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Experiments of in vitro formation of blood vessels show that cells randomly spread on a gel matrix autonomously organize to form a connected vascular network. We propose a simple model which reproduces many features of the biological system. We show that both the model and the real system exhibit a fractal behavior at small scales, due to the process of migration and dynamical aggregation, followed at large scale by a random percolation behavior due to the coalescence of aggregates. The results are in good agreement with the analysis performed on the experimental data.

In colloidal suspensions, at low volume fraction and temperature, dynamical arrest occurs via the growth of elongated structures, that aggregate to form a connected network at gelation. Here we show that, in the region of parameter space where gelation occurs, the stable thermodynamical phase is a crystalline columnar one. Near and above the gelation threshold, the disordered spanning network slowly evolves and finally orders to form the crystalline structure. At higher volume fractions the stable phase is a lamellar one, that seems to have a still longer ordering time.PACS numbers: 82.70. Dd, 64.60.Ak, 82.70.Gg In colloidal suspensions solid (or liquid) mesoscopic particles are dispersed in another substance. These systems, like blood, proteins in water, milk, black ink or paints, are important in our everyday lives, in biology and industry [1,2]. It is crucial, for example, to control the process of aggregation in paint and paper industries [3], or to favour the protein crystallization in the production of pharmaceuticals and photonic crystals [4,5].A practical and exciting feature of colloidal suspensions is that the interaction energy between particles can be well controlled [6][7][8]. In fact particles can be coated and stabilized leading to a hard sphere behaviour, and an attractive depletion interaction can be brought out by adding some non-adsorbing polymers. The range and strength of the potential are controlled respectively by the size and concentration of the polymer [8,9]. Recent experimental works highlighted the presence of a net charge on colloidal particles [7,10] giving rise to a long range electrostatic repulsion in addition to the depletion attraction.The competition between attractive and repulsive interactions produces a rich phenomenology and a complex behavior as far as structural and dynamical properties are concerned. For particular choices of the interaction parameters, the aggregation of particles is favoured but the liquid-gas phase transition can be avoided and the cluster size can be stabilized at an optimum value [11]. Experimentally, such a cluster phase made of small equilibrium monodisperse clusters is observed using confocal microscopy at low volume fraction and low temperature (or high attraction strength) [7,10,12]. Increasing the volume fraction, the system is transformed from an ergodic cluster liquid into a nonergodic gel [10,12], where structural arrest occurs. Using molecular dynamic simulations, we showed that such structural arrest is crucially related to the formation of a long living spanning cluster, providing evidence for the percolation nature of the colloidal gel transition at low volume fraction and low temperature [13,14]. This scenario was confirmed by recent experiments [10] and molecular dynamics simulations [15], where it was shown that increasing the volume fraction clusters coalesce into elongated structures eventually forming a disordered spanning network. A realistic framework for the modelization of these systems is represented by DLVO interaction p...

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