Langmuir 1995,11, 2820-2829
2820
Influence of Evaporation on Contact Angle C. Bourgks-Monnier and M. E. R. Shanahan” Centre National de la Recherche Scientifique, Centre des Matkriaux P.M. Fourt, Ecole Nationale Supkrieure des Mines de Paris, B.P. 87, 91 003 Evry Cedex, France Received January 23, 1995. I n Final Form: April 6, 1995@ The evaporation of sessile drops of water and n-decane placed on various substrates has been studied using a projection method. Drop dimensions and contact angle have been measured as a function of time. The first stage ofthe experiment corresponds to a saturated atmosphere;then, when evaporationis allowed to occur, two or three different stages appear (dependingon the surface roughness). For the first of these, a model is proposed which allows us to calculate the coefficient of diffusion of the liquid vapor in air.
Introduction The variability of a contact angle, or wetting hysteresis, originates from various phenomena, among which roughness1 and chemical heterogeneity2 of the substrate are the most commonly referred to. One can detect the difference between advancing, 6*, and receding, OR, angles by, for example, the Wilhelmy plate or sessile drop methods. Even under static conditions, a “memoryeffect” exists depending on the last motion of the wetting front. With mass transfer during evaporation, the phenomena are analogous. During evaporation from a meniscus, an initially advancing contact angle will tend toward a lower ~ a l u e . Saturation ~,~ of the atmosphere around the drop is therefore necessary to measure a significant value of eA.5
Picknett and Bexon4 followed the mass and profile evolution of organic liquid drops. They observed three distinct evaporation modes: mode 1, during which the solidAiquid interface area remains constant (when hysteresis exists);mode 2, for which the contact angle remains constant (ideal system with no hysteresis a t equilibrium); mode 3, which is a mixed mode. They also observed that evaporation follows the first mode until O = OR, and then, the second mode is initiated. Moreover, they pointed out a decrease of evaporation rate with increasing initial contact angle. Birdi and Vu6-*studied the mass and contact diameter of water and n-octane drops placed on glass and poly(tetrafluoroethylene)(PTFE). Their results showed that the weight loss (glmin)of wetting drops ( 6 < 90’) is linear with time and the contact diameter constant. Conversely,
* Corresponding author: telephone, + (33)160 76 30 21/22; fax, + (33) 1 6 0 76 31 50; e-mail,
[email protected].
Abstract published in Advance ACS Abstracts, J u n e 1, 1995. (1)Johnson, R. E.; Dettre, R. H. Contact Angle Hysteresis. I. Study of an Idealized Rough Surface. Adu. Chem. Ser. 1964,43,112-135. (2) Johnson, R. E.; Dettre, R. H. Contact Angle Hysteresis. 11. Study of an Idealized Heterogeneous Surface. J . Phys. Chem. 1964,68 (7), 1744-1750. (3) Penn, L. S.; Miller, B. Advancing, Receding, and ‘Equilibrium’ Contact Angles. J . Colloid Interface Sci. 1980,77 (21, 574-576. (4) Picknett, R. G.; Bexon, R. The Evaporation of Sessile or Pendant Drops in Still Air. J . Colloid Interface Sci. 1977,61 (2), 336-350. ( 5 ) Holmes-Farley,S. R.; Reamey, R. H.; McCarthy, T. J.; Whitesides, G. Acid-Base Behaviour of Carboxylic Acid Groups Covalently attached at the Surface of PE. Langmuir 1985,1 (61, 725-740. (6) Birdi, K. S.; Vu, D. T.; Winter, A. A Study of the Evaporation Rates of Small Water Drops Placed on a Solid Surface. J . Phys. Chem. 1989,93,3702-3703. (7) Birdi, K .S.; Vu, D. T.; Andersen, S. I.; Winter, A,; Topsoe, H.; Christensen, S. V. Determination of Surface Properties of Porous Solids. In Characterization ofporous Solids II; Elsevier: Amsterdam, 1991;pp 151-160. ( 8 ) Birdi, K. S.; Vu, D. T. Wettability and the Evaporation Rates of Fluids from Solid Surfaces. J . Adhesion Sci. Technol. 1993,7(61,485493.
for nonwetting liquids, linearity is no longer the rule; the contact angle remains constant while the diameter decreases. Kamusewitzgand Yekta-FardlO measured contact angles of drops in different atmospheres and showed 6 to vary with the surrounding phase. Other authors have worked on evaporation effects. Peissl’ followed the evolution of spherical drops and showed the evaporation rate to vary linearly with the radius when it is small but to become a curve with increasing r . Morse obtained similar results12 for the sublimation of iodine spheres and obtained concave (toward the abcissa) mass, m, vs time, t, curves and a linear variation of dmldt with the radius as long as the mass remained smaller than 2 mg. Whytlaw-Gray,13on the other hand, measured a constant value of dmldt for various spherical liquid drops. Finally, Kuz14expressed dmldt as a parabolic function of r for spherical drops under certain conditions. In the present study, we chose the sessile drop method to follow the dimensions and contact angle variations of different liquid drops placed on various substrates in saturated or nonsaturated atmospheres. A preliminary m 0 d e 1 ~successfully ~J~ explained some of our observations. We report here a more developed model that allows us to calculate the diffusion coefficient of the vapor of the liquid considered in air.
Experimental Section The substrates studied were an epoxy resin (914, Ciba Geigy), polyethylene (PE),poly(tetrafluoroethy1ene) (PTFE),and glass microscope slides. The epoxy was used with two different states of surface, rough (Ra = 1.4mm) and polished (Ra = 0.1 mm), and the microscope slides were used as received (Ra = 0.02 mm). As
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(9) Kamusewitz, H.; Possart, W.; Paul, D. Measurements of SolidWater Contact Angles in the Presence of Different Vapors. Int. J . Adhesion Adhesives 1993,13 (41, 243-249. (10) Yekta-Fard, M.; Ponter, A. B. The Influences of Vapor Environment and Temperature on the Contact Angle-Drop Size. J . Colloid Interface Sci. 1988,126 (11, 134-140. (11)Peiss, C. N. Evaporation of Small Water Drops maintained a t Constant Volume. J . Appl. Phys. 1989,65 (121, 5235-5237. (12) Morse, H. W. On Evaporation from the Surface ofa Solid Sphere. Proc. A m . Acad. Arts Sci. 1910,XLV (141, 363-367. (13) Whytlaw-Gray,R.; Patterson, H. S. The EvaporationofDroplets. In Smoke: A Study of Aerial Disperse Systems; Arnold, E., Ed.; 1932; pp 168-179. (14) Kuz, V. A. Evaporation ofsmall Drops. J.Appl. Phys. 1991,lO (15), 7034-7036. (15) Bourges, C.; Shanahan, M. E. R. L’influence de 1’6vaporation sur l’angle de contact des gouttes d’eau. C.R. Acad. Sci. Ser. 11 1993, 316,311-316. (16) Shanahan, M. E. R.; BourgBs, C. Effects of Evaporation on Contact Angles on Polymer Surfaces. Znt. J . Adhesion and Adhesives 1994,14 (31, 201-205
0743-746319512411-2820$09.00/0 0 1995 American Chemical Society
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Influence of Evaporation on Contact Angle
Langmuir, Vol. 11, No. 7, 1995 2821 3.5
m o a 00a 0 00c
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I
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Figure 1. Evolution of drop height, h, contact diameter,d , and contact angle, 8, for a water drop of initial volume 4pL on a polished epoxy surface (sample box closed). for polished epoxy, diamond paste was used to polish the two other polymers (Ra was not measurable because of the pressure applied by the profilometer). Two liquids were used: water and n-decane. The water was triple distilled, and the n-decane was of analytical grade. The range of solid surfaces therefore includesdifferencesin roughness, chemical nature, and polarity, while the liquids correspond to a small polar molecule (water) and a large apolar molecule (ndecane). Before each experiment, the surfaces were immersed in pure ethanol in an ultrasonic cleaner for 10 min and then dried for an hour at 100 "C. The following experimental procedure was adopted for most experiments,although some variants, described below,were also investigated. The solid sample was placed in a hermetic box with two parallel glass slides enabling the drop to be observed by projection (magnification 90 x ) during its evolution. Initial vapor saturation within the box was assured by including an open dish of the relevant liquid at 2 or 3 "C above the ambient temperature (ca. 25 "C). A liquid drop of volume in the range 1-15 p L was placed on the surface in the advancing mode with a microsyringe. The three parameters of interest were drop height, h, contact diameter, d , and the contact angle, 8. After following the evolution of a drop for 20 min in the saturated atmosphere, the vapor source was removed. In the case ofwater as the probe liquid, a dish of silica gel was put in the hermetic box. The three parameters h, d , and 0 were then followed until the drop finally disappeared.
Results and Discussion It was observed that, qualitatively, all solidAiquid pairs behaved in a similar manner. An example ofthe evolution with t of 8, h , and d for a water drop of initial volume 4 yL placed on polished epoxy resin is shown in Figure 1. The initial contact angle, @A, is 61". For water, four stages can be distinguished. Stage I takes place while the surrounding atmosphere is saturated: a small decrease of h and also of 0 can be seen while d remains constant. This small effect can possibly be attributed to a n imperfect saturation of the atmosphere
in the case of a glass substrate. Imbibition of water can also be evoked for polymeric surfaces (the possibility of reorientation due to interactions between water and polar molecules5J7J8 is unlikely because of the slight disappearance of liquid shown by d remaining constant). To check this hypothesis, we repeated the experiment on samples of the epoxy resin initially exposed to a hot dry atmosphere (using silica gel in a n oven at 55 "C) or water (at 55 "C) for 2000 h. The initial contact angle was found to be the same for dried substrates but slightly smaller (-55") for those immersed in water. This presumably corresponds to surface evolution because of the presence of water and absorption. The basic shape of the curves of 8 vs t does not change, but for dried samples, the decrease of 8 is more marked because of slight spreading (simultaneous increase of d and decrease of h ) during the first few minutes of stage I. After stage I, the atmosphere is no longer saturated because of the replacement of water by silica gel. The height and contact angle decrease more quickly than in the first stage, while d still remains constant: this corresponds to stage 11. In the example shown in Figure 1,the cover of the box was replaced immediately after the waterlsilica gel exchange, and thus, there exists a transition before h starts to decrease linearly. We attribute this transition to a "drying time" controlled by the absorption kinetics ofwater vapor by silica gel. Whenever the box remained open, the change of gradient was found to occur more rapidly (Figure 2). The diameter and the contact angle decreased linearly. At the final part of stage 11,O (for water) was found to be ca. 51" on PE, ca. 15" on the rough epoxy, and ca. 18" (17) Andrade,J.D.;Gregonis,G.E.;Smith,L.M.AModelforContact Angle Hysteresis. PhysicochemicalAspects of Polymer Surfaces 1985, 2, 911-922. (18)Shanahan, M. E. R.; Came, A,;Moll, S.; Schultz, J. Une nouvelle
interpretation de l'hysterhse de mouillage des polymhes. J. Chim. Phys. 1986, 83 (5), 351-354.
BourghMonnier and Shanahan
2822 Langmuir, Vol. 11, No. 7, 1995
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Figure 2. As for Figure 1 but with initial drop volume of 3 p L and sample box open.
or 13" on the polished epoxy depending on whether the box was closed or left open, respectively. The difference in the last case is possibly due to some residual water vapor not being absorbed by the silica gel in the time of the experiment with the closed box. These values may be compared with OR, the "conventional"value of the receding contact angle as obtained by mechanical withdrawal of liquid from the drop. &was similar for PE (ca. 55'1, larger for polished epoxy (ca. 30'1, but almost zero for rough epoxy. The reasons for the differences in the last two cases are not entirely clear but perhaps are related to the fact that mechanical movement of the triple line during the withdrawal ofliquid from the drop involves some shear resistance (albeit slight).16 This resistance would favor a slight displacement of the triple line instead of an angle reduction. In the case of evaporation, there is no shear. This could possibly explain the behavior on the polished substrate. Presumably the reverse is occurring on the rough epoxy where asperities correspond to strong anchoring points. In stage 111, h and d decrease simultaneously so that 8 remains almost constant and corresponds to a receding angle. This zone was found only to exist on smooth surfaces (with water: 8 51" (PE), 6 18" (polished epoxy), 8 78" (PTFE), 8 10" (glass)). Figure 3 shows the typical behavior of water on rough epoxy. Finally, h , d , and 8 tend to zero (6 far below OR), with the drop disappearing. The parameters now have very small values, and the speed of evaporation is high; this renders the measurements difficult and stage IV hard to interpret. This stage is probably influenced by the surface state ofthe solid and anchoringeffectsofthe triple line.19,20 In a n attempt to check that observed behavior is not due to artifacts, some supplementary experiments have
-
--
-
(19)Joanny, J. F.; de Gennes, P. G. A Model for Contact Angle Hysteresis. J . Chem. Phys. 1984, 81 (11, 552-562. (20) Shanahan, M. E. R. Meniscus Shape and Contact Angle of a Slightly Deformed Axisymmetric Drop. J . Phys. D 1988, 22, 11281135.
been effected. If the experimental conditions of stage I are maintained for the total time of an experiment (e.g., saturation during 2 x lo3, 3 x lo3,or 4.5 x lo3 s for 1-, 4-, or 12-pL drops respectively on polished epoxy), h, d, and 8 remain constant after a possible small initial decrease. This confirms that the observed phenomena are due to evaporation. It is also possible to try to return to stage I (by reheating the liquid in the closed box) after having started stage 11. In this case, 8 and h continue to decrease a t the same rate for a while during restoration of the atmosphere and then decrease more slowly. The restoration of saturated atmosphere is therefore not rapid enough to prevent evaporation immediately. Another configuration has also been studied. When the substrate faces downward, we place the drop on the underside. The trend does not change. This indicates that convection, although present,21does not represent the main phenomenon. In order to verify if the observed evolution applied only to water or was more general, we used another liquid. n-Decane is a large apolar molecule. Experiments were carried out on PTFE in order to get a significantly large contact angle. The evolution is very similar to that of water; stages I and I1 are essentially the same, but stage I11 presents a new characteristic (Figure 4)and can now be divided into two zones: stage I11 as described before; stage 111' where h, d, and 8 decrease step by step. The decrease of d is accompanied by a slight increase of h (and hence of 8 ) during a step. However, the final angle of a step is always smaller than it was a t the end of the previous step; this implies an overall decreasing tendency. The overall variation of 8, however, remains relatively low during stages I11 and 111', so the average is a receding angle. During stage IV,the drop can disappear either with a (21) O'Brien, R.N.;Saville, P. Investigation of Liquid Drop Evaporation by Laser Interferometry. Langmuir 1987, 3, 41-45.
Influence of Evaporation on Contact Angle w
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t (8)
Figure 4. Evolution of drop parameters h, d, and 8 for a drop of n-decane of initial volume 3 pL, on PTFE.
decreasing or with a stable value of d, but in every case, h (and hence 0) goes on diminishing. Phenomena responsible for stage 111' have not been totally elucidated. The n-decane drop behaves as if it rapidly reaches the limit of stage IV,but then the normal evolution stops; the drop contracts slightly and returns to the last configuration where evaporation starts again in the same conditions as before. We attribute these effects to slight surface roughness.
We can compare our results of stage I1 to those of Picknett4 and Birdi,6-8 notwithstanding any variability possibly attributable to different surface states (glass, in particular, can show significant differences depending on the type and cleaning procedures). We shall define tII as the total time of stage 11: tqwateriglasa) lo3, 1.3 x lo3,and 2 x lo3 s and tII~w.t,,,,,thepory) 600, 2.2 x lo3,and 4 x lo3 s for 1-,5-, and 15-pL water drops, respectively. The initial contact angle on epoxy (ca. 60")
-
-
BourghMonnier and Shanahan
2824 Langmuir, Vol. 11, No. 7, 1995
2.6
0
OO 0 0 0
2
0
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i m m mm m m m i
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Figure 5. Evolution of drop of initial contact angle, 0 (sample box open).
> 90": water
is larger than on glass (35"). Thus, except for the first volume where values of t11are comparable, the tendency is that announced by Picknett: the average evaporation rate decreases with increasing initial contact angle. Birdi and Vu distinguished wetting and nonwetting liquids. Almost all of our experiments belong to the first case (8 < go"), and stage I1 shows effectively a decreasing value ofh a t constant d. However, water evolves on PTFE as on other substrates, and we get a decrease of d a t constant 8 only in stage I11 (Figure 5). The drops studied by Birdi (water, n-octane on PTFE and glass) were small and considered a s spherical caps. He deduced a n evaporation rate, I (g/min), proportional to the solifliquid interface radius. That is reasonable because evaporation occurs by gas diffusion as observed by Whytlaw-Gray.13 So a t constant volume, the smaller 8 is, the faster the rate is. We have already verified this property differently stated by Picknett and Bexon. Rayz2 observed also, although in different experimental conditions, that I increased with decreasing drop radius. For the glasslwater couple, Birdi found
I = -7
10-5
+ 9.2119
10-5
where d is in mm and I in glmin. When we calculate the evaporation rate gradient for our experiments and express it in the same way as Birdi (the radius is measured, so it is possible to calculate a linear regression between I and r = d/2),we get open box
I = -13.13
10-5
+5
10-5d 2
(22)Ray, A. K.; Lee, J.; Tilley, H. L. Direct Measurements of Evaporation Rates of Single Droplets at Large Knudsen Numbers. Langmuir 1988,4, 631-637.
on PTFE (initial drop volume 5 pL,legend as for Figure 1)
closed box
I = -2.32 x
+ 10.05 x 10-5d-2
Figure 6 summarises the results. The experimental results obtained using the closed box show a similar gradient to those obtained by Birdi. However, in the case of the open box, the gradient is only about half that value. It is, however, worth noting that a gravimetric approach is probably more sensitive for this aspect, particularly when small drops andor the later stages of evaporation are considered. Moreover, Birdi found evaporation rates of 11.75 x and 16.27 x g/min for initial volumes of 5 and 10p L , respectively. Our experiments show the same trend, but our values are almost twice as large. We have the following results for open and closed boxes and each initial volume: open box (g/min), (23.4 & 2.4) x 10-5-(26 f 1.9) x closed box (g/min), (24.1 f 0.6) x 10-5-(38.8 f 7.1) x Despite the factor of 2 between the two slopes (closed open box), we do not observe a very large differencebecause 8 and it can be we are examining small radii ( ~ 2 . mm) seen on Figure 6 that for these r values the two lines are not far from each other. Moreover,we get poor correlation coefficients, 0.3 and 0.6, respectively, which can explain the differences. It would seem that we do not get any significant difference depending on whether 8 is larger or smaller than 90". Birdi did not distinguish the different stages we observed, perhaps because he studied the drop in terms of mass rather than contact angle. As far as the evaporation rates are concerned, the order of magnitude is the same. Insofar as we have no details about Birdi's experimental conditions and we did not operate in the same way, the results are in satisfactory agreement.
Influence of Evaporation on Contact Angle
Langmuir, Vol. 11, No. 7,1995 2825
8
- I (Birdi) + lexp closed ++ I regression closed -m- I regression open I lexpopen
Figure 6. Evaporation rate vs solifliquid interface radius. Comparison of experimental results (sample box open and closed) and Birdi's equation.
at dynamic equilibrium). Taking the concentration gradient, C(R),of the vapor to be a function of R, we assume that the atmosphere is saturated a t R,, i.e., C(R,)= CO, and that C = 0 as R With these hypotheses, we may
-
00.
assume that the diffusion equation takes the
where D represents the (constant) diffusion coefficient. Using the above boundary conditions, we have
and the concentration gradient can be expressed as
(3)
Figure 7. Model for stage I1 evaporation: definition of the different parameters.
Model of Behavior in Stage 11. The phenomena describedabove seem to have been studied relatively little. To our knowledge, only Picknett and Bexon4 have published anything directly comparable. As a consequence, we have attempted to model the observed behavior, at least as far as stage I1 is concerned (this stage was always found to be present; the others are, anyway, more delicate to treat). A simple model has been presented previ0 u s 1 y , ~ ~but J ~ the following should be a more realistic approximation to the process. Figure 7 represents a drop of contact radius, r , height, h, and contact angle, e,, which corresponds to a spherical cap belonging to a sphere of radius R,. We shall assume that the diffusion of liquid vapor from the drop into the surrounding atmosphere, corresponding to evaporation, is purely radial (this will not be entirely true but should be a good approximation
where S' = 4nR2is the surface area of a sphere of radius
R. Equation 3 is really only valid for an entirely spherical drop, whereas we are considering a spherical cap. By analogy with eq 3, we write
-
dC - -k ----
dR
S
-12 2nR2(1- COS e)
(4)
(23) Crank, J. Diffusion in a Sphere. In Mathematics ofDifision, 2nd ed.; Oxford University: Oxford, 1975; Chapter 6, pp 89-103. (24) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (25) Slattery,J. C.;Bird, R. B. Calculationof the Diffusion Coefficient of Dilute Gases and of the Self-Diffusion Coefficient of Dense Gases. AIChE J. 1958,4 (21, 137-142.
Bourgks-Monnier and Shanahan
2826 Langmuir, Vol. 11, No. 7, 1995 where k is a constant and S represents the surface of a spherical cap, of angle 8, concentric with the drop (see Figure 7). Since R, cos 8, = R cos 8, we may express eq 4 as dC--
dR
-k 2 n R ( R - R, COS eg)
which is readily integrated to give
C ( R )=
-k 2xR, cos 6 ,
{ l n [ R - Rgcos e,]
- In R }
+ k’
(14) Two methods of application of eq 13 have been used. In the first, the final point of stage I1 was determined from the first sign of diminishing contact diameter, d (i.e., the onset of stage 1111, corresponding to time tf and contact angle f&. We calculate numerically a n equivalent value of J , Jf, using c1 a s obtained experimentally from the start of stage I1 andcz correspondingto cos 0 ~ It. is then possible to calculate
(6) where k’ is an integration constant. By using the above boundary conditions of C(-)= 0 and C(R,)= CO, we obtain the constants k (and 112‘ 1, and eq 5 may be restated a s
and then 8 as a function of time by linear interpolation. In the second approach, a statistical method is used. Using the experimentalvalue ofc1, we may calculate the expected value of Ji,for each experimentally exploited time, ti. We wish to minimize the sum of residuals
TakingA and Vrespectively a s the liquidvapor surface area and volume of the drop (both dependent on time, t ) , the mass balance for the evaporation process may be expressed as where K2 is the equivalent parameter to K1, above, but for the statistical treatment. It can readily be shown that the required value of K2 is given by where Q is the liquid density. Since all the drops considered are sufficiently small for gravitational flattening to be negligible, we have
n
i=l
V=
n r 3 ( l - cos 6,12(2
+ cos 6,)
3 sin3 e,
(10)
It should be recalled that stage I1 occurs a t constant r, so we have (11) Usingeqs 7 and 9- 11in eq 8 together with the fact that r = R, sin e,, we obtain, after simplification
( 1 - c)”’ In( 1 - e ) de, - 2DCo c ( l + cl3l2 dt Qr2
(12)
where c represents 8,. Equation 12 may be written alternatively as
where t is time elapsed since the onset of stage I1 and c1 and c2 represent respectively cos 8, a t t = 0 and at (variable) t. It would appear that the integral o f J i s not analytically possible, and so, this has been effected numerically. Equation 13 has been applied to the experimental results for stage 11. The values of 8, were obtained from averaging the three available measurements (for a given time, t): direct values obtained from the left- and righthand sides of the drop and a calculated value assuming drop sphericity using the equation
For each experiment, we obtain a graph of 8 (experimental), 8(Kl) and 8 ( ~ (where *) the latter are obtained by the methods described above for determining K1 and Kz) vs time, t. It is now possible to calculate the diffusion coefficient of water or n-decane vapors in air from the equation
(17) Since D and COare intrinsic constants for each system, we can hope thatDCo will be volume independent. Figures 8-11 show DCO as a function of V for the different substrates studied. Horizontal lines represent (weighted) mean values of DCO (smaller drops presented larger standard deviations and thus were given less weight in the averaging process). To within the limits of experimental error, we can say that effectively DCOis volume independent. Moreover, the method of calculation (KIor K2)has little influence on the final result (Figure 9). The less satisfactory agreement sometimes observed for smaller drops (see Figure 8) may be attributable to the reduced number of experimental points available (shorter evaporation time). Now that we have evaluated the product DCO, in order to estimate the diffusion coefficient, D, we need the saturation concentration, Co. The literature gives COfor HzO as 23.05 g/m3(at 298 K), but for n-decane, we must calculate the value. The saturation pressure, p s , of n-decane a t 293 Kis 2.7 mmHg orps = 360 Pa. The perfect gas law allows us to express the molar volume as a function of temperature, saturation pressure, and the perfect gas constant R. As the molar mass of n-decane is 142.28 g, it is possible to calculate Co(C1oHzz)= 21.02 g/m3, and as
lnfluence of Evaporation on Contact Angle
Langmuir, Vol. 11, No. 7, 1995 2827
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Figure 8. Product of DCo as a function of initial drop volume, V, for water drops on polished epoxy. 3.8
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Figure 9. As for Figure 8 but solid surface is glass.
a result, we obtain Dexp(H20/air)= (1.04 f 0.15) x m2/sandDexp(CloH2z/air) = (0.36 f0.01) x m2/susing KI andD,,,(HzO/air) = (1.05 f0.15) x loF5m2/sand Dexp(CloHzdair) = (0.38 & 0.01) x m2/s using Kz. We used here an average value ofDCo(H2Olair)obtained from all the various solidlliquid couples. In the case of n-decane, we only have one couple available. The choice of K1 or K2 has little influence on the final value of D. There exist in the literature different formulae which
allow us to calculate the diffusion coefficient for gas mixing. We have used those of Fuller2*and Slattery and Bird25 m2/s (Fuller), and obtained D(H2Olair) = 2.53 x m2/s (Slattery and Bird), D(H2Olair) = 0.78 x m2/s(Fuller), and D(CloH2d D(CloH2dair) = 0.58 x air) = 0.61 x low6m2/s (Slattery and Bird). These values may be compared to those obtained experimentally from our evaporation data. It can be seen that despite some variability the agreement is quite
2828 Langmuir, Vol. 11, No. 7, 1995
BourgBs-Monnier and Shanahan
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Figure 10. As for Figure 8 but solid surface is PE. 1.4
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Figure 11 As for Figure 8 but liquid is n-decane and solid PTFE.
reasonable. In the case ofwater vapor diffusion, our value is between the two obtained from the literature, whereas for n-decane, our value is apparently rather lower. Nevertheless, the order of magnitude is the same, and anyway, these data are really by way of a confirmation of the validity of our approach for explaining sessile drop evaporation and its effects on wetting hysteresis. We are not suggesting this technique as a n accurate method for determining vaporlair diffusion coefficients!
A possible source of error in our work could be insufficient saturation of the surrounding atmosphere. If the calculated value of COis overestimated, then
Dexp = is underestimated.
(DCOLXp ~
CO,calcd
(18)
Langmuir, Vol. 11, No. 7, 1995 2829
Influence of Evaporation on Contact Angle The problem is that we cannot measure COexperimentally. Moreover, our boundary conditions suppose that C = 0 ifR = 00, which is probably in error in the case of water because the surrounding atmosphere is never totally dry. Picknett4 explains that when small drops (respectively big drops) are evaporating. at atmospheric pressure (respectively a t low pressure), the mean free path of molecules becomes significant compared to drop dimensions, d , and h , and then the process is no longer entirely diffusion controlled. To know in which regime the drop is, one must calculate the Knudsen number, which must remain lower than 0.1 for pure diffusion. Unfortunately, the definition of this number includes parameters such as the diffusion tortuosity which remain unknown, so it is impossible to verify ifwe are in the pure diffusion regime. Other authors like K u z , ~ ~ , ~and ~ Peissll also take into account the thermal process linked to evaporation and the resulting temperature difference between the interior and the surface of the drop. Peiss also considers the saturation pressure a t the drop surface. Although we have not taken into accounts these refinements, we nevertheless obtain reasonable values of diffusion coefficients for water and n-decane in air.
Conclusion We have undertaken a study of the influence of the degree of atmosphere saturation in the vapor of a liquid on contact angles, 8. It has been shown that, to all intents and purposes, contact angles remain constant when the surrounding atmosphere is saturated in the vapor of the given liquid but that diminishing the atmospheric vapor content can lead to marked reductions, with, under certain circumstances, 8 tending to zero, as liquid evaporates. The process has been split into four stages. The initial (26) Kuz, V. A. Fluid Dynamic Analysis of Droplet Evaporation. Langmuir 1992,8, 2829-2831. (27) Lou, Y. S. On Nonlinear Droplet Condensation and Evaporation Problem. J . Appl. Phys. 1978,49 (4), 2350-2356.
stage corresponds to the saturated atmosphere, and stage 11, in which the contact radius remains constant while 8 and drop height decrease, ensues after reducing atmospheric vapor content. Stage I11 follows for smooth solids, in which drop height and contact radius diminish concomitantly, thus maintaining contact angle more or less constant. This stage is absent for rough substrates, and step evolution can be apparent in some systems. Stage IV corresponds to final drop disappearance, and although this is little understood (and difficult to study due to the small drop dimensions a t this point), it is probablyrelated to triple line anchoring phenomena on local heterogeneous zones of the substrate. The evaporation regime corresponding to stage I1 has been analyzed. After presenting a n elementary mode115J6 which proved partially successful (semi-quantitatively), here we suggest a more developed theory of the process depending on the diffusion of liquid vapor into the surrounding atmosphere. Not only does theory corroborate experimental results to a satisfactory degree of agreement, but also, as a corollary, values ofthe coefficient of diffusion, D,can be evaluated. Although clearly the method is not intented as a rigorous means of evaluating D,good agreement between the values we obtain and those available from the literature lend further credence to the analysis developed. Finally, although further refinements remain to be made in order to explain all aspects of observed phenomena related to evaporation (particularly on rough substrates), it is emphasized that these results show conclusively the importance of atmospheric conditions in contact angle measurements-a fact seemingly often overlooked in the literature.
Acknowledgment. We thank Dr. H. Burlet for her help in the numerical calculations and the Direction des Recherches, Etudes et Techniques (DRET), for financial aid (C.B.M.). LA950050V
