Atomic diffusion in covalent liquids under pressure from ab initio molecular dynamics

The microscopic mechanisms of atomic diffusion in liquid GeO2 and SrGeO3 are investigated by ab initio molecular-dynamics simulations. We clarify the differences of diffusion mechanism between liquid GeO2 and SrGeO3. In both liquids, non-bridging oxygen double bonded to only one germanate plays a key role in the atomic diffusion mechanism. It is found that, in liquid SrGeO3 which has non-bridging oxygen in the equilibrium state at ambient pressure, atomic diffusion is possible without generating overcoordinated atoms at ambient pressure, while the over coordinated atoms are always needed for the formation of non-bridging oxygens in liquid GeO2. When the pressure increases, only liquid GeO2 has a diffusion maximum, which is given by, the atomic diffusion with concerted reaction gives the diffusion maximum.


Introduction
Transport properties of covalent liquids under pressure are very interesting in the sense that they show unexpected pressure dependence.For a number of covalent liquids, such as SiO 2 , GeO 2, silicates and germanates, abnormal behaviour of the viscosity has been observed, i.e. the viscosity significantly drops with pressure [1,2].This anomalous behaviour of the viscosity is considered to be related to the atomic diffusion in the liquids under pressure.It is, therefore, of particular interest to explore the pressure dependence of the microscopic diffusion mechanism in covalent liquids.
Recently, molecular dynamics studies of liquid GeO 2 under pressure have been reported [3].The diffusivity of atoms was shown to increase with pressure, as in other covalent liquids, such as SiO 2 [4] and B 2 O 3 [[5].It is, however, unclear how the rearrangement process of the covalent bonds is affected by compression.Becuse the calculated results obtained by emprical potentials depend on the choice of the potential form, theoretical investigations based on first-principles theory would be needed to clarify the microscopic mechanism of atomic diffusion in liquid state.
For this purpose, the atomic dynamics in covalent liquids, GeO 2 and SrGeO 3 , under pressure is investigated by ab initio molecular-dynamics simulations in this paper.the structure of crystalline GeO 2 and SrGeO 3 is well known.The high temperature phase of crystalline GeO 2 is quartz type structure which is based on corner-sharing GeO 4 tetrahedron [6].The structure of crystalline SrGeO 3 has layers consisting of Ge 3 O 9 molecules and Sr atoms exist between the layers [7].The coordination number of O to Ge atoms is four.The molecule has two oxygen sites, the bridging oxygen and non-bridging oxygens sites, which have two and one neighbouring Ge atoms, respectively.
The local structures of these materials remain the same upon melting.In liquid SrGeO 3 we intended, there exist non-bridging oxygens while there are few non-bridging oxygens in liquid GeO 2 .For this reason, it is expected that the mechanisms of atomic diffusion in these liquids are different from each other.The purpose of our study is to clarify the microscopic mechanism of atomic diffusion in the liquid GeO 2 and SrGeO 3 .We discuss how Ge-O bonds are exchanged accompanying the diffusion of atoms and the pressure dependence of these mechanisms.

Method of Calculation
The electronic states are calculated using the projector augmented-wave (PAW) method [8,9] within the framework of the density functional theory (DFT) in which the generalized gradient approximation (GGA) [10] is used for the exchange-correlation energy.The plane-wave cutoff energies are 30 and 200 Ry for the electronic pseudo-wave functions and the pseudo-charge density, respectively.The energy functional is minimized using an iterative scheme [10,11].The Γ point is used for Brillouin zone sampling.As the valence electrons, we include the 4s, 4p and 5s electrons of Sr, 4p and 5s electrons of Ge and 2s and 2p electrons of O.Other EPJ Web of Conferences electrons in the lower energy electronic states of each atom were treated with the frozen-core approximation.We use two systems of 120 (24Sr+24Ge+72O) atoms for SrGeO3 and 144 (48Ge+96O) atoms for GeO2 in a cubic supercell under periodic boundary conditions.Using the Nos´e-Hoover thermostat technique [13,14], the equations of motion are solved via an explicit reversible integrator [15] with a time step of Δt = 1.2 fs.To obtain a liquid state of both materials, the same procedure is used.We begin by carrying out an ab initio MD simulation for about 5 ps at a temperature of 4000 K starting from the crystalline structure [6,7].The temperature is selected to be high enough to make the system reach a completely disordered state without the effects of the initial configuration.Then, we decrease the temperature of the system gradually to a target temperature of 2500 K.The number density is determined from zero-pressure condition.The target temperature we have chosen is rather high in order to observe enough number of atomicdiffusion events to analyze the diffusion mechanism in a statistically meaningful way within a limited amount of simulation time.Note that the temperature dependence of physical quantities is not discussed in this paper, and conclusions derived are independent of the selected temperature.The simulation time 7.2 ps, is long enough to achieve good statistics.

Diffusion mechanism at ambient pressure
The local structure of crystal GeO 2 and SrGeO 3 remain even in liquid state, which means that the covalent bonds between Ge and O atoms are preserved in both liquids.It is, however, unclear how Ge-O bonds are exchanged with the diffusion of atoms in the liquids state while retaining the covalent bonds.To clarify the mechanism of atomic diffusion, we investigate the time evolution of bonding nature by utilizing the population analysis [17,18].The bond-overlap populations, which give a semiquantitive estimate of the strength of the covalentlike bonding between atoms, are calculated as a function of time.
In liquid GeO 2 , Ge atoms are mainly coordinated to four O atoms and O atoms bridge two adjacent Ge atoms as in the crystalline phase, even though atoms diffuse in the liquid state.We find that in this system non-bridging O atom double bonded to a Ge atom is always involved with atomic diffusion accompanied by Ge-O bond switching as in liquid B 2 O 3 [5].A typical example of the generation of non-bridging oxygen is shown in figure 1 where the time evolution of the bondoverlap populations associated with the Ge and O atoms of interest is displayed with snapshots of atomic configuration.In the atomic configuration at 0.18 ps (the bottom panel of figure 1) all Ge and O atoms displayed are fourfold-and twofold-coordinated, respectively, to heteroatoms, i.e., there is no bond defect.As shown in the top panel of figure 1, O Ge2−O2 begins to increase at about 0.2 ps, which means that a covalent bond is formed between Ge2 and O2.We can see this new Ge2-O2 bond in the snapshot at 0.27 ps.Due to the formation of the bond, both Ge2 and O2 are overcoordinated.Since the overcoordination is unstable, one of covalent bonds around O2 is broken (in the snapshot at 0.38 ps).Note that Ge1 is coordinated to only three oxygens while Ge2 is still over-coordinated.Finally, the covalent bond between Ge2 and O1 is broken.While fourfold-coordination around Ge2 atom is recovered, O1 is coordinated only one Ge atom (Ge1) as displayed in the snapshot at 0.60 ps.In this way, the nonbridging oxygen is generated. .In crystalline SrGeO 3 , all Ge atoms bond to two bridging oxygens and two non-bridging oxygen.This form is also the most stable in the liquid state as in the crystalline states.When Ge-O bond is broken, the local structure is deformed, for example Ge atom might be bonding to three bridging oxyges and only one nonbridging oxygen.A typical example of the bond breaking is shown in figure 2. In the atomic configuration at 0.05 ps, Ge1 bonds to three bridging oxygens and only one non-bridging oxygen.Since this form is not stable, one of bridging oxygen around Ge1 tries to be non-bridging oxygen.As shown in the top panel in figure 2, O Ge2−O1 begins to decrease at about 0.15 ps. the covalent bond between Ge2 and O1 is broken, as O Ge2−O1 is almost zero for t > 0.35 ps.In the snapshot at 0.35 ps, Ge2 is threefold-coordinated to O atoms, and Ge1 has the most stable form, i.e., bonds to two two bridging oxygens and LAM14 two non-bridging oxygens.The disappearance process of threefold-coordinated Ge atom (Ge2) is shown in figure 3. First, the threefoldcoordinated Ge2 approaches non-bridging oxygen O2 to form a new covalent bond between them.We see that O Ge2−O2 gradually increases for t > 0.12 ps as shown in the top panel of figure 3, which means that the formation of covalent bond between Ge2 and O2 as displayed in the snapshot at 0.37 ps.

Diffusion mechanism under high pressure
Figure 4 shows the diffusion coefficients D α for α = Ge and O atoms as a function of pressure.Clearly, only liquid GeO 2 has a diffusion maximum around 5 GPa, while D α in liquid SrGeO 3 decreases monotonically.In liquid GeO 2 , the atomic diffusion with concerted reaction appears when pressure increases.Figure 5 shows this process at about 4 GPa.The concerted reactions with two over-coordinated Ge atoms can take place because Ge atoms are more easily coordinated to five O atoms under higher pressure.The important point is that non-bridging oxygen is not required in this mechanism while the formation of nonbridging oxygens are always involved in diffusion process at ambient pressure.Since liquid SrGeO 3 has non-bridging oxygens in normal condition, the formation of non-bridging oxygens isn't needed for atomic diffusion at ambient pressure.This is why the diffusivities of liquid SrGeO 3 are much higher than those of liquid GeO 2 which doesn't have non-bridging oxygens at ambient conditions.The diffusion mechanism of liquid SrGeO 3 doesn't change so much even when the pressure increases.Therefore, the diffusivities of liquid SrGeO 3 decrease monotonically with increasing pressure.

Summary
The microscopic mechanism of atomic diffusion in liquid GeO 2 and SrGeO 3 has been investigated by ab initio molecular-dynamics simulations.In liquid GeO 2 , it has been found that the formation of non-bridging oxygen is always necessary for atomic diffusion at ambient pressure.When the pressure increases, the usual concerted reactions become possible.We have found that the atomic diffusion with concerted reaction gives the diffusion maximum under pressure.In liquid SrGeO3, on the other hand, the atomic diffusion without generating non-bridging oxygens is possible at ambient pressure.Even when the pressure increases, the diffusion mechanism of liquid SrGeO 3 doesn't change.Threrefore, liquid SrGeO 3 have no diffusion maximum under pressure.

Fig. 1 .
Fig. 1. (Top panel) The time evolution of bond-overlap populations O ij (t) for (a) i = Ge1, j ∈ O and (b) i = Ge2, j ∈ O in the process of the formation of a non-bridging oxygen.The thick sold and thick dashed lines show O ij (t) associated with the Ge and O atoms of interest as denoted in the figures.The thin lines show O ij (t) between the Ge atoms of interest (labeled as 'Ge1' and 'Ge2') and their neighboring O atoms except O1 and O2.(Bottom panel) Atomic configurations at t = 0.18, 0.27, 0.38 and 0.60 ps.The large and small spheres show Ge and O atoms, respectively.

Fig. 2 .
Fig. 2. (Top panel) The time evolution of bond-overlap populations O i j (t) for (a) i = Ge1, j ∈O and (b) i = Ge2, j ∈ O in the process of the formation of a non-bridging oxygen.The thick sold and thick dashed lines show O ij( t) associated with the Ge and O atoms of interest as denoted in the figures.The thin lines show O ij (t) between the Ge atoms of interest (labeled as 'Ge1' and 'Ge2') and their neighboring O atoms except O1. (Bottom panel) Atomic configurations at t = 0.05 and 0.35 ps.The large, middle and small spheres show Sr, Ge and O atoms, respectively.

Fig. 3 .
Fig. 3. (Top panel) The time evolution of bond-overlap populations O ij (t) for (a) i = Ge3, j∈ O and (b) i = Ge2, j∈O in the process of the formation of a non-bridging oxygen.The thick sold and thick dashed lines show O ij (t) associated with the Ge and O atoms of interest as denoted in the figures.The thin lines show O ij (t) between the Ge atoms of interest (labeled as 'Ge1' and 'Ge2') and their neighboring O atoms except O2. (Bottom panel) Atomic configurations at t = 0.05 and 0.37 ps.The large, middle and small spheres show Sr, Ge and O atoms, respectively.

Fig. 5 .
Fig. 5. (Top panel) The time evolution of bond-overlap populations O ij (t) for (a) i = Ge1, j∈ O and (b) i = Ge2, j∈ O in the process of the formation of a non-bridging oxygen.The thick sold and thick dashed lines show O ij (t) associated with the Ge and O atoms of interest as denoted in the figures.The thin lines show O ij( t) between the Ge atoms of interest (labeled as 'Ge1' and 'Ge2') and their neighboring O atoms except O1 and O2.(Bottom panel) Atomic configurations at t = 0.18, 0.27 and 0.47 ps.The large and small spheres show Ge and O atoms,