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Revision 3610 by gezelter, Wed Jul 14 15:15:46 2010 UTC vs.
Revision 3611 by gezelter, Wed Jul 14 15:20:31 2010 UTC

# Line 431 | Line 431 | data collection under RNEMD.
431   without any momentum flux, $10^5$ steps of stablization with an
432   imposed momentum transfer to create a gradient, and $10^6$ steps of
433   data collection under RNEMD.
434 +
435 + \subsubsection*{Thermal Conductivity}
436  
437   Our thermal conductivity calculations show that the NIVS method agrees
438   well with the swapping method. Four different swap intervals were
# Line 522 | Line 524 | velocity distributions in the two slabs.
524    (curves fit for each distribution).}
525   \label{thermalHist}
526   \end{figure}
527 +
528 +
529 + \subsubsection*{Shear Viscosity}
530 + Our calculations (Table \ref{shearRate}) show that velocity-scaling
531 + RNEMD predicted comparable shear viscosities to swap RNEMD method.  All
532 + the scale method results were from simulations that had a scaling
533 + interval of 10 time steps. The average molecular momentum gradients of
534 + these samples are shown in Figure \ref{shear} (a) and (b).
535 +
536 + \begin{table*}
537 +  \begin{minipage}{\linewidth}
538 +    \begin{center}
539 +      
540 +      \caption{Shear viscosities of Lennard-Jones fluid at ${T^* =
541 +          0.72}$ and ${\rho^* = 0.85}$ using swapping and NIVS methods
542 +        at various momentum exchange rates.  Uncertainties are
543 +        indicated in parentheses. }
544 +      
545 +      \begin{tabular}{ccccc}
546 +        Swapping method & & & NIVS-RNEMD & \\
547 +        \hline
548 +        Swap Interval (fs) & $\eta^*_{swap}$ & & Equilvalent $j_p^*(v_x)$ &
549 +        $\eta^*_{scale}$\\
550 +        \hline
551 +        500  & 3.64(0.05) & & 0.09  & 3.76(0.09)\\
552 +        1000 & 3.52(0.16) & & 0.046 & 3.66(0.06)\\
553 +        2000 & 3.72(0.05) & & 0.024 & 3.32(0.18)\\
554 +        2500 & 3.42(0.06) & & 0.019 & 3.43(0.08)\\
555 +        \hline
556 +      \end{tabular}
557 +      \label{shearRate}
558 +    \end{center}
559 +  \end{minipage}
560 + \end{table*}
561 +
562 + \begin{figure}
563 +  \includegraphics[width=\linewidth]{shear}
564 +  \caption{Average momentum gradients in shear viscosity simulations,
565 +    using (a) ``swapping'' method and (b) NIVS-RNEMD method
566 +    respectively. (c) Temperature difference among x and y, z dimensions
567 +    observed when using NIVS-RNEMD with equivalent exchange interval of
568 +    500 fs.}
569 +  \label{shear}
570 + \end{figure}
571 +
572 + However, observations of temperatures along three dimensions show that
573 + inhomogeneity occurs in scaling RNEMD simulations, particularly in the
574 + two slabs which were scaled. Figure \ref{shear} (c) indicate that with
575 + relatively large imposed momentum flux, the temperature difference among $x$
576 + and the other two dimensions was significant. This would result from the
577 + algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after
578 + momentum gradient is set up, $P_c^x$ would be roughly stable
579 + ($<0$). Consequently, scaling factor $x$ would most probably larger
580 + than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would
581 + keep increase after most scaling steps. And if there is not enough time
582 + for the kinetic energy to exchange among different dimensions and
583 + different slabs, the system would finally build up temperature
584 + (kinetic energy) difference among the three dimensions. Also, between
585 + $y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis
586 + are closer to neighbor slabs. This is due to momentum transfer along
587 + $z$ dimension between slabs.
588 +
589 + Although results between scaling and swapping methods are comparable,
590 + the inherent temperature inhomogeneity even in relatively low imposed
591 + exchange momentum flux simulations makes scaling RNEMD method less
592 + attractive than swapping RNEMD in shear viscosity calculation.
593 +
594  
595   \subsection{Bulk SPC/E water}
596  
# Line 775 | Line 844 | $J_z$ (MW/m$^2$) & $T_{gold}$ (K) & $T_{water}$ (K) &
844   \end{minipage}
845   \end{table*}
846  
778 – \subsection{Shear Viscosity}
779 – Our calculations (Table \ref{shearRate}) shows that scale RNEMD method
780 – produced comparable shear viscosity to swap RNEMD method. In Table
781 – \ref{shearRate}, the names of the calculated samples are devided into
782 – two parts. The first number refers to total slabs in one simulation
783 – box. The second number refers to the swapping interval in swap method, or
784 – in scale method the equilvalent swapping interval that the same
785 – momentum flux would theoretically result in swap method. All the scale
786 – method results were from simulations that had a scaling interval of 10
787 – time steps. The average molecular momentum gradients of these samples
788 – are shown in Figure \ref{shear} (a) and (b).
847  
790 – \begin{table*}
791 – \begin{minipage}{\linewidth}
792 – \begin{center}
793 –
794 – \caption{Calculation results for shear viscosity of Lennard-Jones
795 –  fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale
796 –  methods at various momentum exchange rates. Results in reduced
797 –  unit. Errors of calculations in parentheses. }
798 –
799 – \begin{tabular}{ccccc}
800 – Swapping method & & & NIVS-RNEMD & \\
801 – \hline
802 – Swap Interval (fs) & $\eta^*_{swap}$ & & Equilvalent $j_p^*(v_x)$ &
803 – $\eta^*_{scale}$\\
804 – \hline
805 – 500  & 3.64(0.05) & & 0.09  & 3.76(0.09)\\
806 – 1000 & 3.52(0.16) & & 0.046 & 3.66(0.06)\\
807 – 2000 & 3.72(0.05) & & 0.024 & 3.32(0.18)\\
808 – 2500 & 3.42(0.06) & & 0.019 & 3.43(0.08)\\
809 – \hline
810 – \end{tabular}
811 – \label{shearRate}
812 – \end{center}
813 – \end{minipage}
814 – \end{table*}
815 –
816 – \begin{figure}
817 – \includegraphics[width=\linewidth]{shear}
818 – \caption{Average momentum gradients in shear viscosity simulations,
819 –  using (a) ``swapping'' method and (b) NIVS-RNEMD method
820 –  respectively. (c) Temperature difference among x and y, z dimensions
821 –  observed when using NIVS-RNEMD with equivalent exchange interval of
822 –  500 fs.}
823 – \label{shear}
824 – \end{figure}
825 –
826 – However, observations of temperatures along three dimensions show that
827 – inhomogeneity occurs in scaling RNEMD simulations, particularly in the
828 – two slabs which were scaled. Figure \ref{shear} (c) indicate that with
829 – relatively large imposed momentum flux, the temperature difference among $x$
830 – and the other two dimensions was significant. This would result from the
831 – algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after
832 – momentum gradient is set up, $P_c^x$ would be roughly stable
833 – ($<0$). Consequently, scaling factor $x$ would most probably larger
834 – than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would
835 – keep increase after most scaling steps. And if there is not enough time
836 – for the kinetic energy to exchange among different dimensions and
837 – different slabs, the system would finally build up temperature
838 – (kinetic energy) difference among the three dimensions. Also, between
839 – $y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis
840 – are closer to neighbor slabs. This is due to momentum transfer along
841 – $z$ dimension between slabs.
842 –
843 – Although results between scaling and swapping methods are comparable,
844 – the inherent temperature inhomogeneity even in relatively low imposed
845 – exchange momentum flux simulations makes scaling RNEMD method less
846 – attractive than swapping RNEMD in shear viscosity calculation.
847 –
848   \section{Conclusions}
849   NIVS-RNEMD simulation method is developed and tested on various
850   systems. Simulation results demonstrate its validity in thermal

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