Computer simulation of self-avoiding walks: Testing the scanning method

A combined theoretical and numerical study of end-point probability distributions [P(x)] and [P(y)] for a polymer tethered inside a cylindrical confinement is presented here. Analytical solutions of the diffusion equation (Fourier series and image methods) are benchmarked against large-scale Monte Carlo simulations of three canonical polymer architectures: the Freely Jointed Chain (FJC), Self-Avoiding Walk (SAW), and Worm-Like Chain (WLC). To overcome the severe sampling attrition for off-lattice SAW in confinement, a chain-length-adaptive hybrid sampling strategy has been developed by employing kinetic growth for short chains, an enhanced pruned-enriched Rosenbluth method in the crossover regime, and an off-lattice pivot algorithm for long chains, all implemented with Numba JIT (Just-In-Time) acceleration to reach statistically converged ensembles. The agreement is excellent for the FJC vs the Fourier propagator, while the SAW and the WLC show systematic, model-specific departures driven by excluded volume and bending rigidity. Using information-theoretic (Kullback-Leibler), distributional (Kolmogorov-Smirnov, moments), and several other statistical metrics, three distinct regimes (short-chain, crossover, and long-chain saturation) have been identified. In addition, the longitudinal probability distribution has been analyzed for different parameter regimes, such as the location of the tethering point and persistence lengths. The work delivers a validated, multi-scale computational toolkit along with the development of a Streamlit-based interactive application for interpreting confined-polymer experiments and for selecting appropriate polymer models in biological and nanoconfined settings.

The conformations of semiflexible polymers in two dimensions confined in a strip of width D are studied by computer simulations, investigating two different models for the mechanism by which chain stiffness is realized. One model (studied by molecular dynamics) is a bead-spring model in the continuum, where stiffness is controlled by a bond angle potential allowing for arbitrary bond angles. The other model (studied by Monte Carlo) is a self-avoiding walk chain on the square lattice, where only discrete bond angles (0° and ±90°) are possible, and the bond angle potential then controls the density of kinks along the chain contour. The first model is a crude description of DNA-like biopolymers, while the second model (roughly) describes synthetic polymers like alkane chains. It is first demonstrated that in the bulk the crossover from rods to self-avoiding walks for both models is very similar, when one studies average chain linear dimensions, transverse fluctuations, etc., despite their differences in local conformations. However, in quasi-one-dimensional confinement two significant differences between both models occur: (i) The persistence length (extracted from the average cosine of the bond angle) gets renormalized for the lattice model when D gets less than the bulk persistence length, while in the continuum model it stays unchanged. (ii) The monomer density near the repulsive walls for semiflexible polymers is compatible with a power law predicted for the Kratky-Porod model in the case of the bead-spring model, while for the lattice case it tends to a nonzero constant across the strip. However, for the density of chain ends, such a constant behavior seems to occur for both models, unlike the power law observed for flexible polymers. In the regime where the bulk persistence length ℓp is comparable to D, hairpin conformations are detected, and the chain linear dimensions are discussed in terms of a crossover from the Daoud/De Gennes "string of blobs"-picture to the flexible rod picture when D decreases and/or the chain stiffness increases. Introducing a suitable further coarse-graining of the chain contours of the continuum model, direct estimates for the deflection length and its distribution could be obtained.

Polymers in random media: An introduction

In this study, we show the convergence and new properties of persistence length, , for the self-avoiding random walk model (SAW) using Monte Carlo data. We generate high precision estimates of several conformational quantities with a pivot algorithm for the square, hexagonal, triangular, cubic and diamond lattices with path lengths of 103 steps. For each lattice, we accurately estimate the asymptotic limit , which corroborates the convergence of to a constant value, and allows us to check the universality on the curves. Based on the estimates we make an ansatz for dependency with lattice cell and spatial dimension, we also find a new geometric interpretation for the persistence length.

Wind energy has emerged as a promising renewable energy source and wind turbine technology has developed rapidly in recent years. Improved wind turbine performance depends heavily on the design and optimization of wind blades. This work offers a critical evaluation of the state of the art in the field of numerical modelling and simulation analysis, which have become crucial for the design and optimization of wind blades. The evaluation of the literature includes considerable research on the application of numerical methods for the structural and aerodynamic performance of wind blades under various operating situations, as well as for analysis and optimization of wind blades. The article illustrates how numerical techniques can be used to analyse wind blade performance and maximize design efficiency. The study of blade performance under various wind conditions has also been made possible through the use of simulation analysis, thus enhancing the efficiency and dependability of wind turbines. Improvements in wind turbine efficiency and dependability, and ultimately the move towards a more sustainable energy future, will be greatly helpful for the development of numerical modelling and simulation techniques.

An empirical expression for the angular correlation function (ACF) of charged (Debye–Hückel) wormlike chains (WLC) with excluded volume (EV) is introduced. It reproduces the Monte Carlo (MC) data of a previous study very well. Using this expression analytical calculations for the persistence length (Lp), radius of gyration (Rg), and end-to-end distance (R) are given in the form of Taylor series. It is shown that the above quantities can be expressed as a weighted sum over the corresponding quantities (Lph, Rgh, Rh) of a set of ideal wormlike chains {Ch} h=0,1,… . Both the set {Ch} and the coefficients in the Taylor expansions are defined as functions of three parameters which are determined by fitting the ACF expression to the MC data. A comparison of the calculated Rg and R shows excellent agreement with the corresponding sampled values. The persistence length Lp is in good agreement with the values determined by fitting the sampled scattering functions by model expressions for neutral chains with excluded volume interactions, provided that a contribution due to EV is subtracted from Lp. Moreover, the method here proposed allows one to determine the persistence length of very short chains which is not possible by fitting the scattering function. The new expression for the angular correlation function, as well as the expressions derived for Rg and R are a natural generalization of well known results for ideal WLC, when EV and/or electrostatic interactions are present.

We define a statistic an(w), the size of the atmosphere of a self-avoiding walk, w, of length n, with the property that an(w) → μ as n → ∞, where μ is the growth constant of lattice self-avoiding walks. Both μ and the entropic exponent γ may be estimated to high precision from a(w) using canonical Monte Carlo simulations of self-avoiding walks. Previous Monte Carlo measurements of μ and γ have used grand canonical Monte Carlo simulations. Our simulations indicate that μ → 2.63816 ± 0.00006 and γ = 1.345 ± 0.002. These results, based on a modest computer run, are comparable to the best estimates for μ and γ from (grand canonical) Monte Carlo simulations, and are at most two digits of the best series estimates of μ for self-avoiding walks available in the literature.

The translational and rotational diffusion coefficients and the intrinsic viscosity of semiflexible, randomly broken, and wormlike chains have been obtained by Monte Carlo simulation in the context of the rigid-body treatment. Both approximate and rigorous rigid-body hydrodynamics are used, so that the error introduced by the approximate methods can be evaluated. A randomly broken chain and a wormlike chain having the same contour length and persistence length have the same radius of gyration but different values for any of the hydrodynamic properties. The two types of chains are compared in this regard. Considering that the cross section of the chain is represented by a cylinder better than by a string of spheres, we devise a cylindrical correction to be applied to the results simulated for chains of beads. Application is made to the analysis of experimental data for the translational and rotational coefficients of DNA fragments with up to 10(3) base pairs, obtaining the persistence length for each model. The values for the wormlike chain agree well with model-independent values obtained from radii of gyration and with other literature data at varying ionic strength. The randomly broken chain is equally able to reproduce the experimental length dependence of the properties, but the resulting persistence length may be too high.

Lyotropic mixtures of rodlike and flexible polymers are studied. The rodlike polymer is modelled by a hard rod and the flexible polymer by a self-avoiding walk. The interaction between rodlike and flexible polymers is studied and a tendency for this interaction to force demixing is found. Diameters of the rod which are much larger than the diameter of the segments of the flexible polymer but much smaller than its radius of gyration are considered because of their relevance to mixtures of helical biopolymers and flexible polymers.

Monte Carlo simulations of self-avoiding random walks surrounded by aligned rods on a square lattice and a simple cubic lattice were performed to address the topological constraints involved for dilute solutions of flexible polymers in a highly oriented nematic solvent. The nematic constraint exerted by the solvent is modelled by requiring that the distance between nearest neighbour chain segments in the direction of the director of the nematic solvent is such that a discrete number of rods, representing the perfectly ordered nematic solvent, fits exactly in between. Rod lengths of one or two lattice spacings are used. In this case interesting scaling behaviour for the radius of gyration and the end-to-end distance is found. Perpendicular to the director the coil behaves as a two- (or one- in the case of a square lattice) dimensional self-avoiding walk, whereas parallel to the director it behaves as a random walk. The conformations have a disk-like shape with the radius of gyration perpendicular to the director being considerably larger than the parallel component. These simulations address only one aspect of flexible polymers in nematic solvents. In reality the presence of the coil also reduces the entropy of the solvent which has exactly the opposite effect on the shape of the polymer coil.

Single semi-flexible polymer chains are modeled as self-avoiding walks (SAWs) on the square lattice with every 90° kink requiring an energy eb. While for eb = 0 this is the ordinary SAW, varying the parameter qb = exp(−eb/kBT) allows the variation of the effective persistence length p over about two decades. Using the pruned-enriched Rosenbluth method (PERM), chain lengths up to about N = 105 steps can be studied. In previous work it has already been shown that for contour lengths L = Nb (the bond length b is the lattice spacing) of order p a smooth crossover from rods to two-dimensional self-avoiding walks occurs, with radii R ∝ p1/4L3/4, the Gaussian regime predicted by the Kratky–Porod model for worm-like chains being completely absent. In the present study, confinement of such chains in strips of width D is considered, varying D from 4 to 320 lattice spacings. It is shown that for narrow strips (D < p) the effective persistence length of the chains (in the direction parallel to the confining boundaries) scales like p2/D, and R‖ ∝ L (with a pre-factor of order unity). For very wide strips, D ≫ p, the two-dimensional SAW behavior prevails for chain lengths up to Lcross ∝ p(D/p)4/3, while for L ≫ Lcross the chain is a string of blobs of diameter D, i.e. R‖ ∝ L(p/D)1/3. In the regime D < p, the chain is a sequence of straight sequences with length of the order p2/D parallel to the boundary, separated by sequences with length < D perpendicular to the boundary; thus Odijk's deflection length plays no role for discrete bond angles.

SAXS measurements on dilute solutions of polyisobutene inn‐hexane (Mν= 25,600−3.55 × 106,c&lt; 53 g/liter) were carried out by combining conventional slit collimation with the new cone collimation technique. With a fractionated sample (Mν= 25,600;U&lt; 0.2) the radius of gyration (R= 48 Å), the cross‐sectional radius of gyration (Rq= 3.9 Å), the molecular weight per unit length (M/L= 22.8 Å−1), and the Porod persistence length (a*= 8.1 Å) are found. The persistence length appears to be dependent on the molecular weight. The partial specific volumeνof polyisobutene inn‐hexane also depends on the molecular weight according toν= 1.025 + 105/Mν(cm3/g).

We present the results of Monte Carlo simulation for self-avoiding walks on a percolation cluster for square and simple cubic lattices performed very close to the percolation thresholds and estimate critical exponents \ensuremath{\nu} and \ensuremath{\gamma} defined by the disorder averages of the mean-square radius of gyration and the number of self-avoiding walks, respectively. Our results for \ensuremath{\nu} indicate a behavior rather similar to the self-avoiding walks on fully occupied lattice unlike the large increase in \ensuremath{\nu} reported in the only previous work of this kind by Kremer given for the diamond lattice. Our results for \ensuremath{\gamma} suggest an increase from the full lattice value just at ${p}_{c}$ in three dimensions, while in two dimensions the asymptotic behavior appears to be similar to that of the ordinary self-avoiding walks. We also consider the possible crossover from fractal to Euclidean behavior and discuss the reasons why no crossover scaling is observed for the mean-square radius of gyration in the present calculation.

Aggregation of nanoparticles of given size R induced by addition of a polymer strongly depends on its degree of rigidity. This is shown here on a large variety of silica nanoparticle self-assemblies obtained by electrostatic complexation with carefully selected oppositely charged biopolyelectrolytes of different rigidity. The effective rigidity is quantified by the total persistence length L_{T} representing the sum of the intrinsic (L_{p}) and electrostatic (L_{e}) polyelectrolyte persistence length, which depends on the screening, i.e., on ionic strength due to counterions and external salt concentrations. We experimentally show that the ratio L_{T}/R is the main tuning parameter that controls the fractal dimension D_{f} of the nanoparticles' self-assemblies, which is determined using small-angle neutron scattering: (i) For L_{T}/R<0.3 (obtained with flexible poly-l-lysine in the presence of an excess of salt), chain flexibility promotes easy wrapping around nanoparticles in excess, hence ramified structures with D_{f}∼2. (ii) For 0.3<L_{T}/R≤1 (semiflexible chitosan or hyaluronan complexes), chain stiffness promotes the formation of one-dimensional nanorods (in excess of nanoparticles), in good agreement with computer simulations. (iii) For L_{T}/R>1,L_{e} is strongly increased due to the absence of salt and repulsions between nanoparticles cannot be compensated for by the polyelectrolyte wrapping, which allows a spacing between nanoparticles and the formation of one-dimensional pearl necklace complexes. (iv) Finally, electrostatic screening, i.e., ionic strength, turned out to be a reliable way of controlling D_{f} and the phase diagram behavior. It finely tunes the short-range interparticle potential, resulting in larger fractal dimensions at higher ionic strength.

We have recently suggested an approximate computer simulation technique, based on the scanning method, which enables one to extract the entropy (and hence the free energy) of polymer chains from a relatively small sample. So far this technique has been discussed in terms of chains with excluded volume (EV); in the present work we extend its scope to chains, which also have finite interactions. In order to obtain better approximations for the entropy we utilize the concepts of the generalized Monte Carlo procedure suggested by Schmidt [Phys. Rev. Lett. 51, 2175 (1983)] (and discussed in the preceding paper). We analyze our results from the preceding paper for self-attracting random walks (without EV) on square and simple-cubic lattices, and show that our techniques lead to accuracy which is better than 0.3% for both the free energy and the entropy. For chain models which include EV, we use, in addition to Schmidt's procedure, an alternative one based on ``importance sampling.'' We test these two procedures as applied to self-avoiding walks (SAW's) on a square lattice, where the SAW's are unbounded in space or bounded in relatively small ``boxes.'' Very accurate results for the entropy are also obtained here. It turns out, however, that importance sampling is slightly more efficient than the Schmidt procedure.