We report theoretical and numerical evaluations of the phase diagram for patchy colloidal particles of new generation. We show that the reduction of the number of bonded nearest neighbors offers the possibility of generating liquid states (i.e., states with temperature $T$ lower than the liquid-gas critical temperature) with a vanishing occupied packing fraction ($\varphi$), a case which can not be realized with spherically interacting particles. Theoretical results suggest that such reduction is accompanied by an increase of the region of stability of the liquid phase in the ($T\mathrm{\text{−}}\varphi$) plane, possibly favoring the establishment of homogeneous disordered materials at small $\varphi$, i.e., stable equilibrium gels.

Recently, an increasing experimental effort has been devoted to the synthesis of complex colloidal particles with chemically or physically patterned surfaces and possible specific shapes that are far from spherical. These new colloidal particles with anisotropic interactions are commonly named patchy particles. In this Perspective article, we focus on patchy systems characterized by spherical neutral particles with patchy surfaces. We summarize most of the patchy particle models that have been developed so far and describe how their basic features are connected to the physical systems they are meant to investigate. Patchy models consider particles as hard or soft spheres carrying a finite and small number of attractive sites arranged in precise geometries on the particle's surface. The anisotropy of the interaction and the limited valence in bonding are the salient features determining the collective behavior of such systems. By tuning the number, the interaction parameters and the local arrangements of the patches, it is possible to investigate a wide range of physical phenomena, from different self-assembly processes of proteins, polymers and patchy colloids to the dynamical arrest of gel-like structures. We also draw attention to charged patchy systems: colloidal patchy particles as well as proteins are likely charged, hence the description of the presence of heterogeneously distributed charges on the particle surface is a promising perspective for future investigations.

The authors numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory. This comparison offers a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameters and liquid state correlation functions. The Wertheim theory has not been previously subjected to stringent tests against simulation data for ordering across the polymerization transition. The authors numerically determine many of the thermodynamic properties governing this basic form of self-assembly (energy per particle, order parameter or average fraction of particles in the associated state, average chain length, chain length distribution, average end-to-end distance of the chains, and the static structure factor) and find that predictions of the Wertheim theory accord remarkably well with the simulation results.

We report theoretical and numerical evaluations of the phase diagram for a model of patchy particles. Specifically, we study hard spheres whose surface is decorated by a small number f of identical sites (“sticky spots”) interacting via a short-ranged square-well attraction. We theoretically evaluate, solving the Wertheim theory, the location of the critical point and the gas-liquid coexistence line for several values of f and compare them to the results of Gibbs and grand canonical Monte Carlo simulations. We study both ordered and disordered arrangements of the sites on the hard-sphere surface and confirm that patchiness has a strong effect on the phase diagram: the gas-liquid coexistence region in the temperature-density plane is significantly reduced as f decreases. We also theoretically evaluate the locus of specific heat maxima and the percolation line.

We have investigated the self-assembly scenario of patchy colloidal particles in a two-dimensional system. The energetically most favourable ordered particle arrangements have been identified via an optimization tool that is based on genetic algorithms. Assuming different simple models for patchy colloidal particles, which include binary mixtures as well as attraction and repulsion between the patches, we could identify a broad variety of highly non-trivial ordered structures. The strategies of the systems to self-assemble become evident from a systematic variation of the pressure: (i) saturation of patch bonds at low pressure and close packing at high pressure and (ii) for intermediate pressure values, the strategy is governed by a trade-off between these two energetic aspects. The present study is yet another demonstration of the efficiency and the high reliability of genetic algorithms as versatile optimization tools.

Artistic representation of limited valance units consisting of a soft core (in blue) and a small number of flexible bonding patches (in orange).