Rotational isomeric state chain model

A comparison has been made of rotational isomeric state chains modelled by matrix multiplication methods, as in most of the papers mentioned in this section, with lattice and off-lattice chains obtained by Monte Carlo techniques. excluded volume effects than or Rotational angle flexibility has been expressed in the co-ordinates transform motion matrix, and its influence examined upon , the distribution function for r, and the correlation of the relative orientation of the end bonds of the polymer. ... 

The structurally simplest polymer, and one of the most commercially important, is polyethylene. It consists of a linear chain of CH2 units, which we model as single spherical sites in the single-site homopolymer spirit. There exist well-developed ideal rotational isomeric state chain models where the bond rotational degrees of freedom are represented as discrete trans and gauche isomers. Numerical calculation of the required single-chain structure factor can be achieved via Monte Carlo simulation or using the recently developed computationally convenient approximate methods of McCoy and co-workers. ... 

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

The dimensionless ratios of the form (s2p)q/ s2)pq are easily evaluated from a theoretical model for the distribution function, P(s2), using Equation (1.22). For small p, they can also be calculated for unperturbed rotational isomeric state chains by efficient generator matrix methods... 

The numerical self-consistent (SC) MC/RISM procedure [51,52] employed to solve the matrix polymer RISM equation (4) with the RMMSA closure relation (6)-(7) was used in Ref. [53] to study water-containing Nafion systems. The single-chain MC simulation was based on the realistic rotational-isomeric-state (RIS) model [54], in which the short-range intramolecular interactions depending on the details of chemical structure were taken into account via appropriate matrices of statistical weights [54]. 

Birefringence of the Single-Chain and the Rotational Isomeric State (RIS) Model... 

Whilst some of these cases have been treated previously, Eichinger s method allows the calculation not only of statistical averages, but also of the distribution function itself. For polymer molecules with real bond angles and restricted bond rotation, the rotational isomeric state (RIS) model has proved powerful. Flory has summarized some of the most important results of the treatment, and Mark has considered the applications, particularly to bulk polymers and networks, pioneered by his group. A recent paper uses the RIS model to calculate the distribution function of the end to-end vector distribution for short polymer chains. 

The characteristic ratio Coo characterizes chain flexibility. It depends on the 6 and torsional potential and is determined by the chemical structure of the monomers [20]. The rotational isomeric state (RIS) model, introduced by P.J. Flory [20] is essentially an adaptation of the one-dimensional Ising model of statistical physics to chain conformations. This model restricts each torsion angle to a discrete set of states (e.g., trans, gauche , gauche"), usually defined around the minima of the torsional potential of a single bond, V((f>) (see Fig. 2d). This discretization, coupled with the locality of interactions, permits calculations of the conformational partition... 

Because of the third assumption, these theories are called Rotational Isomeric State (RIS) models. Adopting these approximations, the partition function can be considerably simplified and the statistical mechanics of chain molecules can be mapped onto the well-known ID Potts model with K states. [20]... 

In Method 1 a Monte Carlo algorithm was devised which included chain conformation probabilities given by the rotational isomeric states (RIS) model together with nonbonded interactions between backbone carbons separated by four or more bonds. These structures were then relaxed by energy minimization. The same approach has also been used for polycarbonate, polysulfone, and polyvinylchloride chains. 

The first calculation on atomistic systems of chain molecules near solid surfaces was performed in Ref. 21 modeling the surfaces as two impenetrable walls placed at a distance much greater than the molecular dimensions. The overall chain density of the model has been chosen such that the local density far from the walls was equal to that of the bulk liquid at 300 K. Since all the properties of molecules far from the walls were found to match those of unperturbed chains, as described by the rotational isomeric state (RIS) model at this temperature, the chains in contact with the impenetrable walls can be considered to be in equilibrium with the unperturbed bulk liquid. 

Chain models capture the basic elements of the amphiphilic behaviour by retaining details of the molecular architecture. Ben-Shaul et aJ [ ] and others [ ] explored the organization of tlie hydrophobic portion in lipid micelles and bilayers by retaining the confonuational statistics of the hydrocarbon tail withm the RIS (rotational isomeric state) model [4, 5] while representing the hydrophilic/liydrophobic mterface merely by an... 

A suitable approach to the equilibration of an amorphous polymer system at bulk density becomes much more likely when the fully atomistic model in continuous space is replaced by an equivalent coarse-grained model on a lattice with sufficient conformational flexibility. Different strategies, which seek results at different levels of detail, can be employed to create an appropriate coarse-grained model. Section 4 (Doruker, Mattice) describes an approach which attempts to retain a connection with the covalent bonds in the polymer. The rotational isomeric state (RIS) [35,36] model for the chain is mapped into... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

A theoretical investigation of the use of NMR lineshape second moments in determining elastomer chain configurations has been undertaken. Monte Carlo chains have been generated by computer using a modified rotational isomeric state (RIS) theory in which parameters have been included which simulate bulk uniaxial deformation. The behavior of the model for a hypothetical poly(methylene) system and for a real poly(p-fluorostyrene) system has been examined. Excluded volume effects are described. Initial experimental approaches are discussed. 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

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