**4.5 Group oscillation at very high field**

From figure 13, it is clearly shown that at high electric field, which is 74.3GV/m for the hollandite model, all the sodium ions were driven by the field to move as a group and show a single frequency vibration. The period of the oscillation is 4x10-15s. Since we use specular boundary conditions for the tunnel (i.e. the tunnel ends are reflective) this oscillation period is what would be expected if the sodium ions moved as a whole and were reflected from the boundaries after the first and third time-steps (i.e at 90 degree and 270 degree phases in the cycle). In the case of a smaller field, the sodium ion would just hop between empty sites next to the original site where it belongs. At bigger fields the sodium ions would have enough energy to move to the empty sites located furthest away. Figure 22 shows an example of three sodium ions in a tunnel. When the applied field is small, the sodium ions will hop between the available empty sites next to them. The only available site for Ion1 is the one on its right, whereas Ion2 and Ion3 will have two available sites to go to and this depends on the direction of the force acting on that particular sodium ion at that moment. As shown in Figure 21 increasing the field reduces the number of sodium ions taking part in coupled local group site-motions (reduced peak amplitude) in favour of sodium ion hopping between sites. For this reason a trend towards higher resonance frequencies and broader

Molecular Dynamics Simulation and Conductivity

and *n* becomes

frequency.

**5. Conclusions** 

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 395

Substituting equations 9 and 10 into equation 8, the obtained *fp* is 5.469x1012 Hz. This value is the upper limit for the group oscillations that our simulations show occur in the Hollandite system. The estimated plasma frequency is much lower than the high-field group oscillation frequency (~1/46), but is ten times higher than the frequency of the "peak calculated for lower field. This is consistent with an interpretation of the simulations at low fields in which local site vibrations of the sodium ions (equivalent to dipole librations) are coupled together but cannot extend to all the sodium atoms because hopping of an ion between sites destroys the coupled motion, thus giving oscillation frequencies below the plasma oscillation limit. High fields produce a field driven coupled hopping that oscillates the ions between the reflective boundaries at higher frequencies than the plasma oscillation

We have presented a detailed description of the molecular dynamics computational methods applied to the fast-ion conductor Hollandite together with the fundamental concepts needed for interpreting our results. This study focused on the calculation of the sodium ion positions at a range of temperature and electric fields and the resulting frequency dependence of the conductivity mechanism and susceptibility. In the simulation the lattice surrounding the tunnel was held rigid and only the sodium ions in the tunnel were allowed to move. The ac dielectric response was calculated from the rate of change of polarisation, dP/dt, under the action of a dc step-field (i.e. dielectric response function) of 7.43MV/m to 74.3GV/m, at temperatures between 200K and 373K. Our hollandite model shows that the dielectric response due to the motion of the sodium ions in the tunnels behaves approximately like that of polar liquids in the far-infrared frequency region. The susceptibility shows an absorption peak "(f)peak in the frequency region between 4.5x1010 and 8.8x1010 Hz at 297K. This fitted very well with Poley's prediction of an absorption typically observed in polar liquids in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region at room temperature. The frequency dependence of the real and imaginary susceptibility components ' and " obtained show resonance behaviour. This is due to the vibration of sodium ions coupled together in groups in which the motion of each sodium ion is centred around a local site. This mode of motion is equivalent to the coupled libration of a group of dipoles. Transfer of a sodium ion between sites destroys the coupling for any particular group and acts as a damping on the resonance behaviour associated with its group oscillation. This is in agreement with the prediction by Fröhlich who suggested that the absorption due to displacement of charges bound elastically to an equilibrium position is of resonance character. The absorption peaks in " at the resonance frequency lie between 4.5x1010 and 8.8x1010 Hz at 297K which matches very well with the Poley absorption which is typically observed in polar liquids in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region at room temperature. The resonance frequency and the resonance peak height are independent of temperature. The absorption peak was associated with cooperative motions of the sodium ions

28 3 *n volume* 1 / 1.5536 10 *m* (10)

peaks (i.e. higher damping) can be expected. When the applied field is high enough all the sodium ions will take part in hopping between sites, and the field will force all the sodium ions to move to the furthest available sites as shown in figure 22c. The reflective boundary condition at the two ends of the tunnel cause the sodium ions to vibrate between the two ends and this coherent group oscillation was generated. It can be speculated that this behaviour corresponds to a flow of sodium ions (i.e. a sodium ion dc current) when the boundaries regenerate the sodium ion concentration as would an ion exchange membrane.

Fig. 22. The position of three sodium ions along c-axis (a) without electric field (b) with low electric field (c) with high electric field. The square denotes the available sites for the sodium ions to move to.

The oscillatory motion is of frequency 2.5x1014 Hz and this falls into the infrared frequency region. With a longer tunnel, a higher electric field would be needed to give a coherent group oscillation, as higher force is required to push all the sodium ions to the furthest distance. When all the sodium ions move to one end of the tunnel under the high field, it is just like ALL the dipoles being forced to align in one direction by the electric field, i.e. it is a state of motion that cannot exist without the presence of the field and is thus a non-linear response.

The motion of a group of charges as a whole in the Coulomb field of their counter charges defines a plasma oscillation, and it has a natural frequency that would apply for the sodium ions in an infinitely long tunnel. This plasma oscillation frequency is given in equation 8 below (Ziman, 1960),

$$
\rho o\_p = 2\pi f\_p = \sqrt{\frac{ne^2}{m\_{Na}\varepsilon\_0}}\tag{8}
$$

where *mN*a is the mass for the sodium ion, which is 3.81361x10-26 kg, *e* is the electron charge, *<sup>0</sup>* is the permittivity of free space and *n* is the concentration of sodium ions (number of sodium ions in a volume of 1m3). The plasma oscillation frequency that we could expect for our Hollandite model in the absence of the reflective boundaries can be obtained by calculating *n* as follows: The Hollandite model consists of 60 layers with 24 sodium ions, the volume 1.54475x10-27 m3 is the volume for the 24 sodium ions (Khoo, 2003). The average *volume* for one sodium ion is:

$$volume = \frac{1.54475 \times 10^{-27}}{24} = 6.4365 \times 10^{-29} \text{ m}^3 \tag{9}$$

and *n* becomes

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

peaks (i.e. higher damping) can be expected. When the applied field is high enough all the sodium ions will take part in hopping between sites, and the field will force all the sodium ions to move to the furthest available sites as shown in figure 22c. The reflective boundary condition at the two ends of the tunnel cause the sodium ions to vibrate between the two ends and this coherent group oscillation was generated. It can be speculated that this behaviour corresponds to a flow of sodium ions (i.e. a sodium ion dc current) when the boundaries regenerate the sodium ion concentration as would an ion exchange membrane.

(b) (c)

Fig. 22. The position of three sodium ions along c-axis (a) without electric field (b) with low electric field (c) with high electric field. The square denotes the available sites for the sodium

The oscillatory motion is of frequency 2.5x1014 Hz and this falls into the infrared frequency region. With a longer tunnel, a higher electric field would be needed to give a coherent group oscillation, as higher force is required to push all the sodium ions to the furthest distance. When all the sodium ions move to one end of the tunnel under the high field, it is just like ALL the dipoles being forced to align in one direction by the electric field, i.e. it is a state of motion that cannot exist without the presence of the field and is thus a non-linear

The motion of a group of charges as a whole in the Coulomb field of their counter charges defines a plasma oscillation, and it has a natural frequency that would apply for the sodium ions in an infinitely long tunnel. This plasma oscillation frequency is given in equation 8

2 *p p*

24 *volume m*

 

2

*Na*

<sup>27</sup> 1.54475 10 29 3 6.4365 10

*ne <sup>f</sup> m*

where *mN*a is the mass for the sodium ion, which is 3.81361x10-26 kg, *e* is the electron charge,

*<sup>0</sup>* is the permittivity of free space and *n* is the concentration of sodium ions (number of sodium ions in a volume of 1m3). The plasma oscillation frequency that we could expect for our Hollandite model in the absence of the reflective boundaries can be obtained by calculating *n* as follows: The Hollandite model consists of 60 layers with 24 sodium ions, the volume 1.54475x10-27 m3 is the volume for the 24 sodium ions (Khoo, 2003). The average

0

(8)

(9)

High Field

(a)

Field

ions to move to.

response.

below (Ziman, 1960),

*volume* for one sodium ion is:

Along c-axis

Ion2 Ion3

Ion1

$$m = 1 \text{ / volume} = 1.5536 \times 10^{28} \text{ m}^{-3} \tag{10}$$

Substituting equations 9 and 10 into equation 8, the obtained *fp* is 5.469x1012 Hz. This value is the upper limit for the group oscillations that our simulations show occur in the Hollandite system. The estimated plasma frequency is much lower than the high-field group oscillation frequency (~1/46), but is ten times higher than the frequency of the "peak calculated for lower field. This is consistent with an interpretation of the simulations at low fields in which local site vibrations of the sodium ions (equivalent to dipole librations) are coupled together but cannot extend to all the sodium atoms because hopping of an ion between sites destroys the coupled motion, thus giving oscillation frequencies below the plasma oscillation limit. High fields produce a field driven coupled hopping that oscillates the ions between the reflective boundaries at higher frequencies than the plasma oscillation frequency.

## **5. Conclusions**

We have presented a detailed description of the molecular dynamics computational methods applied to the fast-ion conductor Hollandite together with the fundamental concepts needed for interpreting our results. This study focused on the calculation of the sodium ion positions at a range of temperature and electric fields and the resulting frequency dependence of the conductivity mechanism and susceptibility. In the simulation the lattice surrounding the tunnel was held rigid and only the sodium ions in the tunnel were allowed to move. The ac dielectric response was calculated from the rate of change of polarisation, dP/dt, under the action of a dc step-field (i.e. dielectric response function) of 7.43MV/m to 74.3GV/m, at temperatures between 200K and 373K. Our hollandite model shows that the dielectric response due to the motion of the sodium ions in the tunnels behaves approximately like that of polar liquids in the far-infrared frequency region. The susceptibility shows an absorption peak "(f)peak in the frequency region between 4.5x1010 and 8.8x1010 Hz at 297K. This fitted very well with Poley's prediction of an absorption typically observed in polar liquids in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region at room temperature. The frequency dependence of the real and imaginary susceptibility components ' and " obtained show resonance behaviour. This is due to the vibration of sodium ions coupled together in groups in which the motion of each sodium ion is centred around a local site. This mode of motion is equivalent to the coupled libration of a group of dipoles. Transfer of a sodium ion between sites destroys the coupling for any particular group and acts as a damping on the resonance behaviour associated with its group oscillation. This is in agreement with the prediction by Fröhlich who suggested that the absorption due to displacement of charges bound elastically to an equilibrium position is of resonance character. The absorption peaks in " at the resonance frequency lie between 4.5x1010 and 8.8x1010 Hz at 297K which matches very well with the Poley absorption which is typically observed in polar liquids in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region at room temperature. The resonance frequency and the resonance peak height are independent of temperature. The absorption peak was associated with cooperative motions of the sodium ions

Molecular Dynamics Simulation and Conductivity

Toulouse, France, July 5-9, 2004

Winchester, UK, July 8-13, 2007

119

Vol.116, pp. 296-299

Vol.57, pp. 547-551

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653-663, Amsterdam, Holland

J. Wiley and Sons, NY

11298-11302

http://www.ccp14.ac.uk/tutorial/lmgp/

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LMGP-Suite Suite of Programs for the interpretation of X-ray Experiments, by Jean Laugier

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and Bernard Bochu, ENSP/Laboratoire des Matériaux et du Génie Physique, BP 46. 38042 Saint Martin d'Hères, France, http://www.inpg.fr/LMGP and

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Relaxation in Polar Liquids. *Institute of Radio Engineering and Electronics,* Academy

as suggested by the cluster model. The itinerant oscillator model does not allow for such many-body motions. On increasing the applied field " becomes smaller and, more switches of sodium ions between sites are obtained. This indicates that the higher the field, the fewer ions are involved in coupled libration motions and more in damping through hopping between sites. At very high field, which is 74.3GV/m in our simulation, all the sodium ions were driven by the field to move as a group i.e. to transfer collectively from site to site giving a single frequency vibration due to our reflective boundary conditions.

### **6. References**


as suggested by the cluster model. The itinerant oscillator model does not allow for such many-body motions. On increasing the applied field " becomes smaller and, more switches of sodium ions between sites are obtained. This indicates that the higher the field, the fewer ions are involved in coupled libration motions and more in damping through hopping between sites. At very high field, which is 74.3GV/m in our simulation, all the sodium ions were driven by the field to move as a group i.e. to transfer collectively from site to site giving a single frequency vibration due to our reflective

Byström, A. & Byström, A.M. (1950). The Crystal of Hollandite, the Related Manganese Oxide Minerals, and MnO2. *Acta Crystallographica,* Vol.3, pp. 146-154 Chantry, G.W. (1977). Dielectric Measurements in the Submillimeter Region and a

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boundary conditions.

**6. References** 


Yoshikado, A.; Ohachi, T.; Taniguchi, I.; Onoda, Y.; Watanabe, M. & Fujiki, Y. (1982). AC Ionic Conductivity of Hollandite Type Compounds from 100 Hz to 37.0 GHz. *Solid State Ionics,* Vol.7, pp. 335-344

**19** 

*Poland* 

**MD Simulation of the Ion** 

Ewa Hawlicka and Marcin Rybicki

**Solvation in Methanol-Water Mixtures** 

*Institute of Applied Radiation Chemistry, Technical University of Lodz* 

Sodium, calcium and magnesium ions are essential for the biological activity of many polyelectrolytes. This activity depends on a condensation of the metal ions. There are several clues, which suggest that interactions between the polyelectrolyte and metal ions depend on a hydration of the ion. This accounts for the great interest in the hydration of metal ions, particularly in the systems containing hydrophobic groups. In aqueous solutions the hydration of Na+, Mg2+ and Ca2+ differs. The first hydration shells of Na+ and Mg2+ consists of six water molecules and have octahedral symmetry (Dietz et al, 1982; Hawlicka & Swiatla-Wojcik, 1995). The first shell of Ca2+ is larger; it contains eight or more water molecules and does not show any regularity (Owczarek et al., 2007). X-ray diffraction studies (Tamura et al., 1992, Megyes et al., 2004) have suggested that all these cations are six-

Various experimental techniques can be employed to gain insight into a coordination shell of the ion, but a lack of theory renders even a term 'preferential solvation' misleading. The concept of the preferential solvation has been introduced to explain non-linear changes of solution properties, but now this term is commonly used to emphasise a difference of the compositions of the coordination shell and the bulk solvent. Preferential solvation is usually expected if the ion interacts stronger with one of the solvent components. There are however experimental clues that the selective solvation might be due to a microheterogeneity of the

Methanol-water mixture is a suitable model to study structural aspects of solvation in binary systems, particularly when hydrophobic effects may occur. Both net components are highly associated liquids, but their hydrogen-bonded networks are inconsistent. Water molecules form a 3-dimensional, tetrahedrally coordinated structure, where cavities are filled with monomers (Soper & Phillips, 1986). Extension of the hydrogen bonds over 1 nm causes that liquid water, even at room temperature, behaves like a gel (Dore et al., 2000). Hydrogen-bonded molecules of methanol form zig-zag polymer chains (Narten & Habenschuss, 1984). Though in binary mixture the molecules of methanol and water may form a common hydrogen-bonded network the bulky methyl group causes that methanol molecule cannot simply replace the water molecule in the tetrahedral structure. In consequence the methanol-water mixture may become heterogeneous on the molecular level. Neutron diffraction (Dughan et al., 2004) and X-ray spectroscopy (Guo et al. 2004)

coordinated in methanolic solutions thus their shells are octahedral.

**1. Introduction** 

binary solvent.

Ziman, J.M. (1960). *Electron and Phonons. The theory of Transport Phenomena in Solids,* pp. 161- 162, Clarendon Press, Oxford, UK
