Thermodynamics Of Crystal Morphology Based On Inverse Wulff Construction

Morphologies of nanoparticles are an important factor that defines various nanomaterials'properties and applications.However,design and control of reaction conditions to acquire a specific morphology is a complex and difficult process.In principle,one can always predict the equilibrium morphologies of nanoparticles once the specific surface energies of exposed crystallographic facets become available.Wulff construction is an effective tool to understand and predict the morphologies of nanoparticles.Lack of surface energies,however,poses obstacles to applicability of this method.In recent years,many computational and experimental efforts have been dedicated to determining the surface free energies and morphologies of nanoparticles.In order to obtain the surface energy data,we use the inverse Wulff construction,based on nonlinear optimization and crystal symmetries,to acquire surface energies and edge energies from crystal morphologies.The concrete contents are as follows:?1?In this work,we use a shape-dependent thermodynamic model to study the effect of thermodynamics on the morphologies of YF₃ crystals.Density-functional theory calculations are done on multiple terminations of?020?,?111?,?10????and????02?facets,to compute surface energies of these surfaces.The surface energies are found to be in the order of{020}<{111}<{10???}<{???02}.Based on the Wulff construction and the calculated parameters,we have constructed the thermodynamic equilibrium morphologies of YF₃ crystals.?2?We use the crystal system mmm and m???m to verify numerical surface energies by inverse Wulff construction against analytical results by thermodynamic equilibrium between coexisting surfaces.This method is capable of calculating not only the surface energies of those coexisting surfaces of a particle,but also lower bounds of surface energies of missing surfaces.The underlying Gibbs-Wulff theorem relies on Wulff points or particle centers,which are invalid for non-centrosymmetric crystals.We extend the method of inverse Wulff construction to study surface free energies of non-centrosymmetric crystals.A nonpolar????3m?and a polar crystal system?6mm?that lack of inversion centers are selected to show the application of our method.In addition to analytical and numerical results,we also present a general four-parameter function to simplify calculations of surface free energies from observed micro-or nanoparticle morphologies.?3?We model the relationships between surface energy and growth conditions based on the Langmuir adsorption model,in order to control the practical morphology of the crystal.The Ti O₂ nanoparticles with various crystal faces are used to verify the method under varying experimental conditions.This method is a powerful means to acquire surface energy and to guide experimental design for specific morphologies.?4?Even though edge atoms are known to influence nanoparticle morphologies according to numerical and experimental studies,contributions of edge energies are omitted in classical Wulff constructions.It is generally recognized that nanoparticles'edges have extra energies that are especially important to small nanoparticles,and are responsible for shape transitions during post-nucleation growth of nanoparticles.Edge energies are,however,largely unknown and difficult to measure or compute.In this study we evaluate contributions of edge energies to formation energies of nanoparticles,and propose an effective method for calculating edge energies from observed morphologies.This method is tested for two typical morphologies of nanocrystals of the cubic crystal system and compared with results computed with the density functional theory.It is a practical tool to enrich thermodynamic data of nanomorphologies based solely on experimental observations.The significance of the inverse Wulff construction is that the thermodynamic parameters such as surface energy and edge energy,which are difficult to be measured and calculated,can be obtained by using the morphological parameters that are easily observed from experiments.The obtained results can be used as input parameters for thermodynamic modeling.A morphology-condition map can be set up with a small number of experiments through this method,and can be extrapolated to a large number of unexplored experimental conditions of morphology transformation.This strategy is helpful to understand the dependence of the thermodynamic parameters such as surface energy and edge energy on various experimental parameters.