Nucleation and growth model for {110}- and {111}-truncated nanoparticles Nicholas J. Jonesa),b) Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Raja Swaminathan Intel Corporation, Chandler, Arizona 85226, USA Michael E. McHenryc) and David E. Laughlin Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA (Received 17 April 2015; accepted 18 August 2015) Nanoparticle-sized powders have seen more and more use in many of today’s applications. As particle size decreases, many properties change including the ability to embed the small particles in liquids and other media. With decreasing size, however, surface energy becomes more important and can dictate the final shape of the particle. In applications based on polar molecules attaching to the nanoparticle surface, the surface morphology can become a key design parameter. A nucleation and growth model has been constructed for truncated body-centered cubic derivative materials, along with an update to previously published work on face-centered cubic materials. The model shows that for (110)- and (111)-truncations of a cube with a specified surface energy for each surface, the critical nuclei and equilibrium growth shapes are the same, supporting the theory of self-similar growth that had only been mentioned previously, but never proven. In this analysis, saddle points play an important role in determining the critical nuclei for comparison with the equilibrium growth shapes. I. INTRODUCTION the resulting properties, which are different from bulk. In Nanomaterials research is a growing area that is this paper we describe the theory of nucleation and growth important for many diverse applications.1 Solid-state,2 of (100)- and (110)-faceted surfaces observed in body- liquid-phase,3 and gas-phase4 nucleation and growth the- centered cubic (BCC) MNPs. ories are important in understanding nanostructures and An understanding of the role of surface and interface controlling their properties. There is a large literature on structure and crystallography on properties of MNPs is an synthesis ! structure ! properties ! performance important, evolving area. In face-centered cubic (FCC) relationships in magnetic nanoparticles (MNPs), and the derivative nanocrystalline ferrite MNPs, for example, understanding of MNPs relies on studying the increasingly particle morphologies are observed to be faceted with fi 5–8 important influence of atoms at surfaces and interfaces, identi able crystallographic surface orientations. It has which may contribute differently to properties when been demonstrated that these crystallographic orientations compared with atoms in the bulk material. The theory of lead to distinct magnetic properties for the two different 6 fl MNP nucleation and growth can make important contri- surfaces. The surface morphology is also in uential in butions to the understanding of MNP morphologies and microstructural development in grain coarsening during sintering.9 In BCC FeCo/ferrite core–shell nanoparticles, the core is faceted and orientation relationships between the metal Contributing Editor: Suk-Joong Kang a) core and oxide shell have been determined (the shell can Address all correspondence to this author. 10,11 e-mail: [email protected] also be faceted, but only at low thicknesses). The b)Present Address: 9500 MacArthur Blvd., Naval Surface Warfare surface structure is important in understanding the Center, Carderock Division, Metallurgy and Fasteners Branch, oxidation process, and the core–shell morphology and West Bethesda, MD 20817 terminating surfaces control the functionalization of these c) This author was an editor of this focus issue the review and materials in a variety of applications.10,12–14 decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. Explanations for the evolution of nanoparticle morphol- org/jmr-editor-manuscripts/ ogies in terms of nucleation and growth models are DOI: 10.1557/jmr.2015.270 important for understanding the role of particle morphology Downloaded from http:/www.cambridge.org/coreJ. Mater.. Carnegie Res., Mellon Vol. 30, University No. 20,, on Oct 30 Dec 28, 2016 2015 at 19:34:10Ó Materials, subject to the Research Cambridge Society Core terms 2015 of use, available at http:/www.cambridge.org/core/terms 3011 . http://dx.doi.org/10.1557/jmr.2015.270 N.J. Jones et al.: Nucleation and growth model for {110}- and {111}-truncated nanoparticles on physical properties at different sizes.1 In this paper, we a 2nd power surface area term, so we get a traditional extend the nucleation and growth models of Ref. 4 to nucleation curve, for which the critical nuclei can be found address the observations of (100) and (110) surfaces in by setting the first derivative equal to zero (finding the FeCo MNPs. The theory of self-similar growth for homo- critical points), and solving for when the curvature (or geneous nucleation is also proven, graphically. While this second derivative) is less than zero. has been investigated for heterogeneous nucleation of While we use the maximum in the nucleation equation precipitates, the homogeneous nucleation of nanoparticles to give us the critical nuclei and nucleation energy barrier, is a simple concept (yet unaddressed in the literature) that we instead find the minimum in energy for the equilibrium has been mentioned, but not proven.15–17 An analogous 2D growth shape, minimizing the total surface energy. Since model can analytically show that the critical nuclei and we are maximizing in one case and minimizing in another, equilibrium growth shapes are self-similar, although such both with two different equations, it seems reasonable to a model has not been presented here. assume that the shape of the critical nuclei and equilibrium We have calculated both the critical nuclei and growth shapes will be different, which is contrary to what equilibrium growth shapes of BCC FeCo MNPs with has been previously stated by Christian and Aaronson.15,18 (100) and/or (110) terminating surfaces by considering This assumption is based upon the traditional arguments the Helmholtz free energies of perfect and truncated using a dimension of the shape, whether that be the radius cubes and rhombic dodecahedra. This model considers of a sphere or the edge length of a polyhedra, in a 2D/3D volume and surface energy terms in predicting the critical system. It is inadequate, however, to compare edge lengths nucleus shapes. Equilibrium growth shapes are obtained when dealing with the possibility of different shapes. This by minimizing the total surface energy associated with will be expanded upon below, with a three-dimensional (100) and (110) surfaces while constraining the volume to analysis for both {110} and {111} faceting of cubes, but a constant size. Finally, there is a reanalysis of nucleation Fig. 1 shows this graphically for FCC based structures. and growth for FCC-type materials4 in light of the BCC When plotting the Helmholtz free energy versus the analysis. equivalent edge length of multiple polyhedra [Fig. 1(a)], it seems that there is one shape with a minimum nucle- II. REQUIREMENTS FOR NUCLEATION AND ation energy barrier (the cube), and another which will EQUILIBRIUM GROWTH SHAPES provide the minimum energy growth shape at a certain edge length (the cuboctahedron). While this is surely In order for a material to nucleate, there is a surface easier to plot, this is misleading and incorrect. energy barrier that must be overcome. The creation of To compare different shapes, we need to compare surfaces, which always costs energy, can be considered equivalent volumes, otherwise the energies we are com- an introduction of two-dimensional defects. It is for this paring are due to different amounts of atoms attaching to reason that heterogeneous nucleation is often preferred the nucleus, which will alter the energies being compared. over homogeneous nucleation, because in heterogeneous By comparing volumes, we remove the volume free nucleation there is already a surface present which can energy as a variable and we also change the traditional assist in reducing the energy barrier to nucleate new shape of the nucleation curves [Fig. 1(b)], but we can surfaces. Irrespective of the mode of nucleation, the now do both nucleation and growth analyses at the same shape of the critical nuclei can be determined to be the time. The minimum energy barrier for nucleation will not nuclei shape that has the lowest free energy of formation. change, since we are only transforming the x-axis While this has often been thought to be the same as the dimensions; we don’t change the maxima in energy, but equilibrium growth shape (often termed “self-similar only their lateral position with respect to each other. The growth”), this has not been proven analytically for size of the critical nucleus doesn’t change, but we are three-dimensional objects.15 using the volume to compare the polyhedra to each other. Beginning very simply, we can look at the general By comparing volumes, however, we can set a constant equations for traditional nucleation and growth shown volume and compare the energies for the different shapes, below [Eqs. (1) and (2)]. which is a Wulff Construction, in essence (rather than removing the volume term from the equation, we are just Nucleation : À VDfv þ +rA ; ð1Þ making it a constant).19,20 We can now see that the critical nucleus is not different from the equilibrium Growth : +rA : ð2Þ growth shape but will be the same [Fig. 1(b)], since the curve with the minimum nucleation energy barrier will The nucleation equation has both a volume energy always be less than any other curve. This is the origin of term and a surface energy term, while the equilibrium the difference between this model and the old model,4 and growth shape only has a surface energy term. The thus constitutes an improved formalism for self-similar nucleation equation has a 3rd power volume term, and growth.