We study the steady state of driven elastic strings in disordered media below the depinning threshold. In the low-temperature limit, for a fixed sample, the steady state is dominated by a single configuration, which we determine exactly from the transition pathways between metastable states. We obtain the dynamical phase diagram in this limit. At variance with a thermodynamic phase transition, the depinning transition is not associated with a divergent length scale of the steady state below threshold, but only of the transient dynamics. We discuss the distribution of barrier heights, and check the validity of the dynamic phase diagram at small but finite temperatures using Langevin simulations. The phase diagram continues to hold for broken statistical tilt symmetry. We point out the relevance of our results for experiments of creep motion in elastic interfaces.
We study the creep motion of an elastic string in a two dimensional pinning landscape by Langevin dynamics simulations. We find that the Velocity-Force characteristics are well described by the creep formula predicted from phenomenological scaling arguments. We analyze the creep exponent $μ$, and the roughness exponent $ζ$. Two regimes are identified: when the temperature is larger than the strength of the disorder we find $μ\approx 1/4$ and $ζ\approx 2/3$, in agreement with the quasi-equilibrium-nucleation picture of creep motion; on the contrary, lowering enough the temperature, the values o…
The thermally activated creep motion of an elastic interface weakly driven on a disordered landscape is one of the best examples of glassy universal dynamics. Its understanding has evolved over the last 30 years thanks to a fruitful interplay between elegant scaling arguments, sophisticated analytical calculations, efficient optimization algorithms and creative experiments. In this article, starting from the pioneer arguments, we review the main theoretical and experimental results that lead to the current physical picture of the creep regime. In particular, we discuss recent works unveiling t…
The slow motion of a crack line is studied via an experiment in which sheets of paper are split into two halves in a ``peel-in-nip'' (PIN) geometry under a constant load, in creep. The velocity-force relation is exponential. The dynamics of the fracture line exhibits intermittency, or avalanches, which are studied using acoustic emission. The energy statistics is a power-law, with the exponent $β\sim 1.8 \pm 0.1$. Both the waiting times between subsequent events and the displacement of the fracture line imply complicated stick-slip dynamics. We discuss the correspondence to tensile PIN tests a…
We study the slow stochastic dynamics near the depinning threshold of an elastic interface in a random medium by solving a particularly suited model of hopping interacting particles that belongs to the quenched-Edwards-Wilkinson depinning universality class. The model allows us to compare the cases of uniformly activated and Arrhenius activated hops. In the former case, the velocity accurately follows a standard scaling law of the force and noise intensity with the analog of the thermal rounding exponent satisfying a modified "hyperscaling" relation. For the Arrhenius activation, we find, both…
We discuss the universal dynamics of elastic interfaces in quenched random media. We focus in the relation between the rough geometry and collective transport properties in driven steady-states. Specially devised numerical algorithms allow us to analyze the equilibrium, creep, and depinning regimes of motion in minimal models. The relevance of our results for understanding domain wall experiments is outlined.
We review in these notes some dynamical properties of interfaces in random media submitted to an external force. We focuss in particular to the response to a very small force (so called creep motion) and discuss various theoretical aspects of this problem. We consider in details in particular the case of a one dimensional interface (domain wall).
The dynamics of micrometer-sized magnetic domains in ultra-thin ferromagnetic films is so dramatically slowed down by quenched disorder that the spontaneous elastic tension collapse becomes unobservable at ambient temperature. By magneto-optical imaging we show that a weak zero-bias AC magnetic field can assist such curvature-driven collapse, making the area of a bubble to reduce at a measurable rate, in spite of the negligible effect that the same curvature has on the average creep motion driven by a comparable DC field. An analytical model explains this phenomenon quantitatively.
Granular materials such as sand, powders, and grains are omnipresent in daily life, industrial applications, and earth-science [1]. When unperturbed, they form stable structures that resemble the ones of other amorphous solids like metallic and colloidal glasses [2]. It is commonly conjectured that all these amorphous materials show a universal mechanical response when sheared slowly, i.e., to have an elastic regime, followed by yielding [3]. Here we use X-ray tomography to determine the microscopic dynamics of a cyclically sheared granular system in three dimensions. Independent of the shear …
Elastic systems driven in a disordered medium exhibit a depinning transition at zero temperature and a creep regime at finite temperature and slow drive $f$. We derive functional renormalization group equations which allow to describe in details the properties of the slowly moving states in both cases. Since they hold at finite velocity $v$, they allow to remedy some shortcomings of the previous approaches to zero temperature depinning. In particular, they enable us to derive the depinning law directly from the equation of motion, with no artificial prescription or additional physical assumpti…
Viscoelasticity has been described since the time of Maxwell as an interpolation of purely viscous and purely elastic response, but its microscopic atomic-level mechanism in solids has remained elusive. We studied three model disordered solids: a random lattice, the bond-depleted fcc lattice, and the fcc lattice with vacancies. Within the harmonic approximation for central-force lattices, we applied sum-rules for viscoelastic response derived on the basis of non-affine atomic motions. The latter motions are a direct result of local structural disorder, and in particular, of the lack of inversi…
We study numerically the depinning transition of driven elastic interfaces in a random-periodic medium with localized periodic-correlation peaks in the direction of motion. The analysis of the moving interface geometry reveals the existence of several characteristic lengths separating different length-scale regimes of roughness. We determine the scaling behavior of these lengths as a function of the velocity, temperature, driving force, and transverse periodicity. A dynamical roughness diagram is thus obtained which contains, at small length scales, the critical and fast-flow regimes typical o…
Using a dynamic functional renormalization group treatment of driven elastic interfaces in a disordered medium, we investigate several aspects of the creep-type motion induced by external forces below the depinning threshold $f_c$: i) We show that in the experimentally important regime of forces slightly below $f_c$ the velocity obeys an Arrhenius-type law $v\sim\exp[-U(f)/T]$ with an effective energy barrier $U(f)\propto (f_{c}-f)$ vanishing linearly when f approaches the threshold $f_c$. ii) Thermal fluctuations soften the pinning landscape at high temperatures. Determining the corresponding…
The dynamics of an elastic medium driven through a random medium by a small applied force is investigated in the low-temperature limit where quantum fluctuations dominate. The motion proceeds via tunneling of segments of the manifold through barriers whose size grows with decreasing driving force $f$. In the limit of small drive, at zero-temperature the average velocity has the form $v\propto\exp[-{\rm const.}/\hbar^α f^μ]$. For strongly dissipative dynamics, there is a wide range of forces where the dissipation dominates and the velocity--force characteristics takes the form $v\propto\exp[-S(…
We present a quantitative and comparative study of magnetic field driven domain wall depinning transition in different ferromagnetic ultrathin films over a wide range of temperature. We reveal a universal scaling function accounting for both drive and thermal effects on the depinning transition, including critical exponents. The consistent description we obtain for both the depinning and subthreshold thermally activated creep motion should shed light on the universal glassy dynamics of thermally fluctuating elastic objects pinned by disordered energy landscapes.
We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a $ε=d_c-d$ expansion below the upper critical dimension $d_c$ of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform disp…
Magnetic field-driven domain wall motion in an ultrathin Pt/Co(0.45nm)/Pt ferromagnetic film with perpendicular anisotropy is studied over a wide temperature range. Three different pinning dependent dynamical regimes are clearly identified: the creep, the thermally assisted flux flow and the depinning, as well as their corresponding crossovers. The wall elastic energy and microscopic parameters characterizing the pinning are determined. Both the extracted thermal rounding exponent at the depinning transition, $ψ=$0.15, and the Larkin length crossover exponent, $ϕ=$0.24, fit well with the numer…
Controlled movement of nano-scale stable magnetic objects has been proposed as the foundation for a new generation of magnetic storage devices. Magnetic skyrmions, vortex-like spin textures stabilized by their topology are particularly promising candidates for this technology. Their nanometric size and ability to be displaced in response to an electrical current density several orders of magnitude lower than required to induce motion of magnetic domain walls suggest their potential for high-density memory devices that can be operated at low power. However, to achieve this, skyrmion movement ne…
