Low-Hysteresis Shape Memory Alloy Scale-Up: DSC, XRD and Microstructure Analysis on Heat-Treated Vacuum Induction Melted Ni-Ti-Cu-Pd Alloys

: Quaternary Ni-Ti-Cu-Pd formulations were cast by vacuum induction melting (VIM) with the aim of preparing low-hysteresis shape memory alloys and verifying the applicability of the Co-Factor theory in conventional industrial manufacturing processes. The cast alloys showed lower transformation hysteresis width in DSC measurements than binary Ni-Ti, but struggled to achieve a near zero hysteresis, as predicted by the theoretical framework, despite being close to satisfy the ﬁrst Co-Factor condition (CC I) that foresees minimum hysteresis for formulations in which the middle eigenvalue of the martensitic transformation matrix λ 2 approaches one. The microstructure of the annealed Ni-Ti-Cu-Pd alloys exhibited a considerable amount of mostly sub-micron-sized secondary phases, which distort the matrix composition and prevent it from reaching the optimum stoichiometry for satisfying the CC I. In addition, this class of materials is prone to aging effects, leading to the formation of semi-coherent tetragonal precipitates, which tend to also form at the grain boundaries after low-temperature annealing, further affecting the transformation hysteresis in DSC experiments depending on the thermal history. This work reveals the importance of considering typical casting effects that alter the theoretical λ 2 of ideal materials in the compositional design for the development of high-performance low-hysteresis alloys.


Introduction
Low-hysteresis shape memory alloys (LH-SMAs) are researched for a wide field of actuator applications [1], as a low hysteresis improves the actuation efficiency and is expected to significantly increase the fatigue life of devices [2]. Additionally, in orthodontic applications, low-hysteresis alloys are of interest as they could improve dental occlusion therapies [3]. The addition of Cu to binary Ni-Ti is the most common alloying approach to decrease the thermal hysteresis between the high-temperature austenitic parent-phase and the low-temperature martensitic daughter-phase [4]. By increasing the amount of copper, the orthorhombic B19 martensite becomes continuously more stable over the monoclinic B19 allotropy form, which is the low-temperature phase in binary Ni-Ti. Previous research works indicated that by adding an amount of about 7.5 at. % Cu, the orthorhombic crystal structure occurs and that an altered transformation pathway B2 → B19 → B19 becomes stable during cooling up to a Cu content of approximately 15 at. %, while at even higher Cu contents, a single B2 → B19 transformation is observed [5][6][7].
Presently, there exist several simulation works and experimental case studies, which proved that the amount of Cu in this span correlates well with a reduction in hysteresis. The theoretical background was established by J.M. Ball and R.D. James et al. [8], who introduced the elastic theory of martensite, implying that the thermal hysteresis approaches high-temperature shape memory alloy (HT-SMA) applications are targeted, where thermal cycles significantly above 100 • C are necessary for triggering the transformation.
However, because of the high cost of Pd, there are only a few examples of studies that investigate on the transformation properties of cast ingots with an industrial relevant size. Literature works focus on low-scale thin-film approaches [23] or on arc melting, leading to fine microstructures with a matrix composition closer to the nominal total composition due to faster solidification than alloys prepared for industrially feasible commercial applications own [26]. The aim of this study is to design Ni-Ti-Cu-Pd LH-SMAs based on a density functional theory (DFT) approach considering Co-factor theory and prepare them by a scalable, vacuum-induction melting process, applying a typical post-annealing treatment for investigating on their industrial feasibility. To the knowledge of the authors, this study is the first one using this conventional melting process for the Ni-Ti-Cu-Pd alloy. Further solutions and aging heat treatments are applied to study the effect of precipitation on the microstructure and thermal stability.

Materials and Methods
The alloy design was based on two approaches, and accordingly, the studied alloys were divided into two groups designated by the letters "A" and "B". Regarding group A, three formulations (Ti-rich, Ti 50 at. %, Ti-lean) were inspired by the proposed compositions of patent US20060086432A1 by Z. Zhang and R. D. James, which foresees, based on the Co-Factor theory framework, minimum hysteresis whenever the following conditions are satisfied [29]: Ti = 49. 5  Furthermore, for group B, the formula Ti 50 Ni 38,3−x Cu 11,5 Pd x was used to prepare three alloys (5-Pd, 15-Pd, 20-Pd), investigating the effect of Pd additions to a previously optimized ternary Ni-Ti-Cu LH-SMA for actuation applications [30].
DFT was applied to simulate the atomic structure and evaluate Co-factor conditions of the investigated alloys. A supercell of 128 atoms, obtained by an expansion matrix 4 × 2 × 4, was used to simulate the cubic B2 and orthorhombic B19 cell of the austenitic parent-phase and the martensitic daughter-phase, respectively. The plane-wave-based Quantum ESPRESSO package was used [31]. Atoms were described by utrasoft pseudopotentials, and the Perdew-Burke-Ernzerhof (PBE) functional was adopted for the exchange and correlation interaction. For all calculations, energy cutoffs of 27 Ryd and 270 Ryd (for kinetic energy and charge density expansion, respectively) were implemented. The lattice parameters and the atomic positions were fully relaxed until the achievement of the convergence criteria of 10 −3 a. u. for forces and 10 −6 Ryd for energy within a 2 × 2 × 2 k-point mesh. The Co-Factor theory was applied to calculate the CC I, referring to the λ 2 eigenvalue of the transformation matrix and thus to the lattice compatibility between the austenitic and martensitic phases and the Co-Factor condition II (CC II), referring to the compatibility of the twin systems [9]. While satisfying both conditions leads to a reduction in the elastic transition layer during the transformation, the CC I was experimentally found to be of prior importance to reduce the thermal hysteresis and, therefore, will be discussed preferentially in the scope of this work [10]. The designed alloys and DFT calculation results are presented in Table 1.
The alloys were melted by VIM as cylindrical 400 g ingots with a 25 mm diameter. These ingots were then heat-treated in a vacuum furnace by 900 • C for 24 h in an inert Ar atmosphere for homogenization, followed by slow furnace cooling (cooling rate in order of 1-10 • C/min, hereafter referred to as "HOMO"). Small samples (50-100 mg) were cut and grinded to remove surface oxides for differential scanning calorimetry (DSC) and chemistry analysis. The combustion infrared absorption (CIA) method (using LECO analyzers) was used to guarantee that the amount of residual elements, such as C, N and O, are below 500 ppm. XRF analysis was performed to verify that the actual composition after melting represents the nominal composition with an error at least below 0.2 at. % for each element. DSC measurements were carried out on samples from homogenized ingots, and after exposing them to solution treatment at 900 • C × 30 min, followed by water quenching and with additional aging treatment at 550 • C for 180 min in a muffle furnace (hereafter referred to as "HOMO", "ST" and "AGED", respectively). Cycling recipes and measurements with different thermal scanning rates were applied in the DSC analysis to study the transformation stability of alloys. At least three DSC runs were performed in order to assure that the error on the measured transformation temperature parameters was below 0.5 • C. In this work, the distance between the austenite finish and martensite finish temperatures ∆ f T is considered whenever referring to hysteresis width. Ingot sections were prepared by standard metallurgical cutting, mounting, grinding and polishing procedures. An etching reagent composed of H 2 O, HNO 3 and HF in a ratio of 5:4:1 was used to reveal microstructural features. A microstructure analysis was performed with an optical (OM) and a field-emission scanning electron microscope (FE-SEM). The crystal structure characterization was performed by X-ray diffraction (XRD, Panalytical X'Pert Pro instrument) using a theta-2theta configuration. The instrument was equipped with a X'Celerator 1D sensor and employs CuK (λ = 0.15418 nm) radiation. The resulting spectra were evaluated by comparison to the JCPDS database, and a Pawley refinement was applied for a semi-quantitative evaluation of phases and the determination of lattice parameters. Table 1. The nominal composition of experimentally melted alloys, the compositions for DFT calculations, and the results calculated for CC I and CC II; the closer these values are to zero, the more the conditions are satisfied. Alloys can be divided into two groups: Group A consists of alloys (Ti-rich, equiatomic, Ti-lean) with a content of Pd close to 10 at. % and a Cu/Ni ratio selected to minimize |λ 2 − 1|. Group B is composed by adding Pd in expense of Ni to the reference Ni-Ti-Cu LH-SMA alloy, data from [30].

Alloy Group Code
Nominal

Group A Study-The Effect of Ti Content
First, the transformation of homogenized alloys at 900 • C for 24 h with slow furnace cooling (HOMO), solution-treated alloys with water quenching at 900 • C for 30 min after HOMO (ST) and aged alloys at 550 • C for 180 min after ST (AGED) were studied by  The homogenized VIM cast alloys exhibit transformation temperatures with As ranging from 16 °C to 44 °C, increasing with Ti content, despite their coincidently higher Ni:Ti ratio (Figure 1a). Their transformation hysteresis (Figure 1b) width Af-Mf is significantly lower than in binary Ni-Ti VIM ingots (ΔfT ≈ 55-85 °C) [32] but considerably higher than the results obtained in thin-film studies on such quaternary alloys [23]. After solution treatment, the transformation peaks tend to become broader, and the hysteresis increases. Subsequent aging treatment reduces the hysteresis significantly in the Ti-rich alloy and   The homogenized VIM cast alloys exhibit transformation temperatures with As ranging from 16 • C to 44 • C, increasing with Ti content, despite their coincidently higher Ni:Ti ratio (Figure 1a). Their transformation hysteresis (Figure 1b) width A f − M f is significantly lower than in binary Ni-Ti VIM ingots (∆ f T ≈ 55-85 • C) [32] but considerably higher than the results obtained in thin-film studies on such quaternary alloys [23]. After solution treatment, the transformation peaks tend to become broader, and the hysteresis increases. Subsequent aging treatment reduces the hysteresis significantly in the Ti-rich alloy and slightly in the Ti50 alloy, while it further increases in the Ti-lean alloy. Such effects can be explained by the precipitation behavior, as will be discussed in the following section.
The enthalpy of the transformation (Figure 1c) increases in all alloys after solution treatment, indicating that in the homogenized state, secondary phases are formed during slow furnace cooling and dissolve into the matrix phase after ST, leading to a higher fraction of material that participates actively in the transformation [11]. After the aging treatment, the enthalpy of the transformation decreases in all alloys again due to the formation of new secondary phases. The alloys showed in all conditions, HOMO, ST and AGED, a high stability upon thermal cycling. In ten DSC cycles, which covered the whole transformation, the peaks shifted less than 1 • C, as demonstrated in Figure 1d, representatively for the Ti-rich alloy tested after the homogenization treatment. In order to investigate the dependence of the transformation parameters on the thermal scanning rate and in particular on hysteresis width, alloys were also tested at different DSC scanning rates of 3, 5, 20 and 30 • C/min. (Figure 1e illustrates the enlargement of peaks by increasing the thermal scanning rate.) It was shown before that the hysteresis enlarges in such alloys with the thermal rate, caused by the DSC measurement and intrinsic sample characteristics, affecting our perception of "small hysteresis" [32]. As shown in Figure 1f, the Ti-lean alloy is much more sensitive to a change in the scanning rate, the hysteresis is about 48 • C higher testing at 30 • C/min than observed at a rate of 3 • C/min. On the other hand, the Ti50 alloy exhibits the smallest differences in hysteresis width by testing at fast and slow rates. The reasons for the differences in the sensitivity of alloys to the rate is not fully understood yet, as there does not seem to be a straight correlation to the |λ 2 − 1| = 0 eigenvalue criterion of the CC I. However, it is striking that the rate sensitivity correlates well to the values calculated for the CC II in Table 1 for the three alloys. For CC II approaching zero, the hysteresis width becomes less affected by the heating and cooling velocity. Presumably, a higher compatibility of martensitic twin systems leads to accelerated transformation kinetics and hence a lower sensitivity to thermal delay at fast rates. The presence of coherent precipitates after aging could further support the transformation, as they act as nucleation sites for martensite and twins, further reducing the thermal sensitivity after aging.
The XRD spectra of homogenized and aged alloys measured at room temperature are shown in Figure 2.
The alloys contain several crystallographic phases at room temperature: B19 orthorhombic martensite, B2 cubic austenite and various tetragonal precipitates with different ideal stoichiometries (Pd 5 Ti 3 , CuNiTi, TiPdCu, CuTi). In all three alloys, Pd 5 Ti 3 , TiPdCu and CuTi phases are observed, while CuNiTi is only detected in the Ti-lean one. Due to the low furnace-cooling rate in the homogenization treatment, several precipitates are already present in the alloys prior to aging. However, after the solution treatment and subsequent annealing, the peaks of the tetragonal phases become much more evident, and the fraction of secondary phases significantly augments.. At a first glance, as the B2 [110] reflection is located around 41.8 • , the samples seem to contain a significant amount of B2 austenite phases, and the ratio of B19 martensite to cubic B2 austenite increases after aging in correspondence to the increase in transformation temperatures reported in Table 2. However, this observation is counterintuitive to the DSC results presented in Figure 1, which demonstrated all samples-especially the Ti-rich sample-to be in a martensitic state at room temperature. In addition, a high quantity of residual B2 austenite is improbable, as the high transformation enthalpies values in Figure 1c indicate that samples transform completely and other B2 reflections in XRD spectra seem to be absent. It is known from ternary Ni-Ti-Cu alloys that there is a partial superposition of the B2 [110] and the B19 [020] peak [33]. For these reasons, we expect only a low quantity of residual B2 present in the samples, while the mere quantity of the peak is attributed to the B19 [020] planes reflections. The identification becomes even more difficult, as based on the content of the heavy Pd atoms, peaks are shifted increasingly to lower angles, rendering calibration procedures more complicated. However, it would be out of scope to quantify the B2:B19 ratio in this XRD study, which had the primary target of identifying precipitate phases and excluding the presence of other martensite morphologies (such as B19 ) in the alloys.
dependence of the transformation parameters on the thermal scanning rate an ular on hysteresis width, alloys were also tested at different DSC scanning rat and 30 °C/min. (Figure 1e illustrates the enlargement of peaks by increasing scanning rate.) It was shown before that the hysteresis enlarges in such allo thermal rate, caused by the DSC measurement and intrinsic sample characteri ing our perception of "small hysteresis" [32]. As shown in Figure 1f, the Ti-l much more sensitive to a change in the scanning rate, the hysteresis is about 4 testing at 30 °C/min than observed at a rate of 3 °C/min. On the other hand, th exhibits the smallest differences in hysteresis width by testing at fast and slow reasons for the differences in the sensitivity of alloys to the rate is not fully yet, as there does not seem to be a straight correlation to the |λ 2 − 1| = 0 eig terion of the CC I. However, it is striking that the rate sensitivity correlates values calculated for the CC II in Table 1 for the three alloys. For CC II approa the hysteresis width becomes less affected by the heating and cooling velocity bly, a higher compatibility of martensitic twin systems leads to accelerated tran kinetics and hence a lower sensitivity to thermal delay at fast rates. The presen ent precipitates after aging could further support the transformation, as they a ation sites for martensite and twins, further reducing the thermal sensitivity a The XRD spectra of homogenized and aged alloys measured at room t are shown in Figure 2. The alloys contain several crystallographic phases at room temperature rhombic martensite, B2 cubic austenite and various tetragonal precipitates wi The microstructure of the VIM cast alloys shown in Figure 3 consists of austenitic grains in the 100 µm-1 mm range in which, based on the transformation temperatures (c.f. Table 2), different quantities of martensitic twinned areas are observed at room temperature-the least in Ti-lean alloy (Figure 3a). All alloys contain some Ti-oxides and Ti 2 Ni inclusions, which were not revealed by XRD due to their small, localized amount. The higher resolution and back scattering contrast in SEM analysis reveal differences in the precipitation behavior of the alloys. In the Ti-lean (Figure 3d), only a few micronsized, macroscopic Cu-rich precipitates are observed, which are sparsely distributed in the vicinity of the grain boundaries. The matrix seems free of macroscopic phases that can be revealed at these resolutions. On the other hand, in the Ti50 (Figure 3e) and Ti-rich alloy microstructure (Figure 3f), only finer, homogenously dispersed precipitates are present.
Pd-rich precipitates, such as Pd 5 Ti 3 , which were identified by XRD ( Figure 2) in all alloys, remain at the nanoscale and thus cannot be visualized by FE-SEM.   After the solution treatment and annealing at 550 °C × 3 h, no such semi-coherent macroscopic secondary phases are detected in the matrix, except the incoherent Ti2Ni precipitates, which are not soluble at 900 °C (Figure 4a, c). The solution treatment dissolved all coherent phases, and aging led to their reformation on a low nanometric scale in the matrix. However, close-up micrographs show that a thin layer of Cu-rich phase with nmrange thickness formed along the grain boundary in all samples (Figure 4d, f).

Group B Study-The Effect of Pd Content
Like for the alloys of group A, the transformation temperatures of the alloys with different Pd contents from group B were measured by DSC after homogenization at 900 • C for 24 h with slow furnace cooling, a solution treatment with water quenching at 900 • C for 30 min and aging at 550 • C for 180 min after the solution treatment. The transformation temperatures of the investigated alloys are shown in Table 3. Similar to ternary Ni-Ti-Pd alloys [4], up to an elevated substitution of Ni for Pd, the transformation temperatures remain in a narrow range, then they start to increase continuously, as observed here for the alloy 20-Pd (Figure 5a,b). All alloys show a rather narrow transformation hysteresis, which does not lack behind the alloys of group A, despite the higher calculated CC I values (cf. Table 1). The DSC curves in Figure 5a and the effect of heat treatment on hysteresis in Figure 5c demonstrate a clear trend: after the solution treatment, the peaks become wider, and after aging, they decrease again, without showing a significant difference in width with respect to the homogenized transformations. The enthalpy of the transformation increases upon the solution treatment and returns to smaller dimensions after 550 • C for the 3 h treatment, very similar to that observed for the alloys of group A. Additionally, in this case, the changes in enthalpy are related to the dissolution and reformation of secondary phases. The cycling stability is high, as shown in Figure 5e for cycling ten times the homogenized alloy 20-Pd, even though small shifts are observed, which could also be related to the cycling at higher temperatures, as A f is around 135 • C. The sensitivity to the thermal rate is investigated by cycling the alloys at rates ranging from 3-30 • C/min and demonstrated representatively for the solution-treated alloy 20-Pd (Figure 5f).
The spectra and designation of crystallographic phases of group B alloys are shown in Figure 6, as revealed by XRD analysis. All the alloys exhibit an almost fully B19 orthorhombic matrix structure as the analysis is performed significantly below their M f temperature, and only in 5-Pd, a tiny quantity of residual B2 is detected. Due to the much bigger Pd atom, which substitutes for Ni, the crystals unit cell expands, and the peaks sustain a significant shift to lower angles with an increasing Pd amount. Besides the matrix phase, in group B alloys, precipitated secondary phases are also present after 900 • C for 24 h treatment and furnace cooling. Moreover, with a higher amount of Pd, the intensity of reflections associated with precipitates in the homogenized samples increases. Most secondary phases in the three alloys are tetragonal and rich in Cu; however, in the 15-Pd and 20-Pd alloys, a peak at 39.8 • C associated with Pd 5 Ti 3 is measured, which grows significantly in size after a heat treatment at 550 • C for 3 h, in particular for the latter alloy.
The microstructure of group B alloys is presented in Figure 7. The optical micrographs illustrate an evident change in the microstructure morphology of the almost completely martensitic samples (Figure 7a,c). The microstructure of 5-Pd resembles those of the group A alloys with fine twin features, and the primary austenite grain boundaries are clearly visible. Increasing the Pd content, the twin structure becomes coarser, in particular in the 20-Pd sample. Unlike in group A alloys, macroscopic matrix precipitates were not observed. The microstructures appear monophasic with a low amount of oxides and Ti 2 Ni inclusions. Only in 20-Pd, a low amount of Cu-rich precipitates are present. The spectra and designation of crystallographic phases of group B alloys are shown in Figure 6, as revealed by XRD analysis. All the alloys exhibit an almost fully B19 orthorhombic matrix structure as the analysis is performed significantly below their Mf temperature, and only in 5-Pd, a tiny quantity of residual B2 is detected. Due to the much bigger Pd atom, which substitutes for Ni, the crystals unit cell expands, and the peaks sustain a significant shift to lower angles with an increasing Pd amount. Besides the matrix phase, in group B alloys, precipitated secondary phases are also present after 900 °C for 24 h treatment and furnace cooling. Moreover, with a higher amount of Pd, the intensity of reflections associated with precipitates in the homogenized samples increases. Most secondary phases in the three alloys are tetragonal and rich in Cu; however, in the 15-Pd and 20-Pd alloys, a peak at 39.8 °C associated with Pd5Ti3 is measured, which grows significantly in size after a heat treatment at 550 °C for 3 h, in particular for the latter alloy. The microstructure of group B alloys is presented in Figure 7. The optical micrographs illustrate an evident change in the microstructure morphology of the almost completely martensitic samples (Figure 7a,c). The microstructure of 5-Pd resembles those of the group A alloys with fine twin features, and the primary austenite grain boundaries are clearly visible. Increasing the Pd content, the twin structure becomes coarser, in particular in the 20-Pd sample. Unlike in group A alloys, macroscopic matrix precipitates were not observed. The microstructures appear monophasic with a low amount of oxides and Ti2Ni inclusions. Only in 20-Pd, a low amount of Cu-rich precipitates are present.  After the solution heat treatment followed by aging at 550 • C for 3 h, no significant changes in the matrix are observed in the alloy 5-Pd (Figure 8a) and alloy 15-Pd (Figure 8b). However, in these alloys, precipitates are observed at the grain boundaries after aging (Figure 8d,f). In the case of the 5-Pd alloy, where CuNiTi and Cu 2 Ti precipitates were predominantly found by XRD analysis, they show a unique bright contrast. On the other hand, in the 15-Pd and 20-Pd alloys, at least two types of submicron precipitates are revealed (Figure 8e,f): one Pd-rich type Pd 5 Ti 3 and Cu-rich ones, which appear merely with a dark contrast compared to the matrix with a high amount of heavy Pd atoms. The formation of Pd 5 Ti 3 precipitates causes a local depletion of Pd and thus an enrichment in Cu; for this reason, they are found closely attached to Cu-rich precipitates in alternating patterns. Rehman et al. found similar structures in higher Pd-containing alloys, in which Ti 2 Pd and TiPdCu were the main precipitates present [28].

Discussion
The industrialization of higher alloyed shape memory alloys requires a minute control over the microstructure to guarantee processability, reliable mechanical properties, transformation behavior, excellent fatigue life and to fully exploit the low-hysteresis potential of these materials. It is known that not only the CC I Co-factor condition, which takes into account the chemistry, but also the microstructure, i.e., grain size, grain boundary orientation, presence of secondary phases etc. has an impact on the transformation properties of shape memory alloys [34]. In this regard, the use of conventional melting techniques, such as VIM, poses challenges to the realization of novel low-hysteresis alloys as solidification rates are much slower than in frequently explored low-scale synthetization processes, such as VAR, magneton sputtering or additive manufacturing. These techniques own far higher solidification rates than VIM, leaving most elements in solid solution of a fine-grained microstructure after synthesis. In order to estimate the solidification time of the VIM cast alloys of this work, Chvorinov's rule [35] was applied: where t is the solidification time, V the ingot volume, A the contact area between the melt and n a constant, which is empirically assumed to be 2 [35]. B is the mold constant, which can be calculated according to Equation (6): where ρ and ρm are the density of the mold and the metal, respectively, T and Tm are the temperature of the mold before pouring and the melting temperature of the liquid, k is the thermal conductivity of the mold, L is the latent heat of fusion, c and cm are the specific heat capacity of the mold and the metal, and ΔTs is the superheat of the liquid metal. A few approximations have been made, as the precise determination of these parameters is not trivial: the mold was assumed to be a perfect metallic carbon steel cylinder and the liquid metal in contact with the base area and the lateral surface. The mold constant was

Discussion
The industrialization of higher alloyed shape memory alloys requires a minute control over the microstructure to guarantee processability, reliable mechanical properties, transformation behavior, excellent fatigue life and to fully exploit the low-hysteresis potential of these materials. It is known that not only the CC I Co-factor condition, which takes into account the chemistry, but also the microstructure, i.e., grain size, grain boundary orientation, presence of secondary phases etc. has an impact on the transformation properties of shape memory alloys [34]. In this regard, the use of conventional melting techniques, such as VIM, poses challenges to the realization of novel low-hysteresis alloys as solidification rates are much slower than in frequently explored low-scale synthetization processes, such as VAR, magneton sputtering or additive manufacturing. These techniques own far higher solidification rates than VIM, leaving most elements in solid solution of a fine-grained microstructure after synthesis. In order to estimate the solidification time of the VIM cast alloys of this work, Chvorinov's rule [35] was applied: where t is the solidification time, V the ingot volume, A the contact area between the melt and n a constant, which is empirically assumed to be 2 [35]. B is the mold constant, which can be calculated according to Equation (6): where ρ and ρ m are the density of the mold and the metal, respectively, T and T m are the temperature of the mold before pouring and the melting temperature of the liquid, k is the thermal conductivity of the mold, L is the latent heat of fusion, c and c m are the specific heat capacity of the mold and the metal, and ∆T s is the superheat of the liquid metal. A few approximations have been made, as the precise determination of these parameters is not trivial: the mold was assumed to be a perfect metallic carbon steel cylinder and the liquid metal in contact with the base area and the lateral surface. The mold constant was calculated using values for binary equiatomic NiTi. The used parameters are listed in Table 4. Table 4. The parameters used to calculate the solidification time by means of Chvorinov's rule. The parameters were extracted from technical data sheets and scientific literature references [36,37]. The calculation results in a mold constant of B = 3964 min/cm 2 and a solidification time of t = 307 s. Depending on the alloy composition, which affects c m , ρ m , T m , ∆T s , an error of up to ±20% for the mold constant and solidification time is expected. However, even though the exact solidification time might vary depending on composition and simplifications of the system, the order of magnitude is in the range of several hundreds of seconds and thus much higher than all small-scale manufacturing processes, such as VAR, magnetron sputtering or processes that use a laser beam source for solidification of the alloy. For this reason, and due to the necessary high-temperature homogenization treatment, VIM ingots tend to be affected by the thermal history, which translates into different grain sizes and the presence of various secondary phases of different quantities and morphologies, as demonstrated here by XRD and SEM analysis, revealing 100 µm sized grains and several tetragonal precipitates. From a technical point of view, these microstructural features may have many beneficial effects, e.g., precipitates can act as defect pinning sites and stabilize the transformation, as also observed here in the excellent DSC cycling behavior (cf. Figures 1d and 5e). The approach and benefits of designing low-hysteresis SMA with excellent cycling behavior based on a high number of coherent nm-sized precipitates was demonstrated already in previous works, reporting on the enhancement of shape memory properties, cycling stability and fatigue life [2,10,38].
On the other hand, precipitation effects pose challenges to the design of LH-SMAs, as larger precipitates can act as crack initiators during cycling and secondary phases have a different composition than the matrix, thus modifying the composition of the transforming phase. Considering the high sensitivity of the Co-factor conditions, it seems impossible to satisfy them neatly in materials that "lose" matrix elements by the formation of incoherent secondary phases. These issues are amplified when the control over the precipitate presence and size distribution is limited, which becomes inherently more difficult in scaled-up alloy melting. In fact, the best thermal and mechanical cycling stability with near zero hysteresis was only demonstrated so far for alloys produced by thin-scale manufacturing methods, such as sputtering [2,13,14]. Another challenge in VIM microstructures arises from achieving a homogenous microstructure. The texture, distribution of grain size and tiny chemical inhomogeneities due to segregation at the grain boundaries are expected to occur on a higher level in comparison to the aforementioned technics, in particular in highly alloyed shape memory alloys [39]. Such effects might contribute to the slightly higher transformation hysteresis of the here-studied alloys than what was achieved in previous works [12]. However, depending on the amount of post-processing, these effects might become insignificant when working the material down to a semi-finished product, inducing reorientation of texture and grain refinement [34].
While the here-presented alloys, which according to the nominal composition are close to fulfilling the Co-factor conditions (especially group A), exhibit very promising transformation characteristics, particularly for larger-scale cast material, they do not reach a near-zero hysteresis due to their more complex microstructure. This is also reflected by the sensitivity of the DSC results on heat treatments: by solution treatment, precipitates dissolve into the matrix and cause tiny local differences in the composition, which enlarge the hysteresis. After aging, precipitates form and compensate for the compositional effects. Furthermore, the newly formed nano-sized precipitates can act as nucleation sites for the martensite transformation and, in this way, further reduce the hysteresis slightly. However, the formation of grain boundary precipitates after aging has to be seen critically, as they may cause embrittlement upon the subsequent processing of the alloys by forging or drawing.
The only alloy, which demonstrated an opposite trend, was the Ti-lean one. The reason, therefore, is assumed to lie in its different precipitation behavior, owing to a Ti content below 50 at. %, which led to the formation of several micron-sized Cu-rich CuNiTi precipitates. Their dissolution causes higher compositional gradients in the matrix and enlarges the hysteresis upon aging.
Lattice parameters, as measured by XRD, for alloys of group A and group B are shown in Figure 9a,b, respectively. Despite group A alloys being designed for fulfilling precisely the cofactor condition CC I, the geometries of the B19 cell vary significantly, which is another way to illustrate the sensitivity of the system to the composition. On the other hand, in the case of group B alloys, where Pd substitutes for Ni, clear trends for lattice parameters are observed as the cell volume increases with the content of Pd.
The discrepancy between the ideal crystal structure for fulfilling the λ 2 -criteria and the actual alloy crystal structure also becomes evident by comparing the lattice parameters of the cubic B2 austenite and orthorhombic B19 martensite cells calculated by DFT and the results measured by XRD. The lattice expansion by Cu and Pd inclusion has been computationally evaluated by fully optimizing a supercell of 128 atoms starting from NiTi lattice parameters. Cu and Pd atoms have been randomly placed in the Ni-sublattice with atomic compositions close to the experimental ones, as reported in Table 1. The evolution of the difference of the measured and calculated lattice parameters with increasing amounts of Pd (group B alloys) is shown in Figure 9c. It is evident that the empirically determined lattice parameters differ from the predictions made by DFT. While XRD and DFT results for the cubic B2 lattice parameter coincide to ±0.1 Å, the difference becomes significantly bigger for the B19 parameters. The calculation tends to predict a smaller value for the short edge b and slightly larger values for the longest edge a. The error seems to become smaller and invert with a higher Pd content. One reason for the trend could be that secondary precipitates are predominantly Cu-and Pd-rich, and the more of these phases are present, the bigger becomes the deviation from the calculations, which, in DFT, assume that all elements are dissolved in the matrix phase.
One last aspect, which should be highlighted here, is the effect of satisfying the CC II criteria, which got little attention in the literature up to today. No evidence was found in this work that there is a notable correlation between CC II and hysteresis width; however, it was observed that those alloys with a low CC II showed very stable cycling behavior, and the criteria seems also correlated to the microstructure observations. The alloys that are further from CC II = 0 show a more coarsely twinned martensitic structure with clearly visible needles. It is hypothesized that fulfilling these criteria supports twin accommodation, hence accelerating transformation kinetics and reducing the thermal offset in DSC cycling. However, more investigation is needed here, as experiments were not designed to primarily investigate on this effect. Furthermore, applying phase field studies that take into account multi-variants of martensite during the transformation, as they are emerging recently [40], could be taken in consideration to obtain a deeper understanding of the peculiarities in the B2-B19 phase transformation of Ni-Ti-Cu-Pd alloys.
precipitates. Their dissolution causes higher compositional gradients in the matrix and enlarges the hysteresis upon aging.
Lattice parameters, as measured by XRD, for alloys of group A and group B are shown in Figure 9a,b, respectively. Despite group A alloys being designed for fulfilling precisely the cofactor condition CC I, the geometries of the B19 cell vary significantly, which is another way to illustrate the sensitivity of the system to the composition. On the other hand, in the case of group B alloys, where Pd substitutes for Ni, clear trends for lattice parameters are observed as the cell volume increases with the content of Pd. The discrepancy between the ideal crystal structure for fulfilling the λ2-criteria and the actual alloy crystal structure also becomes evident by comparing the lattice parameters of the cubic B2 austenite and orthorhombic B19 martensite cells calculated by DFT and the results measured by XRD. The lattice expansion by Cu and Pd inclusion has been computationally evaluated by fully optimizing a supercell of 128 atoms starting from NiTi lattice parameters. Cu and Pd atoms have been randomly placed in the Ni-sublattice with atomic compositions close to the experimental ones, as reported in Table 1. The evolution of the difference of the measured and calculated lattice parameters with increasing amounts of Pd (group B alloys) is shown in Figure 9c. It is evident that the empirically determined

Conclusions
This research sheds light on the transformation properties of Ni-Ti-Cu-Pd alloys, as prepared by a conventional vacuum induction melting and annealing approach for LH-SMA applications. The technical feasibility of such an approach depends on whether the low-hysteresis, high-thermal cycling stability and high-temperature capabilities of these alloys known from small-scale preparation can be maintained at a bigger ingot scale. This work illustrates the challenges in controlling the microstructure to achieve the expected lowhysteresis behavior. Satisfying the Co-factor conditions in the preparation of larger scale alloys seems to be the most challenging, which, due to the process nature and necessary heat treatments, have a non-ideal, multiphase microstructure. However, despite failing to replicate the same excellent transformation behavior achieved in thin-film studies, the here-presented results give rise to optimism that by considering the melting and annealing effects in alloy design, high-performance Ni-Ti-Cu-Pd alloys can be VIM manufactured based on a Co-factor approach. In the following, a summary of the main conclusions:

•
Homogenized VIM cast Ni-Ti-Cu-Pd alloys exhibit a non-equilibrium microstructure of a B2/B19 matrix phase with the unavoidable presence of secondary phases. XRD and microscopy analysis revealed the presence of different secondary phases in the melted ingots. Low-melting point tetragonal precipitates were found in all Ni-Ti-Cu-Pd alloys. Most frequently observed were semi-coherent, nanometer-sized Pd 5 Ti 3 , Cu 2 Ti, CuTi and Cu-rich CuTiNi/CuTiPd precipitates. Furthermore, small amounts of Ti-oxides were present in all formulations, which act as nucleation sites for macroscopic (several µm), incoherent Ti 2 Ni and Ti 2 Cu precipitates with a high melting point above 1100 • C.
• Despite low cooling rates (1-10 • C/min) of furnace cooling after the homogenization treatment, the content of the secondary phases still increased after the solution treatment and aging at 550 • C × 3 h, confirming a metastable state of homogenized VIM cast alloys. • This metastable state causes significant deviations in the matrix chemistry from the overall composition. A non-proportional amount of the elements will be dissolved in high melting point precipitates and oxides and deteriorate the matrix composition. Another quantity of elements is distributed in semi-coherent precipitates, which form and can be dissolved by applying heat treatments. Therefore, the highly sensitive Co-Factor condition |λ 2 − 1| = 0 is difficult to match, and the correlation with calculated/measured λ 2 and thermal hysteresis in the here-tested alloys was low. New models that predict the change in the matrix composition due to the formation and precipitation of non-shape memory phases needs to be developed and applied to counterbalance the described effect in the design of alloys.

•
In VIM cast alloys, the solution treatment with subsequent aging seemed an effective strategy for most formulations to further reduce hysteresis. However, precipitates tend to enrich also at the grain boundaries, which is expected to impact the workability of the alloys. In particular, in the high Pd-containing alloys, where two different types of precipitates were observed at the grain boundaries, the processability could be compromised after low-temperature annealing. • An apparent correlation between CC II, martensitic morphology and thermal cycling stability was observed. The here-presented results hint at finer martensite features and higher thermal cycling stability when CC II approaches 0. However, these observations need to be further confirmed as the experiments were not designed to principally investigate this hypothesis.
The next step for future investigation is to convert these materials by processing into semi-finished products, such as wire, sheets or springs, to assess their workability and feasibility as actuation devices.