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Nonaxisymmetric drop shape analysis and its application for determination of the local contact angles
Journal of Colloid and Interface Science 301 (2006) 677–684 www.elsevier.com/locate/jcis
Nonaxisymmetric drop shape analysis and its application for determination of the local contact angles Stanimir Iliev, Nina Pesheva ∗ Institute of Mechanics, Bulgarian Academy of Sciences, G. Bonchev street, block 4, 1113 Sofia, Bulgaria Received 20 December 2005; accepted 28 May 2006 Available online 21 June 2006
Abstract We describe here a numerical method for finding the local contact angles of a drop in the case of partial wetting for given values of the drop volume and capillary length when there are data available for the whole contact line of the drop. There are no special restrictions imposed on the type of the contact line: the solid substrate on which the drop rests can be heterogeneous or rough or both, it can be horizontal or tilted. The method is intended to be used in conjunction with experimental results similarly to the axisymmetric drop shape analysis-diameter (ADSA-D) and analysis-profile (ADSA-P) methods. The numerical method is essentially an iterative minimization procedure based on the local variations approach. It allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. Several examples of applying the method are described for a drop on: flat, horizontal but heterogeneous substrate; flat, tilted substrate, and grooved horizontal substrate. © 2006 Elsevier Inc. All rights reserved. Keywords: Liquid drops; Nonaxisymmetric drop shape analysis; Local contact angles
1. Introduction The contact angle that a liquid drop forms with a solid is an important measure of the wettability of the solid surface. Measurements of the contact angles in many research and industrial laboratories are among the most used ways to characterize the surfaces. The basic techniques used for measuring contact angles have been reviewed in [1–5]. The most often used techniques for measurements of the contact angles are the sessile drop and the captive bubble techniques. Their widespread use is due to their simplicity and the small amount of liquid and surfaces needed. These techniques are currently based on taking high resolution photographs and are commonly used in combination with numerical methods to improve the precision of the obtained results since the real surfaces are heterogeneous and/or rough. Such numerical methods are the axisymmetric drop shape analysisdiameter (ADSA-D) and axisymmetric drop shape analysisprofile (ADSA-P) method. In the first case a top view of the * Corresponding author.
E-mail address: [email protected] (N. Pesheva). 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.05.067
sessile drop and in the second a side view of the drop is used. These methods provide a contact angle by finding a numerical axisymmetric solution of the Young–Laplace equation with the given experimental parameters (drop volume, density, diameter, height, capillary length, etc.) and then calculating the slope of the tangent to the drop surface at the liquid–solid– vapor interface line (see, e.g., [6–11]). For more detailed account of recent advancement and modifications of ADSA-D and ADSA-P methods (see [12,13] and references therein). These techniques can be expected to give an accurate result only when axisymmetric approximation for the drop shape can be used. At present the development of new more precise experimental techniques for determining the profile and the contact line of a liquid drop as well as the ever growing demand for going to smaller and smaller scales from the industry requires the development of new numerical methods which will go beyond the axisymmetric approximation for the shape of a drop. Thus currently there is a necessity to determine the contact angles on real surfaces which are heterogeneous and/or rough for various reasons, e.g., due to contamination, the presence of impurities, cavities, uneven oxide layer, “hot” sites, etc. On the other
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hand, chemically heterogeneous solid substrates with specially designed structures are increasingly used, e.g., in surface modification, textile industry, biomedical research, surfactant chemistry, etc., for engineering of desired nano- and micro-patterns in thin films (see, e.g., [14–17]). On such surfaces the drops are not axisymmetric anymore, and the contact angle varies along the contact line. We describe here a numerical method for finding the local contact angles along the contact line of a drop in the case of partial wetting. There are no restrictions imposed on the type of the contact line: the solid substrate on which the drop rests can be heterogeneous or rough or both, it can be horizontal or tilted. The method is intended to be used in conjunction with experimental results (similarly to ADSA-D and ADSA-P methods) when there are data available for the whole contact line of a drop and for given experimental values of the drop volume and capillary length. The method allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. In Section 2 more precise definition of the problem is given. In Section 3 we describe the method in detail, in Section 4 we illustrate how it works on several examples and Section 5 is devoted to our conclusions. 2. Problem formulation Let us consider a 3D equilibrium sessile liquid drop with volume V resting on a solid surface Σs at equilibrium. Both the drop and the substrate on which it is placed are immersed in an ambient fluid and it is assumed that the liquid and fluid are mutually immiscible. The solid substrate can be heterogeneous or rough or both, it can be horizontal or tilted. The surface of the drop Σ forms on the solid surface Σs a line L called contact line or common line. The goal is to determine the shape of a drop and the local contact angles along the contact line for given a priori contact line L, volume V , and capillary length a (see Ref. [1]) a 2 = σ/(ρl − ρf )g,
(1)
where the density of the droplet and the ambient fluid are denoted by ρl and ρf correspondingly; σ is the liquid–fluid surface tension, g is the gravity acceleration constant. After finding the drop surface, the calculation of the local contact angles θ (P ) at point P along the contact line L is straightforward: θ (P ) = arccos nΣ (P ) · nΣs (P ) , (2) where nΣ , nΣs are the outward unit vectors at point P normal to the surfaces Σ and Σs , respectively. The equilibrium drop shape Σ , we are looking for, is formed as a result of the action of the surface tension σ and of the gravitational forces acting in both phases, the liquid and the fluid. This surface is described as a Laplacian surface and as a minimal surface. In the first case a differential description is used and in the second integral. In this work we use the integral approach. That is, in a Cartesian coordinate system (x, y, z) whose z-axis is taken to be in direction opposite to that of the gravity acceleration g, i.e., z = −g/g, the equilibrium drop shape
Σ is the one which minimizes the energy functional (see Refs. [18–20]): dΣ + (ρl − ρf )gz dΩ E=σ (3) Σ
Ω(Σ,Σs )
and the boundary of this surface coincides with the given contact line L and also the surface Σ together with the substrate surface Σs encloses the prescribed volume V , i.e., Ω(Σ, Σs ) = V .
(4)
In the usual way, the energy functional is expressed in terms of the liquid–fluid surface tension σ , i.e., we consider U = E/σ . Thus the variational problem is reduced to minimizing the following functional: U = dΣ + (5) z/a 2 dΩ. Σ
Ω(Σ,Σs )
Equivalently, in the differential description instead of minimizing the renormalized expression (5), one has to solve the Laplace equation to find the equilibrium drop surface Σ : (κ1 + κ2 ) = z/a 2 + c,
(6)
where c is a constant, κ1 and κ2 are the principal curvatures of the surface Σ . In quite the same way one can solve also the above problem, where the condition of a given a priori volume (Eq. (4)) is replaced by the requirement that the surface of the drop Σ passes through a given point P ∈ / Σs or has a maximal distance h from the surface of the substrate Σs . For a horizontal substrate the last condition is equivalent to a given a priori height of the drop. The above formulation of the problem is appropriate to be used for the sessile drop measurements. It can be applied also to pendent bubble measurements of the contact angle. 3. Numerical method We solve the defined in Section 2 problem numerically. We find solutions for the equilibrium drop shapes on heterogeneous or rough substrates by minimization of the renormalized functional (5) subject to the two constraints—that of a fixed volume (4) and prescribed contact line L. First, we find the spherical cap approximation of the drop shape Σ0 with the given volume V and capillary length a which has a circular contact line. After that we perform an iterative procedure which transforms the contact line gradually while the volume is kept fixed and the surface is maintained Laplacian until the desired contact line L is obtained. The numerical method for obtaining drop surfaces was first developed for homogeneous surfaces in Refs. [21,22] and then extended to treating heterogeneous surfaces and line tension effects in Ref. [23]. The numerical method employed is essentially an iterative minimization procedure based on the local variations approach [24] combined with an additional procedure for gradual transformation of the contact line toward the
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desired (target) contact line. The numerical method for obtaining drop surfaces is similar to the numerical methods [25,26], and the one used in the public domain software “Evolver” [27]. The drop shape is approximated by a triangulation Σ¯ N using N points (see, e.g., Fig. 1). The renormalized energy functional (5) is expressed in terms of the coordinates of these N points. This is a standard procedure in the numerical methods for discretization of surfaces. The N points are represented by radius-vectors, p1 , p2 , . . . , pN , which end at the corresponding point. Thus any surface Σ considered in the minimization of the renormalized energy functional (5) is represented as a union of M triangles. Part of the nodes of the triangles are inner for the surface Σ¯ N , the rest, NL nodes, are border nodes: pN −NL +1 , pN−NL +2 , . . . , pN . The renormalized energy functional U , given by Eq. (5), thus becomes a function of the points p1 , p2 , . . . , pN , U Σ¯ N = U p1 , p2 , . . . , pN . (7) Then the problem of finding the surface which minimizes (5) in this N -point approximation can be formulated as follows. For fixed ε 1 (ε gives the precision with which the local surface variations are made) find points p1∗ , p2∗ , . . . , pN ∗ , defining NL +2 L +1 , p , . . . , pN a surface Σ ∗ , whose boundary points, pN ∗ ∗ ∗ , belong to the given a priori contact line L. The surface Σ ∗ must be also such that together with the substrate surface Σs encloses the given volume V and for which U ∗ = U p1∗ , p2∗ , . . . , pN ∗ L L +1 , pN−NL +2 , . . . , pN , pN−N U p1 , p2 , . . . , pN−N ∗ ∗ ∗ (8) for any other surface Σ close to Σ ∗ defined by the points 1 2 L L +1 , pN−NL +2 , . . . , pN , p , p , . . . , pN−N , pN−N ∗ ∗ ∗ where pi = pi∗ + εi Ei , εi > ε,
Ei = 1,
i = 1, . . . , N − NL .
(9)
Condition (8) defines the ε-minimum (given by the surface Σ ∗ ) of the function (7) with respect to the set of possible finite ε variations which is the condition for equilibrium. We use a Monte Carlo scheme for choosing the node points which we will try to move since in this way the algorithm works faster as compared to a fixed order of regular sampling of all the points defining the drop surface. At every iteration step the drop shape is changed in such a way so that the renormalized energy expression (7) is decreased while the drop volume and the NL points of the contact line are kept fixed. The correctness of the obtained solution at every iteration step is monitored by keeping track of the accuracy with which the coordinates of the points of the drop surface satisfy the Laplace condition (6). For ensuring a better work of the minimization procedure, a regular check and readjustment of the surface mesh is performed in order to keep the approximation of the liquid/gas interface uniform. As it will be seen in Section 4, the final drop shape satisfies the Laplace equation with high precision. The realization of this step is described in detail in [28], it is tested also
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in [22] by comparing of the obtained equilibrium shapes with those in [29]. Let the prescribed contact line L be given as a sequence of points {I1 , I2 , . . . , Ik }. We start the numerical process with a spherical cap approximation Σ0 of the drop shape with a circular contact line C(O(L), r(L)) with a center O(L) and a radius r(L) defined in the following way: pO =
I1 + I2 + · · · + Ik , k
r(L) ∈ (rmin , rmax ).
(10)
pO is the radius-vector of the center O(L) and rmin and rmax are respectively, the minimal and the maximal distance from the center O(L) to the points of the contact line L ≡ {I1 , I2 , . . . , Ik }. Then this surface is triangulated using N -points, Σ¯ 0 = Σ¯ 0 (p1 , . . . , pN ), in a coordinate system whose origin O coincides with the center of mass O(L) of the points of the contact line. Next, an iterative procedure which transforms the contact line gradually is implemented in a way very similar to the approach used in [30] where a quasi-static relaxation of a drop was simulated: At every point of the contact line, pN−NL +1 , pN−NL +2 , . . . , pN , instead of moving the contact line at a distance proportional to its velocity (as it was done in [30]), the points of the contact line are moved at small fixed distance in the appropriate direction. The simplest possible choice is to move a point along the radius passing through the origin O(L) and that point in direction toward the prescribed contact line L (see the two thick points and the arrows in Fig. 2). This process is accompanied by a change of the drop volume. In order to compensate for the change in the volume at every iteration step also a sequence of displacements of the inner points p1 , p2 , . . . , pN−NL is performed so that the initial volume V is restored. When the substrate surface is rough then the problem can be solved in two stages. First, one finds a solution for a contact line Lp which is a projection of the original contact line on a flat surface. Then a gradual transformation of the “flat” contact line Lp into the original contact line is performed. This is described in more detail in example 4.3 in Section 4. As a last iteration step we perform final minimization procedure with a fixed contact line L and a fixed volume V . After the desired drop shape is obtained, the determination of the contact angles along the contact line is straightforward. From the approximating triangles at every point of the contact line one determines the normal to the drop surface and therefore the contact angle at that point. The precision with which one can obtain the contact angles depends on the number of points N used to approximate the drop surface, on the ε-parameter of the algorithm, the form of the triangulation net, on the realization of the re-mesh procedure. It depends also on the peculiarities of the contact line, on the roughness of the substrate, etc. In Section 4 we give the specific values of some of these parameters with which the contact angles (on both hydrophilic and hydrophobic substrates) can be determined with a high precision, of the order of θ ∼ 0.1◦ , for a drop on a flat surface, and θ ∼ 1◦ , for a drop on a rough surface.
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4. Numerical examples We applied the numerical method described in Section 3 for several cases: flat, horizontal but heterogeneous substrate (examples 4.1 (A1 and A2); flat, tilted substrate (example 4.2), and grooved horizontal substrate (example 4.3). We compare our results with numerical solutions for drops obtained on substrates with given heterogeneity pattern (structure) and given surface tensions (examples 4.1 (A1 and A2) and 4.3), and with experimental results (example 4.2). In the example 4.2 we work with the real dimensional variables and in the rest of the examples dimensionless variables are used. Let us remind here that the initial contact line L is given as a sequence of points {I1 , I2 , . . . , Ik }. In some cases we need to perform a spline interpolation on these points so that we can choose the appropriate number NL of points −NL +1 N−NL +2 pN , p∗ , . . . , pN ∗ ∗ ∈ L needed for the algorithm. Their number NL depends on the parameters of the model, i.e., the total number of points N used for the triangulation of the surface and the precision ε of the algorithm. Here we apply the numerical algorithm with N = 12781 points, in examples 4.1, (A1 and A2) and 4.2, and N = 16381 points in example 4.3, approximating the surface of the drop, NL = 360 of these are used to approximate the contact line. 4.1. Drop placed on a horizontal, flat but heterogeneous substrate Example A1 We choose as given a priori the equilibrium contact line obtained numerically in [23] and shown in Fig. 3a there (see also Fig. 3b). We use here dimensionless spatial variables {x , y , z } defined by dividing by the capillary length a (Eq. (1)), i.e., x = x/a, y = y/a, z = z/a. The given contact line is smooth and with regular shape. It corresponds to a drop with dimension less volume V = 0.475 (rmean = 0.7858) sitting on a heterogeneous substrate with radial symmetry consisting of 20 sectors with alternating surface tensions with sharp boundaries. The obtained in [23] values of the contact angles on the two types of sectors are θ = 41◦ and 89◦ . We take here the same value for V as in Ref. [23]. (These parameters correspond approximately to a real water drop with radius rmean = 0.217 cm—a system similar to the one studied experimentally in Refs. [31,32]). From the given contact line we obtain the point O(L) (Eq. (10)) which we take as origin of our coordinate system. Then we obtain the average distance from the contact L line points to the origin: rmean = (1/NL ) N i=1 ri . We start our iteration procedure with a spherical cap approximation with dimensionless radius rmean = 0.7858 and dimensionless vol ume V = 0.475. This approximation gives a contact angle θ = 63.19◦ . In Fig. 1 we show a part of the obtained solution in the vicinity of the contact line with ε = 0.4 × 10−3 . For the inner points of the surface, {pi }, i = 1, . . . , N − NL , we check how well the Laplace equation (6), 2κ(pi ) + pzi = c, is satisfied. For every point pi = {pxi , pyi , pzi } we compute the mean curvature κ(pi ) on the base of the commonly used formula (see Ref. [33]) and
Fig. 1. A part of the obtained drop shape (in dimensionless units) in the vicinity of the contact line is shown with ε = 0.4 × 10−3 considering as given the equilibrium contact line obtained numerically in Ref. [23].
from there we calculate 2κ(pi ) + pzi = ci . We find that the values of ci at the different points are very close to each other with a mean cmean = 2.93 and a standard error of the mean c = 0.00055. The contact angles (see Eq. (2)) of the obtained solution practically coincide with the ones obtained in Ref. [23], the difference being less than 0.1◦ . The precision with which we determine the contact angles depends on the number of points N with which we approximate the surface, especially in the neighborhood of the contact line. After the solution is obtained one can refine the triangulation mesh by increasing the number of points N approximating the surface of the drop and by decreasing the ε-parameter of the algorithm. Then one can apply the minimization procedure again and attain a higher precision. The solution shown in Fig. 1 requires approximately 6 min of cpu time on a Pentium 4 (3 GHz). We perform here also parametric analysis. If one changes the value of 1/a 2 with ±5%, then the contact angle at every point increases or decreases respectively with less than 0.2◦ . The change of the volume V with ±5% leads to a change of the contact angle with ∼1.5◦ at the points where the contact angle is ∼41◦ and with ∼3◦ at the points where the contact angle is ∼89◦ . The drop dimensionless height increases with increasing the volume from h = 0.4619 to h = 0.4793 and decreases to h = 0.4437 when the volume is decreased. This analysis shows that the accurate determination of the contact angles also depends on the precision with which the volume is known. The parametric analysis can be combined with comparison with experimentally obtained drop profiles. This will allow determination of some of the parameters of the drop which are not known with high precision by fitting the experimental curves. This can be done similarly to the procedure described in [13] for determination of the surface tension in the case of axisymmetric drop shape analysis. One can use the numerical method from Ref. [23] for parametric analysis of the surface tensions of the solid substrate. This variant of the method does not need the contact line as boundary condition, instead it needs a given structure of the heterogeneity and given surface tensions of the solid surface to obtain the equilibrium drop shape. The determination of the contact angle with high precision (∼0.1◦ ) requires high precision approximation of the volume, of the action of the gravity and of the drop surface. In the algorithm, we use here, the volume does not change, since we seek unconditional extremum only in the class of variations preserv-
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ing the volume. This results in a more accurate approximation also of the gravitation energy in the minimization process. In the public domain software “Evolver” the volume conservation is incorporated by the method of Lagrange multiplier so that certain variation of the volume throughout the minimization is unavoidable (and therefore also of the approximation of the gravitation energy). This may increase substantially the time required for calculating the equilibrium drop shape when
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the surfaces need to be approximated uniformly with more than 104 points. Example A2 In this example we apply the method again to a drop on a flat, horizontal and heterogeneous substrate, but the given a priori contact line does not have the regular and smooth shape as in the previous case. It has a rather irregular shape and is shown with a thick solid line in Fig. 2. Such experimental contact lines with irregular shapes are found in many experimental works (see, e.g., [34]). We use here again dimensionless variables and the same values of the dimensionless volume V = 0.475 as in example 4.1 (A1). The contact line shown in Fig. 2 is produced in the following way. = 0.7858 and in the We take as in the previous case rmean (ϕ, r )-coordinate system we generate 120 points {I1 , . . . , I120 } (Mi + from the contact line by setting Ii = {(i − 1)π/60, rmean 0.2 cos(iπ/30)), i = 1, . . . , 120}, where Mi is a random number uniformly distributed in the interval [−0.05, 0.05]. Then we perform a spline-interpolation using the {I1 , . . . , I120 } points to determine the final contact line shown in Fig. 2. In Fig. 2 it is also shown how the contact line gradually transforms from the spherical cap approximation (a circle with dimensionless radius = 0.7858) toward the final contact line at regular intervals rmean during the iteration procedure. We also show how a part of the drop surface in the neighborhood of the contact line (the part enclosed in the square in Fig. 2) evolves during the iteration procedure (see Figs. 3a–3e). For the drop with irregular contact line the corresponding contact angle variation is shown in Fig. 4. The average contact angle is θmean = 62.87◦ , shown with the straight thin solid line
Fig. 2. A contact line with irregular shape (in dimensionless units) is shown with = 0.7858 thick solid line. It is generated from a circular contact line with rmean by drawing uniformly distributed random numbers according to the procedure, described in the text. Also the initial (circular) contact line as well as several contact lines taken at regular intervals of iteration steps are shown with thinner solid lines. The square encloses a part of the contact line corresponding to the part of the drop shape shown consecutively in Figs. 3a–3e. The two thick points and the arrows illustrate the simplest possible choice to move a point along the radius passing through the origin O(L) and that point in direction toward the prescribed contact line L.
(a)
(b)
(c)
(d)
(e) Fig. 3. (a)–(e) 3D images (in dimensionless units) of the obtained drop shape with irregular contact line, corresponding to the part of the contact line enclosed in the square in Fig. 2 above, are shown consecutively at regular intervals of iteration steps illustrating the transformation of the drop surface during the iteration procedure.
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Fig. 4. The final contact angle θ (in degrees) is shown as function of the polar contact angle ϕ (in degrees) of the drop with the irregular contact line (see Fig. 2). The thin straight line corresponds to the average contact angle θmean = 62.87◦ . Fig. 6. The final equilibrium 3D drop shape is displayed in Cartesian coordinate system (x, y, z in cm) for a drop on inclined surface with the contact line shown in Fig. 5. The underlying substrate is flat and is tilted at α = 30◦ .
Fig. 5. A noncircular contact line is reproduced in polar coordinate system (r in cm, ϕ in degrees) of a drop resting on inclined surface (obtained numerically in Ref. [36]).
in Fig. 4. One can clearly see the existing correlation between the changes of the drop contact line and the changes of the contact angle. We find that the values of ci (see Laplace equation (6)) at the different points are very close with a mean cmean = 2.9698 and the standard error of the mean c = 0.0019. 4.2. Flat, inclined surface Example B Here we apply the method to a drop on a flat, inclined surface with a contact line which is not circular. It is not difficult to obtain the drop shape in the case of a circular contact line (see, e.g., [29]). In Refs. [35,36] the drop shapes of drops on inclined surfaces with noncircular contact lines are obtained numerically. It is shown there also that the solution compares well with experimental results. We consider as given the contact line shown in Fig. 4a in Ref. [36] in real dimensional coordinates. Our approximation of this contact line is shown in Fig. 5. The substrate is flat and is tilted at α = 30◦ . The drop volume is V = 0.0093 cm3 (see
Ref. [36]) and we have taken a standard value for the capillary length for water a = 0.27 cm. We start the iteration procedure with a spherical cap approximation with rmean = 0.4 cm, which corresponds to a contact angle θ = 78.04◦ . The final drop shape is shown in Fig. 6. The profile of this solution compares well with the one in Ref. [36]. The contact angles θh , θl at the highest and the lowest points are θh = 62.4◦ and θl = 119.01◦ respectively, as compared to 61.52◦ and 119.57◦ , obtained in Ref. [36]. Here the contact angles θh and θl at the highest and the lowest points are obtained by averaging the contact angle over 20 points of the contact line in the neighborhood of the highest and lowest points. 4.3. Horizontal, homogeneous, and rough surface Example C We use here the dimensionless variables as in the examples 4.1 (cases A1 and A2) and the same dimensionless drop volume V = 0.475. The drop is resting on a grooved surface Σs described by the equation Σs ≡ {z = 0.03 cos(12πx ) + 0.03}. We take as a priori given contact line—the contact line L whose p projection Lp on the surface Σs ≡ {z = 0.06} is shown in Fig. 7. The contact line L on the grooved surface was obtained previously by solving the opposite problem, i.e., that of finding the equilibrium drop shape of a drop resting on a rough surface with volume V = 0.475 and surface tension which on a flat surface would produce a contact angle θ = 60◦ . We find the solution in two stages. First, we find a solution p for a drop resting on a flat, horizontal surface Σs with the given projection contact line Lp (shown in Fig. 7). The volume of this drop is reduced to the part of the initial p volume V (see Fig. 8) which is above the plane Σs (above the level of the roughness). This part is solved as in the previous
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Fig. 7. The projection contact line Lp (in dimensionless units) on the surface p Σs ≡ {z = 0.06} is depicted of the contact line L of a drop resting on a grooved substrate.
Fig. 8. A part of the final drop surface (in dimensionless units) is shown with the projection contact line Lp (displayed in Fig. 7) on the grooved substrate at the end of the first stage of the procedure.
examples, i.e., by starting from a spherical cap approximation with circular contact line C. Then we allow the points from the circular contact line C to move toward the target contact line Lp . This part is realized in a manner different from the previous cases since the simplest possible way by moving the points along the radius does not work here. Instead we move every point of the contact line along the line normal to the current contact line at this point and pointing toward the target contact line Lp . This process is repeated at small steps until all the points from the current contact line coincide with the points from the target contact line Lp . This is illustrated in Fig. 9 for a part of the contact line. We show a part of the final drop shape at the end of the first step in Fig. 8, together with the part of the rough surface Σs on which the final drop should be resting. At the second stage of the problem solution the points from the projection contact line Lp are moved down vertically toward the rough surface and the final contact line L. At the same time also the given volume V is restored. Then a sequence of minimization procedures alternating with readjustments of the triangulating mesh is performed without changing the contact line L. This process stops when a drop surface is obtained which satisfies the Laplace equation with the desired precision and whose contact line is the given contact line L. This is the solution we are seeking of our problem. A part of this final solution (corresponding to the part displayed in Fig. 8 obtained at the previous stage) is shown in Fig. 10.
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Fig. 9. A part of the target contact line (in dimensionless units) is displayed, which in this case is the projection contact line Lp (given by the thick solid line) together with several contact lines (given by solid square-solid lines), recorded at regular intervals of iteration steps.
Fig. 10. A part of the final solution is shown for the drop surface (in dimensionless units), corresponding to the part displayed in Fig. 8 obtained in the previous step, for a drop resting on the rough surface.
5. Conclusion Measurements of the contact angles are among the most used ways to characterize the surfaces. At present there is a necessity to determine the contact angles on real surfaces which are heterogeneous and/or rough for various reasons. In addition the solid surfaces can also be horizontal or tilted. This requires the development of new numerical methods which go beyond the axisymmetric approximation for the drop shape. Here we have described a numerical method for determining the local contact angles when there are data available for the whole contact line of a drop for given values of the drop volume, and capillary length. The method allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. This method could also be used to study the relaxation of the contact angle whenever a qausi-static approximation for the drop relaxation is applicable [37]. The numerical method employed uses the variational approach, i.e., it minimizes the energy functional of the capillary theory, subject to two constraints—a fixed volume and prescribed contact line. It is based on previously developed by the authors numerical method for obtaining the equilibrium drop shapes on solid substrates (homogeneous [21,22] or heterogeneous [23]) for given values of the drop volume, capillary length and given values of the surface tensions. Here this method is combined with an additional procedure for gradual transformation of the contact line toward the desired (target)
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contact line. A similar approach can be developed using also other numerical techniques [25,26,29,35,38–41] for obtaining equilibrium drop shapes Similarly to the ADSA-D and ADSA-P methods, this method is intended to be used in conjunction with experimental data. The application of the described method does not have limitations with respect to the value of the contact angle, i.e., it is not limited to contact angles less than 90◦ and flat substrates where a top view photograph of the drop is sufficient to obtain the contact line experimentally. It can be applied also for contact angles greater than 90◦ provided it is possible to obtain the contact line experimentally (see, e.g., [42–47]). When the substrate is rough information about the roughness profile is also necessary and can be obtained experimentally (see, e.g., [48,49]). Several examples are considered to illustrate how the method works. We have applied it to a drop on: flat, horizontal, but heterogeneous substrate; flat, inclined and heterogeneous substrate, and grooved horizontal substrate. It will be interesting to apply the method to different sets of experimental data and compare the results for the local and averaged contact angles with those obtained by ADSA-D and ADSA-P methods. Acknowledgment S.I. has received financial support from the NSF-Bulgaria under grant number VU-MI-02/05. References [1] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, sixth ed., Wiley, New York, 1997. [2] A.W. Neumann, R.J. Godd, in: R.J. Good, R.R. Stromberg (Eds.), Techniques of Measuring Contact Angles, Surface and Colloid Science, in: Experimental Methods, vol. II, Plenum, New York, 1979, p. 31. [3] M.J. Jaycock, G.D. Parfitt, Chemistry of Interfaces, Wiley, New York, 1981. [4] J.C. Berg, Wettability, Dekker, New York, 1993. [5] M.E. Schrader, G. Loeb, Modern Approach to Wettability, Plenum, New York, 1992. [6] D. Li, A.W. Neumann, Colloids Surf. 43 (1990) 195. [7] D. Li, P. Cheng, A.W. Neumann, Adv. Colloid Interface Sci. 39 (1992) 347. [8] R.M. Prokop, O.I. del Rio, N. Niyakan, A.W. Neumann, Can. J. Chem. Eng. 74 (1996) 534. [9] P. Chen, D.Y. Kwok, R.M. Prokop, O.I. del Rio, S.S. Susnar, A.W. Neumann, in: D.D. Mobius, R. Miller (Eds.), Drops and Bubbles in Interfacial Research, Elsevier, Amsterdam, 1998, p. 61, chap. 2.
[10] D.Y. Kwok, T. Gietzelt, K. Grundke, H.-J. Jacobasch, A.W. Neumann, Langmuir 13 (1997) 2880. [11] D. Duncan, D. Li, J. Gaydos, A.W. Neumann, J. Colloid Interface Sci. 169 (1995) 256. [12] N.M. Dingle, K. Tjiptowidjojo, O.A. Basaran, M.T. Harris, J. Colloid Interface Sci. 286 (2005) 647. [13] N.M. Dingle, M.T. Harris, J. Colloid Interface Sci. 286 (2005) 670. [14] H. Gau, S. Herminghaus, P. Lenz, R. Lipowsky, Science 283 (1999) 46. [15] D.E. Kataoka, S.M. Troyan, Nature 402 (1999) 794. [16] C.S. Chen, M. Mrksich, S. Huang, G.M. Whitesides, D.E. Ingber, Science 276 (1997) 1425. [17] E. Delamarche, A. Bernard, H. Schmidt, B. Michel, H.A. Biebuyck, Science 276 (1997) 779. [18] R. Finn, Equilibrium Capillary Surfaces, Springer, Berlin, 1986. [19] L.D. Landau, E.M. Lifshitz, Mechanics, Elsevier, Burlington, 2003. [20] A.D. Myshkis, V.G. Babskii, N.D. Kopachevskii, L.A. Slobozhanin, A.D. Tyuptsov, Low-Gravity Fluid Mechanics, Springer-Verlag, Berlin, 1987. [21] S.D. Iliev, J. Colloid Interface Sci. 213 (1999) 1. [22] S.D. Iliev, J. Colloid Interface Sci. 194 (1997) 287. [23] S.D. Iliev, N.Ch. Pesheva, Langmuir 19 (2003) 9923. [24] F.L. Chernousko, J. Comput. Math. Math. Phys. 4 (1965) 749 [in Russian]. [25] F. Milinazzo, M. Shinbrot, J. Colloid Interface Sci. 121 (1988) 254. [26] U. Hornung, H.D. Mittelmann, J. Colloid Interface Sci. 133 (1989) 409. [27] K. Brakke, Exp. Math. 1 (1992) 141. [28] S. Iliev, Comput. Methods Appl. Mech. Eng. 126 (1995) 251. [29] R.A. Brown, F.M.Jr. Orr, L.E. Scriven, J. Colloid Interface Sci. 73 (1980) 76. [30] S. Iliev, N. Pesheva, V.S. Nikolayev, Phys. Rev. E 72 (2005) 011606. [31] D. Li, Colloids Surf. A 116 (1996) 1. [32] J. Drelich, J.D. Miller, A. Kumar, G.M. Whitesides, Colloids Surf. A 93 (1994) 1. [33] M. Loewenberg, E. Hinch, J. Fluid Mech. 321 (1996) 395. [34] M.A. Rodríguez-Valverde, M.A. Cabrerizo-Vílchez, P. Rosales-López, A. Páez-Dueñas, R. Hidalgo-Álvarez, Colloids Surf. A 206 (2002) 485. [35] P. Dimitrakopoulos, J.J.L. Higdon, J. Fluid Mech. 395 (1999) 181. [36] Y. Rotenberg, L. Boruvka, A.W. Neumann, J. Colloid Interface Sci. 102 (1984) 424. [37] B. Lavi, A. Marmur, Colloids Surf. A 250 (2004) 409. [38] R.A. Brown, J. Comput. Phys. 33 (1979) 217. [39] X. Li, C. Pozrikidis, J. Fluid Mech. 307 (1996) 167. [40] S. Brandon, N. Haimovich, E. Yeger, A. Marmur, J. Colloid Interface Sci. 263 (2003) 237. [41] G. Wolansky, A. Marmur, Langmuir 14 (1998) 5292. [42] V. Srinivasan, V.K. Pamula, K. Divakar, M.G. Pollack, J.A. Izatt, R.B. Fair, in: Proceedings of the Seventh International Conference on Miniaturized Chemical and Biochemical Analysis Systems, Lake Tahao, October 2003. [43] N. Zhang, D.F. Chao, J. Flow Visual. Image Proc. 8 (2001) 303. [44] N. Zhang, D.F. Chao, F. David, Opt. Laser Technol. 34 (2002) 243. [45] A.H. Barber, S.R. Cohen, H.D. Wagner, Phys. Rev. Lett. 92 (2004) 186103. [46] E. Rio, A. Daerr, B. Andreotti, L. Limat, Phys. Rev. Lett. 94 (2005) 024503. [47] E. Rio, A. Daerr, L. Limat, J. Colloid Interface Sci. 269 (2004) 164. [48] R. Seemann, W. Monch, S. Herminghaus, Europhys. Lett. 55 (2001) 698. [49] R. Seemann, M. Brinkmann, E.J. Kramer, F.F. Lange, R. Lipowsky, Proc. Nat. Acad. Sci. 102 (2005) 1848.
Nonaxisymmetric drop shape analysis and its application for determination of the local contact angles Stanimir Iliev, Nina Pesheva ∗ Institute of Mechanics, Bulgarian Academy of Sciences, G. Bonchev street, block 4, 1113 Sofia, Bulgaria Received 20 December 2005; accepted 28 May 2006 Available online 21 June 2006
Abstract We describe here a numerical method for finding the local contact angles of a drop in the case of partial wetting for given values of the drop volume and capillary length when there are data available for the whole contact line of the drop. There are no special restrictions imposed on the type of the contact line: the solid substrate on which the drop rests can be heterogeneous or rough or both, it can be horizontal or tilted. The method is intended to be used in conjunction with experimental results similarly to the axisymmetric drop shape analysis-diameter (ADSA-D) and analysis-profile (ADSA-P) methods. The numerical method is essentially an iterative minimization procedure based on the local variations approach. It allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. Several examples of applying the method are described for a drop on: flat, horizontal but heterogeneous substrate; flat, tilted substrate, and grooved horizontal substrate. © 2006 Elsevier Inc. All rights reserved. Keywords: Liquid drops; Nonaxisymmetric drop shape analysis; Local contact angles
1. Introduction The contact angle that a liquid drop forms with a solid is an important measure of the wettability of the solid surface. Measurements of the contact angles in many research and industrial laboratories are among the most used ways to characterize the surfaces. The basic techniques used for measuring contact angles have been reviewed in [1–5]. The most often used techniques for measurements of the contact angles are the sessile drop and the captive bubble techniques. Their widespread use is due to their simplicity and the small amount of liquid and surfaces needed. These techniques are currently based on taking high resolution photographs and are commonly used in combination with numerical methods to improve the precision of the obtained results since the real surfaces are heterogeneous and/or rough. Such numerical methods are the axisymmetric drop shape analysisdiameter (ADSA-D) and axisymmetric drop shape analysisprofile (ADSA-P) method. In the first case a top view of the * Corresponding author.
E-mail address: [email protected] (N. Pesheva). 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.05.067
sessile drop and in the second a side view of the drop is used. These methods provide a contact angle by finding a numerical axisymmetric solution of the Young–Laplace equation with the given experimental parameters (drop volume, density, diameter, height, capillary length, etc.) and then calculating the slope of the tangent to the drop surface at the liquid–solid– vapor interface line (see, e.g., [6–11]). For more detailed account of recent advancement and modifications of ADSA-D and ADSA-P methods (see [12,13] and references therein). These techniques can be expected to give an accurate result only when axisymmetric approximation for the drop shape can be used. At present the development of new more precise experimental techniques for determining the profile and the contact line of a liquid drop as well as the ever growing demand for going to smaller and smaller scales from the industry requires the development of new numerical methods which will go beyond the axisymmetric approximation for the shape of a drop. Thus currently there is a necessity to determine the contact angles on real surfaces which are heterogeneous and/or rough for various reasons, e.g., due to contamination, the presence of impurities, cavities, uneven oxide layer, “hot” sites, etc. On the other
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hand, chemically heterogeneous solid substrates with specially designed structures are increasingly used, e.g., in surface modification, textile industry, biomedical research, surfactant chemistry, etc., for engineering of desired nano- and micro-patterns in thin films (see, e.g., [14–17]). On such surfaces the drops are not axisymmetric anymore, and the contact angle varies along the contact line. We describe here a numerical method for finding the local contact angles along the contact line of a drop in the case of partial wetting. There are no restrictions imposed on the type of the contact line: the solid substrate on which the drop rests can be heterogeneous or rough or both, it can be horizontal or tilted. The method is intended to be used in conjunction with experimental results (similarly to ADSA-D and ADSA-P methods) when there are data available for the whole contact line of a drop and for given experimental values of the drop volume and capillary length. The method allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. In Section 2 more precise definition of the problem is given. In Section 3 we describe the method in detail, in Section 4 we illustrate how it works on several examples and Section 5 is devoted to our conclusions. 2. Problem formulation Let us consider a 3D equilibrium sessile liquid drop with volume V resting on a solid surface Σs at equilibrium. Both the drop and the substrate on which it is placed are immersed in an ambient fluid and it is assumed that the liquid and fluid are mutually immiscible. The solid substrate can be heterogeneous or rough or both, it can be horizontal or tilted. The surface of the drop Σ forms on the solid surface Σs a line L called contact line or common line. The goal is to determine the shape of a drop and the local contact angles along the contact line for given a priori contact line L, volume V , and capillary length a (see Ref. [1]) a 2 = σ/(ρl − ρf )g,
(1)
where the density of the droplet and the ambient fluid are denoted by ρl and ρf correspondingly; σ is the liquid–fluid surface tension, g is the gravity acceleration constant. After finding the drop surface, the calculation of the local contact angles θ (P ) at point P along the contact line L is straightforward: θ (P ) = arccos nΣ (P ) · nΣs (P ) , (2) where nΣ , nΣs are the outward unit vectors at point P normal to the surfaces Σ and Σs , respectively. The equilibrium drop shape Σ , we are looking for, is formed as a result of the action of the surface tension σ and of the gravitational forces acting in both phases, the liquid and the fluid. This surface is described as a Laplacian surface and as a minimal surface. In the first case a differential description is used and in the second integral. In this work we use the integral approach. That is, in a Cartesian coordinate system (x, y, z) whose z-axis is taken to be in direction opposite to that of the gravity acceleration g, i.e., z = −g/g, the equilibrium drop shape
Σ is the one which minimizes the energy functional (see Refs. [18–20]): dΣ + (ρl − ρf )gz dΩ E=σ (3) Σ
Ω(Σ,Σs )
and the boundary of this surface coincides with the given contact line L and also the surface Σ together with the substrate surface Σs encloses the prescribed volume V , i.e., Ω(Σ, Σs ) = V .
(4)
In the usual way, the energy functional is expressed in terms of the liquid–fluid surface tension σ , i.e., we consider U = E/σ . Thus the variational problem is reduced to minimizing the following functional: U = dΣ + (5) z/a 2 dΩ. Σ
Ω(Σ,Σs )
Equivalently, in the differential description instead of minimizing the renormalized expression (5), one has to solve the Laplace equation to find the equilibrium drop surface Σ : (κ1 + κ2 ) = z/a 2 + c,
(6)
where c is a constant, κ1 and κ2 are the principal curvatures of the surface Σ . In quite the same way one can solve also the above problem, where the condition of a given a priori volume (Eq. (4)) is replaced by the requirement that the surface of the drop Σ passes through a given point P ∈ / Σs or has a maximal distance h from the surface of the substrate Σs . For a horizontal substrate the last condition is equivalent to a given a priori height of the drop. The above formulation of the problem is appropriate to be used for the sessile drop measurements. It can be applied also to pendent bubble measurements of the contact angle. 3. Numerical method We solve the defined in Section 2 problem numerically. We find solutions for the equilibrium drop shapes on heterogeneous or rough substrates by minimization of the renormalized functional (5) subject to the two constraints—that of a fixed volume (4) and prescribed contact line L. First, we find the spherical cap approximation of the drop shape Σ0 with the given volume V and capillary length a which has a circular contact line. After that we perform an iterative procedure which transforms the contact line gradually while the volume is kept fixed and the surface is maintained Laplacian until the desired contact line L is obtained. The numerical method for obtaining drop surfaces was first developed for homogeneous surfaces in Refs. [21,22] and then extended to treating heterogeneous surfaces and line tension effects in Ref. [23]. The numerical method employed is essentially an iterative minimization procedure based on the local variations approach [24] combined with an additional procedure for gradual transformation of the contact line toward the
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desired (target) contact line. The numerical method for obtaining drop surfaces is similar to the numerical methods [25,26], and the one used in the public domain software “Evolver” [27]. The drop shape is approximated by a triangulation Σ¯ N using N points (see, e.g., Fig. 1). The renormalized energy functional (5) is expressed in terms of the coordinates of these N points. This is a standard procedure in the numerical methods for discretization of surfaces. The N points are represented by radius-vectors, p1 , p2 , . . . , pN , which end at the corresponding point. Thus any surface Σ considered in the minimization of the renormalized energy functional (5) is represented as a union of M triangles. Part of the nodes of the triangles are inner for the surface Σ¯ N , the rest, NL nodes, are border nodes: pN −NL +1 , pN−NL +2 , . . . , pN . The renormalized energy functional U , given by Eq. (5), thus becomes a function of the points p1 , p2 , . . . , pN , U Σ¯ N = U p1 , p2 , . . . , pN . (7) Then the problem of finding the surface which minimizes (5) in this N -point approximation can be formulated as follows. For fixed ε 1 (ε gives the precision with which the local surface variations are made) find points p1∗ , p2∗ , . . . , pN ∗ , defining NL +2 L +1 , p , . . . , pN a surface Σ ∗ , whose boundary points, pN ∗ ∗ ∗ , belong to the given a priori contact line L. The surface Σ ∗ must be also such that together with the substrate surface Σs encloses the given volume V and for which U ∗ = U p1∗ , p2∗ , . . . , pN ∗ L L +1 , pN−NL +2 , . . . , pN , pN−N U p1 , p2 , . . . , pN−N ∗ ∗ ∗ (8) for any other surface Σ close to Σ ∗ defined by the points 1 2 L L +1 , pN−NL +2 , . . . , pN , p , p , . . . , pN−N , pN−N ∗ ∗ ∗ where pi = pi∗ + εi Ei , εi > ε,
Ei = 1,
i = 1, . . . , N − NL .
(9)
Condition (8) defines the ε-minimum (given by the surface Σ ∗ ) of the function (7) with respect to the set of possible finite ε variations which is the condition for equilibrium. We use a Monte Carlo scheme for choosing the node points which we will try to move since in this way the algorithm works faster as compared to a fixed order of regular sampling of all the points defining the drop surface. At every iteration step the drop shape is changed in such a way so that the renormalized energy expression (7) is decreased while the drop volume and the NL points of the contact line are kept fixed. The correctness of the obtained solution at every iteration step is monitored by keeping track of the accuracy with which the coordinates of the points of the drop surface satisfy the Laplace condition (6). For ensuring a better work of the minimization procedure, a regular check and readjustment of the surface mesh is performed in order to keep the approximation of the liquid/gas interface uniform. As it will be seen in Section 4, the final drop shape satisfies the Laplace equation with high precision. The realization of this step is described in detail in [28], it is tested also
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in [22] by comparing of the obtained equilibrium shapes with those in [29]. Let the prescribed contact line L be given as a sequence of points {I1 , I2 , . . . , Ik }. We start the numerical process with a spherical cap approximation Σ0 of the drop shape with a circular contact line C(O(L), r(L)) with a center O(L) and a radius r(L) defined in the following way: pO =
I1 + I2 + · · · + Ik , k
r(L) ∈ (rmin , rmax ).
(10)
pO is the radius-vector of the center O(L) and rmin and rmax are respectively, the minimal and the maximal distance from the center O(L) to the points of the contact line L ≡ {I1 , I2 , . . . , Ik }. Then this surface is triangulated using N -points, Σ¯ 0 = Σ¯ 0 (p1 , . . . , pN ), in a coordinate system whose origin O coincides with the center of mass O(L) of the points of the contact line. Next, an iterative procedure which transforms the contact line gradually is implemented in a way very similar to the approach used in [30] where a quasi-static relaxation of a drop was simulated: At every point of the contact line, pN−NL +1 , pN−NL +2 , . . . , pN , instead of moving the contact line at a distance proportional to its velocity (as it was done in [30]), the points of the contact line are moved at small fixed distance in the appropriate direction. The simplest possible choice is to move a point along the radius passing through the origin O(L) and that point in direction toward the prescribed contact line L (see the two thick points and the arrows in Fig. 2). This process is accompanied by a change of the drop volume. In order to compensate for the change in the volume at every iteration step also a sequence of displacements of the inner points p1 , p2 , . . . , pN−NL is performed so that the initial volume V is restored. When the substrate surface is rough then the problem can be solved in two stages. First, one finds a solution for a contact line Lp which is a projection of the original contact line on a flat surface. Then a gradual transformation of the “flat” contact line Lp into the original contact line is performed. This is described in more detail in example 4.3 in Section 4. As a last iteration step we perform final minimization procedure with a fixed contact line L and a fixed volume V . After the desired drop shape is obtained, the determination of the contact angles along the contact line is straightforward. From the approximating triangles at every point of the contact line one determines the normal to the drop surface and therefore the contact angle at that point. The precision with which one can obtain the contact angles depends on the number of points N used to approximate the drop surface, on the ε-parameter of the algorithm, the form of the triangulation net, on the realization of the re-mesh procedure. It depends also on the peculiarities of the contact line, on the roughness of the substrate, etc. In Section 4 we give the specific values of some of these parameters with which the contact angles (on both hydrophilic and hydrophobic substrates) can be determined with a high precision, of the order of θ ∼ 0.1◦ , for a drop on a flat surface, and θ ∼ 1◦ , for a drop on a rough surface.
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4. Numerical examples We applied the numerical method described in Section 3 for several cases: flat, horizontal but heterogeneous substrate (examples 4.1 (A1 and A2); flat, tilted substrate (example 4.2), and grooved horizontal substrate (example 4.3). We compare our results with numerical solutions for drops obtained on substrates with given heterogeneity pattern (structure) and given surface tensions (examples 4.1 (A1 and A2) and 4.3), and with experimental results (example 4.2). In the example 4.2 we work with the real dimensional variables and in the rest of the examples dimensionless variables are used. Let us remind here that the initial contact line L is given as a sequence of points {I1 , I2 , . . . , Ik }. In some cases we need to perform a spline interpolation on these points so that we can choose the appropriate number NL of points −NL +1 N−NL +2 pN , p∗ , . . . , pN ∗ ∗ ∈ L needed for the algorithm. Their number NL depends on the parameters of the model, i.e., the total number of points N used for the triangulation of the surface and the precision ε of the algorithm. Here we apply the numerical algorithm with N = 12781 points, in examples 4.1, (A1 and A2) and 4.2, and N = 16381 points in example 4.3, approximating the surface of the drop, NL = 360 of these are used to approximate the contact line. 4.1. Drop placed on a horizontal, flat but heterogeneous substrate Example A1 We choose as given a priori the equilibrium contact line obtained numerically in [23] and shown in Fig. 3a there (see also Fig. 3b). We use here dimensionless spatial variables {x , y , z } defined by dividing by the capillary length a (Eq. (1)), i.e., x = x/a, y = y/a, z = z/a. The given contact line is smooth and with regular shape. It corresponds to a drop with dimension less volume V = 0.475 (rmean = 0.7858) sitting on a heterogeneous substrate with radial symmetry consisting of 20 sectors with alternating surface tensions with sharp boundaries. The obtained in [23] values of the contact angles on the two types of sectors are θ = 41◦ and 89◦ . We take here the same value for V as in Ref. [23]. (These parameters correspond approximately to a real water drop with radius rmean = 0.217 cm—a system similar to the one studied experimentally in Refs. [31,32]). From the given contact line we obtain the point O(L) (Eq. (10)) which we take as origin of our coordinate system. Then we obtain the average distance from the contact L line points to the origin: rmean = (1/NL ) N i=1 ri . We start our iteration procedure with a spherical cap approximation with dimensionless radius rmean = 0.7858 and dimensionless vol ume V = 0.475. This approximation gives a contact angle θ = 63.19◦ . In Fig. 1 we show a part of the obtained solution in the vicinity of the contact line with ε = 0.4 × 10−3 . For the inner points of the surface, {pi }, i = 1, . . . , N − NL , we check how well the Laplace equation (6), 2κ(pi ) + pzi = c, is satisfied. For every point pi = {pxi , pyi , pzi } we compute the mean curvature κ(pi ) on the base of the commonly used formula (see Ref. [33]) and
Fig. 1. A part of the obtained drop shape (in dimensionless units) in the vicinity of the contact line is shown with ε = 0.4 × 10−3 considering as given the equilibrium contact line obtained numerically in Ref. [23].
from there we calculate 2κ(pi ) + pzi = ci . We find that the values of ci at the different points are very close to each other with a mean cmean = 2.93 and a standard error of the mean c = 0.00055. The contact angles (see Eq. (2)) of the obtained solution practically coincide with the ones obtained in Ref. [23], the difference being less than 0.1◦ . The precision with which we determine the contact angles depends on the number of points N with which we approximate the surface, especially in the neighborhood of the contact line. After the solution is obtained one can refine the triangulation mesh by increasing the number of points N approximating the surface of the drop and by decreasing the ε-parameter of the algorithm. Then one can apply the minimization procedure again and attain a higher precision. The solution shown in Fig. 1 requires approximately 6 min of cpu time on a Pentium 4 (3 GHz). We perform here also parametric analysis. If one changes the value of 1/a 2 with ±5%, then the contact angle at every point increases or decreases respectively with less than 0.2◦ . The change of the volume V with ±5% leads to a change of the contact angle with ∼1.5◦ at the points where the contact angle is ∼41◦ and with ∼3◦ at the points where the contact angle is ∼89◦ . The drop dimensionless height increases with increasing the volume from h = 0.4619 to h = 0.4793 and decreases to h = 0.4437 when the volume is decreased. This analysis shows that the accurate determination of the contact angles also depends on the precision with which the volume is known. The parametric analysis can be combined with comparison with experimentally obtained drop profiles. This will allow determination of some of the parameters of the drop which are not known with high precision by fitting the experimental curves. This can be done similarly to the procedure described in [13] for determination of the surface tension in the case of axisymmetric drop shape analysis. One can use the numerical method from Ref. [23] for parametric analysis of the surface tensions of the solid substrate. This variant of the method does not need the contact line as boundary condition, instead it needs a given structure of the heterogeneity and given surface tensions of the solid surface to obtain the equilibrium drop shape. The determination of the contact angle with high precision (∼0.1◦ ) requires high precision approximation of the volume, of the action of the gravity and of the drop surface. In the algorithm, we use here, the volume does not change, since we seek unconditional extremum only in the class of variations preserv-
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ing the volume. This results in a more accurate approximation also of the gravitation energy in the minimization process. In the public domain software “Evolver” the volume conservation is incorporated by the method of Lagrange multiplier so that certain variation of the volume throughout the minimization is unavoidable (and therefore also of the approximation of the gravitation energy). This may increase substantially the time required for calculating the equilibrium drop shape when
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the surfaces need to be approximated uniformly with more than 104 points. Example A2 In this example we apply the method again to a drop on a flat, horizontal and heterogeneous substrate, but the given a priori contact line does not have the regular and smooth shape as in the previous case. It has a rather irregular shape and is shown with a thick solid line in Fig. 2. Such experimental contact lines with irregular shapes are found in many experimental works (see, e.g., [34]). We use here again dimensionless variables and the same values of the dimensionless volume V = 0.475 as in example 4.1 (A1). The contact line shown in Fig. 2 is produced in the following way. = 0.7858 and in the We take as in the previous case rmean (ϕ, r )-coordinate system we generate 120 points {I1 , . . . , I120 } (Mi + from the contact line by setting Ii = {(i − 1)π/60, rmean 0.2 cos(iπ/30)), i = 1, . . . , 120}, where Mi is a random number uniformly distributed in the interval [−0.05, 0.05]. Then we perform a spline-interpolation using the {I1 , . . . , I120 } points to determine the final contact line shown in Fig. 2. In Fig. 2 it is also shown how the contact line gradually transforms from the spherical cap approximation (a circle with dimensionless radius = 0.7858) toward the final contact line at regular intervals rmean during the iteration procedure. We also show how a part of the drop surface in the neighborhood of the contact line (the part enclosed in the square in Fig. 2) evolves during the iteration procedure (see Figs. 3a–3e). For the drop with irregular contact line the corresponding contact angle variation is shown in Fig. 4. The average contact angle is θmean = 62.87◦ , shown with the straight thin solid line
Fig. 2. A contact line with irregular shape (in dimensionless units) is shown with = 0.7858 thick solid line. It is generated from a circular contact line with rmean by drawing uniformly distributed random numbers according to the procedure, described in the text. Also the initial (circular) contact line as well as several contact lines taken at regular intervals of iteration steps are shown with thinner solid lines. The square encloses a part of the contact line corresponding to the part of the drop shape shown consecutively in Figs. 3a–3e. The two thick points and the arrows illustrate the simplest possible choice to move a point along the radius passing through the origin O(L) and that point in direction toward the prescribed contact line L.
(a)
(b)
(c)
(d)
(e) Fig. 3. (a)–(e) 3D images (in dimensionless units) of the obtained drop shape with irregular contact line, corresponding to the part of the contact line enclosed in the square in Fig. 2 above, are shown consecutively at regular intervals of iteration steps illustrating the transformation of the drop surface during the iteration procedure.
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Fig. 4. The final contact angle θ (in degrees) is shown as function of the polar contact angle ϕ (in degrees) of the drop with the irregular contact line (see Fig. 2). The thin straight line corresponds to the average contact angle θmean = 62.87◦ . Fig. 6. The final equilibrium 3D drop shape is displayed in Cartesian coordinate system (x, y, z in cm) for a drop on inclined surface with the contact line shown in Fig. 5. The underlying substrate is flat and is tilted at α = 30◦ .
Fig. 5. A noncircular contact line is reproduced in polar coordinate system (r in cm, ϕ in degrees) of a drop resting on inclined surface (obtained numerically in Ref. [36]).
in Fig. 4. One can clearly see the existing correlation between the changes of the drop contact line and the changes of the contact angle. We find that the values of ci (see Laplace equation (6)) at the different points are very close with a mean cmean = 2.9698 and the standard error of the mean c = 0.0019. 4.2. Flat, inclined surface Example B Here we apply the method to a drop on a flat, inclined surface with a contact line which is not circular. It is not difficult to obtain the drop shape in the case of a circular contact line (see, e.g., [29]). In Refs. [35,36] the drop shapes of drops on inclined surfaces with noncircular contact lines are obtained numerically. It is shown there also that the solution compares well with experimental results. We consider as given the contact line shown in Fig. 4a in Ref. [36] in real dimensional coordinates. Our approximation of this contact line is shown in Fig. 5. The substrate is flat and is tilted at α = 30◦ . The drop volume is V = 0.0093 cm3 (see
Ref. [36]) and we have taken a standard value for the capillary length for water a = 0.27 cm. We start the iteration procedure with a spherical cap approximation with rmean = 0.4 cm, which corresponds to a contact angle θ = 78.04◦ . The final drop shape is shown in Fig. 6. The profile of this solution compares well with the one in Ref. [36]. The contact angles θh , θl at the highest and the lowest points are θh = 62.4◦ and θl = 119.01◦ respectively, as compared to 61.52◦ and 119.57◦ , obtained in Ref. [36]. Here the contact angles θh and θl at the highest and the lowest points are obtained by averaging the contact angle over 20 points of the contact line in the neighborhood of the highest and lowest points. 4.3. Horizontal, homogeneous, and rough surface Example C We use here the dimensionless variables as in the examples 4.1 (cases A1 and A2) and the same dimensionless drop volume V = 0.475. The drop is resting on a grooved surface Σs described by the equation Σs ≡ {z = 0.03 cos(12πx ) + 0.03}. We take as a priori given contact line—the contact line L whose p projection Lp on the surface Σs ≡ {z = 0.06} is shown in Fig. 7. The contact line L on the grooved surface was obtained previously by solving the opposite problem, i.e., that of finding the equilibrium drop shape of a drop resting on a rough surface with volume V = 0.475 and surface tension which on a flat surface would produce a contact angle θ = 60◦ . We find the solution in two stages. First, we find a solution p for a drop resting on a flat, horizontal surface Σs with the given projection contact line Lp (shown in Fig. 7). The volume of this drop is reduced to the part of the initial p volume V (see Fig. 8) which is above the plane Σs (above the level of the roughness). This part is solved as in the previous
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Fig. 7. The projection contact line Lp (in dimensionless units) on the surface p Σs ≡ {z = 0.06} is depicted of the contact line L of a drop resting on a grooved substrate.
Fig. 8. A part of the final drop surface (in dimensionless units) is shown with the projection contact line Lp (displayed in Fig. 7) on the grooved substrate at the end of the first stage of the procedure.
examples, i.e., by starting from a spherical cap approximation with circular contact line C. Then we allow the points from the circular contact line C to move toward the target contact line Lp . This part is realized in a manner different from the previous cases since the simplest possible way by moving the points along the radius does not work here. Instead we move every point of the contact line along the line normal to the current contact line at this point and pointing toward the target contact line Lp . This process is repeated at small steps until all the points from the current contact line coincide with the points from the target contact line Lp . This is illustrated in Fig. 9 for a part of the contact line. We show a part of the final drop shape at the end of the first step in Fig. 8, together with the part of the rough surface Σs on which the final drop should be resting. At the second stage of the problem solution the points from the projection contact line Lp are moved down vertically toward the rough surface and the final contact line L. At the same time also the given volume V is restored. Then a sequence of minimization procedures alternating with readjustments of the triangulating mesh is performed without changing the contact line L. This process stops when a drop surface is obtained which satisfies the Laplace equation with the desired precision and whose contact line is the given contact line L. This is the solution we are seeking of our problem. A part of this final solution (corresponding to the part displayed in Fig. 8 obtained at the previous stage) is shown in Fig. 10.
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Fig. 9. A part of the target contact line (in dimensionless units) is displayed, which in this case is the projection contact line Lp (given by the thick solid line) together with several contact lines (given by solid square-solid lines), recorded at regular intervals of iteration steps.
Fig. 10. A part of the final solution is shown for the drop surface (in dimensionless units), corresponding to the part displayed in Fig. 8 obtained in the previous step, for a drop resting on the rough surface.
5. Conclusion Measurements of the contact angles are among the most used ways to characterize the surfaces. At present there is a necessity to determine the contact angles on real surfaces which are heterogeneous and/or rough for various reasons. In addition the solid surfaces can also be horizontal or tilted. This requires the development of new numerical methods which go beyond the axisymmetric approximation for the drop shape. Here we have described a numerical method for determining the local contact angles when there are data available for the whole contact line of a drop for given values of the drop volume, and capillary length. The method allows finding drop shapes which are not axially symmetric. The contact angles are then determined from the obtained shape of the drop. This method could also be used to study the relaxation of the contact angle whenever a qausi-static approximation for the drop relaxation is applicable [37]. The numerical method employed uses the variational approach, i.e., it minimizes the energy functional of the capillary theory, subject to two constraints—a fixed volume and prescribed contact line. It is based on previously developed by the authors numerical method for obtaining the equilibrium drop shapes on solid substrates (homogeneous [21,22] or heterogeneous [23]) for given values of the drop volume, capillary length and given values of the surface tensions. Here this method is combined with an additional procedure for gradual transformation of the contact line toward the desired (target)
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contact line. A similar approach can be developed using also other numerical techniques [25,26,29,35,38–41] for obtaining equilibrium drop shapes Similarly to the ADSA-D and ADSA-P methods, this method is intended to be used in conjunction with experimental data. The application of the described method does not have limitations with respect to the value of the contact angle, i.e., it is not limited to contact angles less than 90◦ and flat substrates where a top view photograph of the drop is sufficient to obtain the contact line experimentally. It can be applied also for contact angles greater than 90◦ provided it is possible to obtain the contact line experimentally (see, e.g., [42–47]). When the substrate is rough information about the roughness profile is also necessary and can be obtained experimentally (see, e.g., [48,49]). Several examples are considered to illustrate how the method works. We have applied it to a drop on: flat, horizontal, but heterogeneous substrate; flat, inclined and heterogeneous substrate, and grooved horizontal substrate. It will be interesting to apply the method to different sets of experimental data and compare the results for the local and averaged contact angles with those obtained by ADSA-D and ADSA-P methods. Acknowledgment S.I. has received financial support from the NSF-Bulgaria under grant number VU-MI-02/05. References [1] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, sixth ed., Wiley, New York, 1997. [2] A.W. Neumann, R.J. Godd, in: R.J. Good, R.R. Stromberg (Eds.), Techniques of Measuring Contact Angles, Surface and Colloid Science, in: Experimental Methods, vol. II, Plenum, New York, 1979, p. 31. [3] M.J. Jaycock, G.D. Parfitt, Chemistry of Interfaces, Wiley, New York, 1981. [4] J.C. Berg, Wettability, Dekker, New York, 1993. [5] M.E. Schrader, G. Loeb, Modern Approach to Wettability, Plenum, New York, 1992. [6] D. Li, A.W. Neumann, Colloids Surf. 43 (1990) 195. [7] D. Li, P. Cheng, A.W. Neumann, Adv. Colloid Interface Sci. 39 (1992) 347. [8] R.M. Prokop, O.I. del Rio, N. Niyakan, A.W. Neumann, Can. J. Chem. Eng. 74 (1996) 534. [9] P. Chen, D.Y. Kwok, R.M. Prokop, O.I. del Rio, S.S. Susnar, A.W. Neumann, in: D.D. Mobius, R. Miller (Eds.), Drops and Bubbles in Interfacial Research, Elsevier, Amsterdam, 1998, p. 61, chap. 2.
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