deletions | additions       diff --git a/Using_the_framework_of_transition__.tex b/Using_the_framework_of_transition__.tex  index e974359..ffc50a0 100644  --- a/Using_the_framework_of_transition__.tex  +++ b/Using_the_framework_of_transition__.tex 
...
\end{equation}  where $\textbf{x}$ is the 3\textit{N}-dimensional configuration of the \textit{N}-particle system, $\beta = 1/k_B T$, $m$ is the effective mass, $V$ is the potential energies, $\Theta_A$ has a value of one if the system is in state $\textbf{A}$ and is zero otherwise, $\delta^{\dagger}_{AB}$ is the delta function defining the location of the dividing hypersurface and it is considered to reside within the domain of $\textbf{A}$, and the factor of $1/2$ limits the flux to trajectories that are exiting from $\textbf{A}$. Neglecting the recrossings of the dividing surface once the trajectories exits from $\textbf{A}$ is the central approximation made in transition-state theory.  For our simulation, the dividing hypersurface is the energy barrier, state $\textbf{A}$ is the local minimum on the right side of the energy barrier, state $\textbf{B}$ is the global minimum on the left side of the energy barrier, $\textbf{x}$ is the reaction coordinate, $m$ is the mass of one Ag atom, and $V$ is the potential of mean force. The domain space of state $A$ is the region within one $RT$ from local minimum on the right side of the energy barrier. We obtained the rate constant of atom flux towards Ag111 and Ag100 to be $25.5 ns^{-1}$ and $12.2 ns^{-1}$, respectively. The ratio of rate constant of atom flux towards \{111\} over \{100\} facets $\frac{k_{111}}{k_{100}}$ is calculated to be $2.10$. Using Fig. \ref{fig:kinetic-wulff}, our calculations indicate that cubes will be form from this relative flux of deposition. The atom flux calculated in this section is one order-of-magnitude larger than the atom flux calculated by atom deposition. This is likely to be a consequence from the assumption of unity transmission coefficient causing the over-estimation of the atom flux by the transition state theory. Studies have found that the transmission coefficient can be as low as 0.1 \cite{Pritchard_2005}, especially for our system where the energy barrier is only $2$ to $4 k_B T$. The accuracy of the relative flux $\frac{F_{111}}{F_{100}}$ is more important because it can be used to define the shape of the grown NCs.  To calculate the relative flux $\frac{F_{111}}{F_{100}}$, we also need to calculate the ratio of trapping coefficent $\frac{P_{111}}{P_{100}}$. This feature can be observed from comparing PMF profile for Ag100 and Ag111 surfaces, where the PMF of Ag100 surface reachs plateau at a shorter distance than the PMF of Ag111 surface. We use absorbing Markov's chain to model the trapping effect of PVP on the Ag atom. The absorbing states are when the Ag atom is bound by the PVP chain and when the Ag atom is in bulk solution. The absorbing Markov's chain for the Ag100 system is longer than for the Ag111 system, distance difference of $2.81 \dot A$ is determined by the difference distance between the descending point of the PMF profile. The exact location of the descending point is obtained from the first maxima from the bulk solution of the first derivative of the PMF profile. The distance between the absorbing states is determined from the atom deposition simulation, which is the distance from the PVP monolayer to the position that was defined to be the bulk solution. We obtained the trapping coefficent $\frac{P_{111}}{P_{100}}$ to be 1.21 from our absorbing Markov's chain model of the PMF profile. The relative flux $\frac{F_{111}}{F_{100}}$ is obtained from $\frac{k_{111}}{k_{100}} \times \frac{P_{111}}{P_{100}}$ to be $2.541$.