Monte Carlo precise determination of the end-to-end distribution function of self-avoiding walks on the simple-cubic lattice

Abstract

Chains have been generated on the simple-cubic lattice to determine, by Monte Carlo simulation, the end-to-end distribution function of self-avoiding walks. The modulus r of the end-to-end distribution vector r, the square of this modulus, and the interactions of all orders were recorded for each chain. The Alexandrowicz dimerization procedure has been used to circumvent attrition and thus obtain statistically significant samples of large chains. This made it possible to obtain samples involving 12 000–16 000 chains, within ‘‘windows’’ of width Δρ=0.2, where ρ=r/Nν, N being the number of steps in the walk and ν the scaling exponent. It was found that the mean value 〈r〉=aNν, with ν=0.5919 and the prefactor a very close (perhaps strictly equal) to unity. The above value of ν is slightly larger than that calculated by Le Guillou and Zinn-Justin, but in accord with the Wilson ε=4−d expansion, where d is the space dimensionality. Also, 〈r 2〉=bN2ν, with b=1.136±0.05. An attempt was made to fit the data to the Fisher–McKenzie–des Cloizeaux distribution, P(r=Nνρ)=N−νdA ρθ exp(−βρδ), where θ, β, and δ are adjustable parameters. Taking δ=(1−ν)−1=2.45, in accordance with the Fisher law, it was found that plots for various N values of ln L(ρ)+βρδ vs ln ρ, where L(ρ) is the number of chains falling within subwindows of width dρ=0.02, were linear (except for the lowest values of ρ), if β was taken equal to 1.07. From the slope of the linear plot it was found that θ=0.27, in accord with theory. Finally, in three dimensions the end-to-end distribution function P(r) can be expressed by the relation P(r=Nνρ)=N−νd 0.2393ρ0.27 exp(−1.07ρ2.45). This relation is in good agreement with that proposed by des Cloizeaux and Jannink. A brief discussion of the two-dimensional case as well as fluctuations of the end-to-end distance is also presented.

The distribution function of internal distances of a single polymer chain with excluded volume in two and three dimensions: A Monte Carlo study

“…͑16͒ and ͑17͒. Our result for B 0 in dϭ3 agrees with the results of Dayantis et al 45 and Eizenberg et al 46 In general the constants B s are very similar to the constants k s of the short range regime ͑see Table II͒. This confirms the procedure to calculate B s by the normalization conditions Eqs.…”

Off-lattice Monte Carlo simulation of the discrete Edwards model

“…From studies of the SAW model it has been found that with increasing N the distribution function ᏼ N (R e ) converges more slowly toward its asymptotic scaling form than its first and second moments converge toward their asymptotic values. 61 However, the comparison with the results of Everaers et al 66 for the SAW model with N up to 240 as well as with the results of Wittkop et al 67 for the bondfluctuation model with N up to 200 indicates that the convergence of the discrete Edwards chain is somewhat slower.…”

Geometrical properties of two-dimensional interacting self-avoiding walks at the θ-point

“…In order to average them out, we will employ a procedure already used in this context in Refs. [40,41,44].…”

“…One of them, the mean asphericity, is an essential ingredient in theoretical studies of the hydrodynamic behavior of dilute polymer solutions [29][30][31]. The EEDF has been extensively studied in three-dimensions, because of its theoretical interest [32][33][34][35][36][37][38][39][40][41][42][43][44]. Under the mapping of SAWs onto the n → 0 σ model (or λφ 4 theory), it corresponds to the spin-spin correlation function, which is the basic object of field-theoretical calculations.…”