Active viscoelasticity of odd materials

The mechanical response of active media ranging from biological gels to living tissues is governed by a subtle interplay between viscosity and elasticity. In this Letter, 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.

Materials at the macroscopic scales are often described either as a fluid or as a solid. Such idealized behaviors are insufficient to describe materials that exhibit more complex mesoscopic organization. This is for example the case for liquid crystals or gels, where the structure of matter is more intricate due to elongation or chirality of the constituents. In addition, dissipative or active processes may enter the description of microscopic building blocks. As a result, the macroscopic description of material includes both fluid-and solid-like features -the interplay of these two elements is known as viscoelasticity (see e.g. [1]). Viscoelasticity is a common phenomenon, described by rheology, that can be observed in polymer systems [2] and various biological media [3,4].
From the swimming strokes of sperm cells to intracellular flows -biological systems present a wide variety of cases where chiral symmetry is broken [5][6][7][8][9][10]. Additionally, biological systems are often driven away from thermodynamic equilibrium by chemical reactions that render the matter active at the molecular level [11][12][13]. Recent work on chiral active matter has shown that the presence of activity and chirality, breaking essential microscopic symmetries, leads to novel response functions and transport coefficients in active fluids and solids [14][15][16][17][18]. In the simplest incarnation, the two coefficients were dubbed odd viscosity [19][20][21][22][23][24][25][26] and odd elasticity [16]. The main goal of the present work is to combine these two descriptions to investigate novel viscoelastic properties of chiral active systems.
Basic viscoelastic models.−Viscoelasticity emerges as a consequence of complex phenomena at different scales. It is usually not possible to have a first principle analysis and various phenomenological simplifications are employed. General viscoelastic response in rheology is defined as where σ ij is the stress tensor and u kl corresponds to strains. We always assume a summation over repeated indices. Causality requires that the time dependent response tensor η ijkl (t) is zero for negative times. The above relation can be inverted introducing the so-called creep tensor c ijkl (t): Response and creep tensors respect the symmetries of the material. Since viscoelasticity can be viewed as a transient phenomenon to a time-independent state we need to make some assumptions about the long time behavior of our viscoelastic system. It can be either fluid or solid. These two distinct limiting cases of viscoelastic behavior are commonly described by the Maxwell and Kelvin-Voigt models respectively. These models can be presented by a spring and a viscous damper in series (for Maxwell materials) or in parallel (for Kelvin-Voigt materials). The Kelvin-Voigt model typically defines a viscoelastic solid and captures strain relaxation. Maxwell materials define viscoelastic fluids, representing stress relaxation. Viscoelastic solids are materials with a definite reference shape that deform under applied external forces. They can store elastic energy that is obtained from external forces, however, they return to the initial shape in the absence of strain. On the other hand viscoelastic fluids are materials that do not have a definite shape and flow irreversibly under the action of external forces, still having transient elastic memory effects. The information about the material properties is encoded in the constitutive relations. We are interested in linear viscoelasticity that provides a simplified treatment assuming that the stress at the current time depends only on the current strain and strain rate. As a simple illustrative example, we note that, for the Kelvin-Voigt model of a solid, the relation between strain and stress is given by where, σ ij is the stress tensor, u ij is the strain tensor, u ij is the strain rate tensor, κ ijkl is the elasticity tensor, arXiv:2002.12564v1 [cond-mat.soft] 28 Feb 2020 and η ijkl is the viscosity tensor. Here, σ ij and u ij are symmetric and κ ijkl and η ijmn are symmetric in the first two and last two indices. This approach was first proposed by Maxwell in his spring and damper model and later complemented by Kelvin, Voigt and also Meyer [27] (see also [28][29][30] for modern developments). In this Letter, following the same logic, we employ linearized viscoelasticity to construct parity-breaking generalized Kelvin-Voigt and Maxwell models in two dimensions that also break time-reversal symmetry, which we use to investigate the odd viscoelastic response in solids and fluids. In the present case activity manifests itself through a non-zero value of the odd elastic coefficient, which encapsulates non-conservative microscopic interactions that violate mechanical reciprocity. This means that in a cyclic process, the net elastic work σ ij du ij can be nonzero in the presence of odd elasticity and the sign of the net elastic work changes when the cyclic process is reversed. Therefore odd elasticity [16] violates time reversal symmetry and only exists in active systems [15,16]. Note that odd viscosity requires parity breaking but can exist in a passive system as it is consistent with time reversibility [31].
Relaxation of parity-odd viscoelastic materials in two dimenions.−The relaxation times of a viscoelastic system tell us how long does it take for the material to return from a deformed state to its equilibrium state. In two dimensions there are two distinct types of stresses one can apply to the material: shear and compression. As we shall see, parity breaking does not modify the compressional response. However, the response to shear receives contributions from odd transport coefficients. We are interested in determining the explicit form of the relaxation times. For the Kelvin-Voigt model we rewrite the constitutive equation in the following form where R ijkl = κ −1 ijmn η mnkl and i, j, k, l = {1, 2}, which we refer to as the relaxation times tensor. In the Kelvin Voigt model u ij corresponds to the deformation from a specific reference state. Analogously for the Maxwell model in the presence of a flow given by the velocity v the constitutive relation reads where, is the symmetric part of velocity gradient. In general the corotational part should also be included [11] but we omit it for simplicity. Fourindex tensors possess a unique product of Kronecker delta functions that gives an identity operator Id ijkl = δ ik δ jl .
In order to define κ −1 we need to construct a tensor that contracted with the elasticity tensor κ mnkl satisfies κ −1 ijmn κ mnkl = Id ijkl . We note that in the classical elasticity the elastic tensor is only partially invertible and we construct the inverse in the invertible subspace. Assuming that there is no memory in the system, we can use the symmetries to extract the structure of the relaxation times tensor for an isotropic two-dimensional material in terms of the three coefficients {τ 1 , τ 2 , τ o }, the Kronecker and antisymmetric tensors denoted by δ and . On symmetry grounds the system exhibits three relaxation times. These times depend on the transport coefficients present in the system. In order to get a better physical understanding of this, it is convenient to use a basis of two-dimensional matrices In this basis the stress and displacements are defined as where α = {0, 2, 3}, see Ref. [16] for details. The absence of s 1 = 0 −1 1 0 stems from the assumption that stress and deformation tensors are symmetric. In this representation the elastic tensor is given by a matrix κ with In order to see the explicit form of the elastic tensors in these two bases we start with the most general form of the elastic tensor and the viscosity tensor consistent with the symmetries and apply (9a) to get Now the inversion requires to find an inverse of a 3 × 3 matrix. Given that the relaxation tensor in the basis of two-dimensional matrices has a simple form R αβ = κ −1 αγ η γβ . It is also convenient to introduce an inverse of the relaxation times tensor that we call relaxation rate tensor. Again we can determine in the matrix representation Λ αβ = η −1 αγ κ γα = R −1 αβ . Modified Kelvin-Voigt material.−The first viscoelastic material we want to investigate is an active, chiral, viscoelastic solid (see Figure 1). Such a material should have parity breaking viscous and elastic response transport coefficients together with the conventional parity preserving responses. It has been shown that odd elastic solids require some dissipation mechanism to make them stable. Such a mechanism can be provided by a small viscosity. This type of viscoelastic solid was studied in [16]. In the present study we want to focus on the interplay between odd elastic and odd viscous effects. To do that one can extend the Kelvin-Voigt model to account for parity breaking responses. We start with the constitutive equation (3) written in the matrix representation: σ α = κ αβ u β + η αγuγ (15) In this model the stresses due to viscosity and elasticity are additive and they can be represented by a spring and a damper in parallel. Odd elastic solids have been modeled by using point masses connected by springs that have linear but non-central forces [16,32,33], the odd Kelvin-Voigt model is the simplest extension of such solids to include damping or viscous effects. The resulting odd response in the continuum limit is perpendicular to the applied force, In the case of viscoelastic solids modelled by the Kelvin-Voigt equations (15) we are interested in the relaxation of displacement. To understand this we have to analyze the eigenvalues of the relaxation time tensor where . We get one time that corresponds to compression that is real and does not depend on parity breaking coefficients and two complex times that determine the relaxation of the deviatoric perturbations We conclude that in the parity breaking viscoelastic solids shear perturbations create damped oscillating waves noticed in [16] without odd or Hall viscosity. We also note that in the limit with κ o = 0 the waves still exist and the imaginary part is given by the product of odd viscosity and an even elastic coefficient µ. Modified Maxwell material.−Soft materials like dilute biopolymer solutions can also exhibit viscoelastic behavior [34][35][36][37]. They can transiently store elastic energy but they can also flow like viscous fluids.
We would like to have a model that can serve as a description of chiral active polymer solutions with odd viscoelastic terms. We fluid read (see also Fig. 2): where ρ is the mass density. We can rewrite the above equations using the inverse of the relaxation times tensor (14): We set ρ = 1. This model together with the relaxation times analysis for broken parity and time reversal sym-metries is a central result of our study. We perform analytic studies in a simplifying limit. Before we do that let us analyze the properties of the relaxation rate tensor. This task is easier in the two-dimensional representation, although in the case of Maxwell materials the equations have to be written using fourth rank tensors because the velocity equations do not simplify in the matrix representation. The explicit form of the relaxation rate tensor can be written as where . The structure is again consistent with the symmetries and we have three independent parameters with dimensions of inverse time. The fourth rank tensor can be obtained from the expression Λ ijmn = (s β ) ij Λ αβ (s α ) −1 mn . We can now proceed to the analysis of the dynamics. We will consider two analytically tractable limits. The first is the small elastic coefficients κ ijkl limit, while keeping the Λ ijmn fixed. In this limit, the velocity is decoupled from the shear and we get the following set of equations for the stress In order to extract the information about relaxation times we have to diagonalize the matrix of coefficients.
The resulting eigenvalues are rates of relaxatioñ The relaxation times are given again by the expressions (18) and (19). In a complete analogy with Kelvin-Voigt material we see that changing the volume of the material relaxes without any influence from activity. On the other hand area preserving shear deformations lead to oscillating, damped shear waves. Compressible Maxwell fluids.−Parity odd signatures affect pressure disturbances in a significant way. As a result we want to study the compressible regime of the odd Maxwell model. In such a regime ramifications of odd viscoelasticity should be most pronounced. Let us now look at the equations linearized around a solution with no velocity and ρ = ρ 0 governing the dynamics of an odd viscoelastic fluid: where ∂ * i = ij ∂ j and v * i = ij v j . In the above equation we have considered an equation of state p = χ(ρ − ρ 0 ) = δρ. In order to make analytic progress we take a limit η o /η → 0 and ζ/η → 0 simultaneously. In this limit the fluid needs a very long time to relax. Next we rewrite the resulting equations using vorticity Ω = ij ∂ i v j and the divergence Θ = ∂ k v k as variables where λ eff = λ + ρ 0 χ is the effective compressional coefficient. These equations allow vorticity and divergence sound waves. Now we study plain-wave perturbations Ω(t, x) = Ω(ω, k)e i(ωt+ k· x) and Θ(t, x) = Θ(ω, k)e i(ωt+ k· x) . In the usual viscoelastic theory, longitudinal waves corresponding to the divergence distur-bances are decoupled from the transverse waves that correspond to shear perturbations [38]. Odd viscoelasticity couples these two collective disturbances. These linear waves have dispersion relations given by: We obtain two modes for each equation. The eigenvalues are plotted in Figure 3. Now, considering the extreme case of κ o λ eff , µ we have: The above dispersion relation gives us waves that have speeds proportional to √ κ o and also linear damping proportional to √ κ o in the stable case with κ o > 0. They are distinct from Avron waves in fluids with odd viscosity [20], which are of diffusive nature. As such they provide a novel manifestation of parity breaking excitations in hydrodynamics.
Discussion.−This Letter provides crucial steps towards a systematic construction of a viscoelastic theory that accounts for parity breaking effects in systems that are active and break chiral symmetry. It is shown that active odd viscoelasticity, in an analogy with pervious developments in rheology, can lead to two different classes of materials at long times -solids and liquids. In order to elucidate active odd solids and fluids we phenomenologically extend two building blocks of viscoelasticity, namely the Kelvin-Voigt and Maxwell models, to include chirality. This is the main result of this Letter. We show that such an extension leads to several novel relaxation mechanisms , which are controlled by relaxation times given by Eq. (19). In the case of the odd Kelvin-Voigt model the relaxation of perpendicular displacement is modified by the presence of odd viscosity and activity. In the odd Maxwell model, the shear stress relaxation depends on the parity breaking coefficients given by (25) -an effect not considered before in active viscoelastic fluids. Our construction of the Maxwell model allows one to study the realm of parity breaking viscoelastic fluids. To lay the ground for such studies we derive the governing fluid equations. We also study collective modes for these odd viscoelastic fluids in a simplifying elastic limit, in which the stress relaxation in a viscoelastic compressible fluid is large.