Marangoni spreading on liquid substrates in new media art.

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 nonlinear rheological properties of the liquid substrate, coupled with the Laplacian nature of Marangoni spreading, can overshadow the local kinetic roughening of the droplet interface.

Modelling of surface reactions and diffusion mediated by bulk diffusion.

We develop a continuum framework applicable to solid-state hydrogen storage, cell biology and other scenarios where the diffusion of a single constituent within a bulk region is coupled via adsorption/desorption to reactions and diffusion on the boundary of the region. We formulate content balances for all relevant constituents and develop thermodynamically consistent constitutive equations. The latter encompass two classes of kinetics for adsorption/desorption and chemical reactions-fast and Marcelin-De Donder, and the second class includes mass action kinetics as a special case. We apply the framework to derive a system consisting of the standard diffusion equation in bulk and FitzHugh-Nagumo type surface reaction-diffusion system of equations on the boundary. We also study the linear stability of a homogeneous steady state in a spherical region and establish sufficient conditions for the occurrence of instabilities driven by surface diffusion. These findings are verified through numerical simulations which reveal that instabilities driven by diffusion lead to the emergence of steady-state spatial patterns from random initial conditions and that bulk diffusion can suppress spatial patterns, in which case temporal oscillations can ensue. We include an extension of our framework that accounts for mechanochemical coupling when the bulk region is occupied by a deformable solid. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Early stages of polycrystalline diamond deposition: laser reflectance at substrates with growing nanodiamonds.

The chemical vapor deposition of polycrystalline diamond (PCD) films is typically done on substrates seeded with diamond nanoparticles. Specular laser reflectance has been used in tandem with a continuous film model to monitor the thickness of these films during their deposition. However, approaches to gain information on properties that strongly affect film morphology, such as the areal density of seeds, remain largely unexplored. This work outlines a strategy for using laser reflectance measurements to refine the monitoring of film thickness during deposition, estimate the mean equivalent radii and the areal density of seeds, and estimate growth incubation periods. We present a general model based on the Rayleigh theory of scattering for laser reflectance at substrates with growing nanoparticles that captures the early stages of PCD deposition. We test our model experimentally by depositing diamond under identical conditions on silicon substrates with various seed densities and by comparing seed densities obtained by scanning electron microscopy to those determined by our strategy. We also explore the different deposition stages for which our model and a continuous film model can be used safely. In addition to providing guidelines for characterizing PCD deposition, this work may also advance the general understanding of nanoparticle growth and formation.

Block copolymer-nanodiamond coassembly in solution: towards multifunctional hybrid materials.

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.

Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve.

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ. The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies |k| ≤ κ, and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k.

Entropic factors and structural motifs of triblock-terpolymer-based patchy nanoparticles.

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.

Single degree of freedom everting ring linkages with nonorientable topology.

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.

Continuum balances from extended Hamiltonian dynamics.

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.

Influence of surface tension in the surfactant-driven fracture of closely-packed particulate monolayers.

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.

Solution of the Kirchhoff-Plateau Problem.

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.

Stability of vertical magnetic chains.

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.

Möbius bands, unstretchable material sheets and developable surfaces.

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.

Crystallization kinetics of binary colloidal monolayers.

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.

Influence of material stretchability on the equilibrium shape of a Möbius band.

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.

Extending the Transport Theorem to Rough Domains of Integration.

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.

Motility versus fluctuations in mixtures of self-motile and passive agents.

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.

Microphysical derivation of the Canham-Helfrich free-energy density.

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.

Statistical foundations of liquid-crystal theory: II: Macroscopic balance laws.

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.

Multidomain and ground state configurations of two-phase vesicles.

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.