Polar swimmers induce different phases in active nematics
PPolar swimmers induce different phases in active nematics
Pranay Bimal Sampat ∗ and Shradha Mishra † Department of Physics, Indian Institute of Technology (BHU), Varanasi, U.P. India - 221005
Swimming bacteria in passive nematics in the form of lyotropic liquid crystals are defined as aclass of active matter known as living liquid crystal in recent studies. It has also been shown thatliquid crystal solutions are promising candidates for trapping and detecting bacteria. Motivated bysuch studies, we developed a mixed model of polar swimmers in active nematics. It is found thatsuch a mixture is highly sensitive to the presence of polar swimmers, and shows the formation oflarge scale defects for relative swimmer density of the order of 0.1%. Upon increasing the densityof swimmers, different phases of active nematics are found and it is observed that the system showstwo phase transitions. The first phase transition is a first order transition from a quasi-long rangedordered state to disordered active nematics with larger scale defects. On further increasing densityof swimmers, the system transitions to a third phase, where swimmers form large, mutually alignedclusters. These clusters sweep the whole system like a comb in entangled hair and enforce localorder in the the nematic rods. In the third phase, nematic ordering increases with an increase in thedensity of polar swimmers. However, due to the presence of small, unaligned stray clusters, defectsin the active nematic still remain.
I. INTRODUCTION
Systems observed in nature can often be describedas inhomogenous mixtures of multiple species. Theinteraction between agents of different species may leadto states that are not seen in pure, homogenous systemsof the constituent species[1–3]. Studies investigatingthe interaction between different types of active matterwith different kinetics and alignment tendencies haveshown interesting dynamics and steady state features.The introduction of a foreign species of active particlescan significantly influence the dynamics and ordering insystems of other active particles [4–15]. Such techniqueshave not just been explored in theoretical studies,but they have also been verified by their experimentalcounterparts [16].Recent studies have shown the introduction of polaractive matter in passive liquid crystals leads to aninteresting class of liquid crystal called as living liquidcrystals (LLC) [17]. Such systems have been studiedextensively[18–20]. They seem to be good candidatesfor several applications. One such application is inthe form of detectors for the presence of swimmingbacteria, through changes in the electrical and opticalproperties of the liquid crystal due to the presence ofthese bacteria.[21].Several studies have also confirmed that active nematicalignment is observed in biological systems of varyingscales and types [22–27]. Motivated by the discoveryof these systems, we seek to study the effect of polarswimmers in a system of active nematic alignment. Todo so, we model a mixed system of active nematics, inthe form of apolar rods, and polar swimmers. ∗ [email protected] † [email protected] The aim of this study is to understand and ennumeratehow the introduction of polar swimmers to a systemof active nematics affects the ordering and structuralproperties of the active nematics. We tune the densityof polar swimmers in the background of a dense systemof apolar rods. The primary result of this study is thata very small density of polar swimmers is enough tobreak the quasi-long ranged ordered state seen in pureactive nematics. A non-monotonic change in nematicordering, relative to the density of polar swimmers,is observed. Such a mixture can broadly be dividedinto three phases, depending on the concentrationof the polar SPPs. Very interestingly, higher orderdefect structures are observed in the mixture, whichare generally not present in pure active nematics [28–30].The first phase of the mixture occurs when the concen-tration of polar swimmers is much lower than the con-centration of apolar rods. The ordering and structuralproperties of the system in this phase are identical tothat of a system consisting purely of apolar rods. It ischaracterized by high nematic ordering, giant numberfluctuations, quasi-long-range order in apolar rods andthe absence of clustering in polar particles. The systemtransitions to a second phase when the concentration ofpolar SPPs is increased. The second phase is character-ized by a breakdown of nematic order, and large scalestructural defects caused by the presence of multiple un-correlated clusters of polar SPPs. Higher order defectsin nematic rods are also observed in this phase. Thesystem again transitions to a third phase of the mixturewhen the density of polar SPPs is further increased. Inthe third phase, polar particles are able to form largeclusters which are collectively aligned and strongly cor-related with respect to time. The ordering of nematicrods in this phase increases as the density of polar SPPsis increased. However, the presence of small, stray uncor-related clusters of polar swimmers still hinder nematicordering and lead to short range nematic ordering, with a r X i v : . [ c ond - m a t . s o f t ] J a n (a) (b) (c)FIG. 1. A cartoon showing the different kind of interactions between particles in the mixture. Polar swimmers are representedas blue quivers, and the instantaneous direction of their is represented through the direction in which the tip of the quiver ispointing. Apolar rods are represented by the red figures. The pink arrows represent the instantaneous velocity of the rods.These cartoons were taken from a simulation, and the dotted circle surrounding the figures represent their circles of interaction.(a) Interaction between apolar rods and polar swimmers. (b) Interactions between polar swimmers (c) Interactions betweenapolar rods. lower order defects. II. MODEL AND NUMERICAL DETAILS
Our system consists of a mixture of active apolar rodsand polar swimmers moving on a two-dimensional sub-strate of friction coefficient µ . Each particle is driven byan internal force F acting along the long axis of the par-ticle. The ratio of the force F to the friction coefficientgives a constant self-propulsion speed v = Fµ . Each par-ticle (apolar/polar) is defined by a position vector r a/p ( t )and orientation θ a/p ( t ). The motion of apolar rods issymmetric about its head and tail, whereas polar swim-mers can move along their direction of orientation only.Both types of particles align with their neighbours whoseradial distance from them is less than a fixed radius ofinteraction r . Polar swimmers align ferromagneticallywith the velocity of polar swimmers and with the instan-taneous velocity direction of apolar rods. The alignmentinteraction is similar to the minimal model suggested byVicsek et al in [31]. Apolar rods align their orientationnematically with the polar swimmers and apolar rods intheir neighbourhood[32, 33]. The cartoon of the threetypes of interactions: (i) apolar-polar, (ii) polar-polarand (iii) apolar-apolar is shown in Fig. 1(a-c) respec-tively.The system is studied on a L × L square geometry and aperiodic boundary condition is used in both directions.The orientation and position of the apolar rods are up-dated at a unit time interval as follows: (cid:126)v i,a ( t ) = R i ( t ) v (cos( θ i,a ( t ))ˆ x + sin( θ i,a ( t ))ˆ y ) (1) (cid:126)r i,a ( t + 1) = (cid:126)r i,a ( t ) + (cid:126)v i,a ( t ) (2) θ i,a ( t + 1) = 12 arg( < e i θ ( t ) > (cid:126)r i,a ( t ) ,r ) + η i,t (3) R i ( t ) is randomly chosen for each particle at each time-step from the set [ − ,
1] with uniform probablity. The argument of the term < e i θ ( t ) > (cid:126)r i,a ( t ) ,r is a measureof the average orientation of both nematic rods and polarswimmers within a circle centered at (cid:126)r i,a ( t ) and withina neighbourhood radius r . < e i θ ( t ) > (cid:126)r i,a ( t ) ,r = (cid:88) | (cid:126)r j,a − (cid:126)r i,a | 4. The density ofnematic rods ( N a L ) is fixed as 1 . 0. The density of polarswimmers ( N p L ), denoted by ρ p , is the primary variableof the study, and was varied in the range [0 − . . − . (a) (b) (c)FIG. 2. The real space snapshots of apolar rods (red) and polar swimmers (blue) for η = 0 . L = 160 in the (a) first phase( ρ p = 0 . ρ p = 0 . 05) and (c) third phase ( ρ p = 0 . N ( θ p ) (top) and probability distribution function (PDF) of apolar rods P ( θ a )(bottom). We start with a random homogeneous distribution ofapolar rods and polar swimmers and their position andorientation is updated as in eqs. 1-8. One simulationtime-step is counted as an update of all the particles( N a + N p ) in parallel. At least 8 ensembles (realisations)of each configuration of ( L, η, ρ p ) were simulated for bet-ter averaging. Each ensemble was simulated for 2 × timesteps, of which the first 1 . × were used to allowthe system to reach the steady state, and the data fromthe remaining time-steps were used to calculate the re-sults in the study. In the critical region, at low densitiesof ρ p , 10 additional ensembles of the system were simu-lated for 4 × time-steps, of which the first 1 . × were used to allow the system to reach the steady state. III. RESULTSA. Nematic ordering ρ p S ( η , ρ p ) ρ p S ( η , ρ p ) η=0.1 η=0.2 η=0.3 I I II III FIG. 3. Variaton of S ( η, ρ p ) with ρ p for η = 0 . L =160. Three clear phases are observed. (Inset) The same non-monotonic behaviour is observed for all values of η = 0 . , . . We first measure the variation in global nematic orderingwith the density of polar swimmers. The ordering inthe system is measured by calculating the scalar nematicorder parameter S ( η, ρ p ), which is defined as: S ( η, ρ p ) = | N a (cid:88) j(cid:15) e i θ j,a | η,ρ p (9)The value of S ( η, ρ p ) is (cid:39) (cid:39) S ( η, ρ p ) vs. ρ p , for three different noise strengths η = 0 . , . . 3. The plot shows the non-monotonicvariation of S ( η, ρ p ) with ρ p , and three distinct phases.The real space snapshots of apolar rods (red) andpolar swimmers (blue) are also shown in Fig. 2. Thedistribution of the orientation of apolar rods is shownby plotting the probability distribution function (PDF) P ( θ a ) and the number distribution (ND) of polar swim-mers by N ( θ p ). The y − axis of N ( θ p ) is intentionallyleft unnormalised. The y − axis shows the number ofpolar swimmers participating in each peak of N ( θ p ). Itgives a measure of size of the clusters of polar swimmers.The first phase is observed at very low values of ρ p ,and exhibits strong nematic ordering. The value of thenematic order parameter S does not deviate from thevalue measured in a system consisting only of apolarrods with the same density. Snapshots of the systemin the steady state in this phase also show that thesystem forms dense bands of apolar rods, which evolveover large timescales. The PDF P ( θ a ) in this phasealso shows a normal distribution around a sharp peak,confirming the presence of a sharp director of nematicordering. These features are similar to those seen in asystem consisting purely of apolar rods, as confirmedby Chate et al in [32]. This suggests that the collectiveproperties of apolar rods are not affected by the presenceof a very small density of polar swimmers.We see a sharp drop in S as ρ p is increased beyond athreshold value. This drop is followed by a flat regionwhere S is invariant to further increases in ρ p . This flatregion in the plot for S vs ρ p is described as the secondphase of the mixture. The transition from the firstphase to the second phase is identified as a first ordertransition through the probability distribution function S( ρ p ) P ( S , ρ p ) ρ p =0.001 ρ p =0.002 ρ p =0.0025 ρ p =0.003 ρ p =0.005 (a)(b)(c)FIG. 4. (a) The steady state probability distribution P ( S )of S in the steady state for different values of ρ p =0 . , . , . , . 003 and 0 . 005 near the critical pointof first and second phase. There is a a clear bimodal distri-bution for ρ p = 0 . ρ p = 0 . ρ p = 0 . L = 160, ρ p = 0 . η = 0 . (PDF) of scalar order parameter P ( S ) in Fig. 4(a).The bimodal distribution of P ( S ) for ρ p in the criticalrange of ρ p = 0 . 002 to 0 . 003 confirms the first ordertransition. The two peaks of bimodal distributionscorrespond to two distinct phases: ordered and disor-dered, which coexist for the same system conditions.Whereas in the first and second phase ( ρ p = 0 . . P ( S ) shows one peak at S close to 1 and0 . P ( θ a ),uniformly ordered nematic rods with bands and random spikes in N ( θ p ). Whereas for the disorderedphase, the polar swimmers form a cluster consisting ofa few polar swimmers, with one major peak in N ( θ p ).These clusters break the backgroud nematic orderingand lead to the formation of defects and hence a broaddistribution of P ( θ a ). Similarly, steady state snaphotsof the system in the second phase in Fig.2(b) show thatthe system is disordered, with the presence of largescale defects in nematic ordering. The P ( θ a ) also showsa uniform distribution with no clear peak and multipledistinct peaks in N ( θ p ). The second phase is alsocharacterized by the presence of circular defects, thesize of which is much larger than the interaction range r .As the the density of polar swimmers is further in-creased, a third phase is observed in Fig. 3. In thisphase, the nematic order parameter increases with ρ p .Snapshots of the system in this phase show the presenceof weak nematic ordering, with much smaller defectsthan those seen in the second phase of the mixture. ThePDF of θ a also shows the emergence of a global directorof nematic ordering, but not one as strong as the oneseen in the first phase. S ( N , η , ρ P ) ρ p =0.001; χ =0.007 ρ p =0.05; χ =0.366 ρ p =0.1; χ =0.343 ρ p =0.3; χ =0.221 ρ p =0.5; χ =0.160 FIG. 5. S ( N, ρ p ) vs N for η = 0 . 2. The dashed line in thefigure represents the fitted form S ( N ) ∼ N − χ . The valueof χ obtained through the fitting for the given parameters ismentioned in the legend. Thus, we find that nematic rods exist in the threedistinct phases in the presence of polar swimmers. Tocharacterise the range of order in these phases we alsomeasured the variation of S with N in the first, secondand third phases( ρ p = 0 . , . , . , . . S ( N ) to the form S ( N ) ∼ N − χ , we see thatthe first phase exhibits quasi-long-range order in itsorientation, with χ < , and the third phase exhibitsshort-range order with χ > . In the second phase,the apolar rods are disordered. Hence on increasing ρ p , the apolar rods undergo a non-monotonic variationof ordering from a QLRO state to disorder and thento a short-range ordered state. The plot of S ( N ) vs. N in the first, second and third phase is shown in Fig. 5.To better understand the effect of polar swimmers onthe active apolar rods, we measure the characteristics ofpolar swimmers in the bulk nematic. t C ( ρ p , t) ρ p =0.001 ρ p =0.01 ρ p =0.05 ρ p =0.1 ρ p =0.5 ρ p ( ρ p ) τ(ρ p )/1000P ( ρ p ) (a) (b)FIG. 6. (a) Polar velocity time auto-correlation function (VACF) C ( ρ p , t ) for different densities of polar swimmers ρ p . (b)Variation of C ( ρ p ), τ ( ρ p ), and P ( ρ p ) with ρ p . B. Characteristics of polar swimmers The characteristics of the polar swimmers in the mixturewere quantified using the following parameters:1) The polar order parameter for polar swimmers P ( ρ p ),given as: P ( ρ p ) = 1 v N p | N p (cid:88) j(cid:15) (cid:126)v j,p | (10)The polar order parameter is a measure of the alignmentof polar swimmers in the system. In a well ordered sys-tem of polar swimmers, where all swimmers are movingin the same direction, the value of P ( ρ p ) is (cid:39) 1. Alter-natively, in a poorly aligned system of polar swimmers,where the swimmers move randomly, the value of P ( ρ p )is (cid:39) C ( ρ p , t ) of polar swimmers, given as: C ( ρ p , t ) = 1 v N p N p (cid:88) j(cid:15) (cid:126)v j,p ( t ) · (cid:126)v j,p ( t + t ) (11)The C ( ρ p , t ) is a measure of the deviation of the directionof polar swimmers with time. In other words it gives ameasure of how long a particle remembers its orientation.3) The mean-squared displacement of the polar swim-mers ∆ p ( t ):∆ p ( t ) = 1 N p N p (cid:88) j(cid:15) | (cid:126)r j,p ( t + t ) − (cid:126)r j,p ( t ) | (12) The mean-squared displacement of the particle is anindicator for the dynamics properties of the particle.When fitted to the form ∆( t ) ∼ t β ( t ) , the exponent β ( t ) = log p (2 t ))∆ p ( t ) can be used to identify the type ofmotion that a particle is undergoing. β ( t ) = 1 for dif-fusive motion, 1 < β ( t ) < β ( t ) = 2 for ballistic motion. 10 100 1000 t ∆ ( ρ p , t) ρ p =0.001 ρ p =0.0025 ρ p =0.005 ρ p =0.05 ρ p =0.1 ρ p =0.5 0 100 200 300 400 500 t β (t) FIG. 7. Variation of mean square displacement (MSD)∆( ρ p , t ) with t for different values of ρ p at η = 0 . L = 160. (Inset) Variation of β ( t ) with ρ p . In the first phase of the mixture, the polar swimmershave a low polar order parameter, with a sharp ex-ponentially decaying velocity auto-correlation whichdecays to 0 with a very short tail as shown in Fig. 6, anddiffusive motion with ∆ p ( ρ p , t ) ∼ t and β ( t ) ∼ θ p in the steady state (Fig. 2(a)), which shows a uniformdistribution of random spikes in N ( θ p ).In the second phase of the mixture, the polar swimmersshow a moderate value of P ( ρ p ). The VACF of the FIG. 8. Snapshots of the mixture in the second phase, showing the action of polar swimmers in the formation of a higherorder +1 defect in nematic ordering at (15,90). The colours of the particles are the same as in Fig2. polar swimmers decays exponentially to 0, but with amuch larger tail than that seen for the polar swimmersin the first phase, as seen in Fig. 6. The exponent of∆ p ( ρ p , t ) also shows that the motion of the swimmersis super-diffusive β ( t ) > ρ p = 0 . P ( ρ p ) ∼ 1, approaching a perfectly ordered state.They also show an exponential decay in C ( ρ p , t ), butthe function decays to a non-zero constant value C ( ρ p ),rather than 0, as seen in the earlier two phases. Thecorrelation time τ ( ρ p ) approaches a finite value in thisphase. The motion of the polar swimmers in this phasecan also be described as ballistic, with the exponent β ( t ) (cid:39) ?? . Snapshots in Fig. 2(c) show that the polar swim-mers form large, mutually aligned clusters which are per-sistent in their motion, and move along the direction ofnematic alignment. Given the properties of C ( ρ p , t ), wefit it to the form C ( ρ p , t ) = (1 − C ( ρ p )) e − tτ ( ρp ) + C ( ρ p ).The variation of τ ( ρ p ) and C with ρ p is given in Fig. 6. C. Structural defects in active nematics The first phase of the the mixture largely shows thesame properties as a system consisting purely of nematicrods. Given the noise strength, strongly ordered, densebands of apolar rods are observed which are parallelto the direction of bulk apolar alignment. The motionof polar swimmers in this phase can be described as abiased random walk in two dimensions, with a highertendency to move along the axis of alignment, but theycan also move perpendicularly. There is an absence of clustering in polar swimmers.As ρ p is increased, and approaches the critical rangefor the transition from the first phase to the second, itis observed that polar swimmers begin clustering. Theclusters formed when the system is in the second phaseare not mutually aligned, and can strongly influence theorientation of apolar rods in regions where the densityof apolar rods is lower than the density of the polarcluster. This leads to defects in nematic ordering, whichin turn lead to high-density regions of apolar rods.Once formed, the dense regions of apolar rods bendthe motion of the polar clusters that pass them. If thesize of the polar cluster is small, they get rotated inthe direction of the defects. However, if cluster of polarswimmers is large enough, then it can further distortthe defect structure. This dynamics keeps reocurringin the system. Hence, the steady state of the systemshows the formation and reformation of large higherorder defects in apolar rods. As we get deeper into thesecond phase, multiple uncorrelated clusters of polarswimmers are formed. Two or more of such clusterscumulatively cause higher order defects in nematicordering, whose size is much larger than the size ofthe clusters themselves. A timeseries of snapshots forprocess of the formation of higher order defects innematic rods in the second phase for ρ p = 0 . 005 isshown in Fig. 8.As ρ p is further increased, the polar swimmers form largeclusters which are mutually aligned in a polar mannerand show a high time auto-correlation. These large per-sistent clusters sweep the lattice repeatedly, and theyhave a comb-like effect on the apolar rods. They tendto enforce a global field for nematic alignment which isparallel to their own motion. Smaller stray clusters ofpolar swimmers still exist, and they prevent perfect ne-matic ordering of apolar rods. However, the sweepingaction of larger clusters prevents the formation of large-scale higher order defects. The real space snapshots atdifferent times are shown in Fig. 9. FIG. 9. Snapshots of the mixture in the third phase, illustrating the alignment and persistent motion of large clusters of polarswimmers. The colours of the particles are the same as in Fig2. D. Number fluctuations 10 100 1000 N ∆ N ( ρ p ) ρ p =0 ρ p =0.001 ρ p =0.01 ρ p =0.1 ρ p =0.5 ρ p h ( ρ p ) FIG. 10. Variation of number fluctuation ∆ N with N for η = 0 . L = 160. The dashed line represents the fittedfunction as given in Eq. 13. (Inset) Variation of h with ρ p The formation of high-density bands of apolar rodsleads to giant number fluctuations in active nematics,and similar fluctuations are seen in the first phase ofthe mixture. The fluctuations are partially supressedin the second phase of the mixture, which does notshow large bands. However the large scale defectswhich lead to dense arrangements of apolar rods arestill a cause for fluctuations. The third phase shows thesmallest fluctuations, and even within the third phasefluctuations are further supressed with an increase in ρ p . The effect of ordered polar swimmers cluster can beassumed as an ordered field and the result of numberfluctuation is compared with the previous linearisedcalculation of ∆ N for active nematic with an externalfield [34]. ∆ NN = N ( a + N h ) (13)In the above expression Eq. 13, a is a system depen-dent constant, which in the pure active nematic sys-tem depends on the diffusivity of the system and h is the strength of the external field. In Fig. 10 we plot the number fluctuation ∆ N vs. N for densities ρ p = 0 , . , . , . , . 5, and show the comparison tothe analytical expression derived for the active nematicwith external field. For all densities the data fits wellwith the analytical expression. From the fitted expres-sion in Eq. 13 we find that a remains almost unchangedon increasing ρ p and h increases with ρ p as shown in Fig.10(inset). The increase in h with ρ p supports the hypoth-esis that the large clusters formed in the third phase actas a sweeping orienting field for the active nematics. IV. DISCUSSION To conclude, we have studied a mixture of self-propelledapolar rods and polar swimmers on a two-dimensionalsubstrate. The density of polar swimmers is tuned fromvery small to moderate values. The density of apolarrods is always kept fixed and the polar swimmers arein minority. We find three distinct phases. The firstphase is observed when the density of polar swimmersin the system is very low ρ p ∼ . ρ p . Further increasing the density of polarswimmers, it is observed that beyond a threshold densityof polar swimmers, clusters of polar swimmers are ableto exhibit high polar order. The large, mutually alignedclusters of polar swimmers formed in this phase arefound to contribute to nematic ordering, and suppressthe giant number fluctuations seen in a pure system ofapolar rods and the first phase of the mixture. However,stray clusters of polar swimmers in this phase still leadto a markedly lower level of nematic ordering in theapolar rods. The transition from the second to the thirdphase is a continuous transition, where the scalar apolarorder parameter changes continuously from disorderedto short ranged ordered state.Interestingly in the second phase we find the formationof large scale and higher ordered defects in apolarrods. The size of these defects are much larger thanthe interaction range of the particles. The steadystate is characterised by the repeated formation anddestruction of these defects. The presence of clustersof polar swimmers are responsible for the formationand reformation of such defects. Such a dynamicalsteady state is absent when self-propelled apolar rodsare replaced by passive apolar rods or the backgroudapolar is in equilibirum, or for the LLC systems [17, 18].Hence our predictions can be used for the very earlydetection of polar swimmers in the self-propelled apolarmixture. For example, bacteria in the epithilial cells[35]. Also it can also used for trapping of the swimmersat the core of the defects.While the highest density of polar swimmers investi-gated in this study is smaller than the density of apolar rods, preliminary investigation of higher densities of po-lar swimmers suggest that stray clusters of polar swim-mers remain even when the population of both speciesis equal. This will have to be confirmed through a morerigorous study. It presents an interesting problem, be-cause it is expected that a system where the density ofpolar swimmers is much higher than the density of ap-olar rods should show similar traits to a pure systemof polar swimmers, which might lead to another phasetransition at that extremum. V. ACKNOWLEDGEMENT The support and the resources provided by PARAMShivay Facility under the National Supercomputing Mis-sion, Government of India at the Indian Institute ofTechnology, Varanasi are gratefully acknowledged by Dr.Shradha Mishra and Pranay Sampat. SM thanks DST-SERB India, ECR/2017/000659 for financial support.SM and PS also thank the Centre for Computing andInformation Services at IIT (BHU), Varanasi. VI. REFERENCES [1] C. Bechinger, R. Di Leonardo, H. Lowen, C. Reichhardt,G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006(2016).[2] Demian Levis and Benno Liebchen Phys. Rev. E 100,012406.[3] M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B.Liverpool, J. Prost, Madan Rao, R. Aditi Simha, Rev.Mod. 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