Testing of Frank's hypothesis on a containerless packing of macroscopic soft spheres and comparison with mono-atomic metallic liquids

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Abstract

It is well-known that metallic liquids can exist below their equilibrium melting temperature for a considerable time. To explain this, Frank [1] proposed that icosahedral ordering, incompatible with crystalline long-range order, is prevalent in the atomic structure of these liquids, stabilizing them and enabling them to be supercooled. Some studies of the atomic structures of metallic liquids using Beam-line Electrostatic Levitation (BESL; containerless melting), and other techniques, support this hypothesis [2], [3]. Here we examine Frank's hypothesis in a system of macroscopic, monodisperse deformable spheres obtained by containerless packing under the influence of centripetal force. The local structure of this packing is analyzed and compared with atomic ensembles of liquid transition metals obtained by containerless melting using the BESL method.

Highlights

► Testing of Frank's hypothesis for Centripetal Packing (CP) has been proposed. ► It is shown that CP is an idealized model for Monatomic Supercooled Liquid (MSL). ► The CP is fit for comparing with studies on MSL in a containerless environment. ► We measure local orders in CP by HA and BOO methods for the first time. ► It is shown that icosahedral order is greater in CP than MSL and reasons explored.

Introduction

Fahrenheit [4] was the first to demonstrate that water can be maintained in the liquid phase below its melting temperature (supercooling). Turnbull's [5] demonstration that liquid metals can also be supercooled was surprising, given the similarity in density and coordination numbers of the liquid and crystalline phases and their simple symmetric structural units, in contrast to the complex molecular structures in water, organic compounds, proteins, etc. To explain the supercooling of metallic liquids, Frank [1] proposed that the local order in the supercooled liquid must be quite different from that in the solid, despite the similar densities and coordination numbers. He observed that there are actually three distinct close packing schemes that give a local coordination of 12 (i.e. when a central atom is surrounded by 12 neighbors). These three distinct packings are face-centered cubic (FCC), hexagonal close packed (HCP) and icosahedrons. Here, two packings are considered distinct if a transformation from one to the other is not possible without an atom losing contact with the central one. Only for icosahedral packing among these three, the surrounding atoms do not touch each other.

If the atoms are deformable (a condition which will be argued for later), the atoms can deform to form denser packing locally and reduce the overall energy of the cluster [1]. The icosahedral structure is incompatible with long-range crystalline (translational) periodicity but is locally stable, thus rearrangement to a crystal structure is energetically costly in local scale but favorable on longer length scale, giving rise to the observed barrier for nucleation of the stable crystal phase. Based on these considerations, Frank proposed that icosahedral short-range order (ISRO) is dominant in the local structures of metallic liquids. The first experimental proof of this hypothesis was obtained recently [2] during an investigation of the supercooling of a Ti–Zr–Ni alloy. To avoid heterogeneous nucleation, samples were containerlessly processed under high vacuum and simultaneous in situ synchrotron scattering measurements were performed using the Beam-line Electrostatic Levitation (BESL) technique [2], [6]. A metastable quasicrystal formed directly from the melt, due to lower nucleation barrier (compared to that of the competing stable C14 crystal structure) arising from local icosahedral order. Analysis of the atomic structures of the supercooled liquid prior to quasicrystal formation also demonstrated a growth of icosahedral short-range order upon cooling, confirming Frank's hypothesis [2].

As a first approximation, atomic interactions can be modeled using hard (rigid) spheres. While this method has been applied often for atomic liquids [7], [8], [9], [10], [11], hard-sphere models do not allow the compressibility commonly observed in liquid metals. Soft-sphere systems facilitate more meaningful studies which provide an attractive way to study the mechanical response of atomic ensembles and gain structural and dynamical insights. A particularly successful example is the use of bubble raft and similar models to study the atomistic nature via nanoindentation [12], [13]. It is also known that the electronic interaction among atoms causes considerable distortion in electron density distribution [14], [15], allowing the atoms to deform. The deformability of the atoms can lead to denser packing which reduces the local energy of the clusters in a given structure, and a soft-sphere model is more appropriate for modeling such a system. It is therefore interesting to use a soft-sphere system as a structural model for supercooled liquids, and the attempt is made in this study to develop a macroscopic model system of this type by using deformable spherical particles.

Section snippets

Methods

A so-called ‘centripetal packing’ (CP) [16] of macroscopic particles is used as a model system. It is produced by computer experiments using discrete element modeling [16], [17], [18]. Initially, 20,000 soft spherical particles of radius 1 cm are randomly distributed in a three-dimensional cubic box, each side of it measuring 75 cm. From the start of this computer experiment (t = 0) all the particles experience a hypothetical force, whose magnitude equals the gravitational force but in a direction

Results and discussion

The pair distribution function of the centripetal packing is shown in Fig. 1. A pronounced split in the second peak is observed, and this feature is more pronounced in the pair distribution function of CP than in that of the monoatomic liquids: Ti, Zr and Ni in supercooled states (T/Tm  0.8), as shown in [19].

It is interesting to analyze the local structure in more detail in terms of the HA and BOO parameters. To avoid surface effects in the CP structure, the central part of the packing is cut

Conclusions

It was demonstrated that icosahedral-type (DICOS + ICOS) ordering is dominant in a containerless centripetal packing of macroscopic spheres. The simulated CP structure was compared with those generated from experimentally measured data for transition metal liquids (Ti, Zr, Ni), which were studied during containerless processing near the melting point and in the supercooled state. The icosahedral-type ordering in the metallic liquids is always less than that of the CP. This is probably because of

Acknowledgements

Support from the Swiss National Science Foundation under grant number 200021-125143 (JFL) and from the US National Science Foundation under grant numbers DMR-0606065 and DMR-0856199 (KFK) is gratefully acknowledged.

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