Hydrodynamics of active polar systems in a (Visco)elastic background

The motion of the active entities is described by the macroscopic velocity v ₁. The passive background gives rise to another velocity v ₂. The kinetic energy, therefore, can be written as with α = ρ ₁ ρ ₂/ρ = ϕ(1 − ϕ)ρ, where ρ ₁ = ϕ ρ and ρ ₂ = (1 − ϕ)ρ are the mass densities of the active and passive part, respectively, with ρ = ρ ₁ + ρ ₂ the total mass density and ϕ the active mass fraction. Accordingly, g = ρ v is the total momentum density defining the mean velocity v. Of the relative velocity F = v ₁ − v ₂ only its modulus enters (1), since F ² = F ². The orientation of F, described by the unit vector f = F/F, is not energetically fixed and describes the spontaneous breaking of rotational symmetry due to the ordered motion of the active entities. This is the appropriate hydrodynamic variable to describe polar dynamic order. Note that it is a true velocity, e.g., it changes sign under time reversal, opposite to the polarization that is used in the static polar case (Brand et al. 2006).

In contrast to a passive two-fluid description (Pleiner and Harden 2003), with an equilibrium state g = 0 = F, the active state is characterized by a finite F > 0 due to the active motion. In particular, we assume a stationary state F _s = c o n s t. > 0 (and g = 0) as the basic state. It has a finite, non-vanishing active velocity F _s = F _s f. In the co-moving frame, there is v = 0, v ₁ = (1 − ϕ)F _s f, and v ₂ = −ϕ F _s f.

Generally, there can be multiple stationary states that might depend on space, or even non-stationary, time dependent ones, but we will not deal with these cases here. We will also not consider a possible transition F _s → 0 to the passive state (cf., e.g., Vicsek et al. 1995; Svenšek et al. 2013).

The existence of a finite active velocity is due to some internal, very often chemical driving force that, e.g., is described by some nonlinear diffusion-reaction equations. This part of the dynamics is system-specific, while we concentrate on the universal aspects. Therefore, we take the finite F _s as given, which is technically achieved by an energetic coupling (Brand et al. 2013) to a fictional “active force density” F ^{a c t} = α F _s f. Since 𝜖 _{k i n} and 𝜖 _{a c t} are the only parts of the total energy density 𝜖 that contain F, the conjugate is and vanishes in the stationary state. A dissipative coupling \(\dot F \sim - \xi ^{\prime } m\) describes the relaxation of F to F _s .

In an Eulerian description, a temporal change of a variable (a field in space and time) can be caused by the advection of that quantity in a velocity field. For vectorial and tensorial fields also convection takes place. As an example, we take the concentration dynamics, which is generally of the form (Pleiner and Brand 1996) with the phenomenological current \(j^{\phi }_{i} \).

The advection contribution is reversible, since it shows the same time reversal behavior as \(\dot \phi \). In the entropy production, it is compensated (together with other transport contributions) by the static isotropic pressure in the diagonal part of the stress tensor (Pleiner and Brand 1996). In a usual 1-fluid system, this zero entropy requirement is sufficient to fix this term and no material dependence is possible.

In a two-fluid system, it is not clear a priori, which velocity has to be used for advection and the zero entropy production requirement is not enough to decide this question. A reasonable procedure (Pleiner and Harden 2003) is to use the mean velocity for all transport terms, since in that case the zero entropy requirement is fulfilled. Nevertheless, the actual advection velocities can be different due to the phenomenological currents. In particular, \(j^{\phi }_{i} \) contains a contribution (for the full expression cf. “Macroscopic equations” section) with the phenomenological reversible transport parameter γ _∥ that leads to the effective, material-dependent transport velocity v _{t r} = v + γ _∥(1 − 2ϕ)(F − F _s f). In the stationary state, this velocity vanishes in the co-moving frame. In a laboratory frame, on the other hand, where the active velocity is v ₁ = F _s f, and the passive one v ₂ = 0, there is a non-vanishing transport velocity v _{t r} = ϕ F _s f rendering the transport term a linear one. Indeed, the linearized active concentration dynamics in the laboratory frame reads with \(j^{\phi ,\prime }_{i} \) containing the phenomenological couplings to other degrees of freedom.

For deviations from the stationary state, \(\Delta \boldsymbol {v}_{1,2} \equiv \boldsymbol {v}_{1,2} - \boldsymbol {v}_{1,2}^{stat}\), there is an additional, frame-independent contribution to the transport velocity in Eq. 4 given by Δv _{t r} = γ′Δv ₁ + γ″ Δ v ₂, with γ′ = ϕ + γ _∥(1 − 2ϕ) and γ″ = 1 − ϕ − γ _∥(1 − 2ϕ), which is material dependent, but only enters the nonlinear dynamics.

The existence of a finite velocity in the stationary state is a clear indication of the non-equilibrium situation due to the internal driving force. The active transport term reflects non-equilibrium, but, like any transport term, cannot drive the system out of equilibrium, since it is reversible with zero entropy production. In the literature (Giomi and Marchetti 2012; Maitra et al. 2014) (and in the Toner-Tu model as presented in Marchetti et al. 2013), dynamic polar order is often described by a polarization vector, p, rather than by a velocity-type of variable. The active transport is thereby written as v + w ₁ p where v is the passive velocity and p is used as the active velocity, with a phenomenological coefficient w ₁. Clearly, the second part does not transform as a velocity rendering it dissipative and leading to a non-zero entropy production that drives the system. The physical origin of this additional, inadvertent driving force is dubious.

Macroscopic descriptions of active polar or nematic systems show a direct coupling of the stress tensor with the order parameter, often called “active term”. Recently, we showed (Brand et al. 2014) that one cannot decide, neither from the symmetry properties of the macroscopic variables involved, nor from the structure of such a cross-coupling, whether the system studied is active or passive. Rather, that depends on whether the variables that give rise to those cross-couplings in the stress tensor are driven internally or externally or not at all. Such terms are reversible and come with counter terms that ensure zero entropy production. Therefore, they cannot drive the system. On the other hand, if the counter terms are neglected, the “active term” leads to non-zero entropy production and acts as an additional, albeit dubious driving force.

In the rheology literature viscoelasticity is often described by an ad hoc generalization of Newton’s linear relation between viscous stress and velocity gradients, involving nonlinearities, time derivatives, and more (Oldroyd 1950, 1961; Coleman and Noll 1961; Truesdell and Noll 1965; Giesekus 1966, 1982; Bird et al. 1977; Johnson and Segalman 1977, 1978; Larson 1988). Such an approach not only immediately becomes unwieldy and intractable when applied to more complicated systems, like the two-fluid and active and ordered case considered here, it is also rather opaque concerning the compliance with general thermodynamic requirements (Pleiner et al. 2005).

Here, we will use the genuine hydrodynamic approach to describe elasticity by employing the strain tensor ε _{i j} as macroscopic variable (Martin et al. 1972; Pleiner and Brand 1991). In crystals, it is related to the displacement vector u _i , the hydrodynamic variable associated with spontaneous breaking of translational symmetry, by 2ε _{i j} = ∇ _i u _j + ∇ _j u _i , if linearized. In general (already in crystals exhibiting transport of defects), the strain tensor is the appropriate variable. We will restrict ourselves to linear elasticity, assuming the elastic deformations caused by the active particles to be small.

The linear dynamic elastic equation reads in Eulerian description (Pleiner et al. 2004) where A _{i j} = (∇ _i v _j +∇ _j v _i )/2 is the “rate-of-strain” tensor and W _{i j} a phenomenological current. For the transport terms, we have kept seemingly nonlinear expressions, since in an active system those terms have linear parts as discussed in the previous section.

For ordinary, permanent elasticity W _{i j} contains diffusion-like contributions, called vacancy diffusion in crystals, which are usually neglected. For viscoelasticity or transient elasticity, the current W _{i j} contains the relaxation of the strains, where Ξ _{i j} = ∂ 𝜖/∂ ε _{i j} is the elastic stress tensor, the conjugate to the strain tensor, that is obtained as a partial derivative of the the total energy density 𝜖. Since Ξ _{i j} contains a contribution ∼ ε _{i j} , elastic strains vanish on a finite time scale. Further contributions to Ξ _{i j} , describing static couplings to other degrees of freedom, are listed in “Macroscopic equations” section.

In addition to relaxation, there are reversible and irreversible contributions to W _{i j} that show in a lucid manner the dynamic couplings of the elastic degree of freedom to the other ones. Using standard hydrodynamic procedures (Pleiner and Brand 1996) for their derivation, one guarantees proper thermodynamic behavior, like positivity of entropy production for dissipative processes and zero entropy production for reversible ones. The full expression for W _{i j} will be presented in Section “Macroscopic equations”. Here, we only want to discuss the (reversible) seemingly nonlinear contributions with the phenomenological material parameters β ₇ and β ₈. In Eq. 7, the advection and convective terms are written with the mean velocity v, but with Eq. 9, these velocities effectively are v + β ₈ α(F − F _s ) and v + β ₇ α(F − F _s ), for advection and convection, respectively. As a result, they are material dependent. For example, the convection of elastic deformations takes place with the active velocity, v ₁, if β ₇ = 1/ρ ₁, and with the passive one, v ₂, if β ₇ = −1/ρ ₂, but generally it will be something in-between. In the stationary state, which is homogeneous, the convection vanishes. For advection, there is a non-vanishing stationary velocity ϕ F _s f in the laboratory frame.

The generalization of this hydrodynamic approach to nonlinear viscoelasticity (transient elasticity) using the Eulerian strain tensor has been given in Temmen et al. (2000, 2001), Pleiner et al. (2000, 2004), and Grmela (2002), and its practical usefulness in describing many experiments has been demonstrated, recently (Müller et al. 2016a; b). An alternative way of describing viscoelasticity is based on the use of a relaxing orientational order parameter tensor (Doi and Edwards 1986; Pleiner et al. 2002) with rather equivalent results (Pleiner et al. 2004).

Rotations of an elastic structure are related to the displacement vector by \(\Omega _{ij} = \tfrac 12 (\nabla _{i} u_{j} - \nabla _{j} u_{i})\). In isotropic elastomers, this variable is irrelevant. If there is an independent preferred direction present, rotations of the preferred direction relative to the elastic medium, \(\tilde {\Omega }_{i} \), are additional independent macroscopic degrees of freedom. They have been introduced for nematic liquid crystal elastomers by deGennes (1980). This concept has been included into the macroscopic dynamic description of liquid crystalline elastomers (Brand and Pleiner 1994). The nonlinear generalization of relative rotations (Menzel et al. 2007) is vital in describing and explaining the reorientation behavior of the nematic director under an external mechanical force (Menzel et al. 2009a, b; Urayama et al. 2007; Rogez and Martinoty 2011).

In the present case, the preferred direction is f _i and is perpendicular to f _i , since \(f_{i} \tilde \Omega _{i}=0\). The dynamic equation can be written as Relative rotations are not truly hydrodynamic variables, since they are neither conserved nor related to any spontaneously broken continuous symmetry. Therefore, they relax in a finite time and the dissipative part of Z _i contains where the relative torque, \(\Sigma _{i} =\partial \epsilon / \partial \tilde \Omega _{i}\), is the conjugate to the relative rotations and is obtained as a partial derivative of the the total energy density 𝜖. The transverse Kronecker symbol, \(\delta _{ij}^{\perp } = \delta _{ij} - f_{i} f_{j}\), reflects the transverse nature of \(\tilde \Omega \), Σ _i and Z _i .

The full expression for Z _i will be given below. In contrast to the nematic case, \(\tilde {\Omega }_{i}\) has the transformation properties of a velocity and, therefore, shows in Z _i rather different types of couplings to the other variables.

We are now in a position to present the full macroscopic dynamic equations for active polar order in a viscoelastic background. In addition to the variables already introduced in “Key aspects” section, total energy density 𝜖, active concentration ϕ, total mass density ρ, total momentum density g, polar order F and preferred direction f _i , elastic strain tensor ε _{i j} , and relative rotations \(\tilde \Omega _{i}\), there is the entropy density σ. The latter is related to all other variables by the Gibbs relation with the thermodynamic conjugates as prefactors, temperature T, osmotic pressure π, chemical potential μ, the mean velocity v, the order conjugate m, the ’molecular field’ h _i , the elastic stress tensor Ξ _{i j} , and the relative torque Σ _i . They all follow from the total energy density as partial derivatives. Note that h _i and Σ _i have to be transverse (f _i h _i = 0 = f _i Σ _i ) and Ξ _{i j} = Ξ _{j i} symmetric.

In the absence of any orienting field, the orientation of f _i is energetically not fixed and only ∇ _j f _i enters the Gibbs relation. It is therefore useful to make this explicit in the Gibbs relation with \(h_{i} = h_{i}^{\prime } - F \nabla _{j} \Psi _{ij}\) and m = m′ − f _i ∇ _j Ψ _{i j} .

To guarantee rotational invariance of the energy, there must be the symmetry

To be meaningful, we need a definite expression for the energy density Restricting ourselves to the harmonic approximation, symmetry allows for the expressions where 𝜖 _kin and 𝜖 _act have been given in Eqs. 1 and 2, respectively. The energy 𝜖 is not specific for active dynamic polar order and is the same as for a passive nematic elastomer (replacing f _i by the director n _i everywhere).

The second rank tensors are of the uniaxial symmetric form (with \(\delta ^{\bot }_{ij}\equiv \delta _{ij} - f_{i} f_{j}\)), e.g., with two generalized susceptibilities, each. The third-order tensors are odd under time reversal and spatial inversion, e.g., and contain three phenomenological static parameters, each. The three fourth-order tensors have different symmetries and and containing six generalized Frank coefficients, five elastic moduli, and four mixed-type parameters, respectively. We do not write down here the explicit expressions for the conjugate quantities, since taking the appropriate derivatives is straightforward. We provide those expressions when necessary (e.g., in “Simple solutions” section). The order conjugate m is given in Eq. 3.

The variables introduced above follow local dynamic equations, either conservation laws or balance equations. Using the symmetry requirements, we get with the vorticity tensor \(\omega _{ij} = \tfrac 12 (\nabla _{i} v_{j} - \nabla _{j} v_{i})\). The dynamic equations for ϕ, ε _{i j} , and \(\tilde \Omega _{i}\) (with the phenomenological currents \(j_{i}^{\phi }\), W _{i j} , Σ _i ) are given in Eqs. 4, 7, and 11. These phenomenological currents, and the other ones, \(j_{i}^{\sigma }\), Ξ _{i j} , X, and Y _i , can all be split into a reversible part (superscript R) and a dissipative part (superscript D) according to their time reversal behavior.

As discussed in “Active transport and stress” section, we have written the transport terms using the mean velocity, while additions are provided by the reversible parts of the phenomenological currents given below. For the origin of the rather lengthy non-phenomenological expression in brackets in Eq. 29 cf. Pleiner et al. (1996, 2004). The dynamic equation for F is Ḟ _i + v _j ∇ _j F _i − F _j ω _{i j} + X _i = 0 with X _i = f _i X + F Y _i .

The source term in Eq. 28, the entropy production R/T, is zero for reversible processes and positive for dissipative ones. With the help of the Gibbs relation, we get for R as a bilinear function of dissipative currents and generalized forces Within linear irreversible thermodynamics, currents, and forces are linearly related, which allows us to use the dissipation function, R, written as a harmonic function of the forces alone, as a potential. In particular, using symmetry arguments again, we find and the dissipative currents are obtained as partial or functional derivatives, e.g., X ^D = ∂ R/∂ m, \(j_{i}^{\sigma ,D} = - \partial R / \partial \nabla _{i} T\) or \(W_{ij}^{D} = {\delta R/\delta \Xi _{ij}} = {\partial R/\partial \Xi _{ij}} -\nabla _{k}\partial R/\partial \nabla _{k}\Xi _{ij}\) etc.. We do not write down here the explicit expressions for the dissipative currents, and only provide them when necessary.

The second rank tensors have the form of Eq. 22, the third rank ones are as in Eq. 23, the fourth-order tensors ν _{i j k l} and τ _{i j k l} have the same form as the elasticity tensor Eq. 25 and the fourth rank tensors \(\xi _{ijkl}^{T}\) and \(\xi _{ijkl}^{\Pi }\) take the form given in Eq. 24. The structure of the fifth rank tensor \(\mu _{ijklm}^{\Xi }\) and the sixth rank tensor ξ _{i j k l p q} will be elucidated in Appendix B.

The dissipation function contains the relaxations of the order parameter (ξ′), of elastic strains (τ _{i j k l} ), and of relative rotations (τ ^D ), already discussed in “Key aspects” section, as well as heat conduction (κ _{i j} ), active concentration diffusion (D _{i j} ), viscosity (ν _{i j k l} ), and rotational viscosity of the polar direction (b ^D ). There are various interesting cross-couplings, thermo-diffusion (\(D_{ij}^{T}\)); thermo-strain diffusion (\(\xi _{ijkl}^{T}\)); solutal-strain diffusion \(\left (\xi _{ijkl}^{\Pi }\right )\); couplings of flow to temperature (\(\mu _{ijk}^{T}\)), to osmotic pressure \(\left (\mu _{ijk}^{\Pi }\right )\), and to strain diffusion \(\left (\mu _{ijklm}^{\Xi }\right )\); couplings of strain relaxation to temperature \(\left (\tau _{ij}^{T}\right )\), to concentration \(\left (\tau _{ij}^{\Pi }\right )\), and to order parameter \(\left (\tau _{ij}^{m}\right )\); and a coupling between relative rotations and polar orientations (ξ). The dissipative flow couplings \(\left (\mu _{ijk}^{T,\Pi }\right )\) and \(\left (\mu _{ijklm}^{\Xi }\right )\) are specific for a dynamic polar system and are absent in, e.g., passive nematic elastomers. The strain diffusion (ξ _{i j k l p q} ) is a sixth rank tensor with 16 independent coefficients whose detailed structure will be given in Appendix B.

The reversible parts of the phenomenological currents cannot be derived from a potential. Rather, one writes down all terms that are possible by symmetry and makes sure that appropriate cross-coupling terms cancel each other in the entropy production Neglecting some higher order gradient terms, we find with the second rank tensors of the form of Eq. 22, and the generalized flow alignment tensors are Relative rotations couple reversibly to all other variables (\(\beta _{\perp }^{\Omega },\gamma _{\perp }^{\Omega },\lambda _{ijk}^{\Omega },\beta _{1}^{\Omega },\beta ^{\Omega }_{kl}\)), except to reorientations of the preferred direction. The latter couple to the elastic degree of freedom (\(\beta ^{W}_{kl}\)) as does the order parameter (\(\beta _{1kl}^{W}\)). All other terms are already present in systems with dynamic polar order in a simple fluid background (Brand et al. 2013). Among them is the so-called active term (a _{i j} in the stress term) mentioned in “Active transport and stress” section and discussed in detail in (Brand et al. 2014).

The nonlinear term \(W_{ij}^{Rnl}\) has been used in ‘‘Viscoelasticity” section to show that it modifies the transport velocities of the elastic strain tensor. The counter terms are ∼β ₇ and ∼β ₈ in Eq. 45. Similarly, and β ₅ adds to the advective velocity of relative rotations in Eq. 11, while β ₃ and β ₄ contribute to the advective and convective velocity of f _i , Eq. 31 respectively.

The set of macroscopic equations derived above contains three variables that relax to a certain stationary value, even when the deviations are homogeneous in space: Elastic strains ε _{i j} relax to zero, since elasticity is transient only for a visco-elastic medium, the order parameter of the active polar order δ F relaxes to its non-zero stationary value F _s , and the relative rotations \(\tilde \Omega _{i}\) relax to zero, since they constitute a non-hydrodynamic degree of freedom.

Actually, these three types of variables are coupled and so are their relaxations. The free energy (21) shows the coupling between relative rotations and elastic strains (via D ₂), well-known already from nematic elastomers, while the active polar order couples dynamically to elasticity via the very last term (\(\sim \tau ^{m}_{ij}\)) in the dissipation function (33). In addition, elastic strains are related dynamically to flow, A _{i j} , Eq. 7, and statically to changes of entropy or temperature, density and concentration, Eq. 20. In the homogeneous case, these variables do not have their own dynamics and are simply slaved by the strains. Thus, for homogeneous relaxations, we are left with with \(\Xi _{ij} = c_{ijkl} \varepsilon _{kl} + D_{2} (f_{j} \delta _{ik}^{\perp } + f_{i} \delta _{jk}^{\perp }) \tilde \Omega _{k}\).

We notice that relative rotations only couple to shear strains (ε _{i j} off-diagonal with respect to f), while the active polar order couples to compressional strains (ε _{i j} diagonal with respect to f). The former case is also present for transient nematic elastomers (nematic polymers), although, to our knowledge, it has not been considered before. It is not related to active order and will be discussed only briefly at the end of this section. We will concentrate here on the coupling to polar active order. To reduce the algebraic complexity of the equations, and since it is a rather good approximation for polymers, we will assume incompressibility in the strains, ε _{i i} = 0. The remaining two equations are (with f along the z-direction) with the abbreviations ζ ≡ τ ₃(c ₃ − c ₄ )+ 2τ ₄(c ₄ − c ₁ − c ₂), c _m ≡ c ₁ + c ₂ + c ₃ − 2c ₄, and \( \tau _{m} \equiv \tau _{\parallel }^{m} = -2 \tau _{\perp }^{m}\). The latter relation is necessary to maintain incompressibility for all times, \(\dot \varepsilon _{ii}=0\). There are appropriate relations for combinations of τ’s and c’s, which we will not show here.

The general solution is a coupled sum of two exponential decays with the inverse relaxation times Although the coupling constant τ _m is bound from above, \({\tau _{m}^{2}} < \tau _{3} \xi ^{\prime }\), by the positivity requirement of the dissipation function, it is not guaranteed that the λ’s are both positive. This is achieved by the requirement of the additional relation τ ₃ = −2τ ₄.

The A- and B-amplitudes are coupled, e.g., α τ _m B _1,2 = −(λ _1,2 + ζ)A _1,2, and any relaxation of one variable triggers the relaxation of the other. Assuming initially F = F _s and a finite deformation ε _{z z} (t = 0) = A, the latter not only relaxes to zero with the amplitudes but also induces a transient change in the active polar order starting with the slope B(λ ₂ − λ ₁) and finally relaxing to zero with the longest relaxation time. Vice versa, an initial order parameter deviation from the stationary value, F(t = 0) − F _s = B leads to the appropriate result of an induced transient compression

The combined relaxation of relative rotations and shear strains, e.g. \(\tilde \Omega _{x}\) and ε _{x z} , is described by the sum of two exponentials (similar to Eq. 52) with the inverse relaxation times again indicating that relaxing relative rotations induce shear strains and vice versa.

In the preceding section, we concentrated on homogeneously relaxing strains. Here, we will consider the case that either elasticity is permanent (the relaxation times diverge) or that experiments are done on a time scale small compared to the relaxation times. We concentrate on sound-like excitations, i.e., propagating plane waves (∼ expi(ω t − k ⋅ r)) characterized by a frequency ω and a wave vector k. We only consider effects to order ω ∼ k and neglect dissipative effects.

In nematic and polar nematic systems, there is no additional propagating excitation in order ω ∼ k, since the nematic order is related to broken rotational symmetry and the flow alignment coupling (λ _{i j k} ) only leads to ω ∼ k ². With active polar order the situation is completely different (Brand et al. 2013), since the preferred direction is a second velocity. It has been shown that the reversible couplings of entropy, concentration and the stress tensor to the polar order, γ _∥, β _∥, a _∥ in Eqs. 35‐37, lead to a second, generally propagating, longitudinal sound. It is anisotropic, depending on the orientation of k relative to f, and couples to the ordinary sound such that the latter also becomes anisotropic. In addition, it reflects the broken time reversal symmetry due to the non-equilibrium nature of the active polar state (cf. eq. (76) of Brand et al. 2013).

In passive elastomers and crystals elasticity allows for additional propagating sound-like excitation. The direct relation between strain dynamics and flow, Eq. 7, and the elastic stress contribution to the stress tensor, Eq. 29, lead to transverse, shear-elastic sound, and an elastic contribution to the longitudinal sound velocity. Only the elastic moduli c _{i j k l} in Eq. 20 are involved as material parameters.

In the present system, we have all these three excitations together, and, of course, they are coupled. In addition, there are extra excitations beyond those of the individual subsystems. In particular, the direct reversible coupling between elasticity and active polar order, provided by the reactive transport parameter \({\beta _{1}^{W}}\) in Eqs. 39 and 41, results in an additional propagating mode with that also couples to the previously discussed modes. It involves compressional, ε _{z z} , and shear strains ε _{x z} + ε _{y z} . This mode can be weakly damped due to the relaxation of F.

There are other reversible cross-coupling terms, like β ₁ in Eqs. 38 and 39, and β ^W in Eqs. 38 and 41. They only contribute to the order ω∼k ², since in both cases reorientations of the active polar direction are involved and f _i is a variable associated with a spontaneously broken continuous symmetry, of which only gradients enter the free energy. In principle, also the reversible cross-couplings in Eq. 40, \(\lambda _{ijk}^{\Omega }\), \(\beta _{1}^{\Omega }\), and β ^Ω, of relative rotations with, respectively, flow, Eq. 37, polar order, Eq. 39, and elasticity, Eq. 41, lead to sound-like excitations; in particular to ω ² = (D ₁/ρ)(λ ^Ω)² k ², \(\omega ^{2} = \alpha D_{1} (\beta _{1}^{\Omega })^{2} k_{\perp }^{2}\) and \(\omega ^{2} = D_{1} (\beta ^{\Omega })^{2} ([c_{1} + 2c_{2}] k_{\perp }^{2} + 4 c_{5} k_{\parallel }^{2})\). The probably strong relaxation of the relative rotations could damp these modes rather quickly.