Star-branched polymers, diffusion

Fig. 5. Diffusion of C PsCl (open symbols) and cispolyisoprene (filled symbols) in solution at SO °C, from dilute solution to the melt. Left linear polymer, M =- 10 right eight-armed star-branched polymer, M = 4x 10 (Ref.32 >, with permission).

It should also be noted that a similar treatment is possible for the translational hydrodynamic radius, Rhj, obtained from measurements of translational diffusion coefficients or sedimentation coefficients of branched polymers. One may define a parameter gn = Rh,fb/Rhji - the ratio of the hydrodynamic radius of the branched polymer relative to that of a linear polymer of the same molecular weight. Again, it is expected that gH < 1. For star polymers with uniform subchain lengths having... 

Discriminating branched and star polymers from linear ones can always be achieved by measuring the properties in dilute solution. In fact, molecules having the same molar mass but different macromolecular architectures exhibit different transport and light scattering properties. More specifically, a branched macromolecule is more compact than a linear molecule having the same molar mass, and therefore it will display less friction and will diffuse more easily in the solvent. Viscometry can be used to detect branched structures, since the Mark-Houwink-Sakurada exponent (Eq. 2.23) for branched and star-shaped polymers is lower tiian that for the corresponding linear chain. Unfortunately, in order to measure the difference, one must have a sample made exclusively... 

How can a branch point move The repertoire of polymer movements that we have considered up to now reptation, primitive path fluctuations, and Rouse motion within the tube do not allow for branch-point motion, at least not directly. Yet, clearly, the branch points do move, for if they did not, branched polymers, including stars, would have zero center-of-mass diffusivity. 

Sikorski and Romiszowski455 study confined branched star polymers by on-lattice MC simulation. Attractive forces are excluded and only excluded volume accounted for, thus making the simulations relevant for chains in a good solvent. Contrary to expectation, they find that the diffusion constant is very similar for either moderate or highly confined chains and scales approximately as A 1, though a more accurate representation is suggested by... 

Linear polymers move a distance of order of their own size during their relaxation time, leading to a diffusion coefficient D R /r [Eq. (9.12)]. However, the diffusion of entangled stars is different because at the time scale of successful arm retraction, the branch point can only randomly hop between neighbouring entanglement cells by a distance of order one tube diameter a. For this reason, diffusion of an entangled star is much slower than diffusion of a linear polymer with the same number of monomers ... 

Jordan, E.A. Donald, A.M. Fetters, L.J. Klein, J. Transition from linear to star-B branched diffusion in entangled polymer melts. Polym. Prep. 1989, 30 (1), 63-64. 

The result is that diffusion in branched-chain polymers is much slower than in linear chains. For rings, diffusion is even more sluggish, because the ring is forced to collapse into a quasihnear conformation in order to have center-of-mass motion. Since many commercial polymers are branched or star-shaped, the self-diffusion of the polymer is correspondingly decreased, and the melt viscosity increased. 

All these observed characteristics of viscoelasticity for star polymers are natural consequences of the tube model. As suggested by the sketch in Fig. 3.49, the presence of even one long branch would surely suppress reptation [53]. There is no longer any direction for the star to move freely into new positions and conformations, and accordingly relaxation and diffusion must occur by some other motion. The Pearson-Helfand theory for stars based on tube-length fluctuations alone [72]... 

Tpre exp[(q - 2) R Z3 ], which, for large q and large Z3, would be an enormously long time. However, the center of mass diffusivity of a star polymer, measured by Shull et al. [52], is only modestly dependent on the number of arms in the star, decreasing by a factor of around 40 as the number of arms (q) in a polystyrene star increases from 3 to 12. This modest dependence of star diffusivity on the number of branches shows that the above naive picture of branchpoint motion must be wrong. 

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