Active Viscoelasticity of Odd Materials

PHYSICAL REVIEW LETTERS 126, 138001 (2021)

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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. below. τ1 and τ2 capture the relaxation of compression and Therefore odd elasticity [18] breaks time-reversal symmetry shear, respectively. The relaxation time τo is associated with and exists only in active systems [17,18]. Note that odd the relaxation of perpendicular responses depicted in Fig. 1 viscosity requires parity breaking but can exist in a passive for the Kelvin-Voigt solid and in Fig. 2 for the Maxwell fluid. system, as it is consistent with time reversibility [45]. We note, however, that in general all of these relaxation

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(a) (a)

odd response of the strain stress σ0 odd response of the stress (b) odd spring strain u0

(b) κo odd damper odd spring

ηo

ηo κo

odd damper FIG. 2. (a) Sketch of the odd Maxwell model. A finite deformation will result in a transverse stress. Its relaxation will FIG. 1. (a) Sketch of the odd Kelvin-Voigt model. Upon be captured by odd transport coefficients. (b) Depiction of the odd application of a constant stress the resulting deformation will response in odd Maxwell materials. It is modeled by a spring and be perpendicular to the stress direction. The decay of the a damper connected in series. We emphasize that this collective deformation will be captured by odd transport coefficients. representation does not imply that odd Maxwell fluids can be (b) Depiction of the odd response in odd Kelvin-Voigt materials. constructed out of springs and dampers. It is modeled by a transverse spring and a damper connected in parallel. ηijkl ¼ ηðδikδjl þ δilδjkÞþζδklδij o processes are coupled, leading to an oscillatory relaxation η þ ðδikϵjl þ δilϵjk þ δjkϵil þ δjlϵikÞ; ð9Þ dynamics. 2 In order to get a better physical understanding of the symmetry structure, it is convenient to use a basis of two- which is consistent with spatial isotropy and the minor dimensional matrices symmetry of indices. Using the representation in Eq. (7) we obtain 0 1 10 10 01 αβ s0 ¼ ;s2 ¼ ;s3 ¼ : ð7Þ λ þ μ 00 01 0 −1 10 B C καβ ¼ 2@ 0 μ −κo A ð10Þ 0 κo μ In this basis the stress and displacements are defined as α α α α σ ¼ sijσij, u ¼ sijuij, where α ¼f0; 2; 3g; see Ref. [18] and for details. The absence of s1 ¼ð0 −1Þ stems from the 1 0 0 1 assumption that stress and deformation tensors are sym- ζ þ η 00αβ metric. In this representation the elastic tensor is given by a αβ B o C αβ 1 β −1 α η ¼ 2@ 0 η −η A : ð11Þ matrix with elements κ ¼ 2 ðs Þij κijmnsmn. The original 0 ηo η form can be obtained using the formula κijmn ¼ 1 β αβ α −1 ðs Þijκ ðs Þmn. In order to see the explicit form of the 2 Now the inversion requires an inverse of a 3 × 3 matrix. elastic tensors in these two bases, we start with the most The relaxation tensor in Eq. (6) has the simple form general form of the elastic tensor, 1 −1 Rαβ ¼ 2 καγ ηγβ. It is also convenient to introduce an inverse of the relaxation-time tensor κijkl ¼ μðδikδjl þ δilδjkÞþλδklδij κo Λ η−1 κ δ ϵ δ ϵ δ ϵ δ ϵ ; ijkl ¼ ijmn mnkl ð12Þ þ 2 ð ik jl þ il jk þ jk il þ jl ikÞ ð8Þ that we call the relaxation rate tensor, whose matrix 1 −1 −1 and the viscosity tensor, representation is Λαβ ¼ 2 ηαγ κγα ¼ Rαβ .

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— D S Modified Kelvin-Voigt material. The first viscoelastic ukl ¼ ukl þ ukl, and supplement it with parity-breaking material we want to investigate is an active, chiral, terms. The superscripts D and S refer to a damper and a viscoelastic solid (see Fig. 1). Such a material shall have spring, respectively. In the Maxwell model the strain is parity-breaking viscous and elastic response transport additive and modeled by a spring and a damper in series (as coefficients together with the conventional parity preserv- opposed to being parallel in the Kelvin-Voigt model), as ing responses. It has been shown that odd elastic solids presented in Fig. 2. The constitutive equation together with require some dissipation mechanism to make them sta- the momentum conservation equation for the Maxwell fluid ble [18]. Such a mechanism can be provided by a small reads (see also Fig. 2) viscosity. This type of viscoelastic solid was studied in d Ref. [18]. In this Letter we want to focus on the interplay v η−1 σ κ−1 σ ; kl ¼ ijkl ij þ ijkl dt ij ð16aÞ between odd elastic and odd viscous effects. To do that one d can extend the Kelvin-Voigt model to account for parity- ρ v ∂ σ ; breaking responses. We start with the constitutive equa- dt i ¼ j ij ð16bÞ tion (3) written in the matrix representation: where ρ is the mass density. We can rewrite the above σα ¼ καβuβ þ ηαγu_ γ: ð13Þ equations using the inverse of the relaxation-time tensor (12). This model together with the relaxation-time analysis In this model the stresses due to viscosity and elasticity are for broken parity and time-reversal symmetries is a central additive and they can be represented by a spring and a result of our Letter. We perform analytic studies in a damper in parallel. Odd elastic solids have been modeled simplifying limit. We consider an analytically tractable by using point masses connected by springs that have linear limit, taking small elastic coefficients κijkl while keeping but noncentral forces [18,46,47], the odd Kelvin-Voigt the relaxation rates Λijmn fixed. The relaxation times are model is the simplest extension of such solids to include given again by the expressions (14) and (15). In a complete damping or viscous effects. analogy with Kelvin-Voigt material we see that changing The resulting odd response in the continuum limit is the volume of the material relaxes without any influence perpendicular to the applied force. In the case of visco- from activity. On the other hand, area preserving shear elastic solids modeled by the Kelvin-Voigt equation (13) deformations lead to oscillating, damped shear waves. we are interested in the relaxation of displacement. To Compressible Maxwell fluids.—Parity breaking affects understand this we have to analyze the eigenvalues of the pressure in a significant way [33]. We can investigate this in relaxation-time tensor. We get one time that corresponds to the compressible regime of the odd Maxwell model. Let us compression that is real and does not depend on parity- now look at Eq. (16a) linearized around a solution with no breaking coefficients, velocity and ρ ¼ ρ0 governing the dynamics of an odd viscoelastic Maxwell fluid: 1 η þ ζ τ˜ ; 14 ¼ 2 μ λ ð Þ ∂ þ σ μ ∂ v ∂ v λδ ∂ v ∂t ij ¼ ð i j þ j iÞþ ij k k ð17aÞ and two complex times that determine the relaxation of the κo deviatoric perturbations, ∂v ∂ v ∂v ∂ v − Λ σ ; þ 4 ð i j þ i j þ j i þ j i Þ ijmn mn 1 ηoκo þ ημ iðηoμ − ηκoÞ ∂ τ˜1;2 ¼ : ð15Þ ρ0 vi ¼ ∂jσij; ð17bÞ 2 μ2 þðκoÞ2 ∂t ∂ We conclude that in the parity-breaking viscoelastic solids δρ ¼ −ρ0∂kvk; ð17cÞ shear perturbations create damped oscillating waves ∂t noticed in Ref. [18] without odd or Hall viscosity. We ∂ ϵ ∂ v ϵ v ρ ρ δρ p κo 0 where i ¼ ij j and i ¼ ij j, ¼ 0 þ ,and ¼ also note that in the limit with ¼ the waves still exist, 1 i − 2 σi (see also Ref. [52]). In order to make analytic progress and the imaginary part is given by the product of odd o viscosity and an even elastic coefficient μ. we take a limit η =η → 0 and ζ=η → 0 simultaneously. Next Modified Maxwell material.—Soft materials like dilute we rewrite the resulting equations using vorticity Ω ¼ biopolymer solutions can also exhibit viscoelastic behavior ϵij∂ivj and the divergence Θ ¼ ∂kvk as variables: [48–51]. They can transiently store elastic energy, but they ∂2 can also flow like viscous fluids. 2 o 2 ρ0 Ω ¼ μ∇ Ω − κ ∇ Θ; ð18aÞ We would like to have a model that can serve as a ∂t2 description of chiral active polymer solutions with odd ∂2 viscoelastic terms. We follow a phenomenological path by ρ Θ 2μ λ ∇2Θ κo∇2Ω: D S 0 2 ¼ð þ Þ þ ð18bÞ taking a Maxwell model, in which σij ¼ σij ¼ σij, ∂t

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important example of such an active chiral system in biology is the cell cortex. This is a thin layer of an active gel formed by actin filaments and many other protein components which is located below the cell membrane. This gel layer has viscoelastic properties because filaments turn over in about one minute, giving its elastic properties on short timescales and viscous behaviors at long times [49]. It has been shown that this system generates active stresses in the layer and active chiral processes give rise to asymmetric flows [54–56]. Other examples of two-dimensional viscoelastic systems include interfaces between complex active fluids, thin layers of viscoelastic substances placed between plates, and solid metamaterials constructed with active or nonreciprocal microscopic interactions, as described in Ref. [47], where the Kelvin-Voigt model of viscoelasticity is directly applicable. Odd active viscoelasticity can, in principle, be tested in microrheological experiments, where viscoelastic properties of complex fluids can be determined from the motion of FIG. 3. Real and imaginary parts of the frequency eigenvalues. embedded colloidal particles [57]. Based on symmetry (a),(b) Real and imaginary parts, respectively, of the first arguments, the dynamics of a tracer particle in an odd eigenvalue. (c),(d) Real and imaginary parts, respectively, of viscoelastic medium would be described by an equation of the second eigenvalue. These are obtained for the parameters the form k ¼ 1, λ ¼ 0, and ρ0 ¼ 1=2. Z t η t−t0 −ηo t−t0 _ 0 ð Þ ð Þ 0 mViðtÞ¼Fi − dt Vjðt Þ: ð21Þ ηo t−t0 η t−t0 These equations allow vorticity and divergence sound 0 ð Þ ð Þ ij waves.Nowwestudytheplain-waveperturbationsΩðt; x⃗Þ¼ V⃗ m F⃗ ⃗ i ωt k⃗x⃗ ⃗ i ωt k⃗x⃗ Here is the velocity of the bead of mass , denotes an Ω ω; k e ð þ · Þ and Θ t; x⃗ Θ ω; k e ð þ · Þ.Inthe o ð Þ ð Þ¼ ð Þ applied force, and the memory kernels ηðtÞ and η ðtÞ usual viscoelastic theory, longitudinal waves corresponding capture normal and odd viscoelastic responses, respec- to the compressional disturbances are decoupled from the tively. In a parity-breaking viscoelastic fluid the drift of a transverse waves that correspond to shear perturbations [53]. particle should be governed by odd transport coefficients. Odd viscoelasticity couples these two collective disturbances. This suggests that signatures of effects associated with odd These linear waves have dispersion relations given by viscoelasticity could be observed in the dynamics of 1 particles embedded in chiral active gels. ω ¼k pffiffiffiffiffiffiffi ½ðλ þ 3μÞððλ þ μÞ2 − 4κo2Þ1=21=2: ð19Þ 2ρ0 P. S. was supported by the DFG through the Leibniz Program, the cluster of excellence ct.qmat (EXC 2147, We obtain two modes for each equation. The eigenvalues are Project No. 390858490), and the National Science Centre plotted in Fig. 3. Now, considering the extreme case of (NCN) Sonata Bis Grant No. 2019/34/E/ST3/00405. V. V. κo ≫ λ, μ,wehave was supported by the Complex Dynamics and Systems Program of the Army Research Office under Grant ffiffiffiffiffi 1 p o 1=2 No. W911NF-19-1-0268 and the Simons Foundation. ω ¼k pffiffiffiffiffi κ ðiÞ : ð20Þ This work was partially supported by the University of ρ0 Chicago Materials Research Science and Engineering Center, which is funded by National Science Foundation The above dispersion relation yields waves that have speeds pffiffiffiffiffi under Grant No. DMR-2011854. proportional to κo and also linear damping proportional to pffiffiffiffiffi κo in the stable case with κo > 0. They are distinct from

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