To describe many-particle systems suspended in incompressible low-Reynolds-number fluids, effective hydrodynamic interactions can be introduced. Here, we consider particles embedded in elastic media. The effective elastic interactions between spherical particles are calculated analytically, inspired by the approach in the fluid case. Our experiments on interacting magnetic particles confirm the theory. In view of the huge success of the method in hydrodynamics, we similarly expect many future applications in the elastic case, e.g. for elastic composite materials.
Embedding rigid inclusions into elastic matrix materials is a procedure of high practical relevance, for instance for the fabrication of elastic composite materials. We theoretically analyze the following situation. Rigid spherical inclusions are enclosed by a homogeneous elastic medium under stick boundary conditions. Forces and torques are directly imposed from outside onto the inclusions, or are externally induced between them. The inclusions respond to these forces and torques by translations and rotations against the surrounding elastic matrix. This leads to elastic matrix deformations, a…
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 fluid-like flow or of completely reversible damped elas…
Interactions between rigid inclusions in continuous three-dimensional linearly elastic solids and low-Reynolds-number viscous fluids have largely been quantified during 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 disk-like inclusions remain well separated from each other and do not get into contact. Specifically, we demonstrate a…
Photoelasticity is employed to investigate the stress state near stiff rectangular and rhombohedral inclusions embedded in a 'soft' elastic plate. Results show that the singular stress field predicted by the linear elastic solution for the rigid inclusion model can be generated in reality, with great accuracy, within a material. In particular, experiments: (i.) agree with the fact that the singularity is lower for obtuse than for acute inclusion angles; (ii.) show that the singularity is stronger in Mode II than in Mode I (differently from a notch); (iii.) validate the model of rigid quadrilat…
We study the collective behavior of inclusions inducing local anisotropic curvatures in a flexible fluid membrane. The N-body interaction energy for general anisotropic inclusions is calculated explicitly, including multi-body interactions. Long-range attractive interactions between inclusions are found to be sufficiently strong to induce aggregation. Monte Carlo simulations show a transition from compact clusters to aggregation on lines or circles. These results might be relevant to proteins in biological membranes or colloidal particles bound to surfactant membranes.
Membrane shape fluctuations induce attractive interactions between rigid inclusions. Previous analytical studies showed that the fluctuation-induced pair interactions are rather small compared to thermal energies, but also that multi-body interactions cannot be neglected. In this article, it is shown numerically that shape fluctuations indeed lead to the dynamic separation of the membrane into phases with different inclusion concentrations. The tendency of lateral phase separation strongly increases with the inclusion size. Large inclusions aggregate at very small inclusion concentrations and …
Using Brownian Dynamics simulations, we study effective interactions mediated between two identical and impermeable disks (inclusions) immersed in a bath of identical, active (self-propelled), Brownian rods in two spatial dimensions, by assuming that the self-propulsion axis of the rods may generally deviate from their longitudinal axis. When the self-propulsion is transverse (perpendicular to the rod axis), the accumulation of active rods around the inclusions is significantly enhanced, causing a more expansive steric layering (ring formation) of the rods around the inclusions, as compared wi…
We consider an elastic composite material containing particulate inclusions in a soft elastic matrix that is bounded by a rigid wall, e.g., the substrate. If such a composite serves as a soft actuator, forces are imposed on or induced between the embedded particles. We investigate how the presence of the rigid wall affects the interactions between the inclusions in the elastic matrix. For no-slip boundary conditions, we transfer Blake's derivation of a corresponding Green's function from low-Reynolds-number hydrodynamics to the linearly elastic case. Results for no-slip and free-slip surface c…
Interactions mediated by the cell membrane between inclusions, such as membrane proteins or antimicrobial peptides, play important roles in their biological activity. They also constitute a fascinating challenge for physicists, since they test the boundaries of our understanding of self-assembled lipid membranes, which are remarkable examples of two-dimensional complex fluids. Inclusions can couple to various degrees of freedom of the membrane, resulting in different types of interactions. In this chapter, we review the membrane-mediated interactions that arise from direct constraints imposed …
In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end-effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, new shape solutions are found when the membrane is curvature unstabl…
We consider how membrane fluctuations can modify the miscibility of lipid mixtures, that is to say how the phase diagram of a boundary-constrained membrane is modified when the membrane is allowed to fluctuate freely in the case of zero surface tension. In order for fluctuations to have an effect, the different lipid types must have differing Gaussian rigidities. We show, somewhat paradoxically, that fluctuation-induced interactions can be treated approximately in a mean-field type theory. Our calculations predict that, depending on the difference in bending and Gaussian rigidity of the lipids…
We study steady-state properties of a bath of active Brownian particles (ABPs) in two dimensions in the presence of two fixed, permeable (hollow) disklike inclusions, whose interior and exterior regions can exhibit mismatching motility (self-propulsion) strengths for the ABPs. We show that such a discontinuous motility field strongly affects spatial distribution of ABPs and thus also the effective interaction mediated between the inclusions through the active bath. Such net interactions arise from soft interfacial repulsions between ABPs that sterically interact with and/or pass through permea…
Orientational ordering driven by mechanical distortion of soft substrates plays a major role in material transformation processes such as elastocapillarity and surface anchoring. We present a theoretical model of the orientational response of anisotropic rods deposited onto a surface of a soft, elastic substrate of finite thickness. We show that anisotropic rods exhibit a continuous isotropic-nematic phase transition, driven by orientational interactions between surface deposited rods. This interaction is mediated by the deformation of the underlying elastic substrate, and is quantified by the…
Soft elastic composite materials containing particulate rigid inclusions in a soft elastic matrix are candidates for developing soft actuators or tunable damping devices. The possibility to reversibly drive the rigid inclusions within such a composite together to a close-to-touching state by an external stimulus would offer important benefits. Then, a significant tuning of the mechanical properties could be achieved due to the resulting mechanical hardening. For a long time, it has been argued whether a virtual touching of the embedded magnetic particles with subsequent detachment can actually…
In a system of colloidal inclusions suspended in a thermalized bath of smaller particles, the bath engenders an attractive force between the inclusions, arising mainly from entropic origins, known as the depletion force. In the case of active bath particles, the nature of the bath-mediated force changes dramatically from an attractive to a repulsive one, as the strength of particle activity is increased. We study such bath-mediated effective interactions between colloidal inclusions in a bath of self-propelled Brownian particles, being confined in a narrow planar channel. Confinement is found …
Colloidal inclusions suspended in a bath of smaller particles experience an effective bath-mediated attraction at small intersurface separations, which is known as the depletion interaction. In an active bath of nonchiral self-propelled particles, the effective force changes from attraction to repulsion; an effect that is suppressed, when the active bath particles are chiral. Using Brownian Dynamics simulations, we study the effects of channel confinement and bath chirality on the effective forces and torques that are mediated between two inclusions that may be fixed within the channel or may …
We study the dynamics of extended rod-like bodies in (or associated with) membranes and films. We demonstrate a striking difference between the mobilities in films and bulk fluids, even when the dissipation is dominated by the fluid stress: for large inclusions we find that rotation and motion perpendicular to the rod axis exhibits purely local drag, in which the drag coefficient is algebraic in the rod dimensions. We also study the dynamics of the {\em internal} modes of a semiflexible inclusion and find two dynamical regimes in the relaxation spectrum.
We study the hydrodynamic interactions between colloids suspended in a compressible fluid inside a rigid channel. Using lattice-Boltzmann simulations and a simplified hydrodynamic theory, we find that the diffusive dynamics of density perturbations (sound) in the confined fluid give rise to particle correlations of exceptionally long spatial range and algebraic temporal decay. We examine the effect of these sound-mediated correlations on two-particle dynamics and on the collective dynamics of a quasi-one-dimensional suspension.
We analytically study membrane mediated interactions between inclusions embedded in a tubular membrane. We model inclusions as constraints coupled to the curvature tensor of the membrane tube. First, as special test cases, we analyze the interaction between ring and rod shaped inclusions. Using Monte Carlo simulations, we further show how point-like inclusions interact to form linear aggregates. Our results reveal that depending on the hard-core radius of the inclusions, they arrange into either lines or rings to globally minimize the curvature energy of the membrane.
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 rigid particles in a deformed elastic medium. Additionally, the displaceability and rotateability matrices are calculated up to (including) sixth order in inverse particle separation d…
The shapes of cell membranes are largely regulated by membrane associated, curvature active, proteins. We use a numerical model of the membrane with elongated membrane inclusions, recently developed by us, which posses spontaneous directional curvatures that could be different along and perpendicular to its long axis. We show that, due to membrane mediated interactions these curvature inducing membrane nematogens can oligomerize spontaneously, even at low concentrations, and change the local shape of the membrane. We demonstrate that for a large group of such inclusions, where the two spontane…
Three-dimensional DNA networks, composed of tri- or higher valent nanostars with sticky, single-stranded DNA overhangs, have been previously studied in the context of designing thermally responsive, viscoelastic hydrogels. In this work, we use linker-mediated gels, where the sticky ends of two trivalent nanostars are connected through the complementary sticky ends of a linear DNA duplex. We can design this connection to be either rigid or flexible by introducing flexible, non-binding bases. The additional flexiblity provided by these non-binding bases influences the effective elasticity of the…
We calculate the elastic field mediated interaction between macroscopic particles in a columnar hexagonal phase. The interaction is found to be long-ranged and non-central, with both attractive and repulsive parts. We show how the interaction modifies the particle correlations and the column fluctuations. We also calculate the interaction of particles with the topological defects of the columnar phase. The particle-defect interaction reduces the mobility of the defects.
The fluctuation-induced interaction between two rod-like, rigid inclusions in a fluid vesicle is studied by means of canonical ensemble Monte Carlo simulations. The vesicle membrane is represented by a triangulated network of hard spheres. Five rigidly connected hard spheres form rod-like inclusions that can leap between sites of the triangular network. Their effective interaction potential is computed as a function of mutual distance and angle of the inclusions. On account of the hard-core potential among these, the nature of the potential is purely entropic. Special precaution is taken to re…
Motion of an object near a soft wall with intervening fluid is a canonical problem in elastohydrodynamics, finding presence in subjects spanning biology to tribology. Particularly, motion of a sphere towards a soft substrate with intervening fluid is often encountered in the context of scanning probe microscopy. While there have been fundamental theoretical studies on this setup, they have focussed on specific applications and thus enforced suitable simplifications. Here we present a versatile semi-analytical framework for studying the elastohydrodynamics of axisymmetric loading of a rigid sph…
Although Casimir forces are inseparable from their fluctuations, little is known about these fluctuations in soft matter systems. We use the membrane stress tensor to study the fluctuations of the membrane-mediated Casimir-like force. This method enables us to recover the Casimir force between two inclusions and to calculate its variance. We show that the Casimir force is dominated by its fluctuations. Furthermore, when the distance d between the inclusions is decreased from infinity, the variance of the Casimir force decreases as -1/d^2. This distance dependence shares a common physical origi…
When particulate suspensions are sheared, perturbations in the shear flows around the rigid particles increase the local energy dissipation, so that the viscosity of the suspension is effectively higher than that of the solvent. For bulk (three-dimensional) fluids, understanding this viscosity enhancement is a classic problem in hydrodynamics that originated over a century ago with Einstein's study of a dilute suspension of spherical particles. \cite{Einstein1} In this paper, we investigate the analogous problem of the effective viscosity of a suspension of disks embedded in a two-dimensional …
Considering two rigid conical inclusions embedded in a membrane subject to lateral tension, we study the membrane-mediated interaction between these inclusions that originates from the hat-shaped membrane deformations associated with the cones. At non-vanishing lateral tensions, the interaction is found to depend on the orientation of the cones with respect to the membrane plane. The interaction of inclusions of equal orientation is repulsive at all distances between them, while the inclusions of opposite orientation repel each other at small separations, but attract each other at larger ones.…
Up-down asymmetric inclusions impose a local, spontaneous curvature to an elastic membrane. When several of them are inserted in a same membrane, they feel effective forces mediated by the membrane, both of elastic and entropic nature. Following an approach initiated by Dommersnes and Fournier in the vanishing tension case [Eur. Phys. J. B 12, 9 (1999)], and also using a pseudo-analytical micellization theory, we derive the statistical mechanics of asymmetric inclusion assemblies when they are also subject to an additional short-range, attractive interaction. Our main conclusion is that generi…
We discuss the fluctuation-induced force, a finite-temperature analog of the Casimir force, between two inclusions embedded in a fluid membrane under tension. We suggest a method to calculate this Casimir interaction in the most general case, where membrane fluctuations are governed by the combined action of surface tension, bending modulus, and the Gaussian rigidity. We find that the surface tension strongly modifies the power law in the separation dependence of the Casimir interaction. This results in a strong suppression of the Casimir force at separations beyond a characteristic length, wh…
The aim of this work is to revisit the phenomenological theory of the interaction between membrane inclusions, mediated by the membrane fluctuations. We consider the case where the inclusions are separated by distances larger than their characteristic size. Within our macroscopic approach a physical nature of such inclusions is not essential, however we have always in mind two prototypes of such inclusions: proteins and RNA macromolecules. Because the interaction is driven by the membrane fluctuations, and the coupling between inclusions and the membrane, it is possible to change the interacti…
Inclusions dissolved in an anisotropic quasi-2D membrane acquire new types of interactions that can drive assembly of complex structures and patterns. We study colloidal membranes composed of a binary mixture of long and short rods, such that the length ratio of the long to short rods is approximately two. At very low volume fractions, short rods dissolve in the membrane of long rods by strongly anchoring to the membrane polymer interface. At higher fractions, the dissolved short rods phase separate from the background membrane, creating a composite structure comprised of bilayer droplets enri…
We investigate the forces exerted on embedded inclusions by a fluid medium with long-range correlations, described by an effective scalar field theory. Such forces are the basis for the medium-mediated Casimir-like force. To study these forces beyond thermal average, it is necessary to define them in each microstate of the medium. Two different definitions of these forces are currently used in the literature. We study the assumptions underlying them. We show that only the definition that uses the stress tensor of the medium gives the sought-after force exerted by the medium on an embedded incl…
We calculate the induced elastic-interaction between pointwise membrane inclusions that locally interact up to quadratic order with the membrane curvature tensor. For isotropic inclusions, we recover the usual interaction proportional to the inverse fourth power of the separation, however with a prefactor showing a non-trivial dependence on the rigidity $Γ$ of the quadratic potential. In the large $Γ$ limit, corresponding to ``hard'' inclusions, we recover the standard prefactor first obtained by Goulian et al. [Europhys. Lett. \textbf{22}, 145 (1993)]. In the small $Γ$ limit, corresponding to…
We study a physical model for the interaction between general inclusions bound to fluid membranes that possess finite tension, as well as the usual bending rigidity. We are motivated by an interest in proteins bound to cell membranes that apply forces to these membranes, due to either entropic or direct chemical interactions. We find an exact analytic solution for the repulsive interaction between two similar circularly symmetric inclusions. This repulsion extends over length scales of order tens of nanometers, and contrasts with the membrane-mediated contact attraction for similar inclusions …
We study effective two- and three-body interactions between non-active colloidal inclusions in an active bath of chiral or non-chiral particles, using Brownian Dynamics simulations within a standard, two-dimensional model of disk-shaped inclusions and active particles. In a non-chiral active bath, we first corroborate previous findings on effective two-body repulsion mediated between the inclusions by elucidating the detailed non-monotonic features of the two-body force profiles, including a primary maximum, and a secondary hump at larger separations that was not previously reported. We then s…
The phase behavior of rodlike molecules with polydisperse length and solvent-mediated attraction and repulsion is described by an extension of the Onsager theory for rigid rods. A phenomenological square-well potential is used to model these long-range interactions, and the model is used to compute phase separation and length fractionation as a function of well depth and rod concentration. The model closely captures experimental data points for isotropic/liquid crystalline phase coexistence of single-walled carbon nanotubes (SWNTs) in superacids. The model also predicts that the isotropic-biph…
Integral membrane proteins deform the surrounding bilayer creating long-ranged forces that influence distant proteins. These forces can be attractive or repulsive, depending on the proteins' shape, height, contact angle with the bilayer, as well as the local membrane curvature. Although interaction energies are not pairwise additive, for sufficiently low protein density, thermodynamic properties depend only upon pair interactions. Here, we compute pair interaction potentials and entropic contributions to the two-dimensional osmotic pressure of a collection of noncircular proteins. In contrast …
The wrapping equilibria of one and two adsorbing cylinders are studied along a semi-flexible filament (polymer) due to the interplay between elastic rigidity and short-range adhesive energy between the cylinder and the filament. We show that statistical mechanics of the system can be solved exactly using a path integral formalism which gives access to the full effect of thermal fluctuations, going thus beyond the usual Gaussian approximations which take into account only the contributions from the minimal energy configuration and small fluctuations about this minimal energy solution. We obtain…
Internalization of particles by cells plays a crucial role for adsorbing nutrients and fighting infection. Endocytosis is one of the most important mechanisms of the particles uptake which encompass multiple pathways. Although endocytosis is a complex mechanism involving biochemical signaling and active force generation, the energetic cost associated to the large deformations of the cell membrane wrapping around the foreign particle is an important factor controlling this process, which can be studied using quantitative physical models. Of particular interest is the competition between membran…
