Molecular dynamics simulation of high frequency (10/sup 10/ to 10/sup 12/ Hz) dielectric absorption in the hollandite Na/sub x/(Ti/sub 8-x/Cr/sub x/)O/sub 16/
Abstract: The charge-compensating sodium ions that reside interstitially in the one-dimensional tunnels of the hollandite Na/sub x/(Ti/sub 8-x/Cr/sub x/)O/sub 16/ are used as a simple model for a fluid. Molecular dynamics are used to calculate the motions of the ions at a range of temperatures between 200 K and 373 K. The polarization response of the system to a step-up electric field is calculated for field strengths between 7.43 MV/m and 74.3 GV/m, and converted to an ac susceptibility. Resonance absorption is found, peaking at frequencies between 4.5/spl times/10/sup 10/ and 8.8/spl times/10/sup 10/ Hz at 297 K. The origin of the response is shown to be the anharmonically coupled ion vibrations damped by ion hopping to neighbouring sites. The relationship of the result to the experimentally observed Poley absorption is explored, and a brief comparison of the calculated dynamics to previous theoretical models is made.
Summary (2 min read)
- In these materials a fraction of the cations M are replaced by a cation in a different oxidation state N, with charge neutrality maintained by the presence of interstitial monovalent cations A. A special feature of their structure is the existence of one-dimensional (1-D) tunnels within which the interstitial ions are located at specific binding sites [1,2].
- The ions in the lattice cage produce a potential surface with minima at the binding sites about which the sodium ions vibrate.
- Dissado and Hill  have presented a theory for dielectric relaxation that takes into account such cooperative many-body motions and it has been shown  that the theory predicts an additional absorption peak at frequencies of 10 10 to 10 12 Hz.
- Here the authors use Molecular Dynamic (MD) simulations to see if such motions do indeed produce the predicted Poley absorption.
- The MD simulation has been carried out using programs that the authors have written specifically for the purpose.
- The potential between two ions is taken to have the form: ij ji p ij ijij r ezz r rV 0 2 4 )( (1) where zi and zj are the formal charge of ith and jth ions respectively, e is the unit charge, rij is the distance between the ith and jth ions.
- Electrostatic interactions between the sodium ions along the tunnel lead to a fluctuating potential environment for the motion of the sodium ions.
- Initial positions for the ions were taken from the results of the x-ray analysis structure refinement .
- Ten simulations were carried out initiating with different sets of Na + velocities, but with the same average kinetic energy, i.e. temperature.
- During the simulation the Na + ions may hop back and forth between neighbouring binding sites when they are unoccupied and vibrate in the binding site in which they reside, as shown in Fig.
- The frequency dependence of ‟, see Figs 3(a) and 3(b), show clearly that the response has the form of a broadened resonance in the frequency region around 5x10 10 Hz.
- This is the case for all temperatures and fields investigated.
- The many body nature of the forces in the Na + system imply that the Lorenztian should be the better description.
- The hopping of charges between alternative sites is usually assumed to yield a relaxation peak at the hopping frequency .
- Ionic vibrations are expected to take place at a much higher frequency and give a resonance behaviour .
- The ion hopping can be taken to be equivalent to the friction between annulus and disc in the Itinerant Oscillator model , however this model treats the coupling between the oscillator and the vibrations as random impulses and does not include the anharmonic coupling the authors have shown to be important, as implied in .
- Furthermore Johari  found that the resonance peak height increased with temperature whereas Noskova et al  found and absorption peak that decreased with the increase in temperature.
- Although the changes are small and possibly not detectable in their calculation it is therefore not certain that the authors can equate their calculated behaviour with the Poley absorption in liquids.
- Anharmonic coupling between vibrating ions extends the vibration modes to lower frequencies and gives rise to a broad resonance in the frequency region between 4.5x10 10 and 8.8x10 10 Hz at 297K.
- The „damping‟ associated with the resonance absorption was the result of ion hopping to unoccupied neighbouring sites, causing the ion to be disconnected from its original group vibration and connected to another.
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Q1. What are the contributions in this paper?
In this paper, the authors used molecular dynamics to calculate the motions of the ions at a range of temperatures between 200 K and 373 K.