Effect of the Excluded Volume on the Dynamics of a Polymer Chain

Abstract

ABSTRACT:The effect of the excluded volume on the dynamics of a polymer chain is studied, using a stochastic jump model introduced by Orwoll and Stockmayer. The relaxation time of the normal mode of the averaged position vectors is obtained as 'tp-1=1:p101-1(1-Cpz) where 1:p1o1 is the relaxation time in the absence of the excluded volume effect, z is the usual excluded volume parameter, and Cp is a constant depending on the mode number p. The slowing down is not so large as in the computer simulation by Verdier. Further, the autocorrelation function of the normal mode of the averaged position vectors is shown to be a simple exponential function, in contrast to the nonexponential form used by Verdier. These differences are due partly to the particularity of the lattice model used by Verdier and partly to the approximations we have used.KEY WORDS Excluded Volume Effect/ Relaxation Time / Slowing Down / Freely Jointed Chain Model / Lattice Model / A polymer molecule in solution suffers frictional as well as random forces from surrounding solvent molecules and moves irregularly. To investigate the effect of such Brownian motion on the dynamics of a polymer molecule, a bead-and-spring model was proposed by Rouse 1 in 1953 and independently by Bueche 2 in 1954. This model was improved so as to take account of the hydrodynamic interaction 3 -7 and of the excluded volume effect. 5 -7 In regard to the excluded volume effect, however, the theory is still not satisfactory, as will be mentioned later. On the other hand, Verdier and Stockmayer 8 presented a stochastic jump model on a cubic lattice to study the dynamics of a polymer in solution by computer simulation. Later Verdier 9 investigated this model precisely and found that in the absence of the excluded volume effect the relaxation curve of the autocorrelation function of the end-to-end distance is quite similar to that of the Rouse model. Verdier 10 confirmed this agreement by calculating numerically the autocorrelation function of the square of the normal mode of the position vectors. This agreement was also explained analytically by Iwata and Kurata 11 for the lattice model and by Orwoll and Stockmayer 12 and by Verdier 13 for the freely jointed chain. The latter model will be explained in the next section.The effect of the excluded volume on the dynamics of a polymer is not as well understood as is the effect of the excluded volume on the equilibrium polymer configurations. The excluded volume effect on the polymer dynamics was investigated by Tschoegl 4 in the Rouse model only indirectly, through the introduction of a nongaussian distribution of the lengths between consective beads. Verdier and his coworkers8-10 · 14 made a computer simulation of the dynamics of a lattice polymer by taking account of the excluded volume effect in such a way that a bead cannot make a jump into a position which is already occupied by another bead. The purpose of this paper is to investigate analytically the effect of the excluded volume on the dynamics of a polymer in solu-601

Segmental Mobility of Ring Polymers in Good and Poor Solvents

“…In this article, the influence of the repulsive excluded volume and attractive interactions on the orientational relaxation dynamics of ring polymers in dilute solutions is investigated within the framework of the Rouse–Zimm theory. In this context, two different theoretical models of EVI are employed, one recently developed by Biswas , (model I) and another motivated by a study on the linear polymers by Miyakawa and Saitô (model II). To the best of our knowledge, this is the first theoretical study which characterizes the relaxation dynamics of the ring polymers of various sizes with both repulsive and attractive interactions over a wide range of length scales ranging from the local segmental motion to the diffusion of the entire polymer.…”

“…Model I , represents the excluded volume interaction as the harmonic nonbonded potential where the strength of EVI is evaluated self-consistently from Flory’s mean-field approach. Miyakawa and Saitô model analytically investigated the effect of the excluded volume interactions on the dynamics of linear polymers. Model II denotes the generalized model for the ring polymers.…”

“…This is motivated by the Miyakawa and Saitô model for linear polymers that analytically investigated the effect of the excluded volume interactions on their dynamics in solutions, using a stochastic jump model. They derived the equations of motion expressed in matrix form as where q i = ⟨ R i ⟩ represents the averaged position vectors and α is a constant independent of the position vectors.…”

“…The parameter β in the matrix [ B ′] is the dimensionless volume parameter defined in terms of the excluded volume, v , as . This excluded volume is expressed in terms of the Flory temperature, θ, and bead diameter, a as .…”

“…This study examines the effect of the repulsive excluded volume interactions and attractive interactions on the segmental relaxation of ring polymers in dilute solutions that captures the essence of the Rouse–Zimm theory. − The term “dilute” implies that the system under study comprises an isolated ring polymer in solution. This work theoretically predicts the experimentally relevant quantities, i.e., spectral density, spin–lattice, and spin–spin relaxation rates, while comparing two different models of EVI in ring polymers: (i) a recently developed model I , and (ii) model II motivated by the Miyakawa and Saitô model for linear polymers. This enables the characterization of relaxation dynamics of ring polymers over a wide range of time scales spanning from the segmental length scale to that of the entire polymer.…”