Correction: Shape-controlled anisotropy of superparamagnetic micro-/nanohelices.

Correction for 'Shape-controlled anisotropy of superparamagnetic micro-/nanohelices' by Alexander M. Leshansky et al., Nanoscale, 2016, 8, 14127-14138, https://doi.org/10.1039/C6NR01803C.

Can the Thermophoretic Mobility of Uncharged Colloids Be Predicted?

The thermophoretic motion of nonionic colloids in an inhomogeneous temperature field is due to the solvent-colloid dispersion interactions. The latter form an attractive near-particle "gravity" field that leads to sinking of the colder solvent layers toward a colloid. The spatial extension of this microconvective motion is comparable to the size of the colloids, which prove to be small enough to observe their own regular thermophoretic drift to the cold. The Boussinesq equations of convection are augmented by the boundary conditions at the characteristic molecular distance dividing the immovable and motile solvent layers. For organic liquids, this distance proves to be a property of pure solvent. The thermophoretic mobilities are found for colloids with and without surfacted layers. They are determined by the bulk properties of substances and the Hamaker constant of the solvent-solute interactions. The mobilities weakly (logarithmically) depend on the size of colloids and tend to a universal value in the limiting case of strongly asymmetrical mixtures. This is the first report that shows a prediction of the thermophoretic velocities of uncharged colloids. The relation between the thermophoretic mobility of colloids and the Hamaker constant of the solute-solvent interactions enables an experimental determination of the latter quantity from thermophoresis data.

Brownian Particle Thermal Drift Caused by Hydrodynamic Fluctuations.

We have studied the thermophoretic drift of a Brownian particle within the framework of the theory of fluctuating hydrodynamics of Landau and Lifshitz. The stochastic motion of a dispersed particle results from the fluctuating part of the stress tensor of the liquid. In the presence of a temperature gradient, the mean volume force acting on the liquid assumes a finite value due to non-vanishing quadratic terms in the fluctuations, which originate from the temperature dependence of the viscosity. An analytical expression is derived for the drift velocity and compared to experimental results.

Towards focusing of a swarm of magnetic micro/nanomotors.

Magnetic micro- and nanoparticles propelled by a rotating magnetic field provide a new technology for targeted drug delivery. The therapeutic effect of the technology is achievable with the collective action of large groups (swarms) of the motors. Narrowing of the swarm width before it reaches the target or, vice versa, its expansion prior to deposition on the channel walls are of high importance. Here we show how such swarm tuning can be achieved using a combination of the in-plane rotating and static magnetic field acting in the same plane. Although for this asymmetric actuating field the steady synchronous solutions of the problem do not materialize, the developed approach admits an analytical consideration for the average motor orientation and propulsion. The unique property of an average motion in the actuation by an asymmetric magnetic field is the emergence of propeller's net drift in plane of the field rotation. The drift velocity can be comparable in magnitude to the propulsion velocity along the axis of the field rotation. The field-induced drift is studied in detail for achiral V-shaped and chiral helical propellers, depending on their magnetization. Finally, we suggest spatial configuration of the constant magnetic field capable of focusing/defocusing swarms of magnetic motors.

Modeling Propulsion of Soft Magnetic Nanowires.

The emergent interest in artificial nanostructures that can be remotely navigated a specific location in a fluidic environment is motivated by the enormous potential this technology offers to biomedical applications. Originally, bio-inspired micro-/nanohelices driven by a rotating magnetic field were proposed. However, fabrication of 3D helical nanostructures is complicated. One idea to circumvent complex microfabrication is to use 1D soft magnetic nanowires that acquire chiral shape when actuated by a rotating field. The paper describes the comprehensive numerical approach for modeling propulsion of externally actuated soft magnetic nanowires. The proposed bead-spring model allows for arbitrary filament geometry and flexibility and takes rigorous account of intra-filament hydrodynamic interactions. The comparison of the numerical predictions with the previous experimental results on propulsion of composite two-segment (Ni-Ag) nanowires shows an excellent agreement. Using our model we could substantiate and rationalize important and previously unexplained details, such as bidirectional propulsion of three-segment (Ni-Ag-Au) nanowires.

Photonics of Template-Mediated Lattices of Colloidal Clusters.

The recent progress in microfluidic microfabrication enables mass production of "colloidal molecules" with a preprogrammed geometry (e.g., dumbbells, tetrahedrons, etc.). Such colloids can be used as elementary building blocks in the fabrication of colloidal crystals with unique optical properties. Anisotropic clusters, however, cannot be readily assembled into regular lattices. In this paper, we study photonic properties of compact cubic templates of microdrops encapsulating complex "colloidal molecules". Because monodisperse droplets can be easily packed into dense cubic lattices and encapsulation techniques (e.g., using microfluidics) are well developed, such a material is experimentally feasible. The rationale behind such a methodology is that for a particular alignment of the encapsulated "colloidal molecules" (e.g., by applying an external magnetic or electric field), the resulting structures resemble a diamond lattice, which is known to exhibit a wide complete photonic band gap. The photonic properties of two cubic templates encapsulating dumbbells (symmetric and asymmetric) and tetrahedrons are investigated numerically. In particular, we show the emergence of the complete 3D band gap (∼8% wide for the dielectric contrast ε = 14) for symmetric dumbbells embedded within a face-centered cubic template and oriented along the space diagonal of the elementary cubic cell.

Geometric constraints and optimization in externally driven propulsion.

Micro/nanomachines capable of propulsion through fluidic environments provide diverse opportunities in important biomedical applications. In this paper, we present a theoretical study on micromotors steered through liquid by an external rotating magnetic field. A purely geometric tight upper bound on the propulsion speed normalized with field frequency, known as propulsion efficiency, δ, for an arbitrarily shaped object is derived. Using this bound, we estimate the maximum propulsion efficiency of previously reported random magnetic aggregates. We introduce a complementary definition of the propulsion efficiency, δ*, that ranks propellers according to their maximal speed in body lengths per unit time and that appears to be preferable over the standard definition in a search for fastest machines. Using a bead-based hydrodynamic model combined with genetic algorithms, we determine that δ*-optimal propeller deviates strongly from the bioinspired slim helix and has a surprising chubby skew-symmetric shape. It is also shown that optimized propellers with preprogrammed shape are substantially more efficient than random magnetic aggregates. We anticipate that the results of the present study will provide guidance toward prospective experimental design of more efficient magnetic micro/nanomachines.

Highly Efficient Freestyle Magnetic Nanoswimmer.

The unique swimming strategies of natural microorganisms have inspired recent development of magnetic micro/nanorobots powered by artificial helical or flexible flagella. However, as artificial nanoswimmers with unique geometries are being developed, it is critical to explore new potential modes for kinetic optimization. For example, the freestyle stroke is the most efficient of the competitive swimming strokes for humans. Here we report a new type of magnetic nanorobot, a symmetric multilinked two-arm nanoswimmer, capable of efficient "freestyle" swimming at low Reynolds numbers. Excellent agreement between the experimental observations and theoretical predictions indicates that the powerful "freestyle" propulsion of the two-arm nanorobot is attributed to synchronized oscillatory deformations of the nanorobot under the combined action of magnetic field and viscous forces. It is demonstrated for the first time that the nonplanar propulsion gait due to the cooperative "freestyle" stroke of the two magnetic arms can be powered by a plane oscillatory magnetic field. These two-arm nanorobots are capable of a powerful propulsion up to 12 body lengths per second, along with on-demand speed regulation and remote navigation. Furthermore, the nonplanar propulsion gait powered by the consecutive swinging of the achiral magnetic arms is more efficient than that of common chiral nanohelical swimmers. This new swimming mechanism and its attractive performance opens new possibilities in designing remotely actuated nanorobots for biomedical operation at the nanoscale.

Shape-controlled anisotropy of superparamagnetic micro-/nanohelices.

Micro-/nanopropellers can be actuated remotely by a rotating magnetic field and steered at high precision through various fluidic environments. Recent progress comprises microfabrication of superparamagnetic microhelices not possessing remanent magnetization, but rather magnetized by an applied magnetic field. In this article we present a numerical approach for computing, from first principles, the effective susceptibility of polarizable helical micro-/nanopropellers. We show that nanopropeller geometry, in particular, filament cross-section elongation and orientation, play a central role in determining its magnetic anisotropy and polarizability. The numerical predictions are in qualitative agreement with the previously reported experiments, showing that tight polarizable helices are propulsive. The numerical results are also supported by the approximate slender-body theory. Finally, we propose a semi-quantitative energy criterion to rank polarizable helices with different geometries of the filament by their propulsive capacity and also estimate their maximal propulsion speed.

Swimming by reciprocal motion at low Reynolds number.

Biological microorganisms swim with flagella and cilia that execute nonreciprocal motions for low Reynolds number (Re) propulsion in viscous fluids. This symmetry requirement is a consequence of Purcell's scallop theorem, which complicates the actuation scheme needed by microswimmers. However, most biomedically important fluids are non-Newtonian where the scallop theorem no longer holds. It should therefore be possible to realize a microswimmer that moves with reciprocal periodic body-shape changes in non-Newtonian fluids. Here we report a symmetric 'micro-scallop', a single-hinge microswimmer that can propel in shear thickening and shear thinning (non-Newtonian) fluids by reciprocal motion at low Re. Excellent agreement between our measurements and both numerical and analytical theoretical predictions indicates that the net propulsion is caused by modulation of the fluid viscosity upon varying the shear rate. This reciprocal swimming mechanism opens new possibilities in designing biomedical microdevices that can propel by a simple actuation scheme in non-Newtonian biological fluids.

Dynamics and polarization of superparamagnetic chiral nanomotors in a rotating magnetic field.

Externally powered magnetic nanomotors are of particular interest due to their potential use for in vivo biomedical applications. Here we develop a theory for dynamics and polarization of recently fabricated superparamagnetic chiral nanomotors powered by a rotating magnetic field. We study in detail various experimentally observed regimes of the nanomotor dynamic orientation and propulsion and establish the dependence of these properties on polarization and geometry of the propellers. Based on the proposed theory we introduce a novel "steerability" parameter γ that can be used to rank polarizable nanomotors by their propulsive capability. The theoretical predictions of the nanomotor orientation and propulsion speed are in excellent agreement with available experimental results. Finally, we apply slender-body approximation to estimate the polarization anisotropy and orientation of the easy-axis of superparamagnetic helical propellers.

Nanopropellers and their actuation in complex viscoelastic media.

Tissue and biological fluids are complex viscoelastic media with a nanoporous macromolecular structure. Here, we demonstrate that helical nanopropellers can be controllably steered through such a biological gel. The screw-propellers have a filament diameter of about 70 nm and are smaller than previously reported nanopropellers as well as any swimming microorganism. We show that the nanoscrews will move through high-viscosity solutions with comparable velocities to that of larger micropropellers, even though they are so small that Brownian forces suppress their actuation in pure water. When actuated in viscoelastic hyaluronan gels, the nanopropellers appear to have a significant advantage, as they are of the same size range as the gel's mesh size. Whereas larger helices will show very low or negligible propulsion in hyaluronan solutions, the nanoscrews actually display significantly enhanced propulsion velocities that exceed the highest measured speeds in Newtonian fluids. The nanopropellers are not only promising for applications in the extracellular environment but small enough to be taken up by cells.

Thermophoresis of polymers: nondraining vs draining coil.

Present theories for the thermophoretic mobility of polymers in dilute solution without long-ranged electrostatic interaction are based on a draining coil model with short-ranged segment-solvent interaction. We show that the characteristic thermophoretic interaction decays as r(-2) with the distance from the chain segment, which is of much longer range than the underlying rapidly decaying binary van der Waals interaction (∝ r(-6)). As a consequence, thermophoresis on the monomer level is governed by volume forces, resulting in hydrodynamic coupling between the chain segments. The inner parts of the nondraining coil do not actively participate in thermophoresis. The flow lines penetrate only into a thin surface layer of the coil and cause tangential stresses along the surface of the entire coil, not the individual segments. This model is motivated by recent experimental findings for thermoresponsive polymers and core-shell particles, and it explains the well-known molar mass independent thermophoretic mobility of polymers in dilute solution.

The chiral magnetic nanomotors.

Propulsion of chiral magnetic nanomotors powered by a rotating magnetic field is in the focus of the modern biomedical applications. This technology relies on strong interactions of dynamic and magnetic degrees of freedom of the system. Here we study in detail various experimentally observed regimes of the helical nanomotor orientation and propulsion depending on the actuation frequency, and establish the relation of these two properties to the remanent magnetization and geometry of the helical nanomotors. The theoretical predictions for the transition between the regimes and nanomotor orientation and propulsion speed are in excellent agreement with available experimental data. The proposed theory offers a few simple guidelines towards the optimal design of the magnetic nanomotors.

Cytoskeleton fluidization versus resolidification: prestress effect.

The differential elastic modulus of an active actomyosin network is computed as a function of applied stress, taking into account both thermal and motor contributions to filament compliance in the low-frequency domain. It is shown that, due to a dual nature of motor activity, increasing motor concentration may either stiffen the network due to stronger prestress or soften it due to motor agitation, in accordance with experimental data. Prestress anisotropy, which may be induced by redistribution of motors triggered by external force, causes anisotropy of the elastic moduli. This helps to explain the contradictory phenomena of cell fluidization and resolidification in response to transient stretch observed in recent experiments.

Motor-driven effective temperature and viscoelastic response of active matter.

We consider dynamic response of a cytoskeletal network to both thermal and motor-induced fluctuations. The latter are viewed in two independent ways, as either additive or multiplicative colored noise. Due to a natural upper frequency limit of the motor agitation, the response of a living cell is similar to that of an equilibrium system in the high-frequency domain. At lower frequencies, the role of motor agitation manifests itself in intensified network fluctuations, which is equivalent to effective growth of the environment temperature. The effective temperature becomes frequency dependent, which signifies violation of the conventional fluctuation-dissipation theorem. The motor action affects the dynamic shear modulus in two opposite ways: by stiffening the network through filament prestress and softening it through increased agitation. The latter tendency is isolated when only single-headed motors are present. The theory is in good agreement with experimental measurements of the amplitude of the shear modulus under these conditions.

Soret effect in molecular mixtures.

A theoretical approach to the description of the Soret effect in binary nonpolar liquids is proposed. The temperature gradient of the partial pressure is determined as the driving force of thermal diffusion. The hard-sphere fluid is chosen as a reference system and an explicit relation for the Soret coefficient is found. Two additive contributions owing to steric repulsions and attractive interactions form the so-called chemical contribution to S_{T} [C. Debuschewitz and W. Köhler, Phys. Rev. Lett. 87, 055901 (2001)]. The parameters of interparticle interactions are defined with the help of the solvation theory. In particular, the van der Waals constant of cross interactions is expressed via the excess volume of mixture. The proposed theory is applied to the benzene-cyclohexane system. A reasonable agreement of theoretical and experimental results is revealed for the Soret coefficient and its temperature dependence.

The dielectric virial expansion and the models of dipolar hard-sphere fluid.

The virial expansion technique to determine the dielectric constant epsilon of dipolar hard-sphere fluid is developed. It is shown that the formalism allows to bring into agreement the results of Debye's, Onsager's, and Langevin's to the problem. The third virial coefficient of epsilon is considered as a series over dipolar parameter lambda=m(2)d(3)kT. The terms up to O(lambda(11)) are calculated analytically providing a correct description of the third virial coefficient for small and intermediate values of lambda (0