Shivendra Pandey

Algorithmic design of self-folding polyhedra

Self-assembly has emerged as a paradigm for highly parallel fabrication of complex three-dimensional structures. However, there are few principles that guide a priori design, yield, and defect tolerance of self-assembling structures. We examine with experiment and theory the geometric principles that underlie self-folding of submillimeter-scale higher polyhedra from two-dimensional nets. In particular, we computationally search for nets within a large set of possibilities and then test these nets experimentally. Our main findings are that (i) compactness is a simple and effective design principle for maximizing the yield of self-folding polyhedra; and (ii) shortest paths from 2D nets to 3D polyhedra in the configuration space are important for rationalizing experimentally observed folding pathways. Our work provides a model problem amenable to experimental and theoretical analysis of design principles and pathways in self-assembly.

Origami Inspired Self-assembly of Patterned and Reconfigurable Particles

There are numerous techniques such as photolithography, electron-beam lithography and soft-lithography that can be used to precisely pattern two dimensional (2D) structures. These technologies are mature, offer high precision and many of them can be implemented in a high-throughput manner. We leverage the advantages of planar lithography and combine them with self-folding methods1-20 wherein physical forces derived from surface tension or residual stress, are used to curve or fold planar structures into three dimensional (3D) structures. In doing so, we make it possible to mass produce precisely patterned static and reconfigurable particles that are challenging to synthesize. In this paper, we detail visualized experimental protocols to create patterned particles, notably, (a) permanently bonded, hollow, polyhedra that self-assemble and self-seal due to the minimization of surface energy of liquefied hinges21-23 and (b) grippers that self-fold due to residual stress powered hinges24,25. The specific protocol described can be used to create particles with overall sizes ranging from the micrometer to the centimeter length scales. Further, arbitrary patterns can be defined on the surfaces of the particles of importance in colloidal science, electronics, optics and medicine. More generally, the concept of self-assembling mechanically rigid particles with self-sealing hinges is applicable, with some process modifications, to the creation of particles at even smaller, 100 nm length scales22, 26 and with a range of materials including metals21, semiconductors9 and polymers27. With respect to residual stress powered actuation of reconfigurable grasping devices, our specific protocol utilizes chromium hinges of relevance to devices with sizes ranging from 100 μm to 2.5 mm. However, more generally, the concept of such tether-free residual stress powered actuation can be used with alternate high-stress materials such as heteroepitaxially deposited semiconductor films5,7 to possibly create even smaller nanoscale grasping devices.

Building polyhedra by self-assembly: Theory and experiment

We investigate the utility of a mathematical framework based on discrete geometry to model biological and synthetic self-assembly. Our primary biological example is the self-assembly of icosahedral viruses; our synthetic example is surface-tension-driven self-folding polyhedra. In both instances, the process of self-assembly is modeled by decomposing the polyhedron into a set of partially formed intermediate states. The set of all intermediates is called the configuration space, pathways of assembly are modeled as paths in the configuration space, and the kinetics and yield of assembly are modeled by rate equations, Markov chains, or cost functions on the configuration space. We review an interesting interplay between biological function and mathematical structure in viruses in light of this framework. We discuss in particular: (i) tiling theory as a coarse-grained description of all-atom models; (ii) the building game—a growth model for the formation of polyhedra; and (iii) the application of these models to the self-assembly of the bacteriophage MS2. We then use a similar framework to model self-folding polyhedra. We use a discrete folding algorithm to compute a configuration space that idealizes surface-tension-driven self-folding and analyze pathways of assembly and dominant intermediates. These computations are then compared with experimental observations of a self-folding dodecahedron with side 300 μm. In both models, despite a combinatorial explosion in the size of the configuration space, a few pathways and intermediates dominate self-assembly. For self-folding polyhedra, the dominant intermediates have fewer degrees of freedom than comparable intermediates, and are thus more rigid. The concentration of assembly pathways on a few intermediates with distinguished geometric properties is biologically and physically important, and suggests deeper mathematical structure.

Self-assembly of mesoscale isomers: the role of pathways and degrees of freedom.

The spontaneous self-organization of conformational isomers from identical precursors is of fundamental importance in chemistry. Since the precursors are identical, it is the multi-unit interactions, characteristics of the intermediates, and assembly pathways that determine the final conformation. Here, we use geometric path sampling and a mesoscale experimental model to investigate the self-assembly of a model polyhedral system, an octahedron, that forms two isomers. We compute the set of all possible assembly pathways and analyze the degrees of freedom or rigidity of intermediates. Consequently, by manipulating the degrees of freedom of a precursor, we were able to experimentally enrich the formation of one isomer over the other. Our results suggest a new approach to direct pathways in both natural and synthetic self-assembly using simple geometric criteria. We also compare the process of folding and unfolding in this model with a geometric model for cyclohexane, a well-known molecule with chair and boat conformations.

A cellular architecture for self-assembled 3D computational devices

To overcome physical size limitations in scaling transistors in inherently two-dimensional geometries, efforts are being directed at wafer stacking to implement more quasi three-dimensional (3D) architectures. However, significant and unprecedented gains in terms of packing and speed can be achieved if CMOS components can be integrated in truly 3D cellular porous architectures. In this paper, we present our initial results to create prototype 3D cellular computational devices by self-assembly. We first describe the cellular computational architecture based on Cell Matrix, an inherently defect and fault-tolerant architecture that is self-configurable, and therefore is ideally suited for ultra large-scale integration (ULSI). We then show first prototypes of functional polyhedral computational integrated devices at the centimeter and millimeter scales as a step toward self-folding porous crystal structures at the nanoscale. Our approach is rooted in the synergy between experiments, computation, and theory. It has the potential to address the major challenges of 3D integration: self-assembly, self-configuration, defect-tolerance, and cooling.

Assembly of a 3D Cellular Computer Using Folded E-Blocks

The assembly of integrated circuits in three dimensions (3D) provides a possible solution to address the ever-increasing demands of modern day electronic devices. It has been suggested that by using the third dimension, devices with high density, defect tolerance, short interconnects and small overall form factors could be created. However, apart from pseudo 3D architecture, such as monolithic integration, die, or wafer stacking, the creation of paradigms to integrate electronic low-complexity cellular building blocks in architecture that has tile space in all three dimensions has remained elusive. Here, we present software and hardware foundations for a truly 3D cellular computational devices that could be realized in practice. The computing architecture relies on the scalable, self-configurable and defect-tolerant cell matrix. The hardware is based on a scalable and manufacturable approach for 3D assembly using folded polyhedral electronic blocks (E-blocks). We created monomers, dimers and 2 × 2 × 2 assemblies of polyhedral E-blocks and verified the computational capabilities by implementing simple logic functions. We further show that 63.2% more compact 3D circuits can be obtained with our design automation tools compared to a 2D architecture. Our results provide a proof-of-concept for a scalable and manufacture-ready process for constructing massive-scale 3D computational devices.

Patterned soft-micropolyhedra by self-folding and molding

It is extremely challenging to mass-produce polymeric microstructures with patterns in all three dimensions (3D). In this paper, we describe a highly parallel strategy to create such patterned soft-micropolyhedra with a range of polymers and even live mammalian cell-laden hydrogels. Specifically, we first patterned metallic micropolyhedra using photolithography and capillary force assisted self-folding. Inverse molds were created in the elastomer polydimethylsiloxane (PDMS); interestingly, the patterns were imprinted on the side walls of the molds. The molds were then filled with pre-polymers such as NOA73, PEGDA, and pNIPAM-AAc and cell-laden hydrogel solutions and cross-linked to form soft patterned micropolyhedra. Our results suggest that the combination of self-folding and molding could be used to create a variety of smart particles and building blocks with precisely patterned surfaces for applications in colloidal science, self-assembly, drug delivery and tissue engineering.