Physical Review E 78, 011402 (2008)

Viscous Decoupling Transitions For Individually Dragged Particles in Systems with Quenched Disorder

C. J. Olson Reichhardt and C. Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 11 April 2008; published 10 July 2008)

We show that when an individual particle is dragged through an assembly of other particles in the presence of quenched disorder, a viscous decoupling transition occurs between the dragged particle and the surrounding particles which is controlled by the quenched disorder. A counterintuitive consequence of this transition is that the velocity of the dragged particle can be increased by increasing the strength or density of the quenched disorder. The decoupling transition can also occur when the external drive on the dragged particle is increased, and is observable as a clear signature in the velocity-force response.

Recently, several experiments have demonstrated the effectiveness of a local probing technique in which an individual particle is dragged through an assembly of other particles [1,2,3,4]. The velocity-force response of the driven particle was measured for a single magnetic colloid pulled through a background of nonmagnetic colloids in Ref. [1], while in other experiments, optical tweezers were used to drive a single colloid through a collection of charge stabilized colloids [3,4]. Several numerical works have predicted that a rich variety of dynamical regimes can occur in these types of systems depending on whether the probe particle is driven through a glassy media [5,6,7,8,9], a crystal [10], or an assembly of rods [11]. The colloidal dragging experiments are performed on the microscale; however, it is now becoming possible to conduct similar experiments on the meso- and nanoscales. For example, recent experiments have shown that individual vortices in type-II superconductors can be manipulated using a magnetic force microscope tip [12,13]. This technique could be used to study vortex entanglement [14] and the depinning of vortices from extended defects [15]. A simpler initial study would be to measure the drag forces on a single driven vortex in the presence of other vortices and quenched disorder from the sample. Currently, no predictions exist for the response in this regime.

In previous studies of single colloids driven through colloidal assemblies, the driven or probe colloid interacted only with the surrounding colloids. In the superconducting vortex system or in colloidal systems containing random quenched disorder, both the probe particle and the surrounding particles interact with the underlying quenched disorder which acts as a pinning potential. The addition of pinning might be expected merely to increase the overall drag on the probe particle; however, in this work we show that pinning can induce counterintuitive changes in the probe particle motion since the pinning couples to both the probe and the surrounding particles. For example, increasing the strength or density of pinning sites can induce a decoupling transition which sharply reduces the effective drag on the probe particle. The same decoupling transition occurs when the driving force on the probe particle is increased, and appears as a clear jump in the velocity-force curves. In the strongly damped regime, the probe particle couples strongly to the surrounding particles and induces irreversible plastic deformations. When the pinning strength is increased, the surrounding particles are trapped by pins and cannot respond to the probe particle, so the probe particle decouples from the surrounding particles and the effective damping is reduced. Similarly, when the drive is sufficiently large, the probe particle is unable to induce topological rearrangements in the surrounding particles and a decoupling transition occurs. We specifically examine a system with screened Coulomb particle-particle interactions. Experimentally, this corresponds to a two-dimensional colloidal system with quenched disorder, such as in Ref. [16]. We also study interactions appropriate for vortices in type-II superconductors.

We consider N particles in a two-dimensional sample of size L ×L with periodic boundary conditions in the x and y directions. The overdamped equation of motion for particle i is

dr_i

= Fⁱ_s + Fⁱ_p + Fⁱ_ext + F_i^T

(1)

where η is the damping constant which we set to η = 1. The force from particle-particle interactions is Fⁱ_s=−F₀∑_{i ≠ j}^N∇V(r_ij), where F₀ = Z^*2/(4πϵϵ₀), Z^* is the unit of charge, ϵ is the dielectric constant of the medium, and the distance between particles located at r_i and r_j is r_ij = |r_i − r_j|. We use a Yukawa potential, V(r_ij) = exp(−κr_ij)/r_ij, with screening length 1/κ = 2. The quenched disorder is modeled as N_p randomly placed parabolic pinning traps with pinning force Fⁱ_p=∑_k^N_pf_p(r_ik/r_p)Θ(r_p−r_ik)∧r_ik, where f_p is the pinning strength, r_p=0.25 is the pinning radius, Θ is the Heaviside step function, r_ik=|r_i−r^p_k|, r_k^p is the location of pin k, and ∧r_ik=(r_i−r^p_k)/r_ik. The pinning density is n_p = N_p/L² and the particle density is n=N/L². We have examined several different system sizes and present results for L = 48. For the probe particle, the external driving force Fⁱ_ext=F_ext∧x, and for all remaining particles, Fⁱ_ext=0. We construct a velocity-force curve by measuring the time-averaged velocity of the probe particle, v=〈v·∧x〉, as a function of F_ext. The thermal force F^T_i arises from random Langevin kicks with the properties 〈F^T_i〉 = 0 and 〈F^T_i(t)F^T_j(t^′)〉 = 2ηk_BTδ(t − t^′)δ_ij. Unless otherwise noted, we set T=0. We also consider the case of vortices in type-II superconductors, modeled as in Ref. [17] where Fⁱ_s = ∑^N_{i ≠ j}f₀K₁(r_ij/λ)∧r_ij. Here K₁ is a modified Bessel function, f₀ = ϕ₀/(2πμ₀λ³), ϕ = h/2e is the flux quantum, and λ is the London penetration depth. The probe particle in our system recrosses the same pinned region repeatedly during the course of a measurement; to verify that this does not affect the results, we tested driving forces applied at low angles to the x-axis which cause the probe to cross a different portion of the pinned region with each pass through the sample. We find the same results for all driving directions.

Figure 1: The velocity-force (v−F) curves for a single driven probe particle in a system with n_p = 0.85 and f_p = 2.0 at particle densities n = 0, 0.181, 0.292, 0.375, 0.484, 0.53, and 0.59 (from top to bottom). Inset: The transition region for n = 0.375 (bottom) along with the n = 0.181 curve (top).

We first study a system of Yukawa interacting particles and examine velocity force (v−F) relations for the probe particle in the absence of thermal fluctuations for varying particle densities n. Figure 1 shows the v−F curves for fixed n_p = 0.85 and f_p = 2.0 at particle densities ranging from n=0 to n=0.59. In the limit n=0, where the probe particle interacts only with the pinning, there is an initial pinned phase at low drive followed by a transition to a moving state at a critical external drive F^c_ext=1.7. In the moving state the particle velocity is proportional to the damping, v ∝ η, representing Ohmic type behavior. For n < 0.292, the v−F curves are similar to the n=0 single particle limit, with only a slight decrease in v in the moving state for increasing n. For n ≥ 0.292, in addition to the pinned regime at low drive and the Ohmic regime at high drive, we find an intermediate non-Ohmic regime where the velocity undergoes pronounced fluctuations and is reduced below the Ohmic value. Between the non-Ohmic and Ohmic regimes there is a distinctive jump in the velocity at a drive F_ext^t indicating a transition into a moving state with lower damping and reduced velocity fluctuations. The inset of Fig. 1 shows a blowup of the transition region for n = 0.375; the n = 0.181 curve is also presented for comparison. As n increases, the velocity of the probe particle in the non-Ohmic regime decreases and the transition drive F_ext^t increases.

Figure 2: Images of a 38 ×12 region of a sample with n_p=0.85, f_p=2.0, and F_ext=8.0. Large red dot: driven probe particle; black dots: unperturbed surrounding particles; open circles: surrounding particles perturbed by the passage of the probe particle; lines: particle trajectories. (a) The decoupled moving regime at n=0.3 has weak damping. (b) The coupled moving regime at n=0.5 has high damping and significant plastic distortions of the surrounding particles.

To illustrate the origin of the different damping in the two moving states, in Fig. 2 we show the trajectories of the probe particle and the surrounding particles in the weakly damped regime at n=0.3 and in the strongly damped regime at n=0.5. For the weakly damped regime in Fig. 2(a), only small distortions of the surrounding particles occur as the probe particle moves through the sample. In contrast, Fig. 2(b) indicates that in the strongly damped regime, the surrounding particles are significantly disturbed by the passage of the probe particle and undergo irreversible plastic distortion events. In order for a plastic rearrangement to occur, a portion of the probe particle energy must be transferred to the surrounding particles, increasing the effective damping on the probe particle. The plastic distortions produce large fluctuations in the probe particle velocity. In the weakly damped regime, the probe particle does not create plastic distortions in the surrounding particles. Thus, the transition we observe between the weakly and strongly damped regimes occurs due to a viscous decoupling transition between the probe particle and the surrounding particles.

The ability of the probe particle to create plastic distortions is affected by the external drive, particle density, pinning strength, and pinning density. In general, a plastic rearrangement occurs when a surrounding particle is both depinned and displaced by a distance a₀/2, where a₀=n^−1/2 is the average distance between particles. The probe particle spends an average time δt=a₀/F_ext interacting with a surrounding particle, and exerts a force F_d=−F₀∇V(a₀/2) on the particle. In the absence of pinning, plastic distortions would occur when −F₀∇V(a₀/2)/F_ext > 1/2, and thus F_ext^t ∝ −∇V(1/(2√n)). The v−F curves in Fig. 1 indicate that a transition between the high damping, plastic motion regime and the low damping, decoupled regime occurs when the driving force F_ext increases. When F_ext is large, F_ext ≥ F_ext^t, the probe particle passes the surrounding particles so rapidly that it can only induce small displacements due to the short interaction time δt. These displacements are too small to permit plastic rearrangements to occur, and the probe particle decouples from the surrounding particles. For F^c_ext < F_ext < F_ext^t, plastic distortions occur and the probe is in the high damping, coupled motion regime. The lower limit of the coupled motion regime is determined by the fact that the driving force F_ext must exceed the critical force F^c_ext so that the probe particle itself remains depinned.

Figure 1 shows that the transition drive F_ext^t increases with increasing particle density n. This is a result of the increasing strength of the interaction F_d between the probe particle and the surrounding particles with increasing n. At low n, the average distance a_o between particles is large and F_d is low, so the transition to decoupled motion F_ext^t falls at low values of F_ext. As n increases, F_d increases due to the decrease in a_o, and the plastic distortions persist up to higher values of F_ext, corresponding to a higher value of F_ext^t. In Fig. 3(a) we present a phase diagram for F_ext versus n taken from a series of v−F curves. The three phases, pinned (P), decoupled moving (DM), and coupled moving (CM), are identified based on the values of F_ext^c and F_ext^t. The range of driving forces over which coupled motion can occur increases with increasing n. For fixed drive, a velocity drop occurs when n is increased above the coupling-decoupling transition, as illustrated in the inset of Fig. 3(a) for F_ext=10.

Figure 3: (a) Phase diagram F_ext versus n at f_p = 2.0 and n_p = 0.85 showing the pinned (P), coupled moving (CM), and decoupled moving (DM) regimes. Squares: depinning threshold F_ext^c. Circles: decoupling threshold F_ext^t. Inset: v versus n at F_ext = 10.0, showing a drop in v above the transition density of n=0.3. (b) Phase diagram for F_ext versus f_p at n_p = 0.85 and n = 0.292. Inset: v versus f_p at F_ext = 12.0 shows that v can increase with increasing f_p. (c) Phase diagram for F_ext versus n_p for f_p = 3.0 and n = 0.292. Inset: v versus n_p for F_ext = 12.0. (d) Phase diagram for F_ext versus T with n_p = 0.85, f_p = 3.0, and n = 0.292. Left inset: v vs T for F_ext = 16.0. Right inset: v vs F_ext for superconducting vortices with n=0.44 and n_p=0.46. Lower right curve: f_p = 0.25; upper right curve: f_p = 1.5.

To determine the influence of the pinning on the coupling-decoupling transition, we vary the pinning force f_p in a system with fixed n = 0.292 and n_p = 0.85, and plot the resulting phase diagram in Fig. 3(b). The depinning force increases linearly with f_p, F_ext^c ∝ f_p. For very low f_p the system responds as if there were no pinning, a coupled motion regime which was previously explored in Ref. [5]. At low f_p, the probe particle is strongly coupled to the surrounding particles and generates significant plastic distortions. Since the pinning force tends to localize the particles and therefore competes with F_d in determining whether the surrounding particles can undergo plastic motion, we expect plastic distortions to occur in the presence of pinning when (F_d−f_p)δt > a₀/2, giving F_ext^t ∝ (F_d−f_p). This is in agreement with Fig. 3(b), where F^t_ext decreases linearly with f_p. The coupled motion disappears for f_p > 2.0 when the probe particle can no longer depin the surrounding particles.

For fixed drive, the probe particle velocity v initially decreases with increasing f_p, as shown in the inset of Fig. 3(b) for F_ext=12.0. The pinning increases the effective viscosity in the coupled moving regime compared to the f_p=0 case by increasing the number of irreversible events that occur. As f_p increases, a surrounding particle that has been depinned and displaced by the probe particle can more easily be repinned at a new pinning site, rather than returning to its previous position in a reversible event. When f_p is further increased, a sharp increase in v occurs at the decoupling transition where the plastic distortions are lost. We note that v in the decoupled regime is higher than the value of v at f_p = 0. These results show that increasing the strength of the pinning can cause a counterintuitive increase in the velocity (or decrease in the damping) of the probe particle. At large enough f_p, the probe particle itself becomes pinned.

In Fig. 3(c) we show the phase diagram for F_ext versus pinning density n_p for a system with f_p = 3.0 and n = 0.292. There is little change in F_ext^c with n_p, while F_ext^t decreases with n_p until the coupled motion regime disappears for n_p > 0.75. When n_p increases, the average spacing between adjacent pinning sites decreases, and a surrounding particle that has been depinned by the probe particle can be trapped by a new pinning site before it has moved far enough to allow a plastic rearrangement to occur. The inset of Fig. 3(c) illustrates the velocity of the probe particle versus n_p at fixed F_ext = 12.0. For very low n_p the behavior is similar to the pin-free case. The effective viscosity increases sharply at n_p ≈ 0.125 when the system enters the strongly damped, coupled motion regime. For n_p > 0.5, the viscosity drops sharply at the onset of the decoupled motion regime, and then monotonically decreases for increasing n_p. Increasing the pinning density can counterintuitively increase the velocity of the probe particle v above the value at n_p = 0.

The phase diagram for F_ext vs temperature T is shown in Fig. 3(d) for a system with f_p = 3.0, n_p = 0.85, and n = 0.292. At T = 0 the probe particle undergoes decoupled motion for F_ext > F_ext^c. As T increases, the effectiveness of the pinning is reduced, F_ext^c decreases, and the probe particle recouples to the surrounding particles. This produces a velocity drop with increasing T, as shown in the left inset of Fig. 3(d) for F_ext=16.0. A transition back to the decoupled moving regime occurs for F_ext > F_ext^t. The overall structure of the phase diagram in Fig. 3(d) is very similar to the dynamical phase diagram for vortices in type-II superconductors moving over random disorder, where a transition from plastic to elastic flow occurs as the driving force on the vortex lattice is increased [18]. Our results indicate that a similar effect can occur even when only a single particle is driven.

We have performed similar simulations for vortices in type-II superconductors. In the right inset of Fig. 3 we plot the v−F curves from a vortex sample with n_p = 0.46, n_v = 0.44, and r_p = 0.2 in the strong pinning regime with f_p=1.5 and in the weak pinning regime with f_p=0.25. The v−F curves have the same trend seen in Fig. 1 and Fig. 3(b). The f_p=0.25 sample exhibits coupled motion with high damping, while the f_p=1.5 sample is in the low damping, decoupled motion regime. A decoupling transition occurs with increasing driving force for the weakly pinned sample. This suggests that our results should be generic to single driven probe particles moving through a background of repulsively interacting particles in the presence of quenched disorder.

In summary, we show that when a single probe particle is driven through an assembly of other particles in the presence of quenched disorder, a novel viscous decoupling transition can occur between the probe particle and the surrounding particles which is controlled by the strength and density of the quenched disorder. This transition is from a highly damped regime, where the probe particle depins the surrounding particles and produces irreversible plastic distortions, to a low damping state, where the probe particle does not couple to the surrounding particles. Increasing the pinning strength or density reduces the coupling between the probe and background particles, producing the counterintuitive result that increasing the strength or density of the quenched disorder increases the velocity of the probe particle. The decoupling transition appears as a clear signal in the effective damping on the probe particle and produces a distinct feature in the velocity force curve. In addition to colloids with Yukawa interactions, our results should be general to other systems of interacting particles with quenched disorder, including vortices in type-II superconductors.

This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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