| 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 |
| 83 |
> |
transformation to adjust the box volume. As long as the material in |
| 84 |
> |
the simulation box is essentially a bulk-like liquid which has a |
| 85 |
> |
relatively uniform compressibility, the standard affine transform |
| 86 |
|
approach provides an excellent way of adjusting the volume of the |
| 87 |
|
system and applying pressure directly via the interactions between |
| 88 |
< |
atomic sites. |
| 88 |
> |
atomic sites. |
| 89 |
|
|
| 90 |
< |
The problem with these approaches becomes apparent when the material |
| 90 |
> |
The problem with this approach becomes apparent when the material |
| 91 |
|
being simulated is an inhomogeneous mixture in which portions of the |
| 92 |
|
simulation box are incompressible relative to other portions. |
| 93 |
|
Examples include simulations of metallic nanoparticles in liquid |
| 96 |
|
atomic coordinates will either cause numerical instability when the |
| 97 |
|
sites in the incompressible medium collide with each other, or lead to |
| 98 |
|
inefficient sampling of system volumes if the barostat is set slow |
| 99 |
< |
enough to avoid collisions in the incompressible region. |
| 99 |
> |
enough to avoid the instabilities in the incompressible region. |
| 100 |
|
|
| 101 |
|
\begin{figure} |
| 102 |
|
\includegraphics[width=\linewidth]{AffineScale2} |
| 110 |
|
\label{affineScale} |
| 111 |
|
\end{figure} |
| 112 |
|
|
| 113 |
< |
Additionally, one may often wish to simulate explicitly non-periodic |
| 114 |
< |
systems, and the constraint that a periodic box must be used to |
| 113 |
> |
One may also wish to avoid affine transform periodic boundary methods |
| 114 |
> |
to simulate {\it explicitly non-periodic systems} under constant |
| 115 |
> |
pressure conditions. The use of periodic boxes to enforce a system |
| 116 |
> |
volume either requires effective solute concentrations that are much |
| 117 |
> |
higher than desirable, or unreasonable system sizes to avoid this |
| 118 |
> |
effect. For example, calculations using typical hydration shells |
| 119 |
> |
solvating a protein under periodic boundary conditions are quite |
| 120 |
> |
expensive. [CALCULATE EFFECTIVE PROTEIN CONCENTRATIONS IN TYPICAL |
| 121 |
> |
SIMULATIONS] |
| 122 |
|
|
| 123 |
< |
Explicitly non-periodic systems |
| 123 |
> |
There have been a number of other approaches to explicit |
| 124 |
> |
non-periodicity that focus on constant or nearly-constant {\it volume} |
| 125 |
> |
conditions while maintaining bulk-like behavior. Berkowitz and |
| 126 |
> |
McCammon introduced a stochastic (Langevin) boundary layer inside a |
| 127 |
> |
region of fixed molecules which effectively enforces constant |
| 128 |
> |
temperature and volume (NVT) conditions.\cite{Berkowitz1982} In this |
| 129 |
> |
approach, the stochastic and fixed regions were defined relative to a |
| 130 |
> |
central atom. Brooks and Karplus extended this method to include |
| 131 |
> |
deformable stochastic boundaries.\cite{iii:6312} The stochastic |
| 132 |
> |
boundary approach has been used widely for protein |
| 133 |
> |
simulations. [CITATIONS NEEDED] |
| 134 |
|
|
| 135 |
< |
Elastic Bag |
| 135 |
> |
The electrostatic and dispersive behavior near the boundary has long |
| 136 |
> |
been a cause for concern. King and Warshel introduced a surface |
| 137 |
> |
constrained all-atom solvent (SCAAS) which included polarization |
| 138 |
> |
effects of a fixed spherical boundary to mimic bulk-like behavior |
| 139 |
> |
without periodic boundaries.\cite{king:3647} In the SCAAS model, a |
| 140 |
> |
layer of fixed solvent molecules surrounds the solute and any explicit |
| 141 |
> |
solvent, and this in turn is surrounded by a continuum dielectric. |
| 142 |
> |
MORE HERE. WHAT DID THEY FIND? |
| 143 |
|
|
| 144 |
< |
Spherical Boundary approaches |
| 144 |
> |
Beglov and Roux developed a boundary model in which the hard sphere |
| 145 |
> |
boundary has a radius that varies with the instantaneous configuration |
| 146 |
> |
of the solute (and solvent) molecules.\cite{beglov:9050} This model |
| 147 |
> |
contains a clear pressure and surface tension contribution to the free |
| 148 |
> |
energy which XXX. |
| 149 |
|
|
| 150 |
+ |
Restraining {\it potentials} introduce repulsive potentials at the |
| 151 |
+ |
surface of a sphere or other geometry. The solute and any explicit |
| 152 |
+ |
solvent are therefore restrained inside this potential. Often the |
| 153 |
+ |
potentials include a weak short-range attraction to maintain the |
| 154 |
+ |
correct density at the boundary. Beglov and Roux have also introduced |
| 155 |
+ |
a restraining boundary potential which relaxes dynamically depending |
| 156 |
+ |
on the solute geometry and the force the explicit system exerts on the |
| 157 |
+ |
shell.\cite{Beglov:1995fk} |
| 158 |
+ |
|
| 159 |
+ |
Recently, Krilov {\it et al.} introduced a flexible boundary model |
| 160 |
+ |
that uses a Lennard-Jones potential between the solvent molecules and |
| 161 |
+ |
a boundary which is determined dynamically from the position of the |
| 162 |
+ |
nearest solute atom.\cite{LiY._jp046852t,Zhu:xw} This approach allows |
| 163 |
+ |
the confining potential to prevent solvent molecules from migrating |
| 164 |
+ |
too far from the solute surface, while providing a weak attractive |
| 165 |
+ |
force pulling the solvent molecules towards a fictitious bulk solvent. |
| 166 |
+ |
Although this approach is appealing and has physical motivation, |
| 167 |
+ |
nanoparticles do not deform far from their original geometries even at |
| 168 |
+ |
temperatures which vaporize the nearby solvent. For the systems like |
| 169 |
+ |
the one described, the flexible boundary model will be nearly |
| 170 |
+ |
identical to a fixed-volume restraining potential. |
| 171 |
+ |
|
| 172 |
+ |
The approach of Kohanoff, Caro, and Finnis is the most promising of |
| 173 |
+ |
the methods for introducing both constant pressure and temperature |
| 174 |
+ |
into non-periodic simulations.\cite{Kohanoff:2005qm,Baltazar:2006ru} |
| 175 |
+ |
This method is based on standard Langevin dynamics, but the Brownian |
| 176 |
+ |
or random forces are allowed to act only on peripheral atoms and exert |
| 177 |
+ |
force in a direction that is inward-facing relative to the facets of a |
| 178 |
+ |
closed bounding surface. The statistical distribution of the random |
| 179 |
+ |
forces are uniquely tied to the pressure in the external reservoir, so |
| 180 |
+ |
the method can be shown to sample the isobaric-isothermal ensemble. |
| 181 |
+ |
Kohanoff {\it et al.} used a Delaunay tessellation to generate a |
| 182 |
+ |
bounding surface surrounding the outermost atoms in the simulated |
| 183 |
+ |
system. This is not the only possible triangulated outer surface, but |
| 184 |
+ |
guarantees that all of the random forces point inward towards the |
| 185 |
+ |
cluster. |
| 186 |
+ |
|
| 187 |
+ |
In the following sections, we extend and generalize the approach of |
| 188 |
+ |
Kohanoff, Caro, and Finnis. The new method, which we are calling the |
| 189 |
+ |
``Langevin Hull'' applies the external pressure, Langevin drag, and |
| 190 |
+ |
random forces on the facets of the {\it hull itself} instead of the |
| 191 |
+ |
atomic sites comprising the vertices of the hull. This allows us to |
| 192 |
+ |
decouple the external pressure contribution from the drag and random |
| 193 |
+ |
force. Section \ref{sec:meth} |
| 194 |
+ |
|
| 195 |
|
\section{Methodology} |
| 196 |
+ |
\label{sec:meth} |
| 197 |
|
|
| 198 |
|
We have developed a new method which uses a constant pressure and |
| 199 |
|
temperature bath. This bath interacts only with the objects that are |
| 264 |
|
\begin{equation} |
| 265 |
|
{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
| 266 |
|
\end{equation} |
| 267 |
< |
and $\Xi_f(t)$ is a ($3 \times 3$) hydrodynamic tensor that depends on |
| 268 |
< |
the geometry and surface area of facet $f$ and the viscosity of the |
| 269 |
< |
fluid (See Appendix A). The hydrodynamic tensor is related to the |
| 270 |
< |
fluctuations of the random force, $\mathbf{R}(t)$, by the |
| 271 |
< |
fluctuation-dissipation theorem, |
| 267 |
> |
and $\Xi_f(t)$ is an approximate ($3 \times 3$) hydrodynamic tensor |
| 268 |
> |
that depends on the geometry and surface area of facet $f$ and the |
| 269 |
> |
viscosity of the fluid (See Appendix A). The hydrodynamic tensor is |
| 270 |
> |
related to the fluctuations of the random force, $\mathbf{R}(t)$, by |
| 271 |
> |
the fluctuation-dissipation theorem, |
| 272 |
|
\begin{eqnarray} |
| 273 |
|
\left< {\mathbf R}_f(t) \right> & = & 0 \\ |
| 274 |
|
\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\ |
| 297 |
|
random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to |
| 298 |
|
have the correct properties required by Eq. (\ref{eq:randomForce}). |
| 299 |
|
|
| 300 |
+ |
Our treatment of the hydrodynamic tensor must be approximate. $\Xi$ |
| 301 |
+ |
for a triangular plate would normally be treated as a $6 \times 6$ |
| 302 |
+ |
tensor that includes translational and rotational drag as well as |
| 303 |
+ |
translational-rotational coupling. The computation of hydrodynamic |
| 304 |
+ |
tensors for rigid bodies has been detailed |
| 305 |
+ |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun2008} |
| 306 |
+ |
but the standard approach involving bead approximations would be |
| 307 |
+ |
prohibitively expensive if it were recomputed at each step in a |
| 308 |
+ |
molecular dynamics simulation. |
| 309 |
+ |
|
| 310 |
+ |
We are utilizing an approximate hydrodynamic tensor obtained by first |
| 311 |
+ |
constructing the Oseen tensor for the interaction of the centroid of |
| 312 |
+ |
the facet ($f$) with each of the subfacets $j$, |
| 313 |
+ |
\begin{equation} |
| 314 |
+ |
T_{jf}=\frac{A_j}{8\pi\eta R_{jf}}\left(I + |
| 315 |
+ |
\frac{\mathbf{R}_{jf}\mathbf{R}_{jf}^T}{R_{jf}^2}\right) |
| 316 |
+ |
\end{equation} |
| 317 |
+ |
Here, $A_j$ is the area of subfacet $j$ which is a triangle containing |
| 318 |
+ |
two of the vertices of the facet along with the centroid. |
| 319 |
+ |
$\mathbf{R}_{jf}$ is the vector between the centroid of facet $f$ and |
| 320 |
+ |
the centroid of sub-facet $j$, and $I$ is the ($3 \times 3$) identity |
| 321 |
+ |
matrix. $\eta$ is the viscosity of the external bath. |
| 322 |
+ |
|
| 323 |
+ |
\begin{figure} |
| 324 |
+ |
\includegraphics[width=\linewidth]{hydro} |
| 325 |
+ |
\caption{The hydrodynamic tensor $\Xi$ for a facet comprising sites $i$, |
| 326 |
+ |
$j$, and $k$ is constructed using Oseen tensor contributions |
| 327 |
+ |
between the centoid of the facet $f$ and each of the sub-facets |
| 328 |
+ |
($i,f,j$), ($j,f,k$), and ($k,f,i$). The centroids of the sub-facets |
| 329 |
+ |
are located at $1$, $2$, and $3$, and the area of each sub-facet is |
| 330 |
+ |
easily computed using half the cross product of two of the edges.} |
| 331 |
+ |
\label{hydro} |
| 332 |
+ |
\end{figure} |
| 333 |
+ |
|
| 334 |
+ |
The Oseen tensors for each of the sub-facets are summed, and the |
| 335 |
+ |
resulting matrix is inverted to give a $3 \times 3$ hydrodynamic |
| 336 |
+ |
tensor for translations of the triangular plate, |
| 337 |
+ |
\begin{equation} |
| 338 |
+ |
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}. |
| 339 |
+ |
\end{equation} |
| 340 |
|
We have implemented this method by extending the Langevin dynamics |
| 341 |
< |
integrator in our group code, OpenMD.\cite{Meineke2005,openmd} |
| 341 |
> |
integrator in our group code, OpenMD.\cite{Meineke2005,openmd} There |
| 342 |
> |
is a moderate penalty for computing the convex hull at each step in |
| 343 |
> |
the molecular dynamics simulation (HOW MUCH?), but the convex hull is |
| 344 |
> |
remarkably easy to parallelize on distributed memory machines (see |
| 345 |
> |
Appendix B). |
| 346 |
|
|
| 347 |
|
\section{Tests \& Applications} |
| 348 |
+ |
\label{sec:tests} |
| 349 |
|
|
| 350 |
|
\subsection{Bulk modulus of gold nanoparticles} |
| 351 |
|
|
| 384 |
|
\label{compWater} |
| 385 |
|
\end{figure} |
| 386 |
|
|
| 387 |
< |
We initially used the classic compressibility formula |
| 387 |
> |
The volume of a three-dimensional point cloud is not an obvious property to calculate. In order to calculate the isothermal compressibility we adapted the classic compressibility formula so that the compressibility could be calculated using information about the local density instead of the total volume of the convex hull. |
| 388 |
|
|
| 389 |
|
\begin{equation} |
| 390 |
|
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right )_{T} |
| 391 |
|
\end{equation} |
| 392 |
|
|
| 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. |
| 393 |
|
|
| 394 |
< |
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 |
| 394 |
> |
Assuming a uniform density, we can use the relationship $\rho = \frac{N}{V}$ to rewrite the isothermal compressibility formula as |
| 395 |
|
|
| 396 |
|
\begin{equation} |
| 397 |
< |
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle V \right \rangle ^{2}}{V \, k_{B} \, T} |
| 397 |
> |
\kappa_{T} = \frac{1}{N} \left ( \frac{\partial N}{\partial P} \right )_{T} |
| 398 |
|
\end{equation} |
| 399 |
|
|
| 400 |
< |
Thus, the compressibility of each simulation run can be calculated entirely independently from all other trajectories. However, the resulting compressibilities were still as much as an order of magnitude larger than the reference values. The effect was particularly pronounced at the low end of the pressure range. At ambient temperature and low pressures, there exists an equilibrium between vapor and liquid phases. Vapor molecules are naturally more diffuse around the exterior of the cluster, causing artificially large cluster volumes. Any compressibility calculation that relies on the hull volume will suffer these effects. |
| 400 |
> |
Isothermal compressibility values calculated using this modified expression are in good agreement with the reference values throughout the 1 - 1000 atm pressure regime. Regardless of the difficulty in obtaining accurate hull volumes at low temperature and pressures, the Langevin Hull NPT method provides reasonable isothermal compressibility values for water through a large range of pressures. |
| 401 |
|
|
| 402 |
< |
In order to calculate the isothermal compressibility without being hindered by hull volume issues, we adapted the classic compressibility formula so that the compressibility could be calculated using information about the local density instead of the volume of the convex hull. We calculated the $g_{OO}(r)$ for a 1 nanosecond simulation of a cluster of 1372 SPC/E water molecules and spherically integrated the function over the bounds 0 to $r'$. In all cases, the value of $r'$ was 17.26216 $\AA$. The value of the total integral between these bounds is essentially the number (N) of molecules within volume $\frac{4}{3}\pi r'^{3}$ at a given pressure. To yield an actual molecule count, N must be scaled by an ideal density. However, even in the absence of an ideal density, we can use the relationship $\rho = \frac{N}{V}$ to rewrite the isothermal compressibility formula as |
| 402 |
> |
We initially used the classic compressibility formula 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. |
| 403 |
|
|
| 404 |
+ |
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 |
| 405 |
+ |
|
| 406 |
|
\begin{equation} |
| 407 |
< |
\kappa_{T} = \frac{1}{N} \left ( \frac{\partial N}{\partial P} \right )_{T} |
| 407 |
> |
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle V \right \rangle ^{2}}{V \, k_{B} \, T} |
| 408 |
|
\end{equation} |
| 409 |
|
|
| 410 |
< |
Isothermal compressibility values calculated using this modified expression are in good agreement with the reference values throughout the 1 - 1000 atm pressure regime. Regardless of the difficulty in obtaining accurate hull volumes at low temperature and pressures, the Langevin Hull NPT method provides reasonable isothermal compressibility values for water through a large range of pressures. |
| 410 |
> |
Thus, the compressibility of each simulation run can be calculated entirely independently from all other trajectories. However, the resulting compressibilities were still as much as an order of magnitude larger than the reference values. The effect was particularly pronounced at the low end of the pressure range. At ambient temperature and low pressures, there exists an equilibrium between vapor and liquid phases. Vapor molecules are naturally more diffuse around the exterior of the cluster, causing artificially large cluster volumes. Any compressibility calculation that relies on the hull volume will suffer these effects. |
| 411 |
|
|
| 412 |
+ |
|
| 413 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 414 |
|
|
| 415 |
|
In order for non-periodic boundary conditions to be widely applicable, they must be constructed in such a way that they allow a finite, usually small, simulated system to replicate the properties of an infinite bulk system. Naturally, this requirement has spawned many methods for inserting boundaries into simulated systems [REF... ?]. Of particular interest to our characterization of the Langevin Hull is the orientation of water molecules included in the geometric hull. Ideally, all molecules in the cluster will have the same orientational distribution as bulk water. |
| 454 |
|
|
| 455 |
|
\section{Appendix A: Hydrodynamic tensor for triangular facets} |
| 456 |
|
|
| 337 |
– |
\begin{figure} |
| 338 |
– |
\includegraphics[width=\linewidth]{hydro} |
| 339 |
– |
\caption{Hydro} |
| 340 |
– |
\label{hydro} |
| 341 |
– |
\end{figure} |
| 342 |
– |
|
| 343 |
– |
\begin{equation} |
| 344 |
– |
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1} |
| 345 |
– |
\end{equation} |
| 346 |
– |
|
| 347 |
– |
\begin{equation} |
| 348 |
– |
T_{if}=\frac{A_i}{8\pi\eta R_{if}}\left(I + |
| 349 |
– |
\frac{\mathbf{R}_{if}\mathbf{R}_{if}^T}{R_{if}^2}\right) |
| 350 |
– |
\end{equation} |
| 351 |
– |
|
| 457 |
|
\section{Appendix B: Computing Convex Hulls on Parallel Computers} |
| 458 |
|
|
| 459 |
|
\section{Acknowledgments} |
