Bead-rod model

Fig. 1. Freely jointed bead-rod model of a chain formed by(N + 1) beads and N rigid links of length Ip...

Fig. 19. Monte Carlo result for the phase diagrams of an off-lattice bead rod model of a symmetric binary polymer mixture with N=20, in the plane of variables reduced temperature T =kBT/ AA and volume fraction of component A, denoted here as xx. Data are for bulk systems (full dots), and for confined films of thicknesses D=10.5a (squares) and 5a (triangles), respectively. Dashed curves represent fits to xx—xlc oc T-Tc fil, where the Ising model exponent [229,230] was chosen as Pj=l/3. From Kumar et al. [39]...

The kink-jump technique applies random local rotational jumps along the chain. An algorithm for the bead-rod model would have, for example, the following steps ... 

Note that although the bead-rod model predicts that ( yy-TzJ is zero during the steady shear flow (t < 0), it is nonzero during the recoil process and furthermore it differs in sign from (rxx — r)l),)+. 

This process has been examined theoretically by a number of authors (29-31), who derived constitutive equations based upon finitely extendable nonlinear elastic (FENE) dumbbell models (29), bead-rod models (30), and bead-spring models (31). There is general agreement that a large increase in elongational viscosity should be expected. 

As we have seen, there exist a number of treatments of the increase in extensional viscosity of the solution corresponding to the coil-stretch transition. In particular, the Warner FENE dumbbell model (29) and the Kramers bead-rod model (30) predict increases in normalized extensional viscosity of the order of N, the number of statistical segment units in the flexible chain. The normalized extensional viscosity (T)e ) compares the increase in extensional viscosity due to the polymer to three times (the Trouton ratio) the corresponding increase in the simple shear viscosity and is given by... 

The scission of isolated molecules in extensional flow fields has been the subject of a number of investigations 11-15, 53). The scission occurs very precisely at the center of the polymer chain and at strain rates that broadly correspond to expectations for extended bead-rod models and covalent C-C bonds. 

Keywords Bead-rod model Complex topology Mechanophore Polymer mechanochemistry... 

Mechanical Stability of Star-Shaped Polymers Bead-Rod Model. 172... 

Odell and Keller first proposed a bead-rod model to explain the degradation of an isolated polymer chain in QSSF [27,28]. This is also the basis for further discussing the effect of chain topology in the following sections. The key assumptions are ... 

Fig. 14 Parabolic distribution of tensile force along a fully extended chain (bead-rod model)...

Odell and Keller [97] developed a thermally activated barrier to scission (TABS) model. They incorporated the expression of F ((11), bead-rod model) into (17) and accounted for the nonuniform distribution of tension along the contours of the... 

Agarwal and Mashelkar first analyzed contradictory reports from Kim [190], Gryte [191], and Singh [192, 193], and proposed a simple mechanistic model [209]. In stark contrast to the concept of preferential scission of side chains, their model reveals decreased shear stability by grafting side chains. They extended Odell and Keller s bead-rod model [27]. The backbone was modeled as a fully extended rod with Ni = 2m+ beads (Fig. 25). p grafted bead-spring chains having g beads with... 

Fig. 25 Agarwal and Mashelkar s modified bead-rod model for grafted chain in elongational flow [209]. The backbone has p uniformly distributed side chains. Each side chain is modeled as a random coil and has g beads with bead size 2c. The hydrodynamic force exerted on the coils adds additional force to the backbone so the force along the linear backbone consists of two part

Agarwal and co-workers adopted Odell and Keller s bead-rod model to predict the shear stability of star-shaped polymers [228]. As illustrated in Fig. 27, they adopted a key assumption that all arms are fully stretched before any scission event. For a linear polymer, the tensile force built at the midpoint is tr id (Eq. 13). For a six-arm star molecule bearing the same N, the force at the base of each arm ffaim is... 

Fig. 27 Schematic drawing of bead-rod model for a star-shaped molecule with/a = 6 arms and N total beads [228]. Each bead experiences a hydrodynamic ftyrcefi. The accumulated tensile force at the 0th bead depends rat the structure of the core (a) a linear core, 1I3N (b) a fused...

The bead-rod model implicitly assumes other bonds in the star molecule are as strong as those at the midpoint of the core and favors a core-fracture mechanism. Practically, the linkages which coimect the arms to the core may be weaker than the bonds in the core. The fracture may proceed by an arm-loss mechanism. 

Neelov IM, Adolf DB (2003) Brownian dynamics simulations of dendrimers under elongational flow bead-rod model with hydrodynamic interactions. Macromolecules 36 6914-6924... 

Models with arbitrary rigid arrays of equally spaced centers of frictional resistance (rigid bead-rod models) that are axisymmetric, i.e., have two equal moments... 

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