--- trunk/langevinHull/langevinHull.tex	2010/10/21 15:27:57	3660
+++ trunk/langevinHull/langevinHull.tex	2010/10/21 15:43:22	3662
@@ -217,7 +217,18 @@ simulation.
 defined. The pressure and temperature bath interacts {\it directly}
 with the atoms on the edge and not with atoms interior to the
 simulation.
+
+
+\begin{figure}
+\includegraphics[width=\linewidth]{hullSample}
+\caption{The external temperature and pressure bath interacts only
+  with those atoms on the convex hull (grey surface).  The hull is
+  computed dynamically at each time step, and molecules dynamically
+  move between the interior (Newtonian) region and the Langevin hull.}
+\label{fig:hullSample}
+\end{figure}
 
+
 Atomic sites in the interior of the point cloud move under standard
 Newtonian dynamics,
 \begin{equation}
@@ -354,8 +365,8 @@ bulk properties that depend on the external applied pr
 the Langevin Hull on a crystalline system (gold nanoparticles), a
 liquid droplet (SPC/E water), and a heterogeneous mixture (gold
 nanoparticles in a water droplet).  In each case, we have computed
-bulk properties that depend on the external applied pressure.  Of
-particular interest is the bulk modulus,
+properties that depend on the external applied pressure.  Of
+particular interest for the single-phase systems is the bulk modulus,
 \begin{equation}
 \kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right
 )_{T}.
@@ -375,9 +386,9 @@ the middle of the cluster.  This radius is arbitrary, 
 \label{eq:BMN}
 \end{equation}
 The region we pick is a spherical volume of 10 \AA radius centered in
-the middle of the cluster.  This radius is arbitrary, and any
-bulk-like portion of the cluster can be used to compute the bulk
-modulus.
+the middle of the cluster.  The geometry and size of the region is
+arbitrary, and any bulk-like portion of the cluster can be used to
+compute the bulk modulus.
 
 One might also assume that the volume of the convex hull could be
 taken as the system volume in the compressibility expression
@@ -449,8 +460,8 @@ geometries which include large volumes of empty space.
   phase, and isolated molecules can detach from the liquid droplet.
   This is expected behavior, but the reported volume of the convex
   hull includes large regions of empty space.  For this reason,
-  compressibilities should be computed using local number densities
-  rather than hull volumes.}
+  compressibilities are computed using local number densities rather
+  than hull volumes.}
 \label{fig:coneOfShame}
 \end{figure}