PHYSICAL REVIEW LETTERS 126, 138001 (2021) Editors' Suggestion Active Viscoelasticity of Odd Materials Debarghya Banerjee ,1,2 Vincenzo Vitelli,3,4,5 Frank Jülicher ,2,6 and Piotr Surówka 2,7,8,* 1Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany 2Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 3James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA 4Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA 5Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, Illinois 60637, USA 6Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany 7Department of Theoretical Physics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland 8Würzburg-Dresden Cluster of Excellence ct.qmat, Germany (Received 18 March 2020; revised 17 February 2021; accepted 19 February 2021; published 29 March 2021) The mechanical response of active media ranging from biological gels to living tissues is governed by a subtle interplay between viscosity and elasticity. We generalize the canonical Kelvin-Voigt and Maxwell models to active viscoelastic media that break both parity and time-reversal symmetries. The resulting continuum theories exhibit viscous and elastic tensors that are both antisymmetric, or odd, under exchange of pairs of indices. We analyze how these parity violating viscoelastic coefficients determine the relaxation mechanisms and wave-propagation properties of odd materials. DOI: 10.1103/PhysRevLett.126.138001 Materials at the macroscopic scales are often described the two coefficients were dubbed odd viscosity [30–39] and either as a fluid or as a solid. Such idealized behaviors are odd elasticity [18]. The main goal of this Letter is to combine insufficient to describe materials that exhibit more complex these two formulations through a systematic description of mesoscopic organization. This is, for example, the case for chiral active systems based on symmetry principles. This liquid crystals or gels, where the structure of matter is more leads to a hydrodynamic theory of active odd viscoelastic intricate due to elongation or chirality of the constituents. In solids and liquids, distinguished by a long-time response to addition, dissipative or active processes may enter the static and dynamic deformations. Odd responses are absent description of microscopic building blocks. As a result, in standard Kelvin-Voigt and Maxwell models. the macroscopic description of materials includes both Basic viscoelastic models.—Viscoelasticity emerges as a fluid- and solidlike features—the interplay of these two consequence of complex phenomena at different scales. It is elements is known as viscoelasticity (see, e.g., Ref. [1]). usually not possible to have a first principle analysis, and Viscoelasticity is a common phenomenon, described by various phenomenological simplifications are employed. rheology, that can be observed in polymer systems [2], Since viscoelasticity can be viewed as a transient phenome- metamaterials [3,4], and various biological media [5,6]. non to a time-independent state, we need to make some From the swimming strokes of sperm cells to intracellular assumptions about the long-time behavior of our viscoelastic flows, biological systems present a wide variety of cases system. It can be either fluid or solid. These two distinct where chiral symmetry is broken [7–12]. Additionally, limiting cases of viscoelastic behavior are commonly biological systems are often driven away from thermody- described by the Maxwell and Kelvin-Voigt models, respec- namic equilibrium by chemical reactions that render the tively. These models can be presented by a spring and a matter active at the molecular level [13–15]. Recent work on viscous damper in series (for Maxwell materials) or in chiral active matter has shown that the presence of activity parallel (for Kelvin-Voigt materials). The Kelvin-Voigt and chirality, breaking essential microscopic symmetries, model typically defines a viscoelastic solid and captures leads to novel response functions and transport coefficients in strain relaxation. Maxwell materials define viscoelastic active fluids and solids [16–29]. In the simplest incarnation, fluids, representing stress relaxation. A general viscoelastic response in rheology is defined for small deformations as Z t du σ t η t − τ kl dτ; Published by the American Physical Society under the terms of ijð Þ¼ ijklð Þ dτ ð1Þ the Creative Commons Attribution 4.0 International license. −∞ Further distribution of this work must maintain attribution to σ u ’ where ij is the stress tensor and kl corresponds to strains. the author(s) and the published article s title, journal citation, η t and DOI. Open access publication funded by the Max Planck ijklð Þ is the generalization of the elasticity tensor to Society. viscoelastic systems with a memory kernel. It is crucial to 0031-9007=21=126(13)=138001(6) 138001-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 126, 138001 (2021) keep the tensor structure when symmetries are broken. Relaxation of parity-odd viscoelastic materials in two Causality requires that ηijklðtÞ is zero for negative t. dimensions.—The relaxation times of a viscoelastic system ηijklðtÞ captures the response to a constant shear rate applied tell us how long it takes for the material to return from a to the material, in such a way that symmetries are preserved. deformed state to its equilibrium state. In two dimensions We always imply a summation over repeated indices, which there are two distinct types of stresses that one can apply to run over spatial x and y coordinates as we focus on two- the material: shear and compression. As we shall see, parity dimensional systems. Solids and liquids differ by the limiting breaking does not modify the compressional response. value of the stress under a deformation, which, at long times, However, the response to shear receives contributions from tends to zero for fluids and to a constant value for solids. As odd transport coefficients. We are interested in determining an equivalent description of viscoelastic response, one can the explicit form of the relaxation times. For the Kelvin- consider the stress evolution as an input using the creep Voigt model we rewrite the constitutive equation in the compliance tensor cijklðtÞ in the following way [40]: following form: Z t dσ −1 kl Rijklu_ kl ¼ −uij þ κijklσkl; ð4Þ uijðtÞ¼ cijklðt − τÞ dτ: ð2Þ −∞ dτ −1 where Rijkl ≡ κijmnηmnkl and i; j; k; l ¼f1; 2g, which we Creep is a progressive deformation of a material under refer to as the relaxation-time tensor. In the Kelvin-Voigt constant stress. If it approaches a finite shear, the material model uij corresponds to the deformation from a specific is said to be solid, and if it increases linearly after a long time, reference state. Analogously, for the Maxwell model in the the material is a fluid. presence of a flow given by the velocity v⃗the constitutive We are interested in linear viscoelasticity and focus on the relation reads simple case where the stress at the current time depends only on the current strain and strain rate. As a simple illustrative d R σ −σ η v ; example, we note that, for the Kelvin-Voigt model of a solid, ijkl dt kl ¼ ij þ ijkl kl ð5Þ the relation between strain and stress is given by where d=dt ¼ ∂t þ vi∂i is the convective derivative and σij ¼ κijklukl þ ηijmnu_ mn; ð3Þ 1 vkl ¼ 2 ð∂kvl þ ∂lvkÞ is the symmetric part of the velocity where σij is the stress tensor, uij is the strain tensor, u_ ij is the gradients. In general the corotational part should also be strain rate tensor, κijkl is the elasticity tensor, and ηijkl is the included [13], but we omit it for simplicity. Four-index δ δ viscosity tensor. Here σij and uij are symmetric, and κijkl and tensors possess a unique identity operator Idijkl ¼ ik jl, δ ηijmn are symmetric in the first two and last two indices. where ij denotes the Kronecker symbol. In order to define −1 Considering linear viscoelasticity and isotropy reduces the κ we need to construct a tensor that, contracted with the κ κ−1 κ number of entries in κijkl and ηijmn to two in the parity-even elasticity tensor mnkl,satisfies ijmn mnkl ¼ Idijkl.Wenote case. These are the first and second Lam´e parameters λ and μ that in classical elasticity the elasticity tensor is only partially as well as two viscosities, bulk viscosity ζ and shear viscosity invertible and we construct the inverse in the invertible η. This approach was first proposed by Maxwell in his spring subspace. Assuming that there is no memory in the system, and damper model and later complemented by Kelvin, we can use the symmetries to extract the structure of the Voigt, and also Meyer [41] (see also [42–44] for modern relaxation-time tensor for an isotropic two-dimensional developments). material In this Letter, following the same logic, we employ linearized viscoelasticity to construct parity-breaking gen- Rijkl ¼ τ1ðδikδjl þ δilδjkÞþτ2δklδij eralized Kelvin-Voigt and Maxwell models in two dimen- τo δ ϵ δ ϵ δ ϵ δ ϵ sions that also break time-reversal symmetry, which we use þ 2 ð ik jl þ il jk þ jk il þ jl ikÞð6Þ to investigate the parity-odd viscoelastic response in solids o and fluids. In the present case the parity breaking manifests in terms of the three coefficients fτ1; τ2; τ g. Here ϵij ¼ −ϵji itself through a nonzero value of the odd viscosity ηo and the denotes the antisymmetric tensor with odd parity. On odd elastic coefficient κo. Odd elasticity encapsulates non- symmetry grounds the system exhibits three relaxation times. o o o 2 2 conservative microscopic interactions.H This means that in a These times, τ1¼ðη κ þημÞ=f4½ðκ Þ þμ g, τ2 ¼ðζ þ ηÞ= o o o o 2 2 cyclic process the net elastic work σijduij can be nonzero in ½4ðλ þ μÞ − τ1,andτ ¼ðη μ − ηκ Þ=f4½ðκ Þ þ μ Þg, the presence of odd elasticity and that the sign of the net depend on the transport coefficients ζ, λ, η,andμ defined elastic work changes when the cyclic process is reversed.