22,247 research outputs found
Optimal self-assembly of finite shapes at temperature 1 in 3D
Working in a three-dimensional variant of Winfree's abstract Tile Assembly
Model, we show that, for an arbitrary finite, connected shape , there is a tile set that uniquely self-assembles into a 3D
representation of a scaled-up version of at temperature 1 in 3D with
optimal program-size complexity (the "program-size complexity", also known as
"tile complexity", of a shape is the minimum number of tile types required to
uniquely self-assemble it). Moreover, our construction is "just barely" 3D in
the sense that it only places tiles in the and planes. Our
result is essentially a just-barely 3D temperature 1 simulation of a similar 2D
temperature 2 result by Soloveichik and Winfree (SICOMP 2007)
Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D
We investigate the power of the Wang tile self-assembly model at temperature
1, a threshold value that permits attachment between any two tiles that share
even a single bond. When restricted to deterministic assembly in the plane, no
temperature 1 assembly system has been shown to build a shape with a tile
complexity smaller than the diameter of the shape. In contrast, we show that
temperature 1 self-assembly in 3 dimensions, even when growth is restricted to
at most 1 step into the third dimension, is capable of simulating a large class
of temperature 2 systems, in turn permitting the simulation of arbitrary Turing
machines and the assembly of squares in near optimal
tile complexity. Further, we consider temperature 1 probabilistic assembly in
2D, and show that with a logarithmic scale up of tile complexity and shape
scale, the same general class of temperature systems can be simulated
with high probability, yielding Turing machine simulation and
assembly of squares with high probability. Our results show a sharp
contrast in achievable tile complexity at temperature 1 if either growth into
the third dimension or a small probability of error are permitted. Motivated by
applications in nanotechnology and molecular computing, and the plausibility of
implementing 3 dimensional self-assembly systems, our techniques may provide
the needed power of temperature 2 systems, while at the same time avoiding the
experimental challenges faced by those systems
Dimensionality and design of isotropic interactions that stabilize honeycomb, square, simple cubic, and diamond lattices
We use inverse methods of statistical mechanics and computer simulations to
investigate whether an isotropic interaction designed to stabilize a given
two-dimensional (2D) lattice will also favor an analogous three-dimensional
(3D) structure, and vice versa. Specifically, we determine the 3D ordered
lattices favored by isotropic potentials optimized to exhibit stable 2D
honeycomb (or square) periodic structures, as well as the 2D ordered structures
favored by isotropic interactions designed to stabilize 3D diamond (or simple
cubic) lattices. We find a remarkable `transferability' of isotropic potentials
designed to stabilize analogous morphologies in 2D and 3D, irrespective of the
exact interaction form, and we discuss the basis of this cross-dimensional
behavior. Our results suggest that the discovery of interactions that drive
assembly into certain 3D periodic structures of interest can be assisted by
less computationally intensive optimizations targeting the analogous 2D
lattices.Comment: 22 pages (preprint version; includes supplementary information), 5
figures, 3 table
Stable Frank-Kasper phases of self-assembled, soft matter spheres
Single molecular species can self-assemble into Frank Kasper (FK) phases,
finite approximants of dodecagonal quasicrystals, defying intuitive notions
that thermodynamic ground states are maximally symmetric. FK phases are
speculated to emerge as the minimal-distortional packings of space-filling
spherical domains, but a precise quantitation of this distortion and how it
affects assembly thermodynamics remains ambiguous. We use two complementary
approaches to demonstrate that the principles driving FK lattice formation in
diblock copolymers emerge directly from the strong-stretching theory of
spherical domains, in which minimal inter-block area competes with minimal
stretching of space-filling chains. The relative stability of FK lattices is
studied first using a diblock foam model with unconstrained particle volumes
and shapes, which correctly predicts not only the equilibrium {\sigma} lattice,
but also the unequal volumes of the equilibrium domains. We then provide a
molecular interpretation for these results via self-consistent field theory,
illuminating how molecular stiffness regulates the coupling between
intra-domain chain configurations and the asymmetry of local packing. These
findings shed new light on the role of volume exchange on the formation of
distinct FK phases in copolymers, and suggest a paradigm for formation of FK
phases in soft matter systems in which unequal domain volumes are selected by
the thermodynamic competition between distinct measures of shape asymmetry.Comment: 40 pages, 22 figure
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