Nathann Teixeira Rodrigues

Universidade Federal Fluminense

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Universidade Federal de Viçosa

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Nathann Teixeira Rodrigues has a total of 8 co-authors. Here are the top co-authors by number of shared publications:

Tiago J. Oliveira
Universidade Federal de Viçosa
14 shared publications
Thomas Prellberg
School of Mathematical Sciences Queen Mary University of London Mile End Road E1 4NS London UK
6 shared publications
Fabio David Alves Aarão Reis
Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói, RJ, Brazil
5 shared publications
Aleksander L. Owczarek
University of Melbourne
4 shared publications
Ismael S. S. Carrasco
Universidade Federal de Viçosa
2 shared publications
Vaughan R. Voller
Department of Civil, Environmental, and Geo‐Engineering Saint Anthony Falls Laboratory University of Minnesota Minneapolis MN USA
2 shared publications

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- 23 June 2026
- Published by American Physical Society (APS) in Physical Review E
Random walks on disordered lattices: The role of effective porosity, blocked domain width, and spatial correlations
by Nathann Teixeira Rodrigues, Fábio D. A. Aarão Reis
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- 23 June 2026
- Published by American Physical Society (APS) in Physical Review E
Random walks on disordered lattices: The role of effective porosity, blocked domain width, and spatial correlations
by Nathann Teixeira Rodrigues, Fábio D. A. Aarão Reis
Open Access
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Cited by 2
- 28 November 2024
- Published by MDPI AG in Minerals
Mineral Deposition on the Rough Walls of a Fracture
by Nathann Teixeira Rodrigues, Ismael S. S. Carrasco, Vaughan R. Voller, Fábio D. A. Aarão Reis
Modeling carbonate growth in fractures and pores is important for understanding carbon sequestration in the environment or when supersaturated solutions are injected into rocks. Here, we study the simple but nontrivial problem of calcite growth on fractures with rough walls of the same mineral using kinetic Monte Carlo simulations of attachment and detachment of molecules and scaling approaches. First, we consider wedge-shaped fracture walls whose upper terraces are in the same low-energy planes and show that the valleys are slowly filled by the propagation of parallel monolayer steps in the wedge sides. The growth ceases when the walls reach these low-energy configurations so that a gap between the walls may not be filled. Second, we consider fracture walls with equally separated monolayer steps (vicinal surfaces with roughness below 1 nm) and show that growth by step propagation will eventually clog the fracture gap. In both cases, scaling approaches predict the times to attain the final configurations as a function of the initial geometry and the step-propagation velocity, which is set by the saturation index. The same reasoning applied to a random wall geometry shows that step propagation leads to lateral filling of surface valleys until the wall reaches the low-energy crystalline plane that has the smallest initial density of molecules. Thus, the final configurations of the fracture walls are much more sensitive to the crystallography than to the roughness or the local curvature. The framework developed here may be used to determine those configurations, the times to reach them, and the mass of deposited mineral. Effects of transport limitations are discussed when the fracture gap is significantly narrowed.
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- 28 November 2024
- Published by MDPI AG in Minerals
Open Access
Mineral Deposition on the Rough Walls of a Fracture
by Nathann Teixeira Rodrigues, Ismael S. S. Carrasco, Vaughan R. Voller, Fábio D. A. Aarão Reis
Modeling carbonate growth in fractures and pores is important for understanding carbon sequestration in the environment or when supersaturated solutions are injected into rocks. Here, we study the simple but nontrivial problem of calcite growth on fractures with rough walls of the same mineral using kinetic Monte Carlo simulations of attachment and detachment of molecules and scaling approaches. First, we consider wedge-shaped fracture walls whose upper terraces are in the same low-energy planes and show that the valleys are slowly filled by the propagation of parallel monolayer steps in the wedge sides. The growth ceases when the walls reach these low-energy configurations so that a gap between the walls may not be filled. Second, we consider fracture walls with equally separated monolayer steps (vicinal surfaces with roughness below 1 nm) and show that growth by step propagation will eventually clog the fracture gap. In both cases, scaling approaches predict the times to attain the final configurations as a function of the initial geometry and the step-propagation velocity, which is set by the saturation index. The same reasoning applied to a random wall geometry shows that step propagation leads to lateral filling of surface valleys until the wall reaches the low-energy crystalline plane that has the smallest initial density of molecules. Thus, the final configurations of the fracture walls are much more sensitive to the crystallography than to the roughness or the local curvature. The framework developed here may be used to determine those configurations, the times to reach them, and the mass of deposited mineral. Effects of transport limitations are discussed when the fracture gap is significantly narrowed.
Cited by 2
Open Access
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Cited by 1
- 26 October 2024
- Published by MDPI AG in Preprints.org
Mineral Deposition on the Rough Walls of a Fracture
by Nathann Teixeira Rodrigues, Ismael S. S. Carrasco, Vaughan R. Voller, Fábio D. A. Aarão Reis
Modeling carbonate growth in fractures and pores is important for understanding carbon sequestration in the environment or when supersaturated solutions are injected into rocks. Here we study the simple but nontrivial problem of calcite growth on fractures with rough walls of the same mineral using kinetic Monte Carlo simulations of attachment and detachment of molecules and scaling approaches. First we consider wedge-shaped fracture walls whose upper terraces are in the same low energy planes and show that the valleys are slowly filled by propagation of parallel monolayer steps in the wedge sides. The growth ceases when the walls reach these low energy configurations, so that a gap between the walls may not be filled. Secondly we consider fracture walls with equally separated monolayer steps (vicinal surfaces with roughness below $1$~nm) and show that growth by step propagation will eventually clog the fracture gap. In both cases, scaling approaches predict the times to attain the final configurations as a function of the initial geometry and of the step propagation velocity in the chosen supersaturation. The same reasoning applied to a random wall geometry shows that step propagation leads to lateral filling of surface valleys until the wall reaches the low-energy crystalline plane that has the smallest initial density of molecules. Thus, the final configurations of the fracture walls are much more sensitive to the crystallography than to the roughness or to the local curvature. The framework developed here may be used to determine those configurations, the times to reach them, and the mass of deposited mineral.
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- 26 October 2024
- Published by MDPI AG in Preprints.org
Open Access
Mineral Deposition on the Rough Walls of a Fracture
by Nathann Teixeira Rodrigues, Ismael S. S. Carrasco, Vaughan R. Voller, Fábio D. A. Aarão Reis
Modeling carbonate growth in fractures and pores is important for understanding carbon sequestration in the environment or when supersaturated solutions are injected into rocks. Here we study the simple but nontrivial problem of calcite growth on fractures with rough walls of the same mineral using kinetic Monte Carlo simulations of attachment and detachment of molecules and scaling approaches. First we consider wedge-shaped fracture walls whose upper terraces are in the same low energy planes and show that the valleys are slowly filled by propagation of parallel monolayer steps in the wedge sides. The growth ceases when the walls reach these low energy configurations, so that a gap between the walls may not be filled. Secondly we consider fracture walls with equally separated monolayer steps (vicinal surfaces with roughness below $1$~nm) and show that growth by step propagation will eventually clog the fracture gap. In both cases, scaling approaches predict the times to attain the final configurations as a function of the initial geometry and of the step propagation velocity in the chosen supersaturation. The same reasoning applied to a random wall geometry shows that step propagation leads to lateral filling of surface valleys until the wall reaches the low-energy crystalline plane that has the smallest initial density of molecules. Thus, the final configurations of the fracture walls are much more sensitive to the crystallography than to the roughness or to the local curvature. The framework developed here may be used to determine those configurations, the times to reach them, and the mass of deposited mineral.
Cited by 1
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Cited by 6
- 31 May 2024
- Published by Elsevier in Geochimica et Cosmochimica Acta
Step propagation controls pore shape evolution when mineral walls dissolve under saturation gradients
by Nathann Teixeira Rodrigues, Ismael S.S. Carrasco, Fábio D.A. Aarão Reis
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- 31 May 2024
- Published by Elsevier in Geochimica et Cosmochimica Acta
Step propagation controls pore shape evolution when mineral walls dissolve under saturation gradients
by Nathann Teixeira Rodrigues, Ismael S.S. Carrasco, Fábio D.A. Aarão Reis
Cited by 6
Open Access
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Cited by 10
- 17 May 2024
- Published by American Chemical Society (ACS) in Langmuir
Adsorption of Diffusing Tracers, Apparent Tortuosity, and Application to Mesoporous Silica
by Nathann Teixeira Rodrigues, Fábio David Alves Aarão Reis
The apparent tortuosity due to adsorption of diffusing tracers in a porous material is determined by a scaling approach and is used to analyze recent data on LiCl and alkane diffusion in mesoporous silica. The slope of the adsorption isotherm at small loadings is written as β = q_A/q_G, where q_A is the adsorption-desorption ratio and q_G = ϵ/(as) - 1 is a geometrical factor depending on the range a of the tracer-wall interaction, the porosity ϵ, and the specific surface area s. The adsorption leads to a decrease of effective diffusion coefficient, which is quantified by multiplying the geometrical tortuosity factor τ_geom by an apparent tortuosity factor τ_app. In wide pores or when the adsorption barrier is high, τ_app = β + 1, as obtained in previous works, but in narrow pores there is an additional contribution from frequent adsorption-desorption transitions. These results are obtained in media with parallel pores of constant cross sections, where the ratio between the effective pore width ϵ/s and the actual width is ≈0.25. Applications to mesoporous silica samples are justified by the small deviations from this ideal ratio. In the analysis of alkane self-diffusion data, the fractions of adsorbed molecules predicted in a recent theoretical work are used to estimate τ_geom of the silica samples, which is ≫1 only in the sample with the narrowest pores (nominal 3 nm). The application of the model to Li⁺ ion diffusion leads to similar values of τ_geom and to a difference of energy barriers of desorption and adsorption for those ions of ∼0.06 eV. Comparatively, alkane self-diffusion provides the correct order of magnitude of τ_geom, with adsorption playing a less important role, whereas adsorption effects on Li⁺ diffusion are much more important.
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- 17 May 2024
- Published by American Chemical Society (ACS) in Langmuir
Open Access
Adsorption of Diffusing Tracers, Apparent Tortuosity, and Application to Mesoporous Silica
by Nathann Teixeira Rodrigues, Fábio David Alves Aarão Reis
The apparent tortuosity due to adsorption of diffusing tracers in a porous material is determined by a scaling approach and is used to analyze recent data on LiCl and alkane diffusion in mesoporous silica. The slope of the adsorption isotherm at small loadings is written as β = q_A/q_G, where q_A is the adsorption-desorption ratio and q_G = ϵ/(as) - 1 is a geometrical factor depending on the range a of the tracer-wall interaction, the porosity ϵ, and the specific surface area s. The adsorption leads to a decrease of effective diffusion coefficient, which is quantified by multiplying the geometrical tortuosity factor τ_geom by an apparent tortuosity factor τ_app. In wide pores or when the adsorption barrier is high, τ_app = β + 1, as obtained in previous works, but in narrow pores there is an additional contribution from frequent adsorption-desorption transitions. These results are obtained in media with parallel pores of constant cross sections, where the ratio between the effective pore width ϵ/s and the actual width is ≈0.25. Applications to mesoporous silica samples are justified by the small deviations from this ideal ratio. In the analysis of alkane self-diffusion data, the fractions of adsorbed molecules predicted in a recent theoretical work are used to estimate τ_geom of the silica samples, which is ≫1 only in the sample with the narrowest pores (nominal 3 nm). The application of the model to Li⁺ ion diffusion leads to similar values of τ_geom and to a difference of energy barriers of desorption and adsorption for those ions of ∼0.06 eV. Comparatively, alkane self-diffusion provides the correct order of magnitude of τ_geom, with adsorption playing a less important role, whereas adsorption effects on Li⁺ diffusion are much more important.
Cited by 10
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- 30 August 2023
- Published by American Physical Society (APS) in Physical Review E
Universality class of the special adsorption point of two-dimensional lattice polymers
by Nathann T. Rodrigues, Tiago J. Oliveira, Thomas Prellberg
In recent work [Rodrigues et al., Phys. Rev. E 100, 022121 (2019)], evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a nonuniversal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $ϕ^{(s)}$ and $1 / δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS), on which the trails are forbidden from having two consecutive steps, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1 / δ^{(s)}$ exponents at the SAP, namely, $1 / δ^{(s)} \approx 0.44$ , whereas $1 / δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $ϕ^{(s)} \approx 1 / δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $ϕ^{(s)}$ is about 10% smaller than $1 / δ^{(s)}$ . These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.
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- 30 August 2023
- Published by American Physical Society (APS) in Physical Review E
Universality class of the special adsorption point of two-dimensional lattice polymers
by Nathann T. Rodrigues, Tiago J. Oliveira, Thomas Prellberg
In recent work [Rodrigues et al., Phys. Rev. E 100, 022121 (2019)], evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a nonuniversal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $ϕ^{(s)}$ and $1 / δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS), on which the trails are forbidden from having two consecutive steps, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1 / δ^{(s)}$ exponents at the SAP, namely, $1 / δ^{(s)} \approx 0.44$ , whereas $1 / δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $ϕ^{(s)} \approx 1 / δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $ϕ^{(s)}$ is about 10% smaller than $1 / δ^{(s)}$ . These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.
Open Access
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- 16 May 2023
- Published by arXiv in arXiv
On the universality class of the special adsorption point of two-dimensional lattice polymers
by Nathann T. Rodrigues, Tiago J. Oliveira, Thomas Prellberg
In recent work [PRE 100, 022121 (2019)] evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a non-universal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $φ^{(s)}$ and $1/δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a previously unseen surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS) where the trails are forbidden of having two consecutive steps along it, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1/δ^{(s)}$ exponents at the SAP, being $1/δ^{(s)} \approx 0.44$, whilst $1/δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $φ^{(s)} \approx 1/δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $φ^{(s)}$ is about $10$\% smaller than $1/δ^{(s)}$. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.
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Google Scholar
- 16 May 2023
- Published by arXiv in arXiv
Open Access
On the universality class of the special adsorption point of two-dimensional lattice polymers
by Nathann T. Rodrigues, Tiago J. Oliveira, Thomas Prellberg
In recent work [PRE 100, 022121 (2019)] evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a non-universal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $φ^{(s)}$ and $1/δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a previously unseen surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS) where the trails are forbidden of having two consecutive steps along it, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1/δ^{(s)}$ exponents at the SAP, being $1/δ^{(s)} \approx 0.44$, whilst $1/δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $φ^{(s)} \approx 1/δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $φ^{(s)}$ is about $10$\% smaller than $1/δ^{(s)}$. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.
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Cited by 5
- 26 August 2022
- Published by American Physical Society (APS) in Physical Review E
Semianalytical solutions of Ising-like and Potts-like magnetic polymers on the Bethe lattice
by Nathann T. Rodrigues, Tiago J. Oliveira
We study magnetic polymers, defined as self-avoiding walks where each monomer $i$ carries a “spin” $s_{i}$ and interacts with its first neighbor monomers, let us say $j$ , via a coupling constant $J (s_{i}, s_{j})$ . Ising-like $[s_{i} = \pm 1$ , with $J (s_{i}, s_{j}) = ɛ s_{i} s_{j}]$ and Potts-like $[s_{i} = 1, ..., q$ , with $J (s_{i}, s_{j}) = ɛ (s_{i}) δ (s_{i}, s_{j})]$ models are investigated. Some particular cases of these systems have recently been studied in the continuum and on regular lattices and are related to interesting applications. Here, we solve these models on Bethe lattices of branching number $σ$ , focusing on the ferromagnetic case in zero external magnetic field. In most cases, the phase diagrams present a nonpolymerized (NP) and two polymerized phases: a paramagnetic (PP) and a ferromagnetic (FP) one. However, quite different thermodynamic properties are found depending on $q$ in the Potts-like polymers and on whether one uses the Ising or Potts coupling in the two-state systems. For instance, for the standard Potts model [where $ɛ (1) = \dots = ɛ (q)]$ with $q = 2$ , beyond the $θ$ -point (where the critical and discontinuous NP-PP transition lines meet), a second tricritical point connecting a critical and a discontinuous transition line between the PP-FP phases is found in the system. A triple point where the NP-PP, NP-FP, and PP-FP first-order transition lines meet is also present in the phase diagram. For $q \geq 3$ the PP-FP transition is always discontinuous, and the scenario with the triple and $θ$ points appears for $q \leq 6$ . Interestingly, for $q \geq 7$ , as well as for the Ising-like model the $θ$ point becomes metastable and the critical NP-PP transition line ends at a critical end point (CEP), where it meets the NP-FP and PP-FP coexistence lines. Importantly, these results indicate that when $q \leq 6$ the spin ordering transition is preceded by the polymer collapse transition, whereas for $q \geq 7$ and in the Ising case these transitions happen together at the CEPs. Some interesting nonstandard Potts models are also studied, such as the lattice version of the model for epigenetic marks in the chromatin introduced by Michieletto et al. [ Phys. Rev. X 6, 041047 (2016)]. In addition, the solution of the dilute Ising and dilute Potts models on the Bethe lattice are also presented here, since they are important to understand the PP-FP transitions.
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- 26 August 2022
- Published by American Physical Society (APS) in Physical Review E
Semianalytical solutions of Ising-like and Potts-like magnetic polymers on the Bethe lattice
by Nathann T. Rodrigues, Tiago J. Oliveira
We study magnetic polymers, defined as self-avoiding walks where each monomer $i$ carries a “spin” $s_{i}$ and interacts with its first neighbor monomers, let us say $j$ , via a coupling constant $J (s_{i}, s_{j})$ . Ising-like $[s_{i} = \pm 1$ , with $J (s_{i}, s_{j}) = ɛ s_{i} s_{j}]$ and Potts-like $[s_{i} = 1, ..., q$ , with $J (s_{i}, s_{j}) = ɛ (s_{i}) δ (s_{i}, s_{j})]$ models are investigated. Some particular cases of these systems have recently been studied in the continuum and on regular lattices and are related to interesting applications. Here, we solve these models on Bethe lattices of branching number $σ$ , focusing on the ferromagnetic case in zero external magnetic field. In most cases, the phase diagrams present a nonpolymerized (NP) and two polymerized phases: a paramagnetic (PP) and a ferromagnetic (FP) one. However, quite different thermodynamic properties are found depending on $q$ in the Potts-like polymers and on whether one uses the Ising or Potts coupling in the two-state systems. For instance, for the standard Potts model [where $ɛ (1) = \dots = ɛ (q)]$ with $q = 2$ , beyond the $θ$ -point (where the critical and discontinuous NP-PP transition lines meet), a second tricritical point connecting a critical and a discontinuous transition line between the PP-FP phases is found in the system. A triple point where the NP-PP, NP-FP, and PP-FP first-order transition lines meet is also present in the phase diagram. For $q \geq 3$ the PP-FP transition is always discontinuous, and the scenario with the triple and $θ$ points appears for $q \leq 6$ . Interestingly, for $q \geq 7$ , as well as for the Ising-like model the $θ$ point becomes metastable and the critical NP-PP transition line ends at a critical end point (CEP), where it meets the NP-FP and PP-FP coexistence lines. Importantly, these results indicate that when $q \leq 6$ the spin ordering transition is preceded by the polymer collapse transition, whereas for $q \geq 7$ and in the Ising case these transitions happen together at the CEPs. Some interesting nonstandard Potts models are also studied, such as the lattice version of the model for epigenetic marks in the chromatin introduced by Michieletto et al. [ Phys. Rev. X 6, 041047 (2016)]. In addition, the solution of the dilute Ising and dilute Potts models on the Bethe lattice are also presented here, since they are important to understand the PP-FP transitions.
Cited by 5
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Cited by 7
- 22 February 2022
- Published by American Physical Society (APS) in Physical Review E
Entropy of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit
by Nathann T. Rodrigues, Jürgen F. Stilck, Tiago J. Oliveira
Although hard rigid rods ( $k$ -mers) defined on the square lattice have been widely studied in the literature, their entropy per site, $s (k)$ , in the full-packing limit is only known exactly for dimers ( $k = 2$ ) and numerically for trimers ( $k = 3$ ). Here, we investigate this entropy for rods with $k \leq 7$ , by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size $L$ , where $L$ defines the level of approximation to the square lattice. Due to an $L$ -parity effect, by increasing $L$ we obtain two systematic sequences of values for the entropies $s_{L} (k)$ for each type of cluster, whose extrapolations to $L \to \infty$ provide estimates of these entropies for the square lattice. For dimers, our estimates for $s (2)$ differ from the exact result by only $0.03 %$ , while that for $s (3)$ differs from best available estimates by $3 %$ . In this paper, we also obtain a new estimate for $s (4)$ . For larger $k$ , we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for $s (k)$ . In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels $L$ that increase with $k$ . However, it is very challenging computationally to advance to solve the problem for large values of $L$ and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for $k \geq 4$ , in contrast to the results for the Bethe lattice wherein it is nematic.
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- 22 February 2022
- Published by American Physical Society (APS) in Physical Review E
Entropy of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit
by Nathann T. Rodrigues, Jürgen F. Stilck, Tiago J. Oliveira
Although hard rigid rods ( $k$ -mers) defined on the square lattice have been widely studied in the literature, their entropy per site, $s (k)$ , in the full-packing limit is only known exactly for dimers ( $k = 2$ ) and numerically for trimers ( $k = 3$ ). Here, we investigate this entropy for rods with $k \leq 7$ , by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size $L$ , where $L$ defines the level of approximation to the square lattice. Due to an $L$ -parity effect, by increasing $L$ we obtain two systematic sequences of values for the entropies $s_{L} (k)$ for each type of cluster, whose extrapolations to $L \to \infty$ provide estimates of these entropies for the square lattice. For dimers, our estimates for $s (2)$ differ from the exact result by only $0.03 %$ , while that for $s (3)$ differs from best available estimates by $3 %$ . In this paper, we also obtain a new estimate for $s (4)$ . For larger $k$ , we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for $s (k)$ . In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels $L$ that increase with $k$ . However, it is very challenging computationally to advance to solve the problem for large values of $L$ and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for $k \geq 4$ , in contrast to the results for the Bethe lattice wherein it is nematic.
Cited by 7
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Cited by 10
- 29 March 2021
- Published by American Physical Society (APS) in Physical Review E
Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases
by Nathann T. Rodrigues, Tiago J. Oliveira
Although lattice gases composed of particles preventing up to their $k th$ nearest neighbors from being occupied (the $k NN$ models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with $2 L (L + 1)$ sites, for $L ⩽ 7$ . The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential $μ_{2, c} (L)$ and particle density $ρ_{2, c} (L)$ to $L \to \infty$ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters $μ_{1, c}$ and $ρ_{1, c}$ —for the fluid-solid transition in this model on the square lattice—are found from extrapolations of data for $L ⩽ 6$ . The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields $β$ and $ν$ differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of $μ_{2, c}$ used to calculate them. Notwithstanding, our results suggest that $β$ and $ν$ are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.
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- 29 March 2021
- Published by American Physical Society (APS) in Physical Review E
Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases
by Nathann T. Rodrigues, Tiago J. Oliveira
Although lattice gases composed of particles preventing up to their $k th$ nearest neighbors from being occupied (the $k NN$ models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with $2 L (L + 1)$ sites, for $L ⩽ 7$ . The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential $μ_{2, c} (L)$ and particle density $ρ_{2, c} (L)$ to $L \to \infty$ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters $μ_{1, c}$ and $ρ_{1, c}$ —for the fluid-solid transition in this model on the square lattice—are found from extrapolations of data for $L ⩽ 6$ . The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields $β$ and $ν$ differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of $μ_{2, c}$ used to calculate them. Notwithstanding, our results suggest that $β$ and $ν$ are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.
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