Erratum: Large-scale kinetic roughening behavior of coffee-ring fronts [Phys. Rev. E 106, 044801 (2022)].

This corrects the article DOI: 10.1103/PhysRevE.106.044801.

Universal interface fluctuations in the contact process.

We study the interface representation of the contact process at its directed-percolation critical point, where the scaling properties of the interface can be related to those of the original particle model. Interestingly, such a behavior happens to be intrinsically anomalous and more complex than that described by the standard Family-Vicsek dynamic scaling Ansatz of surface kinetic roughening. We expand on a previous numerical study by Dickman and Muñoz [Phys. Rev. E 62, 7632 (2000)10.1103/PhysRevE.62.7632] to fully characterize the kinetic roughening universality class for interface dimensions d=1,2, and 3. Beyond obtaining scaling exponent values, we characterize the interface fluctuations via their probability density function (PDF) and covariance, seen to display universal properties which are qualitatively similar to those recently assessed for the Kardar-Parisi-Zhang (KPZ) and other important universality classes of kinetic roughening. Quantitatively, while for d=1 the interface covariance seems to be well described by the KPZ, Airy_{1} covariance, no such agreement occurs in terms of the fluctuation PDF or the scaling exponents.

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, ß≃1, and z≃1.2. For 00.01 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

Spreading fronts of wetting liquid droplets: Microscopic simulations and universal fluctuations.

We have used kinetic Monte Carlo (kMC) simulations of a lattice gas to study front fluctuations in the spreading of a nonvolatile liquid droplet onto a solid substrate. Our results are consistent with a diffusive growth law for the radius of the precursor layer, R∼t^{δ}, with δ≈1/2 in all the conditions considered for temperature and substrate wettability, in good agreement with previous studies. The fluctuations of the front exhibit kinetic roughening properties with exponent values which depend on temperature T, but become T independent for sufficiently high T. Moreover, strong evidence of intrinsic anomalous scaling has been found, characterized by different values of the roughness exponent at short and large length scales. Although such a behavior differs from the scaling properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class, the front covariance and the probability distribution function of front fluctuations found in our kMC simulations do display KPZ behavior, agreeing with simulations of a continuum height equation proposed in this context. However, this equation does not feature intrinsic anomalous scaling, at variance with the discrete model.

Surface nanopatterning by ion beam irradiation: compositional effects.

Surface nanopatterning induced by ion beam irradiation (IBI) has emerged as an effective nanostructuring technique since it induces patterns on large areas of a wide variety of materials, in short time, and at low cost. Nowadays, two main subfields can be distinguished within IBI nanopatterning depending on the irrelevant or relevant role played by the surface composition. In this review, we give an up-dated account of the progress reached when surface composition plays a relevant role, with a main focus on IBI surface patterning with simultaneous co-deposition of foreign atoms. In addition, we also review the advances in IBI of compound surfaces as well as IBI systems where the ion employed is not a noble gas species. In particular, for the IBI with concurrent metal co-deposition, we detail the chronological evolution of these studies because it helps us to clarify some contradictory early reports. We describe the main patterns obtained with this technique as a function of the foreign atom deposition pathway, also focusing in those systematic studies that have contributed to identify the main mechanisms leading to the surface pattern formation and development. Likewise, we explain the main theoretical models aimed at describing these nanopattern formation processes. Finally, we address two main special features of the patterns induced by this technique, namely, the enhanced pattern ordering and the possibility to produce both morphological and chemical patterns.

Concurrent segregation and erosion effects in medium-energy iron beam patterning of silicon surfaces.

We have bombarded crystalline silicon targets with a 40 keV Fe+ ion beam at different incidence angles. The resulting surfaces have been characterized by atomic force, current-sensing and magnetic force microscopies, scanning electron microscopy, and x-ray photoelectron spectroscopy. We have found that there is a threshold angle smaller than 40° for the formation of ripple patterns, which is definitely lower than those frequently reported for noble gas ion beams. We compare our observations with estimates of the value of the critical angle and of additional basic properties of the patterning process, which are based on a continuum model whose parameters are obtained from binary collision simulations. We have further studied experimentally the ripple structures and measured how the surface slopes change with the ion incidence angle. We explore in particular detail the fluence dependence of the pattern for an incidence angle value (40°) close to the threshold. Initially, rimmed holes appear randomly scattered on the surface, which evolve into large, bug-like structures. Further increasing the ion fluence induces a smooth, rippled background morphology. By means of microscopy techniques, a correlation between the morphology of these structures and their metal content can be unambiguously established.

Morphological stabilization and KPZ scaling by electrochemically induced co-deposition of nanostructured NiW alloy films.

We have assessed the stabilizing role that induced co-deposition has in the growth of nanostructured NiW alloy films by electrodeposition on polished steel substrates, under pulsed galvanostatic conditions. We have compared the kinetic roughening properties of NiW films with those of Ni films deposited under the same conditions, as assessed by Atomic Force Microscopy. The surface morphologies of both systems are super-rough at short times, but differ at long times: while a cauliflower-like structure dominates for Ni, the surfaces of NiW films display a nodular morphology consistent with more stable, conformal growth, whose height fluctuations are in the Kardar-Parisi-Zhang universality class of rough two-dimensional interfaces. These differences are explained by the mechanisms controlling surface growth in each case: mass transport through the electrolyte (Ni) and attachment of the incoming species to the growing interface (NiW). Thus, the long-time conformal growth regime is characteristic of electrochemical induced co-deposition under current conditions in which surface kinetics is hindered due to a complex reaction mechanism. These results agree with a theoretical model of surface growth in diffusion-limited systems, in which the key parameter is the relative importance of mass transport with respect to the kinetics of the attachment reaction.

Universal behavior of crystalline membranes: Crumpling transition and Poisson ratio of the flat phase.

We revisit the universal behavior of crystalline membranes at and below the crumpling transition, which pertains to the mechanical properties of important soft and hard matter materials, such as the cytoskeleton of red blood cells or graphene. Specifically, we perform large-scale Monte Carlo simulations of a triangulated two-dimensional phantom network which is freely fluctuating in three-dimensional space. We obtain a continuous crumpling transition characterized by critical exponents which we estimate accurately through the use of finite-size techniques. By controlling the scaling corrections, we additionally compute with high accuracy the asymptotic value of the Poisson ratio in the flat phase, thus characterizing the auxetic properties of this class of systems. We obtain agreement with the value which is universally expected for polymerized membranes with a fixed connectivity.

Fully nonlinear dynamics of stochastic thin-film dewetting.

The spontaneous formation of droplets via dewetting of a thin fluid film from a solid substrate allows materials nanostructuring. Often, it is crucial to be able to control the evolution, and to produce patterns characterized by regularly spaced droplets. While thermal fluctuations are expected to play a role in the dewetting process, their relevance has remained poorly understood, particularly during the nonlinear stages of evolution that involve droplet formation. Within a stochastic lubrication framework, we show that thermal noise substantially influences the process of droplets formation. Stochastic systems feature a smaller number of droplets with a larger variability in size and space distribution, when compared to their deterministic counterparts. Finally, we discuss the influence of stochasticity on droplet coarsening for asymptotically long times.

Macroscopic response to microscopic intrinsic noise in three-dimensional Fisher fronts.

We study the dynamics of three-dimensional Fisher fronts in the presence of density fluctuations. To this end we simulate the Fisher equation subject to stochastic internal noise, and study how the front moves and roughens as a function of the number of particles in the system, N. Our results suggest that the macroscopic behavior of the system is driven by the microscopic dynamics at its leading edge where number fluctuations are dominated by rare events. Contrary to naive expectations, the strength of front fluctuations decays extremely slowly as 1/logN, inducing large-scale fluctuations which we find belong to the one-dimensional Kardar-Parisi-Zhang universality class of kinetically rough interfaces. Hence, we find that there is no weak-noise regime for Fisher fronts, even for realistic numbers of particles in macroscopic systems.

Pattern-wavelength coarsening from topological dynamics in silicon nanofoams.

We report the experimental observation of a submicron cellular structure on the surface of silicon targets eroded by an ion plasma. Analysis by atomic force microscopy allows us to assess the time evolution and show that the system can be described quantitatively by the convective Cahn-Hilliard equation, found in the study of domain coarsening for a large class of driven systems. The space-filling trait of the ensuing pattern relates it to evolving foams. Through this connection, we are actually able to derive the coarsening law for the pattern wavelength from the nontrivial topological dynamics of the cellular structure. Thus, the study of the topological properties of patterns in nonvariational spatially extended systems emerges as complementary to morphological approaches to their challenging coarsening properties.

Independence of interrupted coarsening on initial system order: ion-beam nanopatterning of amorphous versus crystalline silicon targets.

Interrupted coarsening (IC) has recently been identified as an important feature for the dynamics of the typical length-scale in pattern-forming systems on surfaces. In practice, it can be beneficial to improve pattern ordering since it combines a certain degree of defect suppression with a limited increase in the typical pattern wavelength. However, little is known about its robustness with respect to changes in the preparation of the initial system for cases with potential applications. Working in the context of nano-scale pattern formation by ion-beam sputtering (IBS), we prove that IC properties do not depend on sample preparation. Specifically, interface dynamics under IBS is quantitatively compared on virgin amorphous and crystalline silicon surfaces, using 1 keV Ar(+) ions at normal incidence where nanodot pattern formation is triggered by concurrent co-deposition of Fe atoms during processing. Atomic force microscopy shows that dot patterns with similar spatial order and dynamics are obtained in both cases, underscoring the key dynamical role of the amorphous surface layer produced by irradiation. Both systems have been quantitatively described by an effective interface equation. We employ a new procedure based on the linear growth of the initial surface correlations to accurately estimate the equation coefficients. Such a method improves the predictive power of the interface equation with respect to previous studies and leads to a better description of the experimental pattern and its dynamical features.

Pattern formation in stromatolites: insights from mathematical modelling.

To this day, computer models for stromatolite formation have made substantial use of the Kardar-Parisi-Zhang (KPZ) equation. Oddly enough, these studies yielded mutually exclusive conclusions about the biotic or abiotic origin of such structures. We show in this paper that, at our current state of knowledge, a purely biotic origin for stromatolites can neither be proved nor disproved by means of a KPZ-based model. What can be shown, however, is that whatever their (biotic or abiotic) origin might be, some morphologies found in actual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a process modelled exclusively in terms of the KPZ equation and acting over sufficiently large times. This suggests the need to search for alternative mathematical approaches to model these structures, some of which are discussed in this paper.

One-dimensional pattern of Au nanodots by ion-beam sputtering: formation and mechanism.

Highly ordered one-dimensional arrays of nanodots, or nanobeads, are fabricated by forming nanoripples and nanodots in sequence, entirely by ion-beam sputtering (IBS) of Au(001). This demonstrates the capability of IBS for the fabrication of sophisticated nanostructures via hierarchical self-assembly. The intricate nanobead pattern ideally serves to identify the governing mechanisms for the pattern formation: nonlinear effects, especially local redeposition and surface-confined transport, are essential both for the formation and the preservation of the one-dimensional order of the nanobead pattern.

Single-phase-field model of stepped surfaces.

We formulate a phase-field description of step dynamics on vicinal surfaces that makes use of a single dynamical field, at variance with previous analogous works in which two coupled fields are employed, namely, a phase-field proper plus the physical adatom concentration. Within an asymptotic sharp interface limit, our formulation is shown to retrieve the standard Burton-Cabrera-Frank model in the general case of asymmetric attachment coefficients (Ehrlich-Schwoebel effect). We confirm our analytical results by means of numerical simulations of our phase-field model. Our present formulation seems particularly well adapted to generalization when additional physical fields are required.

Phase transition in tensionless surfaces.

We study the critical behavior of the Laplacian roughening model, which describes the growth of tensionless surfaces. This type of growth phenomena is very important, for instance, in biological membranes and in molecular beam epitaxy. We summarize previous analytical and numerical results and point out their contradictions and differences, thus making clear the context of our work. Our contribution, achieved through large-scale numerical simulations, is the confirmation that the model exhibits a single continuous phase transition: the transition takes place between a continuum massless (i.e., with infinite correlation length) bilaplacian behavior and a massive propagator (finite correlation length).

Transients due to instabilities hinder Kardar-Parisi-Zhang scaling: a unified derivation for surface growth by electrochemical and chemical vapor deposition.

We propose a unified moving boundary problem for surface growth by electrochemical and chemical vapor deposition, which is derived from constitutive equations into which stochastic forces are incorporated. We compute the coefficients in the interface equation of motion as functions of phenomenological parameters. The equation features the Kardar-Parisi-Zhang (KPZ) nonlinearity and instabilities which, depending on surface kinetics, can hinder the asymptotic KPZ scaling. Our results account for the universality and the experimental scarcity of KPZ scaling in the growth processes considered.

Variational mean-field study of a continuum model of crystalline tensionless surfaces.

We study analytically the equilibrium and near-equilibrium properties of a model of a d-dimensional surface relaxing via linear surface diffusion and subject to a lattice potential. We employ the variational mean-field formalism introduced by Saito for the study of the sine-Gordon model. In equilibrium, our variational theory predicts a first-order roughening transition between a flat low-temperature phase and a rough high-temperature phase with the properties of the linear molecular-beam epitaxy equation. Moreover, the study of a Gaussian approximation to the Langevin dynamics of the system indicates that the surface shows hysteresis when temperature is continuously tuned. Out of equilibrium, these approximate Langevin dynamics show that the surface mobility can have different behaviors as a function of a driving flux. Some considerations are made regarding different underlying lattices, and connections are drawn to related models or different approaches to the same model we study.

Multiparticle biased diffusion-limited aggregation with surface diffusion: A comprehensive model of electrodeposition

We present a complete study of the multiparticle biased diffusion-limited aggregation (MBDLA) model supplemented with surface diffusion (SD), focusing on the relevance and effects of the latter transport mechanism. By comparing different algorithms, we show that MBDLA + SD is a very good qualitative model for electrodeposition in essentially the whole range of current intensities provided one introduces SD in the model in the proper fashion. We have found that the correct procedure involves simultaneous bulk diffusion and SD, introducing a time scale arising from the ratio of the rates of the two processes. We discuss in detail the different morphologies obtained and compare them to the available experimental data with very satisfactory results. We also characterize the aggregates thus obtained by means of the dynamic scaling exponents of the interface height, allowing us to distinguish several regimes in the mentioned interface growth. Our asymptotic scaling exponents are again in good agreement with recent experiments. We conclude by discussing a global picture of the influence and consequences of SD in electrodeposition.