Methane hydrate: towards a quantum-induced phase transition 

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Proton di‚usion mechanism

In a recent study50 a proton di‚usion process scheme was proposed for the case of Brucite minerals. ‘is process requires two mechanisms to be present: a proton reorientation in the (a,b) plane, and a proton dissociation out of that plane. Indeed, if a di‚usion process, that is, non-spatially limited motion of the protons from site to site, is to occur, both mechanisms must atoms are colored in red, hydrogen atoms in grey. be present. Dissociation is necessary as it allows hopping from one oxygen atom to another, but if no reorientation happens, the proton will only be able to return to its original position: reorientation enables the proton to move on to another O-O segment and thus to hop to yet another oxygen atom (Figure 1.2). Proton di‚usion in Brucite minerals is, therefore, a two-step compound process, di‚erent from a standard Gro‹huss mechanism.51,39 In practice, the above mentioned P3 con€guration generates an e‚ective triple-well potential for reorientations within the (a,b) plane: this we refer to as “in-plane” motion. ‘e proton dynamics on these sites describes the reorientation motion. On the other hand, the “out of plane” dissociation mechanism involves an e‚ective double well potential along the O-O direction characterizing the covalent and hydrogen bonds similar to the ice case.1 It has been shown42 that only weak hydrogen bonds could be present in Brucite. However, as we will discuss later, taking into account nuclear quantum e‚ects, a double-well potential is found along the O-O direction at low pressure suggesting the H-bond interaction, which can be enhanced by the pressure-induced creation of a quasi-2D hydrogen layer in the structure.
Finally, the thermal activation of the reorientation motion was assumed50 to be a limiting factor for di‚usion, while nuclear quantum e‚ects are suggested as being a facilitating factor for the dissociation mechanism. However, proton dissociation has not been observed yet and nuclear quantum e‚ects for the la‹er mechanisms have been neglected up to now. ‘erefore, in the following, we undertake to unravel the complex and quantum driven proton di‚usion mechanisms in Brucites by including NQE: €rst, we will discuss both the reorientation and dissociation mechanism upon compression, for which we will compare thermal and quantum e‚ects. In the second part, we will address the evolution of the di‚usion process upon compression. Finally, a comparison with Portlandite will be provided.

In plane reorientation

First, we discuss in this part the reorientation mechanism. As already described, within the P3 structure, protons move in-plane between the 6i sites. ‘erefore, this motion can be e‚ectively described by the azimuthal angle ‘, as shown in the sketch of Figure 1.3. From the probability distribution of the la‹er Pr¹’º, we extracted the Gibbs free-energy pro€le G = 􀀀kBT log Pr¹’º, which includes both thermal and quantum e‚ects.
As shown in Figure 1.3, the proton e‚ective potential along this coordinate has a three-fold 2 3 symmetry with equivalent barrier heights between the three wells, as expected from the symmetry of the P3 con€guration.
‘e barrier width between the wells of this potential is essentially proportional to 2 3 dO􀀀H cos .
being the zenith polar angle, that is, how far the O-H bond slants away from the c axis; dO􀀀H being the covalent O-H bond distance.
Upon compression, we observe that the free energy barrier heights increase, from 20meV at 30GPa up to 100meV at 90GPa, revealing a pressure induced con€nement of the proton along this coordinate. ‘is stems essentially from the fact that the average polar angle increases with pressure so that the separation of the wells also increases. ‘us, as the overall atom-atom separations decrease, mainly through the compression of the layers along the ®c axis, the H-H repulsive interaction increases and the reorientational dynamical disorder, thermally activated at low pressure, tends to slow down, eventually to halt. It can be noted that the classical simulations, not including NQE, yield almost identical distributions meaning that the quantum behavior is in this case limited within the pressure range that was explored. ‘e e‚ect of pressure contrasts with that of temperature, which tends to allow the proton to explore equivalently all the wells50 by usual thermal activation.

Out of plane dissociation

In this section, we now focus on the out of plane dissociation by proton hopping between the two oxygen layers. Upon compression, the hydrogen planes get closer, due to the important compressibility of the system along the ®c axis. ‘is, in turn, can favor proton hopping from one oxygen atom to another, that is dissociation and thus out-of-plane delocalization.
• †antum quasi 2D hydrogen plane Figure 1.4 shows the probability distribution of the hydrogen nuclei along the c direction. Initially, each hydrogen atom belongs to either the upper or lower plane. During the simulation, the “lower layer” distribution refers to the hydrogen atom initially in the lower layer while the “upper layer” refers to those initially in the upper layer. We distinguish both the upper and lower hydrogen layers distributions from the overall one. As pressure is increased, the overall distribution width decreases as the two hydrogen layers distributions get closer, meaning that the pressure tends to merge the two hydrogen layers. At the two lower pressures, 30 and 50GPa, bo‹om and top layer protons can be distinguished, although hopping does occur, as the protons return to their original layer by a second hop in reverse: the protons initially situated on one layer will remain on that layer with, from time to time, a short exploration of the other.
At 70GPa, lower and upper layers are not distinguishable as reverse hopping does not always follow: the protons do not belong to one particular layer. ‘is speci€c case will be discussed later. At the highest pressure, 90GPa, the overall distribution becomes narrower, but the lower
and upper protons can again be distinguished as they return to their initial layer, despite a signi€cant distribution overlap.
‘e characterization of the two hydrogen layers is however highly dependent on the quantum spread of protons. Indeed, for a delocalization of the same order of magnitude as the separation between the two layers, one cannot distinguish upper from lower layer due to quantum indetermination. In this context, the RPMD simulations account for NQE by representing each particle by a set of replicas (or beads), thus, the spread of these replicas provides insight into the quantum delocalization of the particles. ‘erefore, we investigated the standard deviation, in each cartesian coordinate, of the proton replicas with respect to the instantaneous centroid position. As shown in Figure 1.5 the in-plane delocalization a and b of the particle along the ®a and ® b axes are similar and both decrease upon compression. In contrast, the out-of-plane delocalization along the ®c axis, described by c is less than in the two other directions but increases upon compression. Finally, at high pressure (90GPa) the delocalization z is of the order of the distance between the two hydrogen layers. ‘is indicates that the two hydrogen planes merge through quantum indetermination into a so-called “antum 2D proton layer”. Such con€guration allows the protons to form covalent bonds either with the upper or lower oxygen layer, thus easing the proton hopping.

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Proton di‚usion sweet spot

the dissociation one. However, as described earlier, the proton di‚usion in Brucite minerals requires both. ‘erefore, due to competition between pressure e‚ects, a pressure sweet spot exists, allowing both reorientation and dissociation mechanisms, enhancing the proton di‚usion in Brucite minerals. ‘us, we studied the maximum probability of di‚usion, considering both the reorientation and the dissociation mechanism to be independent. ‘is approximation was checked by calculating the correlation between the two probabilities. Under this assumption, the maximum di‚usion probability is given by the product of the reorientation and dissociation maximum probabilities Pr and Pd which reduces to the sum of the two barrier heights, Gr and Gd , in terms of free energy:
P = Pr Pd (1.3.2).
G = 􀀀kbT log¹Pr Pd º (1.3.3).
= Gr + Gd (1.3.4).
‘e evolution of the free energy barrier heights upon compression is given in Figure 1.7. For the case of Brucite, we observe that the barrier height for dissociation decreases from 0:11 eV at 30GPa to 0:01 eV at 90GPa. In contrast, the reorientation barrier increases from 0.03 eV to 0.1 eV within the same pressure range. ‘us, the two curves cross at 70GPa, giving rise to a minimum of di‚usion free energy barrier G. Hopping rates for the two processes have the same order of magnitude at this pressure, while reorientation dominates at lower pressures and dissociation does so at higher pressures. ‘erefore, 70GPa represents the sweet spot for a maximum di‚usion probability.
In order to give a rough estimate of the di‚usion reaction rate , one can use the di‚usion free energy barrier obtained above in the chemical kinetics Eyring-Polanyi equation: = kBT h e􀀀 G kBT (1.3.5).
As shown in Table 1, the estimation of this reaction rate in Brucite naturally follows the same trend as the free energy evolution upon compression. It decreases by a factor of four between 30 and 70GPa, for which the characteristic time evaluation is 􀀀1 = 8ps, and then increases by a factor of three at 90GPa.

Comparison with Portlandite

Finally, we close our discussion of proton di‚usion in Brucite minerals by a comparison with Portlandite (Ca(OH)2) which presents the same structure for pressures up to approximately 15GPa. ‘e same analysis as for the Brucite was done systematically for Portlandite. In Figure 1.7, we present the evolution of free energy barriers of the proton reorientation and dissociation mechanisms. We observe that in Portlandite the reorientational barrier at 10GPa is close to that of Brucite at 50GPa. However, the pressure e‚ect on the la‹er barrier is more important in Portlandite as shown by the larger increase rate of 2.8 meV/GPa while it is evaluated to be 1.6 meV/GPa in Brucite. ‘is rapid increase can be understood from the fact that the Van der Waals radius of calcium atoms is larger than for magnesium and thus tends to expand the la‹ice in the (a,b) plane. Indeed, our calculations at 10GPa give the in-plane €rst neighbor O-O distances to be 3.45°A in Portlandite while 3.06°A in Brucite. In addition, the out-of-plane €rst neighbor O-O distances are equivalent in both systems. ‘is implies larger polar angles in Portlandite, that eciently hinder the reorientation mechanism, including at relatively low pressures. On the other hand, the dissociation barriers in Portlandite are greater than in Brucite but decrease much faster with a decay of 14 meV/GPa as compared to 2meV/GPa in Brucite. ‘is derives from the larger compressibility of the Portlandite with respect to the Brucite structure, as demonstrated in recent work.40 It has to be noticed that the large value of the dissociation barrier in Portlandite implies a long simulation duration in order to be observed, well beyond the scope of path integral methods, to address properly the statistic of the la‹er, and thus barrier height evaluation. Nevertheless, some events are detected as some of the replicas of the RPMD simulations do occasionally reach the top giving rise to estimation provided here. Finally, the crossing point of the two barriers in Portlandite should occur beyond 20GPa with a di‚usion barrier comparable to that of Brucite at 70GPa. However, a transition towards an amorphous phase is reported between 10 and 15GPa52,53 and our own simulations reveal the instability of the system at 20GPa. ‘erefore, as shown in Figure 1.9, no di‚usion was observed for Portlandite within the time scale of our simulations. Indeed, the reaction rate estimates, given in Table 1, yield much longer times than in Brucite.
‘is comparison suggests that Brucite could be a particular case for proton di‚usion within its mineral family. Among the other systems sharing the same structure, our €rst analysis of ‘eophrastite (Ni(OH)2) indicates that this system should present the same mechanism at approximately the same pressure, due to comparable Van der Waals radii between magnesium and nickel atoms.

Table of contents :

Abbreviations
Nomenclature
Introduction
I State of the art 
1 ‡eoretical framework 
1.1 Introduction
1.2 Density matrix
1.2.1 Pure and mixed states
1.2.2 Time evolution
1.2.3 ‘ermal equilibrium
1.3 Linear Response theory
1.3.1 Response function
1.3.2 Generalized susceptibility
1.3.3 Fluctuations
1.3.4 Relaxation
1.3.5 Dissipation
1.3.6 Fluctuation-Dissipation theorem
1.4 Ab-Initio Molecular Dynamics
1.4.1 ‘e Born-Oppenheimer approximation
1.4.2 ‘e Density Functional ‘eory
2 Modelisation of Nuclear quantum e‚ects 
2.1 Introduction
2.2 Langevin methods
2.2.1 Langevin equation
2.2.2 antum ‘ermal Bath
2.3 Path integral formalism
2.3.1 Path Integral Molecular Dynamics
2.3.2 Ring Polymer Molecular Dynamics
2.4 Conclusion
3 Phase transitions description 
3.1 Introduction
3.2 Mininum energy path sampling with Nudged Elastic Band
3.3 Free energy sampling with Metadynamics
3.3.1 Path collective variables and the path invariant vector
3.4 Conclusion
II Investigation of Nuclear †antum E‚ects 
1 †antum driven proton di‚usion in brucite minerals
1.1 Introduction
1.2 Brucite mineral structure
1.3 Proton di‚usion mechanism
1.3.1 In plane reorientation
1.3.2 Out of plane dissociation
1.3.3 Proton di‚usion sweet spot
1.4 Comparison with Portlandite
1.5 Conclusion
2 Methane hydrate: towards a quantum-induced phase transition 
2.1 Introduction
2.1.1 MH-III structure
2.2 MH-III under pressure
2.2.1 Spectral analysis
2.2.2 Methane ordering and locking-in
2.2.3 Methane-Water interaction
2.3 ‘e methane hydrate IIIs
2.3.1 From MH-III to MH-IIIs
2.3.2 Nuclear quantum e‚ects and isotopic substitution
2.4 ‘e methane hydrate IV
2.4.1 Structural properties
2.4.2 Hydrogen bond symmetrization: MH-IVs
2.4.3 Vibrationnal properties
2.5 Transition description
2.5.1 Transition stages
2.5.2 Stability of the ice frame
2.5.3 Transition path
2.6 Conclusion
3 ‡e †antum equilibrium structure of sodium hydroxide 
3.1 Introduction
3.2 Sodium hydroxide structure
3.3 antum mechanical description of structural properties
3.4 Towards a dynamical paraelectric state
3.5 From temperature- to pressure-induced transition
3.6 Conclusion
Conclusion
Appendices 
A Brucite minerals 
A.1 Computational details
B Methane hydrate 
B.1 Computational details
B.2 Methane rotation characterization
C Sodium hydroxide 
C.1 Computational details
C.2 Volume optimization
Communications
Bibliography 

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