Classical MD and MTD simulations on NaZr2(PO4)3 and Na4Zr2(SiO4)3are carried out in the canonical ensemble (NVT) at 500 K, employing the Vashishta-Rahman form of the interaction potential [38],
U(ri j)= 1 4π²0
qiqj
ri j +Ai j(σi+σj)ni j rni ji j −Ci j
r6
i j
, (4.1)
where qi and qj are the partial charges, σi and σj are the ionic radius of ith and jthspecies respectively. Ai j andCi j are respectively the overlap repulsive energy and dispersion energy between ithand jthions. Listed in Table 4.1, the potential parameters employed are taken from a previous study by Kumar and Yashonath [39, 40].
The initial structure of the NASICONs are taken from the X-ray study of Boilotet al. [41]. The simulation box is made from 3×3×1 unit cells having rhomohedral (R3C)
CHAPTER 4. ACCESSING SLOW DIFFUSION IN SOLIDS EMPLOYING METADYNAMICS SIMULATION symmetry. The simulation super-cell thus includes a total of 972 ions (54-Na, 108-Zr, 162-P and 648-O) for NaZr2(PO4)3 and 1134 ions (216-Na, 108-Zr, 162-P and 648-O) for Na4Zr2(SiO4)3. The initial X-ray structures [41] of both systems are relaxed under isothermal-isobaric (NPT) ensemble at 500 K. The dimension of the relaxed simulation box area=b=26.989 Å,c=22.672 Å for NaZr2(PO4)3anda=b=27.680 Å, c=22.379 Å for Na4Zr2(SiO4)3. At an integration time-step of 2 fs, the production run for NaZr2(PO4)3 was 100 ns long, while that for Na4Zr2(SiO4)3was 200 ns long for better convergence of properties.
Molecular dynamics that powers the MTD calculation, is carried out employing the software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [42].
MTD simulation of the system is invoked by interfacing PLUMED [43] with LAMMPS.
In chapter 2, a brief description on MTD is given. Now the specific details about the implementation of it in the current study shall be discussed. Our implementation of the techniques for the expedition of the ion transport process in the systems employs coordinates of a single, arbitrarily chosen, N a+ ion, ¯rs=(xs,ys,zs), as the ‘collective variables’ (CV). The net external potential imposed on this tagged ion or ‘walker’ at any instant, t, is the sum of the Gaussian ‘hills’ deposited at regular intervals (100 MD steps in the present work), during the period oft=0 to t=t0, and is given by,
V( ¯rs,t)=
t0≤t
X
t0
H(t0) exp Ã
−
¡r¯s(t)−r¯s(t0)¢2
2w2
!
, (4.2)
wherew is the width of the Gaussian, kept constant at a value of 0.35 Å.H(t0) is the height of the Gaussian imposed at an MTD step, t=t0. In the ‘well tempered’ MTD formalism [25] the Gaussian heights are tuned as,
H(t0)=H0exp µ
−V( ¯rs,t0) kB∆T
¶
, (4.3)
whereT is the instantaneous temperature of the system, and∆T=(γ−1)T. The initial height H0 is chosen as 0.006 eV, andγ=6 following Barducci et al. [25] However, to validate the choice ofγwe have carried out additional runs – withγ=3 and withγ=9.
These additional results are presented in theAppendix A.
Free Energy Calculation: One of the most useful information from MTD simulation is the mechanism of the process and its underlying free energy landscape. It shall be noted that the free energy of a ‘process’ (except for aninsignificant additive termdue to the reference state) as a function of CV (in the present case, the position of the tagged
particle, ¯rs), is given by [25],
F( ¯rs)= −lim
t0→∞
T+∆T
∆T V( ¯rs,t0). (4.4) In the present work the free energy profile for the N a+along the migration channel is calculated by mapping theF( ¯rs), as a function of the distance of the location, ¯rs, with respect toany twoof its nearest Na1-sites,λ, under periodic boundary conditions. Note that the mapping of the F( ¯rs), from space coordinates, ¯rs, to F(λ), is in the spirit of introducing a ‘reaction coordinate’, which reduces the dimensionality of the information, and helps to visualize the free energy profile along the migration channel.
Below we shall note certain technicalities of the mapping procedure:
The separation between two neighboring N a1 sites (∼6.4 Å), is divided into fine bins (of size 0.1 Å). The distance ¯rs1(i)=r¯s−r¯1(i), where ¯r1(i) is the location of ith N a1 site, is searched over all Na1 sites in the simulation cell. Whenever this distance is less than 6.4 Å under PBC, theF( ¯rs) is cumulated in an appropriate bin, say indexed j, which spans the reaction coordinate,λ. The quantity is then averaged over the counts registered in each bin. That is,
F(λj)= <F(PBC( ¯rs1(i)))>j, i f r¯s1(i)< 6.4 ˚A (4.5) where < · · · >j represent the average over the trajectory in a given bin j. It shall be noted that, the migration channel being the one connecting two neighboring N a1 (which shall be demonstrated in the later sections), for every hill-drop locations, ¯rs, there would be two i values (that is, N a1 sites) which would meet the above criterion. It shall be remarked further that the above mapping procedure is not unique and any other method ofminingtheF( ¯rs) data onto repeating segments of trajectories may also be employed.
The free-energy profile from MTD simulation is compared with other compatible quantities, such as the potential energy profile of individual N a+ ions, and potential energy barrier employing nudged-elastic-band (NEB) calculations, detailed as bellow:
1. The potential energy profile: the average potential energy of individualN a+ions at a given location, ¯ri, due toallthe rest of the ions in the system,
U( ¯ri)= XN
j=1,j6=i
U( ¯ri j) (4.6)
where N is the total number of ions (including mobile as well as those of the framework) in the system andU( ¯ri j) is the pair-wise interaction potential as in
CHAPTER 4. ACCESSING SLOW DIFFUSION IN SOLIDS EMPLOYING METADYNAMICS SIMULATION eq. (4.1). The quantity,U( ¯ri), computed from the MTD trajectory, is mapped on to reaction coordinate,λi (under periodic boundary conditions), as discussed earlier.
TheU(λi), thus calculated is averaged over all the N a+ions and over the entire MD trajectory.
2. Fromclimbing-image nudged elastic band(CI-NEB) [44, 45] for a single N a+ion crossing over from one N a1 site to its neighbouring site.
Figure 4.1: Dynamic RDFs (g(r)) of select ion pairs are shown along with that of the corre- sponding X-ray structure (bars) from both MTD and MD simulations at 500 K. Top panel: for NaZr2(PO4)3. Bottom panel: for Na4Zr2(SiO4)3. The nearest-neighbour coordination numbers, C(n), of Zr-O, P-O, and Si-O are also marked in the respective, sub-panels. The plots are system- atically shifted-up, along the y- axis, for clarity.