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# Statistical Physics of Soft Matter and Biological Systems

We work on a broad spectrum of projects from statistical physics of soft condensed matter and biological systems. Besides problems in thermal equilibrium, we investigate systems that are kept in nonequilibrium either by external fields or

by an internal propulsion mechanism such as active Brownian particles.

The highly topical fields of microfluidics and (bio)fluiddynamics inspire us. One current focus in our research is active motion of artificial and biological systems. We aim at understanding design principles for generating active

motion, its generic properties, and the emergent collective dynamics. Our topics are:

- Soft matter: such as liquid crystals, colloidal dispersions, (bio)polymers
- Structure formation of soft matter at interfaces
- Soft Matter and biological systems in nonequilibrium, microfluidics
- Biofluiddynamics and locomotion of microorganisms, active motion
- Biomimetic materials and biomechanics

# News

Topics und methods of the statistical mechanics of soft matter and biological systems

Introduction on Wednesday,**04.11.2020****Please register in the SAP system.****Q&A with Prof. Holger Stark, Editor-in-Chief of EPJE**

Prof. Stark offers the following

**examination dates**:**Friday, 26.02.2021**

Please register in the SAP system.

- Brennpunkt-Artikel im Physik Journal:
**Die Körperform macht's**

Holger Stark

Physik Journal

**Multi-Particle Collision Dynamics with a Non-Ideal Equation of State: Part I**

**Arne Zantop and Holger Stark**

The method of multi-particle collision dynamics (MPCD) and its different implementations are commonly used in the field of soft matter physics to simulate fluid flow at the micron scale. Typically, the coarse-grained fluid particles are described by the equation of state of an ideal gas, and the fluid is rather compressible. This is in contrast to conventional fluids, which are incompressible for velocities much below the speed of sound, and can cause inhomogeneities in density. We propose an algorithm for MPCD with a modified collision rule that results in a non-ideal equation of state and a significantly decreased compressibility. It allows simulations at less computational costs compared to conventional MPCD algorithms. We derive analytic expressions for the equation of state and the corresponding compressibility as well as shear viscosity. They show overall very good agreement with simulations, where we determine the pressure by simulating a quiet bulk fluid and the shear viscosity by simulating a linear shear flow and a Poiseuille flow.

J. Chem. Phys. 154, 024105 (2021)

**Effective Langevin equations for a polar tracer in an active bath**

**by Milos Knezevic and Holger Stark**

We study the motion of a polar tracer, having a concave surface, immersed in a two-dimensional suspension of active particles. Using Brownian dynamics simulations, we measure the distributions and auto-correlation functions of force and torque exerted by active particles on the tracer. The tracer experiences a finite average force along its polar axis, while all the correlation functions show exponential decay in time. Using these insights we construct the full coarse-grained Langevin description for tracer position and orientation, where the active particles are subsumed into an effective self-propulsion force and exponentially correlated noise for both translations and rotations. The ensuing mesoscopic dynamics can be described in terms of five dimensionless parameters. We perform a thorough parameter study of the mean squared displacement, which illustrates how the different parameters influence the tracer dynamics, which crosses over from a ballistic to diffusive motion. We also demonstrate that the distribution of tracer displacements evolves from a non-Gaussian shape at early stages to a Gaussian behavior for sufficiently long times. Finally, for a given set of microscopic parameters, we establish a procedure to estimate the matching parameters of our effective model, and show that the resulting dynamics is in a very good quantitative agreement with the one obtained in Brownian dynamics simulations.

New J. Phys. **22**, 113025 (2020)

**Active open-loop control of elastic turbulence**

**by Reinier van Buel and Holger Stark**

We demonstrate through numerical solutions of the Oldroyd-B model in a two-dimensional Taylor–Couette geometry that the onset of elastic turbulence in a viscoelastic fluid can be controlled by imposed shear-rate modulations, one form of active open-loop control. Slow modulations display rich and complex behavior where elastic turbulence is still present, while it vanishes for fast modulations and a laminar response with the Taylor–Couette base flow is recovered. We find that the transition from the laminar to the turbulent state is supercritical and occurs at a critical Deborah number. In the state diagram of both control parameters, Weissenberg versus Deborah number, we identify the region of elastic turbulence. We also quantify the transition by the flow resistance, for which we derive an analytic expression in the laminar regime within the linear Oldroyd-B model. Finally, we provide an approximation for the transition line in the state diagram introducing an effective critical Weissenberg number in comparison to constant shear. Deviations from the numerical result indicate that the physics behind the observed laminar-to-turbulent transition is more complex under time-modulated shear flow.

Sci. Rep. **10**, 15704 (2020)

**Particle pairs and trains in inertial microfluidics**

**by Christian Schaaf and Holger Stark**

Staggered and linear multi-particle trains constitute characteristic structures in inertial microfluidics. Using lattice-Boltzmann simulations, we investigate their properties and stability, when flowing through microfluidic channels. We confirm the stability of cross-streamline pairs by showing how they contract or expand to their equilibrium axial distance. In contrast, same-streamline pairs quickly expand to a characteristic separation but even at long times slowly drift apart. We reproduce the distribution of particle distances with its characteristic peak as measured in experiments. Staggered multi-particle trains initialized with an axial particle spacing larger than the equilibrium distance contract non-uniformly due to collective drag reduction. Linear particle trains, similar to pairs, rapidly expand toward a value about twice the equilibrium distance of staggered trains and then very slowly drift apart non-uniformly. Again, we reproduce the statistics of particle distances and the characteristic peak observed in experiments. Finally, we thoroughly analyze the damped displacement pulse traveling as a microfluidic phonon through a staggered train and show how a defect strongly damps its propagation.

Eur. Phys. J. E **43**, 50 (2020).

selected for front cover picture

**Optimal Control of Colloidal Trajectories in Inertial Microfluidics Using the Saffman Effect**

**by Felix Rühle, Christian Schaaf, and Holger Stark**

In inertial microfluidics colloidal particles in a Poiseuille flow experience the Segré-Silberberg lift force, which drives them to specific positions in the channel cross section. An external force applied along the microchannel induces a cross-streamline migration to a new equilibrium position because of the Saffman effect. We apply optimal control theory to design the time protocol of the axial control force in order to steer a single particle as precisely as possible from a channel inlet to an outlet at a chosen target position. We discuss the influence of particle radius and channel length and show that optimal steering is cheaper than using a constant control force. Using a single optimized control-force protocol, we demonstrate that even a pulse of particles spread along the channel axis can be steered to a target and that particles of different radii can be separarted most efficiently.

**Squirmer rods as elongated microswimmers: flow fields and confinement**

**by Arne W. Zantop and Holger Stark**

Microswimmers or active elements, such as bacteria and active filaments, have an elongated shape, which determines their individual and collective dynamics. There is still a need to identify what role long-range hydrodynamic interactions play in their fascinating dynamic structure formation. We construct rods of different aspect ratios using several spherical squirmer model swimmers. With the help of the mesoscale simulation method of multi-particle collision dynamics we analyze the flow fields of these squirmer rods both in a bulk fluid and in Hele-Shaw geometries of different slab widths. Based on the hydrodynamic multipole expansion either for bulk or confinement between two parallel plates, we categorize the different multipole contributions of neutral as well as pusher-type squirmer rods. We demonstrate how confinement alters the radial decay of the flow fields for a given force or source multipole moment compared to the bulk fluid.

**Emergent collective dynamics of bottom-heavy squirmers under gravity**

**by Felix Rühle and Holger Stark**

We present the results of hydrodynamic simulations using the method of multi-particle collision dynamics for a system of squirmer microswimmers moving under the influence of gravity at low Reynolds numbers. In addition, the squirmers are bottom-heavy so that they experience a torque which aligns them along the vertical. The squirmers interact hydrodynamically by the flow fields of a stokeslet and rotlet, which are initiated by the acting gravitational force and torque, respectively, and by their own flow fields. By varying the ratio of swimming to bulk sedimentation velocity and the torque, we determine state diagrams for the emergent collective dynamics of neutral squirmers as well as strong pushers and pullers. For low swimming velocity and torque we observe conventional sedimentation, while the sedimentation profile becomes inverted when their values are increased. For neutral squirmers we discover convective rolls of circulating squirmers between both sedimentation states, which sit at the bottom of the system and are fed by plumes made of collectively sinking squirmers. At larger torques porous clusters occur that spawn single squirmers. The two latter states can also occur transiently starting from a uniform squirmer distribution and then disappear in the long-time limit. For strong pushers and pullers only weak plume formation is observed.

Eur. Phys. J. E **43**, 26 (2020)

see also: *'Bottom-heavy' squirmers adopt characteristic group behaviours*, article as EPJE Highlight and at EurekAlert

**Capillary condensation in an active bath**

**by Milos Knezevic and Holger Stark**

e study capillary condensation in a bath of active Brownian particles (ABPs) and the forces acting on the capillary close to the motility-induced phase separation (MIPS). The capillary is modeled as two parallel rods, which are fixed in space. We consider a bath of ABPs having a self-propulsion speed much larger than the critical speed necessary for MIPS to occur.We gradually increase the packing fraction of ABPs, starting from a dilute phase of ABPs and going towards the binodal of MIPS. In stark contrast to conventional capillary condensation, we do not observe any hysteresis in the capillary packing fraction and attribute this to strong temporal fluctuations in the capillary packing fraction . Depending on the packing fraction of ABPs and capillary width, we find that the effective force between the capillary rods can be either attractive or repulsive. In fact,with increasing width it shows damped oscillations as long as capillary condensation occurs. We analyze them in detail by studying the distribution of particle distances from the inner and outer wall of the capillary, respectively. In addition, we examine the capillary in the active bath close to the critical point. We do not observe signs of the presence of long-range Casimir interactions.

Europhys. Lett. **128**, 40008 (2020)

**A flowing pair of particles in inertial microfluidics**

**by Christian Schaaf, Felix Rühle and Holger Stark**

A flowing pair of particles in inertial microfluidics gives important insights into understanding and controlling the collective dynamics of particles like cells or droplets in microfluidic devices. They are applied in medical cell analysis and engineering. We study the dynamics of a pair of solid particles flowing through a rectangular microchannel using lattice Boltzmann simulations. We determine the inertial lift force profiles as a function of the two particle positions, their axial distance, and the Reynolds number. Generally, the profiles strongly differ between particles leading and lagging in flow and the lift forces are enhanced due to the presence of a second particle. At small axial distances, they are determined by viscous forces, while inertial forces dominate at large separations. We identify cross-streamline pairs as stable fixed points in the lift force profiles and argue that same-streamline configurations are only one-sided stable. Depending on the initial conditions, the two-particle lift forces in combination with the Poiseuille flow give rise to three types of unbound particle trajectories, called moving-apart, passing, and swapping, and one type of bound trajectory, where the particles perform damped oscillations towards the cross-stream line configuration. The damping rate scales with Reynolds number squared, since inertial forces are responsible for driving the particles to their steady-state positions.

Soft Matter **15**, 1988 (2019)

**Traveling concentration pulses of bacteria in a generalized Keller–Segel model**

**by Maximilian Seyrich, Andrzej Palugniok, and Holger Stark**

We formulate a Markovian response theory for the tumble rate of a bacterium moving in a chemical field and use it in the Smoluchowski equation. Based on a multipole expansion for the one-particle distribution function and a reaction-diffusion equation for the chemoattractant field, we derive a polarization extended model, which also includes the recently discovered angle bias. In the adiabatic limit we recover a generalized Keller–Segel equation with diffusion and chemotactic coefficients that depend on the microscopic swimming parameters. Requiring the tumble rate to be positive, our model introduces an upper bound for the chemotactic drift velocity, which is no longer singular as in the original Keller–Segel model. Solving the Keller–Segel equations numerically, we identify traveling bacterial concentration pulses, for which we do not need a second, signaling chemical field nor a singular chemotactic drift velocity as demanded in earlier publications. We present an extensive study of the traveling pulses and demonstrate how their speeds, widths, and heights depend on the microscopic parameters. Most importantly, we discover a maximum number of bacteria that the pulse can sustain—the maximum carrying capacity. Finally, by tuning our parameters, we are able to match the experimental realization of the traveling bacterial pulse.

New J. Phys. **21, **103001 (2019)

**Chemotaxis in a binary mixture of active and passive particles**

**by Julian Stürmer , Maximilian Seyrich , and Holger Stark**

Mixtures of active and passive colloids show an intriguing dynamics of self-assembling, which is driven by the active component. Self-phoretic active colloids generate sinks in a chemical concentration field that cause passive colloids to drift toward active colloids by diffusiophoresis. The strength of this effective attraction is governed by the diffusiophoretic parameter, which determines the drift velocity. Simulating the Langevin dynamics of the colloids, we determine the state diagram for increasing diffusiophoretic strength and fixed active velocity. Three main states are distinguished. For weak attraction, passive particles are first scattered in the simulation box and then form a colloidal cloud around its center. Increasing the diffusiophoretic parameter further, passive particles oscillate between the cloud and a compact cluster, which embeds active colloids. Ultimately, in the third state, all particles collapse into a single stable cluster. In the collapse regime, the clustering dynamics of the largest cluster follows a logistic function and the mean cluster velocity vs cluster size decays with a power law. Throughout this article, we discuss our simulation results with regard to the experiments of Singh et al., Adv. Mater. **29** (32), 1701328 (2017).

J. Chem. Phys. **150**, 214901 (2019)

**Optimal steering of a smart active particle**

**by Elias Schneider and Holger Stark**

We formulate the theory for steering an active particle with optimal travel time between two locations and apply it to the Mexican hat potential without brim. For small heights the particle can cross the potential barrier, while for large heights it has to move around it. Thermal fluctuations in the orientation strongly affect the path over the barrier. Then we consider a smart active particle and apply reinforcement learning. We show how the active particle learns in repeating episodes to move optimally. The optimal steering is stored in the optimized action-value function, which is able to rectify thermal fluctuations.

**Collective Dynamics in a Monolayer of Squirmers Confined to a Boundary by Gravity**

**by Jan-Timm Kuhr, Felix Rühle, and Holger Stark**

We present a hydrodynamic study of a monolayer of squirmer model microswimmers confined to aboundary by strong gravity using the simulation method of multi-particle collision dynamics. Thesquirmers interact with each other via their self-generated hydrodynamic flow fields and thereby form avariety of fascinating dynamic states when density and squirmer type are varied. Weak pushers, neutralsquirmers, and pullers have an upright orientation. With their flow fields they push neighbors away andthereby form a hydrodynamic Wigner fluid at lower densities. Furthermore, states of fluctuating chainsand trimers, of kissing, and at large densities a global cluster exist. Finally, pushers at all densities can tiltagainst the wall normal and their in-plane velocities align to show swarming. It turns into chaoticswarming for strong pushers at high densities. We characterize all these states quantitatively.

Soft Matter **15**, 5685 (2019)

**Artificial Chemotaxis of Self-Phoretic Active Colloids: Collective Behavior**

**by Holger Stark**

Microorganisms use chemotaxis, regulated by internal complex chemical pathways, to swim along chemical gradients to find better living conditions. Artificial microswimmers can mimic such a strategy by a pure physical process called diffusiophoresis, where they drift and orient along the gradient in a chemical density field. Similarly, for other forms of taxis in nature such as photo- or thermotaxis the phoretic counterpart exists.

In this Account, we concentrate on the chemotaxis of self-phoretic active colloids. They are driven by self-electro- and diffusiophoresis at the particle surface and thereby acquire a swimming speed. During this process, they also produce nonuniform chemical fields in their surroundings through which they interact with other colloids by translational and rotational diffusiophoresis. In combination with active motion, this gives rise to effective phoretic attraction and repulsion and thereby to diverse emergent collective behavior. A particular appealing example is dynamic clustering in dilute suspensions first reported by a group from Lyon. A subtle balance of attraction and repulsion causes very dynamic clusters, which form and resolve again. This is in stark contrast to the relatively static clusters of motility-induced phase separation at larger densities.

To treat chemotaxis in active colloids confined to a plane, we formulate two Langevin equations for position and orientation, which include translational and rotational diffusiophoretic drift velocities. The colloids are chemical sinks and develop their long-range chemical profiles instantaneously. For dense packings, we include screening of the chemical fields. We present a state diagram in the two diffusiophoretic parameters governing translational, as well as rotational, drift and, thereby, explore the full range of phoretic attraction and repulsion. The identified states range from a gaslike phase over dynamic clustering states 1 and 2, which we distinguish through their cluster size distributions, to different types of collapsed states. The latter include a full chemotactic collapse for translational phoretic attraction. Turning it into an effective repulsion, with increasing strength first the collapsed cluster starts to fluctuate at the rim, then oscillates, and ultimately becomes a static collapsed cloud. We also present a state diagram without screening. Finally, we summarize how the famous Keller−Segel model derives from our Langevin equations through a multipole expansion of the full one-particle distribution function in position and orientation. The Keller−Segel model gives a continuum equation for treating chemotaxis of microorganisms on the level of their spatial density.

Our theory is extensible to mixtures of active and passive particles and allows to include a dipolar correction to the chemical field resulting from the dipolar symmetry of Janus colloids.

Acc. Chem. Res. **51**, 2681 (2018)

**Elastic turbulence in two-dimensional Taylor-Couette flows **

**by R. van Buel, C. Schaaf, and H. Stark**

We report the onset of elastic turbulence in a two-dimensional Taylor-Couette geometry using numerical solutions of the Oldroyd-B model generated with the program OpenFOAM®. Beyond a critical Weissenberg number, an elastic instability causes a supercritical transition from the laminar Taylor-Couette flow to a turbulent flow. The order parameter, the time average of secondary-flow strength, follows the scaling law Φ∝ Wi -Wi_{c}^{γ} with Wi_{c}=10 and γ= 0.45 . Additionally, the flow resistance increases beyond Wi_{c} . The temporal power spectra of the velocity fluctuations show a power-law decay with a characteristic exponent in the range 2< α < 4 , which strongly depends on the radial position. The characteristic exponent β for the spatial power spectra obeys the necessary condition β > 3 , associated with elastic turbulence, for all Wi > Wi_{c} .

Europhys. Lett. **124,** 14001 (2018)

**Feedback control of photoresponsive fluid interfaces**

**by Josua Grawitter and Holger Stark**

Photoresponsive surfactants provide a unique microfluidic driving mechanism. Since they switch between two molecular shapes under illumination and thereby affect surface tension of fluid interfaces, Marangoni flow along the interface occurs. To describe the dynamics of the surfactant mixture at a planar interface, we formulate diffusion–advection–reaction equations for both surfactant densities. They also include adsorption from and desorption into the neighboring fluids and photoisomerization by light. We then study how the interface responds when illuminated by spots of light. Switching on a single light spot, the density of the switched surfactant spreads in time and assumes an exponentially decaying profile in steady state. Simultaneously, the induced radial Marangoni flow reverses its flow direction from inward to outward. We use this feature to set up specific feedback rules, which couple the advection velocities sensed at the light spots to their intensities. As a result two neighboring spots switch on and off alternately. Extending the feedback rule to light spots arranged on the vertices of regular polygons, we observe periodic switching patterns for even-sided polygons, where two sets of next-nearest neighbors alternate with each other. A triangle and pentagon also show regular oscillations, while heptagon and nonagon exhibit irregular oscillations due to frustration. While our findings are specific to the chosen set of parameters, they show how complex patterns at photoresponsive fluid interfaces emerge from simple feedback coupling.

Soft Matter **14**, 1856 (2018)