Crossover between Linear and Nonlinear Elastic Behaviors in Random and Self-Avoiding Walks

Abstract

In order to reexamine earlier studies on the elastic behavior of a single-chain molecule precisely, we performed a large-scale Monte Carlo simulation of a random walk (RW) and a self-avoiding walk (SAW) on a simple cubic lattice with lattice constant a under an external force X in the x direction. We use the important-sampling enrichment algorithm generating 1 000 000 samples of chain length N up to 300. We first confirmed that the mean stretching distance behaves exactly as ⟨xN⟩RW = Na(eβXa − e−βXa)/(eβXa + e−βXa + 4) for RW (β = 1/kBT) and as ⟨xN⟩SAW ∼ const × N2νβXa (for very small X) and ∼const × X2/3 (for 0 ≪ X ≪ kBT/a) for SAW in accordance with the scaling theory by de Gennes (Scaling Concepts in Polymer Physics; Cornell University Press Ithaca, NY, 1979) and Pincus (Macromolecules 1976, 9, 386). However, for SAW, the crossover between these two behaviors is rather smooth in contrast to the earlier Monte Carlo result by Webman et al. (Phys. Rev. A 1981, 23, 316) showing an abrupt transition. The present study resolves the long-standing difficulty concerning if the crossover is abrupt or smooth, which has been raised by Oono et al. (Macromolecules 1981, 14, 880) in their renormalization-group study. The crossover point ηc = βXcR0 is accurately determined in the present study. On the other hand, we found that the relation ⟨xN⟩ ∼ R02X/3kBT holds up to the proportional constant for very small X irrespectively for SAW and RW, or for any real chains (R02 is the mean-square end-to-end distance at X = 0), as first suggested by Webman et al. and reconfirmed by Oono et al.