Elsevier

Chemical Physics Letters

Volume 612, 18 September 2014, Pages 273-279
Chemical Physics Letters

Implementation of an alternative method to determine the critical cooling rate: Application in silver and copper nanoparticles

https://doi.org/10.1016/j.cplett.2014.08.044Get rights and content

Highlights

  • Structural and electronic properties of nanoparticles depend on the cooling rate and size.

  • The studied physical quantities in the nanoparticles have scaling symmetry.

  • The threshold size of nanoparticle behavior is independent of the cooling rate.

  • Our values for critical cooling rate are within the range of those expected for metal pures.

Abstract

An alternative method to determine the critical cooling rate of materials has been developed by explaining the size and cooling rate dependences of physical properties of metallic nanoparticles through the scaling theory. This method has been applied to silver and copper nanoparticles which have been obtained by molecular dynamics simulations. The results reveal that our values for critical rate are close for each studied physical quantity. Thus, by taking the average among them, we obtain 6.2(8) × 1012 K/s for silver and 8.9(5) × 1012 K/s for copper. We have also found the threshold size of nanoparticle behavior is independent of the cooling rate.

Introduction

Recently, the research field of solidification of materials has shown an impressive development in both experimental [1], [2], [3], [4], [5], [6] and theoretical [7], [8], [9], [10], [11], [12] studies. This has been possible due to the ability to manipulate the physical properties by controlling the solidification process, which strongly depends on the cooling rate. In fact, when liquid is cooled, the system can solidify in two very different ways [1], [2]. If the material is quenched fast enough, the end product will be an amorphous solid (a glass). At contrary, if the system is slowly cooled, the end product will be a crystalline solid. Hence, there is a critical cooling rate separating glass forming and crystal forming, i.e., to start the crystallization process it is necessary a minimum cooling rate [11], [12], [13]. The determination of this critical parameter in pure metals and alloys has been extensively studied. For instance, Liu et al. [11] have found, employing a time–temperature–transformation (TTT) diagram, that the critical cooling rate for copper is about 7.0 × 1012 K/s, while Liu et al. [9] have reported, for the same element but applying the pair analysis technique and the angular distribution function, that this rate is 80.5–35.8 × 1012 K/s. In a similar analysis for silver, it has been obtained that the critical rate for crystal-forming is about 1.0 × 1013 K/s [12]. Shimono and Onodera [10] have also shown that the critical rate of binary Ti–Al alloys depends on the Al concentration. Corresponding experimental values for Pd–Pt–Cu–P [2], Zr–Ti–Cu–Ni–Be [3], and Zr–Al–Cu–Ni [6] alloys have also been found by using differential thermal analysis (DTA) and continuous-cooling-transformation (CCT) curves. It is worth mentioning that, due to the experimental difficulties to reach such high cooling rates, the most of works related to solidification of pure metals have been carried out employing molecular dynamics (MD) simulations [14], [15], [16], [17].

Nevertheless, despite the large amount of research on determination of the critical cooling rate of pure metals and alloys, currently, there are discrepancies among the results obtained for this critical parameter, as indicated above. For this reason, in the present work, in order to give an alternative solution, we propose a method to determine the critical cooling rate of materials starting from a different (not thermodynamic) point of view. In general, our method considers two important features of the nanoparticles, which are promising materials for advanced magnetic, catalytic, optical, electronic and bio-sensor applications [18], [19], [20]. Thus, we take into account that their physical properties (e.g. cohesive energy [21], melting temperature [22], [23], atomic structure [24], [25], and electronic density of states [26]) depend strongly on the system size and on the cooling rate (similar as in the bulk phase [8], [9]). Hence, when the cooling rate increases [27], [28], a transition from crystalline to a distorted fcc-structure and finally to an amorphous-like silver nanoparticle has been observed employing MD simulations. For slower cooling processes, the external geometry of nanoparticles presents high symmetry [25], [29], [30]. The former phenomenon have also been observed in other fcc-metal nanoparticles [17], [31]. Besides, Lobato et al. [28] have found that the cooling rate employed to obtain silver nanoparticles influences the electronic density of states.

Inside the context described above, we introduce an alternative method to determine the critical cooling rate of materials. This method is applied in silver and copper nanoparticles that contain atoms ranging from 13 to 5083 (these numbers belong to the icosahedral magic numbers sequence [32]). The solidification process of the nanoparticles are carried out by MD simulations using a suitable and effective semi-empirical potential for each element: the tight-binding (TB) potential for silver [33] and the Johnson potential (based on the Embedded Atom Method, EAM) for copper [34], which accurately reproduce the thermodynamic and structural properties of most transition and noble metals. Additionally, following a previous work [26], we determine that the threshold size of nanoparticle behavior is independent of the cooling rate for both silver and copper nanoparticles. The letter is organized as follows: in Section 2, the simulation procedure, structural and electronic analysis methods are described. In Section 3 we introduce our alternative method to determine the critical cooling rate. The results are presented and discussed in Section 4. Finally, the conclusions are given in Section 5.

Section snippets

Simulation procedure

In order to obtain results that can be compared with experiments, the molecular dynamics calculations were carried out employing different (but the most effective) semi-empirical potentials for each chemical element. Hence, for silver nanoparticles we use the many-body potential developed by Cleri and Rosato [33] on the basis of the second-moment approximation to the tight-binding model (SMA-TB). This potential has been successfully used in several studies such as the atomic and electronic

Modeling of critical cooling rate

It is well-known that there are currently several experimental and theoretical methods to determine the critical cooling rate, kc, of macroscopic materials such as differential thermal analysis (DTA) [2], [3], time–temperature–transformation (TTT) diagram [4], [11] and continuous-cooling-transformation (CCT) curve [2], [6], [16]. However, the most of these methods only consider as main factor the energy changes in the system, i.e., thermodynamic processes that due to the instability of some

External geometry

It can be observed in Figure 1 that silver and copper nanoparticles cooled at rates less than 5.0 × 1012 K/s (slow cooling) present facets which are expected in ordered systems at room temperature (300 K) [31]. Whereas, for fast cooling rates (k> 1.0 × 1013 K/s), the crystallization process is suppressed and, as a consequence, these facets cannot be formed [28]. Besides, it can be seen that the crystallization of copper nanoparticles starts at a higher cooling rate than for silver, which indicates

Conclusions

In conclusion, we have shown that taking advantage of the size and cooling rate dependence of the physical properties of silver and copper nanoparticles (with atoms numbers ranging from 13 to 5083) and employing concepts from the scaling theory, it can be possible to implement an alternative method to determine the critical cooling rate of these materials. The results for kc are 6.2(8) × 1012 K/s for silver and 8.9(5) × 1012 K/s for copper. These values are within the range of critical rates expected

Acknowledgements

L.R. Medrano is grateful to CONCYTEC for the financial support for the development of part of this work through the Post-graduate Scholarship Program 2010 for Peruvian Universities.

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