Treating random sequential addition via the replica method.

While many physical processes are non-equilibrium in nature, the theory and modeling of such phenomena lag behind theoretical treatments of equilibrium systems. The diversity of powerful theoretical tools available to describe equilibrium systems has inspired strategies that map non-equilibrium systems onto equivalent equilibrium analogs so that interrogation with standard statistical mechanical approaches is possible. In this work, we revisit the mapping from the non-equilibrium random sequential addition process onto an equilibrium multi-component mixture via the replica method, allowing for theoretical predictions of non-equilibrium structural quantities. We validate the above approach by comparing the theoretical predictions to numerical simulations of random sequential addition.

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.

Inverse methods for design of soft materials.

Functional soft materials, comprising colloidal and molecular building blocks that self-organize into complex structures as a result of their tunable interactions, enable a wide array of technological applications. Inverse methods provide a systematic means for navigating their inherently high-dimensional design spaces to create materials with targeted properties. While multiple physically motivated inverse strategies have been successfully implemented in silico, their translation to guiding experimental materials discovery has thus far been limited to a handful of proof-of-concept studies. In this perspective, we discuss recent advances in inverse methods for design of soft materials that address two challenges: (1) methodological limitations that prevent such approaches from satisfying design constraints and (2) computational challenges that limit the size and complexity of systems that can be addressed. Strategies that leverage machine learning have proven particularly effective, including methods to discover order parameters that characterize complex structural motifs and schemes to efficiently compute macroscopic properties from the underlying structure. We also highlight promising opportunities to improve the experimental realizability of materials designed computationally, including discovery of materials with functionality at multiple thermodynamic states, design of externally directed assembly protocols that are simple to implement in experiments, and strategies to improve the accuracy and computational efficiency of experimentally relevant models.

Structure and phase behavior of polymer-linked colloidal gels.

Low-density "equilibrium" gels that consist of a percolated, kinetically arrested network of colloidal particles and are resilient to aging can be fabricated by restricting the number of effective bonds that form between the colloids. Valence-restricted patchy particles have long served as one archetypal example of such materials, but equilibrium gels can also be realized through a synthetically simpler and scalable strategy that introduces a secondary linker, such as a small ditopic molecule, to mediate the bonds between the colloids. Here, we consider the case where the ditopic linker molecules are low-molecular-weight polymers and demonstrate using a model colloid-polymer mixture how macroscopic properties such as the phase behavior as well as the microstructure of the gel can be designed through the polymer molecular weight and concentration. The low-density window for equilibrium gel formation is favorably expanded using longer linkers while necessarily increasing the spacing between all colloids. However, we show that blends of linkers with different sizes enable wider variation in microstructure for a given target phase behavior. Our computational study suggests a robust and tunable strategy for the experimental realization of equilibrium colloidal gels.

The role of pressure in inverse design for assembly.

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 simulations are performed in the isothermal-isobaric ensemble. The benefits of this approach are twofold. First, the structure and the thermodynamics associated with the optimized interaction can be controlled simultaneously. Second, by varying the pressure in the optimization, a family of interparticle potentials that all self-assemble the same structure can be systematically discovered, allowing for a deeper understanding of self-assembly of a given target structure and providing multiple assembly routes for its realization. Selecting an appropriate simulation ensemble to control the thermodynamic properties of interest is a general design strategy that could also be used to discover interaction potentials that self-assemble structures having, for example, a specified chemical potential.

Communication: From close-packed to topologically close-packed: Formation of Laves phases in moderately polydisperse hard-sphere mixtures.

Particle size polydispersity can help to inhibit crystallization of the hard-sphere fluid into close-packed structures at high packing fractions and thus is often employed to create model glass-forming systems. Nonetheless, it is known that hard-sphere mixtures with modest polydispersity still have ordered ground states. Here, we demonstrate by computer simulation that hard-sphere mixtures with increased polydispersity fractionate on the basis of particle size and a bimodal subpopulation favors the formation of topologically close-packed C14 and C15 Laves phases in coexistence with a disordered phase. The generality of this result is supported by simulations of hard-sphere mixtures with particle-size distributions of four different forms.

Gelation of plasmonic metal oxide nanocrystals by polymer-induced depletion attractions.

Gelation of colloidal nanocrystals emerged as a strategy to preserve inherent nanoscale properties in multiscale architectures. However, available gelation methods to directly form self-supported nanocrystal networks struggle to reliably control nanoscale optical phenomena such as photoluminescence and localized surface plasmon resonance (LSPR) across nanocrystal systems due to processing variabilities. Here, we report on an alternative gelation method based on physical internanocrystal interactions: short-range depletion attractions balanced by long-range electrostatic repulsions. The latter are established by removing the native organic ligands that passivate tin-doped indium oxide (ITO) nanocrystals while the former are introduced by mixing with small PEG chains. As we incorporate increasing concentrations of PEG, we observe a reentrant phase behavior featuring two favorable gelation windows; the first arises from bridging effects while the second is attributed to depletion attractions according to phase behavior predicted by our unified theoretical model. Our assembled nanocrystals remain discrete within the gel network, based on X-ray scattering and high-resolution transmission electron microscopy. The infrared optical response of the gels is reflective of both the nanocrystal building blocks and the network architecture, being characteristic of ITO nanocrystals' LSPR with coupling interactions between neighboring nanocrystals.

Inverse design of multicomponent assemblies.

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 binary assemblies to those obtained from optimization of analogous single-component systems. This comparison reveals that self-interactions act as a "primer" to position particles at approximately correct coordination shell distances, while cross interactions act as the "binder" that refines and locks the system into the desired configuration. For simpler binary targets, it is possible to successfully design self-assembling systems while restricting one of these interaction types to be a hard-core-like potential. However, optimization of both self- and cross interaction types appears necessary to design for assembly of more complex or open structures.

Inverse Design of Self-Assembling Frank-Kasper Phases and Insights Into Emergent Quasicrystals.

We discuss how a machine learning approach based on relative entropy optimization can be used as an inverse design strategy to discover isotropic pair interactions that self-assemble single- or multicomponent particle systems into Frank-Kasper phases. In doing so, we also gain insights into the self-assembly of quasicrystals.

Interactions and design rules for assembly of porous colloidal mesophases.

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 inverse methods of statistical mechanics to design model isotropic pair potentials that form porous mesophases. We characterize the resulting porous structures, correlating features of the pair potential with the targeted pore size and the particle packing fraction. The former is primarily encoded by the amplitude and range of the repulsive barrier of the designed pair potential and the latter by the attractive well depth. We observe a trade-off with respect to the packing fraction of the targeted morphology: greater values support more spherical and monodisperse pores that themselves organize into periodic structures, while lower values yield more mobile pores that do not assemble into ordered structures but remain stable over a larger range of packing fraction. We conclude by commenting on the limitations of targeting a specific pore diameter within the present inverse design approach as well as by describing future directions to overcome these limitations.

Insights into the Electronic Structure of Ozone and Sulfur Dioxide from Generalized Valence Bond Theory: Addition of Hydrogen Atoms.

Ozone (O3) and sulfur dioxide (SO2) are valence isoelectronic species, yet their properties and reactivities differ dramatically. In particular, O3 is highly reactive, whereas SO2 is chemically relatively stable. In this paper, we investigate serial addition of hydrogen atoms to both the terminal atoms of O3 and SO2 and to the central atom of these species. It is well-known that the terminal atoms of O3 are much more amenable to bond formation than those of SO2. We show that the differences in the electronic structure of the π systems in the parent triatomic species account for the differences in the addition of hydrogen atoms to the terminal atoms of O3 and SO2. Further, we find that the π system in SO2, which is a recoupled pair bond dyad, facilitates the addition of hydrogen atoms to the sulfur atom, resulting in stable HSO2 and H2SO2 species.

Assembly of nothing: equilibrium fluids with designed structured porosity.

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 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 of microphase-separated morphologies including pore- or particle-based lattices, pore- or particle-based columns, and bicontinuous or lamellar structures. Sensitivity of pore assembly to the form of the designed interaction potential is explored.

Insights into the Electronic Structure of Molecules from Generalized Valence Bond Theory.

In this article we describe the unique insights into the electronic structure of molecules provided by generalized valence bond (GVB) theory. We consider selected prototypical hydrocarbons as well as a number of hypervalent molecules and a set of first- and second-row valence isoelectronic species. The GVB wave function is obtained by variationally optimizing the orbitals and spin coupling in the valence bond wave function. The GVB wave function is a generalization of the Hartree-Fock (HF) wave function, lifting the double occupancy restriction on a subset of the HF orbitals as well as the associated orthogonality and spin coupling constraints. The GVB wave function includes a major fraction (if not all) of the nondynamical correlation energy of a molecule. Because of this, GVB theory properly describes bond formation and can answer one of the most compelling questions in chemistry: How are atoms changed by molecular formation? We show that GVB theory provides a unified description of the nature of the bonding in all of the above molecular species as well as contributing new insights into the well-known, but poorly understood, first-row anomaly.

Linking Semiconductor Nanocrystals into Gel Networks through All-Inorganic Bridges.

For colloidal semiconductor nanocrystals (NCs), replacement of insulating organic capping ligands with chemically diverse inorganic clusters enables the development of functional solids in which adjacent NCs are strongly coupled. Yet controlled assembly methods are lacking to direct the arrangement of charged, inorganic cluster-capped NCs into open networks. Herein, we introduce coordination bonds between the clusters capping the NCs thus linking the NCs into highly open gel networks. As linking cations (Pt(2+)) are added to dilute (under 1 vol %) chalcogenidometallate-capped CdSe NC dispersions, the NCs first form clusters, then gels with viscoelastic properties. The phase behavior of the gels for variable [Pt(2+)] suggests they may represent nanoscale analogues of bridged particle gels, which have been observed to form in certain polymer colloidal suspensions.

Insights into the Electronic Structure of Ozone and Sulfur Dioxide from Generalized Valence Bond Theory: Bonding in O3 and SO2.

There are many well-known differences in the physical and chemical properties of ozone (O3) and sulfur dioxide (SO2). O3 has longer and weaker bonds than O2, whereas SO2 has shorter and stronger bonds than SO. The O-O2 bond is dramatically weaker than the O-SO bond, and the singlet-triplet gap in SO2 is more than double that in O3. In addition, O3 is a very reactive species, while SO2 is far less so. These disparities have been attributed to variations in the amount of diradical character in the two molecules. In this work, we use generalized valence bond (GVB) theory to characterize the electronic structure of ozone and sulfur dioxide, showing O3 does indeed possess significant diradical character, whereas SO2 is effectively a closed shell molecule. The GVB results provide critical insights into the genesis of the observed difference in these two isoelectronic species. SO2 possesses a recoupled pair bond dyad in the a"(π) system, resulting in SO double bonds. The π system of O3, on the other hand, has a lone pair on the central oxygen atom plus a pair of electrons in orbitals on the terminal oxygen atoms that give rise to a relatively weak π interaction.

Insights into the electronic structure of disulfur tetrafluoride isomers from generalized valence bond theory.

Sulfur and fluorine can participate in a variety of bonding motifs, lending significant diversity to their chemistry. Prior work has identified three distinct minima for disulfur tetrafluoride (S2F4) compounds: two FSSF3 isomers and one SSF4 species. We used a combination of sophisticated explicitly correlated coupled cluster calculations and generalized valence bond (GVB) theory to characterize the electronic structure of these species as well as additional stationary points on the potential energy surface with F2SSF2 connectivity. On the singlet surface, the two stationary points considered in this work with an F2SSF2 structure are first- or second-order saddle points and not minima. However, on the triplet surface, we discovered a novel C2 symmetric F2SSF2 minimum that was anticipated from the structure of an excited state ((3)B1) of SF2. Analysis using the GVB wave function in conjunction with the recoupled pair bonding model developed by our group provides a straightforward explanation of the bonding in all of the S2F4 structures considered here. In addition, the model predicted the existence of the F2SSF2((3)B) minimum.

Effects of ligand electronegativity on recoupled pair bonds with application to sulfurane precursors.

Recoupled pair bonds (RPBs) are conditional bonds-they only form for selected central atoms and ligands. A complete theoretical description of RPBs requires an understanding of the properties of the central atom and ligands that enable such bonds to be formed. In this work, we show that ligand electronegativity is positively correlated with recoupled pair bond strength for a variety of ligands interacting with the 3p(2) pair of sulfur. We also describe substituent (X) effects on the SF(a(4)Σ(-)) state by investigating X2SF species. These effects generally mirror those observed for covalently bound analogues, but we found that recoupled pair bonding can lead to breakdowns in the expected relationships among bond length, strength, and force constant for some of these species. Finally, we compare the properties of two molecules of practical interest that are bound by recoupled pair bonds: the dimethyl sulfur fluoride and hydroxide radicals (DMS-F and DMS-OH).

Electronic structure of H2S, SF2, and HSF and implications for hydrogen-substituted hypervalent sulfur fluorides.

It is well known that hypervalent molecules are more stable with very electronegative ligands such as fluorine. For example, while SF6 is uniquely stable and experimentally well characterized and many of the features of SF4 have been characterized, neither H4S nor H6S has been observed. Furthermore, no hypervalent sulfur species with mixed hydrogen and fluorine ligands have been experimentally characterized to date. In this work, we present detailed calculations of the electronic structure of H2S, SF2, and HSF. While all three compounds have similar bent singlet ground states, the potential energy surfaces of various low lying electronic states as a function of bond angle reveal very different behaviors, in particular for linear geometries. We use the disparate bonding motifs of the low-lying triplet states to rationalize the differences between SF4 and the hypothetical H4S molecule. We also make predictions about the effects of hydrogen substitution on the energetics and geometries of hypervalent sulfur fluoride compounds.

Bonding in Sulfur-Oxygen Compounds-HSO/SOH and SOO/OSO: An Example of Recoupled Pair π Bonding.

The ground states (X(2)A″) of HSO and SOH are extremely close in energy, yet their molecular structures differ dramatically, e.g., re(SO) is 1.485 Å in HSO and 1.632 Å in SOH. The SO bond is also much stronger in HSO than in SOH: 100.3 kcal/mol versus 78.8 kcal/mol [RCCSD(T)-F12/AVTZ]. Similar differences are found in the SO2 isomers, SOO and OSO, depending on whether the second oxygen atom binds to oxygen or sulfur. We report generalized valence bond and RCCSD(T)-F12 calculations on HSO/SOH and OSO/SOO and analyze the bonding in all four species. We find that HSO has a shorter and stronger SO bond than SOH due to the presence of a recoupled pair bond in the π(a″) system of HSO. Similarly, the bonding in SOO and OSO differs greatly. SOO is like ozone and has substantial diradical character, while OSO has two recoupled pair π bonds and negligible diradical character. The ability of the sulfur atom to form recoupled pair bonds provides a natural explanation for the dramatic variation in the bonding in these and many other sulfur-oxygen compounds.