Scale-invariant avalanche dynamics in the temperature-driven martensitic transition of a Cu-Al-Be single crystal

We have combined high sensitivity, extra-low differential temperature scanning rate calorimetry, and acoustic emission (AE) measurements to study avalanches during the cubic ↔ 18R martensitic transition of a Cu-Al-Be single crystalline shape memory alloy. Both AE and calorimetry corroborate a good power-law behavior for cooling with an exponent $\varepsilon \simeq 1.6$. For heating, a slope is observed in the maximum likelihood curves, which confirms that our data are affected by an exponential cutoff. An effective energy exponent, $\varepsilon \sim 1.85$, and a cutoff, ${\lambda }^{-1}=0.115\left(38\right)\times {10}^{-3}\mathrm{aJ}$, were determined by fits of power-laws with exponential damping. The long tail observed in the low-temperature region by calorimetric measurements suggests the existence of significant elastic effects that constrain the progress of the transformation at low temperatures. While thermodynamic features such as transformation enthalpy and entropy are those expected for Cu-based shape-memory alloys undergoing a cubic ↔ 18R transition, the critical behavior deviates from the corresponding behavior expected from this symmetry change. These deviations are a consequence of the elastic hardening induced by the interplay of the transformation with dislocation jamming, which has the effect of effectively reducing the number of pathways connecting the parent and martensitic phase.

Figure 1

Scaled representation of the AE activity for cooling (a) and heating (b) runs.

Figure 2

Histograms representing the distribution of energies for the forward (a) and reverse (b) transitions corresponding to selected values of the temperature rate.

Figure 3

Fitted exponent as function of the cutoff $E_{\min }$ for AE experiments for cooling (a) and heating (b) runs at selected temperature rates. Cooling runs yield power law behavior with an exponent close to 1.6. Heating data (b) are clearly affected by an exponential damping.

Figure 4

Heat flux Φ as a function of temperature for heating (top, endothermic curves) and cooling (bottom, exothermic curves) runs at several temperature rates. A series of spikes is evident on both the reverse and forward transformations indicating the jerky behavior of the transition.

Figure 5

Normalized heat flux $\mathrm{\Delta }\mathrm{\Phi }/r$ as a function of temperature for heating (top, endothermic) and cooling (bottom, exothermic) at two temperature rates. The area subtended by each run above the base line (dotted lines) equals the excess enthalpy.

Figure 6

Excess enthalpy computed by integrating of the normalized heat flux for data shown in Fig. 5. Hysteresis is evident with both curves displaced $\mathrm{\Delta }T=8$ K and different rounding mechanisms.

Figure 7

Cumulative thermal avalanche energy for cooling and heating runs, at selected scanning rates. Monotonous increasing avalanche energy with increasing the rate is observed both for the forward and reverse transitions. Total avalanche energy, $E_{\mathrm{th}}$, is five times larger in the forward than in the reverse transition. This difference is also apparent in Figs. 4 and 5.

Figure 8

Histograms representing the distribution of thermal avalanche energies for the reverse (top) and forward (bottom) transformations at several rates. Vertical axis displays counts per unit bin length scaled by the minimum observed energy $E_0=14\mu \mathrm{J}$. Data were distributed in $N=20$ bins logarithmically spaced from $E_0$ to the highest event. For the sake of comparison axes kept the ranges displayed in Fig. 2.

Figure 9

Maximum likelihood plots giving the fitted exponent as function of the cutoff energy for calorimetric experiments.