Inertial Effects on the Stress Generation of Active Fluids
Suspensions of self-propelled bodies generate a unique mechanical stress owing to their motility that impacts their large-scale collective behavior. For microswimmers suspended in a fluid with negligible particle inertia, we have shown that the virial ‘swim stress’ is a useful quantity to understand the rheology and nonequilibrium behaviors of active soft matter systems. For larger self-propelled organisms like fish, it is unclear how particle inertia impacts their stress generation and collective movement. Here, we analyze the effects of finite particle inertia on the mechanical pressure (or stress) generated by a suspension of self-propelled bodies. We find that swimmers of all scales generate a unique ‘swim stress’ and ‘Reynolds stress’ that impacts their collective motion. We discover that particle inertia plays a similar role as confinement in overdamped active Brownian systems, where the reduced run length of the swimmers decreases the swim stress and affects the phase behavior. Although the swim and Reynolds stresses vary individually with the magnitude of particle inertia, the sum of the two contributions is independent of particle inertia. This points to an important concept when computing stresses in computer simulations of nonequilibrium systems—the Reynolds and the virial stresses must both be calculated to obtain the overall stress generated by a system.
Active matter systems constitute an intriguing class of materials whose constituents have the ability to self-propel, generate internal stress, and drive the system far from equilibrium. Because classical concepts of thermodynamics do not apply to nonequilibrium active matter, recent work has focused on invoking the mechanical pressure (or stress) as a framework to understand the complex dynamic behaviors of active systems (Takatori14, ; Takatori16a, ; Ginot15, ; Yang14, ; Mallory14, ; Solon15, ). In particular, active swimmers exert a unique “swim pressure” as a result of their self propulsion (Takatori14, ; Yang14, ). A physical interpretation of the swim pressure is the pressure (or stress) exerted by the swimmers as they interact with the surrounding boundaries that confine them, similar to molecular or colloidal solutes that collide into the container walls to exert an osmotic pressure. The swim pressure is a purely entropic, destabilizing quantity (Takatori15a, ) that can explain the self-assembly and phase separation of a suspension of self-propelled colloids into dilute and dense phases that resemble an equilibrium gas-liquid coexistence (Theurkauff12, ; Palacci13, ; Redner13, ; Bialke13, ; Stenhammar13, ; Cates10, ).
Existing studies of the pressure of active systems (Takatori14, ; Takatori14b, ; Takatori15a, ; Yang14, ; Mallory14, ; Ginot15, ; Solon15, ) have focused on overdamped systems where swimmer inertia is neglected (i.e, the particle Reynolds number is small). However, the swim pressure has no explicit dependence on the body size and may exist at all scales including larger swimmers (e.g., fish, birds) where particle inertia is not negligible (Takatori14, ). The importance of particle inertia is characterized by the Stokes number, , where is the particle mass, is the hydrodynamic drag factor, and is the reorientation time of the active swimmer. Here we analyze the role of nonzero on the mechanical stress exerted by a system of self-propelled bodies and provide a natural extension of existing pressure concepts to swimmers with finite inertia. We maintain negligible fluid inertia so that the fluid motion satisfies the steady Stokes equation.
We consider a suspension of self-propelled spheres of radii that translate with an intrinsic swim velocity and tumble with a time in a continuous Newtonian fluid. The random tumbling results in a diffusive process for time with in 3D. An isolated active particle generates a swim pressure , where is the number density of active particles. We do not include the effects of hydrodynamic interactions, and there is no macroscopic polar order of the swimmers or any large-scale collective motion (e.g., bioconvection).
From previous work (Takatori16b, ; Takatori16a, ; Yan15b, ; Ezhilan15, ; Tailleur09, ), we know that geometric confinement of overdamped active particles plays a significant role in their dynamics and behavior. Confinement from potential traps, physical boundaries, and collective clustering can reduce the average run length and swim pressure of the particles. We have shown experimentally (Takatori16b, ) that active Brownian particles trapped inside a harmonic well modifies the swim pressure to in 2D, where is a ratio of the run length, , to the size of the trap, . For weak confinement, , we obtain the ‘ideal-gas’ swim pressure of an isolated swimmer. For strong confinement, , the swim pressure decreases as . Confinement reduces the average distance the swimmers travel between reorientation events, which results in a decreased swim pressure.
In this work, we find that particle inertia plays a similar role as confinement by reducing the correlation between the position and self-propulsive swim force of the swimmers. In addition to the swim pressure, active swimmers exert the usual kinetic or Reynolds pressure contribution associated with their average translational kinetic energy. For swimmers with finite particle inertia we find that the sum of the swim and Reynolds pressures is the relevant quantity measured by confinement experiments and computer simulations. We also study systems at finite swimmer concentrations and extend our existing mechanical pressure theory to active matter of any size or mass.
An important implication of this work pertains to the computation of mechanical stresses of colloidal suspensions at the appropriate level of analysis. Consider the Brownian osmotic pressure of molecular fluids and Brownian colloidal systems, , where is the number density of particles and is the thermal energy. At the Langevin-level analysis where mass (or inertia) is explicitly included, the Reynolds stress is the source of the Brownian osmotic pressure, , where is the density and is the velocity fluctuation. The virial stress or the moment of the Brownian force, , is identically equal to zero—the position and Brownian force are uncorrelated at the Langevin level. However, at the overdamped Fokker-Planck or Smoluchowski level where inertia is not explicitly included, the position and Brownian force are correlated, and the virial stress is the source of the Brownian osmotic pressureBrady93 (), whereas the Reynolds stress is zero. The important point is that the sum of the Reynolds and virial stresses gives the correct Brownian osmotic pressure at both levels of analysis, as it must be since the osmotic pressure of colloidal Brownian suspensions is whether or not inertia is explicitly included in the analysis.
We report in this work that an identical concept applies for active swimmers. The virial stress arising from the correlation between the particle position and its ‘internal’ force, , is a term that is separate and in addition to the Reynolds stress associated with their average translational kinetic energy. Interestingly, the mechanical stress generated by active swimmers has a nonzero contribution from both the Reynolds and virial stresses because the internal force associated with self-propulsion has an autocorrelation that is not instantaneous in time and instead decays over a finite timescale modulated by the reorientation time of the active swimmer, : . A distinguishing feature of active swimmers compared with passive Brownian particles is that their direction of self-propulsion can relax over large timescales, and that the ‘internal’ swim force autocorrelation cannot in general be described by a delta-function in time.
Ii Swim stress
All self-propelled bodies exert a swim pressure, a unique pressure associated with the confinement of the active body inside a bounded domain. The swim pressure is the trace of the swim stress, which is defined as the symmetric first moment of the self-propulsive force, , where is the number density of swimmers, is the position, is the swimmer’s self-propulsive swim force, and denotes the symmetric part of the tensor. As we noted previouslyTakatori17a (), the swim stress is properly defined as the symmetric force moment since the force arises from the fluid which can only generate symmetric stresses. For the active Brownian particle model, the swim force can be written as where is a unit vector specifying the swimmer’s direction of self-propulsion. For a dilute suspension of active particles with negligible particle inertia the “ideal-gas” swim stress is given by , where we define as the swimmer’s “energy scale” (force () distance ()). The swim pressure (or stress) is entropic in origin and is the principle destabilizing term that facilitates a phase transition in active systems (Takatori15a, ).
In the absence of any external forces, the motion of an active Brownian particle is governed by the Langevin equations:
where and are the particle mass and moment-of-inertia, and are the translational and angular velocities, is the hydrodynamic drag factor coupling angular velocity to torque, and are the Brownian translational force and rotational torque, respectively, and are unit random normal deviates, is the reorientation timescale set by rotational Brownian motion, and is the Stokes-Einstein-Sutherland translational diffusivity. The translational diffusivity and the reorientation dynamics are modeled with the usual white noise statistics, and . The swimmer orientation is related to the angular velocity by the kinematic relation . The translational and angular velocities may be combined into a single vector, , and similarly for the force and torque, , to obtain a general solution to the system of ordinary differential equations (Hinch75, ). Although a general solution is available for any particle mass and moment-of-inertia, inclusion of nonzero moment-of-inertia leads to calculations that are analytically involved. For convenience and to make analytical progress, here we summarize the case of zero moment-of-inertia () and focus on finite mass () to elucidate the effects of inertia on the dynamics of active matter.
where we have taken times and , is the swimmer momentum relation time, energy scale , and is the Stokes number. For we recover the “ideal-gas” swim pressure for an overdamped system: (Takatori14, ). This is precisely the mechanical force per unit area that a dilute system of confined active micro-swimmers exert on its surrounding container (Takatori14, ; Yang14, ; Mallory14, ).
Notice in the other limit as , vanishes. Physically, the magnitude of the swim stress decreases because inertia may translate the swimmer in a trajectory that is different from the direction of the swim force, reducing the correlation between the moment arm and the orientation-dependent swim force . Our earlier work (Takatori16b, ) showed that active particles confined by an acoustic trap exert a swim pressure that is reduced by a factor of , where is the degree of confinement of the swimmer run length relative to the size of the trap . Equation 3 has a reduction in the swim pressure by a similar factor, , suggesting that particle inertia may play a similar role as confinement by reducing the correlation between the position and self-propulsive direction of the swimmers.
Particle inertia may be interpreted as imposing a confinement effect on the swimmers because their effective run length between reorientation events decreases. The average run length of the swimmers with inertia reduces to , where is the distance over which inertia translates the swimmer along a trajectory that is independent of the direction of its swim force, and the time over which this occurs scales with the inertial relaxation time, . Substituting these terms into the virial expression for the swim pressure, we obtain . Using the definition of the Stokes number and for small , we can rewrite the swim pressure as . Aside from the factor of in the denominator (which arises from spatial dimensionality), this scaling argument agrees with Eq 3 and shows that particle inertia plays a confining role in the swim pressure, analogous to the physical confinement of swimmers in a potential well(Takatori16b, ).
Iii Reynolds stress
For systems with finite particle inertia, an additional stress contribution arises owing to particle acceleration: the Reynolds stress. This term is seen in Bernoulli’s equation and is associated with the average translational kinetic energy of a particle, . In atomic or molecular systems this is often referred to as the kinetic stress. This contribution was not included in previous studies since overdamped active systems have no particle mass, (i.e., ). As shown in the Appendix, we can use the solution to Eqs 1 and 2 to obtain the Reynolds stress for arbitrary , given by
which is a sum of the Brownian osmotic stress and a self-propulsive contribution that depends on . For an overdamped system where , the self-propulsive contribution to the Reynolds stress vanishes, justifying the neglect of this term in previous studies of overdamped systems. Mallory et al (Mallory14, ) analytically calculated the expression of the Reynolds stress, Eq 4, but not the swim stress, Eq 3; the mechanical pressure that is measured from the walls of an enclosing container is the sum of the Reynolds and swim pressures, however.
Notice that the Brownian osmotic stress, , arises solely from the Reynolds stress and not from taking the virial moment of the Brownian force, . As stated earlier, this is precisely because of the delta-function statistics imposed for the Brownian force in the Langevin-level analysis where mass is explicitly included: . In contrast, the active stress has a nonzero contribution from both the Reynolds and swim stresses because the swim force autocorrelation is a decaying exponential modulated by the reorientation timescale : . If one were to model the Brownian force autocorrelation as one that relaxes over a finite solvent relaxation timescale, , then we would obtain nonzero contributions from both the Reynolds and virial stresses, with their sum equal to for all .
Remarkably, the Stokes number disappears. The magnitude of the swim pressure that decreases with increasing cancels exactly the increase in the magnitude of the Reynolds stress. This verifies that swimmers of all scales exert the pressure, , regardless of their mass and inertia. We conducted simulations where the dynamics of active Brownian particles were evolved following Eqs 1 and 2 using the velocity verlet algorithm (Allen89, ), and the results are shown in Fig 1. Results from the simulations agree with our theoretical predictions in Eqs 3 - 5. In an experiment or simulation, the average mechanical pressure exerted on a confining boundary gives the sum of the swim and Reynolds pressures, and not their separate values.
We have shown previously that random motion gives rise to a micromechanical stress via the relationship , where is the effective translational diffusivityTakatori14 (). For overdamped active systems, the self-propulsive contribution to the diffusivity is described solely through the swim stress, . With finite particle inertia, the particle diffusivity is unaltered and the stress-diffusivity relationship still applies but the stress is now a sum of two independent contributions, as shown in Eq 5.
In the presence of a nonzero moment-of-inertia, there is another dimensionless Stokes number, , which is a ratio of the inertial reorientation timescale, , and the swimmers’ intrinsic reorientation timescale, . Similar to the translational Stokes number that does not appear in Eq 5, the moment-of-inertia is not expected to appear explicitly in the total stress generated by an active swimmer; at long times the random walk diffusive motion is unaffected by the particle inertia, whether that be translational or rotational.
Iv Finite concentrations
The results thus far are for a dilute suspension of active swimmers. At finite concentrations, experiments and computer simulations have observed unique phase behavior and self-assembly in active matter (Theurkauff12, ; Palacci13, ; Redner13, ; Bialke13, ; Stenhammar13, ; Cates10, ). Recently a new mechanical pressure theory was developed to provide a phase diagram and a natural extension of the chemical potential and other thermodynamic quantities to nonequilibrium active matter (Takatori15a, ).
At finite concentrations of swimmers, dimensional analysis shows that the nondimensional swim and Reynolds stresses depend in general on , where is the volume fraction of active swimmers and is the reorientation Péclet number—the ratio of the swimmer size to its run length . The ratio quantifies the magnitude of the swimmers’ activity () relative to the thermal energy ; this ratio can be a large quantity for typical micro-swimmers.
From previous work on overdamped active systems with negligible particle inertia (Takatori14, ), we know that is a key parameter controlling the phase behavior of active systems. For large the swimmers reorient rapidly and take small swim steps, behaving as Brownian walkers; the swimmers thus do not clump together to form clusters and the system remains homogeneous. For small the swimmers obstruct each others’ paths when they collide for a time until they reorient. This decreases the run length of the swimmers between reorientation events and causes the system to self-assemble into dense and dilute phases resembling an equilibrium liquid-gas coexistence.
As reported previously for (Takatori14, ; Takatori15a, ), for small the swim pressure decreases with increasing swimmer concentration. To verify how finite particle inertia affects the swim pressure at larger concentrations, we conducted simulations by evolving the motion of active particles following Eqs 1 and 2, with an additional hard-sphere interparticle force that prevents particle overlaps using a potential-free algorithm (Heyes93, ). Care was taken to ensure that the simulation time step was small enough to preclude unwanted numerical errors associated with resolution of particle collisions. We varied the simulation time step from and found a negligible difference in our results. As shown in Fig 2A, for finite the data from our simulations are well described by the expression , which is simply a product of a volume fraction dependence and a Stokes number dependence of Eq 3. The volume fraction dependence was used previously to model the phase behavior of active matter (Takatori15a, ).
The clustering of swimmers reduces their translational velocity autocorrelation, , and hence decreases the Reynolds pressure. As shown in Fig 2B, our simulations show that the Reynolds pressure decreases with concentration, increases with , and is well described by the expression .
The sum of the swim and Reynolds stresses is given by
which again has no dependence on (nor for considered here), even at finite . Equation 6 is corroborated by our simulations as shown in Fig 3. This result implies that the existing mechanical pressure theory (Takatori15a, ) developed for overdamped systems can be used directly for swimmers with finite Stokes numbers, as long as we include the Reynolds stress contribution into the active pressure. Inclusion of the Reynolds stress is critical, as confinement experiments and computer simulations measure the total active pressure, including both the swim and Reynolds contributions.
In addition to the swim and Reynolds stresses, interparticle interactions between the swimmers at finite concentrations give rise to an interparticle stress, . For repulsive interactions, the interparticle (or collisional) pressure, , increases monotonically with concentration and helps to stabilize the system. As shown in Fig 4, we find that the expression agrees with the simulation data for a fixed value of , where is the pair distribution function at particle contact, and is a parameter obtained from the interparticle pressure of hard-sphere molecular fluids (Takatori15a, ). We can add to Eq 6 to construct phase diagrams of a system of inertial swimmers, which are qualitatively similar to those presented in (Takatori15a, ) for small values of . Adding particle inertia shifts the stablization to larger .
As shown by Batchelor (Batchelor70, ), there may be an additional contribution to the particle stress arising from local fluctuations in acceleration, , and is given by , where is the volume of the suspension (fluid plus particles), is the volume of an individual particle, is the uniform density of the particle, is a position vector (or the moment arm) from the particle center, and the summation is over the number of particles in the volume . For a dilute system of rigid particles, this term arises only from solid body rotation of the particle and takes the form , where is the average angular velocity of the rigid particle of size . Here the active swimmers have no average angular velocity, so there is no stress arising from local fluctuations in acceleration for dilute active systems of rigid particles.
In addition to using a potential-free algorithm to model hard-sphere particles, we have also tested a short-ranged, repulsive Weeks-Chandler-Andersen potential with an upper cut-off at particle separation distances of . Using this softer potential, the swim pressure does not exhibit a concentration dependence of because the effective radius of the particles decreases as the system becomes denser, meaning that colliding particles exhibit increasingly large overlaps. Increasing particle inertia (i.e., larger ) also changed the effective particle size. As previously stated (Stenhammar14, ), one must use care when soft potentials are used to model hard-sphere particle collisions because the effective particle size may depend on system parameters (like and ).
Here we presented a mechanical pressure theory for active Brownian particles with finite inertia. We neglected hydrodynamic interactions between the swimmers, which may contribute additional terms (like the “hydrodynamic stresslet” (Saintillan08a, )) to the active pressure. The ratio of the magnitudes of the hydrodynamic stress to the swim stress is . The hydrodynamic stress contribution becomes negligible when phase-separation occurs at low .
We assumed that the surrounding fluid obeys the steady Stokes equations, which may not be true for larger swimmers that propel themselves using fluid inertia. However, the concepts of the swim and Reynolds stress apply for swimmers with a nonlinear hydrodynamic drag factor, , where is the magnitude of the swimmer velocity. For example, a self-propelled body may experience a fluid drag that is quadratic in the velocity , where is the fluid density and is the characteristic size of the body. The nondimensional Langevin equations would become , where is the orientation vector of the swimmer and is the relevant quantity that must be varied.
The authors thank Luis Nieves-Rosado for his contributions to this work as part of the Undergraduate Research Program at the California Institute of Technology. S.C.T. acknowledges support by the Gates Millennium Scholars fellowship and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469. This work was also supported by NSF Grant CBET 1437570.
Integrating Eq 1 twice in time, we obtain the position of the swimmer,
where and are the arbitrary initial position and velocity, respectively, is the momentum relaxation time, is the intrinsic swimmer velocity, is the unit orientation vector of the swimmer, and is a unit random deviate. Integrating Eq 2 and using the kinematic relation, , we obtain
where is the initial angular velocity, is the angular momentum relaxation time, and is a unit random deviate. Equation 8 is of the form , where is a coefficient matrix. The general solution of Eq 8 is , where is an arbitrary initial orientation of the swimmer.
We are interested in the orientation autocorrelation
where is the coefficient matrix, and is the unit alternating tensor. In the limit of small angular momentum relaxation time, , we obtain , and
Notice that as the autocorrelation becomes a delta function and the swimmer reorients rapidly and behaves as a Brownian walker.
where we have used that . Taking times and , we obtain Eq 3 of the main text.
Following a similar procedure, the Reynolds stress is given by
Taking times and , we obtain the Reynolds stress as given in Eq 4 of the main text.
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