deletions | additions       diff --git a/Using_the_framework_of_transition__.tex b/Using_the_framework_of_transition__.tex  index da02801..2edd684 100644  --- a/Using_the_framework_of_transition__.tex  +++ b/Using_the_framework_of_transition__.tex 
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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 recrossings to successful crossings as high as 10 has been shown in the literature \cite{Pritchard_2005}, which is possible for our system where the energy barrier is only 2 to 4 $k_B T$. We focus more on the accuracy of the relative flux $\frac{F_{111}}{F_{100}}$ 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 obtain the ratio of trapping coefficient $\frac{P_{111}}{P_{100}}$. Higher mean fluffiness of the PVP layer adsorbed on the Ag111 surface associates with higher trapping coefficient, which is reflected by the further distance for the PMF to reach the maxima plateau. We use the absorbing Markov chain \cite{kemeny1976finite} to model how the difference in PVP layer fluffiness affects the ratio of trapping coefficient. The length from the top PVP layer to the bulk solution is divided into discrete Markov states. The absorbing states are at the top PVP layer and at the bulk solution. The difference in PVP layer fluffiness is reflected by a longer Markov chain for the Ag100 system than the Ag111 system. We calculate the ratio of trapping coefficient $\frac{P_{111}}{P_{100}}$ to be 1.21 from our Markov chain model of the PMF profile. The obtained  relative flux $\frac{F_{111}}{F_{100}}$ from $\frac{k_{111}}{k_{100}} \times \frac{P_{111}}{P_{100}}$ is 2.541, which predicts that the kinetic Wulff shape is a cube as shown in Fig. \ref{fig:kinetic-wulff}. Slight discrepancy from the relative flux calculated by \textit{in-silico} deposition is likely a consequence of recrossing frequency difference for the Ag100 and Ag111 surface because the smaller energy barrier permits more recrossing possibilities.