Percolation clusters dynamical scaling

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

It is obvious that the dynamical scaling hypothesis is not valid through the dynamical percolation-to-cluster transition. This is because the pattern changes from the bicontinuous periodic structure to the cluster of spheres. In fact the scaled structure factor was found to substantially broaden before and after the transition. 

Vojta and Sknepnek also performed analogous calculations for the quantum percolation transition at p = pp, J < 0.16/ and the multicritical point 2itp=pp,J = 0.16/. A summary of the critical exponents for all three transitions is found in Table 3. The results for the percolation transition are in reasonable agreement with theoretical predictions of a recent general scaling theory of percolation quantum phase transitions P/v = 5/48, y/v = 59/16 and a dynamical exponent oi z = Df = (coinciding with the fractal dimension of the critical percolation cluster). 

P — where is the dynamical scaling exponent. In order to introduce the idea of polymeric fractals we have to extend the considerations of ordinary fractals, i.e. those created on a lattice. It has been shown that at least three fractal dimensions are necessary to characterize crudely a fractal object. Indeed in random fractals like percolation clusters or diffusion-limited... 

In a recent paper [9], the present author has considered a model for the dynamics of percolation clusters with the aim of harmonizing the two independent results obtained in references [4] and [5] for the connection-disconnection probability Ngv(s) of a finite cluster of size s. These two results are in mutual agreement in d=2 but not in d > 2 and the only way to obtain coherent results is to consider that the scaling of the anti-red bonds (or reconnecting bonds) is different from that of the red bonds (or disconnecting bonds). The general expression, supposed valid in any case is,... 

Non-mean field corrections can be treated by renormalization group theory,which is not discussed here. In order to leave the tree approximation we turn to percolation description and model the microgels as finite clusters. The percolation theory itself is not essential for this description, but only the fractal character of the clusters on their scale of extension. Polymeric fractals have been discussed already in Section 8.2.6 and we use the properties here as well. Indeed most of the results may be applied here. In Ref. 123 the dynamics of the sol phase is discussed extensively, but we do not want to go in these details here. 

Dynamic simulation approaches to model kinetic percolation are difficult to implement because of the inherent complexity of the problem, which requires intensive computation. As with any kinetic modd, the duration of the simulation must be commensruate with the critical timescales of the experiments. An early study to investigate the effeas of interactions on the percolation threshold was conducted by Bug et al. Here, a continuum Monte Carlo algorithm was used to modd a small system of 500 spherical particles undergoing Brownian motion. More recently, advanced simulation approaches such as Dissipative Particle Dynamics (DPD) have been applied to study kinetic percolation in composite sys-tems. " DPD is an off-lattice simulation technique similar to molecular dynamics, but applied to the supramolecular scale. Here, the larger-scale dynamics of a system are studied by monitoring the motion of particle clusters in response to pairwise, dissipative, and random forces. ... 

---