Monte Carlo branching

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

In tfiis chapter we address first the electrochemical application of the more familiar method of molecular (or atom) dynamics, and later turn to consider Monte Carlo methods, in each case giving a short introduction that should motivate the reader to pursue reading more specific works. Although the present research field is relatively new, the investigations are already too extensive to review in detail in a single chapter. For this reason, we discuss here the more extended research branches in the field and present a few representative examples. The application of simulations applied to nanostructuring problems is discussed in Chapter 36 liquid-liquid interfaces have been addressed by I. Benjamin (1997). 

Local sensitivity analysis is of limited value when the chemical system is non-linear. In this case global methods, which vary the parameters over the range of their possible values, are preferable. Two global uncertainty methods have been used in this work, a screening method, the so-called Morris One-At-A-Time (MOAT) analysis and a Monte Carlo analysis with Latin Hypercube Sampling (Saltelli et al., 2000 Zador et al., submitted, 20041). The analyses were performed by varying rate parameters, branching ratios and constrained concentrations within their uncertainty interval,... 

Figure 11.5 Two-dimensional representation of a D3G4 [D = spacers (3), G = dendrimer generation (4)] dendrimer. Typical snapshots of various generations of codendrimers, from Monte Carlo simulations, with the same number of spacers (D = 2) between each branch point. Inner H and outer H topologies are compared for three different degrees of branching (G=l, 5, or 7). Reprinted from Connelly et al. (2004). Copyright 2004 American Chemical Society.

Monte Carlo methods comprise that branch of experimental mathematics which is concerned with experiments on random numbers (Addnl Ref N)... 

The presence of bulky, (3-branched side chains can be helix promoting or destabilizing depending on the environment. 100 The role of hydrophobic residues in helix stabilization has been studied in Ala-based peptides 106 as well as through Monte Carlo calculations. 107 The positioning of hydrophobic residues is also important. In amphiphilic helices, hydrophobic residues repeat approximately every three to four residues, such that one side of the helix is hydrophilic and one side hydrophobic. The amphiphilicity makes the peptide susceptible to helix formation in the presence of lipid-water interfaces. 108 109 ... 

The Monte Carlo method is a very powerful numerical technique used to evaluate multidimensional integrals in statistical mechanics and other branches of physics and chemistry. It is also used when initial conditions are chosen in classical reaction dynamics calculations, as we have discussed in Chapter 4. It will therefore be appropriate here to give a brief introduction to the method and to the ideas behind the method. 

Mansfield 79 performed Monte Carlo calculations on model dendrimers and determined that as a result of the unique architecture of the branches, even when similar chemically, they are well segregated. Further, he concluded that dendrimers are fractal (D ranges from 2.4 to 2.8) and self-similar only over a rather narrow scale of lengths. 

S. T. Cui, P. T. Cummings, and H. D. Cochran, Fluid Phase Equilibria, 141, 45 (1997). Configurational Bias Gibbs Ensemble Monte Carlo Simulation of Vapor-Liquid Equilibria of Linear and Short-Branched Alkanes. 

Essential progress has been made recently in the area of molecular level modeling of capillary condensation. The methods of grand canonical Monte Carlo (GCMC) simulations [4], molecular dynamics (MD) [5], and density functional theory (DFT) [6] are capable of generating hysteresis loops for sorption of simple fluids in model pores. In our previous publications (see [7] and references therein), we have shown that the non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts desorption branches of hysteretic isotherms of nitrogen and argon on reference MCM-41 samples with pore channels narrower than 5 nm. 

Caillol, J.M. A Monte Carlo study of the dielectric constant of the restricted primitive model of electrolytes on the vapor branch of the coexistence line. J. Chem. Phys., 1995, 102, p. 5471-5479. 

Configurationally biased Monte Carlo techniques [63-65] have made it possible to compute adsorption isotherms for linear and branched hydrocarbons in the micropores of a siliceous zeolite framework. Apart from Monte Carlo techniques, docking techniques [69] have also been implemented in some available computer codes. Docking techniques are convenient techniques that determine, by simulated annealing and subsequent freezing techniques, local energy minima of adsorbed molecules based on Lennard-Jones-or Buckingham-type interaction potentials. 

Freire and coworkers [285,286] studied the case of miktoarm star copolymers of the type AxBf x, where f is the total functionality of the star copolymer. The conformational characteristics of these kinds of molecules were investigated as a function of molecular weight and number of the different branches, as well as the thermodynamic cross interactions between the arms and the solvent medium. Calculations based on the renormalization group and Monte Carlo methods allowed the estimation of the dimensions of each arm and of the whole molecule and the mean square distance between the two centers of mass of the different homopolymers. From these estimations different expansion factors relative to the homopolymer precursors could be calculated (Fig. 2). Different degrees of agreement were obtained by the two methods depending on the property under consideration. 

---