Smart Materials Research

Volume 2016, Article ID 7512642, 12 pages

http://dx.doi.org/10.1155/2016/7512642

## Atomistic Study on Size Effects in Thermally Induced Martensitic Phase Transformation of NiTi

Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721, USA

Received 23 July 2016; Revised 7 September 2016; Accepted 18 September 2016

Academic Editor: Yong Liu

Copyright © 2016 Sourav Gur and George N. Frantziskonis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The atomistic study shows strong size effects in thermally induced martensitic phase transformation evolution kinetics of equiatomic NiTi shape memory alloys (SMAs). It is shown that size effects are closely related to the presence of free surfaces; thus, NiTi thin films and nanopillars are studied. Quasi-static molecular dynamics simulations for several cell sizes at various (constant) temperatures are performed by employing well-established interatomic potentials for NiTi. The study shows that size plays a crucial role in the evolution of martensite phase fraction and, importantly, can significantly change the phase transformation temperatures, which can be used for the design of NiTi based sensors, actuators, or devices at nano- to microscales. Interestingly, it is found that, at the nanometer scale, Richard’s equation describes very well the martensite phase fraction evolution in NiTi thin films and nanopillars as a function of temperature.

#### 1. Introduction

Shape memory alloys (SMAs) form a particular class of materials that can recover a high level of inelastic or residual strain (up to 8%) through martensitic phase transformation upon heating. SMAs such as NiTi also show superelasticity associated with mechanical loading above a certain critical temperature, that is, the austenite finish temperature , where deformation can be fully recovered upon unloading without any application of heat. Among different SMAs, NiTi is a popular one due to its unique shape recovery capability and high energy dissipation potential through the reversible phase transformation process. Phase transformation in NiTi is diffusionless and displacive in nature [1–4], where one particular type of crystal structure (austenite) transforms to another (martensite) from the application of heat and/or stress/strain. Due to the aforementioned aspects NiTi has found applications in a wide range of fields and at scales ranging from nano- to macrocontinuum level [1, 5–7].

It has been well-established that the behavior of NiTi SMA depends highly on the instantaneous martensite phase fraction and its growth rate. Within its typical operating temperature range, NiTi consists of two phases: (a) the high temperature phase of high symmetry cubic B2 crystal structure, called austenite, and (b) the low temperature phase of a combination of different low symmetry monoclinic variants (up to 12 variants) of B19′ crystal structure, called martensite [8–12]. The martensitic phase transformation process () involves shear driven lattice distortion of B2 crystals, and occurs via an underlying phonon instability due to the softening of a specific transverse acoustic (TA_{2}) branch in its phonon dispersion band structure [1–4]. During the process of displacive transformation, NiTi B2 crystals undergo shear lattice distortion due to the combination of shear along the basal (on (110) []) and a nonbasal (on (001) []) planes and transforms into the monoclinic B19′ (martensite) phase [1, 13–17]. Such transformation occurs over a range of temperatures, where, during the cooling process, the transformation temperature decreases continuously (specifically below the martensite start temperature) [1, 13–17]. In terms of thermodynamics, whenever B2 transform into B19′, the total free energy of the system () increases as the developed internal strain increases the strain energy. Thus, to advance the transformation process, the temperature needs to decrease, and this reduces the total free energy of the system, thereby increasing the driving force on the remaining B2 to transform into B19′ [18, 19]. Thus, to understand the martensitic phase transformation in NiTi SMA, it is very important to have knowledge of the evolution kinetics of various phases at the atomic level.

The crystal structure of the NiTi SMA austenite phase (B2) is cubic, whereas, martensitic (B19′) crystals can have different orientation directions, forming the so-called variants. The phase transition from austenite (B2) to martensite (B19′) results in the formation of several martensitic variants, up to 12 for NiTi [8–12], and the assembly of martensitic variants can exist in the form of twinned martensite (i.e., martensite variant compatible in terms of deformation gradient) [11, 20–22]. These twins are classified as type I twin, type II twin and compound twin [11, 20–22]. In type I the twin plane is a rational crystallographic plane, whereas in type II the twin shear is a rational crystal direction, and in the compound twin both the twin plane and twin shear are rational [11, 20–24]. Evolution of twin martensite variants strongly depends on the grain size. In coarse-grained NiTi type I and II twins are more noticeable than compound twins, which is due to deformation compatibility requirements for the habit planes that is, in between the interface of austenite and different martensite variants [11, 20–24]. These compound twins are typically observed in nanocrystalline NiTi, where they are generally present at the nanoscale. It has been observed from different studies at the nanometer level [11, 20–25] that martensitic phase transformation kinetics are strongly influenced by the nanostructural size, twin boundary energy, grain boundary energy, and the elastic energy of twin variants and of surrounding matrix [11, 20–25].

From the above, it is clear that at the atomic level (nanometer scale) the martensite phase fraction evolution and phase transformation kinetics (nucleation and growth of different B19′ variants) [11, 20–25] depend strongly on the microstructure of NiTi SMA. Further, the complexity in the phase transformation process increases when free surfaces are introduced at the atomic scale. Free surfaces act as defects, thus changing the total free energy of the NiTi system per unit volume, thereby helping the phase transformation activation process. These, known as surface effects, affect the phase transformation process and result in size effects [26–29]. Recent experimental studies show that the presence of free surfaces at the nano- to micrometer scale in NiTi and other SMAs [20–25, 30, 31] substantially change the phase transformation kinetics (i.e., evolution of different phases) and thus the overall material properties as compared to its properties at bulk scales. Examples include (a) evolution of unique martensite microstructure in terms of compound twins through entire NiTi nanograins [21, 22]; (b) NiTi nanograins exhibiting superelasticity with small hysteresis loops [24, 32]; and (c) CuAlNi thin films and nanopillars showing two-way shape memory effect and high damping capacity [33]. In addition to experiments, classical MD simulations have been performed at the nanometer scale, on different NiTi structural components. Studies show, at such scales, remarkable differences in transformation dynamics and phase evolution kinetics compared to at bulk scales [34, 35] and that material properties depend strongly on the NiTi nanostructure or morphology [21, 22, 30, 36, 37]. Examples include (a) equiatomic NiTi thin films [30, 36, 38] and nanowires [24, 39] showing high possibility of evolution (nucleation and growth) of unexpected stable crystal structures during phase transformation; (b) multiple transformation paths from B2 to B19′ [21, 22, 25]; and (c) highly stable square nanowires against torsional buckling, as compared to circular ones [39]. Therefore, at nano- to micrometer level, length scales in conjunction with the presence of free surfaces tailor the microstructure evolution, thereby changing the phase transformation temperature and the temperature dependency of the phase fraction evolution process. Variations in the phase transformation process at the atomic level can finally alter the properties of NiTi at macroscales.

The microstructure or morphology based unique behavior of NiTi at nano- or micrometer length scales makes it an attractive choice in the field of sensors, actuators, composites, or devices for medical and other applications [1, 5–7, 40–43], where certain functionalities at certain length scales are required. Several studies address the surface and size effects at the atomic scale in the NiTi phase transformation process, attempting to provide more physics-based material models. Only studies relevant to this work are cited herein. A very recent study based on classical MD shows size effects at the atomic scale, in both the austenitic and martensitic phase transformation process of NiTi nanoparticles [25, 31]. Experimental studies at the nanometer and micrometer length scales, for different structures such as thin films [30, 36], nanoparticles [25, 31], and nanopillars [23, 24, 44], report the presence of strong size effects on the thermally induced martensitic phase transformation process. They mainly address surface and/or size effects in (a) transformation induced morphologies or microstructure of NiTi and (b) phase transformation process. Further, experimental studies on NiTi thin films show the effect of thickness on thermally induced phase transformation [38]. Other studies report a critical system size below which phase transformation will not occur in NiTi [25].

*Motivation, Objective, and Brief Description of the Present Work*. As mentioned above, extensive computational and experimental studies address the thermally induced martensitic phase transformation process. Of particular interest, some of those studies demonstrate the effect of size on the phase transformation process at the nanometer scale. Studies based on MD simulations [11, 24, 25, 31, 45] involve thermal cycles, where the temperature changes continuously with certain temperature rate (1 to 4 K/ps). Experimental studies [46–49] show that the temperature rate can cause systematic shift in the phase transformation temperatures. Martensite start and finish temperatures decrease with increasing temperature rate, thereby changing the temperature dependency of martensite phase fraction evolution. Other experimental studies [37, 38, 50–52] (in case of NiTi thin films) report that formation of surface oxide and interfacial diffusion layers between NiTi and a silicon substrate surface constrain the phase transformation process. Thus, while previous studies at the nanometer scale provide valuable information about the phase transformation process (temperature dependency of phase fraction evolution and transformation temperatures) in NiTi SMA, they do not solely study size effects, since they include parameters such as temperature rate in size effects. The present study isolates (from temperature rates and other parameters) size effects in thermally induced martensitic phase transformation process of NiTi SMA and also reports the effects of size on the transformation temperature. For this, quasi-static atomistic simulations are performed at constant temperature (so that no temperature rate is involved) using a well-established interatomic potential for NiTi [53] based on a classical MD simulation platform [54, 55]. Two components, thin films and nanopillars, are examined, which are also found in many applications [1, 5–7, 40–43, 56].

#### 2. Methods

Important to the simulations is materials characterization, that is, the method used to identify spatial domains of austenite and martensite during the thermally induced phase transformation. This section describes the materials characterization process used within the MD simulation process adopted herein and executed in LAMMPS [46, 47].

##### 2.1. Interatomic Potential

An effective MD simulation framework for NiTi binary alloy based on the Finnis-Sinclair many-body interatomic potential has been proposed and examined in detail [11, 23, 24, 31, 39, 44, 45, 53, 57–59]. For self-containment, a brief and relevant description of this semiempirical interatomic MD potential is provided here, yet more detailed information can be found elsewhere [11, 23, 24, 31, 39, 44, 45, 53, 57–59]. Lai and Liu [53] developed this semiempirical interatomic potential based on experimental data and first principal calculations. It is based on the embedded-atom method (EAM) considering a second-moment approximation of tight-binding theory (TBSMA). The TBSMA is used for Ni-Ni and Ti-Ti interaction and mixed rule is used for the NiTi interaction. All the parameters in the Lai and Liu [53] interatomic potential are calculated from lattice parameters, cohesive energy, elastic constants, and vacancy formation energy of B2 structure [11, 53]. Employing this potential, reversible phase transformation between austenite (B2) and martensite (B19′) structures is observed. It has been confirmed that structurally stable B2 and B19′ phases are obtained with low potential energy. In particular the cohesive energy of B19′ variants is found to be more stable than that of B2 [57, 58].

According to this potential, the total energy of the system over all atoms is expressed aswhere the first term in the right hand side indicates the pairwise interaction energy due to Born-Mayer repulsion, which stabilizes the crystal, and the second term represents many-body effects, considering second-moment approximation for the electron density. Recently the many-body interaction part of Lai and Liu potential has been modified by Zhong et al. [11]. In particular, a smooth cutoff behavior is considered to avoid the diverging forces (generated from large atomic displacements) during simulations. Such modification ensures the continuity of the second term in (1) and its first derivative at cutoff radius and hence provides smooth behavior in energies and interatomic forces molecular static and dynamic simulations. Different parameters of this modified interatomic potential are determined by fitting properties, obtained from first principles calculations, of the B2 phase at 0 K, with the potential cutoff radius . The terms in (1) are expressed aswhere and denote atoms and denote type of atoms (Ni or Ti) and . A detailed description of the parameters involved in this interatomic potential is provided in Zhong et al. [11, 23].

##### 2.2. Simulation Details

Quasi-static MD simulations are performed in LAMMPS [54, 55], for temperature driven martensitic phase transformation in NiTi. Two types of simulation cells are considered: (a) 2D square thin films and (b) 3D nanopillars, with free (nonperiodic) boundary conditions in all directions. For the thin films, seven different sizes are simulated, that is, of side 2, 4, 8, 16, 32, and 64 nm, and the film thickness is kept constant at 2 nm for all sizes. Similarly, seven different sizes of 3D nanopillars were simulated, that is, of diameters 2, 4, 8, 16, 32, and 64 nm and length 8 nm with nonperiodic boundaries in all directions. To demonstrate the effect of free surfaces on the transformation kinetics, the thin films and nanopillars simulation results are compared to bulk simulation results of cell size 8 nm × 8 nm × 8 nm (76532 atoms) and periodic boundary conditions in all three directions.

For all simulations, initially, pristine austenite B2 NiTi samples with crystallographic orientation [] (vertical axis), [], and [] (horizontal axis) are considered. Simulations are performed within the temperature range from 50 K to 750 K, with an interval of 10 K within the temperature range from 250 K to 550 K and with 50 K interval for the remaining temperature range. Such large temperature range is considered so that an accurate estimation of the size dependent transformation temperatures, that is, martensite start and finish and phase fraction evolution, is obtained, which is important for obtaining, as detailed subsequently, the overall martensite phase fraction evolution in NiTi thin films and nanopillars as a function of temperature.

First, a pristine austenite B2 NiTi sample is relaxed using a stress controlled conjugate gradient energy minimization method. Then, the relaxed sample is given a velocity distribution corresponding to a particular temperature and held for 360 ps, under an NPT stabilization where the system pressure is kept constant at 0 bar. Then, the NPT stabilized sample is subjected to constant volume (NVT) conditions for an additional 240 ps. In each case of NPT or NVT stabilization, the adopted time step is 0.0005 ps. During simulation the temperature is controlled by a Nose-Hoover thermostat and Berendsen barostat. In order to remove the effects of thermal fluctuations, the atomic coordinates were averaged over 20 ps. It was verified that the systems were well-equilibrated and the fluctuations in total energy and temperature were smaller than 0.001 eV/atom and 0.50 K, respectively.

##### 2.3. Structural Characterization

To distinguish austenite B2 phase from martensite B19′ phase and its variants, the bond length based order parameter proposed by Mutter and Nielaba [31, 44] was adopted. It can be expressed aswhere different bond length parameters of pure austenite phase and martensite phase are , and . The value of this order parameter becomes +1.0 for the martensite B19′ phase when and . For the austenite B2 phase and thus the value of this order parameter becomes −1.0. However, after simulation, lattice parameters of B2 and B19′ phases change because of the linear thermal expansion or contraction and because of the internal strains that develop during the phase transformation process. Thus, instead of showing distinct values (like delta function) at +1.0 (for B19′ phase) or −1.0 (for B2 phase), this bond length based order parameter for B19′ and B2 phase exhibits a narrow Gaussian distribution around +1.0 and −1.0, respectively [31, 44, 59]. Therefore, to distinguish between the austenite (B2) phases from martensite (B19′) phase, atoms with order parameter greater than +0.75 correspond to martensite B19′ phase and less than −0.75 to austenite B2 phase. It has been observed that this order parameter shows Gaussian distribution around a mean of and a standard deviation . At every temperature and size, the martensite phase fraction was calculated by taking the ratio of intensity and spread in the Gaussian distributions. Further detailed description of the martensite phase fraction calculation based on this order parameter is provided in Mutter and Nielaba [31, 44].

##### 2.4. Phase Transformation Kinetics

Phase fraction evolution kinetics in NiTi SMA is a very complex process that involves the competition between developed internal strain and phonon dispersion instability [1–4]. The rather limited studies reported in the literature mostly address the behavior of single crystals at small scales (quantum or atomic level). There are significant size effects present at small scales [11, 21–25, 31, 39, 44] that add to the complexity and saturate at length scales in the range of several nanometers. Several works based on continuum thermodynamics address the macroscale constitutive behavior of NiTi and other SMAs subjected to thermomechanical loading [60–62]. In those studies, the phase fraction evolution is obtained directly from experimental results or is assumed empirically. Recently Zotov et al. [35] reported extensive experimental data on thermally induced martensitic phase fraction for bulk NiTi. Based on these experimental data on polycrystalline (mainly [] B2 crystal with some [] B2 crystal) and annealed (at 750°C) NiTi SMA, Zotov et al. [35] proposed a phenomenological model and verified it with respect to additional experimental data reported in the literature for NiTi of different concentrations of Ni and Ti [34, 35]. In the Zotov et al. [35] study, the experiments are performed with 20°C/min cooling and heating rate. In this model, the austenite transformation rate at any given temperature is assumed to be proportional to the nucleation rate of austenite per unit volume. It is also assumed that the size of austenite nucleus increases linearly with the austenite phase fraction. This phenomenological model captures adequately the kinetics of the observed martensitic phase transformation rate as a function of the martensite phase fraction in the reported experiments. According to the model, the temperature dependent martensite phase fraction evolution is described through the Richards equation; that is,where denotes the temperature dependent martensite phase fraction, the temperature of interest in degrees Kelvin, and the temperature corresponding to maximum transformation rate in degrees Kelvin, while (unit less) and (in 1/K) are fit parameters which dictate the transformation or growth rate of the martensite phase. Here, the nonlinear curve fitting tool box in MATLAB 2014b has been used to obtain the optimal value of different parameters in (4), where a nonlinear least square minimization technique is employed.

In the present study, for estimation of the martensite start and martensite finish temperatures, a range between 5% and 95% is considered. This is because, below 5% and beyond 95% of martensite phase fraction, it is observed that the standard deviation associated with the estimated fraction of the minor phase is of the order of phase fraction of the major phase itself. Thus, using these limits in martensite phase fraction and (4), the estimated values for transformation temperatures, that is, and , are expressed asA larger temperature range is included in this study in order to obtain accurate parameters for Richard’s equation (i.e., , , and ), which describes the evolution of martensite phase fraction with temperature. Since Richard’s equation (i.e., (4)) is highly nonlinear, to obtain accurate values for its parameters, nonlinear least square minimization is performed, which requires a large number of data points for precise evaluation of the parameters. Further, the accurate evaluation of Richard’s equation parameters is important for determining the martensite start () and finish () temperatures, in (5a) and (5b).

#### 3. Results and Discussions

##### 3.1. NiTi Thin Films

As mentioned above, quasi-static MD simulations are performed on NiTi thin films with free boundary conditions in all three directions for side sizes of 2, 4, 8, 16, 32, and 64 nm with thickness 2 nm for all sizes. Simulations are performed within the temperature range from 50 K to 750 K, with an interval of 10 K within the temperature range of 250 to 550 K and with 50 K interval for the remaining temperature range.

Figure 1 shows the microstructure of NiTi thin film at three different temperatures, 200, 350, and 500 K, for 8 nm size. The microstructure at 200 K, Figure 1(a), shows a single-phase martensite with (001) compound twins, and this is in good agreement with earlier work reported by Zhong et al. [11, 23]. As shown in Figure 1(b), at 350 K the microstructure shows a two-phase mixture wherein the (001) plane forms the interface (i.e., the habit plane) between the austenite and martensite phases, as reported in Zhong et al. [11, 23]. Further, the formation of the (001) habit plane is attributed to a combination of shear along the basal (on (110) []) plane and a nonbasal (on (001) []) plane in austenite, in agreement with what is reported in literature [1, 14–17]. Figure 1(c) shows a single-phase austenite at 500 K temperature.