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A.
Libál
*^{a},
S.
Stepanov
^{b},
C.
Reichhardt
^{c} and
C. J. O.
Reichhardt
*^{c}
^{a}Mathematics and Computer Science Department, Babes-Bolyai University, Cluj 400084, Romania
^{b}Physics Department, Babes-Bolyai University, Cluj 400084, Romania
^{c}Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. E-mail: cjrx@lanl.gov; Tel: +1 505 665 1134

Received
3rd August 2023
, Accepted 29th September 2023

First published on 29th September 2023

We consider a two-dimensional system of elongated particles driven over a landscape containing randomly placed pinning sites. For varied pinning site density, external drive magnitude, and particle elongation, we find a wide variety of dynamic phases, including random structures, stripe or combed phases with nematic order, and clogged states. The different regimes can be identified by examining nematic ordering, cluster size, number of pinned particles, and transverse diffusion. In some regimes we find that the pinning can enhance the particle alignment, producing a nonmonotonic signature in the nematic ordering with a maximum at a particular combination of pinning density and drive. The optimal nematic occurs when a sufficient number of particles can be pinned, generating a local shear and leading to what we call a combing effect. At high drives, the combing effect is reduced when the number of pinned particles decreases. For stronger pinning, the particles form a heterogeneous clustered or clogged state that depins into a fluctuating state with high diffusion.

A wide variety of systems can be modeled effectively as an assembly of interacting particles driven over random quenched disorder,

In most studies for driven particles over quenched disorder, the particle–particle interactions are isotropic; however, there are also numerous examples of systems where the particles have anisotropic interactions or are elongated, such as the rods or ellipses found in different kinds of granular matter,^{28–34} colloidal systems,^{35–39} and active matter systems.^{40–43} Anisotropic interactions can also arise in systems with longer range interactions such as magnetic colloids in tilted fields,^{44,45} superconducting vortex or electron liquid crystals,^{46–50} and certain types of magnetic skyrmion systems.^{51,52} Despite the number of rod-like particles or assemblies with one-dimensional anisotropy that have been realized, there are almost no studies of what happens when such systems are driven over quenched disorder, although there has been work on the diffusive motion of stiff filaments through disordered backgrounds.^{53–56}

In this work, we consider a two-dimensional system of elongated particles modeled as five to nine connected disks that also interact with N_{p} randomly placed pinning sites. We apply an increasing driving force and measure the number of pinned particles, the number of particles in the largest cluster, the nematic alignment, and the transverse diffusion. We find a wide variety of dynamic phases, including a random phase at low pinning densities and several varieties of what we call combed phases at intermediate densities, where a portion of the particles are pinned and produce local shearing of the mobile particles, resulting in the emergence of alignment with the driving direction. At higher drives where all the particles are moving, both the combing effect and the smectic ordering are reduced. For high pinning densities, a clogged or arrested phase appears in which the particle density is heterogeneous. This clogged state depins into a disordered phase with intermediate nematic order. Another interesting effect we observe is that the nematic ordering is strongly nonmonotonic as a function of pinning density and drive, with random flow at low pinning density, maximal nematic ordering for intermediate pinning densities and drives, and the reemergence of disorder at high pinning densities.

To obtain the interaction between two particles i and j, we calculate the pairwise repulsion f^{(i,α;j,β)}_{dd} where α ranges over all disks belonging to particle i and β ranges over all disks belonging to particle j. These interactions are given by a short-range stiff harmonic spring, so we obtain where R_{(i,α;j,β)} = r_{i,α} − r_{j,β}, R_{(i,α;j,β)} = |R_{(i,α;j,β)}|, _{(i,α;j,β)} = R_{(i,α;j,β)}/R_{(i,α;j,β)}, Θ is the Heaviside step function, and the elastic constant k = 10.0. Each of the constituent disks also interacts with a substrate modeled as N_{p} randomly placed truncated parabolic attractive sites of range R_{p} = 0.5 and maximum strength F_{p} = 1.0: , where R_{(i,α;k)} = r_{i,α} − r^{(p)}_{k}, R_{(i,α;k)} = |R_{(i,α;k)}|, and _{(i,α;k)} = R_{(i,α;k)}/R_{(i,α;k)}. The radius of the pinning site is chosen to be small enough to prevent multiple disks from being trapped by a single pinning site. Each disk is also subjected to an external driving force f_{ext} = F_{D}. The pinning forces and driving forces are applied at the center of each individual disk.

To update the position and orientation of a particle, we determine the forces and torques exerted on the center of mass of the particle by the constituent disks,

f^{α}_{i} = f^{(i,α)}_{dd} + f^{(i,α)}_{pin} + f_{ext} | (1) |

(2) |

(3) |

Next we update the center of mass and angular orientation of the particle according to:

R_{i}(t + 1) = R_{i}(t) + f_{i}Δt/η | (4) |

θ_{i}(t + 1) = θ(t) + τΔt/η | (5) |

We initialize the system by placing randomly oriented elongated particles at randomly chosen locations subject to the constraint that there is no overlap between constituent disks belonging to two different particles. Our desired disk density is ϕ = 0.4, but the constraint method gives a maximum possible density below this value. Thus, in order to prepare denser samples, we start from a lower density constrained state, allow it to evolve for a period of time under the equations of motion described above, and then insert additional particles into the free spaces created by the tendency of the particles to cluster. For example, in the case of n = 5, we initialize the sample with N = 600 particles and then introduce an additional 30 particles after each t = 100000 simulation time steps until we reach the desired density ϕ ≃ 0.4 with N = 1200. To obtain the same density in all systems, for n = 7 we use a total of N = 900 particles, and for n = 9 we use N = 760. We do not perform any measurements until after all particles are present in the sample and have been given a chance to move under the applied drive for 4 × 10^{6} simulation time steps. After this time interval, when the system has reached a steady state, we take measurements during the next 4 × 10^{6} simulation time steps. The maximum distance traveled by the particles during this time interval depends upon the value of F_{D} and ranges from 0.25S_{x} for small F_{D} values to 8.75S_{x} for the largest values of F_{D} that we consider. The number of pinning sites in the sample ranges from N_{p} = 0 to 600, and the external drive ranges from F_{D} = 0.01 to 0.5.

One possible physical realization of the system we consider would be to use appropriately shaped colloidal particles in the low Reynolds number limit moving over trapping sites such as the gravitational potential traps employed in ref. 57 For such a system, our dimensionless simulation units would map to physical units as follows: the length scale R_{d} = 1.0 becomes 5.15 μm, the force scale F_{p} = 1.0 becomes 1.758 pN, and the viscosity coefficient η = 1.0 becomes 1.729 pN s μm^{−1}. The simulation time unit is given by t_{0} = ηR_{d}/F_{p}, which for the example above becomes t_{0} = 5.06 s, giving Δt = 5.06 ms.

Fig. 2(d) shows Phase IV or the tooth combed state at N_{p} = 30 and F_{D} = 0.07. There are now some permanently pinned particles and the pinning density is large enough to induce strong nematic ordering. In the combed channel flow phase VI, illustrated in Fig. 2(e) at N_{p} = 600 and F_{D} = 0.12, a number of particles are pinned and there is some local nematic ordering, but the flow occurs plastically through channels. Fig. 2(f) shows phase VII or the clogged state at N_{p} = 600 and F_{D} = 0.02, where now most of the particles are permanently pinned in local clusters, giving a heterogeneous state composed of coexisting regions of high density and low density, similar to the clogged states observed for monodisperse individual disks moving through random obstacle arrays.^{58} For high N_{p} and high F_{D}, all of the particles are moving and we observe phase V (not shown) where the particle positions are uniformly random, similar to phase I, but due to the collisions with the pinning sites, the particles gradually diffuse with respect to each other. For phase I at low N_{p}, there is almost no diffusion.

We can characterize the different states by measuring the mean square displacement D_{y} in the direction perpendicular to the drive, the alignment S of the particles, and the fraction C_{L} of particles in the largest contiguous cluster. We obtain D_{y} from the first order linear fit coefficient to the time evolution of the displacement , where y_{i}(t) is the y position of the center of mass of particle i at time t. The particle alignment is given^{59} by . Particles are defined to belong to a single cluster if they are in direct force contact with each other, and we find the average size of the largest cluster efficiently by using the neighbor lookup table method described in ref. 60.

In Fig. 3(a) and (b) we plot D_{y}versus F_{D} for the n = 5 sample from Fig. 2 for N_{p} = 5, 50, and 600. Fig. 3(c) and (d) shows the corresponding S versus F_{D} measures, while in Fig. 3(e) and (f) we plot C_{L}versus F_{D}. We also highlight the locations of phases I through VII. For N_{p} = 5, D_{y}, S, and C_{L} are low, only phases II and I occur, and there is a small decrease of S in phase I. There is a small number of pinned particles in phase II that nucleate localized nematic ordering, and the drop in S upon entering phase I occurs when all the particles begin to move and the local combing effect is lost. For N_{p} = 50, D_{y} decreases with increasing F_{D} and S passes through a maximum near F_{D} = 0.11. Phase IV is associated with a large value of S and intermediate values of C_{L} and D_{y}, while in phase III where all of the particles are moving, C_{L} and D_{y} are small. S is lower in phase III than in phase IV due to the reduced combing effect. There is a small feature in C_{L} near F_{D} = 0.15 at the transition between phases IV and III. For N_{p} = 600, D_{y} and S are both low in the clogged phase VII, but C_{L} is large since the system forms a large pinned cluster. As the drive increases, the particles depin and C_{L} decreases, while S and D_{y} pass through a local peak in phase VI and then decrease as the drive is further increased. At the higher drives, all the particles are moving and S and C_{L} are low, but D_{y} remains finite in phase V due to the collisions with the pinning sites.

In Fig. 4 we plot D_{y}, S, and C_{L}versus F_{D} for particles with n = 9 moving over landscapes with N_{p} = 5, 50, and 600, where we observe similar trends as in the n = 5 system. Here, S shows a strong drop for N_{p} = 600 at F_{D} = 0.2 where a transition occurs from phase VI to phase V, while at the same drive for N_{p} = 50, S is still large and the system is in phase III. The diffusion D_{y} in phase V for N_{p} = 600 is high at F_{D} = 0.35 but S and C_{L} are small, while for N_{p} = 5 and N_{p} = 50, the diffusion is low at the same drive. For particles of length n = 7, we also observe similar behavior (not shown).

From measurements of D_{y}, S, and C_{L}versus F_{D}, we can construct dynamic phase diagrams for the different particle lengths. Fig. 5(a)–(c) shows heat diagrams of the cluster size C_{L}, alignment S, and traverse diffusion D_{y} as a function of F_{D}versus N_{p} for the n = 5 system from Fig. 2. The resulting schematic phase diagram in Fig. 5(d) highlights the regions where phases I through VII appear as a function of F_{D}versus N_{p}. The boundary separating phases I and II from phases III and IV is obtained by identifying points where S = 0.4. The separation between phases IV and VI is given by points where D_{y} = 1.5 × 10^{−5}. The boundary between phases I and II follows the location of a local minimum in the value of S, while the boundary between phases III and IV follows a local maximum in the value of S. Particle motion drops to zero in phase VII, and the orientation S drops to zero in phase V. When N_{p} < 7, phase I occurs for F_{D} ≥ 0.07, where C_{L} and S are small and there is no diffusion, while phase II appears for F_{D} < 0.07, where some of the particles are permanently pinned and there is weak alignment. In the clogged phase VII, which exists in the range N_{p} > 100 and F_{D} < 0.03, C_{L} is high while S and D_{y} remain small. The tooth combed phase IV for intermediate N_{p} and F_{D} < 0.16, where some of the particles can become pinned temporarily, has high S. Fig. 5(b) illustrates that S is nonmonotonic as a function of both F_{D} and N_{p}, with the highest value occurring in phase IV. In phase III, S is still large but there are no pinned particles present. The combed channel phase VI occurs for 0.02 < F_{D} < 0.25, above the depinning transition from the clogged state, and has high D_{y}, low C_{L}, and high S. Phase V is where the drive is strong enough that all the particles are moving and the alignment is lost; however, there is still some weak diffusion due to the collisions with the pinning sites.

Fig. 5 (a)–(c) Heat maps as a function of F_{D}vs. N_{p} for the n = 5 system from Fig. 2. (a) C_{L}. (b) S. (c) D_{y}. (d) Schematic phase diagram constructed from the above measures showing the locations of phases I (random ballistic), II (locally combed ballistic), III (point combed), IV (tooth combed), V (uniform random), VI (combed channel), and VII (clogged). |

In Fig. 6, the heat maps of C_{L}, S, and D_{y} along with the phase diagram as a function of F_{D}versus N_{p} for a sample with n = 7 show the same general trends found in Fig. 5 for the n = 5 particles. Fig. 7 is for the n = 9 particles and shows the same quantities of C_{L}, S, and D_{y} plotted as heat maps as a function of F_{D}versus N_{p} along with a schematic phase diagram. The tooth combed phase IV becomes wider as n increases since the longer particles can be combed more effectively by the pinning sites. The uniform random phase V also becomes more extensive in size with increasing n since the pointlike pinning sites decouple at lower drives from the more elongated particles. Overall, these results indicate that the generic phases we observe remain robust for a range of particle lengths.

In the results presented in this work, we made the assumption that thermal fluctuations could be neglected. This will be the case for sufficiently large colloidal particles or for systems in which friction from a substrate dominates over the thermal energy scale. We expect the combed phases to remain robust against introduction of modest thermal fluctuations, but sufficiently strong thermal fluctuations would be likely to fluidize the entire system. This would be an interesting direction for future study.

Here we considered an intermediate particle density. For much smaller particle densities, individual particles would contact each other so infrequently that the combing process would become inefficient and the combed phases would be lost. For particle densities much higher than what we consider here, the optimal packing would be aligned rods; in that case, the system could act like a rigid solid, and the behavior would strongly depend on how the system is initially prepared. It could also be interesting to consider mixtures of rods of different lengths or monodisperse disks interspersed with rods, where different regimes involving phase separation could arise. We concentrated on varying the pinning density over a range of drives; however, since the combed phases occur when some particles are permanently pinned, this suggests that some of the phases could also be realized by introducing obstacles or elongated obstacles instead of pinning sites. It is likely that if obstacles were used, there would be much more extended regions of clogged phase.

- D. S. Fisher, Phys. Rep., 1998, 301, 113–150 CrossRef.
- C. Reichhardt and C. J. O. Reichhardt, Rep. Prog. Phys., 2017, 80, 026501 CrossRef CAS PubMed.
- C. Reichhardt and C. J. Olson, Phys. Rev. Lett., 2002, 89, 078301 CrossRef CAS PubMed.
- A. Pertsinidis and X. S. Ling, Phys. Rev. Lett., 2008, 100, 028303 CrossRef PubMed.
- T. Bohlein and C. Bechinger, Phys. Rev. Lett., 2012, 109, 058301 CrossRef PubMed.
- P. Tierno, Soft Matter, 2012, 8, 11443–11446 RSC.
- C. Reichhardt, C. J. Olson, I. Martin and A. R. Bishop, Europhys. Lett., 2003, 61, 221–227 CrossRef CAS.
- H. J. Zhao, V. R. Misko and F. M. Peeters, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 88, 022914 CrossRef CAS PubMed.
- S. Bhattacharya and M. J. Higgins, Phys. Rev. Lett., 1993, 70, 2617–2620 CrossRef CAS PubMed.
- A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett., 1994, 73, 3580–3583 CrossRef CAS PubMed.
- T. Giamarchi and P. Le Doussal, Phys. Rev. Lett., 1996, 76, 3408–3411 CrossRef CAS PubMed.
- F. Pardo, F. de la Cruz, P. L. Gammel, E. Bucher and D. J. Bishop, Nature, 1998, 396, 348–350 CrossRef CAS.
- C. J. Olson, C. Reichhardt and F. Nori, Phys. Rev. Lett., 1998, 81, 3757–3760 CrossRef CAS.
- L. Balents, M. C. Marchetti and L. Radzihovsky, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 57, 7705–7739 CrossRef CAS.
- C. Reichhardt, C. J. Olson, N. Grønbech-Jensen and F. Nori, Phys. Rev. Lett., 2001, 86, 4354–4357 CrossRef CAS PubMed.
- C. Reichhardt, D. Ray and C. J. O. Reichhardt, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 104426 CrossRef.
- A. Vanossi, N. Manini, M. Urbakh, S. Zapperi and E. Tosatti, Rev. Mod. Phys., 2013, 85, 529–552 CrossRef CAS.
- A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot and D. Bartolo, Nature, 2013, 503, 95–98 CrossRef CAS PubMed.
- C. Sándor, A. Libál, C. Reichhardt and C. J. Olson Reichhardt, Phys. Rev. E, 2017, 95, 032606 CrossRef PubMed.
- Y. Fily, E. Olive, N. Di Scala and J. C. Soret, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 134519 CrossRef.
- N. Di Scala, E. Olive, Y. Lansac, Y. Fily and J. C. Soret, New J. Phys., 2012, 14, 123027 CrossRef.
- C. Reichhardt, C. J. Olson and F. Nori, Phys. Rev. Lett., 1997, 78, 2648–2651 CrossRef CAS.
- J. Gutierrez, A. V. Silhanek, J. Van de Vondel, W. Gillijns and V. V. Moshchalkov, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 140514 CrossRef.
- Y. Yang, D. McDermott, C. J. O. Reichhardt and C. Reichhardt, Phys. Rev. E, 2017, 95, 042902 CrossRef CAS PubMed.
- D. McDermott, Y. Yang, C. J. O. Reichhardt and C. Reichhardt, Phys. Rev. E, 2019, 99, 042601 CrossRef CAS PubMed.
- D. McDermott, C. J. O. Reichhardt and C. Reichhardt, Phys. Rev. E, 2020, 101, 042101 CrossRef CAS PubMed.
- C. Sándor, A. Libál, C. Reichhardt and C. J. O. Reichhardt, Phys. Rev. E, 2017, 95, 012607 CrossRef PubMed.
- V. Narayan, S. Ramaswamy and N. Menon, Science, 2007, 317, 105–108 CrossRef CAS PubMed.
- J. Deseigne, O. Dauchot and H. Chaté, Phys. Rev. Lett., 2010, 105, 098001 CrossRef PubMed.
- A. Kudrolli, G. Lumay, D. Volfson and L. S. Tsimring, Phys. Rev. Lett., 2008, 100, 058001 CrossRef PubMed.
- T. Börzsöenyi and R. Stannarius, Soft Matter, 2013, 9, 7401–7418 RSC.
- T. Börzsönyi, B. Szabó, G. Törös, S. Wegner, J. Török, E. Somfai, T. Bien and R. Stannarius, Phys. Rev. Lett., 2012, 108, 228302 CrossRef PubMed.
- D. B. Nagy, P. Claudin, T. Börzsönyi and E. Somfai, Phys. Rev. E, 2017, 96, 062903 CrossRef PubMed.
- K. To, Y.-K. Mo, T. Pongó and T. Börzsönyi, Phys. Rev. E, 2021, 103, 062905 CrossRef CAS PubMed.
- H. Löwen, J. Chem. Phys., 1994, 100, 6738–6749 CrossRef.
- S. Sacanna and D. J. Pine, Curr. Opin. Colloid Interface Sci., 2011, 16, 96–105 CrossRef CAS.
- Z. Zheng, F. Wang and Y. Han, Phys. Rev. Lett., 2011, 107, 065702 CrossRef PubMed.
- A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield and E. Sloutskin, Phys. Rev. Lett., 2011, 107, 238301 CrossRef CAS PubMed.
- Y. Chen, X. Tan, H. Wang, Z. Zhang, J. M. Kosterlitz and X. S. Ling, Phys. Rev. Lett., 2021, 127, 018004 CrossRef CAS PubMed.
- W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. Angelo, Y. Y. Cao, T. E. Mallouk, P. E. Lammert and V. H. Crespi, J. Am. Chem. Soc., 2004, 126, 13424–13431 CrossRef CAS PubMed.
- N. Kumar, H. Soni, S. Ramaswamy and A. K. Sood, Nat. Commun., 2014, 5, 4688 CrossRef CAS PubMed.
- M. Bär, R. Großmann, S. Heidenreich and F. Peruani, Ann. Rev. Condens. Matter Phys., 2020, 11, 441–466 CrossRef.
- P. Arora, A. K. Sood and R. Ganapathy, Phys. Rev. Lett., 2022, 128, 178002 CrossRef CAS PubMed.
- C. Eisenmann, U. Gasser, P. Keim and G. Maret, Phys. Rev. Lett., 2004, 93, 105702 CrossRef CAS PubMed.
- V. A. Froltsov, C. N. Likos, H. Löwen, C. Eisenmann, U. Gasser, P. Keim and G. Maret, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 71, 031404 CrossRef CAS PubMed.
- E. W. Carlson, A. H. Castro Neto and D. K. Campbell, Phys. Rev. Lett., 2003, 90, 087001 CrossRef CAS PubMed.
- C. Reichhardt and C. J. Olson Reichhardt, Europhys. Lett., 2006, 75, 489–495 CrossRef CAS.
- E. J. Roe, M. R. Eskildsen, C. Reichhardt and C. J. O. Reichhardt, New J. Phys., 2022, 24, 073029 CrossRef.
- S. A. Kivelson, E. Fradkin and V. J. Emery, Nature, 1998, 393, 550–553 CrossRef CAS.
- M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett., 1999, 82, 394–397 CrossRef CAS.
- S.-Z. Lin and A. Saxena, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 92, 180401 CrossRef.
- T. Nagase, M. Komatsu, Y. G. So, T. Ishida, H. Yoshida, Y. Kawaguchi, Y. Tanaka, K. Saitoh, N. Ikarashi, M. Kuwahara and M. Nagao, Phys. Rev. Lett., 2019, 123, 137203 CrossRef CAS PubMed.
- F. Höfling, T. Munk, E. Frey and T. Franosch, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, 060904 CrossRef PubMed.
- A. K. Tucker and R. Hernandez, J. Phys. Chem. A, 2010, 114, 9628–9634 CrossRef CAS PubMed.
- Z. Mokhtari and A. Zippelius, Phys. Rev. Lett., 2019, 123, 028001 CrossRef CAS PubMed.
- S. Mandal, C. Kurzthaler, T. Franosch and H. Löwen, Phys. Rev. Lett., 2020, 125, 138002 CrossRef CAS PubMed.
- A. Libál, D. Y. Lee, A. Ortiz-Ambriz, C. Reichhardt, C. J. O. Reichhardt, P. Tierno and C. Nisoli, Nat. Commun., 2018, 9, 4146 CrossRef PubMed.
- H. Peter, A. Libál, C. Reichhardt and C. J. O. Reichhardt, Sci. Rep., 2018, 8, 10252 CrossRef CAS PubMed.
- H. Chaté, F. Ginelli and R. Montagne, Phys. Rev. Lett., 2006, 96, 180602 CrossRef PubMed.
- S. Luding and H. J. Herrmann, Chaos, 1999, 9, 673–681 CrossRef PubMed.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3sm01034a |

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