Langmuir 1992,
Shear-Dependent Slippage at
8, 3033-3037
a
3033
Polymer/Solid Interface
F. Brochará* and P. G. de Gennes*1 PSI, Instituí Curie,
11 rue
Pierre et Marie Curie, 75231 Paris Cedex 05, France, and College de France, 75231 Paris Cedex 05, 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 < * we expect a strong friction, analyzed in ref l.1 Above a certain critical shear * 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 a > *. 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 flows along a solid surface (Figure 1), under a shear stress , there may exist a nonzero flow velocity, V, at the surface. The ratio k = /Vis the friction coefficient. Equivalently one may describe the flow pattern in terms of a slippage length
7
b
\
I
/
·/ I
b
where
=
=
n/k
(1)
Figure 1. An idealized view of shear flow near a surface, assuming that the viscosity of the liquid ( ) is the same at all scales. The
is the melt viscosity.
shear stress is the same at all distances y; the shear rate S = dv/dy is also constant. There is a finite velocity at the surface
In ideal conditions, with a smooth solid surface, and no chains attached to it, one expects that the friction k is k = km. comparable to what it is in a fluid of monomers: On the other hand the viscosity of an entangled melt is (1)
V(y=0)
through local attachment) some chains are bound to the wall, we expect no slippage in slow flows, in agreement or
12
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 flows was discussed in detail by Alexander and Rabin.9 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.10 At this time, a computer simulation (or an ideal grafted chain plus solvent) exhibited the transition.11 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),
interfaces). (2) However, in most usual situations, no slippage is observed.7 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 (p chains per unit area) is grafted on the solid.1·8 We found that in the low velocity regime (V— O) slippage is strongly suppressed; the slippage length b being reduced to
W1
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 coil/stretch transition induced by polymer flow is more spectacular, because the grafted chains disentangle at the transition point; the friction in the disentangled regime is a Rouse friction,12 much weaker than the entangled friction. In section II, we consider a simplified system, with no wall and a chain of Z units hooked at one end onto a fixed point (Figure 2). The chain is immersed in a polymer melt (JV units per chain) flowing at a velocity V, and it exhibits a transition at a certain critical velocity V*. We constantly assume N > Z.
(2)
where Ro = Zl!2a is the coil size of the grafted chains (Z is the number of monomers per grafted chain and a is a monomer size). Equation 2 gives b values which are small f
PSI, Instituí Curie. Collége de France. (1) Brochará, F.; de Gennes, P. G.; Pincus, P. C.R. Seances Acad. Sci.
1
1992 314 873—878 (2) de Gennes, P. G. CM. Acad. Sci. 1979, 288B, 219. (3) Galt, J.; Maxwell, B. Modern Plastics; McGraw-Hill: New York, 1964. (4) Burton, R.; Folkes, M.; Karin, N.; Keller, A. J. Mater. Sci. 1983, 18, 316. (5) Redon, C. Private communication. (6) Miroshnikov, A. Vysokomol. Soedin., Ser. A 1987, 29, 579-82. (7) See the following reviews: (a) Meissner, J. Polym. Test. 1983, 3, 291. (b) Meissner, J. Anna. Rev. Fluid Mech. 1986, 17, 45. (8) de Gennes, P. G. Simple Views on Condensed Matter, World Scientific: London, 1992.
0743-7463/92/2408-3033$03.00/0
V= Sb.
(~ 10 nm); thus, in all cases where (through a glassy layer
huge, and b can become very large (~ 100 am),* There are some experiments which do suggest strong slippage: (a) observations in a transparent extruder;3 (b) rheological studies on molten polystyrene in a plate-plate geometry with small gaps;4 (c) studies on thin films in wetting or dewetting processes;6 (d) measurements on multilayer extrusions6 (where we probe a number of melt/melt
b0 s
=
(9) Rabin, I.; Alexander, S. Europhys. Lett. 1990,13, 49-64. (10) de Gennes, P. G. J. Chem. Phys. 1974, 60, 5030. (11) Mavrantzas, V. G.; Beris, A. N. J. Rheol. 1992, 36,175-213. (12) For a general presentation of this concept, see de Gennes, P. G.
Introduction to polymer dynamics; Lezioni Lincee; Academia Nazionale dei Lincei: Rome, 1990; pp 1-57. ©
1992 American Chemical Society
Langmuir, Vol. 8, No.
3034
Brochará and de Gennes
12, 1992
V one end at a fixed point 0—immersed in a moving polymer melt, of velocity V. When V is not too low (V V*) the chain elongates significantly.
Figure
2.
A tethered chain—attached by ~
V
4. Elongation L versus flow velocity V for a tethered chain in a melt: (E) = entangled regime; (R) = Rouse regime (disentangled).
Figure +
+
+
+
+
the shear stress exceeds is strongly stretched.
a
+
+
Figure 3. A grafted chain under shear flow in
melt. Where low threshold * (eq 22), the chain a
In section III, 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 = i¡S. In this situation it will turn out that, beyond threshold, the slippage length 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 s. To find
exact coefficients in the entangled regime would represent a very heavy program.
II. One Tethered Chain in a Flowing Melt 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 chains12·13 1.
F
3kT
=
kT "5"
,
fio)? Let us extend the argument of ref 1 and first count the number m of matrix chains which are entangled with the (Z) coil. The volume of the elongated D2L. Each of the m chains has a number object is g of monomers in the volume , and crosses it over a length ~D. Thus g D2/a2. The volume is mainly filled with matrix chains, and thus ~
~
where a3 is
a monomer m
=
(5)
azmg
volume. Thus
L/a
(L
»fi0)
(6)
now
~
TS
fi
m^Nv2
a friction coefficient for a monomer. 6 this can be cast in the form
where
of eq
=
is
(13) Pincus, P. Macromolecules 1976, 9, 386-391. (14) de Gennes, P. G. MRS Bull. 1991, 20-21.
(8)
is the reptation viscosity of the melt. Equating TS to the product of force by velocity FV we arrive at the basic
formula
F
tjLV
s
(L»R0)
(9)
(c) To interpolate between the low elongation case (eq 4) and the high elongation case (eq 9), we shall use the following formula
F
vV(R2 +
=
LY2
(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 (10), we obtain a relation between elongation L and velocity V of the form
V/V* where
we
L
=
(fi02 +
LY2
(11)
have defined
V*
kT
kT
7?fi02
j)Za2
=
(12)
Typically for = 104 P, Z = 1000, a = 3 Á, V* = 0.05 µ /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 finite extensibility of the Z chains. But we shall see that the physical cut off is different. 3. Marginal State, (a) The preceding discussion assumed that the Z 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 entanglement length Ne. The limiting point corresponds to
estimate the dissipation TS due to the m chains reptating following the ideas of refs 1 and 14. Each of them has a curvilinear velocity v (N/Ne)V (where Ne is the entanglement distance) and
We
tiV*L
£
where
(3)
(a) At very low velocities (L < fio), our chain is a spheroid of size fio, and we know from ref 1 that the friction law, relating F and V, is (surprisingly) similar to a Stokes law
~
TS
(7)
By use
D
=
D*
L
=
L*s R2/D*
s Ne1/2a
(13)
Whenever Ne« Z, this elongation L* is much larger than fio, and from the plot of Figure 4, we see that the elongation L* is reached at a velocity practically equal to V*. (b) What happens if we impose a velocity V somewhat larger than V*? At first sight we might think that the Z chain disentangles completely. However, if we go to this regime, we expect a friction force much weaker than eq 9. From the Rouse model we would write the friction as a sum of independent contributions from each monomer
Shear Flows of
a
Polymer Melt
F
ZfjV
=
Langmuir, Vol. 8, No. 12,1992 3035 (Rouse)
(14)
Equating (14) to the elastic force (3), we would obtain a much shorter elongation Lr (where R stands for Rouse)
)
z^R02v
L^-kTr
=
The quantity
=
kTL*-
(15)
L*^
D*. Then it must entangle ~
Thus we are led to expect that, at fixed velocity in this interval, the Z chain stays in a marginal state, with D = D* and L = L*. Ultimately, at V > Vi, we should go to a disentangled state, with L = Lr > L*. The overall plot of elongation L versus velocity then looks as shown on Figure 4, with a long plateau. It is nearly equivalent to plot the force F as a function of the conjugate variable V, since the force is proportional to L (eq 3) in the domain of interest (L L* « Zo). Note finally that if the force F is monitored, we expect an abrupt jump of velocity (from V = V* to V = Vi) when breaches a threshold value
h0
a
0)
(19)
=_ ___L
=2L
km + vr)R0
k0
in agreement with eq 2. (ii) In the marginal state (L fixed value F* (eq 17) and =
(on)
“
I>R0
L*) the force F
=
has a
kmV + vF*
(21)
Here again the km term is usually negligible and essentially constant
*
=
vF*
s
=
is
(22)
~¡j¿-
Ne1/2a
In this regime the slippage length b (V) is increasing linearly with V vV/a* (23) (iii) In the disentangled state F( V)—given by eq 14—is b(V)
=
linear in V and the slippage length is now independent of V
(17)
III. Grafted Chains in Strong Shear Flows 1. General Program. We now return to the situation depicted in Figure 2, with v grafted chains per unit area, and
-*·
since the km term is usually negligible (Note that ko s here). The corresponding slippage length is
~
kT
(
~
again!
kTL*
=
km + vi¡R0
=
k0
a_1fi is something like the viscosity of
^
V/S
=
2.
where we have introduced another characteristic velocity
V1
;"1); (b) using the friction equations (10 or 14), we eliminate F and find from eq 18 a relation between and V, from which we extract the slippage length
b
Here km
_V
f>„
km +
vh Z is not negligible, and in fact is dominant k + v^z
low coverage (pfio2 < 1). The imposed shear stress
=
—[1 + vZa2]
~
=
a
is
(24)
—[1 + vRq] a
tJl
^
=
=
t;S
kmV + vF
=
(18)
Here S = du/dz is the shear rate. The term kmVdescribes the weak friction due to monomer wall interactions
fi°"2 Via1 where m is something like the viscosity of a fluid of The second term vF is proportional to the monomers. elastic force F present on the stretched Z chains, given by eq 3. The main point here is that the Z chains stretch near the wall very much as they do in the infinite matrix of section II. The formulas for the diameter D and the length L remain valid (within coefficients). In particular we retain the friction eqs 9 (in the entangled regime) and 14 (in the Rouse regime). The meaning of V is now the slippage velocity at the surface. It will turn out that the slip length b is always much larger than the chain diameter D\ this implies that the velocity profile is essentially flat at the scales probed by the grafted chain km
~
~
V(z=D) -V _SD _D V V b(Z) " Z(V)
kT —
D*
(35)
Let us choose for instance a flat distribution extending from z = 1 to z = P (with P » Ne) (1
P)
Inserting (36) in (35) we find two regions: (a) for V < VoIP, no molecule is marginal and originally linearly in V a
~
(36)
increases
yP^^apV
(b) for V > VoIP, some grafted chains are marginal and
_£___WM1/2 pkT~
µß \ )
(37)
The second term on the right-hand side of eq 37 is due to the marginal chains, and is rapidly dominant. Thus the ( V) plot has lost the exact plateau of Figure 6, but does reach * as soon as V » VoIP, i.e. when V is larger than the characteristic V* of the longer chains; in practice we still expect a large plateau. 3. Coil Stretch versus Debonding. If we observe a sharp transition between nonslippage to slippage in increasing shear stresses, , may we conclude that this is the coil stretch transition discussed in the present paper? We must be quite sure that the Z chains are not torn out from the surface. (a) If we are dealing with grafted chains, it is easy to check whether the plot of 6( ) is reversible or not. (b) If we are dealing with a nongrafted surface, onto which a few chains from the matrix do bind (with a
Shear Flows of
a
Polymer Melt
distribution of loop sizes extending up to
some maximum then we must forces. In the marginal compare length P), the bound force chains is /* = the by state, experienced kTI (Nell2a). We must compare this to the minimum force for tearing out one chain/surface bond, which we call /b. Our discussion will hold only if /b > f*.
Langmuir, Vol.
8, No.
12,1992
3037
Acknowledgment. This work was initiated by discussions with H. Hervet and L. Léger. It was performed at the NATO ASI 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.
