A novel strategy for determining line tension from the shape of a liquid meniscus near a stripwise heterogeneous wall

Colloids and Surfaces, 43 (1990) 307-326 Elsevier Science Publishers B .V ., Amsterdam - Printed in The Netherlands

307

A Novel Strategy for Determining Line Tension from the Shape of a Liquid Meniscus near a Stripwise Heterogeneous Wall L . BORUVKA . J . GAYDOS and A .W . NEUMANN Department of Mechanical Engineering, University (Canada)

of

Toronto, Toronto, Ontario M5S 1A4

(Received 28 December 1988; accepted 12 June 1989)

ABSTRACT According to the classical theory of capillarity, which does not consider the influence of line tension, a liquid meniscus in contact with a stripwise, heterogeneous wall will experience a high curvature situation att the ends of the four-phase (i .e ., solid-solid-liquid-vapour) contact line . For capillary systems with finite line tension the effect on the contact line will be most pronounced at those locations where the curvature is large . Usingthese basic physical ideas, a strategy isproposed for using a stripwise, heterogeneous wall to evaluate the magnitude of the line tension based on the shape of the contact line . The strategy uses a combination of incremental loading coupled with the Newton-Raphson method to generate a series of non-zero line tension solutions from an initial analytical solution that corresponds to the zero line tension case .

A STRATEGY FOR MEASURING LINE TENSIONS

Introduction Several investigators have considered the influence of line tension on the shape of both the liquid surface and of the line of contact which is formed when at least three phases meet . Various experimental arrangements have been setup . The most direct methods involved an investigation of the size dependence of either liquid lenses at fluid-fluid interfaces or sessile drops on solid surfaces [1-10] . Our investigations to date fall into this category [11] . We have measured the effect of line tension on the contact angle of sessile drops on polymer solids using axisymmetric drop shape analysis 112,13 ] . The deformation of two fluid particles which meet in a third phase have also been investigated 114171, along with the impact of line tension on heterogeneous nucleation [1822 ] . Finally, a number of studies have been conducted which consider the influence of the linear phase on the composite energetics of thin soap films [2335] . Line tension values spanning several orders of magnitude have been reported with both positive and negative signs [361 . At present very few inves-

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tigators are examining line tensions for liquids in contact with solids primarily because of the inherent difficulties with the preparation of suitable solid surfaces. In this paper we propose an additional strategy and arrangement for measuring the size of line tension forces . In our arrangement a liquid in equilibrium with its vapour makes contact with a solid wall that is constructed of two different but compatible materials (see Fig . 1) . Both solid materials forming the stripwise, heterogeneous wall are rigid and do not chemically react in any fashion with the liquid . They are positioned such that the line of contact between the solids runs in a direction perpendicular to the level of the liquid-vapour interface. The wetting liquid forms a unique contact angle on each strip of homogeneous material which forms the wall . The contact angles on adjacent strips are not identical . Thus, as shown in Fig . 1, the liquid surface in close proximity to the solid wall will have a wavy pattern that arises because of this difference in the contact angles on each strip . As one moves away from the wall

Fig . 1 . Artist's conception of a liquid meniscus in contact with a stripwise heterogeneous wall.

309

(i.e., in the positive z-direction as shown in Fig . 3) one finds that the amplitude of the wave pattern dies down and the liquid-vapour surface approaches the Cassie configuration [371 (see Fig . 2) . In the presence of an external gravitational field, the Cassie configuration is the true or actual shape of the meniscus far away from the wall . Near to the wall, the actual shape of the meniscus is dependent upon which strip is touched by the liquid-vapour interface . When the capillary rise height is not the same on adjacent solid strips, the Cassie configuration, near to the wall, is positioned between the upper and lower menisci. The Cassie plane is the tangent plane to the Cassie configuration defined at the position of the wall . When the strips are sufficiently narrow, that is, they are much less than the the capillary constant a, then the Cassie plane is a good approximation of the Cassie configuration at the wall and within several strip widths away from the wall (as illustrated by the first two schematics of Fig . 2) . For this case, one finds that the wave pattern of the meniscus dies away over a short distance from the wall and that this distance is of the order of one strip width . The middle frame shows the superposition of the angle of contact STRIPWISE WALL (SIDE VIEW}

CASSIE CONFIGURATION

CASSIE CONFIGURATION

CASSIE PLANE

Fig. 2 . Schematic of the side view of the meniscus in contact with a stripwise, heterogeneous wall .



:3 10

of the meniscus on two adjacent strips of the wall . With strip width a, corresponding to contact angle 0, and strip width a 2 corresponding to contact angle 0, the Cassie contact angle, 0,,, which is the angle that the Cassie plane intersects the wall is cos 0,

=_a: a"+a-,-cos

0, +

¢z -Cos 02 a, +a 2

The lower diagram illustrates a situation in which our approximation that the largest strip width is much less than the capillary constant, a, would not apply and the method of solution discussed herein would not apply . A positive line tension operating along the boundary tends to constrain the contact line and to reduce its length . This is analogous to the effect of positive surface tension which attempts to constrain or reduce the amount of surface present in a capillary system . In comparison with a capillary system with vanishingly small line tension, one finds that a positive line tension will dampen the magnitude of the liquid surface waves and shorten the length of the four-phase contact line which follows the boundary between the solid strips. If the line tension is sufficiently large one finds that the contact line crosses the solid-solid strip boundary at an angle of less than ninety degrees (see Fig . 7) . For infinite line tension, the contact line would be straight and the Cassie angle would be the contact angle of the liquid on the wall . Whenever a liquid surface comes into contact with a solid phase there are several associated phenomena like interfacial tensions, line tension and gravity which contribute to both the final equilibrium shape of the liquid-vapour interface and the position of the contact line, The minimization of the total free energy of the capillary system yields both Euler's equation which is the stationary value of the first variation of the total free energy and the corresponding transversality condition which describes the variation of the boundary of the surface . These two relations are the mechanical equilibrium conditions of the capillary system . It may be shown that Euler's equation is a surface condition, commonly known as the Laplace equation of capillarity, which governs the shape of the liquidvapour meniscus, while the transversality condition, commonly known as Young's equation of capillarity, yields the behaviour of the line of contact formed by the liquid meniscus at those points where it touches the wall., If the capillary constant a, which we shall define by the equality a 2 =y,,,/ (dpg), is significantly larger than either of the strip widths a, and a, [i .e., a >> w=max(a,,a2 ) ) then the wave pattern dies out sufficiently near the wall that the Cassie configuration may be well approximated by a plane (see Fig. 2) . In that case, the influence of gravity is small and the pressure difference in the Laplace's equation of capillarity may be neglected as follows AP _ 1 + 1 -0 Y (1) R R2



311

where AP is the pressure difference across the meniscus, pictured in Fig . 1, y,,, is the liquid-vapour surface tension, zip is the difference in density between the liquid and vapour, g is the local value of gravity and R, and R 2 are the principal radii of curvature for the liquid-vapour surface . A surface for which the sum of the principal curvatures equals zero everywhere is known as a minimal surface. If we represent the liquid-vapour surface by the function z=z (x,y ) then it is possible to express the principal radii of curvature in terms of z and its partial derivatives so that one may write Eqn (1) as AP Yw

(1+z,v)z„-2z,z,z,,.+(l+zr)z,,, (1+z 2 +z 2 ) 31z i

0

(2)

or (1+zj)z„-2z,z,,z,,,+(1+zz)z,,,=0

(3)

where z, and z,, represent the first partial derivatives of the surface function z (x,y) with respect to the independent variables x and y respectively. The second-order partial derivatives of z are represented by z,,, z,,, and z,,. Equation (3) represents the Laplace equation of capillarity for a minimal surface [38] . The corresponding Young equation of capillarity (with line tension) for the contact line boundary is given by [39-41 ] a,,v cos B= cos B,-

Kg .

( 4)

Yiv

where B, is the contact angle that the liquid would form on an infinitely wide solid strip of either material i = 1 or material i=2 . In our case, the contact angle is defined as the smallest angle formed between the tangent plane of the liquidvapour surface and the tangent plane of the solid-liquid surface determined at the point where all three phases intersect to form the contact line (see Fig . 2), In addition, B is the actual contact angle, aG ,,, is the line tension and Kg . is the geodesic curvature of the contact line evaluated in the tangent plane of the solid boundary at each point along the contact line . The solution for the shape of both the surface and the contact line for nonvanishing line tension are obtained via numerical methods that will be explained below. However, to understand the rationale for the numerical approach is it necessary to review previous analytical (zero line tension) results . The zero line tension case

In a previous paper [41 ] an analytical solution was found for the exact shape of the liquid surface (without considering line tension) for the special arrangement depicted in Fig . 1 . The Laplace equation of capillarity governed the equilibrium shape of the liquid-vapour interface, while the classical Young equation of capillarity (i.e., Eqn (4) without the line tension term) described the



312

9

r ,a

Fig. 3 . Top left and right figures are schematics of the front and side views of a liquid-vapour interface in contact with an equal width stripwise wall . The contact line is indicated by a thick line labelled S S' N' N while the thinner lines representt the liquid-vapour surface att various positions out from the wall . The lower figure is a schematic of the position of the contact line after the mapping to the a- 13 domain .

equilibrium condition at the boundary of the liquid-vapour interface (i.e., at the line of contact which the liquid forms with the wall) . The liquid-vapour surface was a minimal surface with zero mean curvature as described above . The method of characteristics was employed to transform the Laplace equation of capillarity for a minimal surface into standard canonical form [42 ] 2Z

a 1Z

as ~aflb °

( 5)

by the coordinate transformation cos a az sin a and ax sinhfl dy=sinh/i az _

(6)

In conjunction with this transformation it was found that the domain of the problem shown in Figs 1 and 3 was transformed into a rectangle (' see Fig. 3 for an example) . Thus, the coordinate transformation given by Eqn (6) carries or maps the x-y domain shown in Fig . 1 and illustrated in Fig . 3 (top) into the a-fl domain shown in Fig. 3 (bottom) in which Eqn (5) applies . These two new variables have a straightforward physical interpretation . It may be shown that /i is related directly to the contact angle . It achieves a minimum value on the strip with the larger contact angle and a maximum value on the strip with the smaller contact angle . The relationship is given by



313

jt=-Inl tang !

(7)

where 0 represents the contact angle (see Ref. [411 for details) . When the line tension vanishes and the contact angle across each strip is a constant /9 will only change in value along the four-phase contact line formed by the two solids and the liquid meniscus . A direct physical interpretation also applies to a . If one bisects the liquid surface by a plane which is parallel to the original heterogeneous, stripwise wall (i .e ., the plane is positioned at a constant value of z) then the liquid surface meeting the plane will form a wavy pattern . The amplitude of the wavy pattern is larger the closer one positions the bisecting plane to the wall . A tangent line to this wavy pattern drawn in the plane of the bisecting plane at any point will make an angle with the corresponding horizontal and the angle measured counter-clockwise from the horizontal represents a . Thus, we may describe a as the turning angle of the liquid surface at a specified constant distance from the wall . With the aid of Fig . 4 we can trace out the a-fl domain. Beginning with the point denoted by S where the contact angle is a maximum and the slope of the liquid surface is zero we proceed to the point S' . At S' the contact angle is still identical in value to that at S (for the case of zero line tension) so f remains unchanged, but a has changed from zero to n/2 . Along the vertical section of contact line a remains fixed at the constant value of n/2 while (f changes from its lower constant value on one strip to its higher constant value on the adjacent strip . If we follow the progression of the contact line as it curves off of the four-phase contact line at N' one discovers that (I now remains fixed while a decreases in value from 7r/2 back to zero at point N . Tracing along the curve from N to smaller values of

N- 1112

P-Po Y 3=R,

S N b

Fig. 4 .The fill a-J# domain that corresponds to the liquid-vapour meniscus of Fig . 3 . The closed curves shown inside this square domain correspond to the thin curves shown in Fig . 3 . Curves that are positioned closer to the origin of the a-/J domain are positioned further from the wall in the x-y domain (i .e., they are at greater z) .

314

the coordinate y yields the remaining half of the rectangular domain . Recognizing that a will achieve both its maximum and minimum values of ±z/2 along the four-phase contact lines we realize that all other points in the domain will lie within the rectangle defined by tracing out the contact line of the liquid surface on the wall . Thus, all other points on the liquid-vapour surface will occur at some point within the boundaries of the rectangular domain . The near circular curves in the interior of the rectangle of Fig . 4 illustrate the manner in which the wave patterns of the liquid-vapour surface vary at different positions away from the wall (see Fig. 3) are mapped into the interior of the a/S domain . The next step in the solution was the introduction of the complex variable w= a+i/f. Application of a Schwarz-Christoffel transformation (i.e., these transformations map half-planes to polygons or vice versa) mapped the rectangular co-domain into the complex upper (-half-plane . A subsequent bilinear transformation mapped this upper half-plane of ( onto the interior of a unit circle in the w-plane . With these two mappings, each point on the perimeter of the rectangle was mapped to a corresponding point on the circumference of the circle . Likewise, every point in the interior of the rectangle was mapped into the interior of the circle . The coordinates of the circumference of the circle were p = 1 for the radius of a unit circle and 0 < 0 < 2rt for the angular rotation about the origin . The essential purpose of these mappings was to change the problem from a domain where the solution is unknown to one of a standard number of domains in which a solution is easier to obtain . Once the solution is obtained one then carries the solution function back via reverse mappings, The analytical solution could be written down directly in the complex co-plane . This solution was then reverse mapped to the (o-plane where integration was performed directly to return to the original x-y domain (see Eqn (6) sad Ref . [41] for details) . With zero line tension one discovers that the contact line follows the solidsolid boundary when the liquid-vapour interface changes from its constant angle on one strip to its contact angle on the adjacent strip . A high curvature situation occurs at the ends of this four-phase (i.e ., solid-solid-liquid-vapour) contact line because of the rapid. bending of the contact line as it crosses the boundary between the solid strips forming the wall . At these four-phase points the curvature of the contact line is infinite (see Fig . 5). However, the presence of these infinite curvatures does not in any way violate the classical Laplace or Young equations . What is violated is our intuition that natural boundaries should be both continuous and smooth, the assumption of moderate curvature upon which Gibbs' derivation of the theory of capillarity is based [431 and our concepts connected with contact angle hysteresis, as explained below . It has been suggested [41,44,45 ] that contact angle hysteresis vanishes when the amplitude of the contact line contortions, be they due to heterogeneity or roughness, fall within the diffuse zone of a real liquid-vapour interface . When

315

S N

S 8

-

Fig . 5 . Schematic of the contact angle and the contact line curvature along the length of the contact line for the case of zero line tension that is illustrated in Fig . 3 . The curvature of the contact tine approaches infinity as the contact line approaches the points N' and S' from NandS respectively and it is zero between these points .

the effect of line tension is ignored one finds that the amplitude of the contortions of the contact line are of the same order of magnitude as the characteristic wavelength or width of the strips which from the wall . In this argument we consider the width of the strips as a measure of the characteristic size of the heterogeneous patches . This interpretation would lead one to the conclusion that the lower limit for patchwise heterogeneity to produce contact angle hysteresis when line tension vanishes would be of the order of 10 -y m. As it is unlikely that actual solid surfaces are homogeneous on such a small scale, one must either conclude that all solid surfaces will show contact angle hysteresis or that another effect (i.e ., line tension) is operational . However, the conclusion that all surfaces show hysteresis is at odds with experimental studies which show the absence of contact angle hysteresis on certain well-prepared solid surfaces . Consequently, the assumption of a positive line tension seems inevitable ; thus one may expect that the magnitude of the amplitude of the contact line is dampened to such a degree that it is possible to support much larger patches of heterogeneity without the appearance of contact angle hysteresis .

The non-zero line tension case It may prove useful to consider a few essential characteristics of the zero line tension arrangement before delving into the case of finite line tension . Figure 5 illustrates the manner in which the curvature of the contact line varies as one proceeds across the stripwise wall . When the line tension, a,,,, is zero there will be certain points along the contact line at which the curvature will be zero and other points at which it will be infinite . For the case of zero line tension it can readily be seen from Eqn (4) that the actual contact angle which the meniscus forms on a given strip of the wall will be exactly equal to the intrinsic



316

contact angle of that liquid on that particular material . Thus, with as ,,.=0 we find from Eqn (4) that O=0, where i=1,2 . This straight-forward situation changes when the line tension becomes finite . When line tension is non-zero one must consider the effect of the term a s ,,,Kgs upon the over-all force balance at the contact line . For a planar solid wall the geodesic curvature Kg, is equal to the curvature of the contact line, as denoted by c, in Fig . 5 . When the curvature of the contact line is small, the geodesic curvature Ks, will also be small so that the effect of the term as , vKgs on the actual contact angle 0 will not be appreciable . However, at points on the contact line where the curvature is significant, even a small line tension will produce a noticable influence on the contact angle at those locations . These locations occur near the solid-solid strip boundary where the curvature of the contact line is extremely large (see Fig. 5) . The shape of the a-fl domain is also influenced when the line tension is finite . When the line tension is zero we know from our previous discussion that /3 [which is related to the contact angle by Eqn (7) ] only changes in value along the solid-solid-liquid-vapour four-phase contact line . However, when the line tension is finite changes in fi occur in regions away from the contact line and the cr-/f domain experiences rounding at its corners (contrast the lower diagrams in Figs 3 and 6) . As the magnitude of the line tension increases /3 changes in a non-linear fashion . In addition, the length of contact line which

N~ ' N' ,V-_'

l

y

S

S

X

M Fig . 6. Schematic of the contact line in the x-y domain and its corresponding mapping into the a-(3 domain for a small, positive line tension- For this case, there is still a portion of the contact line with a=n/2 which follows the solid-solid boundary (i .e ., the section of the line between points S' and N') .

317

follows the solid-solid boundary "shrinks", thus reducing the vertical dimension of the rectangular a-fl domain . This behaviour occurs because positive line tension tends to pull the contact angles on the two adjacent strips closer together and to dampen the amplitude of the wavy pattern so that the turning angle a is pulled toward zero and /I is brought closer to the value fe that corresponds to the Cassie plane (see Figs 6, 7 and 8) . The presence of these complexities requires a numerical approach . Our solution for the shape of the surface in the case of positive line tension begins with the canonical form of the Laplace equation of capillarity [see Eqns (5) and (6) ] . The unknown shape of the a-)3 domain prevents a solution via an analytical transformation . Instead, the mapping between the co-domain and the unit circle is performed directly by a complex power series . The series is given by

W-c

0=ai-i(lf-~c)=-i c k w k

(8)

k~l

where fe corresponds to the Cassie angle 8, , co is the complex representation of the coordinates (a,#), w represents the same coordinates after both a Schwarz--Christoffel and a bilinear mapping ( i.e., the end-result domain in this case is a unit circle) and ck represent the unknown complex coefficients .

N

a

Fig. 7 . Schematic of the contact line in the x-y domain and its corresponding mapping into the a- fl domain for a larger line tension value . In this case, the contact line crosses the solid-solid boundary at only one point and a is less than x/2 .



318

Fig. 8 . Schematic of the full contact line mapped in to the a-/f domain . As the line tension increases in magnitude (in the manner illustrated by Figs 6 and 7) the contact line curve is pulled away from the boundaries of the rectangular domain and "shrinks" towards an elliptic-like domain . Larger positive values of the line tension result in smaller, concentric ellipses .

Recognizing that w maybe expressed as w=pexp (iy) and that c k =ak+ib k permits one to express a and /f in the form of Fourier series a=Re(w-wo )= > (akp k sin k¢+bkp k cos k¢) k=1

(9)

00

/3-/I0=Im(w-wo)=- 1] (akp k cos k¢-bk p k sin k¢) k=1

where a k and bk are the real coefficients which need to be determined from the contact line boundary conditions along the perimeter of the unit circle (i.e., where p=1) . The boundary condition, given by Eqn (4), may also be represented by y,,. (cos 0-cos O,)-vsiv Kg,=0

(10)

where 0 ;, represents the intrinsic contact angle of the ith strip . If both cos 0 and xk, are expressed in terms of a and /I Eqn (10) becomes sinh(f-/3)

+ay , cosh

fl,aa=0

(11)

where c is proportional to the strip widths a, and a z by the relation c-

a, +a, 27E cosh ft.

(12)

In this problem, both c and the intrinsic contact angles for each strip are used as input data. The contact angle Oi enters Eqn (11) through its relationship with i3, [see Eqn (7) ] . For simplicity, we consider the case where the solid--



319 solid boundary is perpendicular to the Cassie plane and the strips are of equal width (see Fig. 3) . These restrictions may be relaxed without difficulties, leading to expressions more general than those given below . With these assumptions, Eqn (11) may be split into three relations . Each relation represents the equation for a particular section of the contact line (i .e ., one relation for the contact line across the first strip, another relation for the contact line along the four-phase boundary and a final relation for the contact line across the second, adjacent strip) . The set of relations are sinh(T+f31-f%o)+c1tsinh(Q1-fso) -=0,0<0<¢1 n=0101 <0<02 2 sinh(T+)62-f3o)+c2tsinh(fl2-/30)d~=0, 02 <¢
ak sin k¢ k=1

and c, = a, +a2 _ where i=1,2 The parameter t, which is discussed below, incorporates the influence of the line tension into the boundary condition . The expression for this parameter is given by ; cosh (fl,-fio) cosh to t- 2nas,va,_ Y1v(a1+a2)2 (-1)'sinh(l=-/lo)

(14)

The case of larger line tension, when the contact line crosses the solid-solid boundary at one point (see Figs 7 and 8) requires a slight modification to Eqn (13) because the contact line is composed of just two sections rather than three . In particular, the modified relations are sinh (T+/~ /f1 -Po)+c, t sinh(ff, -/3o)a¢=0, 0<¢<¢s (13a) d3 sinh(T+J2-lo)+c2tsinh(32-lo)d=0, ¢3 <¢« ¢ The numerical solution for this case (not given here) would be a simple extension of the method given below . To reiterate, each relation shown in Eqn (13)



320

corresponds to a particular section of the contact line . These sections are given by the length of contact line on the first strip from S to S', the second strip from N' to N as well as the length of the boundary or contact line along the four-phase line from S' to N' (see either Fig. 3 or 6) . When the a-(3 domain is rectangular (see Fig . 3) each section is represented by a side of the rectangle . Thus, the bottom horizontal line represents a constant contact angle on one strip while the uppermost horizontal line represents the constant contact angle on the adjacent strip . The two vertical line segments correspond to the fourphase lines which separate the two strips . Along the solid-solid boundary we know that a, the turning angle, must be equal to rr/2 since the four-phase line runs in a vertical direction . Thus, the Fourier series which represents a must equal n/2 along this section of the contact line [see middle relation in Eqn (13) ] . On the other two strips, Eqn (11) is the governing relation that must be satisfied with (3, input as a known quantity . By requiring that the difference between the analytic function for the rectangular domain and the Fourier series used to approximate it are minimized (in the least squares sense [42] ) along all regions of the contact line it is possible to determine the coefficients in the series which give the best agreement .. When the line tension vanishes the set of equations given in E (13) simplify to

91 -fio +Y_a k cosk¢=0 , 0<0<0 1 k=1 7E

~a,,sinkg9--=0 2

, 01


(15)

k=1

J32 -#0 +Ea k coshO=0 , 02 <¢<71 k-1

The least squares problem becomes linear and can be solved by matrix inversion . The solution yields the coefficients, ak, for the case were the line tension is zero. When the line tension is non-zero the initial set of Fourier coefficients (corresponding to the zero line tension case discussed above) are used as a springboard to other adjacent coefficients which represent surfaces (and contact line boundaries) with finite line tensions . The numerical method of optimizing the new Fourier coefficients for the non-linear least squares problem which results when the line tension is non-zero involves a combination of incremental loading coupled with the Newton-Raphson method [46-48 ] . This strategy is useful since the powerful Newton-Raphson method depends upon a good initial approximation to the solution . In fact, since the very first set of Fourier coefficients corresponds to the zero line tension analytical solution, the initial approximation is quite good . Incremental loading provides a means of obtaining



321

a series of solutions for a set of increasing values of line tension . The parameter, t, defined by Eqn (14) serves as the loading parameter, varying typically from 0 to 0.2 in 50 steps . Once the Fourier coefficients are known, the corresponding shape of the a-/1 domain can be obtained from Eqn (9) . The steps for obtaining the actual shape of the surface (i .e., of getting x, y and z from a and /3) are the same as those described in Ref . [41 ] . Numerical results

Figures 9 through 13 present characteristic plots of the boundary and surface profiles along with the magnitude of the line tension which produces their shape. In Fig. 9 we have two equal width strips with characteristic wetting angles of 145 .0° and 35 .0°, respectively. The four profiles shown represent different positions of the liquid meniscus in contact with the wall for four different values of the dimensionless ratio (a t,) / [Y]„ (a, +al ) ] . Thus, if one knew beforehand that a given liquid of known surface tension y,,, gave equilibrium wetting angles of 145 .0° and 35 .0° on two different solid materials then one could assemble a stripwise wall of known strip width and obtain an estimate of the line tension by comparing the shape of the experimental profile with those profiles generated via computer . Figure 10 shows the same solid wall arrangement as in Fig. 9. The curves in this case, however, show the manner in which the wavy nature of the liquid-vapour surface dies out as one moves away from the wall . The line tension is equal to zero for all curves in Fig . 10. Analogous plots for as,,. > 0 show similar trends with distance out from the wall . Physically, these computer results suggest that the effect of positive line ten-

-0 .10

0 .10

0 .30

0 .50

0 .70

0 .90

1 .10

Y

Fig . 9 . Computer generated shape of the contact line for four values of the ratio (a,,,,)/ { p,J a, +a,)] . The intrinsic contact angles of the two strips are 35 .0` and 145 .0' . The value of the ratio for each curve is : (1) 0 .0; (2) 0 .2 . 10 -4; (3) 0 .3 . 10 -4 , and (4) 0 .4 . 10 -4 .



322

Fig . 10, Computer generated shape of the liquid-vapour surface at various distances out from the stsipwise, heterogeneous wall . The heavy curve, which is denoted by curve (1) in Fig . 9, represents the contact line for the case when the line tension is zero . Al) other curves represent sections of this meniscus at different values of z (the coordinate which measures the distance out from the wall) .

0 00

D,30

0.

ca

0 .90

1 .20

1,50

1 .60

a Fig . 11 . Curve (1) represents the mapping of the zero line tension contact line, illustrated as curve (1) in Fig, 9, onto the a-/3 domain . Curve (1) is the best least squares approximation to a rectangular domain using 64 Fourier coefficients . Curves (2) through (4) correspond to curves (2) through (4) in Fig . 9 and indicate the fashion in which the a-jf domain "shrinks" with increasing line tension .

sion will be two-fold . It will both reduce the overall length and hence the amplitude of the contact line and it . will shift the two adjacent contact angles closer to the Cassie angle . In the limit of very large line tension the two contact angles would approach the contact angle of the Cassie plane .



323 02 = 23 .5°

0, = 156 .5° a,

n2-

%-

-0 . :0

0 .10

0 .30

0 .50 Y

0 .70

0 .90

: .10

Fig . 12. Computer generated shape of the contact tine for various values of the ratio (asst/ (y, (u,Ta ,2 ) )_ The intrinsic wetting angles of the two strips are 23 .5' and 156 .5' . The value of the ratio for each curve is : (1) 0 .0 : (2) 0 .2 . 10 -4 ; ( 3) 0 .4'10 -4 ; ( 4) 0 .5 . 10 -° ;and (5) 0 .6 . 10 -4 .

a

O

C .00

0 .3C

O .EG

0 .90

1 .20

1 .50

1 .60

a Fig . 13 . Curve (1) represents the mapping of the zero line tension contact line, illustrated as curve (1) in Fig . 12, onto the a-fldomain . Once again, curve (1) is the best least squares approximation to a rectangular domain using 64 Fourier coefficients . Curves (2) through (5) correspond to curves (2) through (5) in Fig. 12 and indicate the manner in which the a-l domain "shrinks" with increasing line tension .

The horizontal and vertical scales are not dimensionalized in either Fig . 9 or 10 since the curves are independent of length scale . However, since we have assumed that the influence of gravity was negligible in close proximity to the wall through our assumption that the maximum strip width w, defined above, is significantly smaller than the capillary constant a, also defined above, we must choose a length scale such that gravitation effects are not significant . It can be shown that the approximation is accurate at the wall and to within



324

several strip widths away from the wall provided the characteristic strip width, w, is sufficiently small [411 . With a liquid whose surface tension, y, v , is given as 72 mJ m 2 and whose density difference between liquid and vapour, dp, is 1 g cm - ''the capillary constant, a, is approximately 3 mm . Our approximation will be valid provided that the largest strip width, w, is much less than this capillary constant . If Fig . 9 illustrated the specific case of water in contact with a stripwise wall where the horizontal scale is in millimetres then the strip widths would both equal 0 .5 mm and the condition w << a would be satisfied . Figure 9 shows four values of the ratio (a,,v)/[ ;,V(a,+a2) 1 . Using the surface tension of water, as given above, with a,+a 2 =1 mm the line tension for each curve is shown in Table 1 . The line tension values shown in Table l are three orders of magnitude smaller than the values obtained recently from the dropsize dependence measurements of contact angles [11,49] . If these much larger values of line tension are correct, then we would expect that physically the contact line would cross the solid-solid boundary with a turning angle of less than ?T/2 and would not follow this boundary for a finite distance . Such a situation will become tractable once Eqns (13) and (13a) are solved . Figure 11 shows only the right-hand portion of the symmetric rectangular cx-/3 domain that corresponds to the contact line profiles shown in Fig . 9 . Curve number one of Fig. 9 (see Fig . 3 for comparison) represents the best (in the least squares sense) Fourier coefficient approximation to the zero line tension (rectangular) domain. The oscillations that arise near the vertical line a=it/ 2 are characteristic of any kind of Fourier approximation to a sharp corner or step function. The over-shoot decreases with increasing number of coefficients, but never completely vanishes [50] . Figure 11 shows the calculated behaviour of the a-f domain boundary as the line tension is progressively increased above zero . The four curves illustrated in Fig . 9 and labelled one through four are likewise labelled in Fig . 11. Figure 12 shows two equal width strips with characteristic wetting angles of 165.5' and 23 .5°, respectively, together with five curves corresponding to five TABLE 1 An example of the magnitude of line tension effects for the case of water (see discussion in text) ae„ 7w (a,+a 9 1

as ,

(1 . 10 -4 )

(1 . 10-9 )

0 .0 0 .2 0 .3 0 .4

0.0 1 .4 2 .2 2 .9

»a

325

different values for the ratio ((T,,,,)/ [y,,,(a1+a2) ] . Once again the same physical interpretation applies for this case as for the case shown in Fig . 9 . Figure 13 is analogous to Fig . 11 and represents the calculated boundary profile of the a-J3 domain for various magnitudes of the line tension . These results represent preliminary numerical solutions to the Laplace and Young equations for a symmetric stripwise wall when line tension is finite . Positive line tension tends to flatten out the contortions of the contact line and reduce the amplitude of the surface wave pattern . Our numerical method could be used to determine the magnitude of the line tension for a specified experimental set-up . Necessary input data would include the equilibrium contact angles on the two strips and the shape of the contact line . The latter information can be readily obtained by means of recently developed digital image analysis strategies [51,52] . While universally accepted numerical values for line tensions are still not available at this time, it is clear that progress is forthcoming through direct measurements of the kind mentioned at the beginning of this article . Greater consensus will undoubtably occur via this route, or through investigations of contact angle hysteresis [53-58] . Regardless, we anticipate that the numerical scheme suggested here will render information on both the magnitude of the line tension required to eliminate the infinite curvature difficulty with the con tact line while at the same time yielding an estimate of the critical patch size required for the generation of contact angle hysteresis .

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