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Articles 1 - 25 of 25
Full-Text Articles in Physical Sciences and Mathematics
Mechanics Of Large Folds In Thin Interfacial Films, Vincent Demery, Benny Davidovitch, Christian Santangelo
Mechanics Of Large Folds In Thin Interfacial Films, Vincent Demery, Benny Davidovitch, Christian Santangelo
Christian Santangelo
A thin film confined to a liquid interface responds to uniaxial compression by wrinkling, and then by folding, that has been solved exactly before self-contact. Here, we address the mechanics of large folds, i.e., folds that absorb a length much larger than the wrinkle wavelength. With scaling arguments and numerical simulations, we show that the antisymmetric fold is energetically favorable and can absorb any excess length at zero pressure. Then, motivated by puzzles arising in the comparison of this simple model to experiments on lipid monolayers or capillary rafts, we discuss how to incorporate film weight, self-adhesion, or energy dissipation.
Nonuniform Growth And Topological Defects In The Shaping Of Elastic Sheets, Nakul Bende, Ryan C. Hayward, Christian Santangelo
Nonuniform Growth And Topological Defects In The Shaping Of Elastic Sheets, Nakul Bende, Ryan C. Hayward, Christian Santangelo
Christian Santangelo
We demonstrate that shapes with zero Gaussian curvature, except at singularities, produced by the growth-induced buckling of a thin elastic sheet are the same as those produced by the Volterra construction of topological defects in which edges of an intrinsically flat surface are identified. With this connection, we study the problem of choosing an optimal pattern of growth for a prescribed developable surface, finding a fundamental trade-off between optimal design and the accuracy of the resulting shape which can be quantified by the length along which an edge should be identified.
Self-Assembly On A Cylinder: A Model System For Understanding The Constrain Of Commensurability, D. A. Wood, Christian Santangelo, A. D. Dinsmore
Self-Assembly On A Cylinder: A Model System For Understanding The Constrain Of Commensurability, D. A. Wood, Christian Santangelo, A. D. Dinsmore
Christian Santangelo
A crystal lattice, when confined to the surface of a cylinder, must have a periodic structure that is commensurate with the cylinder circumference. This constraint can frustrate the system, leading to oblique crystal lattices or to structures with a chiral seam known as a ‘line slip’ phase, neither of which is stable for isotropic particles in equilibrium on flat surfaces. In this study, we use molecular dynamics simulations to find the steady-state structure of spherical particles with short-range repulsion and long-range attraction far below the melting temperature. We vary the range of attraction using the Lennard-Jones and Morse potentials and …
Nambu-Goldstone Modes And Diffuse Deformations In Elastic Shells, Christian Santangelo
Nambu-Goldstone Modes And Diffuse Deformations In Elastic Shells, Christian Santangelo
Christian Santangelo
I consider the shape of a deformed elastic shell. Using the fact that the lowest-energy, small deformations are along infinitesimal isometries of the shell's mid-surface, I describe a class of weakly stretching deformations for thin shells based on the Nambu–Goldstone modes associated with those isometries. The main result is an effective theory to describe the diffuse deformations of thin shells that incorporate stretching and bending energies. The theory recovers previous results for the propagation of a “pinch” on a cylinder. A cone, on the other hand, has two length scales governing the persistence of a pinch: one governing the relaxation …
The Shape And Mechanics Of Curved Fold Origami Structures, Marcelo A. Dias, Christian Santangelo
The Shape And Mechanics Of Curved Fold Origami Structures, Marcelo A. Dias, Christian Santangelo
Christian Santangelo
We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed. In the case of no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures, we derive continuum equations, valid in the limit of vanishing spacing between folds, to describe the smooth surface intersecting all the mountain folds. We find that this surface has negative Gaussian curvature with magnitude equal to the square of the fold's torsion. A series of open folds with …
Slack Dynamics On An Unfurling String, J. A. Hanna, Christian Santangelo
Slack Dynamics On An Unfurling String, J. A. Hanna, Christian Santangelo
Christian Santangelo
An arch will grow on a rapidly deployed thin string in contact with a rigid plane. We present a qualitative model for the growing structure involving the amplification, rectification, and advection of slack in the presence of a steady stress field, validate our assumptions with numerical experiments, and pose new questions about the spatially developing motions of thin objects.
Smectic Pores And Defect Cores, Elisabetta A. Masumoto, Randall D. Kamien, Christian Santangelo
Smectic Pores And Defect Cores, Elisabetta A. Masumoto, Randall D. Kamien, Christian Santangelo
Christian Santangelo
Riemann's minimal surfaces, a one-parameter family of minimal surfaces, describe a bicontinuous lamellar system with pores connecting alternating layers. We demonstrate explicitly that Riemann's minimal surfaces are composed of a nonlinear sum of two oppositely handed helicoids.
Developed Smectics: When Exact Solutions Agree, Gareth P. Alexander, Randall D. Kamien, Christian Santangelo
Developed Smectics: When Exact Solutions Agree, Gareth P. Alexander, Randall D. Kamien, Christian Santangelo
Christian Santangelo
In the limit where the bending modulus vanishes, we construct layer configurations with arbitrary dislocation textures by exploiting a connection between uniformly spaced layers in two dimensions and developable surfaces in three dimensions. We then show how these focal textures can be used to construct layer configurations with finite bending modulus.
Geometric Mechanics Of Curved Crease Origami, Marcelo A. Dias, Levi H. Dudte, L. Mahadevan, Christian Santangelo
Geometric Mechanics Of Curved Crease Origami, Marcelo A. Dias, Levi H. Dudte, L. Mahadevan, Christian Santangelo
Christian Santangelo
Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold, and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases …
Packing Squares In A Torus, D. W. Blair, Christian Santangelo, J. Machta
Packing Squares In A Torus, D. W. Blair, Christian Santangelo, J. Machta
Christian Santangelo
The densest packings of N unit squares in a torus are studied using analytical methods as well as simulated annealing. A rich array of dense packing solutions are found: density-one packings when N is the sum of two square integers; a family of 'gapped bricklayer' Bravais lattice solutions with density N/(N + 1); and some surprising non-Bravais lattice configurations, including lattices of holes as well as a configuration for N = 23 in which not all squares share the same orientation. The entropy of some of these configurations and the frequency and orientation of density-one solutions as N -> infinity …
Frustrated Order On Extrinsic Geometries, Badel L. Mbanga, Gregory M. Grason, Christian Santangelo
Frustrated Order On Extrinsic Geometries, Badel L. Mbanga, Gregory M. Grason, Christian Santangelo
Christian Santangelo
We study, numerically and theoretically, defects in an anisotropic liquid that couple to the extrinsic geometry of a surface. Though the intrinsic geometry tends to confine topological defects to regions of large Gaussian curvature, extrinsic couplings tend to orient the order along the local direction of maximum or minimum bending. This additional frustration is generically unavoidable, and leads to complex ground-state thermodynamics. Using the catenoid as a prototype, we show, in contradistinction to the well-known effects of intrinsic geometry, that extrinsic curvature expels disclinations from the region of maximum curvature above a critical coupling threshold. On catenoids lacking an “inside-outside” …
Programmed Buckling By Controlled Lateral Swelling In A Thin Elastic Sheet, M. A. Dias, J. A. Hanna, Christian Santangelo
Programmed Buckling By Controlled Lateral Swelling In A Thin Elastic Sheet, M. A. Dias, J. A. Hanna, Christian Santangelo
Christian Santangelo
Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.
Soft Spheres Make More Mesophases, Mattew A. Glaser, Gregory M. Granson, Randall D. Kamien, A. Kosmrlj, Christian Santangelo, P. Ziherl
Soft Spheres Make More Mesophases, Mattew A. Glaser, Gregory M. Granson, Randall D. Kamien, A. Kosmrlj, Christian Santangelo, P. Ziherl
Christian Santangelo
We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard core and repulsive, soft shoulder. Despite the purely repulsive and isotropic interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and soft-shoulder length scales, including fluid and crystalline phases with micellar, lamellar, and inverse micellar morphology. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. …
Triply-Periodic Smectic Liquid Crystals, Christian Santangelo, Randall D. Kamien
Triply-Periodic Smectic Liquid Crystals, Christian Santangelo, Randall D. Kamien
Christian Santangelo
Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a π∕2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk’s first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and …
Membrane Fluctuations Around Inclusions, Christian Santangelo, Oded Farago
Membrane Fluctuations Around Inclusions, Christian Santangelo, Oded Farago
Christian Santangelo
The free energy of inserting a protein into a membrane is determined by considering the variation in the spectrum of thermal fluctuations in response to the presence of a rigid inclusion. Both numerically and through a simple analytical approximation, we find that the primary effect of fluctuations is to reduce the effective surface tension, hampering the insertion at low surface tension. Our results, which should also be relevant for membrane pores, suggest (in contrast to classical nucleation theory) that a finite surface tension is necessary to facilitate the opening of a pore.
Geometric Theory Of Columnar Phases On Curved Substrates, Christian Santangelo, Vincenzo Vitelli, Randall D. Kamien, David R. Nelson
Geometric Theory Of Columnar Phases On Curved Substrates, Christian Santangelo, Vincenzo Vitelli, Randall D. Kamien, David R. Nelson
Christian Santangelo
We study thin self-assembled columns constrained to lie on a curved, rigid substrate. The curvature presents no local obstruction to equally spaced columns in contrast with curved crystals for which the crystalline bonds are frustrated. Instead, the vanishing compressional strain of the columns implies that their normals lie on geodesics which converge (diverge) in regions of positive (negative) Gaussian curvature, in analogy to the focusing of light rays by a lens. We show that the out of plane bending of the cylinders acts as an effective ordering field.
Undulated Cylinders Of Charged Diblock Copolymers, Gregory M. Grason, Christian Santangelo
Undulated Cylinders Of Charged Diblock Copolymers, Gregory M. Grason, Christian Santangelo
Christian Santangelo
We study the cylinder to sphere morphological transition of diblock copolymers in aqueous solution with a hydrophobic block and a charged block. We find a metastable undulated cylinder configuration for a range of charge and salt concentrations which, nevertheless, occurs above the threshold where spheres are thermodynamically favorable. By modeling the shape of the cylinder ends, we find that the free-energy barrier for the transition from cylinders to spheres is quite large and that this barrier falls significantly in the limit of high polymer charge and low solution salinity. This suggests that observed undulated cylinder phases are kinetically trapped structures.
Smectic Liquid Crystals: Materials With One-Dimensional, Periodic Order, Randall D. Kamien, Christian Santangelo
Smectic Liquid Crystals: Materials With One-Dimensional, Periodic Order, Randall D. Kamien, Christian Santangelo
Christian Santangelo
Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Although smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers becomes nontrivial. We discuss various topological and geometrical aspects of these materials and present recent progress on finding some exact layer configurations. We also exhibit connections to the study of certain embedded minimal surfaces and briefly summarize some important open problems.
Elliptic Phases: A Study Of The Nonlinear Elasticity Of Twist-Grain Boundaries, Christian Santangelo, Randall D. Kamien
Elliptic Phases: A Study Of The Nonlinear Elasticity Of Twist-Grain Boundaries, Christian Santangelo, Randall D. Kamien
Christian Santangelo
We develop an explicit and tractable representation of a twist-grain-boundary phase of a smectic-A liquid crystal. This allows us to calculate the interaction energy between grain boundaries and the relative contributions from the bending and compression deformations. We discuss the special stability of the π/2 grain boundaries and discuss the relation of this structure to the Schwarz D surface.
Computing Counterion Densities At Intermediate Coupling, Christian Santangelo
Computing Counterion Densities At Intermediate Coupling, Christian Santangelo
Christian Santangelo
By decomposing the Coulomb interaction into a long-distance component appropriate for mean-field theory, and a non-mean-field short distance component, we compute the counterion density near a charged surface for all values of the counterion coupling parameter. A modified strong-coupling expansion that is manifestly finite at all coupling strengths is used to treat the short-distance component. We find a nonperturbative correction related to the lateral counterion correlations that modifies the density at intermediate coupling.
Pore Formation In Fluctuating Membranes, Oded Farago, Christian Santangelo
Pore Formation In Fluctuating Membranes, Oded Farago, Christian Santangelo
Christian Santangelo
We study the nucleation of a single pore in a fluctuating lipid membrane, specifically taking into account the membrane fluctuations, as well as the shape fluctuations of the pore. For large enough pores, the nucleationfree energy is well-described by shifts in the effective membrane surface tension and the pore line tension. Using our framework, we derive the stability criteria for the various pore formation regimes. In addition to the well-known large-tension regime from the classical nucleation theory of pores, we also find a low-tension regime in which the effective line and surface tensions can change sign from their bare values. …
Effects Of Counterion Fluctuations In A Polyelectrolyte Brush, Christian Santangelo, A.W.C. Lau
Effects Of Counterion Fluctuations In A Polyelectrolyte Brush, Christian Santangelo, A.W.C. Lau
Christian Santangelo
We investigate the effect of counterion fluctuations in a single polyelectrolyte brush in the absence of added salt by systematically expanding the counterion free energy about Poisson-Boltzmann mean-field theory. We find that for strongly charged brushes, there is a collapse regime in which the brush height decreases with increasing charge on the polyelectrolyte chains. The transition to this collapsed regime is similar to the liquid-gas transition, which has a first-order line terminating at a critical point. We find that, for monovalent counterions, the transition is discontinuous in theta solvent, while for multivalent counterions, the transition is generally continuous. For collapsed …
Distribution Of Counterions Near Discretely Charged Planes And Rods, M. L. Henle, Christian Santangelo, D. M. Patel, P. A. Pincus
Distribution Of Counterions Near Discretely Charged Planes And Rods, M. L. Henle, Christian Santangelo, D. M. Patel, P. A. Pincus
Christian Santangelo
Realistic charged macromolecules are characterized by discrete (rather than homogeneous) charge distributions. We investigate the effects of surface charge discretization on the counterion distribution at the level of mean-field theory using a two-state model. Both planar and cylindrical geometries are considered; for the latter case, we compare our results to numerical solutions of the full Poisson-Boltzmann equation. We find that the discretization of the surface charge can cause enhanced localization of the counterions near the surface; for charged cylinders, counterion condensation can exceed Oosawa-Manning condensation.
Bogomol'nyi, Prasad And Sommerfield Configurations In Smectics, Christian Santangelo, Randall D. Kamien
Bogomol'nyi, Prasad And Sommerfield Configurations In Smectics, Christian Santangelo, Randall D. Kamien
Christian Santangelo
It is typical in smectic liquid crystals to describe elastic deformations with a linear theory when the elastic strain is small. In smectics, certain essential nonlinearities arise from the requirement of rotational invariance. By employing the Bogomol’nyi, Prasad, and Sommerfield decomposition and relying on boundary conditions and geometric invariants, we have found a large class of exact solutions. We introduce an approximation for the deformation profile far from a spherical inclusion and find an enhanced attractive interaction at long distances due to the nonlinear elasticity, confirmed by numerical minimization.
Coiling Instabilities Of Multilamellar Tubes, Christian Santangelo, P. Pincus
Coiling Instabilities Of Multilamellar Tubes, Christian Santangelo, P. Pincus
Christian Santangelo
Myelin figures are densely packed stacks of coaxial cylindrical bilayers that are unstable to the formation of coils or double helices. These myelin figures appear to have no intrinsic chirality. We show that such cylindrical membrane stacks can develop an instability when they acquire a spontaneous curvature or when the equilibrium distance between membranes is decreased. This instability breaks the chiral symmetry of the stack and may result in coiling. A unilamellar cylindrical vesicle, on the other hand, will develop an axisymmetric instability, possibly related to the pearling instability.