The “critical limits for crystallinity” in nanoparticles of the elements: A combined thermodynamic and crystallographic critique

Highlights

• Size dependence of amorphous/crystalline Gibbs energy difference in nanoparticles.

• Gibbs energy method using the lattice-stability concept and Calphad-type modeling.

• Critical comparison with the Loeff, Weeber, Miedema amorphous phase approach.

• Crystallographic estimate of the height of a monolayer of atoms at the surface.

Abstract

The theme of the present work is the procedure for evaluating the minimum size for the stability of a crystalline particle with respect to the same group of atoms but in the amorphous state. A key goal of the study is the critical analysis of an extensively quoted paper by F.G. Shi [J. Mater. Res. 9 (1994) 1307–1313], who presented a criterion for evaluating a “crystallinity distance” ($h$) through its relation with the “critical diameter” ($d_C$) of a particle, i.e., the diameter below which no particles with the crystalline structure are expected to exist at finite temperatures. Key assumptions of Shi's model are a direct proportionality relation between $h$ and $d_C$, and a prescription for estimating $h$ from crystallographic information. In the present work the accuracy of the Shi model is assessed with particular reference to nanoparticles of the elements. To this end, an alternative way to obtain $h$, that better realizes Shi's idea of this quantity as “the height of a monolayer of atoms on the bulk crystal surface”, is explored. Moreover, a thermodynamic calculation of $d_C$, which involves a description of the bulk- and the surface contributions to the crystalline/amorphous relative phase stability for nanoparticles, is performed. It is shown that the Shi equation does not account for the key features of the $h$ vs. $d_C$ relation established in the current work. Consequently, it is concluded that the parameter $h$ obtained only from information about the structure of the crystalline phase, does not provide an accurate route to estimate the quantity $d_C$. In fact, a key result of the current study is that $d_C$ crucially depends on the relation between bulk- and surface contributions to the crystalline/amorphous relative thermodynamic stability.

Introduction

There is ample evidence that the sintering and alloying ability, mechanical strength, critical temperatures for phase transitions, catalytic properties and other physicochemical properties of nanoparticles are strongly size-dependent [see, for example [1], [2], [3], [4], [5], [6]]. More specifically, it is often hypothesized that the differences between the properties of the nanoparticles and the macroscopic material can be understood in terms of the surface-to-volume ratio, which is a measure of the amount of atoms located at the surface compared to that in the bulk [7], [8]. A further, conceptually related issue, is that of the minimum size for a stable crystalline nanoparticle. Since crystallinity is a long-range characteristic of the material, when the fraction of the total number of atoms located at its surface is sufficiently large, a non-crystalline (in the following “an amorphous”) phase might become more stable [9]. In a pioneering and extensively quoted paper, Shi [10] suggested that such critical condition would be realized in a spherical nanoparticle of diameter $d_C$, for which all the atoms are accommodated as if they were located at the surface. This idea was quantitatively expressed by introducing the distance $h$, which was defined by Shi as “the height of a monolayer of atoms on the bulk crystal surface” [10]. By equating the volume of the spherical nanoparticle of diameter $d_C$ with that of a thin spherical shell of the same diameter and width $h$, the following relation was established [10].

$$\frac{4}{3}\pi {\left(\frac{d_C}{2}\right)}^3=4\pi {\left(\frac{d_C}{2}\right)}^{2}h$$

The so-determined critical diameter, which is directly related to the distance $h$ through the relation

$$d_C=6h$$

was adopted by Shi to represent a “crystallinity limit”, i.e., the size below which no particles with the crystalline structure are expected to exist at finite temperatures [10].

In order to apply eq. (2), Shi [10] assumed that $h$ was related to the lattice parameter $a$ of the crystalline material. Specifically, it was postulated without further arguments that $h=a/2$ and $h=a/4$ for the face centered cubic and diamond structures, respectively.

A survey of the standard citation databases indicates that the Shi paper has been extensively quoted (223 times according to Scopus). In particular, the concept of a critical distance has been included in theoretical analyses of the size-dependence of the melting temperature of nanoparticles [11], [12], [13], and Shi's values for $h$ have been used to interpret experiments on the solid/liquid transition [14]. Contrasting with the ample use of this approach, it is noteworthy that a critical evaluation of its accuracy has not yet been reported. Such an assessment has been performed in the present work, which has been motivated by the following critical issues.

The first issue concerns the Shi prescription to estimate $h$. At first glance, one would have expected that the distance between close-packed planes in the face centered cubic structures ($\sqrt{3}a/3$) could be a better estimate for “the height of a monolayer of atoms on the bulk crystal surface”. Such alternative crystallographic criterion would yield a new set of $h$ values.

The second issue concerns the need for an independent method to determine $d_C$. Since this critical radius expresses the crystalline/amorphous relative stability, it is natural to expect that a thermodynamic approach would provide additional insight on the $d_C$ values.

Both issues will be addressed in the following work by using information on the elements. Once the new, theoretically based $h$ and $d_C$ values had been determined, a critical discussion of the Shi [10] relation for the crystallinity limit, which is expressed by eq. (2) will be performed.