**Molecular Dynamics Simulation and Conductivity Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16**

Kien Ling Khoo1,2 and Leonard A. Dissado2 *1Invion Technologies Sdn Bhd 2Engineering Department, University of Leicester, Leicester 1Malaysia 2UK* 

#### **1. Introduction**

370 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

Zhao, X. & Vanderbilt, D. (2002) Phys. Rev. B Vol. 65, pp. 075105.

Fast ion conductors are keystone materials in the development of high performance solid oxide fuel cells and solid electrolytes. In spite the significant contributions in this area a fundamental understanding of the correlation between crystal structure and ionic conductivity is still lacking. In this chapter we report our recent computer simulation results on Hollandite ionic crystals. The objective of this work is to provide the structural parameters which will lead to the design and synthesis of high performance ionic conductors.

Hollandites are ionic crystals of a rather unusual kind, in which the ions of one type are in a disordered and highly mobile state (Dixon & Gillan, 1982). Such materials often have rather special crystal structures in that there are open tunnels or layers through which the mobile ions may move (West, 1988). Their crystal structure corresponds to a family of compounds of general formula AxM4-xNyO8. The basic formula of the Hollandite structure used in the research is Nax(Ti8-xCrx)O16, (Michiue & Watanabe, 1995a, 1996). The chromium and titanium ions are randomly placed in unit cells according to the relative proportions of titanium and chromium ions with a corresponding amount of sodium ions to compensate for the smaller charge on the chromium ions (+3) compared to the titanium ions (+4).

The main interest of this Na-priderite is its structure as a promising (1-D) Na ion conductor (Michiue & Watanabe, 1995b). Priderites, titania-based hollandites, are typical onedimensional ion conductors in which the 1-D tunnels are available for transport of cations in the tunnel (hereafter the "tunnel ion") (Michiue & Watanabe, 1995a). Priderites are generally represented by AxMyTi8-yO16, where A is the alkali or alkaline earth ions and M, di- or trivalent cations (Michiue & Watanabe, 1995a; A. Byström & A.M. Byström, 1950). The host structure for the hollandite being used mainly consists of titanium ions and oxygen ions. These titanium ions are each octahedrally bonded to six oxygen ions. Two such octahedra are joined by sharing an edge, and these doubled groups share further edges above and below to form extended double strings parallel to the growth axis. Four such double strings are joined by corner sharing to form a unit tunnel. The sodium ions are situated in the

Molecular Dynamics Simulation and Conductivity

flexible displacement in the surrounding crystal.

**2. Molecular dynamics (MD) simulation** 

frequency range around 1.2-70 cm-1 (3.6x1010 – 2.1x1012 Hz).

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 373

permittivity for K-hollandites around 100-300 cm-1 (3x1012 – 9x1012 Hz) is due to the vibration of the framework structure. Not much work has been done in investigation for the

Dissado and Hill (Dissado & Hill, 1983) predicted that particles moving in a flexible local environment experienced a cooperative interaction between the particle and their environment leading to a specific form of dielectric response, the constant phase angle response, '( ) "( ) *<sup>p</sup> C f C f f* . Dissado and Alison (Dissado & Alison, 1993) showed that this form of response included a Poley absorption peak in the far infra-red region of the spectrum. This form of absorption peak was observed by Poley (Poley, 1955), Davies (Davies et al, 1969) and some others (Johari, 2002; Chantry, 1977) in both liquids and solids and was named after Poley. The "liquid-like" hollandite structure is similar in form to the materials in which the Poley peak is observed, and hence such a peak can be expected. The theories (Dissado & Alison, 1993; Poley, 1955) suggest that this peak will be caused by the cooperative librations of the dipole produced by the sodium ions and their environmental counter-charges as the sodium ions displace under the many-body interactions of one another. It is the intention here to use molecular dynamics simulations to see if the predicted Poley absorption will be produced by such motions even in the absence of vibrations and

Molecular dynamics (MD) simulations technique has been used in this research to perform the atomistic calculations, of frequency dependent electrical conductivity in Nax(Ti8-xCrx)O16, (x = 1.7). MD simulation is generally carried out to compute the motions of individual molecules in models of solids, liquids and gases. The key idea is motion, which describes how positions, velocities, and orientation change with time (Haile, 1992). The MD simulation method is carried out in such a way that atoms are represented by point particles and the classical (Newton) equations of motion, "force equals mass times acceleration or F = ma" are integrated numerically. The motions of these large numbers of atoms are governed by their mutual interatomic interaction. MD simulations are limited largely by the speed and storage constraints of available computers. Hence, simulations are usually done on system containing 100-1000 particles, with a time step of 1x10-15 s. This technique for simulating the motions of a system of particles when applied to biological macromolecules gives the fluctuations in the relative positions of the atoms in a protein or in DNA as a function of time. Knowledge of these motions provides insights into biological phenomena. MD is also being used to determine protein structure from NMR, to refine protein X-ray crystal structures faster from poorer starting models, and to calculate the free energy

The construction of the molecular dynamics model involved mainly model development and the use of the Molecular Dynamics simulation technique to solve the equation of motion iteratively. In model development, firstly the interaction between ions and the interaction between ion and environment have to be defined. The interaction between ions comprises three components: Lennard-Jones potential, Coulomb potential and Van der Waals' attraction. The rigid lattice approximation is being used in the simulation. This describes the interaction between the ions and the environment. In the rigid lattice approximation only the tunnel ions are allowed to displace from their equilibrium position. The environment

changes resulting from mutation in proteins (Karpus & Petsko, 1990).

tunnels, and each is ionically bonded to eight oxygen ions of the host structure at the corners of a slightly distorted cube. These cubes form a string of ion cages along the axis of a tunnel, and since not all cages have a chromium ion in the place of a titanium ion not all cages contain a sodium ion. The availability of vacant sites into which the sodium ion can move is what allows ion transport to be possible. A typical hollandite structure is shown in figure 1*.*

Fig. 1. A typical hollandite structure projected along the c-axis.

The behaviour of the tunnel ion is usually complicated because it interacts not only with the ions in the lattice sites but also with the other tunnel ions. The main two interactions are Lennard-Jones potential for the short range and Coulomb potential for the long range. Although the ions at the lattice sites give a potential surface for the tunnel ions, which the sodium ions move on, the interactions with other sodium ions in the tunnel is a many body problem like a liquid.

Herein, we investigate the response of the sodium ions under the effect of electric field. The dielectric behaviour of the hollandites is typically studied in the low frequency region [102- 109 Hz]. Dryden and Wadsley (Dryden & Wadsley, 1958), in the first publication on the structure and electrical properties of a hollandite, reported the existence of a dielectric relaxation in several BaMg hollandites, which had a maximum absorption at room temperature in the radio frequency region of the spectrum (105-108 Hz). They reported that the dielectric absorption was only detected when the electric field was in the direction of the tunnel. The activation energy for the peak frequency representing the energy barrier to be overcome for ion displacement was found to be 0.17eV. Cheary and Dryden (Cheary & Dryden, 1991) reported that the mobility of the tunnel ion in Ba hollandite is very low except in limited regions and the activation energy that they obtained is 0.24eV. Activation energies of the order 0.05eV claimed by others was only detected in some samples with the presence of impurity TiO2 as a second phase. Since then there have been further experimental work on the dielectric properties of hollandites published by Singer et al (Singer et al., 1973). Yoshikado et al (Yoshikado et al., 1982) emphasize the loss factor (") at low frequencies which increases with decreasing frequency and usually depends on frequency as f-n where n<1. Jonscher (Jonscher, 1983) concluded that the response of fast ion conductors follows the "universal law" for the charge carriers. He showed that strong low-frequency dispersion occurs at high temperature, 373K, in the hollandite (K1.8Mg0.9Ti7.1O16). The behaviour in the high frequency region is not well understood. Michiue and Watanabe (Michiue & Watanabe, 1999) reported that the strong response observed for the imaginary part of the dielectric

tunnels, and each is ionically bonded to eight oxygen ions of the host structure at the corners of a slightly distorted cube. These cubes form a string of ion cages along the axis of a tunnel, and since not all cages have a chromium ion in the place of a titanium ion not all cages contain a sodium ion. The availability of vacant sites into which the sodium ion can move is what allows ion transport to be possible. A typical hollandite structure is shown in figure 1*.*

The behaviour of the tunnel ion is usually complicated because it interacts not only with the ions in the lattice sites but also with the other tunnel ions. The main two interactions are Lennard-Jones potential for the short range and Coulomb potential for the long range. Although the ions at the lattice sites give a potential surface for the tunnel ions, which the sodium ions move on, the interactions with other sodium ions in the tunnel is a many body

Herein, we investigate the response of the sodium ions under the effect of electric field. The dielectric behaviour of the hollandites is typically studied in the low frequency region [102- 109 Hz]. Dryden and Wadsley (Dryden & Wadsley, 1958), in the first publication on the structure and electrical properties of a hollandite, reported the existence of a dielectric relaxation in several BaMg hollandites, which had a maximum absorption at room temperature in the radio frequency region of the spectrum (105-108 Hz). They reported that the dielectric absorption was only detected when the electric field was in the direction of the tunnel. The activation energy for the peak frequency representing the energy barrier to be overcome for ion displacement was found to be 0.17eV. Cheary and Dryden (Cheary & Dryden, 1991) reported that the mobility of the tunnel ion in Ba hollandite is very low except in limited regions and the activation energy that they obtained is 0.24eV. Activation energies of the order 0.05eV claimed by others was only detected in some samples with the presence of impurity TiO2 as a second phase. Since then there have been further experimental work on the dielectric properties of hollandites published by Singer et al (Singer et al., 1973). Yoshikado et al (Yoshikado et al., 1982) emphasize the loss factor (") at low frequencies which increases with decreasing frequency and usually depends on frequency as f-n where n<1. Jonscher (Jonscher, 1983) concluded that the response of fast ion conductors follows the "universal law" for the charge carriers. He showed that strong low-frequency dispersion occurs at high temperature, 373K, in the hollandite (K1.8Mg0.9Ti7.1O16). The behaviour in the high frequency region is not well understood. Michiue and Watanabe (Michiue & Watanabe, 1999) reported that the strong response observed for the imaginary part of the dielectric

Fig. 1. A typical hollandite structure projected along the c-axis.

problem like a liquid.

permittivity for K-hollandites around 100-300 cm-1 (3x1012 – 9x1012 Hz) is due to the vibration of the framework structure. Not much work has been done in investigation for the frequency range around 1.2-70 cm-1 (3.6x1010 – 2.1x1012 Hz).

Dissado and Hill (Dissado & Hill, 1983) predicted that particles moving in a flexible local environment experienced a cooperative interaction between the particle and their environment leading to a specific form of dielectric response, the constant phase angle response, '( ) "( ) *<sup>p</sup> C f C f f* . Dissado and Alison (Dissado & Alison, 1993) showed that this form of response included a Poley absorption peak in the far infra-red region of the spectrum. This form of absorption peak was observed by Poley (Poley, 1955), Davies (Davies et al, 1969) and some others (Johari, 2002; Chantry, 1977) in both liquids and solids and was named after Poley. The "liquid-like" hollandite structure is similar in form to the materials in which the Poley peak is observed, and hence such a peak can be expected. The theories (Dissado & Alison, 1993; Poley, 1955) suggest that this peak will be caused by the cooperative librations of the dipole produced by the sodium ions and their environmental counter-charges as the sodium ions displace under the many-body interactions of one another. It is the intention here to use molecular dynamics simulations to see if the predicted Poley absorption will be produced by such motions even in the absence of vibrations and flexible displacement in the surrounding crystal.
