Pinning-depinning Transition Research Articles

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

Pinning-depinning transition of droplets on inclined substrates with a three-dimensional topographical defect.

Droplets on inclined substrates can depin and slide freely above a critical substrate inclination angle. Pinning can be caused by topographical defects on the substrate, and understanding the influence of defect geometry on the pinning-depinning transition is important for diverse applications such as fog harvesting, droplet-based microfluidic devices, self-cleaning surfaces, and inkjet printing. Here, we develop a lubrication-theory-based model to investigate the motion of droplets on inclined substrates with a single three-dimensional Gaussian-shaped defect that can be in the form of a bump or a dent. A precursor-film/disjoining-pressure approach is used to capture contact-line motion, and a nonlinear evolution equation is derived which describes droplet thickness as a function of the position along the substrate and time. The evolution equation is solved numerically using an alternating direction implicit finite-difference scheme to study how the defect geometry influences the critical inclination angle and the shape of a pinned droplet. It is found that the critical substrate inclination angle increases as the defect becomes taller/deeper or wider along the direction lateral to the droplet-sliding direction. However, the critical inclination angle decreases as the defect becomes wider along the sliding direction. Below the critical inclination angle, the advancing contact line of the droplet at the droplet centerline is pinned to the defect at the point having maximum negative slope. Simple scaling relations that reflect the influence of defect geometry on the droplet retention force arising from surface tension are able to account for many of the trends observed in the numerical simulations.

Large-scale kinetic roughening behavior of coffee-ring fronts.

We have studied the kinetic roughening behavior of the fronts of coffee-ring aggregates via extensive numerical simulations of the off-lattice model considered for this context [Dias et al., Soft Matter 14, 1903 (2018)1744-683X10.1039/C7SM02136D]. This model describes ballistic aggregation of patchy colloids and depends on a parameter r_{AB} which controls the affinity of the two patches, A and B. Suitable boundary conditions allow us to elucidate a discontinuous pinning-depinning transition at r_{AB}=0, with the front displaying intrinsic anomalous scaling, but with unusual exponent values α≃1.2,α_{loc}≃0.5,β≃1, and z≃1.2. For 0<r_{AB}≤1, comparison with simulations of standard off-lattice ballistic deposition indicates the occurrence of a morphological instability induced by the patch structure. As a result, we find that the asymptotic morphological behavior is dominated by macroscopic shapes. The intermediate time regime exhibits one-dimensional KPZ exponents for r_{AB}>0.01 and the system suffers a strong crossover dominated by the r_{AB}=0 behavior for r_{AB}≤0.01. A detailed analysis of correlation functions shows that the aggregate fronts are always in the moving phase for 0<r_{AB}≤1 and that their kinetic roughening behavior is intrinsically anomalous for r_{AB}≤0.01.

Localized states with nontrivial symmetries: Localized labyrinthine patterns.

The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with nontrivial symmetries such as labyrinthine patterns are observed in different physical contexts. Here we report stable localized disordered patterns in spatially extended dissipative systems. These two- and three-dimensional localized structures consist of an isolated labyrinth embedded in a homogeneous steady state. Their partial bifurcation diagram allows us to explain this phenomenon as a manifestation of a pinning-depinning transition. We illustrate our findings on the Swift-Hohenberg-type of equationsand other well-established models for plant ecology, nonlinear optics, and reaction-diffusion systems.

Oscillatory motion of dissipative solitons induced by delay-feedback in inhomogeneous Kerr resonators

We investigate the destabilization mechanisms of dissipative solitons in inhomogeneous nonlinear resonators subjected to injection and to time-delayed feedback. We consider the paradigmatic Lugiato-Lefever model describing inhomogeneous driven nonlinear optical resonator. We analyze the pinning-depinning transition of dissipative solitons by introducing a potential induced by the inhomogeneity. Further, we identify conditions under which these structures are destabilized and describe different bifurcation scenarios. We show that the combined influence of inhomogeneities and delayed feedback induces an Andronov-Hopf-bifurcation that leads to oscillations of the dissipative soliton around the inhomogeneity. Finally, we show that for large values of the feedback strength, the dissipative solitons escapes from the potential well and starts to drift.

Noise-induced kink propagation in shallow granular layers

Abstract Out of equilibrium systems are characterized by exhibiting the coexistence of domains with complex spatiotemporal dynamics. Here, we investigate experimentally the noise-induced domain wall propagation on a one-dimensional shallow granular layer subjected to an air flow oscillating in time. We present results of the appearance of an effective drift as a function of the inclination of the experimental cell, which can be understood using a simple Langevin model to describe the dynamical evolution of these solutions via its pinning-depinning transition. The statistical characterization of displacements of the granular kink position is performed. The dynamics of the stochastic model shows a fairly good agreement with the experimental observations.

Pinning and depinning of Wenzel-state droplets around inclined steps

Droplets moving over inclined heterogenous substrates may pin around rough spots. A simplified model of one of these spots having a backward facing step geometry over a chemically homogeneous inclined substrate reveals that the step is responsible from the pinning-depinning of the leading edge of the droplet in Wenzel-state. We consider a two-dimensional partially wetting droplet and model its motion by the evolution equation with a precursor film model. The phase diagrams formed define pinning-depinning transition curves determined by a simple force balance of retention forces due to hysteresis, step and gravitational force. For fixed Bond number and inclination, we show that it is the slope of the step that determines transition. The existence of the multiple steps, however, alters this transition as the receding contact line may pin at the steps and leave residual droplets behind.

Front depinning by deterministic and stochastic fluctuations: A comparison.

Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.

Self-Organized Criticality of Domain Walls and Magnetization Curve

We propose a kind of Ginzburg--Landau equation with quenched randomness. There is a pinning--depinning transition in the system when the external magnetic force is changed. The transition is self-organized when the external magnetic field is slowly changed under the demagnetizing effect. The total magnetization increases stepwise and the probability distribution of the increase in the total magnetization approximately obeys a power law. A hysteresis loop is obtained when the external magnetic field is changed reciprocally. In our model, the coercivity in the magnetization curve is expressed as the critical value for the pinning-depinning transition.

Numerical simulations of critical dynamics in anisotropic magnetic films with the stochastic Landau-Lifshitz-Gilbert equation.

With the stochastic Landau-Lifshitz-Gilbert (sLLG) equation, critical dynamic behaviors far from equilibrium or stationary around the order-disorder and pinning-depinning phase transitions in anisotropic magnetic films are investigated. From the dynamic relaxation with and without an external field, the Curie temperature and critical exponents of the order-disorder phase transition are accurately determined. For the pinning-depinning phase transition induced by quenched disorder, the nonstationary creep motion of domain wall activated by finite temperatures is simulated, and the thermal rounding exponent is extracted. The results show that the dynamic universality class of the sLLG equation is different from those of the Monte Carlo dynamics and quenched Edwards-Wilkinson equation, and it may lead to alternative understanding of experiments. The dynamic approach shows its great efficiency for the sLLG equation.

Thermodynamic formulation of the barrier for heterogeneous pinned nucleation: Implication to the crossover scenarios associated with barrierless and homogeneous nucleation.

The effect of contact line pinning on nucleation is reported using continuum thermodynamics. Based on the principle of the free-energy maximization, closed-form expressions in the dimensionless form for the free-energy of the three-phase metastable system and the thermodynamic barrier are formulated with respect to the system geometry and the substrate wettability. The condition of maximality limits the dynamic contact angle within the cluster-phase-phobic regime. The dimensionless nucleation barrier or the potency factor can be divided into two components related to the system geometry and the pinning effect. Depending on the relative value of the equilibrium and the critical dynamic contact angle, the contact line pinning can either have favorable or adverse effects. Associated pinning-depinning transition can also lead to the crossovers related to barrierless and homogeneous nucleation. Contact line tension is found to have a considerable effect during these transitional scenarios. Complete wetting transition associated with barrierless nucleation can take place due to the presence of tensile (negative) line tension. On the other hand, complete drying transition related to homogeneous nucleation can occur when line tension is compressive (positive) in nature. The pinning has a favorable effect only when the substrate wettability is within the cluster-phase-philic regime. There can be favorable, adverse, or no pinning effects when the substrate wettability is within the cluster-phase-phobic regime. Although the contact line is pinned, the minimum value of the potency factor is obtained when equilibrium and dynamic contact angles are equal.

Splitting in the pinning-depinning transition of fronts in long-delayed bistable systems.

We investigate the formation of localized domains through front pinning in a periodically forced, bistable semiconductor laser with long-delayed optoelectronic feedback. At difference with 1D spatially extended systems, the transition from the pinning to the propagation regime occurs via two separated bifurcations, each corresponding to the unpinning of one of the fronts surrounding the localized domain. The bifurcation splitting is systematically explored, unveiling the crucial role played by the forcing frequency. The experimental results are reproduced and interpreted by means of a prototypical model of our system.

Octagon to Square Wetting Area Transition of Water-Ethanol Droplets on a Micropyramid Substrate by Increasing Ethanol Concentration.

The wettability and evaporation of water-ethanol binary droplets on the substrate with micropyramid cavities are studied by controlling the initial ethanol concentrations. The droplets form octagonal initial wetting areas on the substrate. As the ethanol concentration increases, the side ratio of the initial wetting octagon increases from 1.5 at 0% ethanol concentration to 3.5 at 30% ethanol concentration. The increasing side ratio indicates that the wetting area transforms from an octagon to a square if we consider the octagon to be a square with its four corners cut. The droplets experience a pinning-depinning transition during evaporation. The pure water sessile droplet evaporation demonstrates three stages from the constant contact line (CCL) stage, and then the constant contact angle (CCA) stage, to the mixed stage. An additional mixed stage is found between the CCL and CCA stages in the evaporation of water-ethanol binary droplets due to the anisotropic depinning along the two different axes of symmetry of the octagonal wetting area. Droplet depinning occurs earlier on the patterned surface as the ethanol concentration increases.

Pinning-depinning transition in a stochastic growth model for the evolution of cell colony fronts in a disordered medium.

We study a stochastic lattice model for cell colony growth, which takes into account proliferation, diffusion, and rotation of cells, in a culture medium with quenched disorder. The medium is composed of sites that inhibit any possible change in the internal state of the cells, representing the disorder, as well as by active medium sites that do not interfere with the cell dynamics. By means of Monte Carlo simulations we find that the velocity of the growing interface, which is taken as the order parameter of the model, strongly depends on the density of active medium sites (ρ_{A}). In fact, the model presents a (continuous) second-order pinning-depinning transition at a certain critical value of ρ_{A}^{crit}, such as, for ρ_{A}>ρ_{A}^{crit}, the interface moves freely across the disordered medium, but for ρ_{A}<ρ_{A}^{crit} the interface becomes irreversible pinned by the disorder. By determining the relevant critical exponents, our study reveals that within the depinned phase the interface can be rationalized in terms of the Kardar-Parisi-Zhang universality class, but when approaching the critical threshold, the nonlinear term of the Kardar-Parisi-Zhang equation tends to vanish and then the pinned interface belongs to the quenched Edwards-Wilkinson universality class.

Dynamic effects of quenched disorder on domain wall motion in magnetic nanowires

The domain wall dynamics in magnetic nanowires is numerically studied with the Landau-Lifshitz-Gilbert equation. Below the Walker breakdown threshold, the domain wall presents a stable propagation, while above the threshold where the retrograde mode dominates, the oscillation period is controlled by the external field and anisotropy. More importantly, the dynamic effects of quenched disorder on the domain wall motion are explored. A continuous pinning-depinning phase transition is detected. The dynamic scaling form is analyzed with the data collapse of the domain wall velocity, and both the static and dynamic critical exponents are extracted.

Experimental study of stable imbibition displacements in a model open fracture. I. Local avalanche dynamics

We report the results of an experimental investigation of the spatiotemporal dynamics of stable imbibition fronts in a disordered medium, in the regime of capillary disorder, for a wide range of experimental conditions. We have used silicone oils of various viscosities μ and nearly identical oil-air surface tension and forced them to slowly invade a model open fracture at different constant flow rates v. In this first part of the study we have focused on the local dynamics at a scale below the size of the quenched disorder. Changing μ and v independently, we have found that the dynamics is not simply controlled by the capillary number Ca∼μv. Specifically, we have found that the wide statistical distributions of local front velocities, and their large spatial correlations along the front, are indeed controlled by the capillary number Ca. However, local velocities exhibit also very large temporal correlations, and these correlations depend more strongly on the mean imposed velocity v than on the viscosity μ of the invading fluid. Correlations between local velocities lead to a burstlike dynamics. Avalanches, defined as clusters of large local velocities, follow power-law distributions-both in size and duration-with exponential cutoffs that diverge as Ca→0, the pinning-depinning transition of stable imbibition displacements. Large data sets have led to reliable statistics, from which we have derived accurate values of critical exponents of the relevant power-law distributions. We have investigated also the dependence of their cutoffs on μ and v and related them to the autocorrelations of local velocities in space and time.

Limiting mechanism for critical current in topologically frustrated Josephson junctions

Eutectic Sr$_2$RuO$_4$-Ru samples with $\mu$m-sized Ru-metal inclusions support inhomogeneous superconductivity above the bulk transition of Sr$_2$RuO$_4$ in the so-called 3-Kelvin phase. In Pb/Ru/Sr$_2$RuO$_4$ Josephson junctions as realized by Maeno et al., a Pb film is indirectly coupled to the superconductor Sr$_2$RuO$_4$ mediated by the proximity-induced superconducting Ru-inclusions, yielding an extended Josephson contact through the interface between Ru and Sr$_2$RuO$_4$. Motivated by this experimental setup, we formulate a sine-Gordon model for the Josephson phase of the interface, assuming a simple cylindrical shape for the Ru-inclusion hosting the proximity-induced $s$-wave superconducting phase. Considering the Sr$_2$RuO$_4$ as a chiral $p$-wave superconductor, we discuss two types of Josephson junctions, a frustrated one due to the nature of the order parameter in Sr$_2$RuO$_4$, and an unfrustrated one for the topologically trivial 3-Kelvin phase. While the latter situation displays standard junction behavior, the former yields an unusual limiting mechanism for the critical current, based on a pinning-depinning transition of a spontaneously induced magnetic flux driven by an externally applied current. We analyze different coupling limits and show that different critical currents can arise for the two topologies. This concept fits well to recent experimental data obtained for the above setup showing an anomalous temperature dependence of the critical current at the transition temperature $T_c $ of bulk Sr$_2$RuO$_4$.

Water Drop Friction on Superhydrophobic Surfaces

To investigate water drop friction on superhydrophobic surfaces, the motion of water drops on three different superhydrophobic surfaces has been studied by allowing drops to slide down an incline and capturing their motion using high-speed video. Two surfaces were prepared using crystallization of an alkyl ketene dimer (AKD) wax, and the third surface was the leaf of a Lotus (Nelumbo Nucifera). The acceleration of the water droplets on these superhydrophobic surfaces was measured as a function of droplet size and inclination of the surface. For small capillary numbers, we propose that the energy dissipation is dominated by intermittent pinning-depinning transitions at microscopic pinning sites along the trailing contact line of the drop, while at capillary numbers exceeding a critical value, energy dissipation is dominated by circulatory flow in the vicinity of the contacting disc between the droplet and the surface. By combining the results of the droplet acceleration with a theoretical model based on energy dissipation, we have introduced a material-specific coefficient called the superhydrophobic sliding resistance, b(sh). Once determined, this parameter is sufficient for predicting the motion of water drops on superhydrophobic surfaces of a general macroscopic topography. This theory also infers the existence of an equilibrium sliding angle, β(eq), at which the drop acceleration is zero. This angle is decreasing with the radius of the drop and is in quantitative agreement with the measured tilt angles required for a stationary drop to start sliding down an incline.

Pinning-depinning transition of fronts between standing waves

Dynamic behaviors of fronts connecting standing waves, such as the locking phenomenon, pinning-depinning transitions, propagation, and front interactions, are studied. Two systems are considered, a vertically driven pendulum chain and a generalized φ(4) model. Both models exhibit in an appropriate region of parameters bistability between standing waves. In the driven pendulum chain, using a Galerkin expansion we characterize the region of bistability between subharmonic waves for the upright and the upside-down pendulum states. We derive analytically the front dynamics in the generalized φ(4) model, showing regions where fronts are oscillatory or propagative. We also characterize the mechanism of the pinning-depinning transition of fronts between standing waves. Using front interactions we predict the emergence of dissipative localized waves supported on a standing wave and characterize their corresponding homoclinic snaking bifurcation diagrams. All these analytical predictions are confirmed by numerical simulations with quite good agreement.

Symmetry-induced pinning-depinning transition of a subharmonic wave pattern

The stationary to drifting transition of a subharmonic wave pattern is studied in the presence of inhomogeneities and drift forces as the pattern wavelength is comparable with the system size. We consider a pinning-depinning transition of stationary subharmonic waves in a tilted quasi-one-dimensional fluidized shallow granular bed driven by a periodic air flow in a small cell. The transition is mediated by the competition of the inherent periodicity of the subharmonic pattern, the asymmetry of the system, and the finite size of the cell. Measurements of the mean phase velocity of the subharmonic pattern are in good agreement with those inferred from an amplitude equation, which takes into account asymmetry and finite-size effects of the system, emphasizing the main ingredients and mechanism of the transition.