Etude de la fléxoélectricité de nanosystèmes par le développement d'algorithmes mêlant approche atomistique et mécanique des milieux continus

The flexoelectricity tensor of a material characterizes its ability to polarize under the action of a deformation gradient. The phenomenon is still rarely used though it exists in every material, because the effects are usually very weak. However, for nanoscale systems, flexoelectricity can be largely enhanced because of a possibly much greater gradient. Thus, the aim of this PHD thesis is to build a model that would allow us to compute the characteristic tensors of flexoelectricity in order to design a nanosytem in which huge flexoelectric effects could be used for energy conversion. For that purpose, we have studied the flexion of several semi-conducting Single-Wall Carbon NanoTubes (SWCNT), considered either as continuous cylinders or as a discrete network of carbon atoms. In the continuum point of view, we have applied the principle of virtual powers and classical thermodynamics to systematically obtain the constitutive equations of a semi-conducting, electro-magnetic deformable continuum, including the effects of the deformation, polarization and magnetization gradients. Meanwhile, we have improved an atomistic model with distributed permanent and induced dipoles to simulate the inverse flexoelectric effect on the SWCNTs. Using homogenization hypothesis, we have coupled these two approaches by obtaining the equations binding the atomistic quantities computed in the numerical simulations, with the corresponding macroscopic quantities used in the previously obtained constitutive equations. The first numerical results seem to show a notable variation of the elements of the flexoelectric tensors as a function of the radius and length of the SWCNT.La flexoélectricité d’un matériau est sa capacité à se polariser électriquement sous l’effet d’un gradient de déformation. Bien qu’il existe dans tous les matériaux, ce phénomène est encore rarement utilisé car il est en général de très faible amplitude. Cependant, à l’échelle du nanomètre, la flexoélectricité est fortement augmentée. Le défi de ce travail est donc de proposer une modélisation multi-échelle permettant, d’une part, de caractériser et de quantifier les propriétés flexoélectriques et, d’autre part, de dimensionner un nanosystème mettant en évidence des effets flexoélectriques importants. Pour cela, nous avons choisi de nous intéresser à un nanosystème constitué d’un nanotube de carbone mono-paroi semi-conducteur. Dans le cadre des milieux continus, nous avons tout d’abord appliqué le principe des puissances virtuelles et la thermodynamique des milieux continus pour obtenir de façon systématique les équations constitutives d’un matériau aux comportements couplées semi-conducteur élastique électro-magnétique, en prenant en compte les gradients de déformation, de polarisation électrique et d’aimantation. En parallèle, dans le cadre d’une approche atomistique, nous avons développé un modèle numérique afin de simuler l’effet flexoélectrique inverse de nano-objets tels que des nanotubes de carbone décrits atome par atome, avec des dipôles électriques permanents et induits sur chaque atome. Moyennant quelques hypothèses d’homogénéisation, nous avons couplé ces deux approches et obtenu les équations reliant les quantités atomistiques, calculées dans la simulation, aux quantités macroscopiques correspondantes, utilisées dans les équations constitutives des milieux continus préalablement déterminées. Les premiers résultats mettent en évidence une variation importante des éléments de l’un des tenseurs de flexoélectricité en fonction du rayon et de la longueur du nanotube

Study of flexoelectric nanosystems through the development of multi-scale algorithms mixing an atomistic approach and continuum mechanics

La flexoélectricité d’un matériau est sa capacité à se polariser électriquement sous l’effet d’un gradient de déformation. Bien qu’il existe dans tous les matériaux, ce phénomène est encore rarement utilisé car il est en général de très faible amplitude. Cependant, à l’échelle du nanomètre, la flexoélectricité est fortement augmentée. Le défi de ce travail est donc de proposer une modélisation multi-échelle permettant, d’une part, de caractériser et de quantifier les propriétés flexoélectriques et, d’autre part, de dimensionner un nanosystème mettant en évidence des effets flexoélectriques importants. Pour cela, nous avons choisi de nous intéresser à un nanosystème constitué d’un nanotube de carbone mono-paroi semi-conducteur. Dans le cadre des milieux continus, nous avons tout d’abord appliqué le principe des puissances virtuelles et la thermodynamique des milieux continus pour obtenir de façon systématique les équations constitutives d’un matériau aux comportements couplées semi-conducteur élastique électro-magnétique, en prenant en compte les gradients de déformation, de polarisation électrique et d’aimantation. En parallèle, dans le cadre d’une approche atomistique, nous avons développé un modèle numérique afin de simuler l’effet flexoélectrique inverse de nano-objets tels que des nanotubes de carbone décrits atome par atome, avec des dipôles électriques permanents et induits sur chaque atome. Moyennant quelques hypothèses d’homogénéisation, nous avons couplé ces deux approches et obtenu les équations reliant les quantités atomistiques, calculées dans la simulation, aux quantités macroscopiques correspondantes, utilisées dans les équations constitutives des milieux continus préalablement déterminées. Les premiers résultats mettent en évidence une variation importante des éléments de l’un des tenseurs de flexoélectricité en fonction du rayon et de la longueur du nanotube.The flexoelectricity tensor of a material characterizes its ability to polarize under the action of a deformation gradient. The phenomenon is still rarely used though it exists in every material, because the effects are usually very weak. However, for nanoscale systems, flexoelectricity can be largely enhanced because of a possibly much greater gradient. Thus, the aim of this PHD thesis is to build a model that would allow us to compute the characteristic tensors of flexoelectricity in order to design a nanosytem in which huge flexoelectric effects could be used for energy conversion. For that purpose, we have studied the flexion of several semi-conducting Single-Wall Carbon NanoTubes (SWCNT), considered either as continuous cylinders or as a discrete network of carbon atoms. In the continuum point of view, we have applied the principle of virtual powers and classical thermodynamics to systematically obtain the constitutive equations of a semi-conducting, electro-magnetic deformable continuum, including the effects of the deformation, polarization and magnetization gradients. Meanwhile, we have improved an atomistic model with distributed permanent and induced dipoles to simulate the inverse flexoelectric effect on the SWCNTs. Using homogenization hypothesis, we have coupled these two approaches by obtaining the equations binding the atomistic quantities computed in the numerical simulations, with the corresponding macroscopic quantities used in the previously obtained constitutive equations. The first numerical results seem to show a notable variation of the elements of the flexoelectric tensors as a function of the radius and length of the SWCNT

Preliminary results concerning the flexoelectricity of carbon nanotubes

International audienceUnlike piezoelectricity (proportionality between a uniform stress or strain and the electric polarization of a material), flexoelectricity (proportionality between stress or strain gradient and electric polarization of a material) is rarely considered for electromechanical transduction and energy harvesting at the sub-micron scale, though flexoelectricity does not require that the material unit cell be non-centrosymmetric (as volume piezoelectricity does). Indeed, the challenge is to find a material with flexoelectric effects strong enough to be interesting for applications. One strategy could be to use the increase of the deformation gradient for a reduction of scale and the corresponding transition from bulk effects to surface effects to allow for new ways to select materials at the nanoscale and get an interesting conversion ratio between electric and mechanical energies. In order to study these phenomena, we have improved on previous molecular dynamics simulations of the bending of various carbon single-wall nanotubes by non-longitudinal static electric fields [1,2] and compared the results with the previsions of a continuum mechanics model for the bending of a flexoelectric beam by an external electric field [3], in order to extract the relevant flexoelectric tensor components from a fit. Preliminary results will be given

Principle of virtual power applied to deformable semiconductors with strain, polarization, and magnetization gradients

International audienceThe aim of this article is to generalize previous works in order to provide a systematic method to derive the equilibrium equations and the constitutive ones for deformable semiconductors accounting for first-order strain, polarization, and magnetization gradients. This is done by use of the “principle of virtual power” subject to the objectivity requirement (i.e., translational and rotational invariances) to which we add the first and the second laws of thermodynamics associated with the conservation of energy and the entropy production. This leads to a generalized expression of the Clausius–Duhem inequality, from which constitutive equations are derived. The interactions of the electromagnetic fields with the deformable and the semiconducting continua appear naturally by generalized non-symmetric stress tensors and the body and surface forces of electromagnetic origin. A comparison with some previous works is made putting emphasis on flexoelectricity that will be dealt with in a future work. Finally, special attention is given to particular cases relative to dissipative phenomena associated with semiconducting properties. In order to be close to what is nowadays done by physicists, the SI units have replaced the Lorentz–Heaviside units often used in previous works

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