Inverse Wulff construction for surface energies of coexisting and missing surfaces of crystal particles

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 $m\overline{3}m$ 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.

Introduction

Industrial crystallization processes, especially for nanomaterials, are aimed to grow micro- or nano-sized crystals and structures of specific sizes, shapes, and agglomeration that are suitable for specific applications [1], [2], [3], [4]. It is generally accepted that the shape of crystals under equilibrium or near-equilibrium conditions is determined by thermodynamics of surfaces. Wulff construction, based on minimization of free energy associated with all enclosing surfaces of a free-standing particle, is a reliable tool to understand and predict equilibrium shapes of nanoparticles [5], [6], [7], [8], [9], [10]. For substrate-supported nanoparticles, extension to the Wulff construction has led to the Winterbottom construction, which defines equilibrium shapes of particles that sit on a substrate [11]. Following the lines of Wulff and Winterbottom, Zia [12] exploited scale invariance to find equilibrium shapes of crystals in contact with more than one flat substrates, and called it Summertop construction. Ringe et al. [8], [13] derived the alloy Wulff construction that further includes variables of alloy compositions at the bulk and surfaces of an alloy particle, and a major consequence of this is that the equilibrium shapes of alloy nanoparticles become size dependent.

At present, considerable efforts have been focused on investigating morphology via Wulff construction and its extensions. In most cases deviations from the Wulff shape are minor for micro- and nano-sized particles of single crystals. Advances have also been made for those particles that have internal interfaces. In this respect, Marks and coworkers proposed the modified Wulff construction in modeling structures of many complicated twinned particles [14], [15], [16]. Ringe et al. [17] presented a kinetic model which was used to successfully explain the plethora of different shapes obtained for twinned materials in face-centered cubic (FCC) structure. For the modified Wulff construction and the kinetic Wulff construction, atomistic calculations are often used for small clusters. Many computer programs that implement Wulff construction are also available, including Wulffman [18], VESTA [19], SOWOS [20] and so on. These programs focus on visualizing Wulff shapes based on input surface energies, and the SOWOS program is special by its capability of outputting numerical results of surface areas. For these programs, surface energies are input variables that are usually obtained by computations. Surface-energy ratios may be estimated, if an equilibrium shape is known, by visual comparison between computed and experimental shapes. This process not only requires multiple trials and adjustments of surface-energy ratios, but is inaccurate.

The Wulff construction and all its extensions and modified forms require surface energies to be known. Other thermodynamic models may also require surface energies of materials as input parameters. But, in fact, measuring surface energies is generally a difficult task even for simple cases. A surface energy of a single-element solid is dependent on temperature, vapor pressure, and surface relaxations or reconstructions. The dependence becomes more complicated for compound materials because constituents’ chemical potentials may change independently, and numbers of probable surface structures and compositions typically are large. For most cases surface energies are unknown experimentally, and they generally are calculated using computation techniques at electronic or atomistic levels. But, even for calculations, it is extremely challenging to deal with a large amount of probable surface terminations, clean or contaminated, in a solution or atmosphere that exchanges ions, atoms, or molecules with the surfaces. With simplifications a limited set of surface energies have been obtained from computations, and applied to Wulff constructions for a number of materials. Lack of reliable data of surface energies still is, however, a big obstacle to Wulff constructions and other thermodynamic models that involve material surfaces.

Experimentally observed particle shapes should be close to the equilibrium shapes under the experimental conditions. We can characterize indices of the exposing surfaces, and measure the edge lengths, surface areas, and volumes of the particles, and this work is especially easy when the particle shapes are polyhedrons enclosed by facets. This geometric information of the particle shapes is adequate to backtrack surface energies through the inverse Wulff construction method [21], [22], [23]. The resultant surface energies are thus experimental values of the surfaces under the specific conditions. They can serve as reliable input parameters of thermodynamic models for the experimental conditions. The above-mentioned programs for Wulff construction may be used for estimating surface energies by measuring distances between surfaces and center of a particle, because by Wulff construction the ratios of the distances are equal to the ratios of the respective surface energies. However, locating a particle center is not always feasible because non-centrosymmetric crystal particles lack definite centers, and even centrosymmetric ones may show no centers due to external interfaces or internal planner defects. Instead, the exposed surfaces’ areas are always measurable and unequivocal, and should be used as input parameters of particle shapes. We develop a new program that is capable of computing (not estimating) surface-energy ratios from geometric information of observed particle shapes, therefore overcomes the shortcoming of those programs mentioned above. This feature set our program apart from those programs that focus on visualizing Wulff constructions.

In this paper we demonstrate that this method of inverse Wulff construction is viable as long as geometric information of particle shapes is available, because in the formulation of the original Wulff construction, surface areas and surface energies are essentially symmetric, and each set of values may be derived from the other set through minimization. In the following sections of this paper we first address formulation of the inverse Wulff construction and iterative algorithms of minimization (Section 2), and show numerical and analytical results of surface energies of a free-standing single-crystal particle of the $m\overline{3}m$ crystal class. Selecting this crystal class as the test case is because the shapes are highly symmetric, and the analytical forms are simple for the functions of their surface areas.