Time reversal symmetry breaking and odd viscosity in active fluids: Green-Kubo and NEMD results
Cory Hargus, Katherine Klymko, Jeffrey M. Epstein, Kranthi K. Mandadapu
TTime reversal symmetry breaking and odd viscosity in active fluids:Green-Kubo and NEMD results
Cory Hargus, ∗ Katherine Klymko, Jeffrey M. Epstein, and Kranthi K. Mandadapu
1, 4, † Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA, USA Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA Department of Physics, University of California, Berkeley, CA, USA Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Active fluids, which are driven at the microscale by non-conservative forces, are known to ex-hibit novel transport phenomena due to the breaking of time reversal symmetry. Recently, Epsteinand Mandadapu [1] obtained Green-Kubo relations for the full set of viscous coefficients govern-ing isotropic chiral active fluids, including the so-called odd viscosity, invoking Onsager’s regressionhypothesis for the decay of fluctuations in active non-equilibrium steady states. In this Communica-tion, we test these Green-Kubo relations using molecular dynamics simulations of a canonical modelsystem consisting of actively torqued dumbbells. We find the resulting odd and shear viscosityvalues from the Green-Kubo relations to be in good agreement with values measured independentlythrough non-equilibrium molecular dynamics (NEMD) flow simulations. This provides a rigoroustest of the Green-Kubo relations, and validates the application of the Onsager regression hypothesisin relation to viscous behaviors of active matter systems.
Introduction.
Statistical physics has traditionally beenconcerned with systems at equilibrium. A natural gen-eralization pursued by Onsager, Prigogine, de Groot andMazur, and others is to consider systems that are globallyout of equilibrium but that obey the local equilibrium hy-pothesis [2–6]. Such systems model transport phenomenaallowing linear laws, such as those of Fourier and Fick, tobe derived from the principles of equilibrium thermody-namics and statistical mechanics [4, 5, 7]. The physicalorigin of the non-equilibrium nature of these systems isdriving at boundaries, as in a rod heated from one endor a channel connecting regions of different solute con-centration.A more radical departure from equilibrium is achievedin active matter systems, in which equilibrium is brokenat the local level by non-conservative microscopic forces.Such activity is known to modify existing phase behavioras well as give rise to qualitatively new dynamical phases,as in motility-induced phase separation [9, 10]. Similarly,activity not only modifies existing transport coefficients,but can lead to entirely new coefficients, such as the odd(or Hall) viscosity appearing in chiral active fluids [1, 11–14].Recent work by Epstein and Mandadapu [1] revealsthat odd viscosity arises in two-dimensional chiral activefluids due to the breaking of time reversal symmetry atthe level of stress correlations. This is demonstrated by aset of Green-Kubo relations derived through the applica-
Authors’ note:
Independent and concurrently released work byHan et al. [8] measures transport coefficients including the oddviscosity in a model system consisting of frictional granular par-ticles, upon obtaining identical Green-Kubo equations presentedin [1] through a projection operator formalism. Together with thepresent work, this confirms the robustness and applicability of theGreen-Kubo relations. tion of the Onsager regression hypothesis [4, 5, 7]. In thisCommunication, we evaluate these Green-Kubo relationsusing molecular dynamics simulations of a model systemcomposed of microscopically torqued dumbbells, find-ing them to be in good agreement with non-equilibriummolecular dynamics (NEMD) flow simulations across awide range of densities and activities (Fig. 4).
Theory.
We begin by reviewing the continuum theoryin [1] for two-dimensional viscous active fluids with inter-nal spin. This provides the setting for the derivation ofGreen-Kubo relations for viscosity coefficients in fluidsbreaking time reversal symmmetry. Because the chiralactive dumbbell model considered in this paper is capa-ble of storing angular momentum in the form of molec-ular or internal spin, we anticipate coupling between avelocity field v i and a spin field m . These satisfy balanceequations for linear and angular momentum, as proposedby Dahler and Scriven [15]: ρ ˙ v i = T ij,j + ρg i , (1) ρ ˙ m = C i,i − (cid:15) ij T ij + ρG . (2) T ij denotes the stress tensor and C i the spin flux,which accounts for transfer of internal angular momen-tum across surfaces. The variables g i and G denote bodyforces and body torques, respectively. Finally note thatthe balance of angular momentum includes a term inwhich the two-dimensional Levi-Civita tensor (cid:15) ij is con-tracted with the stress, so that the antisymmetric compo-nent of the stress may be nontrivial. We use the notation a ,i = ∂a/∂x i .The most general isotropic constitutive equations forviscous fluids relating T ij and C i to v i , m and theirderivatives up to first order in two-dimensional systems a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b are given by T ij = η ijkl v k,l + γ ij m − pδ ij + p ∗ (cid:15) ij , (3) C i = α ij m ,j , (4)where η ijkl , γ ij and α ij are the viscous transport coeffi-cients [1]. Here, p and p ∗ are hydrostatic contributionsand are not constitutively related to v i and m . Isotropyfurther allows the transport coefficients to be constrainedto have the form η ijkl = (cid:88) n =1 λ n s ( n ) ijkl , (5) γ ij = γ δ ij + γ (cid:15) ij , (6) α ij = α δ ij + α (cid:15) ij . (7)This follows from a general representation theorem stat-ing that any isotropic tensor can be expressed in a basisconsisting of contractions of Kronecker tensors δ ij andLevi-Civita tensors (cid:15) ij and that, consequently, there ex-ist no isotropic tensors of odd rank in two dimensions;see Ref. [1] for a detailed discussion and Appendix TableI for the definitions of tensors s ( n ) ijkl .The coefficients γ i and α i indicate the responses of thestress and spin flux tensors to spin and spin gradients. λ and λ are the typical bulk and shear viscosities. λ is the rotational viscosity indicating resistance to vor-ticity and giving rise to an anti-symmetric stress, while λ is the so-called odd viscosity quantifying response toshear with a tension or compression in the orthogonaldirection. λ and λ correspond to an anti-symmetricpressure from compression and isotropic pressure fromvorticity, respectively. Note that non-vanishing λ or λ violates objectivity (independence of stress from vortic-ity), while non-vanishing λ or λ violates symmetry ofthe stress tensor.Using the conservation and constitutive equations (1)-(2) and (5)-(7), Ref. [1] obtains a set of Green-Kubo re-lations for γ n and λ n via invocation of the Onsager re-gression hypothesis: γ = 12 ρ ν δ ij (cid:15) kl T ijkl , (8) γ = 12 ρ ν (cid:15) ij (cid:15) kl T ijkl , (9) λ + 2 λ + λ − γ π µ + γ τ µ = 12 ρ µ δ ik δ jl T ijkl , (10) λ + λ + λ − γ τ µ − γ π µ = 14 ρ µ (cid:15) ik δ jl T ijkl , (11) λ − γ π µ = 18 ρ µ (cid:15) ij δ kl T ijkl , (12) λ + γ τ µ = 14 ρ µ (cid:15) ij (cid:15) kl T ijkl . (13) T ijkl is the integrated stress correlation function given by T ijkl = (cid:90) ∞ dt (cid:104) δT ij ( t ) δT kl (0) (cid:105) . (14) µ , ν , τ , and π are static correlation functions in the non-equilibrium steady state given by µδ ij = 1 L (cid:90) (cid:10) δv i ( x ) δv j ( y ) (cid:11) d x d y , (15) π = 1 L (cid:90) ( y i − x i ) (cid:10) δv i ( x ) δm ( y ) (cid:11) d x d y , (16) τ = 1 L (cid:90) (cid:15) kr ( y r − x r ) (cid:10) δm ( x ) δv k ( y ) (cid:11) d x d y , (17) ν = 1 L (cid:90) (cid:104) δm ( x ) δm ( y ) (cid:105) d x d y , (18)respectively. In particular, µ and ν can also be regardedas the effective translation and spin temperatures in thesteady state. For equilibrium systems, equipartition im-plies µ = ν and π = τ = 0. Lastly, the above Green-Kuborelations show that two of the transport coefficients λ and γ are related by 2 λ = γ ( ν − τ ) /µ . Note that thestress tensor in (14) is defined as a spatial average, as inthe following section, unlike the velocity and spin fieldsin (15)-(18).For the chiral active dumbbell fluid, the situation isfurther simplified. As we will show in the followingsections, the absence of alignment interactions betweendumbbells results in γ = γ = 0, effectively decouplingthe velocity from the spin field and also setting λ = 0.Moreover, symmetry and objectivity of the stress tensorsets two more of the viscosity coefficients to zero, leaving η ijkl = λ (cid:0) δ ij δ kl (cid:1) + λ (cid:0) δ ik δ jl − (cid:15) ik (cid:15) jl (cid:1) + λ (cid:0) (cid:15) ik δ jl + (cid:15) jl δ ik (cid:1) . (19)These simplifications also allow us to write simplifiedGreen-Kubo expressions for the shear and odd viscosi-ties λ = 14 ρ µ (cid:90) ∞ dt (cid:104) ( δT ( t ) − δT ( t ))( δT (0) − δT (0)) (cid:105) , (20)and λ = 14 ρ µ (cid:90) ∞ dt (cid:104) (cid:104) δT ( t ) δT (0) (cid:105) − (cid:104) δT (0) δT ( t ) (cid:105) + (cid:104) δT ( t ) δT (0) (cid:105) − (cid:104) δT (0) δT ( t ) (cid:105) (cid:21) , (21)(see Appendix II for separating the coefficient λ from(10)). Equation (21) shows that non-vanishing odd vis-cosity, i.e., λ (cid:54) = 0 requires breaking time reversal sym-metry at the level of stress correlation functions, thusbreaking the Onsager reciprocal relations [1, 5]. Notethat (20) is not the typical Green-Kubo expression usedto calculate the shear viscosity. However, it can also berewritten for isotropic systems in the typical form, whichare invariant under rotation as λ = 1 ρ µ (cid:90) ∞ dt (cid:104) δT (cid:48) ( t ) δT (cid:48) (0) (cid:105) , (22)using a transformation T (cid:48) = R T T R corresponding to arotation tensor R of angle π/
4, for which T (cid:48) = ( T − T ). The form in (20) is a result of the theory for thechoice of the representation theorem for viscous transportcoefficients using the basis s ( n ) ijkl .In what follows, we evaluate the shear and odd vis-cosity Green-Kubo expressions at various densities anddriving forces by using molecular simulations of chiralactive dumbbells in a non-equilibrium steady state. Wethen subject the dumbbell system to imposed periodic-Poiseulle simulations [16] resulting in non-uniform shear-ing flows, and evaluate the shear and viscous coefficientsindependently. Such an analysis will provide support toboth the application of Onsager’s regression hypothesisto fluctuations in active non-equilibrium steady statesand the ensuing Green-Kubo relations for viscous behav-iors of active systems. f FIG. 1. A two-dimensional fluid composed of chiral activedumbbells. In addition to interacting with its neighbors, eachdumbbell is rotated counterclockwise by equal and oppositeactive forces f αi . Microscopic model–Chiral active dumbbells.
Weconsider a fluid composed of dumbbells subject to ac-tive torques [17], as shown in Fig. 1. Each dumbbell iscomposed of two particles of unit mass connected by aharmonic spring. The system evolves according to un-derdamped Langevin dynamics˙ x αi = v αi , ˙ v αi = (cid:88) jβ F αβij + f αi + g αi − γ v αi + (cid:112) γk B T d W αi dt , (23) with indices i, j ∈ [1 , N ] and α, β ∈ { , } running overdumbbells and particles, respectively. Variables x αi and v αi represent atom positions and velocities. γ is the dis-sipative substrate friction and T the substrate temper-ature determining the variance of the thermal fluctua-tions d W αi dt , modeled as Gaussian white noise. Particlesin different dumbbells interact through a pairwise WCApotential [18], resulting in interaction forces F αβij . Theparticles in a dumbbell are subjected to equal and op-posite non-conservative active forces f αi , which satisfy f i = − f i := f i , and are always perpendicular to thebond vector d i = x i − x i . This imposes an active torqueat the level of individual dumbbells. Finally, g αi = g ( x αi )is an optional externally imposed body force, and will beemployed later in Poisueille flow simulations to test theGreen-Kubo relations.Previous work [17] used the Irving-Kirkwood proce-dure to coarse-grain the microscopic equations (23) andderive the equations of hydrodynamics, including balanceof mass, linear momentum and angular momentum, asalso employed in the context of measuring odd viscosityby [14]. This coarse-graining procedure yields expres-sions for the stress tensor in terms of molecular variablesand active forces. In particular, it is found that apply-ing active couple forces at the microscale results in anasymmetric stress tensor given by T = T K + T V + T A , (24)where T K = − A (cid:88) i,α m αi v αi ⊗ v αi , (25) T V = − A (cid:88) i,j,α,β F αβij ⊗ x αβij , (26) T A = − A (cid:88) i f i ⊗ d i , (27)denote the kinetic, virial, and active contributions, re-spectively.The active force vector f i is related to the unit bondvector ˆ d i by a rotation matrix R of angle π/ i.e., f i = f R ˆ d i (28)For positive (negative) f , the dumbbells rotate counter-clockwise (clockwise). We find that the steady state timeaverage of T A is (cid:104) T A (cid:105) = − ρ (cid:104) f ⊗ d (cid:105) = − ρ f d (cid:104) R ˆ d ⊗ ˆ d (cid:105) = ρ f d (cid:20) (cid:21) , (29)where d = (cid:104)| d |(cid:105) is the average bond length. Becausethe dumbbells rotate with no preferred alignment, theantisymmetry of (cid:104) T A (cid:105) follows from replacing the timeaverage with a uniformly weighted average over angles ofrotation θ . For example, (cid:104) R ˆ d ⊗ ˆ d (cid:105) = (cid:104) ˆ d ˆ d (cid:105) = 12 π (cid:90) π dθ cos ( θ ) = 12 (30)while the diagonal elements are zero. This shows thatthe antisymmetric hydrostatic-like term p ∗ introduced in(3) arises in a non-equilibrium steady state of the activedumbbell model due to the presence of active rotationalforces, and has the magnitude p ∗ = ρ f d/
2. We furtherrelate p ∗ to a non-dimensional P´eclet number describingthe ratio of active rotational forces to thermal fluctua-tions due to the substrate bathPe = 2f dρ µ = 4 p ∗ µρ . (31)We use Pe as defined in (31) to vary the activity in thesystem when evaluating the transport coeffcients. Green-Kubo calculations.
Steady-state molecular dy-namics simulations allow direct measurement of the in-tegrated stress correlation functions T ijkl defined in (14),which are required for evaluation of the viscous transportcoefficients using the Green-Kubo equations (8)-(13). Wefind that several of these coefficients vanish in the non-equilibrium steady states at all simulated activities anddensities due to cancellations of the correlation functions(see Appendix Fig. 5). In particular, (cid:15) ij (cid:15) kl T ijkl = δ ij (cid:15) kl T ijkl = (cid:15) ij δ kl T ijkl = 0 . (32)This immediately implies γ = γ = λ = λ = λ = 0,so that the stress tensor is symmetric and objective. Itnow remains to evaluate the two non-trivial transportcoefficients λ and λ using (20) and (21).Figure 2 shows the stress correlators (cid:104) δT ( t ) δT (0) (cid:105) and (cid:104) δT (0) δT ( t ) (cid:105) for various Pe. These are typicallyzero for systems in equilibrium, and become non-zerowhenever Pe (cid:54) = 0. In particular, we find these correla-tion functions to be equal and opposite, and thereforeadd constructively yielding a non-vanishing odd viscos-ity from the Green-Kubo relation (21). In general, forthe chiral active dumbbell fluid, we find (cid:104) δT ( t ) δT (0) (cid:105) = −(cid:104) δT (0) δT ( t ) (cid:105) = −(cid:104) δT (- t ) δT (0) (cid:105) , (33)where the final equality is due to stationarity. The anal-ogous equations are satisfied by (cid:104) δT ( t ) δT (0) (cid:105) .Figure 4 shows the Green-Kubo estimates for λ and λ for various activities and for a range of low to highdensities. We find that the shear viscosity increases withdensity as well as with activity. The dependence of theodd viscosity on activity, while apparently linear at lowdensity, becomes increasingly sigmoidal at high density.Note that the odd viscosity, as a non-dissipative trans-port coefficient, may be negative without introducing aninconsistency with the second law of thermodynamics. FIG. 2. Stress correlation functions contributing to the oddviscosity ( ρ = 0 . (cid:54) = 0 these correlation functionsadd constructively, yielding a nonzero odd viscosity. Poiseuille flow NEMD simulations.
To verify thevalues computed from the Green-Kubo formulas, (20)and (21), we measure λ and λ independently via non-equilibrium molecular dynamics simulations. To thisend, we simulate plane Poiseuille-like flow via the inclu-sion of a nonzero body force g in (23) according to theperiodic Poiseuille method [16]. We apply equal and op-posite body forces as shown in Fig. 3, of magnitude g in the x direction compatible with periodic boundaryconditions.This setup represents a non-trivial boundary valueproblem, which not only yields non-uniform flows andnon-uniform stresses, but also provides a stringent testfor the expected constitutive behaviors of the activedumbbell fluid and the estimates of the transport coef-ficients obtained from Green-Kubo formulas. The veloc-ity profile and pressure profiles for such an applied body FIG. 3. A schematic of the periodic Poiseuille non-equilibriummolecular dynamics (NEMD) simulation method. The tophalf of the system is subjected to a uniform body force tothe left, and the bottom half to a uniform body force of equalmagnitude to the right. This yields a parabolic velocity profileand, for odd viscous fluids, an atypical normal stress.
FIG. 4. Comparison of shear viscosity ( λ ) and odd viscosity ( λ ) values obtained from the Green-Kubo relations with thoseobtained from periodic Poiseuille NEMD simulations, showing agreement across a range of activities and densities. force can be solved analytically and are given by v ( x ) = ρ g λ x ( L − x ) , (34)and p ( x ) = λ λ ρ g x + p , (35)respectively, where p is an arbitrary reference pressure(see Appendix IV for the solution to the correspond-ing boundary value problem). Our simulations of ac-tive dumbbell fluids are consistent with these profiles forvarious densities and activities. Given the velocity andpressure profiles in (34) and (35), the shear and odd vis-cosities can be computed from the expressions λ = ρ g L v , (36) λ = T , v , = − λ T , ρ g , (37)respectively, where ¯ v = 1 L (cid:82) L dx v ( x ); see AppendixIV. The slope of the stress component T can be identi-fied in molecular simulations using the Irving-Kirkwoodexpression (24)-(27).The shear and odd viscosities calculated using thisNEMD approach are in excellent agreement with theGreen-Kubo predictions for a wide range of densities andP´eclet numbers, see Fig. 4. Discussion.
In this work, we have validated the non-equilibrium Green-Kubo formulas derived in [1], showingthat the odd viscosity results directly from the breakingof time reversal symmetry at the level of stress fluctua-tions in the steady state. In doing so, we also providesupport for the application of the Onsager regression hy-pothesis to fluctuations about the non-equilibrium steady state, which was used to derive these equations. Futurework entails understanding the microscopic origins of thefunctional dependence of the viscosities with density andactivity.
Acknowledgements.
C.H. is supported by the NationalScience Foundation Graduate Research Fellowship Pro-gram under Grant No. DGE 1752814. K.K.M is sup-ported by Director, Office of Science, Office of Basic En-ergy Sciences, of the U.S. Department of Energy undercontract No. DEAC02-05CH11231. ∗ [email protected] † [email protected][1] J. M. Epstein and K. K. Mandadapu, arXiv:1907.10041(2019).[2] S. R. de Groot, Thermodynamics of Irreversible Processes (Interscience Publishers Inc., New York, 1951).[3] S. R. de Groot and P. Mazur,
Non-Equilibrium Thermo-dynamics (Dover, New York, 1984).[4] L. Onsager, Physical review , 405 (1931).[5] L. Onsager, Physical review , 2265 (1931).[6] I. Prigogine, Introduction to Thermodynamics of Irre-versible Processes (Wiley-Interscience, New York, 1967).[7] R. Kubo, M. Yokota, and S. Nakajima, Journal of phys-ical society of Japan , 1203 (1957).[8] M. Han, M. Fruchart, C. Scheibner, S. Vaikuntanathan,W. Irvine, J. de Pablo, and V. Vitelli, arXiv preprintarXiv:2002.07679 (2020).[9] J. Tailleur and M. E. Cates, Physical review letters ,218103 (2008).[10] M. E. Cates and J. Tailleur, Annual reviews of condensematter physics , 219 (2015).[11] D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli,Nature communications , 1573 (2017).[12] S. Ganeshan and A. G. Abanov, Physical review fluids ,094101 (2017).[13] A. Souslov, K. Dasbiswas, M. Fruchart, S. Vaikun- tanathan, and V. Vitelli, Physical review letters ,128001 (2019).[14] Z. Liao, M. Han, M. Fruchart, V. Vitelli, and S. Vaikun-tanathan, The Journal of chemical physics , 194108(2019).[15] J. Dahler and L. Scriven, Nature , 36 (1961).[16] J. A. Backer, C. P. Lowe, H. C. Hoefsloot, and P. D. Iedema, The journal of chemical physics (2005).[17] K. Klymko, D. Mandal, and K. K. Mandadapu, TheJournal of chemical physics , 194109 (2017).[18] J. D. Weeks, D. Chandler, and H. C. Andersen, Thejournal of chemical physics , 5237 (1971).[19] S. J. Plimpton, J. Comp. Phys. , 1 (1995), see alsohttp://lammps.sandia.gov/. Appendix
I. Simulation Details
To investigate the viscous behavior of a fluid composed of self-spinning dumbbells, we perform molecular dynamicssimulations in LAMMPS [19], implementing our own modifications to impose microscopic driving forces and computethe active stress T A . Particles interact with their non-bonded neighbors through a Weeks-Chandler-Andersen [18]potential defined by V WCA ij ( r ) = (cid:15) (cid:20)(cid:0) σ/r (cid:1) − (cid:0) σ/r (cid:1) (cid:21) + (cid:15) r < / σ r ≥ / σ . Here, σ , (cid:15) and particle mass m are the characteristic length, energy, and mass scales, which are used to define theLennard-Jones units system. All numerical settings and results in this Communication are reported in Lennard-Jonesunits. The two particles in a single dumbbell are held together by a harmonic potential V ( r ) = k ( r − r ) withspring constant k = 100 and reference length r = 1.Dynamics are evolved according to underdamped Langevin dynamics (23) with bath temperature T = 1 . γ = 0 .
5. We apply the Langevin bath interactions only along the x direction, so as not to impede flowin the x direction, and employ these conditions in both Green-Kubo and periodic Poiseuille simulations. We notethat imposing bath interactions selectively along x may lead to a violation of isotropy by aligning dumbbells alonga preferred axis. In all simulations, however, we check that dumbbells have no preferred alignment by measuring thedeparture of the bond angle of a dumbbell projected onto [0 , π/
2] from the reference value of π/ δθ + i = arctan (cid:18) | d i · e || d i · e | (cid:19) − π . (38)We find that in all simulations, max( |(cid:104) δθ + i (cid:105)| ) < .
01 radians, where angle brackets indicate averaging in time andthe maximum is over all dumbbells. We also confirm that the density is indeed uniform in all periodic Poiseuillecalculations. The relative spatial variation in the density is bounded in all simulations by (cid:0) (cid:104) ( δρ ) (cid:105) / (cid:104) ρ (cid:105) (cid:1) / < . II. Green-Kubo Formula for Shear Viscosity
In Table I, we provide the basis for the 2D viscosity tensor η ijkl derived in [1]. We also perform a derivation toobtain separate expressions for the shear and bulk viscosities. To this end, we begin with the following equation (alsoequation (127) in SI of [1] in the absence of internal spin): k j k l η ijkl = 1 ρ µ k j k l (cid:90) ∞ dt (cid:104) δT ij k ( t ) δT kl − k (0) (cid:105) = 1 ρ µ k j k l T k ijkl , (39)where T k ijkl = (cid:90) ∞ dt (cid:104) δT ij k ( t ) δT kl − k (0) (cid:105) . (40)Following [1], we can obtain an equation for λ and λ λ + 2 λ = 12 ρ µ δ ik δ jk T k ijkl , (41)in the limit of k → . Our simulation and analysis code is publicly available athttps://github.com/mandadapu-group/active-matter
Basis Tensor Components i ↔ j k ↔ l ij ↔ kl P s (1) δ ij δ kl + + + + s (2) δ ik δ jl − (cid:15) ik (cid:15) jl + + + + s (3) (cid:15) ij (cid:15) kl - - + + s (4) (cid:15) ik δ jl + (cid:15) jl δ ik + + - - s (5) (cid:15) ik δ jl − (cid:15) jl δ ik + (cid:15) ij δ kl + (cid:15) kl δ ij - + N/A - s (6) (cid:15) ik δ jl − (cid:15) jl δ ik − (cid:15) ij δ kl − (cid:15) kl δ ij + - N/A - TABLE I. Basis for the isotropic rank four tensors in two dimensions corresponding to viscous transport coefficients η ijkl . Wealso indicate the nature of the tensors under index permutations (A) i ↔ j indicating the symmetry of the stress tensor, (B) k ↔ l indicating objectivity, (C) i ↔ k , j ↔ l , indicating symmetry with respect to Onsager reciprocal relations, and finally(D) the mirror transformation x (cid:55)→ − x , x (cid:55)→ x , also known as parity transformation (P). Reproduced from [1]. To separate λ from λ we return to (39) and contract both sides with k i k k to obtain k i k j k k k l η ijkl = 1 ρ µ k i k j k k k l T k ijkl . (42)The resulting equation holds independently for any choice of k in the limit k →
0. Now, we set k = k ( e + e ) and k = k ( e − e ) in (42) and sum the resulting equations to obtain4 λ + 4 λ = 1 ρ µ (cid:0) T k + T k + T k + T k + T k + T k + T k + T k (cid:1) , (43)which cannot be written in compact form as a contraction of Kronecker and Levi-Civita tensors with T k ijkl . Subtracting(43) from twice (41) and invoking the symmetry of the stress fluctuations gives λ = 14 ρ µ ( T k − T k − T k + T k + T k − T k − T k + T k )= 14 ρ µ ( T k − T k − T k + T k ) . (44) FIG. 5. The sixteen stress correlation functions computed at ρ = 0 .
4, Pe = 12. Due to symmetries present in the chiral activedumbbell model, many of the correlation functions are identical, and are grouped as such. From this grouping, it is possible toascertain that certain viscosity coefficients defined in (8)-(13) will vanish. For example, λ depends on a sum of the correlationfunctions T − T − T + T . Here we see that these four correlation functions are identical, hence their sum will bezero. We further observe that the correlation functions contributing to the odd viscosity λ go to zero in the static limit t → Finally, returning to the definition of T k ijkl in (39), and taking the zero wavevector limit k → λ = 14 ρ µ (cid:90) ∞ dt (cid:104) ( δT ( t ) − δT ( t ))( δT (0) − δT (0)) (cid:105) = 1 ρ µ (cid:90) ∞ dt (cid:104) δT ( t ) δT (0) (cid:105) , (45)where, in obtaining the last equality, we use material isotropy to make the stress transformation T (cid:48) = R T T R corresponding to a two-dimensional rotation tensor R of angle π/
4, for which T (cid:48) = ( T − T ). The last equalityin (45) is the standard Green-Kubo relation for the shear viscosity. One may evaluate either of these expressions tocompute the shear viscosity λ . III. Decomposed contributions to the viscosity coefficients from the Irving-Kirkwood stress tensor
FIG. 6. Components of the stress contributing to Green-Kubo and Poiseuille calculations of the shear and odd viscosity at ρ = 0 . λ and λ , respectively, fromGreen-Kubo calculations according to the decompositions in (46) and (47). Here, λ A ∗ = λ AK + λ AV + λ AA and similarly λ ∗ A = λ KA + λ VA + λ AA . Figures (c) and (d) are the component-wise contributions to the λ and λ , respectively, in periodicPoiseuille calculations. The solid black line indicates the total viscosity coefficient, obtained by adding the shaded areas above y = 0 and subtracting those below y = 0. The Irving-Kirkwood procedure provides a natural decomposition of the stress tensor into kinetic, virial, and activemolecular contributions (24). In Fig. 6, we examine the component-wise stress contributions to the shear and oddviscosity in both Green-Kubo and periodic Poiseuille calculations. The stress appears twice in the correlation functionsentering the Green-Kubo equations via (14), thus there are nine components contributing to the Green-Kubo viscositycoefficients, which we label λ KK , λ KV , λ KA , λ VK , λ VV , λ VA , λ AK , λ AV and λ AA .From (20), we define a decomposed shear viscosity as λ XY = 1 ρ µ (cid:90) ∞ dt (cid:104) δT X ( t ) δT Y (0) (cid:105) , (46)where X, Y ∈ { K , V , A } indicate the kinetic, virial and active parts. Similarly, the odd viscosity from (21) may bedecomposed as λ XY = 14 ρ µ (cid:90) ∞ dt (cid:104) δT Xij ( t ) δT Ykl (0) (cid:105) (cid:15) ik δ jl . (47)0For periodic Poiseuille calculations, the decompositions contributing to the viscous coefficients simply involve thechoice of whether to use T K , T V , or T A in (36) and (37), corresponding to λ K , λ V , and λ A , respectively. We observethat the active stress T A plays a small but not insignificant role in both λ and λ at Pe (cid:54) = 0. Notably, the dominantGreen-Kubo contributions to λ are λ KK and λ VV while the cross correlations λ KV and λ VK are dominant in λ . IV. Periodic Poiseuille Simulation
Non-equilibrium molecular dynamics simulations allow measurement of viscosity coefficients in direct analogy toexperimental viscometry. For the chiral active dumbbell fluid, γ = γ = λ = λ = λ = 0, resulting in decouplingof the linear and angular momentum balances and leading to modified Navier-Stokes equations ρ ˙ v i = λ v k,ki + λ v i,jj + λ (cid:15) ik v k,jj − p ,i + (cid:15) ij p ∗ ,j + ρg i , (48)with bulk viscosity λ , shear viscosity λ , odd viscosity λ , pressure p , and body force g i .In the periodic Poiseuille simulations, we subject the system to equal and opposite body forces along the x directionas shown in Fig. 3. In this case, we verify that the density ρ is well-approximated as constant for small shear rates,as described in Appendix I. Therefore, we assume incompressible flow, v i,i = 0 . (49)and obtain the simplified constitutive and Navier-Stokes equations: T ij = λ (cid:0) v i,j + v j,i (cid:1) + λ (cid:0) (cid:15) ik v k,j + (cid:15) jk v i,k (cid:1) − pδ ij + p ∗ (cid:15) ij , (50)and ρ v i,j v j = λ v i,jj + λ (cid:15) ik v k,jj − p ,i + (cid:15) ij p ∗ ,j + ρ g i . (51)where ρ is the uniform reference density.We now seek a steady state analytical solution for the velocity and pressure profiles of a fluid between two platesseparated by a distance L , subjected to a body force g = ( g , g is uniform in space. The solution isanalogous to that of a planar Poiseuille flow, with boundary conditions v i = 0 at x = 0 and x = L . Using the ansatz v = v ( x ), v = 0, p = p ( x ), and p ∗ = const, one may find the steady state solution to be v ( x ) = ρ g λ x ( L − x ) , (52)and p ( x ) = λ λ ρ g x + p , (53)where p is an arbitrary reference pressure.We see that the steady state velocity profile is identical to the usual solution for planar Poiseuille flow, remainingunaffected by odd viscosity. In fact it is always true that odd viscosity does not appear in the velocity profile inincompressible flows with no-slip boundary conditions [12]. The odd viscosity does appear, however, in a pressuregradient arising in the x -direction to maintain the no-penetration condition at the walls, i.e. to prevent flow in the x -direction. Our active dumbbell fluid simulations show parabolic velocity profiles consistent with (52) and (53)when subjected to equal and opposite body forces as shown in Fig. 3.Integrating the velocity profile to get an average velocity ¯ v = 1 L (cid:82) L v ( x ) dx , we obtain a convenient expressionfor computing the shear viscosity λ in molecular simulations: λ = ρ g L v . (54)As noted above, λ does not appear in the velocity but in the stress (50). For the velocity profile (52), T = − p + λ v , , (55)1which results in T , = − p , + λ v , . (56)Using (51) in the x -direction, one may reduce (56) to T , = 2 λ v , = − λ ρ g λ . (57)Finally, rearranging (57), λ is obtained in terms of the slope of T as λ = T , v , = − λ T , ρ g . (58)where T11