10/28/2015

# Kinetic Wulff Plot

Away from equilibrium, the NC shape is governed by the kinetics of inter- and intrafacet atom diffusion, as well as by the kinetics of deposition to various facets. At nonequilibrium growth conditions, the resulting shapes are expected to be different from the thermodynamic shapes. Examples of well-known kinetic shapes include nanowires and highly branched (bi- and tripods) structures (Xiong 2007). When NCs grow beyond a critical size, the relative atom deposition rate to various facets becomes a major influence in the NC shape. In this kinetically-controlled growth regime, the kinetic Wulff construction can predict the shape evolution of faceted crystal growth based on the surface kinetics (Du 2005, Frank 1958, Osher 1997). Using 3-dimensional shape evolution calculation method (Zhang 2006), we correlate the relative flux of Ag atom deposition to {111} and {100} facets $$\frac{F_{111}}{F_{100}}$$ and the resulting kinetic Wulff shape in the reversible octahedron-to-cube transformation. This transformation is observed in the seed-mediated growth of Ag NCs (Xia 2012), in which the shape-controlling parameter is the concentration of poly(vinylpyrrolidone) (PVP) in the solution. The constructed kinetic Wulff plot is shown in Fig. \ref{fig:kinetic-wulff}. The construction of the kinetic Wulff plot is described in the supporting information. When the relative flux to {111} facets is less than half of the flux to {100} facets, the octahedra is predicted as the kinetic Wulff shape. As $$\frac{F_{111}}{F_{100}}$$ increases, we observe a shape progression from octahedra to cubo-octahedra, then to truncated cubes, and eventually to cubes at $$\frac{F_{111}}{F_{100}} \geq \sqrt{3}$$.

To study the mechanism by which SDAs impart shape selectivity, we use the seed-mediated Ag polyol synthesis in the presence of PVP (Xia 2012) as our model. We utilize large-scale MD simulations to quantify $$F_{100}$$ and $$F_{111}$$ using in-silico deposition and potential of mean force calculation.

\label{fig:kinetic-wulff} The correlation between the fraction of surface covered by 100 facets and the required relative flux in a reversible octahedron-to-cube transformation. Examples of predicted shapes are given, with the 100 facets are colored in green and the 111 facets are colored in orange.

# Findings from Potential of Mean Force Calculations

<<<<<<< HEAD We explored the potential of mean force (PMF) along the absorption path of Ag atoms from solution phase to the Ag NC surface, with the goal of gaining quantitative insight of the influence of the adsorbed PVP layer. To calculate the PMF of the Ag atom, we use umbrella sampling (Kästner 2011) with harmonic bias potential on the canonical molecular dynamics simulation of the previously described system for the in-silico deposition shown in Fig \ref{fig:sim-setup}. Umbrella sampling is used to enhance the sampling because the free energy barrier of absorption is greater than $$k_B T$$. Umbrella integration (Kästner 2005) is used to combine data from individual windows sampled, also yielding a statistical error of the PMF calculated (Kästner 2006). The reaction coordinate of the PMF is the orthogonal axis of the Ag slab, with the origin at the surface layer of the bottom slab. Further description of the PMF calculation methods can be found in the supporting information. In this section, we will present our result of the PMF profile of the Ag atom and calculate the relative atom flux to {111} and {100} facets $$\frac{F_{111}}{F_{100}}$$ using the framework of transition-state theory (Hänggi 1990).

The calculated PMF profile of the Ag atom along the orthogonal axis of the Ag slab with Ag100 and Ag111 surfaces is shown in Fig. \ref{fig:pmf}. The Ag atom approaching the surface goes through the PMF profile from the right to left. On the far right, the PMF is a flat maxima, which is where the Ag atom is in bulk solvent. As the Ag atom move closer to the surface, it interacts with the PVP monolayer which causes the PMF to decline from the flat maxima. The PMF declines until it reaches a local basin trapped by an energy barrier, which is caused by the hindering effect of the network of PVP anchored on the surface as observed in the in-silico deposition trajectories. Once the Ag atom overcomes the energy barrier, it reaches an energy minimum where the Ag atom is absorbed onto the surface.

Using the framework of transition-state theory (Hänggi 1990), we can obtain the rate constant of atom flux from the calculated PMF profile. Methods are described in the supporting information. The rate constant of atom flux towards Ag111 and Ag100 is calculated to be 25.5 ns^-1 and 12.2 ns^-1, respectively. From the rate constants calculate, the ratio of rate constants $$\frac{k_{111}}{k_{100}}$$ is 2.10. The atom flux calculated by transition-state theory is one order-of-magnitude larger than the atom flux calculated by 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 (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 a linear absorbing Markov chain (Kemeny 1976) 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. Further description of the absorbing Markov chain is in the supporting information. 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 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. ======= We explored the potential of mean force (PMF) along the absorption path of an Ag atom from solution phase to the Ag NC surface, with the goal of gaining quantitative insight of the influence of the adsorbed PVP layer. To calculate the PMF profile of the Ag atom, we use umbrella sampling (Kästner 2011) with a harmonic bias potential on the canonical molecular dynamics simulation. We use the same system as for the in-silico deposition, which is shown in Fig \ref{fig:sim-setup}. Umbrella sampling is used to enhance the sampling because the free energy barrier of absorption is greater than $$k_B T$$. We use the umbrella integration method (Kästner 2005) to combine data from individual windows sampled. The advantage of the umbrella integration method over the conventional weight-histogram analysis method (WHAM) is the independence of the number of grid points and one can obtain the statistical error directly through umbrella integration (Kästner 2006). The reaction coordinate of the PMF is the orthogonal axis of the Ag slab, with the origin at the surface layer of the bottom slab. Further description of the PMF calculation methods can be found in the supporting information. In this section, we present our result of the PMF profile of the Ag atom and calculate the relative atom flux to {111} and {100} facets $$\frac{F_{111}}{F_{100}}$$ using transition-state theory. >>>>>>> parent of 55b9e94... edited section_Findings_from_PMF.tex