Rouse Dynamics of a Dendrimer Model in the ϑ Condition

Abstract

We studied the dependence of the dynamic properties on the generation G of a star-burst dendrimer model. The Rouse approximation for the mobility matrix was used for simplicity. We categorized all normal-mode displacements and considered various time-dependent correlation functions and the intrinsic viscosity. We determined three different time scales corresponding to (1) the diffusion of the center of mass through a distance equal to the dendrimer size, (2) the relaxation of the position of the center of mass relative to the central core monomer, and (3) the rotational and internal elastic motions of the molecule. These relaxation times depend differently on the generation G for large G: the first time is approximately τD = 6G2 G ζ/k, the second time is approximately τs = 5.8ζ/k, and the third time is approximately τr = τe = 2 G +1ζ/k, where ζ is the friction coefficient for a single monomer and k = 3k B T/a 2 with k B being the Boltzmann constant, T the temperature, and a the average distance between the monomers. A linear dependence of the intrinsic viscosity on the generation was also found. These findings form the basis for further generalization of the theory to include hydrodynamic and excluded-volume interactions in order to model the actual systems more realistically.

Relaxation of a Single Dendrimer

“…For the Gaussian dendrimer with P ¼ 1, the slowest relaxation time has been proven to behave aŝ 1;Gauss $ ð f À 1Þ Gþ1 =ð f À 2Þ 2 for G ) 1. 19,20) Thus, the estimated dependence of 1;Gauss on f c and G agrees with that of^1 ;Gauss . This agreement supports the assumption (ii) that the relaxation of a single dendrimer proceeds hierarchically from the outermost generation to the central core.…”

“…The relaxation modes of a single dendrimer can be classified into two types: 19,20) Symmetric and antisymmetric modes 21) (in other literatures, these modes are often called mobile and immobile core classes, respectively). The symmetric modes involve the displacements of all the segments and the segments with the same topological distance from the core move in the same way.…”

“…; G represents the depth of the generation measured from the outermost generation. In many cases, the slowest relaxation mode, the most important mode, coincides with the slowest antisymmetric mode 19,20) corresponding to the relative displacement of two branches grafted to the core.…”

“…2,6,[13][14][15][16][17][18][19][20][21][22][23] The relaxation modes fX p g satisfy hX p ðtÞX q ð0Þi / p;q expðÀ p tÞ, where hX p ðtÞX q ð0Þi denotes the equilibrium correlation of X p at time t and X q at time 0. The quantity p is called the relaxation rate of X p and the reciprocal of p gives the corresponding relaxation time, i.e., characteristic time of the pth mode.…”

“…6) The relaxation modes and related properties of the Gaussian chains have been studied in detail for branched polymers such as stars or dendrimers. 6,[18][19][20][21] Moreover effects of hydrodynamic interaction on such Gaussian chains were studied in terms of preaveraging approximation. 22,23) However the excluded Fig.…”