Nanocrystals Definition

[Shanta Plansinis]

Unlikein the bulk, at the nanoscale shape dictates properties. The imperative to understand and predict nanocrystal shape led to the development, over several decades, of a large number of mathematical models and, later, their software implementations. In this review, the various mathematical approaches used to model crystal shapes are first overviewed, from the century-old Wulff construction to the year-old (2020) approach to describe supported twinned nanocrystals, together with a discussion and disambiguation of the terminology. Then, the multitude of published software implementations of these Wulff-based shape models are described in detail, describing their technical aspects, advantages and limitations. Finally, a discussion of the scientific applications of shape models to either predict shape or use shape to deduce thermodynamic and/or kinetic parameters is offered, followed by a conclusion. This review provides a guide for scientists looking to model crystal shape in a field where ever-increasingly complex crystal shapes and compositions are required to fulfil the exciting promises of nanotechnology.

Nanocrystals, defined as crystalline particles of size ranging from 1 to 1000 nm, have found a myriad of applications across science and engineering, for instance in optical devices [1, 2], chemical catalysis [3, 4], drug delivery [5,6,7], and biological sensors [8,9,10], to name a few. Unlike in the bulk, the size and shape of nanocrystals has a profound influence on their properties, driving interest and effort in precise control strategies for both top-down fabrication and bottom-up synthesis approaches. In bottom-up approaches, thermodynamic and kinetic mechanisms dictate atomic assembly. Understanding such effects, possible through the various crystal growth models developed over the last century, is key to rationalizing and predicting the shape of crystalline nanoparticles.

This short review starts with a brief overview and disambiguation of the various mathematical models and terminologies used to model crystal shapes, from the century-old Wulff construction to the year-old (2020) approach to describe supported twinned nanocrystals. Next, we explore the multitude of published software implementations of Wulff-based shape models, describing for each their technical aspects, advantages and limitations. Finally, a discussion of the scientific applications of shape models to either predict shape or use shape to deduce thermodynamic and/or kinetic parameters is offered, followed by a conclusion.

In this section, the classic thermodynamic Wulff construction and its mathematical basis will first be introduced, followed by an overview of derivative but mathematically distinct Wulff-related constructions for (nano)crystals under various constraints such as twinned crystals, crystals governed by kinetic growth, alloys and supported crystals. We aim to keep this brief and the reader is directed to several more detailed publications for further details, if needed [11, 21,22,23]. This section concludes with a discussion of the often-confusing terminology used to describe the Wulff construction and its variants.

This thermodynamic approach, while limited to single crystals, has been abundantly used to understand and describe the shape of nanocrystals. In the common FCC system, adopted by Au, Ag, Cu, Al, and many others, thermodynamic crystal shapes lay somewhere in between a cube and an octahedron, i.e. a cuboctahedron (Fig. 3), because this shape exposes the close-packed 111 and the densely packed 100 facets, both of low surface energy [29, 30]. Meanwhile, single crystal of hexagonal close-packed (HCP) elements such as Mg, where the 0001 plane is close-packed, form hexagonal prisms and related structures as shown in Fig. 3.

Nanocrystal shapes in the single crystal thermodynamic/kinetic, thermodynamic modified and kinetic modified Wulff constructions for FCC and HCP crystal structures. The FCC structures are twinned along the (111) plane, while the twinning plane varies for HCP as described in ref. [31] Facets are colour coded according to legend and shapes were created in Crystal Creator [32]

The introduction of internal planar defects, namely twin boundaries formed at the nucleation and early stage of growth, leads to different underlying symmetry and potentially more complex crystal shapes. Particles with a single twin boundary or parallel sets of twin boundaries are referred to as singly-twinned or lamellar twinned particles (LTPs) [16], while those with more than one twin boundary are often called multiply twinned particles (MTPs) [16]. Twinned crystals can be regarded as linked single crystal subunits, joined along specific twin boundary(ies) and closed, for MTP, at the cost of lattice strain [11, 33]. For instance, the decahedral particles common in FCC systems are MTPs consisting of five tetrahedra each sharing a face and all sharing a common edge, while icosahedral particles are an assembly of 20 tetrahedra each sharing a face.

where \(S_mn^t\) is the bounding twin surface of subunit m to subunit n. Interestingly, whilst single crystals and subunits of a twinned structure require convex shapes, LTPs and MTPs can have concave, re-entrant surfaces which appear odd but do minimize total surface energy. For instance, the thermodynamic shape of a small FCC MTP, the Marks decahedron, exposes such grooves, as does a MTP with more stable 111 facets (Fig. 3). The thermodynamic shape of a singly twinned bipyramid exposing 100 facets also contains these features. Similarly, re-entrant corners are expected in some of the thermodynamic HCP shapes, as shown in Fig. 3. These are not always observed experimentally because of kinetic effects, as explained below.

In the case of twinned crystals grown in kinetic conditions, a treatment similar to that of the thermodynamic approach can be applied by forming individual kinetically grown crystals (Eq. 6) and assembling them as previously (Eq. 3), assuming zero growth velocity for the twin plane(s) [21]. This yields the modified kinetic Wulff construction. However, additional kinetic effects are responsible for the shapes observed, such as preferential growth in concave facets and enhanced growth along twin boundaries [21]. These effects are well established [36] and hold true for twin boundaries, re-entrant surfaces and disclination lines.

The kinetic single crystal Wulff construction allows for shapes beyond the thermodynamic equilibrium by taking into account the growth-directing effects of surfactants, underpotential deposition, or other reaction additives and conditions [37,38,39,40,41,42]. For instance, Ag underpotential deposition on Au nanoparticles blocks the growth of low-index facets, allowing for rather exotic shapes such as 310-bound truncated ditetragonal prisms and 720-bound concave cubes, while poly(vinyl pyrrolidone) (PVP) directs the growth of Ag to preferentially form nanocubes with 100 facets rather than the thermodynamically expected cuboctahedra [43, 44].

The ability to add growth enhancements to re-entrant facets as well as disclinations and twin boundaries enabled the modified (twinned) Wulff construction to model a host of observed but previously unexplained nanocrystal shapes, i.e. shapes impossible to obtain by simply changing the surface energies/growth velocities (Fig. 3). These include sharp decahedra and bipyramids in FCC crystals [21], and sharp folded structures in HCP materials, to name a few [31].

The ability of homogeneous alloy particles to form more stable structures via surface segregation, and the resulting change in shape, has been incorporated in the alloy Wulff construction [17] and can be written as:

where \(C_n^S\) and \(C_n^V\) are the surface and volume fractional concentration of element n, respectively, and \(\gamma _\left(i, C_1^S, C_1^V,C_2^S,C_2^V\dots \right)\) is the face- and composition-dependent surface energy. \(\Lambda\) is a Lagrangian multiplier with which energy is minimized with respect to changes in bulk and surface segregation and \(\Delta G\) is the difference between the free energy per unit volume of the final bulk composition after surface segregation and that of an initial composition where the surface and bulk concentrations are assumed to be equal. The lowering of overall surface energy thus competes with the resulting changes in bulk energy due to changes in bulk concentration. The resulting nanoparticle shape and surface composition is dictated by the composition-dependent bulk and surface energy curves as well as the particle size, an unusual feature for the typically size-independent Wulff approach. While experimental, systematic studies of this effect are scarce, several observations and predictions of size-dependent segregation have been reported and are summarized in ref [45].

Nanocrystal applications may require or benefit from support on a substrate, for instance in catalysis where this can improve performance and reduce sintering [46,47,48]. The thermodynamic Winterbottom construction (Fig. 4), developed in 1967 [18], allows for the determination of crystal shape grown directly on a flat substrate (rather than deposited) by adding an interface with positive or negative adhesion energy \(\gamma _A\). This term is added to the surface free energy \(\gamma _i\) of the facet i being replaced by the interface to yield \(\gamma _int\), an effective interfacial energy term:

The thermodynamic Winterbottom construction has been used or invoked in multiple contexts, often related to supported catalysts, including the epitaxial growth and resulting orientation of La1.3Sr1.7Mn2O7 on LaAlO3 [49] and the shape of Pt nanoparticles on SrTiO3 [46].

The case of a supported decahedral particle of FCC has been explored in detail [19] and matches well with experimental evidence. Extensions to other twinned shapes and comparison with experimental data should be straightforward as long as one considers that the approach may only yield local, rather than global, minima.

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