2803
J. Phys. Chem. 1995, 99, 2803-2806
Line Tension and the Shape of a Sessile Drop? B. Widom Department of Chemistry, Baker Laboratory, Come11 University, Ithaca, New York, 14853
Received: May 27, 1994; In Final Form:August 8, 1994@
The effect of line tension on the contact angle between a small sessile drop (radius a few tens or a hundred pm) and a solid substrate is analyzed. When the line tension is great enough, it induces a wetting (“drying”) transition, in which the contact angle jumps discontinuously to 0. The droplet is then completely detached from the surface, which remains in contact only with the vapor. Another result is that even if the drop would have wet (spread on) the substrate when the drop was macroscopically large, it will not do so when it is a small droplet if the line tension is positive.
Introduction The contact angles in three-phase equilibrium can be affected by the tension of the line in which the three phases meet.‘ A simple case, in which one of the three phases ( y ) is a nondeformable solid substrate while the other two (a and p) are fluids, is shown in Figure 1. The phases /?and a may be a liquid and its equilibrium vapor, respectively, or (in a multicomponent system) may both be liquid phases, of which p is the denser. Assume the radius R of the droplet @ (Figure 1) to be much less than the capillary length of the gS interface:
Y (b)
(0)
Figure 1. Droplet of /3 immersed in a resting on a nondefomable substrate y. The contact angle measured through the a phase is 8: (a) 0 d 2 ; (b) 0 > d 2 . The radius of the circle in which the three phases meet is r; the radius of the spherical surface of the droplet is R. The center of the sphere is above the ay plane in (a) and below it in (b).
where uag is the gS interfacial tension, A@the difference in density of the ,8 and a phases, and g the acceleration due to gravity. Then the j? droplet is not distorted by gravity; it is a truncated sphere (Figure la) or a spherical cap (Figure lb).2 The radius R may be a few tens or a hundred micrometers while the capillary length is typically one or a few millimeters. It is only for such small drops that the effects to be discussed are significant. The radius R is nevertheless large compared with the size of the molecules or the range of the intermolecular forces, so the interfacial tensions 048, etc., may be assumed to be independent of R. The three phases meet in a circle of common contact, of radius r (Figure 1). The excess free energy per unit length of this contact line is the line tension, t.3 If e is the contact angle between the droplet and the substrate, measured through the surrounding a phase (Figure l), then 8 depends not only on the three interfacial tensions uag, up,, and Gay but also on z; it is smaller for larger z and larger for smaller z. This will be apparent in the formulas that follow but is readily understood as the effect of tightening or loosening the “collar” (the threephase line) that bounds the droplet. The object of this article is to calculate the effect of the line tension z on the contact angle e. The same question has been addressed independently by Clarke4 and Nijmeijer.5 The case in which y , too, is a deformable liquid has also been analyzed,’ but then one must distinguish two independent contact angles and the results are less transparent. One of the main results of the present analysis is that when z increases to some critical value, 8 jumps discontinuously to 0. This is thus a line-tension-driven wetting transition (some+ @
Dedicated to Stuart A. Rice. Abstract published in Advance ACS Abstracts, February 1, 1995.
0022-365419512099-2803$09.00/0
times called a “drying” transition6 when a is a vapor phase). As a result of it the droplet p is detached from the solid surface, so that both the liquid /?(now a full sphere) and the solid y are in contact only with a. The latter is then said to “wet”, or spread at, the Py interface. Another result of the present analysis is that even if j? as a macroscopically large phase (with line tension thus negligible) would have wet (spread at) the ay interface, it will not do so as a small droplet if the line tension is positive.
Analysis The volume V of the p droplet (truncated sphere or spherical cap), the area A of the droplet’s spherical surface that is in contact with the a phase, and the radius r of the circle of threephase contact are expressible in terms of the radius R of the droplet and the contact angle 6’ (Figure 1) by
v = -31~ R ~+( COS I e12(2 - COS e) A = 2nR2(1
+ cos 8)
r = R sin 8
(2)
(3) (4)
For a given volume V only one of the four quantities R, r, A, and 8 is independently variable; the remaining three are related to that one by eqs 2-4. With 048, up,, and Gay again the tensions of the three interfaces and z the tension of the three-phase line, the free energy F is
For a given fixed volume V of the B droplet, the equilibrium 8 (or r or R or A, these being all related by eqs 2-4) is that which 0 1995 American Chemical Society
Widom
2804 J. Phys. Chem., Vol. 99, No. 9, 1995
minimizes F. For a local extremum in F at some 8 in the range 0 < 8 < n,one must have dFld8 = 0. If A and r in eq 5 are expressed in terms of 8 and the fixed V by eqs 2-4, the condition dFld8 = 0 becomes -ass cos 8
+ up, - uay+ d r = 0
(6)
which is the expression of the mechanical balance of forces on the line (circle) of three-phase ~ o n t a c t .Without ~ the line tension term zIr it would be the classical Young’s e q ~ a t i o n . ~ The 8 determined by eq 6 is not always that at which F is minimal; the solution of eq 6 might correspond to a local maximum rather than minimum of F, or even when it is a local minimum, the global minimum may occur at one of the extremes 8 = 0 or n,at which the three-phase line has disappeared. It is the competition between a local minimum determined by eq 6 and the value of F at 8 = 0 that is responsible for the wetting (“drying”) transition anticipated above. Let = cos e
Figure 2. Equilibrium x (=cos 0) vs T for various fixed xo (=cos 00).
To find the global minimum, one must compare eq 13 with the values of 4 at the limits x = 1 (8 = 0) and x = - 1 (8 = n). From eq 11, 4x=1 = 2-
(7)
and denote by 80 and x g = cos 8 0 the values 8 and x would have if the line tension were 0; Le., from eq 6,
Define a dimensionless line tension t related to T by
(9)
Note that this becomes negligibly small in the limit of a macroscopically large droplet volume V. That, ultimately, is why line tension significantly affects the contact angle only of small droplets. For R = 1pm, It1 = dyn (which, although greater than a typical line tension, is within the range of measured values’), and a@ = 1 dydcm (a low but not ultralow interfacial tension), we would have It1 = 0.063. Finally, define a dimensionless free energy 4 related to F by
1
+w
4x=-1-
-
1/3
(14)
if xo > -1 (15)
+=ifx,=-landt>O -wifx,= -1 a n d f < 0
+
The condition xo > - 1, by eq 8, means Uay < uag up,, which is the condition that when the drop is macroscopically large and the effect of line tension consequently negligible, the p phase does not wet (spread at) the ay interfa~e.~Then, according to the first of eqs 15, even for a small droplet and a non-negligible ‘G, whether T be positive or negative, the free energy at the local extremum is certainly less than that at x = -1, so the equilibrium contact angle is not n;i.e., the p droplet still does not wet the ay interface. Correspondingly, the curves for xo > - 1 in Figure 2 lie above x = - 1 and approach x = -1 only asymptotically as t --. The condition no = -1, by contrast, means uay= uag opu,so that a macroscopically large p phase does indeed wet the ay interface. Then according to the second of eqs 15, if t > 0, the equilibrium contact angle is again not n (curve = - 1 in Figure 2); a positive line tension always prevents the p drop from spreading indefinitely at (wetting) the ay interface (although the effect would be noticeable only for very small drops). Finally, from the last of eqs 15, one sees that if a macroscopic p phase would wet the ay interface (i.e., if xo = -1, so Gay = crag up,) and if the line tension is negative, then the equilibrium contact angle is certainly n (baseline at x = -1 for t -= 0 in Figure 2). This says that if the /?phase would have spread at the ay interface even when the line tension was negligible (xo = -I), it will certainly do so when the line tension is appreciably negative, which is intuitively obvious. (The free energy in this case does not actually become -00; once the /?film has spread to molecular thickness, the thermodynamic analysis ceases to apply.) Consider next the competition between hocextI in eq 13 and @xx=l in eq 14. Taking account of eq 12, one finds these free energies to be equal when
-+
+
Then from eqs 2-4 and 7-9, the expression (eq 5) for the free energy becomes
4 = (1 + x)-1/3(2 - x)-2/3[ 1 + t(1
+
X)+6(1
- X)lI2(2 - x)-1’3 (11)
while the force-balance relation (eq 6) becomes
+
t = ( x - x o ) ( ~ x)-1/6(2 - x)-1/3(1
- x)1/2
(12)
This relation between t and x is shown for various fixed xo as the solid curves in Figure 2. The relation (eq 12) between line tension and contact angle at given xo (i.e., at given values of the interfacial tensions, according to eq 8), is the condition for a local extremum of F. From eqs 11 and 12 one has at the local extremum
with x given implicitly as a function of t by eq 12.
This and eq 12 are a pair of parametric equations (parameter xo) for a locus of wetting (“drying”) transitions in the x , t plane.
This locus is shown as the dashed curve in Figure 2. The solid curves of x vs t in Figure 2, which display the force-balance relation (eq 12) for various xo, give the value of the contact angle at which, for given xo and t,the free energy has a local extremum. Everywhere along any of the solid curves in Figure 2, up to the dashed wetting-transition locus (and even
Line Tension and the Shape of a Sessile Drop
J. Phys. Chem., Vol. 99, No. 9, 1995 2805 2*r
Figure 3. x vs t at fixed xo for local extremum of the free energy (solid curve) and wetting-transition locus (dashed curve); schematic. Below P on the solid curve the local free energy extremum is a minimum; beyond P it is a maximum. The equilibrium wetting transition occurs at Q.
for some way beyond, on the curve's analytic continuation), the local extremum is a local minimum, as one may verify; while up to the transition locus hoc < #JX=l; so the solid curves correspond to global minima of the free energy. To the high-7 side of the dashed curve in Figure 2, hW extI > so the minimum free energy there is at x = 1. Therefore, when, with increasing t,the solid curve in Figure 2 for any given xg intersects the dashed curve, the equilibrium x jumps discontinuously to 1 (the contact angle 8 jumps discontinuously to 0), as shown in the figure. This is a sudden wetting of the By interface by the a phase. (Clarke4 found what is essentially this same wetting phenomenon although differing in some details.) There is a discontinuous change in the shape of the p droplet, which changes suddenly from being a truncated sphere (Figure la) or spherical cap (Figure lb) resting on the substrate y to being a full sphere entirely separated from y by a layer of
a. If one analytically continued any of the solid curves in Figure 2 beyond the wetting-transition locus, i.e., to higher t,the states represented by it would still for some way be local (although no longer global) minima of the free energy. That would be so up to the point at which dtldx on the analytically continued curve became negative, beyond which those states would then be local maxima of the free energy rather than minima. This is summarized schematically, for one fixed XO, in Figure 3. The part of the solid curve that lies between the wetting-transition locus (dashed curve) and the point P represents metastable states in which the contact angle 8 remains positive (x < 1) while the true equilibrium contact angle is 0 ( x = 1). The point P, at which dttdx on the solid curve is 0 and where the local free energy minima become maxima, is the limit of possible existence of such metastable states at that XQ. Beyond that point, where dtldx on the solid curve is negative, an increase in the line tension would cause the contact angle to increase rather than decrease (recall that x = cos e), which is a physical impossibility in a stable state and implies a mechanical instability. The discontinuous jump to 8 = 0 (x = 1) must therefore occur before that point is reached. Figure 4 is similar to Figure 3, but now the circumference 2nr of the contact circle is plotted against the line tension t (again schematically). The points marked P and Q in the two figures correspond. In Figure 4 the point Q, where the equilibrium wetting transition occurs, is such that the two shaded areas are equal. The tension t and circumference 2nr are a conjugate pair, and the equilibrium transition is thus determined by this Maxwell construction, as may be verified from eqs 5 and 6 . The objective of this work, which was to determine the effect of line tension on contact angle, has now been essentially achieved with the results contained in Figure 2. It is still of
0
Figure 4. Circumference 2rcr of the circle of contact vs line tension t. Points P and Q correspond to those in Figure 3. The two shaded areas are equal.
-
interest, however, to examine more closely the behavior as t 0 at the two extremes xo = -1 (80 = n, where a macroscopically large phase /3 would wet the ay interface) and xo = 1 ( 8 0 = 0, where a wets the /3y interface) and the behavior at t = 0 as xo -1 or 1. It is small t that are of greatest practical interest because of the factor V-1/3 in eq 9. On the curve xo = -1 in Figure 2
-
(17)
+(9/8)1/5T66/5( x o = - 1 , 7 - O )
x--1
near 7 = 0. The curve thus comes into x = -1 at 7 = 0 with vanishing slope, so the cosine of the contact angle is not very sensitive to line tension there. The contact angle 8 itself, however, is more sensitive to it:
-
e n - 61/573'5 (x, = -1,
7
-
0)
-
(18)
implying an infinite rate of change of 8 with t as t 0 at xo = -1 (8, = n). By a similar calculation one finds from eq 12 that the rate of change of 8 with t at t = 0, near but not at the bulk wetting transition, Le., for 80 close to but less than n, is
showing how sensitive the contact angle 8 is to the line tension near the bulk wetting transition. These results have been obtained also by Clarke4 and Nijmeijer.5 Clarke4 remarks in addition that the sensitivity of 8 to t is greater the smaller DM is, because of the relation (eq 9) of t to t. It may be of even greater interest and importance to know how the contact angle is affected by line tension at the other bulk wetting transition, where xo = 1 (80 = 0) and where it is then the a phase that wets the interface between the p droplet and the substrate y (cf. Figure la). On the curve xo = 1 in Figure 2, where 7 is negative,
-
1- 2 1 / 9 ( - p
(xo = 1, T
-
0)
(20)
near t = 0. The curve thus comes into x = 1 at 7 = 0 with infinite slope, so at this bulk wetting point even the cosine of the contact angle is sensitive to 7. The contact angle 8 itself is even more sensitive to it:
From eq 12 again, one finds for the rate of change of 8 with t at t = 0, near but not at the bulk wetting transition, i.e., for 8 0 close to but greater than 0,
2806 J. Phys. Chem., Vol. 99, No. 9, 1995
Widom
showing how sensitive the contact angle is to the line tension near this bulk wetting transition. This increases more rapidly with increasing proximity to the bulk wetting point (60 0) than is the case at the other bulk wetting point (60 n),as found in eq 19.
--
Summary The results in eqs 17-22 give magnified views of the immediate neighborhoods of the two points x = 1, 7 = 0 and x = -1, T = 0 in Figure 2. With the rest of that figure, they provide a detailed account of the effects of line tension on the equilibrium shapes of small sessile drops. Two of the main features that have been noticed are (i) the discontinuous transition to wetting of the ,!3y interface by a as the dashed curve in Figure 2 is crossed in the direction of increasing 7 and (ii) the nonwetting of the ay interface by ,!3 (the presence of a line of three-phase contact, x > -1,O .c x) when T > 0, even when ,f? as a macroscopically large phase (line tension negligible) would have wet that interface (XO = -1, 60 = n).
Acknowledgment. The author acknowledges with gratitude the inspiring model of a scientist Stuart Rice has been for so many of us. In connection with the present work the author is grateful for helpful conversations and correspondence with A. S. Clarke and M. J. P. Nijmeijer. Support by the U.S. National Science Foundation and the Come11 University Materials Science Center is also acknowledged. References and Notes (1) Toshev, B. V.; Platikanov, D.; Scheludko, A. Lnngmuir 1988, 4 , 489. (2) Miller, C. A.; Neogi, P. Interfacial Phenomena; Marcel Dekker: New York, 1985; Chapter I, section 6. (3) Rowliison, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford University Press: Oxford, 1982; section 8.6. (4)Clarke, A. S. Private communication. ( 5 ) Nijmeijer, M. J. P. Private communication. ( 6 ) van Swol, F.; Henderson, J. R. Phys. Rev. A 1991, 43, 2932. (7) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464.
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