Optimizing the formation of colloidal compounds with components of different shapes

Inverse design of triblock Janus spheres for self-assembly of complex structures in the crystallization slot via digital alchemy

The digital alchemy framework is an extended ensemble simulation technique that incorporates particle attributes as thermodynamic variables, enabling the inverse design of colloidal particles for desired behavior. Here, we extend the digital alchemy framework for the inverse design of patchy spheres that self-assemble into target crystal structures. To constrain the potentials to non-trivial solutions, we conduct digital alchemy simulations with constant second virial coefficient. We optimize the size, range, and strength of patchy interactions in model triblock Janus spheres to self-assemble the 2D kagome and snub square lattices and the 3D pyrochlore lattice, and demonstrate self-assembly of all three target structures with the designed models. The particles designed for the kagome and snub square lattices assemble into high quality clusters of their target structures, while competition from similar polymorphs lower the yield of the pyrochlore assemblies. We find that the alchemically designed potentials do not always match physical intuition, illustrating the ability of the method to find nontrivial solutions to the optimization problem. We identify a window of second virial coefficients that result in self-assembly of the target structures, analogous to the crystallization slot in protein crystallization.

Effect of particle anisotropy on the thermodynamics and kinetics of ordering transitions in hard faceted particles

Monte Carlo simulations were used to study the influence of particle aspect ratio on the kinetics and phase behavior of hard gyrobifastigia (GBF). First, the formation of a highly anisotropic nucleus shape in the isotropic-to-crystal transition in regular GBF is explained by the differences in interfacial free energies of various crystal planes and the nucleus geometry predicted by the Wulff construction. GBF-related shapes with various aspect ratios were then studied, mapping their equations of state, determining phase coexistence conditions via interfacial pinning, and computing nucleation free-energy barriers via umbrella sampling using suitable order parameters. Our simulations reveal a reduction of the kinetic barrier for isotropic-crystal transition upon an increase in aspect ratio, and that for highly oblate and prolate aspect ratios, an intermediate nematic phase is stabilized. Our results and observations also support two conjectures for the formation of the crystalline state from the isotropic phase: that low phase free energies at the ordering phase transition correlate with low transition barriers and that the emergence of a mesophase provides a steppingstone that expedites crystallization.

On the calculation of free energies over Hamiltonian and order parameters via perturbation and thermodynamic integration

In this work, complementary formulas are presented to compute free-energy differences via perturbation (FEP) methods and thermodynamic integration (TI). These formulas are derived by selecting only the most statistically significant data from the information extractable from the simulated points involved. On the one hand, commonly used FEP techniques based on overlap sampling leverage the full information contained in the overlapping macrostate probability distributions. On the other hand, conventional TI methods only use information on the first moments of those distributions, as embodied by the first derivatives of the free energy. Since the accuracy of simulation data degrades considerably for high-order moments (for FEP) or free-energy derivatives (for TI), it is proposed to consider, consistently for both methods, data up to second-order moments/derivatives. This provides a compromise between the limiting strategies embodied by common FEP and TI and leads to simple, optimized expressions to evaluate free-energy differences. The proposed formulas are validated with an analytically solvable harmonic Hamiltonian (for assessing systematic errors), an atomistic system (for computing the potential of mean force with coordinate-dependent order parameters), and a binary-component coarse-grained model (for tracing a solid-liquid phase diagram in an ensemble sampled through alchemical transformations). It is shown that the proposed FEP and TI formulas are straightforward to implement, perform similarly well, and allow robust estimation of free-energy differences even when the spacing of successive points does not guarantee them to have proper overlapping in phase space.

Inverse design of equilibrium cluster fluids applied to a physically informed model

Inverse design strategies have proven highly useful for the discovery of interaction potentials that prompt self-assembly of a variety of interesting structures. However, often the optimized particle interactions do not have a direct relationship to experimental systems. In this work, we show that Relative Entropy minimization is able to discover physically meaningful parameter sets for a model interaction built from depletion attraction and electrostatic repulsion that yield self-assembly of size-specific clusters. We then explore the sensitivity of the optimized interaction potentials with respect to deviations in the underlying physical quantities, showing that clustering behavior is largely preserved even as the optimized parameters are perturbed.

Shape-driven entropic self-assembly of an open, reconfigurable, binary host-guest colloidal crystal

Entropically driven self-assembly of hard anisotropic particles, where particle shape gives rise to emergent valencies, provides a useful perspective for the design of nanoparticle and colloidal systems. Hard particles self-assemble into a rich variety of crystal structures, ranging in complexity from simple close-packed structures to structures with 432 particles in the unit cell. Entropic crystallization of open structures, however, is missing from this landscape. Here, we report the self-assembly of a two-dimensional binary mixture of hard particles into an open host-guest structure, where nonconvex, triangular host particles form a honeycomb lattice that encapsulates smaller guest particles. Notably, this open structure forms in the absence of enthalpic interactions by effectively splitting the structure into low- and high-entropy sublattices. This is the first such structure to be reported in a two-dimensional athermal system. We discuss the observed compartmentalization of entropy in this system, and show that the effect of the size of the guest particle on the stability of the structure gives rise to a reentrant phase behavior. This reentrance suggests the possibility for a reconfigurable colloidal material, and we provide a proof-of-concept by showing the assembly behavior while changing the size of the guest particles in situ. Our findings provide a strategy for designing open colloidal crystals, as well as binary systems that exhibit co-crystallization, which have been elusive thus far.

Inverse Design of Self-Assembling Diamond Photonic Lattices from Anisotropic Colloidal Clusters

Colloidal nanoparticles with anisotropic interactions are promising building blocks for the fabrication of complex functional materials. A challenge in the self-assembly of colloidal particles is the rational design of geometry and chemistry to program the formation of a desired target structure. We report an inverse design procedure integrating Langevin dynamics simulations and evolutionary algorithms to engineer anisotropic patchy colloidal clusters to spontaneously assemble into a cubic diamond lattice possessing a complete photonic band gap. The combination of a tetrahedral cluster geometry and optimized placement of a single type of anisotropic interaction patch results in a colloidal building block predicted to assemble a cubic diamond lattice with more than 82% yield. This design represents an experimentally viable colloidal building block capable of high-fidelity assembly of a cubic diamond lattice.

Effect of shape on liquid-vapor coexistence and surface properties of parallelepiped molecules

Liquid-vapor coexistence is calculated via molecular dynamics for a variety of parallelepiped shaped molecules. Models are constructed as an array of tangential hard spheres interacting with an attractive square-well potential. Each shape is formed by varying the number of spheres in their three sides. The initial density of the system is chosen close to the critical density of a SW fluid to obtain an equilibrated liquid-vapor coexistence curve by the process of spinodal decomposition. A pattern that relates the geometry of the molecular models and the existence or non-existence of a liquid-vapor orthobaric curve is shown.

Inverse design of self-assembling colloidal crystals with omnidirectional photonic bandgaps

Open colloidal lattices possessing omnidirectional photonic bandgaps in the visible or near-visible regime are attractive optical materials the realization of which has remained elusive. We report the use of an inverse design strategy termed landscape engineering that rationally sculpts the free energy self-assembly landscape using evolutionary algorithms to discover anisotropic patchy colloids capable of spontaneously assembling pyrochlore and cubic diamond lattices possessing complete photonic bandgaps. We validate the designs in computer simulations to demonstrate the defect-free formation of these lattices via a two-stage hierarchical assembly mechanism. Our approach demonstrates a principled strategy for the inverse design of self-assembling colloids for the bottom-up fabrication of desired crystal lattices.