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\begin{abstract} |
| 66 |
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|
| 67 |
< |
We have adapted the Velocity Shearing and Scaling Reverse Non-Equilibium Molecular Dynamics (VSS-RNEMD) method for use with aperiodic system geometries. This new method is capable of creating stable temperature and angular velocity gradients in heterogeneous non-periodic systems while conserving total energy and angular momentum. To demonstrate the method, we have computed the thermal conductivities of a gold nanoparticle and water cluster, the shear viscosity of a water cluster, the interfacial thermal conductivity of a solvated gold nanoparticle and the interfacial friction of solvated gold nanostructures. |
| 67 |
> |
We have adapted the Velocity Shearing and Scaling Reverse Non-Equilibium Molecular Dynamics (VSS-RNEMD) method |
| 68 |
> |
for use with non-periodic system geometries. This new method is capable of creating stable temperature and |
| 69 |
> |
angular velocity gradients in heterogeneous non-periodic systems while conserving total energy and angular |
| 70 |
> |
momentum. To demonstrate the method, we have computed the thermal conductivities of a gold nanoparticle and |
| 71 |
> |
water cluster, the shear viscosity of a water cluster, the interfacial thermal conductivity of a solvated gold |
| 72 |
> |
nanoparticle and the interfacial friction of solvated gold nanostructures. |
| 73 |
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|
| 74 |
|
\end{abstract} |
| 75 |
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|
| 82 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 83 |
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\section{Introduction} |
| 84 |
|
|
| 85 |
+ |
Non-equilibrium Molecular Dynamics (NEMD) methods impose a temperature or velocity {\it gradient} on a |
| 86 |
+ |
system,\cite{ASHURST:1975tg,Evans:1982zk,ERPENBECK:1984sp,MAGINN:1993hc,Berthier:2002ij,Evans:2002ai,Schelling:2002dp,PhysRevA.34.1449,JiangHao_jp802942v} and use linear response theory to connect the resulting thermal or |
| 87 |
+ |
momentum flux to transport coefficients of bulk materials. However, for heterogeneous systems, such as phase |
| 88 |
+ |
boundaries or interfaces, it is often unclear what shape of gradient should be imposed at the boundary between |
| 89 |
+ |
materials. |
| 90 |
|
|
| 91 |
+ |
Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an unphysical {\it flux} between different |
| 92 |
+ |
regions or ``slabs'' of the simulation box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} The system |
| 93 |
+ |
responds by developing a temperature or velocity {\it gradient} between the two regions. The gradients which |
| 94 |
+ |
develop in response to the applied flux are then related (via linear response theory) to the transport |
| 95 |
+ |
coefficient of interest. Since the amount of the applied flux is known exactly, and measurement of a gradient |
| 96 |
+ |
is generally less complicated, imposed-flux methods typically take shorter simulation times to obtain converged |
| 97 |
+ |
results. At interfaces, the observed gradients often exhibit near-discontinuities at the boundaries between |
| 98 |
+ |
dissimilar materials. RNEMD methods do not need many trajectories to provide information about transport |
| 99 |
+ |
properties, and they have become widely used to compute thermal and mechanical transport in both homogeneous |
| 100 |
+ |
liquids and solids~\cite{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as well as heterogeneous |
| 101 |
+ |
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
| 102 |
+ |
|
| 103 |
+ |
The strengths of specific algorithms for imposing the flux between two |
| 104 |
+ |
different slabs of the simulation cell has been the subject of some |
| 105 |
+ |
renewed interest. The original RNEMD approach used kinetic energy or |
| 106 |
+ |
momentum exchange between particles in the two slabs, either through |
| 107 |
+ |
direct swapping of momentum vectors or via virtual elastic collisions |
| 108 |
+ |
between atoms in the two regions. There have been recent |
| 109 |
+ |
methodological advances which involve scaling all particle velocities |
| 110 |
+ |
in both slabs. Constraint equations are simultaneously imposed to |
| 111 |
+ |
require the simulation to conserve both total energy and total linear |
| 112 |
+ |
momentum. The most recent and simplest of the velocity scaling |
| 113 |
+ |
approaches allows for simultaneous shearing (to provide viscosity |
| 114 |
+ |
estimates) as well as scaling (to provide information about thermal |
| 115 |
+ |
conductivity). |
| 116 |
+ |
|
| 117 |
+ |
To date, however, the RNEMD methods have only been usable in periodic |
| 118 |
+ |
simulation cells where the exchange regions are physically separated |
| 119 |
+ |
along one of the axes of the simulation cell. This limits the |
| 120 |
+ |
applicability to infinite planar interfaces. |
| 121 |
+ |
|
| 122 |
+ |
In order to model steady-state non-equilibrium distributions for |
| 123 |
+ |
curved surfaces (e.g. hot nanoparticles in contact with colder |
| 124 |
+ |
solvent), or for regions that are not planar slabs, the method |
| 125 |
+ |
requires some generalization for non-parallel exchange regions. In |
| 126 |
+ |
the following sections, we present the Velocity Shearing and Scaling |
| 127 |
+ |
(VSS) RNEMD algorithm which has been explicitly designed for |
| 128 |
+ |
non-periodic simulations, and use the method to compute some thermal |
| 129 |
+ |
transport and solid-liquid friction at the surfaces of spherical and |
| 130 |
+ |
ellipsoidal nanoparticles, and discuss how the method can be extended |
| 131 |
+ |
to provide other kinds of non-equilibrium fluxes. |
| 132 |
+ |
|
| 133 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 134 |
|
% **METHODOLOGY** |
| 135 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 136 |
< |
\section{Methodology} |
| 136 |
> |
\section{Velocity Shearing and Scaling (VSS) for non-periodic systems} |
| 137 |
|
|
| 138 |
+ |
The periodic VSS-RNEMD approach uses a series of simultaneous velocity |
| 139 |
+ |
shearing and scaling exchanges between the two slabs.\cite{Kuang2012} |
| 140 |
+ |
This method imposes energy and momentum conservation constraints while |
| 141 |
+ |
simultaneously creating a desired flux between the two slabs. These |
| 142 |
+ |
constraints ensure that all configurations are sampled from the same |
| 143 |
+ |
microcanonical (NVE) ensemble. |
| 144 |
+ |
|
| 145 |
+ |
\begin{figure} |
| 146 |
+ |
\includegraphics[width=\linewidth]{figures/npVSS} |
| 147 |
+ |
\caption{Schematics of periodic (left) and non-periodic (right) |
| 148 |
+ |
Velocity Shearing and Scaling RNEMD. A kinetic energy or momentum |
| 149 |
+ |
flux is applied from region B to region A. Thermal gradients are |
| 150 |
+ |
depicted by a color gradient. Linear or angular velocity gradients |
| 151 |
+ |
are shown as arrows.} |
| 152 |
+ |
\label{fig:VSS} |
| 153 |
+ |
\end{figure} |
| 154 |
+ |
|
| 155 |
+ |
We have extended the VSS method for use in {\it non-periodic} simulations, in which the ``slabs'' have been |
| 156 |
+ |
generalized to two separated regions of space. These regions could be defined as concentric spheres (as in |
| 157 |
+ |
figure \ref{fig:VSS}), or one of the regions can be defined in terms of a dynamically changing ``hull'' |
| 158 |
+ |
comprising the surface atoms of the cluster. This latter definition is identical to the hull used in the |
| 159 |
+ |
Langevin Hull algorithm. |
| 160 |
+ |
|
| 161 |
+ |
We present here a new set of constraints that are more general than the VSS constraints. For the non-periodic |
| 162 |
+ |
variant, the constraints fix both the total energy and total {\it angular} momentum of the system while |
| 163 |
+ |
simultaneously imposing a thermal and angular momentum flux between the two regions. |
| 164 |
+ |
|
| 165 |
+ |
After each $\Delta t$ time interval, the particle velocities ($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two |
| 166 |
+ |
shells ($A$ and $B$) are modified by a velocity scaling coefficient ($a$ and $b$) and by a rotational shearing |
| 167 |
+ |
term ($\mathbf{c}_a$ and $\mathbf{c}_b$). |
| 168 |
+ |
|
| 169 |
+ |
\begin{displaymath} |
| 170 |
+ |
\begin{array}{rclcl} |
| 171 |
+ |
& \underline{~~~~~~~~\mathrm{scaling}~~~~~~~~} & & |
| 172 |
+ |
\underline{\mathrm{rotational~shearing}} \\ \\ |
| 173 |
+ |
\mathbf{v}_i $~~~$\leftarrow & |
| 174 |
+ |
a \left(\mathbf{v}_i - \langle \omega_a |
| 175 |
+ |
\rangle \times \mathbf{r}_i\right) & + & \mathbf{c}_a \times \mathbf{r}_i \\ |
| 176 |
+ |
\mathbf{v}_j $~~~$\leftarrow & |
| 177 |
+ |
b \left(\mathbf{v}_j - \langle \omega_b |
| 178 |
+ |
\rangle \times \mathbf{r}_j\right) & + & \mathbf{c}_b \times \mathbf{r}_j |
| 179 |
+ |
\end{array} |
| 180 |
+ |
\end{displaymath} |
| 181 |
+ |
Here $\langle\mathbf{\omega}_a\rangle$ and $\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular |
| 182 |
+ |
velocities of each shell, and $\mathbf{r}_i$ is the position of particle $i$ relative to a fixed point in space |
| 183 |
+ |
(usually the center of mass of the cluster). Particles in the shells also receive an additive ``angular shear'' |
| 184 |
+ |
to their velocities. The amount of shear is governed by the imposed angular momentum flux, |
| 185 |
+ |
$\mathbf{j}_r(\mathbf{L})$, |
| 186 |
+ |
\begin{eqnarray} |
| 187 |
+ |
\mathbf{c}_a & = & - \mathbf{j}_r(\mathbf{L}) \cdot |
| 188 |
+ |
\overleftrightarrow{I_a}^{-1} \Delta t + \langle \omega_a \rangle \label{eq:bc}\\ |
| 189 |
+ |
\mathbf{c}_b & = & + \mathbf{j}_r(\mathbf{L}) \cdot |
| 190 |
+ |
\overleftrightarrow{I_b}^{-1} \Delta t + \langle \omega_b \rangle \label{eq:bh} |
| 191 |
+ |
\end{eqnarray} |
| 192 |
+ |
where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia tensor for each of the two shells. |
| 193 |
+ |
|
| 194 |
+ |
To simultaneously impose a thermal flux ($J_r$) between the shells we use energy conservation constraints, |
| 195 |
+ |
\begin{eqnarray} |
| 196 |
+ |
K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle |
| 197 |
+ |
\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a |
| 198 |
+ |
\rangle \right) + \frac{1}{2} \mathbf{c}_a \cdot \overleftrightarrow{I_a} |
| 199 |
+ |
\cdot \mathbf{c}_a \label{eq:Kc}\\ |
| 200 |
+ |
K_b + J_r \Delta t & = & b^2 \left(K_b - \frac{1}{2}\langle |
| 201 |
+ |
\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b |
| 202 |
+ |
\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh} |
| 203 |
+ |
\end{eqnarray} |
| 204 |
+ |
Simultaneous solution of these quadratic formulae for the scaling coefficients, $a$ and $b$, will ensure that |
| 205 |
+ |
the simulation samples from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ is the instantaneous |
| 206 |
+ |
translational kinetic energy of each shell. At each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
| 207 |
+ |
$\mathbf{c}_b$, subject to the imposed angular momentum flux, $j_r(\mathbf{L})$, and thermal flux, $J_r$, |
| 208 |
+ |
values. The new particle velocities are computed, and the simulation continues. System configurations after the |
| 209 |
+ |
transformations have exactly the same energy ({\it and} angular momentum) as before the moves. |
| 210 |
+ |
|
| 211 |
+ |
As the simulation progresses, the velocity transformations can be performed on a regular basis, and the system |
| 212 |
+ |
will develop a temperature and/or angular velocity gradient in response to the applied flux. Using the slope of |
| 213 |
+ |
the radial temperature or velocity gradients, it is quite simple to obtain both the thermal conductivity |
| 214 |
+ |
($\lambda$), interfacial thermal conductance ($G$), or rotational friction coefficients ($\Xi^{rr}$) of any |
| 215 |
+ |
non-periodic system. |
| 216 |
+ |
|
| 217 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 218 |
+ |
% **COMPUTATIONAL DETAILS** |
| 219 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 220 |
+ |
\section{Computational Details} |
| 221 |
+ |
|
| 222 |
+ |
The new VSS-RNEMD methodology for non-periodic system geometries has been implemented in our group molecular |
| 223 |
+ |
dynamics code, OpenMD.\cite{openmd} We have used the new method to calculate the thermal conductance of a gold |
| 224 |
+ |
nanoparticle and SPC/E water cluster, and compared the results with previous bulk RNEMD values, as well as |
| 225 |
+ |
experiment. We have also investigated the interfacial thermal conductance and interfacial rotational friction |
| 226 |
+ |
for gold nanostructures solvated in hexane as a function of nanoparticle size and shape. |
| 227 |
+ |
|
| 228 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 229 |
|
% FORCE FIELD PARAMETERS |
| 230 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 231 |
|
\subsection{Force field parameters} |
| 232 |
|
|
| 233 |
< |
We have chosen the SPC/E water model for these simulations. There are many values for physical properties from previous simulations available for direct comparison. |
| 233 |
> |
Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The QSC |
| 234 |
> |
parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and |
| 235 |
> |
include zero-point quantum corrections. |
| 236 |
|
|
| 237 |
< |
Gold-gold interactions are described by the quantum Sutton-Chen (QSC) model. The QSC parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and include zero-point quantum corrections. |
| 237 |
> |
We have chosen the SPC/E water model for these simulations, which is particularly useful for method validation |
| 238 |
> |
as there are many values for physical properties from previous simulations available for direct |
| 239 |
> |
comparison.\cite{Bedrov:2000, Kuang2010} |
| 240 |
|
|
| 241 |
< |
Hexane molecules are described by the TraPPE united atom model. This model provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. |
| 241 |
> |
Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good |
| 242 |
> |
computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model, sites are |
| 243 |
> |
located at the carbon centers for alkyl groups. Bonding interactions, including bond stretches and bends and |
| 244 |
> |
torsions, were used for intra-molecular sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 245 |
> |
potentials were used. We have previously utilized both united atom (UA) and all-atom (AA) force fields for |
| 246 |
> |
thermal conductivity,\cite{kuang:AuThl,Kuang2012} and since the united atom force fields cannot populate the |
| 247 |
> |
high-frequency modes that are present in AA force fields, they appear to work better for modeling thermal |
| 248 |
> |
conductivity. |
| 249 |
|
|
| 250 |
< |
Metal-nonmetal interactions are governed by parameters derived from Luedtke and Landman. |
| 250 |
> |
Gold -- hexane nonbonded interactions are governed by pairwise Lennard-Jones parameters derived from Vlugt |
| 251 |
> |
\emph{et al}.\cite{vlugt:cpc2007154} They fitted parameters for the interaction between Au and CH$_{\emph x}$ |
| 252 |
> |
pseudo-atoms based on the effective potential of Hautman and Klein for the Au(111) surface.\cite{hautman:4994} |
| 253 |
|
|
| 101 |
– |
|
| 254 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 255 |
|
% NON-PERIODIC DYNAMICS |
| 256 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 257 |
< |
\subsection{Dynamics for non-periodic systems} |
| 257 |
> |
% \subsection{Dynamics for non-periodic systems} |
| 258 |
> |
% |
| 259 |
> |
% We have run all tests using the Langevin Hull non-periodic simulation methodology.\cite{Vardeman2011} The |
| 260 |
> |
% Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different |
| 261 |
> |
% compressibilities, which are typically problematic for traditional affine transform methods. We have had |
| 262 |
> |
% success applying this method to several different systems including bare metal nanoparticles, liquid water |
| 263 |
> |
% clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal |
| 264 |
> |
% compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous |
| 265 |
> |
% theoretical and experimental results. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex |
| 266 |
> |
% hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled |
| 267 |
> |
% to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are |
| 268 |
> |
% subject to standard Newtonian dynamics. |
| 269 |
|
|
| 270 |
< |
We have run all tests using the Langevin Hull nonperiodic simulation methodology. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are subject to standard Newtonian dynamics. Thermal coupling to the bath was turned off to avoid interference with any imposed kinetic flux. Systems containing liquids were run under moderate pressure ($\sim$ 5 atm) to avoid the formation of a substantial vapor phase, which would have a hull created out of a relatively small number of molecules. |
| 270 |
> |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 271 |
> |
% SIMULATION PROTOCOL |
| 272 |
> |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 273 |
> |
\subsection{Simulation protocol} |
| 274 |
|
|
| 275 |
+ |
In all cases, systems were fully equilibrated under non-periodic isobaric-isothermal (NPT) conditions -- using |
| 276 |
+ |
the Langevin Hull methodology\cite{Vardeman2011} -- before any non-equilibrium methods were introduced. For |
| 277 |
+ |
heterogeneous systems, the gold nanoparticles and ellipsoid were first created from a bulk lattice and |
| 278 |
+ |
thermally equilibrated before being solvated in hexane. Packmol\cite{packmol} was used to solvate previously |
| 279 |
+ |
equilibrated gold nanostructures within a spherical droplet of hexane. |
| 280 |
+ |
|
| 281 |
+ |
Once fully equilibrated, a thermal or angular momentum flux was applied for 1 - 2 |
| 282 |
+ |
ns, until a stable temperature or angular velocity gradient had developed. Systems containing liquids were run |
| 283 |
+ |
under moderate pressure (5 atm) and temperatures (230 K) to avoid the formation of a substantial vapor phase at the boundary of the cluster. Pressure was applied to the system via the non-periodic convex Langevin Hull. Thermal coupling to the external temperature and pressure bath was removed to avoid interference with any |
| 284 |
+ |
imposed flux. |
| 285 |
+ |
|
| 286 |
+ |
To stabilize the gold nanoparticle under the imposed angular momentum flux we altered the gold atom at the |
| 287 |
+ |
designated coordinate origin to have $10,000$ times its original mass. The nonbonded interactions remain |
| 288 |
+ |
unchanged and the heavy atom is excluded from the angular momentum exchange. For rotation of the ellipsoid about |
| 289 |
+ |
its long axis we have added two heavy atoms along the axis of rotation, one at each end of the rod. We collected angular velocity data for the heterogeneous systems after a brief VSS-RNEMD simulation to initialize rotation of the solvated nanostructure. Doing so ensures that we overcome the initial static friction and calculate only the \emph{dynamic} interfacial rotational friction. |
| 290 |
+ |
|
| 291 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 292 |
< |
% NON-PERIODIC RNEMD |
| 292 |
> |
% THERMAL CONDUCTIVITIES |
| 293 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 294 |
< |
\subsection{VSS-RNEMD for non-periodic systems} |
| 294 |
> |
\subsection{Thermal conductivities} |
| 295 |
|
|
| 296 |
< |
The adaptation of VSS-RNEMD for non-periodic systems is relatively |
| 297 |
< |
straightforward. The major modifications to the method are the addition of a rotational shearing term and the use of more versatile hot / cold regions instead |
| 116 |
< |
of rectangular slabs. A temperature profile along the $r$ coordinate is created by recording the average temperature of concentric spherical shells. |
| 296 |
> |
Fourier's Law of heat conduction in radial coordinates yields an expression for the heat flow between the |
| 297 |
> |
concentric spherical RNEMD shells: |
| 298 |
|
|
| 299 |
< |
At each time interval, the particle velocities ($\mathbf{v}_i$ and |
| 300 |
< |
$\mathbf{v}_j$) in the cold and hot shells ($C$ and $H$) are modified by a |
| 301 |
< |
velocity scaling coefficient ($c$ and $h$), an additive linear velocity shearing |
| 302 |
< |
term ($\mathbf{a}_c$ and $\mathbf{a}_h$), and an additive angular velocity |
| 122 |
< |
shearing term ($\mathbf{b}_c$ and $\mathbf{b}_h$). Here, $\langle |
| 123 |
< |
\mathbf{v}_{i,j} \rangle$ and $\langle \mathbf{\omega}_{i,j} \rangle$ are the |
| 124 |
< |
average linear and angular velocities for each shell. |
| 299 |
> |
\begin{equation} |
| 300 |
> |
q_r = - \frac{4 \pi \lambda (T_b - T_a)}{\frac{1}{r_a} - \frac{1}{r_b}} |
| 301 |
> |
\label{eq:Q} |
| 302 |
> |
\end{equation} |
| 303 |
|
|
| 304 |
< |
\begin{displaymath} |
| 305 |
< |
\begin{array}{rclcl} |
| 128 |
< |
& \underline{~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~} & & |
| 129 |
< |
\underline{\mathrm{rotational \; shearing}} \\ \\ |
| 130 |
< |
\mathbf{v}_i $~~~$\leftarrow & |
| 131 |
< |
c \, \left(\mathbf{v}_i - \langle \omega_c |
| 132 |
< |
\rangle \times r_i\right) & + & \mathbf{b}_c \times r_i \\ |
| 133 |
< |
\mathbf{v}_j $~~~$\leftarrow & |
| 134 |
< |
h \, \left(\mathbf{v}_j - \langle \omega_h |
| 135 |
< |
\rangle \times r_j\right) & + & \mathbf{b}_h \times r_j |
| 136 |
< |
\end{array} |
| 137 |
< |
\end{displaymath} |
| 304 |
> |
where $\lambda$ is the thermal conductivity, and $T_{a,b}$ and $r_{a,b}$ are the temperatures and radii of the |
| 305 |
> |
two RNEMD regions, respectively. |
| 306 |
|
|
| 307 |
< |
\begin{eqnarray} |
| 308 |
< |
\mathbf{b}_c & = & - \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_c^{-1} + \langle \omega_c \rangle \label{eq:bc}\\ |
| 309 |
< |
\mathbf{b}_h & = & + \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_h^{-1} + \langle \omega_h \rangle \label{eq:bh} |
| 310 |
< |
\end{eqnarray} |
| 307 |
> |
A thermal flux is created using VSS-RNEMD moves, and the temperature in each of the radial shells is recorded. |
| 308 |
> |
The resulting temperature profiles are analyzed to yield information about the interfacial thermal conductance. |
| 309 |
> |
As the simulation progresses, the VSS moves are performed on a regular basis, and the system develops a thermal |
| 310 |
> |
or velocity gradient in response to the applied flux. Once a stable thermal gradient has been established |
| 311 |
> |
between the two regions, the thermal conductivity, $\lambda$, can be calculated using a linear regression of |
| 312 |
> |
the thermal gradient, $\langle \frac{dT}{dr} \rangle$: |
| 313 |
|
|
| 314 |
< |
The total energy is constrained via two quadratic formulae, |
| 314 |
> |
\begin{equation} |
| 315 |
> |
\lambda = \frac{r_a}{r_b} \frac{q_r}{4 \pi \langle \frac{dT}{dr} \rangle} |
| 316 |
> |
\label{eq:lambda} |
| 317 |
> |
\end{equation} |
| 318 |
|
|
| 319 |
< |
\begin{eqnarray} |
| 320 |
< |
K_c - J_r \; \Delta \, t & = & c^2 \, (K_c - \frac{1}{2}\langle \omega_c \rangle \cdot I_c\langle \omega_c \rangle) + \frac{1}{2} \mathbf{b}_c \cdot I_c \, \mathbf{b}_c \label{eq:Kc}\\ |
| 148 |
< |
K_h + J_r \; \Delta \, t & = & h^2 \, (K_h - \frac{1}{2}\langle \omega_h \rangle \cdot I_h\langle \omega_h \rangle) + \frac{1}{2} \mathbf{b}_h \cdot I_h \, \mathbf{b}_h \label{eq:Kh} |
| 149 |
< |
\end{eqnarray} |
| 319 |
> |
The rate of heat transfer between the two RNEMD regions is the amount of transferred kinetic energy over the |
| 320 |
> |
length of the simulation, t |
| 321 |
|
|
| 322 |
< |
the simultaneous |
| 323 |
< |
solution of which provide the velocity scaling coefficients $c$ and $h$. Given an |
| 324 |
< |
imposed angular momentum flux, $\mathbf{j}_{r} \left( \mathbf{L} \right)$, and/or |
| 325 |
< |
thermal flux, $J_r$, equations \ref{eq:bc} - \ref{eq:Kh} are sufficient to obtain |
| 326 |
< |
the velocity scaling ($c$ and $h$) and shearing ($\mathbf{b}_c,\,$ and $\mathbf{b}_h$) at each time interval. |
| 322 |
> |
\begin{equation} |
| 323 |
> |
q_r = \frac{KE}{t} |
| 324 |
> |
\label{eq:heat} |
| 325 |
> |
\end{equation} |
| 326 |
> |
|
| 327 |
> |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 328 |
> |
% INTERFACIAL THERMAL CONDUCTANCE |
| 329 |
> |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 330 |
> |
\subsection{Interfacial thermal conductance} |
| 331 |
|
|
| 332 |
+ |
\begin{figure} |
| 333 |
+ |
\includegraphics[width=\linewidth]{figures/NP20} |
| 334 |
+ |
\caption{A gold nanoparticle with a radius of 20 \AA$\,$ solvated in TraPPE-UA hexane. A thermal flux is applied between the nanoparticle and an outer shell of solvent.} |
| 335 |
+ |
\label{fig:NP20} |
| 336 |
+ |
\end{figure} |
| 337 |
+ |
|
| 338 |
+ |
For heterogeneous systems such as a solvated nanoparticle, shown in Figure \ref{fig:NP20}, the interfacial |
| 339 |
+ |
thermal conductance at the surface of the nanoparticle can be determined using an applied kinetic energy flux. |
| 340 |
+ |
We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each |
| 341 |
+ |
shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or |
| 342 |
+ |
multiple interfaces) it is useful to consider the shells as resistors wired in series. The resistance of the |
| 343 |
+ |
shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza |
| 344 |
+ |
resistance. The thermal resistance of each shell is |
| 345 |
+ |
|
| 346 |
+ |
\begin{equation} |
| 347 |
+ |
R_K = \frac{1}{q_r} \Delta T 4 \pi r^2 |
| 348 |
+ |
\label{eq:RK} |
| 349 |
+ |
\end{equation} |
| 350 |
+ |
|
| 351 |
+ |
making the total resistance of two neighboring shells |
| 352 |
+ |
|
| 353 |
+ |
\begin{equation} |
| 354 |
+ |
R_{total} = \frac{1}{q_r} \left [ (T_2 - T_1) 4 \pi r^2_1 + (T_3 - T_2) 4 \pi r^2_2 \right ] = \frac{1}{G} |
| 355 |
+ |
\label{eq:Rtotal} |
| 356 |
+ |
\end{equation} |
| 357 |
+ |
|
| 358 |
+ |
This series can be expanded for any number of adjacent shells, allowing for the calculation of the interfacial |
| 359 |
+ |
thermal conductance for interfaces of considerable thickness, such as self-assembled ligand monolayers on a |
| 360 |
+ |
metal surface. |
| 361 |
+ |
|
| 362 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 363 |
+ |
% INTERFACIAL ROTATIONAL FRICTION |
| 364 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 365 |
+ |
\subsection{Interfacial rotational friction} |
| 366 |
+ |
|
| 367 |
+ |
The interfacial rotational friction, $\Xi^{rr}$, can be calculated for heterogeneous nanostructure/solvent |
| 368 |
+ |
systems by applying an angular momentum flux between the solvated nanostructure and a spherical shell of |
| 369 |
+ |
solvent at the boundary of the cluster. An angular velocity gradient develops in response to the applied flux, |
| 370 |
+ |
causing the nanostructure and solvent shell to rotate in opposite directions about a given axis. |
| 371 |
+ |
|
| 372 |
+ |
\begin{figure} |
| 373 |
+ |
\includegraphics[width=\linewidth]{figures/E25-75} |
| 374 |
+ |
\caption{A gold prolate ellipsoid of length 65 \AA$\,$ and width 25 \AA$\,$ solvated by TraPPE-UA hexane. An angular momentum flux is applied between the ellipsoid and an outer shell of solvent.} |
| 375 |
+ |
\label{fig:E25-75} |
| 376 |
+ |
\end{figure} |
| 377 |
+ |
|
| 378 |
+ |
Analytical solutions for the rotational friction coefficients for a solvated spherical body of radius $r$ under ideal ``stick'' boundary conditions can be estimated using Stokes' law |
| 379 |
+ |
|
| 380 |
+ |
\begin{equation} |
| 381 |
+ |
\Xi^{rr}_{stick} = 8 \pi \eta r^3 |
| 382 |
+ |
\label{eq:Xisphere}. |
| 383 |
+ |
\end{equation} |
| 384 |
+ |
|
| 385 |
+ |
where $\eta$ is the dynamic viscosity of the surrounding solvent. An $\eta$ value for TraPPE-UA hexane under |
| 386 |
+ |
these particular temperature and pressure conditions was determined by applying a traditional VSS-RNEMD linear |
| 387 |
+ |
momentum flux to a periodic box of solvent. |
| 388 |
+ |
|
| 389 |
+ |
For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact |
| 390 |
+ |
solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids under ideal ``stick'' conditions. For simplicity, we define |
| 391 |
+ |
a Perrin Factor, $S$, |
| 392 |
+ |
|
| 393 |
+ |
\begin{equation} |
| 394 |
+ |
S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. |
| 395 |
+ |
\label{eq:S} |
| 396 |
+ |
\end{equation} |
| 397 |
+ |
|
| 398 |
+ |
For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements |
| 399 |
+ |
\begin{equation} |
| 400 |
+ |
\Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S} |
| 401 |
+ |
\label{eq:Xia} |
| 402 |
+ |
\end{equation}\vspace{-0.45in}\\ |
| 403 |
+ |
\begin{equation} |
| 404 |
+ |
\Xi^{rr}_{b,c} = \frac{32 \pi}{2} \eta \frac{ \left( a^4 - b^4 \right)}{ \left( 2a^2 - b^2 \right)S - 2a}. |
| 405 |
+ |
\label{eq:Xibc} |
| 406 |
+ |
\end{equation} |
| 407 |
+ |
|
| 408 |
+ |
corresponding to rotation about the long axis ($a$), and each of the equivalent short axes ($b$ and $c$), respectively. |
| 409 |
+ |
|
| 410 |
+ |
Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown |
| 411 |
+ |
that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} |
| 412 |
+ |
investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained |
| 413 |
+ |
numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ as a function of $\tau$, the |
| 414 |
+ |
ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid |
| 415 |
+ |
shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. Under ``slip'' conditions, |
| 416 |
+ |
$\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate ellipsoid about its long axis approaches |
| 417 |
+ |
$0$, as no solvent is displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for rotation of the |
| 418 |
+ |
prolate ellipsoid about its short axis is $35.9\%$ of the analytical $\Xi^{rr}_{\mathit{stick}}$ result, |
| 419 |
+ |
accounting for the reduced interfacial friction under ``slip'' boundary conditions. |
| 420 |
+ |
|
| 421 |
+ |
The effective rotational friction coefficient, $\Xi^{rr}_{\mathit{eff}}$ at the interface can be extracted from non-periodic VSS-RNEMD simulations quite easily using the applied torque ($\tau$) and the observed angular velocity of the gold structure ($\omega_{Au}$) |
| 422 |
+ |
|
| 423 |
+ |
\begin{equation} |
| 424 |
+ |
\Xi^{rr}_{\mathit{eff}} = \frac{\tau}{\omega_{Au}} |
| 425 |
+ |
\label{eq:Xieff} |
| 426 |
+ |
\end{equation} |
| 427 |
+ |
|
| 428 |
+ |
The applied torque required to overcome the interfacial friction and maintain constant rotation of the gold is |
| 429 |
+ |
|
| 430 |
+ |
\begin{equation} |
| 431 |
+ |
\tau = \frac{L}{2 t} |
| 432 |
+ |
\label{eq:tau} |
| 433 |
+ |
\end{equation} |
| 434 |
+ |
|
| 435 |
+ |
where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation. |
| 436 |
+ |
|
| 437 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 438 |
|
% **TESTS AND APPLICATIONS** |
| 439 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 444 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 445 |
|
\subsection{Thermal conductivities} |
| 446 |
|
|
| 447 |
< |
Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at different kinetic energy flux values are shown in Table \ref{table:goldconductivity}. For these calculations, the hot and cold slabs were excluded from the linear regression of the thermal gradient. The measured linear slope $\langle dT / dr \rangle$ is linearly dependent on the applied kinetic energy flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 - 1.26 W m$^{-1}$ K$^{-1}$ [cite NIVS paper], though still significantly lower than the experimental value of 320 W m$^{-1}$ K$^{-1}$, as the QSC force field neglects significant electronic contributions to heat conduction. The small increase relative to previous simulated bulk values is due to a slight increase in gold density -- as expected, an increase in density results in higher thermal conductivity values. The increased density is a result of nanoparticle curvature relative to an infinite bulk slab, which introduces surface tension that increases ambient density. |
| 447 |
> |
Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at |
| 448 |
> |
different kinetic energy flux values are shown in Table \ref{table:goldTC}. For these calculations, the hot and |
| 449 |
> |
cold slabs were excluded from the linear regression of the thermal gradient. |
| 450 |
|
|
| 451 |
|
\begin{longtable}{ccc} |
| 452 |
|
\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.} |
| 453 |
|
\\ \hline \hline |
| 454 |
< |
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
| 455 |
< |
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 456 |
< |
3.25$\times 10^{-6}$ & 0.11435 & 1.9753 \\ |
| 457 |
< |
6.50$\times 10^{-6}$ & 0.2324 & 1.9438 \\ |
| 458 |
< |
1.30$\times 10^{-5}$ & 0.44922 & 2.0113 \\ |
| 459 |
< |
3.25$\times 10^{-5}$ & 1.1802 & 1.9139 \\ |
| 460 |
< |
6.50$\times 10^{-5}$ & 2.339 & 1.9314 |
| 454 |
> |
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ |
| 455 |
> |
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline |
| 456 |
> |
3.25$\times 10^{-6}$ & 0.11435 & 1.0143 \\ |
| 457 |
> |
6.50$\times 10^{-6}$ & 0.2324 & 0.9982 \\ |
| 458 |
> |
1.30$\times 10^{-5}$ & 0.44922 & 1.0328 \\ |
| 459 |
> |
3.25$\times 10^{-5}$ & 1.1802 & 0.9828 \\ |
| 460 |
> |
6.50$\times 10^{-5}$ & 2.339 & 0.9918 \\ |
| 461 |
> |
\hline |
| 462 |
> |
This work & & 1.0040 |
| 463 |
|
\\ \hline \hline |
| 464 |
< |
\label{table:goldconductivity} |
| 464 |
> |
\label{table:goldTC} |
| 465 |
|
\end{longtable} |
| 182 |
– |
|
| 183 |
– |
SPC/E Water Cluster: |
| 466 |
|
|
| 467 |
+ |
The measured linear slope $\langle \frac{dT}{dr} \rangle$ is linearly dependent on the applied kinetic energy |
| 468 |
+ |
flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 -- 1.26 W / m $\cdot$ K\cite{Kuang2010}, though still significantly lower than the experimental value |
| 469 |
+ |
of 320 W / m $\cdot$ K, as the QSC force field neglects significant electronic contributions to |
| 470 |
+ |
heat conduction. |
| 471 |
+ |
|
| 472 |
+ |
Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table |
| 473 |
+ |
\ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were |
| 474 |
+ |
excluded from the $\langle \frac{dT}{dr} \rangle$ fit. |
| 475 |
+ |
|
| 476 |
|
\begin{longtable}{ccc} |
| 477 |
< |
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
| 477 |
> |
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and 5 atm.} |
| 478 |
|
\\ \hline \hline |
| 479 |
< |
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
| 480 |
< |
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 479 |
> |
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ |
| 480 |
> |
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline |
| 481 |
> |
1$\times 10^{-5}$ & 0.38683 & 0.8698 \\ |
| 482 |
> |
3$\times 10^{-5}$ & 1.1643 & 0.9098 \\ |
| 483 |
> |
6$\times 10^{-5}$ & 2.2262 & 0.8727 \\ |
| 484 |
> |
\hline |
| 485 |
> |
This work & & 0.8841 \\ |
| 486 |
> |
Zhang, et al\cite{Zhang2005} & & 0.81 \\ |
| 487 |
> |
R$\ddot{\mathrm{o}}$mer, et al\cite{Romer2012} & & 0.87 \\ |
| 488 |
> |
Experiment\cite{WagnerKruse} & & 0.61 |
| 489 |
|
\\ \hline \hline |
| 490 |
< |
\label{table:waterconductivity} |
| 490 |
> |
\label{table:waterTC} |
| 491 |
|
\end{longtable} |
| 492 |
|
|
| 493 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 494 |
< |
% SHEAR VISCOSITY |
| 495 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 496 |
< |
\subsection{Shear viscosity} |
| 493 |
> |
Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average |
| 494 |
> |
calculated thermal conductivity from this work, $0.8841$ W / m $\cdot$ K, compares very well to |
| 495 |
> |
previous non-equilibrium molecular dynamics results\cite{Romer2012, Zhang2005} and experimental |
| 496 |
> |
values.\cite{WagnerKruse} |
| 497 |
|
|
| 199 |
– |
SPC/E Water Cluster: |
| 200 |
– |
|
| 498 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 499 |
|
% INTERFACIAL THERMAL CONDUCTANCE |
| 500 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 501 |
|
\subsection{Interfacial thermal conductance} |
| 502 |
|
|
| 503 |
< |
The interfacial thermal conductance, $G$, is calculated by defining a temperature difference $\Delta T$ across a given interface. |
| 503 |
> |
Calculated interfacial thermal conductance ($G$) values for three sizes of gold nanoparticles and Au(111) |
| 504 |
> |
surface solvated in TraPPE-UA hexane are shown in Table \ref{table:G}. |
| 505 |
|
|
| 506 |
+ |
\begin{longtable}{ccc} |
| 507 |
+ |
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous simulation results for a Au(111) interface in TraPPE-UA hexane.} |
| 508 |
+ |
\\ \hline \hline |
| 509 |
+ |
{Nanoparticle Radius} & {$G$}\\ |
| 510 |
+ |
{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline |
| 511 |
+ |
20 & {47.1} \\ |
| 512 |
+ |
30 & {45.4} \\ |
| 513 |
+ |
40 & {46.5} \\ |
| 514 |
+ |
\hline |
| 515 |
+ |
Au(111) & {30.2} |
| 516 |
+ |
\\ \hline \hline |
| 517 |
+ |
\label{table:G} |
| 518 |
+ |
\end{longtable} |
| 519 |
+ |
|
| 520 |
+ |
The introduction of surface curvature increases the interfacial thermal conductance by a factor of |
| 521 |
+ |
approximately $1.5$ relative to the flat interface. There are no significant differences in the $G$ values for |
| 522 |
+ |
the varying nanoparticle sizes. It seems likely that for the range of nanoparticle sizes represented here, any |
| 523 |
+ |
particle size effects are not evident. The simulation of larger nanoparticles may demonstrate an approach to the $G$ value of a flat Au(111) slab but would require prohibitively costly numbers of atoms. |
| 524 |
+ |
|
| 525 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 526 |
|
% INTERFACIAL FRICTION |
| 527 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 528 |
|
\subsection{Interfacial friction} |
| 529 |
|
|
| 530 |
< |
The interfacial friction coefficient, $\kappa$, can be calculated from the solvent dynamic viscosity, $\eta$, and the slip length, $\delta$. The slip length is measured from a radial angular momentum profile generated from a nonperiodic VSS-RNEMD simulation, as shown in Figure X. |
| 530 |
> |
Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold |
| 531 |
> |
nanoparticles and a prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied |
| 532 |
> |
between the A and B regions defined as the gold structure and hexane molecules beyond a certain radius, |
| 533 |
> |
respectively. |
| 534 |
|
|
| 215 |
– |
Table \ref{table:interfacialfriction} shows the calculated interfacial friction coefficients $\kappa$ for a spherical gold nanoparticle and prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied between the A and B regions defined as the gold structure and hexane molecules included in the convex hull, respectively. The resulting angular velocity gradient resulted in the gold structure rotating about the prescribed axis within the solvent. |
| 216 |
– |
|
| 217 |
– |
Analytical solutions for the rotational friction coefficients for a solvated spherical body of radius $r$ under ``stick'' boundary conditions can be estimated using Stokes' law |
| 218 |
– |
|
| 219 |
– |
\begin{equation} |
| 220 |
– |
\Xi = 8 \pi \eta r^3 \label{eq:Xi}. |
| 221 |
– |
\end{equation} |
| 222 |
– |
|
| 223 |
– |
where $\eta$ is the dynamic viscosity of the surrounding solvent and was determined for TraPPE-UA hexane under these condition by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of solvent. |
| 224 |
– |
|
| 225 |
– |
For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids. For simplicity, we define a Perrin Factor, $S$, |
| 226 |
– |
|
| 227 |
– |
\begin{equation} |
| 228 |
– |
S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. \label{eq:S} |
| 229 |
– |
\end{equation} |
| 230 |
– |
|
| 231 |
– |
For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements |
| 232 |
– |
|
| 233 |
– |
\begin{eqnarray} |
| 234 |
– |
\Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S} \label{eq:Xia}\\ |
| 235 |
– |
\Xi^{rr}_{b,c} = \frac{32 \pi}{2} \eta \frac{ \left( a^4 - b^4 \right)}{ \left( 2a^2 - b^2 \right)S - 2a}. \label{eq:Xibc} |
| 236 |
– |
\end{eqnarray} |
| 237 |
– |
|
| 238 |
– |
However, the friction between hexane solvent and gold more likely falls within ``slip'' boundary conditions. Hu and Zwanzig investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor as a function of $\tau$, the ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. According to the values tabulated by Hu and Zwanzig, the sphere friction coefficient approaches $0$, while the ellipsoidal friction coefficient must be scaled by a factor of $0.880$ to account for the reduced interfacial friction under ``slip'' boundary conditions. |
| 239 |
– |
|
| 240 |
– |
Another useful quantity is the friction factor $f$, or the friction coefficient of a non-spherical shape divided by the friction coefficient of a sphere of equivalent volume. The nanoparticle and ellipsoidal nanorod dimensions used here were specifically chosen to create structures with equal volumes. |
| 241 |
– |
|
| 535 |
|
\begin{longtable}{lccccc} |
| 536 |
< |
\caption{Calculated ``slip'' interfacial friction coefficients ($\kappa$) and friction factors ($f$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 536 |
> |
\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' ($\Xi^{rr}_{\mathit{stick}}$) conditions and effective ($\Xi^{rr}_{\mathit{eff}}$) rotational friction coefficients of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 537 |
|
\\ \hline \hline |
| 538 |
< |
{Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$} & {$f_{VSS}$} & {$f_{calc}$}\\ |
| 539 |
< |
{} & {} & {\small($10^4$ Pa s m$^{-1}$)} & {\small($10^4$ Pa s m$^{-1}$)} & {} & {}\\ \hline |
| 540 |
< |
{Sphere} & {$x = y = z$} & {} & {} & {1} & {1}\\ |
| 541 |
< |
{Prolate Ellipsoid} & {$x = y$} & {} & {} & {} & {}\\ |
| 542 |
< |
{Prolate Ellipsoid} & {$z$} & {} & {} & {} & {}\\ \hline \hline |
| 543 |
< |
\label{table:interfacialfriction} |
| 538 |
> |
{Structure} & {Axis of Rotation} & {$\Xi^{rr}_{\mathit{slip}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ |
| 539 |
> |
{} & {} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & \\ \hline |
| 540 |
> |
Sphere (r = 20 \AA) & {$x = y = z$} & 0 & {2386} & {3314} & {0.720}\\ |
| 541 |
> |
Sphere (r = 30 \AA) & {$x = y = z$} & 0 & {8415} & {11749} & {0.716}\\ |
| 542 |
> |
Sphere (r = 40 \AA) & {$x = y = z$} & 0 & {47544} & {34464} & {1.380}\\ |
| 543 |
> |
Prolate Ellipsoid & {$x = y$} & {1792} & {3128} & {4991} & {0.627}\\ |
| 544 |
> |
Prolate Ellipsoid & {$z$} & 0 & {1590} & {1993} & {0.798} |
| 545 |
> |
\\ \hline \hline |
| 546 |
> |
\label{table:couple} |
| 547 |
|
\end{longtable} |
| 548 |
|
|
| 549 |
+ |
The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold |
| 550 |
+ |
structures solvated by hexane do not exist in the ``slip'' boundary condition. At this length scale, the |
| 551 |
+ |
nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience |
| 552 |
+ |
increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius |
| 553 |
+ |
nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of |
| 554 |
+ |
the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short |
| 555 |
+ |
axis, which at first seems counterintuitive. However, the `propellor' motion caused by rotation about the |
| 556 |
+ |
short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the |
| 557 |
+ |
gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius) |
| 558 |
+ |
appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of |
| 559 |
+ |
surface features or anomalous solvent behaviors that are not fully understood at this time. |
| 560 |
+ |
|
| 561 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 562 |
|
% **DISCUSSION** |
| 563 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 564 |
|
\section{Discussion} |
| 565 |
|
|
| 566 |
+ |
We have demonstrated a novel adaptation of the VSS-RNEMD methodology for the application of thermal and angular momentum fluxes in explicitly non-periodic system geometries. Non-periodic VSS-RNEMD preserves the virtues of traditional VSS-RNEMD, namely Boltzmann thermal velocity distributions and minimal thermal anisotropy, while extending the constraints to conserve total energy and total \emph{angular} momentum. We also still have the ability to impose the thermal and angular momentum fluxes simultaneously or individually. |
| 567 |
|
|
| 568 |
+ |
Most strikingly, this method enables calculation of thermal conductivity in homogeneous non-periodic geometries, as well as interfacial thermal conductance and interfacial rotational friction in heterogeneous clusters. The ability to interrogate explicitly non-periodic effects -- such as surface curvature -- on interfacial transport properties is an exciting prospect that will be explored in the future. |
| 569 |
+ |
|
| 570 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 571 |
|
% **ACKNOWLEDGMENTS** |
| 572 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 573 |
|
\section*{Acknowledgments} |
| 574 |
|
|
| 575 |
< |
We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for |
| 576 |
< |
this project was provided by the National Science Foundation under grant |
| 266 |
< |
CHE-0848243. Computational time was provided by the Center for Research |
| 575 |
> |
We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for this project was provided by the |
| 576 |
> |
National Science Foundation under grant CHE-0848243. Computational time was provided by the Center for Research |
| 577 |
|
Computing (CRC) at the University of Notre Dame. |
| 578 |
|
|
| 579 |
|
\newpage |
