Langmuir 1991, 7, 2756-2763
2756
Motion of Marangoni-ContractedWater Drops across Inclined Hydrophilic Surfaces J. A. M.Huethorst' and J. Marra Philips Research Laboratories, P.O. Box 80 000,5600 J A Eindhoven, The Netherlands Received March 4, 1991 A theoretical model is developed for describing the movement of Marangoni-contracted water droplets across inclined hydrophilic surfaces in a 1-methoxy-2-propanol vapor atmosphere. Due to Marangoni flows, these droplets assume a macroscopic static contact angle 0, = 14' on a horizontal surface,irrespective of the surfaceroughness.1 Based on the lubrication approximation, expressions are derived for the viscous force F, and the surface tension force F,.that oppose droplet movement when the surface is inclined. They can be evaluated when both the droplet shape and the velocity dependence of the dynamic receding and advancing contact angles 6dJ and 6 d R are known. In the limit of slight droplet deformations at small droplet speeds, the expressions simpiify and can be evaluated analytically. In particular, they allow the droplet speed u to be predicted as a function of experimentally measurable parameters. The equations are successfully applied for droplet movement on silicon surfaces of different roughnesses. The results shed more light on the hydrodynamic relevance of dynamic contact angles. Furthermore, they illustrate the role of the surface roughness with respect to droplet movement and Marangoni drying. 1. Introduction
In many industrially important processes, substrates have to be dried after a wet processing sequence. Hydrophilic surfaces, such as silicon wafers or glass plates, are usually dried with spin drying. The preceding paper' has outlined how this drying may be improved by withdrawing the substrates from water while a watersoluble organic vapor is directed onto the water meniscus against the substrate surface. Absorption of organic vapor induces a surface tension gradient within the meniscus. This gradient gives rise to a Marangoni flow from the meniscus perimeter, which causes the meniscus to contract and assume a finite contact angle. Meniscus contraction can conveniently be observed on a sessile droplet (see Figure 1). It is the surface tension gradient that enables substrate drying during withdrawal. Regarding the existence of an (apparent) contact angle, one may conclude that hydrophilic surfaces in an organic vapor environment become similar to hydrophobic (partially wetted) surfaces in behavior. It is known2that such surfaces can be dried at a given withdrawal speed from a liquid provided that the dynamic receding contact angle ed,R remains nonzero. In this respect, the maximum dewetting speed denotes the withdrawal speed where ed,R = 0. Similar principles apply when dealing with the removal of droplets from partially wetted surfaces. The minimum force Fmin required to remove droplets from such surfaces has been a subject of several experimental3* and theoretical7* studies. Investigations were carried out either by varying the angle of tilt of the substrates (gravity (1) Marra, J.; Huethorst, J. A. M. Langmuir, preceding paper in this issue. (2) Blake, T. D.; Ruschak, K. J. Nature 1979,282,489-491. (3) Wolfram, E.; Fauet, R. In Wetting, Spreading and Adhesion, Padday, J. F., Ed.;Academic Press: New York, 1978; pp 213-222. (4) Furmidge, C. G. L. J. Colloid Sci. 1962, 17, 309-324. (5) Extrand, C. W.; Gent, A. N. J. Colloid Interface Sci. 1990, 138, 431-442. (6) Goodwin, R.; Rice, D.; Middleman, S. J. Colloid Interface Sci. 1988,125, 162-169. (7) Duesan, E. B.;Chow, R. T.-P. J. Fluid Mech. 1983,137, 1-29. (8) Brown, R. A.; Orr, F. M.: Scriven. L. E. J. Colloid Interface Sci. 1980, 73,7647. (9) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J.Colloid Interface Sci. 1984, 102,424-434.
Spread droplet
I
fA\
Organic vapour ,Be
Figure 1. Water dropson a hydrophilicsurface. (A) Fullyspread water drop in air takes the shape of a thin pancake. (B)The initially spreadwater drop contacts when a water-solubleorganic vapor is allowed to absorb into the water. Due to a Marangoni flow from the droplet perimeter, the drop assumes an equilibrium contact angle 8,. force) or by varying the speed of a rotating platter (centrifugal force). Some confusion exists about the precise state of the droplet just before it starts to movess However, most experimental data are found to be consistent with the equation which was first formally derived by Dussan and Chow.' Here, ~ L isV the surface tension of the liquid and w is the maximum width of the deformed droplet perpendicular to the direction of the applied force Fmin. Bs,R and dah denote the static receding and advancing contact angles, respectively. Equation 1 shows that the reason for the necessity of applying a finite force F m h to initiate droplet movement lies in the existence of a contact angle hysteresis &,A - os,^. In particular for water, contact angle hysteresis is virtually always encountered on all partially wetted surfaces of practical interest. Its origin lies in the presence of chemical inhomogeneities and/or roughness profiles on the surfaces. Essentially, eq 1 gives a surface tension force (Young force) that arises from the nonequilibrium curvature of the droplet surface near its perimeter. Unfortunately, it is not a predictive equation since the width w of the droplet is not known a priori. A theoretical study into this matter
0743-7463/91/2407-2756$02.50/00 1991 American Chemical Society
Marangoni-Contracted Water Drop Motion
was initiated by Dussan et al.7JO Once droplet movement is initiated, its shape becomes determined by, among other factors, dynamic contact angles (Le., dynamic wetting pheomena) instead of static contact angles. The droplet loses its integrity and leaves a trail of liquid as soon as 6d.R becomes Despite considerable efforts, the dynamic behavior of moving sessile droplets across surfaces remains poorly understood. Research into this areal1-l5has faced major difficulties: Dynamic wetting involves both hydrodynamic and molecular forces. Besides, these wetting phenomena occur within a very small scale, which makes them almost inaccessible to direct observation. As yet, no agreement exists among researchers as to whether an observed contact angle is apparent or real.15 The foregoing paper' demonstrated that wetting phenomena on hydrophilic surfaces moreover depend on Marangoni effects: Water droplets with an apparent finite contact angle arise on these surfaces when a stream of a water-solubleorganic vapor is directed onto a water-wetted substrate. It is expected that the effectivity and the speed with which such droplets can be removed will also in this case depend on dynamic wetting phenomena (in particular, the magnitude of &a).An interesting aspect of Marangonicontracted water droplets on hydrophilic surfaces is that the magnitude of their static contact angle is independent of the chemical nature of the surface or its roughness. This inference is relevant for the drying of hydrophilic surfaces in an organic vapor environment, because of its implications for the contact angle hysteresis. The present paper is mainly concerned with dynamic contact angles of water droplets on hydrophilic silicon surfaces in a 1-methoxy-2-propanol vapor atmosphere. This vapor was found to induce relatively large contact angles. An experimental study was undertaken on the dependence Of Bd,R and ed,A on the droplet volume, droplet speed, and surface roughness when the droplets slide across a tilted silicon surface under the influence of gravity. Based on theoretical work in refs 16 and 17 that successfully employed the hydrodynamic lubrication approximation to describe the spreading process of fully wetting droplets, mathematical expressions are derived that relate the surface tension forces and viscous forces, which counteract droplet motion, to experimentally measurable data. An evaluation of the derived set of equations in terms of experimental data is useful to investigate the hydrodynamical significance of the macroscopic contact angle of a Marangoni-contracted water droplet.
2. Theory Figure 2 schematically depicts a Marangoni-contracted water droplet on an inclined hydrophilic substrate. The droplet with mass m moves with a constant speed u down the substrate surface. The magnitude of u must follow from the balance of the driving gravitation force Fg and the resisting viscous and surface tension forces Fr that arise from the liquid flow within the droplet. The directions of the forces Fgand Fr are indicated in Figure 2. A mathematical expression for each of these forces will be derived by considering only flows within the droplet (10) Dussan, E. B. J. Fluid Mech. 1985, 151, 1-20. (11) Dussnn, E. B.; Davis, G. H. J. Fluid Mech. 1974,65, 71-95. (12) Duesan, E. B. Annu. Rev. Fluid Mech. 1979,11, 371-400. (13) Hansen, R. J.; Toong, T. Y. J. Colloid Interface Sci. 1971, 37, 196-207. (14) Mu-, W.; Hens, J.; Boiy, L. AIChE J. 1989,35,1521-1526, nnd
references listad therein. (15) Ngnn, C. G.; Dusean, E. B. J. Fluid Mech. 1982,118, 27-52. (16) De Gennes, P. G. Reu. Mod. Phva. 1986.57.827-863. (17) Brochnrd, F. Langmuir 1989,5,-432-438.
Langmuir, Vol. 7, No. 11, 1991 2757
I'
Figure 2. Marangoni-contracted water droplet moving with a speed u across an inclined surface in the x direction. 4 denotes the angle of inclination. The driving force for movement is the resolved component of the gravity force Fgalong the substrate surface. Droplet movement is opposed by a resistance force Fr, which comprises the sum of the viscous force F, and the surface tension force F,.. Receding and advancing perimeters are located at XI and xp, respectively, with dynamic contact angles B u and The local thickness of the droplet h(x) is indicated.
that arise from droplet motion. Superimposed Marangoni flows are not explicitly accounted for. (A) Gravity Force. The magnitude of the gravity force Fg depends on the inclination angle 4 of the substrate according to
Fg = mg sin 4 = Vpg sin 4 (2) whereg is the gravitation acceleration, p the liquid density, and V the droplet volume. (B) Viscous Force. The viscous force F, results from flow resistance within the droplet due to the liquid visc0sity.1~It is equal to the total viscous stress exerted onto the solid/liquid interface a t z = 0 in the direction opposite to droplet movement. To derive an equation for F, for a three-dimensional droplet, it is instructive to first consider the viscous forcefv on the two-dimensional droplet in the ( x , z ) plane in Figure 2. This moving sheet of water may also be considered as the cross section of a liquid strip (along the y direction) on the surface. The viscous force fv per unit length in the y direction follows from f, = L y u x y ( x z, = 0) dx
(3)
where x 2 - x1 represents the length of the droplet (see Figure 2). The viscous stress uxUon the x , y plane is a function of the flow pattern within the droplet. For small contact angles, the flow pattern can easily be derived if the lubrication approximation is used. According to ref 17, this approximation yields uzy(x,z = 0) = 377u/h(x)
(4)
with h(x) the height of the droplet along the twodimensional contour of the droplet and 7 the liquid viscosity. Thus
Equation 5 shows that the viscous force becomes infinite at the droplet perimeter where h(x) = 0. To remove this singularity, an ad hoc hydrodynamic plane of shear must be introduced at h ( z )= h-. The magnitude of h h must be comparable to molecular dimensions. Only that part of the droplet where h(x)> hmin can move with respect to the substrate. f v can be split up according to (6) = fv,per + fv,bulk where fv,per denotes the viscous stress near the droplet perimeter, and fv,bdk denotes the viscous stress in the bulk fv
Huethorst and Marra
2758 Langmuir, Vol. 7,No. 11, 1991
K
of the droplet. To derive an expression for fv,per, h(x)near the receding perimeter is written as
h(x) = ( x -Xl)ed,R x1 5 5 + xm, (7a) (with x- the distance from the perimeter where the macroscopic dynamic receding contact angle 8d,R is measured) and h(X) = ( X 2 - x ) 8 d A
X2
- Xmax 5 X 5 X 2
(7b)
near the advancing contact line. Hydrodynamic slip occurs at some small distance Xmin = hmin/B from the contact line for both 8 = 8d,R and 8 = e d A . Thus, fv,per follows from the equation xi+%-
fvmr
dx
sz-x-xmi.
dx
(xz -
-
+ s,,U4U
Generally, both 8d,R and 8 d A will be a function of u. In case no contact angle hysteresis exists, 8d;R = 8 d A = 8,;i.e., the static contact angle for u = 0. Usually, the prefactor In (XmaJXmjn) is assumed17J9 to be a constant of the order 10-15. The viscous stress in the bulk of the droplet fv,bdk can be calculated approximately by considering a flat drop (Le., a puddle) of thickness ho = 2 ~ - sin l (6,/2). Here, K is the Laplace length: K~ = pg/y, with p the liquid density and y the surface tension. For water, K - ~ = 2.7 mm. If 8, remains small, ho = K-18, and
It follows from eqs 8 and 9 that f,, gives the dominant contribution to fv as long as (x2 - x,p”) (16) The total surface tension force Fyon a three-dimensional droplet in the direction opposite to Fg is found by integrating the Young components fy cos cp along the total contour length 1 of the droplet perimeter according to the contour integral fy e
7L.V
Fy =
f, cos cp dl
(17)
Since dl = dyjcos cp, eq 17 combined with eq 16 reduces to
In fact, the physical origin of Fyfor a Marangoni-contracted drop must lie in the departure of the surface tension gradients around the moving droplet perimeter from "equilibrium" as found during vapor absorption in a motionless droplet. For a drop moving with a constant speed, the balance between the forces must be
Fg= F, + Fy
(19)
from which, in principle, the droplet speed u can be calculated.
3. Experimental Section All experiments on the motion of water droplets on inclined hydrophilic surfaces were carried out on 6411. silicon wafers (Wacker Chemitronic Co., Burghausen, F.R.G.). These wafers were cleaned in a 25% NHdOH/BO% H202/H20 solution (Merck, MOS-Selectipur) in a 1:1:5 volume ratio at 60 "C, rinsed with filtered, deionized (DI) water, and dried in a Semitool spin dryer. They were additionally treated in a UV/ozone reactor (UVP Inc.; PR-100) for 10 min just prior to use to ensure that the surfaceswere truly hydrophilic. This latter treatment was carried out after each experiment. For the experiments, a silicon wafer was placed inside a small Perspex chamber (17 X 17 X 6 cms) on a tilted table (see Figure 4). The angle of tilt of this table could be varied between 1.5' and 15'. When a larger tilt was required, the complete chamber was tilted. To obtain an organic vapor atmosphere inside the chamber, nitrogen gas (with a flow of 10 L/min) was bubbled through an aspirator bottle containing 1-methoxy-2-propanol liquid. The emerging N2/l-methoxy-2-propanol vapor mixture was directed via an inlet into the bottom part of the chamber. This indirect vapor supply avoids local fluctuations of the organic vapor flow. Vapor was able to escape the chamber only via an outlet in the center of the lid, thus ensuring an optimal vapor flow through the chamber across the silicon surface. After 1
min, when the vapor pressure inside the chamber was nearly saturated, the vapor flow was reduced to 1 L/min. This small flow rate was necessary to eliminate local flow fluctuations across the siliconsurface,whichwerenoticedto often disturb the droplet motion. At this stage, a water droplet (with a volume V = 1,2,5,10, or 25 pL) was placed with an Eppendorf pipet on the top end of thewafer viaanother holein the lid. The water droplet contracted due to a Marangoni flow and moved at constant speed down the inclined surface. After an initially covered distance of 1cm, the average droplet speed was measured over a length of 3 cm. Generally, this distance was covered in less than 1min. Within this time span, no significant uptake of 1-methoxy-2-propanol vapor into the water is expected to occur.' Therefore, the surface tension of the droplet should remain close to 65 mN/m. Droplet motioh and shape were measured by viewing the droplet both from the side and from a h v e (see Figure 4) with a Philips CCD camera connected to a U-Matic video cassette recorder. Dynamic contact angles 8~ and 8 d . as ~ a function of the droplet speed u were measured with a protractor (to an accuracy of -1') from video prints. To study the influence of surface roughness, droplet speeds as a function of the tilt angle were measured on both smooth and rough silicon surfaces. The degree of surface roughness was quantified with a Talysurf 5-120 (Taylor-Hobson) roughness meter. This instrument enables a surfaceroughness measurement in terms of the parameter R., which denotes the averaged absolute value of the height deviations across the surface with respect to ita average height level. The minimum attainable wavelength within the roughness profiles was 1pm. Polished front sides of silicon wafers had typical R, values of 2 nm, whereas two rough back sides used had R. values of 214 f 13 and 723 f 64 nm. As mentioned before, the surface tension of a water droplet briefly exposedto 1-methoxy-2-propanolvapor is close to 65 mN/ m.l The latter value is consistent with an interfacial 1-methoxy2-propanol concentration of 0.25 M.1 Such a concentration must exist near the droplet perimeter where, according to eq 12, the local viscosity q must be known. This viscosity was measured with a Couette viscometer (Contraves, Low-shear 30) on a bulk solution of 0.25 M 1-methoxy-2-propanol yeilding 7 = 1.055 X 10-3 N s m-2.
4. Results and Discussion (A) Dynamic Characteristics of Moving Droplets. Typical video prints of moving Marangoni-contracted water droplets in a 1-methoxy-2-propanol vapor atmosphere, taken from the side and from above are shown in Figure 5. Because silicon has a reflective surface, video pictures from the side also show the mirror image of the droplet. Figure 5 demonstrates that the top view of the droplet contour on a solid surface loses ita circular symmetry when OdAand d d a become different. At very small droplet speeds (Figure 5A,B), the observed droplet shape qualitatively agrees with theoretically predicted shapes of Rotenberg et al.9 In this case,the advancing perimeter became slightly narrower than the receding perimeter. Such shapes were also reported by Bikermansla At higher speeds and/or on rough surfaces, an extensive elongation in the direction of movement takes place with a contracting receding perimeter and an expanding advancing perimeter. Moving droplets with parallel sides between the advancing and recedingperimeters, which have occasionally been reported to occur on hydrophobic surfaces in air,4J8 were not observed. According to theoretical work of Dussan and Chow,' the presence of parallel sides (in droplets with small contact angles) is necessary to obtain a transition from the static advancing contact angle &,A to the static receding contact angle O18 when going from the advancing t o the receding perimeter. Contact angles along the parallel sides are static because their perimeter does not move along the solid surface. Thus, an absence of parallel
2760 Langmuir, Vol. 7, No. 11, 1991
Huethorst and Marra
A
B
C
D
E
Figure 5. Side view and top view video prints of moving water droips on a smootn surface \fin= L nm, n-ul allu wll a rough surface (R,= 723 nm; E). The drops are moving from the left to the right: (A) V = 5 pL; 4 = 2.5"; U = 0.8 mm/s; 8dA = 15"; = 14'. (B) v = 25 pb 4 = 2.5'; u = 1.8 "/S; 8d,A = 16"; 8d.R = 14". (c)v = 25 pL; 4 = 10"; u = 7.4 mm/s; = 20"; dd,R = 7". (D)V = 25 NL;4 = 15'; U = 13.5 "/Si 8d,A = 21"; dd,R = 3'. (E)v = 25 @L;4 = 15'; u = 9 mm/s; = 23"; Bd,R = 0".
sides indicates that no contact angle hysteresis exists with Marangoni-contracteddroplets. Figures 6-8 give droplet velocities as a function of the angle of inclination sin 4 for different droplet volumes and surface roughnesses. Even on rough surfaces, droplets can be set in motion with only a slight inclination angle. Thus, in terms of eq 1,essentially no critical minimum force F m b appears to exist for these droplets, which confirms that no contact angle hysteresis exists. For small values of u, u increases linearly with sin
4 for all droplet volumes on all surfaces. Larger droplets move faster than small droplets a t the same value of 4 but the speed always decreases on rougher surfaces. The velocity dependence of 6d,A and 6d,R as a function of the surface roughness is shown in Figures 9-11. A common feature in these figures is that all measured contact angles converge to an "equilibrium" angle 6, = 1 4 O when u 0. This reconfirmsthe absence of contact angle hysteresis. Evidently, the extent of surface roughness does
-
Langmuir, Vol. 7, No. 11,1991 2761
Marangoni-Contracted Water Drop Motion 25
0.1
I
t
Ra=2nm
“0
I
02 sin 0
03
0.4 0.1
Figure 6. Dependence of the droplet speed on the angle of
inclination sin 4 for three drop volumes: ( 0 )2, (A)5, and ( 0 ) 25 pL. Surface roughness parameter R. = 2 nm. Solid curves denote theoretically predicted droplet speeds according to eq 25. 20
/ a
0
1
10
v (mmls)
Figure 9. Dependence of the dynamic advancing contact angle
(open symbols) and receding contact angle 8dp (closed symbols) on the droplet speed u for different droplet volumes: (0)1, ( 0 )2, (A)5, (‘7) 10, and ( 0 )25 pL. Surface roughness parameter R, = 2 nm.
I9dL
Ra = 214 nm
15
-E
10
5
-
Vcrit
v (mm/s)
Figure 10. Same aa Figure 9 but for R. = 214 nm. 15
eE
Ra = 723 nm
I
I
10
I
-
0
E >
25 UI 5
0
0.1
0.2
0.3
0.4
sin 4
Figure 8. Same aa Figure 6 but for R, = 723 nm. not affect 8, (aslong as the surfaces are hydrophilic). This contrasta with common experience on hydrophobic surfaces, where contact angle hysteresis is always found to strongly depend on the surface roughness.16 Clearly, the Marangoni-induced contact angle only reflects physical processes occurring within the liquid of the droplet
v (mmls)
Figure 11. Same as Figure 9 but for R. = 723 nm. perimeter. Dynamic contact angles, on the other hand, do depend on both surface roughness and droplet speed but not on the droplet volume. Apparently, not only hy-
Huethorst and Marra
2762 Langmuir, Vol. 7, No. 11, 1991
R.,O
Table I. Relation between B and v (mm/s) nm Ban, deg kiz,deg
2 214 723 0
14 + 0 . 7 ~ ' . ~ 14 + i.iuo.9 14 + 1.2uo.9
14 - 0 . 9 ~ ' , ~ 14 - 1.4~l.O 14 - 4.5u0.'
Roughness parameter.
drodynamical but also surface-induced factors are important with respect to the dynamic behavior. The difference &,A- 8 d p increases with the droplet speed and the surface roughness. As soon as ed,R approached zero, a trail of water was left behind which tended to break up into smaller droplets. This occurred at lower droplet speeds on rough surfaces than on smooth surfaces. The latter observation is consistent with the lower Marangoni drying speeds Umm on rough surfaces that were reported in the foregoing paper.' For both the rough and the polished surfaces, experimentally determined dynamic contact angles could be fitted with mathematical expressions such as
+ (1.0'
8d,A(u)
= 8,
8d.R(u)
= 8, - au5
Equation 24 can be substituted in eqs 21 and 23. When the resulting expressions are combined with eqs 2 and 19, explicit approximate expressions for the droplet speed u as well as for the forces F, and Fyare obtained according to
and
(20)
where a, b, CY,and Bare positive constants whose magnitude depends on the surface roughness. Values for these constants are given in Table I. (B) Comparison with Theory. Our aim is to provide theoretical estimates for the droplet speed u and the resisting forcesF, and Fyas a function of sin 4. For moving droplets of arbitrary shape, eqs 12 and 18 must be evaluated 0, the droplet numerically. However, in the limit u retains its circular footprint contour (Figure 5A) and 8d,A N 8d,R. Therefore, it appears a natural starting point to evaluate eqs 12 and 18 for droplets that only experience a minimum of deformation. In that case, y = yo sin cp in Figure 3 for both cp = (CA and cp = I ~ R .yo become the radius of the droplet footprint on the substrate. With cos (m)/ 8d,R + cos ((PA)/&,A = 2 cos (@)/Os, eq 12 transforms according to
-
Regarding eq 18 for the Young force Fy,an expansion of the cosine terms can be made for small values of the contact angles. With 8d,A(U cos (PA) + 8d,R(U cos (PR) 28,, and (PA = (OR = cp (i.e., a circular contact line), eq 18 becomes
ed,R(V COS (P)] COS
CP dv (22)
in which eq 20 must be substituted for &,A and ed,R. The resulting expression can only be evaluated analytically when the exponents b and ,6 in eq 20 are integers. Table I shows that to a good approximation b 0 = 1. Consequently, eq 22 combined with eq 20 yields
F~ N 27.4yLvyoes(CY +a
angle Os and volumes Vnot exceeding those of the droplets used in the present study may to a good approximation be considered identical in shape to the cap of a sphere. yo may then be written as
) ~
(23)
where the numerical factor has been calculated such that the (a, a) from Table I can directly be substituted, while the other parameters should be expressed in SI units. According to theoretical calculations by, for example, Goodwin et a1.,6 sessile water drops with a small contact
The straight lines in Figures 6-8 give u as a function of sin 4 accoxding to eq 25 with In (X-/Xmb) = 12 and ~ L = V 65 mN m. Agreement with experimental droplet speeds is exce lent up to some approximately determined speed u = U a i t . Moreover, the predicted proportionality u = W3 is well obeyed. Above u = Ucrit, u increases more rapidly with sin 4 than accordin to eq 25. In Figures 6-8, this only pertained to the 25-pL ro lets. The good agreement holds as long as the droplets o not ex erience more than a slight amount of deformation (see i ure 5A-C). ucrit decreases with increasing surface roug ness but is always consistent with a speed where 8d,R N- 5'. At u = BUcrit, 8d,R = 0 and the droplet leaves a trail of smaller droplets behind. Note that the maximum drying speed umlu on polished silicon surfaces (as introduced in the accompanyingpaper) is well below the droplet speed where ed,R 0. Apparently, small amounts of water can be left behind by Marangoni-contracted water meniscuses even when 8d,R > 0 although this clearly becomes less likely when 8d,R increases. A rigorous derivation of u from eqs 12 and 18-20 would involve an iterative procedure. This was not persued in the present study. Instead, experimentally measured droplet shapes and speeds were used to explicitly evaluate the magnitudes of F, and Fyaccording to eqs 12 and 18 in order to check the consistency with eq 19. Results are given in Figures 12-14 on surfaces of different roughness as a function of sin 4. The figures show that both the viscous force and the Young force must be accounted for to predict the hydrodynamicdroplet behavior. Their magnitude is comparable but F, becomes relatively more important on smoother surfaces. Fyis the dominating resisting force on very rough surfaces where the dynamic contact angles show a pronounced dependence on the droplet speed. The sum of the explicitly calculated forces F, and Fyalways agrees with the gravity force Fgto within 10%. This is also true at higher droplet speeds and/or strong droplet deformations (see Figure 14) where eq 25 breaks down. The good agreement supports a posteriori the validity of eqs 12 and 18. For the 2-pL droplets in Figure 12, the calculated values for F, and Fyagree well with the approximatelydetermined forces. This is according to expectation because of the small speeds and the limited extent of droplet deformation. For the 5- and 25-pL droplets, the agreement holds only
i
%B
-
Pg
Langmuir, Vol. 7, No. 11, 1991 2763
Marangoni-Contracted Water Drop Motion 80 600 2 MI
1
/ FQ'
1
25 pl
60 -
s.
1
h
0
40
U
20
OO
0.1
0.3
0.2
0.4
0.5
s1n $
sin 0
Figure 12. Dependence of the driving force Fg(according to eq 2) and the resisting forces F, and Fyon the angle of inclination sin 4 for 2-pL water dropleta in a 1-methoxy-2-propanolvapor environment. The drawn curvesdenoteapproximatelycalculated forcesaccording to eq 26 for F,and eq 27 for Fy.Open and closed symbolsrepresentrigorouslycalculatedforceson silicon surfaces R, = 2 nm; (A)R. = 214 nm; ( 0 ) with different roughnesses: (0) R. = 723 nm. Encircled symbols represent the sum of the magnitudes of the open and closed symbols. 160 -
t
Figure 14. Same as Figure 12 but for 25-pL droplets.
In respect of the latter, the dependence of Od,R and Od,A on the droplet speed u presents a key issue. For all investigated droplets, the differences - 6d.R and &,A Os are approximately proportional to u. The proportionality coefficients increase with the surface roughness, in particular the coefficient with regard to 0.38 (see Table I). This finding illustrates that Marangoni drying of hydrophilic surfaces is adversily affected by the surface roughness. No drying is possible at meniscus speeds where dd,R 0. Although uncertainty exists in the literature about the concept of a dynamic contact angle and how it should be measured,ls the present study indicates that a mere knowledge about macroscopic dynamic contact angles is sufficient to account for the hydrodynamics of a moving droplet. No contact angle hysteresis exists for Marangonicontracted droplets on hydrophilic surfaces irrespective of the surface roughness. This implies that no finite minimum force needs to be exceeded before the droplet can be set in motion. Surface tension forces appear to be well accounted for when the macroscopically measured static contact angle Os is taken to be identical with the equilibrium contact angle Or according to Young's law. Extensive droplet elongation at high speeds results mainly from the viscous force that exists in the receding perimeter when 8d,R 0. Generally speaking, the extent of droplet deformation is proportional to the difference - 8d,R as measured at the advancing and receding perimeters. For droplets that are not or are only slightly deformed, the droplet speed is proportional to VI3sin 4. No evidence exists that Marangoni flows near a droplet perimeter have to be explicitly accounted for to describe the droplet hydrodynamics. Nevertheless, the origin of the existence of a surface tension force and a static contact angle 0, lies in a surface tension gradient within the droplet perimeter induced by vapor absorption.' The surface tension force arises as a response to the departure of the contact angle from its static value 8,.
-
5 VI
/v
Ra=2nm R. = 214 nm
"
'Fv
-
R. = 723 nm ,
0.4
0.5
sin $
Figure 13. Same as Figure 12 but for 5-pL droplets.
for small values of sin 4. At larger angles of inclination, eq 26 underestimates F, whereas eq 27 overestimates Fy' Evidently, a large contribution of F, will come from the receding perimeter when 8d,R is small. Droplet elongation must also be due to the latter. 5. Summary and Conclusions The hydrodynamic behavior of small Marangonicontracted water droplets on inclined hydrophilic silicon surfaces in a 1-methoxy-%-propanolatmosphere is well accounted for when the lubrication approximation is used for the flow profile within the droplet perimeter with flow lines normal to the contact line. Steady droplet motion occurs when the driving gravity force is balanced by counteracting viscous and surface tension forces. These can be separately calculated once the droplet speed, the droplet shape, and the velocity dependence of the dynamic contact angles are known.
-
Registry No. Si, 7440-21-3;1-methoxy-Bpropanol,107-98-2.
