deletions | additions       diff --git a/Using_the_framework_of_transition__.tex b/Using_the_framework_of_transition__.tex  index e3ebea7..e1e4583 100644  --- a/Using_the_framework_of_transition__.tex  +++ b/Using_the_framework_of_transition__.tex 
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For our model, the dividing hypersurface is at the energy barrier, state $\textbf{A}$ is the local basin on the right side of the energy barrier, state $\textbf{B}$ is the energy 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 $k_B T$ from the local basin. We calculated the rate constant of atom flux towards Ag111 and Ag100 to be 25.5 ns^{-1} and 12.2 ns^{-1}, respectively. From the rate constants calculate, the ratio $\frac{k_{111}}{k_{100}}$ is 2.10.  The atom flux calculated by transition-state theory is one order-of-magnitude larger than the atom flux calculated by \textit{in-silico} deposition. This is likely to be a consequence from the neglecting recrossings, which causes the over-estimation of the atom flux by the transition state theory. The ratio of Studies have found that the transmission coefficient can be recrossings to successful crossings  as low high  as 0.1 10 has been shown in the literature  \cite{Pritchard_2005}, especially which is possible  for our system where the energy barrier is only $2$ 2  to $4 k_B 4 $k_B  T$. The We focus more on the  accuracy of the relative flux $\frac{F_{111}}{F_{100}}$ is more important by sufficient sampling of the domain space  because it can be used to define the kinetic Wulff  shape of the grown NCs. To calculate the relative flux $\frac{F_{111}}{F_{100}}$, we also need to calculate obtain  the ratio of trapping coefficent coefficient  $\frac{P_{111}}{P_{100}}$. This feature can be observed from comparing PMF profile for Ag100 and Ag111 surfaces, where the PMF Higher mean fluffiness  of Ag100 the PVP layer adsorbed on the Ag111  surface reachs plateau at a shorter associates with higher trapping coefficient, which is reflected by the further  distance than for  the PMF of Ag111 surface. to reach the maxima plateau.  We use the  absorbing Markov's Markov  chain \cite{kemeny1976finite}  to model how  the trapping effect difference in PVP layer fluffiness affects the ratio  of trapping coefficient. The length from the top  PVP on layer to  the Ag atom. bulk solution is divided into discrete Markov states.  The absorbing states are when the Ag atom is bound by at  the top  PVP chain layer  and when at  theAg atom is in  bulk solution. The absorbing Markov's difference in PVP layer fluffiness is reflected by a longer Markov  chain for the Ag100 systemis longer  thanfor 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. This distance defined to be $13.0 \dot A$ from  the Ag111 surface. system.  We obtained calculate  the ratio of  trapping coefficent coefficient  $\frac{P_{111}}{P_{100}}$ to be 1.21 from our absorbing Markov's Markov  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 $2.541$. Using Fig. \ref{fig:kinetic-wulff}, our calculations indicate is 2.541, which predicts  that cubes will be form from this relative flux of deposition. the kinetic Wulff shape is a cube as shown in Fig. \ref{fig:kinetic-wulff}.  Slight discrepancy from the relative flux calculated by the atom \textit{in-silico}  depositionsimulation  is likely to be due to the difference in a consequence of  recrossing frequency difference  for the Ag100 and Ag111 surface because the smaller energy barrier permits more recrossing possibilities.