Lattice Monte Carlo Simulations of Polymer Melts

We use Monte Carlo simulations to study polymer melts consisting of fully flexible and moderately stiff chains in the bond fluctuation model at a volume fraction $0.5$. In order to reduce the local density fluctuations, we test a pre-packing process for the preparation of the initial configurations of the polymer melts, before the excluded volume interaction is switched on completely. This process leads to a significantly faster decrease of the number of overlapping monomers on the lattice. This is useful for simulating very large systems, where the statistical properties of the model with a marginally incomplete elimination of excluded volume violations are the same as those of the model with strictly excluded volume. We find that the internal mean square end-to-end distance for moderately stiff chains in a melt can be very well described by a freely rotating chain model with a precise estimate of the bond-bond orientational correlation between two successive bond vectors in equilibrium. The plot of the probability distributions of the reduced end-to-end distance of chains of different stiffness also shows that the data collapse is excellent and described very well by the Gaussian distribution for ideal chains. However, while our results confirm the systematic deviations between Gaussian statistics for the chain structure factor $S_c(q)$ [minimum in the Kratky-plot] found by Wittmer et al.~\{EPL {\bf 77} 56003 (2007).\} for fully flexible chains in a melt, we show that for the available chain length these deviations are no longer visible, when the chain stiffness is included. The mean square bond length and the compressibility estimated from collective structure factors depend slightly on the stiffness of the chains.Comment: 15 pages, 12 figure

2-Dimensional Polymers Confined in a Strip

Single two dimensional polymers confined to a strip are studied by Monte Carlo simulations. They are described by N-step self-avoiding random walks on a square lattice between two parallel hard walls with distance 1 << D << N^\nu (\nu = 3/4 is the Flory exponent). For the simulations we employ the pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. We measure the densities of monomers and of end points as functions of the distance from the walls, the longitudinal extent of the chain, and the forces exerted on the walls. Their scaling with D and the universal ratio between force and monomer density at the wall are compared to theoretical predictions.Comment: 5 pages RevTex, 7 figures include

Polymers Confined between Two Parallel Plane Walls

Single three dimensional polymers confined to a slab, i.e. to the region between two parallel plane walls, are studied by Monte Carlo simulations. They are described by $N$-step walks on a simple cubic lattice confined to the region $1 \le z \le D$. The simulations cover both regions $D > R_F$ (where $R_F \sim N^\nu$ is the Flory radius, with $\nu \approx 0.587$), as well as the cross-over region in between. Chain lengths are up to $N=80,000$, slab widths up to D=120. In order to test the analysis program and to check for finite size corrections, we actually studied three different models: (a) Ordinary random walks (mimicking $\Theta$-polymers); (b) Self-avoiding walks (SAW); and (c) Domb-Joyce walks with the self-repulsion tuned to the point where finite size corrections for free (unrestricted) chains are minimal. For the simulations we employ the pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. In addition to the partition sum (which gives us a direct estimate of the forces exerted onto the walls), we measure the density profiles of monomers and of end points transverse to the slab, and the radial extent of the chain parallel to the walls. All scaling laws and some of the universal amplitude ratios are compared to theoretical predictions.Comment: 8 pages, 14 figures include

Detailed analysis of Rouse mode and dynamic scattering function of highly entangled polymer melts in equilibrium

We present large-scale molecular dynamics simulations for a coarse-grained model of polymer melts in equilibrium. From detailed Rouse mode analysis we show that the time-dependent relaxation of the autocorrelation function (ACF) of modes $p$ can be well described by the effective stretched exponential function due to the crossover from Rouse to reptation regime. The ACF is independent of chain sizes $N$ for $N/p<N_e$ ($N_e$ is the entanglement length), and there exists a minimum of the stretching exponent as $N/p \rightarrow N_e$. As $N/p$ increases, we verify the crossover scaling behavior of the effective relaxation time $\tau_{{\rm eff},p}$ from the Rouse regime to the reptation regime. We have also provided evidence that the incoherent dynamic scattering function follows the same crossover scaling behavior of the mean square displacement of monomers at the corresponding characteristic time scales. The decay of the coherent dynamic scattering function is slowed down and a plateau develops as chain sizes increase at the intermediate time and wave length scales. The tube diameter extracted from the coherent dynamic scattering function is equivalent to the previous estimate from the mean square displacement of monomers.Comment: 8 pages, 7 figures, to be published in EPJST special issue on "Phase transitions and critical phenomena" (2017

Monte Carlo Protein Folding: Simulations of Met-Enkephalin with Solvent-Accessible Area Parameterizations

Treating realistically the ambient water is one of the main difficulties in applying Monte Carlo methods to protein folding. The solvent-accessible area method, a popular method for treating water implicitly, is investigated by means of Metropolis simulations of the brain peptide Met-Enkephalin. For the phenomenological energy function ECEPP/2 nine atomic solvation parameter (ASP) sets are studied that had been proposed by previous authors. The simulations are compared with each other, with simulations with a distance dependent electrostatic permittivity $\epsilon (r)$, and with vacuum simulations ($\epsilon =2$). Parallel tempering and a recently proposed biased Metropolis technique are employed and their performances are evaluated. The measured observables include energy and dihedral probability densities (pds), integrated autocorrelation times, and acceptance rates. Two of the ASP sets turn out to be unsuitable for these simulations. For all other sets, selected configurations are minimized in search of the global energy minima. Unique minima are found for the vacuum and the $\epsilon(r)$ system, but for none of the ASP models. Other observables show a remarkable dependence on the ASPs. In particular, autocorrelation times vary dramatically with the ASP parameters. Three ASP sets have much smaller autocorrelations at 300 K than the vacuum simulations, opening the possibility that simulations can be speeded up vastly by judiciously chosing details of the forceComment: 10 pages; published in "NIC Symposium 2004", eds. D. Wolf at el. (NIC, Juelich, 2004

Growth Algorithms for Lattice Heteropolymers at Low Temperatures

Two improved versions of the pruned-enriched-Rosenbluth method (PERM) are proposed and tested on simple models of lattice heteropolymers. Both are found to outperform not only the previous version of PERM, but also all other stochastic algorithms which have been employed on this problem, except for the core directed chain growth method (CG) of Beutler & Dill. In nearly all test cases they are faster in finding low-energy states, and in many cases they found new lowest energy states missed in previous papers. The CG method is superior to our method in some cases, but less efficient in others. On the other hand, the CG method uses heavily heuristics based on presumptions about the hydrophobic core and does not give thermodynamic properties, while the present method is a fully blind general purpose algorithm giving correct Boltzmann-Gibbs weights, and can be applied in principle to any stochastic sampling problem.Comment: 9 pages, 9 figures. J. Chem. Phys., in pres

Computer simulation of bottle brush polymers with flexible backbone: Good solvent versus Theta solvent conditions

By Molecular Dynamics simulation of a coarse-grained bead-spring type model for a cylindrical molecular brush with a backbone chain of $N_b$ effective monomers to which with grafting density $\sigma$ side chains with $N$ effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range $5 \le N \le 40$, backbone chain lengths are in the range $50 \le N_b \le 200$, and we perform a comparison to results for the bond fluctuation model on the simple cubic lattice (for which much longer chains are accessible, $N_b \le 1027$, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of side chains and the backbone chain and discuss their $N$-dependence in terms of power laws and the associated effective exponents. We show that even at the Theta point the side chains are considerably stretched, their linear dimension depending on the solvent quality only weakly. Effective persistence lengths are extracted both from the orientational correlations and from the backbone end-to-end distance; it is shown that different measures of the persistence length (which would all agree for Gaussian chains) are not mutually consistent with each other, and depend distinctly both on $N_b$ and the solvent quality. A brief discussion of pertinent experiments is given.Comment: 30 pages, 13 figures, 1 tabl