Hydrodynamics of a disk in a thin film of weakly nematic fluid subject to linear friction.

To make progress towards the development of a theory on the motion of inclusions in thin structured films and membranes, we here consider as an initial step a circular disk in a two-dimensional, uniaxially anisotropic fluid layer. We assume overdamped dynamics, incompressibility of the fluid, and global alignment of the axis of anisotropy. Motion within this layer is affected by additional linear friction with the environment, for instance, a supporting substrate. We investigate the induced flows in the fluid when the disk is translated parallel or perpendicular to the direction of anisotropy. Moreover, expressions for corresponding mobilities and resistance coefficients of the disk are derived. Our results are obtained within the framework of a perturbative expansion in the parameters that quantify the anisotropy of the fluid. Good agreement is found for moderate anisotropy when compared to associated results from finite-element simulations. At pronounced anisotropy, the induced flow fields are still predicted qualitatively correctly by the perturbative theory, although quantitative deviations arise. We hope to stimulate with our investigations corresponding experimental analyses, for example, concerning fluid flows in anisotropic thin films on uniaxially rubbed supporting substrates.

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.

Vortex Pattern Stabilization in Thin Films Resulting from Shear Thickening of Active Suspensions.

The need for structuring on micrometer scales is abundant, for example, in view of phononic applications. We here outline a novel approach based on the phenomenon of active turbulence on the mesoscale. As we demonstrate, a shear-thickening carrier fluid of active microswimmers intrinsically stabilizes regular vortex patterns of otherwise turbulent active suspensions. The fluid self-organizes into a periodically structured nonequilibrium state. Introducing additional passive particles of intermediate size leads to regular spatial organization of these objects. Our approach opens a new path toward functionalization through patterning of thin films and membranes.

Density functional approach to elastic properties of three-dimensional dipole-spring models for magnetic gels.

Magnetic gels are composite materials consisting of a polymer matrix and embedded magnetic particles. Those are mechanically coupled to each other, giving rise to the magnetostrictive effects as well as to a controllable overall elasticity responsive to external magnetic fields. Due to their inherent composite and thereby multiscale nature, a theoretical framework bridging different levels of description is indispensable for understanding the magnetomechanical properties of magnetic gels. In this study, we extend a recently developed density functional approach from two spatial dimensions to more realistic three-dimensional systems. Along these lines, we connect a mesoscopic characterization resolving the discrete structure of the magnetic particles to macroscopic continuum parameters of magnetic gels. In particular, we incorporate the long-range nature of the magnetic dipole-dipole interaction and consider the approximate incompressibility of the embedding media and relative rotations with respect to an external magnetic field breaking rotational symmetry. We then probe the shape of the model system in its reference state, confirming the dependence of magnetostrictive effects on the configuration of the magnetic particles and on the shape of the considered sample. Moreover, calculating the elastic and rotational coefficients on the basis of our mesoscopic approach, we examine how the macroscopic types of behavior are related to the mesoscopic properties. Implications for real systems of random particle configurations are also discussed.

Stochastic motion under nonlinear friction representing shear thinning.

We study stochastic motion under a nonlinear frictional force that levels off with increasing velocity. Specifically, our frictional force is of the so-called Coulomb-tanh type. At small speed, it increases approximately linearly with velocity, while at large speed, it approaches a constant magnitude, similarly to solid (dry, Coulomb) friction. In one spatial dimension, a formal analogy between the associated Fokker-Planck equation and the Schrödinger equation for a quantum mechanical oscillator in a nonharmonic Pöschl-Teller potential is revealed. Then, the stationary velocity statistics can be treated analytically. From such analytical considerations, we determine associated diffusion coefficients, which we confirm by agent-based simulations. Moreover, from such simulations and from numerically solving the associated Fokker-Planck equation, we find that the spatial distribution function, starting from an initial Gaussian shape, develops tails that appear exponential at intermediate timescales. At small magnitudes of stochastic driving, the velocity distribution resembles the case of linear friction, while at large magnitudes, it rather approaches the case of solid (dry, Coulomb) friction. The same is true for diffusion coefficients. In a certain sense thus interpolating between the two extreme cases of linear friction and solid (dry, Coulomb) friction, our approach should be useful to describe several cases of practical relevance. For instance, a reduced increase in friction with increasing relative speed is typical of shear-thinning behavior. Therefore, driven motion in shear-thinning environments is one specific example to which our description may be applied.

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.

Variations in the thermal conductivity of magnetosensitive elastomers by magnetically induced internal restructuring.

Magnetosensitive elastomers respond to external magnetic fields by changing their stiffness and shape. These effects result from interactions among magnetized inclusions that are embedded within an elastic matrix. Strong external magnetic fields induce internal restructuring, for example the formation of chain-like aggregates. However, such reconfigurations affect not only the overall mechanical properties of the elastomers but also the transport through such systems. We concentrate here on the transport of heat, that is thermal conductivity. For flat, thin model systems representing thin films or membranes and modeled by bead-spring discretizations, we evaluate the internal restructuring in response to magnetization of the particles. For each resulting configuration, we evaluate the associated thermal conductivity. We analyze the changes in heat transport as a function of the strength of magnetization, particle number, density of magnetizable particles (at fixed overall particle number), and aspect ratio of the system. We observe that varying any one of these parameters can induce pronounced changes in the bulk thermal conductivity. Our results motivate future experimental and theoretical studies of systems with magnetically tunable thermal but also electric conductivity-both of which have only rarely been addressed so far.

When low-order expansions fail and all higher-order contributions matter-basic example of the mean squared displacement for Brownian motion.

Hardly any theoretically formulated realistic problem can be solved exactly. Therefore, as a standard, we resort to approximations. In this context, expansions play a major role. We are used to relying on lowest-order expansions and confining our point of view accordingly. However, one should always bear in mind that such considerations may fail at some point. Here, we address a very common example situation, namely, the motion of a Brownian particle. We know that the associated mean-squared displacement in the long term increases linearly in time. Yet, when we take the Fokker-Planck approach in combination with a low-order expansion, the direct route towards this result fails. That is, in the expansion the term linear in time vanishes. Instead, the treatment requires consideration of all higher-order contributions. Together, they restore the linear increase in time. In this way, we stress that care is always mandatory when resorting to low-order expansions, and we present in a traceable way a route to solving the considered problem.

Statistics for an object actively driven by spontaneous symmetry breaking into reversible directions.

Propulsion of otherwise passive objects is achieved by mechanisms of active driving. We concentrate on cases in which the direction of active drive is subject to spontaneous symmetry breaking. In our case, this direction will be maintained until a large enough impulse by an additional stochastic force reverses it. Examples may be provided by self-propelled droplets, gliding bacteria stochastically reversing their propulsion direction, or nonpolar vibrated hoppers. The magnitude of active forcing is regarded as constant, and we include the effect of inertial contributions. Interestingly, this situation can formally be mapped to stochastic motion under (dry, solid) Coulomb friction, however, with a negative friction parameter. Diffusion coefficients are calculated by formal mapping to the situation of a quantum-mechanical harmonic oscillator exposed to an additional repulsive delta-potential. Results comprise a ditched or double-peaked velocity distribution and spatial statistics showing outward propagating maxima when starting from initially concentrated arrangements.

Circular motion subject to external alignment under active driving: Nonlinear dynamics and the circle map.

Hardly any real self-propelling or actively driven object is perfect. Thus, undisturbed motion will generally not follow straight lines but rather bent or circular trajectories. We here address self-propelled or actively driven objects that move in discrete steps and additionally tend to migrate towards a certain direction by discrete angular adjustment. Overreaction in the angular alignment is possible. This competition implies pronounced nonlinear dynamics including period doubling and chaotic behavior in a broad parameter regime. Such behavior directly affects the appearance of the trajectories. Furthermore, we address collective motion and effects of spatial self-concentration.

Stimuli-responsive twist actuators made from soft elastic composite materials-linking mesoscopic and macroscopic descriptions.

Very recently, the construction of twist actuators from magnetorheological gels and elastomers has been suggested. These materials consist of magnetizable colloidal particles embedded in a soft elastic polymeric environment. The twist actuation is enabled by a net chirality of the internal particle arrangement. Upon magnetization by a homogeneous external magnetic field, the systems feature an overall torsional deformation around the magnetization direction. Starting from a discrete minimal mesoscopic model setup, we work toward a macroscopic characterization. The two scales are linked by identifying expressions for the macroscopic system parameters as functions of the mesoscopic model parameters. In this way, the observed behavior of a macroscopic system can, in principle, be mapped to and illustratively be understood from an appropriate mesoscopic picture. Our results apply equally well to corresponding soft electrorheological gels and elastomers.

Dynamical Crystallites of Active Chiral Particles.

One of the intrinsic characteristics of far-from-equilibrium systems is the nonrelaxational nature of the system dynamics, which leads to novel properties that cannot be understood and described by conventional pathways based on thermodynamic potentials. Of particular interest are the formation and evolution of ordered patterns composed of active particles that exhibit collective behavior. Here we examine such a type of nonpotential active system, focusing on effects of coupling and competition between chiral particle self-propulsion and self-spinning. It leads to the transition between three bulk dynamical regimes dominated by collective translative motion, spinning-induced structural arrest, and dynamical frustration. In addition, a persistently dynamical state of self-rotating crystallites is identified as a result of a localized-delocalized transition induced by the crystal-melt interface. The mechanism for the breaking of localized bulk states can also be utilized to achieve self-shearing or self-flow of active crystalline layers.

Towards an analytical description of active microswimmers in clean and in surfactant-covered drops.

Geometric confinements are frequently encountered in the biological world and strongly affect the stability, topology, and transport properties of active suspensions in viscous flow. Based on a far-field analytical model, the low-Reynolds-number locomotion of a self-propelled microswimmer moving inside a clean viscous drop or a drop covered with a homogeneously distributed surfactant, is theoretically examined. The interfacial viscous stresses induced by the surfactant are described by the well-established Boussinesq-Scriven constitutive rheological model. Moreover, the active agent is represented by a force dipole and the resulting fluid-mediated hydrodynamic couplings between the swimmer and the confining drop are investigated. We find that the presence of the surfactant significantly alters the dynamics of the encapsulated swimmer by enhancing its reorientation. Exact solutions for the velocity images for the Stokeslet and dipolar flow singularities inside the drop are introduced and expressed in terms of infinite series of harmonic components. Our results offer useful insights into guiding principles for the control of confined active matter systems and support the objective of utilizing synthetic microswimmers to drive drops for targeted drug delivery applications.

Frequency-dependent higher-order Stokes singularities near a planar elastic boundary: Implications for the hydrodynamics of an active microswimmer near an elastic interface.

The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future for biomedical and technological applications. These microswimmers move autonomously through aqueous media, where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than those due to shear resistance. Bending can enhance (suppress) the velocities resulting from higher-order singularities whereas the shear related contribution decreases (increases) the velocities. Most prominently, we find that near an elastic interface of only energetic resistance toward shear deformation, such as that of an elastic capsule designed for drug delivery, a swimming bacterium undergoes rotation of the same sense as observed near a no-slip wall. In contrast to that, near an interface of only energetic resistance toward bending, such as that of a fluid vesicle or liposome, we find a reversed sense of rotation. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.

Magnetostriction in magnetic gels and elastomers as a function of the internal structure and particle distribution.

Magnetic gels and elastomers are promising candidates to construct reversibly excitable soft actuators, triggered from outside by magnetic fields. These magnetic fields induce or alter the magnetic interactions between discrete rigid particles embedded in a soft elastic polymeric matrix, leading to overall deformations. It is a major challenge in theory to correctly predict from the discrete particle configuration the type of deformation resulting for a finite-sized system. Considering an elastic sphere, we here present such an approach. The method is in principle exact, at least within the framework of linear elasticity theory and for large enough interparticle distances. Different particle arrangements are considered. We find, for instance, that regular simple cubic configurations show elongation of the sphere along the magnetization if oriented along a face or space diagonal of the cubic unit cell. Contrariwise, with the magnetization along the edge of the cubic unit cell, they contract. The opposite is true in this geometry for body- and face-centered configurations. Remarkably, for the latter configurations but the magnetization along a face or space diagonal of the unit cell, contraction was observed to revert to expansion with decreasing Poisson ratio of the elastic material. Randomized configurations were considered as well. They show a tendency of elongating the sphere along the magnetization, which is more pronounced for compressible systems. Our results can be tested against actual experiments for spherical samples. Moreover, our approach shall support the search of optimal particle distributions for a maximized effect of actuation.

Classical density functional theory for a two-dimensional isotropic ferrogel model with labeled particles.

In this study, we formulate a density functional theory (DFT) for systems of labeled particles, considering a two-dimensional bead-spring lattice with a magnetic dipole on every bead as a model for ferrogels. On the one hand, DFT has been widely studied to investigate fluidlike states of materials, in which constituent particles are not labeled as they can exchange their positions without energy cost. On the other hand, in ferrogels consisting of magnetic particles embedded in elastic polymer matrices, the particles are labeled by their positions as their neighbors do not change over time. We resolve such an issue of particle labeling, introducing a mapping of the elastic interaction mediated by springs onto a pairwise additive interaction (pseudosprings) between unlabeled particles. We further investigate magnetostriction and changes in the elastic constants under altered magnetic interactions employing the pseudospring potential. It is revealed that there are two different response scenarios in the mechanical properties of the dipole-spring systems: While systems at low packing fractions are hardened as the magnetic moments increase in magnitude, at high packing fractions softening due to diminishing effects from the steric force, associated with increases in the volume, is observed. The validity of the theory is also verified by Monte Carlo simulations with both real springs and pseudosprings. We expect that our DFT approach may promote our understanding of materials with particle inclusions.

Creeping motion of a solid particle inside a spherical elastic cavity: II. Asymmetric motion.

An analytical method is proposed for computing the low-Reynolds-number hydrodynamic mobility function of a small colloidal particle asymmetrically moving inside a large spherical elastic cavity, the membrane of which is endowed with resistance toward shear and bending. In conjunction with the results obtained in the first part (A. Daddi-Moussa-Ider, H. Löwen, S. Gekle, Eur. Phys. J. E 41, 104 (2018)), in which the axisymmetric motion normal to the surface of an elastic cavity is investigated, the general motion for an arbitrary force direction can now be addressed. The elastohydrodynamic problem is formulated and solved using the classic method of images through expressing the hydrodynamic flow fields as a multipole expansion involving higher-order derivatives of the free-space Green's function. In the quasi-steady limit, we demonstrate that the particle self-mobility function of a particle moving tangent to the surface of the cavity is larger than that predicted inside a rigid stationary cavity of equal size. This difference is justified by the fact that a stationary rigid cavity introduces additional hindrance to the translational motion of the encapsulated particle, resulting in a reduction of its hydrodynamic mobility. Furthermore, the motion of the cavity is investigated, revealing that the translational pair (composite) mobility, which linearly couples the velocity of the elastic cavity to the force exerted on the solid particle, is solely determined by membrane shear properties. Our analytical predictions are favorably compared with fully-resolved computer simulations based on a completed-double-layer boundary integral method.

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.

An efficient route to the displacement field around a rigid spherical inclusion in an infinitely extended homogeneous elastic medium is presented in a slightly alternative way when compared to some common textbook methods. Moreover, two Faxén relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on a rigid spherical particle in a deformed elastic medium. As a consequence, additional contributions to the distortions of the deformed elastic medium are identified that are absent to lower order. Furthermore, the displaceability and rotateability matrices for an ensemble of rigid spheres are calculated up to (including) sixth order in inverse particle separation distance. These matrices describe the interactions mediated between the rigid embedded particles by the elastic environment. In this way, additional coupling effects are identified that are absent to lower order, particularly when rotations and torques are involved. All methods and results can formally be transferred to the corresponding case of incompressible hydrodynamic low-Reynolds-number Stokes flow by considering the limit of an incompressible environment. The roles of compressibility of the embedding medium and of the here additionally derived higher-order contributions are highlighted by some selected example configurations.

Membrane penetration and trapping of an active particle.

The interaction between nano- or micro-sized particles and cell membranes is of crucial importance in many biological and biomedical applications such as drug and gene delivery to cells and tissues. During their cellular uptake, the particles can pass through cell membranes via passive endocytosis or by active penetration to reach a target cellular compartment or organelle. In this manuscript, we develop a simple model to describe the interaction of a self-driven spherical particle (moving through an effective constant active force) with a minimal membrane system, allowing for both penetration and trapping. We numerically calculate the state diagram of this system, the membrane shape, and its dynamics. In this context, we show that the active particle may either get trapped near the membrane or penetrate through it, where the membrane can either be permanently destroyed or recover its initial shape by self-healing. Additionally, we systematically derive a continuum description allowing us to accurately predict most of our results analytically. This analytical theory helps in identifying the generic aspects of our model, suggesting that most of its ingredients should apply to a broad range of membranes, from simple model systems composed of magnetic microparticles to lipid bilayers. Our results might be useful to predict the mechanical properties of synthetic minimal membranes.

Memory-based mediated interactions between rigid particulate inclusions in viscoelastic environments.

Many practically relevant materials combine properties of viscous fluids and elastic solids to viscoelastic behavior. Our focus is on the induced dynamic behavior of damped finite-sized particulate inclusions in such substances. We explicitly describe history-dependent interactions that emerge between the embedded particles. These interactions are mediated by the viscoelastic surroundings. They result from the flows and distortions of the viscoelastic medium when induced by the rigid inclusions. Both viscoelastic environments of terminal fluidlike flow and of completely reversible damped elastic behavior are covered. For illustration and to highlight the role of the formalism in potential applications, we briefly address the relevant examples of dragging a rigid sphere through a viscoelastic environment together with subsequent relaxation dynamics, the switching dynamics of magnetic fillers in elastic gel matrices, and the swimming behavior of active microswimmers in viscoelastic solutions. The approach provides a basis for more quantitative and extended investigations of these and related systems in the future.