Dynamic entanglement interactions

When polymer chains become sufficiently concentrated, entanglement interactions lead to cooperative relaxation phenomena that can dominate the dynamical response. The relaxation times dramatically slow down with concentration and nonlinear flow phenomena... 

Aranguren et al. (1992) examined the rheology of PDMS filled with untreated and silane-treated fused silica. It was found that the dynamic moduli increase with increasing concentration of silanol groups (as present on the untreated silica) on the surface, due to the increase in interactions with the PDMS. Direct interactions and indirect entanglement interactions are thought to be the types of interactions occurring. 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

In the perspective discussed in the present contribution, bundle formation occurs within the amorphous phase and in undercooled polymer solutions. It does not imply necessarily a phase separation process, which, however, may occur by bundle aggregation, typically at large undercoolings [mode (ii)]. In this case kinetic parameters relating to chain entanglements and to the viscous drag assume a paramount importance. Here again, molecular dynamics simulations can be expected to provide important parameters for theoretical developments in turn these could orient new simulations in a fruitful mutual interaction. 

One tool for working toward this objective is molecular mechanics. In this approach, the bonds in a molecule are treated as classical objects, with continuous interaction potentials (sometimes called force fields) that can be developed empirically or calculated by quantum theory. This is a powerful method that allows the application of predictive theory to much larger systems if sufficiently accurate and robust force fields can be developed. Predicting the structures of proteins and polymers is an important objective, but at present this often requires prohibitively large calculations. Molecular mechanics with classical interaction potentials has been the principal tool in the development of molecular models of polymer dynamics. The ability to model isolated polymer molecules (in dilute solution) is well developed, but fundamental molecular mechanics models of dense systems of entangled polymers remains an important goal. 

The other important physical assumption is that the friction is local (hydro-dynamic interactions are screened in the melt [2]) so that D -(N /N)D with the diffusion constant in the melt of an unentangled chain of segments. Now the characteristic relaxation (Rouse) time of an entanglement segment % is just a /D. so that... 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

Differing from Bitsanis et al. who neglected the intramolecular hydro-dynamic interaction and equated DL0 to D[0, Doi et al. incorporated this interaction and set D,0 equal to D 0/2. The increase of the transverse diffusivity should enhance the disengagement of the rotation of the test rod from hindrances and tend to diminish the entanglement effect. 

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