Rouse model fractals

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

Rouse model of randomly branched polymers exhibits fractal dynamics the relaxation time r(g) of a polymer section of g monomers has the same dependence on the number of monomers g as the whole chain [Eq. (8.144)] ... 

Calculate the stress relaxation modulus G(t), valid for all times longer than the relaxation time of a monomer, for a monodisperse three-dimensional melt of unentangled flexible fractal polymers that have fractal dimension V <1. Assume complete hydrodynamic screening. Hint Keep the fractal dimension general and make sure your result coincides with the Rouse model for V — 2. 

Here, the discussion of the viscoelastic exponent n in relation to the assumed gelation model (e.g. electrical analogy percolation or Rouse model), as well as the fractal dimension dfof the critical gel (Muthukumar and Winter 1986 Muthukumar 1989) will be ignored. The reader is referred e.g. to (Adam and Lairez 1996 Martin and Adolf 1991). It has been also shown, that stoichiometry, molecular weight and concentration have an impact on the critical gel properties (Winter and Mours... 

Our review starts with the general formulation of the GGS model in Sect. 2. In the framework of the GGS approach many dynamical quantities of experimental relevance can be expressed through analytical equations. Because of this, in Sect. 3 we outline the derivation of such expressions for the dynamical shear modulus and the viscosity, for the relaxation modulus, for the dielectric susceptibility, and for the displacement of monomers under external forces. Section 4 provides a historical retrospective of the classical Rouse model, while emphasizing its successes and limitations. The next three sections are devoted to the dynamical properties of several classes of polymer networks, ranging from regular and fractal networks to network models which take into account structural heterogeneities encountered in real systems. Sections 8 and 9 discuss dendrimers, dendritic polymers, and hyperbranched polymers. 

In the Rouse model, for which the hydrodynamic interactions are totally screened, the relaxation time, >1, of a cluster of mass, M is proportional to M (20,22). The surroundings of a molecule of mass Mare based on smaller fractals, which relax at times, and of larger clusters considered as... 

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