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%!TEX root = /Users/charles/Documents/chuckDissertation/dissertation.tex |
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\chapter{\label{chap:metalglass}COMPARING MODELS FOR DIFFUSION IN SUPERCOOLED LIQUIDS: THE EUTECTIC COMPOSITION OF THE AG-CU ALLOY} |
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\chapter{\label{chap:metalglass}COMPARING MODELS FOR DIFFUSION IN SUPERCOOLED LIQUIDS: THE EUTECTIC COMPOSITION OF THE Ag-Cu ALLOY} |
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Solid solutions of silver and copper near the eutectic point have been |
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historical curiosities since Roman emperors used them to cut the |
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correlation functions that decay according to the the famous |
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Kohlrausch-Williams-Watts ({\sc kww}) law, |
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\begin{equation} |
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C(t) \approx A \exp\left[-(t/\tau)^{\beta}\right]. |
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C(t) \approx A \exp\left[-(t/\tau)^{\beta}\right], |
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\label{eq:kww2} |
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\end{equation} |
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where $\tau$ is the characteristic relaxation time for the system. |
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Kob and Andersen observed stretched exponential decay of the van Hove |
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correlation function in a system comprising an 80-20 mixture of |
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particles with different well depths $(\epsilon_{AA} \neq |
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$F_{s}(k,t)$.\cite{Hansen86} |
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In single component Lennard-Jones systems, the stretching parameters |
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appear to be somewhat lower. Angelani {\em et al.} have reported |
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appear to be somewhat lower. Angelani {\em et al.}\cite{Angelani98} have reported |
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$\beta \approx 1/2$ for relatively low temperature Lennard-Jones |
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clusters, and Rabani {\em et al.} have reported similar values for the |
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decay of correlation functions in defective Lennard-Jones crystals. |
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decay of correlation functions in defective Lennard-Jones crystals.\cite{Rabani99} |
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There have been a few recent studies of amorphous metals using fairly |
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realistic potentials and molecular dynamics methodologies. Gaukel and |
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whether their stretching behavior is similar to that observed in the |
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two-component Lennard-Jones systems. |
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Additionally, bi-metallic alloys present an ideal opportunity for us |
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to apply the cage-correlation function methodology that one of us |
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Additionally, bi-metallic alloys present an ideal opportunity |
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to apply the cage-correlation function methodology which was |
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developed to study the hopping rate in supercooled |
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liquids.\cite{Rabani97,Gezelter99,Rabani99,Rabani2000} In particular, |
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we want to use it to test two models for diffusion, both of which use |
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it will be used to test two models for diffusion, both of which use |
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hopping times which are easily calculated by observing the long-time |
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decay of the cage correlation function. The two models are Zwanzig's |
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model,\cite{Zwanzig83} which is based on the periodic interruption of |
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used to derive transport properties from a random walk on a regular |
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lattice where the time between jumps is non-uniform. |
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\section{THEORY} |
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\section{Theory} |
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In this section we give a brief introduction to the models for |
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In this section a brief introduction will be given to the models for |
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diffusion that we will be comparing, as well as a brief description of |
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how one can use the the cage-correlation function to obtain hopping |
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rates in liquids. |
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\subsection{ZWANZIG'S MODEL} |
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\subsection{Zwanzig's Model} |
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In his 1983 paper~\cite{Zwanzig83} on self-diffusion in liquids, |
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Zwanzig proposed a model for diffusion which consisted of ``cells'' or |
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basins in which the liquid's configuration oscillates until it |
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Zwanzig does not explicitly derive the inherent structure normal modes |
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from the potential energy surface (he used the Debye spectrum for |
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$\rho(\omega)$. Moreover, the theory avoids the |
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$\rho(\omega)$.) Moreover, the theory avoids the |
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problem of how to estimate the lifetime $\tau$ for cell jumps that |
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destroy the coherent oscillations in the sub-volume. Nevertheless, the |
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model fits the experimental results quite well for the self-diffusion |
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of tetramethylsilane (TMS) and benzene over large ranges in |
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temperature.\cite{Parkhurst75a,Parkhurst75b} |
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\subsection{THE {\sc ctrw} MODEL} |
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\subsection{The {\sc ctrw} Model} |
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In the {\sc ctrw} |
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model,\cite{Blumen83,Klafter94,Klafter96,Shlesinger99} random walks |
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take place on a regular lattice but with a distribution of waiting |
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\label{eq:CTRW_diff} |
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D = \frac {\sigma_{0}^{2}} {6\tau} |
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\end{equation} |
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This behavior is also suggested by our estimates of $\tau$ in |
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This behavior is also suggested by estimates of $\tau$ in |
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$\mbox{CS}_{2}$ and in Lennard-Jones systems using the cage |
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correlation function.\cite{Gezelter99,Rabani99} |
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They conclude that in order to observe anomalous transport, waiting |
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time distributions with pathological long-time tails are required. |
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\subsection{THE CAGE CORRELATION FUNCTION} |
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\subsection{The Cage Correlation Function} |
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In a recent series of |
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papers,\cite{Gezelter97,Rabani97,Gezelter98a,Gezelter99,Rabani99,Rabani2000} |
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one of us has been investigating approaches to calculating the hopping |
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Gezelter {\it et al.} have investigated approaches to calculating the hopping |
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rate ($k_{h} = 1/\tau$) in liquids, supercooled liquids and defective |
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crystals. To obtain this estimate, we introduced the {\it cage |
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crystals. To obtain this estimate, they introduced the {\it cage |
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correlation function} which measures the rate of change of atomic |
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surroundings, and associate the long-time decay of this function with |
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the basin hopping rate for diffusion. The details on calculating the |
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cage correlation function can be found in Refs. \citen{Rabani97} and |
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\citen{Gezelter99}, but we will briefly review the concept here. |
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cage correlation function can be found in Refs. \cite{Rabani97} and |
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\cite{Gezelter99}, but the concept will be briefly reviewed here. |
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An atom's immediate surroundings are best described by the list of |
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other atoms in the liquid that make up the first solvation shell. |
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itself to include the original members. Only those events which |
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result in irreversible changes to the surroundings will cause the cage |
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to de-correlate at long times. The mathematical formulation of the |
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cage correlation function is given in Refs. \citen{Rabani97} and |
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\citen{Gezelter99}. |
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cage correlation function is given in Refs. \cite{Rabani97} and |
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\cite{Gezelter99}. |
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Averaging over all atoms in the simulation, and studying the decay of |
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the cage correlation function gives us a way to measure the hopping |
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rates directly from relatively short simulations. We have used the |
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rates directly from relatively short simulations. Gezelter {\it et al.} have used the |
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cage correlation function to predict the hopping rates in |
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atomic~\cite{Rabani97} and molecular~\cite{Gezelter99} liquids, as |
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well as in defective crystals.\cite{Rabani99,Rabani2000} In the |
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defective crystals, we found that the cage correlation function, after |
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defective crystals, they found that the cage correlation function, after |
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being corrected for the initial vibrational behavior at short times, |
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decayed according to the {\sc kww} law (Eq. (\ref{eq:kww2})) with a |
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stretching parameter $\beta \approx 1/2$. Angelani {\em et al.} have |
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to perform the simulations. Section \ref{metglass:sec:results} contains our |
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results, and section \ref{metglass:sec:discuss} concludes. |
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\section{COMPUTATIONAL DETAILS} |
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\section{Computational Details} |
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\label{metglass:sec:details} |
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We have chosen the Sutton-Chen potential with the same |
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parameterization as Qi {\it et al.}\cite{Qi99}. Details of this potential can be found in section \ref{introSec:tightbind}. This model was chosen over {\sc EAM} because of better experimental agreement with the heat of solution for Ag and Cu alloys and for better prediction of melting point and liquid state properties. This particular parametarization has been shown to reproduce the experimentally available heat of mixing data for both FCC solid solutions of Ag-Cu and the high-temperature liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does not reproduce the experimentally observed heat of mixing for the liquid alloy.\cite{MURRAY:1984lr} |
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The Sutton-Chen potential with the same |
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parameterization as Qi {\it et al.}\cite{Qi99} has been chosen for this set of simulations. Details of this potential can be found in section \ref{introSec:tightbind}. This model was chosen over {\sc EAM} because of better experimental agreement with the heat of solution for Ag and Cu alloys and for better prediction of melting point and liquid state properties. This particular parametarization has been shown to reproduce the experimentally available heat of mixing data for both FCC solid solutions of Ag-Cu and the high-temperature liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does not reproduce the experimentally observed heat of mixing for the liquid alloy.\cite{MURRAY:1984lr} |
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In order to study the long-time portions of the correlation functions |
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in this system without interference from the simulation methodology, |
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we carried out molecular dynamics simulations in the constant-NVE |
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ensemble. The density of the system was taken to be $8.742 |
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a set of molecular dynamics simulations in the constant-NVE |
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ensemble was conducted. The density of the system was taken to be $8.742 |
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\mbox{g}/\mbox{cm}^3$. This density was chosen immediately to the |
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liquid side of the melting transition from the constant thermodynamic |
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tension simulations of Qi {\it et al.}. Their simulations gave |
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tension simulations of Qi {\it et al.}.\cite{Qi99} Their simulations gave |
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excellent estimates of phase and structural behavior, and should be |
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seen as a starting point for investigations of these materials. |
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50 K increments to 400 K. At each temperature increment, the systems |
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re-sampled velocities from a Maxwell-Boltzmann distribution every ps |
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for 20 ps, then were allowed to equilibrate for 50 ps. Following the |
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equilibration period, we collected particle positions and velocities |
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equilibration period, particle positions and velocities were collected |
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every ps for 250 ps. The lower temperature runs (375 K, 350 K, 325 K, |
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and 300 K) were sampled for 500 ps, 1 ns, 1 ns and 7 ns respectively |
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to accumulate more accurate long-time statistics. Cooling in this |
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through experimental methods which typically are on the order of |
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10$^{6}$ or 10$^{7}$ K/s depending on the quenching method.\cite{duwez:1136} |
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\section{RESULTS} |
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\section{Results} |
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\label{metglass:sec:results} |
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The simulation results were analyzed for several different structural |
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is the ratio of the magnitude of the first minimum, $g_{min}$ to |
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the first maximum, $g_{max}$ in the radial distribution function. |
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According to their estimates, when the value of $R_{WA}$ reaches 0.14, |
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the system has passed through the glass transition.\cite{Wendt78} We |
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observed a $T_{g}^{WA}$ of 547 K given a cooling rate of $1.56 \times |
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10^{11}$ K/s. Goddard, {\it et al.} observed a $T_g^{WA}\approx 500 |
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K$ for a cooling rate of $2\times 10^12$ K/s in constant temperature, |
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the system has passed through the glass transition.\cite{Wendt78} A $T_{g}^{WA}$ of 547 K was observed given a cooling rate of $1.56 \times |
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10^{11}$ K/s. Goddard, {\it et al.} observed a $T_g^{WA}\approx 500 |
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K$ for a cooling rate of $2\times 10^{12}$ K/s in constant temperature, |
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constant thermodynamic tension (TtN) simulations\cite{Qi99}. |
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We note that the split second peak in $g(r)$ appears at $T_g^{WA}$, a |
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temperature for which the diffusion constant is still an appreciable |
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fraction of it's value in the liquid phase. We also note that |
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It should be noted that the split second peak in $g(r)$ appears at $T_g^{WA}$ (Figure \ref{fig:Wendt_abraham}), a temperature for which the diffusion constant is still an appreciable |
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fraction of it's value in the liquid phase. In addition, the |
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diffusive behavior continues at the lowest simulated temperature of |
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300 K indicating that glass transition for our system lies below 300 |
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K. Taking the operational definition of the glass transition to be the |
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\end{figure} |
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In order to compute diffusion constants using Zwanzig's model |
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(Eq. (\ref{eq:zwanz}), one must obtain an estimate of the density of |
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vibrational states ($\rho(\omega)$) of the liquid. We obtained the |
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$\rho(\omega)$ in two different ways, first by quenching twenty |
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(Eq. (\ref{eq:zwanz})), one must obtain an estimate of the density of |
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vibrational states ($\rho(\omega)$) of the liquid. $\rho(\omega)$ has been obtained in two different ways, first by quenching twenty |
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high-temperature (1350 K) configurations to the nearest local minimum |
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on the potential energy surface. These structures correspond to |
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inherent structures on the liquid state potential energy surface. |
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\rho_0(\omega) = \int_{-\infty}^\infty \left \langle \mathbf{v}(t) \cdot |
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\mathbf{v}(0) \right \rangle e^{-i \omega t} dt |
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\end{equation} |
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for trajectories which hover just above the inherent structures. To |
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calculate the power spectrum, a small amount of kinetic energy (8 K) |
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for trajectories which hover just above the inherent structures. These are displayed in Figure \ref{fig:rho} |
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To calculate the power spectrum, a small amount of kinetic energy (8 K) |
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was given to each of the twenty inherent structures and the system was |
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allowed to equilibrate for 30 ps. After equilibration, the velocity |
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autocorrelation functions were calculated from relatively short (30 |
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($\rho_{0}(\omega)$) recovers these modes, but gives a much noisier |
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estimate for the density of states. |
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We determined that with a radial cutoff of $2 \alpha_{ij}$ the |
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It was determined that with a radial cutoff of $2 \alpha_{ij}$ the |
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potential (and forces) exhibited a large number of discontinuities |
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which made the minimization into the inherent structures somewhat |
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challenging. In this type of potential, the discontinuities at the |
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larger system size, and so for the minimizations and the calculation |
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of the densities of states, we used a total of 1372 atoms. |
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We note that one possible way to provide a surface without |
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It should be noted that one possible way to provide a surface without |
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discontinuities would be to use a shifted form of the density, |
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\begin{equation} |
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\rho_{i}=\sum_{j\neq |
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re-parameterization of $c_{i}$ and $D_{ii}$, tasks which are outside |
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the purview of the current work. |
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\subsection{DIFFUSIVE TRANSPORT AND EXPONENTIAL DECAY} |
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\subsection{Diffusive Transport and Exponential Decay} |
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Translational diffusion constants were calculated via the Einstein |
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$\rho(\omega)$,\cite{Gezelter99} so it is no surprise that the choice |
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of the density of states gives a large variation in the predicted |
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results. The agreement with the diffusion constants is better at lower |
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temperatures, but we observe an obviously incorrect temperature |
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dependence in the higher temperature liquid regime. |
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temperatures, but an obviously incorrect temperature |
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dependence is observed in the higher temperature liquid regime. |
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The {\sc ctrw} model with $\gamma=1$ and an assumption of fixed jump |
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distances gives much better agreement in the liquid regime, and the |
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trend with changing temperature seems to be in excellent agreement |
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with the Einstein relation. |
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|
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Delving a bit more deeply into the {\sc ctrw} predictions, we can |
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assume that the distribution of hopping times is well behaved |
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Delving a bit more deeply into the {\sc ctrw} predictions, it can be |
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assumed that the distribution of hopping times is well behaved |
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(i.e. $\gamma=1$) but that the jump {\it distance} is temperature |
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dependent. To obtain estimates of the jump distance as a function of |
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temperature, $\sigma_0(T)$, we can invert (Eq.(\ref{eq:CTRW_diff})) by |
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multiplying the cage-correlation hopping time by the self-diffusion |
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constant. We show the temperature-dependent hopping distances in |
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Fig. \ref{fig:r0_ctrw}. Note that these assumptions would lead us to |
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constant. The temperature-dependent hopping distances is shown in |
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Fig. \ref{fig:r0_ctrw}. Note that these assumptions would lead one to |
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believe that the average jump distance is increasing sharply as one |
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approaches the glass transition, which could be an indicator of motion |
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dominated by Levy flights.\cite{Klafter96,Shlesinger99} |
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\end{figure} |
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|
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\subsection{NON-DIFFUSIVE TRNASPORT AND NON-EXPONENTIAL DECAY} |
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\subsection{Non-Diffusive Transport and Non-Exponential Decay} |
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A much more realistic scenario is that the distribution of hopping |
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{\it times} changes while the jump distance remains the same at all |
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temperatures. In Figs. \ref{fig:exponent} and |
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\ref{fig:tauhop}, we show the effects of relaxing the linear fits of |
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\ref{fig:tauhop}, the effects of relaxing the linear fits of |
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$\langle r^{2}(t) \rangle$ and of the long time portion of |
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$ln[C_{cage}(t)]$. To fit the mean square displacements, we performed |
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$ln[C_{cage}(t)]$ are shown. To fit the mean square displacements, a |
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weighted non-linear least-squared fits using the {\sc ctrw} ($\gamma < |
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1$) expression for the mean square displacement |
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(Eq. (\ref{eq:ctrw_msd})). The only free parameter in the fit is the |
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jump distance, which we fixed at a value of 1.016 \AA, the optimal |
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jump distance for the $\gamma=1$ case. These fits allow us to obtain |
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estimates of $\gamma$ and the hopping times $\tau$ directly from the |
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(Eq. (\ref{eq:ctrw_msd})) was performed. The only free parameter in the fit is the |
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jump distance, which is fixed at a value of 1.016 \AA, the optimal |
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jump distance for the $\gamma=1$ case. These fits allow estimates of $\gamma$ and the hopping times $\tau$ to be obtained directly from the |
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mean square displacements. |
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\begin{figure}[htbp] |
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\label{fig:tauhop} |
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\end{figure} |
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|
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In Fig. \ref{fig:tauhop}, we show the hopping times for both |
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models. As expected, the hopping times diverge as the temperature is |
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In Fig. \ref{fig:tauhop}, the hopping times for both |
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models are shown. As expected, the hopping times diverge as the temperature is |
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lowered closer to the glass transition. There is relatively good |
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agreement between the hopping times calculated via {\sc ctrw} approach |
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with those calculated via the cage correlation function, although the |
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error is as much as a factor of 3 discrepancy at the lowest |
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temperatures. |
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|
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\section{DISCUSSION} |
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\section{Discussion} |
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\label{metglass:sec:discuss} |
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|
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It is relatively clear from using the cage correlation function to |
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cannot be fit with a sticking probability that is commensurate with |
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Eq. (\ref{eq:ctrw}). |
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|
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Instead, we have been able to fit the long-time decay of the cage |
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correlation function with the more familiar Kohlrausch-Williams-Watts |
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Instead, the long-time decay of the cage |
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correlation function has been fit with the more familiar Kohlrausch-Williams-Watts |
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law (Eq. (\ref{eq:kww2})), which appears to be a more accurate model |
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for the sticking probability. This point has been addressed by Ngai |
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and Liu.\cite{Ngai81} It is of some interest that there is a |
