Showing papers by "Albert-László Barabási published in 1995"

Fractal Concepts in Surface Growth

Abstract: This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html

Dynamic scaling of ion-sputtered surfaces.

Abstract: We derive a stochastic nonlinear equation to describe the evolution and scaling properties of surfaces eroded by ion bombardment. The coefficients appearing in the equation can be calculated explicitly in terms of the physical parameters characterizing the sputtering process. We find that transitions may take place between various scaling behaviors when experimental parameters, such as the angle of incidence of the incoming ions or their average penetration depth, are varied.

Scaling properties of driven interfaces in disordered media

Abstract: We perform a systematic study of several models that have been proposed for the purpose of understanding the motion of driven interfaces in disordered media. We identify two distinct universality classes. (i) One of these, referred to as directed percolation depinning (DPD), can be described by a Langevin equation similar to the Kardar-Parisi-Zhang equation, but with quenched disorder. (ii) The other, referred to as quenched Edwards-Wilkinson (QEW), can be described by a Langevin equation similar to the Edwards-Wilkinson equation but with quenched disorder. We find that for the DPD universality class, the coefficient \ensuremath{\lambda} of the nonlinear term diverges at the depinning transition, while for the QEW universality class, either \ensuremath{\lambda}=0 or \ensuremath{\lambda}\ensuremath{\rightarrow}0 as the depinning transition is approached. The identification of the two universality classes allows us to better understand many of the results previously obtained experimentally and numerically. However, we find that some results cannot be understood in terms of the exponents obtained for the two universality classes at the depinning transition. In order to understand these remaining disagreements, we investigate the scaling properties of models in each of the two universality classes above the depinning transition. For the DPD universality class, we find for the roughness exponent ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$=0.63\ifmmode\pm\else\textpm\fi{}0.03 for the pinned phase and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{M}}$=0.75\ifmmode\pm\else\textpm\fi{}0.05 for the moving phase.For the growth exponent, we find ${\mathrm{\ensuremath{\beta}}}_{\mathit{P}}$=0.67\ifmmode\pm\else\textpm\fi{}0.05 for the pinned phase and ${\mathrm{\ensuremath{\beta}}}_{\mathit{M}}$=0.74\ifmmode\pm\else\textpm\fi{}0.06 for the moving phase. Furthermore, we find an anomalous scaling of the prefactor of the width on the driving force. A new exponent ${\mathit{cphi}}_{\mathit{M}}$=-0.12\ifmmode\pm\else\textpm\fi{}0.06, characterizing the scaling of this prefactor, is shown to relate the values of the roughness exponents on both sides of the depinning transition. For the QEW universality class, we find that ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\alpha}}}_{\mathit{M}}$=0.92\ifmmode\pm\else\textpm\fi{}0.04 and ${\mathrm{\ensuremath{\beta}}}_{\mathit{P}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\beta}}}_{\mathit{M}}$=0.86\ifmmode\pm\else\textpm\fi{}0.03 are roughly the same for both the pinned and moving phases. Moreover, we again find a dependence of the prefactor of the width on the driving force. For this universality class, the exponent ${\mathit{cphi}}_{\mathit{M}}$=0.44\ifmmode\pm\else\textpm\fi{}0.05 is found to relate the different values of the local ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$ and global roughness exponent ${\mathrm{\ensuremath{\alpha}}}_{\mathit{G}}$\ensuremath{\approxeq}1.23\ifmmode\pm\else\textpm\fi{}0.04 at the depinning transition. These results provide us with a more consistent understanding of the scaling properties of the two universality classes, both at and above the depinning transition. We compare our results with all the relevant experiments.

Avalanches and the directed percolation depinning model: Experiments, simulations, and theory.

Abstract: We study the recently-introduced directed percolation depinning (DPD) model for interface roughening with quenched disorder for which the interface becomes pinned by a directed percolation (DP) cluster for $d = 1$, or a directed surface (DS) for $d > 1$. The mapping to DP enables us to predict some of the critical exponents of the growth process. For the case of $(1+1)$ dimensions, the theory predicts that the roughness exponent $\alpha$ is given by $\alpha = u_{\perp} / u_{\parallel}$, where $ u_{\perp}$ and $ u_{\parallel}$ are the exponents governing the divergence of perpendicular and parallel correlation lengths of the DP incipient infinite cluster. The theory also predicts that the dynamical exponent $z$ equals the exponent $d_{\rm min}$ characterizing the scaling of the shortest path on a isotropic percolation cluster. The exponent $\alpha$ decreases monotonically with $d$ but does not seem to approach zero for any dimension calculated ($d \le 6$), suggesting that the DPD model has no upper critical dimension for the static exponents. On the other hand, $z$ appears to approach $2$ as $d \rightarrow 6$, as expected by the result $z = d_{\rm min}$, suggesting that $d_c = 6$ for the dynamics. We also perform a set of imbibition experiments, in both $(1+1)$ and $(2+1)$ dimensions, that can be used to test the DPD model. We find good agreement between experimental, theoretical and numerical approaches. Further, we study the properties of avalanches in the context of the DPD model. We relate the scaling properties of the avalanches in the DPD model to the scaling properties for the self-organized depinning (SOD) model, a variant of the DPD model. We calculate the exponent characterizing the avalanches distribution $\tau_{\rm aval}$ for $d = 1$ to $d = 6$, and compare our results with recent theoretical

Why are computer simulations of growth useful

Abstract: We show how computer simulations can give unique information on the growth of nanostructures and thin films. Specifically, they can predict the morphologies and the island size distributions corresponding to different growth mechanisms. This information cannot be obtained from other approaches such as mean-field mathematical theories or scaling analysis. Special attention is given to the effects of small cluster mobility on experimental results.

Dynamics of Random Networks: Connectivity and First Order Phase Transitions

Abstract: The connectivity of individual neurons of large neural networks determine both the steady state activity of the network and its answer to external stimulus. Highly diluted random networks have zero activity. We show that increasing the network connectivity the activity changes discontinuously from zero to a finite value as a critical value in the connectivity is reached. Theoretical arguments and extensive numerical simulations indicate that the origin of this discontinuity in the activity of random networks is a first order phase transition from an inactive to an active state as the connectivity of the network is increased.

Fractal and Non-Fractal Surfaces in Ion Sputtering

Abstract: Recently a number of experimental studies focusing on the scaling properties of surfaces eroded by ion bombardment provided apparently contradictory results. A number of experiments report the observation of self-affine fractal surfaces, while others provide evidence about the development of a non-fractal periodic ripple structure. To explain these discrepancies, here we derive a stochastic nonlinear equation that describes the evolution and scaling properties of surfaces eroded by ion bombardment. The coefficients appearing in the equation can be calculated explicitly in terms of the physical parameters characterizing the sputtering process. We find that transitions may take place between various scaling behaviors when experimental parameters, such as the angle of incidence of the incoming ions or their average penetration depth, are varied.

A fractal model for the first stages of thin film growth

Abstract: In this paper we briefly review a model that describes the diffusion-controlled aggregation exhibited by particles as they are deposited on a surface. This model allows to understand many experiments of thin film deposition. In the first part, we describe the model, which incorporates deposition, particle and cluster diffusion, and aggregation. In a second part, we study the dynamical evolution of the model. Finally, we analyze the effects of small cluster mobility, and we show that the introduction of cluster diffusion dramatically affects the dynamics of film growth. Some of these effects can be tested experimentally.