Volume preserving and non-preserving strains
3 Branched entangled polymers
3.5 General chain architectures
The picture of hierarchical retraction dynamics with dynamic dilution can be generalised in a straightforward way to arbitrarily branched polymer 'trees'. For structures with many branch points a simplification is to treat the relaxation in discrete stages, calculating the time scales at which arm retraction has penetrated to each layer of the tree. At each stage the effective geometry of the molecule simplifies, as faster relaxing (outer) segments cease to be part entangled network, but instead dilute the current value of Ne.
For example, the Cayley tree of n layers and functionality f [13], contains f" segments in its outermost layer and (fn+' - f)/(f - 1) segments altogether. (In such a tree, each stem branches into f stems, with no dead ends until the nth layer is reached, at which point all stems terminate.) The effective concentration of unrelaxed segments after m levels have relaxed is C ( m ) = (f"-"+l
-
f)/(fns'-
f ) N f-" when n is large. Solving the retraction problem from level m to level m+
1 (with the approximation that the effective concentration at level m is valid throughout that stage of the hierarchy) gives the recurrence relation7,+1 = 7m e x ~ [ 4 N ~ / ~ + ~ ~ ) f - " l , (62) with solution
Rheology of linear and branched polymers 109
Here IJ = 15/8 and N , the number of monomers between branch points on the tree. At this level of approximation we may also take G ( t ) N Go[C(m(t))12. This leads to the logarithmic form of stress relaxation
G ( t ) = Go In
(*)
t e,
where T , , , ~ is a (finite) limiting relaxation time ( T ~ as m
-+
CO), and 8 is a branching- dependent exponent with a value of 2 in this case. It turns out that other, less regular, tree-like structures also have this form of G ( t ) but with different values for 8. For example, the ensemble of randomly-branched trees predicted to occur at the classical mean-field gelation point has t9 = 4 [14].Experimental verification for more general architectures with well-controlled materials has so far only proceeded to two-level branching. A number of groups have studied polymers shaped like the letter H in the melt; extra arms can be added at the same two junctions to make a ‘pom-pom’ polymer [15][16]. In the H-polymer case the frequency- dependent rheology directly reflects the chain structure (once the polymers are well- entangled) with features in G” ( U ) arising from both outer arms and the central ‘cross-bar’
(at lower frequencies). These are shown in Figure 20. The theory (solid curves) does A76
6.1 6.25
& 6
z
-!! 4.75
;
4.54.25 4
-a Log w
Figure 20. Linear viscoelastic data for a n H-polyisoprene melt with molecular weights for a m s of Ma = 20000 and the cross-bar of M, = 111000 (synthesised by J. Allgaier 1151). Solid and dashed lines are the theory with and without polydispersity respectively.
indeed grasp the quantitative form of the rheology, once it is realised that the cross-bar motion is actually reptation, in spite of the branched nature of the polymer! For at long time scales when the outer arms have completely relaxed, only the cross-bar sections of the molecules remain topologically ‘active’ and so behave as linear polymers in tubes with a diameter set by their mutual entanglements only (see Figure 21).
The sharpness of the peak in G”(w) at low frequencies in the H-polymer data arises precisely from the narrowness of the mode distribution in reptation of a linear chain (Equa-
110 Tom McLeish
.
*e. a..:*
... - ... .. ...” . . . . ;
Figure 21. Successive stages in the configurational relaxation of an entangled H-polymer.
In the final stage it reptates as a linear chain.
tion 49). One additional important insight arose from studies on these highly monodis- perse model materials-the role of residual polydispersity. Even the small (10% level) variation in molecular weights of the arms in the specially polymerised polyisoprene H- molecules had a significant quantitative effect on the linear relaxation spectrum. This is because the diffusion constants of the branch points depend exponentially on the arm molecular weight. This greatly amplifies the contribution of the few arms that are sig- nificantly longer than the average, incresing relaxation times overall. The calculation is straightforward for small polydispersity, and produces the full curves in Figure 20 (dashed curves are for the purely monodisperse case).
3.6 Conclusions
Although at first sight very complex, the system of entangled flexible branched polymers seems to give rise to a rather simple picture of hierarchical dynamics for configurational relaxation, needing only two parameters for each chemistry of polymer. The remarkable slowness of trapped dynamics results in molecular relaxation times on a time scales of seconds and even hours, for moderate molecular weights. Hierarchical dynamics can produce features very well separated in time scale even though they arise from relatively close molecular ‘neighbours’ along the chain backbone. An exciting challenge for the next few years lies in the application of this branch of soft condensed matter physics to the full complexity of industrial materials.
Rheology of linear and branched polymers 111
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