Distribution functions for Gaussian molecules. II. Reduction of the Kirchhoff matrix for large molecules

Abstract

The eigenvalues of the Kirchhoff matrix for a molecule consisting of m chains and μ multifunctional junctions can be calculated from a reduced matrix of dimension μ×μ. Many of the eigenvalues for complicated molecules can be obtained exactly, and the remainder are accessible by numerical methods. We find small eigenvalues for trifunctional networks which can be associated with very slowly relaxing oscillations in gels. In addition, networks have μ high frequency modes, which are located in unique regions of the spectrum.

Hydrodynamic effects on the extension of stars and dendrimers in external fields

“…We follow the usual development of the theory, [5,6,8,[12][13][14][25][26][27] while paying particular attention to the extension of the GGS in external fields. As indicated above, recent optical and mechanical developments allow one to realise micromanipulations of such GGS in solution.…”

“…Gaussian models are extremely important here, because they allow one to treat the arising problems in a linear algebraic framework, which permits a relatively convenient treatment of complex topologies, such as branched geometries or networks. Extending the perennial Rouse model [1][2][3][4][5][6][7][8][9][10][11][12][13][14] by including hydrodynamic interactions in a very general manner is fraught with difficulties, as amply demonstrated in the literature, [9][10][11][15][16][17][18][19] where it was shown that in a range of parameters non-physical behavior, such as negative diffusion coefficients and related instabilities can occur. On the other hand, typical generalized Rouse-Zimm-models, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][20][21][22][23][24][25][26][27][28][29] when used in a reasonable ...…”

Langevin dynamics of a polymer with internal distance constraints

“…The first step in deriving a general expression for R(ω) for hard crosslinks is to find a way to perform the limit z → 0 in equation (12). This is an interesting problem in its own right which so far could only been handled by introducing a finite cutoff at z = 1 and successive crude variational estimates.…”

“…However, there is still a huge calculational advantage with (25). For a polymer network we have in general M ≪ N. Equation (25) requires "only" the inverse of an M ×M matrix [7] and not of the complete N ×N connectivity matrix as is commonly believed in the polymer literature [12,13]. An analytic approach to the network problem would be to perform the quenched average of the resolvent over the crosslink positions C. The latter problem is a key problem in current network research and has not been analytically solved even for the static problem.…”

Anomalous Diffusion and Relaxation

“…In this section we focus on GGS based on SWNs, the so-called SWRN; recently such SWRNs were used to model disordered crosslinked polymers [29] by employing the GGSextension of the Rouse model [26][27][28][47][48][49]. Such SWRNs are of great theoretical interest, since they interpolate between linear Rouse chains and disordered polymer networks.…”