Molten Polymers in Strong Flows: A Nonclassical Proposal
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("retraction process"). I propose a différent view: Upon stretching, the tube becomes thinner so the chain is squeezed and tends to span a longer length of tube; there is no retraction. This idea might explain certain neutron experiments on labeled chains in an elongated sample, 2 where the retraction process was sought but not found. In the classical theory, the viscosity
U„(z)
T)(y) at high shear rates (y) should scale like (y)~3/2. In the présent scheme, r\{y) appears to be independent of y (for a monodisperse melt). A Simplified Picture of the Viscosity For non-entangled polymer chains, a simple argument explains the viscosity: 3 in a flow field, Ux = y z, the monomers of one chain (C) move with a velocity v ~ (Jx (where z is measured from the center of gravity of the chain). The dissipation per monomer is {$)*, where £t is a friction coefficient, and the dissipation per unit volume is:
r s ^ T,Rouser2 = ^
k
T7RcH.se
a3
11~^{Z2)^-N a
(2)
(3)
where d = N e " 2 a is the mesh size of the entanglement network, and N c (—100) is the number of monomers per entanglement point. We know N0 from the elastic modulus E of the network, (measured at frequencies high enough so that the chains do not disentangle).
—^j
c
r
• ^
•v Xd
Figure 1. A melt in simple shear: (a) in weak shears (yT, < 1), the entanglement network is nearly undisturbed. The monomer M of chain (C) will jump to position M ' when the ambient chain (A) has slipped (in its own tube) by an amount comparable to its tube length L,. This imposes a high tube velocity v to the chain (A); (b) in strong shears (yT, >1), the network is strongly distorted. The monomer M will jump to a more distant position M ' when the blocking chain (A) has moved out.
20
(i)
where N is the number of monomers per chain, and each coil is a s s u m e d gaussian (( z2) ~ Na 2 ). It is possible to find a similar argument for entangled Systems (Figure la). Again we focus our attention on one particular coil (C). A monomer M of (C) must move with a velocity ~ Ux. But it is now hindered by one of the ambient chains (A). To liberate M, the A chain must move, by a length comparable to its total length L„ along its own tube. If v is the tube velocity of the chain, the jump rate at point M is thus:
d
1
= p(z V
(where a3 is a monomer volume). Thus the "Rouse viscosity" TJROUSC for non-entangled Systems scales like:
v
-p
2
kT AU 3
(4)
Having defined Ne and d, we can also specify the tube length L, of our polymer: along one tube we hâve N/N c subunits, each of size d. Thus:
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