Langmuir 1991, 7, 335-338
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Spreading of Nonvolatile Liquids in a Continuum Picture FranCoise Brochard-Wyart’ Structure et Rkactivits aux Interfaces, Bhtiment de Chimie Physique, Universitb Paris 6, 11 rue P. et M . Curie, 75231 Paris Cedex 05, France
Jean-Marc di Meglio, David QuBrB, and Pierre-Gilles de Gennes Physique de la MatiBre Condensbe, CollBge de France, 75231 Paris Cedex 05, France Received March 26, 1990. In Final Form: June 25, 1990 We discuss wetting criteria for solid/liquid pairs where the long-range interaction is not oscillating and is described by a Hamaker constant A . The key parameters are then A and the spreading coefficient S. Since S contains contributions from short range interactions, S and A are independent variables and can be of either sign. Discussing the resulting four possibilities, we expect three fundamental regimes: (1) complete wetting, a small droplet spreads to become a flat “pancake” surrounded by a dry solid; (2) pseudo partial wetting, a droplet forms a spherical cap with a finite contact angle 0 but the surrounding solid is wet, the drop is in equilibrium with a molecular film; (3) partial wetting, the contact angle 0 is nonzero and now the solid around the drop is dry. The pseudo partial wetting regime may explain some surprising observations concerning the spreading of silicone droplets.
I. Fundamental Parameters of the Continuum Picture The spreading of nonvolatile liquids on ideal smooth solid surfaces is often discussed in terms of the free energy F (per unit area) of a film of thickness e1 F(e) = ySl+
+Pk)
(1)
where ys1(y)are the solid/liquid (liquid/air) interfacial tensions. A t large e , P ( e ) tends to zero. When e is larger than the molecular size ao, P ( e ) is controlled by long range van der Waals forces
P(e) = A/123re2
a,
0, A > 0 This case is pictured in Figure 1; the free energy F(e) per unit area of a flat film of thickness e is given by eqs 1-3, where we take both A and S to be positive. P ( e ) defined in eq 1 is related to the disjoining pressure n ( e ) introduced by Derjaguin”
n ( e ) = -dP/de (4) F(e) may be a monotonically decreasing function (Figure la), or F(e) may have a maximum (Figure lb,c) at e = e,. Then, for e < e,, n ( e ) becomes negative. In any case, the free energy minimum corresponds to a solid, wet by a thick film (e m). This complete wetting regime is studied in detail in ref 3. The main conclusions are summarized below.
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(6)Wayner, P. C. J . Colloid Interface Sci. 1982,88,294-295. (7)Deryaguin, B.V.; Churaev, N. V.; Muller, V. M. In Surface Forces; Plenum Press: New York, 1987. (8)Dietrich, S. In Phase Transition and Critical Phenomena; Domb, C., Lebowitz, J., Eds.; Academic Press: London, 1988;Vol. 12. (9)Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989,338,640-641. (10)Churaev, N. V. Reu. Phys. Appl. 1988,23,975-987. (11)Deryaguin, B. V. Kolloid Zh. 1955,17, 205-216.
0743-746319112407-0335$02.50/0 0 1991 American Chemical Society
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336 Langmuir, Vol. 7, No. 2, 1991
S = e n(e) + P(e) (6) The solution e, of eq 6 can be graphically constructed as shown in Figure 1; e, is given by the tangent to the curve F(e),which intercepts t h e y axis a t yso. It corresponds to a minimum of 9(e) if F”(e) > 0. Equation 6 has one root for cases a and b and two roots for case c corresponding to a maximum (e = e,(”)) and a minimum (e = e,(min))of 3(e). Inserting eq 2 in eq 6 leads to5 e , = ( A / ~ T S ) ’=/ a(3y/2S)li2 ~
(7)
where a is a microscopic length defined by
14(/67r= y a 2
(8) The thickness of the wetting layer can be large if A I S is large. In this limit, the continuum theory is well justified to derive the shape of the pancake. (2) Profile at thecontact Line. The profile is deduced from the minimum of the free energy per unit length
where we have assumed that dz/dx is small. PO is the Lagrange multiplier associated with the constraint of fixed volume Q = S z dx. The minimization of eq 9 with respect to z leads to -yz”
+ P = Po
(10)
which has a first integraP
In the flat part of the drop, eq 10 leads to PO= -We,) and eq 11 becomes (c1
Figure 1. Qualitative plot of the free energy F/cm2 of a film versus thickness e for both A and S positive (a) F(e) is monotonously decreasing, (b) F(e) has a maximum and one inflection point, (c) F ( e ) has two inflection points. In all cases, F(e) is minimal for e a;this corresponds to the total wetting situation. We have also shown the construction of the tangent t o derive the thickness e, of the pancake shown in Figure 2. The solution ea(min) corresponds to a minimum (a maximum) of the energy 3 ( e ) .
-
DISTAL
PROXIMAL
+
+
+
+
Figure 2. Final “pancake” in the case of complete wetting ( A > 0, S > 0). If S is small (while A remains of order k T ) , the final thickness e, is larger than the molecular size ao. Near the contact line, the profile is parabolic in the proximal region controlled by van der Waals energies and ends with a microtip reaching the solid surface with a horizontal slope.
( 1 ) Thickness of the Spreading Droplet. A small droplet put in contact with a flat solid surface spreads out and becomes the thin “pancake”pictured in Figure 2. The thickness of the wetting film e, results from a competition between long-range forces, which tend to thicken the film, and S, which favors a large wet region. The free energy of the droplet of volume Q is
9(e) = 9, - A S + H e ) A (5) where A is the surface of the wet area. The equilibrium thickness corresponds to the minimum of 3(e) with constraint s2 = &e. Equation 5 leads to
Near the contact line defined by x = XL, the first two terms are dominant and z ( x ) has a parabolic shape z2 = 2a(x - x L )
(13)
One final remark, eq 13 holds for e, >> z >> a. If we want to go down to smaller sizes ( z of order a ) ,short-range forces come into play and eq 12 implies dz/dx = 0. Thus the pancake ultimately ends with a molecular tip as shown in Figure 2. But at these microscopic scales, the whole continuum picture breaks down, and this tip is thus not very significant. Conclusion. For S and A positive, a droplet spreads and achieves a thin “pancake” surrounded by a dry solid. 111. Pseudo Partial Wetting: A C 0, S > 0, or S < 0 (1) S > 0, A C 0. We have pictured in Figure 3 the free energy F(e) of a liquid film for S > 0 and A C 0. In this
case the free energy must have a minimum a t a certain value, em,of the film thickness. Two regimes may arise depending on the volume of the droplet. (a) A microscopic droplet ( Q Asem,where A, is the total area of the solid) has two possible modes of spreading: (i) If the construction of Figure 3a applies, the final state will be a “pancake” of finite thickness. (ii) In the case depicted in Figure 3b, the final state corresponds to e = 0, i.e. to a very dilute (bidimensional) gas of molecules expanding indefinitely on the solid. (12) The choice of the integration constant in eq 11may be understood by considering the case of S negative (and small). Then, at large z, we have P(z) = 0 and with a large droplet (weak curvature) PO= 0, and this gives -S = 1/2y92 = y ( l -cos e), where 8 is the equilibrium contact angle.
Spreading of Nonvolatile Liquids
Langmuir, Vol. 7, No. 2, 1991 337
Fk)t
HYPERBOLIC TRUNCATED
\ WET
1
\
PROFILE
x eE
em1
1
Figure 5. Profile of a droplet wedge in the case of pseudo partial wetting. The droplet and the film join with a hyperbolic profile. We have also represented the truncated wedge discussed by
‘4’ ’ \
I /
,’
‘\
Wayner.6 increased d A of the surface a t constant volume 0 = A e : d 3 = ( P ( e )+ eII(e))d A ) . A simplified model assumes that 3 ( e ) has a sharp minimum at e = em and that 3 ( e ) retains the van der Waals form (eq 2) a t all thicknesses e larger than e,. This leads to
\
cos 0 = 1 - a2/2em2 (15) The profile of the cross over region between the film and the macroscopic cap is also given by eqs 10 and 11 but the Here constant of integration is now y - ye instead of (3). we have PO= n ( e m )= 0, expressing the fact that the film is in equilibrium with a macroscopic droplet. Equation 11 gives
The profile deduced from the simplified model, i.e. from eq 16, has the hyperbolic form z 2 ( x ) = em2+ a2/em2(x- xL)2
(C) (d1 Figure 3. Free energy corresponding to pseudo partial wetting, A < 0: (a, b) S > 0 and (c, d) S < 0.
/
DROPLET
Figure 4. In pseudo partial wetting conditions, the final equilibrium state of a large droplet is a spherical cap and a film of thickness e,,, corresponding to the minimum of F ( e ) shown in Figure 3.
(b)For a larger droplet (Q> Asem),the final equilibrium state is a film of thickness e , in equilibrium with a residual droplet (Figure 4). What is found is very similar to the Plateau border of soap films.13 The contact angle 0 of the droplet at equilibrium is given by a balance of forces acting on the contact line
and is qualitatively depicted in Figure 5. For S > 0 but A < 0, we find that the final state is a drop with a finite contact angle and a solid completely coated with a film of thickness e,. We call this regime pseudo partial wetting, because the solid surrounding the droplet is wet. This regime may explain some results observed with poly(dimethylsi1oxane) spreading on either silicon wafers or water: the precursor film surrounding the spreading droplet is of molecular size and a residual droplet resists spreading, while S is clearly positive.gJ4J5 (2) S < 0, A < 0. The pseudo partial wetting regime may also be found for S < 0, if F(e) presents a minimum as shown in Figure 3c,d. The contact angle is still given by eq 13 and the profile of the drop reaching the wet solid surface is given by eq 17. This case with S < 0 and A < 0 was first considered by Wayner? who described the contact line as a truncated liquid wedge (Figure 5). In fact the wedge is smoothly continued by a liquid film without any discontinuity. Conclusion. The pseudo partial regime describes a droplet spreading with a finite contact angle, but the solid surrounding the residual droplet is wet by a film of thickness e , of order ale, which can be relatively thick in the limit of small contact angles: this is the most interesting limit for the continuum theory. A microscopic droplet, which cannot achieve the film of thickness e , may either form a cohesive spot of liquid (Figure 3a,c) or spread to infinity (Figure 3b,d).
cos 0 (14) where ye = y + e n + P is the effective surface tension of the microscopic film. For e = e,, n = 0 and ye = y + P. Our definition of ye can be deduced from the variation of d 3 of the free energy of the liquid film associated with an
IV. Partial Wetting We call partial wetting all situations where the liquid droplet makes a finite contact angle on a dry solid. (1) S < 0, A > 0. We have pictured in Figure 6a F ( e )
(13) De Fejter, J. A,; Frij, A. J. ElectroanaL Chem. Interfacial Electrochem. 1972, 37, 9-20.
(14) Langevin, D.,private communication. (15) Daillant, J.; Benattar, J. J.; Bosio, L.; LBger, L. Europhys. Lett. 1988,6, 431-436.
ye = y
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338 Langmuir, Vol. 7, No. 2, 1991
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Pso
1:
( b)
Figure 6. Free energy F ( e )correspondingto partial wetting: (a) A > 0, S < 0; (b) A < 0, S < 0. F(e) has no minimum at finite
thickness.
(b)
Figure 7. Final equilibrium of a droplet in the case of partial uwtting. The solid around the droplet is dry: the profile is hyperbolic and curved downward if A > 0 (a) and upward if A < 0 (b).
for S < 0 and A > 0. There is a nonzero contact angle 6 given by the Young equation
Yso = YSL + Y cos e (18) The profile near the contact line deduced from eq 11with PO= 0 (eq 10 assuming a large droplet (weak curvature)) is given by 112 Y(dz/dx)2 = 112 y@+ 112 y a 2 / z 2 (19) The profile (Figure 7) is hyperbolic and curved positively z2 =
- a2/@
(b)
Figure 8. Examples of an oscillating free energy F(e) (resulting from short-range interactions). For A > 0 our discussion must be modified, if F(e) has a minimum at finite thickness.
(20)
(2) Cases of Partial Wetting (No Film) with S < 0, A C 0. This occurs if F(e) has no minimum and is monotonously increasing (Figure 6b). The corresponding profile is shown in Figure 7b and has a negative curvature, because the sign of&) in eq 11is negative. In the proximal
region z2 = + a2/02 (21) V. Concluding Remarks Contrary to common belief the condition for complete wetting is not only S L 0. The sign of A has to be specified: the wetting criterion for nonvolatile liquids depends on two parameters: (1) the sign of the spreading coefficient and (2) the sign of the Hamaker constant A = A,I - An. The main conclusion is the existence of a pseudo partial wetting regime if 3 ( e )has an absolute minimum a t finite thickness. In all our discussion, we have assumed plots of energy versus distances F(e)which interpolate smoothly between the long range van der Waals tail and the dry limit (P(0) = S ) controlled by short-range contributions. Actually, at short scales (for molecular diameters) oscillations of F(e)due to steric interactions may be important as shown in Figure 8. This may lead to an absolute minimum of F(e)at finite thickness and again to a pseudo partial regime. This effect may modify our criteria in the case A > 0 but not for A C 0. Acknowledgment. We have benefited from discussions with J. Daillant and L. Leger and we thank Claire Wyart for her help. Appendix Structure of the Effective Hamaker Constant A. Adding the pairwise interactions for the liquid film inserted between the solid and the gas, one obtains a quadratic function of the three polarizabilities a],as,and ag(aiis the polarizability per unit volume of phase i). Since P ( e ) vanishes for a1 = a,or for a1 = ag,the structure of P(e) has to be W e ) = K(aI- a,)(al- ag) (Al) K is derived from the simple case of free liquid film under vacuum ( a , = ag = 0)
P ( e ) = K a12= -Al,/127re2 Hence giving for a film on a solid substrate
(A2)
P ( e ) = -A/12ne2 643) with A = A11 - A,1+ A,, - ASg.For nonvolatile liquids, a~ is negligible compared to a1 and cyg, which gives for the effective Hamaker constant A = A,I - AI](see eq 2).
