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Low-index surfaces and morphologies of cobalt spinel (Co 3 O 4 ) are studied using density functional theory calculations. For each of the surfaces (100), (110), and (111), a set of terminations with different surface compositions and atomic structures are assessed to study their thermodynamic stability under different environmental conditions. Surface energies are calculated as functions of oxygen's chemical potential, which in turn is a function of temperature and partial pressure. We select seven typical shapes of cobalt oxide nanoparticles and compute their free energies of formation using the surface energies of all the surrounding facets. The results show that the shapes of octahedron and cuboctahedron have the lowest Gibbs free energies among the seven shapes for all the environmental conditions being explored. [ABSTRACT FROM AUTHOR]

Highlights • Inverse Wulff construction is used to backtrack surface energies from experimentally observed particle shapes. • The method can obtain exact surface-energy ratios of coexisting surfaces, and lower bounds for missing surfaces. • A general four-parameter function is proposed to simplify calculations of surface-energy ratios. • The method is applicable to non-centrosymmetric crystal particles for which symmetry centers are undefined. Abstract Morphologies of nanoparticles are an important factor that defines various nanomaterials' properties and applications. 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. We use inverse Wulff construction, based on nonlinear optimization and crystal symmetries, to acquire surface energies from Wulff shapes. 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. We use the m 3 ¯ m crystal class to verify numerical surface energies by optimization against analytical results by thermodynamic equilibrium between coexisting surfaces. We further propose a general function for the surface-energy ratios. The method can be used to study evolution of surface energies from particles' shapes for thermodynamic modeling that takes surface energies as input parameters. [ABSTRACT FROM AUTHOR]

• Multiple terminations of surfaces (10 1 ¯ 0) , (10 1 ¯ 1) , (11 2 ¯ 0) , and (0001) of Y(OH) 3 are established. • Effects of temperature and partial pressure of water and oxygen are included through first-principles thermodynamics. • The equilibrium morphology is of elongated hexagonal bipyramid enclosed by stoichiometric surfaces. Yttrium hydroxide Y(OH) 3 is an important precursor for yttrium oxides, and its morphologies are inherited by the yttrium oxides, thus affecting the products' properties and performance. Hydrothermally synthesized Y(OH) 3 particles are typically elongated hexagonal bipyramid, but thermodynamic parameters are inadequate for quantitative modeling of the morphology. We calculate surface energies and surface stresses of low-index orientations using density functional theory, and present a thermodynamic morphology model based on these parameters. We find that the thermodynamically stable surface terminations are always stoichiometric, and the surface energies are in the order of γ (10 1 ¯ 0) < γ (10 1 ¯ 1) < γ (11 2 ¯ 0) < γ (0001). The equilibrium morphology is of hexagonal prism capped by hexagonal pyramid enclosed by (10 1 ¯ 1) and (10 1 ¯ 0) surfaces. Because these terminations are stoichiometric, varying temperature or pressure is ineffective for modifying Y(OH) 3 particles' morphology, whereas surfactants should be used for this purpose in hydrothermal synthesis. Image, graphical abstract [ABSTRACT FROM AUTHOR]