Langmuir 1992,8, 3033-3031
3033
Shear-DependentSlippage at a Polymer/Solid Interface F. Brochardt and P. G. de Gennes'J PSI, Institut Curie, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France, and Collage de France, 75231 Paris Cedex OS, France Received July 1, 1992 We discuss shear flows of a polymer melt near a solid surface onto which a few chains (chemically identical to the melt) have been grafted. At low shear rates u < u* we expect a strong friction, analyzed in ref 1.1 Above a certain critical shear u* the grafted chains should undergo a coil stretch transition. In the stretched state, they are not entangled with the melt, and a significant slippage is expected when u > u*. This transition may be important in the processing of polymers, where a few chains from the melt can be bound on an extruder wall and play the role of the grafted chains.
I. Principles When a polymer melt flowsalong a solid surface (Figure l),under a shear stress u, there may exist a nonzero flow velocity, V, at the surface. The ratio k = ulVis the friction coefficient. Equivalently one may describe the flow pattern in terms of a slippage length
b l I 1" I
Figure 1. An idealized view of shear flow near a surface, assuming that the viscosity of the liquid ( 7 ) is the same at all scales. The shear stress u is the same at all distances y; the shear rata S = dvfdy is also constant. There is a finite velocity at the surface
where 7 is the melt viscosity. (1)In ideal conditions, with a smooth solid surface, and no chains attached to it, one expects that the friction k is comparable to what it is in a fluid of monomers: k = k,. On the other hand the viscosity q of an entangled melt is huge, and b can become very large (- 100pmh2 There are some experiments which do suggest strong slippage: (a) observations in a transparent e ~ t r u d e r (b) ; ~ rheological studies on molten polystyrene in a plats-plate geometry with small gaps;* (c) studies on thin films in wetting or dewetting processe~;~ (d) measurements on multilayer extrusions6 (where we probe a number of meltlmelt interfaces). (2) However, in most usual situations, no slippage is ~bserved.~ This may be due to various effects: (a) a thin layer of polymer near the wall may be glassy; (b) some chains from the melt can bind strongly to special sites on the solid surface. Thus the solid behaves really like a fluffy carpet, and this suppresses slippage. We recently investigated a model example, where a small number of chains ( v chains per unit area) is grafted on the solid.'I8 We found that in the low velocity regime (V- 0) slippage is stronglysuppressed;the slippage length b being reduced
V(y=O)=
(- 10 nm); thus, in all cases where (through a glassy layer or through local attachment) some chains are bound to the wall, we expect no slippage in slow flows,in agreement with ref 7. (3) In the present paper, we investigate stronger flows, and their effect on a weakly grafted layer ("weakly" meaning the "mushroom" regime, where different grafted chains do not overlap, vRo2 < 1). The opposite case of a strong "brush" (vRo2>> 1)under strong flowswas diecussed in detail by Alexander and However, one of us (P.G.)pointed out a year ago that the mushroom regime might be more interesting, become the grafted chains will undergo a "coil stretch transition" under flow.1° At thie time, a computer simulation (or an ideal grafted chain plus solvent) exhibited the transition.ll We realized progressively that the central problem is different: Instead of a simple solvent, we must have a polymer melt (chemically identical to the grafted chain), then the discussion becomes relevant to many practical systems, such as extruders, where the solid surface may often attach a few chains. Also, on the theoretical side, we shall see that the coillstretch transition induced by polymer flow is more spectacular, because the grafted chains disentangle at the transition point; the friction in the disentangledregime is a Rouse friction,12much weaker than the entangled friction. In section 11, we consider a simplified system, with no wall and a chain of 2 units hooked at one end onto a fmed point (Figure 2). The chain is immersed in a polymer melt (N units per chain) flowing at a velocity V, and it exhibits a transition at a certain critical velocity V*. We constantly assume N > 2.
to bo
(vR,)-'
where Ro = @I2a is the coil size of the grafted chains (2 is the number of monomers per grafted chain and a is a monomer size). Equation 2 gives b values which are small t PSI,Institut Curie.
*(1)Collage de France. Brochard, F.; de &Mea, P. G.;Pincus, P. C.R. Seances Acad. Sci. 1992,314,873-878. (2) de &Me& P. G. CJI. Acad. Sci. 1979, N E , 219. (3) Galt, J.; Maxwell, B. Modern Plastics; McGraw-Hill: New York, 1964. (4) Burton, R.; Folkea, M.;Karm, N.; Keller, A. J. Mater. Sci. 1983, 18, 315. (5) W o n , C. Private communication. (6) Miroshnikov, A. Vysokomol. Soedin., Ser. A 1987,29, 519-82. (7) See the following reviews: (a) Meisener, J. Polym. Test. 1983,3, 291. (b) Meieaner, J. Annu. Rev. Fluid Me&. 1986, 17,45. (8) de Gennee, P. G. Simple Views on Condensed Matter; World Scientific: London, 1992.
0143-7463/92/ 2408-3033$03.Q~/ 0
v = Sb.
(9) Rabin, I.; Alexander, S. Europhys. Lett. 1990, 13, 49-54. (10) de Gennee, P. G. J. Chem. Phys. 1974,60,5030. (11) Mavrantzaa, V. G.;Beris, A. N. J. Rheol. 1992,36,175-213. (12) For a general presentation of thie concept, we de G~Mw,P. G. Introduction to polymer dynamics;Lezioni Lincee; Academia Nazionale dei Lincei: Rome, 1% pp 1-57. (9
1992 American Chemical Society
Brochard and de Gennes
3034 Langmuir, Vol. 8, No. 12, 1992 L4
I
Figure 2. A tethered chain-attached by one end at a fixed
-
point 0-immersed in amovingpolymer melt, of velocity V.When V is not too low (V V*) the chain elongates significantly.
-
V'
V
Dj
+ + + + + + Figure 3. A grafted chain under shear flow in a melt. Where +
Vl
v
Figure 4. Elongation L versus flow velocity V for a tethered chain in a melt: (E) = entangled regime; (R) = Rouse regime (disentangled).
the shear stress u exceeds a low threshold u* (eq 22), the chain is strongly stretched.
In section 111,we investigate the chain plus wall (Figure 3). Here, what is imposed is not the velocity at the surface V , but the shear rate S or, equivalently, the shear stress u = qS. In this situation it will turn out that, beyond threshold, the slippagelength b is velocity dependent, and thus the velocity at the surface V = bS is not proportional to S; thus the discussion is slightly more complicated. Possible consequences, limitations, and extensions of these ideas are listed in section IV. All our discussion is restricted to the level of scaling laws; in most formulas, we ignore numerical coefficients and replace the equality signs by the symbol =. To find exact coefficients in the entangled regime would represent a very heavy program.
11. One Tethered Chain in a Flowing Melt 1. Friction in the Entangled Regime. Our chain is pictured in Figure 2 as an elongated object of length L and diameter D. The elastic force F or the hook is derived from standard discussions on ideal chains12J3
(3)
where
is the reptation viscosity of the melt. Equating T$ to the product of force by velocity FV we arrive at the basic formula
F qLV ( L >> Ro) (9) (c) To interpolate between the low elongation case (eq 4) and the high elongation case (eq 91,we shall use the following formula
F = qV(R;
+ L')"'
VlV* =
+ L')"'
(10) Other interpolatioins could be used-they are all equivalent at our scaling level. 2. Critical Velocity v*. Equating the elastic force (eq 3) and the viscous force (101, we obtain a relation between elongation L and velocity V of the form L
(R:
(a)At very low velocities ( L< Ro),our chain is a spheroid of size Ro, and we know from ref 1 that the friction law, relating F and V , is (surprisingly)similar to a Stokes law
where we have defined
F g qRoV (4) (b) How is eq 4 modified when we have a strong elongation ( L > Ro)? Let us extend the argument of ref 1 and first count the number m of matrix chains which are coil. The volume of the elongated entangled with the (2) object is Q D'L. Each of the m chains has a number g of monomers in the volume Q , and crosses it over a length D. Thus g D2/a2. The volume 52 is mainly filled with matrix chains, and thus
Typically for q = 104 P, = 1000,a = 3 A, V* = 0.06 pm/sec. The plot of L(V) is shown on Figure 4. In our approximation L diverges for V = V*. Of course, this divergence is formally suppressed by the finiteextensibility of the 2 chains. But we shall see that the physical cut off is different. 3. Marginal State. (a) The preceding discussion assumed that the 2 chains are entangled with the surrounding melt. This requires that the number, g, of monomers spanning the cross sectional diameter D, be larger than the entanglementlengthNe. The limiting point corresponds to
-
-
-
-
Q a3mg where a3 is a monomer volume. Thus
(5)
m = L/a ( L >> Ro) (6) We now estimate the dissipation T$ due to the m chains reptating following the ideas of refs 1 and 14. Each of (NINe)V (where Ne them has a curvilinear velocity u is the entanglement distance) and
-
T$ = mJ;Nv' (7) where {I is a friction coefficient for a monomer. By use of eq 6 this can be cast in the form ~~
(13) Pincus, P.Macromolecules 1976,9, 386-391. (14) de Gennea, P.G. MRS Bull. 1991,20-21.
D = D*
Ne'f2a
(13)
L = L* z Rt/D* Whenever Ne fc.
Langmuir, Vol. 8, No. 12,1992 3037 Acknowledgment. This work was initiated by discussions with H. Hervet and L. LBger. It was performed at the NATO AS1 workshop on Interfacial Interactions in Polymeric Composites (June 1992). We wish to thank Professor G. Akovali for his hospitality on this occasion and X.Olympos for various exchanges.
