Langmuir 1996,11, 627-634
627
From Droplet Growth to Film Growth on a Heterogeneous Surface: Condensation Associated with a Wettability Gradient Hong Zhao and Daniel Beysens" Service de Physique de l'Etat Condensk, Centre &Etudes de Saclay, F-91191 Gif-sur-Yvette Cedex, France Received July 25, 1994. I n Final Form: October 18, 1994@ Condensation experiments on a cold solid substrate with a gradient of contact angle (0) are reported. The gradient is sufficiently small so that the driving force due to the imbalance of the interfacial tensions at the contact line is negligible throughout most stages of the experiment. The condensation proceeds by forming spherical droplets near the hydrophobic side (0 100') and by forming quasi-films near the hydrophilic side (0 0"). In the middle part is the crossover region, where the condensation proceeds by forming islands of nonspherical shapes. It is found that the hysteresis of contact angle due to the pinning of contact lines by surface heterogeneityplays an important role in the growth of the condensation pattern. The effects of the wettability gradient and the contact-angle hysteresis on the nucleation rate, the growth rate of islands, the dynamics of coalescence, and the growth morphology are studied and discussed. An important result is that the growth remains self-similarwith the samegrowth law as for perfect hemispherical drops (with negligible pinning force), although the droplets exhibit complicated shapes.
-
I. Introduction Condensation of vapor on a cold surface is a daily experience to all of us. The phenomenon is also very important to engineering and technical problems, such as thin-film growth,'r2 heat t r a n ~ f e rand , ~ recovery of atmospheric ~ a t e r . For ~ , ~this reason, heterogeneous condensation (including nucleation and growth) has been a n active field of study for a long time.6 Because the condensing liquid keeps contact with the solid in heterogeneous condensation, the surface properties of the substrate play a crucial role in the process of condensation. One of the very important surface properties is the wettability of the solid with the condensed liquid. The effect of the wettability on heterogeneous nucleation is manifested through the contact-angle dependence of the nucleation energy barrier.6 This in turn results in the contact-angle dependence of the nucleation rate. Another interesting effect in heterogeneous condensation is the change of growth morphology when the wetting condition is changed. It has been known that, depending on the surface condition, condensation of liquid can happen by forming either droplets or thin films.7 The main object of this work is to study the effects of surface properties on heterogeneous nucleation and growth. It has been known that if the liquid wets the solid completely, condensation proceeds by forming thin films. Otherwise, condensation proceeds by forming droplet^.^ Droplet growth on a partial-wetting substrate where the contact angle is around 90"has been studied extensively in the past few years by several groups7-11 and is now Abstract published in Advance ACS Abstracts, December 15, 1994. (1)Lewis, B., Anderson, J. C., Eds. Nucleation and Growth ofThin Films; Academic Press: New York, 1978. (2) Venables, J . A.; Spiller, G. D. T., Handbucken, M. Rep. Prog. Phys. 1984,47,339. (3)Rose, J . W.,Glicksman, L. R. Znt. J. Heat Mass Transfer 1973, 16. 411. (4)Nikolayev, V.; Beysens, D.; Milimouk, I.; Gioda, A.; Katiushin, E.; Morel, J.-P., preprint, 1994. (5)Jumikis, A. R. Soil Sci. 1965,100, 83. (6)Sigsbee, R. A. in Nucleation; Zettlemoyer, A. C., Eds.; Marcel Dekker: New York, 1969. (7)Beysens, D., Knobler, C. M. Phys. Rev. Lett. 1986,57,1433. (8)Beysens, D.,Steyer, A., Guenoun, P., Fritter, D., Knobler, C. M. Phase Transitions 1991,31,219,and reference therein.
relatively well-understood. The condensation is governed by the growth of individual droplets and the coalescence of droplets when they touch. When a balance between the two processes is reached, the droplet pattern reaches a temporal self-similar regime in which the average surface coverage is a universal constant which only depends on the dimensionality of the system. The observed value for 3D droplets on a 2D substrate is approximately 0.55,* which is close to the value of the 2Djamming limit obtained from simulations of random sequential adsorption.12J3The typical size of droplets in this interesting regime ranges from a few pm to about 3 0 0 ~ m Due . ~ to the large contact angle and small radius, the restoring (capillary) force is strong enough to keep the droplets spherical even though the defects on the substrate tend to pin the perimeters of the drops when they fuse. In some previous work on "breath figure" (a term used to describe the condensation pattern on a partial-wetting substrate), the effect of wetting condition was studied by comparing condensation growth on substrates of different ~ e t t a b i l i t i e s . ~ JBut ~J~ as far as we know, the effect of surface heterogeneity in terms of contact-line pinning on a gradient surface has not been studied for the breath-figure problem. When the complete-wetting limit is approached, the pinning of contact lines cannot be ignored. Motion of contact lines (including the spontaneous spreading of a wetting drop) on a heteorgeneous surface has drawn considerable interest in the past But the study of the macroscopic effect of surface heterogeneity on the condensation phenomenon has been lacking. As we will see in the following, the growth and coalescence of droplets in condensation are strongly affected by the presence of
@
(9) Fritter, D., Knobler, C. M., Beysens, D. A. Phys. Rev. A 1991,43, 2858. (10)Briscoe, B. J., Galvin, K. P. Phys. Rev. A 1991,43,1906. (11)Family, F., Meakin, P. Phys. Rev. A 1989,40,3836. (12)Hinrichsen, E. L.,Feder, J.,Jgssang, T. J.Stat. Phys. l986,44, 793. Schaaf, P., Talbot, J. Phys. Rev. Lett. 1989,62, 175. di Meglio, J.-M., Gandeboeuf, P. J.Chim. Phys. 1992,89,1357. de la Parra, R. E. Microscopy Res. Technique 1993,25,362. de Gennes, P. G. Rev. Mod. Phys. 1985,57,827. Joanny, J. F., de Gennes, P. G. J. Chem. Phys. 1984,81,552. Robbins, M. O.,Joanny, J. F. Europhys. Lett. 1987,3, 729. r, J. F., Robbins, M. 0. J. Chem. Phys. 1990,192, 3206. Nadkarni, G. D.,Garoff, S. Europhys. Lett. 1992,20,523. di Meglio, J.-M. Europhys. Lett. 1992,17,607.
0743-746319512411-0627$09.00/00 1995 American Chemical Society
628 Langmuir, Vol. 11, No. 2, 1995
surface heteogeneity. Condensation on a substrate with a continuous change of wettability displays a crossover region in which the hysteretic effect of contact-line motion dominates t h e process. The interesting question is, how does t h e transition from droplet growth to film growth happens on a heterogeneous surface with a wettability gradient? What roles does the hysteretic effect play in determining the dynamics and morphology of the growth pattern? It is our purpose to perform experiments to obtain some preliminary answers to these questions. Under a gradient of contact angle, the interfacial tensions near the contact line of a drop along t h e direction of the gradient are imbalanced. There is a net force acting on the droplet in the direction of decreasing contact angle. A drop will move along this direction if certain conditions a r e met. The situation has been theoretically studied by BrochardZ1without considering the pinning force due to surface heterogeneity and experimentally demonstrated by Chaudhury and Whitesides.22 But as will be shown below, the droplets do not move at all during most stages of a condensation experiment (with t h e only exception being at an extremely late stage when the size of the droplets at the nonwetting side is close to 0.5 mm). Since t h e driving force (which is the net force of the imbalanced interfacial tensions at the contact line and the dissipative viscous force) is proportional to the drop size,21it is not hard to understand why movement is observed only when a drop is large enough. Since we are interested in stages when the drop size is normally less than a few hundred microns, we will assume that the droplets are stationary in the following discussions except when particularly mentioned.
11. Experimental Section The setup for the condensation experiment is typical for studying breath figures and has been described in details el~ewhere.~,g Here we only outline some basic features. The condensaticn assembly consists of a Peltier-element thermostat (to lower the temperature of the substrate) in a closed Plexiglas chamber. Filtered Nz gas saturated with water at room temperature (typically22 "C)is sent to the chamber in a quiescent way (to avoid complex hydrodynamic effect) at a controlled flow rate. The substrate is placed on the thermostat with a thermocouple next to it to monitor the temperature. The experimental procedure consists of cooling the substrate down t o the desired temperature with the chamber filled with dry N2 first and then sending in the saturated vapor to condense on the substrate. Each experiment was performed under a constant flow rate and constant (100%) supersaturation. The growth of condensation pattern was observed by an optical microscope and recorded by a CCD camera on video tapes. The video data are analyzed by a digital-image-processing system. The solid substrates that we used were oxidized silicon wafers. The methods of generating a gradient surface were proposed by several p e ~ p l e We . ~ followed ~ ~ ~ ~ a method similar to that used by Chaudhury and Whitesides.22 The principle is to expose the substrate to the diffusing front of a silane vapor and let the silane molecules react with the substrate. The resulting substrate is coated with a layer of silane molecules with a gradient of concentration, which in turn results in a wettability gradient with water. A wafer was cut into the dimensions 10 x 20 mm and cleaned thoroughly24before silanization. A piece of Teflon block with a trough of dimension 33 x 3 x 2 mm was used to hold (21)Brochard, F.Langmuir 1989,5, 432. (22)Chaudhury, M.K.,Whitesides, G. M. Sciences 1992,256,1539. Welin, S., Askendal, A,, Nilsson, U., Lunstr6m, I. (23)Elwing, H., J . Colloid Interface Sci. 1987,119, 203. (24)The procedures we used for cleaning the Si wafers are (1) in trichlorethylene at 60 "C for 5 min, (2)in acetone for 5 min, (3)in ethanol for 5 min. (Steps 1-3 are performed in an ultrasonic cleaner), and (4)in a newly made solution of HzOz and HzS04 (2:8)for 30 min. At the end of each step, the wafer is rinsed thoroughly in running deionized-distilled water. The final step is to dry the wafer by blowing clean NP gas on it.
Zhao and Beysens
solid circles: ea open circles: Or
lo
, -
4
d
0
0.7
0.1
50-d
0
0.8
n
I
I
I
I
I
2
4
6
8
10
12
v 14
G 16
0 18
0.0 20
Figure 1. Advancing contact angle (solid circles) and receding contact angle (open circles)vs the distance,X, to the hydrophobic edge of the substrate (sessile-drop measurement for the silicon wafer silanized for 1 min). The squares are the hysteresis strength, H, as defined in eq 6. the silane and the wafer. Fifty microliters of decyltrichlorosilane [C13Si(CH2)&H31was put in the trough. Acleanwafer was placed 0.5 mm from the edge of the trough. The width of the wafer (10 mm) was parallel t o the length of the trough and they had the same middle line. Decyltrichlorosilane is volatile. The vapor diffuses and reacts with the wafer surface, on which a concentration profile of silane molecules is imprinted. The whole assembly was covered in a 10 cm polystyrene Petri dish to prevent the silane vapor from being disturbed. The allowed time of exposure was from 1to 5 min. After silanization, the wafer was rinsed in warm distilled water and dried in blowing Nz gas. The contact angle ofthe silanized silicon wafers was measured by the sessile-drop method: a small drop of water (-0.05-0.5 pL) was introduced onto the wafer surface by the tip of a microsyringe and observed by a video camera with a macrolens. On the hydrophobic side, the droplet was held stationary by the syringe tip during the measurement (otherwise the droplet will move in the direction of decreasing contact angle due to the imbalanced interfacial tensions.22r25 The advancing (receding) contact angle was obtained by adding (removing)water to (from) the droplet through the microsyringe. The advancing and receding contact angles were actually measured when the contact line was just about to move (Le. under quasi-static condition). Figure 1shows the results ofthe contact-angle measurement for a silicon wafer silanized for 1 min. It is clear to see that there is a smooth but nonlinear change of contact angle (from around 90" to a few degrees) over a distance of approximately 12 mm. The gradient of contact angle near the hydrophobic side was approximately 1lo/" and decreases gradually toward the hydrophilic side. In general, the gradient of contact angle decreases with increasing silanization time (for 5 min silanization, it was approximately 7"/mm),but the contact angle increases as the silanization time is increased (for example, the advancing and receding angles at the hydrophobicedge for 5 min silanization were 115" and 95",respectively). Notice also that the hysteresis in contact angle always exists (normally between 10" and 17") even though we have been very careful in preparing our gradient wafers. Ellipsometry measurements showed that the average layer thickness of the silane molecule was about 3 A near the hydrophobic side of the wafer and dropped to a value below 1A (at which the ellipsometer cannot detect reliably) about 6 mm from the hydrophobic side of the wafer silanized for 1min. The average silane-layer thickness varies from 10 to 2 A over a 10 mm distance for the 5 min silanization. Since the length of a silane molecule is -17 A, the result indicates that the density of the silane molecules is below that of a monolayer and the gradient of contact angle is due to the gradient of silane density. Atomic-force microscopy indicated that there was no apparent
Langmuir, Vol. 11,No. 2, 1995 629
Droplet Growth to Film Growth
0
1 minute
5 minutes
14 minutes
Figure 2. Typical examples of growth patterns of water condensing on a cold silicon wafer with a gradient of contact angle (1 min silanization). Each column represents the patterns observed at four different locations on the substrate a t roughly the same time as indicated (within 15 s). The angles marked in the graph are the average contact angles at each location. The width of the photos is 385 pum. island of silane molecules of lateral size greater than 500 A on all the silanized silicon wafers. We found that the growth of condensation pattern on a gradient substrate was very sensitive to the surface condition (e.g., the contamination by fat). Although we were very careful in cleaning the wafers and keeping them from contamination, we discovered that the surface was contaminated after a condensation experiment. So we only used a newly processed (cleaned and silanized followingthe same procedures) silicon wafer in each experiment to insure that the surface condition was approximately the same. After the experiment, the wafer was discarded. We also did the contact-angle measurement from newly processed wafers.
111. Results and Discussion A. General Description of Observation. In order to observethe whole length of the substrate with the optical microscope,we have to scan our sample (actually the whole condensation assembly) under the microscope objective during a condensation experiment. Since the condensation process that we studied was rather slow, each scan represents only a small time span over the time scale of the experiment. For other purposes, we fmed at a particular location and followed the condensation process. We observed and recorded the growth pattern from the beginning of an experiment to the very late stage when water started to flow under the influence of gravity. Heterogeneous nucleation theory6 shows that the nucleation energy barrier is a function of the surface energies. In terms of the wetting property, the nucleation rate of droplets of critical size increases with decreasing
contact angle. As we will see in detail later, we did observe that more droplets nucleated near the wetting side of our gradient wafers than the nonwetting side at the start of a n experiment. Accurate measurement of the nucleation rate can thus provide a method for characterizing the wetting condition. Following the initial nucleation, the growth of the condensation pattern proceeds very differently at different regions on a gradient surface. Figure 2 shows examples of typical patterns obtained on a gradient silicon wafer after 1min silanization. Different morphologiesof growth patterns are clearly visible at different regions on the substrate: (a)spherical drops grow near the hydrophobic edge, (b)increasingly nonspherical and irregular drops develop as one goes toward the hydrophilic side, and (c) growth becomes filmlike near the hydrophilic side. These basic features remain qualitatively unchanged with time as the condensation process goes on in an experiment. Now we discuss the above features in more details. 1. Droplet Growth. Near the hydrophobic side (typically 8 > 70°), spherical droplets are formed. They grow and coalesce following the same rules as if the wettability gradient did not exist during most time of the experiment. When two or more droplets coalesce, a larger spherical drop forms at the same center of mass. As we will see later, the growth of droplets follows the well-known l13 power law with time at the early stage and changes to linear growth with time later.8 The local surface coverage rate saturates at a value around 0.55 when a self-similar
Zhao and Beysens
630 Langmuir, Vol. 11,No. 2, 1995 regime is reached (just as that observed on substrates with 90"contact angle without a gradient8). The droplets did not move at all during most stages of a n experiment, except immediately after a coalescence event at the very late stage. The immobility of droplets indicates that the driving force due to the gradient is balanced by the pinning force due to surface heterogeneity. We can qualitatively understand why a droplet of size below a certain value is immobile by comparing the difference between the advancing and receding contact angles (normally 2 10")and the difference in contact angles a droplet would have over its length along the gradient direction if there was no hysteresis. For a typical wafer we used, the gradient of contact angle was 10"/mm. Then a droplet of size less than 1mm only sees a contact-angle difference which is certainly too small to overcome the contact-angle hysteresis. Contact-angle hysteresis is thus an important factor when considering the mobility of droplets on a gradient surface. 2. Filmlike Growth. At the region where 8 < lo", the condensation proceeds by forming quasi-films. By quasifilms we mean that the pattern starts out at individual nucleated sites and grows as very thin pancakes until a real hole-free film is eventually formed. Fringes of thinfilm interference are clearly visible at early stage in this region. The film thickness estimated from the fringe distance was 0.4-1.0pm at the early stage of growth. The contact angle was also estimated from the interference pattern to be in the range of 0.6-6", in agreement with the results of the sessile-drop measurement. The growth of a film is very irregular and unsteady. Some parts apparently grow faster than the others. As the film grows, the contact angle apparently increases with time (as can be seen from the decrease of fringe distance). So the thickness of the film grows faster than its lateral size. This indicates again that the contact lines were not able to move freely due to the pinning force of surface heterogeneity. 3. Hysteresis-Dominated Cross-Over Region. In the intermediate region (typically 70" > 8 > 10") is the crossover from the spherical droplet growth to filmlike growth. In this crossover region, the effect of contact-line pinning is manifested mainly through the courses of coalescence, the main features of which are a fast partial coalescence followed by a slow relaxation process. When two drops touch, the liquid near the contact point feels a high capillary pressure (because of a small radius of curvature). So the liquid rearranges itself locally to reduce the capillary pressure by forming a n elongated island. This process is very fast (normally within the time scale of a video frame, i.e. l/25 s). This process causes a reduction in contact angles at the rest of the contact line since the total volume is conserved. Although the drop is still in a nonequilibrium configuration, the longated island does not relax immediately because of the pinning force on the contact line. But instead, the elongated island relaxes slowly (on the time scale of many seconds) until the capillary force is balanced by the pinning force. This means that the aspect ratio of a n island resulting from a coalescence process reduces slowly after coalescence. Figure 3 shows a n example of the features associated with a coalescence event in the crossover region. The top photo shows two droplets (A1 and A2) just before coalescence. The middle photo shows the coalesced droplet (A) 40 ms later. The bottom photo shows the same droplet 36s later. After several successive coalescence events, the shape of the drop becomes irregular, but the aspect ratio still remains low because of the slow relaxation process. Of course, the overall shape of the coalesced drops varies with the position along the gradient direction. The
t =34s
Figure 3. An example of partial coalescencein the hysteresisdominated crossover region. This is a time sequence from top to bottom. The droplets A1 and A2 in the top photo were just about to coalesce. A in the middle photo 1/25 s (onevideo frame) after the top one was the coalescenced droplet. A' was the same one 34s later when the aspect ratio stops to decrease any further. The width of the photos is 267 pm. morphology change of condensation pattern thus provides a macroscopic manifestation of the microscopic effect of contact-line pinning due to the surface heterogeneity. B. The Effects of Wettability Gradient. 1. Nucleation Rate. The energy barrier for heterogeneous nucleation depends on the wetting condition of the solid substrate. As Shown by Sigsbee,6the nucleation rate is a function of the contact angle through the following expression:
where A is a constant,
=
2 - 3 COS e 4
+ cos3 e
and the other parameter in the exponential is given by
B=
16m?
3(AGJ2kT
(3)
Droplet Growth to Film Growth ' ~ ' ~ ' " 1
I
'06
3
.
loo
_.*.'
'.,
I 1 0.0
I
1
I
I
I
0.1
0.2
0.3
0.4
0.5
where o is the surface tension of the condensing liquid, K is the Boltzman constant, T is the temperature of the substrate, and AGv is the free energy change per unit volume assocated with condensation, which is given by
u
P pe
-
10'
9
kT
,
1
-
10
0.6
Figure 4. Local number density of droplets (M at the early stage of condensation (circles) vs 4. The solid lines are the locally fitted curves according t o eqs 1-4. The dashed line has the same slope as predicted by eq 1. The dotted-dashed line is 0 vs 4. Note the right ordinate (0) is only read with the dotted-dashed line.
AGv=--1n-
" ' ~ ' ' ' 1
k-
,,/"
,,." ,/'
1o2
'
e=so:
triangles: 8=400 squares: e = i 5
lo3
_.I'
circles:
(4)
where u is the molecular volume of liquid water, and P and P, are the water vapor pressures a t the' steam temperature (far away from the substrate) and substrate temperature, respectively. Note that when 8 = 0")4 = 0 and no energy barrier to nucleation is present. In fact, as noted by de Gennes,16one has then to take into account the energy of the contact line (line tension). Owing to the limitation of the resolution of the video microscopy (the size of the smallest droplet that we can discern was about 5 pm), we could not observe the real nucleation period of condensation. Instead of measuring the real nucleation rate, we actually measured the number of droplets per unit area ( N ) at the early period of condensation in order to compare it with the above theory. The legitimacy of doing this is that the number of droplets does not change significantly a t the early stage until coalescence becomes important (we restrict our measurement to 8 > 15" because coalescence happens early a t small contact angle). According to eq 1, In N = In A B+(8). In Figure 4 we plot N in a logarithmic scale versus 4. Our results indicate that the relation between In N and 4 is not linear in the whole &range. It seems that there is a sharp change of slope around 8 = 40". We fitted the high 8 and the low 8 parts separately and found the slopes to be -48 and -6.4, respectively. These values are 3 and 2 orders of magnitude higher than the prediction N m-l, T = 283 K, u = from eq 3 (taking o = 73 x 3x m3,P = 20 mmHg (22 "C),P, = 15 mmHg (15"C), we get -B = -0.05). Our measurement indicates that the nucleation rate may not behave as simply as predicted by eqs 1-4. It is striking that the nucleation rate ( N ) a t low contact angle seems overestimated or N a t large contact angle underestimated. This goes against the influence of heterogeneity or contact-angle hysteresis on 4, which should only increase N a t large 8. This result
102
1 o3 t (sec)
1 o4
Figure 5. Equivalent diameter,D,of islands at three different contact angles vs time. Reference lines with slopes and 1 are drawn for visual aid.
suggests that the above theory6 cannot describe properly the role of surface heterogeneity. In addition, one cannot rule out some subtle influence of the gradient in 8, which is ignored in the above theory.6 2. Growth Rate. After the initial nucleation period, the droplets begin to grow. In order to investigate the growth law of the condensation pattern, we measured the droplet size as a function of time a t different locations, i.e. at different contact angles. For the spherical droplets near the hydrophobic side, it is easy to measure their diameters. For the nonspherical droplets at low contact angles, we measure their equivalent diameter (diameter of gyration) according to D2 = (4/n)C,r?,where r, is the distance of a n area element to center of mass and n is the number of the area element inside a n island. Figure 5 shows the measured average equivalent diameter ( D ) versus time a t three different contact angles. The reference lines in the figure have slopes 1 and V 3 , respectively. At large contact angles, we can easily identify two growth regimes: the individual growth regime with power exponent V3 and the coalescence growth regime with power exponent 1. But at low contact angles, because of the high nucleation rate and small aspect ratio (vertical vs lateral dimensions), coalescence happens a t a much earlier time. For example, we can still see the trend of crossover from the slow growth regime to the fast growth regime for 8 = 40")but for 8 = 15")only the fast growth regime is observable from the graph. Notice that the same power law exponent of the fast growth regime is obtained with and without hysteresis, which indicates that selfsimilar growth still proceeds although the drop shape is complicated (see below). 3. Motion of Droplets. We did observe some effect of the gradient a t the very late stage when the size of the larger droplets a t the hydrophobic side was getting close to 0.5 mm. Even at this stage, individual droplets still did not move,25but as two droplets coalesce, the newlyformed droplet did not form at the location of the same center of mass. Instead, the center ofmass always moved toward the hydrophilic side, the correct direction as predicted by the imbalanced interfacial tensions.21 The coalescence process looks as if the drop which is closer to the hydrophilic side remains still while the other one which (25) A droplet of size around 1 mm in diameter did move with a speed of a few millimeters per second on our gradient wafers when introduced near the hydrophobic side by a syringe, as was first observed in ref 22.
632 Langmuir, Vol. 11, No. 2, 1995
Zhao and Beysens
>
i t
I
I
I
I
i I
Figure 6. An example of asymmetrical coalescence on a gradient wafer a t a very late stage of condensation. The droplets A1 and AZin the top photo were just about to coalesce. After they coalesce, the newly formed droplet (A)(bottom photo) had its center of mass moved toward the hydrophilic direction (to the left). Notice also that the droplets A2 and B were the results of two previous coalescence events with their centers of mass moved to the left also. The width of the photos is 3.1 mm.
size) surpassingthe time of growth (which is proportional to the droplet size).27 In our case of water condensation, we never observed the morphology change until the size approaches the capillary length (-2.7 mm for water), when gravity becomes influential. We purposely increased the flow rate and the temperature supercooling to the highest available values (120 L/h and 20 "C,respectively), but we still could not see the morphological transition. The long coalescencetime in the metal case is due to the low surf'acediffusion rate, as the coalescence was interpreted as the result of the capillarity-induced surface diffu~ion.~~-~O In the water case, the coalescence time is much shorter than the growth time due to high capillary flow. In fact, we did not see any observable increase in the coalescence time as the droplet size increased. Although at low contact angles (e.g. 30")coalescence is hindered by the presence of contact-line pinning, there is a very fast partial coalescence followed by a slow relaxation. So the aspect ratios of the coalescedislands always remain low compared to the percolation structure. In the following, we discuss some morphological properties of the growth pattern and their relations with the contact-angle hysteresis. 1. Droplet Shape. As we have already seen in the preceding discussion, the droplet shape changes from spherical at the hydrophobic side to increasingly nonspherical as the contact angle is lowered. One way of characterizing a nonspherical shape is to calculate its circularity defined as
where P is the perimeter and A is the area. For a circle, this quantity is unity, and for a n irregular shape, its value decreases as the shape becomes more nonspherical. The most interesting part is the middle crossover region where the islands are irregularly shaped. The circularity decreases after a coalescence event because the contact line is not free to move. After successive coalescence events, the circularity of the island decreases further, but because of the partial coalescence and the slow relaxation, the decrease of circularity stops at a certain value which depends on the location. So the final value of circularity is a n incomplete memory of the coalescenceevents, which depends on the competition between coalescence and contact-line pinning. Only the first generation of droplets was used in the calculation of the circularity. An average has been taken over a narrow band whose width along the gradient direction is shorter than the length over which the circularity changes significantly. The decrease of circularity with respect to 8 is a result of the increasing contact-line pinning. Although we do not know how to quantify the strength of the hysteretic effect accurately, we can follow the spirit of de Gennes16 by defining the following quantity as the strength of hysteresis:
is further from the hydrophilic side bumps into it and forms a larger drop at the same position as the first one. Figure 6 shows an example of the asymmetrical coalescence of two large droplets. The center of mass of droplets A1 and A2 in the top photo moved to the hydrophilic direction after the coalescence and formation of the new drop, A. The droplets A2 and B were the results of two previous coalescence events with their centers of mass shifted to the left (notice the wipe-clean regions on the right side of each droplet). Of course, if the two initial droplets are lined up perpendicular to the gradient direction, we observed the usual symmetrical coalescence; i.e. the position of the center of mass remains unchanged. C. PatternMorphologyandthe HystereticEffect. It is known that, for metal-film growth on a nonwetting substrate, the growth morphology changes from an earlystage breath-figure morphology26to a percolation morphology when a critical droplet size is r e a ~ h e d . ~ The ~.~~ where 8, is the advancing contact angle and 8, is the morphology transition was interpreted as the coalescence receding contact angle. It is clear that the hysteretic effect time (which is proportional to the 4th power of the droplet becomes stronger as one goes from the hydrophobic side to the hydrophilic side since 8, - 8, does not change very (26) Beysens, D., Knobler, C.M., Schaffar, H. Phys. Rev. B 1990,41, 9814. much with position while 8, 8, decreases continuously. (27) Jeffers, G., Dubson, M. A., Duxbury, P. M. J.Appl. Phys. 1994, Figure 7 shows the inverse of the circularity versus the 75, 1; Duxbury, P. M., Dubson, M., Yu, X., Joffers, G. Europhys. Lett. ~~
+
1994,26, 601. (28) Yu, X., Duxbury, P. M., Jeffers, G., Dubson, M. A. Phys. Reu. B 1991,44, 13163.
(29) Nichols, F. A. J. Appl. Phys. 1966,37, 2805. (30) Brailsford, A. D., Gjostein, N. A. J.Appl. Phys. 1975,46,2390.
Droplet Growth to Film Growth 5
I
I
I
I
I
Langmuir, Vol. 11, No. 2, 1995 633 ~
-
l
1 .o
I
I
1 t-+-+lo'
0.9 4
0.8 0.7
T
3 3
I
u
0.6
"w
2
-3
0.5
0.4 0.3
1
0.2
0.1 I
I
I
I
I
I
I
I
I
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
I 0
I
I
I
I
I
2000
4000
6000
8000
10000
H Figure 7. The inverse of the average local circularityofislands
vs the hysteresis strength as defined in eq 6.
above defined hysteresis strength. It is clear that the circularity decreases a s H increases, but how to quantitatively relate the observed pattern morphology with the hysteretic phenomenon needs better characterization of the surface heterogeneity (both mechanical and chemical), which was outside the scope of the present study. Nevertheless, we have shown that by studying the variation of the shape of growth patterns, one can obtain a t least some macroscopic representation of the microscopic contact-line-pinning effect. 2. Surface Coverage and Self-SimilarRegime. One of the most interesting things in the breath-figure problem is the presence of a self-similar-growth regime when a balance between growth and coalescence is When there is a strong hysteresis effect, such as in our experiment, the coalescence is hindered and becomes incomplete. Whether the self-similar-growth regime can still be reached in this case is a n interesting question. From the above visual observation of the growth pattern (e.g. Figure 2) and the measurement of the growth laws (Figure 51, the self-similar growth was highly plausible. In order to verify it, we calculated the average local surface coverage rate, c2, as a function ofthe local average contact angle a t different times. We calculated e2 in a narrow band whose width (along the direction of the gradient) is shorter than the length over which the pattern morphology changes substantially. If there were no hysteresis, this quantity would be independent of the contact angle, which means that we should recover the universal saturation value 0.55. But when there exists hysteresis in contact angle, contact lines are pinned and full coalescence is prevented. Thus the surface coverage rate is increased. Figure 8 shows c2 as a function of condensation time at several different locations, i.e. contact angles. It clearly indicates that, no matter what the contact angle is, the surface coverage always saturates at a value (which depends on the hysteresis strength) after some time. This is the hallmark of the temporal self-similar growth. The increased surface coverage is certainly a direct result of the contact-line-pinning effect. Although the hysteretic effect makes the drop shape very complicated, it appears that self-similar growth is preserved as long as the coalescence of domains leaves a free surface which balances the growth of the islands. Figure 9 shows the long-time (saturated)surface coverage, cm2,as a function of contact angle, 8. Included in the figure is also the value
t (second) Figure 8. Local surface coverage, €2, vs time at different
locations (contact angles). 1 .o
" 8
w
0.7
t " " " " I
1
P P
simulation - no hysteresis ................................ 0
10
20
30
40 50 60 70 80
0
090 100
(degree)
Figure 9. Long-time (saturated) surface coverage, vs contact angle, 8. The dashed line is the result of simulation
without contact-anglehysteresis (from ref 9). Note that 8
H-' since ea - er is approximately a constant.
-
from computer simulation which does not consider the hysteretic e f f e ~ t . Since ~ 8, - 8, is approximately a constant, we have 8 H-l. Note that the limiting values em2should be unity a t large H and 0.55 a t small H. Our results obviously agree with these limits. The variation of em2between these limiting values seems to obey cm2 In H. We have no obvious reasons for this behavior. We have tried to construct a simplified geometrical model to describe our findings. We assumed that the droplets were spherical caps with radius R and contact angle 8 which cover 55% of the substrate in the ideal case, i.e. when there is no hysteresis. The increase of surface coverage is due to the reduction of 0 and thus the incrase of R (because ofvolume conservation) in the presence of surface heterogeneity. This oversimplified model only produces the correct trend of the variation of cW2but not the exact form. Our work suggests a challenge for future work on the breath-figure problem. Notice also the time a t which saturation is reached (Figure 8): the hydrophilic side saturates faster than the hydrophobic side because of the lower contact angle.
-
634 Langmuir, Vol. 11, No. 2, 1995
IV. Conclusions We performed heterogeneous condensation experiments on silicon wafers which displayed a continuous change of contact angle from beyond 90" to a few degrees and a hysteresis in contact angle (10-17"). The transition from spherical droplet growth near the hydrophobic side to filmlike growth near the hydrophilic side is governed by the contact-line-pinning effect mainly through the process of coalescence,which is characterized by a slow relaxation following a fast, imperfect coalescence. The imperfect coalescence produces the special pattern morphology. We showed that the variation of the shapes of the condensedwater islands to related to the strength of the contactangle hysteresis. We demonstrated that even though the dynamics of coalescence and thus the growth morphology are strongly affected by the presence of the hysteretic effect, the regime of the temporal self-similar growth is
Zhao and Beysens preserved although the saturated surface coverage is significantly increased depending on the hysteresis strength. In addition, our measurement of the droplet density a t early time indicates that the heterogeneous nucleation theory needs to properly incorporate the effect of surface heterogeneity and the subtle influence of the contact-angle gradient in order to correctly explain the experimental results.
Acknowledgment. We are particularly indebted to C. M. Knobler for suggesting this work to us. We have benefited from helpful discussions with J.-P. Bouchaud and J.-M. di Meglio. We are grateful to G . Zalczer for the ellipsometry measurement, D. Chatenay for the atomicforce-microscopemeasurement, and C. Chaleil for providing the silicon wafers and the clean-room facility. L4940588L
