Entanglement constraints

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

The radius of the beads and the interactions, Eqs. (8,9), have been chosen such that the chains may not intersect themselves or each other in the course of their movement within the box. Thus entanglement constraints are obeyed automatically and need not be enforced by extra (time consuming ) control. The chains are treated as fully flexible and a potential for bond angles is not considered, although an extension of the model to allow for semi-flexibihty of the chains is straightforward. 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Entanglement Constraints in Long-Chain Polymer Melts... 

The presence of entanglement constraints is expected to show itself (1) by a reduction in the time decay of S(Q,t) compared to the Rouse dynamic structure factor and (2) by the systematic Q-dependent deviation from the Rouse scaling properties. 

Fig. 2.50 An entangled diblock copolymer chain in a lamellar morphology with a sinusoidal composition profile (Lodge and Dalvi 1995). Entanglement constraints are indicated by x. Coupling between the thermodynamic forces produced by the composition gradients and replation dynamics leads to an anisotropic diffusion coefficient.

Richter D, Farago B, Butera R, Fetters LJ, Huang JS, Ewen B (1993) On the origin of entanglement constraints. Macromolecules 26(4) 795—804 Rice SA, Gray P (1965) Statistical mechanics of simple liquids. Wiley, New York Ronca G (1983) Frequency spectrum and dynamic correlation of concentrated polymer liquids. J Chem Phys 79(2) 1031-1043... 

Here X ,ax is the single chain limiting extension ratio in the isotropic, unoriented polymer with the same entanglement weight. A complication is that the process of orientation above Tg may result in a loss of entanglement constraint, effectively increasing and One of the ways this loss can take place in the current versions of the tube model is by tube relaxation whereby the process of tube... 

Another important relaxation process in entangled melts is constraint release, depicted in Fig. 3-27. When an end of a surrounding chain moves past a test chain, an entanglement constraint restricting the motion of the test chain is released, and a portion of the latter is freed to reorient (Graessley 1982 Montfort et al. 1986 Pearson 1987 Viovy et al. 1991). Constraint release can only be completely neglected for the case of an isolated chain... 

Reptation and tube length fluctuations of surrounding chains release some of the entanglement constraints they impose on a given chain and lead to Rouse-like motion of its tube, called constraint release. Constraint release modes are important for stress relaxation, especially in polydisperse entangled solutions and melts. 

Predictions for the Parameters k and The parameter f is not far from zero, which is to he expected since the surroundings of jimctions cause their deformation to be nearly affine with the macroscopic strain. The primary parameter ic is defined as the ratio of the mean-square junction fluctuations in the equivalent phantom network, ie, in the absence of constraints, to the mean-square jimction fluctuations about the centers of domains of entanglement constraints (in the absence of the network) in the isotropic state. Thus in a phantom network, the absence of constraints leads to /c = 0. In an affine one, the complete suppression of fluctuations is equivalent to /c = oo. It has been proposed that k should be proportional to the degree of interpenetration of chains and junctions (165). Since an increasing number of junctions in a volume pervaded by a chain leads to stronger constraints on these jimctions, k was taken to be... 

The basic process associated with this word was introduced in Ref. 14 and is shown in Fig. VIII. 10, where we see one entanglement constraint being altered when one of the ambient chains (F) has an extremity in the immediate vicinity of the test chain (F,), the relative positions of (F) and (Fj) may change qualitatively in a very short time. This may be viewed as a modification of the tube. 

Finally, we also mention models that are in some way intermediate between the SAW and the bond fluctuation model. For example, Shaffer [53,54] has used a model where the monomer takes only a lattice site like the SAW, but the bond length can be 1, /2, and /3 lattice spacings, and then moves can be allowed where bonds cross each other, junctions can move, etc. This model is useful to separate the effect of excluded volume (site occupancy) from entanglement constraints (noncrossability of chains) in the dynamics [53,54]. Since because of the chain crossability a rather fast... 

RUBBEH ELASTICITY, ENTANGLEMENT CONSTRAINTS AND THE MEMORY-LATTICE MODEL... 

Two monomer units cannot occupy the same space, and two chains cannot pass through each other. These entanglement constraints are not easy to introduce into the theory of rubber elasticity though many attempts have been made. 

Fig. 1.9. Sketches of various choices for the locations of entanglement constraints.

Fig. 3.49. A depiction of a branched polymer in its tube of entanglement constraints.

Fig. 14.10. (a) Schematic representation of entanglement constraints (b) the tube model. 

These should he compared to the dependencies on molar mass predicted hy the Rouse model (Eq. 2.34). As indicated in Fig. 2.21, experiments reveal that for entangled polymers the zero-shear viscosity scales with The subtle difference in exponent compared to the predictions of reptation theory indicates additional effects not considered in the original model. These include the release of the entanglement constraints and fluctuation-driven stretchings and contractions of the chain along the tube. 

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