Dynamics_of_concentrated_polymer_systems._Part_4.Rheological_properties1

In the previous papers (to be referred to as IY1 I1 and 111),3 a rheological constitutive equation was derived for polymeric liquids (polymer melts and concentrated solutions). In this paper we apply the constitutive equation to two typical flows, i.e., simple shear flow and uniaxial elongational flow, and examine the stress response for various histories of deformation. The detailed argument for deriving the constitutive equation is described in I, I1 and 1 1 1 , but it may be worthwhile to review the main points briefly for the discussion of this paper. The basic idea is that, in the system of highly entangled polymers, the molecular motion of a single chain can be divided into two types (i) the small scale wriggling motion which does not alter the topology of th entanglement, and (ii) the large scale diffusive motion which changes the topology. The time scales of these two motions are quite separated : the time scale of the first motion, denoted by Tes,is essentially the Rouse relaxation time and is proportional to the square of the molecular weight M . The time scale of the second motion, denoted by Td, is a renewal time of the topology of a single chain, and was shown to be proportional to M3.1* The present theory is concerned only with the motion of the second type, i.e., the motion whose characteristic time is much longer than Teq. In such motion it is convenient to replace the real chain by a more basic chain which represents only the bare topological structure and does not involve the small scale fluctuation. For that purpose the primitive chain is introduced, If the topology of the entanglement is, for simplicity, represented by slip-link junctions through which the chain can pass freely, the primitive chain is defined as the broken line joining these slip-links (see fig. 1). In this model the stress tensor a,b(a, /3 = x, y , z) is calculated from the formula of the rubber elasticity ; 5
Oag
= (c Fiarig) +P6ab*
i
(1 1)
Here rf is the vector joining the neighbouring slip-links, Fi = 3kTri/nib2is the force acting between these slip-links (where k is Boltzmann’s constant, Tis the temperature, b the bond length and ni is the number of monomers included between these sliplinks) and P is the hydrodynamic pressure.
A 0
FIG. 1 . 4 ~ Entanglement ) junctions are represented by small rings, A, B, .F through which the chrrin can pass lively. The dottad line is the primitive chain. (b) If the chain s r i through the as shown here, the junction bttween A and B i s destroyed and a new jullction FG i s created.
Downloaded by Yale University on 25 May 2011 Published on 01 January 1979 on | doi:10.1039/F29797500038
where Go ( N kT x number density of the entanglement points), is a constant with the dimension of the rigidity modulus, L the arc length of the primitive chain and u(s) is the unit vector tangent to the primitive chain at the point of arc length ~ ( 0 5 s L). [Eqn (1.2) is obtained from eqn (1.1) by rewriting Fhria as Feaua(s)ug(s)ds].
.
deformation changes a unit vector u(s), which is distributed isotropically, to E Y(s)/ (E u(s)I, the stress after the deformation is given by
Based on these conceptq the stress reponse of the polymeric liquid is understodTt as follows.2 Suppose that 8 macroscopic deformation is applied to a system at rest and that this deformation changes a vector dx;, embedded in the material, to dx, = +?Z4dxs, w h e r e the tensor E is called the deformation tensor. Since this
where (
View Online
Dynamics of Concentrated Polymer Systems
P a r t 4.-Rheological
Properties
BY MASAO Do1 AND S. F. EDWARDS*
Downloaded by Yale University on 25 May 2011 Published on 01 January 1979 on | doi:10.1039/F29797500038
Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE Received 19th May, 1978
The constitutive equation for polymer melts and concentrated solutions derived in the previous papers is applied to two typical rheometrical flows : steady and transient shear flow, oscillatory shear flow superposed on steady flow, steady and transient uniaxial elongational f l o w . The stress responses predicted are qualitatively i n good agreement with experiments except for one case (the first normal stress i n the transient shear flow). A particularly interestingresult is that the constitutive equation suggests instability i n steady shear flow and uniaxial elongational f l o w . 1. INTRODUCTION