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Carrasco ISS, Oliveira TJ. Dimensional crossover in Kardar-Parisi-Zhang growth. Phys Rev E 2024; 109:L042102. [PMID: 38755819 DOI: 10.1103/physreve.109.l042102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2023] [Accepted: 04/08/2024] [Indexed: 05/18/2024]
Abstract
Two-dimensional (2D) Kardar-Parisi-Zhang (KPZ) growth is usually investigated on substrates of lateral sizes L_{x}=L_{y}, so that L_{x} and the correlation length (ξ) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates L_{x}≠L_{y} and, thus, the surfaces can become correlated in a single direction, when ξ∼L_{x}≪L_{y}. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as W∼t^{β_{2D}} for t≪t_{c} and W∼t^{β_{1D}} for t≫t_{c}, where t_{c}∼L_{x}^{1/z_{2D}}. The height distributions (HDs) also cross over from the 2D flat (cylindrical) HD to the asymptotic Tracy-Widom Gaussian orthogonal ensemble (Gaussian unitary ensemble) distribution. Moreover, 2D to one-dimensional (1D) crossovers are found also in the asymptotic growth velocity and in the steady-state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as L_{y}/L_{x} increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.
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Affiliation(s)
- Ismael S S Carrasco
- International Center of Physics, Institute of Physics, University of Brasilia, 70910-900 Brasilia, Federal District, Brazil
| | - Tiago J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900, Viçosa, Minas Gerais, Brazil
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Carrasco ISS, Oliveira TJ. One-point height fluctuations and two-point correlators of (2+1) cylindrical KPZ systems. Phys Rev E 2023; 107:064140. [PMID: 37464689 DOI: 10.1103/physreve.107.064140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2023] [Accepted: 04/20/2023] [Indexed: 07/20/2023]
Abstract
While the one-point height distributions (HDs) and two-point covariances of (2+1) Kardar-Parisi-Zhang (KPZ) systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat (2+1) KPZ systems, while the latter resembles the Airy_{2} covariance of circular (1+1) KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance A(t/t_{0}) decays asymptotically as A(x)∼x^{-λ[over ¯]} with an exponent λ[over ¯]=β+d^{*}/z, where d^{*} is the number of interface sides kept fixed during the growth (being d^{*}=1 for the systems analyzed here). Overall, these results complete the picture of the main statistics for the (2+1) KPZ class.
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Affiliation(s)
- Ismael S S Carrasco
- University of Brasilia, International Center of Physics, Institute of Physics, 70910-900 Brasilia, Federal District, Brazil
| | - Tiago J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil
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Mozo Luis EE, Oliveira FA, de Assis TA. Accessibility of the surface fractal dimension during film growth. Phys Rev E 2023; 107:034802. [PMID: 37073068 DOI: 10.1103/physreve.107.034802] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Accepted: 03/06/2023] [Indexed: 04/20/2023]
Abstract
Fractal properties on self-affine surfaces of films growing under nonequilibrium conditions are important in understanding the corresponding universality class. However, measurement of the surface fractal dimension has been intensively investigated and is still very problematic. In this work, we report the behavior of the effective fractal dimension in the context of film growth involving lattice models believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. Our results, which are presented for growth in a d-dimensional substrate (d=1,2) and use the three-point sinuosity (TPS) method, show universal scaling of the measure M, which is defined in terms of discretization of the Laplacian operator applied to the height of the film surface, M=t^{δ}g[Θ], where t is the time, g[Θ] is a scale function, δ=2β, Θ≡τt^{-1/z}, β, and z are the KPZ growth and dynamical exponents, respectively, and τ is a spatial scale length used to compute M. Importantly, we show that the effective fractal dimensions are consistent with the expected KPZ dimensions for d=1,2, if Θ≲0.3, which include a thin film regime for the extraction of the fractal dimension. This establishes the scale limits in which the TPS method can be used to accurately extract effective fractal dimensions that are consistent with those expected for the corresponding universality class. As a consequence, for the steady state, which is inaccessible to experimentalists studying film growth, the TPS method provided effective fractal dimension consistent with the KPZ ones for almost all possible τ, i.e., 1≲τ<L/2, where L is the lateral size of the substrate on which the deposit is grown. In the growth of thin films, the true fractal dimension can be observed in a narrow range of τ, the upper limit of which is of the same order of magnitude as the correlation length of the surface, indicating the limits of self-affinity of a surface in an experimentally accessible regime. This upper limit was comparatively lower for the Higuchi method or the height-difference correlation function. Scaling corrections for the measure M and the height-difference correlation function are studied analytically and compared for the Edwards-Wilkinson class at d=1, yielding similar accuracy for both methods. Importantly, we extend our discussion to a model representing diffusion-dominated growth of films and find that the TPS method achieves the corresponding fractal dimension only at steady state and in a narrow range of the scale length, compared to that found for the KPZ class.
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Affiliation(s)
- Edwin E Mozo Luis
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Rua Barão de Jeremoabo s/n, 40170-115, Salvador, BA, Brazil
| | - Fernando A Oliveira
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Rua Barão de Jeremoabo s/n, 40170-115, Salvador, BA, Brazil
- Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340, Niterói, RJ, Brazil
| | - Thiago A de Assis
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Rua Barão de Jeremoabo s/n, 40170-115, Salvador, BA, Brazil
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340, Niterói, RJ, Brazil
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