Inhomogeneous activity enhances density phase separation in active model B
IInhomogeneous activity enhances density phase separation in active model B
Ajeya Krishna ∗ and Shradha Mishra † Indian Institute of Technology (BHU), Varanasi, U.P. India - 221005
I. ABSTRACT
We study the binary phase separation in active model B, on a two-dimensional substratewith inhomogeneous activity. Activity is introduced with a maximum value at the center ofthe box and spread as a Bivariate-Gaussian distribution as we move away from the center.The system is studied for three different intensities of the distribution. Towards the boundaryof the box, activity is zero or the model is similar to the passive model B. We start from therandom homogeneous distribution of density of particles, system evolve towards a structureddistribution of density. With time, density starts to phase separate with maximum densityat the center of the box and decreases as we move away from the center of the box. Thewidth of the density profile at the center increases as a power law with respect to time andreaches a maximum at the late time. The power law exponent increases with an increasein the intensity of the activity. Hence our result shows the response of density in an activebinary system with respect to the patterned substrate. It can be used to design the devicesuseful for the trapping and segregation of active particles.
II. INTRODUCTION
Ranging from small organisms like bacteria [1], algae to higher organisms like fish [2], birds [3],animals self-organizes themselves to form complex structures [4]. This topic has been under activeresearch for the past decade [5]. Emergence of meso-scale[6–9] turbulent motion was a forwardstep in research on non-equilibrium biological systems. Dynamics of active colloidal particles suchas natural microorganisms like bacteria or algae [4, 10], or synthetic swimmers, active Brownianparticles (ABP) [11–15] described by having a non-trivial dependence of active current to the localcurvature of the underlying density profile. This leads to an activity term in standard binary phaseseparation in equilibrium system or also called as passive model B [16–18]. The correspondingactive model is called as active model B (AMB) [19]. The study of passive model B [16–18] isuseful to understand the phase separation in equilibrium binary systems [16, 17], whereas active ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s o f t ] F e b model B, gives the understanding of phase separation in many natural and biological systems.Also, many artificially designed active Janus particles in the lab are also useful candidates fortechnological and pharmaceutical applications [20–22]. Active Brownian particles predominantlyshow short-range steric repulsion [23]. The presence of activity shows fascinating behavior likecoherent motion, phase separation without any external parameter or quenching in temperature[4, 24]. The phase separation of systems with self-propelled particles (active particles) is studiednumerically [23, 25–27] and to some extent in experimental studies [28, 29].In this study, we consider a conserved scalar order parameter φ which is linearly associatedto the local density of the particles. The motivation of selecting a scalar order parameter isModel B [19, 30]. The resulting phase-separation kinetics of passive colloidal particles is bestdescribed by mean field theories involving a conserved scalar order parameter φ (continuous localdensity parameter). Simplifying the free energy to quadratic polynomial in φ using general diffusionmechanisms, gives a theory of “Model B”. Model B or φ field theories are the simplest form ofCahn-Hilliard equation [16–18].The kinetics of phase separation of passive and active systems are extremely distinct but coarse-graining of active systems at large scale demonstrates some relation between the two cases. Therelation was first observed in models of swimming bacteria with discrete reorientations [31] andlater extended to ABPs [32].In our model, we are adding a non-integrable gradient term to passive Model B [16–18]. Theadded gradient term simply breaks detailed balance in Passive Model B, which suggest that activemodel B cannot be derived from a free-energy functional [19]. In this active model, we add λ |∇ φ | term to the derivative of free-energy functional or simply chemical potential. The λ here determinesthe strength of activity in the system. In previous studies, [19, 33], the activity λ is considered aconstant value or uniform over the system. In this work, we consider λ inhomogeneous in spaceand take it as Bivariate-Gaussian distributed in the system. We consider a two-dimensional squarebox with periodic boundary condition, where the activity is chosen maximum at the center of thebox and decay as a Bivariate-Gaussian distribution as we go away from the center. The systemis studied for three different values of maximum intensity at the center. We call the model asinhomogeneous active model B (IAMB). For comparison we also studied the constant activitymodel, where λ remains constant in the whole system. Below we report the results of steady stateand kinetics of IAMB. We observed that on the increasing activity at the center of the box, lead totrapping of the particles at the center, hence higher the activity more the value of local density. Welater studied the growth kinetics of density in the middle of the box for three different intensitiesand find the density grow as a power law with time. The power law exponent varies from 2 / / III. MODEL AND NUMERICAL DETAILS
We consider a conserved scalar order parameter field φ ( r , t ) at position r and time t in twodimensions. The variable φ is linearly related to the local number density ρ ( r, t ) of active particlesby the transformation φ = (2 ρ − ρ H − ρ L )( ρ H − ρ L ) , where ρ H and ρ L are the high and low local densities co-existing phases respectively. The dynamical equation for rate of change of φ is given by continuityequation [34, 35]: ∂φ∂t = −∇ · J (1) J = −∇ µ (2) µ = − φ + φ − ∇ φ + λ |∇ φ | (3)The expression in eq. 1 represents the conservation of φ and the expression in eq. 2 expressesthe relation between the mean current J and the non-equilibrium chemical potential µ . The meancurrent J is proportional to the negative gradient of the chemical potential µ .The chemical potential µ = µ + µ is the sum of bulk and gradient contributions. In equilibrium µ can be obtained by the variation of free energy functional. The bulk part µ can be obtained fromthe polynomial terms in the standard φ free energy functional, µ = δδφ f , where f = − φ + φ .Hence, µ = − φ + φ .The gradient term µ has two terms, µ = µ p + µ a . The passive gradient term is µ p = −∇ φ ,which can be obtained by variation of gradient term in Landau-Ginzburg free energy functional [36].The µ a is the simplest addition to chemical potential, that cannot be derived from a free-energyfunctional. The active gradient term is the same as obtained for active model B, µ a = λ |∇ φ | [19].In our present study it is inhomogeneous λ , instead of constant value. We write λ as Bivariate-Gaussian Distribution [37] which is given as eq. 4 , λ ( x, y ) = λ × πσ X σ Y (cid:112) − ρ exp( −
12 [ ( x − µ X ) σ X − ρ ( x − µ X )( y − µ Y ) σ X σ Y + ( y − µ Y ) σ Y ]) (4)In eq. 4, µ X , µ Y are the mean along x and y directions respectively. In our case it is the centerof the system. σ X = σ Y = σ represents variance along x and y directions. In our case it is *(size of system). In eq. 4, ρ refers to correlation of distribution in x and y . In our case, there isno correlation i.e., ρ ≈ x and y in eq. 4, represents x and y co-ordinates of the system. λ determines the maximum intensity of the distribution.We studied the model for three different cases G = [ λ , σ ]: G = [1 . , G = [2 . , G = [3 . , Intensity plots for diferent λ :- Here, we graphically represent the different intensities ofGaussian distribution of activity. We included 2D plots in Fig. 1, in which center peak has highestvalue and value decreases confirming the Gaussian distribution. (a) (b) (c) FIG. 1: Graphical representation of the distribution λ ( x, y ). (a) λ = 1, i.e., G = [1 . , λ = 2, i.e., G = [2 . , λ = 3, i.e., G = [3 . , Numerical Details:
We have performed numerical analysis on our model using numericalmethods to solve differential eqs. 1, 2, 3. We randomly initialized φ and calculated chemicalpotential ( µ ) which is given as in eq. 3. Computing chemical potential ( µ ), we can calculateflux( J ) which is given as in eq. 2. Then the φ is updated using flux ( J ) from eq. 2 and inserting itin eq. 1. The stochastic differential equations of this model were solved by using Euler’s numericalmethod [38]. The essential parameters we have considered are, dx = 1 . dy = 1 . dt = 0 .
01. Weconsidered a critical system i.e., number of higher density particles equals number of lower densityparticles. The system was iterated for 5000 times i.e., t = 5000. The scalar order parameter φ ( r, t ) was randomly initialized, with highest value as 0 . − .
5. The mean andvariance of this distribution were 0 . . × λ , the essential parameters are µ X , µ Y , σ , and λ . Parameters µ X and µ Y determines the meanof the distribution. In this model, center of system is considered as origin, so µ X = 0 and µ Y = 0.Parameters σ determines the variance of our model. In our model, variance is *(size of system)which is 32. So variance along x and y directions σ = 32. Parameter λ determines the intensityof distribution. Our system is studied for 3 different intensities λ = 1 , , IV. RESULTS
Now we discuss our results in detail: We integrate the nonlinear partial differential equa-tion for local density with mean density φ = 0 . G = [1 . , G = [2 . , G = [3 . , Active model B (AMB) :- In this section, we will discuss the results of active model B.This model has a constant activity ( λ = 1 .
0) all over the system with box size L = 64. Thisstudy is preformed to understand the comparison between the constant activity parameter (AMB)and Gaussian distribution of activity parameter (IAMB). We observe interesting difference in thekinetics of domain formation and steady state structure of domains. Fig. 2 shows the real spacesnapshot of density at time steps = 0, 1000, 2000, 3000, 4000, 5000. Starting from randomhomogeneous density, as time progress, high density regions starts to phase separate and phaseseparation happens with the formation of isolated domains [19]. The size of the domains increaseswith time. (a) (b) (c)(d) (e) (f) FIG. 2: Density evolution snapshots of the Active model B . (a) at t=0, (b) at t=1000, (c) att=2000, (d) at t=3000, (e) at t=4000, (f) at t=5000. The color bar represents the value of localdensity φ . A. Time evaluation of density for inhomogeneous activity (IAMB)
In this section, we will discuss the results of Gaussian activity of intensity, i.e. , G = [1 . , G = [2 . , G = [3 . , Case 1: Gaussian Distribution of activity with λ = 1 : In this case we consider thelowest intensity distribution i.e., ( λ = 1) which is represented graphically in Fig. 1a. In Fig. 3,we attached snapshots of density evolution. (a) (b) (c)(d) (e) (f) FIG. 3: Evolution snapshots of the active model B with Gaussian distribution of activity with λ = 1 .
0. (a) at t=0, (b) at t=1500, (c) at t=2500, (d) at t=3500, (e) at t=4000, (f) at t=5000.The color bar have the same meaning as in Fig. 2In Fig. 3, we have snapshots of density evolution with time. When the intensity of distributionis λ = 1 .
0, we observe small-scale accumulation of high density particles at the center of the box.In Fig. 3a, particles are randomly distributed in system, so we observe scattered density. Withtime, we can see the accumulation of higher density particles at the center of system. Dropletsof lower density particles are also formed besides the accumulation of higher density particles inactivity region.
Case 2: Gaussian Distribution of activity with λ = 2 : In this case we consider theintensity distribution which is represented graphically in Fig., 1b. In Fig. 4 we show snapshots ofdensity evolution of the system. (a) (b) (c)(d) (e) (f)
FIG. 4: Evolution snapshots of the active model B with Gaussian distribution of activity with λ = 2 .
0. (a) at t=0, (b) at t=1500, (c) at t=2500, (d) at t=3500, (e) at t=4000, (f) at t=5000.The color bar have the same meaning as in Fig. 2In Fig. 4 we can observe the evolution of the system. When the intensity of distribution is λ = 2 .
0, we again observe more accumulation of particles at the center where there is maximumvalue for activity. Traversing from the center of system to the walls of system, there is decreasein particle accumulation and we observe lower density particles forms connected domains outsidethe activity domain. Droplets of lower density particles are also formed besides accumulation ofhigher density particles.
Case 3: Gaussian Distribution of activity with λ = 3 : In this case we consider theintensity distribution which is represented graphically in Fig. 1c. In Fig. 5 we show snapshots ofdensity evolution of the system. (a) (b) (c)(d) (e) (f) FIG. 5: Evolution snapshots of the active model B with Gaussian distribution of activity with λ = 3 .
0. (a) at t=0, (b) at t=1500, (c) at t=2500, (d) at t=3500, (e) at t=4000, (f) at t=5000.The color bar have the same meaning as in Fig. 2.In Fig. 5 we can observe the evolution of the system. When the intensity of distribution is λ = 3 . L ( t )(2 σ ) vs Time ( t ) for different intensities of distribution. The comparisonis for scaled length i.e., t = 5000. B. Kinetics of growing domains in high activity region
We now estimate the kinetics of growing high density domain at the center of the box, fordifferent intensities G = [1 . , G = [2 . , G = [3 . , φ to binary density ψ = +1 if φ > φ and ψ = − φ < φ , then calculate themean size of domains L ( t ) of high density in the center of the box. Mean is calculated over 20independent realisations. As time progress the domain size increases L ( t ) at the middle of the box.We estimate the length of the growing domains at the center of the box and plot the scaled length L ( t )2 σ vs. time t for three different intensities. In Fig. 6, the plot is shown for the three cases. Veryclearly growth is faster for larger intensities and saturates at late times. The L ( t ) grows with time1as, L ( t ) (cid:39) t α , where α ∼ / / V. DISCUSSION
We have studied the steady state and dynamics of phase separation of active-particles: whichexperience the inhomogeneous activity on a two-dimensional substrate. Activity parameter is dis-tributed as Bivariate-Gaussian distribution at the center of the system, with standard deviation of *(size of the system). Unlike model with constant activity parameter we have observed that accu-mulation of particles with variable densities are formed. The accumulation increases by increasingthe intensity of distribution. In the region outside the distribution or away from the distribution,density phase separation is same as for passive model B. [16–18]We also estimated the growth of high density domains in the center of the box with time. Thedomain of high density in the region of high activity grow as a power law with time. The powerlaw exponent increases with increasing intensity of activity distribution.Hence our study gives an interesting steady state of density phase separation of active particles on ainhomogeneous patterned substrate and it can be useful to understand the trapping and transportof active particles in inhomogeneous systems. A detail understanding of density phase separationis required to understand the mechanism of density phase separation in the inhomogeneous system. VI. REFERENCES [1] J. T. Bonner, Proc. Natl. Acad. Sci. U.S.A. 95, 9355 (1998).[2] J. K. Parrish and W. M. Hammer (eds), Animal Group in Three Dimensions (Cambridge: CambridgeUniversity Press) (1997).[3] D. Chen , Y. Wang, G. Wu, M. Kang, Y. Sun, and W. Yu, Chaos29, 113118 (2019).[4] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, Madan Rao, and R. AditiSimha Rev. Mod. Phys. 85, 1143 (2013).[5] H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Lowen, and J.M Yeomans,Proc. Natl. Acad. Sci. U.S.A. , 109(36), 14313 (2012).[6] C. Dombrowski, Cisneros, S. Chatkaew, R. E. Goldstein, J. O. Kessler, Phys. Rev. Lett. 93, 098103(2004). [7] A. Sokolov, I. S. Aranson, J. O. Kessler, R. E. Goldstein, Phys. Rev. Lett. 98, 158102 (2007).[8] L. H. Cisneros, R. Cortez, C. Dombrowski, R. E. Goldstein, J. E. Kessler, Exp. Fluids 43, 753 (2007).[9] C. W. Wolgemuth, Biophys J 95, 1574 (2008).[10] M. E. Cates, Rep. Prog. Phys. 75, 042601 (2012).[11] Jonathan R. Howse, Richard A. L. Jones, Anthony J. Ryan, Tim Gough, Reza Vafabakhsh, and RaminGolestanian, Phys. Rev. Lett. 99, 048102 (2007)[12] S. J. Ebbens and J. R. Howse, Soft Matter 6, 738 (2010).[13] S. Thutupalli, R. Seemann and S. Herminghaus, New J. Phys. 13, 073021 (2011).[14] G. Volpe, I. Buttinoni, D. Vogt, H. K¨ummerer and C. Bechinger, Soft Matter 7, 8815 (2011).[15] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P. M. Chaikin, Science 339, 940 (2013).[16] Ken A Hawick, Daniel P Playne. Modelling and Visualizing the Cahn-Hilliard-Cook Equation, TechnicalReport CSTN-049, 155 (2008).[17] J. W. Cahn and J. E Hilliard. Chem. Phys. 28, 267 (1958).[18] J. W. Cahn and J. E. Hilliard J. Chem. Phys. 31, 688 (1959).[19] R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Nat.Comm. 5, 4351 (2014).[20] S. Pattanayak, R. Das, M. Kumar, and Shradha Mishra, Eur. Phys. J. E 42, 62 (2019).[21] P. Malgaretti and H. Stark, J. Chem. Phys. 146, 174901 (2017).[22] Bao-quan Ai, Qiu-yan Chen, Ya-feng He, Feng-guo Li, and Wei-rong Zhong Phys. Rev. E 88, 062129(2013).[23] P. Dolai, A. Simha, and S. Mishra, Soft Matter, 14(29), 6145 (2018).[24] S. Ramaswamy, Annual Review of Condensed Matter Physics, 1323 (2010).[25] J. P. Singh and S. Mishra, Physica A: Statistical Mechanics and its Applications 544, 123530, (2020).[26] Y. Fily and M. C. Marchetti, Phys. Rev. Lett. 108, 235702 (2012).[27] J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo and M. E. Cates, Phys. Rev. Lett. 111,145702 (2013).[28] Ivo Buttinoni, Julian Bialk´e, Felix K¨ummel, Hartmut L¨owen, Clemens Bechinger, and Thomas Speck,Phys. Rev. Lett. 110, 238301 (2013).[29] Quan-Xing Liu, Arjen Doelman, Vivi Rottsch¨afer, Monique de Jager, Peter M. J. Herman, Max Rietk-erk, and Johan van de Koppel, Proc. Natl Acad. Sci. USA 110, 11910 (2013).[30] Sanjay Puri, Vinod Wadhawan. Kinetics of Phase Transitions, CRC Press (2009).[31] J. Tailleur and M. E. Cates, Phys. Rev. Lett. 100, 218103 (2008).[32] M. E. Cates and J. Tailleur, Europhys. Lett. 101, 20010 (2013).[33] R Das, S Mishra, S Puri, EPL 121 37002 (2018).[34] A. Fick, Annalen der Physik 170, 59 (1855).[35] A. Paul, T. Laurila, V. Vuorinen, S.V. Divinski. Fick’s Laws of Diffusion. In: Thermodynamics, Diffu-sion and the Kirkendall Effect in Solids. Springer, Cham (2014).3