On the bandwidth of stable nonlinear stripe patterns in finite size systems.

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Universal aspects of collective behavior in chemotactic systems.

We investigate the collective dynamics of particles (e.g., microorganisms) interacting via chemotactic gradients. Specifically, we focus on continuum models for chemotaxis that include a damping of the chemical production with increasing local particle density and/or systems where the chemotactic sensitivity is reduced with increasing local concentration of the chemical. Using a recently introduced perturbative method [Phys. Rev. E 98, 020603 (2018)10.1103/PhysRevE.98.020603], we show that the onset of particle clustering in these systems is described by the universal Cahn-Hilliard (CH) model. On the one hand, this establishes particle-conserving models for chemotaxis as a further example for the universal class of nonequilibrium demixing phenomena we call active phase separation. On the other hand, the reduction to the CH model allows an analytical determination of suitable parameter ranges wherein, e.g., the transition to spatial density modulations is continuous and/or undesired blow-up solutions can be avoided. A comparison between the numerical solutions of the chemotaxis model and the derived CH model also provides the parameter range where the basic chemotaxis model behaves like other systems showing active phase separation, including the coarsening behavior in two spatial dimensions. Our approach highlights how basic principles of pattern formation theory allow the identification of common basic properties in different chemotaxis models.

Focusing and splitting streams of soft particles in microflows via viscosity gradients.

Microflows are intensively used for investigating and controlling the dynamics of particles, including soft particles such as biological cells and capsules. A classic result is the tank-treading motion of elliptically deformed soft particles in linear shear flows, which do not migrate across straight streamlines in the bulk. However, soft particles migrate across straight streamlines in Poiseuille flows. In this work we describe a new mechanism of cross-streamline migration by using soft capsules with a spherical equilibrium shape. If the viscosity varies perpendicular to the streamlines then the soft particles migrate across streamlines towards regions of a lower viscosity, even in linear shear flows. An interplay with the repulsive particle-boundary interaction causes then focusing of particles in linear shear flows with the attractor streamline closer to the wall in the low viscosity region. Viscosity variations perpendicular to the streamlines in Poiseuille flows leads either to a shift of the particle attractor or even to a splitting of particle attractors, which may give rise to interesting applications for particle separation. The location of attracting streamlines depend on the particle properties, like their size and elasticity. The cross-stream migration induced by viscosity variations is explained by analytical considerations, Stokesian dynamics simulations with a generalized Oseen tensor and lattice-Boltzmann simulations.

On system-spanning demixing properties of cell polarization.

A number of mathematical models have been suggested to describe cell polarization in eukaryotic cells. One class of models takes into account that certain proteins are conserved on the time scale of cell polarization and may switch between a fast and a slow diffusing state. We raise the question whether models sharing this design feature can be condensed into one system-spanning model. We show exemplarily for the mass-conserved reaction-diffusion model of Otsuji et al. (Otsuji M et al. (2007) PLoS Comput Biol 3(6):e108) that cell polarization can be classified as active phase separation. This includes a fundamental connection between a number of non-equilibrium demixing phenomena such as cell polarization to phase separation. As shown recently, generic properties of active phase separation close to its onset are described by the Cahn-Hilliard model. By a systematic perturbation analysis we directly map the basic cell polarization model to the universal Cahn-Hilliard model. Comparing the numerical solutions of the polarization model and the Cahn-Hilliard equation also provides the parameter range where the basic cell polarization model behaves like other systems showing active phase separation. Polarization models of the active phase separation type cover essential properties of cell polarization, e.g. the adaptability of cell polarity to the length of growing cells. Our approach highlights how basic principles of pattern formation theory allow the identification of common basic properties in different models for cell polarization.

Systematic extension of the Cahn-Hilliard model for motility-induced phase separation.

We consider a continuum model for motility-induced phase separation (MIPS) of active Brownian particles (ABP) (J. Chem. Phys. 142, 224149 (2015)). Using a recently introduced perturbative analysis (Phys. Rev. E 98, 020604(R) (2018)), we show that this continuum model reduces to the classic Cahn-Hilliard (CH) model near the onset of MIPS. This makes MIPS another example of the so-called active phase separation. We further introduce a generalization of the perturbative analysis to the next higher order. This results in a generic higher-order extension of the CH model for active phase separation. Our analysis establishes the mathematical link between the basic mean-field ABP model on the one hand, and the leading order and extended CH models on the other hand. Comparing numerical simulations of the three models, we find that the leading-order CH model agrees nearly perfectly with the full continuum model near the onset of MIPS. We also give estimates of the control parameter beyond which the higher-order corrections become relevant and compare the extended CH model to recent phenomenological models.

Emerging Attractor in Wavy Poiseuille Flows Triggers Sorting of Biological Cells.

Microflows constitute an important instrument to control particle dynamics. A prominent example is the sorting of biological cells, which relies on the ability of deformable cells to move transversely to flow lines. A classic result is that soft microparticles migrate in flows through straight microchannels to an attractor at their center. Here, we show that flows through wavy channels fundamentally change the overall picture. They lead to the emergence of a second, coexisting attractor for soft particles. Its emergence and off-center location depends on the boundary modulation and the particle properties. The related cross-stream migration of soft particles is explained by analytical considerations, Stokesian dynamics simulations in unbounded flows, and Lattice-Boltzmann simulations in bounded flows. The novel off-center attractor can be used, for instance, in diagnostics, for separating cells of different size and elasticity, which is often an indicator of their health status.

Splitting and separation of colloidal streams in sinusoidal microchannels.

The control of the distribution of colloidal particles in microfluidic flows plays an important role in biomedical and industrial applications. A particular challenge is to induce cross-streamline migration in laminar flows, enabling the separation of colloidal particles according to their size, shape or elasticity. Here we show that viscoelastic fluids can mediate cross-streamline migration of deformable spherical and cylindrical colloidal particles in sinusoidal microchannels at low Reynolds numbers. For colloidal streams focused into the center of the channel entrance this leads to a symmetric stream-splitting and separation into four substreams. The degree of stream splitting and separation can be controlled via the flow rates, viscoelasticity of the focusing fluid, and the spatial microchannel modulation with an upper limit when reaching the microchannel walls. We demonstrate that this effect can be used to separate flexible particles of different size and shape. This methodology of cross-stream migration has thus great potential for the passive separation of colloids and cells in microfluidic channels.

Active phase separation: A universal approach.

We identify active phase separation as a generic demixing phenomenon in nonequilibrium systems with conservation constraints. Examples range from cell polarization to cell populations communicating via chemotaxis, and from self-propelled particle communities to mussels in ecology. We show that system-spanning properties of active phase separation in nonequilibrium systems near onset are described by the classical Cahn-Hilliard (CH) model. This result is rather surprising since the CH equation is famous as a model for phase separation at thermal equilibrium. We introduce a general reduction scheme to establish a unique mathematical link between the generic CH equation and system-specific models for active phase separation. This approach is exemplarily applied to a model for polarization of cells and a model for chemotactic cell communities. For cell polarization, we also estimate the validity range of the CH model.

Effective viscosity of a suspension of flagellar-beating microswimmers: Three-dimensional modeling.

Micro-organisms usually can swim in their liquid environment by flagellar or ciliary beating. In this numerical work, we analyze the influence of flagellar beating on the orbits of a swimming cell in a shear flow. We also calculate the effect of the flagellar beating on the rheology of a dilute suspension of microswimmers. A three-dimensional model is proposed for Chlamydomonas Reinhardtii swimming with a breaststroke-like beating of two anterior flagella modeled by two counter-rotating fore beads. The active swimmer model reveals unusual angular orbits in a linear shear flow. Namely, the swimmer sustains orientations transiently across the flow. Such behavior is a result of the interplay between shear flow and the swimmer's periodic beating motion of flagella, which exert internal torques on the cell body. This peculiar behavior has some significant consequences on the rheological properties of the suspension. We calculate Einstein's viscosity of the suspension composed of such isolated modeled microswimmers (dilute case) in a shear flow. We use numerical simulations based on a Rotne-Prager-like approximation for hydrodynamic interaction between simplified flagella and the cell body. The results show an increased intrinsic viscosity for active swimmer suspensions in comparison to nonactive ones as well as a shear thinning behavior in accordance with previous experimental measurements [Phys. Rev. Lett. 104, 098102 (2010)10.1103/PhysRevLett.104.098102].

Passive swimming in viscous oscillatory flows.

Fluid-based locomotion at low Reynolds number is subject to the constraints of Purcell's scallop theorem: reciprocal shape kinematics identical under a time-reversal symmetry cannot cause locomotion. In particular, a single degree-of-freedom scallop undergoing opening and closing motions cannot swim. Most strategies for symmetry breaking and locomotion rely on direct control of the swimmer's shape kinematics. Less is known about indirect control via actuation of the fluid medium. To address how such indirect actuation strategies can lead to locomotion, we analyze a Λ-shaped model system analogous to Purcell's scallop but able to deform passively in oscillatory flows. Neutrally buoyant scallops undergo no net locomotion. We show that dense, elastic scallops can exhibit passive locomotion in zero-mean oscillatory flows. We examine the efficiency of swimming parallel to the background flow and analyze the stability of these motions. We observe transitions from stable to unstable swimming, including ordered transitions from fluttering to chaoticlike motions and tumbling. Our results demonstrate that flow oscillations can be used to passively actuate and control the motion of microswimmers, which may be relevant to applications such as surgical robots and cell sorting and manipulation in microfluidic devices.

Hierarchical line-defect patterns in wrinkled surfaces.

We demonstrate a novel approach for controlling the formation of line-defects in wrinkling patterns by introducing step-like changes in the Young's modulus of elastomeric substrates supporting thin, stiff layers. Wrinkles are formed upon treating the poly(dimethylsiloxane) (PDMS) substrates by UV/Ozone (UVO) exposure in a uniaxially stretched state and subsequent relaxation. Line defects such as minutiae known from fingerprints are a typical feature in wrinkling patterns. The position where these defects occur is random for homogenous substrate elasticity and film thickness. However, we show that they can be predetermined by using PDMS substrates consisting of areas with different cross-linking densities. While changing the cross-linking density is well known to influence the wrinkling wavelength, we use this parameter in this study to force defect formation. The defect formation is monitored in situ using light microscopy and the mechanical parameters/film thicknesses are determined using imaging AFM indentation measurements. Thus the observed wrinkle-wavelengths can be compared to theoretical predictions. We study the density and morphology of defects for different changes in elasticity and compare our findings with theoretical considerations based on a generalized Swift-Hohenberg-equation to simply emulate the observed pattern-formation process, finding good agreement. The fact that for suitable changes in elasticity, well-ordered defect patterns are observed is discussed with respect to formation of hierarchical structures for applications in optics and nanotechnology.

Effects of a temperature-dependent viscosity on thermal convection in binary mixtures.

We investigate the effect of a temperature-dependent viscosity on the onset of thermal convection in a horizontal layer of a binary fluid mixture that is heated from below. For an exponential temperature dependence of the viscosity, we find, in binary mixtures as a function of a positive separation ratio ψ and beyond a certain viscosity contrast, a discontinuous transition between two stationary convection modes having different wavelengths. In the range of negative values of the separation ratio ψ, a (continuous or discontinuous) transition from an oscillatory to a stationary onset of convection occurs beyond a certain viscosity contrast, and for large values of the viscosity ratio, the oscillatory onset of convection is suppressed.

Self-focusing and jet instability of a microswimmer suspension.

Three-dimensional (3D) numerical simulations are performed on suspensions composed of puller-like microswimmers that are sensitive to light (phototaxis) mimicking microalgae in a Poiseuille flow. Simulations are based on the numerical resolution of the flow equations at low Reynolds numbers discretized on a 3D grid (finite differences). The model reproduces the formation of a central jet of swimmers by self-focusing [Phys. Rev. Lett. 110, 138106 (2013)] but also predicts an instability of the jet, which leads to its fractionation in clusters. We show that this instability is due to hydrodynamic interactions between microswimmers, which attract each other along the flow direction. This effect was not observed in the experiments conducted on dilute suspensions (i.e., where hydrodynamic interactions are weak). This phenomenon is peculiar for pullers for which collective motions are usually not observed on such a time scale. With this modeling, we hope to pave the way toward a better understanding of concentration techniques of algae (a bottleneck challenge in industrial applications).

Periodic structures in binary mixtures enforced by Janus particles.

Phase separation in binary mixtures in the presence of Janus particles has been studied in terms of a Cahn-Hilliard model coupled to the Langevin equations describing the particle dynamics. We demonstrate that the phase separation process is arrested leading to unexpected regular stripe patterns in the concentration field. The underlying pattern forming mechanism has been elucidated: The twofold absorption properties on the surface of Janus particles with respect to the two components of a binary mixture trigger in their neighborhood spatial concentration variations. They result in an effective interaction between the particles mediated by the binary mixture. Our findings open a route to design composite materials with nanoscale lamellar morphologies where the pattern wavelength can be tuned by changing the wetting properties of the Janus particles.

Stability and orientation of lamellae in diblock copolymer films.

The dynamics of microphase separation and the orientation of lamellae in diblock copolymers are investigated in terms of a mean-field model. The formation of lamellar structures and their stable states are explored and it is shown that lamellae are stable not only for the period of the structure corresponding to the minimum of the free energy. The range of wavelengths of stable lamellae is determined by an efficient functional approach introduced with this work. The effects of the interaction of diblock copolymers with two confining substrates on the lamellae orientation are studied by an extensive analysis of the total free energy. By changing the wetting property at one boundary, a transition from a preferentially perpendicular to a parallel lamellar orientation with respect to the confining plates is found, which is rather independent of the distance between the boundaries. Simulations of the dynamics of microphase separation reveal that the time scale of the lamellar orientational order dynamics, which is quantitatively characterized in terms of an orientational order parameter and the structure factor, depends significantly on the properties of the confining boundaries as well as on the quench depth.

Propulsion efficiency of a dynamic self-assembled helical ribbon.

We study the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a dynamic magnetic field. The sedimented ribbon assembles due to time averaged dipolar interactions between the beads. The time dependence of the dipolar interactions together with hydrodynamic interactions cause a twisted ribbon conformation. Domain walls of high twist connect domains of nearly constant orientation and negligible twist and travel through the ribbon. The particular form of the domain walls can be controlled via the frequency and the eccentricity of the modulation. The flux of twist walls-a true ribbon property absent in slender bodies-provides the thrust onto the surrounding liquid that propels this biomimetic flagellum into the opposite direction. The propulsion efficiency increases with frequency and ceases abruptly at a critical frequency where the conformation changes discontinuously to a flat standing ribbon conformation.

Anisotropic particles align perpendicular to the flow direction in narrow microchannels.

The flow orientation of anisotropic particles through narrow channels is of importance in many fields, ranging from the spinning and molding of fibers to the flow of cells and proteins through thin capillaries. It is commonly assumed that anisotropic particles align parallel to the flow direction. When flowing through narrowed channel sections, one expects the increased flow rate to improve the parallel alignment. Here, we show by microfocus synchrotron X-ray scattering and polarized optical microscopy that anisotropic colloidal particles align perpendicular to the flow direction after passing a narrow channel section. We find this to be a general behavior of anisotropic colloids, which is also observed for disk-like particles. This perpendicular particle alignment is stable, extending downstream throughout the remaining part of the channel. We show by microparticle image velocimetry that the particle reorientation in the expansion zone after a narrow channel section occurs in a region with considerable extensional flow. This extensional flow is promoted by shear thinning, a typical property of complex fluids. Our discovery has important consequences when considering the flow orientation of polymers, micelles, fibers, proteins, or cells through narrow channels, pipes, or capillary sections. An immediate consequence for the production of fibers is the necessity for realignment by extension in the flow direction. For fibrous proteins, reorientation and stable plug flow are likely mechanisms for protein coagulation.

Shear-flow-enhanced barrier crossing.

We consider a single Brownian particle confined in a double well potential (DWP) and investigate its response to a linear shear flow by means of the probability density and current determined via numerical solution of the Fokker-Planck equation. Besides a shear-dependent distortion of the probability distribution, we find that the associated current crossing the potential barrier exhibits a convex dependency on the shear rate when the DWP's minima are far apart. With decreasing distance this functional dependency changes from a convex to concave characteristics accompanied with an increase of the probability current crossing the DWP's barrier. Through the difference map of the particle density distribution it is possible to extract the shear-flow-induced contribution to the particle density driving the barrier-crossing current. This may open the possibility to design specific flow profiles to optimize flow-induced activated transport of nanoparticles.

Identifying hydrodynamic interaction effects in tethered polymers in uniform flow.

Using Brownian dynamics simulations, we investigate how hydrodynamic interaction (HI) affects the behavior of tethered polymers in uniform flow. While it is expected that the HI within the polymer will lead to a dependency of the polymer's drag coefficient on the flow velocity, the interchain HI causes additional screening effects. For the case of two polymers in uniform flow with their tether points a finite distance apart, it is shown that the interchain HI not only causes a further reduction of the drag per polymer with decreasing distance between the tether points but simultaneously induces a polymer-polymer attraction as well. This attraction exhibits a characteristic maximum at intermediate flow velocities when the drag forces are of the order of the entropic forces. The effects uniquely attributed to the presence of HI can be verified experimentally.

Convection in nanofluids with a particle-concentration-dependent thermal conductivity.

Thermal convection in nanofluids is investigated by means of a continuum model for binary-fluid mixtures, with a thermal conductivity depending on the local concentration of colloidal particles. The applied temperature difference between the upper and the lower boundary leads via the Soret effect to a variation of the colloid concentration and, therefore, to a spatially varying heat conductivity. An increasing difference between the heat conductivity of the mixture near the colder and the warmer boundary results in a shift of the onset of convection to higher values of the Rayleigh number for positive values of the separation ratio ψ>0 and to smaller values in the range ψ<0. Beyond some critical difference of the thermal conductivity between the two boundaries, we find an oscillatory onset of convection not only for ψ<0, but also within a finite range of ψ>0. This range can be extended by increasing the difference in the thermal conductivity, and it is bounded by two codimension-2 bifurcations.