Microstructure and surface scaling in ballistic deposition at oblique incidence

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 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 Collapse

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Affiliation(s)

B G Barreales Departamento de Física, Universidad de Extremadura, 06006 Badajoz, Spain Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura, 06006 Badajoz, Spain
J J Meléndez Departamento de Física, Universidad de Extremadura, 06006 Badajoz, Spain Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura, 06006 Badajoz, Spain
R Cuerno Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III de Madrid, 28911 Leganés, Spain
J J Ruiz-Lorenzo Departamento de Física, Universidad de Extremadura, 06006 Badajoz, Spain Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura, 06006 Badajoz, Spain Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain

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Evolution in range expansions with competition at rough boundaries

When a biological population expands into new territory, genetic drift develops an enormous influence on evolution at the propagating front. In such range expansion processes, fluctuations in allele frequencies occur through stochastic spatial wandering of both genetic lineages and the boundaries between genetically segregated sectors. Laboratory experiments on microbial range expansions have shown that this stochastic wandering, transverse to the front, is superdiffusive due to the front's growing roughness, implying much faster loss of genetic diversity than predicted by simple flat front diffusive models. We study the evolutionary consequences of this superdiffusive wandering using two complementary numerical models of range expansions: the stepping stone model, and a new interpretation of the model of directed paths in random media, in the context of a roughening population front. Through these approaches we compute statistics for the times since common ancestry for pairs of individuals with a given spatial separation at the front, and we explore how environmental heterogeneities can locally suppress these superdiffusive fluctuations.

Shape of shortest paths in random spatial networks

In the classic model of first-passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order L^{χ}, while the transversal fluctuations, known as wandering, grow as L^{ξ}. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ=3/5 and χ=1/5, or ξ=7/10 and χ=2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel-time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first-passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbor random geometric graphs, where via Monte Carlo simulations we find ξ=0.60±0.01 and χ=0.20±0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β skeletons and the Delaunay triangulation and are characterized by the values ξ=0.70±0.01 and χ=0.40±0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the so-called Kardar-Parisi-Zhang (KPZ) relation χ=2ξ-1 is satisfied for all these models. These results shed some light on the Euclidean first-passage process but also raise some theoretical questions about the scaling laws and the derivation of the exponent values and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large-deviation regime

The following question is the subject of our work: could a two-dimensional (2D) random path pushed by some constraints to an improbable "large-deviation regime" possess extreme statistics with one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though nonuniversal, since the fluctuations depend on the underlying geometry. We consider in detail two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span 〈d(t)〉∼t^{γ} of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ=1/3, i.e., 〈d(t)〉 shares the KPZ statistics for the semicircle, while γ=0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations, and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

Interface Roughness: What is it and How is it Measured?

A panel discussion on interface roughness was held at the Fall 1992 Materials Research Society meeting. We present a summary of the results presented by the invited speakers on the application and interpretation of X-ray reflectivity, atomic force microscopy (AFM), scanning tunneling microscopy (STM), photoluminescence and transmission electron microscopy. A transcript of the moderated discussion is provided in the final section.

Ballistic deposition patterns beneath a growing Kardar-Parisi-Zhang interface

We consider a (1+1)-dimensional ballistic deposition process with next-nearest-neighbor interactions, which belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The focus of our analysis is on the properties of structures appearing in the bulk of a growing aggregate: a forest of independent clusters separated by "crevices." Competition for growth (mutual screening) between different clusters results in "thinning" of this forest, i.e., the number density c(h) of clusters decreases with the height h of the pattern. For the discrete stochastic equation describing the process we introduce a variational formulation similar to that used for the randomly forced continuous Burgers equation. This allows us to identify the "clusters" and crevices with minimizers and shocks in the Burgers turbulence. Capitalizing on the ideas developed for the latter process, we find that c(h) ∼ h(-α) with α=2/3. We compute also scaling laws that characterize the ballistic deposition patterns in the bulk: the law of transversal fluctuations of cluster boundaries and the size distribution of clusters. It turns out that the intercluster interface is superdiffusive: the corresponding exponent is twice as large as the KPZ exponent for the surface of the aggregate. Finally we introduce a probabilistic concept of ballistic growth, dubbed the "hairy" Airy process in view of its distinctive geometric features. Its statistical properties are analyzed numerically.

Ballistic deposition on deterministic fractals: observation of discrete scale invariance

The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height (h_{s}) and width (w_{s}) of the trees of size s obey power laws of the form h_{s} proportional, variants;{nu_{ parallel}} and w_{s} proportional, variants;{nu_{ perpendicular}} , respectively. Also, the tree-size distribution (n_{s}) behaves according to n_{s} approximately s;{-tau} . Here, nu_{ parallel} and nu_{ perpendicular} are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, tau is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.

Strongly anisotropic roughness in surfaces driven by an oblique particle flux

Using field theoretic renormalization, an MBE-type growth process with an obliquely incident influx of atoms is examined. The projection of the beam on the substrate plane selects a "parallel" direction, with rotational invariance restricted to the transverse directions. Depending on the behavior of an effective anisotropic surface tension, a line of second-order transitions is identified, as well as a line of potentially first-order transitions, joined by a multicritical point. Near the second-order transitions and the multicritical point, the surface roughness is strongly anisotropic. Four different roughness exponents are introduced and computed, describing the surface in different directions, in real or momentum space. The results presented challenge an earlier study of the multicritical point.

Numerical study of the development of bulk scale-free structures upon growth of self-affine aggregates

During the last decade, self-affine geometrical properties of many growing aggregates, originated in a wide variety of processes, have been well characterized. However, little progress has been achieved in the search of a unified description of the underlying dynamics. Extensive numerical evidence is given showing that the bulk of aggregates formed upon ballistic aggregation and random deposition with surface relaxation processes can be broken down into a set of infinite scale invariant structures called "trees." These two types of aggregates have been selected because it has been established that they belong to different universality classes: those of Kardar-Parisi-Zhang and Edward-Wilkinson, respectively. Exponents describing the spatial and temporal scale invariance of the trees can be related to the classical exponents describing the self-affine nature of the growing interface. Furthermore, those exponents allow us to distinguish either the compact or noncompact nature of the growing trees. Therefore, the measurement of the statistic of the process of growing trees may become a useful experimental technique for the evaluation of the self-affine properties of some aggregates.

Dynamical scaling behavior in two-dimensional ballistic deposition with shadowing

The dynamical scaling behavior in two-dimensional ballistic deposition with shadowing is studied as a function of the angular distribution of incoming particles and of the underlying lattice structure. Using a dynamical scaling form for the surface box number, results for the scaling of the surface fractal dimension are also presented. Our results indicate that, in addition to the usual self-affine universality class corresponding to vertical deposition, there exist at least two additional universality classes characterized by distinct values of the coarsening and roughening exponents p and beta describing the evolution of the lateral feature size and surface roughness with film thickness, as well as the surface fractal dimension D(f). For the case of a uniform angular distribution corresponding to an anisotropic flux, we find p=beta=1 and D(f) approximately 1.7. However, for ballistic deposition with an isotropic flux (corresponding to a "cosine" angular distribution), we find p approximately 2/3 and D(f) approximately 1.5 while the effective roughening exponent beta approximately 0.52-0.64 was found to be slightly lattice dependent. In both cases, anomalous scaling of the height-height correlation function is also observed. In contrast, vertical deposition leads to a self-affine surface with p=2/3, beta=1/3, and D(f)=1. The asymptotic scaling behavior appears to depend on the behavior of the angular distribution at large angles but does not depend on other details. An analysis that clarifies the relationship between the launch angle distribution used in the simulations and the flux distribution is also presented.

Aggregation and chimney formation during the solidification of ammonium chloride

Experiments study large-scale pattern formation during the growth of ammonium chloride (NH4Cl) from solution in a thin (Hele-Shaw) geometry. In particular a solid-liquid mixture ("mushy layer") forms in which growing solid NH4Cl crystals form a solid network interspersed with liquid. There are different ways that the mushy layer can be formed, however. If the cell is heated from below and cooled from above, thermal convection generates large-scale recirculating flows that carry seed crystals from the upper (cold) boundary to the (warmer) side and bottom boundaries. Ballistic deposition of these seed crystals leads to aggregation patterns with significant voids (filled with liquid) with a wide range of length scales. If the cell is cooled from below with a warm environment, the solid NH4Cl grows dendritically without deposition, resulting in a compact mushy layer. Plume convection within this mushy layer produces one or two well-defined "chimneys." If the environment is cool (comparable to the liquidus temperature of the solution), the mushy layer forms by a combination of dendritic growth and ballistic deposition, resulting in a more permeable mushy layer and enhanced chimney formation. The effects of ballistic deposition are enhanced if the cell is tipped, in which case the voids reappear. Plume convection and chimney formation are dramatically enhanced in this case. Additional experiments are done in which fluid flows in the system are enhanced artificially to verify that enhancements in chimney formation are due primarily to the aggregation process, and not to the increases in fluid flows due to thermal and compositional convection.