--- stokes/stokes.tex	2011/12/13 23:31:35	3785
+++ stokes/stokes.tex	2011/12/14 21:59:12	3786
@@ -28,8 +28,8 @@
 
 \begin{document}
 
-\title{A minimal perturbation approach to RNEMD able to simultaneously
-impose thermal and momentum gradients}
+\title{Velocity Shearing and Scaling RNEMD: a minimally perturbing
+  method for producing temperature and momentum gradients}
 
 \author{Shenyu Kuang and J. Daniel
 Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
@@ -46,18 +46,19 @@ Notre Dame, Indiana 46556}
 \begin{abstract}
   We present a new method for introducing stable nonequilibrium
   velocity and temperature gradients in molecular dynamics simulations
-  of heterogeneous systems. This method conserves the linear momentum
-  and total energy of the system and improves previous Reverse
-  Non-Equilibrium Molecular Dynamics (RNEMD) methods while maintaining
-  thermal velocity distributions. It also avoid thermal anisotropy
-  occured in previous NIVS simulations by using isotropic velocity
-  scaling on the molecules in specific regions of a system. To test
-  the method, we have computed the thermal conductivity and shear
+  of heterogeneous systems. This method conserves both the linear
+  momentum and total energy of the system and improves previous
+  reverse non-equilibrium molecular dynamics (RNEMD) methods while
+  retaining equilibrium thermal velocity distributions in each region
+  of the system.  The new method avoids the thermal anisotropy
+  produced by previous methods by using isotropic velocity scaling and
+  shearing on all of the molecules in specific regions. To test the
+  method, we have computed the thermal conductivity and shear
   viscosity of model liquid systems as well as the interfacial
-  frictions of a series of metal/liquid interfaces. Its ability to
-  combine the thermal and momentum gradients allows us to obtain shear
-  viscosity data for a range of temperatures in only one trajectory.
-
+  friction coeefficients of a series of solid / liquid interfaces. The
+  method's ability to combine the thermal and momentum gradients
+  allows us to obtain shear viscosity data for a range of temperatures
+  from a single trajectory.
 \end{abstract}
 
 \newpage
@@ -69,96 +70,129 @@ Imposed-flux methods in Molecular Dynamics (MD)
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Introduction}
-Imposed-flux methods in Molecular Dynamics (MD)
-simulations\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101}
-can establish steady state systems with an applied flux set vs a
-corresponding gradient that can be measured. These methods does not
-need many trajectories to provide information of transport properties
-of a given system. Thus, they are utilized in computing thermal and
-mechanical transfer of homogeneous bulk systems as well as
-heterogeneous systems such as solid-liquid
+One of the standard ways to compute transport coefficients such as the
+viscosities and thermal conductivities of liquids is to use 
+imposed-flux non-equilibrium molecular dynamics
+methods.\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101}
+These methods establish stable non-equilibrium conditions in a
+simulation box using an applied momentum or thermal flux between
+different regions of the box.  The corresponding temperature or
+velocity gradients which develop in response to the applied flux is
+related (via linear response theory) to the transport coefficient of
+interest.  These methods are quite efficient, in that they do not need
+many trajectories to provide information about transport properties. To
+date, they have been utilized in computing thermal and mechanical
+transfer of both homogeneous liquids as well as heterogeneous systems
+such as solid-liquid
 interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl}
 
-The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that
-satisfy linear momentum and total energy conservation of a system when
-imposing fluxes in a simulation. Thus they are compatible with various
-ensembles, including the micro-canonical (NVE) ensemble, without the
-need of an external thermostat. The original approaches proposed by
-M\"{u}ller-Plathe {\it et
+The reverse non-equilibrium molecular dynamics (RNEMD) methods utilize
+additional constraints that ensure conservation of linear momentum and
+total energy of the system while imposing the desired flux.  The RNEMD
+methods are therefore capable of sampling various thermodynamically
+relevent ensembles, including the micro-canonical (NVE) ensemble,
+without resorting to an external thermostat. The original approaches
+proposed by M\"{u}ller-Plathe {\it et
   al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple
-momentum swapping for generating energy/momentum fluxes, which can
-also be compatible with particles of different identities. Although
-simple to implement in a simulation, this approach can create
-nonthermal velocity distributions, as discovered by Tenney and
-Maginn\cite{Maginn:2010}. Furthermore, this approach is less efficient
-for kinetic energy transfer between particles of different identities,
-especially when the mass difference between the particles becomes
-significant. This also limits its applications on heterogeneous
-interfacial systems.
+momentum swapping moves for generating energy/momentum fluxes.  The
+swapping moves can also be made compatible with particles of different
+identities. Although the swapping moves are simple to implement in
+molecular simulations, Tenney and Maginn have shown that they create
+nonthermal velocity distributions.\cite{Maginn:2010} Furthermore, this
+approach is not particularly efficient for kinetic energy transfer
+between particles of different identities, particularly when the mass
+difference between the particles becomes significant.  This problem
+makes applying swapping-move RNEMD methods on heterogeneous
+interfacial systems somewhat difficult.
 
-Recently, we developed a different approach, using Non-Isotropic
-Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose
-fluxes. Compared to the momentum swapping move, it scales the velocity
-vectors in two separate regions of a simulated system with respective
-diagonal scaling matrices. These matrices are determined by solving a
-set of equations including linear momentum and kinetic energy
-conservation constraints and target flux satisfaction. This method is
-able to effectively impose a wide range of kinetic energy fluxes
-without obvious perturbation to the velocity distributions of the
-simulated systems, regardless of the presence of heterogeneous
-interfaces. We have successfully applied this approach in studying the
-interfacial thermal conductance at metal-solvent
+Recently, we developed a somewhat different approach to applying
+thermal fluxes in RNEMD simulation using a Non-Isotropic Velocity
+Scaling (NIVS) algorithm.\cite{kuang:164101} This algorithm scales all
+atomic velocity vectors in two separate regions of a simulated system
+using two diagonal scaling matrices.  The scaling matrices are
+determined by solving single quartic equation which includes linear
+momentum and kinetic energy conservation constraints as well as the
+target thermal flux between the regions.  The NIVS method is able to
+effectively impose a wide range of kinetic energy fluxes without
+significant perturbation to the velocity distributions away from ideal
+Maxwell-Boltzmann distributions, even in the presence of heterogeneous
+interfaces. We successfully applied this approach in studying the
+interfacial thermal conductance at chemically-capped metal-solvent
 interfaces.\cite{kuang:AuThl}
 
-However, the NIVS approach has limited applications in imposing
-momentum fluxes. Temperature anisotropy could happen under high
-momentum fluxes due to the implementation of this algorithm. Thus,
-combining thermal and momentum flux is also difficult to obtain with
-this approach. However, such combination may provide a means to
-simulate thermal/momentum gradient coupled processes such as Soret
-effect in liquid flows. Therefore, developing improved approaches to
-extend the applications of the imposed-flux method is desirable.
+The NIVS approach works very well for preparing stable {\it thermal}
+gradients.  However, as we pointed out in the original paper, it has
+limited application in imposing {\it linear} momentum fluxes (which
+are required for measuring shear viscosities). The reason for this is
+that linear momentum flux was being imposed by scaling random
+fluctuations of the center of the velocity distributions.  Repeated
+application of the original NIVS approach resulted in temperature
+anisotropy, i.e. the width of the velocity distributions depended on
+coordinates perpendicular to the desired gradient direction.  For this
+reason, combining thermal and momentum fluxes was difficult with the
+original NIVS algorithm.  However, combinations of thermal and
+velocity gradients would provide a means to simulate thermal-linear
+coupled processes such as Soret effect in liquid flows. Therefore,
+developing improved approaches to the scaling imposed-flux methods
+would be useful.
 
-In this paper, we improve the RNEMD methods by proposing a novel
-approach to impose fluxes. This approach separate the means of applying
-momentum and thermal flux with operations in one time step and thus is
-able to simutaneously impose thermal and momentum flux. Furthermore,
-the approach retains desirable features of previous RNEMD approaches
-and is simpler to implement compared to the NIVS method. In what
-follows, we first present the method and its implementation in a
-simulation. Then we compare the method on bulk fluids to previous
-methods. Also, interfacial frictions are computed for a series of
-interfaces.
+In this paper, we improve the RNEMD methods by introducing a novel
+approach to creating imposed fluxes. This approach separates the
+shearing and scaling of the velocity distributions in different
+spatial regions and can apply both transformations within a single
+time step.  The approach retains desirable features of previous RNEMD
+approaches and is simpler to implement compared to the earlier NIVS
+method. In what follows, we first present the Shearing-and-Scaling
+(SS) RNEMD method and its implementation in a simulation. Then we
+compare the SS-RNEMD method in bulk fluids to previous methods.  We
+also compute interfacial frictions are computed for a series of
+heterogeneous interfaces.
 
 \section{Methodology}
-Similar to the NIVS method,\cite{kuang:164101} we consider a
-periodic system divided into a series of slabs along a certain axis
-(e.g. $z$). The unphysical thermal and/or momentum flux is designated
-from the center slab to one of the end slabs, and thus the thermal
-flux results in a lower temperature of the center slab than the end
-slab, and the momentum flux results in negative center slab momentum
-with positive end slab momentum (unless these fluxes are set
-negative). Therefore, the center slab is denoted as ``$c$'', while the
-end slab as ``$h$''.
+In an approach similar to the earlier NIVS method,\cite{kuang:164101}
+we consider a periodic system which has been divided into a series of
+slabs along a single axis (e.g. $z$). The unphysical thermal and
+momentum fluxes are applied from one of the end slabs to the center
+slab, and thus the thermal flux produces a higher temperature in the
+center slab than in the end slab, and the momentum flux results in a
+center slab moving along the positive $y$ axis (see Fig. \ref{cartoon}).
+The applied fluxes can be set to negative values if the reverse
+gradients are desired.  For convenience the center slab is denoted as
+the {\it hot} or {\it ``H''} slab, while the end slab is denoted {\it
+  ``C''} (or {\it cold}).
 
-To impose these fluxes, we periodically apply different set of
-operations on velocities of particles {$i$} within the center slab and
-those of particles {$j$} within the end slab:
+\begin{figure}
+\includegraphics[width=\linewidth]{cartoon}
+\caption{The SS-RNEMD approach we are introducing imposes unphysical
+  transfer of both momentum and kinetic energy between a ``hot'' slab
+  and a ``cold'' slab in the simulation box.  The molecular system
+  responds to this imposed flux by generating both momentum and
+  temperature gradients.  The slope of the gradients can then be used
+  to compute transport properties (e.g. shear viscosity and thermal
+  conductivity).}
+\label{cartoon}
+\end{figure}
+
+To impose these fluxes, we periodically apply a set of operations on
+the velocities of particles {$i$} within the cold slab and a separate
+operation on particles {$j$} within the hot slab.
 \begin{eqnarray}
 \vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c
   \rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\
 \vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h
   \rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right)
 \end{eqnarray}
-where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes
-the instantaneous bulk velocity of slabs $c$ and $h$ respectively
-before an operation is applied. When a momentum flux $\vec{j}_z(\vec{p})$
-presents, these bulk velocities would have a corresponding change
-($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's
-second law:
+where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denote
+the instantaneous average velocity of the molecules within slabs $C$
+and $H$ respectively.  When a momentum flux $\vec{j}_z(\vec{p})$ is
+present, these slab-averaged velocities also get corresponding
+incremental changes ($\vec{a}_c$ and $\vec{a}_h$ respectively) that
+are applied to all particles within each slab.  The incremental
+changes are obtained using Newton's second law:
 \begin{eqnarray}
-M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\
+M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \label{eq:newton1} \\
 M_h \vec{a}_h & = &  \vec{j}_z(\vec{p}) \Delta t
+\label{eq:newton2}
 \end{eqnarray}
 where $M$ denotes total mass of particles within a slab:
 \begin{eqnarray}
@@ -167,138 +201,142 @@ The above operations already conserve the linear momen
 \end{eqnarray}
 and $\Delta t$ is the interval between two separate operations.
 
-The above operations already conserve the linear momentum of a
-periodic system. To further satisfy total energy conservation as well
-as to impose the thermal flux $J_z$, the following equations are
-included as well:
-[MAY PUT EXTRA MATH IN SUPPORT INFO OR APPENDIX]
+The operations in Eqs. \ref{eq:newton1} and \ref{eq:newton2} already
+conserve the linear momentum of a periodic system.  To further satisfy
+total energy conservation as well as to impose the thermal flux $J_z$,
+the following constraint equations must be solved for the two scaling
+variables $c$ and $h$:
 \begin{eqnarray}
 K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c
 \rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\
 K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h
-\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2
+\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2.
+\label{constraint}
 \end{eqnarray}
-where $K_c$ and $K_h$ denotes translational kinetic energy of slabs
-$c$ and $h$ respectively before an operation is applied. These
-translational kinetic energy conservation equations are sufficient to
-ensure total energy conservation, as the operations applied in our
-method do not change the kinetic energy related to other degrees of
-freedom or the potential energy of a system, given that its potential
-energy does not depend on particle velocity.
+Here $K_c$ and $K_h$ denote the translational kinetic energy of slabs
+$C$ and $H$ respectively. These conservation equations are sufficient
+to ensure total energy conservation, as the operations applied in our
+method do not change the kinetic energy related to orientational
+degrees of freedom or the potential energy of a system (as long as the
+potential energy is independent of particle velocity).
 
-The above sets of equations are sufficient to determine the velocity
-scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and
-$\vec{a}_h$. Note that there are two roots respectively for $c$ and
-$h$. However, the positive roots (which are closer to 1) are chosen so
-that the perturbations to a system can be reduced to a minimum. Figure
-\ref{method} illustrates the implementation sketch of this algorithm
-in an individual step.
+Equations \ref{eq:newton1}-\ref{constraint} are sufficient to
+determine the velocity scaling coefficients ($c$ and $h$) as well as
+$\vec{a}_c$ and $\vec{a}_h$. Note that there are usually two roots
+respectively for $c$ and $h$. However, the positive roots (which are
+closer to 1) are chosen so that the perturbations to the system are
+minimal.  Figure \ref{method} illustrates the implementation of this
+algorithm in an individual step.
 
 \begin{figure}
-\includegraphics[width=\linewidth]{method}
-\caption{Illustration of the implementation of the algorithm in a
-  single step. Starting from an ideal velocity distribution, the
-  transformation is used to apply the effect of both a thermal and a
-  momentum flux from the ``c'' slab to the ``h'' slab. As the figure
-  shows, thermal distributions can preserve after this operation.}
+\centering
+\includegraphics[width=5in]{method}
+\caption{Illustration of a single step implementation of the
+  algorithm.  Starting with the velocity distributions for the two
+  slabs in a shearing fluid, the transformation is used to apply the
+  effect of both a thermal and a momentum flux from the ``c'' slab to
+  the ``h'' slab. As the figure shows, gaussian distributions are
+  preserved by both the scaling and shearing operations.}
 \label{method}
 \end{figure}
 
-By implementing these operations at a certain frequency, a steady
-thermal and/or momentum flux can be applied and the corresponding
-temperature and/or momentum gradients can be established.
+By implementing these operations at a fixed frequency, stable thermal
+and momentum fluxes can both be applied and the corresponding
+temperature and momentum gradients can be established.
 
-Compared to the previous NIVS method, this approach is computationally
-more efficient in that only quadratic equations are involved to
-determine a set of scaling coefficients, while the NIVS method needs
-to solve quartic equations. Furthermore, this method implements
-isotropic scaling of velocities in respective slabs, unlike the NIVS,
-where an extra criteria function is necessary to choose a set of
-coefficients that performs a scaling as isotropic as possible. More
-importantly, separating the means of momentum flux imposing from
-velocity scaling avoids the underlying cause to thermal anisotropy in
-NIVS when applying a momentum flux. And later sections will
-demonstrate that this can improve the performance in shear viscosity
-simulations.
+Compared to the previous NIVS method, the SS-RNEMD approach is
+computationally simpler in that only quadratic equations are involved
+to determine a set of scaling coefficients, while the NIVS method
+required solution of quartic equations. Furthermore, this method
+implements {\it isotropic} scaling of velocities in respective slabs,
+unlike NIVS, which required extra flexibility in the choice of scaling
+coefficients to allow for the energy and momentum constraints.  Most
+importantly, separating the linear momentum flux from velocity scaling
+avoids the underlying cause of the thermal anisotropy in NIVS.  In
+later sections, we demonstrate that this can improve the calculated
+shear viscosities from RNEMD simulations.
 
-This approach is advantageous over the original momentum swapping in
-many aspects. In one swapping, the velocity vectors involved are
-usually very different (or the generated flux is trivial to obtain
-gradients), thus the swapping tends to incur perturbations to the
-neighbors of the particles involved. Comparatively, our approach
-disperse the flux to every selected particle in a slab so that
-perturbations in the flux generating region could be
-minimized. Additionally, because the momentum swapping steps tend to
-result in a nonthermal distribution, when an imposed flux is
-relatively large and diffusions from the neighboring slabs could no
-longer remedy this effect, problematic distributions would be
-observed. In comparison, the operations of our approach has the nature
-of preserving the equilibrium velocity distributions (commonly
-Maxwell-Boltzmann), and results in later section will illustrate that
-this is helpful to retain thermal distributions in a simulation.
+The SS-RNEMD approach has advantages over the original momentum
+swapping in many respects. In the swapping method, the velocity
+vectors involved are usually very different (or the generated flux
+would be quite small), thus the swapping tends to create strong
+perturbations in the neighborhood of the particles involved. In
+comparison, the SS approach distributes the flux widely to all
+particles in a slab so that perturbations in the flux generating
+region are minimized. Additionally, momentum swapping steps tend to
+result in nonthermal velocity distributions when the imposed flux is
+large and diffusion from the neighboring slabs cannot carry momentum
+away quickly enough.  In comparison, the scaling and shearing moves
+made in the SS approach preserve the shapes of the equilibrium
+velocity disributions (e.g. Maxwell-Boltzmann).  The results presented
+in later sections will illustrate that this is quite helpful in
+retaining reasonable thermal distributions in a simulation.
 
 \section{Computational Details}
 The algorithm has been implemented in our MD simulation code,
-OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with
-previous RNEMD methods or equilibrium MD (EMD) methods in homogeneous fluids
-(Lennard-Jones and SPC/E water). And taking advantage of the method,
-we simulate the interfacial friction of different heterogeneous
-interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid
-water).
+OpenMD.\cite{Meineke:2005gd,openmd} We will first compare the method
+with previous RNEMD methods and equilibrium MD in homogeneous
+fluids (Lennard-Jones and SPC/E water).  We have also used the new
+method to simulate the interfacial friction of different heterogeneous
+interfaces (Au (111) with organic solvents, Au(111) with SPC/E water,
+and the Ice Ih - liquid water interface).
 
 \subsection{Simulation Protocols}
-The systems to be investigated are set up in orthorhombic simulation
+The systems we investigated were set up in orthorhombic simulation
 cells with periodic boundary conditions in all three dimensions. The
-$z$ axis of these cells were longer and set as the temperature and/or
-momentum gradient axis. And the cells were evenly divided into $N$
-slabs along this axis, with various $N$ depending on individual
-system. The $x$ and $y$ axis were of the same length in homogeneous
-systems or had length scale close to each other where heterogeneous
-interfaces presents. In all cases, before introducing a nonequilibrium
-method to establish steady thermal and/or momentum gradients for
-further measurements and calculations, canonical ensemble with a
-Nos\'e-Hoover thermostat\cite{hoover85} and microcanonical ensemble
-equilibrations were used before data collections. For SPC/E water
-simulations, isobaric-isothermal equilibrations\cite{melchionna93} are
-performed before the above to reach normal pressure (1 bar); for
-interfacial systems, similar equilibrations are used to relax the
-surface tensions of the $xy$ plane.
+$z$-axes of these cells were typically quite long and served as the
+temperature and/or momentum gradient axes.  The cells were evenly
+divided into $N$ slabs along this axis, with $N$ varying for the
+individual system. The $x$ and $y$ axes were of similar lengths in all
+simulations. In all cases, before introducing a nonequilibrium method
+to establish steady thermal and/or momentum gradients, equilibration
+simulations were run under the canonical ensemble with a Nos\'e-Hoover
+thermostat\cite{hoover85} followed by further equilibration using
+standard constant energy (NVE) conditions.  For SPC/E water
+simulations, isobaric-isothermal equilibrations\cite{melchionna93}
+were performed before equilibration to reach standard densities at
+atmospheric pressure (1 bar); for interfacial systems, similar
+equilibrations with anisotropic box relaxations are used to relax the
+surface tension in the $xy$ plane.
 
-While homogeneous fluid systems can be set up with rather random
-configurations, our interfacial systems needs a series of steps to
-ensure the interfaces be established properly for computations. The
-preparation and equilibration of butanethiol covered gold (111)
-surface and further solvation and equilibration process is described
-in details as in reference \cite{kuang:AuThl}.
+While homogeneous fluid systems can be set up with random
+configurations, interfacial systems are typically prepared with a
+single crystal face presented perpendicular to the $z$-axis. This
+crystal face is aligned in the x-y plane of the periodic cell, and the
+solvent occupies the region directly above and below a crystalline
+slab.  The preparation and equilibration of butanethiol covered
+Au(111) surfaces, as well as the solvation and equilibration processes
+used for these interfaces are described in detail in reference
+\cite{kuang:AuThl}.
 
-As for the ice/liquid water interfaces, the basal surface of ice
-lattice was first constructed. Hirsch {\it et
-  al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice
-lattices with all possible proton order configurations. We refer to
-their results and choose the configuration of the lowest energy after
-geometry optimization as the unit cell for our ice lattices. Although
+For the ice / liquid water interfaces, the basal surface of ice
+lattice was first constructed. Hirsch and Ojam\"{a}e
+\cite{doi:10.1021/jp048434u} explored the energetics of ice lattices
+with all possible proton ordered configurations. We utilized Hirsch
+and Ojam\"{a}e's structure 6 ($P2_12_12_1$) which is an orthorhombic
+cell giving a proton-ordered version of Ice Ih.  The basal face of ice
+in this unit cell geometry is the $\{0~0~1\}$ face.  Although
 experimental solid/liquid coexistant temperature under normal pressure
-should be close to 273K, Bryk and Haymet's simulations of ice/liquid
-water interfaces with different models suggest that for SPC/E, the
-most stable interface is observed at 225$\pm$5K.\cite{bryk:10258}
-Therefore, our ice/liquid water simulations were carried out at
-225K. To have extra protection of the ice lattice during initial
-equilibration (when the randomly generated liquid phase configuration
-could release large amount of energy in relaxation), restraints were
-applied to the ice lattice to avoid inadvertent melting by the heat
-dissipated from the high enery configurations.
-[MAY ADD A SNAPSHOT FOR BASAL PLANE]
+are close to 273K, Bryk and Haymet's simulations of ice/liquid water
+interfaces with different models suggest that for SPC/E, the most
+stable interface is observed at 225$\pm$5K.\cite{bryk:10258}
+Therefore, our ice/liquid water simulations were carried out at an
+average temperature of 225K.  Molecular translation and orientational
+restraints were applied in the early stages of equilibration to
+prevent melting of the ice slab.  These restraints were removed during
+NVT equilibration, well before data collection was carried out.
 
 \subsection{Force Field Parameters}
-For comparison of our new method with previous work, we retain our
+For comparing the SS-RNEMD method with previous work, we retained
 force field parameters consistent with previous simulations. Argon is
 the Lennard-Jones fluid used here, and its results are reported in
-reduced unit for direct comparison purpose.
+reduced units for purposes of direct comparison with previous
+calculations.
 
-As for our water simulations, SPC/E model is used throughout this work
-for consistency. Previous work for transport properties of SPC/E water
-model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so
-that unnecessary repetition of previous methods can be avoided.
+For our water simulations, we utilized the SPC/E model throughout this
+work.  Previous work for transport properties of SPC/E water model is
+available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so direct
+comparison with previous calculation methods is possible.
 
 The Au-Au interaction parameters in all simulations are described by
 the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The
@@ -310,36 +348,38 @@ butanethiol and liquid hexane and toluene. The United-
 
 For our gold/organic liquid interfaces, the small organic molecules
 included in our simulations are the Au surface capping agent
-butanethiol and liquid hexane and toluene. The United-Atom
+butanethiol as well as liquid hexane and liquid toluene. The
+United-Atom
 models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes}
-for these components were used in this work for better computational
+for these components were used in this work for computational
 efficiency, while maintaining good accuracy. We refer readers to our
 previous work\cite{kuang:AuThl} for further details of these models,
 as well as the interactions between Au and the above organic molecule
 components.
 
 \subsection{Thermal conductivities}
-When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to
-impose kinetic energy transfer, the method can be used for thermal
-conductivity computations. Similar to previous RNEMD methods, we
-assume linear response of the temperature gradient with respect to the
-thermal flux in general case. And the thermal conductivity ($\lambda$)
-can be obtained with the imposed kinetic energy flux and the measured
+When the linear momentum flux $\vec{j}_z(\vec{p})$ is set to zero and
+the target $J_z$ is non-zero, SS-RNEMD imposes kinetic energy transfer
+between the slabs, which can be used for computation of thermal
+conductivities. Similar to previous RNEMD methods, we assume that we
+are in the linear response regime of the temperature gradient with
+respect to the thermal flux.  The thermal conductivity ($\lambda$) can
+be calculated using the imposed kinetic energy flux and the measured
 thermal gradient:
 \begin{equation}
 J_z = -\lambda \frac{\partial T}{\partial z}
 \end{equation}
 Like other imposed-flux methods, the energy flux was calculated using
 the total non-physical energy transferred (${E_{total}}$) from slab
-``c'' to slab ``h'', which is recorded throughout a simulation, and
-the time for data collection $t$:
+``c'' to slab ``h'', which was recorded throughout the simulation, and
+the total data collection time $t$:
 \begin{equation}
-J_z = \frac{E_{total}}{2 t L_x L_y}
+J_z = \frac{E_{total}}{2 t L_x L_y}.
 \end{equation}
-where $L_x$ and $L_y$ denotes the dimensions of the plane in a
+Here, $L_x$ and $L_y$ denote the dimensions of the plane in a
 simulation cell perpendicular to the thermal gradient, and a factor of
-two in the denominator is necessary for the heat transport occurs in
-both $+z$ and $-z$ directions. The average temperature gradient
+two in the denominator is necessary as the heat transport occurs in
+both the $+z$ and $-z$ directions. The average temperature gradient
 ${\langle\partial T/\partial z\rangle}$ can be obtained by a linear
 regression of the temperature profile, which is recorded during a
 simulation for each slab in a cell. For Lennard-Jones simulations,
@@ -347,20 +387,18 @@ Alternatively, the method can carry out shear viscosit
 (${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$).
 
 \subsection{Shear viscosities}
-Alternatively, the method can carry out shear viscosity calculations
-by specify a momentum flux. In our algorithm, one can specify the
-three components of the flux vector $\vec{j}_z(\vec{p})$
-respectively. For shear viscosity simulations, $j_z(p_z)$ is usually
-set to zero. For isotropic systems, the direction of
-$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, but the
-ability of arbitarily specifying the vector direction in our method
-could provide convenience in anisotropic simulations.
+Alternatively, when the linear momentum flux $\vec{j}_z(\vec{p})$ is
+non-zero in either the $x$ or $y$ directions, the method can be used
+to compute the shear viscosity.  For isotropic systems, the direction
+of $\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, but the
+ability to arbitarily specify the vector direction in our method could
+provide convenience when working with anisotropic interfaces.
 
-Similar to thermal conductivity computations, for a homogeneous
-system, linear response of the momentum gradient with respect to the
-shear stress is assumed, and the shear viscosity ($\eta$) can be
-obtained with the imposed momentum flux (e.g. in $x$ direction) and
-the measured gradient:
+In a manner similar to the thermal conductivity calculations, a linear
+response of the momentum gradient with respect to the shear stress is
+assumed, and the shear viscosity ($\eta$) can be obtained with the
+imposed momentum flux (e.g. in $x$ direction) and the measured
+velocity gradient:
 \begin{equation}
 j_z(p_x) = -\eta \frac{\partial v_x}{\partial z}
 \end{equation}
@@ -376,74 +414,78 @@ Although $J_z$ may be switched off for shear viscosity
 viscosities are also reported in reduced units
 (${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$).
 
-Although $J_z$ may be switched off for shear viscosity simulations at
-a certain temperature, our method's ability to impose both a thermal
-and a momentum flux in one simulation allows the combination of a
-temperature and a velocity gradient. In this case, since viscosity is
-generally a function of temperature, the local viscosity also depends
-on the local temperature. Therefore, in one such simulation, viscosity
-at $z$ (corresponding to a certain $T$) can be computed with the
-applied shear flux and the local velocity gradient (which can be
-obtained by finite difference approximation). As a whole, the
-viscosity can be mapped out as the function of temperature in one
-single trajectory of simulation. Results for shear viscosity
-computations of SPC/E water will demonstrate its effectiveness in
-detail.
+Although $J_z$ may be switched off for shear viscosity simulations,
+the SS-RNEMD method allows the user the ability to simultaneously
+impose both a thermal and a momentum flux during a single
+simulation. This can create system with coincident temperature and a
+velocity gradients.  Since the viscosity is generally a function of
+temperature, the local viscosity depends on the local temperature in
+the fluid. Therefore, in a single simulation, viscosity at $z$
+(corresponding to a certain $T$) can be computed with the applied
+shear flux and the local velocity gradient (which can be obtained by
+finite difference approximation). This means that the temperature
+dependence of the viscosity can be mapped out in only one
+trajectory. Results for shear viscosity computations of SPC/E water
+will demonstrate SS-RNEMD's efficiency in this respect.
 
-\subsection{Interfacial friction and Slip length}
-While the shear stress results in a velocity gradient within bulk
-fluid phase, its effect at a solid-liquid interface could vary due to
-the interaction strength between the two phases. The interfacial
-friction coefficient $\kappa$ is defined to relate the shear stress
-(e.g. along $x$-axis) with the relative fluid velocity tangent to the
-interface:
+\subsection{Interfacial friction and slip length}
+Shear stress creates a velocity gradient within bulk fluid phases, but
+at solid-liquid interfaces, the effects of a shear stress depend on
+the molecular details of the interface. The interfacial friction
+coefficient, $\kappa$, relates the shear stress (e.g. along the
+$x$-axis) with the relative fluid velocity tangent to the interface:
 \begin{equation}
-j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface}
+j_z(p_x) = \kappa \left[v_x(fluid) - v_x(solid)\right]
 \end{equation}
-Under ``stick'' boundary condition, $\Delta v_x|_{interface}
-\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for
-``slip'' boundary conditions at the solid-liquid interfaces, $\kappa$
-becomes finite. To characterize the interfacial boundary conditions,
-slip length ($\delta$) is defined using $\kappa$ and the shear
-viscocity of liquid phase ($\eta$):
+where $v_x(fluid)$ and $v_x(solid)$ are the velocities measured
+directly adjacent to the interface.  Under ``stick'' boundary
+condition, $\Delta v_x|_\mathrm{interface} \rightarrow 0$, which leads
+to $\kappa\rightarrow\infty$. However, for ``slip'' boundary
+conditions at a solid-liquid interface, $\kappa$ becomes finite. To
+characterize the interfacial boundary conditions, the slip length,
+$\delta$, is defined by the ratio of the fluid-phase viscosity to the
+friction coefficient of the interface:
 \begin{equation}
 \delta = \frac{\eta}{\kappa}
 \end{equation}
-so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition,
-and depicts how ``slippery'' an interface is. Figure \ref{slipLength}
-illustrates how this quantity is defined and computed for a
-solid-liquid interface. [MAY INCLUDE SNAPSHOT IN FIGURE]
+In ``no-slip'' or ``stick'' boundary conditions, $\delta\rightarrow
+0$, and $\delta$ is a measure of how ``slippery'' an interface is.
+Figure \ref{slipLength} illustrates how this quantity is defined and
+computed for a solid-liquid interface. 
 
 \begin{figure}
 \includegraphics[width=\linewidth]{defDelta}
 \caption{The slip length $\delta$ can be obtained from a velocity
   profile of a solid-liquid interface simulation, when a momentum flux
-  is applied. An example of Au/hexane interfaces is shown, and the
-  calculation for the left side is illustrated. The calculation for
-  the right side is similar to the left.}
+  is applied. The data shown is for a simulated Au/hexane interface.
+  The Au crystalline region is moving as a block (lower dots), while
+  the measured velocity gradient in the hexane phase is discontinuous
+  a the interface.}
 \label{slipLength}
 \end{figure}
 
-In our method, a shear stress can be applied similar to shear
-viscosity computations by applying an unphysical momentum flux
-(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as
-shown in Figure \ref{slipLength}, in which the velocity gradients
-within liquid phase and velocity difference at the liquid-solid
-interface can be measured respectively. Further calculations and
-characterizations of the interface can be carried out using these
-data.
+Since the method can be applied for interfaces as well as for bulk
+materials, the shear stress is applied in an identical manner to the
+shear viscosity computations, e.g. by applying an unphysical momentum
+flux, $j_z(\vec{p})$.  With the correct choice of $\vec{p}$ in the
+$x-y$ plane, one can compute friction coefficients and slip lengths
+for a number of different dragging vectors on a given slab.  The
+corresponding velocity profiles can be obtained as shown in Figure
+\ref{slipLength}, in which the velocity gradients within the liquid
+phase and the velocity difference at the liquid-solid interface can be
+easily measured from saved simulation data.
 
 \section{Results and Discussions}
 \subsection{Lennard-Jones fluid}
-Our orthorhombic simulation cell of Lennard-Jones fluid has identical
-parameters to our previous work\cite{kuang:164101} to facilitate
-comparison. Thermal conductivitis and shear viscosities were computed
-with the algorithm applied to the simulations. The results of thermal
-conductivity are compared with our previous NIVS algorithm. However,
-since the NIVS algorithm could produce temperature anisotropy for
-shear viscocity computations, these results are instead compared to
-the momentum swapping approaches. Table \ref{LJ} lists these
-calculations with various fluxes in reduced units.
+Our orthorhombic simulation cell for the Lennard-Jones fluid has
+identical parameters to our previous work\cite{kuang:164101} to
+facilitate comparison. Thermal conductivities and shear viscosities
+were computed with the new algorithm applied to the simulations. The
+results of thermal conductivity are compared with our previous NIVS
+algorithm. However, since the NIVS algorithm produced temperature
+anisotropy for shear viscocity computations, these results are instead
+compared to the previous momentum swapping approaches. Table \ref{LJ}
+lists these values with various fluxes in reduced units.
 
 \begin{table*}
   \begin{minipage}{\linewidth}
@@ -462,7 +504,7 @@ calculations with various fluxes in reduced units.
         \multicolumn{2}{c}{$\lambda^*$} &
         \multicolumn{2}{c}{$\eta^*$} \\
         \hline
-        Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ &
+        Swap Interval & Equivalent $J_z^*$ or $j_z^*(p_x)$ &
         NIVS\cite{kuang:164101} & This work & Swapping & This work \\
         \hline
         0.116 & 0.16  & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\
@@ -498,7 +540,7 @@ applying a thermal flux). In this sense, this method a
 or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha
 P^\alpha$) would not achieve this effect unless thermal flux vanishes
 (i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which do not contribute to
-applying a thermal flux). In this sense, this method aids to achieve
+applying a thermal flux). In this sense, this method aids in achieving
 minimal perturbation to a simulation while imposing a thermal flux.
 
 \subsubsection{Shear viscosity}
