deletions | additions       diff --git a/section_Supporting_Information_subsection_Kinetic__.tex b/section_Supporting_Information_subsection_Kinetic__.tex  index 722878b..367d66e 100644  --- a/section_Supporting_Information_subsection_Kinetic__.tex  +++ b/section_Supporting_Information_subsection_Kinetic__.tex 
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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 where the PMF starts to decline 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. The exact location of the declining point of the PMF is obtained from the first derivative of the PMF profile. Moving from right to left, the first maxima of the PMF profile's first derivative is defined as the declining point. The declining point for the Ag100 system and the Ag111 system is compared, and the distance between its location is 2.81 Angstrom. This distance is divided into 50 discrete Markov states, defining the distance between consecutive Markov states to be 0.0562 Angstrom. From the \textit{in-silico} deposition, trajectories with the Ag atom moving 13.0 Angstrom away from the top PVP layer is treated as if the Ag atom is in bulk solution because it will take a very long time for the Ag atom to diffuse back to the PVP layer. Therefore we used 13.0 Angstrom as the length of Markov chain of the Ag111 system, which represents the distance from the top PVP layer absorbed to Ag111 surface to the bulk solution.  The transition matrix $P$ which holds the transition probabilities between all Markov states is constructed as following; all states except for the absorbing states have a transition probability of 0.5 to each of their adjacent state, and absorbing states have a transition probability to itself of unity. The transition matrix$P$  have the form \begin{equation}  \label{eqn:markov}  \begin{array}{c|c}  1 P =  \begin{pmatrix}  I_m  & 2 0  \\ \hline  3 R  & 4  \end{array} Q  \end{pmatrix}  \end{equation}