Untitled

Abstract

In this study a general bead-spring model is used for predicting some rheological properties of a cubic bead-spring structure of arbitrary size immersed in a Newtonian solvent. The topology of this bead-spring structure is based upon the well-known cubic crystals (SC, BCC or FCC) and it consists of equal Hookean springs and beads with equal friction coefficients, while hydrodynamic interaction is not included. An appropriate combination of the equations of motion, the expression for the stress tensor and the equation of continuity leads to an explicit constitutive equation with three sets of relaxation times belonging to the three types of bead-spring cubes (SC, BCC or FCC). For small-amplitude oscillatory shear flow it is found that the three relaxation spectra, which are significantly different, result in dynamic moduli which differ mainly in one aspect: the characteristic SC, BCC and FCC time scales are different. The BCC and FCC time scales can be obtained by multiplication of the SC time scale by the ratios M sc /M bcc and M sc /M fcc respectively, where M sc , M bcc and M fcc denote the number of springs in the three types of cubic bead-spring structures.

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

“…Let¯ g and g the sets of nodes generated at iteration g or earlier, respectively. Then S tot (g) can be recast as S tot (g) = i∈¯ g , j∈ g d ij (g)+ i∈¯ g , j∈¯ g d ij (g)+ i∈ g , j∈ g d ij (g), (9) It is evident that the third term on the right-hand side (rhs) of Eq. (9) is exactly S tot (g − 1), i.e.,…”

“…Thus far, the Laplacian eigenvalues for some classes of graphs have been determined exactly, including regular hypercubic lattices, 1,9 dual Sierpinski gaskets, 10,11 Vicsek fractals, 12,13 dendrimer also known as Cayley tree, 14 and Husimi cacti. 15,16 Recent empirical research indicated that some real-life networks (e.g., power grid) display small-world behavior.…”

A colloidal crystal modeled by bead–spring cubes

“…In a previous paper 80 we have already shown analytically that the obtained frequency dependency of the dynamic moduli GЈ() and GЉ() belonging to Hookean bead-spring cubes in a Newtonian fluid does not really depend upon the specific cubic topology, i.e., the obtained linear viscoelastic results belonging to the SC, BCC, and FCC cases are essentially the same. We therefore expect that the SC results obtained in this paper shall provide a good qualitative description of the linear viscoelastic behavior of a real colloidal crystal consisting of crystallites of the BCC or FCC type.…”

“…͑57͒ and ͑58͒, respectively. By using a standard method 80,85 we obtained for each region asymptotic expressions for the reduced moduli GЈ and GЉ given by Eqs. ͑68͒ and ͑69͒, respectively.…”

“…To keep the analysis as simple as possible we have decided to model the crystallites in the polycrystalline sample by bead-spring cubes of the SC type, which is in contrast with the experimental observation that the crystallites in a colloidal crystal are of the BCC or FCC type. The results presented in a previous paper 80 suggest that it allowed us to make the assumption that the frequency dependency of the dynamic moduli GЈ() and GЉ() belonging to a colloidal crystal does not really depend upon the specific cubic topology. This assumption appears to be not a bad one; experimental observations on a colloidal crystal can be very well described by the SC results obtained in this paper.…”