1
|
Zhan L, Zhang Z, Zheng F, Liu W, Zhang Y, Sha J, Chen Y. Ion Concentration-Dependent Surface Charge Density Inside a Nanopore. J Phys Chem Lett 2023; 14:11536-11542. [PMID: 38095320 DOI: 10.1021/acs.jpclett.3c02319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2023]
Abstract
Surface charges shape the electrical double layer (EDL) structure at solid-liquid interfaces, critically influencing the performance of energy storage and micro/nanofluidic devices. However, accurately measuring surface charge density in nanoconfined spaces continues to be a challenge. Here, we introduce a methodology via solid-state nanopores that can investigate the dependence of surface charge density on salt concentrations and nanopore diameters. Measurements, complemented by a theoretical model, reveal that the surface charge density decreases as both the salt concentration in bulk solutions and the nanopore sizes are reduced. Notably, when the salt concentration in the bulk solution drops below 10-3 M, protons dominate ion conductance in a nanopore, resulting in a constant surface charge density. This study introduces an effective approach to surface charge characterization and may serve in the design of electrokinetically driven nanofluidic systems.
Collapse
Affiliation(s)
- Lijian Zhan
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Zhenyu Zhang
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Fei Zheng
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Wei Liu
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Yin Zhang
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Jingjie Sha
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| | - Yunfei Chen
- Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189, China
| |
Collapse
|
2
|
Zhou T, Zhao J, He X, Shi L, Wen L. Effect of brush roughness on volume charge density. J Electroanal Chem (Lausanne) 2023. [DOI: 10.1016/j.jelechem.2023.117368] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/29/2023]
|
3
|
Hachim D, Zhao J, Bhankharia J, Nuñez-Toldra R, Brito L, Seong H, Becce M, Ouyang L, Grigsby CL, Higgins SG, Terracciano CM, Stevens MM. Polysaccharide-Polyplex Nanofilm Coatings Enhance Nanoneedle-Based Gene Delivery and Transfection Efficiency. SMALL (WEINHEIM AN DER BERGSTRASSE, GERMANY) 2022; 18:e2202303. [PMID: 35770803 PMCID: PMC7615482 DOI: 10.1002/smll.202202303] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Indexed: 06/15/2023]
Abstract
Non-viral vectors represent versatile and immunologically safer alternatives for nucleic acid delivery. Nanoneedles and high-aspect ratio nanostructures are unconventional but interesting delivery systems, in which delivery is mediated by surface interactions. Herein, nanoneedles are synergistically combined with polysaccharide-polyplex nanofilms and enhanced transfection efficiency is observed, compared to polyplexes in suspension. Different polyplex-polyelectrolyte nanofilm combinations are assessed and it is found that transfection efficiency is enhanced when using polysaccharide-based polyanions, rather than being only specific for hyaluronic acid, as suggested in earlier studies. Moreover, results show that enhanced transfection is not mediated by interactions with the CD44 receptor, previously hypothesized as a major mechanism mediating enhancement via hyaluronate. In cardiac tissue, nanoneedles are shown to increase the transfection efficiency of nanofilms compared to flat substrates; while in vitro, high transfection efficiencies are observed in nanostructures where cells present large interfacing areas with the substrate. The results of this study demonstrate that surface-mediated transfection using this system is efficient and safe, requiring amounts of nucleic acid with an order of magnitude lower than standard culture transfection. These findings expand the spectrum of possible polyelectrolyte combinations that can be used for the development of suitable non-viral vectors for exploration in further clinical trials.
Collapse
Affiliation(s)
- Daniel Hachim
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Juzhi Zhao
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Jash Bhankharia
- Department of Bioengineering, Imperial College London, London SW7 2AZ, UK
| | - Raquel Nuñez-Toldra
- National Heart and Lung Institute, Imperial College London, London SW3 6LY, UK
| | - Liliana Brito
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Hyejeong Seong
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Michele Becce
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Liliang Ouyang
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | - Christopher L. Grigsby
- Department of Medical Biochemistry and Biophysics, Karolinska Institute, Stockholm 171 65, Sweden
| | - Stuart G. Higgins
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| | | | - Molly M. Stevens
- Department of Materials, Imperial College London, London SW7 2AZ, UK
| |
Collapse
|
4
|
An R, Laaksonen A, Wu M, Zhu Y, Shah FU, Lu X, Ji X. Atomic force microscopy probing interactions and microstructures of ionic liquids at solid surfaces. NANOSCALE 2022; 14:11098-11128. [PMID: 35876154 DOI: 10.1039/d2nr02812c] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Ionic liquids (ILs) are room temperature molten salts that possess preeminent physicochemical properties and have shown great potential in many applications. However, the use of ILs in surface-dependent processes, e.g. energy storage, is hindered by the lack of a systematic understanding of the IL interfacial microstructure. ILs on the solid surface display rich ordering, arising from coulombic, van der Waals, solvophobic interactions, etc., all giving near-surface ILs distinct microstructures. Therefore, it is highly important to clarify the interactions of ILs with solid surfaces at the nanoscale to understand the microstructure and mechanism, providing quantitative structure-property relationships. Atomic force microscopy (AFM) opens a surface-sensitive way to probe the interaction force of ILs with solid surfaces in the layers from sub-nanometers to micrometers. Herein, this review showcases the recent progress of AFM in probing interactions and microstructures of ILs at solid interfaces, and the influence of IL characteristics, surface properties and external stimuli is thereafter discussed. Finally, a summary and perspectives are established, in which, the necessities of the quantification of IL-solid interactions at the molecular level, the development of in situ techniques closely coupled with AFM for probing IL-solid interfaces, and the combination of experiments and simulations are argued.
Collapse
Affiliation(s)
- Rong An
- Herbert Gleiter Institute of Nanoscience, School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China.
| | - Aatto Laaksonen
- Energy Engineering, Division of Energy Science, Luleå University of Technology, 97187 Luleå, Sweden.
- Department of Materials and Environmental Chemistry, Arrhenius Laboratory, Stockholm University, SE-10691 Stockholm, Sweden
- Center of Advanced Research in Bionanoconjugates and Biopolymers, "Petru Poni" Institute of Macromolecular Chemistry, Iasi 700469, Romania
- State Key Laboratory of Materials-Oriented and Chemical Engineering, Nanjing Tech University, Nanjing 211816, China
| | - Muqiu Wu
- Herbert Gleiter Institute of Nanoscience, School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China.
| | - Yudan Zhu
- State Key Laboratory of Materials-Oriented and Chemical Engineering, Nanjing Tech University, Nanjing 211816, China
| | - Faiz Ullah Shah
- Chemistry of Interfaces, Luleå University of Technology, 97187 Luleå, Sweden
| | - Xiaohua Lu
- State Key Laboratory of Materials-Oriented and Chemical Engineering, Nanjing Tech University, Nanjing 211816, China
| | - Xiaoyan Ji
- Energy Engineering, Division of Energy Science, Luleå University of Technology, 97187 Luleå, Sweden.
| |
Collapse
|
5
|
Yang Y, Zhang X, Tian Z, Deissmann G, Bosbach D, Liang P, Wang M. Thermodiffusion of ions in nanoconfined aqueous electrolytes. J Colloid Interface Sci 2022; 619:331-338. [DOI: 10.1016/j.jcis.2022.03.077] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2022] [Revised: 03/08/2022] [Accepted: 03/17/2022] [Indexed: 10/18/2022]
|
6
|
Alan BO, Barisik M. Size and roughness dependent temperature effects on surface charge of silica nanoparticles. Colloids Surf A Physicochem Eng Asp 2021. [DOI: 10.1016/j.colsurfa.2021.127407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
|
7
|
Abulaiti A, Salai A, Sun X, Yibulayin W, Gao Y, Gopinath SCB, Sun W. Nano-silica embedded polydimethylsiloxane on interdigitated sensor as adhesive polymer for detecting lung cancer mutation. Biotechnol Appl Biochem 2021; 69:451-460. [PMID: 33576539 DOI: 10.1002/bab.2122] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/02/2021] [Accepted: 02/05/2021] [Indexed: 12/22/2022]
Abstract
Non-small cell lung cancer (NSCLC) incited by epidermal growth factor receptor (EGFR) mutation makes up ∼85% of lung cancer diagnosed and death cases worldwide. The presented study introduced an alternative approach in detecting EGFR mutation using nano-silica integrated with polydimethylsiloxane (PDMS) polymer on interdigitated electrode (IDE) sensor. A 400 μm gap-sized aluminum IDE was modified with nano-polymer layer, which was made up of silica nanoparticles and PDMS polymer. IDE and PDMS-coated IDE (PDMS/IDE) were imaged using electron microscopes that reveals its smooth and ideal sensor morphology. The nano-silica-integrated PDMS/IDE surface was immobilized with EGFR probe and target to specify the lung cancer detection. The sensor specificity was justified through the insignificant current readouts with one-base mismatch and noncomplementary targets. The sensitivity of nano-silica-integrated PDMS/IDE was examined with mutant target spiked in human serum, where the resulting current affirms the detection of EGFR mutation. Based on the slope of the calibration curve, the sensitivity of nano-silica-integrated PDMS/IDE was 2.24E-9 A M-1 . The sensor recognizes EGFR mutation lowest at 1 aM complementary mutant target; however, the detection limit obtained based on 3σ calculation is 10 aM with regression value of 0.97.
Collapse
Affiliation(s)
- Abulimiti Abulaiti
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| | - Adili Salai
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| | - Xiaohong Sun
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| | - Waresijiang Yibulayin
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| | - Yunfei Gao
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| | - Subash C B Gopinath
- Faculty of Chemical Engineering Technology, Universiti Malaysia Perlis (UniMAP), Arau, Perlis, 02600, Malaysia.,Institute of Nano Electronic Engineering, Universiti Malaysia Perlis (UniMAP), Kangar, Perlis, 01000, Malaysia
| | - Wei Sun
- Department of Thoracic Surgery, Xinjiang Tumor Hospital, Urumqi, People's Republic of China
| |
Collapse
|
8
|
Physical methods for controlling bacterial colonization on polymer surfaces. Biotechnol Adv 2020; 43:107586. [DOI: 10.1016/j.biotechadv.2020.107586] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2020] [Revised: 06/05/2020] [Accepted: 07/06/2020] [Indexed: 02/06/2023]
|
9
|
Ozcelik HG, Satiroglu E, Barisik M. Size dependent influence of contact line pinning on wetting of nano-textured/patterned silica surfaces. NANOSCALE 2020; 12:21376-21391. [PMID: 33078810 DOI: 10.1039/d0nr05392a] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Wetting behavior on a heterogeneous surface undergoes contact angle hysteresis as the droplet stabilized at a metastable state with a contact angle significantly different from its equilibrium value due to contact line pinning. However, there is a lack of consensus on how to calculate the influence of pinning forces. In general, the pinning effect can be characterized as (i) microscopic behavior when a droplet is pinned and the contact angle increases/decreases as the droplet volume increases/decreases and (ii) macroscopic behavior as the pinning effects decrease and ultimately, disappear with the increase of the droplet size. The current work studied both behaviors using molecular dynamics (MD) simulation with more than 300 different size water droplets on silica surfaces with three different patterns across two different wetting conditions. Results showed that the contact angle increases linearly with increasing droplet volume through the microscopic behavior, while the droplet is pinned on top of a certain number of patterns. When we normalized the droplet size with the corresponding pattern size, we observed a "wetting similarity" that linear microscopic contact angle variations over different size heterogeneities continuously line up. This shows that the pinning force remains constant and the resulting pinning effects are scalable by the size ratio between the droplet and pattern, independent of the size-scale. The slope of these microscopic linear variations decreases with an increase in the droplet size as observed through the macroscopic behavior. We further found a universal behavior in the variation of the corresponding pinning forces, independent of the wetting condition. In macroscopic behavior, pinning effects become negligible and the contact angle reaches the equilibrium value of the corresponding surface when the diameter of the free-standing droplet is approximately equal to 24 times the size of the surface structure. We found that the pinning effect is scalable with the droplet volume, not the size of the droplet base.
Collapse
Affiliation(s)
- H Gokberk Ozcelik
- Department of Mechanical Engineering, Izmir Institute of Technology, Izmir, 35430, Turkey.
| | - Ezgi Satiroglu
- Department of Energy Systems Engineering, Izmir Institute of Technology, Izmir, 35430, Turkey
| | - Murat Barisik
- Department of Mechanical Engineering, Izmir Institute of Technology, Izmir, 35430, Turkey.
| |
Collapse
|
10
|
Bakhshandeh A, Frydel D, Levin Y. Charge regulation of colloidal particles in aqueous solutions. Phys Chem Chem Phys 2020; 22:24712-24728. [PMID: 33104140 DOI: 10.1039/d0cp03633a] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.
Collapse
Affiliation(s)
- Amin Bakhshandeh
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil.
| | | | | |
Collapse
|
11
|
Sen T, Barisik M. Slip Effects on Ionic Current of Viscoelectric Electroviscous Flows through Different Length Nanofluidic Channels. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2020; 36:9191-9203. [PMID: 32635731 DOI: 10.1021/acs.langmuir.0c01457] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
The pressure driven slip flow of an electrolyte solution is studied through different nanofluidic channel lengths at varying salt concentrations. The viscous-thickening due to the electrostatic interactions within the electric double layer and the reverse ionic transport due to the streaming potential are developed. The influence of the Navier slip boundary condition is described under both electroviscous and viscoelectric effects with a surface charge regulation (CR) model while the observed behavior is compared and validated with molecular dynamic (MD) calculations from multiple studies. Results show that electroviscous and viscoelectric effects decrease transport. Earlier studies at the no slip boundary presented an increase of ionic current by increasing salt concentration and decreasing channel length. In contrast, our study found that the ionic current occurred almost independent of both salt concentration and channel length, except for very short channels and very low salt concentrations, when electroviscous and viscoelectric effects were considered. In the case of the constant slip length condition, ionic conduction was enhanced, but velocity slip developing on surfaces showed significant variation based on the salt concentration and channel length. This is due to the natural CR behavior enhancing the surface charge and consequential near surface electrohydrodynamics as a result of increase in salt concentration and/or decrease of channel length. Considering that the electroviscous effect alone creates up to 70% lower velocity slips than Poiseuille flow predictions, while further including the viscoelectric effect, results in an almost no-slip condition at high salt concentrations and/or short channels. As a result, the ionic current of a viscoelectric electroviscous slip flow is found to be equal to 1/3 of an electroviscous slip flow and to decrease with a decrease in the channel length.
Collapse
Affiliation(s)
- Tumcan Sen
- Department of Mechanical Engineering, Izmir Institute Of Technology, Izmir 35430, Turkey
| | - Murat Barisik
- Department of Mechanical Engineering, Izmir Institute Of Technology, Izmir 35430, Turkey
| |
Collapse
|
12
|
Winter H, Neufeld MJ, Makotamo L, Sun C, Goforth AM. Synthesis of Radioluminescent CaF 2:Ln Core, Mesoporous Silica Shell Nanoparticles for Use in X-ray Based Theranostics. NANOMATERIALS (BASEL, SWITZERLAND) 2020; 10:E1447. [PMID: 32722132 PMCID: PMC7466269 DOI: 10.3390/nano10081447] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/24/2020] [Revised: 07/19/2020] [Accepted: 07/22/2020] [Indexed: 12/29/2022]
Abstract
X-ray radiotherapy is a common method of treating cancerous tumors or other malignant lesions. The side effects of this treatment, however, can be deleterious to patient quality of life if critical tissues are affected. To potentially lower the effective doses of radiation and negative side-effects, new classes of nanoparticles are being developed to enhance reactive oxygen species production during irradiation. This report presents the synthesis and radiotherapeutic efficacy evaluation of a new nanoparticle formulation designed for this purpose, composed of a CaF2 core, mesoporous silica shell, and polyethylene glycol coating. The construct was additionally doped with Tb and Eu during the CaF2 core synthesis to prepare nanoparticles (NPs) with X-ray luminescent properties for potential application in fluorescence imaging. The mesoporous silica shell was added to provide the opportunity for small molecule loading, and the polyethylene glycol coating was added to impart aqueous solubility and biocompatibility. The potential of these nanomaterials to act as radiosensitizers for enhancing X-ray radiotherapy was supported by reactive oxygen species generation assays. Further, in vitro experiments indicate biocompatibility and enhanced cellular damage during X-ray radiotherapy.
Collapse
Affiliation(s)
- Hayden Winter
- Department of Chemistry, Portland State University, 1719 SW 10th Ave., Portland, OR 97201, USA; (H.W.); (L.M.)
| | - Megan J. Neufeld
- Department of Pharmaceutical Sciences, College of Pharmacy, Oregon State University, Portland, OR 97201, USA;
| | - Lydia Makotamo
- Department of Chemistry, Portland State University, 1719 SW 10th Ave., Portland, OR 97201, USA; (H.W.); (L.M.)
| | - Conroy Sun
- Department of Pharmaceutical Sciences, College of Pharmacy, Oregon State University, Portland, OR 97201, USA;
- Department of Radiation Medicine, 3181 S. W. Sam Jackson Park Rd, Oregon Health & Science University, Portland, OR 97239, USA
| | - Andrea M. Goforth
- Department of Chemistry, Portland State University, 1719 SW 10th Ave., Portland, OR 97201, USA; (H.W.); (L.M.)
| |
Collapse
|
13
|
Bakhshandeh A, Frydel D, Diehl A, Levin Y. Charge Regulation of Colloidal Particles: Theory and Simulations. PHYSICAL REVIEW LETTERS 2019; 123:208004. [PMID: 31809122 DOI: 10.1103/physrevlett.123.208004] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2019] [Indexed: 06/10/2023]
Abstract
To explore charge regulation (CR) in physicochemical and biophysical systems, we present a model of colloidal particles with sticky adsorption sites which account for the formation of covalent bonds between the hydronium ions and the surface functional groups. Using this model and Monte Carlo simulations, we find that the standard Ninham and Parsegian (NP) theory of CR leads to results which deviate significantly from computer simulations. The problem with the NP approach is traced back to the use of a bulk equilibrium constant to account for surface chemical reactions. To resolve this difficulty we present a new theory of CR. The fundamental ingredient of the new approach is the sticky length, which is nontrivially related to the bulk equilibrium constant. The theory is found to be in excellent agreement with computer simulations, without any adjustable parameters. As an application of the theory we calculate the effective charge of colloidal particles containing carboxyl groups, as a function of pH and salt concentration.
Collapse
Affiliation(s)
- Amin Bakhshandeh
- Programa de Pós-Graduação em Física, Instituto de Física e Matemática, Universidade Federal de Pelotas, Caixa Postal 354, CEP 96010-900 Pelotas, Rio Grande do Sul, Brazil
| | - Derek Frydel
- Department of Chemistry, Federico Santa Maria Technical University, Campus San Joaquin, 7820275 Santiago, Chile
| | - Alexandre Diehl
- Departamento de Física, Instituto de Física e Matemática, Universidade Federal de Pelotas, Caixa Postal 354, CEP 96010-900 Pelotas, Rio Grande do Sul, Brazil
| | - Yan Levin
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970 Porto Alegre, Rio Grande do Sul, Brazil
| |
Collapse
|