Rods, modelling

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

But what was the meaning of the pictorial formulas What did they explain Were they theoretical models What was the aim of Hofmann s croquet balls, Dewar s brass-and-rod models, Kekule s wooden sausages, and van t Hoffs cardboard tetrahedrons 86... 

Fig. 8a,b. Off-lattice representations of a three-functional star a Bead and Rod model b Bead and Spring model... 

Usually, MD methods are applied to polymer systems in order to obtain short-time properties corresponding to problems where the influence of solvent molecules has to be explicitly included. Then the models are usually atomic representations of both chain and solvent molecules. Realistic potentials for non-bonded interactions between non-bonded atoms should be incorporated. Appropriate methods can be employed to maintain constraints corresponding to fixed bond lengths, bond angles and restricted torsional barriers in the molecules [117]. For atomic models, the simulation time steps are typically of the order of femtoseconds (10 s). However, some simulations have been performed with idealized polymer representations [118], such as Bead and Spring or Bead and Rod models whose units interact through parametric attractive-repulsive potentials. 

A very simple model that predicts lyotropic phase transitions is the hard-rod model proposed by Onsager (Friberg, 1976). This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder where the... 

Now at present we have no means of applying a known force to a particular chain and measuring its extension. What can be done is to measure the vibrations of chains in various situations and relate these directly via normal mode analysis, or indirectly via continuum (rod) models to a set of stiffnesses. 

Modifications of the rod model to account for end conditions in, for example, lamella crystals do not seem to be worthwhile except as curve-fitting exercises, for the rod can have no physical meaning. 

A number of continuous network, jointed-rod models for the structures of the Sic, Vi, and V2 phases have been proposed by Luzzati and his collaborators (10, 11, 12) on the basis of x-ray diffraction measurements. In these models, the individual rods are close to isodimensional and thus represent globular micelles, but these are pictured, not as rotating at the lattice points but as jointed into continuous interpenetrating networks so as to confer rigidity on the structure. Perhaps the main objection to these models is that, in contrast to rotational plastic... 

Various models are available [20] to describe semiflexible chains. Some are based on expansions either close to the gaussian coil or to the rigid rod limits, while others interpolate between these two chain stiffness limits. One of these, the sliding rod model [19], is described here because of its inherent simplicity. 

A) Onsager s rigid-rod model (1949) was the first correct model of an athermal Isotropic —> Nematic phase transition. It is a relatively simple model that predicts phase transitions. It is based on excluded volume between two rigid rods, which reads... 

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]...

Fig. 30. Interlocking rod model for the Im3m (A) and Ia3d (B) cubic phases and the infinite periodic minimum surface model for Im3m (C).

We see that the magnitude of ij/o in the rod model is larger than that predicted from... 

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 ... 

Here, Sff and off are the potential well depth between N2 molecules and the effective diameter, respectively. The used LJ parameters for an N2 molecule are /// ke = 104.2 K and Off = 0.3632 nm. The molecule-pore interaction was approximated by the smoothed graphitic wall function for the structure-less tube. [4,5] Even the molecule-pore interaction for the corn part was approximated by the spinning fishing rod model. [6] Here, the fee / ks = 30.14 K and Ooc - 0.3416 nm were used for a carbon... 

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