Molecular Dynamics is a formidable tool for investigating matter at the atomic scale. It involves numerically simulating the evolution of a system of particles as a function of time, with the aim of predicting and understanding experimental results. It makes it possible to highlight structural arrangements or dynamic phenomena that are still inaccessible to current experimental observation methods (EXAFS, NMR, Atom Probe Tomography—APT), especially for amorphous materials such as glasses. This move into the digital world requires time to be discretised in order to solve Newton’s equations governing the motion of each particle. The principle of Molecular Dynamics is then to integrate these discretised equations, under various physical constraints, using various algorithms which can be found in various reference books such as that by Griebel et al. 12, a clear work containing examples and practical applications, useful for anyone wanting to start writing their own Molecular Dynamics programme. The methods presented are those used in this thesis. After an explanation of the “Velocity-Störmer-Verlet” algorithm, which strikes a balance between robustness, practicality and performance, a presentation of the Nosé-Hoover thermostat and barostat shows how the temperature and pressure of a material can be controlled in a fully integrated way. A key point in the simulations is the appropriate parameterisation of interatomic interactions. The interaction potential used must above all take account of known properties in order to be able to predict those that are not, or to be able to explain a phenomenon that is still poorly understood. Calculating these interactions during simulation, particularly electrostatic interactions, is the step that requires the most numerical computing resources. The method developed by Wolf overcomes this limitation and opens up the field of simulations on much larger size and time scales. These methods and algorithms are finally being applied to the modelling of silica materials, in particular glasses, where certain properties and experimental behaviours are reproduced with acuity.

a) Algorithms for integrating Newton’s equations [8]

In Classical Molecular Dynamics, each particle is considered as a point mass interacting at a distance with others via an effective interaction potential. In an ensemble of N interacting particles, the motion of particle i of mass m_i (with i = 1 to N) is governed by the equation :

m i d 2 x i d t 2 = F i (1)

with the convention x i = x i → for the position vector, and F i is the vector resulting from all the forces applied to particle i.

b) Standard Störmer-Verlet method [9] [10]

The basic numerical method for solving the equations of motion is to perform a 3rd order Taylor series expansion of the position x around the date t, i.e. at t ± δ t (with δ t small):

{ x ( t + δ t ) = x ( t ) + δ t d x d t + 1 2 δ t 2 d 2 x d t 2 + 1 6 δ t 3 d 3 x d t 3 x ( t − δ t ) = x ( t ) − δ t d x d t + 1 2 δ t 2 d 2 x d t 2 − 1 6 δ t 3 d 3 x d t 3 (2)

By adding up and grouping the terms, we obtain:

d 2 x d t 2 = 1 δ t 2 [ x ( t + δ t ) − 2 x ( t ) + x ( t − δ t ) ] (3)

Time is discretised using a time step δ t (of the order of femto seconds in Molecular Dynamics). At iteration n, the date is expressed by t n = n δ t , the position vector by x i n , the velocity vector by v i n and the force vector by F i n . After this discretisation and digitisation, the previous equation becomes, for the date t n + 1 = t n + δ t :

m i 1 δ t 2 ( x i n + 1 − 2 x i n + x i n − 1 ) = F i n (4)

In the same way, but subtracting the terms this time, we obtain the expression for speed:

d 2 x d t 2 = x ( t + δ t ) − x ( t − δ t ) 2 δ t = v i n = x i n + 1 − x i n − 1 2 δ t (5)

This is the standard form of the Störmer-Verlet method for integrating Newton’s equations:

{ x i n + 1 = 2 x i n − x i n − 1 + δ t 2 m i F i n v i n = x i n + 1 − x i n − 1 2 δ t (6)

Interatomic potentials are of vital importance for simulations that model the properties of materials. The basis of these potentials is density function theory (DFT), which postulates that energy is a function of electron density. By knowing the electron density of an entire system, we can determine the potential energy of a system:

U = [ ρ ( r ) ] (7)

E [ ρ ( r ) ] = T s [ ρ ( r ) ] + J [ ρ ( r ) ] + E x c [ ρ ( r ) ] + E e x t [ ρ ( r ) ] + E i i [ ρ ( r ) ] (8)

where E is the total energy, Ts is the kinetic energy of the single particle, J is the Hartree electron-electron energy, Exc is the exchange correlation function, Eext is the electron-ion coulombic interaction, and E_ii is the ion-ion energy.

On this basis, the Embedded Atom Method (EAM) was created by assuming that an atom can be embedded in a homogeneous electron gas and that the change in potential energy is a function of the electron density of the embedded atom which can be approximated by an embedding function. In a crystal, however, the electron density is not homogeneous, so the EAM potential replaces the background electron density with the electron densities of each atom and supplements the embedding energy with a repulsive pair potential to represent the core-core interactions of the atoms.

With a simple linear superposition of the electron densities of the atoms as the background electron density, the EAM is governed by the following equations:

R i i = | r i − r j | (9)

ρ ¯ i = ∑ ​ ( R i j ) j (10)

U = ∑ ​ ( ρ ¯ i ) i + 1 2 ∑ ϕ ( R i j ) i , j (11)

where R i j is the distance between atoms i and j, r i , j is the position between atoms i and j, ρ ¯ i is the fundamental electron density, and ϕ is the pair interaction potential.

However, EAM does not do an excellent job of simulating materials with significant directional binding, which includes most metals. In order to correctly simulate metals, the modified embedded atom method was created, which allows the background electron density to depend on the local environment instead of assuming a linear superposition.

In the MEAM formalism, we consider a set of atoms forming a cluster. Each atom is immersed in the electron density created by the other atoms. The total energy depends on two factors: the immersion potential and the pair interaction potential:

E = ∑ i F i ( ρ ¯ i ) + 1 2 ∑ j ≠ 1 ϕ i j ( R i j ) (12)

For an atom i, F i is the immersion potential, ρ ¯ i is the fundamental electron density. ϕ i j ( R i j ) is the pair interaction potential between two atoms i and j, at distance R i j .

The immersion potential is calculated as follows:

F i ( ρ ¯ i ) = A E c ρ ¯ i ρ 0 L n ( ρ ¯ i ρ 0 ) (13)

where:

A is a parameter that can be adjusted according to the experimental data;

E c is the sublimation energy;

ρ 0 the electron density in the reference structure;

ρ ¯ i the electron density in the real structure.

In MEAM1, interactions in the reference structure are limited to the first neighbourhood. Under these conditions, the atomic positions and bond directions are fixed. The immersion potential depends only on the distance to the first neighbourhood and the number of first neighbours. Consequently, the energy of an atom can be written as a function of R as follows:

E u ( R ) = F [ ρ ¯ i 0 ( R ) ] + Z 1 2 ϕ ( R ) (14)

where Z 1 is the number of first neighbours of the atom.

By calculating E u ( R ) from Rose’s equation of state, we can derive the expression for the interaction potential of the pairs as follows:

ϕ ( R ) = 2 Z 1 [ E u ( R ) − F [ ρ ¯ i 0 ( R ) ] ] (15)

In MEAM2, we consider second-neighbour interactions in the reference structure and this can be done by adding a screen parameter S. From this, the energy of an atom in a reference structure can then be written:

E u ( R ) = F [ ρ ¯ i 0 ( R ) ] + Z 1 2 ϕ ( R ) + Z 2 S 2 ϕ ( a R ) (16)

where:

Z 1 is the number of first neighbours in the reference structure;

Z 2 is the number of second neighbours in the reference structure;

a is the ratio of the distances of the second and first neighbours a = R₂/R₁;

S is the screen function. For a given reference structure, the screen factor S is constant.

The above equation can be written as:

E u ( R ) = F [ ρ ¯ i 0 ( R ) ] + Z 1 2 ψ ( R ) (17)

With

ψ ( R ) = ϕ ( R ) + Z 2 S Z 1 ϕ ( a R ) (18)

From the value ψ ( R ) the pair interaction potential is calculated iteratively using the following formula:

ϕ ( R ) = ψ ( R ) + ∑ n = 1 ( − 1 ) n ( Z 2 S Z 1 ) n ψ ( a n R ) (19)

For a MEAM interatomic potential that describes the relationship for alloys with two or more components, each component needs 13 individual adjustable parameters. In addition, each binary interaction requires at least 14 adjustable parameters. These parameters are used in the calculation of the potential energy described in Equation (2-7) and govern the forces acting on the atoms. These parameters are listed below (see library: parameter).

The simulation of the cell begins by determining the nature of the elements to be used as electrodes (anode and cathode) and their melting points. For simplicity’s sake, we have chosen alkalis as the anode and their oxides as the cathode; their electronic properties are shown in Table 1.

This table shows lithium as the best basic element for electronics. Caesium, which has a similar melting point at room temperature, was quickly eliminated from our calculations.

After choosing the composite elements for the anode and cathode, we selected the structures to be used thanks to the “materials projects” website, which offers a wide range of structures for a single element, as well as a database of the physical and chemical properties of these elements, see Figure 1.

For the anode and cathode, we arbitrarily chose BCC structures in order to be able to compare them without worry. These basic structures were processed using OVITO software to generate “atomic position” data files that could be accessed by our LAMMPS calculation code.

These atomic position data files then underwent secondary processing to generate an order of multiplicity 4 × 4 × 4 and others in which we remove an atom to form the ion of the element treated in the vicinity of the melting point.

The choice of potentials defining the nature of the systems remains essential for any simulation and any physical phenomenon to be interpreted. The Table 2 below shows a set of potentials used in LAMMPS and the nature of the systems used.

Simple elements

The potential file, in Table 3, used in our LAMMPS calculation code is divided into two essential parts: a librairy file and a parameter file, each containing specific types of fixed and adjustable parameters.

File-library

# DATE: 2012-06-29 DATE: 2007-06-11 CONTRIBUTOR: Greg Wagner, gjwagne@sandia.gov CITATION: Baskes, Phys Rev B, 46, 2727-2742 (1992)

# meam data from vax files fcc, bcc, dia 11/4/92

# elt lat z ielement atwt alpha b0 b1 b2 b3 alat esub asub

# t0 t1 t2 t3 rozero ibar

Li’ ‘bcc’ 8 3 6.939 2.97244804 1.425 1.00 1.00169907 1.00 3.509 1.65 0.87

1.0 0.26395017 0.44431129 −0.2 1. 0

Na’ ‘bcc’ 8 11 22.9898 3.64280541 2.313 1.00 1.00173951 1.00 4.291 1.13 0.9

1.0 3.55398839 0.68807569 −0.2 1. 0

K’ ‘bcc’ 8 19 39.102 3.90128376 2.687 1.00 1.00186667 1.00 5.344 0.94 0.92

1.0 5.09756981 0.69413264 −0.2 1. 0

The diffusion coefficients were calculated (generated) by gnuplot. Between two consecutive states, the activation energy reflecting the behaviour of the atoms was calculated using the following expression:

{ ln D 1 = ln D 0 − E a k B T 1 ln D 2 = ln D 0 − E a k B T 2 (20)

E a = k B T 2 T 1 T 1 − T 2 ln D 1 D 2 (21)

With D_i diffusion coefficient at the time T_i;

E_a activation energy;

k_B boltzman constant.

The activation energy of a chemical reaction is closely linked to its speed. The higher the activation energy, the slower the chemical reaction. This is because the molecules can only complete the reaction once they have reached the top of the activation barrier.

a) Validation of potential parameters

Here we have calculated the cohesion energies for BCC structures of the elements to be used as alloys at the anode. Table 4 shows the cohesive energy of the structures selected in the BCC, FCC and HCP crystallographic structures. This was also done for 4*4*4 and 6*6*6 multiplicities for a triclinic BCC phase.

For each of these elements, the hcp form remains the most stable, although transitions between structures are possible with temperature variation. Lithium remains the most stable, followed by sodium and potassium.

The transition order is given as follows: ∆_{(bcc_fcc):Li ∆ (bcc_hcp):Li ; ∆ (fcc_hcp):Li lithium has the lowest energy for each of its transitions compared to sodium and potassium.}

The calculated transition energies show that lithium transits preferentially from the BCC structure to the hexagonal structure, increasing from 2 to 6 (gaining 4 atoms per transition) atoms per cell, while sodium transits favourably from the CFC structure to the hexagonal structure, increasing from 4 to 6 atoms. Finally, the transition of potassium from the BCC structure to the CFC structure is the most difficult to achieve.

b) Mesh parameters

The crystal parameters giving the size of each of these structures and their multiplicities were calculated. The results are shown in Table 5.

Potassium has a considerable volume compared to sodium and lithium, and this order of magnitude is maintained whatever the crystallographic structure.

These results were in line with the theory, so we proceeded to simulate the battery on the anode.

Choice of operating temperatures

Our study requires the tracking of particles around the melting temperature, on the understanding that at this temperature several phenomena become interesting, particularly the diffusion of particles in a system.

For this reason, we have arbitrarily chosen three (3) consecutive temperatures to monitor the behaviour of the physical quantities. These temperatures are shown in the Table 6 below:

Allures of physical quantities

All the physical quantities studied (of all the elements), see Figure 2, have the same appearance whatever the order of multiplicity 4*4*4 and 6*6*6, so we present here that of lithium to explain the diffusion phenomenon observed.

a) Lithium curves

This Figure 2 shows us the behaviour of the temperature drop between 450 K and about 200 K, at which temperature the kinetic energy becomes lower and

lower, reflecting the low activity of the particles between (200 − 100,000) × 0.001 s = 99.8 s. This immobility is reflected by the increase in the interaction potential between the particles, which is retained more and more around 200 K.

This suggests that the closer we are to the melting temperature, the greater the extent of diffusion.

These results, in Table 7, show us how many times lithium remains the order of multiplicity for diffusion, since the greatest kinetic energy is reached at the highest temperature close to fusion.

b) Potassium

For potassium, we studied the behaviour of physical quantities for multiplicities without and with a gap. The results are presented in Table 8.

We can say that the 4 × 4 × 4 structures with a gap and without a gap are almost similar, although some differences are observed in favour of one over the other:

➢ (Ec, Pre): reaches 5.417 with a pressure of 8014.3164 at a temperature of 330 K compared with (5.374, 7499.723) around 315 K for 4 × 4 × 4;

➢ (Ec, Pre): (18.384, 7844.761) around 315 as much as (18.342, 7849.907) for the 6x6x6 multiplicity.

c) Sodium

The table below shows the physical quantities for sodium for multiplicities with a gap.

(Ec, Pre) is reached (24.484, 14500.312) around 330 K for 4 × 4 × 4 and (87.858, 14873.777) around350 K for 6 × 6 × 6. This information, in Table 9, tells us that around the melting point the scattering behaviour becomes increasingly important as the multiplicity increases.

a) the case of potassium

The behaviour shown in this Figure 3, from Table 10, shows us how often structures with a gap follow an order of magnitude, whereas structures without a gap are as if random over a considerable temperature range (15 K for potassium). However, the value of the diffusion coefficient is considerable in structures with a gap.

As a result of these observations, we reduced our work to structures with gaps for the other study elements, sodium and lithium.

b) case of sodium

This Figure 4, from Table 11, clearly shows the dual behaviour of diffusion at low multiplicity, which tends to disappear as the multiplicity increases. However, for sodium, diffusion is greater for 4 × 4 × 4 than for 6 × 6 × 6.

c) the case of lithium

This Figure 5, from Table 12, for lithium suggests that around non-consecutive temperatures, the behaviour of diffusion remains disturbed.

On the whole, it is preferable to carry out this study for close temperatures.

Our overall analysis leads us to the 6*6*6 multiplicity structures, which show the most interesting results of all the structures.

Table 13 shows that it is preferable to use sodium as an anode because of its lower activation energy value, which results in greater chemical reduction than lithium and potassium.