41 |
|
applies an external pressure to the facets comprising the convex |
42 |
|
hull surrounding the objects in the system. Additionally, a Langevin |
43 |
|
thermostat is applied to facets of the hull to mimic contact with an |
44 |
< |
external heat bath. This new method, the ``Langevin Hull'', |
45 |
< |
performs better than traditional affine transform methods for |
46 |
< |
systems containing heterogeneous mixtures of materials with |
47 |
< |
different compressibilities. It does not suffer from the edge |
48 |
< |
effects of boundary potential methods, and allows realistic |
49 |
< |
treatment of both external pressure and thermal conductivity to an |
50 |
< |
implicit solvents. We apply this method to several different |
51 |
< |
systems including bare nano-particles, nano-particles in explicit |
52 |
< |
solvent, as well as clusters of liquid water and ice. The predicted |
53 |
< |
mechanical and thermal properties of these systems are in good |
54 |
< |
agreement with experimental data. |
44 |
> |
external heat bath. This new method, the ``Langevin Hull'', performs |
45 |
> |
better than traditional affine transform methods for systems |
46 |
> |
containing heterogeneous mixtures of materials with different |
47 |
> |
compressibilities. It does not suffer from the edge effects of |
48 |
> |
boundary potential methods, and allows realistic treatment of both |
49 |
> |
external pressure and thermal conductivity to an implicit solvent. |
50 |
> |
We apply this method to several different systems including bare |
51 |
> |
nanoparticles, nanoparticles in an explicit solvent, as well as |
52 |
> |
clusters of liquid water and ice. The predicted mechanical and |
53 |
> |
thermal properties of these systems are in good agreement with |
54 |
> |
experimental data. |
55 |
|
\end{abstract} |
56 |
|
|
57 |
|
\newpage |
75 |
|
system geometry. An affine transform scales both the box lengths as |
76 |
|
well as the scaled particle positions (but not the sizes of the |
77 |
|
particles). The most common constant pressure methods, including the |
78 |
< |
Melchionna modification\cite{melchionna93} to the |
79 |
< |
Nos\'e-Hoover-Andersen equations of motion, the Berendsen pressure |
80 |
< |
bath, and the Langevin Piston, all utilize coordinate transformation |
81 |
< |
to adjust the box volume. |
78 |
> |
Melchionna modification\cite{Melchionna1993} to the |
79 |
> |
Nos\'e-Hoover-Andersen equations of |
80 |
> |
motion,\cite{Hoover85,ANDERSEN:1980vn,Sturgeon:2000kx} the Berendsen |
81 |
> |
pressure bath,\cite{ISI:A1984TQ73500045} and the Langevin |
82 |
> |
Piston,\cite{FELLER:1995fk,Jakobsen:2005uq} all utilize coordinate |
83 |
> |
transformation to adjust the box volume. |
84 |
|
|
85 |
+ |
As long as the material in the simulation box is essentially a bulk |
86 |
+ |
liquid which has a relatively uniform compressibility, the standard |
87 |
+ |
approach provides an excellent way of adjusting the volume of the |
88 |
+ |
system and applying pressure directly via the interactions between |
89 |
+ |
atomic sites. |
90 |
+ |
|
91 |
+ |
The problem with these approaches becomes apparent when the material |
92 |
+ |
being simulated is an inhomogeneous mixture in which portions of the |
93 |
+ |
simulation box are incompressible relative to other portions. |
94 |
+ |
Examples include simulations of metallic nanoparticles in liquid |
95 |
+ |
environments, proteins at interfaces, as well as other multi-phase or |
96 |
+ |
interfacial environments. In these cases, the affine transform of |
97 |
+ |
atomic coordinates will either cause numerical instability when the |
98 |
+ |
sites in the incompressible medium collide with each other, or lead to |
99 |
+ |
inefficient sampling of system volumes if the barostat is set slow |
100 |
+ |
enough to avoid collisions in the incompressible region. |
101 |
+ |
|
102 |
|
\begin{figure} |
103 |
|
\includegraphics[width=\linewidth]{AffineScale2} |
104 |
|
\caption{Affine Scaling constant pressure methods use box-length |
111 |
|
\label{affineScale} |
112 |
|
\end{figure} |
113 |
|
|
114 |
+ |
Additionally, one may often wish to simulate explicitly non-periodic |
115 |
+ |
systems, and the constraint that a periodic box must be used to |
116 |
|
|
96 |
– |
Heterogeneous mixtures of materials with different compressibilities? |
97 |
– |
|
117 |
|
Explicitly non-periodic systems |
118 |
|
|
119 |
|
Elastic Bag |
122 |
|
|
123 |
|
\section{Methodology} |
124 |
|
|
125 |
< |
A new method which uses a constant pressure and temperature bath that |
126 |
< |
interacts with the objects that are currently at the edge of the |
127 |
< |
system. |
125 |
> |
We have developed a new method which uses a constant pressure and |
126 |
> |
temperature bath. This bath interacts only with the objects that are |
127 |
> |
currently at the edge of the system. Since the edge is determined |
128 |
> |
dynamically as the simulation progresses, no {\it a priori} geometry |
129 |
> |
is defined. The pressure and temperature bath interacts {\it |
130 |
> |
directly} with the atoms on the edge and not with atoms interior to |
131 |
> |
the simulation. This means that there are no affine transforms |
132 |
> |
required. There are also no fictitious particles or bounding |
133 |
> |
potentials used in this approach. |
134 |
|
|
135 |
< |
Novel features: No a priori geometry is defined, No affine transforms, |
136 |
< |
No fictitious particles, No bounding potentials. |
135 |
> |
The basics of the method are as follows. The simulation starts as a |
136 |
> |
collection of atomic locations in three dimensions (a point cloud). |
137 |
> |
Delaunay triangulation is used to find all facets between coplanar |
138 |
> |
neighbors. In highly symmetric point clouds, facets can contain many |
139 |
> |
atoms, but in all but the most symmetric of cases one might experience |
140 |
> |
in a molecular dynamics simulation, the facets are simple triangles in |
141 |
> |
3-space that contain exactly three atoms. |
142 |
|
|
143 |
< |
Simulation starts as a collection of atomic locations in 3D (a point |
144 |
< |
cloud). |
145 |
< |
|
146 |
< |
Delaunay triangulation finds all facets between coplanar neighbors. |
147 |
< |
|
148 |
< |
The Convex Hull is the set of facets that have no concave corners at a |
119 |
< |
vertex. |
120 |
< |
|
121 |
< |
Molecules on the convex hull are dynamic. As they re-enter the |
122 |
< |
cluster, all interactions with the external bath are removed.The |
123 |
< |
external bath applies pressure to the facets of the convex hull in |
124 |
< |
direct proportion to the area of the facet. Thermal coupling depends on |
125 |
< |
the solvent temperature, friction and the size and shape of each |
126 |
< |
facet. |
143 |
> |
The convex hull is the set of facets that have {\it no concave |
144 |
> |
corners} at an atomic site. This eliminates all facets on the |
145 |
> |
interior of the point cloud, leaving only those exposed to the |
146 |
> |
bath. Sites on the convex hull are dynamic. As molecules re-enter the |
147 |
> |
cluster, all interactions between atoms on that molecule and the |
148 |
> |
external bath are removed. |
149 |
|
|
150 |
+ |
For atomic sites in the interior of the point cloud, the equations of |
151 |
+ |
motion are simple Newtonian dynamics, |
152 |
|
\begin{equation} |
153 |
< |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U |
153 |
> |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U, |
154 |
> |
\label{eq:Newton} |
155 |
|
\end{equation} |
156 |
< |
|
156 |
> |
where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the |
157 |
> |
instantaneous velocity of site $i$ at time $t$, and $U$ is the total |
158 |
> |
potential energy. For atoms on the exterior of the cluster |
159 |
> |
(i.e. those that occupy one of the vertices of the convex hull), the |
160 |
> |
equation of motion is modified with an external force, ${\mathbf |
161 |
> |
F}_i^{\mathrm ext}$, |
162 |
|
\begin{equation} |
163 |
< |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext} |
163 |
> |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
164 |
|
\end{equation} |
165 |
|
|
166 |
+ |
The external bath interacts directly with the facets of the convex |
167 |
+ |
hull. Since each vertex (or atom) provides one corner of a triangular |
168 |
+ |
facet, the force on the facets are divided equally to each vertex. |
169 |
+ |
However, each vertex can participate in multiple facets, so the resultant |
170 |
+ |
force is a sum over all facets $f$ containing vertex $i$: |
171 |
|
\begin{equation} |
172 |
|
{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\ |
173 |
|
} f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\ {\mathbf |
174 |
|
F}_f^{\mathrm ext} |
175 |
|
\end{equation} |
176 |
|
|
177 |
+ |
The external pressure bath applies a force to the facets of the convex |
178 |
+ |
hull in direct proportion to the area of the facet, while the thermal |
179 |
+ |
coupling depends on the solvent temperature, friction and the size and |
180 |
+ |
shape of each facet. The thermal interactions are expressed as a |
181 |
+ |
typical Langevin description of the forces, |
182 |
|
\begin{equation} |
183 |
|
\begin{array}{rclclcl} |
184 |
|
{\mathbf F}_f^{\text{ext}} & = & \text{external pressure} & + & \text{drag force} & + & \text{random force} \\ |
185 |
|
& = & -\hat{n}_f P A_f & - & \Xi_f(t) {\mathbf v}_f(t) & + & {\mathbf R}_f(t) |
186 |
|
\end{array} |
187 |
|
\end{equation} |
188 |
< |
|
188 |
> |
Here, $P$ is the external pressure, $A_f$ and $\hat{n}_f$ are the area |
189 |
> |
and normal vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is |
190 |
> |
the velocity of the facet, |
191 |
> |
\begin{equation} |
192 |
> |
{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
193 |
> |
\end{equation} |
194 |
> |
and $\Xi_f(t)$ is a ($3 \times 3$) hydrodynamic tensor that depends on |
195 |
> |
the geometry and surface area of facet $f$ and the viscosity of the |
196 |
> |
fluid (See Appendix A). The hydrodynamic tensor is related to the |
197 |
> |
fluctuations of the random force, $\mathbf{R}(t)$, by the |
198 |
> |
fluctuation-dissipation theorem, |
199 |
|
\begin{eqnarray} |
150 |
– |
A_f & = & \text{area of facet}\ f \\ |
151 |
– |
\hat{n}_f & = & \text{facet normal} \\ |
152 |
– |
P & = & \text{external pressure} |
153 |
– |
\end{eqnarray} |
154 |
– |
|
155 |
– |
\begin{eqnarray} |
156 |
– |
{\mathbf v}_f(t) & = & \text{velocity of facet} \\ |
157 |
– |
& = & \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i \\ |
158 |
– |
\Xi_f(t) & = & \text{is a hydrodynamic tensor that depends} \\ |
159 |
– |
& & \text{on the geometry and surface area of} \\ |
160 |
– |
& & \text{facet} \ f\ \text{and the viscosity of the fluid.} |
161 |
– |
\end{eqnarray} |
162 |
– |
|
163 |
– |
\begin{eqnarray} |
200 |
|
\left< {\mathbf R}_f(t) \right> & = & 0 \\ |
201 |
|
\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\ |
202 |
< |
\Xi_f(t)\delta(t-t^\prime) |
202 |
> |
\Xi_f(t)\delta(t-t^\prime). |
203 |
> |
\label{eq:randomForce} |
204 |
|
\end{eqnarray} |
205 |
|
|
206 |
< |
Implemented in OpenMD.\cite{Meineke:2005gd,openmd} |
206 |
> |
Once the hydrodynamic tensor is known for a given facet (see Appendix |
207 |
> |
A) obtaining a stochastic vector that has the properties in |
208 |
> |
Eq. (\ref{eq:randomForce}) can be done efficiently by carrying out a |
209 |
> |
one-time Cholesky decomposition to obtain the square root matrix of |
210 |
> |
the resistance tensor, |
211 |
> |
\begin{equation} |
212 |
> |
\Xi_f = {\bf S} {\bf S}^{T}, |
213 |
> |
\label{eq:Cholesky} |
214 |
> |
\end{equation} |
215 |
> |
where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A |
216 |
> |
vector with the statistics required for the random force can then be |
217 |
> |
obtained by multiplying ${\bf S}$ onto a random 3-vector ${\bf Z}$ which |
218 |
> |
has elements chosen from a Gaussian distribution, such that: |
219 |
> |
\begin{equation} |
220 |
> |
\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot |
221 |
> |
{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij}, |
222 |
> |
\end{equation} |
223 |
> |
where $\delta t$ is the timestep in use during the simulation. The |
224 |
> |
random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to |
225 |
> |
have the correct properties required by Eq. (\ref{eq:randomForce}). |
226 |
|
|
227 |
+ |
We have implemented this method by extending the Langevin dynamics |
228 |
+ |
integrator in our group code, OpenMD.\cite{Meineke2005,openmd} |
229 |
+ |
|
230 |
|
\section{Tests \& Applications} |
231 |
|
|
232 |
|
\subsection{Bulk modulus of gold nanoparticles} |
274 |
|
|
275 |
|
to calculate the the isothermal compressibility at each target pressure. These calculations yielded compressibility values that were dramatically higher than both previous simulations and experiment. The particular compressibility expression used requires the calculation of both a volume and pressure differential, thereby stipulating that the data from at least two simulations at different pressures must be used to calculate the isothermal compressibility at one pressure. |
276 |
|
|
277 |
< |
Per the fluctuation dissipation theorem \cite{Debendedetti1986}, the hull volume fluctuation in any given simulation can be used to calculated the isothermal compressibility at that particular pressure |
277 |
> |
Per the fluctuation dissipation theorem \cite{Debenedetti1986}, the hull volume fluctuation in any given simulation can be used to calculated the isothermal compressibility at that particular pressure |
278 |
|
|
279 |
|
\begin{equation} |
280 |
|
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle V \right \rangle ^{2}}{V \, k_{B} \, T} |