BACHELOR’S THESIS
TRANSITORY ULTRASONIC ABSORPTION IN MULTIFERROIC Ni-Mn-Ga ALLOY
Bruno D’Agosto Pons
Degree in Physics Faculty of Science
Academic Year 2020-21
TRANSITORY ULTRASONIC ABSORPTION IN MULTIFERROIC Ni-Mn-Ga ALLOY
Bruno D’Agosto Pons
Bachelor’s Thesis Faculty of Science
University of the Balearic Islands
Academic Year 2020-21
Key words:
High mobility twin boundary, Ni-Mn-Ga, transitory internal friction, transitory damping, twin boundary motion.
Thesis Supervisor’s Name: Sergey Kustov Dolgov
The University is hereby authorized to include this project in its institutional repository for its open consultation and online dissemination, for academic and research purposes only.
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Abstract
In this work we create in 10M Ni-Mn-Ga martensitic samples a special martensitic variant structure consisting of only 3 twins separated by twoa/ctwin boundaries: Type I and Type II, with relatively low and very high mobility, respectively. The domain engineered structure thus created allows us to investigate the dynamics of a single highly mobile a/c twin boundary. We show that temperature variations induce in our samples an intense transitory absorption at ultrasonic frequencies ca. 105 Hz, peaking around 215 K. We also prove the existence of mobile obstacles to the motion of Type II twin boundary under quasistatic thermal stresses. Observation of conventional (structural) transitory damping at a frequency 105 Hz is not allowed, despite rather low strain amplitude used, due to the inverse frequency dependence of this damping component. We argue that the transitory absorption registered at rather high ultrasonic frequencies is associated with rearrangement of magnetic domain structure, associated with motion of Type IIa/c twin boundary. The uncovered transitory internal friction term of magnetic origin is a new category that falls within the classes of magnetomechanical damping and transitory internal friction.
Contents
1 Introduction 5
1.1 Internal friction . . . 5
1.2 Transitory internal friction . . . 5
1.3 Magnetomechanical damping . . . 6
2 Objective of the work 7 3 Material, method and sample preparation 7 3.1 Material . . . 7
3.2 Experimental method and protocol . . . 8
3.3 Samples . . . 10
4 Results and interpretation 10 4.1 Strain amplitude dependence . . . 10
4.2 Temperature dependence . . . 12
4.3 Kinetics and T-dot dependence of damping . . . 15
4.4 Comparison with previous experiments . . . 17
5 Conclusions 18
6 References 19
1 Introduction
1.1 Internal friction
The so called internal friction (IF), damping or ultrasonic absorption in rather high-frequency ex- periments relates to the amount of dissipated mechanical energy by a sample under a periodic stress.
Its measure implies the oscillation of the sample, and thus, it is defined as the ratio between the energy dissipated in a cycle of oscillations ∆W and maximum elastic stored energyW:
IF ∝ ∆W
W (1)
Although having a simple definition, the IF of a sample stems from one or, usually, a combination of many dissipation mechanisms that can be traced back to the microstructural level. A study of the IF of a sample can give insight to, for example, its structure and phase transformations, dynamics of point defects or thermal activation of dislocations jumps, to name a few. Analysis of the IF has been of great usefulness to many fields of physics, especially solid state physics and materials science, see e.g classical textbooks [1,2].
Separation and identification of IF components is desirable to understand the complexity of the microstructure and dynamics of the defects in a sample when it shows many mechanisms of the IF operating at once. This separation can be achieved by changing external variables like strain ampli- tude, frequency of oscillations, temperature, temperature rate, etc.
In this work we encounter different microscopic mechanisms of the IF in a multiferroic functional material - Ni-Mn-Ga ferromagnetic shape memory alloy - and use their different characteristic be- haviour with several control external parameters to comprehend them. The most important one is the transitory internal friction, which is cooling-heating rate or T-dot dependent and has a rather complex mechanism that will be interpreted in a separate section. The second IF term that appears is the isothermal internal friction, which does not depend on cooling/heating rate but takes into consideration the phase, orientation and number of elastic variants of the sample as well as having a dependence with temperature, strain amplitude and frequency of oscillations. The third term shows time dependence under isothermal conditions and is associated with interaction of planar defects (twin boundaries) with mobile obstacles that impede the motion of twin boundaries.
In our experiments the phase composition and the miscrostructure of the sample are maintained.
We also minimize the number of external variables to temperature and heating/cooling rate, keeping the strain amplitude within the range of linear anelastic effects. Since the frequency of oscillations is also essentially unchanged in our experiments, we can pick up only T-dot and temperature depen- dences of the IF from a wide range of important variables.
1.2 Transitory internal friction
The notion oftransitory internal frictionrefers to the component of absorption of mechanical energy of periodic vibrations that emerges when oscillatory stress is coupled with certain external field. The superimposed coupling field results in an extra irreversible strain produced by periodic stress and, hence, additional damping of oscillations. The irreversibility of strain marks the transitory internal friction,IFtr, since in conventional structural internal friction experiments anelastic strain is usually reversible. Physical origins of the additional periodic irreversible strain and, hence, of coupling fields can be different:
• First order structural phase transition in a solid [3]; this transition can be induced by temper- ature, mechanical stress or any other relevant parameter like magnetic field;
• Plastic or even microplastic deformation of crystals during active deformation, creep or tem- perature variation in non-cubic polycrystals [4]; external quasistatic stress is coupled then with the oscillatory stress.
The general formal description of the IFtr remains essentially the same despite apparent dif- ference in potential physical mechanisms. The transitory internal friction is a strong function of strain-coupled external field and parameters of the measuring procedure, like heating-cooling (stress,
magnetic field, etc.) rate, frequency of measurements and oscillatory strain amplitude. During plastic or microplastic deformation, in a first approximation, IFtr is expressed as [4]:
IFtr=K ε˙ f εo
(2) where K is a constant, ˙εis the strain rate,fandεoare the frequency and elastic strain amplitude of mechanical oscillations, respectively. The first order structural transition (FOT) is also accompanied by lattice strain. In the case of FOT induced, for example, by temperature variation, the strain rate in Eq, (2) is substituted for temperature variation rate or T-dot, dTdt ≡T˙:
IFtr=K0 T˙ f εo
(3) withK0 - numerical factor.
Equations (2) and (3) are easily interpreted qualitatively: IFtr is proportional to the trans- formation or deformation rate, since the amount of additional strain per period of oscillations is proportional to these rates. The same reason explains the inverse frequency dependence: the higher is the frequency, the less is the additional strain accumulated during one period of oscillations under constant strain rate or T-dot. The inverse dependence on strain amplitude stems from the definition of the internal friction, eq. (1).
Since the denominator in the ratio, Eq, (3),W ∝ε2o, a simple assumption that ∆W is propor- tional to the elastic strain amplitude yields the inverse proportionality ofIFtr to εo. Many specific refinements of this simple interpretation have been suggested, see e.g. reviews [3,4].
One of the conditions under which the IFtr emerges is the variation of temperature of poly- crystalline materials with non-cubic lattice that possess anisotropy of thermal expansion coefficients, see e.g. [4]. The anisotropy of thermal expansion is responsible for thermal stresses generated in differently oriented grains when the temperature of the polycrystalline aggregate changes. These thermal stresses induce microplastic or even plastic deformation of grains and, hence, the transitory damping emerges [4]. Under these conditions the IFtr is proportional to the cooling-heating rate, T-dot. For the same reason thermal stresses are induced in multiphase materials, for example, pre- cipitation hardened [5] and composite materials [6]. Another group of materials showing transitory damping during thermal cycling are shape memory alloys in the martensitic phase. The latter con- sists of usually self-accommodating twin-related elastic domains (”martensitic variants”) with low lattice symmetry and high anisotropy of thermal expansion [7]. Therefore, similar to polycrystalline aggregates, martensitic phases show transitory damping, even if the material is single crystalline in the high-temperature phase. Transitory damping was observed and studied in the martensitic phase of classical Cu-Al-Ni [8] and NiTi [9] shape memory alloys.
An important property of IFtr is the inverse (or nearly inverse in more refined theories [3]) frequency dependence, Eqs. (2) and (3). Therefore, it is generally accepted that the IFtr can be observed only at low frequencies, f < 103Hz, and it takes negligible values at higher, ultrasonic frequencies [3-5].
1.3 Magnetomechanical damping
The magnetomechanical damping (MMD) is defined as any damping suppressed by a magnetic field [10]. The classical description of MMD considers three additive components, associated with mag- netic domain walls and net magnetization of the sample. Two MMD components are linear in strain (with the damping value independent of strain amplitude εo) and one non-linear, whose value is strain amplitude-dependent [10-12]. Expressed as logarithmic decrement of oscillations, δ, these components are: a linear microeddy current damping,δµ, usually found for low strain amplitudes; a non-linear hysteretic damping,δh(εo), emerging at higher strain amplitudes and a macroeddy current damping component, δM, which is also linear in strain amplitude or strain amplitude independent.
δµ andδh(εo) are ascribed to a short-range reversible and long-range hysteretic motion of magnetic domain walls, respectively.
Macroeddy MMD originates from macroscopic net magnetization of the sample. Although by definition all three MMD terms vanish at saturation, they differ in their dependence on field. δµ
andδh(εo) attain maximum values for demagnetized state of a sample (magnetizationM = 0) then decline monotonously with applied field H, see e.g. [10-13], whereas δM = 0 when M = 0. There- fore, δM(H) has a maximum for H slightly above half saturating field. Thus, only microeddy and hysteretic damping are relevant in the demagnetized samples without external field, the former, only under low oscillation amplitudes and the latter, in the high-amplitude non-linear range. A crucial property of microeddy MMD is linear frequency dependence up to a frequency somewhat lower than the frequency of magnetic domain wall relaxation, 105 - 107 Hz [14,12]. This frequency dependence is a consequence of Faraday’s law controlling the intensity of the dissipation of energy provoked by eddy currents.
Recently, T-dot dependent damping was reported in polycrystalline Dy below the N´eel temper- ature, in the ordered antiferromagnetic state [15] at an ultrasonic frequency close to 90 kHz. Since the T-dot effect was observed only in the magnetically ordered state, it had been attributed to the magnetic domain structure, more specifically, to the magnetic domain walls perpendicular to thec- axis (axis of the 6thorder), which carry magnetic moments. Antiferromagnetic Dy possesses so-called spiral magnetic structure: localized magnetic moments of atoms are parallel and confined in the basal plane, but rotate by a certain angle between the neighbouring planes. This spiral angle changes with temperature between the N´eel and Curie point from ca. 43 to 27 degrees [16]. The ultrasonic T-dot effect had been explained by rearrangement of the magnetic domain wall structure either due to vari- ation of the spiral angle or due to the thermal stresses in highly anisotropic hexagonal structure. The possibility to observe T-dot effect at ultrasonic frequency was associated in [15] with the presumably magnetic origin of losses, which are proportional to frequency in the low-amplitude range (δµ). This proportionality compensates the inverse frequency dependence of the transitory damping, Eq. (3), and makes the IFtr essentially frequency-independent up to a frequency approaching the frequency of domain wall relaxation.
Taking the relaxation frequency ca. 105 - 107 Hz, one gets the range of frequency-independent transitory MMD reaching the range up to 104 - 106Hz. It has to be emphasized that the transitory damping related to change of magnetic domain structure should be classified as a new category of both transitory and magnetomechanical damping.
2 Objective of the work
The objective of this work is to test whether the damping during thermal cycling of ferromagnetic martensites shows similar behaviour as antiferromagnetic polycrystalline Dy, which has been at- tributed to magnetic transient ultrasonic damping. The reasoning for this test is quite simple:
• Martensitic alloys consisting of several differently oriented elastic domains (martensitic variants) show transitory damping due to rearrangement of twin structure [5,6].
• In ferromagnetic martensites magnetic domain structure and structure of martensitic variants are strongly coupled [17-19] due to high magnetocrystalline anisotropy; hence, variations of martensitic variant structure due to thermal stresses will inevitably produce rearrangement of the magnetic domain structure and, probably, magnetic transitory damping term.
3 Material, method and sample preparation
3.1 Material
For several reasons Ni-Mn-Ga ferromagnetic martensitic alloy with layered structure, (so-called 5 layered or 10M martensite) with slightly monoclinically distorted tetragonal lattice was chosen. The differences betweena andblattice parameters is only around 0.002 nm and martensitic transforma- tion temperatures are slightly above 300 K for the alloy with Mn content around 28 at%, which was studied in the present work [20].
First, this type of the martensite is known to possess rather low twinning stress [21,22]. Therefore, it was expected to suffer, during temperature variations, microplastic strains sufficient to produce transitory effects. Second, the structure of this martensite is well documented due to its potential in practical applications. Recent studies show that 10M martensite possesses hierarchical twin structure at micro-, meso- and macroscopic structural levels [23]. At a microscopic level the structure of 10M martensite consists of so-called adaptive nanotwins of 2, 3 or 5 atomic layers thick. The small difference between a and b lattice parameters is, nevertheless, sufficient to create a/b twins at a mesoscopic level. Their thickness ranges from hundreds nanometers to dozens of micron [24-26].
Finally, a/c twins, observable at a macroscopic level, can have dimensions comparable with the sample size. a/ctwins are responsible for so-called magnetostrains: large strains that can be induced in 10M martensite by external magnetic field [27,28]. The difference betweena,b and c axis results in a substantial anisotropy of thermal expansion of the variants joint by a/c twin boundaries [29].
Two types of a/c twin boundaries were found in 10M Ni-Mn-Ga martensite: Type I with (1 0 1) twinning planes and Type II with irrational indices close to (10 1 10). Type I twin boundaries have much lower mobility (twinning stress around 1 MPa) as compared to Type II ones (twinning stress below 0.1 MPa) [30].
3.2 Experimental method and protocol
The piezoelectric ultrasonic composite oscillator technique, PUCOT [31,32], was used to measure elastic and anelastic (damping) properties of the samples at a frequency close to 90 kHz. The PU- COT employs longitudinal resonant oscillations of the oscillator, consisting of two quartz transducers and a sample attached to them. Due to the absence of moving mechanical elements and rather high frequency, the PUCOT is characterized by high resolution and fast data acquisition rate. The fre- quency employed ca. 90 kHz, on one hand, is two orders of magnitude higher than the limiting frequency of the conventional transitory damping and, on the other hand, is somewhat below the characteristic frequency of domain wall relaxation. That is, the PUCOT is perfectly suited to detect expected transitory damping of magnetic origin.
Figure 1: Schematic drawing of the ultrasonic oscillator, consisting of two quartz transducers (drive quartz and gauge quartz) and a sample glued to one end of the gauge quartz transducer.
Ud and Ug are periodic voltages across the drive and gauge transducers, respec- tively.
A schematic drawing of the assembled composite os- cillator is shown in Fig. 1. The two quartz trans- ducers are rectangular bars of 2.8 × 2.5 × 28.5 mm.
They actually are two parts of the same 18.5 X-cut quartz single crystal. This crystallographic orientation of the quartz transducers is a standard one that min- imizes coupling of the longitudinal mode with flexu- ral and torsional ones. One of the transducers (drive quartz) serves to excite the oscillations, while the sec- ond one (gauge quartz) to determine the strain am- plitude of oscillations and to provide a positive feed- back signal to maintain self-oscillations of the reso- nant system. Despite rather low piezoelectric constants, quartz is the optimal piezoelectric material for trans- ducers. Firstly, quartz is not ferroelectric, hence, it lacks domains and its behaviour is highly linear. Sec- ondly, quartz shows great stability of properties over a wide range of temperatures. The transverse piezo- effect is used in the current version of the PUCOT:
two lateral surfaces of each part of the crystal are plated with Ni electrodes (separated by a gap be- tween the drive and gauge parts of the quartz) and a periodic voltage applied across the electrodes, Ud
and Ug, is coupled with the longitudinal strain of the quartz.
The length of each element of the oscillator corresponds to one half of the ultrasonic wavelength of the oscillations, that is, each element vibrates in its fundamental mode. The sample, also of rectangular parallelepiped shape, is glued to one end of the quartz transducer. The length of the
sample is chosen in such a way that its fundamental resonant frequency coincides within 5% with the frequency of quartz transducers. Under this condition, the glue layer between the sample and quartz transducer is close to the strain node of the resonant system and does not contribute significantly to the background damping. Knowing the RMS values of voltages across the quartz transducers at resonance is sufficient to determine the ultrasonic damping (expressed as logarithmic decrement, δ) and strain amplitudeεo of the oscillations [31]:
δ=kδUd Ug
(4)
εo=kεUg (5)
wherekδandkεare constants, depending on parameters of the quartz transducers, input impedance of the measuring circuit, frequency of oscillations, and sample length. The Young’s modulus of the sample E is determined from the resonant frequency of its fundamental mode, fs, length ls and densityρ[32]:
E= 4ρfs2l2s (6)
An important step in the treatment of experimental data is the determination of the properties of the sample (decrementδsand resonant frequencyfs) from the corresponding values registered for the entire oscillator,δandf. These parameters of the sample are calculated from the corresponding values measured for the entire oscillator and for the quartz transducer alone, using a rule of mixture with weight functions equal to the corresponding masses [32]:
δ(mq+ms) =δqmq+δsms (7) f(mq+ms) =fqmq+fsms (8) withmq and msthe masses of the two quartz transducers together and of the sample,δs andδq the damping of the sample and of the quartz transducers alone (without the sample), respectively, and fq is the resonant frequency of the transducer alone. Usually, the damping of the transducer is very low and its resonant frequency is very stable. Therefore, Eqs. (7) and (8) allow us to precisely determine the parameters of the sample.
In order to obtain a precise measurement our setup relies on maintaining resonant conditions.
Adjusting the frequency before each measure is mandatory and achieved automatically by a process which searches for the new resonant frequency. Due to the relatively high frequency of the data acquisition (one point every 5 s) a search in the vicinity of the last resonant frequency by changing the phase shift in the positive feedback loop is sufficient. The new resonant frequency is extrapo- lated from a quadratic regression, finding its maximum. After this process the actual measurement is made, registering values for the two voltages, Ud and Ug, and for the resonant frequency of the oscillatorf at resonance. On top of that, the fully automated experimental setup [32] also allowed us to measure and control the strain amplitude of the oscillations and register the values of absorption and resonant frequency during temperature scans.
Three experimental protocols were employed, consisting ini)measurements ofδandf as a func- tion of strain amplitude in order to determine the ranges of amplitude-independent and amplitude- dependent damping; these ranges correspond to predominantly linear and non-linear dynamics of twins under periodic stress in acoustic experiments;ii)temperature cycling the sample between 290 and 173 K under a constant T-dot; iii) interruptions of cooling/heating scans in several points of the thermal cycles and registration of the kinetics of relaxation of damping under isothermal condi- tions during 2400 s. Measurements of temperature spectra were performed at low strain amplitudes, εo = 10−7, in order to avoid non-linearity of damping due to a large scale motion of elastic and magnetic domain walls.
3.3 Samples
The samples studied had the composition Ni50.0Mn28.4Ga21.6 (±0.3 at. %) and were produced by AdaptaMat Ltd., Finland. The samples were supplied by the Material Physics group, Lappeenranta- Lahti University of Technology (LUT). The samples were bar-shaped, approximately 1.0×1.0×11 mm3, with surfaces parallel to 100 planes of the high-temperature cubic phase. The length of the sample was chosen to fit the resonant frequency of the quartz transducer. Two samples were checked and showed similar results. The samples were mechanically and chemically polished.
An optical image of the two adjacent lateral surfaces of one of the samples is shown in Fig. 2. a/c twin structure of the samples was configured at LUT. The samples contained only three martensitic variants, separated by twoa/c twin boundaries: one with low mobility, Type I, and another one of high mobility, Type II. Such samples are an example of the so-called ”domain engineering” [33]. The variant with c-axis perpendicular to the sample long side was narrow and was always situated in the middle part of the sample. This section of the sample is stress antinode and the shear stress in the plane inclined 45◦ with respect to the sample axis reach maximum values in the narrow central variant. Two big variants withc-axis parallel to the sample length occupied both sides of the samples.
It is important to mention thata/btwinning planes in the two big variants at the ends of the sample are parallel to the sample’s long axis and do not experience shear stress, whereas a/b twins in the central narrow variant are under maximum shear stress and contribute, apart from thea/ctwins, to the measured elastic and anelastic effects. Traces of micron-scale a/b twins are not seen in Figure 2, since their observation in an optical microscope requires higher magnifications and is difficult in general because of their low optical contrast [26].
Figure 2: Two polarized light optical im- ages of adjacent lateral surfaces of one of the studied samples (courtesy of Dr. A. Saren, Lappeenranta-Lahti University of Technology, Finland). The sample consists of three marten- sitic variants separated by twoa/ctwin bound- aries, Type I and Type II. The orientations of the c-axes, perpendicular in the two adjacent variants, are indicated by arrows.
The advantage of the domain structure used in our experiments is the existence of only one highly mobilea/ctwin boundary of Type II. We mention here that a displacement of even relatively low-mobility Type I twin boundary can be detected in the results presented by Straka et al. [23].
In the optical images of Fig. 5, Ref. [23], the position of the Type I twin boundary changes with temperature. This displacement is seen for temperatures 298 and 223 K, Fig. 5 (a) and (d) and becomes obvious if the position of the twin boundary at 223 K is compared with the trace of this boundary in the austenite, cf. Fig. 5 (d) and Fig. 5 (f). The temperature-induced motion of the Type II twin boundary in domain-engineered structure must be much more intense due to an order of magnitude lower twinning stress for Type II boundary as compared with Type I in Ref. [23].
4 Results and interpretation
4.1 Strain amplitude dependence
Figure 3 shows the effect of oscillatory strain amplitude on damping and Young’s modulus defect registered at room temperature. The strain amplitude dependences show a pattern typical for twin boundary (TB) related non-linear anelastic effects when the twin boundaries are pinned by Cottrell- like clouds of mobile pinning points [34]:
• A strong difference between direct run (increasing strain amplitudes) and reverse run (decreas- ing strain amplitude); we will refer to this effect as strain amplitude hysteresis.
Figure 3: Strain amplitude dependence of the damping (a,b) and Young’s modulus defect (c,d) for a Ni50.0Mn28.4Ga21.6 ”domain engineered” sample with one highly mobile (Type II) and one low mobility (Type I)a/c twin boundary. Measurements at room temperature (290 K). Arrows indicate the scans for increasing and decreasing strain amplitude.
• The strain amplitude hysteresis is well reproducible in consecutive strain amplitude scans, as we can see from the resemblance of both runs.
• A very weak non-linearity at low strain amplitudes, below ca. 6.5·10−7 for the direct run, transforms into a very steep increase of both IF and modulus defect.
• The high-amplitude part of the dependence is jerky; it shows abrupt changes of anelasticity and strain amplitude.
All these features reflect interaction of moving twin boundaries with agglomerations of partially mobile pinning points that form Cottrell-like clouds around twin boundaries [34-36]. Low amplitude stage of the strain amplitude dependence corresponds to the motion of TBs within pinning clouds, the sharp increase of anelasticity at the high-amplitude stage represents the de-pinning of TBs from the clouds and the transition to the motion of TBs in the homogeneous spatial distribution of pinners of lower concentration as compared to the one of the Cottrell-like clouds. The amplitude hysteresis reflects, within this model, the redistribution of pinning points in the pinning clouds by TBs, moving in acoustic experiments.
Summary:
• The study of the strain amplitude dependences of damping and modulus defect at room tem- perature points to the existence of pinning clouds of mobile defects around TBs.
• Since a/b TBs are not pinned in the same material [37], it is reasonable to assume that the anelastic effects registered correspond to the motion of highly mobile Type II TB.
• The temperature spectra analyzed in the following section were registered at low strain ampli- tudes (of the order of 10−7) in order to avoid the depinning of TBs induced by oscillatory stresses and thus maintaining linearity of anelasticity with respect to oscillatory strain amplitude.
4.2 Temperature dependence
Figure 4: Continuous temperature spectra of damping registered in a cooling-heating scan between 290 and 173 K for a Ni50.0Mn28.4Ga21.6 domain engineered sample. Cooling- heating rate 1 K/min, oscillatory strain amplitude 10−7.
Figure 5: Examples of temperature (a) and cooling-heating rate T-dot (b) over time. In the shown cooling-heating scan temperature was held constant for 2400 s at 273, 253, 233, 213 and 193 K on cooling and at 193 and 213 K on heating.
Figure 6: Example of the damp- ing temperature spectrum for a 10M Ni50.0Mn28.4Ga21.6”domain en- gineered” sample in a cooling scan between 290 and 173 K interrupted for 2400s at 273, 253, 233, 213 and 193 K. Cooling rate 1 K/min (0.016 K/s), strain amplitude 10−7. The dashed line is derived by con- necting the experimental points ob- tained after isothermal holding at each temperature of interruption.
This dashed line can be therefore considered as an isothermal temper- ature spectrum.
Figure 4 shows the temperature spectra of damping registered during cooling and heating be- tween 175 and 290 K. The most crucial feature of the spectra is a broad maximum around 215 K.
The peak shows a small temperature shift and also a minor difference in peak height between cooling and heating. Another effect is a strong abrupt increase of damping during the initiation of cooling from 290 K and a sharp drop of damping at the lowest temperature of the thermal cycle during
the change from cooling to heating. These sharp variations of damping level are indicative of the transitory, T-dot effect. To confirm this conclusion, several thermal cycles were performed with pro- grammed interruptions of cooling/heating scans at several temperatures and isothermal exposures of the sample during 40 min. An example of time dependence of temperature and its time derivative is shown in Fig. 5 for one of the experiments with interrupted cooling/heating scans.
Experiments show an intense drop of the damping during each interruption of cooling/heating.
Figure 6 exemplifies the results of one of the cooling scans interrupted at several temperatures for 2400 s. The results indicate a nearly tenfold drop of the damping in the vicinity of the damping peak.
The isothermal spectrum, represented by the dashed line, does not show any damping peak. The damping increases monotonously upon cooling, the most intense rise occurs over the temperature range 210-190 K. The distance between the isothermal and general spectrum corresponds at least partly to the transitory term, which depends not only on T-dot but also on temperature. It is worth mentioning also that the damping becomes noisy and shows instabilities close the temperature of the damping maximum. Kinetics of the damping during interruptions of cooling/heating allows one to identify the origin of the strong damping decline upon interruption of cooling. Here, several possible scenarios can be seen:
• the decline is time-dependent and, probably, related to the pinning effect;
• the decline is T-dot dependent and is related to the transitory damping term;
• the decline is a combination of the two above-mentioned processes.
Time dependences of the temperature and T-dot shown in Fig. 5 were registered during the first cooling scan, when the cryostat was being filled with liquid nitrogen during initial part of the scan.
Filling the crysostat with nitrogen results in instabilities of the T-dot control. These occasional ther- mal instabilities, observed for instance during the first interruption of cooling programmed for 273 K, Fig. 5b, also serve as an efficient test to confirm the existence of the T-dot transitory damping term.
The data in Fig. 7 correspond to one of the initial cooling scans. In this experiment, the cooling segment between 290 and 163 K was programmed at a rate of 2 K/min. Figure 7a indicates the transitory damping peak centred around 215 K. The temperature scan was interrupted on cooling at 226 K and on heating at 200 K. The temperature control was not well established during the initial part of cooling down to 220 K, producing those instabilities and suggesting again a big T-dot dependence of IF.
Figure 7b shows the time dependence of the temperature, its derivativedT /dt and the internal friction during heating scan. A gradual IF decrease is clearly seen during isothermal holding at 200 K. This overall trend is combined with a fast IF decline at the moment of the scan interruption. The most indicative is the existence of a sharp minimum that coincides with the zero of dT /dtwhich is provoked by overshoot during temperature stabilization. This fact proves the existence of the tran- sitory damping term, proportional to |dT /dt|. Figure 7c shows similar time dependences registered during interruption of cooling under poor temperature control during the interruption at 226 K. The overall temperature variation during this interruption was ca. 4 K, between 224 and 228 K. No cor- relation between temperature variation and the IF behaviour is seen. On the other hand, excellent correlation is found between the IF and T-dot, Fig. 7c. Thus, the time dependence under isothermal conditions, clearly seen in Fig. 7b, and T-dot dependence point to the coexistence of time and T-dot dependent damping terms in temperature spectra of 10M Ni-Mn-Ga martensite. The former should be associated with pinning by mobile obstacles detected in strain amplitude scans, the latter with the motion of twin boundaries under thermal stresses during temperature variations.
Figure 7: Temperature spectrum of damping for a Ni50.0Mn28.4Ga21.6 ”domain engineered” sample during thermal cycle between 290 and 163 K, interrupted for 2400 s during cooling at 226 K and during heating at 200 K (a) and corresponding time dependences of temperatureT, absolute value of temperature derivative dT /dtand internal frictionIF during the entire heating scan (b) and during interruption of cooling (c). Values of IF and |dT /dt| are of the same magnitude and are plotted together sharing an axis.
The occurrence of this twin boundary motion, although not commented upon by the authors, can be seen in Scanning Electron Microscopy images of Fig. 5 in Ref. [23]. Some of these images are reproduced in Fig. 8. Figures 8 a-c indicate the change of the position of Type I twin boundary, with respect to characteristic surface defects indicated by red numbers 1 and 2, upon changing the temperature between 223 and 333 K. The last temperature corresponds to the austenitic state of the sample, but the ghost interface position is clearly detected on the surface polished in the martensitic
Figure 8: Scanning electron microscopy images of an a/c Type I twin boundary in a Ni50.0Mn28.2Ga21.8 10M martensite at 223, (a), 298 (b) and 333 K (c) (images taken and adapted from Fig. 5 of Ref. [23]). Image (c) corresponds to the high temperature austenitic state and shows the trace of the a/c boundary that existed in the martensite. The sequence of images indi- cates displacement of the twin boundary with respect to the surface marks 1 and 2 with increasing temperatures.
state. The twin boundary displacement can be evaluated in Fig. 8 as around only 0.2µm. However, the twin boundary motion must be much more intense in our domain engineered sample for the following two reasons:
• the overall twin structure of the sample has not been identified in Ref. [23], which was sup- posedly polyvariant; so, twin boundary interaction is expected to substantially inhibit their mutual displacement; in other words, a single TB must be much more mobile than TBs of the polyvariant structure;
• twin boundary shown in Fig. 8 is of Type I type, much less mobile than Type II TB studied in the present work.
4.3 Kinetics and T-dot dependence of damping
As has been shown [15], dependences of damping versus T-dot during interruptions of temperature scans provide more detailed information on the relative role of time and T-dot dependent terms in the IF spectrum. Such dependences are depicted in Fig. 9 for interruptions of cooling at 233 and 193 K. The experimental results follow the same trend as has been reported in polycrystalline Dy [15] and interpreted as due to a superposition of several damping terms.
It is convenient to analyze separately the dependences of damping during the decrease of T-dot and isothermal holding and during re-initiation of cooling. During decrease of T-dot upon inter- ruption of cooling the damping shows a linear decline, proportional to T-dot, associated with the transitory damping term. The linear slope registered at 233 K is higher than at 193 K, reflecting the temperature dependence of the transitory damping. The deviation from this linear trend appears at low T-dot values and transforms into isothermal drop of damping. These two features reflect the time-dependent decrease of damping due to immobilization of twin boundaries. After re-initiation of cooling, the damping first remains nearly constant and after overcooling by ca. 0.2 K recovers the proportionality to T-dot. The initial stage of this transition from isothermal immobilized state of twin boundaries to steady-state cooling process requires overcooling around 0.2 K.
Another argument confirming that the time-dependent decline of the IF is due to the pin- ning/immobilization of TBs is the unusual kinetics of the IF level recovery after resuming the heating scan, Fig. 7b. This figure shows that the T-dot recovery to its stable stationary level is gradual. At the same time, the IF level reaches the stationary level in nearly one jump. This fact is in perfect agreement with the overcooling/overheating necessary to start the motion of TBs after resuming temperature scan. These two observations, combined together, indicate that TBs are immobilized and require extra stress to initiate their motion after isothermal holding. After the threshold stress
Figure 9: Damping versus absolute value of T-dot for a sample of 10M during interruption of cooling of Ni50.0Mn28.2Ga21.8 ”domain engineered” sample at 233 and 193 K. Vertical arrows mark the fast and slow damping componentsδf astandδslow, respectively.
is overcome due to overheating/overcooling the TB moves rapidly, in a jerky mode, under the unbal- anced stress. Moreover, the noisy, jerky type of the temperature spectra around the IF maximum temperature might indicate the existence of the so-called dynamic strain ageing: pinning of the TB by mobile obstacles moving together with the TB. Thus, the kinetics of damping during interrup- tions of cooling/heating proves the existence of a fast, T-dot dependent, and slow, time-dependent components.
Figures 7b and 9 reproduce in detail the pattern reported for polycrystalline Dy [15]. Therefore, we will follow the same phenomenological qualitative interpretation elaborated for antiferromagnetic domain walls in Dy. Assuming additivity (linear independence) of different damping terms, the overall registered damping can be written in the following form [15]:
δ(T,|T˙|, t) =δo(T) +δslow(T, t) +δf ast(T,|T˙|) (9) whereδo(T) is the isothermal temperature spectra, similar to the one shown in Fig. 6. Figure 9 shows the separation of the fast and slow damping components from the overall T-dot and time de- pendent damping terms. These results indicate that the isothermal structural termδo(T) represents only minor part of the overall damping.
Following the explanations and conclusions made in [15] we assume that the ultrasonic transitory term in 10M Ni-Mn-Ga has its origin in the rearrangement of magnetic domain structure. In Dy the specific mechanism has been associated with rearrangement of magnetic moments of domain walls perpendicular to the c-axis. In the case of 10M Ni-Mn-Ga the interpretation is more straightforward.
The c-axis is the axis of easy magnetization. Therefore, any displacement of a/c twin boundary provokes rearrangement of magnetic domains, more specifically, the rotation of magnetization of parts of the magnetic domains adjacent toa/c twin boundaries. This rearrangement during motion of twin boundaries has been directly observed in a Ni53.8Mn23.7Ga22.5 alloy [38] by means of high- resolution Interference-Contrast-Colloid method, a technique that allows to simultaneously observe the ferromagnetic domains and the ferroelastic twin domains up to a high resolution. Thus, in our sample elastic and magnetic domains are coupled: the motion ofa/c twin boundary provokes local changes of magnetic induction flux, and, hence, transitory damping having the same microscopic mechanism as microeddy current damping. Since this damping mechanism results in proportionality to frequency up to the frequency of (10-100) kHz (depending on the frequency of microeddy current relaxation, as discussed in Section 1.3), this proportionality largely compensates the intrinsic inverse
frequency dependence of the transitory damping and makes the transitory damping of magnetic ori- gin observable at ultrasonic frequencies around 100 kHz in agreement with the conclusion of Ref. [15].
4.4 Comparison with previous experiments
Finally, we compare our observations of the damping peak around 220 K with previous reports for Ni-Mn-Ga martensites [39-42]. Previous experiments were performed at low frequencies using Dy- namic Mechanical Analyzers. Unfortunately, the data available are scarce and certain parameters, like oscillatory strain amplitude are not or poorly defined. Due to the unknown contribution of non- linear effects, the direct comparison of the absolute values of damping registered at low frequencies and in our experiments is not feasible. However, certain previous observations and conclusions can be confronted with present results.
We mention first that the peak observed is not related to intermartensitic transitions since it does not show typical very wide temperature hysteresis and, therefore, is different from the max- ima reported by Chernenko et al. [39]. Nevertheless, the maximum found in the martensite of Ni51.2Mn31.1Ga17.7 around 280-300 K might have the same origin as the one found in the present study.
Liu et al. [40] observed a damping peak close to 220 K in Ni50+xMn25−xGa25alloys with differ- ent values of x (between 0-2), which showed the same sequence of phase transitions as in the single crystal used in the present study. They classified it as twin boundary motion peak, which showed a strong decrease with frequency between 0.1 and 1 Hz. The damping level was hardly detectable and, assuming its existence, much less sensitive to frequency between 1 and 5 Hz.
Chang and Wu [41] reported an IF maximum around 215 K in a Ni52.0Mn26.5Ga21.5 alloy be- longing to the same group as present single crystals (Group II). Similar to the present observations, the peak apparently disappeared in the isothermal spectrum, although this property has not been commented upon. The authors described this peak as relaxation associated with twin boundaries.
However, this interpretation contradicts frequency independence of the peak position and its disap- pearance in the isothermal spectrum. Kinetics of the IF relaxation, as we did, were studied by Chang and Wu only over the ranges of phase transitions. They did not register the T-dot dependence of damping and attributed the time dependence exclusively to the transitory damping term, despite very long-term kinetics, especially in Ni53.8Mn26.8Ga19.4 from Group III of Ni-Mn-Ga alloys. We have seen that a more consistent explanation of the long relaxation time is achieved by a separation of the transitory (δf ast) and time-dependent (δslow) terms of the IF.
It is worth mentioning that Wang et al. [42] reported an additional nearly frequency-independent broad peak in the 14M martensite of a quaternary Ni52Mn16Fe8Ga24 alloy, centred around 230 K.
The authors could not suggest any interpretation of the maximum uncovered. We note that a crucial property of the peak was its temperature shift with strain amplitude. The maximum also marked a notable change in the effect of strain amplitude on damping temperature spectra. This property agrees with the important role of pinning process on the twin boundary mobility revealed in the present experiments. Such role was seen in Fig. 3a, where at low strain amplitudes the IF has a linear dependence with amplitude of oscillations. Furthermore, the change of the type of the strain amplitude dependence upon crossing the peak temperature in low-frequency experiments of Ref. [42]
is easily explained by a change from dragging (high temperatures) to depinning (low temperatures) of twin boundaries upon immobilization of obstacles upon cooling. In summary, our experimental results essentially do not contradict previous studies which point to a possible generic nature of the damping maximum investigated in the present study.
Thus, the present results prove the existence of the ultrasonic transitory damping in multiferroic ferromagnetic martensites. This new damping term is a new category of both magnetomechanical and transitory internal friction. One more conclusion relates to the important role of thermal stresses induced in polyvariant martensitic samples. The interpretation of recent observations of a/b nan- otwinning in 10 M martensites [23] is lacking the driving force for this transition to the so-called adaptive structure. We suggest that the driving force for such transitions could be thermal stresses induced during temperature variations in polyvariant samples.
5 Conclusions
• The use of domain engineered sample with only one highly mobile Type II a/c twin bound- ary enables us to attribute transitory anelastic effects in 10M Ni-Mn-Ga martensite during temperature variations to the motion of this boundary under thermal stresses.
• The internal friction maximum in the 10M martensite centred around 215 K does not exist in the isothermal spectrum, thus discarding its relaxational nature.
• The internal friction maximum is formed by a combination of the fast transitory damping proportional to T-dot and the slow time-dependent damping term due to immobilization of the twin boundary by mobile obstacles.
• Observations of transitory damping at rather high ultrasonic frequency of 105 Hz point to a major contribution of rearrangement of magnetic domains and related eddy currents during quasistatic motion of the a/c twin boundary under thermal stresses.
• The observed ultrasonic transitory internal friction in ferromagnetic ferroelastic Ni-Mn-Ga crys- tals is, on one hand, a new category of transitory internal friction having magnetic origin of main dissipation mechanism, and, on the other hand, a new category of magnetomechanical damping associated with irreversible rearrangement of magnetic domain structure.
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