433 research outputs found
A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces
In this paper, we study triply periodic surfaces with minimal surface area
under a constraint in the volume fraction of the regions (phases) that the
surface separates. Using a variational level set method formulation, we present
a theoretical characterization of and a numerical algorithm for computing these
surfaces. We use our theoretical and computational formulation to study the
optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume
fractions of the two phases are equal and explore the properties of optimal
structures when the volume fractions of the two phases not equal. Due to the
computational cost of the fully, three-dimensional shape optimization problem,
we implement our numerical simulations using a parallel level set method
software package.Comment: 28 pages, 16 figures, 3 table
Inverse designing surface curvatures by deep learning
Smooth and curved microstructural topologies found in nature - from soap
films to trabecular bone - have inspired several mimetic design spaces for
architected metamaterials and bio-scaffolds. However, the design approaches so
far have been ad hoc, raising the challenge: how to systematically and
efficiently inverse design such artificial microstructures with targeted
topological features? Here, we explore surface curvature as a design modality
and present a deep learning framework to produce topologies with as-desired
curvature profiles. The inverse design framework can generalize to diverse
topological features such as tubular, membranous, and particulate features.
Moreover, we demonstrate successful generalization beyond both the design and
data space by inverse designing topologies that mimic the curvature profile of
trabecular bone, spinodoid topologies, and periodic nodal surfaces for
application in bio-scaffolds and implants. Lastly, we bridge curvature and
mechanics by showing how topological curvature can be designed to promote
mechanically beneficial stretching-dominated deformation over bending-dominated
deformation.Comment: 23 pages, 12 figure
Strong-Segregation Theory of Bicontinuous Phases in Block Copolymers
We compute phase diagrams for starblock copolymers in the
strong-segregation regime as a function of volume fraction , including
bicontinuous phases related to minimal surfaces (G, D, and P surfaces) as
candidate structures. We present the details of a general method to compute
free energies in the strong segregation limit, and demonstrate that the gyroid
G phase is the most nearly stable among the bicontinuous phases considered. We
explore some effects of conformational asymmetry on the topology of the phase
diagram.Comment: 14 pages, latex, 21 figures, to appear in Macromolecule
Recommended from our members
MODELING CHAIN PACKING IN COMPLEX PHASES OF SELF-ASSEMBLED BLOCK COPOLYMERS
Block copolymer (BCP) melts undergo microphase seperation and form ordered soft matter crystals with varying domain shapes and symmetries. We study the con- nection between diblock copolymer molecular designs and thermodynamic selection of ordered crystals by modeling features of variable sub-domain geometry filled with individual blocks within non-canonical sphere-like and network phases that together with layered, cylindrical and canonical spherical phases forms “natural forms” of self- assembled amphiphilic soft matter at large. First, we present a model to revise our understanding of optimal Frank-Kasper sphere-like morphologies by advancing the- ory to account for varying domain volumes. We then develop generic approaches to quantify local changes to domain thickness or packing frustration using medial sets and show its application to morphologies with arbitrary domain topologies and sym- metries in both theoretical models and experimental data. We further use medial sets as a proxy for terminal boundaries of blocks within different domains and revise thermodynamic models of BCP assembly in the strong segregation limit. Finally, we use this revised model to study effect of elastic stiffness asymmetry on relaxing packing frustration experienced by BCPs in tubular and matrix domains leading to equilibrium double gyroid network morphology in diblock copolymers
Minimality via second variation for a nonlocal isoperimetric problem
We discuss the local minimality of certain configurations for a nonlocal
isoperimetric problem used to model microphase separation in diblock copolymer
melts. We show that critical configurations with positive second variation are
local minimizers of the nonlocal area functional and, in fact, satisfy a
quantitative isoperimetric inequality with respect to sets that are
-close. The link with local minimizers for the diffuse-interface
Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative
estimate, we get new results concerning periodic local minimizers of the area
functional and a proof, via second variation, of the sharp quantitative
isoperimetric inequality in the standard Euclidean case. As a further
application, we address the global and local minimality of certain lamellar
configurations.Comment: 35 page
Modeling, assessment, and design of porous cells based on schwartz primitive surface for bone scaffolds
The design of bone scafolds for tissue regeneration is a topic of great interest, which involves diferent issues related to geometry of architectures, mechanical behavior, and biological requirements, whose optimal combination determines the success of an implant. Additive manufacturing (AM) has widened the capability to produce structures with complex geometries, which should potentially satisfy the diferent requirements. These architectures can be obtained by means of refned methods and have to be assessed in terms of geometrical and mechanical properties. In this paper a triply periodic minimal surface (TPMS), the Schwarz's Primitive surface (P-surface), has been considered as scafold unit cell and conveniently parameterized in order to investigate the efect of modulation of analytical parameters on the P-cell geometry and on its properties. Several are the cell properties, which can afect the scafold performance. Due to the important biofunctional role that the surface curvature plays in mechanisms of cellular proliferation and diferentiation, in this paper, in addition to properties considering the cell geometry in its whole (such as volume fraction or pore size), new properties were proposed. Tese properties involve, particularly, the evaluation of local geometrical-diferential properties of the P-surface. Te results of this P-cell comprehensive characterization are very useful for the design of customized bone scafolds able to satisfy both biological and mechanical requirements. A numerical structural evaluation, by means of fnite element method (FEM), was performed in order to assess the stifness of solid P-cells as a function of the changes of the analytical parameters of outer surface and the thickness of cell. Finally, the relationship between stifness and porosity has been analyzed, given the relevance that this property has for bone scafolds design
Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers
Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers
CAD-Based Porous Scaffold Design of Intervertebral Discs in Tissue Engineering
With the development and maturity of three-dimensional (3D) printing technology over the past decade, 3D printing has been widely investigated and applied in the field of tissue engineering to repair damaged tissues or organs, such as muscles, skin, and bones, Although a number of automated fabrication methods have been developed to create superior bio-scaffolds with specific surface properties and porosity, the major challenges still focus on how to fabricate 3D natural biodegradable scaffolds that have tailor properties such as intricate architecture, porosity, and interconnectivity in order to provide the needed structural integrity, strength, transport, and ideal microenvironment for cell- and tissue-growth. In this dissertation, a robust pipeline of fabricating bio-functional porous scaffolds of intervertebral discs based on different innovative porous design methodologies is illustrated. Firstly, a triply periodic minimal surface (TPMS) based parameterization method, which has overcome the integrity problem of traditional TPMS method, is presented in Chapter 3. Then, an implicit surface modeling (ISM) approach using tetrahedral implicit surface (TIS) is demonstrated and compared with the TPMS method in Chapter 4. In Chapter 5, we present an advanced porous design method with higher flexibility using anisotropic radial basis function (ARBF) and volumetric meshes. Based on all these advanced porous design methods, the 3D model of a bio-functional porous intervertebral disc scaffold can be easily designed and its physical model can also be manufactured through 3D printing. However, due to the unique shape of each intervertebral disc and the intricate topological relationship between the intervertebral discs and the spine, the accurate localization and segmentation of dysfunctional discs are regarded as another obstacle to fabricating porous 3D disc models. To that end, we discuss in Chapter 6 a segmentation technique of intervertebral discs from CT-scanned medical images by using deep convolutional neural networks. Additionally, some examples of applying different porous designs on the segmented intervertebral disc models are demonstrated in Chapter 6
Bending Frustration of Lipid-Water Mesophases Based on Cubic Minimal Surfaces
Inverse bicontinuous cubic phases are ubiquitous in lipid-water mixtures and
consist of a lipid bilayer forming a cubic minimal surface, thereby dividing
space into two cubic networks of water channels. For small hydrocarbon chain
lengths, the monolayers can be modeled as parallel surfaces to a minimal
midsurface. The bending energy of the cubic phases is determined by the
distribution of Gaussian curvature over the minimal midsurfaces which we
calculate for seven different structures (G, D, P, I-WP, C(P), S and F-RD). We
show that the free-energy densities of the structures G, D and P are
considerably lower than those of the other investigated structures due to their
narrow distribution of Gaussian curvature. The Bonnet transformation between G,
D, and P implies that these phases coexist along a triple line, which also
includes an excess water phase. Our model includes thermal membrane
undulations. Our qualitative predictions remain unchanged when higher order
terms in the curvature energy are included. Calculated phase diagrams agree
well with the experimental results for 2:1 lauric acid/dilauroyl
phosphatidylcholine and water.Comment: Revtex, 23 pages with 9 postscript figures included, to appear in
Langmui
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