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[K. Niitsu](https://orcid.org/0000-0002-0430-8868), Y. Yano, R. Kainuma, H. Inui

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[Viscosity of superelasticity: A comprehensive interpretation of nonreciprocal isothermal dynamics, kinetic arrest, and nonergodic anelastic strain based on thermal activation of martensitic transformations](https://mdr.nims.go.jp/datasets/28f8c61d-3a64-488a-a79f-523b8f4b2ef7)

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1  Viscosuperelasticity: A comprehensive interpretation of non-reciprocal isothermal 1 dynamics, kinetic arrest, and non-ergodic anelastic strain based on thermal activation of 2 martensitic transformations 3 K. Niitsua,b*, Y. Yanob, R. Kainumac, H. Inuib,d 4  5 a National Institute for Materials Science, Tsukuba 305-0047, Japan 6 b Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, 7 Japan 8 c Department of Materials Science, Graduate School of Engineering, Tohoku University, 9 Sendai 980-8579, Japan 10 d Center for Elements Strategy Initiative for Structure Materials (ESISM), Kyoto 11 University, Kyoto 606-8501, Japan 12  13 *Corresponding author. Tel.: +81-29-859-2566; e-mail: NIITSU.Kodai@nims.go.jp 14  15  16   17 mailto:NIITSU.Kodai@nims.go.jp2  ABSTRACT 1 Much of the impact of thermal activation on the low-temperature dynamics of the 2 thermoelastic martensitic transformation (MT) remains unresolved. This study introduces 3 a new concept of “viscosuperelasticity” to comprehensively interpret the slow dynamics 4 of stress-induced MT at low temperatures. The thermal activation dynamics underlying 5 this concept allow for deriving the time-temperature-transformation (TTT) diagrams at 6 any holding stress from the temperature- and strain-rate-dependent superelastic stress-7 strain curves. Experimental results for isothermal forward and reverse MTs 8 comprehensively agree with the derived TTT diagrams, disclosing the non-reciprocity in 9 the dynamics between the forward and reverse MT paths. One of the representative 10 manifestations of “viscosuperelasticity” is the difference in the cooling/heating rate 11 dependence of the forward/reverse MT starting and finishing temperatures (TMs, TMf, TAs, 12 and TAf). A significant decrease in TMs and TMf with increasing cooling rate explains 13 kinetic arrest, and a gradual increase in TAs and TAf with increasing heating rate explains 14 the difficulty in detecting the isothermal evolution of the reverse MT. The TTT diagrams 15 also enable the impact of thermal activation upon dynamic cooling/heating to be 16 evaluated quantitatively, explaining the non-ergodic thermal-history dependence of 17 anelastic transformation strain, which is believed to be an essential signature of strain 18 glass. The concept of “viscosuperelasticity” offers a comprehensive interpretation of non-19 reciprocal isothermal dynamics, kinetic arrest, and non-ergodicity in the anelastic strain 20 as manifestations of thermal activation of MTs. 21 3   1 Keywords: Thermally activated process; Martensitic transformation; Kinetics; Shape 2 memory alloys; Isothermal martensitic transformation 3   4 4  1. INTRODUCTION 1 Since the discovery of the isothermal martensitic transformation (MT) in Fe-Mn-2 C [1] and Fe-Ni-Mn alloys [2], numerous studies have fostered a consensus that MT 3 dynamics have both athermal and thermal nature [3–6]. Athermal dynamics, which 4 depend on intensive variables such as temperature, stresses, magnetic fields, and chemical 5 potential, essentially exist in any first-order phase transformation and give rise to finite 6 transformation hysteresis for driving transforming interfaces. In contrast, isothermal 7 (thermally activated) dynamics depend on time along with the aforementioned intensive 8 variables and are more pronounced at low temperatures where thermal excitations are 9 stochastically suppressed [7]. These two natures should be independently considered for 10 nucleation and growth, the processes by which MTs develop. Earlier studies on thermally-11 induced MTs have concluded that the rate-controlling process is martensite nucleation 12 (followed by auto-catalytic ultrafast growth) rather than growth [3–5]. This conclusion 13 seems reasonable because, in the case of thermally-induced MTs, a large part of the 14 excess energy could be consumed to overcome the nucleation barrier, and once nucleation 15 occurs, the nuclei instantaneously grow into martensite variants. A similar scenario could 16 be the case for magnetic-field-induced MTs [8]. 17 Meanwhile, a detailed study on the kinetics of thermally-induced MT in a Cu-18 Zn-Al alloy revealed that the MT process is governed by continuous growth, with only 19 5% involving jerks which may be responsible for nucleation [9]. Furthermore, unlike 20 thermally- and magnetic-field-induced MTs, stress-induced MTs should be primarily 21 5  governed by the growth process because a reduced number of variants with higher 1 resolved shear stresses is preferred for activation. Macroscopically smooth propagation 2 of the interface (habit plane) is observed during the evolution of stress-induced MTs. As 3 a result, the nucleation process is typically not manifested in superelastic stress-strain 4 curves [10]. 5 The earlier conclusion was drawn on nucleation, thus is inadequate to interpret 6 stress-induced MTs that proceed predominantly by growth. Following a study by Ghosh 7 and Raghavan, which reported the effect of superimposed stresses [11], they stated that 8 isothermal MT could significantly develop even under stress. They discussed this within 9 the framework of earlier models based on martensite nucleation [3,12–16]. This 10 conceptualization has led to the classification of “athermal” and “isothermal” dynamics 11 as being promoted by long-range stress fields and by short-range interactions of point 12 defects, respectively, which enlightens the phenomenological similarity to the dynamics 13 of slip deformation [17,18]. Keeping this in mind, Kajiwara claimed the microscopic role 14 of dislocations on the dynamics of isothermal MTs [19–21], according to which the 15 generation of dislocations leading to the formation of martensite is a rate-controlling 16 process. Since the thermally activated dislocation motion underlying these two events 17 governs the isothermal MT dynamics, the possibility of isothermal macroscopic growth 18 (which may be microscopically constituted by avalanches [22]) of the martensite variants, 19 which was not considered earlier, can not be ruled out. Furthermore, this hypothesis 20 suggests the possibility of an isothermal “reverse” MT driven by habit plane propagation 21 6  rather than nucleation. As demonstrated later, this study clarifies that the isothermal 1 reverse MT indeed occurs, enlightening the predominant role of the reversible growth 2 process in isothermal MTs. 3 The most representative feature of the isothermal forward MT is the presence of 4 a C-shaped contour curve in the time–temperature–transformation (TTT) diagram [6]. 5 Some models based on the thermal excitation of martensite nuclei have been proposed to 6 explain this behavior [23–25]. Although these models reproduce a C-shaped contour 7 curve for the isothermal forward MT, there remains a concern that the underlying 8 assumption of the models may be applicable to only a limited aspect of the overall MT 9 evolution. More specifically, the homogeneous nucleation assumed in some models may 10 not be suitable for real systems as nucleation is highly inhomogeneous. Besides, the 11 models were proposed for nucleation and did not assume isothermal growth, which could 12 constitute a large portion of the MT evolution. In this sense, the growth-driven reverse 13 MT is out of the scope of existing models. More importantly, no model currently includes 14 prerequisites to account for non-reciprocity in the isothermal forward and reverse MT 15 paths. The existing models are not applicable to the reverse MT, or even if possible, the 16 predicted reverse dynamics would be exactly reciprocal to the forward one, calling into 17 question the implication of “nucleation”. 18 In addition to isothermal dynamics, the possible impact of thermal activation on 19 the MT dynamics under the dynamic sweep of external fields is of practical interest 20 because the MT-starting/finishing temperatures, stresses, and magnetic fields, are usually 21 7  determined under the sweep of corresponding fields. Isothermal dynamics should be 1 superimposed even during the dynamic sweep, but its impact has not yet been clarified. 2 As mentioned above, interpreting the overall dynamics of MTs is not 3 straightforward. Various theoretical or phenomenological models have been proposed, 4 but they can only shape limited aspects. To resolve this disruption, a comprehensive 5 picture encompassing the phenomenology of MT dynamics, including the 6 athermal/isothermal, nucleation/growth, and forward/reverse concepts, is necessary. The 7 authors consider that the stress-induced MTs provide an appropriate platform for studying 8 growth dynamics, as other cases (thermally- and magnetically-induced MTs) involve both 9 nucleation and growth and thus are likely to be complex to analyze. 10 Recent studies on the low-temperature behavior of stress-induced MTs have 11 revealed that the temperature and strain-rate dependence of transformation hysteresis 12 becomes more pronounced at lower temperatures (typically below 200 K) for many alloy 13 systems [26–32]. This tendency was interpreted as a manifestation of the thermal 14 activation of habit plane propagation. For modeling this tendency, a mathematical 15 description that mimics the thermally-activated dynamics of dislocation glides was 16 proposed [18,31,33–35] (see ref. [31] for a detailed explanation of this equation): 17 1/1/01 ln2pqhys Beff TASEk TQµσ εσ σ σε       ≈ = + −             (1) 18 where σµ and σTA represent the athermal component and thermal activation offset at 0 K 19 for the overloading/underloading stresses required to drive the forward/reverse MTs, 20 respectively; Q the activation energy at 0 K; kB the Boltzmann constant; 0ε  a pre-21 8  exponential factor; SEε  the superelastic strain rate; and p and q are fitting parameters that 1 characterize the curvature of the temperature and strain-rate dependence of σeff. It is 2 empirically known that the combination of p = 1/2 and q = 3/2 is the most reproducible. 3 According to the original Seeger’s formulation, this combination shapes the Peierls 4 potential by misfit stresses [33]. Equation 1 indicates that the effective stress (σeff) 5 required to drive the habit planes, usually approximated as half of the transformation 6 hysteresis (σhys), can be decomposed into two components of an athermal term and a 7 thermal activation term. Accordingly, if the proposed form can explicitly depict the 8 impact of thermal activation on growth, the isothermal dynamics could also be 9 comprehended with an extension of this form, as the strain rate involves the dimension of 10 time. 11 In this study, this idea was verified by examining its applicability to both the 12 isothermal forward and reverse MTs. This examination was demonstrated for a 13 prototypical Ni-rich Ti-Ni superelastic alloy, which is known to exhibit a remarkable 14 dependence of transformation stress hysteresis on temperature and strain rate [28,31]. In 15 addition to modeling the macroscopic responses, the isothermal growth of the martensite 16 variants was directly observed by in situ transmission electron microscopy (TEM) under 17 uniaxial tensile stress. The impact of thermal activation on the MT behaviors under the 18 dynamic cooling/heating was quantitatively evaluated based on the isothermal nature. 19 The possible origin of kinetic arrest and non-ergodic behavior of anelastic transformation 20 strains, which is often regarded as a signature of strain glass [36,37], is discussed in terms 21 9  of the underlying thermal activation nature. While it has been previously considered that 1 these phenomena have different origins, it is found that they are commonly related to the 2 slow dynamics of MTs governed by thermal activation. To provide a comprehensive 3 interpretation of these phenomena, a new concept of “viscosuperelasticity” is introduced 4 following the concept of viscoplasticity.  5  6 2. EXPERIMENTAL 7 A Ti-51.8Ni (at.%) alloy was fabricated using arc melting. At this composition, 8 the thermally-induced MT is completely suppressed without the help of stress [26]. The 9 obtained polycrystalline button was homogenized at 1173 K for 24 h, followed by 10 quenching in water. A 3.5 × 3.5 × 10.5 mm specimen was cut from the button for 11 compression testing. The temperature dependence of the compressive stress-strain curves 12 at 30–220 K was examined. Isothermal stress relaxations were examined at 82 K, and 13 isothermal strain evolutions (isothermal MTs) were recorded at various temperatures 14 under holding compressive stresses (σh) of 500 and 200 MPa for the forward and reverse 15 MT paths, respectively. The isothermal reverse MT was recorded by interrupting the 16 unloading sequence at 200 MPa after reversing the forward stressing sequence at a 17 superelastic strain (εSE) of 0.0450, which is approximately 82% of the full superelastic 18 strain (εfull) (= 0.0550). The temperature oscillation during isothermal holding was within 19 ±2 K. All mechanical tests were performed on one specimen, and the total number of tests 20 was < 50, within which cyclic fatigue is insignificant [22]. It was also confirmed that the 21 10  strain relaxation for 1 × 106 s of the specimen and compression jigs under isothermal 1 holding at 10 K and σh = 300 MPa was less than 0.0007. In situ TEM observations were 2 conducted at 95 and 220 K under uniaxial tensile stress using a combination of a TEM 3 and a single-tilt liquid nitrogen cooling straining holder [38]. The observation area was 4 thinned by electrochemical polishing in a solution of 72% acetic acid, 12% ethanol, 8% 5 ethylene glycol, and 8% perchloric acid. 6  7 3. RESULTS 8 3.1. DYNAMIC PHASE DIAGRAM 9  Figure 1a shows the temperature dependence of the superelastic behavior. As 10 previously reported for this alloy system [26,30,31], the transformation hysteresis 11 increases significantly with decreasing temperature. This is a typical manifestation of the 12 thermal activation of stress-induced MTs [26]. The equilibrium stress (σ0) is typically 13 approximated as the midpoint between the forward/reverse MT starting/finishing stresses 14 (σM and σA) [39]. The temperature dependence of σ0 was drawn to follow the Clausius-15 Clapeyron relationship wherein the transformation entropy change obeys the third law of 16 thermodynamics [40]. The effective overloading/underloading stress required to drive the 17 forward/reverse MTs, σeff, is shown in Fig. 1c as a function of temperature. The curve 18 fitted using Eq. 1 was in close agreement with the experimental plots (Fig. 1b and c; 19 numerical values of the parameters are given in the caption). Equation 1 can be converted 20 to the Arrhenius form: 21 11  0 0ln ln 1 ln ( , )qpeffSE SEB TAQ QZ Tk Tµσ σε ε ε εσ −  = − − ≡ −        ,  (2) 1 which enables the evaluation of Q and 0ε  from the gradient and intercept of the fitting 2 line, respectively (see inset of Fig. 1c). 3  4 Fig. 1. Temperature dependence of (a) the superelastic stress-strain curves, (b) σM, σA, 5 and σ0, and (c) σeff. Inset in (c) represents the Arrhenius plot of Eq. 2. The tests were 6 carried out at a constant strain rate of 3.3 × 10−4 s−1. The numerical values of the 7 parameters in Eq. 1 were as follows: σµ = 63 MPa, σTA = 1073 MPa, Q = 0.47 eV, 0ε  = 8 8.0 × 108 s-1 (with an error of 2.7 × 107–5.9 × 1010), p = 1/2, and q = 3/2. 9 As Eq. 1 implicitly includes the dimension of time in SEε , this contribution is 10 made explicit. In the simple framework of viscoplasticity [41], the strain rate is 11 decomposed into the parts of elasticity (εe) and viscoplasticity (εvp), that is, 12 vpet tεεε∂∂= +∂ ∂ . The former part is given by 1eAt E tε σ∂ ∂=∂ ∂ referring to Hooke’s law, with 13 the elastic modulus of the austenite phase (EA). Considering the stress relaxation condition, 14 12  the total strain rate ε  is zero. Hence we have 1vpAt E tε σ∂ ∂= −∂ ∂. As mentioned, Eq. 1 is 1 derived by mimicking Seeger’s formulation for thermal activation of dislocation glides. 2 It is thus a generic scheme that εvp corresponds to εSE in the context of 3 “viscosuperelasticity” and that the stress component exhibiting temperature and time 4 dependence is σeff −  σµ. Therefore, for the stress relaxation condition, Eq. 2 can be 5 converted to 6 ( )0exp ln 1qpeff effAB TAQEt k Tµ µσ σ σ σεσ  ∂ − −   = − − −  ∂       .      (3) 7 Here, one might question the applicability of Hooke’s law to the superelastic 8 regime. Eq. 3 should be logically valid up to the upper limit of the elastic regime. 9 However, the solution of Eq. 3 remains phenomenologically valid at any stage of the 10 stress-induced MTs; otherwise the transformation hysteresis would change as the MTs 11 progress. Equation 3 is the time differential equation of σeff − σµ (that is, the thermal 12 activation component of effective stresses). Therefore, the time dependence of σeff at fixed 13 temperature and strain can be derived by solving this equation; a prototypical solution is 14 shown in Fig. 2a. The significant decrease of σeff over time is a straightforward 15 consequence of its thermal activation nature. Instead, σ0 up- or downshifts in response to 16 the transformation hardening as the forward/reverse MTs progress, as schematically 17 illustrated in Fig. 2a. The thick black curve corresponds to the dynamic stress-strain 18 curves showing experimentally accessible superelastic behavior. This figure tells us how 19 the superelastic stress-strain curve is captured under dynamic stressing; the thick black 20 13  curve traces the σ0 + σeff surface depending on time as well as VM, thus exhibiting a strain-1 rate dependence. This trend is experimentally confirmed by the isothermal stress 2 relaxation tests, as shown in Fig. 2b and c. Despite the change in stress level, the 3 magnitude of stress relaxation was almost unchanged with respect to VM and followed the 4 relaxation curve derived from Eq. 3. As demonstrated later, the TTT diagram is derived 5 for the strain relaxation condition. Under this condition, isothermal forward and reverse 6 MT progresses depending on how far the calculated relaxation stress falls below and 7 exceeds σh, respectively. 8 Figure 2 illustrates the impact of thermal activation on transforming stresses. By 9 incorporating this contribution into the observed phase diagram (Fig. 1b), the so-called 10 dynamic phase diagram can be derived, as shown in Fig. 3. This diagram unveils the 11 substantial yet obscured influence of time on MT dynamics. The experimentally 12 determined phase diagram (Fig. 1b) is now interpreted to represent a specific (but not 13 precisely constant) time section of Fig. 3. 14  15 Fig. 2. (a) Typical solution of Eq. 3 for the forward MT at 100 K. The gray, green, and 16 magenta curves represent σ0, σ0 + σµ, and σ0 + σeff, respectively, drawn for the VM interval 17 of 0.2. The thick black curve is a schematic drawing of the stress-strain curve observed 18 under dynamic stressing. Superelasticity at 82 K interrupted at several superelastic strains 19 14  is shown in the (b) stress-strain and (c) stress-time coordinates. The holding time is 1 overlaid with the inserted color scale. The strain rate to reach the holding strain was 3.3 2 × 10−3 s−1. 3  4 Fig. 3. Dynamic phase diagram as functions of temperature, stress, and time. The blue 5 and red curves are time-dependent σM (= σ0 +σeff) and σA  (= σ0 −σeff), respectively. σeff is 6 calculated from Eq. 3. 7  8 3.2. ISOTHERMAL TRANSFORMATION DYNAMICS 9 Figure 3 represents the time dependence of σM and σA, both of which are 10 originally defined as the endpoint of the elastic deformation regime of the austenite phase 11 (that is, VM = 0). As demonstrated in Fig. 2, however, similar diagrams are derivable for 12 any VM by simply shifting σ0. To explicitly visualize the time evolution of MTs, namely, 13 the isothermal transformation dynamics, these convoluted dynamic phase diagrams were 14 cross-sectioned at given σh; see Fig. 4a and b for the isothermal forward and reverse MTs, 15 respectively. The derived TTT diagrams disclose some critical characteristics of the 16 isothermal transformation dynamics. In the isothermal forward MT, the well-known C-17 15  shaped contour lines are reproduced and are found to be analogous across arbitrary σh 1 sections and VM contours. Additionally, the projection of the nose temperature on the 2 contours (Fig. 4c) shows a non-monotonic variation over time, indicating that the nose 3 temperature is also time-dependent. In contrast, the contour lines of the isothermal reverse 4 MT are characterized by the lower half of the C-shaped curves; hence the nose 5 temperature is missing. 6  7 Fig. 4. Stress-sectioned TTT diagrams for the isothermal (a) forward and (b) reverse MTs. 8 White contour lines are drawn for the VM interval of 0.2. (c) Nose temperature for the 9 isothermal forward MT as a function of time. Schematics of the isothermal dynamics for 10 the (d) forward and (e) reverse MTs. Note that ∆g and δ in the energy diagrams of (d) and 11 (e) are the thermodynamic and non-thermodynamic (thermally surmountable) energy 12 gaps, respectively, but illustrated in a unique scheme for simplicity. 13 It is now apparent that the isothermal dynamics of the forward and reverse MTs 14 are not reciprocal. The origin of such non-reciprocal dynamics is outlined in the schematic 15 diagrams of Fig. 4d and e. Considering the forward MT with fixed stress (σh) and 16 temperature (TL, TM, and TH) as shown in the right panel of Fig. 4d, the remaining energy 17 barrier ∆ (= δ – ∆g, where δ is the time-dependent thermal activation barrier for driving 18 16  MTs and ∆g is an offset given by σh in the thermal activation regime) that must be 1 overcome by the thermal activation process is very high at a low temperature (TL), 2 resulting in a considerable stagnation of the isothermal evolution. ∆ takes a minimum at 3 an intermediate temperature (TM), and the isothermal transformation proceeds most easily. 4 The higher temperature becomes, the lower δ is (see T = TH); however, the athermal 5 barrier becomes pronounced as the σ0 + σµ curve approaches the σh line, again increasing 6 ∆ (see the middle panel of Fig. 4d). As illustrated in the left panel of Fig. 4d, the transition 7 probability (P ∝ ∆−1) draws a C-shaped curve with a nose temperature at approximately 8 TM, and the upper-temperature limit of the isothermal forward MT is defined as the 9 intersection of the σh line and σ0 + σµ curve. In contrast, ∆ in the reverse MT (Fig. 4e) 10 decreases monotonically with increasing temperature as ∆g increases and δ decreases (see 11 the middle panel of Fig. 4e). This means that P increases with increasing temperature and 12 reaches 1 at the intersection of the σh line and σ0 − σeff curve (see the right panel of Fig. 13 4e), above which the reverse transformation proceeds without incubation; therefore, there 14 is no upper limit on the TTT curves for the isothermal reverse MT. The thermodynamic 15 σ0 curve, which has not been accounted for in any existing kinetic models, plays a critical 16 role in making the isothermal dynamics of the forward and reverse MTs non-reciprocal. 17 The derived TTT diagrams were verified by performing isothermal compression 18 tests at various combinations of σh and T, the results of which are presented in the stress-19 strain diagram in Fig. 5. VM is also shown in this figure, which was estimated by the 20 mathematical fitting of the superelastic stress-strain curves (refer to Appendix A for the 21 17  detailed protocol). Notably, isothermal transformation occurs in both the forward and 1 reverse MT paths. Figure 6 compares the experimental and calculated results. The right 2 panels in Fig. 6a and b show the time evolution of VM in the isothermal forward and 3 reverse MT paths, respectively; the locations of the monitored segments are overlaid on 4 the corresponding TTT diagrams (left panels of Fig. 6a and b). Despite the calculated 5 curves of the forward path showing non-trivial variations with time and temperature, they 6 closely reproduce the overall observed isothermal dynamics. The dynamics are 7 significantly suppressed at lower (such as 32 K) and higher (such as 160 K) temperatures 8 and become more pronounced at intermediate temperatures (such as 100 and 115 K) 9 within the monitored time scale. Similarly, the calculations for the reverse path 10 successfully capture the overall observed trend. Nearly full reverse transformation is 11 observed isothermally at 100 K, dispelling doubts about the existence of the isothermal 12 reverse MT [42]. It should be noted that the small spikes are due to temperature 13 fluctuations and thus are not intrinsic. 14 18   1 Fig. 5. Isothermal compressive test results for (a) forward path at σh = 500 MPa and (b) 2 reverse path at σh = 200 MPa. Along with the stress-strain response, VM is depicted in 3 gray. The holding time at σh is overlaid with the inserted color scale. The strain rate to 4 reach the holding stress was 3.3 × 10−3 s−1. 5 19   1 Fig. 6. Comparison of the calculated and experimental isothermal transformation 2 dynamics for the (a) forward and (b) reverse transformation paths. The left panels show 3 the calculated TTT diagrams at σh = (a) 500 and (b) 200 MPa, respectively. The right 4 panels show the calculated and experimental variation in VM as a function of time at 5 several fixed temperatures. The masked regions reflect a temperature error of ±2 K in the 6 calculations. Corresponding sections are overlaid onto their respective TTT diagrams with 7 the same color. 8  9 3.3. IN-SITU OBSERVATION OF ISOTHERMAL GROWTH 10 The isothermal evolution of martensite was confirmed by in situ TEM 11 observations under tensile stress. As soon as martensite nucleation was observed, 12 dynamic elongation was interrupted and the isothermal growth under the holding 13 20  elongation was monitored. This observation was carried out at 220 and 95 K. The strain 1 applied was estimated to be approximately 2.0% at 220 K and 0.8% at 95 K. Figure 7a 2 and b presents time-series snapshots of several martensite variants nucleated near 3 cleavages, respectively. The sketches of the evolution of the martensite edges (rightmost 4 panels of Fig. 7a and b) show that the growth is remarkable at 95 K but stagnant at 220 5 K. The growth curve (Fig. 7c) captures the essential characteristics of the temperature 6 dependence of the isothermal forward MT dynamics presented in Fig. 6a. It should be 7 noted that the spatial stress distribution and magnitude are, of course, likely to be 8 inhomogeneous with respect to location and time; therefore it is premature to discuss the 9 detailed dynamics from these observations. However, they support that the remarkable 10 temperature dependence of isothermal MT evolution shown in Fig. 6a is more or less 11 attributable to the growth of martensite variants. 12  13 Fig. 7. Time-series snapshots of nucleated martensite variants at (a) 220 and (b) 95 K. 14 The time scale was set from when the nucleation was confirmed. The evolution of the 15 martensite edges over time is sketched in the rightmost figure of (a) and (b). (c) Changes 16 in the area of a particular martensite variant over time. Scale bars, 500 nm. 17  18 3.4. TRANSFORMATION DYNAMICS UNDER COOLING/HEATING 19 21  The critical roles of thermal activation and thermodynamic equilibrium in the 1 dynamics of isothermal MTs are explicitly visualized in the TTT diagrams. The TTT 2 diagrams are convertible to continuous cooling/heating transformation (CCT/CHT) 3 diagrams. The VM change under continuous cooling can be derived by solving the path 4 integral on the forward TTT diagrams, that is, 5 0( ) ( )( ) ( ) M MM MV VV V dt dTt T∂ ∂= + + Ψ∂ ∂∫ ∫r rr rr r  6 where 1 ( ( ) / 0)0 ( ( ) / 0)MMwhen V Twhen V T− ∂ ∂ <Ψ =  ∂ ∂ ≥rr  (4) 7 where ( , )T t=r  (in dimensionless units) represents the temperature sweep route. Let the 8 starting point set 0 (250,1)=r  for the calculation of the CCT diagram. Whereas /MV t∂ ∂  9 is always positive, /MV T∂ ∂  becomes positive below the nose temperature, indicating the 10 occurrence of the reverse MT. However, this case should not be involved in this path 11 integral because the reverse MT is kinetically arrested in the low-temperature region; 12 hence, Ψ is applied to ( )MV dTT∂∂∫rr . The derivation of VM change during the MT under 13 continuous heating is analogous but somewhat complicated. Considering the calculation 14 of the CHT diagram with a finite σh starting from 0 (0,1)=r  after zero-field (σh = 0) 15 cooling (hereafter referred to as ZFC-FH), the VM changes differently depending on the 16 temperature range. When T < TµM (TµM is the temperature at which σh = σ0 + σµ), the 17 following path integral is performed on the forward TTT diagrams: 18 0( ) ( )( ) ( ) M MM MV VV V dt dTt T∂ ∂′= + + Ψ∂ ∂∫ ∫r rr rr r  19 where 1 ( ( ) / 0)0 ( ( ) / 0)MMwhen V Twhen V T∂ ∂ ≥′Ψ =  ∂ ∂ <rr. (5) 20 22  When TµM ≤ T < TµA (TµA is the temperature at which σh = σ0 − σµ), neither the 1 forward nor reverse MTs proceeds, thus 2 ( ) ( ) 0M MV Vt T∂ ∂= =∂ ∂r r .  (6) 3 Finally, the reverse MT progresses when TµA ≤ T. The following path integral is 4 performed on the reverse TTT diagrams: 5 ( ) ( )( ) ( , )A A M MM MV VV V T t dt dTt Tµ µ∂ ∂′′= + + Ψ∂ ∂∫ ∫r rr rr  6 where 1 ( ( ) / 0)0 ( ( ) / 0)MMwhen V Twhen V T∂ ∂ <′′Ψ =  ∂ ∂ ≥rr. (7) 7 where tµA represents the time at T = TµA. When VM reaches 1 and 0, further forward and 8 reverse MTs do not proceed, respectively. It is important to note that TµM and TµA are VM-9 dependent. 10 Solving Eqs. 4, 5, 6, and 7 for various cooling/heating rates yields the CCT and 11 CHT diagrams. Figure 8a and b shows representative CCT and CHT diagrams at σh = 500 12 MPa, respectively. These diagrams demonstrate that the MT dynamics are not simple, 13 even under dynamic temperature sweep. 14  15 23  Fig. 8. (a) CCT and (b) CHT diagrams at σh = 500 MPa. The CHT diagram is derived for 1 the ZFC-FH process. The overlaid lines are the temperature route defined in Fig. 10a. 2  3 4. DISCUSSION 4 4.1. MANIFESTATIONS OF THERMAL ACTIVATION 5 The isothermal TTT and dynamic CCT/CHT diagrams clarify the impact of time 6 on both the isothermal and anisothermal MT dynamics. The findings will help interpret 7 low-temperature MT dynamics, which are not comprehensively understood within the 8 existing frameworks. More practically, elucidating the temperature sweep rate ( 𝑇̇𝑇 ) 9 dependence of the forward MT starting/finishing temperatures (TMs and TMf) and the 10 reverse MT starting/finishing temperatures (TAs and TAf), is illustrative. Their 𝑇̇𝑇 11 dependence can be derived from Fig. 4, as shown in Fig. 9a. These parameters exhibit a 12 non-monotonic dependence on 𝑇̇𝑇 as well as σh. This poses a fundamental concern that the 13 commonly employed dynamic identifications of these parameters make sense only under 14 the given conditions. TAs and TAf moderately increase with increasing 𝑇̇𝑇 and downshift 15 with decreasing σh, and the transformation span (= TAf − TAs) narrows with decreasing σh. 16 In contrast, TMs and TMf exhibit a pronounced depression at a certain 𝑇̇𝑇 range as can be 17 observed at σh = 600 MPa for TMf, and σh = 400 MPa for TMs. At σh = 0 and 300 MPa, TMs 18 and TMf can no longer be defined within the temperature window shown. A similar 𝑇̇𝑇 19 dependence has been reported in a Ti-50.2Ni alloy (see the lowest panel of Fig. 9a), 20 although resolving the subtle difference in their 𝑇̇𝑇  dependencies is difficult [43]. 21 Following the basic concept of Eq. 1, the characteristic temperatures (TMs, TMf, TAs, and 22 24  TAf) can be decomposed into the 𝑇̇𝑇-independent athermal and 𝑇̇𝑇-dependent thermal terms. 1 The latter term is difficult to capture at elevated temperatures where the thermal 2 excitations occur very quickly; therefore, the measured values are less dependent on 𝑇̇𝑇. 3 This term can be captured at lower temperatures (typically below 200 K). This situation 4 allows for isothermal MT to drive even above the experimentally determined TMs, as has 5 been reported to occur [2,44]. 6 The 𝑇̇𝑇 dependence of TMs and TMf is particularly noteworthy as their pronounced 7 depression means that the occurrence of kinetic arrest is controllable by changing 𝑇̇𝑇 and 8 σh. The cooling rate dependence of the volume of the kinetically arrested austenite phase 9 (1 − VM) is an informative and experimentally tractable measure. As shown in Fig. 9b, VM 10 at 0 K, as estimated from Fig. 8a for σh = 500 MPa, exhibits a non-monotonic dependence 11 on the cooling rate. Indeed, a similar cooling rate dependence has been experimentally 12 observed in a different type of 1st-order transformation with a similar temperature 13 dependence of the transformation hysteresis [45]. The kinetic arrest can also be observed 14 in magnetic-field-induced MT systems [46]. The same principle holds analogously when 15 magnetic fields or chemical driving forces are applied. 16 25   1 Fig. 9. (a) Temperature sweep rate dependence of TMs, TMf, TAs, and TAf at various σh, 2 where TMs/TAf are defined as the temperatures at which VM = 0.1 upon cooling/heating 3 and TMf/TAs are defined as the temperatures at which VM = 0.7 upon cooling/heating, 4 respectively. The plot in the lowest panel was obtained from ref. [43]. Masked regions 5 26  indicate the transformation spans of TMs − TMf and TAf − TAs. (b) Cooling rate dependence 1 of VM at 0 K at various σh. 2  3 4.2. INTERPRETATION OF NON-ERGODICITY 4 The glass-like thermal history dependence of strain is referred to as non-5 ergodicity and is believed to be an essential signature of strain glass [36,37]. In light of 6 the concept of viscosuperelasticity, the thermal history dependence of VM can be 7 discussed based on the derived CCT and CHT diagrams. As an example, the changes in 8 VM were calculated via Eqs. 4, 5, 6, and 7 for the following three cooling/heating routes 9 overlaid in Fig. 8a and b, namely, at a cooling rate of 0.04 K/s with σh = 500 MPa (FC); 10 at a heating rate of 0.04 K/s with σh = 500 MPa after FC (FC-FH); and at a heating rate 11 of 0.04 K/s with σh = 500 MPa after quenching to 0 K without stress (ZFC-FH). The 12 results are presented in Fig. 10a as a function of temperature. The thermal history 13 dependence of VM is evidence that the observed MT dynamics are not ergodic. In the 14 ZFC-FH curve, VM initially increases in response to the heating-induced forward MT, 15 reaches a stable value between TµM and TµA, and then converges to zero due to the reverse 16 MT upon further heating. A similar trend is observed in the FC-FH curve, but the initial 17 increase is less significant because the forward MT is largely completed during the FC 18 process. For reference, the hypothetical ZFC-FH and FC-FH curves assuming TµA = TµM 19 (that is, without hysteresis) are shown by the dotted lines in Fig. 10a. The hypothetical 20 ZFC-FH curve, in comparison to the actual case, exhibits a narrower peak near the nose 21 temperature and exhibits very little thermal history dependence above this temperature. 22 27  Additionally, the hypothetical FC-FH curve exhibits high reproducibility with the FC 1 curve. These trends mimic the typical non-ergodic behavior in spin glass and relaxor 2 systems [47,48]. However, such a feature would not be observed in MTs because the 3 transformation hysteresis must exist. The contrasting behavior of the modeled (filled 4 lines) and hypothetical (dotted lines) cases shows that the origin of non-ergodic strains 5 can be inferred from the observed thermal history dependence. A broader hump in the 6 ZFC-FH curve and the apparent history dependence above the nose temperature manifest 7 the activation of MTs rather than what would be expected in the strain glass state. 8 Figure 10b and c presents examples with a significantly lower σh together with 9 reference data [36,49]. The difference between the experimental and calculated curves, 10 especially the sharpness of the reverse MT, deserves additional remarks. Small σh 11 corresponds to a small VM and results in a faint difference between the reverse 12 transformation starting/finishing temperatures (This tendency can be seen in Fig. 9a). 13 Since the calculation does not account for microstructural and chemical heterogeneities, 14 the transformation dynamics are specified to proceed uniformly along the CCT/CHT 15 diagrams. However, in reality, microstructural and chemical heterogeneities locally 16 disturb the transformation temperatures. Besides, the elastocaloric cooling effect may 17 widen the reverse MT span. The ZFC-FH curves of refs. 36 and 49 start to decrease at a 18 temperature apparently higher than the nose temperature. This trend was reproduced in 19 our model, supporting the activation of MTs. Indeed, a small amount of martensite is 20 inducible even under such small stresses, as reported in ref. 50. Such martensite could 21 28  presumably breathe in the vicinity of stress-concentrated regions, such as cracks, grain 1 boundaries, precipitates, and surfaces, contributing to the non-ergodic response. 2 Additionally, it is important to mention another possibility that could mislead to the 3 branding of strain glass. Another type of MT, called the R-phase transformation, may be 4 stress-induced at the temperature and stress condition where the B2-B19’ MT is 5 kinetically arrested [30]. The R-phase transformation accompanies an exceptionally 6 narrow temperature hysteresis and small transformation strain [51,52]. Once this 7 transformation takes place, it could be difficult to discern the signature of hysteretic MTs 8 in the thermal history dependence of VM. 9  10 Fig. 10. The variation in VM change in the FC, FC-FH, and ZFC-FH routes overlaid in 11 Fig. 8a and b: (a) σh = 500 MPa and 𝑇̇𝑇 = 0.04 K/s, (b) σh = 40 MPa and 𝑇̇𝑇 = 0.04 K/s, 12 29  (c) σh = 100 MPa and 𝑇̇𝑇 = 0.04 K/s, and (d) σh = 750 MPa and 𝑇̇𝑇 = 0.004 K/s. The ZFC-1 FH and FC-FH curves assuming TµA = TµM are shown together in (a) with broken lines. 2 The reference data with a comparable 𝑇̇𝑇 is shown together in (b), (c), and (d) [26,36,49]. 3 Here, the reference data were originally given in strain and were converted to VM by 4 dividing by εfull after subtracting the elastic strain referring to the temperature dependence 5 of EA given in ref. 49. Note that 𝑇̇𝑇 was not addressed in ref. 36. Vertical mismatch in VM 6 could be attributed to compositional difference or inaccuracy in measured strain or EA. 7 More apparent signatures of the thermal activation of MTs can be captured at a 8 larger σh, as shown in Fig. 10d. A broad peak in the ZFC-FH curve and a remarkable 9 history dependence above the nose temperature can be observed experimentally [26]. This 10 behavior has also been observed in the magnetic-field-induced MT of a Ni-Co-Mn-In 11 alloy [53]. 12 Kustov also discussed the origin of non-ergodic strain from a different 13 perspective to that of strain glass, stating that a large part of the strain observed in the so-14 called strain glass state is elastic strain, and claiming that the remaining non-ergodic 15 component (in the order of 10-3 in VM) originates from the precursor martensitic 16 nanodomains [49]. However, the detailed reason for the non-ergodic behavior attributed 17 to such nanodomains has yet to be explicitly addressed. The findings in the present study 18 strongly suggest that this non-ergodic behavior originates from the thermal activation of 19 martensite growth rather than the glassy strain state of the austenite phase. This 20 conclusion raises a concern about the unique branding between strain glass and non-21 ergodicity. Apart from the universality of the concept of strain glass, the non-ergodic 22 30  behavior of VM can be interpreted as a manifestation of the isothermal propagations of 1 habit planes, which is now predicated upon the concept of viscosuperelasticity. 2  3 4.3. SIGNIFICANCE OF THERMAL ACTIVATION 4 Following a number of studies on the dynamics of thermoelastic MTs, three 5 different claims on the athermal/isothermal nature have been made. Firstly, two distinct 6 types of athermal and isothermal dynamics exist depending on the alloy system, 7 temperature, and/or directions of the MTs [1,2,4,16–18,42]. Secondly, thermal activation 8 exists to a greater or lesser extent in all thermoelastic MT systems [25,54–57]. Lastly, 9 MTs are always athermal due to their diffusionless/displacive nature [58,59]. The 10 isothermal experimental results in this study (see Figs. 5 and 6) reveal that the isothermal 11 MT proceeds even at 62 K, where atomic diffusion is unlikely to occur, thus negating the 12 possibility of the third claim. Furthermore, as previously claimed by Kajiwara [19–21], 13 the propagation of martensite variants is accompanied by dislocation motion. This claim 14 supports the isothermal growth of MT without the need for atomic diffusion and, more 15 importantly, envisages the validity of our thermal activation model mimicking the 16 classical thermal activation theory for dislocation glides. Kustov [42] performed similar 17 experiments on this alloy system and classified the three transformation paths of B2-B19’, 18 B2-R-B19’, and B2-B19-B19’ as either athermal or isothermal. He therein concluded that 19 the forward B2-B19’ transformation is isothermal but the reverse B19’-B2 transformation 20 is likely athermal, and therefore claimed that the athermal/isothermal distinction is 21 31  attributable to a distinction in strain accommodation, inferred from the difference of 1 thermal hysteresis in relation to the types of MTs. However, this conclusion was drawn 2 based on experiments above 180 K, where the isothermal nature is difficult to capture, 3 and as demonstrated in Fig. 9a, the temperature hysteresis varies with the temperature 4 sweep rate; therefore, it cannot be a unique scale for comparison. In a Cu-Al-Ni system, 5 Kakeshita detected isothermal incubation of the reverse MT. He discussed this in the same 6 way as isothermal incubation of the forward MT based on his nucleation model [55]. 7 However, it is questionable whether the underlying concept of “nucleation” can be 8 applied to the reverse MT and even to the growth process. Like other existing models, 9 Kakeshita’s approach does not account for the role of thermodynamic equilibrium; 10 therefore, there is no way to describe the non-reciprocity in the forward and reverse MT 11 dynamics. Nevertheless, although based on the nucleation concept, Kakeshita’s approach 12 leads to an interpretation that integrates athermal and isothermal dynamics [54,56,57], 13 proposing a sequential schema bridging athermal and isothermal in the TTT diagram for 14 the forward MT. The results in this study basically support his interpretation (that is, the 15 second claim) and quantitatively consolidate it in terms of the growth dynamics with 16 accounting for the thermodynamic equilibrium boundary that forms a spine in the phase 17 diagram. 18 Almost all existing studies have tackled isothermal dynamics by changing only 19 the isothermal holding temperature. However, this approach is most likely to overlook 20 the impact of thermal activation on the reverse MT because, as shown in Fig. 9a, the shifts 21 32  of TAs and TAf are less visible in most cases. Based on our phase diagram shown in Figs. 1 1b and 3, the significance of thermal activation should be assessed by screening external 2 variables, such as stresses and magnetic fields (and also chemical composition), in 3 addition to temperature. 4 The most straightforward measure of the significance of thermal activation is 5 σTA. As shown in Fig. 11, σTA differs by more than 3 orders among the studied alloy 6 systems [27,60–62]. Kustov assumes that the difference should be primarily related to the 7 distinction in strain accommodation [42]. Yet, even accounting for various effects arising 8 from crystallographic differences, σTA in a Ti-Ni system is anomalously large. The origin 9 of this anomaly remains an open question. The authors assume that the precursor 10 commensurate/incommensurate nanodomains [63,64] or the intermartensitic R-phase 11 [51] may be crucial in amplifying σTA in this system. This perspective will be discussed 12 in the near future. 13  14 Fig. 11. Comparison of σTA for the studied systems [27,60–62]. 15  16 5. CONCLUSIONS 17 33  This study presents a new approach for clarifying the impact of thermal 1 activation upon MT evolutions, typically manifested below room temperature. The 2 thermal activation component of transformation hysteresis is shown to vary with time and 3 temperature. The isothermal TTT diagrams for the forward and reverse MTs are non-4 reciprocal. The TTT diagrams for the forward MT show typical C-shaped contour curves, 5 whereas those for the reverse MT show the lower half of the C-shaped contour curves. 6 The TTT diagrams are further converted to the CCT/CHT diagrams. These diagrams 7 explain that the sweep rate dependence of the characteristic temperatures (TMs, TMf, TAs, 8 and TAf) and the non-ergodic thermal-history dependence of anelastic transformation 9 strain are ascribed to the thermally activated sluggish MT dynamics. This raises a concern 10 about the unique branding between non-ergodicity and the concept of strain glass. 11 The proposed scheme offers a unified picture of the dynamics of thermoelastic 12 MTs involving the impacts of athermal and isothermal dynamics. Since the underlying 13 framework is an extension of the thermal activation model of plastic deformation, it sheds 14 light on the similarities between the thermally activated sluggish dynamics of habit plane 15 propagations and dislocation glides. A new concept of “viscosuperelasticity” is 16 introduced following the concept of viscoplasticity. This concept provides a 17 straightforward interpretation; kinetic arrest and strain glass, which were thought to be 18 distinct and unrelated phenomena, are to be interpreted as having the same roots of 19 “viscosuperelastic” dynamics. This concept allows a more comprehensive understanding 20 of various manifestations related to the thermal activation of MTs 21 34    1 35  Data availability 1 The raw data related to this manuscript will be made available upon reasonable request. 2  3 Declaration of Competing Interest 4 The authors declare that they have no known competing financial interests or personal 5 relationships that could have appeared to influence the work reported in this paper. 6  7 Acknowledgments 8 The authors acknowledge fruitful discussions with K. Matsuura and H. Oike. This study 9 was supported by the Mizuho Foundation for the Promotion of Sciences, Kyoto 10 University Foundation, JST PRESTO (Grant Number JPMJPR22Q6), and JSPS 11 KAKENHI (Grant Numbers 19H02418 and 19K22052). 12  13 Appendix 14 A. Mathematical fitting of the superelastic stress-strain curves 15 VM was estimated from the stress-strain curves to analyze the isothermal 16 evolutions of martensite. A mathematical fitting was performed on the dynamic 17 superelastic stress-strain curves to estimate VM as a function of stress. Representative 18 examples of fittings are shown in Fig. A1a and b for the stress-strain curves at 115 and 19 100 K exhibiting the dynamic forward and reverse MTs, respectively. Based on the 20 36  existing constitutive equations [65,66], this study employed the following equation for 1 fitting: 2 mix ME Vσ ε= + Ω      (A.1) 3 where Emix is the effective elastic modulus as a function of VM and Ω is the transformation 4 coefficient; they are given by ( )mix A M M AE E V E E= + −  and SE mixEεΩ = − , respectively. 5 The elastic moduli of austenite and martensite (EA and EM, respectively) were estimated 6 by least-squares fitting to be 17.5 and 18.2 GPa, respectively, at 100 K and 16.6 GPa and 7 17.6 GPa, respectively, at 115 K. Note that these values are considerably lower than the 8 reported values [67] because they include the elastic modulus of the compression jig 9 inside the cooling chamber. The following asymmetric sigmoid function [68] was 10 employed to fit the evolution of VM: 11 [ ]{ }1/1 exp ( )McA BV B υυ λ ε ε−= ++ − −      (A.2) 12 where A and B are the upper and lower limits, respectively, λ is a shape parameter, υ is a 13 parameter characterizing the asymmetric curvature, and εc is the inflection point. The 14 resultant fitting curves (Fig. A1a and b) accurately trace the experimental curves with the 15 evolution of VM (Fig. A1c and d) as estimated using Eq. A.2. Furthermore, polynomial 16 fitting was employed to convert the evolution of VM  into a function of stress (Fig. A1e). 17  18 37   1 Fig. A1. Stress–strain curves at (a) 100 and (b) 115 K; gray and colored curves represent 2 the experimental and fitted data, respectively. Evolution of VM at (c) 100 and (d) 115 K 3 as a function of strain and (e) as a function of stress. The numerical values of the 4 parameters in Eq. 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