Study of the Potential of McMillan and Mayer at long distance

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From interactions of matter to numerical simulations

Motivation of this thesis

This thesis nds its motivation in the recycling and the nuclear fuel cycle as well as in the extraction of rare earths. For radioactive elements, article L. 542-1-1 [10] of the environmental code speci es that « A radioactive substance is a substance that contains radionuclides, natural or arti cial, whose activity or concentration justi es a radiation protection control » A distinction is made between radioactive materials « for which a subsequent use is planned or envisaged », and on the contrary, radioactive waste « requali ed as such by the administrative authority in appli-cation of article L. 542-13-2. ». It is also explained that the management of the latter « includes all activities related to the handling, pre-treatment, treatment, disposal, storage and disposal of radioactive waste ». conditioning, storage and disposal of waste radioactive materials, excluding o -site transport. »
The national inventory [2] lists the origin of waste essentially « according to ve economic sec-tors : the nuclear power sector (which includes in particular nuclear power plants for electricity production and plants dedicated to the manufacture and reprocessing of nuclear fuel and to the recycling of part of the materials extracted from it), the research sector (particularly in the eld of civil nuclear energy and nuclear and particle physics), the defense sector (deterrence force and activities related to the army), the non-nuclear industry sector (including the extraction of rare earths), and the medical sector (diagnostic and therapeutic activities) ». Figure 1 shows the distribution of radioactive waste according to these ve sectors. Among these radioactive wastes are the radioactive isotopes of long-lived actinides, which are known (provided they have a high atomic number and a main oxidation state of +3) to have a chemical behaviour similar to that of the lanthanides, for which speci c potentials have been developed in modelling [55, 60].
The development of computer calculations tends to allow the prediction of the behavior of chemi-cal systems. These calculations are of interest because of their low cost and because experiments on radioelements are particularly cumbersome. The use of mathematical models and numerical simulation is thus a way for the National Agency for Radioactive Waste Management to study the phenomena that exist or will exist in the storage facilities and their natural environments, especially since it must ensure that the solutions it adopts will be safe in the very long term [3].
The study by simulation of these materials is carried out by means of statistical thermodyna-mics. This uses statistical mathematical laws, such as the law of large numbers, to calculate the macroscopic thermodynamic quantities of interest from a representation of the system on a microscopic scale. In chemistry, the most accurate way to describe a system is to use quantum mechanics [41] (which takes into account the electronic cloud of particles). But if we want to describe larger systems, we rather use the so-called classical (or molecular) level, for two main reasons. The rst is the cumbersome nature of quantum simulations, which cannot describe large systems. The second comes from the fact that recycling and extraction are mainly driven by weak, non-covalent interactions, for which a simple classical model is justi ed. In order to take advantage of these two levels of description, the QM/MM method [158,181] has been developed ; it is based on the description of part of the system at the quantum level and the rest at the classical level. In order to study in a simple way our systems of interest in the framework of this thesis, we have described them at the classical molecular scale.
Based on Newton’s second law, molecular dynamics makes it possible to explore the movements of particles, described according to the previously mentioned scales, in order to deduce average properties of the system.
In numerical simulations, in order to reduce surface e ects, it is often necessary to impose perio-dical conditions 1 on the edges of the simulation boxes [29]. However, these can lead to problems of anisotropy, including in the case of short-range interactions [92, 113, 143, 144]. Long-range interactions may be particularly a ected depending on the size of the periodic system conside-red [6, 89]. Hence the importance of trying to correct, according to the period considered, the calculation of macroscopic properties obtained by this bias. In [185], Yeh and Hummer (as we will see speci cally in chapter 6) have for example attempted to make a correction to the calculation of the di usion coe cient for a particle in a free uid, a correction which is a function of the spatial period considered. Through this thesis, we wish to focus on the theoretical research of the in uence of periodic conditions on the calculation of quantities of interest : the Mean Force Po-tential (which leads notably to equilibrium constants and activity coe cients) on the one hand, and the di usion coe cient in con ned media on the other hand. Our studies thus lead us to try to make a correction, according to the given period, for these quantities of interest.

Context of radioactive waste storage

In order to manage radioactive waste, the National Agency for Radioactive Waste Management (ANDRA), a French public institution under the supervision of the Ministries of Research, In-dustry and the Environment, independent of the producers of radioactive waste, classi es it according to its radioactivity and its lifespan. A distinction is made between high level waste, long-lived medium-level waste, low-level waste, short-lived low- and medium-level waste and, nally, very low-level waste.
Each category of waste has its own speci c management method. Figure 3 shows these.
On December 30, 1991, the law no 91-1381 [108] on research on radioactive waste management states that « The management of high level and long life radioactive waste must be ensured in respect of the protection of nature, environment and health, taking into consideration the rights of future generations ». (Article 1)
Then, on June 28, 2006, the law no 2006-739 [109] on the program for the sustainable manage-ment of radioactive materials and waste speci es that concerning « the management of long-lived radioactive waste of high or medium activity, research (…) shall be pursued along the following three complementary lines : separation and transmutation of long-lived radioactive elements, re-versible storage in deep geological strata, and, nally, disposal ». (Article 3)
In order to implement the second axis of the law of June 28, 2006, the project for an Industrial Center for Deep Geological Disposal (Cigeo) [3], in Bure, is planned to store the highly radioac-tive and long-lived waste produced by all current nuclear facilities, until their dismantling, and by the treatment of spent fuel used in nuclear power plants.
It was necessary to nd a stable geological layer, little exposed to earthquakes and erosion, and impermeable. Researchers looked at Callovo-Oxfordian (COx) clay sites. The site is expected to be used for more than a century (the duration of reversibility), for storage over 10,000 years.
The project provides for the digging of 15 km2 of galleries 490 m underground, in order to store 85000 m3 of radioactive waste by 2100, for a bill estimated in 2016 at 25 billion euros by the State (34.5 billion by ANDRA) and nanced according to law by the producers of radioactive waste (EDF, CEA, Areva).
While in June 2018 ANDRA begins work to clear certain parts of the Lejuc woods to allow for the installation of Cigeo’s aeration chimneys, in September 2018 the State announces the launch of a new public debate on the management of all of the country’s radioactive waste, between December 2018 and March 2019, which will include the Bure site. The submission of the project, its application for authorization for its creation, is therefore postponed to 2019, which should push back the start of construction to 2022. Thereafter, the launch of the pilot phase is planned for 2025 : an industrial phase that should be implemented for 10 years. It should allow the storage of dummy then real packages to con rm the choices made in terms of storage and security, the type of site ventilation, storage, etc. The rst packages should be transported between 2030 and 2033. In 2035, routine operations are scheduled to start with the lling of the site, which will last a century (at a rate of 5 to 10 packages per day). Finally, in 2150, the storage facility is scheduled to close and the start of monitoring the site after closure.
Numerous projects have been initiated to study the behavior and properties of the [150] argil-lites present on the site that the project plans to use for the storage of radioactive waste. These come from the deposition of sedimentary particles resulting from the destructuring of rocks of the continental crust. The spatial organization of the minerals in a rock controls the geometry of the pores and thus the geometry of uid circulation within a [50] rock. The argillites of COx are characterized by a clayey matrix consisting mainly of a mixture of illite and illite/smectite interlayers  modeling makes it possible to study the physico-chemical properties of these structures. At the microscopic scale, the pores are thus described at the interpolar, interparticle and intergranular scales [44, 49, 154] (cf gure5).
Porosity is a factor in uencing the macroscopic transfer properties of particles within the clay medium. It is necessary to understand the transport phenomena and the containment capacity of radionuclides in order to prevent their return to the biosphere for as long as possible 3. The pores of argillites are very small (of the order of nanometer), which gives this medium a low permeability. Therefore, the transport of ionic solutes is mainly done by di usion 4.
At the pore scale, the sorption phenomenon implies a (non-uniform) compensation of the negative charges of the sheets by cations. A di use layer is then formed from the surface to the liquid. This leads to the phenomenon of anionic exclusion : the porosity is accessible to cations and neutral species, while the di use layer prevents the passage of anions 5.
At this level of modeling, ab initio calculations [28] or by means of classical molecular dyna-mics [91, 112] allow to account for the di usion, within this geometry, of ions and solvent, the latter having to be considered as con ned between the layers. Other studies are also carried out by Brownian dynamics, a description where the solvent is considered as a continuous medium, which allows a gain in the degrees of freedom [12, 116].
On the macroscopic scale 6, porosity is considered to be uniformly distributed and averaged. For a con ned medium, interactions with the solid surface modify the mechanics and dynamics, including di usion-related properties [77, 157]. This is described at this scale by an e ective di usion coe cient. The latter can be obtained, for example, using the « though-di usion » tech-nique [20,51,124,137], where the di usion of an element (the « tracer ») is established by measuring the concentration gradient between two reservoirs (one upstream which contains the tracer, and the second downstream which does not ; and the sample being placed between these two reser-voirs). Other methods of measuring the liquid phase di usion coe cient in the laboratory consist for example in measuring a ow at the terminals of the sample under consideration or a concen-tration pro le [159].
In an uncon ned (free) uid, the di usion coe cient D is de ned by the Fick law : j = −DrC (3.1)
where j is the molar ux of the di using species and rC is the local gradient of molar concen-tration.
In a saturated porous medium, such as compacted clay, this law is written for the e ective di usion coe cient De 7 : j = −DerC (3.2)
where C concentrations here are macroscopic quantities at the ends of the sample.
In addition, the hydrodynamic models [27] established with the Stokes equations (equations on which we based our studies in chapter 6) can be used to model the transport of a particle in the solvent : ηΔu(x, y, z) = 5p(x, y, z) (3.3) r.u(x, y, z) = 0
The resolution of these equations relates the speed u of the particle with the force it undergoes.
This results in the relation between the mobility µ of the particle and u by the formula : u = µF (3.4)
where F is the external force applied to the particle.
The di usion coe cient can then be calculated from Einstein’s relation [62] : D = µkBT (3.5)
where kB is the Boltzmann constant and T is the temperature.
Multi-scale modeling allows the link between the microscopic and macroscopic scales, in particu-lar by using relevant information at various levels. For example, we can cite the coarse-grained models [53] as well as other techniques based on homogenization methods [131] 8.
As we have seen above, a common computational trick is to impose periodic conditions on the edges of simulation boxes, conditions that can lead to biases in the computation of macroscopic quantities. The phenomenon of di usion being particularly important to quantify, the study of the di usion coe cient, in the simpli ed case of a particle di using in a con ned solvent under non-periodic and then periodic conditions, was the subject of chapter 6 of this thesis. It was here calculated analytically from hydrodynamic models, taking into account the periodic boundary conditions.

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Context of extraction and recycling of rare earths (lanthanides)

Rare earths include 17 metals, including 15 elements in the lanthanides family (see gure 6). They have very interesting properties mainly due to f electrons [84,133], especially optical and magne-tic, particularly useful in the manufacture of permanent magnets, phosphors for the manufacture of light-emitting diodes, or in medicine for Magnetic Resonance Imaging, for example. The family of actinides, which is composed of 15 elements (see gure 6), have physico-chemical properties similar to lanthanides, particularly the oxidation degree +3 in aqueous phase [94,138,183], which is why studies conducted on these two families can be compared.
Because of their growing need for the development of new technologies today, and because they are inhomogeneously distributed around the globe, component separation techniques present a signi cant challenge in order to exploit these resources contained in existing technologies.
Among the methods used in industry, we can mention hydrometallurgy, which consists in dis-solving the material to be recycled in an acidic aqueous medium, but also pyrometallurgy, where the material is notably melted before proceeding to the extraction of the elements, for example by a liquid-liquid extraction process (cf gure 7). The latter, also called solvent extrac-tion, is speci cally exploited in the PUREX process (Plutonium-Uranium Re ning by Extrac-tion) [102,118,133,177] for the recycling of used fuel (which contains uranium (U) and plutonium (Pu) in particular). This process consists beforehand in a « shearing » that allows contact between the fuel and a nitric solution. Next, a key stage aims at a dissolution of the fuel. Uranium dioxide dissolves rapidly in hot nitric acid, unlike plutonium dioxide, which is why it is generally mixed solid solutions of uranium and plutonium (with a lower content than uranium) that are dissolved. Consequently, another important step is the separation and puri cation of uranium and pluto-nium by extraction, with Tributyl Phosphate (TBP) as the extractant 9 (which is e ective only in very acidic solution), diluted in an organic solvent the TetraPropylene-Hydrogen (TPH) 10.

Table of contents :

Remerciements
1 Introduction générale 
2 General introduction 
3 From interactions of matter to numerical simulations 
3.1 Motivation of this thesis
3.1.1 Context of radioactive waste storage
3.1.2 Context of extraction and recycling of rare earths (lanthanides)
3.2 Description of matter and equilibrium models
3.2.1 Description of the interactions of a system
3.2.2 Statistical thermodynamics
3.2.3 Distribution functions
3.3 Simulation methods
3.3.1 Molecular Dynamics
3.3.2 Periodic boundary conditions
3.3.3 Ewald
4 Calculation of free energy diérences 
4.1 Reaction coordinate
4.2 Free energy
4.3 Potential of Mean Force
4.3.1 The Potential of Mean Force
4.3.2 The Potential of McMillan and Mayer
4.3.3 Practical calculation of McMillan and Mayer’s potential
4.4 The Problem of Barriers for the Calculation of the Potential of Mean Force
4.5 Umbrella Sampling Method
4.6 WHAM Method
4.7 Analysis of a spring problem, period and stiness constant
4.7.1 Analysis with Newton’s principles
4.7.2 Analysis with Hamiltonian formulation
4.7.3 Stiness constant, umbrella amplitude and period for our simulations
4.8 Calculation of coecients of interest
4.8.1 The association constant
4.8.2 The osmotic coecient of activity of the solvent
5 Study of the Potential of McMillan and Mayer at long distance
5.1 Mathematical problem
5.2 Numerical results
5.2.1 Size of a simulation box
5.2.2 Description of the AMOEBA model
5.2.3 Study of the potential of McMillan and Mayer for Na-Cl in aqueous phase
5.2.4 Macroscopic study for Na-Cl in aqueous phase
5.2.5 Comparison of our expansion with lanthanide salt potentials
5.3 Conclusion
6 Study of the difusion constant in confned conditions
6.1 Introduction
6.2 Method of calculating the difusion constant
6.3 Study in the non-periodic case
6.4 Study in the case with periodic boundary conditions
6.5 Conclusion
7 Conclusion and perspectives 
Bibliographie 

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