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\begin{abstract} |
| 66 |
|
|
| 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 |
|
|
| 74 |
|
\end{abstract} |
| 75 |
|
|
| 82 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 83 |
|
\section{Introduction} |
| 84 |
|
|
| 85 |
< |
Non-equilibrium Molecular Dynamics (NEMD) methods impose a temperature |
| 86 |
< |
or velocity {\it gradient} on a |
| 87 |
< |
system,\cite{ASHURST:1975tg,Evans:1982zk,ERPENBECK:1984sp,MAGINN:1993hc,Berthier:2002ij,Evans:2002ai,Schelling:2002dp,PhysRevA.34.1449,JiangHao_jp802942v} |
| 88 |
< |
and use linear response theory to connect the resulting thermal or |
| 89 |
< |
momentum flux to transport coefficients of bulk materials. However, |
| 90 |
< |
for heterogeneous systems, such as phase boundaries or interfaces, it |
| 88 |
< |
is often unclear what shape of gradient should be imposed at the |
| 89 |
< |
boundary between materials. |
| 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:2 |
| 87 |
> |
002dp,PhysRevA.34.1449,JiangHao_jp802942v} and use linear response theory to connect the resulting thermal or |
| 88 |
> |
momentum flux to transport coefficients of bulk materials. However, for heterogeneous systems, such as phase |
| 89 |
> |
boundaries or interfaces, it is often unclear what shape of gradient should be imposed at the boundary between |
| 90 |
> |
materials. |
| 91 |
|
|
| 92 |
+ |
Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an unphysical {\it flux} between different |
| 93 |
+ |
regions or ``slabs'' of the simulation box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} The system |
| 94 |
+ |
responds by developing a temperature or velocity {\it gradient} between the two regions. The gradients which |
| 95 |
+ |
develop in response to the applied flux are then related (via linear response theory) to the transport |
| 96 |
+ |
coefficient of interest. Since the amount of the applied flux is known exactly, and measurement of a gradient |
| 97 |
+ |
is generally less complicated, imposed-flux methods typically take shorter simulation times to obtain converged |
| 98 |
+ |
results. At interfaces, the observed gradients often exhibit near-discontinuities at the boundaries between |
| 99 |
+ |
dissimilar materials. RNEMD methods do not need many trajectories to provide information about transport |
| 100 |
+ |
properties, and they have become widely used to compute thermal and mechanical transport in both homogeneous |
| 101 |
+ |
liquids and solids~\cite{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as well as heterogeneous |
| 102 |
+ |
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
| 103 |
+ |
|
| 104 |
+ |
|
| 105 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 106 |
+ |
% **METHODOLOGY** |
| 107 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 108 |
+ |
\section{Velocity Shearing and Scaling (VSS) for non-periodic systems} |
| 109 |
+ |
|
| 110 |
+ |
The VSS-RNEMD approach uses a series of simultaneous velocity shearing and scaling exchanges between the two |
| 111 |
+ |
slabs.\cite{Kuang2012} This method imposes energy and momentum conservation constraints while simultaneously |
| 112 |
+ |
creating a desired flux between the two slabs. These constraints ensure that all configurations are sampled |
| 113 |
+ |
from the same microcanonical (NVE) ensemble. |
| 114 |
+ |
|
| 115 |
|
\begin{figure} |
| 116 |
|
\includegraphics[width=\linewidth]{figures/npVSS} |
| 117 |
|
\caption{Schematics of periodic (left) and non-periodic (right) |
| 122 |
|
\label{fig:VSS} |
| 123 |
|
\end{figure} |
| 124 |
|
|
| 125 |
< |
Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an |
| 126 |
< |
unphysical {\it flux} between different regions or ``slabs'' of the |
| 127 |
< |
simulation |
| 128 |
< |
box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} The |
| 129 |
< |
system responds by developing a temperature or velocity {\it gradient} |
| 106 |
< |
between the two regions. The gradients which develop in response to |
| 107 |
< |
the applied flux are then related (via linear response theory) to the |
| 108 |
< |
transport coefficient of interest. Since the amount of the applied |
| 109 |
< |
flux is known exactly, and measurement of a gradient is generally less |
| 110 |
< |
complicated, imposed-flux methods typically take shorter simulation |
| 111 |
< |
times to obtain converged results. At interfaces, the observed |
| 112 |
< |
gradients often exhibit near-discontinuities at the boundaries between |
| 113 |
< |
dissimilar materials. RNEMD methods do not need many trajectories to |
| 114 |
< |
provide information about transport properties, and they have become |
| 115 |
< |
widely used to compute thermal and mechanical transport in both |
| 116 |
< |
homogeneous liquids and |
| 117 |
< |
solids~\cite{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as |
| 118 |
< |
well as heterogeneous |
| 119 |
< |
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
| 125 |
> |
We have extended the VSS method for use in {\it non-periodic} simulations, in which the ``slabs'' have been |
| 126 |
> |
generalized to two separated regions of space. These regions could be defined as concentric spheres (as in |
| 127 |
> |
figure \ref{fig:VSS}), or one of the regions can be defined in terms of a dynamically changing ``hull'' |
| 128 |
> |
comprising the surface atoms of the cluster. This latter definition is identical to the hull used in the |
| 129 |
> |
Langevin Hull algorithm. |
| 130 |
|
|
| 131 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 132 |
< |
% **METHODOLOGY** |
| 133 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 124 |
< |
\section{Velocity Shearing and Scaling (VSS) for non-periodic systems} |
| 131 |
> |
We present here a new set of constraints that are more general than the VSS constraints. For the non-periodic |
| 132 |
> |
variant, the constraints fix both the total energy and total {\it angular} momentum of the system while |
| 133 |
> |
simultaneously imposing a thermal and angular momentum flux between the two regions. |
| 134 |
|
|
| 135 |
< |
The VSS-RNEMD approach uses a series of simultaneous velocity shearing |
| 136 |
< |
and scaling exchanges between the two slabs.\cite{Kuang2012} |
| 137 |
< |
This method imposes energy and momentum conservation constraints while |
| 129 |
< |
simultaneously creating a desired flux between the two slabs. These |
| 130 |
< |
constraints ensure that all configurations are sampled from the same |
| 131 |
< |
microcanonical (NVE) ensemble. |
| 135 |
> |
After each $\Delta t$ time interval, the particle velocities ($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two |
| 136 |
> |
shells ($A$ and $B$) are modified by a velocity scaling coefficient ($a$ and $b$) and by a rotational shearing |
| 137 |
> |
term ($\mathbf{c}_a$ and $\mathbf{c}_b$). |
| 138 |
|
|
| 133 |
– |
We have extended the VSS method for use in {\it non-periodic} |
| 134 |
– |
simulations, in which the ``slabs'' have been generalized to two |
| 135 |
– |
separated regions of space. These regions could be defined as |
| 136 |
– |
concentric spheres (as in figure \ref{fig:VSS}), or one of the regions |
| 137 |
– |
can be defined in terms of a dynamically changing ``hull'' comprising |
| 138 |
– |
the surface atoms of the cluster. This latter definition is identical |
| 139 |
– |
to the hull used in the Langevin Hull algorithm. |
| 140 |
– |
|
| 141 |
– |
We present here a new set of constraints that are more general than |
| 142 |
– |
the VSS constraints. For the non-periodic variant, the constraints |
| 143 |
– |
fix both the total energy and total {\it angular} momentum of the |
| 144 |
– |
system while simultaneously imposing a thermal and angular momentum |
| 145 |
– |
flux between the two regions. |
| 146 |
– |
|
| 147 |
– |
After each $\Delta t$ time interval, the particle velocities |
| 148 |
– |
($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two shells ($A$ and $B$) |
| 149 |
– |
are modified by a velocity scaling coefficient ($a$ and $b$) and by a |
| 150 |
– |
rotational shearing term ($\mathbf{c}_a$ and $\mathbf{c}_b$). |
| 139 |
|
\begin{displaymath} |
| 140 |
|
\begin{array}{rclcl} |
| 141 |
|
& \underline{~~~~~~~~\mathrm{scaling}~~~~~~~~} & & |
| 148 |
|
\rangle \times \mathbf{r}_j\right) & + & \mathbf{c}_b \times \mathbf{r}_j |
| 149 |
|
\end{array} |
| 150 |
|
\end{displaymath} |
| 151 |
< |
Here $\langle\mathbf{\omega}_a\rangle$ and |
| 152 |
< |
$\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular |
| 153 |
< |
velocities of each shell, and $\mathbf{r}_i$ is the position of |
| 154 |
< |
particle $i$ relative to a fixed point in space (usually the center of |
| 167 |
< |
mass of the cluster). Particles in the shells also receive an |
| 168 |
< |
additive ``angular shear'' to their velocities. The amount of shear |
| 169 |
< |
is governed by the imposed angular momentum flux, |
| 151 |
> |
Here $\langle\mathbf{\omega}_a\rangle$ and $\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular |
| 152 |
> |
velocities of each shell, and $\mathbf{r}_i$ is the position of particle $i$ relative to a fixed point in space |
| 153 |
> |
(usually the center of mass of the cluster). Particles in the shells also receive an additive ``angular shear'' |
| 154 |
> |
to their velocities. The amount of shear is governed by the imposed angular momentum flux, |
| 155 |
|
$\mathbf{j}_r(\mathbf{L})$, |
| 156 |
|
\begin{eqnarray} |
| 157 |
|
\mathbf{c}_a & = & - \mathbf{j}_r(\mathbf{L}) \cdot |
| 159 |
|
\mathbf{c}_b & = & + \mathbf{j}_r(\mathbf{L}) \cdot |
| 160 |
|
\overleftrightarrow{I_b}^{-1} \Delta t + \langle \omega_b \rangle \label{eq:bh} |
| 161 |
|
\end{eqnarray} |
| 162 |
< |
where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia tensor for |
| 178 |
< |
each of the two shells. |
| 162 |
> |
where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia tensor for each of the two shells. |
| 163 |
|
|
| 164 |
< |
To simultaneously impose a thermal flux ($J_r$) between the shells we |
| 181 |
< |
use energy conservation constraints, |
| 164 |
> |
To simultaneously impose a thermal flux ($J_r$) between the shells we use energy conservation constraints, |
| 165 |
|
\begin{eqnarray} |
| 166 |
|
K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle |
| 167 |
|
\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a |
| 171 |
|
\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b |
| 172 |
|
\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh} |
| 173 |
|
\end{eqnarray} |
| 174 |
< |
Simultaneous solution of these quadratic formulae for the scaling |
| 175 |
< |
coefficients, $a$ and $b$, will ensure that the simulation samples |
| 176 |
< |
from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ |
| 177 |
< |
is the instantaneous translational kinetic energy of each shell. At |
| 178 |
< |
each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
| 179 |
< |
$\mathbf{c}_b$, subject to the imposed angular momentum flux, |
| 197 |
< |
$j_r(\mathbf{L})$, and thermal flux, $J_r$, values. The new particle |
| 198 |
< |
velocities are computed, and the simulation continues. System |
| 199 |
< |
configurations after the transformations have exactly the same energy |
| 200 |
< |
({\it and} angular momentum) as before the moves. |
| 174 |
> |
Simultaneous solution of these quadratic formulae for the scaling coefficients, $a$ and $b$, will ensure that |
| 175 |
> |
the simulation samples from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ is the instantaneous |
| 176 |
> |
translational kinetic energy of each shell. At each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
| 177 |
> |
$\mathbf{c}_b$, subject to the imposed angular momentum flux, $j_r(\mathbf{L})$, and thermal flux, $J_r$, |
| 178 |
> |
values. The new particle velocities are computed, and the simulation continues. System configurations after the |
| 179 |
> |
transformations have exactly the same energy ({\it and} angular momentum) as before the moves. |
| 180 |
|
|
| 181 |
< |
As the simulation progresses, the velocity transformations can be |
| 182 |
< |
performed on a regular basis, and the system will develop a |
| 183 |
< |
temperature and/or angular velocity gradient in response to the |
| 184 |
< |
applied flux. Using the slope of the radial temperature or velocity |
| 185 |
< |
gradients, it is quite simple to obtain both the thermal conductivity |
| 207 |
< |
($\lambda$), interfacial thermal conductance ($G$), or rotational friction coefficients ($\Xi^{rr}$) of any nonperiodic system. |
| 181 |
> |
As the simulation progresses, the velocity transformations can be performed on a regular basis, and the system |
| 182 |
> |
will develop a temperature and/or angular velocity gradient in response to the applied flux. Using the slope of |
| 183 |
> |
the radial temperature or velocity gradients, it is quite simple to obtain both the thermal conductivity |
| 184 |
> |
($\lambda$), interfacial thermal conductance ($G$), or rotational friction coefficients ($\Xi^{rr}$) of any |
| 185 |
> |
non-periodic system. |
| 186 |
|
|
| 187 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 188 |
|
% **COMPUTATIONAL DETAILS** |
| 189 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 190 |
|
\section{Computational Details} |
| 191 |
|
|
| 192 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 193 |
< |
% NON-PERIODIC DYNAMICS |
| 194 |
< |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 195 |
< |
\subsection{Dynamics for non-periodic systems} |
| 192 |
> |
The new VSS-RNEMD methodology for non-periodic system geometries has been implemented in our group molecular |
| 193 |
> |
dynamics code, OpenMD.\cite{openmd} We have used the new method to calculate the thermal conductance of a gold |
| 194 |
> |
nanoparticle and SPC/E water cluster, and compared the results with previous bulk RNEMD values, as well as |
| 195 |
> |
experiment. We have also investigated the interfacial thermal conductance and interfacial rotational friction |
| 196 |
> |
for gold nanostructures solvated in hexane as a function of nanoparticle size and shape. |
| 197 |
|
|
| 219 |
– |
We have run all tests using the Langevin Hull nonperiodic simulation methodology.\cite{Vardeman2011} The Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different compressibilities, which are typically problematic for traditional affine transform methods. We have had success applying this method to |
| 220 |
– |
several different systems including bare metal nanoparticles, liquid water clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous theoretical and experimental results. 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. |
| 221 |
– |
|
| 198 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 223 |
– |
% SIMULATION PROTOCOL |
| 224 |
– |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 225 |
– |
\subsection{Simulation protocol} |
| 226 |
– |
|
| 227 |
– |
Systems containing liquids were run under moderate pressure ($\sim$ 5 atm) to avoid the formation of a substantial vapor phase. Thermal coupling to the Langevin Hull external bath was turned off to avoid interference with any imposed flux. |
| 228 |
– |
|
| 229 |
– |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 199 |
|
% FORCE FIELD PARAMETERS |
| 200 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 201 |
|
\subsection{Force field parameters} |
| 202 |
|
|
| 203 |
< |
We have chosen the SPC/E water model for these simulations, which is particularly useful for method validation as there are many values for physical properties from previous simulations available for direct comparison.\cite{Bedrov:2000, Kuang2010} |
| 203 |
> |
Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The QSC |
| 204 |
> |
parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and |
| 205 |
> |
include zero-point quantum corrections. |
| 206 |
|
|
| 207 |
< |
Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The QSC parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and include zero-point quantum corrections. |
| 207 |
> |
We have chosen the SPC/E water model for these simulations, which is particularly useful for method validation |
| 208 |
> |
as there are many values for physical properties from previous simulations available for direct |
| 209 |
> |
comparison.\cite{Bedrov:2000, Kuang2010} |
| 210 |
|
|
| 211 |
< |
Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model, |
| 212 |
< |
sites are located at the carbon centers for alkyl groups. Bonding |
| 213 |
< |
interactions, including bond stretches and bends and torsions, were |
| 214 |
< |
used for intra-molecular sites closer than 3 bonds. For non-bonded |
| 215 |
< |
interactions, Lennard-Jones potentials were used. We have previously |
| 216 |
< |
utilized both united atom (UA) and all-atom (AA) force fields for |
| 217 |
< |
thermal conductivity,\cite{kuang:AuThl,Kuang2012} and since the united |
| 218 |
< |
atom force fields cannot populate the high-frequency modes that are |
| 246 |
< |
present in AA force fields, they appear to work better for modeling |
| 247 |
< |
thermal conductivity. |
| 211 |
> |
Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good |
| 212 |
> |
computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model, sites are |
| 213 |
> |
located at the carbon centers for alkyl groups. Bonding interactions, including bond stretches and bends and |
| 214 |
> |
torsions, were used for intra-molecular sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 215 |
> |
potentials were used. We have previously utilized both united atom (UA) and all-atom (AA) force fields for |
| 216 |
> |
thermal conductivity,\cite{kuang:AuThl,Kuang2012} and since the united atom force fields cannot populate the |
| 217 |
> |
high-frequency modes that are present in AA force fields, they appear to work better for modeling thermal |
| 218 |
> |
conductivity. |
| 219 |
|
|
| 220 |
< |
Gold -- hexane nonbonded interactions are governed by pairwise Lennard-Jones parameters derived from Vlugt \emph{et al}.\cite{vlugt:cpc2007154} They fitted parameters for the interaction between Au and CH$_{\emph x}$ pseudo-atoms based on the effective potential of Hautman and Klein for the Au(111) surface.\cite{hautman:4994} |
| 220 |
> |
Gold -- hexane nonbonded interactions are governed by pairwise Lennard-Jones parameters derived from Vlugt |
| 221 |
> |
\emph{et al}.\cite{vlugt:cpc2007154} They fitted parameters for the interaction between Au and CH$_{\emph x}$ |
| 222 |
> |
pseudo-atoms based on the effective potential of Hautman and Klein for the Au(111) surface.\cite{hautman:4994} |
| 223 |
|
|
| 224 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 225 |
+ |
% NON-PERIODIC DYNAMICS |
| 226 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 227 |
+ |
% \subsection{Dynamics for non-periodic systems} |
| 228 |
+ |
% |
| 229 |
+ |
% We have run all tests using the Langevin Hull non-periodic simulation methodology.\cite{Vardeman2011} The |
| 230 |
+ |
% Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different |
| 231 |
+ |
% compressibilities, which are typically problematic for traditional affine transform methods. We have had |
| 232 |
+ |
% success applying this method to several different systems including bare metal nanoparticles, liquid water |
| 233 |
+ |
% clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal |
| 234 |
+ |
% compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous |
| 235 |
+ |
% theoretical and experimental results. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex |
| 236 |
+ |
% hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled |
| 237 |
+ |
% to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are |
| 238 |
+ |
% subject to standard Newtonian dynamics. |
| 239 |
+ |
|
| 240 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 241 |
+ |
% SIMULATION PROTOCOL |
| 242 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 243 |
+ |
\subsection{Simulation protocol} |
| 244 |
+ |
|
| 245 |
+ |
In all cases, systems were fully equilibrated under non-periodic isobaric-isothermal (NPT) conditions -- using |
| 246 |
+ |
the Langevin Hull methodology\cite{Vardeman2011} -- before any non-equilibrium methods were introduced. For |
| 247 |
+ |
heterogeneous systems, the gold nanoparticles and ellipsoid were first created from a bulk lattice and |
| 248 |
+ |
thermally equilibrated before being solvated in hexane. Packmol\cite{packmol} was used to solvate previously |
| 249 |
+ |
equilibrated gold nanostructures within a droplet of hexane. |
| 250 |
+ |
|
| 251 |
+ |
Once fully equilibrated, a thermal or angular momentum flux was applied for 1 - 2 |
| 252 |
+ |
ns, until a stable temperature or angular velocity gradient had developed. Systems containing liquids were run |
| 253 |
+ |
under moderate pressure (5 atm) and temperatures (230 K) to avoid the formation of a substantial vapor phase. 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 |
| 254 |
+ |
imposed flux. |
| 255 |
+ |
|
| 256 |
+ |
To stabilize the gold nanoparticle under the imposed angular momentum flux we altered the gold atom at the |
| 257 |
+ |
designated coordinate origin to have $10,000$ times its original mass. The nonbonded interactions remain |
| 258 |
+ |
unchanged. The heavy atom is excluded from the angular momentum exchange. For rotation of the ellipsoid about |
| 259 |
+ |
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. |
| 260 |
+ |
|
| 261 |
+ |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 262 |
|
% THERMAL CONDUCTIVITIES |
| 263 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 264 |
|
\subsection{Thermal conductivities} |
| 265 |
|
|
| 266 |
< |
Fourier's Law of heat conduction in radial coordinates yields an expression for the heat flow between the concentric spherical RNEMD shells: |
| 266 |
> |
Fourier's Law of heat conduction in radial coordinates yields an expression for the heat flow between the |
| 267 |
> |
concentric spherical RNEMD shells: |
| 268 |
|
|
| 269 |
|
\begin{equation} |
| 270 |
|
q_r = - \frac{4 \pi \lambda (T_b - T_a)}{\frac{1}{r_a} - \frac{1}{r_b}} |
| 271 |
|
\label{eq:Q} |
| 272 |
|
\end{equation} |
| 273 |
|
|
| 274 |
< |
where $\lambda$ is the thermal conductivity, and $T_{a,b}$ and $r_{a,b}$ are the temperatures and radii of the two RNEMD regions, respectively. |
| 274 |
> |
where $\lambda$ is the thermal conductivity, and $T_{a,b}$ and $r_{a,b}$ are the temperatures and radii of the |
| 275 |
> |
two RNEMD regions, respectively. |
| 276 |
|
|
| 277 |
< |
Once a stable thermal gradient has been established between the two regions, the thermal conductivity, $\lambda$, can be calculated using a linear regression of the thermal gradient, $\langle \frac{dT}{dr} \rangle$: |
| 277 |
> |
A thermal flux is created using VSS-RNEMD moves, and the temperature in each of the radial shells is recorded. |
| 278 |
> |
The resulting temperature profiles are analyzed to yield information about the interfacial thermal conductance. |
| 279 |
> |
As the simulation progresses, the VSS moves are performed on a regular basis, and the system develops a thermal |
| 280 |
> |
or velocity gradient in response to the applied flux. Once a stable thermal gradient has been established |
| 281 |
> |
between the two regions, the thermal conductivity, $\lambda$, can be calculated using a linear regression of |
| 282 |
> |
the thermal gradient, $\langle \frac{dT}{dr} \rangle$: |
| 283 |
|
|
| 284 |
|
\begin{equation} |
| 285 |
|
\lambda = \frac{r_a}{r_b} \frac{q_r}{4 \pi \langle \frac{dT}{dr} \rangle} |
| 286 |
|
\label{eq:lambda} |
| 287 |
|
\end{equation} |
| 288 |
|
|
| 289 |
< |
The rate of heat transfer between the two RNEMD regions is the amount of transferred kinetic energy over the length of the simulation, t |
| 289 |
> |
The rate of heat transfer between the two RNEMD regions is the amount of transferred kinetic energy over the |
| 290 |
> |
length of the simulation, t |
| 291 |
|
|
| 292 |
|
\begin{equation} |
| 293 |
|
q_r = \frac{KE}{t} |
| 299 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 300 |
|
\subsection{Interfacial thermal conductance} |
| 301 |
|
|
| 302 |
< |
A thermal flux is created using VSS-RNEMD moves, and the temperature in each of the radial shells is recorded. The resulting temperature |
| 303 |
< |
profiles are analyzed to yield information about the interfacial thermal |
| 304 |
< |
conductance. As the simulation progresses, the VSS moves are performed on a regular basis, and |
| 305 |
< |
the system develops a thermal or velocity gradient in response to the applied |
| 306 |
< |
flux. We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or multiple interfaces) it is useful to consider the shells as resistors wired in series. The resistance of the shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza resistance. The thermal resistance of each shell is |
| 302 |
> |
\begin{figure} |
| 303 |
> |
\includegraphics[width=\linewidth]{figures/NP20} |
| 304 |
> |
\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.} |
| 305 |
> |
\label{fig:NP20} |
| 306 |
> |
\end{figure} |
| 307 |
|
|
| 308 |
+ |
For heterogeneous systems such as a solvated nanoparticle, shown in Figure \ref{fig:NP20}, the interfacial |
| 309 |
+ |
thermal conductance at the surface of the nanoparticle can be determined using an applied kinetic energy flux. |
| 310 |
+ |
We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each |
| 311 |
+ |
shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or |
| 312 |
+ |
multiple interfaces) it is useful to consider the shells as resistors wired in series. The resistance of the |
| 313 |
+ |
shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza |
| 314 |
+ |
resistance. The thermal resistance of each shell is |
| 315 |
+ |
|
| 316 |
|
\begin{equation} |
| 317 |
|
R_K = \frac{1}{q_r} \Delta T 4 \pi r^2 |
| 318 |
|
\label{eq:RK} |
| 325 |
|
\label{eq:Rtotal} |
| 326 |
|
\end{equation} |
| 327 |
|
|
| 328 |
< |
This series can be expanded for any number of adjacent shells, allowing for the calculation of the interfacial thermal conductance for interfaces of significant thickness, such as self-assembled ligand monolayers on a metal surface. |
| 328 |
> |
This series can be expanded for any number of adjacent shells, allowing for the calculation of the interfacial |
| 329 |
> |
thermal conductance for interfaces of significant thickness, such as self-assembled ligand monolayers on a |
| 330 |
> |
metal surface. |
| 331 |
|
|
| 332 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 333 |
< |
% INTERFACIAL FRICTION |
| 333 |
> |
% INTERFACIAL ROTATIONAL FRICTION |
| 334 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 335 |
< |
\subsection{Interfacial friction} |
| 335 |
> |
\subsection{Interfacial rotational friction} |
| 336 |
> |
|
| 337 |
> |
The interfacial rotational friction, $\Xi^{rr}$, can be calculated for heterogeneous nanostructure/solvent |
| 338 |
> |
systems by applying an angular momentum flux between the solvated nanostructure and a spherical shell of |
| 339 |
> |
solvent at the boundary of the cluster. An angular velocity gradient develops in response to the applied flux, |
| 340 |
> |
causing the nanostructure and solvent shell to rotate in opposite directions about a given axis. |
| 341 |
> |
|
| 342 |
> |
\begin{figure} |
| 343 |
> |
\includegraphics[width=\linewidth]{figures/E25-75} |
| 344 |
> |
\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.} |
| 345 |
> |
\label{fig:E25-75} |
| 346 |
> |
\end{figure} |
| 347 |
|
|
| 348 |
|
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 |
| 349 |
|
|
| 350 |
|
\begin{equation} |
| 351 |
< |
\Xi^{rr} = 8 \pi \eta r^3 |
| 352 |
< |
\label{eq:Xistick}. |
| 351 |
> |
\Xi^{rr}_{stick} = 8 \pi \eta r^3 |
| 352 |
> |
\label{eq:Xisphere}. |
| 353 |
|
\end{equation} |
| 354 |
|
|
| 355 |
< |
where $\eta$ is the dynamic viscosity of the surrounding solvent. An $\eta$ value for UA hexane under these particular temperature and pressure conditions was determined by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of solvent. |
| 355 |
> |
where $\eta$ is the dynamic viscosity of the surrounding solvent. An $\eta$ value for TraPPE-UA hexane under |
| 356 |
> |
these particular temperature and pressure conditions was determined by applying a traditional VSS-RNEMD linear |
| 357 |
> |
momentum flux to a periodic box of solvent. |
| 358 |
|
|
| 359 |
< |
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, $S$, |
| 359 |
> |
For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact |
| 360 |
> |
solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids. For simplicity, we define |
| 361 |
> |
a Perrin Factor, $S$, |
| 362 |
|
|
| 363 |
|
\begin{equation} |
| 364 |
|
S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. |
| 375 |
|
\label{eq:Xibc} |
| 376 |
|
\end{equation} |
| 377 |
|
|
| 378 |
< |
The effective rotational friction coefficient at the interface can be extracted from nonperiodic VSS-RNEMD simulations quite easily using the applied torque ($\tau$) and the observed angular velocity of the gold structure ($\omega_{Au}$) |
| 378 |
> |
The effective rotational friction coefficient 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 ($\omega_{Au}$) |
| 379 |
|
|
| 380 |
|
\begin{equation} |
| 381 |
|
\Xi^{rr}_{\mathit{eff}} = \frac{\tau}{\omega_{Au}} |
| 391 |
|
|
| 392 |
|
where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation. |
| 393 |
|
|
| 394 |
< |
Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ 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. Under ``slip'' conditions, $\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate ellipsoid about its long axis approaches $0$, as no solvent is displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for rotation of the prolate ellipsoid about its short axis is $35.9\%$ of the analytical $\Xi^{rr}_{\mathit{stick}}$ result, accounting for the reduced interfacial friction under ``slip'' boundary conditions. |
| 394 |
> |
Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown |
| 395 |
> |
that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} |
| 396 |
> |
investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained |
| 397 |
> |
numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ as a function of $\tau$, the |
| 398 |
> |
ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid |
| 399 |
> |
shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. Under ``slip'' conditions, |
| 400 |
> |
$\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate ellipsoid about its long axis approaches |
| 401 |
> |
$0$, as no solvent is displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for rotation of the |
| 402 |
> |
prolate ellipsoid about its short axis is $35.9\%$ of the analytical $\Xi^{rr}_{\mathit{stick}}$ result, |
| 403 |
> |
accounting for the reduced interfacial friction under ``slip'' boundary conditions. |
| 404 |
|
|
| 405 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 406 |
|
% **TESTS AND APPLICATIONS** |
| 412 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 413 |
|
\subsection{Thermal conductivities} |
| 414 |
|
|
| 415 |
< |
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:goldTC}. For these calculations, the hot and cold slabs were excluded from the linear regression of the thermal gradient. The measured linear slope $\langle \frac{dT}{dr} \rangle$ 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 {\footnotesize W / m $\cdot$ K}\cite{Kuang2010}, though still significantly lower than the experimental value of 320 {\footnotesize W / m $\cdot$ K}, as the QSC force field neglects significant electronic contributions to heat conduction. |
| 415 |
> |
Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at |
| 416 |
> |
different kinetic energy flux values are shown in Table \ref{table:goldTC}. For these calculations, the hot and |
| 417 |
> |
cold slabs were excluded from the linear regression of the thermal gradient. |
| 418 |
|
|
| 365 |
– |
% 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. |
| 366 |
– |
|
| 419 |
|
\begin{longtable}{ccc} |
| 420 |
|
\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.} |
| 421 |
|
\\ \hline \hline |
| 432 |
|
\label{table:goldTC} |
| 433 |
|
\end{longtable} |
| 434 |
|
|
| 435 |
< |
Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table \ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were excluded from the $\langle \frac{dT}{dr} \rangle$ fit. Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average calculated thermal conductivity from this work, $0.8841$ {\footnotesize W / m $\cdot$ K}, compares very well to previous nonequilibrium molecular dynamics results\cite{Romer2012, Zhang2005} and experimental values.\cite{WagnerKruse} |
| 435 |
> |
The measured linear slope $\langle \frac{dT}{dr} \rangle$ is linearly dependent on the applied kinetic energy |
| 436 |
> |
flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 -- 1.26 |
| 437 |
> |
{\footnotesize W / m $\cdot$ K}\cite{Kuang2010}, though still significantly lower than the experimental value |
| 438 |
> |
of 320 {\footnotesize W / m $\cdot$ K}, as the QSC force field neglects significant electronic contributions to |
| 439 |
> |
heat conduction. |
| 440 |
|
|
| 441 |
+ |
Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table |
| 442 |
+ |
\ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were |
| 443 |
+ |
excluded from the $\langle \frac{dT}{dr} \rangle$ fit. |
| 444 |
+ |
|
| 445 |
|
\begin{longtable}{ccc} |
| 446 |
|
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and 5 atm.} |
| 447 |
|
\\ \hline \hline |
| 459 |
|
\label{table:waterTC} |
| 460 |
|
\end{longtable} |
| 461 |
|
|
| 462 |
+ |
Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average |
| 463 |
+ |
calculated thermal conductivity from this work, $0.8841$ {\footnotesize W / m $\cdot$ K}, compares very well to |
| 464 |
+ |
previous non-equilibrium molecular dynamics results\cite{Romer2012, Zhang2005} and experimental |
| 465 |
+ |
values.\cite{WagnerKruse} |
| 466 |
+ |
|
| 467 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 468 |
|
% INTERFACIAL THERMAL CONDUCTANCE |
| 469 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 470 |
|
\subsection{Interfacial thermal conductance} |
| 471 |
|
|
| 472 |
< |
Calculated interfacial thermal conductance ($G$) values for three sizes of gold nanoparticles and Au(111) surface solvated in TraPPE-UA hexane. The introduction of surface curvature increases the interfacial thermal conductance by a factor of approximately $1.5$ relative to the flat interface. There are no significant differences in the $G$ values for the varying nanoparticle sizes. It seems likely that for the range of nanoparticle sizes represented here, size effects are not evident. |
| 472 |
> |
Calculated interfacial thermal conductance ($G$) values for three sizes of gold nanoparticles and Au(111) |
| 473 |
> |
surface solvated in TraPPE-UA hexane are shown in Table \ref{table:G}. |
| 474 |
|
|
| 475 |
|
\begin{longtable}{ccc} |
| 476 |
< |
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous results for a Au(111) interface in TraPPE-UA hexane, revealing increased interfacial thermal conductance for non-planar interfaces.} |
| 476 |
> |
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous results for a Au(111) interface in TraPPE-UA hexane.} |
| 477 |
|
\\ \hline \hline |
| 478 |
|
{Nanoparticle Radius} & {$G$}\\ |
| 479 |
|
{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline |
| 483 |
|
\hline |
| 484 |
|
Au(111) & {30.2} |
| 485 |
|
\\ \hline \hline |
| 486 |
< |
\label{table:interfacialconductance} |
| 486 |
> |
\label{table:G} |
| 487 |
|
\end{longtable} |
| 488 |
|
|
| 489 |
+ |
The introduction of surface curvature increases the interfacial thermal conductance by a factor of |
| 490 |
+ |
approximately $1.5$ relative to the flat interface. There are no significant differences in the $G$ values for |
| 491 |
+ |
the varying nanoparticle sizes. It seems likely that for the range of nanoparticle sizes represented here, any |
| 492 |
+ |
particle size effects are not evident. |
| 493 |
+ |
|
| 494 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 495 |
|
% INTERFACIAL FRICTION |
| 496 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 497 |
|
\subsection{Interfacial friction} |
| 498 |
|
|
| 499 |
< |
Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold nanoparticles and a 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 beyond a certain radius, respectively. The resulting angular velocity gradient causes the gold structure to rotate about the prescribed axis. |
| 499 |
> |
Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold |
| 500 |
> |
nanoparticles and a prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied |
| 501 |
> |
between the A and B regions defined as the gold structure and hexane molecules beyond a certain radius, |
| 502 |
> |
respectively. |
| 503 |
|
|
| 504 |
|
\begin{longtable}{lccccc} |
| 505 |
< |
\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' conditions ($\Xi^{rr}_{\mathit{stick}}$) and effective rotational friction coefficients ($\Xi^{rr}_{\mathit{eff}}$) of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 505 |
> |
\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.} |
| 506 |
|
\\ \hline \hline |
| 507 |
|
{Structure} & {Axis of Rotation} & {$\Xi^{rr}_{\mathit{slip}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ |
| 508 |
|
{} & {} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & \\ \hline |
| 515 |
|
\label{table:couple} |
| 516 |
|
\end{longtable} |
| 517 |
|
|
| 518 |
< |
The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold structures solvated by hexane do not exist in the ``slip'' boundary conditions. At this length scale, the nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short axis, which may at first seem counterintuitive. However, the `propellor' motion caused by rotation about short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius) appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of surface features or anomalous solvent behaviors that are not fully understood at this time. |
| 518 |
> |
The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold |
| 519 |
> |
structures solvated by hexane do not exist in the ``slip'' boundary conditions. At this length scale, the |
| 520 |
> |
nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience |
| 521 |
> |
increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius |
| 522 |
> |
nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of |
| 523 |
> |
the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short |
| 524 |
> |
axis, which may at first seem counterintuitive. However, the `propellor' motion caused by rotation about the |
| 525 |
> |
short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the |
| 526 |
> |
gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius) |
| 527 |
> |
appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of |
| 528 |
> |
surface features or anomalous solvent behaviors that are not fully understood at this time. |
| 529 |
|
|
| 530 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 531 |
|
% **DISCUSSION** |
| 532 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 533 |
|
\section{Discussion} |
| 534 |
|
|
| 535 |
+ |
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. |
| 536 |
|
|
| 537 |
+ |
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 investigated in the future. |
| 538 |
+ |
|
| 539 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 540 |
|
% **ACKNOWLEDGMENTS** |
| 541 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 542 |
|
\section*{Acknowledgments} |
| 543 |
|
|
| 544 |
< |
We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for |
| 545 |
< |
this project was provided by the National Science Foundation under grant |
| 459 |
< |
CHE-0848243. Computational time was provided by the Center for Research |
| 544 |
> |
We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for this project was provided by the |
| 545 |
> |
National Science Foundation under grant CHE-0848243. Computational time was provided by the Center for Research |
| 546 |
|
Computing (CRC) at the University of Notre Dame. |
| 547 |
|
|
| 548 |
|
\newpage |
