Langmuir 1995,11, 1041-1043
Simple Theory of “Stick-Slip” Wetting Hysteresis Martin E. R. Shanahan Centre National de la Recherche Scientifique, Ecole Nationale Supbrieure des Mines de Paris, Centre des Matbriaux P.M. Fourt, B.P. 87, 91003 EVRY Ckdex, France Received June 22, 1994. In Final Form: November 7, 1994
Contact angle measurement constitutes a sensitive method for the characterization of solid surfaces. While the simple theory describing the equilibrium of a small, axisymmetric sessile drop on a flat, horizontal, smooth, homogeneous, isotropic, and rigid (Le., ideal!) solid is straightforward with Young’s1 equation defining the (unique)value of the contact angle, e,, a t the triple line, it is generally found in practice that a whole range of contact angles, 8,is accessible experimentally. This effect of variability of 8, or wetting hysteresis, has been much studied both experimentally and theoretically over the years.2-8 Recognized causes of wetting hysteresis fit basically into two classes: chemical and physical. In the former class, such effects as chemical heterogeneity of the solid surface and dissolution and/or swelling of the solid by the liquid should be included. As for the latter, surface roughness, the existence of pores and asperities, molecular orientation, and surface strain are to be taken into account. The range of wetting hysteresis, generally corresponding to static measurements of contact angle, goes from the maximum, or advancing value, 8 A , down to the minimum, or recedingvalue, OR, while 8, is somewhere in between. Sometimes dynamic contact angles are considered (sliding drop, Wilhelmy plate, etc.), in which case 8 A is usually larger and 8 R smaller, thus increasing the hysteretic range. However, in the present paper, essentially static (quasi-static) values will be considered, as it is assumed that no motion is imposed on the system. The range 8 A - 8 R may be considerable, but for lowenergy polymeric substrates such as poly(tetraflu0roethylene) (PTFE) or poly(ethylene1 (PE), although the range exists, 8 R remains finite for most usual liquids. As stated above, measurements of 8 A and 8 R are generally effected either in the static mode, i.e. when the system is (at least apparently) a t rest, after motion has been imposed (e.g. after placing a sessile drop on a surface or arresting the motion of a Wilhelmy plate), or in the dynamic mode, in which case advancing and receding angles may be a function of the relative speed ofthe wetting front with respect to the solid surface. Nevertheless, if a sessile drop is simply allowed to remain on a substrate, without experimental precautions being taken, it may be observed that 8 will descend below the accepted static value of OR, tending to zero as the drop disappears by evaporation. This type of behavior is readily seen when a drop of water is spilled onto a (polymeric) table top and left, without being wiped up. Under other circumstances, there can be a significant difference between the accepted (1)Young, T.Phil. Trans. Roy. SOC.(London) 1806,95, 65. (2)Fowkes, F. M.; Harkins, W. D. J.Am. Chem. SOC.1940,62,3377. (3)Johnson, R.E., Jr.; Dettre, R. H. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley-Interscience: New York, 1969,Vol. 2,pp 85153. (4)Eick, J. D.;Good, R. J.; Neumann, A. W. J . Colloid Interface Sci. 1976,53,235. (5)de Gennes, P. G. Reu. Mod. Phys. 1985,57,827. (6)Schwartz, L.W.;Garoff, S.Langmuir 1985,1,219. (7)Pomeau, Y.;Vannimenus, J. J . Colloid Interface Sci. 1986,104, 477. (8)Shanahan, M. E.R. J.Phys. D.: Appl. Phys. 1989,22,1128.
1041
static value of and that observed during e v a p ~ r a t i o n . ~ Two main differences are apparent between the conditions associated with the “conventional”assessment of the static value of 8 R as described above and those pertinent to a free drop. Firstly, when motion of the triple line is imposed, even a t slow speeds, some shear resistance to flow must occur in the liquid mass, but more importantly, locally near the wetting front. Secondly, external energy is injected into the system when motion is imposed. Occasionally observed during the evolution of a drop is behavior of the triple line which may be termed “stickslip”, in which the wetting front remains static for most of the time, but from time to time moves quite abruptly. This may be observed, for example, with the slowly moving triple line of a sliding drop or during volume change by adding or removing liquid. The same type of behavior can sometimes be seen when liquid evaporates from a sessile drop, causing gradual lowering of the drop height, h, and contact angle, 8,while the contact radius, r, remains constant. In the last case, used as a basis for the model below, this reduction of h and 8 occurs smoothly over a (relatively) long time until B reaches a minimal value (possibly below the conventional eR) and then, quite abruptly, motion of the triple line causes r to decrease and both 8 and h to increase (at essentially constant drop volume since the process is rapid). The cycle is then repeated with a (relatively) long period a t constant (smaller) r followed by a quick jump of the triple line to a still lower value of r. The purpose of the present note is to propose a tentative explanation of this behavior and of why, under certain conditions, contact angles less than 8 R may be apparent. In the proposed model, certain simplifymg assumptions are made to render the mathematics tractable, although, in principle, the same physical arguments could be applied to other meniscusholid systems. An axisymmetric sessile drop of contact radius r is considered on a n essentially flat and smooth solid surface (any surface roughness leading to hysteresis discussed below is assumed to be on a scale small compared with r ) . The range of contact angles, 8, to be considered is less than 90” and the drop is assumed to be sufficiently small for gravitational potential energy to be negligible, so we may assume the drop to take the form of a spherical cap. Provided that the drop has dimensions smaller than the capillary length, K - ~= [ y / ( ~ g ) ] lwith ’ ~ , y , e, a n d g respectively being liquid surface tension, density, and gravitational acceleration, this assumption is reasonable (e.g., K - ~x 2.7 mm for water). The drop, depicted in Figure l a , starts to lose liquid by evaporation, as shown schematically in Figure l b et seq. Although the spherical cap form is retained, resulting changes in shape (viz. contact angle) lead to lack of capillary equilibrium and thus excess free energy. The anchoring, or hysteresis, sources leading to this effect may be chemical or physical, but it is taken that they are omnipresent and that the resulting scale of heterogeneity is small compared to drop dimensions. (If solid strain leads to hysteresis,lOJ1the distribution of anchoring effects may actually become continuous!) It will be assumed in the following that absence of capillary equilibrium is the sole source of energy giving rise to subsequent triple line motion. (In the case of a sliding drop, a drop whose volume is being modified by addition or removal of liquid, or any situation in which mechanical energy is being injected (9)Bourghs, C.; Shanahan, M. E. R. Compt.Rendus Acad. Sci. (Paris) 1993,316 (ZI), 311. (10)Shanahan, M. E. R. J.Phys. D.: Appl. Phys. 1988,21,981. (11)Shanahan, M.E. R.; Carr6, A. Langmuir 1994,10,1647.
0743-7463/95/2411-1041$09.00/00 1995 American Chemical Society
1042 Langmuir, Vol. 11, No.3, 1995
Notes
energy associated with line tension could be added to eq 3, but since its effect is not always evident, and in order not to overcomplicate the followingdevelopment, this will be neglected. Using eq 1, it may be shown that, a t constant drop volume:
-d _e - -sin dr
(4
(d)4b)
Figure 1. Schematic representation of “stick-slip” behavior of a drop during evaporation. From the initial state (a),both
the height, h, and contact angle, 8, decrease as drop volume diminishes,while the contact radius,r, remains constant.When a lower limit to 8 is obtained, the triple line “jumps”,as in part c, from radius r to radius r - dr, with concomitant increases in 9 and h. The cycle then repeats itself (d), with the new initial contact radius r - 6r. into the system, extra energy source(s) for triple line motion will be potentially available. This would complicate the treatment considerably. It is also worthy of mention that, in these “mechanical” cases, energy dissipation is likely due to the viscous motion of the liquid. This will be virtually absent during evaporation both since internal flow will be negligible and triple line motion will, in general, be much slower.) Bearing in mind the above hypotheses, we commence with two expressions for the drop volume, W, and the liquidhapor (spherical) interfacial area, A:
W = - nr3 (1 - COS el2 ( 2 3 sin3 e
+ COS e)
(1)
To within an additive constant, the Gibbs free energy of the drop, G, due entirely to interfacial free energies, may be written as
(3) where y = ~ L V the , liquidlvapor free energy, and y s and ~ ysv are the equivalent, average values for the solid/liquid and solidhapor interfacial free energies. (By “average”, we mean values taken with respect to “large” solid expanses, comparable to the contact area of the drop, while local features leading to hysteresis are ignored. If surface roughness is a t the heart of the hysteresis phenomenon, ~ S and L ~ S should V correspond to values corrected to allow for the difference between real and geometric areas.) The term involving ~ S and L ysv is simplified by using Young’s equation for the equilibrium contact angle, eo. It has been reported in the literature that line or pseudoline t e n ~ i o n ’ ~(allowing J~ for effects of surface imperfections) may play a role in that, in some systems a t least, the contact angle appears to be a function of the drop contact radius. In principle, a term including the (12)Ponter, A. B.; Boyes, A. P. Can. J. Chem. 1972,50, 2419. (13)Good, R. J.; Koo, M. N. J . Colloid Interface Sci. 1979,71, 283. (14)Gaydos, J.;Neumann, A. W. J . Colloid Interface Sci. 1987,120, 76. (15) Drelich, J.; Miller, J. D. Colloid Surf. 1992,69, 35. (16) Drelich, J.; Miller, J. D. Part. Sci. Technol. 1992,10,1
8 (2 r
+ cos e)
(4)
We consider the drop of radius r to be slightly out of equilibrium, such that the radius of contact of a n equilibrated drop of the same volume, r,, is given by r 6r and 00 = 8 + 68 (6r and 68 may be positive or negative, but for our purposes, we take them as positive). By Taylor’s theorem:
G(r)= G(r,
+ dr) =
and by virtue of the fact that [dGldrl,=,, = 0, since r, corresponds to the equilibrium contact radius, we may evaluate the excess free energy of the unequilibrated drop, compared to the equilibrium state, dG, making use of eq 3-5:
6G = G(r)- G(r,) x y n sin2 8, ( 2
+ cos f3,)(dr)2
(6)
As the drop evaporates, it would uprefer” to maintain the equilibrium value of contact angle, 8,) with a smoothly receding contact line, thus minimizing its free energy for its volume at a given moment. However, we may postulate that this is prevented since the triple line is “anchored” a t a value of r corresponding to a n earlier equilibrium situation (immediately following the previous jump, assuming any overshoot effects due to inertia to be negligible). We may associate with this anchoring effect, or hysteresis, a potential energy barrier of value U per unit length of the triple line. The exact nature of the cause of anchoring (chemical heterogeneity of the solid, roughness, etc.) is immaterial t o the present argument, but we assume that the effect may exist virtually anywhere on the solid surface. From eq 6, we may define a n excess of free energy available per unit length of triple line, 66: dzt. x
y sin2 e, ( 2
+ cos e,)(dr)’
2r
(7)
As evaporation progresses, dr, the difference between the actual contact radius, r (constant), and the corresponding equilibrium value, r, (decreasing since W is getting smaller), increases and, as a consequence, 6G increases as (&I2. When 6 6 attains the value of the potential energy barrier, U , sufficient energy is available to overcome the hysteresis barrier effect and the triple line jumps to its next equilibrium position a t r,. It is implicitly assumed here that inertial effects are sufficient to allow the triple line to ride over any possible metastable states existing for contact radii between r and ro,but insufficient to carry the triple line further which would mean raising 6’ above 8,. (In fact, 8, does not necessarily have to correspond to the absolute free energy minimum and may not therefore be ‘(the”equilibrium contact angle of the system. This is however assumed to simplify the mathematics.) In addition, it is assumed that the jump occurs sufficiently rapidly for the process to be considered as taking place a t constant volume.
Langmuir, Vol. 1 2 , No. 3, 1995 1043
Notes
From the above, we may see that
dre
[
y sin2e, (2
+ cos e,)
(8)
We would therefore expect thejumps between neighboring equilibrium states to occur with dr scaling as rv2. Another consequence of eq 8 is that, for a given value of r , the jump distance will depend on Uf2.Thus, for a n ideal solid, dr 0 and a smoothly receding triple line would be expected a t constant 8 = 8,. Conversely, for a system presenting an important hysteretic energy barrier, U,jumps may be large and relatively infrequent: the process may be predominantly “stick, with little “slip”. This last prediction could well explain another phenomenon referred to earlier, viz. that contact angles can sometimes be seen to descend below the classical receding value, OR, as drops evaporate to disappearance. Similarly, drops even of relatively large contact radius on a very hysteretic substrate can be made to exhibit extremely low values of 8, well below the “conventional” 8R. Let us consider a drop which has followed the (idealized) process described above, conserving its axisymmetry, with its contact radius, r, decreasing byjumps down to alimiting value, r,. We shall define rcas the ultimate, finite, radius ofcontact for which both drop height, h, and contact angle, 8, tend to zero during the final stage of evaporation to disappearance. The actual value of rcwill depend on the initial value of r and the way in which the series of its successive values converges, but we may set a n upper limit. (Under some circumstances, often with very rough substances, the jump process is never experimentally observed and 8 0 a t constant r . We may then take the initial contact radius to be r,.) Equation 3 gives us the equivalent total free energy of the drop/solid system, G,.
-
-
G, may be considered to be the excess free energy compared to the state in which the drop has totally vanished, in which case the maximum free energy available per unit length of triple line, G,,is given by
As 8 diminishes, so does 6,. Thus, if 6, U just following the last jump leading to radius r,, then the drop cannot henceforth have sufficient excess free energy to overcome the hysteretic, anchoring barrier retaining the triple line a t r,. The drop will disappear with 8 0 while the contact radius remains at r,. The simple model proposed accounts plausibly for stickslip motion of triple lines sometimes observed and the fact that 8 may tend to zero due to evaporation (or even mechanical retraction of the triple line on very rough surfaces), but clearly some refinements could be made. We have assumed 8, after a jump to be the Young angle, but it is quite feasible that a metastable value of contact angle would be more appropriate. Inertial effects have been overlooked as has the potential influence of liquid viscosity. No allowance has been made for effects occurring due to the possible presence of line tension. In the case of larger drops for which gravitational effects may not be neglected, a term involving the free energy related to the height of the center of gravity should be included in the analysis. All of these may be important in determining whether a given solifliquid system will display noticeable stick-slip behavior or whether an (apparently) smoothly receding triple line is a more appropriate description of observed behavior. Nevertheless, the basis could be used with suitable data to assess typical values for wetting hysteresis energy barriers.
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LA940501W
