Self-assembly of complex structures in colloid-polymer mixtures Erdal C. O˘guz,1, 2, ∗ Aleksandar Mijailovi´c,1, 3 and Michael Schmiedeberg3 1Institut f¨urTheoretische Physik II, Heinrich-Heine Universit¨at, Universit¨atsstr. 1, 40225 D¨usseldorf,Germany 2School of Mechanical Engineering and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel 3Institut f¨urTheoretische Physik 1, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Staudtstr. 7, 91058 Erlangen, Germany (Dated: November 28, 2017) If particles interact according to isotropic pair potentials that favor multiple length scales, in prin- ciple a large variety of different complex structures can be achieved by self-assembly. We present, motivate, and discuss a conjecture for the occurrence of non-trivial (i.e., non-triangular) orderings based on newly-introduced enthalpy-like pair potentials, the capability of which we demonstrate for the specific example of colloid-polymer mixtures. Upon examining the phase behavior of two- dimensional colloid-polymer mixtures, which can also be realized in experiments, we observe that non-trivial structures only occur in the vicinity of selected densities where triangular ordering is suppressed by the pair potential. Close to these densities, a large number of different phases self- assemble that correspond to tilings containing triangular, rhombic, square, hexagonal, and pentag- onal tiles, and including some of the Archimedean tilings. We obtain the ground-state energies by minimizing the corresponding lattice sums with respect to particle positions in a unit cell as well as cell geometry and verify the occurrence of selected phases at finite temperatures by using Brownian Dynamics simulations. All reported phases should be accessible in experiments and, in addition, our work provides a manual on how to find the regions of non-trivial phases in parameter space for complex pair interactions in general. I. INTRODUCTION non-triangular ground-state structures with engineered pair interaction potentials [23{26]. More recently it has Self-assembly is the process by which the system even been shown that complex ground-state structures constituents form large organized functional units via can be stabilized by repulsive and convex pair interac- their mutual interactions without any external influ- tion potentials which rule out the necessity for single- or ence. Since the pioneering work by Whitesides et al. [1] multiple wells in this potential [27{32]. Unfortunately, on molecular systems, self-assembly has been studied in the interactions obtained by the aforementioned inverse great detail in a wide range of length scales ranging from construction methods are most often artificial and in atomic to macroscopic systems and throughout various the consequence hard to realize in experiments. scientific disciplines, including physics, chemistry, ma- A system that is well-studied in theory and simula- terials science, and biology [2{14]. tion and in addition can be realized and investigated ex- In monodisperse systems that do not possess more perimentally is a charge-stabilized colloid-polymer mix- than one characteristic length scale self-assembly usu- ture which intrinsically involves multiple length scales ally occurs in the same way as for hard disks or spheres, as the colloids effectively attract each other close to i.e., the particles form a triangular phase in two or a contact due to depletion interactions while being repul- fcc-crystal in three dimensions and at appropriate con- sive on larger length scales because of screened electro- ditions. Beyond such simple systems, a plethora of self- static repulsions. Experimental and theoretical stud- assembled complex structures can be achieved in mix- ies have been performed to understand the nature of tures of different particle types such as in metallic alloys colloid-polymer mixtures [33{44] some of which exhibit [15] or in binary colloidal suspensions [16{18]. Two fur- glassy states [36, 37], gels [38{42], and cluster formation ther scenarios pave the way for high structural complex- [41, 43, 44]. The phase behavior concerning the gas, liq- arXiv:1711.09431v1 [cond-mat.soft] 26 Nov 2017 ity in one-component systems by requiring i) an inter- uid, and trivial solid phase has been investigated in [45{ action or shape anisotropy and ii) a deliberate choice of 55]. Moreover, the influence of many-body interactions an isotropic pair interaction with multiple length scales on the phase behavior of such mixtures [56, 57] as well that affect the structure formation. Prototypical ex- as confinement effects [58{61] have been analyzed re- amples of the former are hard convex polyhedra pack- cently. However, to the best of our knowledge, detailed ings [19] and patchy colloids [20{22], whereas concern- structural analysis of the crystal phase were not in the ing the latter inverse statistical-mechanical approaches focus of previous studies, nor any non-trivial complex have been undertaken to investigate the self-assembly of ordering have been reported with the exception of local ground-state clusters [41, 44]. In this article, we determine the ground states that occur in a colloid-polymer mixture in two dimensions. ∗ [email protected] The ground-state orderings are calculated by minimiz- 2 ing the energy for structures with one to six particles per ternal energy unit cell. Aside from the trivial triangular phase, we ob- N serve square, rhombic, triangular, honeycomb, Kagome, 1 X U = v(r ) (2) and Archimedean tilings [62{65] as well as further or- 2 ij derings that correspond to tilings with hexagons and i;j i6=j even pentagons. In order to understand why non-trivial structures in at a given reduced density, depletion length, and energy monodisperse systems with isotropic interactions can amplitude. We use a direct lattice summation technique occur in general, we introduce an enthalpy-like pair po- to determine U, and thus to predict the corresponding tential and conjecture that it can be used in order to ground-state structures. In order to examine the sta- identify the parameters such that triangular structures bility of resulting structures at finite temperatures, we are suppressed. Our approach explains why and how extend our studies to T > 0 by means of Brownian even monotonic pair potentials can be used in order to Dynamics computer simulations. In the following, we self-assemble complex phases. provide details for both the lattice summation and the finite-temperature simulations used in this work. II. SYSTEM AND METHODS B. Lattice-sum calculations A. Colloid-polymer mixtures p At each given density ρd, depletion length κd, and energy amplitude V =W , we have performed lat- When immersed in a solvent of relatively small-sized 0 0 tice sum minimizations for a set of candidates of crys- non-adsorbing polymer coils, the larger colloidal par- talline lattices. As possible candidates, we consider two- ticles sense a short-ranged depletion attraction at suf- dimensional crystals with a periodicity in both the spa- ficiently high polymer concentrations. This attractive tial directions x and y whose primitive cell is a par- force arises due to an unbalanced osmotic pressure stem- allelogram containing n particles. This parallelogram ming from the depletion zone in the region between the is spanned by the two lattice vectors a = a(1; 0) and colloids. b = aγ(cos θ; sin θ), where γ denotes the aspect ratio The effective pair interaction potential between (γ = jbj=jaj = b=a), and θ is the angle between a and point-like colloids in the presence of the polymers is b. We consider candidates with primitive cells compris- given by a screened Coulomb repulsion corresponding ing up to 6 particles, i.e., n = 1; ··· ; 6, with no further to a Yukawa-like interaction, and on short lengths by restrictions. depletion attraction where we employ the AO-model At prescribed parameters, the total potential energy [66, 67]. Therefore, the pair interaction potential is per particle u = U=N (cf. Eq. 2) is minimized with re- spect to the particle coordinates of the basis, and the 8 " 3 # exp(−κrij) 3rij rij cell geometry. The latter includes a minimization pro- >V − W 1 − + > 0 0 3 cess of γ and θ. To be specific, we implement the Nelder- <> κrij 2d 2d v(r ) = Mead method (also known as downhill simplex method ij if rij ≤ d; > or amoeba method) to find the minimum of the energy > exp(−κrij) :>V0 if rij > d; functions without calculating their derivatives [68]. As κrij this technique is a heuristic approach, and thus it may (1) not always converge to the global minimum, we use at where rij is the separation distance between colloids i least 200 and at most 1000 different start configurations and j, κ denotes the inverse screening length, and d the depending on the complexity of those functions. depletion length dictating the range of the attractive in- teraction and corresponding to the typical diameter of the polymers. The energy amplitude of the pure elec- C. Enthalpy-like pair potential trostatic Yukawa interaction is given by V0, whereas the strength of the depletion potential is set by W0. The crystalline phase diagrams can therefore be determined We introduce a new pair potential that corresponds to in three-dimensional space spanned by the reduced en- an enthalpy-like quantity h(rij). This enthalpy-like pair p potential should be given such that the total enthalpy ergy amplitude V0=W0, the reduced density ρd, and the reduced depletion length κd corresponding to the is ratio of depletion length divided by screening length. N 1 X We primarily explore the ground state of our model H = h(rij) = U + pA; (3) colloid-polymer mixtures by determining the corre- 2 p i;j;i6=j sponding phase diagrams in the ( ρd, κd)-plane at fixed V0=W0. At zero-temperature, the optimal structures where p is the macroscopic pressure and A the total with N particles are those that minimize the total in- area of the system (in three dimensions the volume has 3 to be used instead).