Langmuir 1993,9, 619-621
619
Modification of the Cassie Equation Jaroslaw Drelich and Jan D. Miller* Department of Metallurgical Engineering, University of Utah, Salt Lake City, Utah 84112 Received May 21,1992. In Final Form: November 17,1992 A modification of the Camie equation has been derived on the basis of the generalized Young equation including the linetension term. It seems that the modified Camie equation can be uaeful for the interpretation of contact angle data reported for heterogeneous surfaces.
Cassie1p2derived an equation describing contact angle changes for composite solid surfaces with varying degrees of heterogeneity: COS
e = EtiCOS ei
(1)
which for a two-component surface is COS
e = f, COS e, + fi COS 6,
(2)
where fi is the fractional area of the surface with a contact angle of ei. The Cassie equation has been found to be very useful in the analysis of heterogeneous solid surfaces. However, the wide scatter in contact angle data often observed for heterogeneous systems is difficult to explain on the basis of the Cassie equation. When Cassie considered the equilibrium three-phase contact angle, the concept and significance of the linetension term were not appreciated. It should be noted that Cassie only predicted the effect of surface heterogeneity on the interfacial free energy of a system. He did not describe the corrugation of the three-phase contact line, as hae been discussed in the literature.*' A portion of a hypothetical three-phase contact l i e is shown in Figure 1,and it is known that the excess energy associated with the triple junction has, in some cases, made a significant contribution to the total energy of the threephase system. In this regard, a better understanding of wetting phenomena was made possible aftar the generalized Young equation (eq 3) was derived by Boruvka and
Neumarm8 ysv, YSL, and YLV are the interfacial tensions for solid/vapor, solid/liquid, and liquidhapor interfaces, respectively,8 is the contact angle, YSLV is the line tension, and K@ is the geodesic curvature of the three-phase contact line which is equal to the reciprocal of the drop base radius (K@ = l/r) for a spherical drop sitting on a flat and homogeneous solid surface. The line tension ( 7 s ~is~ associated ) with the threephase contact line and can be defied thermodynamically as
= (SFI~L),, (4) where F is the free energy of the system and T,V, A, and YSLV
Portion of Ihe lhree-phase contad line:
i'
conlaminant regions
Figure 1. A hypothetical corrugation of the three-phaee contact line.
L are temperature, volume, interfacial area, and length of the triple junction, respectively. The Cassie equation was derived on the basis of the Young equation, and thus modification of the Young equation (eq 3) leads ale0 to modification of the Cassie equation including the linetension term. Cassie expresaed that the energy (E) gained when the liquid spreads over a unit area of heterogeneous surface, composed of i different energetic surface regions, is and the contact angle (0) for the solid surface is as follows: COS e = E / Y ~ ~ (6) If the generalized Young equation is taken into consideration, the modified Cassie equation is For systems with a uniform radius of curvature ( ~=b llri) the modified Cassie equation (eq 7) can be expressed as
e
Cfi
ej - (1/YLv)CViYsLvi/ri)
(7a) and for surfaces composed of two component8 uniformly distributed it can be expressed as COB
(1)Caeeie, A. B. D.; Baxter, S. Trons. Forodoy SOC. 1944,40, 546. (2) Caeeie, A. B. D. Dkcws. Forodoy SOC.1948,9, 11. (3) Good,R. J.; Koo, M.N. J. Colloid Znterfoee Sei. 1979, 71, 283. (4) Boruvka, L.; Gaydoe, J.; Neumnnn, A. W. Colloida Surf. 1990,43,
COS
307. (5) Li,D.;Lin,F. Y.H.;Neumann,A. W. J. ColloidInterfoceSei.1991, 142, 224.
(6) Drelich, J.; Miller, J. D. Colloids Surf. 1992,69, 35. (7) S h a n h , M.E. R. Adv. Colloid Znterfoce Sei. 1992,39, 35. (8) Boruvka, L.; Neumann, A. W.J. Chem. Phye. 1977,66,5464.
(7b) Consideration of the corrugated three-phase contact line
0143-7463/93/2409-0619$04.00/0(B 1993 American Chemical Society
Drelich and Miller
620 Langmuir, Vol. 9, No.2, 1993
for a hypothetical heterogeneous surface composed of two types of alternating stripes led Li et al.5 to suggest an equation similar to eq 7b: cos8 = (rl& cos 8, + rue2cos 8,) '181
- (~/YLv)(YsLv~& - Y S L V Z ~ Z ) + rue2
(7c) where rlB1 and rd32 are the arc lengths for the corrugated three-phase contact l i e for surface phases 1 and 2, respectively. Li et al.6did not mention that ifthe fractional surface areas, expressed as
and
are substituted into-eq 7c, the modified Cassie equation (eq 7b) for this simplified system can be proposed. Of course, Li and co-workers5were close to the modified Cassie equation;however, they considered a rather simple system with the solid surfacecomposed of two types of alternating stripes and uniformly distributed over the surface. The general derivation of the modified Cassie equation, expressed by eq 7, is presented for the f i s t time in this paper. The line-tension values ( 7 s ~ predicted ~) theoretically, on the basis of the energy excess at the triple junction, are very small (from lo-" to 10-10 N as reported in the literaturegJ0). Experimental data show that the line tension can be even 1 order of magnitude larger (-lo+ N),11-13 Thus, it is expected that the corrugation of the three-phase contact line with radii of local deformation of less thanseveralmicrometerscan significantlycontribute to the contact angle, and eq 7 should be useful. On the other hand, for systemswith only the three-phase contact line corrugated at the macroscale with a radius of local curvature of hundreds of micrometers, generally the linetensionterm haenosignificantcon~butiontotheobserved contact angle, and eq 7 is reduced to the Cassie equation (eq 1). It seemsimportant to estimate the maximum dimension of the impurity, which will affect a contact angle through the line-tension term. Let us assume that a contribution of the line-tensionterm in the analysisof the contact angle will be observed when there is a change in the contact angle value of at least lo,A(cos 8) = cos 0 -cos 1 = 0.0002, due to the contribution of the line-tension term
where for simplificationwe assume that the radius of the local curvature for the primary material is much larger than for the impurity, rl >> r2, and also that the radius of (9) H a r k , W . D.J. Chem. phy8. 1987,5, 136. (10)Rowlineon, J. S.;.Widom, B. Molecular Theory of Capillarity; Oxford Science Publications: New York, 1984, p 240. (11)Schulze, H. J. Physico-Chemical Elementary Proce88e8 in Flotation; Eiwvirr: Amtardam 1984, pp 163,165,253. (12)Kralchevsky,P.A.;Nikolov,A. D.;Ivanov,I. B. J. Colloid Interface Sci. 1986. 112. 132. (13)Phtikkov, D.;Nedyalkov, M.; Naeteva, V. J. Colloid Interface Sci. 1980, 76, 620.
curvature is equal to half the dimension of the impurity (r2 = d2/2). Then
Thus, the maximum dimension of an impurity surface region depends on the fractional area of the impurity cfi), the line tension ( ~ S L V Z ) ,and the surface tension of the liquid ( ~ L v ) .Of course, the same dependence is valid when r2 >> rl but with parameters describing the continuous surface region (the primary material). Further, if some realistic parameters are selected, such as the line tension ~SLVZ = 5 X 10-loN,11-13 and fractional areaf2 = 0.1, then for two different surface tensions, say, */LV = 50 mN/m and ' ~ L V= 1 mN/m, the upper limits for the dimension of surface impurities, which would affect the changes in contact angle with respect to a contribution of the linetension term, are d2"
I10 pm for yLv= 50 mN/m
d2-
5 500 pm for yLv= 1 mN/m
For larger-sized impurities the line-tension term would not influence the magnitude of the contact angle as calculated from the Cassie equation (eq 2). Further, it is clear that the surface tension of a liquid in contact with a heterogeneous solid surface is a significant property of the system (in addition to the distribution and size of surface impurities). The lower size limit of the impurity dimension is more difficult to estimate. Neumann14estimated that there is no contact angle difference between the contorted and the smooth three-phase contact line for surfaces with impurity dimensionsof approximately0.1 pm. Also, when the dimension of impurities reaches the molecular dimension, the Caesieequation as well as the modified Cassie equation may not be valid, and Israelachvili and Gee15 postulated that the system should be described by eq 8. (1 + COS e? = fl(i+ COS
+ f2(i+ COS
(8)
Finally, two systems with extremely different distributions of heterogeneitiesare considered (Figure2). First, a smooth solid surface composed of two types of parallel strips,differingin energeticpropertiea (interfacialtension), isshowninFigure 2A. Aliquidincontactwitheachsurface strip forms contact angles B1 and 82, where > 82. This system can be described by the modified Cassie equation (eq 7b). Second, a smooth heterogeneous solid surface is assumed to be composed of two types of material differing in interfacial tension as shown in Figure 2B (the surface region of the primary material, the host material, and the surface region of the second component,shown as circular spots which represent the impurity at the surface). Impurities are randomly distributed over the solid surface, and as in the first model, a liquid in contact with each surface type forms contact angles 81 and 82. Ae shown in Figure 2B, the three-phaee contact line can cross a different number of impurity patches depending on its position. For a situation representedby a three-phase line in position 1,the modified Cassie equation can be expressed as follows: ~
~~
~
~~~
~~
(14)Neumann, A. W. Adv. Colloid Interface Sci. 1974,4, 106. (16)Israelachvili, J. N.; Gee, M. L. Langmuir 1989,5, 288.
Langmuir, Vol. 9, No.2,1993 621
Modification of the Cassie Equation
a
Figure 2. Nature of the three-phase contact line for different models of heterogeneous surfaces.
(9)
where f 1 is the average fractional area of the surface with a contact angle of 61, f 2 is the average fractional area of the surfacewith a contact angle of 02, /I* is the fractionallength of the three-phase contact line crossing the primary material, andfz* is the fractional length of the three-phase contact line crossing impurity patches; f1* < f1 and f2* > f2.
In the case for the three-phase contact line situated at position 2 (Figure 2B), there is no contact of this line with impurities, and thus, the modified Cassie equation is reduced to the original Cassie equation: COS e = fl COS e, + f2COS e, At heterogeneous surfaces with a random distribution of impurities, the three-phase contact line can also cross a smaller number of impurity patches than should be expected from the average fractional area of these contaminanta, and this situation is represented by the contact
line in position 3. The modified Cmie equation e x p r d by eq 9 describesthis case, but heref,* > r2, eq 9 can be reduced further as follows: COS 0 = fl cos e, + f2 cos 8, + f 2 * ~ s L v z ~ ~ (10) Lv~2 Evidently,severalcontact angle values can be measured for such heterogeneous surfaces. Thus, a scatter of experimental data for contact angle measurementa,often reported for contaminated surfaces,16can be explained, at least qualitatively at the present time, on the basis of the modified Cassie equation presented herein. Also,eq 7 can be of importance in the analysis of contact angle hysteresis for some systems. It is not our intention to discuss the applicability of the modified Caseie equation for theoretical prediction of contact angle hysteresis, in this paper, but only to mention this interesting point for the general relationship which has been derived (eq 7). The analysis of the contact angle hysteresis based on the modified Cassie equation requires much extended discussion, and it will be presented in a separate text" as well as in our next paper. By way of example, notice that for the system presented in Figure 2B, three different contact anglescan be measured (evenseveralcontactangles but for simplicity we consider only the three examples presented in Figure 2B), depending on the location of the three-phase contact line, as has been discussed. The smallest contact angle will be observed when the threephase contact line is located at position 1, and the largest when the three-phase contact line is located at position 2. For this hypothetical system the smallest contact angle can correspondto the receding contact angleand the largest to the advancing contact angle, obviously, if there are no other factors affecting the contact angle hysteresis. Evidently, the advancing contact angle is sensitive to the type and fractional area of randomly distributed impurities. On the other hand, the receding contact angle is also sensitive to the impurities' size and distribution. Of course, this brief discussion is quite simplified. The objective of this paper is to present for the first time a general relationship which should be useful in describing three-phase systems having surface heterogeneities and in understanding the scatter of experimentalcontact angle data observed in many laboratories. The validity of the modified Cassie equation presented herein has recently been confiied by thermodynamic considerations of a three-phase system involving a heterogeneous solid surface.lS Acknowledgment. We thank Dr. V. Hlady (Department of Bioengineering, University of Utah) for reading this paper and for making valuablesuggestions. Financial support from the U.S. Department of Energy, Contract Number DE-FC21-89MC26268, and the Polish Science Foundation, Grant Number CPBP 03.08,are gratefully recognized. (16) Drelich, J.; Miller, J. D.; Hupka, J. J. Colloid Interface Sci., in
press.
(17) Drelich,J. Ph.D. Dissertation,Universityof Utah,in preparation. (18) Drelich, J.; Miller, J. D. Part Sei. Technol., in press.
