Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Faxén relations, as well as higher-order displaceability and rotateability matrices

Rotational dynamics of a disk in a thin film of weakly nematic fluid subject to linear friction

Dynamics at low Reynolds numbers experiences recent revival in the fields of biophysics and active matter. While in bulk isotropic fluids it is exhaustively studied, this is less so in anisotropic fluids and in confined situations. Here, we combine the latter two by studying the rotation of a disk-like inclusion in a uniaxially anisotropic, globally oriented, incompressible two-dimensional fluid film. In terms of a perturbative expansion in parameters that quantify anisotropies in viscosity and in additional linear friction with a supporting substrate or other type of confinement, we derive analytical expressions for the resulting hydrodynamic flow and pressure fields as well as for the resistance and mobility coefficients of the rotating disk. It turns out that, in contrast to translational motion, the solutions remain well-behaved also in the absence of the additional linear friction. Comparison with results from finite-element simulations shows very good agreement with those from our analytical calculations. Besides applications to describe technological systems, for instance, in the area of microfluidics and thin cells of aligned nematic liquid crystals, our solutions are important for quantitative theoretical approaches to fluid membranes and thin films in general featuring a preferred direction.

Internal sites of actuation and activation in thin elastic films and membranes of finite thickness

Functionalized thin elastic films and membranes frequently feature internal sites of net forces or stresses. These are, for instance, active sites of actuation, or rigid inclusions in a strained membrane that induce counterstress upon externally imposed deformations. We theoretically analyze the geometry of isotropic, flat, thin, linearly elastic films or membranes of finite thickness, laterally extended to infinity. At the mathematical core of such characterizations are the fundamental solutions for localized force and stress singularities associated with corresponding Green's functions. We derive such solutions in three dimensions and place them into the context of previous two-dimensional calculations. To this end, we consider both no-slip and stress-free conditions at the top and/or bottom surfaces. We provide an understanding for why the fully free-standing thin elastic membrane leads to diverging solutions in most geometries and compare these situations to the truly two-dimensional case. A no-slip support of at least one of the surfaces stabilizes the solution, which illustrates that the divergences in the fully free-standing case are connected to global deformations. Within the aforementioned framework, our results are important for associated theoretical characterizations of thin elastic films, whether supported or free-standing, and of membranes subject to internal or external forces or stresses.

Effect of boundaries on displacements and motion in two-dimensional fluid or elastic films and membranes

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, the resulting flows in an unbounded two-dimensional incompressible low-Reynolds-number fluid or displacements in a two-dimensional linearly elastic solid seem to diverge logarithmically with the distance from the force center, which has led to some debate. Recently, we have demonstrated that such divergences cancel when the total (net) force vanishes. Here, we illustrate that if a net force is present, the boundaries play a prominent role. Already a single no-slip boundary regulates the flow and displacement fields and leads to their decay to leading order inversely in distance from a force center and the boundary. In other words, it is the boundary that stabilizes the system in this situation, unlike the three-dimensional case, where an unbounded medium by itself is able to absorb a net force. We quantify the mobility and displaceability of an inclusion as a function of the distance from the boundary, as well as interactions between different inclusions. In the case of free-slip boundary conditions, a kinked boundary is necessary to achieve stabilization.

Mediated interactions between rigid inclusions in two-dimensional elastic or fluid films

Interactions between rigid inclusions in continuous three-dimensional linearly elastic solids and low-Reynolds-number viscous fluids have largely been quantified in the past. Prime example systems are given by functionalized elastic composite materials or fluid colloidal suspensions. Here, we address the significantly less frequently studied situation of rigid inclusions in two-dimensional elastic or low-Reynolds-number fluid films. We concentrate on the situation in which disklike inclusions remain well separated from each other and do not get into contact. Specifically, we demonstrate and explain that the logarithmic divergence of the associated Green's function is removed in the absence of net external forces on the inclusions, in line with physical intuition. For instance, this situation applies when only pairwise mutual interactions between the inclusions prevail. Our results will support, for example, investigations on membranes functionalized by appropriate inclusions, both of technical or biological origin, or the dynamics of active microswimmers in appropriately prepared thin films.

Collision and separation of nickel particles embedded in a polydimethylsiloxan matrix under a rotating magnetic field: A strong magneto active function

In order to function as soft actuators, depending on their field of use, magnetorheological elastomers (MREs) must fulfill certain criteria. To name just a few, these can include rapid response to external magnetic fields, mechanical durability, mechanical strength, and/or large deformation. Of particular interest are MREs which produce macroscopic deformation for small external magnetic field variations. This work demonstrates how this can be achieved by just a small change in magnetic field orientation. To achieve this, (super)paramagnetic nickel particles of size ≈ 160 μm were embedded in a non-magnetic polydimethylsiloxan (PDMS) (661–1301 Pa) and their displacement in a stepwise rotated magnetic field (170 mT) recorded using a video microscope. Changes in particle aggregation resulting from very small variations in magnetic field orientation led to the observation of a new strongly magneto-active effect. This configuration is characterized by an interparticle distance in relation to the angle difference between magnetic field and particle axis. This causes a strong matrix deformation which in turn demonstrates hysteresis on relaxation. It is shown that the occurrence strongly depends on the particle size, particle distance, and stiffness of the matrix. Choosing the correct parameter combination, the state can be suppressed and the particle-matrix system demonstrates no displacement or hysteresis. In addition, evidences of non-negligible higher order magnetization effects are experimentally ascertained which is qualitatively in agreement with similar, already theoretically described, particle systems. Even at larger particle geometries, the new strongly magneto-active configuration is preserved and could create macroscopic deformation changes.

Frequency-dependent magnetic susceptibility of magnetic nanoparticles in a polymer solution: a simulation study

Magnetic composite materials i.e. elastomers, polymer gels, or polymer solutions with embedded magnetic nanoparticles are useful for many technical and bio-medical applications. However, the microscopic details of the coupling mechanisms between the magnetic properties of the particles and the mechanical properties of the (visco)elastic polymer matrix remain unresolved. Here we study the response of a single-domain spherical magnetic nanoparticle that is suspended in a polymer solution to alternating magnetic fields. As interactions we consider only excluded volume interactions with the polymers and hydrodynamic interactions mediated through the solvent. The AC susceptibility spectra are calculated using a linear response Green-Kubo approach, and the influences of changing polymer concentration and polymer length are investigated. Our data is compared to recent measurements of the AC susceptibility for a typical magnetic composite system [Roeben et al., Colloid Polym. Sci., 2014, 2013-2023], and demonstrates the importance of hydrodynamic coupling in such systems.

Modeling and theoretical description of magnetic hybrid materials—bridging from meso- to macro-scales

Magnetic gels and elastomers consist of magnetic or magnetizable colloidal particles embedded in an elastic polymeric matrix. Outstanding properties of these materials comprise reversible changes in their mechanical stiffness or magnetostrictive distortions under the influence of external magnetic fields. To understand such types of overall material behavior from a theoretical point of view, it is essential to characterize the substances starting from the discrete colloidal particle level. It turns out that the macroscopic material response depends sensitively on the mesoscopic particle arrangement. We have utilized and developed several theoretical approaches to this end, allowing us both to reproduce experimental observations and to make theoretical predictions. Our hope is that both these paths help to further stimulate the interest in these fascinating materials.

Magnetic torque-driven deformation of Ni-nanorod/hydrogel nanocomposites

Nickel (Ni) nanorods were prepared by the anodized aluminum oxide (AAO) template method and dispersed in poly(acrylamide) (PAM) hydrogels. The deformation of the magnetoresponsive composites was studied with particular attention to the consequences of finite magnetic shape anisotropy as compared to rigid dipoles on the field-dependent torque. For comparison with experiments, the composite was described as an elastic continuum with a local magnetic torque density, applied by discrete particles and determined by the local orientation of their magnetic anisotropy axis with respect to the magnetic field. The mean magnetic moment of the single domain particles m and their volume density in the composite φ vol were derived from the static field-dependent optical transmission (SFOT) of linear polarized light. The mechanical coupling between the particles and their viscoelastic environment was retrieved from the rotational dynamics of the nanorods using oscillating field-dependent optical transmission (OFOT) measurements. Field- and orientation-dependent magnetization measurements were analyzed using the Stoner–Wohlfarth (SW) model and a valid parameter range was identified by introducing an effective anisotropy constant K A as a new empirical model parameter. This adapted SW-model for quantitative description of the field- and orientation dependence of the magnetic torque was validated by measuring the local rotation of nanorods in a soft elastic hydrogel. Finally, torsional and bending deformation of thin magnetically textured composite filaments were computed and compared with experiments.

Hydrodynamic mobility reversal of squirmers near flat and curved surfaces

Self-propelled particles have been experimentally shown to orbit spherical obstacles and move along surfaces. Here, we theoretically and numerically investigate this behavior for a hydrodynamic squirmer interacting with spherical objects and flat walls using three different methods of approximately solving the Stokes equations: The method of reflections, which is accurate in the far field; lubrication theory, which describes the close-to-contact behavior; and a lattice Boltzmann solver that accurately accounts for near-field flows. The method of reflections predicts three distinct behaviors: orbiting/sliding, scattering, and hovering, with orbiting being favored for lower curvature as in the literature. Surprisingly, it also shows backward orbiting/sliding for sufficiently strong pushers, caused by fluid recirculation in the gap between the squirmer and the obstacle leading to strong forces opposing forward motion. Lubrication theory instead suggests that only hovering is a stable point for the dynamics. We therefore employ lattice Boltzmann to resolve this discrepancy and we qualitatively reproduce the richer far-field predictions. Our results thus provide insight into a possible mechanism of mobility reversal mediated solely through hydrodynamic interactions with a surface.