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Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives., Comment: 57 pages, 15 figures

Chiral objects have intrigued scientists across several disciplines, including mathematics, crystallography, chemistry, and biology. A M\"obius band, an emblematic chiral structure, can be made by connecting the ends of a strip after applying an odd number of twists. Traditionally, the direction of the twist governs its rotational behaviour during sedimentation in a fluid. Here, we present experimental and computational investigations of triply-twisted M\"obius bands boasting threefold rotational symmetry that challenge this prevailing understanding. We explore three types of bands with different curvatures, each defined by its construction method. Experimental observations reveal that all three types of bands align axially and exhibit rotational motion during sedimentation. Surprisingly, for only one type of band the spinning direction (chiral hydrodynamic response) departs from expectations; it is not solely determined by the twist direction but changes with the aspect ratio of the band. Numerical simulations corroborate this observation, and an in-depth analysis of the resistance tensors of each type of band sheds light on the possible causes of this transition. We propose that modifications in fluid-induced drag, combined with inertial effects, underpin this phenomenon. Our study challenges existing knowledge of chiral object hydrodynamics, enriching our understanding of complex fluid dynamics. Moreover, it offers transformative potential across diverse fields, promising advancements in mixing, separation processes, and innovative passive swimmers.

With the advent of new media art, artists have harnessed fluid dynamics to create captivating visual narratives. A striking technique known as dendritic painting employs mixtures of ink and isopropanol atop paint, yielding intricate tree-like patterns. To unravel the intricacies of that technique, we examine the spread of ink/alcohol droplets over liquid substrates with diverse rheological properties. On Newtonian substrates, the droplet size evolution exhibits two power laws, suggesting an underlying interplay between viscous and Marangoni forces. The leading edge of the droplet spreads as a precursor film with an exponent of 3/8, while its main body spreads with an exponent of 1/4. For a weakly shear-thinning acrylic resin substrate, the same power laws persist, but dendritic structures emerge, and the texture of the precursor film roughens. The observed roughness and growth exponents (3/4 and 3/5) suggest a connection to the quenched Kardar--Parisi--Zhang universality class, hinting at the existence of quenched disorder in the liquid substrate. Mixing the resin with acrylic paint renders it more viscous and shear-thinning, refining the dendrite edges and further roughening the precursor film. At larger paint concentrations, the substrate becomes a power-law fluid. The roughness and growth exponents then approach 1/2 and 3/4, respectively, deviating from known universality classes. The ensuing structures have a fractal dimension of 1.68, characteristic of diffusion-limited aggregation. These findings underscore how the non-linear rheological properties of the liquid substrate, coupled with the Laplacian nature of Marangoni spreading, can overshadow the local kinetic roughening of the droplet interface.

The chemical vapor deposition of polycrystalline diamond (PCD) films is typically done on substrates seeded with diamond nanoparticles. Specular laser reflectance and a continuous film model have been used to monitor the thickness of these films during their deposition. However, most seeds are isolated during the early stages of the deposition, which questions the utility of applying such a continuous film model for monitoring deposition before film formation. In this work, we present a model based on the Rayleigh theory of scattering for laser reflectance at substrates with growing nanodiamonds to capture the early stages of PCD deposition. The reflectance behavior predicted by our model differs from that of a continuous film, which is well-described by the continuous film model. This difference enlarges as the seed density used in our model decreases. We verify this trend experimentally by depositing diamond under identical conditions on substrates with various seed densities. A relation derived from our model is used to fit reflectance data from which seed densities are obtained that are proportional to those found with electron microscopy. We also show that relying on the continuous film model for describing the early stages of deposition can result in falsely deducing the existence of incubation, and that the continuous film model can be used safely beyond the early stages of deposition. Based on these findings, we delineate a robust method for obtaining growth rates and incubation periods from reflectance measurements. This work may also advance the general understanding of nanoparticle growth and formation., Comment: 32 pages, 9 figures, 1 table

Two-dimensional simulations are used to explore topological transitions that occur during the formation of films grown from grains that are seeded on substrates. This is done for a relatively large range of the initial value $\Phi_s$ of the grain surface fraction $\Phi$. The morphology of porous films is captured at the transition when grains connect to form a one-component network using newly developed raster-free algorithms that combine computational geometry and network theory. Further insight on the morphology of porous films and their suspended counterparts is obtained by studying the pore surface fraction $\Phi_p$, the pore over grain ratio, the pore area distribution, and the contribution of pores of certain chosen areas to $\Phi_p$. Pinhole survival is evaluated at the transition when film closure occurs using survival function estimates. The morphology of closed films ($\Phi = 1$) is also characterized and is quantified by measuring grain areas and perimeters. The majority of investigated quantities are found to depend sensitively on $\Phi_s$ and the long-time persistence of pinholes exhibits critical behavior as a function of $\Phi_s$. In addition to providing guidelines for designing effective processes for manufacturing thin films and suspended porous films with tailored properties, this work may advance the understanding of continuum percolation theory.

In this letter, we demonstrate a relation between the boundary curvature $\kappa$ and the wrinkle wavelength $\lambda$ of a thin suspended film under boundary confinement. Experiments are done with nanocrystalline diamond films of thickness $t \approx 184$~nm grown on glass substrates. By removing portions of the substrate after growth, suspended films with circular boundaries of radius $R$ ranging from approximately 30 to 811 $\mu$m are made. Due to residual stresses, the portions of film attached to the substrate are of compressive prestrain $\epsilon_0 \approx 11 \times 10^{-4}$ and the suspended portions of film are azimuthally wrinkled at their boundary. We find that $\lambda$ monotonically decreases with $\kappa$ and present a model predicting that $\lambda \propto t^{1/2}(\epsilon_0 + \Delta R \kappa)^{-1/4}$, where $\Delta R$ denotes a penetration depth over which strain relaxes at a boundary. This relation is in agreement with our experiments and may be adapted to other systems such as plant leaves. Also, we establish a novel method for measuring residual compressive strain in thin films.

Low-cost and robust platforms are key for the development of next-generation 3D micro- and nanodevices. To fabricate such platforms, nanocrystalline diamond (NCD) is a highly appealing material due to its biocompatibility, robustness, and mechanical, electrical, electrochemical, and optical properties, while glass substrates with through vias are ideal interposers for 3D integration due to the excellent properties of glass. However, developing devices that are comprised of NCD films and through glass vias (TGVs) has rarely been attempted due to a lack of effective process strategies. In this work, a low-cost process - free of photolithography and transfer-printing - for fabricating arrays of TGVs that are sealed with suspended portions of an ultra-thin NCD film on one side is presented. These highly transparent structures may serve as a platform for the development of microwells for single-cell culture and analysis, 3D integrated devices such as microelectrodes, and quantum technologies. The process is demonstrated by fabricating TGVs that are sealed with an NCD film of thickness 175 nm and diameter 60 $\mu$m. The technology described can be extended by replacing NCD with silicon nitride or silicon carbide, allowing for the development of complex heterogenous structures on the small scale.

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 a linearly elastic solid. The model approximates the loss in tensile strength of the monolayer as the surfactant concentration increases 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., Comment: 10 pages, 11 figures, and 3 tables

It is demonstrated that non-coalescent droplets of acetone can be formed on liquid substrates. The fluid flows around and in an acetone droplet hovering on water are recorded to shed light on the mechanisms which might lead to non-coalescence. For sufficiently low impact velocities, droplets undergo a damped oscillation on the surface of the liquid substrate but at higher velocities clean bounce-off occurs. Comparisons of experimentally observed static configurations of floating droplets to predictions from a theoretical model for a small non-wetting rigid sphere resting on a liquid substrate are made and a tentative strategy for determining the thickness of the vapor layer under a small droplet on a liquid is proposed. This strategy is based on the notion of effective surface tension. The droplets show self-propulsion in straight line trajectories in a manner which can be ascribed to a Marangoni effect. Surprisingly, self-propelled droplets can become immersed beneath the undisturbed water surface. This phenomenon is reasoned to be drag-inducing and might provide a basis for refining observations in previous work.

We perform a static analysis of a circular cylinder that forms a barrier between surfactant-laden and surfactant-free portions of a liquid$-$gas interface. In addition to determining the general implications of the balances for forces and torques, we quantify how the imbalance $\Delta\gamma=\gamma_a-\gamma_b$ between the uniform surface tension $\gamma_a$ of the surfactant-free portion of the interface and the uniform surface tension $\gamma_b$ of the surfactant-laden portion of the interface influences the load-bearing capacity of a hydrophobic cylinder. Moreover, we demonstrate that the difference between surface tensions on either side of a cylinder with a cross-section of arbitrary shape induces a horizontal force component $f^h$ equal to $\Delta \gamma$ in magnitude, when measured per unit length of the cylinder. With an energetic argument, we show that this relation also applies to rod-like barriers with cross-sections of variable shape. In addition, we apply our analysis to amphiphilic Janus cylinders and we discuss practical implications of our findings for Marangoni propulsion and surface pressure measurements.

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, Nos\'e, 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 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.

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves., Comment: Revised version; 25 pages; 7 figures

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 non-interpenetration 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.

We explore how the stability and buckling behavior of a flexible but inextensible loop spanned by a liquid film in a flat circular configuration is influenced by the intrinsic shape of the fiber from which the loop is made.

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 in 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., Comment: 10 pages, 11 figures

Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particles, we derive meso-scale continuum equations by applying a spatial averaging version of the Irving--Kirkwood--Noll procedure. Since the method does not rely on kinetic theory, the derivation is valid for highly concentrated particle systems. Spatial averaging yields a stochastic continuum equations similar to those of Toner and Tu. However, our theory also involves a constitutive equation for the average fluctuation force. According to this equation, both the strength and the probability distribution vary with time and position through the effective mass density. The statistics of the fluctuation force also depend on the fine scale dissipative force equation, the physical temperature, and two additional parameters which characterize fluctuation strengths. Although the self-propulsion force entering our DPD model contains no explicit mechanism for aligning the velocities of neighboring particles, our averaged coarse-scale equations include the commonly encountered cubically nonlinear (internal) body force density., Comment: To appear in Physics of Fluids

We use a discrete, lattice-based model for two-dimensional materials to show that M\"obius bands made with stretchable materials are less likely to crease or tear. This stems 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 M\"obius 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 M\"obius bands from materials with continuum parameters that can be measured experimentally or estimated by upscale averaging. To take advantage of stretchability in the case of M\"obius bands made of graphene, DNA, and other effectively unstretchable materials, we develop and explore a novel architecture that uses the Chinese finger trap as a fundamental building block and imparts notable stretchability to otherwise unstretchable materials., Comment: 10 pages, 10 figures, 11 pages supplementary information, 8 supplementary figures

A variational model is used to study the stability of a soap film spanning a flexible loop. The film is modeled as a fluid surface endowed with constant tension and the loop is modeled as an elastic rod resistant to both bending and twist. The first and second energy variations of the underlying energy functional are derived, leading to governing equilibrium equations and stability criteria. The latter criteria are employed to explore the stability of flat circular configurations with respect to planar and transverse perturbations. Experiments are performed to study post-buckled configurations and to explore the validity of the theoretical predictions. Imparting a twist to the bounding loop destabilizes flat circular loops which are short-enough to be stable when twist-free. Nevertheless, when the circular disk is perturbed merely radially or merely transversely, the stability condition is indifferent to twist. By the equilibrium equations, the twisting angle of a planar equilibrium configuration should be distributed uniformly along the loop. The stability criteria guarantee that this uniformity is preserved after buckling. The stability with respect to increasing the surface tension of the film or the length of the loop is sensitive to the cross-sectional thickness of the bounding loop. However, stability with respect to increasing the total twist is indifferent to that thickness.

This article is a translation of Michael Sadowsky's original paper "Ein elementarer Beweis f\"ur die Existenz eines abwickelbaren M\"obiusschen Bandes und die Zur\"uckf\"uhrung des geometrischen Problems auf ein Variationsproblem." which appeared in Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 17. Juli 1930, Mitteilung vom 26. Juni, 412-415. Published on September 12, 1930., Comment: 5 pages, 3 figures, to appear in a special volume of the Journal of Elasticity entitled "The Mechanics of Ribbons and M\"obius Bands", citations of this translation should refer also to Sadowsky's original paper, as cited in the Abstract

This article is a translation of Michael Sadowsky's original paper "Die Differentialgleichungen des M\"obiusschen Bandes." in Jahresbericht der Deutschen Mathematiker-Vereinigung 39 (2. Abt. Heft 5/8, Jahresversammlung vom 16. bis 23. September), 49-51 (1929), which is a short version of his paper "Theorie der elastisch biegsamen undehnbaren B\"ander mit Anwendungen auf das M\"obiussche Band" in 3. internationaler Kongress f\"ur technische Mechanik, Stockholm, 1930., Comment: 4 pages, to appear in a special volume of the Journal of Elasticity entitled "The Mechanics of Ribbons and M\"obius Bands", citations of this translation should refer also to Sadowsky's original paper, as cited in the Abstract

This article is a translation of Michael Sadowsky's original paper "Theorie der elastisch biegsamen undehnbaren B\"ander mit Anwendungen auf das M\"obiussche Band" in 3. internationaler Kongress f\"ur technische Mechanik, Stockholm, 1930. The translation is augmented by an Appendix containing an interpretation of the last section of Sadowsky's original paper including figures generated from recent numerical simulations., Comment: 12 pages, 2 original figures, 2 supplementary figures, to appear in a special volume of the Journal of Elasticity entitled "The Mechanics of Ribbons and M\"obius Bands", citations of this translation should refer also to Sadowsky's original paper, as cited in the Abstract

A generalization of the Euler-Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler-Lagrange equations are derived in a parameterized setting and a bifurcation analysis is performed. A pair of dimensionless parameters govern bifurcations from a flat, circular ground state. While one of these is familiar from the Euler-Plateau problem, the other encompasses information about the ratio of the twisting to bending rigidities, twist, and size of the loop. For sufficiently small values of the latter parameter, two separate groups of bifurcation modes are identified. On the other hand, for values greater than the critical twist arising in Michell's problem of the bifurcation of a twisted annular ring, only one bifurcation mode exists. Bifurcation diagrams indicate that a loop with greater twisting rigidity shows more resistance to transverse buckling. However, a twisted and closed filament spanned by a surface endowed with uniform surface tension buckles at a twist less than the critical value for an elastic ring.

Many biological systems consist of self-motile and passive agents both of which contribute to overall functionality. However, there are very few studies of 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 counterrotating vortices. We use numerical simulations to construct a phase diagram and compare the statistical properties of the different phases of the model with 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., Comment: 5 pages, 3 figures

The Canham--Helfrich free-energy density for a lipid bilayer involves the mean and Gaussian curvatures of the midsurface of the bilayer. The splay and saddle-splay moduli $\kappa$ and $\bar\kappa$ regulate the sensitivity of the free-energy density to changes of these curvatures. Seguin and Fried derived the Canham--Helfrich energy by taking into account the interactions between the molecules comprising the bilayer, giving rise to integral representations for the moduli in terms of the interaction potential. In the present work, two potentials are chosen and the integrals are evaluated to yield expressions for the moduli, which are found to depend on parameters associated with each potential. These results are compared with values of the moduli found in the current literature., Comment: 15 pages, 3 figures. arXiv admin note: text overlap with arXiv:1211.0880

Cox & Jones recently devised and studied an interesting variant of the classical Plateau problem, a variant in which a helical soap film is confined to a cylindrical tube with circular cross-section. Through experiments, numerics, and some analysis, they found that the length and (inner) radius of the tube strongly influence the equilibrium shape of the confined soap film. In this paper, an area minimization problem associated with determining the shape of the film is formulated and analyzed to determine which surfaces are local minima. The connection between a functional inequality and the associated eigenvalue problem plays an important role in the analysis. For helical films, a more detailed analysis is carried out and stability conditions consistent with the experimental and numerical results of Cox & Jones are obtained.

A simple model for simulating flows of active suspensions is investigated. The approach is based on dissipative particle dynamics. While the model is potentially applicable to a wide range of self-propelled particle systems, the specific class of self-motile bacterial suspensions is considered as a modeling scenario. To mimic the rod-like geometry of a bacterium, two dissipative particle dynamics particles are connected by a stiff harmonic spring to form an aggregate dissipative particle dynamics molecule. Bacterial motility is modeled through a constant self-propulsion force applied along the axis of each such aggregate molecule. The model accounts for hydrodynamic interactions between self-propelled agents through the pairwise dissipative interactions conventional to dissipative particle dynamics. Numerical simulations are performed using a customized version of the open-source LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) software package. Detailed studies of the influence of agent concentration, pairwise dissipative interactions, and Stokes friction on the statistics of the system are provided. The simulations are used to explore the influence of hydrodynamic interactions in active suspensions. For high agent concentrations in combination with dominating pairwise dissipative forces, strongly correlated motion patterns and a fluid-like spectral distributions of kinetic energy are found. In contrast, systems dominated by Stokes friction exhibit weaker spatial correlations of the velocity field. These results indicate that hydrodynamic interactions may play an important role in the formation of spatially extended structures in active suspensions.

Motivated by experimental and numerical studies revealing that discoidal high-density lipoprotein (HDL) particles may adopt flat elliptical and nonplanar saddle-like configurations, it is hypothesized that these might represent stabilized configurations of initially unstable flat circular particles. A variational description is developed to explore the stability of a flat circular discoidal HDL particle. While the lipid bilayer is modeled as two-dimensional fluid film endowed with surface tension and bending elasticity, the apoA-I belt is modeled as one-dimensional inextensible twist-free chain endowed with bending elasticity. Stability is investigated using the second variation of the underlying energy functional. Various planar and nonplanar instability modes are predicted and corresponding nondimensional critical values of salient dimensionless parameters are obtained. The results predict that the first planar and nonplanar unstable modes occur due to in-plane elliptical and transverse saddle-like perturbations. Based on available data, detailed stability diagrams indicate the range of input parameters for which a flat circular discoidal HDL particle is linearly stable or unstable.

The Euler--Plateau problem, proposed by \cite{gm}, concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler--Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For spanning surface with energy density $\sigma$ and a bounding loop with length $2\pi R$ and bending rigidity $a$, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at $\sigma R^3/a=3$, confirming a result obtained previously via an energy comparison. Other bifurcation solution branches, including those emanating from the flat circular solution branch to nonplanar solution branches, are also shown to be unstable.