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Langmuir 1994,10, 1566-1572
Wetting and Slippage of Polymer Melts on Semi-ideal Surfaces F. Brochard-Wyart,*p+P.-G. de Gennes,t H. Hervert,f and C. Redont Laboratoire de Physico-Chimie des Surfaces et Interfaces,$lnstitut Curie, 1 1 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France, and Laboratoire de Physique de la Matibre Condensbe,l Coll2ge de France, 11 place Marcelin-Berthelot, 75231 Paris Cedex 05, France Received November 11, 1993. In Final Form: February 16, 1994”
We recently analyzed the dynamics of wetting for polymer melts on smooth solid surfaces with a few grafted chains (v chains per unit area). For u = 0 (“ideal” surface), we expect a strong slippage of the interface. For small v (“semi-ideal”surface),recent theoretical and experimentalstudies reveal a transition between a low-velocity, nonslipping regime and a high-velocity, slipping regime, where the shear stress takes a fixed value, u*. We investigate here the consequences of this transition on wetting and dewetting processes. Our discussion concentrates first on partial wetting conditions, which give relatively large flow velocities, and thus allow the slip regime to be reached. For dewetting processes, we find that a dry patch should first grow with a radius of R(t)= t2I3in a strong slippage regime, and then shift to a nonslip regime ( R = t ) when the size R exceeds a certain critical value, R,. A t 2 / 3 law has indeed been observed in experiments by C.R. The limiting radius R, depends strongly on the grafting density and on the contact angle, but may be typically in the millimeter range. We also discuss the case of complete wetting: here slippage is expected to be important only for large dynamic contact angles ed; below a critical dynamical angle, Be, slippages should be suppressed and the shape of the spreading droplet should simply be a sphericalcap. Above B,, we expect a sphericalcap plus a protruding (macroscopic)“foot”,and the precursor film becomes independent of the velocity.
I. Introduction Entangled polymers do not flow like usual liquids. P.G.d.G.l predicted that polymers slip on “smooth, passive” surfaces. The slippage is characterized by the extrapolation length b, defined by the distance to the wall at which the velocity extrapolates to zero (Figure 1). A pressure gradient parallel to the film induces a usual Poiseuille flow in thick films (thickness e >> b), but a plug flow in thin films (e < b): the polymer then moves like a solid, and the viscous dissipation is confined at the solid/liquid (S/L) interface. The conditions required to observe a slippage of the polymer have been studied recently.14 Three regimes are expected, depending upon the structure of the solid surface. (1) Ideal conditions’ (constant b): The solid surface is perfectly smooth and passive. This may occur on a silicon wafer covered by a compact molecular carpet of aliphatic chains. The length b is large, and the polymer is expected to slip;’ i.e., there exists a nonzero flow velocity, VS, at the solid surface. The ratio between the shear stress u at the S/L interface and the surface velocity Vs defines the friction coefficient k: (b)
Figure 1. Profile z ( x ) for advancing (a) and receding (b)contact lines. At n = 0, the contact angle is the equilibrium angle BE, instead of the dynamic contact angle Bd if no slip is present.
The extrapolation length is related to k: + Institut Curie.
College de France. URA-CNRS no. 1379. URA-CNRS no. 0792. Abstract published in Aduance ACS Abstracts, April 1, 1994. (1) de Gennes, P.-G. C. R. Acad. Sci. 1979, 228B, 219. (2)Brochard, F.; de Gennes, P.-G.; Pincus, P. C. R. Acad. Sci. 1992, 314,813. (3) Brochard, F.; de Gennes, P.-G. Langmuir 1992,8,3033. (4) Ajdari,A.;Brochard-Wyart,F.;deGennes,P.-G.;Leibler,L.;Viovy, J.-L.; Rubinstein, M. Submitted for publication to Macromolecules. f
0743-7463/94/2410-1566$04.50/0
where q is the bulk polymer viscosity. Equation 2 is obtained by equating two forms of the shear stress:
For ideal conditions, k = ko = qo/a,where qo is a monomer viscosity and a molecular size. The viscosity q is huge for an entangled melt [q = q0(W/Ne2), where N is the 0 1994 American Chemical Society
Langmuir, Vol. 10, No. 5, 1994 1567
Wetting and Slippage of Polymer Melts polymerization index and Ne the threshold for entanglements (Ne= loo)]. Equation 1 leads to b, = a(N3/N:)
(typically b = 10 pm for N = lo3 and Ne = lo2). A few long polymer chains ( 2 )Semi-ideal have been grafted on the solid surface, or a few chains from the melt can spontaneously bind to some special sites on the solid surface. This may occur (with silicon oils as the melt) after silanization of a silica surface, when some OH groups on the surface remain unreacted and can still bind a silicon monomer. The solid then behaves like a weakly grafted layer (v chains per unit area); slippage is strongly suppressed2 at low shear (a < u * ) , because the grafted chains are entangled with the melt. b reduces to bo z (vRo)-1 (3) where Ro = Z1/2ais the coil size of the grafted chain (2 is the number of monomers per grafted chain). Equation 3 gives b values which are extremely small (-100 A). At a critical shear stress of a* = v(kT/D*)(where D* = Ne1/2ais the distance between entanglements in the bulk melt), the grafted chains undergo a coil/stretch transition, and disentangle from the melt: a strong slippage is expected. At u = a*, the surface velocity jumps from a very low value, V* = kTIqRo2, to a huge value, V2 = (q/ 70)V*. Between V* and V2, the grafted chains are in a “marginal state”.2 The friction force per chain is constant and equal to kT/D*. The stress has a fixed value: u*
= v(kT/D*)
(4)
The friction coefficient k is thus velocity dependent: k = ko + v(kT/D*Vs)
(5)
This gives an extrapolation length, b, nearly linear in velocity:
For Vs = Vz,b( Vs)reaches the ideal surface value b = b,. Direct optical measurements6 of the slippage on the silanized surface have shown the existence of a critical stress, u*, and a dependence of b on surface velocity, b( Vs) 2:
vp.
( 3 )Nonideal conditions: Polymer chains are bound at many sites on the wall. a* and V* become very large; slippage is completely suppressed ( b is constantly small). This is the case (a) with a bare high-energy surface, (b) with silanized surfaces, where the silanization layer is not quite homogeneous, and ( c )where a thin layer of polymer near the wall crystallizes or becomes glassy. The effect of these densely bound chains on the spreading of polymer melts has been studied by Bruinsma.6 Our aim here is to study the dynamics of wetting and dewetting of thin polymer films deposited on ideal or semiideal surfaces. We have previously studied the dynamics of spreading of liquid droplets on an ideal s ~ r f a c eand ,~ the dewetting of thin polymer films assuming a strong ~lippage.~.g We discuss here the more general case where b may depend upon Vs. (5) Migler, K.;Hervet, H.; Lbger, L. Phys. Reu. Lett. 1993,70,287-290. (6) Bruinsma, R. Macromolecules 1990, 23, 276. (7) Brochard, F.; de Gennes, P.-G. J. Phys. Lett. 1984, 45, L-597. (8) Redon, C.; Brochard, F. Macromolecules 1994,27, 468-471. (9) Brochard, F.; Redon, C.; Sykes, C. C. R. Acad. Sci. 1992,314, 19.
Section I1 describes a liquid wedge moving on ideal or semi-ideal surfaces in partial wetting conditions. Section I11applies these results to the dynamics of drying, Le., the growth of dry patches in microscopic films (thickness e = 10 nm to 10 pm). We focus on the main experimental features [growth of the dry patch R(t) and width for the rim Z(t)].The detailed shape of the rim is constructed in the appendix. Section IV describes some dynamic features of complete wetting on semi-ideal surfaces. Our aim is to find out (1)whether the macroscopic foot predicted on an ideal surface exists also on semi-ideal surfaces and (2) how the precursor film is modified by slippage effects. Section V compares these predictions to some recent experiments of dewetting on nearly perfect surfaces. 11. Profile Near One Moving Contact Line Under partial wetting conditions, a liquid wedge at rest is characterized by an equilibrium contact angle, OE. In terms of the interfacial tensions yij (solid/gas, solid/liquid, and liquid/gas), OE is given by the Young relation y cos OE = yso - YSL (where we set YSG = y for simplicity). For a liquid wedge in motion (Figure l),the dynamic contact angle 6d is related to the velocity U of the contact line by Tanner’s law:lo
(7) where V* = y / q is a typical velocity. The logarithmic factor describes the divergence of the viscous dissipation near the contact line. (1) For simple liquids without slippage, xmin is a molecular cutoff a. At small Oe, Xmin is slightly larger @,in = a/&) and is controlled by long-range forces.’O (2) For entangled polymer melts, we shall show that xmin= b( v ) / e d , where b( V) is the slippage length. We shall also study the profile z ( x ) of the wedge in the slippage region [ ~ ( x = ) bl. The motion of the contact line results from a competition between capillary forces and viscous forces. If a,(O) is the viscous stress at the S/L interface, the friction force, due to the solid and acting on the liquid at distance x from the contact line, is
In the lubrication approximation, the flow V,(Z) of the wedge [of contour z(x)l moving at uniform velocity U is a Poiseuille flow and satisfies the conditions
s,’ V,(Z)dZ = ZU(uniform motion)
This leads to
The viscous stress is then
(10) de Gennes, P.-G. Reo. Mod. Phys. 1985,57,827.
(9)
Brochard- Wyart et al.
1568 Langmuir, Vol. 10, No. 5, 1994
i"'" Figure 2. Forces acting on the hatched region. P(z) = J.: ps)dZ,where p ( x , Z ) is the pressure distribution in the liquid.
The driving force on a wedge at distance x from the contact line is Fd
=
COS
e + ysL - yso + y z z r r= (1/2)y(e,2 - zr2)+ yzz"
(12)
where 0 = dz/dx is the slope of the wedge profile at point x . The first terms in eq 12 are the capillary forces and the last term is the Laplace pressure on the hatched region of Figure 2. The balance FV= F d leads to the basic equation for the dynamical wedge profile
Uis negative if 19< eE (receding contact angle) and positive in the opposite case. Equation 13 shows that z' = OE for x = 0. We calculate z to the first order in U/V* [ z = OEX O( U / V * ) ] .It is convenient to set w = z' - OE. Equation 13 leads to
+
0
NATIVE FILM
t
R 4
e
. )
Figure 3. Dewetting by nucleation and growth of a dry region. The liquid of the dry region is collected in the rim.
view, we can use it for melts in two different contexts: (a) ideal surfaces with b large and independent of V and (b) semi-ideal surfaces with b dependent on V. If we apply this to the spreading of a droplet, we find that the extended Tanner's law (eq 7) should hold for both cases-because the length b enters only in logarithmic prefactors: the special features of case b could be revealed only through a careful scrutiny of these weak factors or adetailed observation of the profiles near the contact line. We conclude that (for partial wetting) the spreading of drops is not a good probe of semi-ideal surfaces. 111. Structure of the Rim in a Dewetting Process The situation of interest here is depicted in Figure 3.
A typical example is a hydrophobic surface initially covered by a water film of thickness e. The film is metastable when the thickness is smaller than a certain limit, e,:13 e, = 2K-1 sin(0,/2)
We look for a solution of w = x s ( x ) . Equation 14 then gives
After integration, this leads to 2'
8EX
- 8, = 3u [log7--
v*6E2
(a) In the limit x
-
+ 36
OEX
3b
+ 3b]
OEX (16) log BEx
0, eq 16 reduces to
(20)
K~ = p g / y , where y = density and g = gravitational acceleration. Typically e, = 1mm. Here we are interested in strongly metastable films with e > z (thin rims):
At point A ( x = 01,we expect z = 0 and dris finite. Thus z'I,=O = eE
64.5)
At point B ( x = 1) we again have z" finite, z = e r 0, and z' = 0. This gives us an important relation for the length of the rim 1:
If we now go to dimensionless variables
x =XI1 (1)For V = 0, the equilibrium shape z ( x ) is a portion of a circle, because the Laplace pressure inside the liquid is independent of x . In the small angle approximation, this corresponds to _yz" = const and the static profile is a portion of a parabola (equivalent to a circle at our level):
eq A.4 becomes
X = 1- (dZ/dx)2+ 222" (2) For V # 0, we have discussed in the past the case of no slip (b = 0) in the following way:14 the integral on the left-hand side is logarithmic, and can be roughly treated as constant. Then the solutions of eq A.1 are again parabolic, but because of the added constant the dynamic angle dd is different from eE:
(A.7)
A numerical solution to eq A.8, with the appropriate boundary conditions Z'(0) = 1
Z'(1) = 0
is displayed in Figure 4b. Note the large asymmetry between the two sides. From a practical point of view, the most important numerical result is related to the volume of the rim:
(A.3) The values of & and V are then obtained by matching the solutions near both ends, giving t)d = 8342 and vlv* = (const)O~~. (3) For our present problem, with b # 0, the integral on the left size of eq A.1 is much more dependent on x and
From the plot of Figure 4b we find s 0.1. (4) The case of intermediate thicknesses ( z = b) is more complex and has not been computed-but it can be in principle be tackled numerically starting from eq A.l.
