Order-parameter coupling and strain relaxation behavior of ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ martensites

A group theoretical model is proposed for linear/quadratic coupling between order parameters which arise from electronic and soft-mode instabilities in doped shape memory alloys, together with coupling to symmetry breaking shear strains. This model is tested by using resonant ultrasound spectroscopy (RUS) to follow the elastic and anelastic anomalies which accompany transitions from B2 to B19, 9R, and incommensurate structures in ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ alloy samples ($0\le x\le 12$). The pure soft-mode transition gives rise to an incommensurate structure but without any associated changes in the shear modulus, implying that coupling with shear strains is weak. By way of contrast, the observed pattern of softening ahead of and stiffening below the martensitic transition is typical of pseudoproper ferroelastic behavior and confirms that there is strong bilinear coupling of the tetragonal shear strain to the order parameter associated with irrep ${\mathrm{\Gamma }}_3^{+}$ of the parent space group. The second order parameter has the symmetry properties of ${\mathrm{M}}_5^{-}$ in TiPd or of a point along the Σ line of the Brillouin zone for the 9R and incommensurate structures with high Cr contents. Comparison of shear modulus data for Cr-rich samples obtained by RUS at ${10}^5-{10}^6\mathrm{Hz}$ with previously reported Young's modulus data obtained for different samples by dynamical mechanical analysis at ∼0.1–10 Hz has not revealed the dispersion with frequency that would be expected for a glass transition governed by Vogel-Fulcher dynamics. The two techniques differ in the magnitude of effective applied stress, however, and differences in chemical homogeneity between samples or decomposition during high-temperature measurements might also be a factor.

Figure 1

Variations of $M_{\mathrm{S}}$ for ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{X}_x$ alloys where $x$ is the atomic % of dopant X: Ti, Ref. [5]; V, Refs. [5, 6]; Cr, Refs. [6, 7, 8, 9]; Mn, Refs. [1, 7]; Fe, Refs. [1, 6, 7, 10, 11]; Co, Refs. [6, 7, 12]; Ni, Refs. [13, 14, 15, 16, 17, 18, 19, 20]; Ru, Ir, Ref. [12]. Data point for Pd is average of values from Refs. [1, 5, 6, 13, 14, 21, 22, 23].

Figure 2

Simplified phase diagram for the ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$, system showing the transition temperatures of B2-B19, B2-incommensurate (IC), IC-9R and IC–incommensurate martensite (ICM) transitions, but without showing two-phase fields. The 9R and ICM fields may have coexisting B19 [6]. $T_{\mathrm{o}}$ is the Vogel-Fulcher temperature given by Zhou et al. [24] as marking the zero-frequency freezing point for the development of a strain glass. Vertical dashed lines mark approximate composition limits for different martensitic phases observed at room temperature [8, 25]. Data points from: Enami et al. [8], Enami and Nakagawa [6], Matveeva et al. [26], Vasseur et al. [4], Shapiro et al. [27]). Decomposition of the B2 phase during high-temperature aging to form a precipitate with composition ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{25}{\mathrm{Cr}}_{25}$ based on the open B2 structure has also been reported for alloys with between 8% and 15% Cr [25].

Figure 3

Top: XRD patterns obtained using a D8-advance diffractometer with CuKα radiation for a subset of the samples at room temperature (a) before and (b) after completion of the RUS experiments. The diffraction geometry was Bragg-Brentano and the data were collected from one flat side of each polycrystalline sample. Note that the $y$ axis is intensity in arbitrary units but diffraction patterns have been offset upwards in proportion to their Cr contents and the axis relabeled as values of $x$ in ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$. Bottom: Patterns expected for samples in the B2 and B19 phase, as reproduced from Fig. 1 of Zhou et al. [2] for 5Cr.

Figure 4

Variations of $f^2$ with temperature for the ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ samples. Values of $f^2$ from peak fitting are shown as markers, with semiquantitative data shown as dotted lines. The lists in captions indicate the sequences in which the data were collected. Vertical dashed lines indicate temperatures at which the minimum value of $f^2$ occurred on first heating ($T_{\mathrm{mfh}}$) and cooling ($T_{\mathrm{mfc}}$). The solid vertical line indicates the temperature at which the B2-IC transformation was expected to occur ($T_{\mathrm{IC}}$), as taken from Fig. 2. Values of $T_{\mathrm{IC}}$ for 11Cr and 12Cr are extrapolations from Fig. 2. The vertical line and the number on the left side of each graph show the change in $f^2$ expressed as the ratio of the maximum and minimum values. For example, $f^2$ values for the resonance shown in (a) for the sample with 0% Cr changed by a total of a factor of 2.01. This means that the shear modulus also changes by a factor of ∼2 between maximum and minimum values in the temperature interval ∼10–950 K.

Figure 5

Evolution of $Q^{-1}$ with changing temperature from the resonance peaks for which values of $f^2$ are given in Fig. 4. Note the different scale on the $y$ axis for 10Cr, 11Cr, and 12Cr. The caption order indicates the sequence in which the data were collected. Vertical lines indicate the temperatures $T_{\mathrm{mfh}}$, $T_{\mathrm{mfc}}$, and $T_{\mathrm{IC}}$, as previously shown in Fig. 4. The black curve superimposed on the data for 10Cr is a fit of Eq. (2) with $E_{\mathrm{act}}/r_2\left(\beta \right)=6.1\mathrm{kJ}\mathrm{mol}{\mathrm{e}}^{-1}$.

Figure 6

Ratio of maximum to minimum values of $f^2$ for compositions between 0 and 12% Cr, as shown in Fig. 4. The straight line is a guide to the eye to illustrate the trend of lowering values with increasing Cr content.

Figure 7

Elastic modulus and acoustic loss data from RUS and DMA measurements from 5Cr [2] (a),(c),(e), and 10Cr [24] (b),(d),(f) samples. RUS data are shown with red crosses. DMA data for 5Cr are for a hydrogen-doped sample (purple dashed lines) and a hydrogen-free sample (solid green lines). Data for 10Cr are from a sample not treated in any particular way for hydrogen content. Vertical black dotted lines indicate either $T_{\mathrm{mfc}}$ (RUS) or $T_{\mathrm{g}}$ (DMA). (a),(b) show the evolution of $f^2$ with temperature from RUS measurements. (c),(d) include the storage modulus from DMA measurements at frequencies of 0.2–20 Hz. (e),(f) include tanδ (DMA), and $Q^{-1}$ (RUS) data divided by $\sqrt{3}$.

Figure 8

Comparison of acoustic properties measured for ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ samples using RUS (solid markers) and DMA (hollow markers). DMA data from Zhou et al. [3] for 9Cr, Zhou et al. [24] for 10Cr, Vasseur et al. [4] for 12Cr. ω is the angular frequency of the measurements ($=2\pi f$). $T_{\mathrm{g}}$ was taken to be the temperature of the minimum in the modulus from DMA or the minimum in $f^2$ from RUS. Straight lines through the DMA data are guides to the eye. In spite of the large difference in ω between DMA and RUS, the two set of values of $T_{\mathrm{g}}$ overlap.

Figure 9

Phase diagram for the ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ system including RUS results (black markers) and a temperature region of acoustic loss in DMA data (purple shading) which, in the case of 5Cr was attributed to motion of hydrogen [2]. Grey markers indicate the phase diagram shown in Fig. 2.

Figure 10

Relationships between the unit cell of the 9R phase and that of a convenient pseudo-orthorhombic structure used to calculate spontaneous strains. Displacements of individual planes have been ignored for clarity. The pseudo-orthorhombic cell is not a true unit cell, as the atoms at each of the corners are different.

Figure 11

(a) Symmetry-adapted shear strains for B2-B19 and B2-9R transitions in ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Cr}}_x$ and ${\mathrm{Ti}}_{50}{\mathrm{Pd}}_{50-x}{\mathrm{Fe}}_x$ as a function of composition at room temperature. (b) Relative changes of symmetry-adapted shear strains for 9R and B19 structures.

Figure 12

Peak positions and peak splitting in x-ray-diffraction patterns for 10Cr, as extracted from Fig. 6 of Vasseur et al. [4]. Top: peak positions. Bottom: difference in angle between the 9R and B2 peaks.