Ordering kinetics in active polar fluid Shambhavi Dikshit1and Shradha Mishra1 1Indian Institute of Technology BHU Varanasi India 221005

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Ordering kinetics in active polar ﬂuid

Shambhavi Dikshit1∗and Shradha Mishra1†

1Indian Institute of Technology (BHU), Varanasi, India 221005

(Dated: June 13, 2023)

We model the active polar ﬂuid as a collection of orientable objects supplied with active stresses

and momentum damping coming from the viscosity of bulk ﬂuid medium. The growth kinetics of

local orientation ﬁeld is studied. The eﬀect of active ﬂuid is contractile or extensile depending upon

the sign of the active stress. We explore the growth kinetics for diﬀerent activities. We observe

that for both extensile and contractile cases the growth is altered by a prefactor when compared to

the equilibrium Model A. We ﬁnd that the extensile ﬂuid enhances the domain growth whereas the

contractile ﬂuid supresses it. The asymptotic growth becomes pure algebraic for large magnitudes

of activity. We also ﬁnd that the domain morphology remains unchanged due to activity and system

shows the good dynamic scaling for all activities. Our study provides the understanding of ordering

kinetics in active polar gel.

The systems in which the energy consumption occurs

on individual constituent level and it leads to collective

dynamics are active systems [1, 2]. The existence

of active systems is found from small microscopic

length scale, i.e. interacellular level like cytoskeletal

actin ﬁlaments [3], bacterial colonies [4] etc. to large

macroscopic scale i.e. upto few meters like animal herds

[5], birds ﬂocks [6, 12] etc. Active systems are deﬁned as

wet when coupled to a momentum conserving solvent,

in which solvent mediated hydrodynamic interaction

becomes important [1, 2]. Bacterial swarms in a ﬂuid,

cytoskeleton ﬁlaments, colloidal or nanoscale particles

propelled through a ﬂuid are examples of wet systems

[7], [8]. When no such ﬂuid is present, then system is

called dry. Dry systems include bacteria gliding on a

surface [9], animal herds or vibrated granular particles

and so on [10, 11].

Starting with the seminal work of Vicsek [12], most of

the previous works on active system have focused on

the steady state properties [1, 13–15]. The study of

ordering kinetics in active systems is complex by the

fact that the system relaxes to a nonequilibrium steady

state (NESS). There have been very few studies [16–19]

of the coarsening kinetics from a homogeneous initial

state to the asymptotic NESS, though understanding

it is of great experimental interest. Previous studies of

coarsening or domain growth have primarily focused

upon systems approaching to an equilibrium state

[20–23]. Based on the symmetry and conservation laws

the domain growth is classiﬁed mainly of two types.

The domain growth in systems with conserved order

parameter is named as Model B and with nonconserved

order parameter is called as Model A, follows an alge-

braic growth law with growth exponent z= 3 [22] and

2 [23] respectively. For the systems with scalar order

parameter and nonconserved growth kinetics, the inter-

facial velocity of the growing domain is proportional to

the local curvature of the interface; that leads to the

size of the domain L(t)∝t1/2; Allen and Cahn growth

∗shambhavidikshit.rs.phy18@itbhu.ac.in

†smishra.phy@iitbhu.ac.in

law [24]. Whereas for the systems with conserved

kinetics the interface have to pay a cost due to local

conservation of order parameter. That leads to the size

of the domain grows with time such that L(t)∝t1/3;

Lifshitz-Slyozov-Wagner (LSW) theory [22, 25].

For the systems having symmetries of two-dimensional

XY-model, with nonconserved growth kinetics and

order parameter with more than one components or

vector order parameter, the asymptotic growth law is

still z= 2. The topological defects are vortices and

antivortices and the domain growth is driven by the

annihilation of these defects. The detailed calculation

[26, 27] shows that there is logarithmic correction to the

pure algebraic growth L(t)∝(t/ ln(t))1/2. Equilibrium

liquid crystals, ferromagnetic materials with continuous

symmetry, spin glasses, two-dimensional superconduc-

tors, etc. are some of the examples of systems with

nonconserved vector order parameter.

The domain growth in systems approaching towards a

thermal equilibrium state is very well studied in dry

[22, 23, 26, 27] as well as wet systems with hydrody-

namic eﬀect [28–31]. The understanding of the ordering

kinetics in terms of the number of topological defects is

explored in many of the studies [32–36]. Recently some

studies are performed on the understanding of ordering

of domain growth in dry active systems [16, 18, 19, 37].

But the ordering kinetics in active systems with ﬂuid

is rarely explored. Some recent studies show the eﬀect

of hydrodynamics on the ordering kinetics of apolar

order parameter ﬁeld [38] and the eﬀect of ﬂuid on the

steady state properties of active polar ﬂuid [39–41].

This motivates us to study the ordering kinetics of

active polar systems with ﬂuid or active polar gel. The

examples of active polar gels are bacteria suspensions,

active emulsions and active gels. [42, 43].

The model contains a collection of orientable objects

supplied with active stresses and momentum damping

coming from the viscosity of bulk ﬂuid medium. The

ordering kinetics of the orientational ﬁeld is studied

after a quench from the random disordered state. With

time the system orders and the size of ordered domain

grows with time. We characterise the domain growth

and scaling. The direction of spontaneous ﬂow makes

arXiv:2210.09199v2 [cond-mat.soft] 12 Jun 2023

the system to respond like extensile and contractile

in nature. For the extensile case, particles act like

pushers (pulling ﬂuid inward equatorially and emitting

it axially) and for contractile case they are more likely

pullers [44] (vice versa). We observe that for both

extensile and contractile cases the growth is altered by

a prefactor when compared to the equilibrium Model A.

We ﬁnd that the extensile ﬂuid enhances the domain

growth whereas the contractile ﬂuid supresses it. The

asymptotic growth becomes pure algebraic for large

magnitudes of activity. For all activities, system shows

good dynamic scaling and domain morphology remains

unaﬀected with respect to the activity.

Model A:- The time evolution of system with noncon-

served local order parameter for a collection of orientable

objects is describe by the time-dependent Ginzburg Lan-

dau equation [23, 45]

∂Pα(r, t)

∂t =−Γ0

δF0

δPα(r, t)+θα(r, t) (1)

where Γ0is the mobility. The Ginzburg Landau free

energy F0is

F0=Zddr{1

2aP(r, t)2+λ

2|∇P(r, t)2|+b

4P(r, t)4}

(2)

here P(r, t) is a vector ﬁeld with components Pα(r, t),

and α= 1 and 2 in two-dimensions. The vector

ﬁeld P(r, t) is the local orientation ﬁeld and is deﬁned

by the average orientation of the particles in a small

coarse-grained region. The size of the region is such

that it consists of suﬃcient number of particles to per-

form the statistical averaging. θα(r, t) is Gaussian ran-

dom white noise with properties < θα(r, t)>= 0 and

< θα(r, t)θα(r′, t′)>= 2∆0δ(t−t′)δαα′.a,band λare

constants. (a < 0 ensures the broken symmetry state,

b > 0 and λ > 0 for stability and the strength of noise

∆0= 0). After substituting the form of F0from Eq.

2 in Eq. 1 and performing the functional derivative

of Ginzburg Landau free energy F0, we get the time-

dependent Ginzburg Landau (TDGL) [23] equation for

nonconserved vector ﬁeld

∂P(r, t)

∂t =aΓ0P(r, t)−bΓ0|P(r, t)|2P(r, t)+λ∇2P(r, t)

(3)

The model describe by Eq. 3 is called as Model A

according to the Halprin and Hohenberg [23]. The noise

term present in Eq. 1 is turned oﬀ and we consider the

deterministic part of the TDGL equation as discussed in

[21]. The Gaussian noise in Eq. 1 is purely thermal in

nature and ensures that the system reaches the global

minima at late times. But most of the kinetic theories

are developed for the deterministic TDGL equation and

thermal noise is irrelevant for growth kinetics [21]. The

equation 3 very well explains the ordering kinetics in

magnets with vector order parameter [46] and liquid

crystals [47]. Now we further introduce the eﬀect of

hydrodynamic interaction on the ordering kinetics of

nonconserved ﬁeld.

Active Polar Fluid:- Now we discuss the hydrodnyam-

ics of active polar ﬂuid or polar active poalr gel. We

focus here the active gel deﬁned with collection of ori-

entable objects supplied with active stresses and momen-

tum damping coming from the viscosity of bulk ﬂuid

medium, compare to the friction due to substrate or

medium [32, 33, 41]. The equations are ﬁrst proposed

by [33] for self-propelling objects but later developed by

[41] and [32] for the system of cytoskeleton in living cells

and polar actin ﬁlament, which become active only in

the presence of molecular motors that consumes ATP.

A collection of artiﬁcial Janus rods which gain motility

due to electrophoresis is a good example of active po-

lar gel and can be easily designed in the laboratory [7].

The presence of ﬂuid introduces the hydrodynamic ef-

fect. If the hydrodynamic interaction is turned oﬀ then

the model is purely passive and same as Model A. The

model incorporate the coupling between local order pa-

rameter and ﬂuid.

We model the system by the coupled dynamics of the

orientation order parameter P(r, t) with a solvent local

velocity v(r, t) with additional active stresses. The sys-

tem is modeled by the coarse-grained coupled hydrody-

namic non-linear partial diﬀerential equations of motion

for the two ﬁelds. The ﬂuid is introduced through the

standard Navier-Stokes equation of motion for ﬂuid with

additional coupling to the polarisation, P(r, t) of parti-

cle through active and passive stresses (deviatoric stress)

as introduced in [41].

In the presence of ﬂuid, in addition to the term present

in Eq. 3 coupling to the ﬂuid velocity, hence the Eq. 3

will have convective nonlinearity of type (v.∇)P. Hence

the modiﬁed equation for the P(r, t) will become

∂P(r, t)

∂t +v.∇P(r, t) + ωαβ Pβ+v1vαβPβ=aΓ0P(r, t)

−bΓ0|P(r, t)|2P(r, t) + λ∇2P(r, t)

(4)

with the comoving and corotational derivative of the po-

larisation, P(r, t), where ωαβ =1

2(∂αvβ−∂βvα) and

vαβ =1

2(∂αvβ+∂βvα) are the vorticity and strain-rate

tensor respectively. The coupled velocity ﬁeld is due to

momentum conserving solvent which satisﬁes the condi-

tion of incompressibility, i.e. ∇.v= 0

The equation for ﬂuid velocity, v(r, t), satisfying con-

servation of mass (condition of incompressibility) and

conservation of momentum is given as

∂v

∂t +v.∇v=η∇2v− ∇p+∇.σtotal

αβ (5)

The Eq. 5 is the Navier-Stokes equation with an ad-

ditional force term due to stresses present in the ﬂuid.

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OrderingkineticsinactivepolarfluidShambhaviDikshit1∗andShradhaMishra1†1IndianInstituteofTechnology(BHU),Varanasi,India221005(Dated:June13,2023)Wemodeltheactivepolarfluidasacollectionoforientableobjectssuppliedwithactivestressesandmomentumdampingcomingfromtheviscosityofbulkfluidmedium.Thegrowthkineti...

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Ordering kinetics in active polar fluid Shambhavi Dikshit1and Shradha Mishra1 1Indian Institute of Technology BHU Varanasi India 221005

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