CE46 - Modèles numériques, simulation, applications

Cross-diffusion systems in moving domains – COMODO

Mathématiques pour le Photovoltaïque

This proposal aims at developping appropriate models and efficient accurate numerical methods for the high-performance simulation and the optimization of the fabrication process of thin film solar cells.

Objectives

The production of the thin film inside of which occur the photovoltaic phenomena accounting for the efficiency of the whole solar cell is done via a Physical Vapor Deposition (PVD) process. More precisely, a substrate wafer is introduced in a hot chamber where the different chemical species composing the film are injected under a gaseous form. Molecules deposit on the substrate surface, so that a thin film layer grows. In addition, the different components diffuse inside the bulk of the film, so that the local volumic fractions of each chemical species evolve through time. The efficiency of the final solar cell crucially depends on the final chemical composition of the film, which is freezed once the wafer is taken out of the chamber. A major challenge consists in optimizing the fluxes of the different atoms injected inside the chamber during the process in order for the final local volumic fractions in the layer to be as close as possible to target profiles. Two different phenomena have to be taken into account in order to correctly model the evolution of these local volumic fractions: 1) the cross-diffusion phenomena between the various components occuring inside the bulk of the film; 2) the evolution of the surface of the thin film layer. In the context of a collaboration with IPVF (French Photovoltaic Institute), Virginie Ehrlacher [PI] and a PhD student, proposed a one-dimensional cross-diffusion system defined on a moving domain in order to model the evolution of the local concentrations of the different components inside the film during the PVD process. Comparisons between numerical simulations and experimental measurements yielded encouraging results on the relevance of this approach. However, the model studied by the PI and her student suffers from several limitations. Because of its one-dimensional nature, it is not currently possible to study geometrical effects due to surface tension or surfacic cross-diffusion phenomena which occur at the surface of the film. These phenomena are nevertheless extremely important to take into account, in particular for the production of curved solar cells for building-integrated photovoltaics. There is a crucial need for overcoming these limitations and proposing a multi-dimensional model for the PVD process along with an accurate numerical scheme for the approximation of its solutions, which can be used in order to optimize the production process of such thin film solar cells. This represents a significant scientific advance with respect to the existing models and numerical methods which we wish to address in this proposal.

Four main tasks are identified to tackle this challenging problem: 1) identifying appropriate models for the evolution of the local volumic fractions of the various chemical species inside the film and of its surface. Such models read as cross-diffusion systems defined on a domain with moving boundary, taking into account surface cross-diffusion phenomena; 2) developing numerical schemes for the simulation of such models, which should respect the mathematical properties of the considered systems; 3) parallelizing the obtained numerical schemes, using time-space domain decomposition methods and parareal algorithms; 4) designing accurate and efficient reduced-order models, which will be used for the calibration of the parameters entering the model with experimental data and for the optimization of the PVD process.

WP1 : With Jan-Frederik Pietschmann and Greta Marino (TU Chemnitz), we analyzed a cross-diffusion Cahn-Hilliard system which may be seen as a diffuse-interface model for moving boundary cross-diffusion systems. This lead to the publication of an article in Journal of Differential Equations.
WP2 : Clément Cancès and Benoît Gaudeul (Université de Lille) developped a structure-preserving convergent finite volume scheme for the cross-diffusion system studied by Athmane Bakhta and myself on fixed boundary domains. A related article was published in SIAM SINUM. With the internship of Nicolas Podvin, we extended the latter numerical scheme in order to treat moving-boundary systems in one-dimensional domains in a structure-preserving way. We also extended the latter scheme with Clément Cancès and Laurent Monasse for the simulation of another family of cross-diffusion system (the Maxwell-Stefan system, used in biomedical applications) with structure-preserving finite volumes. The manuscript has been submitted to the IMA Journal of Numerical Analysis and has been accepted up to some minor modifications. The exetnsion of this type of structure-preserving finite volume schemes for general cross-diffusion systems (on fixed boundary domains) is the object of a recent preprint by Ansgar Jüngel and Antoine Zurek.

WP4 : A joint paper with Olga Mula (Université Paris-Dauphine), Damiano Lombardi (INRIA Paris) and François-Xavier Vialard (Univeristé Gustave Eiffel) about the use of Wasserstein barycenters for the reduction of conservative transport-dominated parametric problems. This is important for the current project since the rescaled moving boundary 1D model involves a transport term which has a strong influence on the evolution of the solutions.
Other : In parallel, I am investigating some numerical methods in order to couple microscopic atomistic models together with such macroscopic cross-diffusion systems. A joint paper with Pierre Monmarché (Sorbonne Université) and Tony Lelièvre, is currently under review about a method based on the use of tensors for the acceleration of molecular dynamics simulations.

WP1: We are currently working with Jan-Frederik Pietschmann, Greta Marino and Jean Cauvin-Vila on the study of the sharp-interface limit of this system and on the numerical discretization of the diffuse-interface model. With Jean Cauvin-Vila and Amaury Hayat (Ecole des Ponts), we are investigating the problem of controlling fluxes on the one-dimensional moving boundary cross-diffusion system. We obtained very interesting results using a method developped by Coron and Nguyen on a linearized version of the model, and a paper is in preparation. We intend to extend in a later work these results to the fully nonlinear model.

WP2: With Jad Dabaghi and Christoph Stroessner (EPFL), we initiated a collaboration during a CEMRACS 2021 project on the design of a numerical scheme for the resolution of cross-diffusion systems which are actual hydrodynamic limits of microscopic multi-species exclusion stochastic processes, and imply the computation of the so-called auto-diffusion matrix of the system. A paper is currently in preparation.

WP3 : With Jad Dabaghi, we intend to investigate the potential numerical gain which could be brought by a parareal method on the cross-diffusion systems we are investigating.

WP4: An extension of the model-order reduction method based on Wasserstein barycenters to non-conservative transport-dominated problem was initiated during a CEMRACS 2021 project and is the object of an article in preparation together with Tobias Blickhan (Max-Planck Institut Garching), Olga Mula, Guillaume Enchéry (IFPEN), Damiano Lombardi and Beatrice Battisti (INRIA Bordeaux).
A paper is currently in preparation with Jad Dabaghi on the design of a structure-preserving reduced-basis method for the reduction of a parametrized cross-diffusion system. The mathematical theory, which was initiated with the M2 internship of Tinh Van Gia Nguyen, is now complete, numerical tests are still under investigation.

3 articles have been accepted in international reviews :
-Virginie Ehrlacher, Damiano Lombardi, Olga Mula, François-Xavier Vialard, Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, accepted for publication in ESAIM: M2AN, 2020
-Virginie Ehrlacher, Greta Marino, Jan-Frederik Pietschmann, Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system, Journal of Differential Equations, 286, 2021, p. 578-623
-Clément Cancès, Benoît Gaudeul, A convergent entropy diminishing finite volume scheme for a cross-diffusion system, SIAM Journal on Numerical Analysis 58 (5), 2020, p. 2684-2710

2 have been submitted

4 are in preparation

This proposal aims at developping appropriate models and efficient accurate numerical methods for the high-performance simulation and the optimization of the fabrication process of thin film solar cells.

The production of the thin film inside of which occur the photovoltaic phenomena accounting for the efficiency of the whole solar cell is done via a Physical Vapor Deposition (PVD) process. More precisely, a substrate wafer is introduced in a hot chamber where the different chemical species composing the film are injected under a gaseous form. Molecules deposit on the substrate surface, so that a thin film layer grows. In addition, the different components diffuse inside the bulk of the film, so that the local volumic fractions of each chemical species evolve through time. The efficiency of the final solar cell crucially depends on the final chemical composition of the film, which is freezed once the wafer is taken out of the chamber. A major challenge consists in optimizing the fluxes of the different atoms injected inside the chamber during the process in order for the final local volumic fractions in the layer to be as close as possible to target profiles.

Two different phenomena have to be taken into account in order to correctly model the evolution of these local volumic fractions: 1) the cross-diffusion phenomena between the various components occuring inside the bulk of the film; 2) the evolution of the surface of the thin film layer.
In the context of a collaboration with IPVF (French Photovoltaic Institute), Virginie Ehrlacher [PI] and a PhD student, proposed a one-dimensional cross-diffusion system defined on a moving domain in order to model the evolution of the local concentrations of the different components inside the film during the PVD process. Comparisons between numerical simulations and experimental measurements yielded encouraging results on the relevance of this approach.

However, the model studied by the PI and her student suffers from several limitations. Because of its one-dimensional nature, it is not currently possible to study geometrical effects due to surface tension or surfacic cross-diffusion phenomena which occur at the surface of the film. These phenomena are nevertheless extremely important to take into account, in particular for the production of curved solar cells for building-integrated photovoltaics. There is a crucial need for overcoming these limitations and proposing a multi-dimensional model for the PVD process along with an accurate numerical scheme for the approximation of its solutions, which can be used in order to optimize the production process of such thin film solar cells. This represents a significant scientific advance with respect to the existing models and numerical methods which we wish to address in this proposal.

Four main tasks are identified to tackle this challenging problem:
1) identifying appropriate models for the evolution of the local volumic fractions of the various chemical species inside the film and of its surface. Such models read as cross-diffusion systems defined on a domain with moving boundary, taking into account surface cross-diffusion phenomena;
2) developing numerical schemes for the simulation of such models, which should respect the mathematical properties of the considered systems;
3) parallelizing the obtained numerical schemes, using time-space domain decomposition methods and parareal algorithms;
4) designing accurate and efficient reduced-order models, which will be used for the calibration of the parameters entering the model with experimental data and for the optimization of the PVD process.

Project coordinator

Madame Virginie Ehrlacher (Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

CERMICS Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Help of the ANR 213,810 euros
Beginning and duration of the scientific project: December 2019 - 48 Months

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