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Chan ST, Fried E. Marangoni spreading on liquid substrates in new media art. PNAS NEXUS 2024; 3:pgae059. [PMID: 38725527 PMCID: PMC11079615 DOI: 10.1093/pnasnexus/pgae059] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/12/2023] [Accepted: 01/31/2024] [Indexed: 05/12/2024]
Abstract
With the advent of new media art, artists have harnessed fluid dynamics to create captivating visual narratives. A striking technique known as dendritic painting employs mixtures of ink and isopropanol atop paint, yielding intricate tree-like patterns. To unravel the intricacies of that technique, we examine the spread of ink/alcohol droplets over liquid substrates with diverse rheological properties. On Newtonian substrates, the droplet size evolution exhibits two power laws, suggesting an underlying interplay between viscous and Marangoni forces. The leading edge of the droplet spreads as a precursor film with an exponent of 3/8, while its main body spreads with an exponent of 1/4. For a weakly shear-thinning acrylic resin substrate, the same power laws persist, but dendritic structures emerge, and the texture of the precursor film roughens. The observed roughness and growth exponents (3/4 and 3/5) suggest a connection to the quenched Kardar-Parisi-Zhang universality class, hinting at the existence of quenched disorder in the liquid substrate. Mixing the resin with acrylic paint renders it more viscous and shear-thinning, refining the dendrite edges and further roughening the precursor film. At larger paint concentrations, the substrate becomes a power-law fluid. The roughness and growth exponents then approach 1/2 and 3/4, respectively, deviating from known universality classes. The ensuing structures have a fractal dimension of 1.68, characteristic of diffusion-limited aggregation. These findings underscore how the nonlinear rheological properties of the liquid substrate, coupled with the Laplacian nature of Marangoni spreading, can overshadow the local kinetic roughening of the droplet interface.
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Affiliation(s)
- San To Chan
- Mechanics and Materials Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
| | - Eliot Fried
- Mechanics and Materials Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
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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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Oliveira TJ. Kardar-Parisi-Zhang universality class in (d+1)-dimensions. Phys Rev E 2022; 106:L062103. [PMID: 36671175 DOI: 10.1103/physreve.106.l062103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 12/06/2022] [Indexed: 06/17/2023]
Abstract
The determination of the exact exponents of the KPZ class in any substrate dimension d is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by β_{d}=7/8d+13. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete growth models and real-space renormalization group (RG) calculations for directed polymers in random media (DPRM), up to d=15. The left-tail exponents of the probability density functions for the DPRM energy provide another striking verification of the analytical result above.
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Affiliation(s)
- Tiago J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil
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Oliveira TJ. Height distributions in interface growth: The role of the averaging process. Phys Rev E 2022; 105:064803. [PMID: 35854512 DOI: 10.1103/physreve.105.064803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Accepted: 06/03/2022] [Indexed: 06/15/2023]
Abstract
Height distributions (HDs) are key quantities to uncover universality and geometry-dependence in evolving interfaces. To quantitatively characterize HDs, one uses adimensional ratios of their first central moments (m_{n}) or cumulants (κ_{n}), especially the skewness S and kurtosis K, whose accurate estimate demands an averaging over all L^{d} points of the height profile at a given time, in translation-invariant interfaces, and over N independent samples. One way of doing this is by calculating m_{n}(t) [or κ_{n}(t)] for each sample and then carrying out an average of them for the N interfaces, with S and K being calculated only at the end. Another approach consists in directly calculating the ratios for each interface and, then, averaging the N values. It turns out, however, that S and K for the growth regime HDs display strong finite-size and -time effects when estimated from these "interface statistics," as already observed in some previous works and clearly shown here, through extensive simulations of several discrete growth models belonging to the EW and KPZ classes on one- and two-dimensional substrates of sizes L=const. and L∼t. Importantly, I demonstrate that with "1-point statistics," i.e., by calculating m_{n}(t) [or κ_{n}(t)] once for all NL^{d} heights together, these corrections become very weak, so that S and K attain values very close to the asymptotic ones already at short times and for small L's. However, I find that this "1-point" (1-pt) approach fails in uncovering the universality of the HDs in the steady-state regime (SSR) of systems whose average height, h[over ¯], is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as h(t)=h[over ¯](t)+s_{λ}A^{1/2}L^{α}ζ+⋯-where P(ζ) is the underlying SSR HD-and the fluctuations in h[over ¯] yield S_{1-pt}∼t^{-1/2} and K_{1-pt}∼t^{-1}. Nonetheless, by analyzing P(h-h[over ¯]), the cumulants of P(ζ) can be accurately determined. I also show that different, but universal, asymptotic values for S and K (related, so, to different HDs) can be found from the "interface statistics" in the SSR. This reveals the importance of employing the various complementary approaches to reliably determine the universality class of a given system through its different HDs.
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Affiliation(s)
- Tiago J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900, Viçosa, Minas Gerais, Brazil
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Reis FDAA, Mallio DO, Galindo JL, Huertas R. Scaling of roughness and porosity in thin film deposition with mixed transport mechanisms and adsorption barriers. Phys Rev E 2020; 102:042802. [PMID: 33212663 DOI: 10.1103/physreve.102.042802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2020] [Accepted: 10/13/2020] [Indexed: 06/11/2023]
Abstract
Thin film deposition with particle transport mixing collimated and diffusive components and with barriers for adsorption are studied using numerical simulations and scaling approaches. Biased random walks on lattices are used to model the particle flux and the analogy with advective-diffusive transport is used to define a Peclet number P that represents the effect of the bias towards the substrate. An aggregation probability that relates the rates of adsorption and of the dominant transport mechanism plays the role of a Damkohler number D, where D≲1 is set to describe moderate to low adsorption rates. Very porous deposits with sparse branches are obtained with P≪1, whereas low porosity deposits with large height fluctuations at short scales are obtained with P≫1. For P≳1 in which the field bias is intense, an initial random deposition is followed by Kardar-Parisi-Zhang (KPZ) roughening. As the transport is displaced from those limiting conditions, the interplay of the transport and adsorption mechanisms establishes a condition to produce films with the smoothest surfaces for a constant deposited mass: with low adsorption barriers, a balance of random and collimated flux is required, whereas for high barriers the smoothest surfaces are obtained with P∼D^{1/2}. For intense bias, the roughness is shown to be a power law of P/D, whose exponent depends on the growth exponent β of the KPZ class, and the porosity has a superuniversal scaling as (P/D)^{-1/3}. We also study a generalized ballistic deposition model with slippery particle aggregation that shows the universality of these relations in growth with dominant collimated flux, particle adsorption at any point of the deposit, and negligible adsorbate diffusion, in contrast with the models where aggregation is restricted to the outer surface.
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Affiliation(s)
- Fábio D A Aarão Reis
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói Rio de Janeiro, Brazil
| | - Daniel O Mallio
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói Rio de Janeiro, Brazil
| | - Jose Luis Galindo
- Departament of Optics, TexColImag Group, University of Granada, Granada 18071, Spain
| | - Rafael Huertas
- Departament of Optics, TexColImag Group, University of Granada, Granada 18071, Spain
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Daryaei E. Universality and crossover behavior of single-step growth models in 1+1 and 2+1 dimensions. Phys Rev E 2020; 101:062108. [PMID: 32688564 DOI: 10.1103/physreve.101.062108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2020] [Accepted: 05/15/2020] [Indexed: 06/11/2023]
Abstract
We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter p in 1+1 and 2+1 dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any p<1/2. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function p. The effective nonuniversal parameters are continuously decreasing with p but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for p≠1/2 belongs to the KPZ universality class in 2+1 dimensions.
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Affiliation(s)
- E Daryaei
- Department of Physics, Faculty of Basic Sciences, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran
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Alves SG. Radial restricted solid-on-solid and etching interface-growth models. Phys Rev E 2018; 97:032801. [PMID: 29776046 DOI: 10.1103/physreve.97.032801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2017] [Indexed: 06/08/2023]
Abstract
An approach to generate radial interfaces is presented. A radial network recursively obtained is used to implement discrete model rules designed originally for the investigation in flat substrates. I used the restricted solid-on-solid and etching models as to test the proposed scheme. The results indicate the Kardar, Parisi, and Zhang conjecture is completely verified leading to a good agreement between the interface radius fluctuation distribution and the Gaussian unitary ensemble. The evolution of the radius agrees well with the generalized conjecture, and the two-point correlation function exhibits also a good agreement with the covariance of the Airy_{2} process. The approach can be used to investigate radial interfaces evolution for many other classes of universality.
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Affiliation(s)
- Sidiney G Alves
- Departamento de Física e Matemática, Universidade Federal de São João Del-Rei 36420-000, Ouro Branco, MG, Brazil
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Kelling J, Ódor G, Gemming S. Universality of (2+1)-dimensional restricted solid-on-solid models. Phys Rev E 2016; 94:022107. [PMID: 27627246 DOI: 10.1103/physreve.94.022107] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2016] [Indexed: 11/07/2022]
Abstract
Extensive dynamical simulations of restricted solid-on-solid models in D=2+1 dimensions have been done using parallel multisurface algorithms implemented on graphics cards. Numerical evidence is presented that these models exhibit Kardar-Parisi-Zhang surface growth scaling, irrespective of the step heights N. We show that by increasing N the corrections to scaling increase, thus smaller step-sized models describe better the asymptotic, long-wave-scaling behavior.
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Affiliation(s)
- Jeffrey Kelling
- Department of Information Services and Computing, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany.,Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany
| | - Géza Ódor
- Institute of Technical Physics and Materials Science, Centre for Energy Research of the Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Sibylle Gemming
- Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany.,Institute of Physics, TU Chemnitz, 09107 Chemnitz, Germany
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Alves SG, Ferreira SC. Scaling, cumulant ratios, and height distribution of ballistic deposition in 3+1 and 4+1 dimensions. Phys Rev E 2016; 93:052131. [PMID: 27300853 DOI: 10.1103/physreve.93.052131] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2016] [Indexed: 06/06/2023]
Abstract
We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)PLEEE81539-375510.1103/PhysRevE.90.052405] in d=2+1 dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for d=3+1 and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both d=3+1 and 4+1. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.
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Affiliation(s)
- Sidiney G Alves
- Departamento de Física e Matemática, Universidade Federal de São João Del Rei, 36420-000 Ouro Branco, MG, Brazil
| | - Silvio C Ferreira
- Departamento de Física, Universidade Federal de Viçosa, 36570-000 Viçosa, MG, Brazil
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Carrasco ISS, Oliveira TJ. Width and extremal height distributions of fluctuating interfaces with window boundary conditions. Phys Rev E 2016; 93:012801. [PMID: 26871135 DOI: 10.1103/physreve.93.012801] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2015] [Indexed: 11/07/2022]
Abstract
We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nβ+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]β. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,β and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.
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Affiliation(s)
- I S S Carrasco
- Departamento de Física, Universidade Federal de Viçosa, 36570-900, Viçosa, Minas Gerais, Brazil
| | - T J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900, Viçosa, Minas Gerais, Brazil
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di Caprio D, Aarão Reis FDA. Transition from compact to porous films in deposition with temperature-activated diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012402. [PMID: 26274181 DOI: 10.1103/physreve.92.012402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2015] [Indexed: 06/04/2023]
Abstract
We study a thin-film growth model with temperature activated diffusion of adsorbed particles, allowing for the formation of overhangs and pores, but without detachment of adatoms or clusters from the deposit. Simulations in one-dimensional substrates are performed for several values of the diffusion-to-deposition ratio R of adatoms with a single bond and of the detachment probability ε per additional nearest neighbor, respectively, with activation energies are E(s) and E(b). If R and ε independently vary, regimes of low and high porosity are separated at 0.075≤ε(c)≤0.09, with vanishingly small porosity below that point and finite porosity for larger ε. Alternatively, for fixed values of E(s) and E(b) and varying temperature, the porosity has a minimum at T(c), and a nontrivial regime in which it increases with temperature is observed above that point. This is related to the large mobility of adatoms, resembling features of equilibrium surface roughening. In this high-temperature region, the deposit has the structure of a critical percolation cluster due to the nondesorption. The pores are regions enclosed by blobs of the corresponding percolating backbone, thus the distribution of pore size s is expected to scale as s(-τ̃) with τ̃≈1.45, in reasonable agreement with numerical estimates. Roughening of the outer interface of the deposits suggests Villain-Lai-Das Sarma scaling below the transition. Above the transition, the roughness exponent α≈0.35 is consistent with the percolation backbone structure via the relation α=2-d(B), where d(B) is the backbone fractal dimension.
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Affiliation(s)
- Dung di Caprio
- PSL Research University, Chimie ParisTech-CNRS, Institut de Recherche de Chimie Paris, 75005 Paris, France
| | - F D A Aarão Reis
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil
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Reis FDAA. Kinetic roughening and porosity scaling in film growth with subsurface lateral aggregation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062401. [PMID: 26172719 DOI: 10.1103/physreve.91.062401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2015] [Indexed: 06/04/2023]
Abstract
We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order a at each position. The model generalizes ballisticlike models by allowing attachment to particles below the outer surface. For small values of a, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions confirm Kardar-Parisi-Zhang roughening of the outer surface for a>0. A scaling approach for small a predicts crossover times as a(-2/3) and local height fluctuations as a(-1/3) at the crossover, independent of substrate dimension. These relations are different from all previously studied models with crossovers from UD to correlated growth due to subsurface aggregation, which reduces scaling exponents. The same approach predicts the porosity and average pore height scaling as a(1/3) and a(-1/3), respectively, in good agreement with simulation results in 1+1 and 2+1 dimensions. These results may be useful for modeling samples with desired porosity and long pores.
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Affiliation(s)
- F D A Aarão Reis
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói, Rio de Janeiro, Brazil
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Araújo NAM, Dias CS, Telo da Gama MM. Kinetic interfaces of patchy particles. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:194123. [PMID: 25923051 DOI: 10.1088/0953-8984/27/19/194123] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
We study the irreversible adsorption of patchy particles on substrates in the limit of advective mass transport. Recent numerical results show that the interface roughening depends strongly on the particle attributes, such as, patch-patch correlations, bond flexibility and strength of the interactions, uncovering new absorbing phase transitions. Here, we revisit these results and discuss in detail the transitions. In particular, we present new evidence that the tricritical point, observed in systems of particles with flexible patches, is in the tricritical directed percolation universality class. A scaling analysis of the time evolution of the correlation length for the aggregation of patchy particles with distinct bonding energies confirms that the critical regime is in the Kardar-Parisi-Zhang with quenched disorder universality class.
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Affiliation(s)
- N A M Araújo
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal. Centro de Física Teórica e Computacional, Universidade de Lisboa, Avenida Professor Gama Pinto 2, P-1649-003 Lisboa, Portugal
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