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Wetting in nonhomogeneous situations: Capillary rise experiments
Colloids and Surfaces, 51 (1990) 309-322 Elsevier Science Publishers B.V., Amsterdam
309
Wetting in nonhomogeneous situations: Capillary rise experiments
A.M. Cazabat 1 and F. Heslot Coll~ge de France, Physique de la Mati~re condensde, 11 place MarceUin Berthelot, 75 231 Paris Cedex 05 (France) (Received 18 May 1990; accepted 4 July 1990)
Abstract
Exploratory experiments on wetting in nonideal situations are reported. The capillary rise of a completely wetting, nonvolatile liquid has been investigated in the presence of nonwetting defects, surface contamination or thermal gradients. A qualitative analysis of the results is proposed in each case. The aim of these studies is to provide information for forthcoming theoretical analyses.
INTRODUCTION
Wetting and spreading are now fairly well understood in simple cases: pure liquids, smooth, clean surfaces, etc. [1-4]. However the reality usually differs from these ideal situations. Chemical defects or roughness of the solid surface [5], contamination of the interfaces [6], or thermal gradients [7] may drastically affect the wetting behaviour. Only in very special cases are theories available [8,9] for nonideal situations. We present here an experimental investigation of the effects of such perturbations on the behaviour of a nonvolatile, completely wetting liquid in the capillary rise geometry. We have studied: (i) the deformation of the contact line in the presence of a nonwetting chemical defect; (ii) the influence of the contamination of the surface by a thin layer miscible with the liquid; and (iii) the influence of a vertical thermal gradient along the solid surface. As no quantitative theory is yet available, we shall analyze our results in terms of simple qualitative models. Let us first recall briefly the main characteristics of the nonperturbed system. 1To whom correspondance should be addressed.
0166-6622/90/$03.50
© 1990 - - Elsevier Science Publishers B.V.
310 SUMMARY OF T H E RESULTS IN T H E IDEAL CASE
The capillary rise geometry Figure 1 summarizes the main features of the experimental setup and the symbols we use for the variables. The xy plane coincides with the horizontal free liquid surface and the yz plane with the vertical solid surface. At the macroscopic scale ( x > #m) the meniscus intersects the solid along the contact line z ( x ~ 0,y) which is horizontal ( z = h = c s t ) in the ideal case. At the microscopic scale, a thin film (x < # m ) may grow on top of the contact line (z> h).
Macroscopic scale (ideal case) Let y be the surface tension of the liquid, p its density, g the acceleration of gravity and 0e the equilibrium contact angle of the liquid on the solid. The position of the contact line is, at equilibrium:
z = h , h = h o ( 1 - s i n 0e) 1/2
(1)
where ho = k - 121/2, k - 1 being the capillary length ( ~,/pg ) 1/2. The thickness profile of the meniscus is:
z = h o ( 1 - s i n 0) 1/2
(2)
For 0 < h - z << h, i.e., very close to the contact line, one has the approximate forms at low angles (0e -< 0 << 1 ):
(X/Oe)
For 0e ¢ 0 (partial wetting) z ~ h -
For 0 = 0 (complete wetting ) z ~ ho - ( xho ) z/2
(3)
(4 )
Z solid
ill
'thin film
~e
'~":0
::::~
~
~
~
r macroscopic 7~,,~
~,
~
~
~'
~
meniscus
~
~
~
~w"
liquid
Fig. 1. The capillary rise experiment: summary of the symbols. Macroscopic scale. The y axis is perpendicular to the plane of drawing.
311
Microscopic scale Strictly speaking, at thermodynamic equilibrium, the macroscopic meniscus can be followed by a microscopic film, the thickness e of which is given by the relation:
~(e) =pgz
z> h
Here, ~ is the disjoining pressure [ 10,11 ] which is a relevant parameter for the description of thin films [ 1 ]. The corresponding values of the film thickness are in the ~, range for partially wetting liquids and of the order of a few tens of nanometers for completely wetting ones (z > h). In the present study, we use a nonvolatile liquid, which means that the transport of molecules towards the solid through the atmosphere is negligible. Thus, the growth of this film is very slow [12]. Within one day, even for complete wetting, only a film of molecular thickness grows significantly on top of the meniscus, the length of which is approximately:
L(t)
~ ( D t ) 1/2
For polydimethylsiloxane ( P D M S ) of low molecular weight (M-- 2400) one finds [12]: D = 2 . 8 - 1 0 -1° m 2 s - ' on a silica surface. This molecular film obviously plays no role when the influence of nonwetting defects is investigated, because the duration of the experiment is only a few minutes. This is not so clear in the two other cases and we shall have to address the question further. We shall now discuss the influence of nonwetting defects on the equilibrium properties of the macroscopic meniscus. NONWETTING DEFECTS
The experiment The liquid was PDMS, molecular weight 2400, surface tension 7= 20.6-10-3 N m -1, viscosity q = 2 0 . 1 0 -3 Pa s which completely wets the solid surface, a silicon wafer covered with natural oxide. Defects were vertical ink lines drawn on the solid surface, which were not wetted by P D M S (overhead Staedler projector pens are convenient! ). For the study of the coupling between defects, we drew slightly tilted lines in order to have a variable distance a between them and checked that the shape of the contact line was negligibly modified by the tilt angle. The contact line was studied with a microscope at low magnification. Equal thickness fringes were easily observed on this reflecting substrate and allowed
312
Photograph 1. Microscopic observation of the deformed contact line (yz plane). The vertical defect is the black area on the right. The upper, clear area is the dry wafer. The bottom, dark part is the PDMS macroscopic meniscus. The interference fringes along the contact line are equalthickness fringes (0.21/2m between fringes). The vertical fringes are due to internal reflections inside the optical setup and have no physical meaning. Horizontal size of the photograph 1 mm.
us to draw the thickness profile of the liquid close to the contact line (Photograph I and Fig. 2). The contact angle of PDMS on ink can be deduced from the height of rise, h, on a wide ink line. Here, it was easier to measure ho-h=0.28 mm (ho for PDMS--2 mm) which gave 0e~15 °, in good agreement with the value obtained with drops [ 13 ]. Thus, the contact angle goes from 0 e = 0 ° o n the silica to 0e= 15 ° on the ink. It is clear from Photograph 1 that the thickness profile is practically unchanged except very close to the ink (10 tim) where the curvature changes rapidly. Figure 2 shows that the unperturbed profile is satisfactorily described by Eqn (4) for h - z ~ 6 5 ttm, as expected. The perturbation of the contact line by the defect typically extends over 1.5 mm, i.e., the capillary length k-1. Two regions can be distinguished in the perturbed line: a smooth region ( a ) corresponding to the almost unchanged thickness profile, which is reproduced in Fig. 3; and a very steep region (fl), where the thickness profile changes drastically (both radii of curvature became comparable ). If two defects are present, they become coupled for a < 2 k - 1. In fact, the position of the contact line on the ink does not change (the width of the ink
313
1~z 'm,
:)
I
1
I
2
I
3
Xk IJ ' v
Fig. 2. Experimental profile (xz plane ) of the macroscopic meniscus and comparison with Eqn (4). Origin ofx on the wafer. Arbitrary origin for z.
lines was 0.5 mm). The question is, in which way are the deformations due to the two defects combined. Figure 4 gives a nonambiguous answer: the deformations ho-z(x=O) are additive. Let us note, however, that in this experiment only the a region of the contact line was investigated. If some nonadditive behaviour would have occurred in the fl region, it would not have been observable, because the effects vanish with vanishing a. Thinner ink lines are required to study coupling with curvature terms.
Discussion These experimental observations obviously ask for a model. At the present time, the only theory, to our knowledge, giving an explicit formula for the shape of a deformed contact line is the one proposed by de Gennes and J o a n n y [8] in the case of partial wetting and low contact angles. The deformation ~ (which is the equivalent of ho-z(x=O) in our case) can be written as:
314
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:-5
x
",, e x p
•,
Z
O•
Fig. 3. Smooth (o~) part of the perturbed contact line, zy plane. The origin is taken on the defect for y and at the position of the meniscus on the ink for z: (-~r) position of the contaet line z ( x ~ 0 ,y ) in the zy plane. Units: m m on both axes. An example of the fit according to the loglyl shape is given ( O ) exp z(with different units).
~+SA
.........................................................
z~
~a
7
Fig. 4. Coupling between defects. The deformation ei is measured at the mid-point between defects (a/2). ~ is the maximum deviation. 5 + A is plotted versus a ( • ) and compared to the sum 2~ (a/ 2 ) of the deviations ~, (~+ ~= A), due to each defect, at the distance a/2 { [] ). These deviations are measured in the outer parts of the line. One expects 2~ (a/2) = ~ + A if the additivity is met. Units: m m on both axes.
315
1 A C o c ~ log [y--~
(5)
where A is a cutoff length (A~k -~) and the defect is at y = 0 . This formula is expected to hold in an intermediate range, say k - 1 > y > a few pm. It is tempting to try to fit the a region of our contact line by a log[y[ shape, and in fact the fit z ~ - log [y[ + cst (which can be more precisely done as exp (z) becoming linear in y with a convenient choice of the scales ) is fairly good (see Fig. 3 } in the validity range of Eqn (5). It is not obvious, however, that this procedure makes sense: the thickness profile in de Gennes' theory is a linear function, Eqn (3), and ours is quadratic, Eqn (4). The mathematical structure of the equations in Ref. [8] suggests that log[y[ might still appear in the shape of the contact line, but not necessarily in a linear way. At the present time, the experiment does not contain enough information to suggest a model. It must be checked against a preexisting one. SURFACE CONTAMINATION
The experiment The contaminated surface was obtained by deposition of a thin layer of a liquid which is miscible with P D M S and has a higher surface tension. Good candidates were tetrakis (2-ethyl-hexoxy)silane (~= 26.7.10 -3 N m-1, ~/= 6.8.10- 3 Pa s) and squalane ( ? = 27.6.10- 3 N m - 1, ~/= 21.10 -3 Pa s). T h e y can be deposited by a spin-coating technique but the most efficient way was to wipe the surface, previously covered by a thick layer, with a lens-cleaning paper. Surprisingly enough, the layers obtained in this way, the local thickness of which is measured by ellipsometry, were fairly homogeneous, especially with squalane. Thin layers (30 .it) seemed to be stable over the time scale of the experiment. Thicker layers become unstable {which was expected because neither tetrakis silane nor squalane completely wetted the substrate) and progressively broke into very small droplets ( ~ pm). The contaminated surface was wettable by PDMS. However, the film which grew on top of the meniscus was much thicker and faster than the normal precursor on a clean surface. On thick contamination layers, the P D M S film showed strong instabilities close to the meniscus and a tendency to dendritic growth at the tip, which mainly reflected the breaking process of the layer. Although very reproducible, this behaviour was too complicated to be given more attention in an exploratory study. On thin layers ( ~ 20 .£.) the P D M S film had a much gentler behaviour. Its thickness was approximately constant, of the order of 100-200 nm. Its length
Lmm
316
O
~D
:}.5
t~
•.~r°
,J S ~
Fig. 5. Contamination by squalane. Thickness of squalane layer 21 _+2 A. Length of the main film as a function of the square root of the time and schematic thickness profiles. Typical thickness between 0.5 and 1 fringe (0.1 and 0.2 ]~m ). Various symbols correspond to different places on the wafer.
L (measured from the macroscopic meniscus ) first scaled at t '/2 and then more slowly (Fig. 5). This slowing down corresponds to a slight change in the film profile, schematically indicated in the figure: a minimum in thickness appeared close to the meniscus and the thickness in the flat part decreased progressively. This evolution which is illustrated in Fig. 5 for a thin layer of squalane, (thickness 21 + 2 ~,, the uncertainty corresponding to thickness variations from place to place) was also observed with tetrakis silane. The t '/2 region of the curve can be characterized by a diffusion coefficient D [ L = (Dt)l/2]. For squalane and the condition of Fig. 5 (layer thickness ~ 21 ~, ) one obtains: D = l . 5 . 1 0 -9 m 2 s-1 The same order of magnitude ( D ~ 0 . 5 - 2 - 1 0 -9 m 2 s -1) was observed for tetrakis silane with similar layers. However, probably due to strong lateral interactions between molecules in this case [ 14 ], the tetrakis layers were rather irregular. If more attention is paid, another much faster, very fugitive film can be observed. Its thickness is around 100 nm. Its length Lo cannot be measured precisely, due to the lack of contrast and the limited time of observation (40 s) but it is compatible with a diffusion-like formula:
317
Lo ~ (Do t)
1/2
where Do is about 5-10 - s m 2 s-1 both for squalane and tetrakis silane.
Discussion The growth of films much thicker and faster that the normal precursor film is obviously due to the gradient of surface tension 7. Close to the macroscopic meniscus, 7 is equal to the surface tension of P D M S , while at the tip of the film it is much higher, because some portion of the contamination layer has been solubilized. For a film of constant thickness e, a surface tension gradient dT/dz ( > 0) produces a surface velocity in the z direction: Vs-
e d7 dz
(6)
A Couette flow is obtained, with an average velocity:
e d7 dL U - 2 ~ d z - dt
(7)
If the gradient dT/dz has a constant value, a linear growth L ~ t is expected. If the gradient scales as L - ', a diffusion-like behaviour L ~ t 1/2 will result. This, together with the experimental observation that the film thickness e is practically constant and that L ~ t 1/2 at short times, suggests a simple model: Let us assume that the difference Ay of surface tensions at the tip of the film and at the meniscus is constant during the growth of the film. Then, dT~zb' dz~L
(8)
so that dL e Ay dt ~ 2 ~ / L
(9)
e~7
(10)
Dth ~--
at least if ~ does not change significantly, which is the case if the contaminant is squalane (~SQ~ ~PDMS). Let us put orders of magnitude in this relation: 100 nm~
~/=20"10 -3 Pa s :
7.10-3 N m -1
with zlT=ATmax, e= 150 nm, one obtains:
318
Dth --5.10 - s m 2 s -1 in good agreement with the Do value. Neither Do n o r Dth are precisely known. However, one can certainly conclude that the model gives a satisfactory description of the first, rapid film if A? is taken to be ~/l~max. A further assumption would be that the main film is described by the same model (in the t 1/2 region ) but with a much smaller value of A~,because the first film has solubilized a large part of the squalane layer. From D--1.5.10 -9 m 2 s-1 and e ~ 150 nm one would obtain: A?~0.2.10 -3 N m -1
<<2~max
This model is obviously very crude. It does not describe the mixing process between P D M S and squalane, which has its own characteristic times. Let us consider it as a first attempt towards the description of contamination-driven wetting films. TEMPERATURE GRADIENTS Surface tension gradients d~/dz may also be produced by temperature gradients along z. As ~ usually decreases with increasing temperature, the temperature gradient dT/dz must be negative for d~,/dz to be positive. One has: d? dy d T ~-dT dz
(11)
In this case, Eqns (6) and (7) hold. If d T / d z (thus d~/dz) is constant, and if the film has a constant thickness, it is expected to grow linearly with time.
The experiment The (clean) wafer was pressed against two temperature-controlled brass holders, 7 mm apart. The temperature difference between the holders was 28 ° C. The P D M S was put into contact with the wafer after about 30 min to ensure that the stationary distribution of temperatures in the wafer had been achieved. As in the preceding case, a thick film ( ~ p m ) , much faster than the normal precursor was observed. However, the behaviour of this film was quite different from that of the contamination case. It can be separated into different steps: First a film of approximately constant thickness climbs the wafer (a). At the tip of the film, a bump in the thickness profile develops (b), which ultimately becomes unstable (c) and results in a fingering instability with a rather well defined wavelength (d, Photograph 2). Between the meniscus and the bump (b,c) or fingers (d) the film is flat, with
319 i i
Photograph 2. Fingers in thermal gradients with equal-thickness interference fringes after 15 min. Upper: dry wafer. Bottom: constant-thickness film. Scale: the average distance between fingers is 480 pm. The meniscus is well below and is not visible in the photograph.
constant thickness emin. The fingers also have a flat part (Photograph 2) but are terminated by a drop of larger thickness em~x. The length of the film is well defined in step (a). For the next steps, one must introduce two lengths, one corresponding to the finger tip, the other to the "valleys" (Fig. 6). All these lengths vary linearly with time and can be characterized by a velocity value, V a in step (a), Um~xand Umi~for (b,c,d). One obtains: U~ ~ 3.3; Umax~ 3.6; and Groin ~ 2.6 pm s- 1.
Discussion Step (a) and the wavelength of the fingering instability (which is basically the Rayleigh instability of a cylinder but in a more complex geometry) are satisfactorily described by available theories [15,16]. For example, the expected velocity Ua.th is given by Eqn (7) with emin=0.54/~m and @/dz=0.21 N m -2 (i.e., dT/dz=AT/Az=4.28.103 K m -1 and ~1 =d), 2 . 5 " 1 0 -3 K_ 1 for # - -
PDMS). So Ua,th---- 2.8 ]l S - 1 , while Ua ~3.3 #m s -1. At the present time, there is no available theory for the following steps. The contact line instability clearly results from the Rayleigh-like instability of the bump, because the friction on the surface is less in the thicker parts which, as
320
~L
mm /
/ / e'
J
os
3 I J
.L
/
,t J
2 ,~
~s
18
• 711~
,.p
OJ
,,
5 !
10
15
t
mn IT
Fig. 6. Position of the contact line ( ~ , a) or of the tips and "valleys" ( 0 and ~ , b,c,d) as a function of time. The origin of time is arbitrary.
a consequence, grow faster. Let us note also that the ratio U m a x / U m i n is comparable with the ratio of the thicknesses in the flat part of the fingers ( ~ 3.5 fringes) and in the flat part of the main film (emin~ 2.5 fringes). There are various ways to obtain complementary results. One can change the P D M S viscosity and the thermal gradient [ 16]. Another possibility is to "force" the film to thicken by imposing (transitorily) a higher injection rate at the meniscus. One way is to move the P D M S external reservoir upwards (smoothly!) just when the film starts climbing. This is enough to get a flat film with larger thickness emin. N o t surprisingly, the corresponding fingers are bigger, thicker and faster. There is a good correlation between the values of emin, emax and Uma~. This is illustrated in Fig. 7.
~.
Umax ps
321
eminA
-1 • j/t
Um a x
S
~J[
sS
I0
.
j
,::;• O'
i
1
•-"
j7
,•
2,
o
emax i
|
:
i
5
P "
Fig. 7. Correlation between U ~ , emi, and emaxfor "forced" injection. Thicknesses in/~m, velocities i n # m s -~.
Systematic studies are under way, in order to provide an experimental basis for the analysis of the developed fingering instability. CONCLUSIONS
We have presented new, exploratory experiments in various situations, which are not described by available theories. Each of them, defects, contamination, thermal gradient, is very often met in practical situations and in all cases the observed wetting behaviour strikingly differs from the ideal case. The aim of our studies is to bring enough experimental data to help for an, obviously necessary, theoretical effort. ACKNOWLEDGEMENT
M. Cazabat is gratefully acknowledged for his critical reading of the manuscript.
322 REFERENCES 1 B.V. Derjaguin and N.V. Churaev, Wetting Films, Nauka, Moscow, 1984, in Russian; B.V. Derjaguin, N.V. Churaev and V.M. Muller, Surface forces, Consultant Bureau, New York, 1987; B.V. Derjaguin, N.V. Churaev and Ya.I. Rabinovich, Adv. Colloid Interface Sci., 28 (1988) 197 and references therein. 2 G.F. Teletzke, Ph.D. Thesis, University of Minnesota, 1983; G.F. Teletzke, L.E. Scriven and H.T. Davis, J. Chem. Phys., 77 (1982) 5794; 78 (1983) 1431. 3 P.G. de Gennes, Rev. Mod. Phys., 57 (1985) 828. 4 E.B. DussanandS. Davis,J. Fluid, Mech.,65 (1974) 71;E.B. Dussan, Ann. Rev. Fluid Mech., 11 (1979) 371; G. Ngan and V.E. Dussan, J. Fluid Mech., 118 (1982) 27. 5 R.E. Johnson and R.H. Dettre, Adv. Chem. Ser., 43 (1964) 112; 136; J.F. Oliver and S.G. Mason, J. Mater. Sci., 15 (1980) 431; A.W. Neumann and R.J. Good, J. Colloid Interface Sci., 38 (1972) 341; J.F. Oliver, C. Huh and S.G. Mason, Colloids Surfaces, 1 (1980) 79. 6 W.A. Zisman, in L.H. Lee (Ed.), Adhesion Science and Technology, Vol. 9, Plenum, New York, 1975, p. 55; W.D. Bascom, R.L. Cottington and C.R. Singleterry, Adv. Chem. Ser., 43 (1964} 355. A. Marmur, Adv. Colloid Interface Sci., 19 (1983) 75. 7 C. Marangoni, Nuovo Cimento, 5 (1871) 239; V. Ludviksson and E.N. Lightfoot, AIChE J., 17 ( 1971 ) 1166. T.S. Sorensen (Ed.), Dynamics and Instability of Fluid Interfaces, Lecture Notes in Physics, Vol. 105, Springer, Berlin, 1979. 8 P.G. de Gennes and J.F. Joanny, J. Chem. Phys., 81 (1984) 552. 9 L.W. Shwartz and S. Garoff, Langmuir, 1 (1985) 219. 10 A.W. Churaev, Rev. Phys. Appl., 23 {1988) 975. 11 J. Lyklema, Adv. Colloid Sci., 65/37 (1968); in A.M. Cazabat and M. Veyssi~ (Eds), Colloides et Interfaces, Editions de Physique, Paris, 1984. 12 F. Heslot, A.M. Cazabat and N. Fraysse, J. Phys. Cond. Matter, 1 (1989) 5793. 13 H. Fraaije, M. Cazabat, X. Hua and A.M. Cazabat, Colloids Surfaces, 41 (1989) 77. 14 A.M. Cazabat, C.R. Acad. Sci. Paris, 310 (1990) 107. 15 S.M. Trojan, E. Herboltzheimer, S.A. Safran and J.F. Joanny, Europhys. Lett., 10 {1989) 25. 16 A.M. Cazabat, F. Heslot, S.M. Trojan and P. Carles, Nature, in press.
309
Wetting in nonhomogeneous situations: Capillary rise experiments
A.M. Cazabat 1 and F. Heslot Coll~ge de France, Physique de la Mati~re condensde, 11 place MarceUin Berthelot, 75 231 Paris Cedex 05 (France) (Received 18 May 1990; accepted 4 July 1990)
Abstract
Exploratory experiments on wetting in nonideal situations are reported. The capillary rise of a completely wetting, nonvolatile liquid has been investigated in the presence of nonwetting defects, surface contamination or thermal gradients. A qualitative analysis of the results is proposed in each case. The aim of these studies is to provide information for forthcoming theoretical analyses.
INTRODUCTION
Wetting and spreading are now fairly well understood in simple cases: pure liquids, smooth, clean surfaces, etc. [1-4]. However the reality usually differs from these ideal situations. Chemical defects or roughness of the solid surface [5], contamination of the interfaces [6], or thermal gradients [7] may drastically affect the wetting behaviour. Only in very special cases are theories available [8,9] for nonideal situations. We present here an experimental investigation of the effects of such perturbations on the behaviour of a nonvolatile, completely wetting liquid in the capillary rise geometry. We have studied: (i) the deformation of the contact line in the presence of a nonwetting chemical defect; (ii) the influence of the contamination of the surface by a thin layer miscible with the liquid; and (iii) the influence of a vertical thermal gradient along the solid surface. As no quantitative theory is yet available, we shall analyze our results in terms of simple qualitative models. Let us first recall briefly the main characteristics of the nonperturbed system. 1To whom correspondance should be addressed.
0166-6622/90/$03.50
© 1990 - - Elsevier Science Publishers B.V.
310 SUMMARY OF T H E RESULTS IN T H E IDEAL CASE
The capillary rise geometry Figure 1 summarizes the main features of the experimental setup and the symbols we use for the variables. The xy plane coincides with the horizontal free liquid surface and the yz plane with the vertical solid surface. At the macroscopic scale ( x > #m) the meniscus intersects the solid along the contact line z ( x ~ 0,y) which is horizontal ( z = h = c s t ) in the ideal case. At the microscopic scale, a thin film (x < # m ) may grow on top of the contact line (z> h).
Macroscopic scale (ideal case) Let y be the surface tension of the liquid, p its density, g the acceleration of gravity and 0e the equilibrium contact angle of the liquid on the solid. The position of the contact line is, at equilibrium:
z = h , h = h o ( 1 - s i n 0e) 1/2
(1)
where ho = k - 121/2, k - 1 being the capillary length ( ~,/pg ) 1/2. The thickness profile of the meniscus is:
z = h o ( 1 - s i n 0) 1/2
(2)
For 0 < h - z << h, i.e., very close to the contact line, one has the approximate forms at low angles (0e -< 0 << 1 ):
(X/Oe)
For 0e ¢ 0 (partial wetting) z ~ h -
For 0 = 0 (complete wetting ) z ~ ho - ( xho ) z/2
(3)
(4 )
Z solid
ill
'thin film
~e
'~":0
::::~
~
~
~
r macroscopic 7~,,~
~,
~
~
~'
~
meniscus
~
~
~
~w"
liquid
Fig. 1. The capillary rise experiment: summary of the symbols. Macroscopic scale. The y axis is perpendicular to the plane of drawing.
311
Microscopic scale Strictly speaking, at thermodynamic equilibrium, the macroscopic meniscus can be followed by a microscopic film, the thickness e of which is given by the relation:
~(e) =pgz
z> h
Here, ~ is the disjoining pressure [ 10,11 ] which is a relevant parameter for the description of thin films [ 1 ]. The corresponding values of the film thickness are in the ~, range for partially wetting liquids and of the order of a few tens of nanometers for completely wetting ones (z > h). In the present study, we use a nonvolatile liquid, which means that the transport of molecules towards the solid through the atmosphere is negligible. Thus, the growth of this film is very slow [12]. Within one day, even for complete wetting, only a film of molecular thickness grows significantly on top of the meniscus, the length of which is approximately:
L(t)
~ ( D t ) 1/2
For polydimethylsiloxane ( P D M S ) of low molecular weight (M-- 2400) one finds [12]: D = 2 . 8 - 1 0 -1° m 2 s - ' on a silica surface. This molecular film obviously plays no role when the influence of nonwetting defects is investigated, because the duration of the experiment is only a few minutes. This is not so clear in the two other cases and we shall have to address the question further. We shall now discuss the influence of nonwetting defects on the equilibrium properties of the macroscopic meniscus. NONWETTING DEFECTS
The experiment The liquid was PDMS, molecular weight 2400, surface tension 7= 20.6-10-3 N m -1, viscosity q = 2 0 . 1 0 -3 Pa s which completely wets the solid surface, a silicon wafer covered with natural oxide. Defects were vertical ink lines drawn on the solid surface, which were not wetted by P D M S (overhead Staedler projector pens are convenient! ). For the study of the coupling between defects, we drew slightly tilted lines in order to have a variable distance a between them and checked that the shape of the contact line was negligibly modified by the tilt angle. The contact line was studied with a microscope at low magnification. Equal thickness fringes were easily observed on this reflecting substrate and allowed
312
Photograph 1. Microscopic observation of the deformed contact line (yz plane). The vertical defect is the black area on the right. The upper, clear area is the dry wafer. The bottom, dark part is the PDMS macroscopic meniscus. The interference fringes along the contact line are equalthickness fringes (0.21/2m between fringes). The vertical fringes are due to internal reflections inside the optical setup and have no physical meaning. Horizontal size of the photograph 1 mm.
us to draw the thickness profile of the liquid close to the contact line (Photograph I and Fig. 2). The contact angle of PDMS on ink can be deduced from the height of rise, h, on a wide ink line. Here, it was easier to measure ho-h=0.28 mm (ho for PDMS--2 mm) which gave 0e~15 °, in good agreement with the value obtained with drops [ 13 ]. Thus, the contact angle goes from 0 e = 0 ° o n the silica to 0e= 15 ° on the ink. It is clear from Photograph 1 that the thickness profile is practically unchanged except very close to the ink (10 tim) where the curvature changes rapidly. Figure 2 shows that the unperturbed profile is satisfactorily described by Eqn (4) for h - z ~ 6 5 ttm, as expected. The perturbation of the contact line by the defect typically extends over 1.5 mm, i.e., the capillary length k-1. Two regions can be distinguished in the perturbed line: a smooth region ( a ) corresponding to the almost unchanged thickness profile, which is reproduced in Fig. 3; and a very steep region (fl), where the thickness profile changes drastically (both radii of curvature became comparable ). If two defects are present, they become coupled for a < 2 k - 1. In fact, the position of the contact line on the ink does not change (the width of the ink
313
1~z 'm,
:)
I
1
I
2
I
3
Xk IJ ' v
Fig. 2. Experimental profile (xz plane ) of the macroscopic meniscus and comparison with Eqn (4). Origin ofx on the wafer. Arbitrary origin for z.
lines was 0.5 mm). The question is, in which way are the deformations due to the two defects combined. Figure 4 gives a nonambiguous answer: the deformations ho-z(x=O) are additive. Let us note, however, that in this experiment only the a region of the contact line was investigated. If some nonadditive behaviour would have occurred in the fl region, it would not have been observable, because the effects vanish with vanishing a. Thinner ink lines are required to study coupling with curvature terms.
Discussion These experimental observations obviously ask for a model. At the present time, the only theory, to our knowledge, giving an explicit formula for the shape of a deformed contact line is the one proposed by de Gennes and J o a n n y [8] in the case of partial wetting and low contact angles. The deformation ~ (which is the equivalent of ho-z(x=O) in our case) can be written as:
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:-5
x
",, e x p
•,
Z
O•
Fig. 3. Smooth (o~) part of the perturbed contact line, zy plane. The origin is taken on the defect for y and at the position of the meniscus on the ink for z: (-~r) position of the contaet line z ( x ~ 0 ,y ) in the zy plane. Units: m m on both axes. An example of the fit according to the loglyl shape is given ( O ) exp z(with different units).
~+SA
.........................................................
z~
~a
7
Fig. 4. Coupling between defects. The deformation ei is measured at the mid-point between defects (a/2). ~ is the maximum deviation. 5 + A is plotted versus a ( • ) and compared to the sum 2~ (a/ 2 ) of the deviations ~, (~+ ~= A), due to each defect, at the distance a/2 { [] ). These deviations are measured in the outer parts of the line. One expects 2~ (a/2) = ~ + A if the additivity is met. Units: m m on both axes.
315
1 A C o c ~ log [y--~
(5)
where A is a cutoff length (A~k -~) and the defect is at y = 0 . This formula is expected to hold in an intermediate range, say k - 1 > y > a few pm. It is tempting to try to fit the a region of our contact line by a log[y[ shape, and in fact the fit z ~ - log [y[ + cst (which can be more precisely done as exp (z) becoming linear in y with a convenient choice of the scales ) is fairly good (see Fig. 3 } in the validity range of Eqn (5). It is not obvious, however, that this procedure makes sense: the thickness profile in de Gennes' theory is a linear function, Eqn (3), and ours is quadratic, Eqn (4). The mathematical structure of the equations in Ref. [8] suggests that log[y[ might still appear in the shape of the contact line, but not necessarily in a linear way. At the present time, the experiment does not contain enough information to suggest a model. It must be checked against a preexisting one. SURFACE CONTAMINATION
The experiment The contaminated surface was obtained by deposition of a thin layer of a liquid which is miscible with P D M S and has a higher surface tension. Good candidates were tetrakis (2-ethyl-hexoxy)silane (~= 26.7.10 -3 N m-1, ~/= 6.8.10- 3 Pa s) and squalane ( ? = 27.6.10- 3 N m - 1, ~/= 21.10 -3 Pa s). T h e y can be deposited by a spin-coating technique but the most efficient way was to wipe the surface, previously covered by a thick layer, with a lens-cleaning paper. Surprisingly enough, the layers obtained in this way, the local thickness of which is measured by ellipsometry, were fairly homogeneous, especially with squalane. Thin layers (30 .it) seemed to be stable over the time scale of the experiment. Thicker layers become unstable {which was expected because neither tetrakis silane nor squalane completely wetted the substrate) and progressively broke into very small droplets ( ~ pm). The contaminated surface was wettable by PDMS. However, the film which grew on top of the meniscus was much thicker and faster than the normal precursor on a clean surface. On thick contamination layers, the P D M S film showed strong instabilities close to the meniscus and a tendency to dendritic growth at the tip, which mainly reflected the breaking process of the layer. Although very reproducible, this behaviour was too complicated to be given more attention in an exploratory study. On thin layers ( ~ 20 .£.) the P D M S film had a much gentler behaviour. Its thickness was approximately constant, of the order of 100-200 nm. Its length
Lmm
316
O
~D
:}.5
t~
•.~r°
,J S ~
Fig. 5. Contamination by squalane. Thickness of squalane layer 21 _+2 A. Length of the main film as a function of the square root of the time and schematic thickness profiles. Typical thickness between 0.5 and 1 fringe (0.1 and 0.2 ]~m ). Various symbols correspond to different places on the wafer.
L (measured from the macroscopic meniscus ) first scaled at t '/2 and then more slowly (Fig. 5). This slowing down corresponds to a slight change in the film profile, schematically indicated in the figure: a minimum in thickness appeared close to the meniscus and the thickness in the flat part decreased progressively. This evolution which is illustrated in Fig. 5 for a thin layer of squalane, (thickness 21 + 2 ~,, the uncertainty corresponding to thickness variations from place to place) was also observed with tetrakis silane. The t '/2 region of the curve can be characterized by a diffusion coefficient D [ L = (Dt)l/2]. For squalane and the condition of Fig. 5 (layer thickness ~ 21 ~, ) one obtains: D = l . 5 . 1 0 -9 m 2 s-1 The same order of magnitude ( D ~ 0 . 5 - 2 - 1 0 -9 m 2 s -1) was observed for tetrakis silane with similar layers. However, probably due to strong lateral interactions between molecules in this case [ 14 ], the tetrakis layers were rather irregular. If more attention is paid, another much faster, very fugitive film can be observed. Its thickness is around 100 nm. Its length Lo cannot be measured precisely, due to the lack of contrast and the limited time of observation (40 s) but it is compatible with a diffusion-like formula:
317
Lo ~ (Do t)
1/2
where Do is about 5-10 - s m 2 s-1 both for squalane and tetrakis silane.
Discussion The growth of films much thicker and faster that the normal precursor film is obviously due to the gradient of surface tension 7. Close to the macroscopic meniscus, 7 is equal to the surface tension of P D M S , while at the tip of the film it is much higher, because some portion of the contamination layer has been solubilized. For a film of constant thickness e, a surface tension gradient dT/dz ( > 0) produces a surface velocity in the z direction: Vs-
e d7 dz
(6)
A Couette flow is obtained, with an average velocity:
e d7 dL U - 2 ~ d z - dt
(7)
If the gradient dT/dz has a constant value, a linear growth L ~ t is expected. If the gradient scales as L - ', a diffusion-like behaviour L ~ t 1/2 will result. This, together with the experimental observation that the film thickness e is practically constant and that L ~ t 1/2 at short times, suggests a simple model: Let us assume that the difference Ay of surface tensions at the tip of the film and at the meniscus is constant during the growth of the film. Then, dT~zb' dz~L
(8)
so that dL e Ay dt ~ 2 ~ / L
(9)
e~7
(10)
Dth ~--
at least if ~ does not change significantly, which is the case if the contaminant is squalane (~SQ~ ~PDMS). Let us put orders of magnitude in this relation: 100 nm~
~/=20"10 -3 Pa s :
7.10-3 N m -1
with zlT=ATmax, e= 150 nm, one obtains:
318
Dth --5.10 - s m 2 s -1 in good agreement with the Do value. Neither Do n o r Dth are precisely known. However, one can certainly conclude that the model gives a satisfactory description of the first, rapid film if A? is taken to be ~/l~max. A further assumption would be that the main film is described by the same model (in the t 1/2 region ) but with a much smaller value of A~,because the first film has solubilized a large part of the squalane layer. From D--1.5.10 -9 m 2 s-1 and e ~ 150 nm one would obtain: A?~0.2.10 -3 N m -1
<<2~max
This model is obviously very crude. It does not describe the mixing process between P D M S and squalane, which has its own characteristic times. Let us consider it as a first attempt towards the description of contamination-driven wetting films. TEMPERATURE GRADIENTS Surface tension gradients d~/dz may also be produced by temperature gradients along z. As ~ usually decreases with increasing temperature, the temperature gradient dT/dz must be negative for d~,/dz to be positive. One has: d? dy d T ~-dT dz
(11)
In this case, Eqns (6) and (7) hold. If d T / d z (thus d~/dz) is constant, and if the film has a constant thickness, it is expected to grow linearly with time.
The experiment The (clean) wafer was pressed against two temperature-controlled brass holders, 7 mm apart. The temperature difference between the holders was 28 ° C. The P D M S was put into contact with the wafer after about 30 min to ensure that the stationary distribution of temperatures in the wafer had been achieved. As in the preceding case, a thick film ( ~ p m ) , much faster than the normal precursor was observed. However, the behaviour of this film was quite different from that of the contamination case. It can be separated into different steps: First a film of approximately constant thickness climbs the wafer (a). At the tip of the film, a bump in the thickness profile develops (b), which ultimately becomes unstable (c) and results in a fingering instability with a rather well defined wavelength (d, Photograph 2). Between the meniscus and the bump (b,c) or fingers (d) the film is flat, with
319 i i
Photograph 2. Fingers in thermal gradients with equal-thickness interference fringes after 15 min. Upper: dry wafer. Bottom: constant-thickness film. Scale: the average distance between fingers is 480 pm. The meniscus is well below and is not visible in the photograph.
constant thickness emin. The fingers also have a flat part (Photograph 2) but are terminated by a drop of larger thickness em~x. The length of the film is well defined in step (a). For the next steps, one must introduce two lengths, one corresponding to the finger tip, the other to the "valleys" (Fig. 6). All these lengths vary linearly with time and can be characterized by a velocity value, V a in step (a), Um~xand Umi~for (b,c,d). One obtains: U~ ~ 3.3; Umax~ 3.6; and Groin ~ 2.6 pm s- 1.
Discussion Step (a) and the wavelength of the fingering instability (which is basically the Rayleigh instability of a cylinder but in a more complex geometry) are satisfactorily described by available theories [15,16]. For example, the expected velocity Ua.th is given by Eqn (7) with emin=0.54/~m and @/dz=0.21 N m -2 (i.e., dT/dz=AT/Az=4.28.103 K m -1 and ~1 =d), 2 . 5 " 1 0 -3 K_ 1 for # - -
PDMS). So Ua,th---- 2.8 ]l S - 1 , while Ua ~3.3 #m s -1. At the present time, there is no available theory for the following steps. The contact line instability clearly results from the Rayleigh-like instability of the bump, because the friction on the surface is less in the thicker parts which, as
320
~L
mm /
/ / e'
J
os
3 I J
.L
/
,t J
2 ,~
~s
18
• 711~
,.p
OJ
,,
5 !
10
15
t
mn IT
Fig. 6. Position of the contact line ( ~ , a) or of the tips and "valleys" ( 0 and ~ , b,c,d) as a function of time. The origin of time is arbitrary.
a consequence, grow faster. Let us note also that the ratio U m a x / U m i n is comparable with the ratio of the thicknesses in the flat part of the fingers ( ~ 3.5 fringes) and in the flat part of the main film (emin~ 2.5 fringes). There are various ways to obtain complementary results. One can change the P D M S viscosity and the thermal gradient [ 16]. Another possibility is to "force" the film to thicken by imposing (transitorily) a higher injection rate at the meniscus. One way is to move the P D M S external reservoir upwards (smoothly!) just when the film starts climbing. This is enough to get a flat film with larger thickness emin. N o t surprisingly, the corresponding fingers are bigger, thicker and faster. There is a good correlation between the values of emin, emax and Uma~. This is illustrated in Fig. 7.
~.
Umax ps
321
eminA
-1 • j/t
Um a x
S
~J[
sS
I0
.
j
,::;• O'
i
1
•-"
j7
,•
2,
o
emax i
|
:
i
5
P "
Fig. 7. Correlation between U ~ , emi, and emaxfor "forced" injection. Thicknesses in/~m, velocities i n # m s -~.
Systematic studies are under way, in order to provide an experimental basis for the analysis of the developed fingering instability. CONCLUSIONS
We have presented new, exploratory experiments in various situations, which are not described by available theories. Each of them, defects, contamination, thermal gradient, is very often met in practical situations and in all cases the observed wetting behaviour strikingly differs from the ideal case. The aim of our studies is to bring enough experimental data to help for an, obviously necessary, theoretical effort. ACKNOWLEDGEMENT
M. Cazabat is gratefully acknowledged for his critical reading of the manuscript.
322 REFERENCES 1 B.V. Derjaguin and N.V. Churaev, Wetting Films, Nauka, Moscow, 1984, in Russian; B.V. Derjaguin, N.V. Churaev and V.M. Muller, Surface forces, Consultant Bureau, New York, 1987; B.V. Derjaguin, N.V. Churaev and Ya.I. Rabinovich, Adv. Colloid Interface Sci., 28 (1988) 197 and references therein. 2 G.F. Teletzke, Ph.D. Thesis, University of Minnesota, 1983; G.F. Teletzke, L.E. Scriven and H.T. Davis, J. Chem. Phys., 77 (1982) 5794; 78 (1983) 1431. 3 P.G. de Gennes, Rev. Mod. Phys., 57 (1985) 828. 4 E.B. DussanandS. Davis,J. Fluid, Mech.,65 (1974) 71;E.B. Dussan, Ann. Rev. Fluid Mech., 11 (1979) 371; G. Ngan and V.E. Dussan, J. Fluid Mech., 118 (1982) 27. 5 R.E. Johnson and R.H. Dettre, Adv. Chem. Ser., 43 (1964) 112; 136; J.F. Oliver and S.G. Mason, J. Mater. Sci., 15 (1980) 431; A.W. Neumann and R.J. Good, J. Colloid Interface Sci., 38 (1972) 341; J.F. Oliver, C. Huh and S.G. Mason, Colloids Surfaces, 1 (1980) 79. 6 W.A. Zisman, in L.H. Lee (Ed.), Adhesion Science and Technology, Vol. 9, Plenum, New York, 1975, p. 55; W.D. Bascom, R.L. Cottington and C.R. Singleterry, Adv. Chem. Ser., 43 (1964} 355. A. Marmur, Adv. Colloid Interface Sci., 19 (1983) 75. 7 C. Marangoni, Nuovo Cimento, 5 (1871) 239; V. Ludviksson and E.N. Lightfoot, AIChE J., 17 ( 1971 ) 1166. T.S. Sorensen (Ed.), Dynamics and Instability of Fluid Interfaces, Lecture Notes in Physics, Vol. 105, Springer, Berlin, 1979. 8 P.G. de Gennes and J.F. Joanny, J. Chem. Phys., 81 (1984) 552. 9 L.W. Shwartz and S. Garoff, Langmuir, 1 (1985) 219. 10 A.W. Churaev, Rev. Phys. Appl., 23 {1988) 975. 11 J. Lyklema, Adv. Colloid Sci., 65/37 (1968); in A.M. Cazabat and M. Veyssi~ (Eds), Colloides et Interfaces, Editions de Physique, Paris, 1984. 12 F. Heslot, A.M. Cazabat and N. Fraysse, J. Phys. Cond. Matter, 1 (1989) 5793. 13 H. Fraaije, M. Cazabat, X. Hua and A.M. Cazabat, Colloids Surfaces, 41 (1989) 77. 14 A.M. Cazabat, C.R. Acad. Sci. Paris, 310 (1990) 107. 15 S.M. Trojan, E. Herboltzheimer, S.A. Safran and J.F. Joanny, Europhys. Lett., 10 {1989) 25. 16 A.M. Cazabat, F. Heslot, S.M. Trojan and P. Carles, Nature, in press.