Colloidal systems with competing interactions: from an arrested repulsive cluster phase to a gel Juan Carlos Fernandez Toledano, 1, 2 Francesco Sciortino, 1 and Emanuela Zaccarelli 1 1 Dipartimento di Fisica and INFM-CRS-SOFT, Universit` a di Roma La Sapienza, P.le A. Moro 2, 00185 Roma, Italy 2 Grupo de F´ ısica de Fluidos y Biocoloides, Departamento de F´ ısica Aplicada, Facultad de Ciencias, Campus Fuentenueva S/N, 18071 Granada, Spain We report an extensive numerical study of a charged colloidal system with competing short-range depletion attraction and long-range electrostatic repulsion. By analizing the cluster properties, we identify two distinct regions in the phase diagram: a state composed of stable finite-size clusters, whose relative interactions are dominated by long-range repulsion, and a percolating network. Both states are found to dynamically arrest at low temperatures, providing evidence of the existence of two distinct non-ergodic states in these systems: a Wigner glass of clusters and a gel. I. INTRODUCTION The origin of low-density non-ergodic states in colloidal systems is a matter of continuous debate and ongoing research[1, 2, 3, 4, 5]. Several different mechanisms may concur in the formation of these arrested states, depending on the relative ratio between the thermal and binding energy and on the shape and symmetry of the interaction potential. When particles interact via excluded volume interactions complemented by a spherically symmetric, at- tractive potential, it has been shown that the formation of a gel structure takes place concurrently with a spinodal decomposition process. Gelation results from an arrested phase separation[6]. A different scenario occurs when col- loidal particles have a residual electrostatic charge which builds up an additional long-range repulsion in the effective colloid-colloid interaction. This term is often modeled as a Yukawa potential with Debye screening length ξ to take into account the presence of the solvent and counterions[7]. In apolar solutions or under low salt conditions, when particles are sub-micron sized, ξ can become comparable to the particle dimension. This long-range repulsive term can coexist with a short-range attraction (which can be induced for example via depletion interactions), generat- ing a competition between aggregation driven by the attractive part of the potential and the stabilizing role of the repulsion, which may ultimately suppress the macroscopic phase separation. Indeed, it has been shown that the addition of a long-range repulsion of moderate strength can shift to larger attraction strengths (or lower tempera- tures) the phase separation[8, 9], eventually inhibiting it[10, 11, 12, 13]. In this case a microphase separation into clusters[14, 15, 16, 17, 18] of a preferred cluster size and shape takes place, depending on repulsion parameters[19]. When repulsion is moderately short-ranged, i.e. ξ/σ  0.5 with σ being the diameter of the colloidal particle, it was observed both in experiments and in simulations that elongated clusters are formed at low enough temperature T [20, 21]. The repulsion between such clusters is relatively weak so that they tend to form at low T quasi-ordered columnar structures[22, 23]. At large enough packing fraction, the clusters are found to merge into a percolating network [20, 21]. This network of clusters exists at low enough T and undergoes dynamical arrest, so that a gel state can be properly identified[4]. For cases where the repulsion term is considerably longer-ranged, i.e. ξ/σ  1, simulations at low enough colloidal densities[24] have reported the presence of a Wigner glass of clusters. This corresponds to a disordered state of polydisperse clusters (due to finite T ) which do not percolate and are actually arrested due to the long-range repulsion, in analogy with the Wigner glass reported by Chaikin and coworkers[25, 26] for charged colloidal particles under very dilute conditions, stabilized by the Coulomb repulsion. Recently, a comparison between theory and simulations[27] of Yukawa particles has shown that the ideal Mode Coupling Theory (MCT) provides a quite accurate description of the formation of a particle Wigner glass. The MCT predictions for Yukawa particles have also been exploited for interpreting arrest into a Wigner glasses of clusters, in systems with competing interactions[24, 28]. Indeed, once clusters are assumed spherical and monodisperse (in size), the effective cluster-cluster interactions can be modeled in terms of a Yukawa potential, with the same screening length as the one acting between single particles but with a renormalized amplitude[24]. These earlier works call for additional investigations, in order to further question the existence and the stability of a Wigner glass of clusters, as well as a deeper understanding of cluster-cluster interactions. To this end, it is also relevant to mention a recent simulation study [29] where clusters were observed to arrest, at not-too-low density, by percolation rather than by repulsion. However, differences in the simulation protocol of this work are present with respect to that used in [24], in particular history of quench and quench rate, as well as a shorter cut-off distance for the long-range repulsion potential. Hence, a more comprehensive study of these models in a wide region of packing fraction φ and temperature T , fully accounting for the long range nature of the repulsive interactions, is needed. arXiv:0903.2929v1 [cond-mat.soft] 17 Mar 2009