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Chan CW, Yang Z, Gan Z, Zhang R. Interplay of chemotactic force, Péclet number, and dimensionality dictates the dynamics of auto-chemotactic chiral active droplets. J Chem Phys 2024; 161:014904. [PMID: 38953449 DOI: 10.1063/5.0207355] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2024] [Accepted: 05/31/2024] [Indexed: 07/04/2024] Open
Abstract
In living and synthetic active matter systems, the constituents can self-propel and interact with each other and with the environment through various physicochemical mechanisms. Among these mechanisms, chemotactic and auto-chemotactic effects are widely observed. The impact of (auto-)chemotactic effects on achiral active matter has been a recent research focus. However, the influence of these effects on chiral active matter remains elusive. Here, we develop a Brownian dynamics model coupled with a diffusion equation to examine the dynamics of auto-chemotactic chiral active droplets in both quasi-two-dimensional (2D) and three-dimensional (3D) systems. By quantifying the droplet trajectory as a function of the dimensionless Péclet number and chemotactic strength, our simulations well reproduce the curling and helical trajectories of nematic droplets in a surfactant-rich solution reported by Krüger et al. [Phys. Rev. Lett. 117, 048003 (2016)]. The modeled curling trajectory in 2D exhibits an emergent chirality, also consistent with the experiment. We further show that the geometry of the chiral droplet trajectories, characterized by the pitch and diameter, can be used to infer the velocities of the droplet. Interestingly, we find that, unlike the achiral case, the velocities of chiral active droplets show dimensionality dependence: its mean instantaneous velocity is higher in 3D than in 2D, whereas its mean migration velocity is lower in 3D than in 2D. Taken together, our particle-based simulations provide new insights into the dynamics of auto-chemotactic chiral active droplets, reveal the effects of dimensionality, and pave the way toward their applications, such as drug delivery, sensors, and micro-reactors.
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Affiliation(s)
- Chung Wing Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR
- Thrust of Advanced Materials, and Guangzhou Municipal Key Laboratory of Materials Informatics, The Hong Kong University of Science and Technology (Guangzhou), Guangdong, China
| | - Zheng Yang
- Thrust of Advanced Materials, and Guangzhou Municipal Key Laboratory of Materials Informatics, The Hong Kong University of Science and Technology (Guangzhou), Guangdong, China
- Interdisciplinary Programs Office, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR
| | - Zecheng Gan
- Thrust of Advanced Materials, and Guangzhou Municipal Key Laboratory of Materials Informatics, The Hong Kong University of Science and Technology (Guangzhou), Guangdong, China
- Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR
| | - Rui Zhang
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR
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2
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Farutin A, Misbah C. Singular bifurcations and regularization theory. Phys Rev E 2024; 109:064218. [PMID: 39021029 DOI: 10.1103/physreve.109.064218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2024] [Accepted: 06/06/2024] [Indexed: 07/20/2024]
Abstract
Nonlinear sciences are present today in almost all disciplines, ranging from physics to social sciences. A major task in nonlinear science is the classification of different types of bifurcations (e.g., pitchfork and saddle-node) from a given state to another. Bifurcation analysis is traditionally based on the assumption of a regular perturbative expansion, close to the bifurcation point, in terms of a variable describing the passage of a system from one state to another. However, it is shown that a regular expansion is not the rule due to the existence of hidden singularities in many models, paving the way to a new paradigm in nonlinear science, that of singular bifurcations. The theory is first illustrated on an example borrowed from the field of active matter (phoretic microswimers), showing a singular bifurcation. We then present a universal theory on how to handle and regularize these bifurcations, bringing to light a novel facet of nonlinear sciences that has long been overlooked.
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3
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Yang Q, Jiang M, Picano F, Zhu L. Shaping active matter from crystalline solids to active turbulence. Nat Commun 2024; 15:2874. [PMID: 38570495 PMCID: PMC11258367 DOI: 10.1038/s41467-024-46520-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 02/27/2024] [Indexed: 04/05/2024] Open
Abstract
Active matter drives its constituent agents to move autonomously by harnessing free energy, leading to diverse emergent states with relevance to both biological processes and inanimate functionalities. Achieving maximum reconfigurability of active materials with minimal control remains a desirable yet challenging goal. Here, we employ large-scale, agent-resolved simulations to demonstrate that modulating the activity of a wet phoretic medium alone can govern its solid-liquid-gas phase transitions and, subsequently, laminar-turbulent transitions in fluid phases, thereby shaping its emergent pattern. These two progressively emerging transitions, hitherto unreported, bring us closer to perceiving the parallels between active matter and traditional matter. Our work reproduces and reconciles seemingly conflicting experimental observations on chemically active systems, presenting a unified landscape of phoretic collective dynamics. These findings enhance the understanding of long-range, many-body interactions among phoretic agents, offer new insights into their non-equilibrium collective behaviors, and provide potential guidelines for designing reconfigurable materials.
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Affiliation(s)
- Qianhong Yang
- Department of Mechanical Engineering, National University of Singapore, Singapore, Singapore
| | - Maoqiang Jiang
- School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan, Hubei, PR China
| | - Francesco Picano
- Department of Industrial Engineering and CISAS "G. Colombo", University of Padova, Padova, Italy
| | - Lailai Zhu
- Department of Mechanical Engineering, National University of Singapore, Singapore, Singapore.
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4
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Wang Y, Zheng L, Li G. Self-propulsion of a droplet induced by combined diffusiophoresis and Marangoni effects. Electrophoresis 2024. [PMID: 38528332 DOI: 10.1002/elps.202400005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2024] [Revised: 02/22/2024] [Accepted: 03/12/2024] [Indexed: 03/27/2024]
Abstract
Chemically active droplets display complex self-propulsion behavior in homogeneous surfactant solutions, often influenced by the interplay between diffusiophoresis and Marangoni effects. Previous studies have primarily considered these effects separately or assumed axisymmetric motion. To understand the full hydrodynamics, we investigate the motion of a two-dimensional active droplet under their combined influences using weakly nonlinear analysis and numerical simulations. The impact of two key factors, the Péclet number (P e $Pe$ ) and the mobility ratio between diffusiophoretic and Marangoni effects ( m $m$ ), on droplet motion is explored. We establish a phase diagram in theP e - m $Pe-m$ space, categorizing the boundaries between four types of droplet states: stationary, steady motion, periodic/quasi-periodic motion, and chaotic motion. We find that the mobility ratio does not affect the criticalP e $Pe$ for the onset of self-propulsion, but it significantly influences the stability of high-wavenumber modes as well as the droplet's velocity and trajectory. Scaling analysis reveals that in the highP e $Pe$ regime, the Marangoni and diffusiophoresis effects lead to distinct velocity scaling laws:U ∼ P e - 1 / 2 $U\sim Pe^{-1/2}$ andU ∼ P e - 1 / 3 $U\sim Pe^{-1/3}$ , respectively. When these effects are combined, the velocity scaling depends on the sign of the mobility ratio. In cases with a positive mobility ratio, the Marangoni effect dominates the scaling, whereas the negative diffusiophoretic effect leads to an increased thickness of the concentration boundary layer and a flattened scaling of the droplet velocity.
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Affiliation(s)
- Yuhang Wang
- School of Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, China
| | - Longtao Zheng
- School of Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, China
| | - Gaojin Li
- School of Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, China
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5
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Farutin A, Rizvi SM, Hu WF, Lin TS, Rafai S, Misbah C. Motility and swimming: universal description and generic trajectories. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2023; 46:135. [PMID: 38146033 DOI: 10.1140/epje/s10189-023-00395-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2023] [Accepted: 12/11/2023] [Indexed: 12/27/2023]
Abstract
Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that for a particle driven by a concentration field of an active species, straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the particle or that of suspending medium. Our proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations, leading to spontaneous symmetry breaking in the concentration field driving the particle motility. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium, or influence of bounding walls are not necessary.
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Affiliation(s)
| | - Suhail M Rizvi
- Univ. Grenoble Alpes, CNRS, LIPhy, F-38000, Grenoble, France
- Department of Biomedical Engineering, Indian Institute of Technology Hyderabad, Sangareddy, Telangana, 502285, India
| | - Wei-Fan Hu
- Department of Mathematics, National Central University, 300 Zhongda Road, Taoyuan, 320, Taiwan
| | - Te-Sheng Lin
- Department of Applied Mathematics, National Yang Ming Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan
| | - Salima Rafai
- Univ. Grenoble Alpes, CNRS, LIPhy, F-38000, Grenoble, France
| | - Chaouqi Misbah
- Univ. Grenoble Alpes, CNRS, LIPhy, F-38000, Grenoble, France.
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6
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Decayeux J, Fries J, Dahirel V, Jardat M, Illien P. Isotropic active colloids: explicit vs. implicit descriptions of propulsion mechanisms. SOFT MATTER 2023; 19:8997-9005. [PMID: 37965908 DOI: 10.1039/d3sm00763d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2023]
Abstract
Modeling the couplings between active particles often neglects the possible many-body effects that control the propulsion mechanism. Accounting for such effects requires the explicit modeling of the molecular details at the origin of activity. Here, we take advantage of a recent two-dimensional model of isotropic active particles whose propulsion originates from the interactions between solute particles in the bath. The colloid catalyzes a chemical reaction in its vicinity, which results in a local phase separation of solute particles, and the density fluctuations of solute particles cause the enhanced diffusion of the colloid. In this paper, we investigate an assembly of such active particles, using (i) an explicit model, where the microscopic dynamics of the solute particles is accounted for; and (ii) an implicit model, whose parameters are inferred from the explicit model at infinite dilution. In the explicit solute model, the long-time diffusion coefficient of the active colloids strongly decreases with density, an effect which is not captured by the derived implicit model. This suggests that classical models, which usually decouple pair interactions from activity, fail to describe collective dynamics in active colloidal systems driven by solute-solute interactions.
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Affiliation(s)
- Jeanne Decayeux
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Jacques Fries
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Vincent Dahirel
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Marie Jardat
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
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7
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Kailasham R, Khair AS. Non-Brownian diffusion and chaotic rheology of autophoretic disks. Phys Rev E 2023; 107:044609. [PMID: 37198791 DOI: 10.1103/physreve.107.044609] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2022] [Accepted: 03/29/2023] [Indexed: 05/19/2023]
Abstract
The dynamics of a two-dimensional autophoretic disk is quantified as a minimal model for the chaotic trajectories undertaken by active droplets. Via direct numerical simulations, we show that the mean-square displacement of the disk in a quiescent fluid is linear at long times. Surprisingly, however, this apparently diffusive behavior is non-Brownian, owing to strong cross correlations in the displacement tensor. The effect of a shear flow field on the chaotic motion of an autophoretic disk is examined. Here, the stresslet on the disk is chaotic for weak shear flows; a dilute suspension of such disks would exhibit a chaotic shear rheology. This chaotic rheology is quenched first into a periodic state and ultimately a steady state as the flow strength is increased.
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Affiliation(s)
- R Kailasham
- Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Aditya S Khair
- Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
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8
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Decayeux J, Jardat M, Illien P, Dahirel V. Conditions for the propulsion of a colloid surrounded by a mesoscale phase separation. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2022; 45:96. [PMID: 36459281 DOI: 10.1140/epje/s10189-022-00247-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Accepted: 11/03/2022] [Indexed: 06/17/2023]
Abstract
We study a two-dimensional model of an active isotropic colloid whose propulsion is linked to the interactions between solute particles of the bath. The colloid catalyzes a chemical reaction in its vicinity, that yields a local phase separation of solute particles. The density fluctuations of solute particles result in the enhanced diffusion of the colloid. Using numerical simulations, we thoroughly investigate the conditions under which activity occurs, and we establish a state diagram for the activity of the colloid as a function of the parameters of the model. We use the generated data to unravel a key observable that controls the existence and the intensity of activity: The filling fraction of the reaction area. Remarkably, we finally show that propulsion also occurs in three-dimensional geometries, which confirms the interest of this mechanism for experimental applications.
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Affiliation(s)
- Jeanne Decayeux
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux)), UMR 8234, F-75005 Paris, France
| | - Marie Jardat
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux)), UMR 8234, F-75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux)), UMR 8234, F-75005 Paris, France
| | - Vincent Dahirel
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux)), UMR 8234, F-75005 Paris, France.
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9
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Abstract
The out-of-equilibrium dynamics of chemotactic active matter—be it animate or inanimate—is closely coupled to the environment, a chemical landscape shaped by secretions from the motile agents, fuel uptake, or autochemotactic signaling. This gives rise to complex collective effects, which can be exploited by the agents for colony migration strategies or pattern formation. We study such effects using an idealized experimental system: self-propelled microdroplets that communicate via chemorepulsive trails. We present a comprehensive experimental analysis that involves direct probing of the diffusing chemical trails and the trail–droplet interactions and use it to construct a generic theoretical model. We connect these repulsive autochemotactic interactions to the collective dynamics in emulsions, demonstrating a state of dynamical arrest: chemotactic self-caging. A common feature of biological self-organization is how active agents communicate with each other or their environment via chemical signaling. Such communications, mediated by self-generated chemical gradients, have consequences for both individual motility strategies and collective migration patterns. Here, in a purely physicochemical system, we use self-propelling droplets as a model for chemically active particles that modify their environment by leaving chemical footprints, which act as chemorepulsive signals to other droplets. We analyze this communication mechanism quantitatively both on the scale of individual agent–trail collisions as well as on the collective scale where droplets actively remodel their environment while adapting their dynamics to that evolving chemical landscape. We show in experiment and simulation how these interactions cause a transient dynamical arrest in active emulsions where swimmers are caged between each other’s trails of secreted chemicals. Our findings provide insight into the collective dynamics of chemically active particles and yield principles for predicting how negative autochemotaxis shapes their navigation strategy.
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10
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De Corato M, Pagonabarraga I, Natale G. Spontaneous chiralization of polar active particles. Phys Rev E 2021; 104:044607. [PMID: 34781499 DOI: 10.1103/physreve.104.044607] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 09/26/2021] [Indexed: 11/07/2022]
Abstract
Polar active particles constitute a wide class of active matter that is able to propel along a preferential direction, given by their polar axis. Here, we demonstrate a generic active mechanism that leads to their spontaneous chiralization through a symmetry-breaking instability. We find that the transition of an active particle from a polar to a chiral symmetry is characterized by the emergence of active rotation and of circular trajectories. The instability is driven by the advection of a solute that interacts differently with the two portions of the particle surface and it occurs through a supercritical pitchfork bifurcation.
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Affiliation(s)
- Marco De Corato
- Aragon Institute of Engineering Research (I3A), University of Zaragoza, 50009 Zaragoza, Spain
| | - Ignacio Pagonabarraga
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, C. Martí Franquès 1, 08028 Barcelona, Spain University of Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, 08028 Barcelona, Spain and CECAM, Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne (EPFL), Batochime, Avenue Forel 2, 1015 Lausanne, Switzerland
| | - Giovanniantonio Natale
- Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive NW, Calgary, Canada
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11
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Decayeux J, Dahirel V, Jardat M, Illien P. Spontaneous propulsion of an isotropic colloid in a phase-separating environment. Phys Rev E 2021; 104:034602. [PMID: 34654103 DOI: 10.1103/physreve.104.034602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/24/2021] [Accepted: 08/16/2021] [Indexed: 11/07/2022]
Abstract
The motion of active colloids is generally achieved through their anisotropy, as exemplified by Janus colloids. Recently, there was a growing interest in the propulsion of isotropic colloids, which requires some local symmetry breaking. Although several mechanisms for such propulsion were proposed, little is known about the role played by the interactions within the environment of the colloid, which can have a dramatic effect on its propulsion. Here, we propose a minimal model of an isotropic colloid in a bath of solute particles that interact with each other. These interactions lead to a spontaneous phase transition close to the colloid, to directed motion of the colloid over very long timescales and to significantly enhanced diffusion, in spite of the crowding induced by solute particles. We determine the range of parameters where this effect is observable in the model, and we propose an effective Langevin equation that accounts for it and allows one to determine the different contributions at stake in self-propulsion and enhanced diffusion.
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Affiliation(s)
- Jeanne Decayeux
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux), 4 place Jussieu, 75005 Paris, France
| | - Vincent Dahirel
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux), 4 place Jussieu, 75005 Paris, France
| | - Marie Jardat
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux), 4 place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Laboratoire PHENIX (Physicochimie des Electrolytes et Nanosystèmes Interfaciaux), 4 place Jussieu, 75005 Paris, France
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12
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Sharma V, Azar E, Schroder AP, Marques CM, Stocco A. Active colloids orbiting giant vesicles. SOFT MATTER 2021; 17:4275-4281. [PMID: 33687403 DOI: 10.1039/d0sm02183k] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Living or artificial self-propelled colloidal particles show original dynamics when they interact with other objects like passive particles, interfaces or membranes. These active colloids can transport small cargos or can be guided by passive objects, performing simple tasks that could be implemented in more complex systems. Here, we present an experimental investigation at the single particle level of the interaction between isolated active colloids and giant unilamellar lipid vesicles. We observed a persistent orbital motion of the active particle around the vesicle, which is independent of both the particle and the vesicle sizes. Force and torque transfers between the active particle and the vesicle is also described. These results differ in many aspects from recent theoretical and experimental reports on active particles interacting with solid spheres or liquid drops, and may be relevant for the study of swimming particles interacting with cells in biology or with microplastics in environmental science.
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Affiliation(s)
- Vaibhav Sharma
- Institut Charles Sadron, CNRS UPR22-University of Strasbourg, 23 rue du Loess, Strasbourg, 67034, France.
| | - Elise Azar
- Institut Charles Sadron, CNRS UPR22-University of Strasbourg, 23 rue du Loess, Strasbourg, 67034, France.
| | - Andre P Schroder
- Institut Charles Sadron, CNRS UPR22-University of Strasbourg, 23 rue du Loess, Strasbourg, 67034, France.
| | - Carlos M Marques
- Institut Charles Sadron, CNRS UPR22-University of Strasbourg, 23 rue du Loess, Strasbourg, 67034, France.
| | - Antonio Stocco
- Institut Charles Sadron, CNRS UPR22-University of Strasbourg, 23 rue du Loess, Strasbourg, 67034, France.
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