Langmuir 1992,8, 989-995
989
Lubrication by Molten Polymer Brushes Jean-FranGois Joanny Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, France Received September 25, 1991.In Final Form: November 20, 1991 We study the viscous friction between two polymer molten layers grafted on parallel solid surfaces and rubbed against each other. The dissipation is dominated by the small zone at the contact between the two brushes where they interdigitate. A sliding viscosity is calculated via an analogy with the problem of end adsorption of a polymer brush. As the relative velocity of the two brushes increases, the polymer chains are stretched and the sliding viscosity decreases. For a finite critical value of the velocity, the chains can no longer sustain the viscous stress and disinterpenetrate. The sliding viscosity and the critical velocity are calculated within the framework of both the Rouse model and the reptation model to describe the polymer dynamics;the critical velocity decreases exponentially with molecular weight in the reptation model. For long chains, even at very low shear gradients, this instability could considerably reduce the role of entanglements in the rheology of polymer brushes. Possible applications to the rheology of block copolymer lamellar phases are also briefly discussed. I. Introduction Polymer brusheslJ are formed by tethering long linear polymer chains on a flat surface with a sufficiently high surface density. These structures can be built by chemical grafting of the end monomers of the chains or by adsorption of a diblock copolymer on the surface; one of the blocks serves then as a tether and anchors the other block. Lamellar phases of block copolymers self-assembling in solution or in the melt also provide good examples of polymer brushes. The specific feature of the brush configuration is the high stretching of the polymer chains due to the steric repulsions between monomers. This leads to a very different behavior from that of usual polymer solutions or melts. The statistical mechanics of grafted polymer layers has been extensively studied over the last few years.3 In the limit of very high molecular weights, due to the strong stretching, the total number of configurations accessible by a given chain is highly reduced with respect to that of a free chain, and the properties of the chains are well accounted for if one considers only the most probable configuration. This is the basis of the self-consistent mean field theory that was developed independently by Milner et al.4 and Zhulina et al.5 One of the important predictions of this theory is that in the asymptotic limit of infinite molecular weights, two polymer brushes pressed against each other do not interpenetrate, the chains of one brush do not interdigitate with the chains of the opposing brush. It was recently shown by Witten, Leibler, and Pincus6 that in the real case of finite molecular weights, there exists an interpenetration zone between two opposing polymer brushes with a thickness larger than a molecular size but much smaller than the thickness of the two original brushes. There is thus a slight interdigitation between polymer chains of the two brushes. As noted by Witten, Leibler, and Pincus the interpenetration between adjacent polymer brushes plays a major role in the rheology of polymer systems involving grafted chains. Using the reptation model to describe the motions (1) Milner, S.Science 1991, 251, 905.
(2) Halperin, H.; Tirrell, M.; Lodge, T. Ado. Polym. Sci. 1991, 100. (3) Alexander, S. J. Phys. (Paris) 1977,38,983. (4) Milner, S.; Witten, T.; Cates, M. Macromolecules 1988,21, 2610. (5) Zhulina, Ye.; Pryamtsyn, V.; Borisov, 0. Polym. Sci. USSR 1989, 31, 205. (6) Witten, T.; Leibler, L.; Pincus, P.Macromolecules 1990, 23, 824.
of the polymer chains, they applied these ideas to calculate the viscosity of molten copolymer lamellar phases. A shear stress applied to the lamellar phase with a plane of shear perpendicular to the plane of the lamellae creates a slip between neighboring lamellae. Most of the energy dissipation comes then from the interpenetration zones where the chains of adjacent lamellae are entangled. For sufficiently long chains, the disentanglement times are very long and increase exponentially with some power of the molecular weight; this leads to a sliding viscosity also increasing strongly with molecular weight (qs e ~ p ( N / ~ ) ) . In this paper, we want to develop these ideas further and study the rheological behavior of grafted polymer chains under a finite shear. For simplicity we consider only two molten polymer brushes rubbed against each other at a given velocity U and calculate the shear stress. The number of grafted chains per unit area in each brush 0 is assumed to be a constant throughout the paper; this is a simplification with respect to liquid lamellar phases where the stress can also be relaxed by adjusting the density of junction points. It might however be relevant for diblock copolymer lamellae where one of the blocks is in a glassy state and the other one is in a molten state; the positions of the junction points of the copolymer are then frozen. One can also think of this problem as the lubrication of the motion of the two solid plates by the molten polymer brushes. Our results may also be of some relevance for two polymer brushes in solution rubbed against each other as in some recent experiment^,^,^ but in this case the permeation of the solvent through the network made by the polymer chains strongly contributes to the viscous dissipation and might be the dominant dissipative process? The paper is organized as follows. In the next section we summarize the self-consistent mean field theory for polymer brushes in the melt and discuss the adsorption of a molten brush onto a solid surface by the free end points of the chains. In section I11 an analogy with this adsorption problem is used to study the shear stress created by the relative motion of opposing lamellae using the Rouse model to describe the polymer dynamics. In a steady state, as the shear velocity increases, the polymers stretch more and more and the size of the interpenetration zone decreases; this leads to an instability at a finite velocity
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(7) Klein, J.; Perahia, D.; Warburg, S. Nature, in press. (8) Rabin, Y.; Alexander, S. Europhys. Lett. 1990,13, 49. Barrat, J. L. Macromolecules, in press. (9) Fredrickson, G.; Pincus, P. Langmuir 1991, 7, 786.
0743-746319212408-0989$03.00/0 0 1992 American Chemical Society
Joanny
990 Langmuir, Vol. 8, No. 3, 1992
where the shear stress jumps discontinuously to a lower value. Our model does not however allow predictions for values of the velocity higher than the instability threshold. We also discuss briefly the effect of an oscillatory shear in the linear regime. The following section is devoted to the extension of the steady-state results when the reptation model is used to describe the dynamics of polymer chains. In the last section we present some conclusions and discuss the implications of our work to the role of entanglements in the rheology of copolymer lamellar phases.
and P = N - Q monomers in the complementary region of the brush of size L = Lo d. The pressure field V(z) is constant in the dead zone and is parabolic in the complementary region. The reduced tension of the chains T = dzldn is constant in the dead zone; the melt being incompressible,it is related to the bridging fraction q by T = q d p . In the parabolic region outside the dead zone, the incompressibility condition gives a relation between the size L of the region and x =AP/~N
11. Self-consistent Mean Field Theory 1. Free Polymer Brush. We study a grafted polymer layer in a melt of constant density p on a planar solid surface. The polymers have a degree of polymerization N and the monomers have size a ( p l / a 3 ) . The grafting density u is imposed and is sufficiently large to ensure a strong stretching of the chains. The constant density in the grafted layer imposes the thickness of the layer
Another relation between x and q is obtained from the conservation of the monomers belonging to free chains
-
Lo = Nulp
{ L(&)'+ V ( z ) ] 2a2 dn
(11-2)
The first term on the right-hand side is the elasticity of the chain and the second term is the pressure field, ensuring the incompressibility of the molten layer. As shown first by Semenov,lothe chain ends are distributed all over the layer and chains starting at different places must be in chemical equilibrium and thus have the same chemical potential. This imposes a parabolic form for the pressure field4 V(2)= -(L,2 - 2) 7r2
8#a2
A
2
(11-7) The size of the dead zone and the number of monomers per chain in the dead zone are
(11-8) The dead zone grows with q and becomes identical to the overall layer when q = 1. In order to determine the fraction of the chains forming bridges, we must balance the chemical potentials of the two types of chains. The chemicalpotentials are calculated from eq 11-2adding the adsorption energy for the bridges. The difference k T A between the chemical potentials of the bridges and the chemical potential of the free chains is such that (11-9)
A t thermodynamic equilibrium A vanishes. It i8 convenient to introduce a reduced adsorption strength
A2
(11-4) Nu2 8Na2 8p2a2 The strong stretching approximation is valid if the chemical potential of the chains is much larger than kT or if Nu2/p2a2>> 1. 2. End Adsorption of a Polymer Brush. The brush is put in contact with another solid surface at a distance Lo from the grafting surface. The end monomers can adsorb on this surface with an adsorption energy -kT6; a finite fraction q of the chains makes bridges between the two surfaces and the remaining fraction 1 - 7 stays free. The structure of the layer is very similar to that obtained for the bridging problem in a good solvent discussed in ref 11 and has been studied independently by Zhulina and co-workers.12 The important result is that there exists in the vicinity of the adsorbing solid a dead zone where no chain end can be found.13 We call d the size of this dead zone. Each chain forming a bridge has Q monomers in the dead zone (10) Semenov, A. J.E.T.P. 1985, 61, 733. (11)Johner, A.; Joanny, J. F. Europhys. Lett. 1991, 15, 265. (12) Zhulina, Ye. Private communication. (13) See also Ligoure, C.; et al. Submitted for publication in Macro-
molecules.
(11-5)
q = x cot x [ q + 741 - q ) / 2 x l (11-6) Equations 11-5 and 11-6 allow the calculation of all the properties of the layer as functions of the bridging fraction q. In the following we consider mainly the limit of small bridging fractions. In this limit
(11-3)
The chemical potential of the chains is then p0=-p=-
LIL, = 1 - q + 2qxla
(11-1)
The configuration of each chain is specified by the distance z of the nth monomer from the grafting surface, the zeroth monomer being the free end of the chain. The chemical potential of the chain, i.e. the energy that one has to pay to insert a new chain inside the layer, is plkT = po = JON dn
-
6 being in general of order one (adsorption energy of the order of a few k l J this reduced adsorption strength is smaller than 1when the chains are highly stretched. For small values of 6* the bridging fraction q is small
(26.) 2
q=
113
(11-10)
The bridging fraction q increases with 6* and reaches 1 (total bridging) for a finite value b* = 1/2. 3. Adsorption under Shear.14 When a shearing force
parallel to the solid plates kTf per unit area is applied to the adsorbing surface, the chains forming bridges are stretched further and the fraction of bridges decreases. The position of the nth monomer of a chain is now described by a vector r with two components: x parallel to the force and z normal to the plates. The plates are infinite in the x direction and the problem is translationally invariant along x so that the pressure field V only depends on z and axlan is constant along one chain. (14) This problem has also been studied independently by Ye. Zhulina and co-workers.
Langmuir, Vol. 8, No. 3, 1992 991
Lubrication by Molten Polymer Brushes For a free chain, the end point is not under tension and axtan vanishes; on average the free chains remain thus perpendicular to the solid surfaces. The bridges are under tension; on the adsorbing surface the projection of the force due to their tension parallel to the plates balances the external force so that (11-11)
In the dead zone the chain tension in the normal direction azlan is also constant and the bridging chains make a constant average angle 0 with the normal to the solid plates. The incompressibility condition in the dead zone still imposes azlan = qu/p. If we introduce a reduced force f* = fpa2/u2,we then obtain a relation between the force and the inclination angle of the chains f* = q2tge (11-12) Outside the dead zone the normal tension azlan increases toward the grafting surface and the local inclination angle of the bridges decreases thus toward the grafting surface; the trajectory of a bridge is bent. In the expression of the chemical potential given by eq 11-2, the elasticity of a chain should be replaced by (11 2a2(drldn))2in the presence of a shearing force; it has now two contributions corresponding to the two components of the tension along x and z. The minimization of the chemical potential with respect to the conformation of the chains gives uncoupled equations in the two directions and the projection of the trajectory of the chains along the z axis follows the same laws as in the absence of external force; the relations between L, x , and q are thus still given by eqs 11-5 and 11-6. The in-plane tension of the bridges gives an extra contribution to the chemical potential (11-2). This leads to the followingdifference A of chemical potential between the two types of chains:
T2L2 Low 1 f 2a2 7- N x (11-13) 2pa 2 qu At thermodynamic equilibrium, A vanishes and in the limit of small reduced adsorption 6* A = -6 - QSpa2
+
+
(11-14) In the absence of shearing force, this is the same as eq 11-10 above. For small values of the dimensionless force f*, the bridging fraction q decreases when the force is increased. For a critical value qc, the force reaches a maximum value f*c above which no bridging is possible and the system becomes mechanically unstable. If we call Bc the inclination angle of the chains at this point, the threshold of the instability is obtained from eqs 11-12and 11-14)
(11-15) For a given shearing force f smaller than the critical value, the characteristics of the dead zone are given by eqs 11-8 and 11-14, Therefore, the size of the dead zone decreases as the shearing force is increased. 111. Lubrication by Molten Polymer Brushes:
Rouse Model 1. Steady Shear. We consider two molten polymer
brushes grafted on parallel solid plates at a distance 2Lo
l4- --,I
inte&netration zone
Figure 1. Two molten polymer brushes rubbed against each other with a relative velocity U. The interpenetration zone has a thickness 2d and the total polymer layer a thickness ?LO.
from each other (Figure 1). The two brushes are thus just in contact with one another. In the self-consistent mean field theory, only the most probable configuration of each chain is considered and the two brushes do not interpenetrate. There are however fluctuations around the most probable configuration of each chain that are responsible for some interpenetration between the two brushes. There exists thus around the midplane between the two solid surfaces an interpenetration zone of size 2d where a fraction q of the chains grafted on one side penetrate into the opposing brush. If we draw an imaginary surface in the midplane, these interpenetrating chains form bridges between their grafting surface and the imaginary surface. When one of the solid surfaces is moved at a velocity U with respect to the other one, in the interpenetration zone the moving chains feel a friction force due to the immobile chains that can be measured macroscopically through the viscous stress kTf (tangential force per unit area). The viscous dissipation is essentially concentrated in the interpenetration zone. We characterize here the viscous friction between the two grafted layers by an average macroscopic sliding viscosity qs = k TfLolU.15 As the shear velocity U is increased, the friction force on each chain that enters the interpenetration zone and forms a bridge increases. The bridging chains have a tendency to bend in the direction of the motion and get more and more stretched. The stretching decreases the conformational fluctuations of the chains around their average trajectories and thus has a tendency to decrease the interpenetration. Eventually for high enough velocities, the interpenetration zone is not strong enough to sustain the shear and the viscous friction is strongly reduced. In order to study more quantitatively the friction process, we draw an analogy with the adsorption process of a polymer brush under shear described in the previous section. As in the adsorption process a few of the chains form bridges between the grafting surface and the imaginary midplane surface (that plays here the role of the adsorbing surface). In the adsorption problem, the bridging occurs because a chain forming a bridge gains an adsorption energy -kT6. In the friction problem, the bridges form by conformational fluctuations around the most probable configuration of the chains; the equivalent (15) We look in the following only for scaling laws, some numerical prefactors are thus ignored.
992 Langmuir, Vol. 8, No. 3, 1992
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adsorption energy 6 is then of order 1. In other words, due to thermal fluctuations, the difference in chemical potentials between free chains and chains forming bridges can be of order kT. Although the value of the bridging energy is no longer large compared to kT, we expect that the results obtained for the bridging problem are still valid within numerical factors of order unity. The interpenetration zone corresponds to the dead zone. The size of the interpenetration layer can then be calculated from eq 11-8as a function of the bridging fraction I]. In the limit where the shear velocity vanishes, I] is given by eq 11-10 with 6 = 1 and d
-
L#0-2/3
-
(N/a)'l3
Q
-
Npc1I3
-
(N/u)~'~
(111-1) Similarly the average number of monomers per bridge in the dead zone is (111-2)
These results are similar to those derived by Witten, Leibler, and Pincus6 through a very different argument. If the shear velocity is finite, the shear stress is finite and the relationship between the fraction of chains forming bridges and the shear stress f is given by eq 11-14
We calculate the shear stress by estimating the viscous dissipation in the interpenetration layer and using the Rouse model to describe the hydrodynamics of the chains.16 This is a good description if the chains forming the interpenetration layer are not entangled; i.e. if the average number of monomers per chain Q is smaller than the number of monomers between entanglement Ne. In the Rouse model, the friction is local and the force on a monomer moving a t a velocity U with respect to its environment is {U, where { is a microscopic friction independent of molecular weight. The shear stress is then
f = tldQU/kT The dimensionless force p is
(111-4)
f* = -pa2Nq2{U (111-5) kT?r2a Equations 111-4 and 111-5 allow the calculation both of the fraction of bridging chains I] and of the shear stress f. The sliding viscosity can then be calculated from the bridging fraction as 4
95 = - {MI]2a2 T2P
(111-6)
In the limit of vanishing shear gradient or vanishing shear velocity, the Rouse sliding viscosity is thus
-
-
(111-7) I]s I]oN4/3u2/3 I ] f l p 0 ' / 3 where 90 is a simple liquid viscosity I]@). If the stretching of the chains in the brush is strong clo >> 1,the sliding viscosity is larger than the Rouse viscosity of a polymer melt of chains of molecular weight N. As the velocity U is increased, the reduced force f* increasesand the bridging fraction q decreases. The sliding viscosity smoothly decreases with the velocity. When the reduced force f* reaches the critical value fc where the instability occurs in the adsorption problem (eq 11-15), the bridges are no longer able to sustain the stress and the two opposing brushes disinterpenetrate. Our model
(r -
(16)Doi, M.; Edwards, S.The Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986.
predicts thus a critical velocity
(111-8) The critical velocity has typically a macroscopic value and can be of the order of millimeters per seconds or even larger. A t the threshold of the instability, the sliding viscosity is a finite fraction of the zero shear slidingviscosity given by eq 111-7. The model does not allow any predictions for velocities larger than the critical velocity. It is clear that above this velocity the interpenetration zone plays a far less important role. Two scenarios seem possible: either the two polymer brushes glide on each other with a very small dissipation due to the monomers in contact in the layer of thickness a at the midplane (a solidlike friction) or the friction can be due to a stick-slip motion, the two brushes interpenetrating and disinterpenetrating periodically. A precice study of the stick-slip motion would require a careful analysis of the interpenetration kinetics of the chains, which will be left here for a future study. 2. Oscillatory Shear. We now study an oscillatory relative motion of the two brushes in contact withavelocity U = UOeiwtand consider only the linear regime where the size of the interpenetration zone and the number of monomers per chain in the interpenetration zone are given by eqs 111-1and 111-2). Within the Rouse description of the interpenetration layer, two new effects must be taken into account when the motion is oscillatory:16 The viscous dissipation is due to the pieces of chains penetrating the opposing brushes, these are related to the moving solid surface by the body of the brush which essentially behaves as a purely elastic medium (as a gel). Some of the energy communicated to the solid plate is then not dissipated by viscous friction but is stored in the elastic modes of the brush. A t sufficiently high frequency, WTQ > 1,where TQis the Rouse time of a Gaussian chain of Q monomers (TQ= Q2({az/kT) N4/3u-4/3~,where T {a2/kTisamicroscopic time),not all the Q monomers follow the motion, but only a small fraction proportional to ( W T ) - ~the / ~ ; friction is thus strongly reduced. We consider first the low frequency limit WTQ < 1where all the monomers in the interpenetration layer follow the motion. The main effect there is the elastic response of the polymer brush. In the polymer brush we will consider that the various chains are strongly entangled and that the number of monomers of the chains N is larger than the number of monomers between entanglements in an unstretched melt Ne. The elastic response of the brush can be characterized by a plateau modulus E. As is usually done for polymer melts, we may define from the plateau modulus an entanglement mass Peby
-
-
E = kTp/Pe (111-9) The chains in the brush are stretched and the entanglement mass P e (that is roughly equal to the average number of monomers between entanglements between different chains) can be different from the eetanglement mass Ne of a simple polymer melt of the same polymer. We expect Peto be larger than Ne (stretched chains are less entangled) and to be largest close to the grafting surface. We do not have a precise theory for P e but bounds can be proposed for the entanglement mass close to the grafting surface. The distance between neighboring anchoring points on the solid surface is of order u-ll2, the radius of a chain of P e monomers must thus be larger than this
Langmuir, Vol. 8, No. 3, 1992 993
Lubrication by Molten Polymer Brushes distance in order to allow for one entanglement between neighboring chains, a lower bound is thus Pe = (ua2)-l.In the scaling model for polymer brushes proposed by Alexander, the chains are confined to tubes of diameter a-lI2 perpendicular to the surface; each chain is considered as a string of blobs inside the tube. In the molten state, the number of monomers of each blob, g, first calculated by Brochard and deGennes17is obtained when the stretching energy of a blob (given by the chemical potential NO) is of pal^)^. If g is larger than Ne, there is at order kT, g least one entanglement per blob and an upper bound for the entanglement mass is P, = (pa/ a)2. If g is smaller than Ne,different blobs are not entangled and the mass between entanglements, P,, is of order Ne. I t is shown below that the Rouse model gives a correct description of the chain dynamics when Ne> The blob size is then smaller than Ne and Pe = Ne. If the velocity of the moving solid plate is U , at the level of the midplane the velocity of the monomers linked to this plane has a smaller value Veiuf. The amplitude of the deformation gradient is then ( UO- V)/iw, and the amplitude of the elastic stress of the brush is
the microscopic equations of the Rouse model is given in the Appendix.
IV. Lubrication by Molten Polymer Brushes: Reptation Model The Rouse model describes correctly the motions of polymer chains in the melt if the total number of monomers of the chains N is smaller than the number of monomers between entanglements Ne,if N > Neentanglements plays a major role and the Rouse model strongly underestimates the viscous dissipation. At the edge of each brush the chains are not stretched and in the interpenetration zone the relevant mass between entanglements is that of a simple melt Ne. In the interpenetration zone, entanglementa are thus important if Q > Ne or if N > Ne (Nea2/pfa2)lI2.However, we assume strong stretching of the chains and we also require N > p2a2/a2. Comparing these twoconditions, we obtain a minimum grafting density above which the Rouse model gives a good description of the chain dynamics a > pa/Ne'I2. This imposes very dense brushes that are certainly difficult to make experimentally. For example in copolymer lamellar phases the grafting density scales as u N-lI3; the Rouse model is expected (111-10) kTf,, = E(Uo- W / i w to give a good description of the dynamics of lamellar phases only if N < Ne3l2.At least in all the situations This stress is balanced by the viscous stress given by eq where u < pa/Ne1l2,we must thus use the reptation model 111-4 where the velocity is now V and not U to describe the chains motions. We sketch this approach in this section for a steady-state relative motion of the two kTf = qa{QV (111-4') brushes. These equations allow the calculation of the velocity V In order to use the same approach as in the previous and then of the stress f as a function of the velocity U. We section, we need to determine the friction force on each characterize here the dissipative process by a complex chain. This can be done by looking at the motions of the modulus G* = kTfLoiwIUo or a complex viscosity qs* chains parallel to the solid plates and then using the defined by G = iwqs* fluctuation dissipation theorem. Following the ideas of Witten, Leibler, and Pincus, the motion along the solid (111-11) t s * = q s / ( l + iwTR) surfaces of a chain penetrating the interpenetration layer is possible only by retraction of the end point of the chain where the relaxation time T Ris given by a usual Maxwell outside the layer. This retraction is similar to that involved formula qs = ETR so that in the reptation of star polymers. We call Td the time that T R = T N ~ N ~ ~ ~ (111-12) ~ ~ / ~a chain takes to reptate outside the interpenetration layer. During the time Td,a given chain can move approximately The relaxation time T R is larger than the Rouse time over adistance of the order of its tube diameter b = Ne'I2a. The diffusion constant of the chains parallel to the solid TQof the dead zone since Ne > (pa/a)2. As long as W T R < 1,the viscosity is equal to the static viscosity qs; the real surfaces is thus &h = b2/Tdand the friction constant on a chain dragged by the shearing motion is {ch = kT Tdlb2. part of the complex modulus (the storage modulus) is G' The shear stress created by the relative motion of two =E (~TR)~. opposing brushes at a velocity U is thus If T R -< ~ w < TQ-',the storage modulus is equal to the plateau modulus of the brushes G' = E and the real part f = qaTdU/b2 (IV-1) of the viscosity qs' decreases as The arguments used in the previous section lead then 0s' = qs/(WTR)2 (111-13) to a sliding viscosity at zero shear As a higher frequency when W T Q < 1,only a small fraction of the Q monomers n (oT)-'/~ in the interpenetration zone follows the motion of the solid plate. Qualitatively, the values of the complex viscosity and of the complex When the shear velocity U is increased, the viscosity modulus can be obtained from eq 111-11where in the decreases and as in the Rouse dynamics an instability expression of the static viscosity, the number of monomers occurs at a critical value Q is replaced by n (wr)-ll2. The real part of the viscosity 7s' decreases as (IV-3) qs' qsTg1i2/w3i2T R (111-14) These two equations crossover smoothly to the Rouse This viscosity is as expected independent of Q and results when Td is equal to the Rouse time TQand b is decreases strongly with frequency. The storage modulus equal to the size of the interpenetration layer d. The in this frequency range is the plateau modulus of the brush knowledge of the retraction time Td allows thus the G' = E . A more complete derivation of these results using calculation of the sliding viscosity and the threshold velocity for the instability. Following the arguments used for the retraction of the arms in the reptation of stars18 (17) Brochard, F.; deGennes, P. G. J. Phys. (Paris) 1979, 40, L399.
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994 Langmuir, Vol. 8, No. 3, 1992
Joanny
and ref 6, the retraction time can be written as
Td = Toexp{Q2/(NNe))
(IV-4)
where TOis the time that a free chain would take to disentangle the Q monomers. The grafting point of the chain being fixed on the surface the disentanglement of the Q monomers is made by contracting the chain inside its tube. The entropy penalty given by the exponential factor of eq IV-4 is thus only due to the tube constraint. It is much smaller than in the case of the reptation of star polymers (Q (Nea')', Q2/(NNe) is large. The motion of the retracting chain involvesall its monomers and the reptation dynamics leads to
To = Q2a2/(N$)t)
INTERBEDIATE CHAINS
(IV-5)
where Dt is the tube diffusion constant Dt = kT/NS: In this limit the retraction time is thus
Td = TQ~N/N, exp{Q2/(NNe)l
(IV-6) The sliding viscosity is very large in the high molecular weight limit and increases strongly with the molecular weight of the polymer forming the brushes qs
-
PIN: e ~ p { N ' / ~ / u ~ ' ~ N , ) (IV-7)
this is the result of Witten, Leibler, and Pincus (though with a different preexponential factor). The critical velocity where the instability occurs has a small value that decreases very rapidly with molecular weight
-
U, u ~ / ~ N : / N ' ~e~p(-{N'/~/u~/~N,)) /~ (IV-8) The instability might occur at such a low velocity that it may preempt the role of entanglements in these systems. At intermediate values of the molecular weight, 1 < N/Ne < (Nea2)', the motion of a retracting chain does not involve all the monomers and the Q monomers have a Rouse motion inside their tube; the retraction time is then
Td= 7Q4/Ne2
(IV-9)
The sliding viscosity increases only as a power law of molecular weight qs
-
N'o/3/(N:u4/3)
(IV10)
The critical velocity decreases with the molecular weight as
u,
N:g8/3/N19/6
(IV-11)
These results cross over smoothly to the Rouse results when Q = Ne. The various regimes for the sliding viscosity and the critical velocity are summarized in Figure 2.
V. Discussion We have studied the viscous friction between two polymer brushes rubbed against each other in the molten state. Following the ideas of ref 6, we have assumed that the viscous dissipation is dominated by the small region around the contact where the two opposing brushes interpenetrate. Because of this interdigitation, when a relative velocity is imposed between the two grafting surfaces, the polymer chains bend in the direction of the motion and get more stretched. This in turn reduces the ~~
~
(18) See ref 16, p 215, eq 6.115; the length of the primitive chain that must be retracted is here La - L, = Q/Ne112.
Figure 2. Various regimes for the lubrication by molten polymer brushes. In the Rouse regime the sliding viscosity and the critical velocity are given by eqs 111-6and 111-8,in the reptation regime with short chains by eqs IV-10 and IV-11 and in the reptation regime with long chains by eqs IV-7 and IV-8.
size of the interpenetration layer and induces a decrease of the sliding viscosity. If the shear gradient or the relative velocity is further increased, the polymer chains in the interpenetration layer cannot sustain the stress and the two brushes disentangle. In order to study the disentanglement of the two brushes, we have used an analogy with the end adsorption of polymer brushes. This analogy predicts a discontinuous jump of the sliding viscosity to a much lower value for a finite value of the relative velocity of the two brushes. It should be noted however that this analogy is not a rigorous mapping and that the discontinuous nature of the transition could well be an artifact of the model. We have implicitly ignored for instance the fact that all the chains entering the interpenetration layer do not have the same number of monomers, Q, in this layer. The discontinuity could in reality be rounded by the polydispersity in Q. We believe however that the strong decrease of the viscosity due to the disinterpenetration effect is not an artifact and is at least qualitatively well described by the analogy with end adsorption. Another question that is not addressed by our model is the behavior of the two polymer brushes beyond the threshold. One can conjecture either that they glide on each other with a solidlike friction or that they interpenetrate and disinterpenetrate periodically, showing thus a stick-slip motion. If such a stick-slip mechanism exists, its period could be governed by the kinetics of entrance and extraction of chains from the interpenetration zone. When the chains in the grafted layers are not loo long, their dynamics in the interpenetration layer can be described by the Rouse model and the critical velocity has a macroscopic value (typically mm/s); it decreases as a power of molecular weight. The instability could thus perhaps be studied directly by grafting polymer layers on the mica surfaces of a force measurement apparatus. Similar experiments have been recently performed for
Lubrication by Molten Polymer Brushes polymer brushes swollen by a solvent but if the disinterpenetration effect still exists, it could be masked by the permeation of the solvent through the polymer brush. For larger grafted chains, entanglements become important in the interpenetration zone and the critical velocity decreases exponentially with a power of the molecular weight. At any velocity larger than the critical threshold, the two brushes disinterpenetrate and thus disentangle. The role of entanglements between parallel brushes appears thus as very small for strong enough shear gradients. This result could be relevant for the rheology of block copolymer lamellar phases in the melt.lg Above a critical shear gradient that might in practice be very low, entanglements between lamellae could play no role. Below this critical shear gradient, we do not expect however the viscosity of real block copolymer lamellar phases to be given by eq IV-7. These phases are rarely monocrystals and one can expect the migration of defects to play a dominant role in their rheology.20 Finally, it should be pointed out that the polydispersity in the number of entanglements per chain in the interpenetration zone could play an important role and considerably slow down the mechanical relaxation of copolymer lamellar phases.21 This would be associated with a decrease of the critical shear gradient for disinterpenetration and further reduce the role of entanglement between different lamellae.
Acknowledgment. I thank Ye. Zhulina (Mainz) for correcting an error in a previous version of this work and S. Obukhov (Landau Institute), M. Rubinstein (Kodak Rochester), S. Milner, and G. Grest (EXXON Annandale) for useful discussions. Appendix In the Rouse model, the equation of motion of monomer s in the interpenetration layer parallel to the (19) Rosedale, J.; Bates, F. Macromolecules 1990,23, 2329. (20) Kawasaki, K.; Onuki, A. Phys. Reu. A: Gem Phys. 1990,42,3664. (21) Obukhov, S.; Rubinstein, M. Preprint, 1991.
Langmuir, Vol. 8, No. 3, 1992 995 solid surfaces is (A-1)
where s = 0 is the first monomer penetrating in the interpenetration layer and s ranges from zero to Q; 7-l is microscopictime and F, a random force that will be ignored in the following. The solution of this equation with the boundary condition that the velocity is Veiwtfor s = 0 and vanishes for large s is such that (A-2) In the limit where WTQ >> 1, the friction force on the piece of chain in the interpenetration zone is (A-3) And the shear stress on the plates is
-
kTf = {qc Ve-ir/4(~7)-1/2 (A-4) In the limit where OTQ= 1 (Q ( 0 7 ) - ' / ~ )this crosses over to the zero frequency result quoted in the text (eq 111-4'). Equating the shear stress to the elastic stress (eq III111,we calculate the velocity Vat mid-plane between the two brushes
where TQ= .Q2. The complex viscosity is then
In the limit of large frequencies, this gives back the result obtained in the text from a scaling argument.
