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Nucleation Radius and Growth of a Liquid Meniscus
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
190, 134– 141 (1997 )
CS974855
Nucleation Radius and Growth of a Liquid Meniscus Georges Debregeas 1 and Franc¸oise Brochard-Wyart Institut Curie, Section de recherche, 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France Received November 18, 1996; accepted March 4, 1997
We consider a horizontal solid plate P placed above the free surface of a liquid L separated by a layer of air of thickness e ( Ç0.1 mm) . With suitable P / L pairs this layer of air is metastable for thicknesses e below a certain limit ec ( Ç1 mm). We have found a way of setting up bridges connecting the liquid surface with the plate in a controlled way ( axisymmetric meniscus of horizontal radius R) . The meniscus grows if R is above a certain threshold Rc (e ) . If R õ Rc the meniscus shrinks to zero. Our method allows precise measurements of Rc (e ): We were able to do this using silicone oils and two types of plates P ( with different contact angles ) . Our results are in good agreement with classical calculations by G. I. Taylor and E. Michael ( J. Fluid Mech. 58, 625 ( 1973 ) ). Furthermore, When R ú Rc (e ) , we find that R grows linearly with time t and that
S S DD
dR } e 00.7 1 0 dt
e ec
2
.
q 1997 Academic Press
Key Words: nucleation, capillary bridge, wetting.
I. INTRODUCTION
Our geometry is shown in Fig. 1. In the unperturbed state, we have a layer of air of thickness e separating the free surface of the liquid from the solid plate. If we raise the liquid by e into perfect contact with the plate, we spend (per unit area) a gravitational energy 12rge 2 ( r Å density, g Å gravitational acceleration) . The contact produces a change of interfacial energy S where S is the spreading coefficient defined by S Å gs/ l 0 ( gs/ o / gl / o ) , where the gi / j ’s are the interfacial tensions between partners i and j ( s Å solid, o Å air, l Å liquid) . Thus, if S õ 0, the homogeneous air film is metastable against formation of bridges whenever 1 To whom correspondence should be addressed. Fax: ( 33) 1 40 51 06 36. E-mail: [email protected].
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02S u Å 2k 01 cos , rg 2
[1]
where k 01 is the capillary length ( k 2 Å rg / gl / o ) and u is the equilibrium contact angle. We will be concerned in the following with air layers of thickness e õ ec . One useful approach to display the metastability amounts to ‘‘drilling a hole’’ in the air film, i.e., to bridge (L ) and ( P ) by an axisymmetric meniscus with a certain radius R. Experimentally we achieve this by starting with a liquid contained in a Petri dish. If we raise the dish we can establish contact between ( L ) and ( P ) using a pendant droplet attached to the solid surface. If we then move the dish downward by a distance e, we have achieved the bridge under discussion. This experiment can be related to the study of the stability of a liquid film of thickness e deposited on a substrate when a hole of radius R is made. The part played by the liquid and the air are however inverted. Although this phenomenon is well understood, it has remained difficult to experimentally examine the critical radius Rc . Several reasons account for it: all previous studies on Rc were based on producing liquid sheets of same depth and nucleating holes of various sizes. The nucleation radius is then assumed to be between the upper value of holes which close up and the lowest value of those which open up with many experiments required to get one value Rc (e) . The holes were generally made by an external probe or a jet of air. The process of creating the hole may influence subsequent evolution of the film. In addition, the surface can be slightly damaged by the contact with the probe. This can perturb the experiment. The experimental technique presented here allows us to vary the depth of air during one single experiment and accurately measure Rc (e) . When the solid surface is sufficiently hysteretic, instead of a single equilibrium radius, we find an interval R0 õ R õ R/ where the bridge remains static. In a second part, we present a brief study of the kinetics of growth of the bridge, using the same experimental procedure. When R ú Rc (e) , the liquid bridge is unstable and expands. We monitor R versus time for thicknesses e in the range 40 mm–2 mm. We also investigate the value of the solid/ liquid dynamic contact angle versus e to determine whether the moving liquid meniscus adopts a quasistatic profile. We give
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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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e õ ec Å
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NUCLEATION RADIUS AND GROWTH OF A LIQUID MENISCUS
FIG. 1.
Experimental setup for producing an axisymmetric liquid bridge.
an empirical equation of the thickness dependence of the growth velocity. II. STATICS: NUCLEATION RADIUS
Experimental Setup We use two different solids: ( a) an almost perfect substrate ( silicon wafer) with no hysteresis and ( b) a standard microscope glass slide, both silanized with octadecyltrichlorosilane ( OTS) . The liquid is a polydimethylsiloxane ( PDMS ) of molecular weight 91,700. The solids ( a) and ( b ) exhibit an advancing angle of respectively 11 { 27 and 68 { 27 ( receding angle 97 and 287 ) with the liquid. The solids are supported horizontally above the liquid bath between two identical rectangular metal pieces. The liquid is contained in a Petri dish of diameter 2L Å 38 mm in case ( a) and 2L Å 49 mm in case (b ) and with a depth of 2.5 cm. This vessel has been treated to be hydrophobic so the free surface has a slight downward curvature. The container is put on a support which can be moved vertically by means of a micrometric screw with a precision of 10 mm. A camera placed horizontally makes it possible to study the film of air between the solid and the liquid surface. Before putting the slide on the device, we first deposit a droplet on the side which will come in contact with the
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liquid. This droplet enables us both to nucleate the meniscus and to measure optically the initial thickness of air when the contact is established: we measure the size of this drop with the camera and we raise the surface until it touches lightly the droplet. e is then exactly the height of the droplet. When the contact is just established, we rapidly lower the liquid bath, then search for the stability of the meniscus by raising or lowering the liquid bath. When the meniscus is stable, we raise very slowly the reservoir until the bridge just starts to expand. We measure the radius of the neck with the camera and note down the variation of e from the micrometric screw following the first contact. We let the meniscus grow and repeat the same process for a larger radius (Fig. 2) . Uncertainty in the measure of Rc comes from the difficulty to precisely evaluate the point of stability of the meniscus. In the range of small thicknesses (R õ 300 mm), the bridge is always about to break, leaving a droplet on the slide, or to rapidly expand. In contrast, in the range of large thicknesses, the system is less sensitive to a change in e, since the driving force for expansion or recession gets lower. We used a PDMS of intermediate viscosity (30,000 mPa.s) which makes it possible to explore the whole range with the best precision on Rc . This inaccuracy is actually difficult to estimate. Our procedure consists in defining a criterium of stability, i.e., a minimum time lapse during which the liquid bridge appears steady on the screen. This time lapse is reevaluated for
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DEBREGEAS AND BROCHARD-WYART
FIG. 2.
Side view of three metastable bridges.
each thickness, to take into account the dynamic effect. For a given thickness e, we get a narrow range of radius in which the bridge remains steady. This defines the experimental dispersion which we find equal to {50 mm.
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The sample ( b) being quite hysteretic, we can also measure a lower limit of the radius of the neck for which the contact line starts to recede. Between these values the meniscus is stable. However, the contact line recession being very
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NUCLEATION RADIUS AND GROWTH OF A LIQUID MENISCUS
slow, these measurements are less accurate and cannot be performed at large thicknesses where the driving force tends to zero. Despite the hysteresis, it must be noted that the bridge always remains perfectly axisymmetric. This can be easily checked by observing the contact patch from above. It appears to be very circular with no local pinning of the contact line. The thickness of the air film measured this way must be corrected for the general level of the liquid has been lowered by roughly e ∗ ( R / L) 2 . We finally evaluate the thickness precision to {30 mm, including the measurement of the pendant droplet as well as the precision of the micrometric screw. Stability of a Meniscus Consider a meniscus connecting the solid to the liquid bath. Its shape is given by r( z) ( Fig. 1) where ( r, z) are the cylindrical polar coordinates rescaled using the capillary length k 01 . The origin ( r Å 0, z Å 0 ) of the coordinate system is chosen at the level of the free surface far from the bridge and on the symmetry axis of the meniscus. The static equilibrium between the Laplace and hydrostatic pressures leads to the differential equation for r( z) , where r * Å dr / dz and r 9 Å d 2r/ dz 2 0
r9 1 / Åz ( 1 / r * 2 ) 3 / 2 r( 1 / r * 2 ) 1 / 2
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These analyses can be entirely transposed to the case of liquid menisci and we can actually consider the growing of a meniscus as the dewetting of an air film. For a given contact angle u and a thickness e õ ec ( u ) of the air film, a meniscus of initial radius R will expand if R ú Rc and shrink if R õ Rc . We will focus in the following on the numerical calculation of T&M ( as described in Ref. (1 ) ) to fit our experimental results for Rc ( u, e) . Results and Discussion The data presented in Fig. 3 compares the numerical fit with the observed advancing or receding contact angle. The experimental results are in good agreement with the calculation in nearly the whole range of thickness explored. It must be noted that our experimental approach is signifi-
[2 ]
with the boundary conditions (i) r(0) Å ` and (ii) r * ( z Å e) Å cot( u ), u being the solid/liquid contact angle. Note that for an hysteretic substrate, u Å ua will give the advancing critical radius, whereas u Å ur will give the receding critical radius. This nonlinear equation was used by James ( 2 ) to study the shape of a meniscus connecting a liquid bath to a small cylindrical rod and to derive an asymptotic solution in the limit of small radii. This problem is formally equivalent to that of a hole made in a sheet of liquid deposited on a horizontal surface. This question has been more extensively studied since it is of considerable practical importance in the coating industry. Taylor and Michael (T&M ) ( 1) showed that for a given contact angle u and thickness e õ e c ( u ) defined by Eq. [1] , there exists a unique solution to Eq. [ 2] which can be characterized by the throat radius Rc of the extended profile. They demonstrated that Rc represents the limiting value between holes which would open out ( R ú Rc ) and holes which would close up ( R õ Rc ) . Using a different approach, Sharma and Ruckenstein (3 ) have considered the free energy of the film after hole formation and compared it with the energy of the unbroken film, in order to determine a criterion for the stability of a hole in a sheet of fluid. Their results are mostly in agreement with T& M’s calculations. More recently, other authors ( 4, 5) have independently derived asymptotic expressions of the critical thickness for dewetting using Eq. [ 2] , when a hole of small radius is drilled in a liquid film.
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FIG. 3. Critical radius ( Rc ) versus reduced thickness ( ke ) and the corresponding calculated fit from Eq. [2 ]. (a ) Nonhysteretic silicium wafer, ue Å 117. ( b) Hysteretic glass slide, ua Å 687, ur Å 287.
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DEBREGEAS AND BROCHARD-WYART
cantly different from previous studies (Padday (6), T&M (1), Sykes et al. ( 7)) on the stability of liquid films. In these experiments, holes of various sizes were made in liquid sheets of same depth. The upper value of hole radii which close up and the lowest value of those which open up were asymptotically deduced from a large number of experiments. The holes were invariably made by an external probe or a jet of air. The way the hole is created may influence the evolution of the film. We ourselves observed that at lowest thicknesses, a small perturbation can destabilize the meniscus in one way or another. Moriarty and Schwartz ( 8) have recently underlined the importance of the initial shape of the hole to predict its opening or closing up. Hence, they developed a time-dependent model, including viscous forces, to follow the evolution of the hole profile after initial formation. In addition, the surface can be slightly damaged by the contact with a probe which can perturb the following experiments. Here the depth of the film of air is adjusted during one single experiment until one gets the meniscus steady without any contact with the solid surface. III. DYNAMICS: GROWTH OF A LIQUID MENISCUS
Experimental Setup We still consider air layers of thicknesses e õ ec . Our present aim is to investigate the kinetics of growth of a liquid bridge of radius R @ Rc (e) , for a given value of e.
FIG. 4.
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We perform video measurements of the radius R of the circular contact patch, versus time, from above the sample. We need the solid substrate to be transparent, and therefore we only use the glass slide. The thickness e of the air layer, in the range of 40 mm to 2 mm, is measured prior to nucleation using the method described above. As we indicated in part I, the general level of the liquid gets lower as the meniscus grows. We thus restricted our observations to the range 2.5 õ R õ 6.5 mm and calculated a mean correction that we applied to the initial thickness of the air layer. The first inequality is prescribed to ensure that Rc ! R. The effects of azimuthal curvature on the growing meniscus are therefore negligible. We varied the PDMS viscosities h by three orders of magnitude ( between 10 3 up to 10 6 mPa.s) . Static contact angles ua and ur were found to be almost independent on molecular masses. We also performed measurements of the ( solid/ liquid) dynamic contact angle ud during the growth process. We used a technique developped by Andrieu et al. ( 9) to measure dynamic contact angles in the dewetting of liquid films. It is based upon the optical deformation of a grid induced by a liquid diopter. Here, the grid lays at the bottom of the liquid bath. When filmed from above, the image of the grid is distorted near the contact line ( Fig. 4a ). The deformation d at the line can be related to ud knowing the refractive indexes of the two media ( air, n0 Å 1; PDMS, n1 Å 1.4 )
Measurement of the dynamic contact angle. (a ) Image of the deformed grid. (b ) Sketch of the contact angle measurement technique.
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NUCLEATION RADIUS AND GROWTH OF A LIQUID MENISCUS
FIG. 5. Radius R of a growing liquid meniscus versus time t for three different values of the intercalated layer of air: m (140 mm) , j (300 mm) , l (680 mm) . The liquid viscosity is 10 5 mPa.s.
FIG. 6. Capillary number Ca Å hV/ g versus thickness e fitted by the empirical law: Ca (e) Å c te ∗ e 00.7 (1 0 (e/ ec ) 2 ). l ( h Å 10 3 mPa.s ), l ( h Å 10 4 mPa.s ), j ( h Å 10 5 mPa.s), m ( h Å 10 6 mPa.s ).
D (e) à gs/ o / g 0 gs/ l 0 and the vertical distance h from the grid to the contact line ( Fig. 4b) . This technique gives precise results in the limit of small deformations, i.e., small angles of the air wedge: C Å p 0 ud . In this limit, one finds CÅ
n1 d . ( n 1 0 n0 ) h
1 rge 2 2
S S DD
Å 0S 1 0
e ec
2
.
[ 4]
Concerning the dynamic contact angle, Fig. 7 shows that the supplementary angle C Å p 0 ud depends linearly on ke: C Å a∗ ke, with a Å 1.18.
We could thus measure C for thicknesses ranging from 40 to 250 mm. For thicker air layers, the angle C is no longer in the domain of validity of the technique. Results and Discussion We find that R varies linearly with time t , as shown in Fig. 5, and that the velocity V Å dR/ dt is proportional to the liquid viscosity for a given thickness e. The relevant parameter for the hydrodynamic problem is therefore the Capillary number, Ca Å hV/ g. Its dependence upon the air thickness e is presented in Fig. 6. It appears that Ca(e ) can be fitted by a power law in the form Ca(e) Å cte ∗ e 00.7 ∗ D (e) ,
[3 ]
where D (e) is the driving force of expansion of the triple line per unit length. In the limit considered here, R ú Rc (e) , D (e) can be written simply as the sum of capillary and gravity forces
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FIG. 7. C Å p 0 ud ( ud is the dynamic contact angle) versus reduced thickness ke .
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DEBREGEAS AND BROCHARD-WYART
All these results can be interpreted by the following picture. ( 1) The profile of the meniscus is quasistatic. If we assume that the moving meniscus is quasistatic, then its shape must obey the hydrostatic equilibrium condition [1] . In the limit R @ Rc (e) , the azimuthal Laplace pressure term vanishes, and Eq. [1] yields to 0
r9 Å z. ( 1 / r *2 ) 3/2
thicknesses. However, this result can be generalized a fortiori to the whole domain explored, since the velocity gets lower as we thicken the air layer and the profile becomes static. We can conclude that the thickness of the air layer, which is our controlled parameter, entirely determines the dynamic contact angle ud , according to Eq. [5 ] . ( 2) The growth velocity V Å dR / dt follows the Hoffman Tanner law. The driving force D (e) acting on the quasistatic meniscus is also the uncompensated Young force on the contact line. Using Eqs. [ 4] and [ 5 ] , D (e ) can be written as
This equation is now solvable. After integration, one gets the following relationship between ud and e: k 2e2 cos ud Å 0 1. 2
[5]
Hence, in the limit of small thicknesses e, we have C Å p 0 ud Å ke .
[6 ]
The quasistatic hypothesis is consistent with the linear dependence C Å a∗ ke experimentally observed, with a prefactor a Å 1.18 close to 1. Our experimental procedure enables us to check this behavior only in the limit of small
D(e ) Å gs/ o 0 gs / l 0 g cos ud . The velocity versus dynamic contact angle dependence of a moving contact line has been experimentally studied by Hoffman ( 10 ) over a wide range of velocities and wetting conditions, using silicone oils in capillary tubes. In his experiments, the velocity was externally imposed. He showed that all his data could be described by a master curve, exhibiting a unique function F , so that F( ud ) 0 F ( ua ) Å Ca,
[ 7]
where Ca, ud , and ua have the same meaning as in the present study.
FIG. 8. Dynamic contact angle ud versus Ca / F ( ua Å 687 ). Gray spots: Hoffman’s results for different silicone oils in capillary tubes. Black spots: our results.
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S S DD
This growth law can have a possible link with some peculiar aspects of adhesion: when a solid plane is brought into contact with a soft elastomeric hemisphere ( JKR) , the effective contact surface varies with time, inducing variations of the adhesion energy, before it reaches its equilibrium value ( 12) . Our results can possibly help to understand the dynamics of this phenomenon. ( iii ) Nucleation of one single liquid bridge by the mean of a pendant droplet is a way of achieving perfect contact, with no air trapped at the interface. This may have practical applications for adhesion of polymer liquids ( ‘‘tack’’) or soft rubbers on solid surfaces.
IV. CONCLUDING REMARKS
ACKNOWLEDGMENTS
( i ) We have found a way to measure the limiting radius for which a meniscus, connecting a liquid bath to a horizontal plate, will expand or retreat when the surface is nonhysteretic. In the case of hysteretic samples, the method allows us to measure a range of R and e within which a meniscus is stable. The method presented here has the advantage of reaching these limiting values by adjusting the critical thickness in a quasistatic way, using a nondisturbing nucleating method. ( ii ) Above and below Rc , bridges expand or shrink. We have also studied the dynamics of growth of capillary bridges in the viscous regime. This is to our knowledge the first experimental study of spontaneous wetting at a very high dynamic contact angle. We showed that the liquid meniscus remains in a quasistatic shape and that, for a given thickness of the air layer, the growth velocity V Å dR/ dt is constant and obeys the empirical law
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V}
1 00.7 e 10 h
e ec
2
A theoretical description of the viscous flows was given by Tanner ( 11 ) in the limit of small contact angle ud , leading to the so-called Tanner’s law: ud } Ca 1 / 3 . A good agreement was met with experimental data up to 1007. In the domain of higher contact angle, the situation remains controversial. Considering the resemblance of our three-phase system to Hoffman’s, we will use the empirical relationship [ 7] to analyze our data. We have plotted in black in Fig. 8 the dynamic contact angle ud , deduced from e measurements via Eq. [ 5] , versus Ca / F ( ua Å 687 ) . A good agreement is found with Hoffman’s results displayed in the same graph. This in turn confirms again the quasistatic hypothesis and indicates that the observed Ca(e) dependence [ 3] is compatible with the empirical laws of contact line motion.
.
We thank C. Sykes, A. Buguin, and R. Bruinsma for stimulating discussions, and L. Perthuis for technical assistance.
REFERENCES 1. Taylor, G. I., and Michael, E., J. Fluid Mech. 58, 625 (1973 ). 2. James, D. F., J. Fluid Mech. 63, 657 (1974 ). 3. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 137, 443 (1990). 4. Sykes, C., C. R. Acad. Sci. Paris 313 II, 607 (1991 ). 5. Sharma, A., J. Colloid Interface Sci. 156, 96 (1992 ). 6. Padday, J. F., Spec. Discuss. Faraday Soc. 1, 64 (1971). 7. Sykes, C., Andrieu, C., De´ tappe, V., and Deniau, S., J. Phys. III France 4, 775 (1994 ). 8. Moriarty, J. A., and Schwartz, L. W., J. Colloid Interface Sci. 161, 335 (1993). 9. Andrieu, C., Chatenay, D., and Sykes, C., C. R. Acad. Sci. Paris, 320, 351 (1995 ). 10. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975). 11. Tanner, L. H., J. Phys. D 12, 1473 (1979 ). 12. Michel, F., and Shanahan, E. R., C. R. Acad. Sci. Paris 310, 17 (1990 ).
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190, 134– 141 (1997 )
CS974855
Nucleation Radius and Growth of a Liquid Meniscus Georges Debregeas 1 and Franc¸oise Brochard-Wyart Institut Curie, Section de recherche, 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France Received November 18, 1996; accepted March 4, 1997
We consider a horizontal solid plate P placed above the free surface of a liquid L separated by a layer of air of thickness e ( Ç0.1 mm) . With suitable P / L pairs this layer of air is metastable for thicknesses e below a certain limit ec ( Ç1 mm). We have found a way of setting up bridges connecting the liquid surface with the plate in a controlled way ( axisymmetric meniscus of horizontal radius R) . The meniscus grows if R is above a certain threshold Rc (e ) . If R õ Rc the meniscus shrinks to zero. Our method allows precise measurements of Rc (e ): We were able to do this using silicone oils and two types of plates P ( with different contact angles ) . Our results are in good agreement with classical calculations by G. I. Taylor and E. Michael ( J. Fluid Mech. 58, 625 ( 1973 ) ). Furthermore, When R ú Rc (e ) , we find that R grows linearly with time t and that
S S DD
dR } e 00.7 1 0 dt
e ec
2
.
q 1997 Academic Press
Key Words: nucleation, capillary bridge, wetting.
I. INTRODUCTION
Our geometry is shown in Fig. 1. In the unperturbed state, we have a layer of air of thickness e separating the free surface of the liquid from the solid plate. If we raise the liquid by e into perfect contact with the plate, we spend (per unit area) a gravitational energy 12rge 2 ( r Å density, g Å gravitational acceleration) . The contact produces a change of interfacial energy S where S is the spreading coefficient defined by S Å gs/ l 0 ( gs/ o / gl / o ) , where the gi / j ’s are the interfacial tensions between partners i and j ( s Å solid, o Å air, l Å liquid) . Thus, if S õ 0, the homogeneous air film is metastable against formation of bridges whenever 1 To whom correspondence should be addressed. Fax: ( 33) 1 40 51 06 36. E-mail: [email protected].
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02S u Å 2k 01 cos , rg 2
[1]
where k 01 is the capillary length ( k 2 Å rg / gl / o ) and u is the equilibrium contact angle. We will be concerned in the following with air layers of thickness e õ ec . One useful approach to display the metastability amounts to ‘‘drilling a hole’’ in the air film, i.e., to bridge (L ) and ( P ) by an axisymmetric meniscus with a certain radius R. Experimentally we achieve this by starting with a liquid contained in a Petri dish. If we raise the dish we can establish contact between ( L ) and ( P ) using a pendant droplet attached to the solid surface. If we then move the dish downward by a distance e, we have achieved the bridge under discussion. This experiment can be related to the study of the stability of a liquid film of thickness e deposited on a substrate when a hole of radius R is made. The part played by the liquid and the air are however inverted. Although this phenomenon is well understood, it has remained difficult to experimentally examine the critical radius Rc . Several reasons account for it: all previous studies on Rc were based on producing liquid sheets of same depth and nucleating holes of various sizes. The nucleation radius is then assumed to be between the upper value of holes which close up and the lowest value of those which open up with many experiments required to get one value Rc (e) . The holes were generally made by an external probe or a jet of air. The process of creating the hole may influence subsequent evolution of the film. In addition, the surface can be slightly damaged by the contact with the probe. This can perturb the experiment. The experimental technique presented here allows us to vary the depth of air during one single experiment and accurately measure Rc (e) . When the solid surface is sufficiently hysteretic, instead of a single equilibrium radius, we find an interval R0 õ R õ R/ where the bridge remains static. In a second part, we present a brief study of the kinetics of growth of the bridge, using the same experimental procedure. When R ú Rc (e) , the liquid bridge is unstable and expands. We monitor R versus time for thicknesses e in the range 40 mm–2 mm. We also investigate the value of the solid/ liquid dynamic contact angle versus e to determine whether the moving liquid meniscus adopts a quasistatic profile. We give
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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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e õ ec Å
r
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NUCLEATION RADIUS AND GROWTH OF A LIQUID MENISCUS
FIG. 1.
Experimental setup for producing an axisymmetric liquid bridge.
an empirical equation of the thickness dependence of the growth velocity. II. STATICS: NUCLEATION RADIUS
Experimental Setup We use two different solids: ( a) an almost perfect substrate ( silicon wafer) with no hysteresis and ( b) a standard microscope glass slide, both silanized with octadecyltrichlorosilane ( OTS) . The liquid is a polydimethylsiloxane ( PDMS ) of molecular weight 91,700. The solids ( a) and ( b ) exhibit an advancing angle of respectively 11 { 27 and 68 { 27 ( receding angle 97 and 287 ) with the liquid. The solids are supported horizontally above the liquid bath between two identical rectangular metal pieces. The liquid is contained in a Petri dish of diameter 2L Å 38 mm in case ( a) and 2L Å 49 mm in case (b ) and with a depth of 2.5 cm. This vessel has been treated to be hydrophobic so the free surface has a slight downward curvature. The container is put on a support which can be moved vertically by means of a micrometric screw with a precision of 10 mm. A camera placed horizontally makes it possible to study the film of air between the solid and the liquid surface. Before putting the slide on the device, we first deposit a droplet on the side which will come in contact with the
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liquid. This droplet enables us both to nucleate the meniscus and to measure optically the initial thickness of air when the contact is established: we measure the size of this drop with the camera and we raise the surface until it touches lightly the droplet. e is then exactly the height of the droplet. When the contact is just established, we rapidly lower the liquid bath, then search for the stability of the meniscus by raising or lowering the liquid bath. When the meniscus is stable, we raise very slowly the reservoir until the bridge just starts to expand. We measure the radius of the neck with the camera and note down the variation of e from the micrometric screw following the first contact. We let the meniscus grow and repeat the same process for a larger radius (Fig. 2) . Uncertainty in the measure of Rc comes from the difficulty to precisely evaluate the point of stability of the meniscus. In the range of small thicknesses (R õ 300 mm), the bridge is always about to break, leaving a droplet on the slide, or to rapidly expand. In contrast, in the range of large thicknesses, the system is less sensitive to a change in e, since the driving force for expansion or recession gets lower. We used a PDMS of intermediate viscosity (30,000 mPa.s) which makes it possible to explore the whole range with the best precision on Rc . This inaccuracy is actually difficult to estimate. Our procedure consists in defining a criterium of stability, i.e., a minimum time lapse during which the liquid bridge appears steady on the screen. This time lapse is reevaluated for
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DEBREGEAS AND BROCHARD-WYART
FIG. 2.
Side view of three metastable bridges.
each thickness, to take into account the dynamic effect. For a given thickness e, we get a narrow range of radius in which the bridge remains steady. This defines the experimental dispersion which we find equal to {50 mm.
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The sample ( b) being quite hysteretic, we can also measure a lower limit of the radius of the neck for which the contact line starts to recede. Between these values the meniscus is stable. However, the contact line recession being very
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slow, these measurements are less accurate and cannot be performed at large thicknesses where the driving force tends to zero. Despite the hysteresis, it must be noted that the bridge always remains perfectly axisymmetric. This can be easily checked by observing the contact patch from above. It appears to be very circular with no local pinning of the contact line. The thickness of the air film measured this way must be corrected for the general level of the liquid has been lowered by roughly e ∗ ( R / L) 2 . We finally evaluate the thickness precision to {30 mm, including the measurement of the pendant droplet as well as the precision of the micrometric screw. Stability of a Meniscus Consider a meniscus connecting the solid to the liquid bath. Its shape is given by r( z) ( Fig. 1) where ( r, z) are the cylindrical polar coordinates rescaled using the capillary length k 01 . The origin ( r Å 0, z Å 0 ) of the coordinate system is chosen at the level of the free surface far from the bridge and on the symmetry axis of the meniscus. The static equilibrium between the Laplace and hydrostatic pressures leads to the differential equation for r( z) , where r * Å dr / dz and r 9 Å d 2r/ dz 2 0
r9 1 / Åz ( 1 / r * 2 ) 3 / 2 r( 1 / r * 2 ) 1 / 2
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These analyses can be entirely transposed to the case of liquid menisci and we can actually consider the growing of a meniscus as the dewetting of an air film. For a given contact angle u and a thickness e õ ec ( u ) of the air film, a meniscus of initial radius R will expand if R ú Rc and shrink if R õ Rc . We will focus in the following on the numerical calculation of T&M ( as described in Ref. (1 ) ) to fit our experimental results for Rc ( u, e) . Results and Discussion The data presented in Fig. 3 compares the numerical fit with the observed advancing or receding contact angle. The experimental results are in good agreement with the calculation in nearly the whole range of thickness explored. It must be noted that our experimental approach is signifi-
[2 ]
with the boundary conditions (i) r(0) Å ` and (ii) r * ( z Å e) Å cot( u ), u being the solid/liquid contact angle. Note that for an hysteretic substrate, u Å ua will give the advancing critical radius, whereas u Å ur will give the receding critical radius. This nonlinear equation was used by James ( 2 ) to study the shape of a meniscus connecting a liquid bath to a small cylindrical rod and to derive an asymptotic solution in the limit of small radii. This problem is formally equivalent to that of a hole made in a sheet of liquid deposited on a horizontal surface. This question has been more extensively studied since it is of considerable practical importance in the coating industry. Taylor and Michael (T&M ) ( 1) showed that for a given contact angle u and thickness e õ e c ( u ) defined by Eq. [1] , there exists a unique solution to Eq. [ 2] which can be characterized by the throat radius Rc of the extended profile. They demonstrated that Rc represents the limiting value between holes which would open out ( R ú Rc ) and holes which would close up ( R õ Rc ) . Using a different approach, Sharma and Ruckenstein (3 ) have considered the free energy of the film after hole formation and compared it with the energy of the unbroken film, in order to determine a criterion for the stability of a hole in a sheet of fluid. Their results are mostly in agreement with T& M’s calculations. More recently, other authors ( 4, 5) have independently derived asymptotic expressions of the critical thickness for dewetting using Eq. [ 2] , when a hole of small radius is drilled in a liquid film.
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FIG. 3. Critical radius ( Rc ) versus reduced thickness ( ke ) and the corresponding calculated fit from Eq. [2 ]. (a ) Nonhysteretic silicium wafer, ue Å 117. ( b) Hysteretic glass slide, ua Å 687, ur Å 287.
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DEBREGEAS AND BROCHARD-WYART
cantly different from previous studies (Padday (6), T&M (1), Sykes et al. ( 7)) on the stability of liquid films. In these experiments, holes of various sizes were made in liquid sheets of same depth. The upper value of hole radii which close up and the lowest value of those which open up were asymptotically deduced from a large number of experiments. The holes were invariably made by an external probe or a jet of air. The way the hole is created may influence the evolution of the film. We ourselves observed that at lowest thicknesses, a small perturbation can destabilize the meniscus in one way or another. Moriarty and Schwartz ( 8) have recently underlined the importance of the initial shape of the hole to predict its opening or closing up. Hence, they developed a time-dependent model, including viscous forces, to follow the evolution of the hole profile after initial formation. In addition, the surface can be slightly damaged by the contact with a probe which can perturb the following experiments. Here the depth of the film of air is adjusted during one single experiment until one gets the meniscus steady without any contact with the solid surface. III. DYNAMICS: GROWTH OF A LIQUID MENISCUS
Experimental Setup We still consider air layers of thicknesses e õ ec . Our present aim is to investigate the kinetics of growth of a liquid bridge of radius R @ Rc (e) , for a given value of e.
FIG. 4.
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We perform video measurements of the radius R of the circular contact patch, versus time, from above the sample. We need the solid substrate to be transparent, and therefore we only use the glass slide. The thickness e of the air layer, in the range of 40 mm to 2 mm, is measured prior to nucleation using the method described above. As we indicated in part I, the general level of the liquid gets lower as the meniscus grows. We thus restricted our observations to the range 2.5 õ R õ 6.5 mm and calculated a mean correction that we applied to the initial thickness of the air layer. The first inequality is prescribed to ensure that Rc ! R. The effects of azimuthal curvature on the growing meniscus are therefore negligible. We varied the PDMS viscosities h by three orders of magnitude ( between 10 3 up to 10 6 mPa.s) . Static contact angles ua and ur were found to be almost independent on molecular masses. We also performed measurements of the ( solid/ liquid) dynamic contact angle ud during the growth process. We used a technique developped by Andrieu et al. ( 9) to measure dynamic contact angles in the dewetting of liquid films. It is based upon the optical deformation of a grid induced by a liquid diopter. Here, the grid lays at the bottom of the liquid bath. When filmed from above, the image of the grid is distorted near the contact line ( Fig. 4a ). The deformation d at the line can be related to ud knowing the refractive indexes of the two media ( air, n0 Å 1; PDMS, n1 Å 1.4 )
Measurement of the dynamic contact angle. (a ) Image of the deformed grid. (b ) Sketch of the contact angle measurement technique.
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FIG. 5. Radius R of a growing liquid meniscus versus time t for three different values of the intercalated layer of air: m (140 mm) , j (300 mm) , l (680 mm) . The liquid viscosity is 10 5 mPa.s.
FIG. 6. Capillary number Ca Å hV/ g versus thickness e fitted by the empirical law: Ca (e) Å c te ∗ e 00.7 (1 0 (e/ ec ) 2 ). l ( h Å 10 3 mPa.s ), l ( h Å 10 4 mPa.s ), j ( h Å 10 5 mPa.s), m ( h Å 10 6 mPa.s ).
D (e) à gs/ o / g 0 gs/ l 0 and the vertical distance h from the grid to the contact line ( Fig. 4b) . This technique gives precise results in the limit of small deformations, i.e., small angles of the air wedge: C Å p 0 ud . In this limit, one finds CÅ
n1 d . ( n 1 0 n0 ) h
1 rge 2 2
S S DD
Å 0S 1 0
e ec
2
.
[ 4]
Concerning the dynamic contact angle, Fig. 7 shows that the supplementary angle C Å p 0 ud depends linearly on ke: C Å a∗ ke, with a Å 1.18.
We could thus measure C for thicknesses ranging from 40 to 250 mm. For thicker air layers, the angle C is no longer in the domain of validity of the technique. Results and Discussion We find that R varies linearly with time t , as shown in Fig. 5, and that the velocity V Å dR/ dt is proportional to the liquid viscosity for a given thickness e. The relevant parameter for the hydrodynamic problem is therefore the Capillary number, Ca Å hV/ g. Its dependence upon the air thickness e is presented in Fig. 6. It appears that Ca(e ) can be fitted by a power law in the form Ca(e) Å cte ∗ e 00.7 ∗ D (e) ,
[3 ]
where D (e) is the driving force of expansion of the triple line per unit length. In the limit considered here, R ú Rc (e) , D (e) can be written simply as the sum of capillary and gravity forces
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FIG. 7. C Å p 0 ud ( ud is the dynamic contact angle) versus reduced thickness ke .
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DEBREGEAS AND BROCHARD-WYART
All these results can be interpreted by the following picture. ( 1) The profile of the meniscus is quasistatic. If we assume that the moving meniscus is quasistatic, then its shape must obey the hydrostatic equilibrium condition [1] . In the limit R @ Rc (e) , the azimuthal Laplace pressure term vanishes, and Eq. [1] yields to 0
r9 Å z. ( 1 / r *2 ) 3/2
thicknesses. However, this result can be generalized a fortiori to the whole domain explored, since the velocity gets lower as we thicken the air layer and the profile becomes static. We can conclude that the thickness of the air layer, which is our controlled parameter, entirely determines the dynamic contact angle ud , according to Eq. [5 ] . ( 2) The growth velocity V Å dR / dt follows the Hoffman Tanner law. The driving force D (e) acting on the quasistatic meniscus is also the uncompensated Young force on the contact line. Using Eqs. [ 4] and [ 5 ] , D (e ) can be written as
This equation is now solvable. After integration, one gets the following relationship between ud and e: k 2e2 cos ud Å 0 1. 2
[5]
Hence, in the limit of small thicknesses e, we have C Å p 0 ud Å ke .
[6 ]
The quasistatic hypothesis is consistent with the linear dependence C Å a∗ ke experimentally observed, with a prefactor a Å 1.18 close to 1. Our experimental procedure enables us to check this behavior only in the limit of small
D(e ) Å gs/ o 0 gs / l 0 g cos ud . The velocity versus dynamic contact angle dependence of a moving contact line has been experimentally studied by Hoffman ( 10 ) over a wide range of velocities and wetting conditions, using silicone oils in capillary tubes. In his experiments, the velocity was externally imposed. He showed that all his data could be described by a master curve, exhibiting a unique function F , so that F( ud ) 0 F ( ua ) Å Ca,
[ 7]
where Ca, ud , and ua have the same meaning as in the present study.
FIG. 8. Dynamic contact angle ud versus Ca / F ( ua Å 687 ). Gray spots: Hoffman’s results for different silicone oils in capillary tubes. Black spots: our results.
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S S DD
This growth law can have a possible link with some peculiar aspects of adhesion: when a solid plane is brought into contact with a soft elastomeric hemisphere ( JKR) , the effective contact surface varies with time, inducing variations of the adhesion energy, before it reaches its equilibrium value ( 12) . Our results can possibly help to understand the dynamics of this phenomenon. ( iii ) Nucleation of one single liquid bridge by the mean of a pendant droplet is a way of achieving perfect contact, with no air trapped at the interface. This may have practical applications for adhesion of polymer liquids ( ‘‘tack’’) or soft rubbers on solid surfaces.
IV. CONCLUDING REMARKS
ACKNOWLEDGMENTS
( i ) We have found a way to measure the limiting radius for which a meniscus, connecting a liquid bath to a horizontal plate, will expand or retreat when the surface is nonhysteretic. In the case of hysteretic samples, the method allows us to measure a range of R and e within which a meniscus is stable. The method presented here has the advantage of reaching these limiting values by adjusting the critical thickness in a quasistatic way, using a nondisturbing nucleating method. ( ii ) Above and below Rc , bridges expand or shrink. We have also studied the dynamics of growth of capillary bridges in the viscous regime. This is to our knowledge the first experimental study of spontaneous wetting at a very high dynamic contact angle. We showed that the liquid meniscus remains in a quasistatic shape and that, for a given thickness of the air layer, the growth velocity V Å dR/ dt is constant and obeys the empirical law
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V}
1 00.7 e 10 h
e ec
2
A theoretical description of the viscous flows was given by Tanner ( 11 ) in the limit of small contact angle ud , leading to the so-called Tanner’s law: ud } Ca 1 / 3 . A good agreement was met with experimental data up to 1007. In the domain of higher contact angle, the situation remains controversial. Considering the resemblance of our three-phase system to Hoffman’s, we will use the empirical relationship [ 7] to analyze our data. We have plotted in black in Fig. 8 the dynamic contact angle ud , deduced from e measurements via Eq. [ 5] , versus Ca / F ( ua Å 687 ) . A good agreement is found with Hoffman’s results displayed in the same graph. This in turn confirms again the quasistatic hypothesis and indicates that the observed Ca(e) dependence [ 3] is compatible with the empirical laws of contact line motion.
.
We thank C. Sykes, A. Buguin, and R. Bruinsma for stimulating discussions, and L. Perthuis for technical assistance.
REFERENCES 1. Taylor, G. I., and Michael, E., J. Fluid Mech. 58, 625 (1973 ). 2. James, D. F., J. Fluid Mech. 63, 657 (1974 ). 3. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 137, 443 (1990). 4. Sykes, C., C. R. Acad. Sci. Paris 313 II, 607 (1991 ). 5. Sharma, A., J. Colloid Interface Sci. 156, 96 (1992 ). 6. Padday, J. F., Spec. Discuss. Faraday Soc. 1, 64 (1971). 7. Sykes, C., Andrieu, C., De´ tappe, V., and Deniau, S., J. Phys. III France 4, 775 (1994 ). 8. Moriarty, J. A., and Schwartz, L. W., J. Colloid Interface Sci. 161, 335 (1993). 9. Andrieu, C., Chatenay, D., and Sykes, C., C. R. Acad. Sci. Paris, 320, 351 (1995 ). 10. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975). 11. Tanner, L. H., J. Phys. D 12, 1473 (1979 ). 12. Michel, F., and Shanahan, E. R., C. R. Acad. Sci. Paris 310, 17 (1990 ).
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