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Bedolla-Montiel EA, Lange JT, Pérez de Alba Ortíz A, Dijkstra M. Inverse design of crystals and quasicrystals in a non-additive binary mixture of hard disks. J Chem Phys 2024; 160:244902. [PMID: 38916271 DOI: 10.1063/5.0210034] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2024] [Accepted: 06/07/2024] [Indexed: 06/26/2024] Open
Abstract
The development of new materials typically involves a process of trial and error, guided by insights from past experimental and theoretical findings. The inverse design approach for soft-matter systems has the potential to optimize specific physical parameters, such as particle interactions, particle shape, or composition and packing fraction. This optimization aims to facilitate the spontaneous formation of specific target structures through self-assembly. In this study, we expand upon a recently introduced inverse design protocol for monodisperse systems to identify the required conditions and interactions for assembling crystal and quasicrystal phases within a binary mixture of two distinct species. This method utilizes an evolution algorithm to identify the optimal state point and interaction parameters, enabling the self-assembly of the desired structure. In addition, we employ a convolutional neural network (CNN) that classifies different phases based on their diffraction patterns, serving as a fitness function for the desired structure. Using our protocol, we successfully inverse design two-dimensional crystalline structures, including a hexagonal lattice and a dodecagonal quasicrystal, within a non-additive binary mixture of hard disks. Finally, we introduce a symmetry-based order parameter that leverages the encoded symmetry within the diffraction pattern. This order parameter circumvents the need for training a CNN and is used as a fitness function to inverse design an octagonal quasicrystal.
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Affiliation(s)
- Edwin A Bedolla-Montiel
- Soft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
| | - Jochem T Lange
- Soft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
| | - Alberto Pérez de Alba Ortíz
- Soft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
- Computational Soft Matter Lab, Computational Chemistry Group and Computational Science Lab, van 't Hoff Institute for Molecular Science and Informatics Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
| | - Marjolein Dijkstra
- Soft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
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2
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Smallenburg F. Comment on 'Pseudo hard-sphere viscosities from equilibrium molecular dynamics'. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:228001. [PMID: 38436284 DOI: 10.1088/1361-648x/ad1f8b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2023] [Accepted: 01/17/2024] [Indexed: 03/05/2024]
Abstract
In a recent article, Nicasio-Collazoet al(2023J. Phys.: Condens. Matter35425401) explore the viscosity of the pseudo-hard-sphere (PHS) model. In this comment, we highlight some discrepancies with expected behavior, and compare their results to new simulations of the same model as well as to true hard spheres. In contrast to the results of Nicasio-Collazoet al, our results follow the relation between shear, bulk, and longitudinal viscosity expected for isotropic fluids. Moreover, we observe clear differences in behavior between PHS and true hard sphere, and encourage future hard-sphere studies to focus on the true hard sphere model whenever possible.
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Affiliation(s)
- Frank Smallenburg
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
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3
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Martínez-Rivera J, Villada-Balbuena A, Sandoval-Puentes MA, Egelhaaf SU, Méndez-Alcaraz JM, Castañeda-Priego R, Escobedo-Sánchez MA. Modeling the structure and thermodynamics of multicomponent and polydisperse hard-sphere dispersions with continuous potentials. J Chem Phys 2023; 159:194110. [PMID: 37982478 DOI: 10.1063/5.0168098] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2023] [Accepted: 10/25/2023] [Indexed: 11/21/2023] Open
Abstract
A model system of identical particles interacting via a hard-sphere potential is essential in condensed matter physics; it helps to understand in and out of equilibrium phenomena in complex fluids, such as colloidal dispersions. Yet, most of the fixed time-step algorithms to study the transport properties of those systems have drawbacks due to the mathematical nature of the interparticle potential. Because of this, mapping a hard-sphere potential onto a soft potential has been recently proposed [Báez et al., J. Chem. Phys. 149, 164907 (2018)]. More specifically, using the second virial coefficient criterion, one can set a route to estimate the parameters of the soft potential that accurately reproduces the thermodynamic properties of a monocomponent hard-sphere system. However, real colloidal dispersions are multicomponent or polydisperse, making it important to find an efficient way to extend the potential model for dealing with such kind of many-body systems. In this paper, we report on the extension and applicability of the second virial coefficient criterion to build a description that correctly captures the phenomenology of both multicomponent and polydisperse hard-sphere dispersions. To assess the accuracy of the continuous potentials, we compare the structure of soft polydisperse systems with their hard-core counterpart. We also contrast the structural and thermodynamic properties of soft binary mixtures with those obtained through mean-field approximations and the Ornstein-Zernike equation for the two-component hard-sphere dispersion.
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Affiliation(s)
- Jaime Martínez-Rivera
- División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, 37150 León, Guanjuato, Mexico
| | | | - Miguel A Sandoval-Puentes
- División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, 37150 León, Guanjuato, Mexico
| | - Stefan U Egelhaaf
- Condensed Matter Physics Laboratory, Heinrich Heine University, 40225 Düsseldorf, Germany
| | - José M Méndez-Alcaraz
- Departamento de Física, Cinvestav, Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, 07360 Ciudad de México, Mexico
| | - Ramón Castañeda-Priego
- Departamento de Ingeniería Física, División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, 37150 León, Guanajuato, Mexico
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4
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Nicasio-Collazo LA, Ramírez-Medina CA, Torres-Carbajal A. Pseudo hard-sphere viscosities from equilibrium Molecular Dynamics. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:425401. [PMID: 37451255 DOI: 10.1088/1361-648x/ace7a5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2023] [Accepted: 07/14/2023] [Indexed: 07/18/2023]
Abstract
Transport coefficients like shear, bulk and longitudinal viscosities are sensitive to the intermolecular interaction potential and finite size effects when are numerically determined. For the hard-sphere (HS) fluid, such transport properties are determined almost exclusively with computer simulations. However, their systematic determination and analysis throughout shear stress correlation functions and the Green-Kubo formalism can not be done due to discontinuous nature of the interaction potential. Here, we use the pseudo hard-sphere (PHS) potential to determine pressure correlation functions as a function of volume fraction in order to compute mentioned viscosities. Simulation results are compared to available event-driven molecular dynamics of the HS fluid and also used to propose empirical corrections for the Chapman-Enskog zero density limit of shear viscosity. Moreover, we show that PHS potential is a reliable representation of the HS fluid and can be used to compute transport coefficients. The molecular simulation results of the present work are valuable for further exploration of HS-type fluids or extend the approach to compute transport properties of hard-colloid suspensions.
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Affiliation(s)
- Luz Adriana Nicasio-Collazo
- Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Chapultepec 1570, Privadas de Pedregal 78295 San Luis Potosí, SLP, México
- Tecnológico Nacional de México-Instituto Tecnológico de León, Léon, Guanajuato 37290, México
| | | | - Alexis Torres-Carbajal
- Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 Ciudad de México, México
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5
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C Barbosa M, Benavides AL, Carlevaro M, Kahl G, Lomba E. Special issue on soft matter research in Latin America. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:410301. [PMID: 37449707 DOI: 10.1088/1361-648x/acdebd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Accepted: 06/15/2023] [Indexed: 07/18/2023]
Affiliation(s)
- Marcia C Barbosa
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
| | - Ana Laura Benavides
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Campus León, Mexico
| | - Manuel Carlevaro
- Instituto de Física de Líquidos y Sistemas Biológicos (CONICET-UNLP) and Universidad Tecnológica Nacional Argentina, La Plata, Buenos Aires, Argentina
| | - Gerhard Kahl
- Institute for Theoretical Physics, TU Wien, Vienna, Austria
| | - Enrique Lomba
- Instituto de Quiímica Fisica Blas Cabrera, Consejo Superior de Investigaciones Científicas, Madrid, Spain
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6
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de los Santos Lopez NM, Pérez Ángel G, Castañeda-Priego R, Méndez Alcaraz JM. Determining depletion interactions by contracting forces. J Chem Phys 2022; 157:074903. [DOI: 10.1063/5.0099919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Depletion forces are fundamental for determining the phase behavior of a vast number of materials and colloidal dispersions, and have been used for the manipulation of in- and out-of-equilibrium thermodynamic states. The entropic nature of depletion forces is well understood; however, most theoretical approaches, and also molecular simulations, work quantitatively at moderate size ratios in very diluted systems, since large size asymmetries and high particle concentrations are difficult to deal with. The existing approaches for integrating out the degrees of freedom of the depletant species may fail under these extreme physical conditions. Thus, the main goal of this contribution is to introduce a general physical formulation for obtaining the depletion forces even in those cases where the concentration of all species is relevant. We show that the contraction of the bare forces uniquely determines depletion interactions. Our formulation is tested by studying depletion forces in binary and ternary colloidal mixtures. We report here results for dense systems, with total packing fractions of 45\% and 55\%. Our results open up the possibility of finding an efficient route to determine effective interactions at finite concentration, even at non-equilibrium thermodynamic conditions.
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7
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Sandoval-Puentes MA, Torres-Carbajal A, Zavala-Martínez AB, Castañeda-Priego R, Méndez-Alcaraz JM. Soft representation of the square-well and square-shoulder potentials to be used in Brownian and molecular dynamics simulations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:164001. [PMID: 35108690 DOI: 10.1088/1361-648x/ac5139] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Accepted: 02/02/2022] [Indexed: 06/14/2023]
Abstract
The discrete hard-sphere (HS), square-well (SW), and square-shoulder (SS) potentials have become the battle horse of molecular and complex fluids because they contain the basic elements to describe the thermodynamic, structural, and transport properties of both types of fluids. The mathematical simplicity of these discrete potentials allows us to obtain some analytical results despite the nature and complexity of the modeled systems. However, the divergent forces arising at the potential discontinuities may lead to severe issues when discrete potentials are used in computer simulations with uniform time steps. One of the few routes to avoid these technical problems is to replace the discrete potentials with continuous and differentiable forms built under strict physical criteria to capture the correct phenomenology. The match of the second virial coefficient between the discrete and the soft potentials has recently been successfully used to construct a continuous representation that mimics some physical properties of HSs (Báezet al2018J. Chem. Phys.149164907). In this paper, we report an extension of this idea to construct soft representations of the discrete SW and SS potentials. We assess the accuracy of the resulting soft potential by studying structural and thermodynamic properties of the modeled systems by using extensive Brownian and molecular dynamics computer simulations. Besides, Monte Carlo results for the original discrete potentials are used as benchmark. We have also implemented the discrete interaction models and their soft counterparts within the integral equations theory of liquids, finding that the most widely used approximations predict almost identical results for both potentials.
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Affiliation(s)
- Miguel A Sandoval-Puentes
- División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, 37150 León, Guanajuato, Mexico
- Departamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Gustavo A. Madero, CDMX, Mexico
| | - Alexis Torres-Carbajal
- Instituto de Física 'Manuel Sandoval Vallarta', Universidad Autónoma de San Luis Potosí, Alvaro Obregón 64, 78000 San Luis Potosí, SLP, Mexico
| | - Arantza B Zavala-Martínez
- Departamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Gustavo A. Madero, CDMX, Mexico
| | - Ramón Castañeda-Priego
- División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, 37150 León, Guanajuato, Mexico
| | - José M Méndez-Alcaraz
- Departamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Gustavo A. Madero, CDMX, Mexico
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8
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Perdomo-Pérez R, Martínez-Rivera J, Palmero-Cruz NC, Sandoval-Puentes MA, Gallegos JAS, Lázaro-Lázaro E, Valadez-Pérez NE, Torres-Carbajal A, Castañeda-Priego R. Thermodynamics, static properties and transport behaviour of fluids with competing interactions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:144005. [PMID: 35026739 DOI: 10.1088/1361-648x/ac4b29] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Accepted: 01/13/2022] [Indexed: 06/14/2023]
Abstract
Competing interaction fluids have become ideal model systems to study a large number of phenomena, for example, the formation of intermediate range order structures, condensed phases not seen in fluids driven by purely attractive or repulsive forces, the onset of particle aggregation under in- and out-of-equilibrium conditions, which results in the birth of reversible and irreversible aggregates or clusters whose topology and morphology depend additionally on the thermodynamic constrictions, and a particle dynamics that has a strong influence on the transport behaviour and rheological properties of the fluid. In this contribution, we study a system of particles interacting through a potential composed by a continuous succession of a short-ranged square-well (SW), an intermediate-ranged square-shoulder and a long-ranged SW. This potential model is chosen to systematically analyse the contribution of every component of the interaction potential on the phase behaviour, the microstructure, the morphology of the resulting aggregates and the transport phenomena of fluids described by competing interactions. Our results indicate that the inclusion of a barrier and a second well leads to new and interesting effects, which in addition result in variations of the physical properties associated to the competition among interactions.
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Affiliation(s)
- Román Perdomo-Pérez
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Jaime Martínez-Rivera
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Norma C Palmero-Cruz
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Miguel A Sandoval-Puentes
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Javier A S Gallegos
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Edilio Lázaro-Lázaro
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
| | - Néstor E Valadez-Pérez
- Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, Carretera Emiliano Zapata km 8, 29050 Tuxtla Gutiérrez, Mexico
| | - Alexis Torres-Carbajal
- Unidad Profesional Interdisciplinaria de Ingeniería, Campus Tlaxcala, Instituto Politécnico Nacional. Plaza Bicentenario, Guillermo Valle 11, Centro, 9000 Tlaxcala de Xicohténcatl, Tlaxcala, México
| | - Ramón Castañeda-Priego
- Departamento de Ingeniería Física, División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico
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9
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Cywiak D, Gil-Villegas A, Patti A. Long-time relaxation dynamics in nematic and smectic liquid crystals of soft repulsive colloidal rods. Phys Rev E 2022; 105:014703. [PMID: 35193200 DOI: 10.1103/physreve.105.014703] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2021] [Accepted: 01/04/2022] [Indexed: 06/14/2023]
Abstract
Understanding the relaxation dynamics of colloidal suspensions is crucial to identifying the elements that influence the mobility of their constituents, assessing their macroscopic response across the relevant time and length scales, and thus disclosing the fundamentals underpinning their exploitation in formulation engineering. In this work, we specifically assess the impact of long-ranged ordering on the relaxation dynamics of suspensions of soft repulsive rodlike particles, which are able to self-organize into nematic and smectic liquid-crystalline phases. Rods are modeled as soft repulsive spherocylinders with a length-to-diameter ratio L^{★}=5, interacting via the truncated and shifted Kihara potential. By performing dynamic Monte Carlo simulations, we analyze the effect of translational and orientational order on the diffusion of the rods along the relevant directions imposed by the morphology of the background phases. To provide a clear picture of the resulting dynamics, we assess its dependence on temperature, which can dramatically determine the response time of the system relaxation and the self-diffusion coefficients of the rods. The computation of the van Hove correlation functions allows us to identify the existence of rods that diffuse significantly faster than the average and whose concentration can be accurately adjusted by a suitable choice of temperature.
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Affiliation(s)
- Daniela Cywiak
- División de Ciencias e Ingenierías, Universidad de Guanajuato, Campus León, Mexico
| | | | - Alessandro Patti
- Department of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, United Kingdom
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10
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de Los Santos-López NM, Pérez-Ángel G, Méndez-Alcaraz JM, Castañeda-Priego R. Competing interactions in the depletion forces of ternary colloidal mixtures. J Chem Phys 2021; 155:024901. [PMID: 34266249 DOI: 10.1063/5.0052369] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Depletion interactions between colloidal particles surrounded by smaller depletants are typically characterized by a strong attraction at contact and a moderately repulsive barrier in front of it that extends at distances similar to the size of the depletants; the appearance and height of the barrier basically depend on the concentration and, therefore, the correlation between depletants. From a thermodynamic point of view, the former can drive the system to phase separation or toward non-equilibrium states, such as gel-like states, but its effects on both local and global properties may be controlled by the latter, which acts as a kind of entropic gate. However, the latter has not been entirely analyzed and understood within the context of colloidal mixtures mainly driven by entropy. In this contribution, we present a systematic study of depletion forces in ternary mixtures of hard spherical particles with two species of depletants, in two and three dimensions. We focus the discussion on how the composition of the depletants becomes the main physical parameter that drives the competition between the attractive well and the repulsive barrier. Our results are obtained by means of the integral equation theory of depletion forces and techniques of contraction of the description adapted to molecular dynamics computer simulations.
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Affiliation(s)
| | - Gabriel Pérez-Ángel
- Departamento de Física Aplicada, Cinvestav-Mérida, AP 73 "Cordemex," 97310 Mérida, Yucatán, Mexico
| | - José M Méndez-Alcaraz
- Departamento de Física, Cinvestav, Av. IPN 2508, Col. San Pedro Zacatenco, Gustavo A. Madero, 07360 Ciudad de México, Mexico
| | - Ramón Castañeda-Priego
- División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Guanajuato, Mexico
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11
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Mercado-Uribe H, Guevara-Pantoja FJ, García-Muñoz W, García-Maldonado JS, Méndez-Alcaraz JM, Ruiz-Suárez JC. On the evolution of the exclusion zone produced by hydrophilic surfaces: A contracted description. J Chem Phys 2021; 154:194902. [PMID: 34240904 DOI: 10.1063/5.0043084] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
After exciting scientific debates about its nature, the development of the exclusion zone, a region near hydrophilic surfaces from which charged colloidal particles are strongly expelled, has been finally traced back to the diffusiophoresis produced by unbalanced ion gradients. This was done by numerically solving the coupled Poisson equation for electrostatics, the two stationary Stokes equations for low Reynolds numbers in incompressible fluids, and the Nernst-Planck equation for mass transport. Recently, it has also been claimed that the leading mechanism behind the diffusiophoretic phenomenon is electrophoresis [Esplandiu et al., Soft Matter 16, 3717 (2020)]. In this paper, we analyze the evolution of the exclusion zone based on a one-component interaction model at the Langevin equation level, which leads to simple analytical expressions instead of the complex numerical scheme of previous works, yet being consistent with it. We manage to reproduce the evolution of the exclusion zone width and the mean-square displacements of colloidal particles we measure near Nafion, a perfluorinated polymer membrane material, along with all characteristic time regimes, in a unified way. Our findings are also strongly supported by complementary experiments using two parallel planar conductors kept at a fixed voltage, mimicking the hydrophilic surfaces, and some computer simulations.
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Affiliation(s)
- H Mercado-Uribe
- CINVESTAV-Monterrey, PIIT, 66600 Apodaca, Nuevo León, Mexico
| | | | - W García-Muñoz
- CINVESTAV-Monterrey, PIIT, 66600 Apodaca, Nuevo León, Mexico
| | - J S García-Maldonado
- Departamento de Física, CINVESTAV, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Ciudad de México, Mexico
| | - J M Méndez-Alcaraz
- Departamento de Física, CINVESTAV, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Ciudad de México, Mexico
| | - J C Ruiz-Suárez
- CINVESTAV-Monterrey, PIIT, 66600 Apodaca, Nuevo León, Mexico
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