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Spreading of “heavy” droplets
Spreading of "Heavy" Droplets I. Theory F. BROCHARD-WYART, ~ H. HERVET,* C. REDON, AND F. RONDELEZ Universit~ Pierre et Marie Curie, Laboratoire de Structure et R(activit~ aux Interfaces, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France and *Colldge de France, Laboratoire de Physique de la Matidre Condens£e, 11 place Marcelin-Berthelot, 75231 Paris Cedex 05, France
Received April 30, 1990; accepted August 3, 1990. We study the spreading of a (nonvolatile) wetting liquid on a fiat solid surface. For small droplets, capillary forces drive the spreading and the shape of the spreading liquid is well known to be close to a spherical cap, with a radius R ( t ) ~ t ~/~° (where t is the spreading time). We investigate here the opposite case of "heavy droplets" (R > K-L, the capillary length), for which gravity controls the process. The velocity of spreading may be understood from the rate of conversion of gravitational energy into viscous losses. The latter process can be divided into two contributions, one from the advancing wedge and one from the central part (squashed under gravity). Depending on which dissipation mechanism is dominant, two different shapes of the spreading droplet can be observed: (a) For R < R~ (Re = r - q n ( 1/Ka) ~ few centimeters, where a is the molecular length), the dissipation in the wedge is dominant. The drop has a quasistatic shape, with a large fiat portion of thickness h = K-a0d (where 0d is the dynamical contact angle). (b) For R >> Re, the dissipation in the bulk is dominant. The drop is not fiat; most of the profile is described by a self-similar solution first postulated by Lopez et al. ( 1 ) (except for a "foot" of size K-1 at the droplet edge). The growth of the drop radius obeys a power law R (t) ~ t ~/8 only in the asymptotic regime of very large droplets R >> Re. © 1991AcademicPress,lne.
I. INTRODUCTION
Wetting processes are important for many practical applications involving paints, lubricants, cosmetics, insecticides, etc. The statics and the dynamics of spreading of microscopic droplets ( 10-4/~1) has been extensively studied within the last few years (3-8) down to the molecular level, and the shape and extension of the precursor film have been established. Our aim here is to study the opposite case of very large droplets ( 100 ~zl) extending over several centimeters, called "heavy droplets" because gravity becomes the major driving force. In spite of their practical importance in achieving thick films of thickness of several hundred of micrometers covering large surTo whom correspondence should be addressed.
faces, only a few theoretical ( 1, 2) and experimental (9-11) studies have been reported until recently. On the theoretical side, Tanner (14) has studied the effect of gravity on the profile near the contact line for relatively small drops (KR < 3). Lopez et al. (1) have studied the last stage of spreading, where gravity and fluid viscosity are the chief promoting and resisting spreading factors. In Ref. (1) the curvature effects and the singular behavior of the viscous dissipation in the drop's edge were neglected. Our aim here is to take into account these two factors. We shall see that this leads to a n e w r e g i m e , where the dissipation at the contact line becomes the major resisting force. The pressure equilibrates faster in the central part, which becomes flat. We study first the main results on the statics and dynamics of wetting, which will be useful for the foUowing.
518 0021-9797/91 $3.00 Copyright© 1991by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand InterfaceScience, Vol. 142,No. 2, March 15, 1991
SPREADING OF "HEAVY" DROPLETS II. EQUILIBRIUM SHAPES OF DROPLETS: PARTIAL VERSUS COMPLETE WETTING
When a liquid drop is put into contact with a flat solid, two distinct regimes may be found: partial and complete wetting. The crucial parameter is the spreading coefficient S = 3"so (3"SL + 3"), which measures the difference of surface energy between the bare solid (-rso) and the wet solid (3"SL + 3"), where 3"SLand 3' are the solid/liquid and the liquid/air interfacial tension, respectively. (A) Partial wettingis found for S < 0. The contact angle 0E at the triple line is finite and given by the Young relation -
3"so = 3"SL+ 3" COS 0E.
[ 1]
The classical equilibrium shapes of a liquid drop are shown in Fig. I. If the drop is small, gravity effects are negligible. The pressure in the droplet is uniform, and the shape is a spherical cap. On the other hand, if the drop is larger than the capillary length K-l (K2 = pg/ 3", where p is the weight per unit volume, and g the acceleration due to gravity), it is flattened by gravity. The drop forms a thick "pancake" of thickness h0, which can be derived from the balance of horizontal forces acting on the hatched zone as shown in Fig. lb 3"so = 3" + 3"SL-- (pgh2)/2,
[2]
where the last term comes from hydrostatic pressure integrated over the thickness h0. From Eqs. [1] and [2], we obtain h0 = 2 r - t s i n ( 0 E / 2 ) ~ K-~OE
[3]
in the limit of small contact angles. (B) Complete wetting is found for S > 0. The droplet spreads over to a very thin film of thickness e, resulting from a balance be-
a
b
-9 o
519
tween long range and capillary forces (6, 12). In the present paper, we focus our attention on complete wetting, with emphasis on the macroscopic aspects of the spreading, for which long range forces will not be important. We do not describe the precursor film surrounding the droplet which was first considered in Ref. ( 1 ). III. A REMINDER ON THE SPREADING OF A SMALL DROPLET
In the regime where gravity is negligible (R K-1 ), the macroscopic shape of the spreading droplet is rather close to a spherical cap (Fig. 2) because the pressure equilibrates very fast in the thick region. The apparent contact angle Od(t), the drop thickness h(t) and the radius R(t) are related by h = ( 112)ROd
~2 = (Tr/2)hR 2,
[4]
where ~2, the drop volume, is assumed to be constant (unvolatile liquid). 0a is related to the velocity U = dR/dt of the contact line by the Hoffman-Tanner law (13-15)
U I V * = c t e O 3 (0d ~ 1),
[51
where V* = 3'/7 is a typical velocity; ~/is the fluid viscosity. The cte is found to be of order 10 -2 from Hoffman data (13). From Eqs. (4) and (5), one deduces easily that the radius R increases with time as t 1/1°, since one has R9(dR/dt) = cte V*f]3(4/Tr)3; i.e., R 1° ~ t + to. This has been measured experimentally by several authors ( 11, 15 ). One finds in Ref. (6) two derivations of the Hoffman-Tanner law. In the first one, the total dissipation is expressed as the product of a flux
Y
•
R < K -1
FIG. 1. Equilibrium shape of a liquid drop in the case of partial wetting: (a) with R < K-~ (capillary regime); (b) with R >> K-1 (gravity regime). Journal of Colloid and lnterface Science, Vol. 142, No. 2, March 15, 1991
520
BROCHARD-WYART
/
ET AL.
stage, we shall not need to write explicit equations for the pressure gradient dPIdx: we express our results in terms of the translation velocity U. The total flux matter is
R FIG. 2. Dynamical shape and physical parameters of a spreading droplet in the case of complete wetting for R K-l.
U by a noncompensated Young force. The entropy source per unit length of line is
TS = FyU,
[6]
Uf =
S measures the driving force of the process, but U does not depend on S! As shown by de Gennes (6), S is entirely burnt in the precursor, a microscopic film which surrounds the macroscopic droplet. This explains why the spreading velocity of the macroscopic droplet is independent o r S . Another presentation of this result amounts to say that the macroscopic drop spreads on a thin liquid film and thus sees an apparent spreading coefficient equal to zero. On the other hand, the precursor is very sensitive to S. In the following, we focus our attention on the macroscopic drop only, and we may drop S in the unbalanced Young force. The dissipation in the wedge TSw is then TSw = ~1 3`0~2 U.
[7]
The dissipation in the wedge can easily be calculated in the limit of small 0d by using the "lubrication approximation" to derive the flow pattern in the wedge advancing at constant velocity U. The velocity profile V ( z ) is of the Poiseuille type
nVx(z) = ( I / 2 ) ( d P / d x ) ( z 2 - 2z~'),
[8]
where x is the direction normal to the contact line and ~'(x) is the local fluid thickness. Equation [ 8 ] ensures Vx (z = 0) = 0 at the solid surface and (OVx/Oz) (z = ~') = 0 at the liquid/air interface of coordinate ~'. At this Journal of Colloid and Interface Science, Vol. 142,No. 2, March 15, 1991
[9]
One can then write
Vx(z) = ( 3 U / 2 ~ z ) ( - z 2 + 2 f z ) .
[101
If we now assume a simple wedge shape for the profile (angle 0d), we find the viscous dissipation integrated over the film thickness
where Fy is the Young force Fy = 3'so - 3`sL - 3' cos 0d = S + (1/2)3,02.
Vx(z)dz.
TSw =
xffmnax U 2 U2 3rt ---f- dx = 3rt --~a L,
[11]
in
where L = ln(xm,~,/Xr, i~). x=~., is related to the macroscopic size of the droplet and Xmin is a molecular size (6). Using Eq. [ 7 ] this leads to the H o f f m a n - T a n n e r law (14, 15) in Eq. [ 12]: 0a3 = 6 L
~.
[12]
This simple argument depends on a specific assumption on the profile and leads to an underestimation of the numerical factor as we shall show now. The H o f f m a n - T a n n e r law can be deduced more generally from a direct calculation of the profile (7) near the triple line. The discussion is complex, because one must include the precursor film, which depends on long range forces. From Eqs. [ 81 and [ 9 ], one obtains the pressure balance near the drop edge
U = (f2/3o)(-oe/ox),
[131
where (OP/Ox) = - 3 ` f . . . . (Orr/Ox). r is the disjoining pressure due to long range forces (15). Without this term, Eq. [ 13 ] has no physical solutions ( f is strictly positive). For a simple van der Waals liquid, Eq. [ 13 ] becomes
3 ~ / ( U / ~ "2) = - - T ~ ' ' + ( A / 2 7 r ~ ' 4 ) ~ "',
[141
where A is the Hamaker constant. Equation [ 14 ] has been solved numerically by Hervet
521
SPREADING OF "HEAVY" DROPLETS
and de Gennes (7). They find a special solution corresponding to (OP/Ox) --~ 0 far from the line, in the form
= ( U / V * ) I / 3 x ( 9 ln(x/Xmin)) 1/3,
[15]
where Xmi,~= (a/Of). From Eq. [ 15 ], one can derive the apparent contact angle
03 = 9 L ( U / V * ) .
[16]
Notice the analogy between Eq. [16 ] and [ 12 ]. The numerical coefficient differs by a factor 3 because the dissipation in Eq. [12 ] is cal~, culated by assuming an approximate form of the profile. From Hoffman data, the logarithm L is of the order 11 for dynamic contact angle above several degrees, while lower values, L 4-5, have been reported for 0d smaller than l° (17). IV. SPREADING OF HEAVY DROPLETS: TWO TYPES OF DISSIPATION
Our aim here is to extend the approach of Section II to describe the spreading of large droplets. We shall use two methods based (i) on a calculation of the dissipation and (ii) on a numerical calculation of the profile in Section V. For a very small droplet, the shape is quasistatic, i.e., almost a spherical cap, with a dynamic contact angle 0d instead of an equilibrium contact angle. For a heavy droplet, the quasistatic shape will be a flat drop, of thickness h = K-10d according to Eq. [3]. We shall show that the real profile is quasistatic only in a certain range of sizes (R smaller than a threshold Re). But this regime will turn out to be the most important since most of the experiments encountered in practical situations fall into this class (5, 9, 10). For large drops (R > r -1 ) the spreading rate is controlled by a balance between viscous resistance to flow and the gain of gravitational energy. Let us for a moment assume that most of the drop is flat, with a thickness h and an apparent contact angle 0d. The energy conservation is written as
TS + og(h/2)f2 -- 0,
[17]
where t2 is the volume of the liquid drop. The entropy source TS can be written in terms of the velocity U = d R / d t of the contact line: TS = 27rR3~(U2/0~)L
+ (3/2)~rR2~l(U2/h).
[18]
The first term is the dissipation in the drop edge discussed in Section III, Eq. [ 11 ] and integrated over the drop perimeter. The second term is the dissipation in the flat part: the horizontal velocity field in the flat part at a distance r from the center is simply U(r) = ( r / R ) U . Integrating the dissipation ( f 3~ × ( U 2 / h ) ( r / R ) 2 27rrdr) over the flat part, one gets the second term of Eq. [18]. We must also impose mass conservation
~2 ~- a~rR2h.
[19]
Let us assume for the moment that a is a constant of order unity. From Eqs. [17 ], [18 ], and [ 19 ], one gets
( U/V*)[67r(R/O,~)L + (3/2)zr(R2/h)] = K2f~(h/R).
[20]
This leads to two regimes depending on the dominant mechanism:
(A) Dissipation Controlled by Wedge (R < R c) In that case, the first term of the left hand side in Eq. [20] is dominant and Eq. [20] reduced to
(6L/Od)(U/V*)
= (~2K2/ozer2R4).
[21]
From Section III, we know that 0d is related to the velocity U of the contact line by 03 = 6 ( U~ V* ) L. Equation [ 21 ] then gives 0~1 = o ~ l / 2 r h -~ K h .
[22]
Thus the thickness of the drop h(t) is related to the dynamic contact angle 0d by the static relation (Eq. [3]). We do check that, in this regime, the profile is quasistatic. Inserting Eq. [22] in the Tanner law leads to Journal of Colloid and InterfaceScience, Vol. 142,No. 2, March 15, 1991
522
BROCHARD-WYART
6 L ( U / V * ) = (Kh) 3
[23]
and using Eq. [ 19 ], with a assumed to be conslant, leads to R 7 -~ ( ~ 3 / o ¢ 3 / 2 ) ( K 3 / T r 3 ) ( V * / L ) ( I
q- to).
[241 Equation [23] should hold up to a critical radius Re, at which the dissipation in the bulk and in the wedge are comparable. At the crossover, we may still write 04 = ~h, and Rc deduced from Eq. [20] is Re = 4LK -1
[25]
By taking K-1 - 1.5 mm, and L ~- 12, one expects Rc ~ 7 cm.
(B) Dissipation Controlled by the Bulk ( R > Rc) In this case the second term of the left hand side of Eq. [ 20 ] is dominant, and one gets
(3/2)~r2e~(R4/~)(U/V *) = ~2K2/a~-R3
ET AL.
where the pressure P may be written
P = -,),(O2~/O2r) q- p g f + 7r. The first two terms are the capillary and gravity contributions. The last term is the disjoining pressure (16) due to long range forces (van der Waals, electrostatic, steric) which show up for thin films. This term has to be included to describe the cross-over between the macroscopic drop and the precursor (7). We must add to this the local equation of volume conservation
(1/r)(O/Or)[(r~U(r, t)] + (O~/Ot) = O. (30) From Eq. [29], one can expect four regions (Fig. 3). - - A central region I where gravitational forces dominate: 3n[V(r, t)/~ "2] = -og(O/Or)~.
[31]
- - A meniscus H, where the Laplace pressure becomes relevant:
[261 Eq. [26 ], assuming ~ to be constant, leads to
R 8 _~ ( f ~ 3 / o t 2 ) ( r 2 / T r 3 ) V * ( t + to).
[271
Thus in this limit of very large drops, we recover the result of Ref. ( 1 ). In this regime, 04 is not related to h by the quasistatic relation (Eq. [3]). Taking U f r o m Eq. [26] and using the Tanner law (Eq. [12]) one finds
04 ~- Kh(RdR)
[28]
or equivalently 0d = Khl, with h, = h(Re/R). We shall see in Section V that in this regime the drop is indeed not flat, and is not described by the quasistatic profile of Eq. [ 3 ]. V. P R O F I L E O F T H E S P R E A D I N G HEAVY DROPLETS
3~[ U(r, t)/~ "2] = V(03~-/03r)
--
pg(O~/Or).
- - A proximal region III near the macroscopic triple line of length (a/O~) and height (a/Od), where the long range van der Waals forces (described by r ) dominate (for a more detailed discussion of this region, see Ref. 7); - - A precursor film I V described in Refs. (1, 7). (2) Similarity Solutions (1) Lopez and co-workers (1, 2) dropped the Laplace term and the disjoining pressure in Eq. [29]. They looked for a self-similar solution of Eqs. [ 30 ], [ 31 ] expressed as
(1) Basic Flow Equations
U(r, t) = U(t)g(u)
In the lubrication approximation, the mean radial velocity U(r) is related to the fluid thickness ~-by a Poiseuille equation
~(r, t) = h(t)f(u),
37( U(r, t)/~-2) = -OP/Or,
[29]
Journalof CoUoidand lnterfaceScience,Vol. 142, No. 2, March 15, 1991
[32]
[331
where u = r/R. The functions f a n d g which are solutions of Eqs. [ 30 ], [ 31 ] satisfy the two equations
523
SPREADING OF "HEAVY" DROPLETS
oee~"
u.
°t
I
I
I
I
[
I
~ IV IIII II I I ~'- - ~* " ~ " - ' ~ . . . . . . . . . . . . . . t
I"I"
I i
•I-
L
may be treated as constant U(r) = U. The profile f ( r ) is then given by
L [
3 r / ( U / ~ "2) = "y(~". . . .
K2~").
I
i
ho(t )
We must find a solution of Eq. [38] which matches the behavior near the contact line (Eq. [161)
X1
R
WIDTH
OF
THE
By integration of Eq. [38], x = r - R , we find
DROP
FIG. 3. Schematic profile of a heavy droplet, showing the four regions: (I) Central region where gravitational force dominates; (II) the meniscus where the Laplace pressure is comparable to the gravitational force; (Ill) the proximal region where the long range van der Waals forces show up; (IV) the precursor film where they dominate.
3n
dx =
2u = - 3 f 2 ( d f / d u ) .
rain.
1
---- v ( t '2 - / Y " ) +
[341
[35]
Imposing the volume conservation of the drop leads to 2rcf(u)rdr
= 7rR2h
yo
2 f ( u ) u d u = f~
[36]
with rrR2h = ~2/a = const. ( a = 0.75). From Eqs. [ 32 ] - [ 33 ], one obtains an equation for R(t) R ( t ) ~ ( K 2 Q 3 V * t ) 1/8.
1
2
og/'. [40]
The profile is t h e n f ( u ) = (1 - u2) 1/3 One obtains for the radius of the droplet 2(dR/dt) = (h3/R)og.
and setting
- y ( f f " - ~2~f')dx
in.
g(u) = u
f0
[39]
Od = ~' (X = O) = [ 9 ( U / V * ) L ] ~/3. 0
h
[381
l- - -
[37]
Note that, in this approach, the shape of the drop edge is not well described and the viscous losses in the edge leading to a logarithmic singularity are omitted. We shall now see that this similarity solution is applicable only for R >> Re, where the edge losses are indeed negligible. (3) Approximate Calculation of the Profile (a) The meniscus. In a region of size K-I near the contact line, the radial velocity U(r)
Equation [40] may be interpreted as a balance of horizontal forces on a portion of the liquid edge extending from Xmi.. to X. The viscous force (left hand side of Eq. [ 40 ]) equilibrates the noncompensated Young force [3'so - (3' cos O(x) + "YSL)] = ( 1 / 2 ) 7 ~ "'2 and the hydrostatic contribution, f (P - Po)dz = __,]/~-~-t,' _]_ 1 ~ p g f ,2 where P0 is the external pressure. We shall now describe a very rough construction of the profile, which will, however, be useful for the interpretation of the exact numerical solutions which are discussed later. Approximation (i): Since the left hand side of Eq. [ 40 ] is only logarithmically dependent upon x, we treat it as a constant C (in the region Xmin. < X <~ K - l ) . For x ~ Xmin. C i s given by setting ~ ~ 0 in the fight hand side of Eq. [40]: C = (1/2)702 .
[41a]
For x > K-~, the slope O(x) and the curvature are negligible and C is given by C = ( 1 / 2 ) p g f 2.
[41b]
Approximation (ii): We assume that for x K-l, the thickness is slowly varying, and we write h = hi. Comparing the two forms [41a,b] we see that hi = K-10d. The thickness hi of the Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991
524
BROCHARD-WYART
meniscus corresponds to the quasistatic thickness (Eq. [ 3 ] with 0E = 0a). We now return to Eq. [38] in the region x >- K-1. Here ~'' is negligible, and the slope ~" = 01 is given by
O~ = [3U/V*(~hl):] ~- (Oo/3L).
[42]
Thus, as announced, the slope is small [01 -~ ( 0 j 10)]. We can now proceed to discuss the central region of the drop, with Eq. [ 42 ] acting as a boundary condition at r = R - xl (where xl is defined by ~'(Xl) = hi ). (b) Profile in the central region. In the central region, the profile is obtained by direct integration of Eq. [ 31 ], incorporating only viscous forces and gravity effects. Assuming a radial velocity U(r) = ( r / R ) U, one obtains
3(U/V*)(r/R)
= --K2~2(O~/Or).
[43]
Introducing the height of the drop at the center
ho = ~"(r = 0) we get f = --9( U/V*)(rZ/2RK 2) + h~.
[44]
(c) Complete profile. We now match the edge and center by imposing the boundary condition [42] on Eq. [44]. This occurs at x = xl ; i.e., r = R - xl ----R. The result is (Kh0) 3 = (Khl) 3 + ( 9 U / V * ) ( r R / 2 ) .
[45]
Assuming hi = K-10a, Eq. [45] can be rewritten (Kho) 3 = (9UL/V*)[1 + R/Rc],
[45'1
where Rc = 2 LK-1. We recover the result obtained from the calculation of the dissipation, except for the numerical factor (half of the preceding value). This discrepancy is due to the simplified profile used to estimate the dissipation. Equation [45 ] shows: ( 1 ) that the drop is flat ifR < Rc; the thickness ho of the center of the drop is roughly equal to the height hi of the meniscus. This corresponds to the quasistatic regime. (2) On the other hand, ho becomes much Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991
E T AL.
larger than hi for R > Re. We recover the selfsimilar regime (Eq. [ 35 ] ).
(4) Numerical Calculations of the Drop Profiles A complete calculation of the profile of the spreading drop involves the resolution of the time dependent equations [29 ] and [30 ]. We have simplified the problem by the two following assumptions: (i) The boundary condition at the contact line is provided by the Hoffman-Tanner law (13); (ii) the velocity U(r, t) is taken of the form
U(r, t) = (r/R)U(t).
[46]
This law for the velocity is exact in the asymptotic limit where the similarity solution holds (R > Rc). It is also exact for a film of constant thickness squeezed between two plates, which corresponds to the regime R < Re. We solve Eq. [32 ] numerically starting from the edge (x = 0) with U(r, t) = U[1 + (x/ 2)1:
3(u/v*)[1 + (x/R)] = ~.2(~-. . . . Kq-'). [47] We set ~"= YoZ and x = K-1X, thus Eq. [47] becomes
1 + (X/KR) = +Z2(Z .... Z'),
[481
where we have chosen (KY0) 3 = 3 ( U / V * ) .
[49]
The structure of Eq. [ 48 ] shows directly that one cannot find a self-similar profile, because two different characteristic lengths (R and ~-1) come into play. Numerical integration of Eq. [48]. (i) We take KR as a variable parameter, which we vary from 5 to 100. (ii) We generate the profile starting from the drop edge imposing
Z = X ( 3 L ) 1/3,
[50]
the boundary condition at the edge (X --~ 0), in agreement with Eq. [ 15 ].
S P R E A D I N G OF " H E A V Y " DROPLETS
We choose several discrete values of L, and we focus on L = 11 to fit Hoffman's data. We start at Z~ = 0.09, Z~ = -3.2, and we vary the curvature Z " at this point. Each choice of Z " leads to one solution of the thirdorder equation [48] [see Fig. 4 (solid line)]. For one particular choice of Z " = Z~ the solution has zero slope at the center of the droplet Z ' ( X = --KR) = O.
The original choice of Ze and Z " ensures that Z~ is always <0 and close to zero, that is in the macroscopic part of the drop. This solution is represented on Fig. 5, for various values of KR (ranging from 5 to 100) spanning both regimes KR < KRc and ~R > KRc. In Fig. 5 we also show the self-similar solution corresponding to the same value of h0 (or Zo). We see that for R < R~ the drop is fiat, as expected from our rough arguments. For R > Rc the self-similar solution is correct (except near the edge). A crucial test of our simplified discussion is provided by a plot of Z 03as a function of KR
(Fig. 5). This plot is found to be linear, as predicted by our Eq. [45], and we set (Z0/Zl) 3 = 1 + ( R / R c ) ,
[511
where Z~ is the limit in value of Zo for R ~ R~ --~ 0. We find Z1 = 1.086 I Z " I (while in the crude discussion we had ZI = Z~). From the slope, we find for the critical radius ~Rc = 29. To test the dependence of R~ with L, we have also studied the case L = 6 (Fig. 5 ). We findZ~ = 1.15 [Z~[ a n d K R ~ = 18 for both values of L; Z 03has a linear dependence with ~R, with a slope of-~. This is exactly the result of our rough analysis. Equation [45 ] written in reduced variables becomes Z03 = Z 3 + (3/2)(KR)
[521
which leads to KRc = ( 2 / 3 ) (Z 13). As Z~ is nearly equal to 1.1 ]Z~], R~ can be approximated by KR~ ~ 2 L ( I . 1 ) 3 ~ 2.7L.
i
Z
525
Z
.
[53]
r
t O. o "o
I I
0
lb
s
0
X
I d
0
'2'o
0
4a
10
20
4b
q)
J~
Z
a/.
o) o
.f,
i
0
0
50
0
4c W i d t h
of
the
50 4d
100
d r 0 p (KR)
FIG. 4. C o m p u t e d drop profiles o f heavy droplets in the case of complete wetting. Full line curves represent the numerical solutions of Eq, [47] obtained with L = 11 for R < Rc (a and b) and R > Re (c and d). The dotted curves are the self-similar profiles obtained for the same ho. Journal of Colloid and Interface Science, Voi. 142,No. 2, March 15, 1991
526
B R O C H A R D - W Y A R T ET AL.
200
¢0
H3
~: l o 4
D.
o
e0
i
i
i
i
i
i
i
i
i
I
i
i
i
I
i
i
i
R
~J
~oo
U.
o~ ee-
/ cross \ / over
o
J=
R o
~
Radius
of
I
50 the
drop
100 (K -1)
FiG. 5. Plot of the thickness at the center of the drop, to the third power, in reduced coordinates versus the radius of the drop in ~-~ units. The straight lines represent the best fits obtained for R > 10 K-~ for two values of L: ( + ) L = I I ; ( O ) L = 6,
For each profile, we have calculated the volume ~2 of the drop normalized by the volume of a flat disk of thickness h0 and radius R: [54]
ot = ~2/TrhoR 2.
The variation of a versus R is shown in Fig. 6. We see that a goes through a maximum for R ~ 1.6Re: in the regime where the drop is
1
O~
e
10-10
1027 T I M E (A.U,)
FiG. 7. Log-log plot of the variation of the radius R of a drop versus time. Notice that 5 decades in R correspond to 37 decades in time!
flat, a increases because the correction due to the drop edge decreases as 1/R. Above Rc, a decreases toward the asymptotic limit o~ = 0.75. One must notice that even for KR 100, a is still much larger than a ~ . (5) Numerical Spreading Laws
From the calculated values of Z0 (Fig. 5 ) and the exact relationship between ho and R given by the volume conservation (Eq. [55] and Fig. 6) we are able to calculate the time dependent spreading laws JR(t), U(t)] from the early stages (R < K- 1) up to the long time limit (R >> K-1 ). From Eqs. [49] and [55], we deduce the differential equation relating U = dR~dr to R: U = (V*K3~23)/(37r3ot3(R)Z3o(R)R6).
We have calculated R (t) by integration of Eq. [56] using interpolated values for a ( R ) and Z o ( R ) . The result is shown in Fig. 7 for a time scale extending over 38 decades:
"~ "~" m ®=lUO~o= e " . . . . . . . . . .
0.5
100
0
Radius
of
the
drop
(K -1)
FIG. 6. Plot of the reduced volume ~ = f~/thoR 2 versus the radius R of the drop for two L values: L = 11 (solid line); L = 6 (dotted line), a ~ corresponds to the value obtained for the self-similar model, where ~ is independent of R . Journal of Colloid and Interface Science,
Vol.
142, No.
[55]
2, M a r c h
15, 1991
- - F o r KR < 1, we find the classical capillary regime R "-~ t 1/1°. - - F o r 1 < KR < KRe, we are in a cross-over region. From Eq. [ 56 ], we see that if c~(R) was a constant, R ( t ) should be proportional to t 1/7 . But due to the variation of a ( R ) in this region R ( t ) does not follow a pure power law.
SPREADING OF "HEAVY" DROPLETS
527
- - F o r R > Re, R ( t ) obeys the power law R t 1/8. This is due to the fact that a is almost a constant; also it has not yet reached its a s y m p t o t i c value a ~ = 0.75.
ular size a and the capillary length (K -1)]. T h u s the pure gravity regime (self-similar behavior) does not show up for R > K-1, contrary to a c o m m o n belief, but only for R > Rc >> K-1.
VI. CONCLUSION
ACKNOWLEDGMENT
On a completely wettable, plane, horizontal surface, f r o m our analysis we expect three regimes for the spreading o f a liquid droplet as a function o f the radius R (t): (i) R ( t ) is smaller than the capillary length K-~. Capillary forces are d o m i n a n t and viscous dissipation occurs m a i n l y in the drop edge. T h e droplet has the shape of a spherical cap, and a radius R ( t ) ~ t 1/1°. (ii) K-1 < R ( t ) < Re. G r a v i t y drives the spreading. Dissipation occurs again m a i n l y near the contact line. This leads to a new quasistatic regime: the drop is almost fiat, with a thickness h0 ~ K-10d, where 0d is the d y n a m i c contact angle. T h e spreading law is not a simple p o w e r law. (iii) R ( t ) >>Re. G r a v i t y controls the spreading, but the viscous losses in the central part o v e r c o m e the dissipation in the drop's edge. T h e spreading droplet is not fiat. This is the d o m a i n covered by the pioneering p a p e r of Lopez et al. (1). However, we show that their p r o p o s e d spreading power law R ( t ) t 1/8 is only correct in the limit o f very large drops: R > 10Rc. In practice this regime is far b e y o n d the usual range studied experimentally. It has to be noticed that the critical radius Ro is m u c h larger than K- 1, the capillary length, Rc ~ 3LK-1 [here L is a logarithmic factor, controlled by the ratio (Ka) between a molec-
The authors thank P. G. de Gennes for various discussions and useful comments to manuscript. REFERENCES 1. Lopez, J., Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 56, 460 (1976). 2. Joanny, J. F., Thesis. Le Mouillage, Paris, 1985. 3. Heslot, F., Fraysse, N., and Cazabat, A. M., Nature 338, 640 (1989). 4. Beaglehole, D., J. Phys. Chem. 93, 893 (1989). 5. Daillant, J., Benattar, J. J., Bosio, L., and Lrger, L., Europhys. Lett. 6, 431 (1988). 6. de Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 7. Hervet, H., and de Gennes, P. G., C. R. Acad. Sci. Paris Ser. 2 299, 499 (1984). 8. Ruckenstein, E., and Lee, P. S., Surf. Sci. 50, 597 (1975). 9. Bascom, W. D., Cottington, R. L., and Singleterry, C. R.,Adv. Chem. Ser. 43, 355 (1964). 10. Ogarev, V. A., Ivanova, T. N., Avslanov, V. V., and Trapeznikov, A. A., Izv. Akad Nauk. SSSR Set. Khim. 7, 1467 (1973). 11. Levinson, P., Cazabat, A. M., Cohen Stuart, M. A., Heslot, F., and Nicolet, S., Rev. Phys. Appl. 23, 1009 (1988). 12. Joanny, J. F., and de Gennes, P. G., C. R. Acad. Sci. Paris Ser. 2 299, 605 (1984). 13. Hoffman, R., J. ColloidlnterfaceSci. 50, 228 (1975), 14. Tanner, L., J. Phys, D 12, 1473 (1979). 15. Lelah, M., Marmur, A., J. Colloid Interface Sci. 82, 518 (1981). 16. Deryagin, B., and Churaev, N., Kolloidn. Zh. 38, 438 (1976). 17. Chen, J. D., and Wada, N., Phys. Rev. Lett. 62, 3050 (1988).
Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991
Received April 30, 1990; accepted August 3, 1990. We study the spreading of a (nonvolatile) wetting liquid on a fiat solid surface. For small droplets, capillary forces drive the spreading and the shape of the spreading liquid is well known to be close to a spherical cap, with a radius R ( t ) ~ t ~/~° (where t is the spreading time). We investigate here the opposite case of "heavy droplets" (R > K-L, the capillary length), for which gravity controls the process. The velocity of spreading may be understood from the rate of conversion of gravitational energy into viscous losses. The latter process can be divided into two contributions, one from the advancing wedge and one from the central part (squashed under gravity). Depending on which dissipation mechanism is dominant, two different shapes of the spreading droplet can be observed: (a) For R < R~ (Re = r - q n ( 1/Ka) ~ few centimeters, where a is the molecular length), the dissipation in the wedge is dominant. The drop has a quasistatic shape, with a large fiat portion of thickness h = K-a0d (where 0d is the dynamical contact angle). (b) For R >> Re, the dissipation in the bulk is dominant. The drop is not fiat; most of the profile is described by a self-similar solution first postulated by Lopez et al. ( 1 ) (except for a "foot" of size K-1 at the droplet edge). The growth of the drop radius obeys a power law R (t) ~ t ~/8 only in the asymptotic regime of very large droplets R >> Re. © 1991AcademicPress,lne.
I. INTRODUCTION
Wetting processes are important for many practical applications involving paints, lubricants, cosmetics, insecticides, etc. The statics and the dynamics of spreading of microscopic droplets ( 10-4/~1) has been extensively studied within the last few years (3-8) down to the molecular level, and the shape and extension of the precursor film have been established. Our aim here is to study the opposite case of very large droplets ( 100 ~zl) extending over several centimeters, called "heavy droplets" because gravity becomes the major driving force. In spite of their practical importance in achieving thick films of thickness of several hundred of micrometers covering large surTo whom correspondence should be addressed.
faces, only a few theoretical ( 1, 2) and experimental (9-11) studies have been reported until recently. On the theoretical side, Tanner (14) has studied the effect of gravity on the profile near the contact line for relatively small drops (KR < 3). Lopez et al. (1) have studied the last stage of spreading, where gravity and fluid viscosity are the chief promoting and resisting spreading factors. In Ref. (1) the curvature effects and the singular behavior of the viscous dissipation in the drop's edge were neglected. Our aim here is to take into account these two factors. We shall see that this leads to a n e w r e g i m e , where the dissipation at the contact line becomes the major resisting force. The pressure equilibrates faster in the central part, which becomes flat. We study first the main results on the statics and dynamics of wetting, which will be useful for the foUowing.
518 0021-9797/91 $3.00 Copyright© 1991by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand InterfaceScience, Vol. 142,No. 2, March 15, 1991
SPREADING OF "HEAVY" DROPLETS II. EQUILIBRIUM SHAPES OF DROPLETS: PARTIAL VERSUS COMPLETE WETTING
When a liquid drop is put into contact with a flat solid, two distinct regimes may be found: partial and complete wetting. The crucial parameter is the spreading coefficient S = 3"so (3"SL + 3"), which measures the difference of surface energy between the bare solid (-rso) and the wet solid (3"SL + 3"), where 3"SLand 3' are the solid/liquid and the liquid/air interfacial tension, respectively. (A) Partial wettingis found for S < 0. The contact angle 0E at the triple line is finite and given by the Young relation -
3"so = 3"SL+ 3" COS 0E.
[ 1]
The classical equilibrium shapes of a liquid drop are shown in Fig. I. If the drop is small, gravity effects are negligible. The pressure in the droplet is uniform, and the shape is a spherical cap. On the other hand, if the drop is larger than the capillary length K-l (K2 = pg/ 3", where p is the weight per unit volume, and g the acceleration due to gravity), it is flattened by gravity. The drop forms a thick "pancake" of thickness h0, which can be derived from the balance of horizontal forces acting on the hatched zone as shown in Fig. lb 3"so = 3" + 3"SL-- (pgh2)/2,
[2]
where the last term comes from hydrostatic pressure integrated over the thickness h0. From Eqs. [1] and [2], we obtain h0 = 2 r - t s i n ( 0 E / 2 ) ~ K-~OE
[3]
in the limit of small contact angles. (B) Complete wetting is found for S > 0. The droplet spreads over to a very thin film of thickness e, resulting from a balance be-
a
b
-9 o
519
tween long range and capillary forces (6, 12). In the present paper, we focus our attention on complete wetting, with emphasis on the macroscopic aspects of the spreading, for which long range forces will not be important. We do not describe the precursor film surrounding the droplet which was first considered in Ref. ( 1 ). III. A REMINDER ON THE SPREADING OF A SMALL DROPLET
In the regime where gravity is negligible (R K-1 ), the macroscopic shape of the spreading droplet is rather close to a spherical cap (Fig. 2) because the pressure equilibrates very fast in the thick region. The apparent contact angle Od(t), the drop thickness h(t) and the radius R(t) are related by h = ( 112)ROd
~2 = (Tr/2)hR 2,
[4]
where ~2, the drop volume, is assumed to be constant (unvolatile liquid). 0a is related to the velocity U = dR/dt of the contact line by the Hoffman-Tanner law (13-15)
U I V * = c t e O 3 (0d ~ 1),
[51
where V* = 3'/7 is a typical velocity; ~/is the fluid viscosity. The cte is found to be of order 10 -2 from Hoffman data (13). From Eqs. (4) and (5), one deduces easily that the radius R increases with time as t 1/1°, since one has R9(dR/dt) = cte V*f]3(4/Tr)3; i.e., R 1° ~ t + to. This has been measured experimentally by several authors ( 11, 15 ). One finds in Ref. (6) two derivations of the Hoffman-Tanner law. In the first one, the total dissipation is expressed as the product of a flux
Y
•
R < K -1
FIG. 1. Equilibrium shape of a liquid drop in the case of partial wetting: (a) with R < K-~ (capillary regime); (b) with R >> K-1 (gravity regime). Journal of Colloid and lnterface Science, Vol. 142, No. 2, March 15, 1991
520
BROCHARD-WYART
/
ET AL.
stage, we shall not need to write explicit equations for the pressure gradient dPIdx: we express our results in terms of the translation velocity U. The total flux matter is
R FIG. 2. Dynamical shape and physical parameters of a spreading droplet in the case of complete wetting for R K-l.
U by a noncompensated Young force. The entropy source per unit length of line is
TS = FyU,
[6]
Uf =
S measures the driving force of the process, but U does not depend on S! As shown by de Gennes (6), S is entirely burnt in the precursor, a microscopic film which surrounds the macroscopic droplet. This explains why the spreading velocity of the macroscopic droplet is independent o r S . Another presentation of this result amounts to say that the macroscopic drop spreads on a thin liquid film and thus sees an apparent spreading coefficient equal to zero. On the other hand, the precursor is very sensitive to S. In the following, we focus our attention on the macroscopic drop only, and we may drop S in the unbalanced Young force. The dissipation in the wedge TSw is then TSw = ~1 3`0~2 U.
[7]
The dissipation in the wedge can easily be calculated in the limit of small 0d by using the "lubrication approximation" to derive the flow pattern in the wedge advancing at constant velocity U. The velocity profile V ( z ) is of the Poiseuille type
nVx(z) = ( I / 2 ) ( d P / d x ) ( z 2 - 2z~'),
[8]
where x is the direction normal to the contact line and ~'(x) is the local fluid thickness. Equation [ 8 ] ensures Vx (z = 0) = 0 at the solid surface and (OVx/Oz) (z = ~') = 0 at the liquid/air interface of coordinate ~'. At this Journal of Colloid and Interface Science, Vol. 142,No. 2, March 15, 1991
[9]
One can then write
Vx(z) = ( 3 U / 2 ~ z ) ( - z 2 + 2 f z ) .
[101
If we now assume a simple wedge shape for the profile (angle 0d), we find the viscous dissipation integrated over the film thickness
where Fy is the Young force Fy = 3'so - 3`sL - 3' cos 0d = S + (1/2)3,02.
Vx(z)dz.
TSw =
xffmnax U 2 U2 3rt ---f- dx = 3rt --~a L,
[11]
in
where L = ln(xm,~,/Xr, i~). x=~., is related to the macroscopic size of the droplet and Xmin is a molecular size (6). Using Eq. [ 7 ] this leads to the H o f f m a n - T a n n e r law (14, 15) in Eq. [ 12]: 0a3 = 6 L
~.
[12]
This simple argument depends on a specific assumption on the profile and leads to an underestimation of the numerical factor as we shall show now. The H o f f m a n - T a n n e r law can be deduced more generally from a direct calculation of the profile (7) near the triple line. The discussion is complex, because one must include the precursor film, which depends on long range forces. From Eqs. [ 81 and [ 9 ], one obtains the pressure balance near the drop edge
U = (f2/3o)(-oe/ox),
[131
where (OP/Ox) = - 3 ` f . . . . (Orr/Ox). r is the disjoining pressure due to long range forces (15). Without this term, Eq. [ 13 ] has no physical solutions ( f is strictly positive). For a simple van der Waals liquid, Eq. [ 13 ] becomes
3 ~ / ( U / ~ "2) = - - T ~ ' ' + ( A / 2 7 r ~ ' 4 ) ~ "',
[141
where A is the Hamaker constant. Equation [ 14 ] has been solved numerically by Hervet
521
SPREADING OF "HEAVY" DROPLETS
and de Gennes (7). They find a special solution corresponding to (OP/Ox) --~ 0 far from the line, in the form
= ( U / V * ) I / 3 x ( 9 ln(x/Xmin)) 1/3,
[15]
where Xmi,~= (a/Of). From Eq. [ 15 ], one can derive the apparent contact angle
03 = 9 L ( U / V * ) .
[16]
Notice the analogy between Eq. [16 ] and [ 12 ]. The numerical coefficient differs by a factor 3 because the dissipation in Eq. [12 ] is cal~, culated by assuming an approximate form of the profile. From Hoffman data, the logarithm L is of the order 11 for dynamic contact angle above several degrees, while lower values, L 4-5, have been reported for 0d smaller than l° (17). IV. SPREADING OF HEAVY DROPLETS: TWO TYPES OF DISSIPATION
Our aim here is to extend the approach of Section II to describe the spreading of large droplets. We shall use two methods based (i) on a calculation of the dissipation and (ii) on a numerical calculation of the profile in Section V. For a very small droplet, the shape is quasistatic, i.e., almost a spherical cap, with a dynamic contact angle 0d instead of an equilibrium contact angle. For a heavy droplet, the quasistatic shape will be a flat drop, of thickness h = K-10d according to Eq. [3]. We shall show that the real profile is quasistatic only in a certain range of sizes (R smaller than a threshold Re). But this regime will turn out to be the most important since most of the experiments encountered in practical situations fall into this class (5, 9, 10). For large drops (R > r -1 ) the spreading rate is controlled by a balance between viscous resistance to flow and the gain of gravitational energy. Let us for a moment assume that most of the drop is flat, with a thickness h and an apparent contact angle 0d. The energy conservation is written as
TS + og(h/2)f2 -- 0,
[17]
where t2 is the volume of the liquid drop. The entropy source TS can be written in terms of the velocity U = d R / d t of the contact line: TS = 27rR3~(U2/0~)L
+ (3/2)~rR2~l(U2/h).
[18]
The first term is the dissipation in the drop edge discussed in Section III, Eq. [ 11 ] and integrated over the drop perimeter. The second term is the dissipation in the flat part: the horizontal velocity field in the flat part at a distance r from the center is simply U(r) = ( r / R ) U . Integrating the dissipation ( f 3~ × ( U 2 / h ) ( r / R ) 2 27rrdr) over the flat part, one gets the second term of Eq. [18]. We must also impose mass conservation
~2 ~- a~rR2h.
[19]
Let us assume for the moment that a is a constant of order unity. From Eqs. [17 ], [18 ], and [ 19 ], one gets
( U/V*)[67r(R/O,~)L + (3/2)zr(R2/h)] = K2f~(h/R).
[20]
This leads to two regimes depending on the dominant mechanism:
(A) Dissipation Controlled by Wedge (R < R c) In that case, the first term of the left hand side in Eq. [20] is dominant and Eq. [20] reduced to
(6L/Od)(U/V*)
= (~2K2/ozer2R4).
[21]
From Section III, we know that 0d is related to the velocity U of the contact line by 03 = 6 ( U~ V* ) L. Equation [ 21 ] then gives 0~1 = o ~ l / 2 r h -~ K h .
[22]
Thus the thickness of the drop h(t) is related to the dynamic contact angle 0d by the static relation (Eq. [3]). We do check that, in this regime, the profile is quasistatic. Inserting Eq. [22] in the Tanner law leads to Journal of Colloid and InterfaceScience, Vol. 142,No. 2, March 15, 1991
522
BROCHARD-WYART
6 L ( U / V * ) = (Kh) 3
[23]
and using Eq. [ 19 ], with a assumed to be conslant, leads to R 7 -~ ( ~ 3 / o ¢ 3 / 2 ) ( K 3 / T r 3 ) ( V * / L ) ( I
q- to).
[241 Equation [23] should hold up to a critical radius Re, at which the dissipation in the bulk and in the wedge are comparable. At the crossover, we may still write 04 = ~h, and Rc deduced from Eq. [20] is Re = 4LK -1
[25]
By taking K-1 - 1.5 mm, and L ~- 12, one expects Rc ~ 7 cm.
(B) Dissipation Controlled by the Bulk ( R > Rc) In this case the second term of the left hand side of Eq. [ 20 ] is dominant, and one gets
(3/2)~r2e~(R4/~)(U/V *) = ~2K2/a~-R3
ET AL.
where the pressure P may be written
P = -,),(O2~/O2r) q- p g f + 7r. The first two terms are the capillary and gravity contributions. The last term is the disjoining pressure (16) due to long range forces (van der Waals, electrostatic, steric) which show up for thin films. This term has to be included to describe the cross-over between the macroscopic drop and the precursor (7). We must add to this the local equation of volume conservation
(1/r)(O/Or)[(r~U(r, t)] + (O~/Ot) = O. (30) From Eq. [29], one can expect four regions (Fig. 3). - - A central region I where gravitational forces dominate: 3n[V(r, t)/~ "2] = -og(O/Or)~.
[31]
- - A meniscus H, where the Laplace pressure becomes relevant:
[261 Eq. [26 ], assuming ~ to be constant, leads to
R 8 _~ ( f ~ 3 / o t 2 ) ( r 2 / T r 3 ) V * ( t + to).
[271
Thus in this limit of very large drops, we recover the result of Ref. ( 1 ). In this regime, 04 is not related to h by the quasistatic relation (Eq. [3]). Taking U f r o m Eq. [26] and using the Tanner law (Eq. [12]) one finds
04 ~- Kh(RdR)
[28]
or equivalently 0d = Khl, with h, = h(Re/R). We shall see in Section V that in this regime the drop is indeed not flat, and is not described by the quasistatic profile of Eq. [ 3 ]. V. P R O F I L E O F T H E S P R E A D I N G HEAVY DROPLETS
3~[ U(r, t)/~ "2] = V(03~-/03r)
--
pg(O~/Or).
- - A proximal region III near the macroscopic triple line of length (a/O~) and height (a/Od), where the long range van der Waals forces (described by r ) dominate (for a more detailed discussion of this region, see Ref. 7); - - A precursor film I V described in Refs. (1, 7). (2) Similarity Solutions (1) Lopez and co-workers (1, 2) dropped the Laplace term and the disjoining pressure in Eq. [29]. They looked for a self-similar solution of Eqs. [ 30 ], [ 31 ] expressed as
(1) Basic Flow Equations
U(r, t) = U(t)g(u)
In the lubrication approximation, the mean radial velocity U(r) is related to the fluid thickness ~-by a Poiseuille equation
~(r, t) = h(t)f(u),
37( U(r, t)/~-2) = -OP/Or,
[29]
Journalof CoUoidand lnterfaceScience,Vol. 142, No. 2, March 15, 1991
[32]
[331
where u = r/R. The functions f a n d g which are solutions of Eqs. [ 30 ], [ 31 ] satisfy the two equations
523
SPREADING OF "HEAVY" DROPLETS
oee~"
u.
°t
I
I
I
I
[
I
~ IV IIII II I I ~'- - ~* " ~ " - ' ~ . . . . . . . . . . . . . . t
I"I"
I i
•I-
L
may be treated as constant U(r) = U. The profile f ( r ) is then given by
L [
3 r / ( U / ~ "2) = "y(~". . . .
K2~").
I
i
ho(t )
We must find a solution of Eq. [38] which matches the behavior near the contact line (Eq. [161)
X1
R
WIDTH
OF
THE
By integration of Eq. [38], x = r - R , we find
DROP
FIG. 3. Schematic profile of a heavy droplet, showing the four regions: (I) Central region where gravitational force dominates; (II) the meniscus where the Laplace pressure is comparable to the gravitational force; (Ill) the proximal region where the long range van der Waals forces show up; (IV) the precursor film where they dominate.
3n
dx =
2u = - 3 f 2 ( d f / d u ) .
rain.
1
---- v ( t '2 - / Y " ) +
[341
[35]
Imposing the volume conservation of the drop leads to 2rcf(u)rdr
= 7rR2h
yo
2 f ( u ) u d u = f~
[36]
with rrR2h = ~2/a = const. ( a = 0.75). From Eqs. [ 32 ] - [ 33 ], one obtains an equation for R(t) R ( t ) ~ ( K 2 Q 3 V * t ) 1/8.
1
2
og/'. [40]
The profile is t h e n f ( u ) = (1 - u2) 1/3 One obtains for the radius of the droplet 2(dR/dt) = (h3/R)og.
and setting
- y ( f f " - ~2~f')dx
in.
g(u) = u
f0
[39]
Od = ~' (X = O) = [ 9 ( U / V * ) L ] ~/3. 0
h
[381
l- - -
[37]
Note that, in this approach, the shape of the drop edge is not well described and the viscous losses in the edge leading to a logarithmic singularity are omitted. We shall now see that this similarity solution is applicable only for R >> Re, where the edge losses are indeed negligible. (3) Approximate Calculation of the Profile (a) The meniscus. In a region of size K-I near the contact line, the radial velocity U(r)
Equation [40] may be interpreted as a balance of horizontal forces on a portion of the liquid edge extending from Xmi.. to X. The viscous force (left hand side of Eq. [ 40 ]) equilibrates the noncompensated Young force [3'so - (3' cos O(x) + "YSL)] = ( 1 / 2 ) 7 ~ "'2 and the hydrostatic contribution, f (P - Po)dz = __,]/~-~-t,' _]_ 1 ~ p g f ,2 where P0 is the external pressure. We shall now describe a very rough construction of the profile, which will, however, be useful for the interpretation of the exact numerical solutions which are discussed later. Approximation (i): Since the left hand side of Eq. [ 40 ] is only logarithmically dependent upon x, we treat it as a constant C (in the region Xmin. < X <~ K - l ) . For x ~ Xmin. C i s given by setting ~ ~ 0 in the fight hand side of Eq. [40]: C = (1/2)702 .
[41a]
For x > K-~, the slope O(x) and the curvature are negligible and C is given by C = ( 1 / 2 ) p g f 2.
[41b]
Approximation (ii): We assume that for x K-l, the thickness is slowly varying, and we write h = hi. Comparing the two forms [41a,b] we see that hi = K-10d. The thickness hi of the Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991
524
BROCHARD-WYART
meniscus corresponds to the quasistatic thickness (Eq. [ 3 ] with 0E = 0a). We now return to Eq. [38] in the region x >- K-1. Here ~'' is negligible, and the slope ~" = 01 is given by
O~ = [3U/V*(~hl):] ~- (Oo/3L).
[42]
Thus, as announced, the slope is small [01 -~ ( 0 j 10)]. We can now proceed to discuss the central region of the drop, with Eq. [ 42 ] acting as a boundary condition at r = R - xl (where xl is defined by ~'(Xl) = hi ). (b) Profile in the central region. In the central region, the profile is obtained by direct integration of Eq. [ 31 ], incorporating only viscous forces and gravity effects. Assuming a radial velocity U(r) = ( r / R ) U, one obtains
3(U/V*)(r/R)
= --K2~2(O~/Or).
[43]
Introducing the height of the drop at the center
ho = ~"(r = 0) we get f = --9( U/V*)(rZ/2RK 2) + h~.
[44]
(c) Complete profile. We now match the edge and center by imposing the boundary condition [42] on Eq. [44]. This occurs at x = xl ; i.e., r = R - xl ----R. The result is (Kh0) 3 = (Khl) 3 + ( 9 U / V * ) ( r R / 2 ) .
[45]
Assuming hi = K-10a, Eq. [45] can be rewritten (Kho) 3 = (9UL/V*)[1 + R/Rc],
[45'1
where Rc = 2 LK-1. We recover the result obtained from the calculation of the dissipation, except for the numerical factor (half of the preceding value). This discrepancy is due to the simplified profile used to estimate the dissipation. Equation [45 ] shows: ( 1 ) that the drop is flat ifR < Rc; the thickness ho of the center of the drop is roughly equal to the height hi of the meniscus. This corresponds to the quasistatic regime. (2) On the other hand, ho becomes much Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991
E T AL.
larger than hi for R > Re. We recover the selfsimilar regime (Eq. [ 35 ] ).
(4) Numerical Calculations of the Drop Profiles A complete calculation of the profile of the spreading drop involves the resolution of the time dependent equations [29 ] and [30 ]. We have simplified the problem by the two following assumptions: (i) The boundary condition at the contact line is provided by the Hoffman-Tanner law (13); (ii) the velocity U(r, t) is taken of the form
U(r, t) = (r/R)U(t).
[46]
This law for the velocity is exact in the asymptotic limit where the similarity solution holds (R > Rc). It is also exact for a film of constant thickness squeezed between two plates, which corresponds to the regime R < Re. We solve Eq. [32 ] numerically starting from the edge (x = 0) with U(r, t) = U[1 + (x/ 2)1:
3(u/v*)[1 + (x/R)] = ~.2(~-. . . . Kq-'). [47] We set ~"= YoZ and x = K-1X, thus Eq. [47] becomes
1 + (X/KR) = +Z2(Z .... Z'),
[481
where we have chosen (KY0) 3 = 3 ( U / V * ) .
[49]
The structure of Eq. [ 48 ] shows directly that one cannot find a self-similar profile, because two different characteristic lengths (R and ~-1) come into play. Numerical integration of Eq. [48]. (i) We take KR as a variable parameter, which we vary from 5 to 100. (ii) We generate the profile starting from the drop edge imposing
Z = X ( 3 L ) 1/3,
[50]
the boundary condition at the edge (X --~ 0), in agreement with Eq. [ 15 ].
S P R E A D I N G OF " H E A V Y " DROPLETS
We choose several discrete values of L, and we focus on L = 11 to fit Hoffman's data. We start at Z~ = 0.09, Z~ = -3.2, and we vary the curvature Z " at this point. Each choice of Z " leads to one solution of the thirdorder equation [48] [see Fig. 4 (solid line)]. For one particular choice of Z " = Z~ the solution has zero slope at the center of the droplet Z ' ( X = --KR) = O.
The original choice of Ze and Z " ensures that Z~ is always <0 and close to zero, that is in the macroscopic part of the drop. This solution is represented on Fig. 5, for various values of KR (ranging from 5 to 100) spanning both regimes KR < KRc and ~R > KRc. In Fig. 5 we also show the self-similar solution corresponding to the same value of h0 (or Zo). We see that for R < R~ the drop is fiat, as expected from our rough arguments. For R > Rc the self-similar solution is correct (except near the edge). A crucial test of our simplified discussion is provided by a plot of Z 03as a function of KR
(Fig. 5). This plot is found to be linear, as predicted by our Eq. [45], and we set (Z0/Zl) 3 = 1 + ( R / R c ) ,
[511
where Z~ is the limit in value of Zo for R ~ R~ --~ 0. We find Z1 = 1.086 I Z " I (while in the crude discussion we had ZI = Z~). From the slope, we find for the critical radius ~Rc = 29. To test the dependence of R~ with L, we have also studied the case L = 6 (Fig. 5 ). We findZ~ = 1.15 [Z~[ a n d K R ~ = 18 for both values of L; Z 03has a linear dependence with ~R, with a slope of-~. This is exactly the result of our rough analysis. Equation [45 ] written in reduced variables becomes Z03 = Z 3 + (3/2)(KR)
[521
which leads to KRc = ( 2 / 3 ) (Z 13). As Z~ is nearly equal to 1.1 ]Z~], R~ can be approximated by KR~ ~ 2 L ( I . 1 ) 3 ~ 2.7L.
i
Z
525
Z
.
[53]
r
t O. o "o
I I
0
lb
s
0
X
I d
0
'2'o
0
4a
10
20
4b
q)
J~
Z
a/.
o) o
.f,
i
0
0
50
0
4c W i d t h
of
the
50 4d
100
d r 0 p (KR)
FIG. 4. C o m p u t e d drop profiles o f heavy droplets in the case of complete wetting. Full line curves represent the numerical solutions of Eq, [47] obtained with L = 11 for R < Rc (a and b) and R > Re (c and d). The dotted curves are the self-similar profiles obtained for the same ho. Journal of Colloid and Interface Science, Voi. 142,No. 2, March 15, 1991
526
B R O C H A R D - W Y A R T ET AL.
200
¢0
H3
~: l o 4
D.
o
e0
i
i
i
i
i
i
i
i
i
I
i
i
i
I
i
i
i
R
~J
~oo
U.
o~ ee-
/ cross \ / over
o
J=
R o
~
Radius
of
I
50 the
drop
100 (K -1)
FiG. 5. Plot of the thickness at the center of the drop, to the third power, in reduced coordinates versus the radius of the drop in ~-~ units. The straight lines represent the best fits obtained for R > 10 K-~ for two values of L: ( + ) L = I I ; ( O ) L = 6,
For each profile, we have calculated the volume ~2 of the drop normalized by the volume of a flat disk of thickness h0 and radius R: [54]
ot = ~2/TrhoR 2.
The variation of a versus R is shown in Fig. 6. We see that a goes through a maximum for R ~ 1.6Re: in the regime where the drop is
1
O~
e
10-10
1027 T I M E (A.U,)
FiG. 7. Log-log plot of the variation of the radius R of a drop versus time. Notice that 5 decades in R correspond to 37 decades in time!
flat, a increases because the correction due to the drop edge decreases as 1/R. Above Rc, a decreases toward the asymptotic limit o~ = 0.75. One must notice that even for KR 100, a is still much larger than a ~ . (5) Numerical Spreading Laws
From the calculated values of Z0 (Fig. 5 ) and the exact relationship between ho and R given by the volume conservation (Eq. [55] and Fig. 6) we are able to calculate the time dependent spreading laws JR(t), U(t)] from the early stages (R < K- 1) up to the long time limit (R >> K-1 ). From Eqs. [49] and [55], we deduce the differential equation relating U = dR~dr to R: U = (V*K3~23)/(37r3ot3(R)Z3o(R)R6).
We have calculated R (t) by integration of Eq. [56] using interpolated values for a ( R ) and Z o ( R ) . The result is shown in Fig. 7 for a time scale extending over 38 decades:
"~ "~" m ®=lUO~o= e " . . . . . . . . . .
0.5
100
0
Radius
of
the
drop
(K -1)
FIG. 6. Plot of the reduced volume ~ = f~/thoR 2 versus the radius R of the drop for two L values: L = 11 (solid line); L = 6 (dotted line), a ~ corresponds to the value obtained for the self-similar model, where ~ is independent of R . Journal of Colloid and Interface Science,
Vol.
142, No.
[55]
2, M a r c h
15, 1991
- - F o r KR < 1, we find the classical capillary regime R "-~ t 1/1°. - - F o r 1 < KR < KRe, we are in a cross-over region. From Eq. [ 56 ], we see that if c~(R) was a constant, R ( t ) should be proportional to t 1/7 . But due to the variation of a ( R ) in this region R ( t ) does not follow a pure power law.
SPREADING OF "HEAVY" DROPLETS
527
- - F o r R > Re, R ( t ) obeys the power law R t 1/8. This is due to the fact that a is almost a constant; also it has not yet reached its a s y m p t o t i c value a ~ = 0.75.
ular size a and the capillary length (K -1)]. T h u s the pure gravity regime (self-similar behavior) does not show up for R > K-1, contrary to a c o m m o n belief, but only for R > Rc >> K-1.
VI. CONCLUSION
ACKNOWLEDGMENT
On a completely wettable, plane, horizontal surface, f r o m our analysis we expect three regimes for the spreading o f a liquid droplet as a function o f the radius R (t): (i) R ( t ) is smaller than the capillary length K-~. Capillary forces are d o m i n a n t and viscous dissipation occurs m a i n l y in the drop edge. T h e droplet has the shape of a spherical cap, and a radius R ( t ) ~ t 1/1°. (ii) K-1 < R ( t ) < Re. G r a v i t y drives the spreading. Dissipation occurs again m a i n l y near the contact line. This leads to a new quasistatic regime: the drop is almost fiat, with a thickness h0 ~ K-10d, where 0d is the d y n a m i c contact angle. T h e spreading law is not a simple p o w e r law. (iii) R ( t ) >>Re. G r a v i t y controls the spreading, but the viscous losses in the central part o v e r c o m e the dissipation in the drop's edge. T h e spreading droplet is not fiat. This is the d o m a i n covered by the pioneering p a p e r of Lopez et al. (1). However, we show that their p r o p o s e d spreading power law R ( t ) t 1/8 is only correct in the limit o f very large drops: R > 10Rc. In practice this regime is far b e y o n d the usual range studied experimentally. It has to be noticed that the critical radius Ro is m u c h larger than K- 1, the capillary length, Rc ~ 3LK-1 [here L is a logarithmic factor, controlled by the ratio (Ka) between a molec-
The authors thank P. G. de Gennes for various discussions and useful comments to manuscript. REFERENCES 1. Lopez, J., Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 56, 460 (1976). 2. Joanny, J. F., Thesis. Le Mouillage, Paris, 1985. 3. Heslot, F., Fraysse, N., and Cazabat, A. M., Nature 338, 640 (1989). 4. Beaglehole, D., J. Phys. Chem. 93, 893 (1989). 5. Daillant, J., Benattar, J. J., Bosio, L., and Lrger, L., Europhys. Lett. 6, 431 (1988). 6. de Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 7. Hervet, H., and de Gennes, P. G., C. R. Acad. Sci. Paris Ser. 2 299, 499 (1984). 8. Ruckenstein, E., and Lee, P. S., Surf. Sci. 50, 597 (1975). 9. Bascom, W. D., Cottington, R. L., and Singleterry, C. R.,Adv. Chem. Ser. 43, 355 (1964). 10. Ogarev, V. A., Ivanova, T. N., Avslanov, V. V., and Trapeznikov, A. A., Izv. Akad Nauk. SSSR Set. Khim. 7, 1467 (1973). 11. Levinson, P., Cazabat, A. M., Cohen Stuart, M. A., Heslot, F., and Nicolet, S., Rev. Phys. Appl. 23, 1009 (1988). 12. Joanny, J. F., and de Gennes, P. G., C. R. Acad. Sci. Paris Ser. 2 299, 605 (1984). 13. Hoffman, R., J. ColloidlnterfaceSci. 50, 228 (1975), 14. Tanner, L., J. Phys, D 12, 1473 (1979). 15. Lelah, M., Marmur, A., J. Colloid Interface Sci. 82, 518 (1981). 16. Deryagin, B., and Churaev, N., Kolloidn. Zh. 38, 438 (1976). 17. Chen, J. D., and Wada, N., Phys. Rev. Lett. 62, 3050 (1988).
Journal of Colloid and Interface Science, Vol. 142, No. 2, March 15, 1991