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# 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.

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