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Nanoscale 2021 Jan;13(3):1639-1651

Mathematics, Mechanics, and Materials Unit (MMMU), Okinawa Institute of Science and Technology Graduate University (OIST), 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa 904-0495, Japan.

Polymer-nanodiamond composites are excellent candidates for the fabrication of multifunctional hybrid materials. They integrate polymer flexibility and exceptional properties of nanodiamonds (NDs), such as biocompatibility, mechanical strength, color centers, and chemically-tailored surfaces. However, their development is hindered by the challenge of ensuring that NDs are homogeneously distributed in the composites. Here, we exploit colloidal coassembly between poly(isoprene-b-styrene-b-2-vinyl pyridine) (ISV) block copolymers (BCPs) and NDs to avoid ND self-agglomeration and direct ND spatial distribution. NDs were first air oxidized at 450 °C to obtain stable dispersions in dimethylacetamide (DMAc). By adding ISV into the dispersions, patchy hybrid micelles were formed due to H-bonds between NDs and ISV. The ISV-ND coassembly in DMAc was then used to fabricate nanocomposite films with a uniform sub-50 nm ND distribution, which has never been previously reported for an ND loading (φND) of more than 50 wt%. The films exhibit good transparency due to their well-defined nanostructures and smoothness and also exhibit an improved UV-absorption and hydrophilicity compared to neat ISV. More intriguingly, at a φND of 22 wt%, ISV and NDs coassemble into a network-like superstructure with well-aligned ND strings via a dialysis method. Transmission electron microscopy and dynamic light scattering measurements suggest a complex interplay between polymer-polymer, polymer-solvent, polymer-ND, ND-solvent, and ND-ND interactions during the formation of structures. Our work may provide an important foundation for the development of hierarchically ordered nanocomposites based on BCP-ND coassembly, which is beneficial for a wide spectrum of applications from biotechnology to quantum devices.

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http://dx.doi.org/10.1039/d0nr07441a | DOI Listing |

January 2021

Nanoscale 2020 Nov;12(43):22059-22069

Mathematics, Mechanics, and Materials Unit. Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan.

A prevalent strategy for synthesizing patchy nanoparticles is through the self-assembly of triblock terpolymers in selective solvents. Since the thermodynamic and kinetic factors that govern the morphology of the particles produced in this way are not fully understood, this strategy usually demands trial-and-error methodologies. We investigate the fundamental mechanisms that produce multiple types of patchy nanoparticles and identify the conditions needed to program the shapes of the nanoparticles and predict their assembly. Our findings demonstrate that particle morphology can be described in a generic fashion by accounting for the energetic balance between the conformation of the polymer coils and the formation of interfaces. This allows us to forecast the synthesis of patchy nanoparticles for systems with different triblock terpolymers and solvents. Since the shape, size, and distribution of the patches influence the growth of larger microscale structures, we construct a library of elemental nanoparticles, or building blocks, suitable for the study of hierarchically larger self-assembled aggregates and useful for streamlining the design of functional materials. Our results provide new insights into the intriguing mechanisms that determine the morphology of soft nanoscale objects, whether synthetic or naturally occurring.

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http://dx.doi.org/10.1039/d0nr06192a | DOI Listing |

November 2020

Proc Natl Acad Sci U S A 2019 01 19;116(1):90-95. Epub 2018 Dec 19.

Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

Linkages are assemblies of rigid bodies connected through joints. They serve as the basis for force- and movement-managing devices ranging from ordinary pliers to high-precision robotic arms. Aside from planar mechanisms, like the well-known four-bar linkage, only a few linkages with a single internal degree of freedom-meaning that they can change shape in only one way and may thus be easily controlled-have been known to date. Here, we present "Möbius kaleidocycles," a previously undiscovered class of single-internal degree of freedom ring linkages containing nontrivial examples of spatially underconstrained mechanisms. A Möbius kaleidocycle is made from seven or more identical links joined by revolute hinges. These links dictate a specific twist angle between neighboring hinges, and the hinge orientations induce a nonorientable topology equivalent to the topology of a [Formula: see text]-twist Möbius band. Apart from having many technological applications, including perhaps the design of organic ring molecules with peculiar electronic properties, Möbius kaleidocycles raise fundamental questions about geometry, topology, and the limitations of mobility for closed loop linkages.

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http://dx.doi.org/10.1073/pnas.1809796115 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6320513 | PMC |

January 2019

J Chem Phys 2017 Jun;146(22):224102

Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna, Okinawa 904-0495, Japan.

The classical procedure devised by Irving and Kirkwood in 1950 and completed slightly later by Noll produces counterparts of the basic balance laws of standard continuum mechanics starting from an ordinary Hamiltonian description of the dynamics of a system of material points. Post-1980 molecular dynamics simulations of the time evolution of such systems use extended Hamiltonians such as those introduced by Andersen, Nosé, and Parrinello and Rahman. The additional terms present in these extensions affect the statistical properties of the system so as to capture certain target phenomenologies that would otherwise be beyond reach. We here propose a physically consistent application of the Irving-Kirkwood-Noll procedure to the extended Hamiltonian systems of material points. Our procedure produces balance equations at the continuum level featuring non-standard terms because the presence of auxiliary degrees of freedom gives rise to additional fluxes and sources that influence the thermodynamic and transport properties of the continuum model. Being aware of the additional contributions may prove crucial when designing multiscale computational schemes in which information is exchanged between the atomistic and continuum levels.

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http://dx.doi.org/10.1063/1.4984823 | DOI Listing |

June 2017

Soft Matter 2017 Sep;13(35):5832-5841

Mathematics, Mechanics, and Materials Unit, OIST Graduate University, Onna-son, Okinawa, 904-0495, Japan.

A phase-field model is used to capture the surfactant-driven formation of fracture patterns in particulate monolayers. The model is intended for the regime of closely-packed systems in which the mechanical response of the monolayer can be approximated as that of a linearly elastic solid. The model approximates the loss in tensile strength of the monolayer with increasing surfactant concentration through the evolution of a damage field. Initial-boundary value problems are constructed and spatially discretized with finite element approximations to the displacement and surfactant damage fields. A comparison between model-based simulations and existing experimental observations indicates a qualitative match in both the fracture patterns and temporal scaling of the fracture process. The importance of surface tension differences is quantified by means of a dimensionless parameter, revealing thresholds that separate different regimes of fracture. These findings are supported by newly performed experiments that validate the model and demonstrate the strong sensitivity of the fracture pattern to differences in surface tension.

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http://dx.doi.org/10.1039/c7sm01245d | DOI Listing |

September 2017

J Nonlinear Sci 2017 13;27(3):1043-1063. Epub 2017 Jan 13.

Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna, Okinawa 904-0495 Japan.

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of noninterpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.

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http://dx.doi.org/10.1007/s00332-017-9359-4 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5479363 | PMC |

January 2017

Proc Math Phys Eng Sci 2017 Feb;473(2198):20160703

Okinawa Institute of Science and Technology Graduate University , Onna, Okinawa 904-0495, Japan.

A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.

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http://dx.doi.org/10.1098/rspa.2016.0703 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5332610 | PMC |

February 2017

Proc Math Phys Eng Sci 2016 Aug;472(2192):20160459

Mathematical Soft Matter Unit , Okinawa Institute of Science and Technology Graduate University , Onna, Okinawa 904-0495, Japan.

A Möbius band can be formed by bending a sufficiently long rectangular unstretchable material sheet and joining the two short ends after twisting by 180. This process can be modelled by an isometric mapping from a rectangular region to a developable surface in three-dimensional Euclidean space. Attempts have been made to determine the equilibrium shape of a Möbius band by minimizing the bending energy in the class of mappings from the rectangular region to the collection of developable surfaces. In this work, we show that, although a surface obtained from an isometric mapping of a prescribed planar region must be developable, a mapping from a prescribed planar region to a developable surface is not necessarily isometric. Based on this, we demonstrate that the notion of a rectifying developable cannot be used to describe a pure bending of a rectangular region into a Möbius band or a generic ribbon, as has been erroneously done in many publications. Specifically, our analysis shows that the mapping from a prescribed planar region to a rectifying developable surface is isometric only if that surface is cylindrical with the midline being the generator. Towards providing solutions to this issue, we discuss several alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometrical notion of developability.

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http://dx.doi.org/10.1098/rspa.2016.0459 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5014118 | PMC |

August 2016

Soft Matter 2016 Oct 1;12(37):7735-46. Epub 2016 Aug 1.

Department of Mechanical Engineering and Materials Science, Duke University, Box 90300 Hudson Hall, Durham, NC 27708, USA.

Experiments and simulations are used to study the kinetics of crystal growth in a mixture of magnetic and nonmagnetic particles suspended in ferrofluid. The growth process is quantified using both a bond order parameter and a mean domain size parameter. The largest single crystals obtained in experiments consist of approximately 1000 particles and form if the area fraction is held between 65-70% and the field strength is kept in the range of 8.5-10.5 Oe. Simulations indicate that much larger single crystals containing as many as 5000 particles can be obtained under impurity-free conditions within a few hours. If our simulations are modified to include impurity concentrations as small as 1-2%, then the results agree quantitatively with the experiments. These findings provide an important step toward developing strategies for growing single crystals that are large enough to enable follow-on investigations across many subdisciplines in condensed matter physics.

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http://dx.doi.org/10.1039/c6sm01072e | DOI Listing |

October 2016

Soft Matter 2016 Apr;12(16):3750-9

Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa, Japan 904-0495.

We use a two-dimensional discrete, lattice-based model to show that Möbius bands made with stretchable materials are less likely to crease or tear. This stems from a delocalization of twisting strain that occurs if stretching is allowed. The associated low-energy configurations provide strategic target shapes for the guided assembly of nanometer and micron scale Möbius bands. To predict macroscopic band shapes for a given material, we establish a connection between stretchability and relevant continuum moduli, leading to insight regarding the practical feasibility of synthesizing Möbius bands from materials with continuum parameters that can be measured experimentally or estimated by upscale averaging.

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http://dx.doi.org/10.1039/c5sm02188j | DOI Listing |

April 2016

Appl Mech Rev 2014 Sep 29;66(5):0508021-5080216. Epub 2014 May 29.

Professor Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology , 1919-1 Tancha, Onna-son, Kunigami-gun , Okinawa, Japan 904-0495 e-mail:

Transport theorems, such as that named after Reynolds, are an important tool in the field of continuum physics. Recently, Seguin and Fried used Harrison's theory of differential chains to establish a transport theorem valid for evolving domains that may become irregular. Evolving irregular domains occur in many different physical settings, such as phase transitions or fracture. Here, emphasizing concepts over technicalities, we present Harrison's theory of differential chains and the results of Seguin and Fried in a way meant to be accessible to researchers in continuum physics. We also show how the transport theorem applies to three concrete examples and approximate the resulting terms numerically. Furthermore, we discuss how the transport theorem might be used to weaken certain basic assumptions underlying the description of continua and the challenges associated with doing so.

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http://dx.doi.org/10.1115/1.4026910 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4240036 | PMC |

September 2014

Soft Matter 2014 Dec 10;10(45):9082-9. Epub 2014 Oct 10.

Kamstrup A/S, Industrivej 28, Stilling, 8660 Skanderborg, Denmark.

Many biological systems consist of self-motile and passive agents both of which contribute to overall functionality. However, little is known about the properties of such mixtures. Here we formulate a model for mixtures of self-motile and passive agents and show that the model gives rise to three different dynamical phases: a disordered mesoturbulent phase, a polar flocking phase, and a vortical phase characterized by large-scale counter rotating vortices. We use numerical simulations to construct a phase diagram and compare the statistical properties of the different phases with observed features of self-motile bacterial suspensions. Our findings afford specific insights regarding the interaction of microorganisms and passive particles and provide novel strategic guidance for efficient technological realizations of artificial active matter.

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http://dx.doi.org/10.1039/c4sm01562b | DOI Listing |

December 2014

Arch Ration Mech Anal 2012 Dec;206(3):1039-1072

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.

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http://dx.doi.org/10.1007/s00205-012-0550-3 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3679949 | PMC |

December 2012

Arch Ration Mech Anal 2013 Jan;207(1):1-37

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit formulas for these terms are obtained. Among the list of derived balances appear two, the tensor moment of inertia balance and the mesofluctuation balance, that are not standard in previously proposed macroscopic theories for liquid crystals but which have precedents in other theories for structured media.

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http://dx.doi.org/10.1007/s00205-012-0551-2 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3611664 | PMC |

January 2013

J R Soc Interface 2013 Jun 20;10(83):20130112. Epub 2013 Mar 20.

Department of Mechanical Engineering, McGill University, Montréal, QC, Canada.

A simple model is used to study the equilibrium of lipid domains on two-phase vesicles. Two classes of configurations are considered: multidomain and ground state configurations. For multidomain configurations, the vesicle has a finite number of identical lipid domains. For ground state configurations, the vesicle is fully phase separated into two coexisting domains. Whereas the volume enclosed by a vesicle with multidomains is fixed, the volume enclosed by a vesicle in a ground state is allowed to vary with the osmotic pressure. Guided by experimental observations, all domains are assumed to be spherical caps. In a multidomain configuration, the line tension is found to decrease with the number of domains present, with possible exceptions when the number of domains is very small. The importance of a critical osmotic pressure and a critical excess radius on ground state configurations is explored. Emphasis is placed on understanding the variations of these critical quantities with relevant parameters.

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http://dx.doi.org/10.1098/rsif.2013.0112 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3645419 | PMC |

June 2013

J Math Biol 2014 Feb 7;68(3):647-65. Epub 2013 Feb 7.

Department of Mathematics and Statistics, 805 Sherbrooke St. W., Montreal, QC, H3A 2K6, Canada,

The Canham-Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham-Helfrich free-energy density. Our derivation (1) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (2) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.

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http://dx.doi.org/10.1007/s00285-013-0647-9 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3694997 | PMC |

February 2014

Biomech Model Mechanobiol 2013 Oct 6;12(5):997-1017. Epub 2012 Dec 6.

Department of Mechanical Engineering, McGill University, Montréal, QC, H3A 0C3, Canada.

Continuum mechanical tools are used to describe the deformation, energy density, and material symmetry of a lipid bilayer with spontaneous curvature. In contrast to conventional approaches in which lipid bilayers are modeled by material surfaces, here we rely on a three-dimensional approach in which a lipid bilayer is modeling by a shell-like body with finite thickness. In this setting, the interface between the leaflets of a lipid bilayer is assumed to coincide with the mid-surface of the corresponding shell-like body. The three-dimensional deformation gradient is found to involve the curvature tensors of the mid-surface in the spontaneous and the deformed states, the deformation gradient of the mid-surface, and the transverse deformation. Attention is also given to the coherency of the leaflets and to the area compatibility of the closed lipid bilayers (i.e., vesicles). A hyperelastic constitutive theory for lipid bilayers in the liquid phase is developed. In combination, the requirements of frame indifference and material symmetry yield a representation for the energy density of a lipid bilayer. This representation shows that three scalar invariants suffice to describe the constitutive response of a lipid bilayer exhibiting in-plane fluidity and transverse isotropy. In addition to exploring the geometrical and physical properties of these invariants, fundamental constitutively associated kinematical quantities are emphasized. On this basis, the effect on the energy density of assuming that the lipid bilayer is incompressible is considered. Lastly, a dimension reduction argument is used to extract an areal energy density per unit area from the three-dimensional energy density. This step explains the origin of spontaneous curvature in the areal energy density. Importantly, along with a standard contribution associated with the natural curvature of the lipid bilayer, our analysis indicates that constitutive asymmetry between the leaflets of the lipid bilayer gives rise to a secondary contribution to the spontaneous curvature.

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http://dx.doi.org/10.1007/s10237-012-0459-7 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3695068 | PMC |

October 2013

Biomech Model Mechanobiol 2013 Jun 21;12(3):597-615. Epub 2012 Aug 21.

Department of Civil Engineering, Duke University, Durham, NC 27708, USA.

A chemo-mechanical model is used to capture the formation and evolution of microdomains on the deforming surface of giant unilamellar vesicles. The model is intended for the regime of vesicle dynamics characterized by a distinct difference in time scales between shape change and species transport. This is achieved by ensuring that shape equilibrium holds away from chemical equilibrium. Conventional descriptions are used to define the curvature and chemical contributions to the vesicle energetics. Both contributions are consistently non-dimensionalized. The phase-field framework is used to cast the coupled model in a diffuse-interface form. The resulting fourth-order nonlinear system of equations is discretized using the finite- element method with a uniform cubic spline basis, which satisfies global higher-order continuity. Two-dimensional and axisymmetric numerical examples of domain evolution coupled to vesicle shape deformation are presented. Curvature-dependent domain sorting and shape deformation dominated by line tension are also considered.

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http://dx.doi.org/10.1007/s10237-012-0428-1 | DOI Listing |

June 2013

Annu Rev Biophys 2010 ;39:207-26

Department of Biochemistry and Molecular Biophysics, Washington University School of Medicine, and Department of Physics, Washington University, St. Louis, Missouri 63110, USA.

Lipid bilayer model membranes that contain a single lipid species can undergo transitions between ordered and disordered phases, and membranes that contain a mixture of lipid species can undergo phase separations. Studies of these transformations are of interest for what they can tell us about the interaction energies of lipid molecules of different species and conformations. Nanoscopic phases (<200 nm) can provide a model for membrane rafts, specialized membrane domains enriched in cholesterol and sphingomyelin, which are believed to have essential biological functions in cell membranes. Crucial questions are whether lipid nanodomains can exist in stable equilibrium in membranes and what is the distribution of their sizes and lifetimes in membranes of different composition. Theoretical methods have supplied much information on these questions, but better experimental methods are needed to detect and characterize nanodomains under normal membrane conditions. This review summarizes linkages between theoretical and experimental studies of phase separation in lipid bilayer model membranes.

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http://dx.doi.org/10.1146/annurev.biophys.093008.131238 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3694198 | PMC |

July 2010

Phys Rev E Stat Nonlin Soft Matter Phys 2009 Sep 14;80(3 Pt 1):030301. Epub 2009 Sep 14.

Department of Mechanical, Aerospace & Structural Engineering, Washington University in St Louis, Missouri 63130, USA.

We report on the dynamical and statistical behavior of flowing collections of granular chains confined two-dimensionally (2D) within a rotating tumbler. Experiments are conducted with systems of chains of fixed length, but various lengths are considered. The dynamics are punctuated by cascades of chains along a free-surface cascades, which drive the development of mixed porous/laminar packing arrangements in bulk. We investigate the conformation of the system, as characterized by the porosity of the flow region occupied by the chains and the mean-square end-to-end distance of the chains during flow. Both of these measures show crossover transitions from a 2D self-avoiding walk to a 2D random walk when the chain length becomes long enough to allow self-contact.

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http://dx.doi.org/10.1103/PhysRevE.80.030301 | DOI Listing |

September 2009

J Colloid Interface Sci 2009 Nov 6;339(2):502-10. Epub 2009 Aug 6.

Department of Mechanical and Nuclear Engineering, Pennsylvania State University, 157 Hammond Building, University Park, PA 16802, United States.

We use a recently developed set of boundary conditions to study the stability of an interface between the uniaxial and isotropic phases of a nematic liquid crystal. In particular, we require satisfaction of director- and configurational-momentum balances imposed on the interface. Using linear analysis, we determine a stability condition for the moving interface and analyze the relevant marginal stability curves. We also study the effect of the front velocity and the stabilizing influence of the various dissipative mechanisms entering the theory on the perturbation growth-rate and wave-numbers. Cut-off wave-numbers arising from the analysis provide a short wave-length boundary for growing perturbations. The proposed theory describing instabilities of a uniaxial-isotropic interface in a system without impurities provides the limiting case for diffusion models driven by impurity gradients in the nematic and isotropic phases.

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http://dx.doi.org/10.1016/j.jcis.2009.06.065 | DOI Listing |

November 2009

Phys Rev E Stat Nonlin Soft Matter Phys 2009 Apr 28;79(4 Pt 2):045307. Epub 2009 Apr 28.

Department of Mechanical Engineering, McGill University, Montréal, Quebec, Canada H3A 2K6.

We study the effect of the length scales alpha and beta in the Navier-Stokes- alphabeta equations on the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor in three-dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier-Stokes- alpha and Navier-Stokes equations. A significant increase in the accuracy of the energy spectrum at large wave numbers arises for beta

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http://dx.doi.org/10.1103/PhysRevE.79.045307 | DOI Listing |

April 2009

Phys Rev E Stat Nonlin Soft Matter Phys 2008 Oct 31;78(4 Pt 2):046317. Epub 2008 Oct 31.

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24060-0002, USA.

Lundgren's vortex model for the intermittent fine structure of high-Reynolds-number turbulence is applied to the Navier-Stokes alphabeta equations and specialized to the Navier-Stokes alpha equations. The Navier-Stokes alphabeta equations involve dispersive and dissipative length scales alpha and beta, respectively. Setting beta equal to alpha reduces the Navier-Stokes alphabeta equations to the Navier-Stokes alpha equations. For the Navier-Stokes alpha equations, the energy spectrum is found to obey Kolmogorov's -5/3 law in a range of wave numbers identical to that determined by Lundgren for the Navier-Stokes equations. For the Navier-Stokes alphabeta equations, Kolmogorov's -5/3 law is also recovered. However, granted that beta < alpha, the range of wave numbers for which this law holds is extended by a factor of alphabeta . This suggests that simulations based on the Navier-Stokes alphabeta equations may have the potential to resolve features smaller than those obtainable using the Navier-Stokes alpha equations.

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http://dx.doi.org/10.1103/PhysRevE.78.046317 | DOI Listing |

October 2008

J Colloid Interface Sci 2009 Jan 17;329(1):140-52. Epub 2008 Sep 17.

Department of Mechanical and Nuclear Engineering, Pennsylvania State University, 157 Hammond Building, University Park, PA 16802, USA.

We derive a supplemental evolution equation for a disclination line located on an interface between the uniaxial and isotropic phases of a nematic liquid crystal. This equation provides an additional kinetic relation accounting for the motion of an interfacial disclination line. In our treatment of the problem we neglect fluid motion. Our approach is based on the notion of configurational forces. To illustrate the role of our additional evolution equation, we consider two simple examples. We also identify an expression for the configurational force exerted by the uniaxial phase at the defect located on the phase interface.

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http://dx.doi.org/10.1016/j.jcis.2008.09.032 | DOI Listing |

January 2009

J Colloid Interface Sci 2008 Jan 21;317(1):298-313. Epub 2007 Sep 21.

Department of Mechanical & Aerospace Engineering, Washington University in St. Louis, Campus Box 1185, St. Louis, MO 63130-4899, USA.

We consider the interaction between a disclination line of strength +/-1/2 and an interface between the uniaxial and isotropic phases of a nematic liquid crystal. We apply a recently developed set of interface conditions including a configurational force balance which generalizes the Gibbs-Thomson equation to account for the curvature elasticity of the uniaxial phase and the orientation dependence of the interfacial free-energy density. We consider a rectangular vessel containing both phases and a disclination. We formulate a relevant free-boundary problem and use numerical methods to determine equilibrium shapes of the interface. When the interfacial free-energy is constant, the shape of the interface is insensitive to whether the strength of the defect is +1/2 or -1/2 and to rotations of the director field consistent with the boundary conditions. Accounting for the dependence of the interfacial free-energy density on the angle between the interfacial unit normal field and the director field eliminates these degeneracies. In particular, when such dependence is taken into account, different solution branches are found, indicating the presence of a bifurcation. We find also that, depending on the magnitude of the anisotropic contribution to the interfacial free-energy density, the interaction between the disclination and the interface may be repulsive or attractive. When the interaction is repulsive, the disclination line positions itself at an energetically optimal distance adjacent to the interface. Otherwise, the uniaxial phase expels the disclination to the interface where a cusp forms.

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http://dx.doi.org/10.1016/j.jcis.2007.09.047 | DOI Listing |

January 2008

Phys Rev E Stat Nonlin Soft Matter Phys 2007 May 15;75(5 Pt 2):056306. Epub 2007 May 15.

Department of Mechanical and Aerospace Engineering, Washington University in St. Louis, St. Louis, Missouri 63130-4899, USA.

We present a continuum-mechanical formulation and generalization of the Navier-Stokes alpha theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales alpha and beta. The first of these scales enters the theory through the internal kinetic energy, per unit mass, alpha2|D|2, where D is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When alpha and beta are equal, our flow equation reduces to the Navier-Stokes alpha equation. In contrast to the original derivation of the Navier-Stokes alpha equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale l characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully developed turbulent flow in a rectangular channel of height 2h, we find that alphabeta approximately Re(0.470) and lh approximately Re(-0.772), where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice alpha=beta required to reduce our flow equation to the Navier-Stokes alpha equation is likely to be problematic. The second result evinces the classical scaling relation eta/L approximately Re(-3/4) for the ratio of the Kolmogorov microscale eta to the integral length scale L . The numerical data also suggests that l < or = beta . We are therefore led to conjecture a tentative hierarchy, l < or = beta < alpha , involving the three length scales entering our theory.

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http://dx.doi.org/10.1103/PhysRevE.75.056306 | DOI Listing |

May 2007

Phys Rev E Stat Nonlin Soft Matter Phys 2006 Jun 16;73(6 Pt 1):061601. Epub 2006 Jun 16.

Department of Mechanical and Aerospace Engineering, Washington University in St. Louis, 1 Brookings Drive, Box 1185, St. Louis, Missouri 63130-4899, USA.

We develop a complete set of equations governing the evolution of a sharp interface separating a volatile-solvent/nonvolatile-surfactant solution from a vapor atmosphere. In addition to a sorption isotherm equation and the conventional balances for mass, linear momentum, and energy, these equations include an alternative to the Hertz-Knudsen-Langmuir equation familiar from conventional theories of evaporation and condensation. This additional equation arises from a consideration of configurational forces within a thermodynamical framework. While the notion of configurational forces is well developed and understood for the description of materials that, like crystalline solids, possess natural reference configurations, very little has been done regarding their role in materials, such as viscous fluids, that do not possess preferred reference states. We therefore provide comprehensive developments of configurational forces, the balance of configurational momentum, and configurational thermodynamics. Our treatment does not require a choice of reference configuration. The general evolution equations arising from our theory account for the thermodynamic structure of the solution and the interface and for sources of dissipation related to the transport of surfactant, momentum, and heat in the solution and within the interface along with the transport of solute, momentum, kinetic energy, and heat across the interface. Moreover, the equations account for the Soret and Dufour effects in the solution and on the interface and for observed discontinuities of the temperature and chemical potential across the interface. Due to the complexity of these equations, we provide approximate equations which we compare to equations preexistent in the literature.

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http://dx.doi.org/10.1103/PhysRevE.73.061601 | DOI Listing |

June 2006

J Chem Phys 2006 Jan;124(2):024908

Department of Mechanical and Aerospace Engineering, Washington University in Saint Louis, Saint Louis, Missouri 63130-4899, USA.

In a nematic elastomer the deformation of the polymer network chains is coupled to the orientational order of the mesogenic groups. Statistical arguments have derived the so-called neoclassical free energy that models this coupling. Here we show that the neoclassical model supplemented by the usual Frank energy predicts incompatible network strains associated with the formation of standard nematic textures. The incompatibility is measured by the Riemann curvature tensor, which we find to be nonzero for both radial hedgehog defects and escaped disclinations of strength +1 in circular cylinders. Analogous problems for conventional nonlinearly elastic solids do not possess solutions with such incompatibilities. Compatibility in nematic elastomers would require either more complicated nematic textures in elastomers than in conventional (polymeric and low molecular weight) liquid crystals or a free-energy density more complicated than the neoclassical expression.

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http://dx.doi.org/10.1063/1.2149857 | DOI Listing |

January 2006

J Chem Phys 2005 Jul;123(4):044901

Department of Mechanical and Aerospace Engineering, Washington University in St. Louis, 1 Brookings Drive, Box 1185, St. Louis, Missouri 63130-4899, USA.

Nematic elastomers exhibit large, spontaneous shape changes at the transition from the high-temperature isotropic phase to the low-temperature nematic phase. These finite deformations are studied here in the context of a nonlinear, properly invariant, variational theory that couples the orientational order and elastic deformation. The theory is based on the minimization of a free-energy functional that consists of two contributions: a nematic one due to the interaction of the mesogenic units and an elastic one arising from the stretching of the cross-linked polymer chains. Suitable choices for these two contributions allow for large, reversible, spontaneous shape changes in which the elastic deformation can affect the isotropic-nematic transition temperature. The change in transition temperature as well as the magnitude of the resulting spontaneous deformation is illustrated for various parameter values. The theory includes soft elasticity as a special case but is not restricted to it.

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http://dx.doi.org/10.1063/1.1979479 | DOI Listing |

July 2005

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