This article reviews recent inverse statistical-mechanical methodologies that we have devised to optimize interaction potentials in soft matter systems that correspond to stable "target" structures. We are interested in finding the interaction potential, not necessarily pairwise additive or spherically symmetric, that stabilizes a targeted many-body system by generally incorporating complete configurational information. Unlike previous work, our primary interest is in the possible many-body structures that may be generated, some of which may include interesting but known structures, while others may represent entirely new structural motifs. Soft matter systems, such as colloids and polymers, offer a versatile means of realizing the optimized interactions. It is shown that these inverse approaches hold great promise for controlling self-assembly to a degree that surpasses the less-than-optimal path that nature has provided. Indeed, we envision being able to "tailor" potentials that produce varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. The notion of tailoring potentials that correspond to targeted structures is motivated by the rich fundamental statistical-mechanical issues and questions offered by this fascinating inverse problem as well as our recent ability to identify structures that have optimal bulk properties or desirable performance characteristics. Recent results have already led to a deeper basic understanding of the mathematical relationship between the collective structural behavior of many-body systems and their interactions, as well as optimized potentials that enable self-assembly of ordered and disordered particle configurations with novel structural and bulk properties.
We present a method that systematically simplifies isotropic interactions designed for targeted self-assembly. The uncertainty principle is used to show that an optimal simplification is achieved by a combination of heat kernel smoothing and Gaussian screening. We use this method to design isotropic interactions for self-assembly of complex lattices and of materials with functional properties. The interactions we derive are significantly simpler than those previously published, and it is realistic to discuss explicit experimental implementation of the designed self-assembling components.
We devise an inverse statistical-mechanical methodology to find optimized interaction potentials that lead spontaneously to a target many-particle configuration. Target structures can possess varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. For illustration purposes, our computational technique is applied to yield an optimized isotropic (non-directional) pair potential that spontaneously yields the three-coordinated honeycomb lattice as the ground state structure in two dimension…
We formulate statistical-mechanical inverse methods in order to determine optimized interparticle interactions that spontaneously produce target many-particle configurations. Motivated by advances that give experimentalists greater and greater control over colloidal interaction potentials, we propose and discuss two computational algorithms that search for optimal potentials for self-assembly of a given target configuration. The first optimizes the potential near the ground state and the second near the melting point. We begin by applying these techniques to assembling open structures in two d…
Isotropic pairwise interactions that promote the self assembly of complex particle morphologies have been discovered by inverse design strategies derived from the molecular coarse-graining literature. While such approaches provide an avenue to reproduce structural correlations, thermodynamic quantities such as the pressure have typically not been considered in self-assembly applications. In this work, we demonstrate that relative entropy optimization can be used to discover potentials that self-assemble into targeted cluster morphologies with a prescribed pressure when the iterative simulation…
Self-assembly is the autonomous organization of components into patterns or structures: an essential ingredient of biology and a desired route to complex organization. At equilibrium, the structure is encoded through specific interactions, at an unfavorable entropic cost for the system. An alternative approach, widely used by Nature, uses energy input to bypass the entropy bottleneck and develop features otherwise impossible at equilibrium. Dissipative building blocks that inject energy locally were made available by recent advance in colloidal science but have not been used to control self-as…
Inverse methods of statistical mechanics have facilitated the discovery of pair potentials that stabilize a wide variety of targeted lattices at zero temperature. However, such methods are complicated by the need to compare, within the optimization framework, the energy of the desired lattice to all possibly relevant competing structures, which are not generally known in advance. Furthermore, ground-state stability does not guarantee that the target will readily assemble from the fluid upon cooling from higher temperature. Here, we introduce a molecular dynamics simulation-based, optimization …
The aggregation of attractive colloids has been extensively studied from both theoretical and experimental perspectives as the fraction of solid particles is changed, and the range, type and strength of attractive or repulsive forces between particles varies. The resulting gels consisting of disordered assemblies of attractive colloidal particles, have also been investigated with regards to percolation, phase separation, and the mechanical characteristics of the resulting fractal networks. Despite tremendous progress in our understanding of the gelation process, and the exploration of differen…
Using large deviation theory and principles of stochastic optimal control, we show that rare molecular dynamics trajectories conditioned on assembling a specific target structure encode a set of interactions and external forces that lead to enhanced stability of that structure. Such a relationship can be formulated into a variational principle, for which we have developed an associated optimization algorithm and have used it to determine optimal forces for targeted self-assembly within nonequilibrium steady-states. We illustrate this perspective on inverse design in a model of colloidal cluste…
Steering Active-Colloid Assembly by Biasing Dissipation
University of Science and Technology of China（中国科学技术大学）；Rice University（莱斯大学）；Hefei National Research Center for Physical Sciences at the Microscale（合肥微尺度物质科学国家研究中心）
Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures (Phys. Rev. Lett. 95, 228301 (2005), Phys. Rev. E 73, 011406 (2006)), we present an isotropic pair potential $V(r)$ for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic (SC) lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems f…
In this work we consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts. Specifically, we model polymer self-assembly using Self-Consistent Field Theory and derive in a non-parametric setting the sensitivity of the misfit between desired and actual morphologies to arbitrary perturbations in the guiding pattern shape. The obtained sensitivities are used for optimization of the confining pattern shapes such that the misfit between desired and actual morphologies is minimized. The efficiency and robustness of the pr…
Here, we describe a method for the enhanced self-assembly of triblock Janus colloids targeted to form a kagome lattice. Using computer simulations, we demonstrate that the formation of this elusive structure can be significantly improved by self-propelling or activating the colloids along the axis connecting their hydrophobic hemispheres. The process by which metastable aggregates are destabilized and transformed into the favored kagome lattice is quite general, and we argue this active approach provides a systematic pathway to improving the self-assembly of a large number of colloidal structu…
A new theoretical approach is described for the inverse self-assembly problem, i.e., the reconstruction of the interparticle interaction from a given structure. This theory is based on the variational principle for the functional that is constructed from a free energy functional in combination with Percus's approach [J. Percus, Phys. Rev. Lett. vol.8, 462 (1962)]. In this theory, the interparticle interaction potential for the given structure is obtained as the function that maximizes the functional. As test cases, the interparticle potentials for two-dimensional crystals, such as square, hone…
While colloids are promising building blocks for the self-assembly of materials with novel microstructures, their numerous tunable parameters inhibit brute force searching for appropriate parameter combinations that yield self-assembly of a desired structure. Instead, inverse approaches that invoke a systematic optimization framework can effectively navigate this design space. In this proceeding, we apply one such inverse technique, Relative Entropy Minimization, to discover isotropic pairwise interaction potentials that prompt self-assembly of clusters in silico. The functional form of the pa…
In this study, a variational method for the inverse problem of self-assembly, i.e., a reconstruction of the interparticle interaction potential of a given structure, is applied to three-dimensional crystals. According to the method, the interaction potential is derived as a function that maximizes the free-energy functional of the one- and two-particle density distribution functions. The interaction potentials of the target crystals, including those with face-centered cubic (fcc), body-centered cubic (bcc), and simple hexagonal (shx) lattices, are obtained by numerical maximization of the func…
We study the inverse problem of tuning interaction parameters between charged colloidal particles interacting with a hard-core repulsive Yukawa potential, so that they assemble into specified crystal structures. Here, we target the body-centered-cubic (bcc) structure which is only stable in a small region in the phase diagram of charged colloids and is, therefore, challenging to find. In order to achieve this goal, we use the statistical fluctuations in the bond orientational order parameters to tune the interaction parameters for the bcc structure, while initializing the system in the fluid p…
The synthesis of complex materials through the self-assembly of particles at the nanoscale provides opportunities for the realization of novel material properties. However, the inverse design process to create experimentally feasible interparticle interaction strategies is uniquely challenging. Standard methods for the optimization of isotropic pair potentials tend toward overfitting, resulting in solutions with too many features and length scales that are challenging to map to mechanistic models. Here we introduce a method for the optimization of simple pair potentials that minimizes the rela…
Porous mesophases, where well-defined particle-depleted 'void' spaces are present within a particle-rich background fluid, can be self-assembled from colloidal particles interacting via isotropic pair interactions with competing attractions and repulsions. While such structures could be of wide interest for technological applications (e.g., filtration, catalysis, absorption, etc.), relatively few studies have investigated the interactions that lead to these morphologies and how they compare to those that produce other micro-phase-separated structures, such as clusters. In this work, we use inv…
We report on the self-assembly of inverse patchy colloids (IPC) using Monte Carlo simulations in two-dimensions. The IPC model considered in this work corresponds to either bipolar colloids or colloids decorated with complementary DNA on their surfaces, where only patch and non-patch parts attract. The patch coverage is found to be a dominant factor in deciding equilibrium self-assembled structures. In particular, both regular square and triangular crystals are found to be stable at 0.5 patch coverage. Upon decreasing the patch coverage to 0.33, the regular square crystal is destablized; inste…
The self-assembly of functionalized (patchy) particles with directional interactions into target structures is still a challenge, despite the significant experimental advances on their synthesis. The self-assembly pathways are typically characterized by high energy barriers that hinder the access to stable (equilibrium) structures. A possible strategy to tackle this challenge is to perform annealing cycles. By periodically switching on and off the inter-particle bonds, one expects to smooth-out the kinetic pathways and favor the assembly of targeted structures. Preliminary results have shown t…
One of the challenges of self-assembling finite-sized colloidal aggregates with a sought morphology is the necessity of precisely sorting the position of the colloids at the microscopic scale to avoid the formation of off-target structures. Microfluidic platforms address this problem by loading into single droplets the exact amount of colloids entering the targeted aggregate. Using theory and simulations, in this paper, we validate a more versatile design allowing us to fabricate different types of finite-sized aggregates, including colloidal molecules or core-shell clusters, starting from fin…
In this review we discuss recent advances in the self-assembly of self-propelled colloidal particles and highlight some of the most exciting results in this field with a specific focus on dry active matter. We explore this phenomenology through the lens of the complexity of the colloidal building blocks. We begin by considering the behavior of isotropic spherical particles. We then discuss the case of amphiphilic and dipolar Janus particles. Finally, we show how the geometry of the colloids and/or the directionality of their interactions can be used to control the physical properties of the as…
Particle shape is a critical parameter that plays an important role in self-assembly, for example, in designing targeted complex structures with desired properties. In the last decades an unprecedented range of monodisperse nanoparticle systems with control over the shape of the particles have become available. In contrast, the choice of micron-sized colloidal building blocks of particles with flat facets, i.e., particles with polygonal shapes, is significantly more limited. This can be attributed to the fact that, contrary to nanoparticles, the larger colloids are significantly harder to synt…
Patchy colloids are promising candidates for building blocks in directed self-assembly. To be successful the surface patterns need to both be simple enough to be synthesized, while feature-rich enough to cause the colloids to self-assemble into desired structures. Achieving this is a challenge for traditional synthesis methods. Recently it has been suggested that the surface pattern themselves can be made to self-assemble. In this paper we show that a wide range of functional structures can be made to self-assemble using this approach. More generally we present a design path for hierarchical t…
Multi-component self-assembly mixtures offer the possibility of encoding multiple target structures with the same set of interacting components. Selective retrieval of one of the stored structures has been attempted by preparing an initial state that favours the assembly of the required target, through seeding, concentration patterning or specific choices of interaction strengths. This may not be possible in an experiment where on-the-fly reconfiguration of the building blocks to switch functionality may be required. In this paper, we explore principles of inverse design of a multi-component s…
A major goal in nanoscience and nanotechnology is the self-assembly of any desired complex structure with a system of particles interacting through simple potentials. To achieve this objective, intense experimental and theoretical efforts are currently concentrated in the development of the so called "patchy" particles. Here we follow a completely different approach and introduce a very accessible model to produce a large variety of pre-programmed two-dimensional (2D) complex structures. Our model consists of a binary mixture of particles that interact through isotropic in plane interactions t…
In contrast to most self-assembling synthetic materials, which undergo unbounded growth, many biological self-assembly processes are self-limited. That is, the assembled structures have one or more finite dimensions that are much larger than the size scale of the individual monomers. In many such cases, the finite dimension is selected by a preferred curvature of the monomers, which leads to self-closure of the assembly. In this article, we study an example class of self-closing assemblies: cylindrical tubules that assemble from triangular monomers. By combining kinetic Monte Carlo simulations…
The unique physical and optical properties of nanoparticles make their reliable and rapid assembly an important goal for nanotechnology. To this end, many competing driving forces have been targeted to control the self-assembly, though issues with reliability and efficiency indicate that interfaces on the nanoscale are not yet fully understood. In this letter we report for the first time that the flow of water media from the interface between chemically nanopatterned hydrophobic surfaces and adjacent hydrophilic regions (a Janus interface) can effectively be employed for rapid nanoparticle ass…
We present a direct method for solving the inverse problem of designing isotropic potentials that cause self-assembly into target lattices. Each potential is constructed by matching its energy spectrum to the reciprocal representation of the lattice to guarantee that the desired structure is a ground state. We use the method to self-assemble complex lattices not previously achieved with isotropic potentials, such as a snub square tiling and the kagome lattice. The latter is especially interesting because it provides the crucial geometric frustration in several proposed spin liquids.
Assembly of spherical colloidal particles into extended structures, including linear strings, in the absence of directional interparticle bonding interactions or external perturbation could facilitate the design of new functional materials. Here, we use methods of inverse design to discover isotropic pair potentials that promote the formation of single-stranded, polydisperse strings of colloids "colloidomers" as well as size-specific, compact colloidal clusters. Based on the designed potentials, a simple model pair interaction with a short-range attraction and a longer-range repulsion is propo…
Controlled micro- to meso-scale porosity is a common materials design goal with possible applications ranging from molecular gas adsorption to particle size selective permeability or solubility. Here, we use inverse methods of statistical mechanics to design an isotropic pair interaction that, in the absence of an external field, assembles particles into an inhomogeneous {\em fluid} matrix surrounding pores of prescribed size ordered in a lattice morphology. The pore size can be tuned via modification of temperature or particle concentration. Moreover, modulating density reveals a rich series …
Inverse design can be a useful strategy for discovering interactions that drive particles to spontaneously self-assemble into a desired structure. Here, we extend an inverse design methodology--relative entropy optimization--to determine isotropic interactions that promote assembly of targeted multicomponent phases, and we apply this extension to design interactions for a variety of binary crystals ranging from compact triangular and square architectures to highly open structures with dodecagonal and octadecagonal motifs. We compare the resulting optimized (self and cross) interactions for the…
Motivated by recent experimental findings in chemical synthesis of colloidal particles, we draw an analogy between self-assembly processes occurring in biological systems (e.g. protein folding) and a new exciting possibility in the field of material science. We consider a self-assembly process whose elementary building blocks are decorated patchy colloids of various types, that spontaneously drive the system toward a unique and predetermined targeted macroscopic structure. To this aim, we discuss a simple theoretical model -- the Kern-Frenkel model -- describing a fluid of colloidal spherical …
Self-assembly dynamics in binary surfactant mixtures and structure changes of lipid vesicles induced by detergent solution are studied using coarse-grained molecular simulations. Disk-shaped micelles, the bicelles, are stabilized by detergents surrounding the rim of a bilayer disk of lipids. The self-assembled bicelles are considerably smaller than bicelles formed from vesicle rupture, and their size is determined by the concentrations of lipids and detergents and the interactions between the two species. The detergent-adsorption induces spontaneous curvature of the vesicle bilayer and results…
The transfer of chirality across length-scales is an intriguing and universal natural phenomenon. However, connecting the properties of individual building blocks to the emergent features of their resulting large-scale structure remains a challenge. In this work, we investigate the origins of mesophase chirality in cellulose nanocrystal suspensions, whose self-assembly into chiral photonic films has attracted significant interest. By correlating the ensemble behaviour in suspensions and films with a quantitative morphological analysis of the individual nanoparticles, we reveal an inverse relat…
Protein aggregation is of great importance in biology, e.g., in amyloid fibrillation. The aggregation processes that occur at the cellular scale must be highly stochastic in nature because of the statistical number fluctuations that arise on account of the small system size at the cellular scale. We study the nucleated reversible self-assembly of monomeric building blocks into polymer-like aggregates using the method of kinetic Monte Carlo. Kinetic Monte Carlo, being inherently stochastic, allows us to study the impact of fluctuations on the polymerisation reactions. One of the most important …
We propose a general framework for solving inverse self-assembly problems, i.e. designing interactions between elementary units such that they assemble spontaneously into a predetermined structure. Our approach uses patchy particles as building blocks, where the different units bind at specific interaction sites (the patches), and we exploit the possibility of having mixtures with several components. The interaction rules between the patches is determined by transforming the combinatorial problem into a Boolean satisfiability problem (SAT) which searches for solutions where all bonds are forme…
One emerging approach for the fabrication of complex architectures on the nanoscale is to utilize particles customized to intrinsically self-assemble into a desired structure. Inverse methods of statistical mechanics have proven particularly effective for the discovery of interparticle interactions suitable for this aim. Here we evaluate the generality and robustness of a recently introduced inverse design strategy [Lindquist et al., J. Chem. Phys. 145, 111101 (2016)] by applying this simulated-based, machine learning method to optimize for interparticle interactions that self-assemble particl…
In recent years significant attention has been attracted to proposals which utilize DNA for nanotechnological applications. Potential applications of these ideas range from the programmable self-assembly of colloidal crystals, to biosensors and nanoparticle based drug delivery platforms. In Chapter I we introduce the system, which generically consists of colloidal particles functionalized with specially designed DNA markers. The sequence of bases on the DNA markers determines the particle type. Due to the hybridization between complementary single-stranded DNA, specific, type-dependent interac…
