Oriented Active Solids

TL;DR

This paper analyzes the linearized dynamics of active solids with orientational order, revealing conditions for stability, instabilities, and long-range order in different dimensions, advancing understanding of active matter physics.

Contribution

It provides a comprehensive analysis of active solids with orientational order, including stability criteria and the effects of active forcing, considering rotation invariance.

Findings

Stable active solids with quasi-long-range order in 2D

Long-range order in 3D active solids

Instability of the solid for one sign of active forcing

Abstract

We present a complete analysis of the linearised dynamics of active solids with orientational order, taking into account a hitherto overlooked consequence of rotation invariance. Our predictions include the possibility of stable active solids with quasi-long-range order in two dimensions and long-range order in three dimensions, generic instability of the solid for one sign of active forcing, and the instability of the orientationally ordered phase in momentum-conserved systems for large active forcing irrespective of its sign.

Figures

Equations

$$F={\int }_{\mathbf{x}}\frac{\lambda }{2}{\left(\Phi -\frac{s}{\lambda }\text{Tr}[{\text{Q}}^2]\right)}^2+\mu \text{Tr}{\left[\text{C}-\frac{t}{\mu }\text{Q}\right]}^2+f_Q$$

$${\mathbf{f}}_a={\tilde{\zeta }}_1\nabla \cdot \text{Q}+{\tilde{\zeta }}_2\text{Q}\cdot (\nabla \cdot \text{Q}).$$

$$F=\frac{1}{2}{\int }_{\mathbf{x}}{B}_1{\eta }_{xx}^2+B_2{\eta }_{yy}^2+B_3{\eta }_{xx}{\eta }_{yy}+B_4[{\eta }_{xy}-\beta (\theta -\Omega ){]}^2$$

$${\dot{u}}_x={\nu }_1{\partial }_x^2{u}_x+{\nu }_2{\partial }_y^2{u}_x+{\nu }_3{\partial }_x{\partial }_y{u}_y$$

$${\dot{u}}_y={\nu }_4{\partial }_x^2{u}_y+{\nu }_5{\partial }_y^2{u}_y+{\nu }_6{\partial }_x{\partial }_y{u}_x$$

$$\begin{gathered} \hat{p}\cdot {\mathbf{f}}_p\stackrel{\text{lin}}{=}\Gamma [v_0+(\upsilon +{\upsilon }_1{W}_{xx}^0)\delta p+{\upsilon }_1{p}_0\delta W_{xx}) \\ =\Gamma v_0+a_1{\partial }_x{u}_x+a_2{\partial }_y{u}_y \end{gathered}$$

$${\dot{u}}_x=a_1{\partial }_x{u}_x+a_2{\partial }_y{u}_y+b_1{\partial }_x^2{u}_x+b_2{\partial }_y^2{u}_x+b_3{\partial }_x{\partial }_y{u}_y$$

$${\dot{u}}_y=b_4{\partial }_x^2{u}_y+b_5{\partial }_y^2{u}_y+b_6{\partial }_x{\partial }_y{u}_x$$

$$\omega_{\pm}=\begin{aligned} \begin{rcases*}\left(a_{1}+\frac{a_{2}b_{6}}{|b2-b5|}\right)q_{x}-ib_{2}q_{y}^{2}\\ -\frac{a_{2}b_{6}}{|b2-b5|}q_{x}-ib_{5}q_{y}^{2}\end{rcases*}q_{x}\ll q_{y}^{2}\\ \begin{rcases*}a_{1}q_{x}-ib_{1}q_{x}^{2}-i\left(b_{2}+\frac{a_{2}}{a_{1}}b_{6}\right)q_{y}^{2}\\ -ib_{4}q_{x}^{2}-i\left(b_{5}-\frac{a_{2}}{a_{1}}b_{6}\right)q_{y}^{2}\end{rcases*}q_{x}\gtrsim q_{y}^{2}\end{aligned}$$

$${\color[rgb]{0,0,0}\omega({\bf q}\to{\bf 0})=-i\frac{S_{0}\tilde{\zeta}_{1}}{2\eta}(\beta^{-1}\pm 1)}$$

$$\omega_{\pm}=\frac{i}{8\eta}\big{[}2S_{0}\tilde{\zeta}_{1}-B_{4}(1+4\beta^{2}\Gamma_{\theta}\eta)\pm\\ \sqrt{\{2S_{0}\tilde{\zeta}_{1}-B_{4}(1+4\beta^{2}\Gamma_{\theta}\eta\}^{2}-32B_{4}(\beta-\beta^{2})\Gamma_{\theta}S_{0}\tilde{\zeta}_{1}\eta}\big{]}$$

Full text

Biological systems and their artificial analogues are driven out of equilibrium by a direct and independent supply of free energy to the individual constituent units. The nontrivial coupling of the constituent particles to this power input SRJSTAT ; LPDJSTAT results in macroscopic stresses and currents in the system which have been shown to be responsible for diverse phenomena ranging from coherent intracellular flows Ano_nuc_rot ; Seb1 ; Gold , to defect turbulence in living liquid crystals Dogic1 , to flocking in animal groups giardina . Active hydrodynamics RMP , which extends the traditional continuum dynamics of conserved and broken-symmetry variables by introducing terms that break only the time-reversal symmetry of the corresponding passive systems, offers a successful general framework to describe the macroscopic behaviour of such systems.

Whereas active versions of orientationally ordered fluids have been studied extensively RMP ; Sriram_rev ; Ramaswamy_Toner_Tu ; Curie_rep ; Jacques_Nat , active solids have received much less attention. In particular, spontaneous alignment in the solid phase, relevant for the dynamics of polarised tissues on intermediate timescales, has been somewhat neglected. Epithelial cellular monolayers philip , and the cell shila1 itself, have been modelled as crosslinked active gels isol , and models that couple phase-field crystals with the Toner-Tu equations for a polar flock have been constructed Lowen . Even the studies that consider anisotropic active gels shila1 ; Lowen ; kopf-pismen , however, ignore a symmetry-mandated coupling between orientation and strain in the free energy of distortion whose consequence is that one shear modulus of a uniaxial gel vanishes: true solids with uniaxial orientational order cannot exist in thermal equilibrium. What about an active orientationally-ordered solid? To answer this question and more we present a complete hydrodynamic theory of active solids with uniaxial polar or apolar orientational order, in contact with a substrate or in a momentum conserving permeating fluid medium. The former description is applicable to cell monolayers or tissues in an extracellular matrix while the latter describes the dynamics of active gels such as actomyosin at timescales that are small compared to the unbinding time of crosslinkers. We start with a description in which the polar or apolar order parameter is coupled to an isotropic solid and, in the ordered phase, eliminate the non-hydrodynamic director modes to obtain the effective equations for polar or apolar active gels. Our treatment applies to the systems considered in Ewan in the limit of infinite polymer relaxation time, and to the hydrodynamic limit of a Vicsek model with particles connected by springs huepe .

Here are our main results. (i) Active polar or apolar elastomers, when dynamically stable, have displacement fields whose variance is finite in three dimensions and diverges only logarithmically with system size in two dimensions. Translational order in such systems is quasi-long-range in two dimensions and long-range in three dimensions. This is in contrast to passive orientationally ordered elastomers which, lacking one shear modulus, can support only quasi-long-range order in three and short-range order in two dimensions. (ii) An active force proportional to $\propto{\textsf{{Q}}}\cdot\nabla\cdot{\textsf{{Q}}}$, where Q is the apolar order parameter, always destabilises elastomers on a substrate when its magnitude is larger than the active force $\propto\nabla\cdot{\textsf{{Q}}}$, irrespective of the sign, in contrast to its stabilising role in incompressible active fluids on substrates. (iii) The dynamics of polar motile elastomers, at linear and nonlinear order, is qualitatively distinct from that of a solid driven in an externally imposed direction Rangan and escapes the latter’s transverse buckling instability. (iv) For bulk momentum-conserving systems, ignoring inertia, extensile (contractile) stresses destabilise positively (negatively) uniaxial elastomers. (v) Elastomeric gels are generically destabilised at high activities when a forcing rate given by the ratio of the coefficient of the active stress to the viscosity exceeds the passive orientational relaxation rate. (vi) Like active smectics Tapan ; chen , stable uniaxial active elastomers have finite concentration fluctuations and, in bulk momentum-conserving settings, a “second sound” mode whose speed is nonzero in all directions.

We now demonstrate how we obtain these results. We start our discussion with apolar permanently crosslinked elastomers on a substrate. Our description involves the displacement field $\bar{{\bf u}}({\bf x},t)$ about an isotropic reference state, the apolar order parameter field ${\textsf{{Q}}}({\bf x},t)$, and the velocity field ${\bf v}({\bf x},t)$. We define a linearised strain field ${\textsf{{W}}}={\textsf{{C}}}+\Phi{\textsf{{I}}}$, $W_{ij}=\partial_{j}\bar{u}_{i}+\partial_{i}\bar{u}_{j}$ about the isotropic state, where C and $\Phi$ are the trace-removed and isotropic parts of W and I is the unit tensor. Since the elastomer is permanently crosslinked there is no motion of structure without mass motion and the density field is slaved to dilations of the structure, $\delta\rho/\rho=-\nabla\cdot\bar{{\bf u}}$, implying that the dynamics of the density need not be explicitly considered. Further, since there is no permeation, the evolution equation for the displacement field is $\dot{\bar{{\bf u}}}={\bf v}$.

We assume a purely relaxational dynamics $\dot{{\textsf{{Q}}}}=-\Gamma_{Q}{\delta F}/{\delta{\textsf{{Q}}}}$, for the apolar order parameter ignoring all couplings to velocity FTNT0 . Here

[TABLE]

is the free-energy functional governing the dynamics in the absence of activity, with $f_{Q}=({r}/{2})\text{Tr}[{\textsf{{Q}}}^{2}]+({w}/{4})(\text{Tr}[{\textsf{{Q}}}^{2}])^{2}+(K/2)(\nabla{\textsf{{Q}}})^{2}$. The generic free-energy couplings $s$ and $t$ between the strain and the polarisation lead to the development of a strain anisotropy either parallel ($t>0$) or perpendicular to ($t<0$) Q. The absence of these couplings shila1 ; Lowen would have implied an invariance of the uniaxial solid under independent rotations of the orientation and the elastic network, at least in the passive limit. Their inclusion reduces this symmetry to one under joint rotations of the two.

The force balance equation is $\Gamma{\bf v}=\Gamma\dot{\bar{{\bf u}}}=-{\delta F}/{\delta\bar{{\bf u}}}+{\bf f}_{a}$, where the total active force is

[TABLE]

The first term in (2) is the divergence of the standard active stress while the second is an active force density that is allowed in all systems not constrained by momentum conservation and was first postulated in ano_apol .

Armed with these equations, we write ${{\textsf{{Q}}}}=S({\bf n}{\bf n}-\textsf{{I}}/2)$ where the unit vector ${\bf n}$ and the scalar $S$ are the director and the magnitude of orientational order. We expand about the equilibrium steady-state ${{\textsf{{Q}}}}={\textsf{{Q}}}^{0}$ with $S=S_{0}=\sqrt{2r/w}$, ${\bf n}={\bf n}_{0}=(1,0)$, $\Phi^{0}=({s}/{2\lambda})S_{0}^{2}$ and $\textsf{{C}}=\textsf{{C}}^{0}=({t}/{\mu})\textsf{{Q}}^{0}$, obtained by minimising (1), and assess how active stresses alter it. Expanding the fields in small fluctuations about this steady state, $S=S_{0}+\delta S$, ${\bf n}=(\cos\theta,\sin\theta)\approx(1,\theta)$, $C_{ij}=C^{0}_{ij}+\delta C_{ij}$, and $\Phi=\Phi^{0}+\delta\Phi$, we find that the fluctuations of the apolar order parameter are slaved to the strain field as $\delta S=({2t}/{\bar{w}})\delta C_{xx}+({sS_{0}}/{\bar{w}})\delta\Phi$ and $\theta=({\mu}/{S_{0}t})\delta C_{xy}$, where $\bar{w}=S_{0}s^{2}/\lambda+t^{2}/\mu+r$. Note that due to the minimal couplings between Q and C, director fluctuations are not independently soft and are instead slaved to the strain fluctuations Anderson . Ignoring the free-energy coupling $\mu$ in (1) would have led to the incorrect conclusion that the transverse fluctuations of Q have a relaxation rate that vanishes as wavelength $\to\infty$.

To analyse the displacement fluctuations about this orientationally ordered configuration, we transform to displacement ${\bf u}$ and strain variables $\bm{\eta}$ relative to the anisotropic state from $\bar{{\bf u}}$ and C and $\Phi$ which were defined relative to an isotropic reference space. Upon integrating out the $\delta S$ fluctuations, the free energy (1) in terms of $\theta$ and ${\bf u}$ transforms to

[TABLE]

where $\eta_{ij}=(1/2)(\partial_{i}u_{j}+\partial_{j}u_{i})$, the rotation angle $\Omega=(\partial_{x}u_{y}-\partial_{y}u_{x})/2$. $\beta\in[-1,1]$, whose sign is that of $t$, measures the degree of anisotropy of the solid. $\beta=0$ is the isotropic case, and we will refer to positive and negative anisotropy according to sgn$(\beta)$. The standard analysis leading up to (3) as well as the expression of $B_{i}$ and $\beta$ in terms of previously introduced coefficients are presented in full in the supplement supp . Note that the angle $\theta$ appears with the rotation field of the solid and the shear strain and integrating it out leads to a vanishing of the shear modulus Olmsted ; Ranjan ; Xing ; FTNT4 .

If the dynamics were controlled purely by the free-energy (3) the hydrodynamic equations for $\dot{\bf u}$ would not contain terms of the form $\partial_{y}^{2}u_{x}\hat{x}$ and $\partial_{x}^{2}u_{y}\hat{y}$ upon integrating out the fast $\theta$ field. However, the active force ${\bf f}_{a}=S_{0}(\tilde{\zeta}_{1}+{\tilde{\zeta}_{2}S_{0}}/{2})\partial_{y}\theta\hat{x}+S_{0}(\tilde{\zeta}_{1}-{\tilde{\zeta}_{2}S_{0}}/{2})\partial_{x}\theta\hat{y}$ yields terms of this form even when $\theta$ is eliminated in favour of $\Omega+\beta^{-1}\eta_{xy}$. Putting all of this together and rescaling time by the friction coefficient $\Gamma$, we obtain the hydrodynamic equations for the displacement fields:

[TABLE]

where $\nu_{2}$ and $\nu_{4}$ are purely active with $\nu_{2}=S_{0}(\beta^{-1}-1){(2\tilde{\zeta}_{1}+\tilde{\zeta}_{2}S_{0})}/{4}$ and $\nu_{4}=S_{0}(\beta^{-1}+1){(2\tilde{\zeta}_{1}-\tilde{\zeta}_{2}S_{0})}/{4}$. All other coefficients have both active and passive contributions though $\nu_{3}-\nu_{6}$ is also purely active (see the supplement supp for expressions for all the coefficients). Both $\nu_{2}$ and $\nu_{4}$ have to be positive for stability i.e. for $\beta>0$ stability requires $\tilde{\zeta}_{1}>|(\tilde{\zeta}_{2}S_{0}/2)|$ and for $\beta<0$, $\tilde{\zeta}_{1}<-|(\tilde{\zeta}_{2}S_{0}/2)|$. A negative $\tilde{\zeta}_{1}$ which denotes extensile stresses in this convention, destabilises a positively uniaxial solid with nematogens aligned along the deviatoric strain while a positive $\tilde{\zeta}_{1}$, denoting contractile active stresses, destabilises solid in which the nematogens align perpendicular to the local deviatoric strain anisotropy. This should be compared to the situation in active smectics Tapan and in cholesterics Tapan_chol or columnar systems.. Interestingly, when $|\tilde{\zeta}_{2}|>2\tilde{\zeta}_{1}/S_{0}$, the orientationally ordered state is destabilised i.e., in contrast to incompressible active fluids on substrates, where, depending on the sign, it may play a stabilising role, here $\tilde{\zeta}_{2}$ plays a destabilising role, irrespective of its sign. This establishes our result (ii).

The presence of $\nu_{2}$ and $\nu_{4}$ ensures that the variances of both $u_{x}$ and $u_{y}$ fluctuations scale as $1/q^{2}$ in all directions when the elastomer is stable. In particular, $\langle u_{x}(q,t)u_{x}(-q,t)\rangle|_{q_{x}=0}\propto 1/(\nu_{2}q_{y}^{2})$ and $\langle u_{y}(q,t)u_{y}(-q,t)\rangle|_{q_{y}=0}\propto 1/(\nu_{4}q_{x}^{2})$, which are the directions in which the fluctuations would have been largest ($\propto 1/q^{4}$) in the absence of activity. This implies that an active elastomer with apolar nematic order can support quasi-long-range translational order in two dimensions. This conclusion remains correct even taking into account all symmetry allowed nonlinearities all of which are of the form $\nabla(\nabla{\bf u})^{2}$ and are irrelevant even in two dimensions in contrast to passive nematic elastomers where the absence of $\nu_{2}$ and $\nu_{4}$ terms render them relevant in $d<3$ Xing ; Stenull .

We now turn to polar elastomers on substrates. The ordering, in this case, is characterised by a polarisation vector ${\bf p}({\bf x},t)$, instead of a rank-2 tensor. We again assume a purely relaxational dynamics for the polarisation field ignoring all advective and self-advective nonlinearities FTNT $\dot{{\bf p}}=-\Gamma_{p}{\delta F_{p}}/{\delta{\bf p}},$ where $F_{p}$ is of the form (1) with $\text{Tr}[{\textsf{{Q}}}^{2}]$ being replaced by ${p^{2}}$ and Q by D with components $D_{ij}=p_{i}p_{j}-(1/2)p^{2}\delta_{ij}$ FTNT2 . The force balance equation $\Gamma{\bf v}={\bf f}_{p}+\nabla\cdot\bm{\sigma}-\delta F/\delta\bar{{\bf u}}$ for polar systems contain active propulsive forces ${\bf f}_{p}=(\upsilon\Gamma){\bf p}+(\upsilon_{1}\Gamma){\bf p}\cdot{\textsf{{W}}}$, in addition to the divergence of the active stress which we now take to be $\sigma_{ij}^{a}=\zeta_{1}D_{ij}+\zeta_{2}p^{2}\delta_{ij}$ FTNT6 We assess the effects of the active motilities and stresses on a state with ${\bf p}=p_{0}\hat{x}$, where $p_{0}^{2}=|r|/w$, $C_{ij}^{0}=t/\mu D_{ij}^{0}$, $\Phi^{0}=(s/\lambda)p_{0}^{2}$ and a steady velocity ${\bf v}=v_{0}\hat{x}=(\upsilon p_{0}+\upsilon_{1}p_{0}W_{xx}^{0})\hat{x}$. Expanding the fields in small fluctuations, ${\bf p}\approx(p_{0}+\delta p)\hat{x}+p_{0}\theta\hat{y}$, $C_{ij}=C^{0}_{ij}+\delta C_{ij}$, and $\Phi=\Phi^{0}+\delta\Phi$, about the homogeneously oriented state we again find that both the magnitude $p_{0}\delta p={s\delta\Phi+2t\delta C_{xx}}/{\bar{r}}$, where $\bar{r}=w+(2s^{2}/\lambda)+2(t^{2}/\mu)$, and the angular fluctuations $\theta=[{\mu}/{tp_{0}^{2}}]\delta C_{xy}$ of the polarisation field are slaved to the elastic deformations. An analysis along the lines of the apolar case yields dynamical equations for the displacement field ${\bf u}$ about the anisotropic stretched state. While the calculation for passive terms and terms originating from the active stress parallels the apolar case (see the supplement supp for details), the components of the propulsive force require more care. Projecting the propulsive force along and transverse to the polarisation, we obtain

[TABLE]

where $\stackrel{{\scriptstyle\text{lin}}}{{=}}$ denotes equality to linear order in disturbances, $\hat{p}$ is the unit vector in the direction of polarisation, $a_{1}$ and $a_{2}$ are obtained by replacing $\delta p$ and $\delta W_{xx}$ by their values in terms of the displacement fields (see supp ) and $\hat{p}_{\perp}\cdot{\bf f}_{p}\stackrel{{\scriptstyle\text{lin}}}{{=}}\Gamma\upsilon_{1}p_{0}(\delta W_{xy}-2\theta C_{xx}^{0}),$ where $\hat{p}_{\perp}$ is a unit vector perpendicular to $\hat{p}$. Since $C_{xx}^{0}=(t/2\mu)p_{0}^{2}$ and $\theta=(\mu/tp_{0}^{2})\delta C_{xy}=(\mu/tp_{0}^{2})\delta W_{xy}$, $f_{a\perp}=0$. This implies that there is no propulsive force in the direction transverse to the polarisation in a spontaneously symmetry broken polar elastomer. Physically, since the only anisotropy in the system is due to the broken rotation symmetry, the anisotropy of W aligns with ${\bf pp}$ on a fast timescale and at longer times, ${\textsf{{W}}}\cdot{\bf p}$ is purely parallel to ${\bf p}$. Next, projecting the velocity along and transverse to the direction of polarisation, we obtain $\hat{p}\cdot{\bf v}\stackrel{{\scriptstyle\text{lin}}}{{=}}\dot{u}_{x}$ and $\hat{p}_{\perp}\cdot{\bf v}\stackrel{{\scriptstyle\text{lin}}}{{=}}\dot{u}_{y}-v_{0}\theta$ where the final term appears due to the mean motion of the solid along $\hat{x}$ supp ; ano_mem ; RamTonPro . Therefore, upon transforming to a frame moving with the mean velocity of the solid $v_{0}\hat{x}$,

[TABLE]

where the coefficients $b_{i}$ are explicitly calculated in the supplement supp . This yields the eigenfrequencies

[TABLE]

As in the nematic case, $b_{2}$ and $b_{4}$ are purely active $\propto\zeta_{1}$ and both of them are positive, implying that the polar solid is stable, only when $\zeta_{1}\beta<0$. Motility introduces the possibility of a further instability when the magnitude of $\zeta_{1}$ is small: if the motility coefficients $a_{2}$ and $a_{1}$ have opposite signs then for small enough value of $b_{2}$, the system is destabilised. It is interesting to contrast (7) with the mode structure of an externally driven solid Rangan . There, terms proportional to $\partial_{y}u_{x}$ and $\partial_{x}u_{y}$ appear in the $\dot{u}_{y}$ equation even in the co-moving frame since the propulsion direction is externally determined (by gravity) and, as a result, the eigenfrequencies are either wavelike at small wavevectors or have an instability with a growth rate $\propto q$. However, the direction of motion of crawling solids is chosen spontaneously forbidding this transverse buckling instability and implying that active solids may be more stable than both their passive counterparts and externally driven ones. This establishes our result (iii).

As in apolar elastomers, the static structure factor of both $u_{x}$ and $u_{y}$ fluctuations scale as $1/q^{2}$ in all directions implying QLRO in two dimensions. However, this conclusion of the linear theory may be invalidated by marginally relevant nonlinearities. There are nine possible terms bilinear in $\nabla{\bf u}$ all of which are marginal in two dimensions. Unlike in an externally driven solid Rangan , where all of them may have independent values however, the coefficients of these nonlinearities are significantly constrained by rotational invariance in this case. Despite this, a treatment including these nonlinearities is beyond the scope of this paper. We however note that the presence of two relevant nonlinearities $(\partial_{x}u)^{2}$ and $(\partial_{y}u)^{2}$, where $u$ is the displacement field of a smectic, in polar smectics does not destroy the ordered phase chen when the two nonlinearities have opposite signs and believe that QLRO is possible even in polar solids in two dimensions for at least some range of parameter values.

While we have focussed on two dimensional elastomers, the three-dimensional case may be considered by changing $q_{y}$ and $u_{y}$ to $q_{\perp}$ and $u_{\perp}$ where $\perp$ denotes the two directions ($y,z$) transverse to the ordering direction $x$ and introducing an additional (passive) elastic constant for $\eta_{yz}$ shears in the transverse plane in that case. While passive elastomers only have QLRO in three dimensions – $\langle u_{x}^{2}\rangle$ diverges logarithmically as in smectics while $\langle|{\bf u}_{\perp}|^{2}\rangle$ is finite as in columnar systems – a calculation like that for two dimensions shows that the static structure factor of displacement fluctuations diverges as $1/q^{2}$ in all directions in three-dimensional active elastomers implying long-ranged order. The absence of any relevant nonlinearity in three dimensions implies that this linear result is exact at large distances for both polar and apolar elastomers. Further supp , active biaxial elastomers, whose passive analogues are softer than their uniaxial counterparts, with vanishing energy cost for all shear deformations and $u_{x}$, $u_{y}$ and $u_{z}$ all having smectic-like correlations, also have long-range order in three dimensions.

We now turn to elastomeric gels i.e. (dilute) elastomers frictionally coupled to an incompressible permeating fluid, with conserved total momentum of elastomer and fluid. The velocity of the solid is $\dot{{\bf u}}={\bf v}+{\bf v}_{r}$ where ${\bf v}$ is the joint velocity of the network and the fluid and ${\bf v}_{r}$ is the velocity of the network relative to the fluid. The joint velocity ${\bf v}$ obeys the force balance equation $\eta\nabla^{2}{\bf v}=\nabla\Pi-\tilde{\zeta}_{1}\nabla\cdot{\textsf{{Q}}}+\delta F/\delta{\bf u}$ where $\Pi$ is the pressure that enforces incompressibility, $\eta$ is the viscosity and $F$ is given by (3) FTNT5 . Writing the force balance equation in terms of a stream function $\psi$, such that $v_{x}=\partial_{y}\psi$ and $v_{y}=-\partial_{x}\psi$, to take the incompressibility constraint into account, and defining $u_{t}={(q_{y}u_{x}-q_{x}u_{y})}/{q}$ and $u_{l}={{\bf q}\cdot{\bf u}}/{q}$, we see that $\dot{u}_{t}=i|q|\psi+\mathcal{O}(q^{2})$ and $\dot{u}_{l}\sim\mathcal{O}(q^{2})$. The relaxation rate of $u_{t}$ fluctuations, which are the ones affected by the fluid, scale as $q^{0}$ to leading order in wavevector. In a passive elastomer the $\mathcal{O}(q^{0})$ part of the relaxation rate of $u_{t}$ has to vanish for perturbations either purely along the ordering direction i.e. with wavevectors ${\bf q}=q\hat{x}$ or purely transverse to it with wavevectors ${\bf q}=q\hat{y}$. However, in the active elastomer, the eigenfrequency in these directions is non-zero :

[TABLE]

where the $+$ ($-$) sign is realised for wavevectors along $\hat{x}$ ($\hat{y}$) (see the supplement supp for the expression for all wavevector directions). This implies that when $\beta\tilde{\zeta}_{1}<0$, the gel is unstable i.e. extensile (contractile) stresses destabilise positively (negatively) uniaxial elastomers, establishing our result (iv). Note that while the relaxation rate of $u_{t}$ fluctuations is $\mathcal{O}(q^{0})$ to leading order, its static structure factor scales as $1/q^{2}$ in all directions, when the gel is not destabilised, as can be checked by adding momentum-conserving spatiotemporally white noise to the force balance equation.

We have, till now, assumed a fast relaxation of the angle field to its steady-state value determined by the displacement field. This may however be questionable in a gel – an orientationally ordered fluid is generically unstable with a wavevector-independent growth-rate Aditi1 ; RMP . Therefore, when the growth rate of this generic instability is greater than the relaxation of the angle field to the value dictated by the local strain, the gel should be generically unstable irrespective of the sign of the active stress. We now calculate this upper bound using the coupled equations for the displacement fields and the angle field $\dot{\theta}=({1-\xi\cos 2\phi})q^{2}\psi/2-\Gamma_{\theta}{\delta F}/{\delta\theta},$ with $\xi$ being the flow-alignment parameter and $\Gamma_{\theta}$ being the angular relaxation rate. We demonstrated (8) that a contractile gel with $\tilde{\zeta}_{1}>0$ is stable for $\beta>0$ in a description retaining only the displacement fields. Let us therefore take $\beta>0$, $\tilde{\zeta}_{1}>0$ (i.e., the case that we would predict to be stable based on a calculation purely in terms of displacement fields). Also, for simplicity, let us take $\xi=0$ in which case, an orientationally ordered active fluid (not solid) is maximally unstable along $q_{y}$ i.e. pure splay. We calculate the $\mathcal{O}(q^{0})$ part of the eigenfrequency of the coupled $u_{t}$ and $\theta$ dynamics in this direction ($u_{l}$ dynamics does not contribute at this order):

[TABLE]

Therefore, when $S_{0}\tilde{\zeta}_{1}/\eta$ exceeds the relaxation rate of angular fluctuations in a passive nematic elastomeric gel, $B_{4}/2\eta+2B_{4}\beta^{2}\Gamma_{\theta}$, the orientational order is destabilised even in a positively uniaxial contractile solid since the angular fluctuations grow fast due to activity and can not relax to their preferred orientation dictated by the local strain. This yields an upper limit for the value of active stress above which a description purely in terms of the displacement fields is no longer possible and the orientationally ordered phase is destroyed irrespective of the sign of the active stress establishing our result (v).

We now leave the steady Stokesian regime, restore inertia $\rho_{0}\partial_{t}{\bf v}$ to the force balance equation and project the equations of motion of a momentum conserved stable active elastomer along and transverse to the wavevector direction, ignoring viscosity as is appropriate in the long wavelength limit. Along $q_{x}=0$ and $q_{y}=0$, the equations of motion decouple and the speed of the “second” sound (i.e. the $u_{t}$ mode) vanishes in passive elastomers as a consequence of vanishing restoring force for shear deformations. In contrast, in active systems the second sound speed along $q_{x}=0$ is $\sqrt{S_{0}\tilde{\zeta}_{1}(\beta^{-1}-1)/2\rho_{0}}$ while along $q_{y}=0$, it is $\sqrt{S_{0}\tilde{\zeta}_{1}(\beta^{-1}+1)/2\rho_{0}}$.

Finally, since in elastomeric systems the density fluctuations are slaved to displacement fluctuations $\delta\rho/\rho=\nabla\cdot\bar{{\bf u}}$, and since $\langle uu\rangle\sim 1/q^{2}$ in all directions, the static structure factor of density fluctuations scales as $q^{0}$, i.e. there are no giant density fluctuations.

This concludes our discussion of active solids. Our theory applies to monolayers or tissues in an extracellular matrix and the dynamics of active gels such as actomyosin at timescales that are small compared to the unbinding time of crosslinkers. Our results in this paper can be checked in simulations of Vicsek models in which the “spins” are coupled by harmonic springs huepe or in artificial active solids, for instance, composed of active chemotactic colloids suro .