 mJ m-2; p = 13.55 g cme3. Using eqn (21), the volume of each drop was calculated. It can be seen that the sum of the volumes of the constitutive drops 1-5 as calculated is in good agreement with the value obtained for the final drop 6.The error is of the order of 0.05 mm3, or 0.3%. The good agreement obtained is considered evidence for the validity of eqn (21). Using the analysis of Bashforth and Adams, in which volume intervenes,2 a value of ca. 17.5 mm3 was obtained; however, since clearly the assumption of /? = 1.5 is very approximate, too much importance should not be attached to this figure. Values of /? in table 1 were calculated from eqn (12) and (14) and are probably slightly larger than the actual values. Considering again fig. 3, it can be seen that the perturbation solution tends to overestimate the value of the radius of curvature at the drop apex, b, and any such error is magnified in /? (B = pgb2/yI2). The sense of the discrepancy between the value obtained for drop 6 and the equivalent figure of Bashforth and Adams of 1.5 tends to confirm this.Comparison ofmeasured contact angles, OM, and those calculated from eqn (1 5)-( 17) also shows a good correlation between theory and experiment. The above theory provides a reliable technique for obtaining contact-angle data from axisymmetric drops of 8, > 90° as long as the drop is sufficiently small. Clearly the measurement of lengths (H, R and R,, if possible) is less subjective than the estimation of a tangent at the contact line. A point of interest is that the above does not require physical measurement of the height of the drop at its maximum diameter: the diameter itself suffices. This is useful since it is well known that precise determination of the former is often difficult experimentally.There is no theoretical reason why the above should not be applied to pendant drops simply by changing the sign of c. Although no experimental results were available, a comparison was made using the drawings of pendant-drop profiles of PaddayS3 The agreement was good from the drop apex up to a polar angle, 0, of ca. 140° (depending on the drop), but the perturbation approach broke down near the neck.M. E. R. SHANAHAN 45 CONCLUSIONS The differential equation describing the profile of an axisymmetric sessile drop has been studied in polar coordinates using a form of perturbation theory. An approximate solution has been found which is valid for drops of sufficiently small maximum radius.In the case of mercury, this radius is ca. 2 mm. From measurements of drop height, maximum diameter and diameter at the contact plane with the solid surface, the contact angle can be calculated directly. Even if the diameter at contact is unavailable, a reiterative (or graphical) method may be employed to calculate the contact angle. The method can in principle be applied to pendant drops, although the solution would seem to be inadequate near the drop neck. I this 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 thank one of the referees for useful constructive criticism of the first version of paper. P. S. Laplace, Mkcanique Celeste, Suppl. au X Iivre (Coureier, Paris, 1805). F. Bashforth and J. C. Adams, An Attempt to Test the Theory of Capillary Action (Cambridge University Press and Deighton, Bell and Co, 1892), described by J. F. Padday in Surface and Colloid Science, ed. E. Matijevid. (Wiley, London, 1969), vol. 1, chap. 2. J. F. Padday, Philos. Trans. R. Soc. London, Ser. A, 1971, 269, 265. J. F. Padday, J. Electroanal. Chem., 1972, 37, 3 13. S. Hartland and R. W. Hartley, Axisymmetric Fluih-Liquid Interfaces (Elsevier, Amsterdam, 1976). E. A. Boucher, Rep. Prog. Phys., 1980,43, 497. E. A. Boucher and T. G. J. Jones, J. Chem. Soc., Faraday Trans. I , 1981,77, 1183. E. Pitts, J. Fluid Mech., 1974, 63, 487. E. Pitts, J. Inst. Math. Its Appl., 1976, 17, 387. M. E. R. Shanahan, in Adhesion 6, ed. K. W. Allen (Applied Science Publishers, London, 1982), chap. 5. R. Ehrlich, J. Colloid Interface Sci., 1968, 28, 5 . A. K. Chesters, J. Fluid Mech., 1977, 81, 609. M. E. R. Shanahan, J. Chem. Soc., Faraday Trans. I , 1982,78, 2701. J. P. Roth, personal communication, 198 1. M. E. R. Shanahan, unpublished work. C. Ray Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 4th edn, 1961), chap. 12. V. I. Smirnov, A Course of Higher Mathematics, transl. D. E. Brown (Pergamon Press, Oxford, 3rd edn, 1964), vol. IV, chap. 11. S. Hartland and S. Ramakrishnan, Chimia, 1975, 29, 314. S. Ramakrishnan, P. Scholten and S. Hartland, Ind. J. Pure Appl. Phys., 1976, 14, 633. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), chap. v. J. F. Padday and A. R. Pitt, Proc. R. Soc. London, Ser. A , 1972,329,421. H. S . W. Massey and H. Kestelman, Ancillary Mathematics (Pitman, London, 2nd edn, 1964), chap. 8. (PAPER 3/208)