Pattern formation in particle systems driven by color field

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Particuology 6 (2008) 515–520

Pattern formation in particle systems driven by color field Xiaoxing Liu a,b,1 , Huabiao Qi a,b , Wei Ge a,∗ , Jinghai Li a a

State Key Lab of Multi-phase Complex Systems, Institute of Processing Engineering, Chinese Academy of Sciences, PO BOX 353, Beijing 100080, China b Graduate University of the Chinese Academy of Sciences, Beijing 100049, China Received 10 June 2008; accepted 15 July 2008

Abstract The structural evolution of systems with two kinds of particles driven in opposite directions, i.e., driven by a color field, is investigated by molecular dynamics simulations. Gaussian thermostat, a common treatment to restrict the thermal velocity of the particles in the systems, has been used so as to account for the dissipation of heat and allow the system to reach a steady state. It has been found that with the increase of the strength of driving force (F), the system undergoes an obvious structural transition from an initially random mixing state to a state characterized by separate lanes and in each lane only one kind of particles exists. The analysis shows that the reason for the formation of lane structure is not only the increase of F but also the variation of particle friction coefficient. While using Gaussian thermostat the particle friction coefficient becomes a function of F. Increasing F leads to high particle friction coefficient and inevitably results in lane formation for strong enough driving force. When lifting the effect of F on friction coefficient and choosing a constant friction coefficient, our results show that for a given F there always exists a critical value of friction coefficient higher than which the system will develop into lane structure. © 2008 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. Keywords: Color field; Molecular dynamics (MD); Simulation; Lane formation; Driven system

1. Introduction When external driving force is exerted on particle systems, rich structural variations such as segregation, jamming, surface wave, etc. will occur (Aranson & Tsimring, 2006). In recent years many works have been done about such structural transitions (Schmittmann & Zia, 1995; Marro & Dickman, 1999). In particular, systems with two different kinds of particles were extensively studied, which are driven by external force in opposite directions. Clustering and lane structure, etc. have been found (Aertsens & Naudts, 1991; Schmittmann, Hwang, & Zia, 1992; Vilfan, Zia, & Schmittmann, 1994). Variations in the strength of the driving force and/or particle density are found to induce structural transitions from initial homogeneous mixed state to spatially inhomogeneous segregation structure. Typical examples have been found in binary mixtures of “pos∗

Corresponding author. Tel.: +86 10 8261 6050; fax: +86 10 6255 8065. E-mail addresses: [email protected] (X. Liu), [email protected] (W. Ge). 1 Current address: Laboratoire GPM2, Institut National Polytechnique de Grenoble, UMR CNRS 5010, ENSPG, BP46, 38402, Saint Martin d’Heres, France.

itively and negatively charged” colloidal particles driven by an external field, where the same charged particles will separate into several lanes if the intensity of driving field exceeds a critical value (Dzubiella, Hoffmann, & Löwen, 2002; Löwen & Dzubiella, 2003; Chakrabarti, Dzubiella, & Löwen, 2004). Such laning phenomenon of Brownian particles has also been observed in experiments recently (Leunissen et al., 2005). The simulation results also reveal a reentrance effect in lane formation (Chakrabarti et al., 2004), that is, with increasing particle density under a fixed strong driving field, there is first a transition towards lane formation which is followed by another transition back to a state with no lanes. In most of these simulations the increase of the intensity of driving force (F) is assumed not to strengthen the mechanical noise of system. In colloid system particle inertia is ignored and the amplitude of random forces is constant and is only a function of the friction coefficient and the temperature (Dzubiella et al., 2002; Chakrabarti, Dzubiella, & Löwen, 2003; Löwen & Dzubiella, 2003). In other particle systems driven by external color field Gaussian thermostat has been used and particle friction coefficient is correlated to F (Evans, Lynden-Bell, & Morriss, 1989; Hoover, Boercker, & Posch, 1998; Jepps & Petravic, 2004). It is reported that ordered structure in driven

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doi:10.1016/j.partic.2008.07.021

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X. Liu et al. / Particuology 6 (2008) 515–520

Nomenclature e F m N r S t T v v¯ VD VD,S VD

direction of driving force value of driving force particle mass number of particles inter-particle distance area of simulated system time average kinetic temperature particle velocity spatially averaged velocity drift velocity instant drift velocity temporal fluctuation of VD,S

Greek letters ε characteristic energy of the pair potential ζ friction coefficient ρ particle number density σ particle exclusion diameter φ interaction potential

systems is associated with small noise-to-field ratios (Helbing, Farkas, & Vicsek, 2000) and friction coefficient also has significant effect on lane formation (Santra, Schwarzer, & Herrmann, 1996). Recently we have simulated a particle system driven by color field where particle friction coefficient is chosen as a constant (Liu, Ge, & Li, 2008). Our simulation results show that the system’s structural transitions are different from that occurring in systems with constant mechanical noise or systems controlled by Gaussian thermostat. For a given particle number density, ordered lane structure can only be formed in a special region of F. A different reentrance phenomenon in the lane formation from that occurring in colloid system has also been found: for a low F, with the increase of particle number density, more than two lanes parallel to the driving forces are observed first, followed by a disordered phase with different kinds of particles blocking each other, and then an ordered state with all particles separating into two demixed lanes. The aim of this work is to show that, for a particle system driven by color field, Gaussian thermostat will result in systematic occurrence of lane formation. The reason is that increasing field intensity F will also increase particle friction coefficient, which will restrict the amplitude of particle velocity fluctuations. By choosing constant friction coefficients our simulation results show that the strength is not the only controlling factor leading to the occurrence of lane structure. In fact, the structure in the final steady state of the system is the result of a cooperative effect of external color field and particle friction coefficient. 2. Simulation method We consider a 2D system containing equal number of A and B particles (NA = NB = N/2) in a square area S surrounded by

periodic images of itself. The particle number density is defined as ρ = N/S. The particles interact with a soft potential (Evans et al., 1989) φij (r) = ε((σ/rij )12 − (σ/rcut )12 ) (rij < rcut ),

(1)

φij (r) = 0

(2)

(rij ≥ rcut ),

where r is the inter-particle distance. Normalized values are used throughout this paper by choosing the characteristic energy of the pair potential ε, the particle exclusion diameter σ and particle mass m as the units of energy, length and mass, respectively. All simulations are performed for S = 900, ρ = 0.5 and rcut = 1.5σ. We denote the location of particle j at time t by rj (t) and its velocity drj (t)/dt by vj (t). The equation of motion for particles reads  dvj  r = −∇ φkj (r) + F ej − ζvj . dt

(3)

k= / j

The right hand side includes all forces acting on the particles, namely the inter-particle potential force, the constant driving forces and the friction force. The driving field is symmetric with F = FA = FB > 0. If particle j belongs to species “A”, it is driven in the direction ej = (1, 0) and otherwise in the direction ej = (−1, 0). Except the applied directions of the driving field, the two types of particles have no difference. The friction force in Eq. (3) is treated in two ways in our simulations. Gaussian thermostat is adopted firstly where ζ is the thermostatting multiplier given by Gauss’s principle of least constraint, that is, N

ej j=1 F 

ζ = N

j j=1 v

· vj

· vj

.

(4)

In this expression, ζ is a function of the field intensity F, which leads to lane structure when the F exceeds a critical value (Evans et al., 1989; Hoover et al., 1998; Jepps & Petravic, 2004). Constant ζ values are then adopted to examine directly the effect of ζ on the structural transition in the system. The equations of motion are solved by a Verlet scheme with an adaptive time step ensuring that no particle moves more than 0.001σ per step under the maximum theoretical velocity. No significant change is seen when finer time steps are used, and it is, therefore, small enough. We have also checked that the results are independent of the size of the simulated area. For every F, particles are initially distributed randomly. To relax the system, each particle is assigned a high random velocity initially which is annealed gradually to zero before the driving field is applied and then kept constant. Each simulation is conducted typically for 10,000 time units while the statistics is usually gathered in the last 5000 time units. To quantify the structural transition, in our simulations we introduce a particle transportation property, that is, the drift velocity. It is a scalar representing the average magnitude of the velocity component of each particle in its driving force direction. In the sense of spatial average, an instant drift velocity can

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Fig. 1. The drift velocity and its standard deviation versus the applied field, using Gaussian thermostat.

be defined as VD,S (t) =

N → → rj (t + t/2) − − rj (t − t/2)) · e៝ j 1  (− , N t

(5)

j=1

where the time interval t is typically 1000 time steps. The temporal fluctuation of VD,S will be significant when the system cannot evolve into lane structure, therefore, to better characterize this transport property, the drift velocity is defined as   VD = VD,S (t) , (6) where   means time average over the steady period of the simulation. To account for the temporal fluctuation of VD,S we also computed its standard deviation VD , that is, (VD )2 =

(VD,S (t) − VD )2  . VD2

(7)

It has been demonstrated that the standard deviation of drift velocity is an effective index of the lane structure (Liu et al., 2008). When the system develops into stable lane structure, only few particles can be affected by the other species and the flow is steady. In this case, VD is close to zero. And when random motion or jamming happens, the alternation of blocking and free motion brings remarkable variation of VD,S with time and VD is notably higher than zero. 3. Simulation results and discussion We first present the results obtained by using Gaussian thermostat. The variations of the drift velocity VD and its standard deviation VD are plotted against the field intensity F in Fig. 1. Two different stages can be found from Fig. 1. The first stage takes place for low values of the field (F ≤ 20.0). In this regime, VD increases with the applied field and its standard deviation VD is obviously larger than zero. Examining the snapshots, it is found that steady lane structure cannot be formed in this regime. Depending on the field intensity, the system may be either in random distribution or in jamming. For F < 10.0, the system is in random distribution state, see Fig. 2(a). In this region VD decreases with the increase of F. Increasing F continuously, the system develops into jamming state, as shown in Fig. 2(b). In this region, VD is nearly maintained at a constant value with the increase of F. Since VD increases

Fig. 2. Typical snapshots from the simulations by using Gaussian thermostat: (a) F = 2.0; (b) F = 15.0; (c) F = 25.

with the increase of F, constant VD also means the jamming will become more and more serious with the increase of field intensity. When F is larger than 20.0, the system develops into ordered state, i.e., the system segregates into several lanes of identical particles, each lane moves along the direction of applied field at a constant velocity, see Fig. 2(c). In this regime, the strength of applied field has no effect on the value of the drift velocity and

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Fig. 3. Kinetic temperature versus the driving field, using Gaussian thermostat.

Fig. 4. Variation of friction coefficient ζ with F when Gaussian thermostat used.

VD is maintained at a constant value, at the same time VD is equal to zero. We also examine the variation of system temperature versus field intensity. The system temperature is defined as

reflects the dispersion of the particle velocity vectors relative to their instant (that is, spatial only) average v¯ A(B) (t), while VD reflects that of the instant average of the scalar velocity components in the driving force directions VD,S (t) relative to their time average VD . Therefore, T may characterize the noise better, that is, the randomness of individual particle motion, while VD is more suitable for characterizing the collective fluctuations in the systems, such as jamming and laning. Fig. 3 shows the variation of T with F. It can be found that the value of T remains constant when the system is in a random mixing state. Constant T means that the increase of F has no effect on mechanical noise. T decreases with the increase of F when the system is in jamming state. This stems from the fact that jamming would limit particle random motion, i.e., the mechanical noise T. T finally approaches zero when the system develops into lane structure. From Fig. 3 it can be inferred that increasing

T =

(TA + TB ) , 2.0

where

(8)



TA =

NA ¯ A (t))2 j=1 (vj,A (t) − v

NA v¯ A (t) =

2NA

j=1 vj,A (t)

NA

,

 ,

(9) (10)

with TB defined similarly. Eq. (9) has utilized m = 1 and the Boltzmann constant is set to 1. Note that, for each species, T

Fig. 5. Typical snapshots for different values of ζ under given F: 2.0 (top) and 25.0 (bottom). (a) ζ = 0.09; (b) ζ = 6.5; (c) ζ = 30.0; (d) ζ = 1.5; (e) ζ = 4.5; (f) ζ = 7.5.

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Fig. 5 presents the typical snapshots. It can be found that, by continuously increasing ζ under these two given F, the system starts with a random state, passes through a jamming state, and finally develops into ordered lane structure. Fig. 6 shows the corresponding variations of VD and T versus ζ. Both VD and T decrease with the increase of ζ and finally approach zero when lane structure has formed, i.e., at the stage of ζ ≥ 15.0 for F = 2.0 and ζ ≥ 6.0 for F = 25.0. These results mean that particle friction coefficient is one of the factors affecting the final structure of the system. For a given field intensity there should exist a critical value of friction coefficient larger than which the system can evolved into lane structure. The simulation results also show that the ratio of field intensity to friction coefficient cannot determine the final structure. From Fig. 6 it can be found that lane formation occurs at different ratio of F to ζ(2.0/15.0 and 25.0/6.0), which means that the ratio of F to ζ cannot decide the system structure uniquely. 4. Conclusions Fig. 6. The variations of VD and T versus ζ: (a) F = 2.0; (b) F = 25.0.

field intensity will strengthen particle directional motion, which will result in lane formation. This conclusion seems to be contrary to our former result, that is, increasing field intensity would strengthen particle random motion (Liu et al., 2008). In fact, the difference stems from the fact that, when Gaussian thermostat is adopted, particle friction coefficient, i.e., the thermostatting multiplier ζ in Eq. (3), is correlative to field intensity F. Fig. 4 shows the variation of ζ with F in the steady state. It can be found that ζ will also increase with the increase of F. Particle friction coefficient will have significant effect on system’s final structure since it can restrict particle velocity fluctuation. Because of the inter-particle actions, particles can experience displacement along y-direction. Since there is no driving force acting on the particles in this direction, the friction coefficient determines the particle maximal displacement in y-direction. For a given y-direction velocity, small ζ means that particle velocity deviating from the direction of the applied field cannot be attenuated quickly and the displacement in y-direction is obviously larger than that for higher ζ, which will lead to high noise–field ratio. Thus the system is difficult to develop to a steady lane structure. We argue that adopting Gaussian thermostat is not favorable to explore the mechanism of lane formation in a physical system. It is hard to know which factors can affect the final structure of the system. Is it the strength of external driving field, or particle friction coefficient, or their ratio? For examining the possible factors affecting the system structure, the effect of external field on friction coefficient must be considered separately. In order to explore the effect of the friction coefficient on the final structure independently, we have simulated cases where Gaussian thermostat was replaced by a constant coefficient. Several field intensities have been considered but only two field intensities are discussed in this paper (since the phase transition under these field intensities are the same), i.e., F = 2.0 and 25.0.

In this work, we show that both the intensity of external driving field and particle friction coefficient can affect the phase transition in the system, and the use of Gaussian thermostat in simulations will systematically result in lane formation for particle systems driven by color field. The reason is as follows. When Gaussian thermostat is used, though the increase of field intensity will strengthen particle random motion and then destroy the ordered lane structure, it also increases particle friction coefficient which will restrict particle irregular motion, attenuate the mechanical noise of the system and thus lead to the formation of lane structure. Our simulation results show that the latter is more remarkable. Thus, when Gaussian thermostat is used, there always exists a critical field intensity larger than which lane structure can always be formed. Based on the simulation results it seems that when the mechanism of the formation of lane structure is explored, the effect of external field on particle friction coefficient should not be introduced without a proper physical mechanism. Both our former (Liu et al., 2008) and present results show that at this stage for a given constant particle friction coefficient lane structure can only be formed under low field intensity. Acknowledgements The authors are grateful to the supports on this work from Nation Natural Science Foundation of China under the grants Nos. 20336040, 20490201 and 20221603, and the Chinese Academy of Sciences under the grants KJCX-SW-L08 and KJCX3-SYW-S01. References Aertsens, M., & Naudts, J. (1991). Field-induced percolation in a polarized lattice gas. Journal of Statistical Physics, 62(3/4), 609–630. Aranson, I. S., & Tsimring, L. S. (2006). Patterns and collective behavior in granular media: theoretical concepts. Reviews of Modern Physics, 78, 641–692. Chakrabarti, J., Dzubiella, J., & Löwen, H. (2003). Dynamical instability in driven colloids. Europhysics Letter, 61(3), 415–421.

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