Physical Review E Rapid Communications 75, 040402(R) (2007)

Stripes, Clusters, and Nonequilibrium Ordering for Bidisperse Colloids with Repulsive Interactions

C. Reichhardt and C.J. Olson Reichhardt

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 12 October 2006; published 13 April 2007)

We show that two-dimensional bidisperse assemblies of colloids with strictly repulsive interactions exhibit stripe, cluster, and partially crystallized states when driven over a quenched random substrate. The nonequilibrium states on a substrate are significantly more ordered than equilibrium states both with and without substrates. A minimum substrate strength is necessary to induce the nonequilibrium pattern formation. Our results suggest that a combination of driving and quenched disorder offers a new approach to controlling pattern formation in colloid mixtures.


Understanding how assemblies of particles organize is an outstanding problem in both equilibrium and nonequilibrium systems. The ability to control pattern formation would have a profound impact on applications that use large-scale self-assembly to create specific pattern morphologies. In equilibrium, it is known that particles with competing repulsive and attractive interactions can organize into clusters, stripes, and spongelike textures [1]. Similar patterns can also arise for particles with strictly repulsive potentials that have a two step form [2]. Recently, it was shown in both simulations and experiments that a two-dimensional (2D) bidisperse assembly of colloids interacting via repulsive magnetic dipoles of two different strengths can cluster into spongelike patterns [3]. This suggests that other types of ordering are possible for bidisperse repulsively interacting particles.
Here, we demonstrate that a variety of distinct partially ordered states can occur for a bidisperse system of strictly repulsively interacting colloids driven over a random substrate that is sufficiently strong. There has been extensive work on the ordering of monodisperse charged particles moving over random quenched disorder for systems such as vortices in type-II superconductors [4,5,6], moving Wigner crystals [7] and colloids [8]. In the absence of quenched disorder, a monodisperse system forms a triangular lattice, while strong quenched disorder distorts the lattice and generates numerous topological defects. These defects can be partially annihilated and the order partially regained by driving the system into a nonequilibrium moving smectic state [4,5,6,7,8]. It is not known what type of nonequilibrium configurations would occur for two species of particles driven with identical force over quenched disorder. In contrast to the monodisperse case, for a bidisperse system even the equilibrium states in the absence of a random substrate are intrinsically disordered [3], so one might expect the addition of quenched disorder to further disorder the system. Instead, we find that for certain regimes, nonequilibrium bidisperse colloid assemblies moving over quenched disorder have more topological order than the corresponding equilibrium states, even those without quenched disorder. The quenched disorder must be sufficiently strong for the topological ordering to occur. We argue that for strong quenched disorder, significant plastic deformations of the driven colloid configuration occur due to the fact that the different species move at different average velocities close to the depinning threshold. We show that the system can organize into stripes with triangular ordering within each stripe. When the difference between the two species is small, a moving smectic state forms, whereas when the disparity is very large moving cluster states form. Our results are robust for a wide range of colloid densities, quenched disorder strengths, and system sizes.
We employ Brownian dynamics to simulate a 2D system of size L ×L containing N colloids and Np potential traps with periodic boundary conditions in the x and y directions. The overall density of the system is n=N/L2. The colloids interact via a screened Coulomb potential V(Rij) = (E0/Rij)exp(−κRij), where Ri(j) is the position of colloid i(j), Rij=|Ri−Rj|, E0 = Z*2/(4πϵϵ0a0), Z* is the unit of charge, ϵ is the solvent dielectric constant, and 1/κ is the screening length. The unit of distance in the simulation is a0, and unless otherwise noted, L=48a0. In this work we fix κ = 4a0, which is reasonable for experiments on colloids in nonpolar fluids [9]. Forces are measured in units of F0=E0/a0. The dynamics of colloid i are given by the equation of motion
η dRi
dt
 = −qi N
∑
j = 1 
 qj∇V(Rij) − Np
∑
k = 1 
 ∇Vp(Rik) + FD + FiT
(1)
where η is the damping term, qi(j) is the dimensionless charge of colloid i(j), Rik=|Ri−Rk|, and Rk is the position of trap k. In this model, hydrodynamic effects are neglected; such effects can be strongly screened in a system confined within 2D walls. To introduce bidispersity to the system, half of the colloids have charge qi=qA and the other half have qi=qB, where we fix qB=1. The quenched random substrate is modeled by randomly distributed parabolic traps of density np and radius rp=0.1a0 with Vp(Rik)=−(Fp/2rp)(Rik−rp)2 for Rik ≤ rp and zero interaction for Rik > rp. Here Fp is the maximum pinning force. This model for quenched random disorder has given results comparable with other models in monodisperse particle systems. The externally applied driving force is identical for all particles and is given by FD=FD∧x. A uniform drive of this type could be created electrophoretically [10]. Thermal effects are modeled by random Langevin kicks with 〈FTi〉 = 0 and 〈FTi(t)FTi(t′)〉 = 2ηkBT δijδ(t − t′). The initial colloid configurations are obtained using simulated annealing. We then set FT=0 and gradually increase FD in increments of δFD=0.001 every τ = 5000 simulation time steps. After each drive increment, once the system has reached a stationary state we measure the average velocity 〈Vx〉 = 〈 (1/N)∑i=1Nvi ·∧x〉, where vi is the velocity of colloid i. The depinning threshold FC corresponds to the value of FD at which 〈Vx〉 = 0.04F0. As an example, for a0 = 0.6μm, ϵ = 2, and Z* = 300e, F0 = 27.8 pN.
Fig1.png
Figure 1: A bidisperse system of colloids with qA/qB=3 and n=0.385/a02. (a,c) Real space colloid configurations. Small circles: species A; large circles: species B. (b,d) The corresponding structure factor S(k). (a,b) Equilibrium state with no pinning or driving. (c,d) Moving stripe state for Fp = 1.0, np=4.0/a02, and FD/FC = 3.0.
We first consider a system where the ratio between the charges of the two colloid species is qA/qB = 3. In the absence of quenched disorder, the colloids form a disordered mixed assembly after annealing, illustrated in Fig. 1(a). To characterize the configuration, we determine the structure factor, S(k)=(1/N)|∑i = 1Nexp(−ik·Ri)|2. In Fig. 1(b), S(k) for the equilibrium case with N=864 colloids at density n=0.385/a02 shows a ring structure which is characteristic of disordered systems. We also analyze the fraction of six-fold coordinated colloids P6 obtained from a Voronoi construction, P6=(1/N)∑i=1Nδ(zi−6), where zi is the coordination number of colloid i. In a triangular lattice all the colloids have zi=6, giving P6 = 1.0. For the configuration shown in Fig. 1(a), P6=0.41, indicating that a large fraction of the colloids have zi ≠ 6. If we anneal in the presence of quenched disorder, we find a similar disordered state as measured by S(k) and P6. In general, for stronger quenched disorder the system becomes more disordered, producing increased smearing in the ring structure of S(k) and reducing P6 to P6=0.3.
Fig2.png
Figure 2: (a) Average velocity 〈Vx〉 vs applied drive FD for the system in Fig. 1. Light upper line: sample with Fp=0; dark lower line: sample with Fp=1.0; dashed line: d〈Vx〉/dFD for the Fp=1.0 case. (b) P6 vs FD for the same system. Upper curve: Fp = 1.0; lower curve: Fp = 0.
In Fig. 2(a) we plot the dynamic response of the system given by the average colloid velocity 〈Vx〉 versus external drive FD for Fp=0 (light line), and for a system with traps of strength Fp=1.0 and density np=4.0/a02 (heavy line). For Fp = 0, the velocity response is strictly linear. In the presence of traps, there is a clear depinning threshold FC=0.7 for colloid motion followed by a nonlinear regime which crosses over to an ohmic regime at higher FD. For Fp=1, Fig. 2(a) shows that d〈Vx〉/dFD has a peak in the nonlinear regime at FD=0.75 and flattens in the ohmic region for FD > 2.0. These features in the velocity-force curves are the same as those previously observed for monodisperse systems with quenched disorder [5,7,8], where the onset of the ohmic response was correlated with a dynamical reordering of the system into a moving smectic state.
In Fig. 2(b) we plot the corresponding P6 versus FD curves for the systems with and without quenched disorder. For Fp=0, P6=0.41 for all FD. For Fp=1.0, we still find P6=0.41 in the pinned region at FD/FC < 1. Above the depinning transition, P6 increases over the same range of FD where there is a nonlinear response in the velocity force curves, as seen in Fig. 2(a). Here, the system moves plastically with portions of the colloids remaining temporarily trapped while other colloids move around the trapped colloids. For FD > 2.0, P6 saturates at P6 ≈ 0.7. This result indicates that the moving system with quenched disorder is more ordered than the equilibrium system without quenched disorder. To illustrate the nature of this order, in Fig. 1(c) we show the colloid configuration for the system with Fp=1.0 at FD/FC=3.0, and we plot the corresponding S(k) in Fig. 1(d). Here a stripe ordering occurs where the species have partially segregated. Complete segregation is prevented since this would produce a strong charge inhomogeneity in the system. Local triangular order appears within each stripe. The structure factor has a smecticlike form, indicative of the presence of stripes, as well as second order peaks at smaller scales resulting from the in-stripe triangular ordering.
Fig3.png
Figure 3: (a) The different regimes for the system in Fig. 1 at n=0.385/a02 and np=4.0/a02 as a function of FD and Fp. I: pinned regime; II: moving disordered regime; III: plastic flow regime; IV: moving stripe regime, as in Fig. 1(c). Circles: the depinning threshold FC. Squares: stripe formation threshold. (b) The different regimes for FD vs density n at fixed Fp = 1.0 and np=4.0/a02. Symbols are the same as in (a). (c) P6 versus Fp for n=0.385/a02 and np=4.0/a02 at FD/FC = 3.0. Inset: P6 versus pinning density np for fixed Fp = 1.0 and n = 0.385/a02 at FD/FC=3.0. (d) P6 versus n for fixed Fp=1.0 and np=4.0/a02 at FD/FC = 3.0. Inset: P6 versus system size L for fixed Fp = 1.0, n=0.385/a02, np=4.0/a02, and FD/FC=3.0.
We next show that sufficiently strong quenched disorder must be present for the stripe patterns to occur. In Fig. 3(a) we outline the four distinct regimes that we find as a function of driving force FD and pinning force Fp for a system with n=0.385/a02 and np=4.0/a02. At all Fp, the colloids are pinned in region I for FD < FC. As FD is increased above FC, we find that for Fp < 0.4, the system never orders into a stripe state. Instead, the disordered mixture depins with little or no plastic deformation, and the moving state, marked region II in Fig. 3(a), retains the same disordered configuration as the pinned state. We have tested different values of δFD and find no change in the location of region II. For Fp ≥ 0.4, a plastic flow state (region III) appears above depinning. Here the colloids are repeatedly trapped and escape from the traps but stripes do not completely form. Finally, for high enough values of Fp and FD we find region IV, the moving stripe state. In Fig. 3(c) we plot P6 as a function of Fp for the same system in Fig. 3(a) at fixed FD/FC = 3.0. For Fp > 0.4, P6 saturates near 0.7 and the moving stripe state forms. In contrast, for Fp < 0.4 the moving state has P6 = 0.41, the same value measured in the pinned state. This indicates that there is a critical disorder strength that is necessary for the formation of the stripe state to occur.
In Fig. 3(b) we plot the locations of the four regimes as a function of FD versus colloid density n for fixed Fp=1.0 and np=4.0/a02. The depinning threshold FC marking the end of the pinned region I drops to lower values of FD as n increases since the colloid-colloid interactions become stronger relative to Fp, making the traps less effective at pinning the colloids. A reordering into the moving stripe state (region IV) occurs only for n ≥ 0.2/a02; for smaller values of n, we find the moving disordered region II at high drives. This is confirmed by Fig. 3(d), which shows P6 versus n for the same system in Fig. 3(b) at fixed FD/FC=3.0. For n < 0.2/a02 in the moving region, P6  ∼ 0.4. Here the system does not reorder since at low densities the colloids are far enough apart that they interact only weakly and remain in a disordered state. The pinning density np also plays a role in determining whether the moving stripe state can form. In the inset of Fig. 3(c) we plot P6 versus np for a system with fixed Fp=1.0 and n=0.385/a02 at FD/FC=3.0. For np < 0.5/a02, P6 remains low, indicating that the system does not reorder into the stripe state. This result shows that a critical amount of quenched disorder, as well as a critical strength of disorder, is necessary for the moving system to form the stripe state. To show that there are no finite size effects for the appearance of the four regions, in the inset of Fig. 3(d) we plot P6 vs system size L at FD/FC=3.0 in the moving stripe state. There is a slight increase in P6 for the smallest values of L since the small systems form fewer stripes, resulting in a lower number of topological defects at the interfaces between any two stripes. We find that P6 saturates for L > 20, well below the size studied throughout this work.
The moving stripe state forms when the quenched disorder is strong enough to induce plastic deformations in the colloid configuration. The effective pinning force from the substrate decreases for increasing colloid-colloid interactions in a monodisperse system. In a bidisperse system, local clustering of colloid species occurs, as seen in experiments [3]. Clusters of the weakly charged colloid species B experience a stronger effective pinning force than clusters of the more strongly charged colloid species A. If the quenched disorder is sufficiently strong and dense, the two species will move at different average velocities even though they both couple to the external drive in the same way. Stripes form as the faster moving species begins to collect together. In a co-moving reference frame, this state resembles a system of bidisperse colloids moving in opposite directions, where a laning instability has been shown to occur [11]. In Ref. [11], the two species were driven in opposite directions with no quenched disorder; here, the disparity in velocity between the two species is induced entirely by the quenched disorder. If the quenched disorder is weak, the colloid configuration moves elastically and no velocity difference between the two species occurs, so that no stripes can form. We have verified that the average velocity distribution for the two different species (not shown) is different in region III but the same in region II. The stripe phases show less ordering than would occur in a system with only a single species since topological defects form at the boundaries between the phases.
Fig4.png
Figure 4: (a) P6 versus qA/qB for Fp = 1.0, n = 0.385/a02, np=4.0/a02, and FD/FC = 3.0. MS: moving smectic region; IV: moving stripe (region IV); MC: moving clump region. (b) Real space colloid configuration for the moving clump state at qA/qB = 11. Small circles: species A; large circles: species B. (c) The corresponding structure factor S(k).
We now address the effect of the polydipersity on the moving regimes by varying qA/qB. A system with the same parameters as in Fig. 2(a) and Fp = 1.0 acts like a monodisperse sample for 1 < qA/qB < 1.3, where it reorders into a moving smectic (MS) state as observed in previous simulations [5,7,8]. For 1.4 < qA/qB <~10 the system forms a moving stripe state (region IV), and for qA/qB >~10 we observe a moving clump (MC) state. The onset of these different states appears in Fig. 4(a) where we plot P6 versus qA/qB at FD/FC=3. In the moving smectic state, P6 ≈ 0.82, in the moving stripe state P6 ≈ 0.7, and in the moving clump state P6 is sharply reduced. The real space configuration of the moving clump state is illustrated in Fig. 4(b), and the corresponding structure factor S(k) in Fig. 4(c) has a multiple ring structure characteristic of a disordered system containing more than one length scale. Here, the different length scales are associated with the triangular ordering of the higher charge qA species and the interclump distance. Similar to the moving stripe state, the moving clump state requires sufficient quenched disorder to form. As qA/qB increases, the size of the individual clumps decreases.
In summary, we have shown that a bidisperse system of repulsively interacting colloids driven over quenched disorder can form moving smectic, moving stripe, and moving clump states. The moving states are in general more ordered than the equilibrium states that form in the absence of quenched disorder, in contrast to a monodisperse system. The stripe state arises when the disorder is strong enough to induce different average velocities for the two species, leading to a laning instability similar to that seen for particles driven in opposite directions. In our system, the velocity difference is due entirely to the quenched disorder, and not to different drives on the different colloids. Physical systems in which this model could be realized include driven bidisperse colloidal assemblies with charge or magnetic interactions driven over optical pinning or a rough wall, electron systems with coexisting single and multiple electron bubble states, and superconductors with mixtures of Abrikosov and Josephson vortices.
This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

References

[1]
M. Seul and D. Andelman, Science 267, 476 (1995).
[2]
G. Malescio and G. Pellicane, Nature Mater. 2 97 (2003).
[3]
N. Hoffmann et al., Phys. Rev. Lett. 97, 078301 (2006).
[4]
L. Balents, M.C. Marchetti, and L. Radzihovsky, Phys. Rev. Lett. 78, 751 (1997); P. Le Doussal and T. Giamarchi, Phys. Rev. B 57, 11356 (1998).
[5]
A.E. Koshelev and V.M. Vinokur, Phys. Rev. Lett. 73, 3580 (1994); K. Moon, R.T. Scalettar, and G.T. Zimányi, ibid. 77, 2778 (1996); C.J. Olson, C. Reichhardt, and F. Nori, ibid. 81, 3757 (1998); A.B. Kolton, D. Domínguez, and N. Grønbech-Jensen, ibid. 83, 3061 (1999).
[6]
F. Pardo et al., Nature (London) 396, 348 (1998).
[7]
C. Reichhardt et al., Phys. Rev. Lett. 86, 4354 (2001).
[8]
J. Chen, Y. Cao, and Z. Jiao, Phys. Rev. E 69, 041403 (2004).
[9]
M.F. Hsu, E.R. Dufresne, and D.A. Weitz, Langmuir 21, 4881 (2005).
[10]
N. Garbow et al., J. Phys.: Condens. Matter 16, 3835 (2004); J. Kim et al., Langmuir 21, 10941 (2005); F. Strubbe, F. Beunis, and K. Neyts, J. Colloid Interf. Sci. 301, 302 (2006).
[11]
J. Dzubiella, G.P. Hoffmann, and H. Löwen, Phys. Rev. E 65, 021402 (2002); J. Chakrabarti, J. Dzubiella, and H. Löwen, ibid. 70, 012401 (2004).


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