Roughness of moving elastic lines: crack and wetting fronts

Dynamics of fluctuating thin sheets under random forcing

We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative (athermal) random forcing. In Steinbock et al. [Phys. Rev. Res. 4, 033096 (2022)2643-156410.1103/PhysRevResearch.4.033096] the static structure factor of such a sheet was studied. In this paper we recap the model developed there and investigate its dynamic properties. Using the self-consistent expansion, the time-dependent two-point function of the height profile is determined and found to decay exponentially in time. Despite strong nonlinear coupling, the decay rate of the dynamic structure factor is found to coincide with the effective coupling constant for the static properties, which suggests that the model under investigation exhibits certain quasilinear behavior. Confirmation of these results by numerical simulations is also presented.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Random close packing from hard-sphere Percus-Yevick theory

The Percus-Yevick theory for monodisperse hard spheres gives very good results for the pressure and structure factor of the system in a whole range of densities that lie within the liquid phase. However, the equation seems to lead to a very unacceptable result beyond that region. Namely, the Percus-Yevick theory predicts a smooth behavior of the pressure that diverges only when the volume fraction η approaches unity. Thus, within the theory there seems to be no indication for the termination of the liquid phase and the transition to a solid or to a glass. In the present article we study the Percus-Yevick hard-sphere pair distribution function, g_{2}(r), for various spatial dimensions. We find that beyond a certain critical volume fraction η_{c}, the pair distribution function, g_{2}(r), which should be positive definite, becomes negative at some distances. We also present an intriguing observation that the critical η_{c} values we find are consistent with volume fractions where onsets of random close packing (or maximally random jammed states) are reported in the literature for various dimensions. That observation is supported by an intuitive argument. This work may have important implications for other systems for which a Percus-Yevick theory exists.

Depinning Dynamics of Crack Fronts

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatiotemporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value v_{m} can be predicted from an overdamped equation of motion (v-v_{m})/v_{0}=[G-G_{c}(v_{m})]/G_{c}(v_{m}) involving the characteristic material speed v_{0}=G_{c}(v_{m})/G_{c}^{'}(v_{m}) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.

Stability and roughness of tensile cracks in disordered materials

We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.

Contact angle hysteresis and meniscus corrugation on randomly heterogeneous surfaces with mesa-type defects

The results of a numerical study of the various characteristics of the static contact of a liquid meniscus with a flat but heterogeneous surface, consisting of two types of homogeneous materials, forming regularly and randomly distributed microscopic defects are presented. The solutions for the meniscus shape are obtained numerically using the full expression of the system free energy functional. The goal is to establish how the magnitude and the limits of the hysteresis interval of the equilibrium contact angle, the Cassie's angle, and the contact line (CL) roughness exponent are related to the parameters, characterizing the heterogeneous surface-the equilibrium contact angles on the two materials and their fractions. We compare the results of different ways of determining the averaged contact angle on heterogeneous surfaces. We study the spread of the CL corrugation along the liquid meniscus. We compare our results with the numerical results, obtained using linearized energy functional, and also with experimental results for the CL roughness exponent. The obtained results support the conclusion that some characteristics depends on the type (regular or random) of the heterogeneity pattern.

Exponent inequalities in dynamical systems

In this Letter, we derive exponent inequalities relating the dynamic exponent z to the steady state exponent Γ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating z and Γ. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems, are discussed.

Reducing the surface roughness beyond the pulsed-laser-deposition limit

Here, we outline the theoretical fundamentals of a promising growth kinetics of films from the vapor phase, in which pulsed fluxes are combined with temperature transients to enable short-range surface relaxations (e.g., species rearrangements) and to inhibit long-range relaxations (atomic exchange between species). A group of physical techniques (fully pulsed thermal and/or laser depositions) based on this kinetics is developed that can be used to prepare films with roughnesses even lower than those obtained with pulsed-laser deposition, which is the physical vapor-phase deposition technique that has produced the flattest films reported so far.