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\begin{document} |
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\begin{doublespace} |
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\begin{abstract} |
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|
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We present a new method for introducing stable non-equilibrium |
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velocity and temperature gradients in molecular dynamics simulations |
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of heterogeneous systems. This method extends earlier Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
50 |
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momentum exchange swapping moves. The standard swapping moves can |
51 |
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create non-thermal velocity distributions and are difficult to use |
52 |
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for interfacial calculations. By using non-isotropic velocity |
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scaling (NIVS) on the molecules in specific regions of a system, it |
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is possible to impose momentum or thermal flux between regions of a |
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simulation while conserving the linear momentum and total energy of |
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the system. To test the methods, we have computed the thermal |
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conductivity of model liquid and solid systems as well as the |
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interfacial thermal conductivity of a metal-water interface. We |
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find that the NIVS-RNEMD improves the problematic velocity |
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distributions that develop in other RNEMD methods. |
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\end{abstract} |
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\newpage |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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– |
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|
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\section{Introduction} |
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The original formulation of Reverse Non-equilibrium Molecular Dynamics |
73 |
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(RNEMD) obtains transport coefficients (thermal conductivity and shear |
74 |
|
viscosity) in a fluid by imposing an artificial momentum flux between |
75 |
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two thin parallel slabs of material that are spatially separated in |
76 |
|
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
77 |
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artificial flux is typically created by periodically ``swapping'' either |
78 |
< |
the entire momentum vector $\vec{p}$ or single components of this |
79 |
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vector ($p_x$) between molecules in each of the two slabs. If the two |
80 |
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slabs are separated along the z coordinate, the imposed flux is either |
81 |
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directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
82 |
< |
simulated system to the imposed momentum flux will typically be a |
83 |
< |
velocity or thermal gradient. The transport coefficients (shear |
84 |
< |
viscosity and thermal conductivity) are easily obtained by assuming |
85 |
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linear response of the system, |
77 |
> |
artificial flux is typically created by periodically ``swapping'' |
78 |
> |
either the entire momentum vector $\vec{p}$ or single components of |
79 |
> |
this vector ($p_x$) between molecules in each of the two slabs. If |
80 |
> |
the two slabs are separated along the $z$ coordinate, the imposed flux |
81 |
> |
is either directional ($j_z(p_x)$) or isotropic ($J_z$), and the |
82 |
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response of a simulated system to the imposed momentum flux will |
83 |
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typically be a velocity or thermal gradient (Fig. \ref{thermalDemo}). |
84 |
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The shear viscosity ($\eta$) and thermal conductivity ($\lambda$) are |
85 |
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easily obtained by assuming linear response of the system, |
86 |
|
\begin{eqnarray} |
87 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
88 |
< |
J & = & \lambda \frac{\partial T}{\partial z} |
88 |
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J_z & = & \lambda \frac{\partial T}{\partial z} |
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\end{eqnarray} |
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RNEMD has been widely used to provide computational estimates of thermal |
91 |
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conductivities and shear viscosities in a wide range of materials, |
92 |
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from liquid copper to monatomic liquids to molecular fluids |
93 |
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(e.g. ionic liquids).\cite{ISI:000246190100032} |
90 |
> |
RNEMD has been widely used to provide computational estimates of |
91 |
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thermal conductivities and shear viscosities in a wide range of |
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materials, from liquid copper to both monatomic and molecular fluids |
93 |
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(e.g. ionic |
94 |
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liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
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|
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RNEMD is preferable in many ways to the forward NEMD methods because |
97 |
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it imposes what is typically difficult to measure (a flux or stress) |
98 |
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and it is typically much easier to compute momentum gradients or |
99 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
100 |
< |
to slowly-converging equilibrium methods for measuring thermal |
101 |
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conductivity and shear viscosity (using Green-Kubo relations or the |
102 |
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Helfand moment approach of Viscardy {\it et |
96 |
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\begin{figure} |
97 |
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\includegraphics[width=\linewidth]{thermalDemo} |
98 |
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\caption{RNEMD methods impose an unphysical transfer of momentum or |
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kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
100 |
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simulation box. The molecular system responds to this imposed flux |
101 |
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by generating a momentum or temperature gradient. The slope of the |
102 |
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gradient can then be used to compute transport properties (e.g. |
103 |
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shear viscosity and thermal conductivity).} |
104 |
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\label{thermalDemo} |
105 |
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\end{figure} |
106 |
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|
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RNEMD is preferable in many ways to the forward NEMD |
108 |
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methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
109 |
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because it imposes what is typically difficult to measure (a flux or |
110 |
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stress) and it is typically much easier to compute the response |
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(momentum gradients or strains). For similar reasons, RNEMD is also |
112 |
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preferable to slowly-converging equilibrium methods for measuring |
113 |
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thermal conductivity and shear viscosity (using Green-Kubo |
114 |
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relations\cite{daivis:541,mondello:9327} or the Helfand moment |
115 |
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approach of Viscardy {\it et |
116 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
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computing difficult to measure quantities. |
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|
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typically samples from the same manifold of states in the |
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microcanonical ensemble. |
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|
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Recently, Tenney and Maginn have discovered some problems with the |
126 |
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original RNEMD swap technique. Notably, large momentum fluxes |
127 |
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(equivalent to frequent momentum swaps between the slabs) can result |
128 |
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in "notched", "peaked" and generally non-thermal momentum |
129 |
< |
distributions in the two slabs, as well as non-linear thermal and |
130 |
< |
velocity distributions along the direction of the imposed flux ($z$). |
131 |
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Tenney and Maginn obtained reasonable limits on imposed flux and |
132 |
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self-adjusting metrics for retaining the usability of the method. |
125 |
> |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered some |
126 |
> |
problems with the original RNEMD swap technique. Notably, large |
127 |
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momentum fluxes (equivalent to frequent momentum swaps between the |
128 |
> |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
129 |
> |
momentum distributions in the two slabs, as well as non-linear thermal |
130 |
> |
and velocity distributions along the direction of the imposed flux |
131 |
> |
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
132 |
> |
and proposed self-adjusting metrics for retaining the usability of the |
133 |
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method. |
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|
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In this paper, we develop and test a method for non-isotropic velocity |
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scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
136 |
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scaling (NIVS) which retains the desirable features of RNEMD |
137 |
|
(conservation of linear momentum and total energy, compatibility with |
138 |
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periodic boundary conditions) while establishing true thermal |
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distributions in each of the two slabs. In the next section, we |
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develop the method for determining the scaling constraints. We then |
141 |
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test the method on both single component, multi-component, and |
142 |
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non-isotropic mixtures and show that it is capable of providing |
139 |
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distributions in each of the two slabs. In the next section, we |
140 |
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present the method for determining the scaling constraints. We then |
141 |
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test the method on both liquids and solids as well as a non-isotropic |
142 |
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liquid-solid interface and show that it is capable of providing |
143 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
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in these cases. |
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in all of these cases. |
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|
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\section{Methodology} |
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We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
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system is partitioned into a series of thin slabs along a particular |
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We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
148 |
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periodic system is partitioned into a series of thin slabs along one |
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axis ($z$). One of the slabs at the end of the periodic box is |
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designated the ``hot'' slab, while the slab in the center of the box |
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is designated the ``cold'' slab. The artificial momentum flux will be |
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hot slab. |
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|
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Rather than using momentum swaps, we use a series of velocity scaling |
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moves. For molecules $\{i\}$ located within the cold slab, |
156 |
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moves. For molecules $\{i\}$ located within the cold slab, |
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\begin{equation} |
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\vec{v}_i \leftarrow \left( \begin{array}{c} |
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x \\ |
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y \\ |
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z \\ |
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\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
159 |
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x & 0 & 0 \\ |
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0 & y & 0 \\ |
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0 & 0 & z \\ |
162 |
|
\end{array} \right) \cdot \vec{v}_i |
163 |
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\end{equation} |
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where ${x, y, z}$ are a set of 3 scaling variables for each of the |
165 |
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three directions in the system. Likewise, the molecules $\{j\}$ |
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located in the hot slab will see a concomitant scaling of velocities, |
164 |
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where ${x, y, z}$ are a set of 3 velocity-scaling variables for each |
165 |
> |
of the three directions in the system. Likewise, the molecules |
166 |
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$\{j\}$ located in the hot slab will see a concomitant scaling of |
167 |
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velocities, |
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|
\begin{equation} |
169 |
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\vec{v}_j \leftarrow \left( \begin{array}{c} |
170 |
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x^\prime \\ |
171 |
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y^\prime \\ |
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z^\prime \\ |
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\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
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x^\prime & 0 & 0 \\ |
171 |
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0 & y^\prime & 0 \\ |
172 |
> |
0 & 0 & z^\prime \\ |
173 |
|
\end{array} \right) \cdot \vec{v}_j |
174 |
|
\end{equation} |
175 |
|
|
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|
Conservation of linear momentum in each of the three directions |
177 |
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($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
177 |
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($\alpha = x,y,z$) ties the values of the hot and cold scaling |
178 |
|
parameters together: |
179 |
|
\begin{equation} |
180 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
181 |
|
\end{equation} |
182 |
|
where |
183 |
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\begin{equation} |
184 |
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\begin{array}{rcl} |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha \\ |
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\end{array} |
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\begin{eqnarray} |
184 |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i v_{i\alpha} \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j v_{j\alpha} |
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|
\label{eq:momentumdef} |
187 |
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\end{equation} |
187 |
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\end{eqnarray} |
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Therefore, for each of the three directions, the hot scaling |
189 |
|
parameters are a simple function of the cold scaling parameters and |
190 |
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the instantaneous linear momentum in each of the two slabs. |
190 |
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the instantaneous linear momenta in each of the two slabs. |
191 |
|
\begin{equation} |
192 |
|
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
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\label{eq:hotcoldscaling} |
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|
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Conservation of total energy also places constraints on the scaling: |
202 |
|
\begin{equation} |
203 |
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\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
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\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha. |
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\sum_{\alpha = x,y,z} \left\{ K_h^\alpha + K_c^\alpha \right\} = \sum_{\alpha = x,y,z} |
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\left\{ \left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha \right\} |
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|
\end{equation} |
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where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
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for each of the three directions in a similar manner to the linear momenta |
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(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
209 |
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hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
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we obtain the {\it constraint ellipsoid equation}: |
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where the translational kinetic energies, $K_h^\alpha$ and |
207 |
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$K_c^\alpha$, are computed for each of the three directions in a |
208 |
> |
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
209 |
> |
Substituting in the expressions for the hot scaling parameters |
210 |
> |
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
211 |
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{\it constraint ellipsoid}: |
212 |
|
\begin{equation} |
213 |
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\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0, |
213 |
> |
\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
214 |
> |
c_\alpha \right) = 0 |
215 |
|
\label{eq:constraintEllipsoid} |
216 |
|
\end{equation} |
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|
where the constants are obtained from the instantaneous values of the |
218 |
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linear momenta and kinetic energies for the hot and cold slabs, |
219 |
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\begin{equation} |
187 |
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\begin{array}{rcl} |
219 |
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\begin{eqnarray} |
220 |
|
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
221 |
|
\left(p_\alpha\right)^2\right) \\ |
222 |
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
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c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha \\ |
192 |
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\end{array} |
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c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
224 |
|
\label{eq:constraintEllipsoidConsts} |
225 |
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\end{equation} |
226 |
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This ellipsoid equation defines the set of cold slab scaling |
227 |
< |
parameters which can be applied while preserving both linear momentum |
228 |
< |
in all three directions as well as kinetic energy. |
225 |
> |
\end{eqnarray} |
226 |
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This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
227 |
> |
cold slab scaling parameters which, when applied, preserve the linear |
228 |
> |
momentum of the system in all three directions as well as total |
229 |
> |
kinetic energy. |
230 |
|
|
231 |
< |
The goal of using velocity scaling variables is to transfer linear |
232 |
< |
momentum or kinetic energy from the cold slab to the hot slab. If the |
233 |
< |
hot and cold slabs are separated along the z-axis, the energy flux is |
234 |
< |
given simply by the decrease in kinetic energy of the cold bin: |
231 |
> |
The goal of using these velocity scaling variables is to transfer |
232 |
> |
kinetic energy from the cold slab to the hot slab. If the hot and |
233 |
> |
cold slabs are separated along the z-axis, the energy flux is given |
234 |
> |
simply by the decrease in kinetic energy of the cold bin: |
235 |
|
\begin{equation} |
236 |
|
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
237 |
|
\end{equation} |
238 |
|
The expression for the energy flux can be re-written as another |
239 |
|
ellipsoid centered on $(x,y,z) = 0$: |
240 |
|
\begin{equation} |
241 |
< |
x^2 K_c^x + y^2 K_c^y + z^2 K_c^z = K_c^x + K_c^y + K_c^z - J_z\Delta t |
241 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = \sum_{\alpha = x,y,z} |
242 |
> |
K_c^\alpha -J_z \Delta t |
243 |
|
\label{eq:fluxEllipsoid} |
244 |
|
\end{equation} |
245 |
< |
The spatial extent of the {\it flux ellipsoid equation} is governed |
246 |
< |
both by a targetted value, $J_z$ as well as the instantaneous values of the |
247 |
< |
kinetic energy components in the cold bin. |
245 |
> |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
246 |
> |
both by the target flux, $J_z$ as well as the instantaneous values of |
247 |
> |
the kinetic energy components in the cold bin. |
248 |
|
|
249 |
|
To satisfy an energetic flux as well as the conservation constraints, |
250 |
< |
it is sufficient to determine the points ${x,y,z}$ which lie on both |
251 |
< |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
252 |
< |
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
253 |
< |
the two ellipsoids in 3-dimensional space. |
250 |
> |
we must determine the points ${x,y,z}$ that lie on both the constraint |
251 |
> |
ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
252 |
> |
(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
253 |
> |
ellipsoids in 3-dimensional space. |
254 |
|
|
255 |
< |
One may also define momentum flux (say along the x-direction) as: |
256 |
< |
\begin{equation} |
257 |
< |
(1-x) P_c^x = j_z(p_x)\Delta t |
258 |
< |
\label{eq:fluxPlane} |
259 |
< |
\end{equation} |
260 |
< |
The above {\it flux equation} is essentially a plane which is |
261 |
< |
perpendicular to the x-axis, with its position governed both by a |
262 |
< |
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
263 |
< |
momentum along the x-direction. |
255 |
> |
\begin{figure} |
256 |
> |
\includegraphics[width=\linewidth]{ellipsoids} |
257 |
> |
\caption{Velocity scaling coefficients which maintain both constant |
258 |
> |
energy and constant linear momentum of the system lie on the surface |
259 |
> |
of the {\it constraint ellipsoid} while points which generate the |
260 |
> |
target momentum flux lie on the surface of the {\it flux ellipsoid}. |
261 |
> |
The velocity distributions in the cold bin are scaled by only those |
262 |
> |
points which lie on both ellipsoids.} |
263 |
> |
\label{ellipsoids} |
264 |
> |
\end{figure} |
265 |
|
|
266 |
< |
Similarly, to satisfy a momentum flux as well as the conservation |
267 |
< |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
268 |
< |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
269 |
< |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
270 |
< |
an ellipsoid and a plane in 3-dimensional space. |
266 |
> |
Since ellipsoids can be expressed as polynomials up to second order in |
267 |
> |
each of the three coordinates, finding the the intersection points of |
268 |
> |
two ellipsoids is isomorphic to finding the roots a polynomial of |
269 |
> |
degree 16. There are a number of polynomial root-finding methods in |
270 |
> |
the literature,\cite{Hoffman:2001sf,384119} but numerically finding |
271 |
> |
the roots of high-degree polynomials is generally an ill-conditioned |
272 |
> |
problem.\cite{Hoffman:2001sf} One simplification is to maintain |
273 |
> |
velocity scalings that are {\it as isotropic as possible}. To do |
274 |
> |
this, we impose $x=y$, and treat both the constraint and flux |
275 |
> |
ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
276 |
> |
intersecting-ellipse problem reduces to finding the roots of |
277 |
> |
polynomials of degree 4. |
278 |
|
|
279 |
< |
To summarize, by solving respective equation sets, one can determine |
280 |
< |
possible sets of scaling variables for cold slab. And corresponding |
281 |
< |
sets of scaling variables for hot slab can be determine as well. |
279 |
> |
Depending on the target flux and current velocity distributions, the |
280 |
> |
ellipsoids can have between 0 and 4 intersection points. If there are |
281 |
> |
no intersection points, it is not possible to satisfy the constraints |
282 |
> |
while performing a non-equilibrium scaling move, and no change is made |
283 |
> |
to the dynamics. |
284 |
|
|
285 |
< |
The following problem will be choosing an optimal set of scaling |
286 |
< |
variables among the possible sets. Although this method is inherently |
287 |
< |
non-isotropic, the goal is still to maintain the system as isotropic |
288 |
< |
as possible. Under this consideration, one would like the kinetic |
289 |
< |
energies in different directions could become as close as each other |
290 |
< |
after each scaling. Simultaneously, one would also like each scaling |
291 |
< |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
292 |
< |
large perturbation to the system. Therefore, one approach to obtain the |
293 |
< |
scaling variables would be constructing an criteria function, with |
294 |
< |
constraints as above equation sets, and solving the function's minimum |
295 |
< |
by method like Lagrange multipliers. |
285 |
> |
With multiple intersection points, any of the scaling points will |
286 |
> |
conserve the linear momentum and kinetic energy of the system and will |
287 |
> |
generate the correct target flux. Although this method is inherently |
288 |
> |
non-isotropic, the goal is still to maintain the system as close to an |
289 |
> |
isotropic fluid as possible. With this in mind, we would like the |
290 |
> |
kinetic energies in the three different directions could become as |
291 |
> |
close as each other as possible after each scaling. Simultaneously, |
292 |
> |
one would also like each scaling as gentle as possible, i.e. ${x,y,z |
293 |
> |
\rightarrow 1}$, in order to avoid large perturbation to the system. |
294 |
> |
To do this, we pick the intersection point which maintains the three |
295 |
> |
scaling variables ${x, y, z}$ as well as the ratio of kinetic energies |
296 |
> |
${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
297 |
|
|
298 |
< |
In order to save computation time, we have a different approach to a |
299 |
< |
relatively good set of scaling variables with much less calculation |
300 |
< |
than above. Here is the detail of our simplification of the problem. |
298 |
> |
After the valid scaling parameters are arrived at by solving geometric |
299 |
> |
intersection problems in $x, y, z$ space in order to obtain cold slab |
300 |
> |
scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
301 |
> |
determine the conjugate hot slab scaling variables. |
302 |
|
|
303 |
< |
In the case of kinetic energy transfer, we impose another constraint |
304 |
< |
${x = y}$, into the equation sets. Consequently, there are two |
305 |
< |
variables left. And now one only needs to solve a set of two {\it |
306 |
< |
ellipses equations}. This problem would be transformed into solving |
307 |
< |
one quartic equation for one of the two variables. There are known |
308 |
< |
generic methods that solve real roots of quartic equations. Then one |
309 |
< |
can determine the other variable and obtain sets of scaling |
310 |
< |
variables. Among these sets, one can apply the above criteria to |
311 |
< |
choose the best set, while much faster with only a few sets to choose. |
303 |
> |
\subsection{Introducing shear stress via velocity scaling} |
304 |
> |
It is also possible to use this method to magnify the random |
305 |
> |
fluctuations of the average momentum in each of the bins to induce a |
306 |
> |
momentum flux. Doing this repeatedly will create a shear stress on |
307 |
> |
the system which will respond with an easily-measured strain. The |
308 |
> |
momentum flux (say along the $x$-direction) may be defined as: |
309 |
> |
\begin{equation} |
310 |
> |
(1-x) P_c^x = j_z(p_x)\Delta t |
311 |
> |
\label{eq:fluxPlane} |
312 |
> |
\end{equation} |
313 |
> |
This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
314 |
> |
its position governed both by a target value, $j_z(p_x)$ as well as |
315 |
> |
the instantaneous value of the momentum along the $x$-direction. |
316 |
|
|
317 |
< |
In the case of momentum flux transfer, we impose another constraint to |
318 |
< |
set the kinetic energy transfer as zero. In another word, we apply |
319 |
< |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
320 |
< |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
321 |
< |
of equations on the above kinetic energy transfer problem. Therefore, |
273 |
< |
an approach similar to the above would be sufficient for this as well. |
317 |
> |
In order to satisfy a momentum flux as well as the conservation |
318 |
> |
constraints, we must determine the points ${x,y,z}$ which lie on both |
319 |
> |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
320 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
321 |
> |
ellipsoid and a plane in 3-dimensional space. |
322 |
|
|
323 |
+ |
In the case of momentum flux transfer, we also impose another |
324 |
+ |
constraint to set the kinetic energy transfer as zero. In other |
325 |
+ |
words, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
326 |
+ |
one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
327 |
+ |
set of quartic equations to the above kinetic energy transfer problem. |
328 |
+ |
|
329 |
|
\section{Computational Details} |
276 |
– |
Our simulation consists of a series of systems. All of these |
277 |
– |
simulations were run with the OOPSE simulation software |
278 |
– |
package\cite{Meineke:2005gd} integrated with RNEMD methods. |
330 |
|
|
331 |
< |
A Lennard-Jones fluid system was built and tested first. In order to |
332 |
< |
compare our method with swapping RNEMD, a series of simulations were |
333 |
< |
performed to calculate the shear viscosity and thermal conductivity of |
334 |
< |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
335 |
< |
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
336 |
< |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
337 |
< |
comparison between our results and others. These simulations used |
338 |
< |
Verlet time-stepping algorithm with reduced timestep ${\tau^* = |
339 |
< |
4.6\times10^{-4}}$. |
331 |
> |
We have implemented this methodology in our molecular dynamics code, |
332 |
> |
OpenMD,\cite{Meineke:2005gd,openmd} performing the NIVS scaling moves |
333 |
> |
with a variable frequency after the molecular dynamics (MD) steps. We |
334 |
> |
have tested the method in a variety of different systems, including: |
335 |
> |
homogeneous fluids (Lennard-Jones and SPC/E water), crystalline |
336 |
> |
solids, using both the embedded atom method |
337 |
> |
(EAM)~\cite{PhysRevB.33.7983} and quantum Sutton-Chen |
338 |
> |
(QSC)~\cite{PhysRevB.59.3527} models for Gold, and heterogeneous |
339 |
> |
interfaces (QSC gold - SPC/E water). The last of these systems would |
340 |
> |
have been difficult to study using previous RNEMD methods, but the |
341 |
> |
current method can easily provide estimates of the interfacial thermal |
342 |
> |
conductivity ($G$). |
343 |
|
|
344 |
< |
For shear viscosity calculation, the reduced temperature was ${T^* = |
291 |
< |
k_B T/\varepsilon = 0.72}$. Simulations were run in microcanonical |
292 |
< |
ensemble (NVE). For the swapping part, Muller-Plathe's algorithm was |
293 |
< |
adopted.\cite{ISI:000080382700030} The simulation box was under |
294 |
< |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
295 |
< |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
296 |
< |
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
297 |
< |
to Tenney {\it et al.}\cite{tenneyANDmaginn}, a series of swapping |
298 |
< |
frequency were chosen. According to each result from swapping |
299 |
< |
RNEMD, scaling RNEMD simulations were run with the target momentum |
300 |
< |
flux set to produce a similar momentum flux and shear |
301 |
< |
rate. Furthermore, various scaling frequencies can be tested for one |
302 |
< |
single swapping rate. To compare the performance between swapping and |
303 |
< |
scaling method, temperatures of different dimensions in all the slabs |
304 |
< |
were observed. Most of the simulations include $10^5$ steps of |
305 |
< |
equilibration without imposing momentum flux, $10^5$ steps of |
306 |
< |
stablization with imposing momentum transfer, and $10^6$ steps of data |
307 |
< |
collection under RNEMD. For relatively high momentum flux simulations, |
308 |
< |
${5\times10^5}$ step data collection is sufficient. For some low momentum |
309 |
< |
flux simulations, ${2\times10^6}$ steps were necessary. |
344 |
> |
\subsection{Simulation Cells} |
345 |
|
|
346 |
< |
After each simulation, the shear viscosity was calculated in reduced |
347 |
< |
unit. The momentum flux was calculated with total unphysical |
348 |
< |
transferred momentum ${P_x}$ and simulation time $t$: |
346 |
> |
In each of the systems studied, the dynamics was carried out in a |
347 |
> |
rectangular simulation cell using periodic boundary conditions in all |
348 |
> |
three dimensions. The cells were longer along the $z$ axis and the |
349 |
> |
space was divided into $N$ slabs along this axis (typically $N=20$). |
350 |
> |
The top slab ($n=1$) was designated the ``hot'' slab, while the |
351 |
> |
central slab ($n= N/2 + 1$) was designated the ``cold'' slab. In all |
352 |
> |
cases, simulations were first thermalized in canonical ensemble (NVT) |
353 |
> |
using a Nos\'{e}-Hoover thermostat.\cite{Hoover85} then equilibrated in |
354 |
> |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
355 |
> |
method. |
356 |
> |
|
357 |
> |
\subsection{RNEMD with M\"{u}ller-Plathe swaps} |
358 |
> |
|
359 |
> |
In order to compare our new methodology with the original |
360 |
> |
M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
361 |
> |
first performed simulations using the original technique. At fixed |
362 |
> |
intervals, kinetic energy or momentum exchange moves were performed |
363 |
> |
between the hot and the cold slabs. The interval between exchange |
364 |
> |
moves governs the effective momentum flux ($j_z(p_x)$) or energy flux |
365 |
> |
($J_z$) between the two slabs so to vary this quantity, we performed |
366 |
> |
simulations with a variety of delay intervals between the swapping moves. |
367 |
> |
|
368 |
> |
For thermal conductivity measurements, the particle with smallest |
369 |
> |
speed in the hot slab and the one with largest speed in the cold slab |
370 |
> |
had their entire momentum vectors swapped. In the test cases run |
371 |
> |
here, all particles had the same chemical identity and mass, so this |
372 |
> |
move preserves both total linear momentum and total energy. It is |
373 |
> |
also possible to exchange energy by assuming an elastic collision |
374 |
> |
between the two particles which are exchanging energy. |
375 |
> |
|
376 |
> |
For shear stress simulations, the particle with the most negative |
377 |
> |
$p_x$ in the hot slab and the one with the most positive $p_x$ in the |
378 |
> |
cold slab exchanged only this component of their momentum vectors. |
379 |
> |
|
380 |
> |
\subsection{RNEMD with NIVS scaling} |
381 |
> |
|
382 |
> |
For each simulation utilizing the swapping method, a corresponding |
383 |
> |
NIVS-RNEMD simulation was carried out using a target momentum flux set |
384 |
> |
to produce the same flux experienced in the swapping simulation. |
385 |
> |
|
386 |
> |
To test the temperature homogeneity, directional momentum and |
387 |
> |
temperature distributions were accumulated for molecules in each of |
388 |
> |
the slabs. Transport coefficients were computed using the temperature |
389 |
> |
(and momentum) gradients across the $z$-axis as well as the total |
390 |
> |
momentum flux the system experienced during data collection portion of |
391 |
> |
the simulation. |
392 |
> |
|
393 |
> |
\subsection{Shear viscosities} |
394 |
> |
|
395 |
> |
The momentum flux was calculated using the total non-physical momentum |
396 |
> |
transferred (${P_x}$) and the data collection time ($t$): |
397 |
|
\begin{equation} |
398 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
399 |
|
\end{equation} |
400 |
< |
And the velocity gradient ${\langle \partial v_x /\partial z \rangle}$ |
401 |
< |
can be obtained by a linear regression of the velocity profile. From |
402 |
< |
the shear viscosity $\eta$ calculated with the above parameters, one |
403 |
< |
can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
404 |
< |
(\varepsilon m)^{-1/2}}$. |
400 |
> |
where $L_x$ and $L_y$ denote the $x$ and $y$ lengths of the simulation |
401 |
> |
box. The factor of two in the denominator is present because physical |
402 |
> |
momentum transfer between the slabs occurs in two directions ($+z$ and |
403 |
> |
$-z$). The velocity gradient ${\langle \partial v_x /\partial z |
404 |
> |
\rangle}$ was obtained using linear regression of the mean $x$ |
405 |
> |
component of the velocity, $\langle v_x \rangle$, in each of the bins. |
406 |
> |
For Lennard-Jones simulations, shear viscosities are reported in |
407 |
> |
reduced units (${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$). |
408 |
|
|
409 |
< |
For thermal conductivity calculation, simulations were first run under |
324 |
< |
reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
325 |
< |
algorithm was adopted in the swapping method. Under identical |
326 |
< |
simulation box parameters, in each swap, the top slab exchange the |
327 |
< |
molecule with least kinetic energy with the molecule in the center |
328 |
< |
slab with most kinetic energy, unless this ``coldest'' molecule in the |
329 |
< |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
330 |
< |
slab. According to swapping RNEMD results, target energy flux for |
331 |
< |
scaling RNEMD simulations can be set. Also, various scaling |
332 |
< |
frequencies can be tested for one target energy flux. To compare the |
333 |
< |
performance between swapping and scaling method, distributions of |
334 |
< |
velocity and speed in different slabs were observed. |
409 |
> |
\subsection{Thermal Conductivities} |
410 |
|
|
411 |
< |
For each swapping rate, thermal conductivity was calculated in reduced |
412 |
< |
unit. The energy flux was calculated similarly to the momentum flux, |
413 |
< |
with total unphysical transferred energy ${E_{total}}$ and simulation |
339 |
< |
time $t$: |
411 |
> |
The energy flux was calculated in a similar manner to the momentum |
412 |
> |
flux, using the total non-physical energy transferred (${E_{total}}$) |
413 |
> |
and the data collection time $t$: |
414 |
|
\begin{equation} |
415 |
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
416 |
|
\end{equation} |
417 |
< |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
418 |
< |
can be obtained by a linear regression of the temperature |
419 |
< |
profile. From the thermal conductivity $\lambda$ calculated, one can |
420 |
< |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
421 |
< |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
417 |
> |
The temperature gradient ${\langle\partial T/\partial z\rangle}$ was |
418 |
> |
obtained by a linear regression of the temperature profile. For |
419 |
> |
Lennard-Jones simulations, thermal conductivities are reported in |
420 |
> |
reduced units (${\lambda^*=\lambda \sigma^2 m^{1/2} |
421 |
> |
k_B^{-1}\varepsilon^{-1/2}}$). |
422 |
|
|
423 |
< |
\section{Results And Discussion} |
350 |
< |
\subsection{Shear Viscosity} |
351 |
< |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
352 |
< |
produced comparable shear viscosity to swap RNEMD method. In Table |
353 |
< |
\ref{shearRate}, the names of the calculated samples are devided into |
354 |
< |
two parts. The first number refers to total slabs in one simulation |
355 |
< |
box. The second number refers to the swapping interval in swap method, or |
356 |
< |
in scale method the equilvalent swapping interval that the same |
357 |
< |
momentum flux would theoretically result in swap method. All the scale |
358 |
< |
method results were from simulations that had 10 time steps of scaling |
359 |
< |
interval. The average molecular momentum gradients of these samples |
360 |
< |
are shown in Figures \ref{shearGradSwap} and \ref{shearGradScale} |
361 |
< |
respectively. |
423 |
> |
\subsection{Interfacial Thermal Conductivities} |
424 |
|
|
425 |
< |
\begin{table*} |
426 |
< |
\begin{minipage}{\linewidth} |
427 |
< |
\begin{center} |
425 |
> |
For interfaces with a relatively low interfacial conductance, the bulk |
426 |
> |
regions on either side of an interface rapidly come to a state in |
427 |
> |
which the two phases have relatively homogeneous (but distinct) |
428 |
> |
temperatures. The interfacial thermal conductivity $G$ can therefore |
429 |
> |
be approximated as: |
430 |
|
|
431 |
< |
\caption{Calculation results for shear viscosity of Lennard-Jones |
432 |
< |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
433 |
< |
methods at various momentum exchange rates. Results in reduced |
434 |
< |
unit. Errors of calculations in parentheses. } |
431 |
> |
\begin{equation} |
432 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
433 |
> |
\langle T_{water}\rangle \right)} |
434 |
> |
\label{interfaceCalc} |
435 |
> |
\end{equation} |
436 |
> |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
437 |
> |
transfer and ${\langle T_{gold}\rangle}$ and ${\langle |
438 |
> |
T_{water}\rangle}$ are the average observed temperature of gold and |
439 |
> |
water phases respectively. If the interfacial conductance is {\it |
440 |
> |
not} small, it is also be possible to compute the interfacial |
441 |
> |
thermal conductivity using this method utilizing the change in the |
442 |
> |
slope of the thermal gradient ($\partial^2 \langle T \rangle / \partial |
443 |
> |
z^2$) at the interface. |
444 |
|
|
445 |
< |
\begin{tabular} |
446 |
< |
\hline |
447 |
< |
Name & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
448 |
< |
\hline |
449 |
< |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
450 |
< |
20-1000 & 3.52(0.16) & -\\ |
451 |
< |
20-2000 & - & 3.32(0.18)\\ |
452 |
< |
20-2500 & - & 3.43(0.08)\\ |
453 |
< |
\end{tabular} |
454 |
< |
\label{shearRate} |
455 |
< |
\end{center} |
456 |
< |
\end{minipage} |
445 |
> |
\section{Results} |
446 |
> |
|
447 |
> |
\subsection{Lennard-Jones Fluid} |
448 |
> |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
449 |
> |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
450 |
> |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
451 |
> |
direct comparison between our results and previous methods. These |
452 |
> |
simulations were carried out with a reduced timestep ${\tau^* = |
453 |
> |
4.6\times10^{-4}}$. For the shear viscosity calculations, the mean |
454 |
> |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. For thermal |
455 |
> |
conductivity calculations, simulations were run under reduced |
456 |
> |
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
457 |
> |
ensemble. The simulations included $10^5$ steps of equilibration |
458 |
> |
without any momentum flux, $10^5$ steps of stablization with an |
459 |
> |
imposed momentum transfer to create a gradient, and $10^6$ steps of |
460 |
> |
data collection under RNEMD. |
461 |
> |
|
462 |
> |
\subsubsection*{Thermal Conductivity} |
463 |
> |
|
464 |
> |
Our thermal conductivity calculations show that the NIVS method agrees |
465 |
> |
well with the swapping method. Five different swap intervals were |
466 |
> |
tested (Table \ref{LJ}). Similar thermal gradients were observed with |
467 |
> |
similar thermal flux under the two different methods (Figure |
468 |
> |
\ref{thermalGrad}). Furthermore, the 1-d temperature profiles showed |
469 |
> |
no observable differences between the $x$, $y$ and $z$ axes (Figure |
470 |
> |
\ref{thermalGrad} c), so even though we are using a non-isotropic |
471 |
> |
scaling method, none of the three directions are experience |
472 |
> |
disproportionate heating due to the velocity scaling. |
473 |
> |
|
474 |
> |
\begin{table*} |
475 |
> |
\begin{minipage}{\linewidth} |
476 |
> |
\begin{center} |
477 |
> |
|
478 |
> |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
479 |
> |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
480 |
> |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
481 |
> |
at various momentum fluxes. The original swapping method and |
482 |
> |
the velocity scaling method give similar results. |
483 |
> |
Uncertainties are indicated in parentheses.} |
484 |
> |
|
485 |
> |
\begin{tabular}{|cc|cc|cc|} |
486 |
> |
\hline |
487 |
> |
\multicolumn{2}{|c}{Momentum Exchange} & |
488 |
> |
\multicolumn{2}{|c}{Swapping RNEMD} & |
489 |
> |
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
490 |
> |
\hline |
491 |
> |
\multirow{2}{*}{Swap Interval (fs)} & Equivalent $J_z^*$ or & |
492 |
> |
\multirow{2}{*}{$\lambda^*_{swap}$} & |
493 |
> |
\multirow{2}{*}{$\eta^*_{swap}$} & |
494 |
> |
\multirow{2}{*}{$\lambda^*_{scale}$} & |
495 |
> |
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
496 |
> |
& $j_z^*(p_x)$ (reduced units) & & & & \\ |
497 |
> |
\hline |
498 |
> |
250 & 0.16 & 7.03(0.34) & 3.57(0.06) & 7.30(0.10) & 3.54(0.04)\\ |
499 |
> |
500 & 0.09 & 7.03(0.14) & 3.64(0.05) & 6.95(0.09) & 3.76(0.09)\\ |
500 |
> |
1000 & 0.047 & 6.91(0.42) & 3.52(0.16) & 7.19(0.07) & 3.66(0.06)\\ |
501 |
> |
2000 & 0.024 & 7.52(0.15) & 3.72(0.05) & 7.19(0.28) & 3.32(0.18)\\ |
502 |
> |
2500 & 0.019 & 7.41(0.29) & 3.42(0.06) & 7.98(0.33) & 3.43(0.08)\\ |
503 |
> |
\hline |
504 |
> |
\end{tabular} |
505 |
> |
\label{LJ} |
506 |
> |
\end{center} |
507 |
> |
\end{minipage} |
508 |
|
\end{table*} |
509 |
|
|
510 |
|
\begin{figure} |
511 |
< |
\includegraphics[width=\linewidth]{shearGradSwap.eps} |
512 |
< |
\caption{Average momentum gradients of simulations using swap method.} |
513 |
< |
\label{shearGradSwap} |
511 |
> |
\includegraphics[width=\linewidth]{thermalGrad} |
512 |
> |
\caption{The NIVS-RNEMD method creates similar temperature gradients |
513 |
> |
compared with the swapping method under a variety of imposed |
514 |
> |
kinetic energy flux values. Furthermore, the implementation of |
515 |
> |
Non-Isotropic Velocity Scaling does not cause temperature |
516 |
> |
anisotropy to develop in thermal conductivity calculations.} |
517 |
> |
\label{thermalGrad} |
518 |
|
\end{figure} |
519 |
|
|
520 |
+ |
\subsubsection*{Velocity Distributions} |
521 |
+ |
|
522 |
+ |
During these simulations, velocities were recorded every 1000 steps |
523 |
+ |
and were used to produce distributions of both velocity and speed in |
524 |
+ |
each of the slabs. From these distributions, we observed that under |
525 |
+ |
relatively high non-physical kinetic energy flux, the speed of |
526 |
+ |
molecules in slabs where swapping occured could deviate from the |
527 |
+ |
Maxwell-Boltzmann distribution. This behavior was also noted by Tenney |
528 |
+ |
and Maginn.\cite{Maginn:2010} Figure \ref{thermalHist} shows how these |
529 |
+ |
distributions deviate from an ideal distribution. In the ``hot'' slab, |
530 |
+ |
the probability density is notched at low speeds and has a substantial |
531 |
+ |
shoulder at higher speeds relative to the ideal MB distribution. In |
532 |
+ |
the cold slab, the opposite notching and shouldering occurs. This |
533 |
+ |
phenomenon is more obvious at higher swapping rates. |
534 |
+ |
|
535 |
+ |
The peak of the velocity distribution is substantially flattened in |
536 |
+ |
the hot slab, and is overly sharp (with truncated wings) in the cold |
537 |
+ |
slab. This problem is rooted in the mechanism of the swapping method. |
538 |
+ |
Continually depleting low (high) speed particles in the high (low) |
539 |
+ |
temperature slab is not complemented by diffusions of low (high) speed |
540 |
+ |
particles from neighboring slabs, unless the swapping rate is |
541 |
+ |
sufficiently small. Simutaneously, surplus low speed particles in the |
542 |
+ |
low temperature slab do not have sufficient time to diffuse to |
543 |
+ |
neighboring slabs. Since the thermal exchange rate must reach a |
544 |
+ |
minimum level to produce an observable thermal gradient, the |
545 |
+ |
swapping-method RNEMD has a relatively narrow choice of exchange times |
546 |
+ |
that can be utilized. |
547 |
+ |
|
548 |
+ |
For comparison, NIVS-RNEMD produces a speed distribution closer to the |
549 |
+ |
Maxwell-Boltzmann curve (Figure \ref{thermalHist}). The reason for |
550 |
+ |
this is simple; upon velocity scaling, a Gaussian distribution remains |
551 |
+ |
Gaussian. Although a single scaling move is non-isotropic in three |
552 |
+ |
dimensions, our criteria for choosing a set of scaling coefficients |
553 |
+ |
helps maintain the distributions as close to isotropic as possible. |
554 |
+ |
Consequently, NIVS-RNEMD is able to impose an unphysical thermal flux |
555 |
+ |
as the previous RNEMD methods but without large perturbations to the |
556 |
+ |
velocity distributions in the two slabs. |
557 |
+ |
|
558 |
|
\begin{figure} |
559 |
< |
\includegraphics[width=\linewidth]{shearGradScale.eps} |
560 |
< |
\caption{Average momentum gradients of simulations using scale |
561 |
< |
method.} |
562 |
< |
\label{shearGradScale} |
559 |
> |
\includegraphics[width=\linewidth]{thermalHist} |
560 |
> |
\caption{Velocity and speed distributions that develop under the |
561 |
> |
swapping and NIVS-RNEMD methods at high flux. The distributions for |
562 |
> |
the hot bins (upper panels) and cold bins (lower panels) were |
563 |
> |
obtained from Lennard-Jones simulations with $\langle T^* \rangle = |
564 |
> |
4.5$ with a flux of $J_z^* \sim 5$ (equivalent to a swapping interval |
565 |
> |
of 10 time steps). This is a relatively large flux which shows the |
566 |
> |
non-thermal distributions that develop under the swapping method. |
567 |
> |
NIVS does a better job of producing near-thermal distributions in |
568 |
> |
the bins.} |
569 |
> |
\label{thermalHist} |
570 |
|
\end{figure} |
571 |
|
|
572 |
+ |
|
573 |
+ |
\subsubsection*{Shear Viscosity} |
574 |
+ |
Our calculations (Table \ref{LJ}) show that velocity-scaling RNEMD |
575 |
+ |
predicted comparable shear viscosities to swap RNEMD method. The |
576 |
+ |
average molecular momentum gradients of these samples are shown in |
577 |
+ |
Figure \ref{shear} (a) and (b). |
578 |
+ |
|
579 |
|
\begin{figure} |
580 |
< |
\includegraphics[width=\linewidth]{shearTempScale.eps} |
581 |
< |
\caption{Temperature profile for scaling RNEMD simulation.} |
582 |
< |
\label{shearTempScale} |
580 |
> |
\includegraphics[width=\linewidth]{shear} |
581 |
> |
\caption{Average momentum gradients in shear viscosity simulations, |
582 |
> |
using ``swapping'' method (top panel) and NIVS-RNEMD method |
583 |
> |
(middle panel). NIVS-RNEMD produces a thermal anisotropy artifact |
584 |
> |
in the hot and cold bins when used for shear viscosity. This |
585 |
> |
artifact does not appear in thermal conductivity calculations.} |
586 |
> |
\label{shear} |
587 |
|
\end{figure} |
404 |
– |
However, observations of temperatures along three dimensions show that |
405 |
– |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
406 |
– |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
407 |
– |
increased imposed momentum flux, the temperature difference among $x$ |
408 |
– |
and the other two dimensions were larger. This would result from the |
409 |
– |
scaling method. From Eq. \ref{eq:fluxPlane}, after momentum gradient |
410 |
– |
is set up, $P_c^x$ would be roughly stable ($<0$). Consequently, scaling |
411 |
– |
factor $x$ would most probably larger than 1. Therefore, the kinetic |
412 |
– |
energy in $x$-dimension $K_c^x$ would keep increase after most scaling |
413 |
– |
step. And if there is not enough time for the kinetic energy to |
414 |
– |
exchange among different dimensions and different slabs, the system would finally build up temperature (kinetic energy) difference among the three dimensions. |
415 |
– |
Also, between $y$ and $z$ dimensions in the scaled slabs, temperatures of |
416 |
– |
$z$-axis are closer to neighbor slabs. This is due to momentum |
417 |
– |
transfer along $z$ dimension between slabs. |
588 |
|
|
589 |
+ |
Observations of the three one-dimensional temperatures in each of the |
590 |
+ |
slabs shows that NIVS-RNEMD does produce some thermal anisotropy, |
591 |
+ |
particularly in the hot and cold slabs. Figure \ref{shear} (c) |
592 |
+ |
indicates that with a relatively large imposed momentum flux, |
593 |
+ |
$j_z(p_x)$, the $x$ direction approaches a different temperature from |
594 |
+ |
the $y$ and $z$ directions in both the hot and cold bins. This is an |
595 |
+ |
artifact of the scaling constraints. After the momentum gradient has |
596 |
+ |
been established, $P_c^x < 0$. Consequently, the scaling factor $x$ |
597 |
+ |
is nearly always greater than one in order to satisfy the constraints. |
598 |
+ |
This will continually increase the kinetic energy in $x$-dimension, |
599 |
+ |
$K_c^x$. If there is not enough time for the kinetic energy to |
600 |
+ |
exchange among different directions and different slabs, the system |
601 |
+ |
will exhibit the observed thermal anisotropy in the hot and cold bins. |
602 |
+ |
|
603 |
|
Although results between scaling and swapping methods are comparable, |
604 |
< |
the inherent temperature inhomogeneity makes scaling RNEMD method less |
605 |
< |
attractive than swapping RNEMD in shear viscosity calculation. |
604 |
> |
the inherent temperature anisotropy does make NIVS-RNEMD method less |
605 |
> |
attractive than swapping RNEMD for shear viscosity calculations. We |
606 |
> |
note that this problem appears only when momentum flux is applied, and |
607 |
> |
does not appear in thermal flux calculations. |
608 |
|
|
609 |
< |
\subsection{Thermal Conductivity} |
609 |
> |
\subsection{Bulk SPC/E water} |
610 |
|
|
611 |
+ |
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
612 |
+ |
to the work of Bedrov {\it et al.}\cite{Bedrov:2000}, which employed |
613 |
+ |
the original swapping RNEMD method. Bedrov {\it et |
614 |
+ |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
615 |
+ |
center-of-mass velocities instead of single atom velocities in a |
616 |
+ |
molecule conserves the total kinetic energy and linear momentum. This |
617 |
+ |
principle is also adopted Fin our simulations. Scaling was applied to |
618 |
+ |
the center-of-mass velocities of the rigid bodies of SPC/E model water |
619 |
+ |
molecules. |
620 |
|
|
621 |
+ |
To construct the simulations, a simulation box consisting of 1000 |
622 |
+ |
molecules were first equilibrated under ambient pressure and |
623 |
+ |
temperature conditions using the isobaric-isothermal (NPT) |
624 |
+ |
ensemble.\cite{melchionna93} A fixed volume was chosen to match the |
625 |
+ |
average volume observed in the NPT simulations, and this was followed |
626 |
+ |
by equilibration, first in the canonical (NVT) ensemble, followed by a |
627 |
+ |
100~ps period under constant-NVE conditions without any momentum flux. |
628 |
+ |
Another 100~ps was allowed to stabilize the system with an imposed |
629 |
+ |
momentum transfer to create a gradient, and 1~ns was allotted for data |
630 |
+ |
collection under RNEMD. |
631 |
|
|
632 |
< |
\section{Acknowledgments} |
633 |
< |
Support for this project was provided by the National Science |
634 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
635 |
< |
the Center for Research Computing (CRC) at the University of Notre |
636 |
< |
Dame. \newpage |
632 |
> |
In our simulations, the established temperature gradients were similar |
633 |
> |
to the previous work. Our simulation results at 318K are in good |
634 |
> |
agreement with those from Bedrov {\it et al.} (Table |
635 |
> |
\ref{spceThermal}). And both methods yield values in reasonable |
636 |
> |
agreement with experimental values. |
637 |
|
|
638 |
< |
\bibliographystyle{jcp2} |
638 |
> |
\begin{table*} |
639 |
> |
\begin{minipage}{\linewidth} |
640 |
> |
\begin{center} |
641 |
> |
|
642 |
> |
\caption{Thermal conductivity of SPC/E water under various |
643 |
> |
imposed thermal gradients. Uncertainties are indicated in |
644 |
> |
parentheses.} |
645 |
> |
|
646 |
> |
\begin{tabular}{|c|c|ccc|} |
647 |
> |
\hline |
648 |
> |
\multirow{2}{*}{$\langle T\rangle$(K)} & |
649 |
> |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & |
650 |
> |
\multicolumn{3}{|c|}{$\lambda (\mathrm{W m}^{-1} |
651 |
> |
\mathrm{K}^{-1})$} \\ |
652 |
> |
& & This work & Previous simulations\cite{Bedrov:2000} & |
653 |
> |
Experiment\cite{WagnerKruse}\\ |
654 |
> |
\hline |
655 |
> |
\multirow{3}{*}{300} & 0.38 & 0.816(0.044) & & |
656 |
> |
\multirow{3}{*}{0.61}\\ |
657 |
> |
& 0.81 & 0.770(0.008) & & \\ |
658 |
> |
& 1.54 & 0.813(0.007) & & \\ |
659 |
> |
\hline |
660 |
> |
\multirow{2}{*}{318} & 0.75 & 0.750(0.032) & 0.784 & |
661 |
> |
\multirow{2}{*}{0.64}\\ |
662 |
> |
& 1.59 & 0.778(0.019) & 0.730 & \\ |
663 |
> |
\hline |
664 |
> |
\end{tabular} |
665 |
> |
\label{spceThermal} |
666 |
> |
\end{center} |
667 |
> |
\end{minipage} |
668 |
> |
\end{table*} |
669 |
> |
|
670 |
> |
\subsection{Crystalline Gold} |
671 |
> |
|
672 |
> |
To see how the method performed in a solid, we calculated thermal |
673 |
> |
conductivities using two atomistic models for gold. Several different |
674 |
> |
potential models have been developed that reasonably describe |
675 |
> |
interactions in transition metals. In particular, the Embedded Atom |
676 |
> |
Model (EAM)~\cite{PhysRevB.33.7983} and Sutton-Chen (SC)~\cite{Chen90} |
677 |
> |
potential have been used to study a wide range of phenomena in both |
678 |
> |
bulk materials and |
679 |
> |
nanoparticles.\cite{ISI:000207079300006,Clancy:1992,Vardeman:2008fk,Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
680 |
> |
Both potentials are based on a model of a metal which treats the |
681 |
> |
nuclei and core electrons as pseudo-atoms embedded in the electron |
682 |
> |
density due to the valence electrons on all of the other atoms in the |
683 |
> |
system. The SC potential has a simple form that closely resembles the |
684 |
> |
Lennard Jones potential, |
685 |
> |
\begin{equation} |
686 |
> |
\label{eq:SCP1} |
687 |
> |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
688 |
> |
\end{equation} |
689 |
> |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
690 |
> |
\begin{equation} |
691 |
> |
\label{eq:SCP2} |
692 |
> |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
693 |
> |
\end{equation} |
694 |
> |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
695 |
> |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
696 |
> |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
697 |
> |
the interactions between the valence electrons and the cores of the |
698 |
> |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
699 |
> |
scale, $c_i$ scales the attractive portion of the potential relative |
700 |
> |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
701 |
> |
that assures a dimensionless form for $\rho$. These parameters are |
702 |
> |
tuned to various experimental properties such as the density, cohesive |
703 |
> |
energy, and elastic moduli for FCC transition metals. The quantum |
704 |
> |
Sutton-Chen (QSC) formulation matches these properties while including |
705 |
> |
zero-point quantum corrections for different transition |
706 |
> |
metals.\cite{PhysRevB.59.3527} The EAM functional forms differ |
707 |
> |
slightly from SC but the overall method is very similar. |
708 |
> |
|
709 |
> |
In this work, we have utilized both the EAM and the QSC potentials to |
710 |
> |
test the behavior of scaling RNEMD. |
711 |
> |
|
712 |
> |
A face-centered-cubic (FCC) lattice was prepared containing 2880 Au |
713 |
> |
atoms (i.e. ${6\times 6\times 20}$ unit cells). Simulations were run |
714 |
> |
both with and without isobaric-isothermal (NPT)~\cite{melchionna93} |
715 |
> |
pre-equilibration at a target pressure of 1 atm. When equilibrated |
716 |
> |
under NPT conditions, our simulation box expanded by approximately 1\% |
717 |
> |
in volume. Following adjustment of the box volume, equilibrations in |
718 |
> |
both the canonical and microcanonical ensembles were carried out. With |
719 |
> |
the simulation cell divided evenly into 10 slabs, different thermal |
720 |
> |
gradients were established by applying a set of target thermal |
721 |
> |
transfer fluxes. |
722 |
> |
|
723 |
> |
The results for the thermal conductivity of gold are shown in Table |
724 |
> |
\ref{AuThermal}. In these calculations, the end and middle slabs were |
725 |
> |
excluded in thermal gradient linear regession. EAM predicts slightly |
726 |
> |
larger thermal conductivities than QSC. However, both values are |
727 |
> |
smaller than experimental value by a factor of more than 200. This |
728 |
> |
behavior has been observed previously by Richardson and Clancy, and |
729 |
> |
has been attributed to the lack of electronic contribution in these |
730 |
> |
force fields.\cite{Clancy:1992} It should be noted that the density of |
731 |
> |
the metal being simulated has an effect on thermal conductance. With |
732 |
> |
an expanded lattice, lower thermal conductance is expected (and |
733 |
> |
observed). We also observed a decrease in thermal conductance at |
734 |
> |
higher temperatures, a trend that agrees with experimental |
735 |
> |
measurements.\cite{AshcroftMermin} |
736 |
> |
|
737 |
> |
\begin{table*} |
738 |
> |
\begin{minipage}{\linewidth} |
739 |
> |
\begin{center} |
740 |
> |
|
741 |
> |
\caption{Calculated thermal conductivity of crystalline gold |
742 |
> |
using two related force fields. Calculations were done at both |
743 |
> |
experimental and equilibrated densities and at a range of |
744 |
> |
temperatures and thermal flux rates. Uncertainties are |
745 |
> |
indicated in parentheses. Richardson {\it et |
746 |
> |
al.}\cite{Clancy:1992} give an estimate of 1.74 $\mathrm{W |
747 |
> |
m}^{-1}\mathrm{K}^{-1}$ for EAM gold |
748 |
> |
at a density of 19.263 g / cm$^3$.} |
749 |
> |
|
750 |
> |
\begin{tabular}{|c|c|c|cc|} |
751 |
> |
\hline |
752 |
> |
Force Field & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
753 |
> |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$\\ |
754 |
> |
\hline |
755 |
> |
\multirow{7}{*}{QSC} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
756 |
> |
& & & 2.86 & 1.08(0.05)\\ |
757 |
> |
& & & 5.14 & 1.15(0.07)\\\cline{2-5} |
758 |
> |
& \multirow{4}{*}{19.263} & \multirow{2}{*}{300} & 2.31 & 1.25(0.06)\\ |
759 |
> |
& & & 3.02 & 1.26(0.05)\\\cline{3-5} |
760 |
> |
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
761 |
> |
& & & 4.84 & 0.92(0.05)\\ |
762 |
> |
\hline |
763 |
> |
\multirow{8}{*}{EAM} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
764 |
> |
& & & 2.06 & 1.37(0.04)\\ |
765 |
> |
& & & 2.55 & 1.41(0.07)\\\cline{2-5} |
766 |
> |
& \multirow{5}{*}{19.263} & \multirow{3}{*}{300} & 1.06 & 1.45(0.13)\\ |
767 |
> |
& & & 2.04 & 1.41(0.07)\\ |
768 |
> |
& & & 2.41 & 1.53(0.10)\\\cline{3-5} |
769 |
> |
& & \multirow{2}{*}{575} & 2.82 & 1.08(0.03)\\ |
770 |
> |
& & & 4.14 & 1.08(0.05)\\ |
771 |
> |
\hline |
772 |
> |
\end{tabular} |
773 |
> |
\label{AuThermal} |
774 |
> |
\end{center} |
775 |
> |
\end{minipage} |
776 |
> |
\end{table*} |
777 |
> |
|
778 |
> |
\subsection{Thermal Conductance at the Au/H$_2$O interface} |
779 |
> |
The most attractive aspect of the scaling approach for RNEMD is the |
780 |
> |
ability to use the method in non-homogeneous systems, where molecules |
781 |
> |
of different identities are segregated in different slabs. To test |
782 |
> |
this application, we simulated a Gold (111) / water interface. To |
783 |
> |
construct the interface, a box containing a lattice of 1188 Au atoms |
784 |
> |
(with the 111 surface in the $+z$ and $-z$ directions) was allowed to |
785 |
> |
relax under ambient temperature and pressure. A separate (but |
786 |
> |
identically sized) box of SPC/E water was also equilibrated at ambient |
787 |
> |
conditions. The two boxes were combined by removing all water |
788 |
> |
molecules within 3 \AA radius of any gold atom. The final |
789 |
> |
configuration contained 1862 SPC/E water molecules. |
790 |
> |
|
791 |
> |
The Spohr potential was adopted in depicting the interaction between |
792 |
> |
metal atoms and water molecules.\cite{ISI:000167766600035} A similar |
793 |
> |
protocol of equilibration to our water simulations was followed. We |
794 |
> |
observed that the two phases developed large temperature differences |
795 |
> |
even under a relatively low thermal flux. |
796 |
> |
|
797 |
> |
The low interfacial conductance is probably due to an acoustic |
798 |
> |
impedance mismatch between the solid and the liquid |
799 |
> |
phase.\cite{Cahill:793,RevModPhys.61.605} Experiments on the thermal |
800 |
> |
conductivity of gold nanoparticles and nanorods in solvent and in |
801 |
> |
glass cages have predicted values for $G$ between 100 and 350 |
802 |
> |
(MW/m$^2$/K). The experiments typically have multiple gold surfaces |
803 |
> |
that have been protected by a capping agent (citrate or CTAB) or which |
804 |
> |
are in direct contact with various glassy solids. In these cases, the |
805 |
> |
acoustic impedance mismatch would be substantially reduced, leading to |
806 |
> |
much higher interfacial conductances. It is also possible, however, |
807 |
> |
that the lack of electronic effects that gives rise to the low thermal |
808 |
> |
conductivity of EAM gold is also causing a low reading for this |
809 |
> |
particular interface. |
810 |
> |
|
811 |
> |
Under this low thermal conductance, both gold and water phase have |
812 |
> |
sufficient time to eliminate temperature difference inside |
813 |
> |
respectively (Figure \ref{interface} b). With indistinguishable |
814 |
> |
temperature difference within respective phase, it is valid to assume |
815 |
> |
that the temperature difference between gold and water on surface |
816 |
> |
would be approximately the same as the difference between the gold and |
817 |
> |
water phase. This assumption enables convenient calculation of $G$ |
818 |
> |
using Eq. \ref{interfaceCalc} instead of measuring temperatures of |
819 |
> |
thin layer of water and gold close enough to surface, which would have |
820 |
> |
greater fluctuation and lower accuracy. Reported results (Table |
821 |
> |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
822 |
> |
calculations on homogeneous systems, and thus have larger relative |
823 |
> |
errors than our calculation results on homogeneous systems. |
824 |
> |
|
825 |
> |
\begin{figure} |
826 |
> |
\includegraphics[width=\linewidth]{interface} |
827 |
> |
\caption{Temperature profiles of the Gold / Water interface at four |
828 |
> |
different values for the thermal flux. Temperatures for slabs |
829 |
> |
either in the gold or in the water show no significant differences, |
830 |
> |
although there is a large discontinuity between the materials |
831 |
> |
because of the relatively low interfacial thermal conductivity.} |
832 |
> |
\label{interface} |
833 |
> |
\end{figure} |
834 |
> |
|
835 |
> |
\begin{table*} |
836 |
> |
\begin{minipage}{\linewidth} |
837 |
> |
\begin{center} |
838 |
> |
|
839 |
> |
\caption{Computed interfacial thermal conductivity ($G$) values |
840 |
> |
for the Au(111) / water interface at ${\langle T\rangle \sim}$ |
841 |
> |
300K using a range of energy fluxes. Uncertainties are |
842 |
> |
indicated in parentheses. } |
843 |
> |
|
844 |
> |
\begin{tabular}{|cccc| } |
845 |
> |
\hline |
846 |
> |
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
847 |
> |
T_{water} \rangle$ (K) & $G$ |
848 |
> |
(MW/m$^2$/K)\\ |
849 |
> |
\hline |
850 |
> |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
851 |
> |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
852 |
> |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
853 |
> |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
854 |
> |
\hline |
855 |
> |
\end{tabular} |
856 |
> |
\label{interfaceRes} |
857 |
> |
\end{center} |
858 |
> |
\end{minipage} |
859 |
> |
\end{table*} |
860 |
> |
|
861 |
> |
|
862 |
> |
\section{Conclusions} |
863 |
> |
NIVS-RNEMD simulation method is developed and tested on various |
864 |
> |
systems. Simulation results demonstrate its validity in thermal |
865 |
> |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
866 |
> |
molecule like water and metal crystals. NIVS-RNEMD improves |
867 |
> |
non-Boltzmann-Maxwell distributions, which exist inb previous RNEMD |
868 |
> |
methods. Furthermore, it develops a valid means for unphysical thermal |
869 |
> |
transfer between different species of molecules, and thus extends its |
870 |
> |
applicability to interfacial systems. Our calculation of gold/water |
871 |
> |
interfacial thermal conductivity demonstrates this advantage over |
872 |
> |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
873 |
> |
shear viscosity calculations, but could cause temperature difference |
874 |
> |
among different dimensions under high momentum flux. Modification is |
875 |
> |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
876 |
> |
calculations. |
877 |
> |
|
878 |
> |
\section{Acknowledgments} |
879 |
> |
The authors would like to thank Craig Tenney and Ed Maginn for many |
880 |
> |
helpful discussions. Support for this project was provided by the |
881 |
> |
National Science Foundation under grant CHE-0848243. Computational |
882 |
> |
time was provided by the Center for Research Computing (CRC) at the |
883 |
> |
University of Notre Dame. |
884 |
> |
\newpage |
885 |
> |
|
886 |
|
\bibliography{nivsRnemd} |
887 |
+ |
|
888 |
|
\end{doublespace} |
889 |
|
\end{document} |
890 |
|
|