Active chiral fluids

S. Furthauer 0 1 M. Strempel 0 1 S.W. Grill 0 1 F. Julicher 1 0 Max Planck Institute of Molecular Cell Biology and Genetics , Pfotenhauerstrae 108, 01307 Dresden, Germany 1 Max Planck Institute for the Physics of Complex Systems , Nothnitzer Strae 38, 01187 Dresden, Germany Active processes in biological systems often exhibit chiral asymmetries. Examples are the chirality of cytoskeletal filaments which interact with motor proteins, the chirality of the beat of cilia and flagella as well as the helical trajectories of many biological microswimmers. Here, we derive constitutive material equations for active fluids which account for the effects of active chiral processes. We identify active contributions to the antisymmetric part of the stress as well as active angular momentum fluxes. We discuss four types of elementary chiral motors and their effects on a surrounding fluid. We show that large-scale chiral flows can result from the collective behavior of such motors even in cases where isolated motors do not create a hydrodynamic far field. 1 Introduction Biological matter is driven far from thermodynamic equilibrium by active processes on the molecular scale. These processes are usually driven by the chemical reaction of a fuel and generate spontaneous movements and mechanical stresses in the system. The prototype example are motor molecules which play a key role for dynamic processes in the cytoskeleton [1, 2]. Motors on the molecular scale are involved in many important cellular processes, such as cell locomotion [3], cell division [47], the beating of cilia [8, 9] and the swimming of microorganism [1, 10, 11]. Biological materials driven by molecular motors often exhibit fluid-like behaviors on long time scales and are thus called active fluids [12, 13]. On large length scales, active fluids can exhibit spontaneous flow patterns [1416], active material stresses [17] and unconventional material properties. Many examples of active fluids are found in biology, such as the cellular actomyosin cytoskeleton [18, 19], suspensions of microswimmers [11, 20, 21] or tissues [13, 22] but artificial examples such as granular systems on a vibrating surface [2325] have also been studied. Most biomolecules and the structures they form are chiral. In particular, this includes force-generating processes on the molecular and cellular scale which have been studied both experimentally [8, 11, 26] and theoretically [9, 2730]. As a consequence, microswimmers typically move on chiral trajectories [11, 31] and cilia can exhibit helical beats [9,32,33]. In vertebrates, the chiral beat of cilia generates fluid flows that participate in left-right a e-mail: b e-mail: symmetry breaking, an essential step in the development of the whole organism [3436]. Because chiral processes are ubiquitous in biological systems, a complete description of active fluids should include the effects of chirality [3739]. Recently, several approaches have been developed to describe the physical properties of active fluids both on mesoscopic [40, 41] and macroscopic scales [4244]. The generic features of active fluids are found in the hydrodynamic limit at large length and time scales. In this limit, the dynamics of active fluids can be obtained systematically using non-equilibrium thermodynamics [45], relying only on conservation laws, broken symmetries and local thermodynamic equilibrium. Such theories take into account local polar or nematic order, which allows for the existence of anisotropic active stresses. The corresponding hydrodynamic equations are a generalization of liquid crystal hydrodynamics [4649] to active systems. While there exists a broad understanding of the main properties of active fluids and gels, studies of the effects of chiral asymmetries of active processes are lacking. In this paper, we present a generic description of active fluids that takes into account active chiral processes. Active force generation induces force dipoles in the material. If all forces are internal, the total force and torque vanish as required by the conservation of momentum and angular momentum. The density of force dipoles is an active stress in the material [13, 14, 17]. In addition, active chiral processes allow for the existence of active torque dipoles, which enter the conservation of angular momentum. We start our analysis in sect. 2 by discussing the conservation of momentum and angular momentum for chiral systems. We identify four different types of elementary chiral motors that are generated by distributions of active torque and force dipoles in the fluid. In sect. 3, we present the general theory of active chiral fluids. We keep the spin angular momentum density as a separate hydrodynamic variable generalizing previous work on passive liquid crystals [49] to active systems. We identify conjugate pairs of thermodynamic variables and write general constitutive material relations. A simplified model is discussed in sect. 4, where active antisymmetric stresses and active angular momentum fluxes are considered. Using this model, we discuss two examples: A) the hydrodynamic flow fields generated by isolated elementary motors and B) the force an active chiral fluid exerts on two plates between which it is confined. We conclude the paper with a discussion in sect. 5. 2 Active chiral fluids We express the force and the torque balance in fluids by conservation laws for momentum and angular momentum. Based on simple symmetry arguments, we show that active angular momentum fluxes and antisymmetric stresses exist in active chiral fluids. 2.1 Antisymmetric stress and angular momentum flux Consider a fluid described by many small-volume elements in the continuum limit. The mass density is denoted by and the velocity of the centers of mass of individual volume elements by v. The momentum density g = v obeys the conservation law The total stress tot is the momentum flux and the force density ext describes externally applied forces. Greek indices denote the three spatial coordinates x, y, z. Einsteins summation convention over repeated indices is implied. The total stress can be split into three parts of the total stress and tot,a = (1/2)(tot tot) is the antisymmetric part of the total stress. In the frame of reference characterized by the position vector r, the total angular momentum density ltot is conserved, Here, Mtot denotes the total flux of angular momentum and ext is the externally applied bulk torque. The total angular momentum density ltot consists of an orbital contribution rg r g due to the center-of-mass motion of individual volume elements and a spin contribution l = ltot (rg r g), which describes the angular momentum in the rest frame of local volume elements. The spin angular momentum can thus be written as l = I, where I is the moment of inertia density and is an intrinsic rotation rate of local volume elements. Note that the antisymmetric tensors and l can be equivalently represented (...truncated)