Conformations, transverse fluctuations, and crossover dynamics of a semi-flexible chain in two dimensions

We present a unified scaling description for the dynamics of monomers of a semiflexible chain under good solvent condition in the free draining limit. We consider both the cases where the contour length L is comparable to the persistence length ℓ(p) and the case L ≫ ℓ(p). Our theory captures the early time monomer dynamics of a stiff chain characterized by t(3/4) dependence for the mean square displacement of the monomers, but predicts a first crossover to the Rouse regime of t(2ν/1 + 2ν) for τ¹ ~ ℓ(p)³, and a second crossover to the purely diffusive dynamics for the entire chain at τ2 ∼ L(5/2). We confirm the predictions of this scaling description by studying monomer dynamics of dilute solution of semi-flexible chains under good solvent conditions obtained from our Brownian dynamics (BD) simulation studies for a large choice of chain lengths with number of monomers per chain N = 16-2048 and persistence length ℓ(p) = 1-500 Lennard-Jones units. These BD simulation results further confirm the absence of Gaussian regime for a two-dimensional (2D) swollen chain from the slope of the plot of ⟨R(N)²⟩/2Lℓ(p) ~ L/ℓ(p) which around L/ℓ(p) ∼ 1 changes suddenly from (L/ℓ(p)) → (L/ℓ(p))(0.5), also manifested in the power law decay for the bond autocorrelation function disproving the validity of the worm-like-chain in 2D. We further observe that the normalized transverse fluctuations of the semiflexible chains for different stiffness √(⟨l(⊥)²⟩)/L as a function of renormalized contour length L/ℓ(p) collapse on the same master plot and exhibits power law scaling √(⟨l(⊥)²⟩)/L ~ (L/ℓ(p))(n) at extreme limits, where η = 0.5 for extremely stiff chains (L/ℓ(p) ≫ 1), and η = -0.25 for fully flexible chains. Finally, we compare the radial distribution functions obtained from our simulation studies with those obtained analytically.