Physical Review B 65, 094301 (2002)

Transverse Depinning of a Driven Elastic String in a Disordered Media

C. Reichhardt(1) and C. J. Olson(2)

(1)CNLS and Applied Physics Division,

(2) Theoretical and Applied Physics Divisions,

Los Alamos National Laboratory, Los Alamos, NM 87545

(Received 2 October 2001; published 13 February 2002)

We examine the dynamics of an elastic string interacting with quenched disorder driven parallel to the string. For a moving parallel driven string we show that there is a finite transverse critical depinning force which decreases for increasing parallel drive. The depinning transition is hysteretic and occurs by the formation of running kinks.
The dynamics of driven elastic media interacting with quenched disorder is important in a wide variety of physical systems which include magnetic domain wall motion [1], nonequilibrium growth [2], models of friction [3], vortex lattice motion in superconductors [4], and charge density waves [5]. Recently an intense interest in driven elastic media with quenched disorder has been directed at dynamical ordering where at low applied drives the system is in a highly disordered pinned state and at a critical drive a depinning into a highly disordered moving state occurs, while at high drives the effect of the disorder is reduced and the system can regain a considerable amount of order. Particular systems where this reordering behavior has been studied extensively experimentally [6,7,8,9], theoretically [10,11,12,13], and numerically [10,14,15,16,17,18] include vortices in superconductors as well as charge-density wave (CDW) systems.
In the vortex system if there is no quenched disorder the vortices form an ordered crystalline state; however, in most samples there are defects which attract and pin vortices, disordering the vortex lattice. When a driving force is applied, the initial depinning transition can be plastic, with certain regions of mobile vortices tearing past pinned regions and a large number of defects generated so that the system has only a liquid like structure. For higher drives the vortices regain order; however, the pinning can still affect the structure so the vortices can have a partial anisotropic ordering or smectic structure [12]. An open question is whether the reordering transition is a true transition or a crossover.
An interesting feature of the reordered moving state is that a transverse depinning threshold can exist as originally proposed in Ref. [11]. Here, although the lattice is moving in the longitudinal direction, the effect of the pinning is still present in the transverse direction so that a finite transverse drive must be applied before the lattice will move in the transverse direction. Simulations [14,18] and experiments [9] on vortex lattices as well as simulations for friction models [3] have found evidence for a finite transverse depinning threshold for the longitudinal moving systems. Other theoretical work contends that at finite T no true critical transverse depinning threshold exists; instead, due to the high pinning barriers, a pronounced crossover in the transverse IV curves may appear [12]. An intriguing question is whether the transverse barrier seen in the vortex system occurs in other types of driven elastic media, such as driven interfaces and polymers interacting with quenched disorder. Additionally the dynamics at the transverse depinning are not well understood, such as whether the transition is hysteretic or involves kink formation. An ideal way to model these systems in 2D is with a driven elastic string interacting with quenched disorder.
Previous numerical studies of driven elastic strings with disorder [19,20,21,22] employed drives perpendicular to the string and focused on the dynamics near the depinning threshold. In this case critical behavior is expected, and was observed in the form of broad distributions of avalanche sizes as well as scaling of the velocity near depinning, v = (f − fc)β, as proposed by Fisher [23]. The roughness of the string, as measured by the roughening coefficient α, was also determined. Recently it has been found that at the depinning transition, an elastic string exhibits an anomalous roughening [24], with a global roughness exponent α = 1.25 and a local exponent of α = 1.0 [25,26]. As the applied drive increases beyond depinning the effectiveness of the pinning becomes reduced, the variance in the noise decreases as 1/v, where v is the string velocity, and the roughness of the string decreases [27].
The case of a string driven in the parallel direction has received relatively little study. One example of such a system is a transversely driven CDW. An open question is whether a parallel moving string will exhibit a transverse depinning threshold. Recently Radzihovsky and Toner studied a CDW system in the context of the effect of the quasi-particle current flowing perpendicular to the CDW 2kF ordering wavevector [28]. They found that as the transverse force increases, the longitudinal depinning threshold decreases exponentially. This decrease occurs because the transverse drive straightens out the CDW distortions, preventing the CDW from adjusting to the pinning. Experimental observations by Markovic et al. [29] are consistent with such a behavior; however, more recent theoretical work [30] suggests that geometrical effects may be very pronounced. In the case of the parallel driven string the parallel drive would correspond to the CDW transverse drive. As the parallel drive is increased further the string may straighten, causing the perpendicular depinning threshold to decrease. Theoretical calculations predict that for a parallel driven string the perpendicular depinning force will decrease with decreasing parallel drive as a power law, rather than exponentially as observed in the CDW system.
Another physical system that may be modeled as a parallel driven elastic string is driven vortices in superconductors near depinning. Simulations and experiments have shown that in strongly disordered superconductors, the initial depinning can occur through the formation of a single winding channel of moving vortices [31,32]. As the applied drive is increased, the vortices in the channel move faster, and at still higher drives, additional channels of moving vortices form. The transverse barrier has been shown to be present in the case of strongly driven vortices where all the vortices are moving in coupled channels [14,18,33]; however, the dynamics of the transverse depinning process have not been explored, and it is not known whether the individual channels can straighten for increased drives, or whether a transverse barrier exists for the single channel limit. Individual channels are also observed in Wigner crystal depinning [34], transport in metallic dots [35], and flux flow in Josephson junction arrays [36]. Other possible physical systems similar to the parallel driven string include fluid channels flowing down an inclined plane and parallel driven polymers in random media. In addition recent studies in a frictional system have observed the transverse propagation of a 1D reorientation front between two incommensurate crystals [37], which resembles a parallel driven string under an additional transverse driving bias.
In this work we investigate the transverse depinning for the moving string. In the case of the perpendicular moving string we do not observe a transverse barrier; however, for the parallel driven string a transverse depinning threshold is observed which occurs through the formation of running kinks. We also find that the transverse depinning is hysteretic.
We model an overdamped elastic string in (1 + 1) dimensions. The string is composed of discrete particles connected by springs. The system has periodic boundary conditions in the x and y-directions and the string is connected periodically in the y-direction. The particle positions in the x-y plane are continuous. The equation of motion for each particle on the string is:
γ dr
dt
 = κfs + fp + Fd  .
(1)
Here r is the particle location, γ = 1 is the damping term, κ = 5 is the string elastic constant, fs=(ri+1−ri−rl ∧ri+1,i)+ (ri−ri−1 − rl ∧ri,i−1) is the spring force from the two neighboring particles, rl is the spring equilibrium length, ∧ri,j = (ri−rj)/|ri−rj|, fp is the pinning force and Fd is the uniform applied driving force. The pinning is modeled as attractive parabolic traps scattered randomly through the sample. The pinning force is fp = −∑iNp(fp/rp)(ri−rk(p))Θ(rp − |ri − rk(p)|), where Θ is the Heaviside step function, rk(p) is the location of pinning site k, fp=0.3 is the maximum pinning force, and rp=0.25 is the pinning radius. The results shown here are for strings containing N=500 to 2000 particles, interacting with Np=4.6 ×105 pinning sites in samples of size 160 ×160. We consider two types of initial conditions: one where the string is put down in its unstretched equilibrium position, and a second where the string is annealed at a finite temperature by the addition of a thermal kick. We find that when the applied drive is increased slowly enough, both methods give similar results. We apply a uniform force Fd in the x direction for perpendicular driving or the y-direction for parallel driving. We increase the driving force in increments of 0.001 and spend 30000 simulation steps at each increment, measuring the average string velocity Vx = ∑iNi∧x·vi, and Vy = ∑iNi∧y·vi. To quantify the order in the string we measure the difference ∆L in the length of the string compared to the equilibrium length, ∆L = ∑iN(|ri − ri+1| − rl), where rl=N/Ly is the equilibrium length of each string segment.
The theoretical work in Ref. [11] predicts that for an elastic media moving in the longitudinal direction, a finite depinning threshold should exist for an applied drive perpendicular or transverse to the direction of the moving interface. To test these predictions, we have conducted a series of simulations where we fix the driving force at a constant value after the string is in motion. We then slowly apply an additional driving force in the direction transverse to the initial longitudinal motion. For the perpendicular driven string the applied transverse drive is along the string and for the parallel driven string the applied transverse drive is perpendicular to the string. For the perpendicular driven string we do not observe a transverse barrier for the moving string. For a parallel driven string, in Fig. 1(a) we show the transverse driving force versus the transverse string velocity, indicating that for the case of the parallel moving string there is a finite transverse pinning threshold. In addition, the depinning transition appears to be sharp.
The theory in Ref. [11] predicts that for moving vortices, a finite transverse barrier occurs only when the vortices are moving in well defined channels and the transverse drive is perpendicular to these channels. The theory also predicts that the transverse barrier should decrease as the longitudinal velocity of the channels increases, since the channels straighten at the increased drive and can no longer pass through the optimal pinning sites. The exact functional form of the decrease of the transverse barrier with longitudinal velocity is not known; however, simple arguments obtained by balancing the pinning force with a transverse force on the Larkin domain give an exponential decrease at high drives in 3D. In recent vortex simulations by Fanghor et al. [33], when coupled channels of vortices formed, a finite transverse barrier appeared which decreased in a nonlinear manner with increasing longitudinal vortex velocity; however, the decrease was not fitted to a functional form.
Fig1.png
Figure 1: (a) The transverse string velocity V versus the transverse drive FdT for a parallel driven string (lower curve). The parallel string drive is Fd/Fdp = 1.22. A finite transverse depinning threshold is visible. A plot of V for a system with no quenched disorder is also shown (dashed curve). (b) The transverse depinning threshold FdpT versus longitudinal drive Fd. The line is a guide to the eye.
The case of the parallel moving string may be viewed as a single moving channel. As the longitudinal drive is increased, the string straightens so that a decrease in the transverse depinning force should also be expected once the string depins in the parallel direction. In Fig. 1(b) we show that the transverse barrier for a parallel moving string decreases with increasing longitudinal velocity. The data in Fig. 1(b) is divided into two regions as indicated by the dashed line. The first region is for parallel drives Fd < 0.09, at which the string is pinned in the parallel direction. In the second region, for Fd > 0.09, the string is in motion in the parallel direction and the transverse depinning threshold decreases with increasing longitudinal drive. The straightening of the string in the parallel direction only occurs after the string begins to move in the parallel direction so a decrease in the transverse barrier is not expected for the first region. Indeed, in Fig. 1(b) there is a negligible decrease of the transverse barrier for Fd < 0.09. Above the parallel depinning transition, Fd > Fdp, the transverse barrier decreases with Fd.
Fig2.png
Figure 2: (a) A series of snapshots of a parallel driven string as a function of time with the longitudinal drive in the y-direction and the transverse drive in the x-direction, for a system with Fd = 0.12, just above the transverse depinning threshold. The transverse depinning occurs via formation of a running kink that moves in the direction of the longitudinal drive. (b) The same snapshots offset in the x direction.
We have also investigated the dynamics of the transverse depinning for the parallel driven string, as indicated in Fig. 2, which shows a series of snapshots of the string just above the transverse depinning threshold. The transverse depinning transition is very distinct from the longitudinal depinning of the perpendicular driven string. The transversely driven parallel moving string does not become more disordered as observed at the depinning transition of the strictly perpendicular driven string. Instead, the transverse depinning occurs through the formation of a running kink that moves in the direction of the longitudinal drive as seen in Fig. 2. Although one might expect a kink and an anti-kink to form, we observed that below the kink the string is positioned at a slight angle. The kink moves along the string through the periodic boundary conditions. As the transverse drive is increased more kinks appear and the kinks begin to move more rapidly. Similar kink motion has been observed in interfaces between incommensurate crystals in sliding friction systems [37], a system which closely resembles the parallel driven string with a perpendicular drive.
Fig3.png
Figure 3: Transverse string velocity V versus transverse drive FdT for a parallel driven string showing hysteresis (dark line: ramp up; light line: ramp down) in the transverse depinning threshold with the downward depinning threshold being much lower than the initial depinning threshold.
Fig4.png
Figure 4: Transverse string velocity V versus transverse drive FdT for increasing system size. Lower light line: system size of 80 ×80; middle dark line: system size of 160 ×160; upper dashed line: system of size 240 ×240.
In Fig. 3 we illustrate hysteresis in the transverse depinning threshold. The sharpness of the depinning and the existence of hysteresis strongly suggest that the transverse depinning transition is first order. Hysteresis at the depinning transition has also been observed for elastic strings in periodic substrates where the motion occurs by the formation of running kinks [38]. In Fig. 4 we examine the transverse depinning threshold for varied system sizes. The jump in Vy at depinning becomes sharper for increased system sizes, and is likely less sharp in the smaller systems due to the effects of increased fluctuations.
In conclusion we have numerically investigated the dynamics of a parallel driven string in quenched disorder. For a parallel driven moving string we find that there is a finite transverse critical depinning threshold FdpT, indicating that the theory of Ref. [11] regarding the transverse barrier in a very different system still appears to apply to this system. We see a decrease of FdpT with increasing longitudinal drive, as predicted in Ref. [28] for CDW's. The transverse depinning occurs by the formation of a running longitudinal kink, where the kink moves in the same direction as the parallel driving. The transverse depinning threshold is sharp and hysteretic, suggesting a first order transition. We also find that the longitudinal velocity goes through a minimum at the transverse depinning threshold, implying that the existence of a transverse barrier can be deduced from measurements of the longitudinal velocity. We do not observe a transverse depinning threshold for the perpendicular driven string.
Our predictions for the reordering and the transverse barrier for the parallel driven string may be observable in elastic systems that form channels at depinning, such as vortices in superconductors. Other systems that may have similar physics to the parallel driven string include polymers aligned with the drive in disordered media or a single stream of liquid flowing over a rough surface at various tilt angles. In addition it would also be interesting to see if the transverse depinning transition observed in the highly driven vortex systems is hysteretic.
We thank M. Chertkov, L. Radzihovsky, and G. Zimányi for useful discussions. We also thank T. Germann for sharing with us unpublished data on friction models.This work was supported by the US Department of Energy under contract W-7405-ENG-36.

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