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Towards the Exploration of the NiTi Phase Diagram with a Classical Force Field Christian Kexel, Alexey V Verkhovtsev, Gennady B Sushko, Andrei V. Korol, Stefan Schramm, and Andrey V Solov'yov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07358 • Publication Date (Web): 19 Oct 2016 Downloaded from http://pubs.acs.org on October 21, 2016

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The Journal of Physical Chemistry

Towards the Exploration of the NiTi Phase Diagram with a Classical Force Field Christian Kexel,†,‡ Alexey V. Verkhovtsev,†,¶ Gennady B. Sushko,† Andrei V. Korol,†,§ Stefan Schramm,‡,k and Andrey V. Solov’yov∗,†,⊥ MBN Research Center, Altenhöferallee 3, 60438 Frankfurt am Main, Germany, Department of Physics, Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany, Instituto de F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 113-bis, 28006 Madrid, Spain, Department of Physics, St. Petersburg State Maritime Technical University, Leninsky ave. 101, 198262 St. Petersburg, Russia, Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany, and On leave from A.F. Ioffe Physical-Technical Institute, Russian Academy of Sciences, Politekhnicheskaya str. 26, 194021 St. Petersburg, Russia E-mail: [email protected]

∗

To whom correspondence should be addressed MBN Research Center ‡ Goethe University ¶ Consejo Superior de Investigaciones Cient´ıficas § St. Petersburg State Maritime Technical University k Frankfurt Institute for Advanced Studies ⊥ On leave from A.F. Ioffe Physical-Technical Institute †

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Abstract Classical force fields, used for atomistic modeling of metal materials, are typically constructed to match low-temperature properties obtained in experiments or from quantum-level calculations. However, force fields can systematically fail to reproduce further fundamental parameters, such as the melting point. In this work, we present a modified force field for modeling metallic compounds, which has been implemented in the MBN Explorer software package. It is employed to simulate different regions of the composition-temperature-size phase diagram of nickel-titanium nanoalloys with particular focus on the evaluation of the melting point of Nix Ti1−x (x = 0.45−0.55) systems. A near-equiatomic NiTi alloy is of paramount interest for biomedical and nanotechnology applications due to its shape memory behavior, but experiments and theory are inconsistent regarding its structural ground-state properties. The presented force field is used to predict the ground-state structure of an equiatomic NiTi nanoalloy. We observe that this compound does not possess the shape memory capacity because it stabilizes in the austenite instead of the required martensite crystalline phase. All results of our atomistic approach utilizing molecular dynamics and Monte Carlo techniques are in agreement with respective ab initio calculations and the available experimental findings.

Introduction Almost all contemporary metal materials utilized by humans are alloys. In consideration of the vast array of potential alloys and the resulting complexity, the necessity exists to liaise fundamental theory and practical applications. The framework of phase diagrams is capable of providing a nexus. 1 Concentrating merely on binary systems in thermodynamic equilibrium, their intricate state space can be mapped onto a composition-temperature phase diagram. In the recent review by Calvo, it was pointed out that for many elements these graphs are not known with high accuracy but require extrapolation based on limited data. 2


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Since the chemical bonds between atoms determine the structure, (thermo-)dynamics and eventually the functionality of a material, it represents a formidable scientific problem to model and understand the interatomic interactions and describe subsequently the behavior in portions of the phase diagram as broad as possible. 3 High-performance computing provides the novel means to tackle this challenge 4 and to gain unprecedented insights into complex physical systems. In recent years, much interest in studying alloys on the nanoscale has emerged. 2,5 Alongside composition and temperature, the state space is then defined by the system size. 3,6 The supplementary complexity of nanoalloys facilitates steering their physicochemical properties, such as reactivity or phase transition points. Regarding the solid-liquid transition, the scaling law due to Pawlow, 7

Tmelt (∞) − Tmelt (R) ∝ Rλ ,

(1)

entangles the melting temperature Tmelt (R) of a spherical nanoparticle of radius R with its macroscopic counterpart, where R → ∞. Typically λ ≈ −1 holds, expressing the relative competition between surface and volume energy contributions. A further hallmark of finite systems are appreciable thermodynamic fluctuations, which scale with N −1/2 (where N is the number of atoms) and lead, for example, to the smoothening of phase transitions from a distinct temperature to a broader interval. 2 Due to the associated martensitic phase transition, the near-equiatomic nickel-titanium (NiTi) alloy stands out among bimetallic alloys as the most prominent instance of a shape memory (SM) material. 8 Besides this solid-solid phase transition, the melting phase transition and phase segregation are relevant physical phenomena in the fabrication and processing of NiTi. 9 Conventional NiTi is already of eminent importance as an adaptive material in biomedical devices because of its good corrosion resistance and low stiffness. However, its nanostructured variant, consisting of crystallites with an extent below 100 nm, can exhibit



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yet enhanced thermomechanical properties. 8 Particularly for nanoscale systems, a compelling atomistic theory of the SM effect in NiTi is nonetheless still lacking. 10 Amongst other things, the low-temperature phase is ambiguous. In near-equiatomic NiTi, the B2 structure (we employ the Strukturbericht designation) is widely accepted to correspond to the high-temperature phase, dubbed austenite. 11 For a more detailed discussion of the austenitic NiTi, we refer to the recent paper by Zarkevich and Johnson. 12 Since it is commonly found in experiments, the monoclinic B190 crystal structure has been believed to constitute the ground-state phase, dubbed martensite. 13–15 In contrast, comprehensive calculations on the level of the local density approximation (LDA) advocate an orthorhombic structure to represent the ground state. 16,17 In these theoretical studies, B190 is assumed to be stabilized at the microstructural level by residual forces under the conditions prevailing in the respective experiments. The urgent question whether the SM effect, witnessed in bulk, can be exploited in future nanotechnological applications 18–21 is therefore closely tied to the occurrence of B190 martensite at the nanoscale. The martensite-austenite and its reverse structural phase transition are expected to be crucial for SM behavior in NiTi. One of the possibilities to get insights into the structural and thermodynamic properties of nanoalloys is tied to the use of advanced computer simulations. However, in spite of the extensive amount of research carried out so far on NiTi alloys, the study of thermodynamic properties of NiTi or other Ti-based (nano-)alloys has been very limited. 10 Pasturel et al. have attempted to deduce theoretically the composition-temperature phase diagram of bulk NiTi using LDA calculations with periodic boundary conditions. 22 In contrast, atomistic simulations of melting in Ni-based nanoalloys have been reported. 3,23,24 We are currently not aware of systematic computational atom-level studies considering the melting transition in NiTi nanoalloys. In the present work, we investigate different regions of the composition-temperature-size phase diagram of NiTi by means of our recent modification 25 of the established classical


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Gupta-type many-body force field (FF), which has been implemented in the MBN Explorer software package. 26 The modified FF, which has been applied earlier for several pure metals, including titanium, is further extended to evaluate the melting point of several NiTi nanoalloys. We consider spherical nanoparticles of different size and use these results to evaluate the melting point in the macroscopic bulk limit in order to compare with experimentally obtained phase diagrams for bulk materials. By performing this analysis, we provide atomlevel insights into the structural and thermal properties of NiTi nanoalloys. We demonstrate that the accounting for distant atomic interactions, which are neglected in many other FFs, is crucial for the accurate assessment of melting, as it is largely attributed to the inclusion of the interaction between second-nearest neighbor and more distant Ti atoms. To the best of our knowledge, we present here for the first time a computational study of the binary phase diagram of NiTi at elevated temperatures at this level of detail. The presented FF is also utilized to predict the ground-state structure of an equiatomic nanoscale NiTi alloy. We observe that, contrary to larger NiTi nanocrystals which are studied in experiments, the nanometer-size NiTi compound does not possess shape memory capacity as it is stabilized in the austenite crystalline phase. This analysis contributes to an atomistic understanding of the SM effect in NiTi nanoscale systems. We stress that the methodology utilized in this work is not limited to Ni, Ti or their alloys but can be applied to any other monatomic or binary metallic material, for which parameters of classical force fields can be obtained. The remainder is organized as follows: Section “Theoretical and Computational Methodology” is devoted to an overview of computational methods as well as to the introduction of the employed FF for NiTi. In Section “Numerical Results and Discussion”, we first address the melting of pure Ni systems for the purpose of validating the utilized FF. Then we discuss the results of simulation of the solid-liquid transition in NiTi compounds for different chemical compositions and address the exigent problem of the ground-state crystal structure of the equiatomic nanoalloy. Finally, in Section “Conclusions” a summary and an outlook are provided.



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Theoretical and Computational Methodology Multiscale Modeling The quantitative description of the structure and (thermo-)dynamics of many-particle systems is a multiscale problem. 26,27 Accordingly, the virtual design and simulation of materials should follow a computational multiscale modeling approach. 8 Depending on system complexity and the involved time and length scales, different levels of theory become adequate. Computational ab initio schemes, such as LDA or second-order Møller-Plesset perturbation theory 28 (MP2) incorporate electronic structure with electronic correlation and the quantummechanical exchange interaction. Geometry optimization of a crystal supercell, molecules or atomic clusters containing up to a few hundred atoms can thus be tackled. Complementarily, all-atom molecular dynamics 29 (MD) techniques render the simulation of condensed matter systems comprising a few million particles workable. 30,31 The equations of motion of the nuclei are solved iteratively, leading thereby to detailed physical trajectories as well as to the sampling of a statistical ensemble, typically the canonical N V T or microcanonical N V E ensemble. Here, N, V, T, E correspond to a constant number of particles, volume, temperature and internal energy, respectively. In the classical MD scheme, the interaction between atoms is modeled by a FF, where the potential energy is expressed as an analytical function of the distance rjκ between atoms j and κ. Shortcomings of the classical MD scheme consist in the rather short time evolution which can be sampled, hence the possibly non-ergodic sampling of configuration space. 32 Global optimization 33 as well as canonical simulations 29,34 resting upon the Monte Carlo (MC) approach allow to explore efficiently configuration space of a complex system. The acceptance probability, Π, that one should accept a trial move from a configuration with energy Ei to another configuration having energy Ei+1 , is defined here by the Metropolis criterion, 34 Ei − Ei+1 , Π(i → i + 1) = min 1, exp kT 6 ACS Paragon Plus Environment

(2)

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where k represents Boltzmann’s constant. Advanced approaches, such as parallel tempering, 35,36 which combine MC and MD techniques allow to rectify the deficiency of broken ergodicity.

Classical Force Field FFs for metals, which are commonly referred to as Gupta 37 potentials, embedded-atom method 38 or the Finnis-Sinclair potential, 39 can be grouped under an additive cohesion model, Uorig =

N X

Uirep + Uiattr

(3)

i=1

with the total potential energy, Uorig , being subdivided into the repulsive pairwise and the attractive many-body contribution. The many-body part imitating delocalization of the outer-shell electrons is connected to the local electron density of states. Various quantities in transition metals have been observed to depend merely on the overall shape of the density of states, which can thus be approximated by its first few moments. 40 A well-established empirical expression for the attractive contribution reads

Uiattr

v uX u rij 2 ξαβ exp −2qαβ −1 , = −t D αβ j6=i

(4)

where summation is performed over pairs of atoms i and j. The interatomic interaction between distinct chemical elements is tagged by α and β. The exponential ansatz is considered likewise for the repulsive part,

Uirep

=

X

Aαβ exp −pαβ

j6=i

rij −1 . Dαβ

(5)

Since the parameters q and p dictate the decay and therefore the function’s curvature, both are tied to the elastic constants. The parameter ξ represents an effective orbital-overlap integral, and the nearest-neighbor distance is denoted by D. Lacking a direct physical 7 ACS Paragon Plus Environment


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meaning, the parameter A adjusts cohesive energy. Table 1: Original parametrization 41 alongside the parameters B and C of the novel FF 25 given in Eq. (7). interaction D (nm) A (eV) Ni-Ni 0.2490 0.104 Ti-Ti 0.2950 0.153 Ni-Ti 0.2607 0.300

p ξ (eV) q B (eV/nm) C (eV) 11.198 1.591 2.413 0.000 0.0000 9.253 1.879 2.513 0.114 -0.0595 7.900 2.480 3.002 0.000 0.0000

The above presented computational scheme can be applied to various metallic materials, for which parameters of classical FFs can be obtained. 42 It lies in the nature of classical FFs that considering the interaction of different metal elements requires the use of different parameters (see, e.g., Ref. 40 for details). In particular, the Gupta-type FF, defined by expressions (4) and (5), has been applied in different studies to model the transition metals Ni and Ti as well as their equiatomic alloy NiTi. 10,31,43,44 We note that these studies have been devoted to the analysis of ground-state and low-temperature properties of the systems, such as stability, elastic properties and the martensitic phase transition which takes place at the temperatures of a few hundred kelvin. These studies have employed the parameterization due to Lai and Liu 41 which is given in Table 1. In the cited work, the values have been selected to match bulk ground-state properties, such as cohesive energy, vacancy formation energy as well as lattice and elastic constants determined using LDA calculations of crystal supercells under periodic boundary conditions. Fitting has involved pure Ni, pure Ti and independently NiTi. Hence parameters for the diatomic Ni-Ti interaction do not rely on geometric or arithmetic averages of the monatomic cases. 45 Furthermore, fitting has embraced the cutoff radius, where all the interatomic interactions are truncated, as a free variable. Since no interpolation in the vicinity of the resulting short cutoff, rc = 0.42 nm, has been discussed in Ref. 41, this truncation leads to a severe discontinuity in the interatomic potential energy. For Ti, amongst other monatomic metals, we have been able to mitigate the deficiency of the widely used Gupta-type FF. 25 It fails to assess accurately the solid-liquid phase transition point with the calculated bulk melting temperature underestimating the experimental 8 ACS Paragon Plus Environment

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value by about 300 K. For other metals with accordingly different parameterization or when employing the popular Sutton-Chen FF, 46 the drastic mismatch prevails. By design, FFs mesh with ab initio results for ground states, but describe insufficiently the vibrationally excited states. In general, low-temperature behavior is governed by the nearly harmonic region in the vicinity of the equilibrium position of the FF, whereas the anharmonicity far from the minimum enables phase transitions. The well-known Lindemann rule 47 connects melting with the microscopic interactions: In crystals the thermal atomic vibrations cause the p system to become unstable if the root mean square displacement (RMSD), hδr2 i, exceeds a distinct threshold relative to the characteristic length scale of the lattice. Typically, this critical amplitude lies between 10 - 15 % of the nearest-neighbor distance D. Augmenting the steepness and thus enhancing the repulsive contribution of the FF leads to a rise of the melting point because increased thermal energy is needed to reach the aforementioned vibrational threshold. Similarly, the model FF due to Dzugutov, 48 which mimics amorphous and liquid metals, is characterized by pronounced repulsion at large distances so as to suppress crystallization. Guided by these insights, we have developed a recipe 25 for the modification of the original Gupta-type FF, which is capable of reproducing both the ground state features and the melting temperature. For the modified potential energy between a pair of atoms, the following set of conditions should hold: (i) the position of the minimum should remain quasi unperturbed by the modification in order to conserve the original lattice constants; (ii) the depth should remain largely unaffected so as to keep the cohesive energy; (iii) the curvature in the vicinity of the minimum should coincide with the original curvature because it is proportional to the elastic constants; (iv) in order to avoid a discontinuity, the potential energy should approach zero at a long cutoff; (v) raising thereby the melting temperature, the steepness should be increased. As an illustration of this methodology, the original Gupta-type FF was amended by



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introducing an additional repulsive term, 25

Umod =

N X

Uirep + Uiattr + ∆Ui .

(6)

i=1

This supplementary contribution consists in a linear function

∆Ui =

X

(Bαβ rij + Cαβ ) ,

(7)

j6=i

which allows the novel FF to meet the aforementioned constraints. Here, the slope B > 0 should be selected so as to reproduce the known bulk melting point Tmelt (∞). The term Brij promotes steepness of the potential energy between pairs of atoms, but also slightly changes the depth of the FF. The constant C has thus been introduced to discard the latter effect and to match the original cohesive energy. The resulting parameterization for Ti is given in Table 1. The modification represents a minor change to the potential energy but has nevertheless significant impact on collective system properties, like the melting temperature. To illustrate the minor role of this modification, we have plotted the potential energy profile of a titanium dimer, calculated with the original and the modified force field (see the dashed and the solid blue curves in Fig. 1). The long cutoff, rc = 0.70 nm, has been chosen to diminish the discontinuity which emerges in the original FF. The complete elimination of the discontinuity is desirable and hence the subject of currently ongoing research. For pure Ti, the new FF has yielded 25 lattice constants, vacancy formation and cohesive energy in very good agreement with theoretical and experimental results. A similar amendment of repulsion by a constant can be found in a recent derivation of a Gupta-type FF for the transition metal zirconium. 49


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Figure 1: Potential energy of the monatomic and diatomic interactions in NiTi for original and modified FF. An illustrative cross section through the nanoparticle below (upper left) and above (right) the melting point is depicted; the higher the local Ni concentration the darker the atom.

Numerical Results and Discussion In this work, we have extended the above presented methodology to study thermal and structural properties of more complex systems, namely bimetallic alloys. For this analysis, several algorithms from the scientific multiscale software package MBN Explorer 26 have been used for efficient crystal generation, 50 velocity-quenching local geometry optimization 51 and (micro-)canonical MD simulations of different duration ∆t. In the MD runs, standard velocity-Verlet integration with a typical step size δt = 2 fs has been employed. In the canonical MD calculations, a Langevin thermostat 26 has been used with a typical damping constant τ = 40 fs. The linked-cell algorithm 26 has been used for efficient search of neighbor atoms. For MP2 calculations, the computational quantum chemistry package Gaussian 09 has been employed. 52



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Monatomic Systems Assessing the Ni-Ni Interaction Both the modified 25 and original 41 Gupta-type FFs in Eqs. (4) to (6) rest upon the additive cohesion scheme which models a binary system by considering three distinct interactions: the two monatomic (Ti-Ti, Ni-Ni) as well as the diatomic (Ni-Ti) one. Our recent work 25 has dealt with modifying and improving the Ti-Ti interaction. Here, we assess the Ni-Ni interaction. For that, icosahedral particles comprising from 147 up to about 18000 Ni atoms have been subject to a sequence of canonical MD runs at subsequent thermostat temperatures Ti+1 = Ti + ∆T . We have employed different FFs; the FF with original parametrization 41 and the short cutoff, rc = 0.42 nm, is referred to as a short-cutoff Gupta FF. For the sake of comparison, we have additionally performed simulations of the icosahedral system with the widely used Sutton-Chen 46 and the so-called quantum Sutton-Chen 45 FFs; the latter represents a different parametrization of the original 46 potential. With growing system size, the crystal lattice becomes the most favorable geometry in nanoparticles. Therefore, we have also carried out MD simulations for a spherical Ni nanocrystal (particle radius R = 2 nm, N = 3043 atoms) with cubic lattice (lattice constant a = 0.3524 nm), where the long-cutoff Gupta FF (rc = 0.70 nm) was employed. Both factors, namely the more favorable geometry as well as the long cutoff radius, enforce cohesion and are thus expected to increase the melting point, if they will have any significant effect at all. The duration of the individual simulation runs ∆t = 20 ps and the resulting heating rate ∆T /∆t = 0.1 K/ps have been chosen low enough to not influence the calculated melting point. The heating protocol used in the present work is about 5 - 10 times slower than protocols which have been imposed in related works (see Refs. 10 and 25). Melting is a first order phase transition which involves a latent heat and accordingly exhibits a peak in the heat capacity CV = (∂E/∂T )V . In order to obtain the specific heat from MD simulations, the caloric curves E(T ) were spline-smoothed and then differentiated with


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respect to T . The global maximum of the specific heat is hence taken as the thermodynamic indication of melting. Plotting the melting temperature Tmelt (R) of the finite systems as a function of the inverse particle radius R (or inverse cubic root of N ) according to Eq. (1) allows one to evaluate by linear regression the bulk transition temperature Tmelt (∞), which is given by the graph’s intercept. The results are depicted in Fig. 2.

Figure 2: Calculated melting temperatures of pure NiN (N = 147 − 18000) nanoparticles (symbols) as a function of N −1/3 . The linear least-squares fit to MD results is indicated by a straight line. The short-cutoff FF yields Tmelt (∞) = 1759 K, which is in good agreement with the experimental bulk melting temperature 53 of 1728 K. Throughout this work, we state deviations, for example, between a calculated and an experimentally obtained quantity X as follows: σX = 2

|X calc − X exp | . |X calc + X exp |

(8)

The deviation of the bulk melting temperature is thus 1.8 %. The long cutoff and the crystalline geometry have an insignificant effect on the simulated melting point of Ni. Neither the original Sutton-Chen FF nor its quantum variant yield reasonable melting points. Hence, the short-cutoff as well as the long-cutoff Gupta FF are, already without modification, suitable 13 ACS Paragon Plus Environment


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for the atomistic simulation of the solid-liquid phase transition in monatomic Ni systems.

Monatomic Molecules Although the FF has been parametrized 41 to match bulk properties, previous studies 43,44 have shown that it also describes small NiTi clusters with reasonable accuracy (despite lacking the capability to fully reproduce quantum effects). In the present work, we have calculated the bond length d of pure Ni and Ti molecules (small clusters) at 0 K using MP2 with cc-pVDZ basis set 54 so as to complement previous works 43,44 which have employed the generalized gradient approximation (GGA). Since the classical FF omits, e.g. Jahn-Teller distortions, we have also constrained the ab initio geometry optimization to the highlysymmetric regular molecules which have one unique bond length d. Using the original FF, dorig can be determined analytically for pure regular molecules with N ≤ 4. Due to the small spatial extent, classical bond length results are independent of selection of a cutoff. By design, dmod ≈ dorig holds, because the position of the potential energy minimum remains quasi unperturbed by our modification. The bond length

dorig

= 1−

1 ln q−p

Ap √ N −1 D ξq

(9)

can be derived from Eqs. (3) to (5) with the equilibrium condition ∂Uorig /∂r = 0 at the minimum r = dorig . For dimers as well as for the regular Ti molecules, Du et al. 55 have obtained GGA results. Alongside their results, our analytical and MP2 results are summarized in Table 2. Table 2: Zero-temperature bond length d of regular monatomic molecules (in nm). orig. FF modif. FF MP2 GGA 55

Ni2 0.2152 0.2041 0.2133

Ni3 0.2250 0.2430 -

Ni4 Ti2 Ti3 Ti4 0.2308 0.2423 0.2574 0.2663 0.2425 0.2578 0.2667 0.2354 0.2468 0.2570 0.2908 0.1943 0.2412 0.2515


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In Table 2, the Ti bond lengths dorig and dmod are indeed in excellent agreement. Furthermore, MP2 results for both Ti and Ni are in good overall agreement with our analytical findings: The average deviation is 3.6% and 5.0%, respectively. For the Ni2 molecule, the bond length of Du et al. 55 is very close to the analytical value dorig . Their results for Ti on average deviate considerably from our analytical (11.4%) as well as our MP2 bond lengths (14.9%). Nonetheless, their result for the Ti2 dimer contributes the most to the average deviation. The deviation from dorig for the Ti3 trimer is only 6.5% and 5.7% for the Ti4 tetramer. One should note that the discrepancy between the results of classical optimization and ab initio calculations of small molecules is within the limits typical for the discrepancies arising from the utilization of different ab initio methods, e.g., different exchange-correlation functionals. 44

Solid-liquid Phase Transition for Different Chemical Compositions Equiatomic NiTi Having discussed monatomic systems, in this section we systematically explore the application of the modified FF to equiatomic NiTi. For that purpose, austenitic nanocrystals comprising from 1061 up to about 67000 atoms have been subject to a sequence of canonical MD runs. The maximal heating rate ∆T /∆t = 0.125 K/ps has been chosen low enough to not influence the calculated melting point. We have simulated melting for some instances at even slower heating rates, but the phase transition point remained unchanged. Again, the global maximum of CV was taken as the thermodynamic indication of melting. By plotting the melting temperature as a function of the inverse particle diameter according to Eq. (1), we have evaluated the bulk transition temperature. Here we employed the short-cutoff, longcutoff as well as the modified Gupta FF. This comparison allows us to discriminate between effects steming from cutoff and modification of steepness in the Ti-Ti interaction. The Ni-Ti interaction has been kept unchanged throughout the present work. The simulation results are shown in Fig. 3. 15 ACS Paragon Plus Environment


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Figure 3: Calculated melting temperatures of equiatomic NiTi nanoparticles (symbols) as a function of inverse diameter. The linear fit to MD results towards the bulk limit is indicated by a straight line. The novel FF yields an extrapolated bulk melting temperature Tmelt (∞) = 1564 K, which is in agreement with the experimental bulk value 53 of 1583 K with a deviation of 1.2%. The short-cutoff FF yields a melting temperature, which is in contradiction to the higher value of 1653 K reported in the original publication. 41 Since superheating is already discussed therein, the application of periodic boundary conditions to the simulated supercell is likely to artificially stabilize the crystal. Figure 3 illustrates that both the augmentation of steepness by our modification and the increased cutoff display a pronounced effect on the simulated melting point. In order to better understand its impact, we have probed different cutoff distances rc for the simulation of melting in a representative crystalline nanoparticle having radius R = 2 nm, with the original Gupta FF being employed. To study the connection of the melting point Tmelt (R) with the atomic structure of the crystal, we computed the radial distribution function, g (r) =

hN (r, δr)i . 4πr2 δr ρ


(10)

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Here, hN (r, δr)i is the average number of atoms in a spherical shell, with radius r and thickness δr, around a distinct position, and ρ is the global number density. We have evaluated through g (r) the distribution of atoms of the same chemical element (Ti-Ti, NiNi) as well as the distribution of atoms of different elements (Ni-Ti). The distribution function has been calculated at 600 K. The results are presented in Fig. 4.

Figure 4: Melting temperature of a finite equiatomic nanocrystal (2 nm particle radius) as a function of cutoff distance alongside with the radial distribution function. The dotted line is given to guide the eye. The melting point exhibits a minimum in the vicinity of the short cutoff (rc = 0.42 nm) and eventually converges at the larger distances. The minimum coincides with the inclusion of distinct monatomic interactions: Since the Ni-Ni interaction is significantly weaker than the Ti-Ti interaction and since the inclusion of Ni-Ni interactions beyond the short cutoff shows little effect (see the comparison of short-cutoff and long-cutoff Gupta FFs in Fig. 1), the sensitivity of the melting point in the vicinity of minimum can be largely attributed to the inclusion of the interaction between second-nearest neighbor Ti atoms.



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Near-equiatomic NiTi After studying the equiatomic case, different chemical compositions have been investigated. The NiTi phase diagram is intricate in the vicinity of equiatomic compositions below the melting point. 53 Without the computational burden of full global optimization, we have prepared near-equiatomic nanoparticles at finite temperature by starting from the known austenite B2 crystal structure of equiatomic NiTi.

First, atoms inside the equiatomic

nanoparticle of radius R have been replaced randomly until the desired chemical composition was reached. The nanoparticle was then subject to an equilibration comprising 1000 macrosteps. At each macrostep, (i) a random pair of atoms swaped their positions, (ii) the resulting trial configuration was relaxed via a short microcanonical MD run, and (iii) the trial configuration was accepted based on its internal energy and the Metropolis criterion in Eq. (2) with temperature T taken equal to 600 K. For the simulation of melting, the equilibrated nanoparticles have been subject to a sequence of canonical MD runs with the heating rate ∆T /∆t taken equal to 0.125 K/ps. In this analysis, the maximum of CV was taken as the thermodynamic indication of melting, but also the atomic RMSD at the core (center) of the nanoparticle was monitored as a local structural indication. Since melting proceeds via surface nucleation, the core transforms not until the end of the phase transition. As an illustration, the RMSD at the core of equiatomic nanocrystals is depicted in the left panel of Fig. 5. The RMSD was averaged over several atoms at the core and several MD time steps. Moreover, in Fig. 6 the specific heat for two representative particle sizes (R = 2 nm and R = 3 nm) is shown. Both the RMSD and the heat capacity exhibit some fluctuations which may be eliminated by performing longer simulations with much lower heat rate. However, the main results of this analysis, namely the temperatures at which a sharp increase of RMSD takes place or the maximum of heat capacity is located, should remain unchanged. Plotting the thermodynamic and local melting temperature as a function of the inverse particle radii according to Eq. (1) allows to evaluate by regression the bulk melting point. We have employed the modified Gupta FF in this study. Results are visualized in the 18 ACS Paragon Plus Environment

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right panel of Fig. 5 by symbols.

Figure 5: Left: RMSD at the core of equiatomic nanocrystals of different diameter. The jump in the RMSD indicates locally the phase transition. Right: Calculated bulk melting temperatures as a function of composition. The experimental phase diagram in the background is taken from Massalski et al. 53 The calculated melting temperatures, which are derived from the specific heat (black squares), are positioned in the solid-liquid coexistence portion of the experimental phase diagram. This result is reasonable because the specific heat peak is a global indication lying between the onset and end of the melting transition. On the Ni rich side of the phase diagram, the calculated melting temperatures which are derived from the RMSD (blue triangles) are in good overall agreement with the experimental liquidus. However, on the Ni poor side, the calculated melting temperatures derived from the RMSD are below the experimental liquidus. It is possible that especially for the Ni poor compositions, the aforementioned MC based equilibration method yields configurations which do not corresponds to nearly optimal configurations. Less favorable geometries should lead to decreased melting points. In Fig. 6, the Ni poor nanoparticles exhibit peaks with pronounced shoulders indicating severe atomic rearrangements and premelting.



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Figure 6: Specific heat for a NiTi nanoparticle of 2 nm radius (left panel) and 3 nm radius (right panel) having different compositions. Near-monatomic Systems Similar to the near-equiatomic compositions, we have additionally carried out MD simulations for near-monatomic nanoparticles. Spherical NiTi nanoparticles (R = 2 nm) with Ni concentrations from 0% up to 6% as well as 94% up to 100% have been considered. While omitting the computational burden of simulating different finite sizes and subsequent extrapolation to the bulk case, for the aforementioned representative nanoparticles we have found a typical feature of the experimental bulk phase diagram: 53 The drop of the melting temperature on the Ni poor side is faster with increasing Ni concentration than the decay of the melting temperature with increasing Ti concentraion on the Ni rich side. The comprehensive treatment of the near-monatomic nanoparticles is however beyond the scope of the present work.

Surface Segregation Surface segregation is known to be an obstructive effect in the fabrication and thermal processing of NiTi. 9 Since MD simulations lead to detailed physical trajectories, surface segregation can be studied using the already discussed MD data. The local concentration of chemical elements inside the NiTi nanoparticle can serve as an indication of phase separation. 20 ACS Paragon Plus Environment

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Therefore, we have assigned to each atom in an equiatomic particle (5 nm diameter, overall Ni concentration 50%) a local Ni concentration during the simulation of melting. The local concentration is given by the fraction of Ni atoms among the neighbor atoms in a sphere of radius 0.5 nm around the atom of interest. This temperature-dependent local concentration was then averaged over several subsequent MD time steps so as to improve statistics. For example, from the left panel of Fig. 5 the particle’s melting temperature Tmelt ≈ 1444 K can be extracted based on the jump in the RMSD. In the left panel of Fig. 7, the local atomic Ni concentrations are plotted as a function of the distance from the particle’s center. This data is given below and above the melting point. Above the melting point in the liquid phase the local Ni concentration at the center of the particle decreases on average, whereas the local Ni concentration at the surface increases (solid orange line in the left panel of Fig. 7). This effect is also observed for particles of different overall chemical composition. For the sake of additional analysis and a clearer visualization, we have tried an alternative representation and have discretized the equiatomic nanoparticle into different layers (ranging from its core to the surface), and then evaluated the mean Ni concentration in each layer. The layers have 0.2 nm thickness except for the core layer, which subsumes all the atoms with radial distance smaller than 1.2 nm. The corresponding plot is shown in the right panel of Fig. 7, where the trend for the nickel atoms to segregate to the surface is confirmed. Singh and Sommer 56 have provided, amongst other criteria for segregation in liquid binary alloys, a criterion based on the relative strength of interatomic potential energies. Segregation occurs if the effective potential

Ueff (r) = UNi−Ti (r) −

UNi−Ni (r) + UTi−Ti (r) 2

(11)

is positive at distance r close to the nearest-neighbor distances D. This happens because at elevated temperatures, regions of the interatomic potential where monatomic interactions by



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Figure 7: Local Ni concentrations as a function of the distance from the nanoparticle center (below and above the melting temperature of 1444 K). In the left panel, a linear fit is indicated by a straight line so as to highlight the concentration gradient. In the right panel, the nanoparticle was discretized into different layers and the mean concentration of Ni was evaluated in each layer (see text for details). trend are stronger than the diatomic interaction become accessible to the vibrating atoms. In these regions the atoms favor the monatomic local environment, leading to phase separation. The potential energy for the monatomic and diatomic interactions in NiTi is shown in Fig. 1, which illustrates that the aforementioned criterion is indeed true for NiTi. It is important to stress again that our modification of Ti-Ti interaction appears as a slight correction of the pairwise potential energy but has major implications for collective properties, such as the melting temperature. The insets of Fig. 1 show an illustrative cross section through the nanoparticle below and above the melting point. High local Ni concentrations are colored in black, while low local Ni densities are colored white. The plotted cross sections show that below the melting point, dark and light spots are distributed evenly, while above the melting point the core appears lighter than the surface.


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Ground State Structure of Equiatomic Nanoscale NiTi Global geometry optimization of bimetallic nanoparticles for finding their ground-state structure is an extremely demanding computational challenge. 57 Using global optimization, we have investigated whether monoclinic B190 represents the ground state in nanoscale NiTi, rendering the martensite-austenite transition (and thus SM behavior) possible. A representative equiatomic NiTi nanocrystal of considerable size (N = 16384) has been subject to extensive global optimization employing the basin hopping technique 33 in combination with the short-cutoff, long-cutoff as well as the modified Gupta FF. Therewith, its configuration space given by the lattice constants a, b and c, and the monoclinic angle γ, has been explored. Different monoclinic (γ > 90◦ ) starting configurations have been probed, each leading to an independent search comprising 10000 macrosteps. Each macrostep consisted (i) in the creation of a trial configuration by small random variation of the previous configuration, (ii) in the local optimization of the trial configuration using the velocity-quenching method 26,51 and (iii) in the acceptance based on the potential energy Ei+1 and the Metropolis criterion in Eq. (2). Here, T was treated as an abstract parameter which controls acceptance, thus balancing exploration of the configuration space and determining of local energy minima. The configuration possessing the global energy minimum for a given FF represented the ground state. Results are summarized in Table 3. Table 3: Calculated lattice constants for nanoscale NiTi in comparison to previous results. The results from Ref. 10 correspond to different simulated annealing runs of a nanoparticle with the radius of 3 nm. method modified Gupta FF long-cutoff Gupta FF short-cutoff Gupta FF short-cutoff Gupta FF 10 short-cutoff Gupta FF 10 short-cutoff Gupta FF 41 exp 11 LDA 58 LDA 16

system a = b (nm) c (nm) finite 0.419 0.296 finite 0.421 0.298 finite 0.411 0.290 finite 0.410 0.308 finite 0.426 0.290 bulk 0.426 0.301 bulk 0.426 0.302 bulk 0.421 0.298 bulk 0.417 0.295



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Interestingly, in the nanosystem the three ground states corresponding to the distinct FFs (short-cutoff, long-cutoff and modified) are all lattices with γ = 90◦ and a = b = √ 2c, like the B2 crystal representing bulk austenitic NiTi. The modified and the longcutoff FF yield similar values. Regarding lattice constants a and c, the modified FF (in the nanosystem) deviates from the experimentally observed bulk austenite 11 by only 2.0% and 1.7%, respectively. Moreover, the result of the modified FF for the nanosystem is close to results obtained through LDA calculations 16,58 for bulk. In these LDA studies the B2 lattice does not correspond to the bulk ground state. However, the additional surface energy can turn B2 into the favorable crystal structure in the finite nanosystem. The simulations carried out using the novel, elaborated and validated FF suggest that the nanometer-size NiTi compounds do not exhibit the martensitic transformation, contrary to experimental findings in the macroscopic counterparts. 13–15

Conclusions In conclusion, in this work we have investigated different regions of the composition-temperature-size phase diagram of distinct nickel-titanium alloys by means of extensive computer simulations. For that, we have employed a modified force field, which has been introduced recently for monatomic titanium and other pure metals and implemented in the MBN Explorer software package. In this work, we have further developed this approach to study the shape-memory nickel-titanium material, which is of eminent interest to biomedical and nanotechnology applications. The accounting for distant interactions, especially between Ti atoms, has been found to be crucial for the accurate assessment of melting. The segregation of nickel towards the nanoparticle surface has been shown. Surface segregation is not only relevant as an obstructive effect to the fabrication and processing of metallic materials, but might also be relevant to the modeling of toxicity in nanostructured nickel-titanium alloys. 8 Moreover, we have carried out global optimization so as to determine the ground state of


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equiatomic nickel-titanium nanocrystal, yielding a lattice similar to the high-temperature phase and differing from the martensitic phase, which is expected to be necessary for shapememory behavior on the nanoscale. The distant interactions, which are crucial for the melting assessment, have been reported previously 10 to hinder the simulation of the required martensite-austenite solid-solid transformation. The performed analysis provides novel information towards the atom-level understanding of the shape-memory effect in NiTi alloys.

Acknowledgments A.V.V. acknowledges the support by the European Commission through the FP7 Initial Training Network “ARGENT” (grant agreement no. 608163) and A.V.K. acknowledges the support from Alexander von Humboldt-Foundation. We thank the Center for Scientific Computing Frankfurt for providing the opportunity to perform calculations on the clusters FUCHS and LOEWE-CSC.

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