Langmuir 1996,11, 3560-3564
3560
Rheology of Polymer Brushes: Rouse Model Alexander N. Semenov Physics Department, Moscow State University, Moscow 11 7234, Russia Received February 11, 1994. In Final Form: September 26, 1994@ The Rouse rheology of dense molten polymer brushes is considered theoretically. Within the framework of the Rouse model, the effective linear and nonlinear viscosities (7) of two brushes rubbed against each other are predicted as a function of the molecular weight of graRed chains (M) and of the shear velocity (u). It shows that 9 is proportional to MIuu2 in the nonlinear regime.
I. Introduction Polymer brushes (grafted layers) can be formed by attaching the end parts oflong polymer chains to a surface. These grafted layers are sometimes used for “steric” stabilization of colloidal particles.lP2 If the grafting density u (number of chains per unit area) is high enough, the chains in the brush must be appreciably stretched in the direction normal to the surface (dense b r u ~ h ) . Let ~ ? ~us consider two molten dense brushes grafted on parallel plates and filling the gap between them (Figure 1). The aim of the present contribution is to predict the stress in these brushes caused by a relative motion of the plates. The results are equally applicable also to microphase-separated lamellar AB diblock-copolymermelts in the strong segregation regime (Figure 2; here the AB interfaces between lamellar sheets play the role of the plates). The equilibrium structure of dense brushes corresponds to the minimum of the elastic energy due to the stretching of the polymer chains. This principle dictates that the polymer chains attached to the opposite brushes must be well-separated, so that the interpenetration layer (see Figure 1)between the brushes must be much thinner than the brush height.5g6The rheological behavior of the system crucially depends on whether or not the chains attached to opposite brushes are entangled in the interpenetration layer (to be referred to as &layer below). The entanglement case was considered first by Witten, Leibler, and Pincus6(and also by Joanny’). Here we will assume that entanglements within the &layer are not important so that the dynamics of polymer segments inside the layer is well-described by the Rouse model. This (Rouse) case was also analyzed by Joanny’ using quasiequilibrium arguments which in fact are valid only in the linear regime. The sharp decrease of the nonlinear viscosity at some critical shear rate predicted in ref 7 is in fact an artifact of the “quasiequilibrium”model. Purely dynamical arguments proposed in the present paper lead to a qualitatively different picture implying a smooth power-like decrease of the effective viscosity in the nonlinear domain. In the next section the theory of equilibrium properties of dense brushes and also their linear dynamic behavior are considered. The nonlinear brush rheology is considAbstract published in Advance ACS Abstracts, December 1, 1994. (1)Napper, D. Polymeric Stabilization of Colloid Dispersions; Academic Press, 1983. (2)De Gennes, P.G. Adv. Colloid Interface Sci. 1987,27,189. (3)Milner, S.;Witten, T.; Cates, M. Macromolecules 1988,21,2610. ( 4 ) Zhulina,Ye.; Pryamitsyn, V.; B O ~ ~ B0. OPolym. V, Sci. USSR 1989, 32,205. ( 5 ) Zhulina, Ye. B.; Semenov, A. N. Vysokomol.Soedin., Ser.A 1989, 31,177. ( 6 )Witten, T.;Leibler, L.; Pincus, P.Macromolecules 1990,23,824. (7) Joanny, J.-F. Lungmuir 1092,8, 989. @
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Figure 1. (a) Two dense brushes grafted to opposite plates z = 0 and z = 2h with the interpenetration layer of width 6. (b) The brushes after a shear deformation.
Figure2. Lamellarstructureinameltofstronglyincompatible symmetric diblock-copolymers(AB). Each lamellar sheet can be considered as a pair of brushes.
ered in section I11 (the only really original one) on the basis of new intrinsically dynamic arguments.
11. Equilibrium and Linear Dynamic Properties A. Equilibrium Brush. Let us consider a molten polymer brush grafted on a flat surface with surface density (T. The height, h, of the brush is defined by the incompressibility condition
h = uNv
(1)
whereNis the number oflinks per chain (whichis assumed to be the same for all chains) and u is the volume per link. Obviously the characteristic size of the chains in the direction normal to the surface (the plates) must be of order of h. We assume that this size is much larger than unperturbed (Gaussian)gyration radius of the chains, R = ap.5
where a is the characteristic size of a link ( a = statistical length divided by 6”). The equilibrium properties of the brushes (both molten and in the presence of solvent) were considered by Milnel.3 and Z h ~ l i n a . ~
0743-7463/95/2411-3560$09.00/0Q 1995 American Chemical Society
Rheology of Polymer Brushes
Langmuir, Vol. 11, No. 9, 1995 3561
NA= NB= N
(10)
As is well-known10,8the system forms a lamellar structure (Figure 2) if the interaction parameter of A and B links is large enough: x = 2 SIN. In the strong segregation limit Figure 3. A representation of a g r a h d chain as a highly stretched sequence of blobs (gb) of typical size rt,.
xNs5
The free energyof the brush is primarily due to a stretch (elongation)ofthe chains normal to the surface.8 A typical conformation of a polymer chain can be described as a completely stretched sequence of blobs (Figure 3). The blob size must be slightly dependent on the distance from the surface since the chains are elongated nonuniformly.8 The characteristic (averaged)blob size, Q, can be readily estimated using the scales h, R:
the structure can be described as a periodic sequence of well-defined A and B layers with narrow interfaces between them. Each layer isjust a double brush; the only difference from the “grahd”case is that now the “grafting” density is determined by the X-paramete$
-
rb R2/h= Na2/h
-
N(Na2/h2)
(4)
-
Obviously, Q
