ferroelastic nanostructures and nanoscale transitions ... xiaobing/papers... · symmetry breaking...

AbstractFor decades, a kind of nanoscale microstructure, known as the premartensitic “tweed

structure” or “mottled structure,” has been widely observed in various martensitic orferroelastic materials prior to their martensitic transformation, but its origin has remainedobscure. Recently, a similar nanoscale microstructure also has been reported in highlydoped ferroelastic systems, but it does not change into martensite; instead, itundergoes a nanoscale freezing transition—“strain glass” transition—and is frozen into a nanodomained strain glass state. This article provides a concise review of therecent experimental and modeling/simulation effort that is leading to a unifiedunderstanding of both premartensitic tweed and strain glass. The discussion shows that the premartensitic tweed or strain glass is characterized by nano-sized quasistaticferroelastic domains caused by the existence of random point defects or dopants inferroelastic systems. The mechanisms behind the point-defect-induced nanostructuresand glass phenomena will be reviewed, and their significance in ferroic functionalmaterials will be discussed.

FerroelasticNanostructures and NanoscaleTransitions: Ferroicswith Point Defects

Xiaobing Ren, Yu Wang, Kazuhiro Otsuka, Pol Lloveras, Teresa Castán, Marcel Porta,

Antoni Planes, and Avadh Saxena

this article we shall use both without distinction), are an important class ofmaterials exhibiting interesting functionalproperties such as shape-memory effectand superelasticity.26 All the functionalproperties of these materials are rooted ina displacive transition called ferroelasticor martensitic transition, which involves along-range ordering of lattice strainsbelow a transition temperature Ms. Suchstrain ordering causes a symmetry lower-ing from a high symmetry “parent” phase(normally a cubic phase) into a low sym-metry ferroelastic (or martensite) phase(e.g., the B19' monoclinic phase).25,27 Thesymmetry breaking gives rise to a charac-teristic hierarchical pattern of ferroelasticdomains in the low-temperature phasethat minimizes strain energy generatedduring the transition.28,29 The lattice orunit cell distortion involved can be eithera homogeneous lattice deformation (like shear) or an intra-cell deformation(like shuffle), or both. Most shape- memory alloys, such as TiNi, involveboth shear and shuffle during the marten-sitic transition.

For ideal ferroelastic/martensitic sys-tems containing no defects (i.e., pointdefects such as dopants, chemical disor-der, or impurities), there exists little mystery for their ferroelastic transition,microstructure, and corresponding macro-scopic properties; they are well under-stood.25,26 However, actual ferroelasticmaterials are not perfectly “clean,” theyinvariably contain a certain amount ofdefects, introduced either inadvertently asimpurities or intentionally as dopants or achemical disorder to modify certain prop-erties. The existence of such defects inthese materials creates a long-standingmystery: A nanoscale microstructure1–3

with a cross-hatched tweed or mottledmorphology has been reported in variousferroelastic systems prior to their ferroelas-tic transitions (i.e., in the parent phase);these nanostructures have been foundto persist tens of degrees above the Ms temperature30,31 and are assumed to bea “precursor phase” of the incomingmartensite phase. Information from neu-tron scattering1,30 and transmission elec-tron microscopy (TEM)1,2,30 has indicatedthat these premartensitic nanostructuresare composed of nano-sized, quasistaticferroelastic domains. The premartensiticnanostructure was found to arise asanisotropic tweed or isotropic mottledmorphology depending on the elasticanisotropy of the system. Highlyanisotropic systems such as Ni-Al give riseto a tweed microstructure (Figure 1a),30

whereas more nearly isotropic systemssuch as TiNi-based alloys result in a

IntroductionNanoscale structures (hereafter abbrevi-

ated as nanostructures), in the form ofcross-hatched tweed or mottled morphol-ogy, have been reported in a wide range of important functional materials, includ-ing ferroelastics (or martensites, shape-memory materials),1–3 ferroelectrics,4,5

ferromagnets,6 superconductors,7,8 andcolossal magnetoresistance (CMR) materi-als.9 They are of great interest for under-standing the nature and functionalities ofthese important materials,10–13 but in manycases, a clear and unified understanding islacking. This article focuses on the nano -structures in ferroelastics and provides a possible unified explanation based onrecent experimental14–18 and theoretical19,20

progress on this subject, which advancesthe earlier understanding.21–25 We shall alsodiscuss striking similarities in the nano -structure evolution as a function of defectconcentration or temperature among dif-ferent classes of materials, as well as in thecorresponding physical properties. Thisindicates a similar underlying physics con-trolling the formation of these nanostruc-tures and their behavior. Therefore, theideas used in ferroelastic systems may beapplicable to other ferroic systems, such asferroelectrics and ferromagnets, and maybe extended even to superconductors andCMR manganites.

Ferroelastic materials, or martensiticmaterials in metallurgical terminology (in

838 MRS BULLETIN • VOLUME 34 • NOVEMBER 2009 • www.mrs.org/bulletin

Ferroelastic Nanostructures and Nanoscale Transitions: Ferroics with Point Defects

MRS BULLETIN • VOLUME 34 • NOVEMBER 2009 • www.mrs.org/bulletin 839

mottled microstructure (Figure 1b)2 priorto the martensitic transition. Simulationresults are shown in Figure 1c and 1d forhigh and low anisotropy, respectively.They will be explained later in this article.The biggest mystery about these pre-martensitic nanostructures is why a nano-sized ferroelastic phase can exist in atemperature range (i.e., the parent state),where it is apparently thermodynamicallyunstable. It also remains unclear how elas-tic anisotropy affects the morphology ofthe premartensitic nanostructures.

Early theoretical studies21–25 in the 1990smade an important first step toward anunderstanding of the premartensiticnanostructures. These studies have indi-cated that random point defects areresponsible for the formation of the pre-martensitic tweed. The random distribu-tion of point defects causes a spatialdistribution in the local Ms temperature,

Figure 1. Two forms of premartensitic nanostructures and the relation to elastic anisotropy.(a) Experimentally observed tweed in Ni-Al with high anisotropy.1 (b) Experimentally observedmottled structure in TiNiFe.2 (c) Simulated tweed with high anisotropy. (d) Simulated mottledstructure with low anisotropy.

thereby some local regions can transforminto the martensite at a temperature wellabove the nominal Ms temperature of thewhole system.21 In this way, nano-sizedmartensite domains, which show up asthe tweed, appear above the Ms tempera-ture. These studies also suggested that thetweed pattern results from a frozen stateof ferroelastic strains, such as a spin glass(a frozen disordered spin state), but thesuggested glassy state of the tweed hasnow been confirmed by experiment.18

Recently, a new kind of nanoscale mot-tled microstructure has been reported inthe well-known ferroelastic system,Ti50−xNi50+x, when the point defect (excess Ni)concentration x exceeded a critical value(x > 1.3 mol%).14–18 (Such compositionshad been known as nontransformingcompositions but showed an abnormalnegative temperature coefficient in electri-cal resistivity.32) Although the mottled

structure has a similar appearance to thepremartensitic mottled structure shown inFigure 1b, this new nanostructure doesnot transform into martensite, unlike thepremartensitic nanostructure; instead, itkeeps a mottled nanostructure down tothe lowest temperature achievable.14

Extensive experimental studies on thenew nanostructure system have revealedthat such a system undergoes a freezing oflocal strains at a glass transition tempera-ture Tg.14,16 Such a glass transition istermed “strain glass transition,” and thelow-temperature state of the nanostruc-ture corresponds to a frozen strain glass,which is a frozen state of disordered latticedistortion or simply a nano-sized marten-site.14 Recent numerical simulations of aLandau-Ginzburg type model (a modelcalculating free energy as a function ofstrain order parameter, which varies inspace)19,20 have demonstrated the crucialrole of point defects (or inhomogeneities)in the formation of strain glass, and it further demonstrates that the elasticanisotropy plays a key role in determiningwhether the nanostructure appears astweed or mottled morphology.

Then a central question arises: Is there aunified explanation for both premarten-sitic nanostructure and the similar nano -structure of strain glass? In view of therecent progress in experimental stud-ies14–18 on strain glass and in the modelingof the premartensitic nanostructures,19–24 itseems possible to provide such an answer.In the following, we shall combine bothexperimental findings with recent model-ing and simulation results to provide ageneral understanding of the origin ofnanoscale microstructures. The ideas alsomay be applicable to similar phenomenain a wide range of ferroic materials andother functional materials, such as high-Tcsuperconductors and CMR materials.Strain glass, as a new horizon of ferroelas-tic research, may offer opportunities forboth fundamental and applied studies.The physics of the glass behavior mayenrich the current theory of ferroelasticity.Strain glass has been shown to exhibit anumber of interesting properties, such asshape-memory effect and superelasticity,15

and many others yet to be discovered.

Crossover from MartensiticTransition to Strain Glass Transition:Premartensitic NanostructureVersus Strain Glass Nanostructure?

Figure 2a shows the phase diagram ofTi50−xNi50+x (x is the concentration of excessNi as point defects),14 where microscopicfeatures of all phases are shown in theinset micrographs. The most notable fea-ture of this phase diagram is that there

30 nm50 nm50 nm50 nm

50 nm 50 nm

000 011

110

gg

110110110

110110110

a

c d

b



exists a point-defect-induced crossoverfrom a normal martensitic transition to astrain glass transition (explained later inarticle) at a critical defect concentration xc ~ 1.3. (Note that the most accurate meas-urements from Reference 14 found xc ~ 1.1,

as shown in Figure 2a, but the phase dia-gram is sufficient for illustrative pur-poses.) This phase diagram reveals thedifferent transformation behaviors of pre-martensitic nano structure (for x < xc) andstrain glass nanostructure (for x > xc). Such

a phase diagram is expected to be genericfor many ferroelastic systems, and it canprovide an ideal arena for understandingthe transformation behavior of defect- containing ferroelastic systems, in particu-lar the effect of point defects. In thefollowing, we shall discuss the normalmartensitic transition (for x < xc) and strainglass transition (for x > xc) separately anddescribe their corresponding nanoscalemicrostructures. Based on the experimen-tal observations discussed in the next sec-tion we shall discuss the relation betweenthe two kinds of nanostructures andanswer the key questions raised in theintroduction.

Normal Martensitic Transition for x < xc and the Associated Pre martensitic Nanostructures

For x < xc, the NiTi system undergoesa normal martensitic transition from B2(cubic) parent phase to B19' (monoclinic)martensite. In the parent state (T > Ms), theparent phase is not an ideal, homo geneousB2 phase, rather it shows a mottled orspotted premartensitic nanostructure.During the martensitic transition tempera-ture, the premartensitic nanostructurechanges drastically into large, regularmartensitic twins at T < Ms, as shown inthe corresponding micrographs in Figure2a (for x < xc). The mottled or spotty pre-martensitic nanostructure of Ti-Ni corre-sponds to the very low elastic anisotropy(A ~ 2) of this system.2 Here A = C44/C' andis the ratio of the shear modulus to thedeviatoric modulus. On the other hand,most martensitic systems exhibit highelastic anisotropy,33 and in such cases, thepremartensitic nanostructure is character-ized by a cross-hatched tweed morphol-ogy, as in Figure 1a. The morphologyfeature (tweed or mottled) of premarten-sitic nanostructures and its relation to elas-tic anisotropy will be explained later.

Previous TEM studies1,2,30 have demon-strated that premartensitic nanostructuresare (quasi-) static martensite domains.They may have either a similar structure(but with less lattice distortion) as the low-temperature martensite phase or an alternative candidate structure of the low-temperature martensite.33 For example, inthe Ni-Al system, the premartensitic struc-ture and the low-temperature martensitehave similar structures.30 On the otherhand, in TiNi and TiNiFe systems, the pre-martensitic structure has a rhombohedral(R)-like structure,2,34 being different fromthe B19' (monoclinic) martensite at lowtemperature. Nevertheless, as the R phaseis another candidate martensite phase forthis system,33 it is not surprising to havethis structure as a premartensitic state.

Figure 2. Crossover from a normal martensitic transition to a strain glass transition by pointdefect doping in the Ti50−xNi50+x system, where x is the concentration of point defects(excess Ni). (a) Phase diagram showing the normal martensitic transition for x < xc andstrain glass transition at x > xc.14 The corresponding microstructures (or nanostructures) ofthe high- and low-temperature phases (B2, B19', strain glass) are embedded. B2 is a CsClstructure; B19' is a monoclinic structure. (Image courtesy of S. Sarkar). (b) Schematicmicroscopic picture of unfrozen (T > Tg) and frozen (T < Tg) strain glass. Unfrozen strainglass shows essentially dynamic domain flipping but quasistatic nanodomains also exist.Frozen strain glass exhibits essentially static nanodomains only. (c) Simulatedpremartensitic nanostructure to normal martensite transition at low defect concentration(intermediate anisotropy). (d) Simulated strain glass transition at higher defectconcentration (with the same anisotropy). T, temperature; Tg, glass transition temperature.

Ti50–xNi50+x

Martensitictransition

B2

B19' Martensite

Strain glass

dynamicdomains

staticdomains

[11]

quasistaticdomains

300

200

T (

K)

100

0

0 1x

20 nm

20 nm

20 nm

20 nm

2

B2

Ms

xc

T < Tg

T > TgT > Ms

T < Ms T < Tg

TgMs

Cooling

T > Tg

UnfrozenGlass

FrozenGlass

TgTg

No martensitictransition

Unfrozen strain glassPremartensiticnanostructure

Martensite Frozen strain glass

a

c d

b



Because the premartensitic nanostructuredoes not possess long-range order, the dif-fraction peaks are usually not located at theexact commensurate positions expected for a fully developed long-range orderedmartensite phase. Therefore, the pre-martensitic nanostructure often appears,from the diffraction pattern, as if it is anincommensurate phase. This explains theexperimental reports of the “incommensu-rate” premartensitic state.33,34

It should be noted that the quasistaticpremartensitic nanostructure correspondsto a “central peak” or “elastic peak” (thezero energy-loss peak) in neutron inelasticscattering experiments.30 The temperaturedependence of the central peak in variousmartensitic systems seems to suggest thatthe premartensitic nanostructure and theassociated static lattice distortion gradu-ally fade away with increasing tempera-ture over a temperature range as wide as50–200 K, and the gradual change from apremartensitic state to an ideal B2 (CsClstructure) state is not a thermodynamicphase transition, as there exists no anom-aly in any physical property.18

An intriguing question arises as towhether or not the premartensitic nano -structure also exists in a defect-free sys-tem. Neutron inelastic scattering on Zr hasfound that for pure Zr, the central peakvanishes, but oxygen-doped Zr shows acentral peak.35 As the central peak corre-sponds to the quasistatic nanodomains inthe TEM pictures, the vanishing centralpeak in high-purity samples suggests that

the defect-free parent phase does notshow a quasistatic nanostructure prior tothe martensitic transformation. It has alsobeen found that for the Ti50−xNi50+x system,with x → 0, the stoichiometric TiNi showsno diffuse 1/3 spots (i.e., no R-like nano -domains).36 Therefore, available experi-mental observations seem to suggest thatthere are no static premartensitic nano -structures prior to martensitic transition.

Nanoscale Transition (Strain GlassTransition) for x > xc and theAssociated Strain GlassNanostructures

For x > xc, the martensitic transitionvanishes, and instead a nanoscale transi-tion called the strain glass transition takesplace.14 The strain glass transition is char-acterized by the freezing of nano-sizedferroelastic domains at freezing tempera-ture Tg (shown in the right insets of Figure2a, and schematically shown in Figure 2b),similar to a cluster spin glass transition ora ferroelectric relaxor transition. Thesenanodomains also originate from the ran-dom point defects (or dopants), being thesame as the premartensitic nanostruc-tures, but strain glass contains a higherdefect concentration. The strain glass tran-sition, being different from a martensitictransition, involves two seemingly contra-dictory signatures: (1) invariance of theaverage structure through Tg (Figure 3a)and (2) the existence of a frequency-dependent anomaly at the glass transitiontemperature Tg (Figure 3b) following a

Vogel-Fulcher relation, a relation betweenTg and frequency that describes thedynamic freezing process.14,15 At T > Tg,the nano-sized ferroelastic domainsdynamically flip as a whole (as evidencedby softening of the phonons), but there area number of long-lived, quasistatic nano -domains (Figure 2b, T > Tg). These quasi-static nanodomains can be imaged byTEM as the mottled nanostructure (Figure2a, inset figure in the strain glass regime).With temperature decreasing, the nano -domains grow to some extent but areeventually frozen below Tg; thus, we cansee bigger nanodomains at T < Tg (Figure2b). Therefore, a strain glass transitioninvolves a freezing transition from adynamically disordered strain state (i.e.,strain liquid) to a frozen disordered strainstate (strain glass). This is analogous tothe formation of a jelly (structural glass)from a gelatin water solution (defect- containing liquid) during cooling. A TEMimage of the unfrozen strain glass and itscorresponding frozen glass can be foundin Figure 2a (the inset in the strain glassregime) for the x > xc.

The most important information fromFigure 2a is that point defects can induce acrossover from normal martensitic transi-tion to strain glass transition at a criticalconcentration xc. Below xc, the martensitictransition remains, but the transition tem-perature is lowered with increasing defectconcentration; above xc, the martensitictransition is absent, and instead the systemundergoes a freezing transition of the

Figure 3. Signatures of strain glass transition: (a) invariance in average structure, and (b) anomaly in ac mechanical susceptibility at the glass transitiontemperature Tg. The storage modulus is the inverse of the mechanical susceptibility. Tan δ is the loss; δ is the phase angle between stress and strainduring dynamical mechanical testing. The insets show the microscopic pictures of nanodomains, essentially ergodic at T > Tg and non-ergodic at T < Tg. The different orientations of the parallelograms represent different strain states.15 T, temperature.

30

56,000

52,000

48,000

Sto

rage

Mod

ulus

(M

Pa)

44,000

40,000

36,000

32,000

138 K

263 K

0.2 Hz

20 Hz

188 K unfrozen

frozen

173 K(Tg)

T > Tg

T < Tg

0.0200.0160.012

0.2 Hz1 Hz4 Hz

20 Hz0.0080.0040.000

Tan

δ

40 502q (Degrees)

158 k

203 k

293 k

323 k

373 k

110B2

Inte

nsity

(A

rbitr

ary)

111B2 200B2

211B2

108 k

T <

Tg

(160

K)

T >

Tg

(160

K)

60 70 80 100 150 200Temperature (K)

250 300 350

a

b



nano domains, the strain glass transition.The strain glass transition does not involvea change in average structure, thus it can-not be detected by x-ray diffractometry;but dynamical mechanical measurementcan clearly reveal such a transition.

Historically, it has been found for a longtime that the martensitic transition seemsto vanish suddenly if the doping levelexceeds a certain level, but it was gener-ally assumed that this is caused by thevery low thermodynamic stability ofmartensite at such a high doping level.However, these “non-martensitic” alloys(e.g., Ti50−xNi50+x with x > 1.3) were foundto exhibit abnormal physical properties,such as a negative temperature coefficientof electrical resistivity,32 but the originremained unclear. With the discovery ofthe strain glass transition in such alloys,14

the previous puzzle can now be solved.The “negative temperature coefficient” ofelectrical resistivity is simply a result ofthe gradual growth of the nanodomainswith an R-like structure, which has ahigher specific resistivity.33

A critical proof for the glass transition isthe existence of non-ergodicity, or historydependence of the physical properties(such as strain) in the glass state. This is

effect and superelasticity.15 As a strainglass does not undergo a martensitic tran-sition and looks like a nontransforming,“dead” material, there seems no reason tohave a shape-memory effect, which char-acterizes only a martensitic system.However, as can be seen from Figure 5a, astrain glass (Tg ~ 168 K) can also exhibit ashape- memory effect (when deformed atT < Tg and heated up to T > Tg, the bluecurve), very similar to a normal marten-sitic alloy. It is further found that at T > Tg,a strain glass can also exhibit superelastic-ity (the 173 K and 188 K curves). Theunexpected shape-memory effect ofstrain glass can be explained by the for-mation of long-range strain order(martensite) by external stress and itsrecovery to the initial unfrozen strainglass state.15 Figure 5b shows the simula-tion results about the stress-strain curveof the frozen glass. The details will beexplained in the next section.

Relationship Between thePremartensitic Nanostructure and the Unfrozen Strain GlassNanostructure

In Figure 2a, the two kinds of nano -structures, the premartensitic nanostruc-

usually done with a field-cooling and zero-field-cooling (FC/ZFC) experiment, whichdetects if there is a history dependence ofthe physical properties. Figure 4a showsthe FC/ZFC curves of the strain glassTi48.5Ni51.5.16 It is clear from the deviating FCand ZFC curves below Tg that the system isergodic at T > Tg and becomes non-ergodicat T < Tg. (Strictly speaking, a small devia-tion actually begins above Tg, indicating aslight ergodicity breaking.) This provesthat this system indeed undergoes a strainglass transition. It is noted that the FC/ZFCbehavior of the strain glass shows a strik-ing similarity with that of two other kindsof glass transitions, the relaxor ferroelectrictransition (freezing of local electric dipoles)(Figure 4c) and the cluster spin glass transi-tion (a freezing of magnetic moments)(Figure 4d). As strain glass, relaxor, andcluster spin glass are derived from threeferroic transitions (i.e., ferroelastic transi-tion, ferroelectric transition, and ferromag-netic transition, res pect ively), these threeglasses can be generalized into one biggerclass of glasses—the ferroic glasses. Theyexhibit very similar behavior in their corre-sponding dc and ac properties.

Strain glass exhibits a number of unex-pected properties such as shape-memory

Figure 4. (a) Ergodicity-breaking at strain glasstransition for the strain glass system Ti48.5Ni51.5, asevidenced by the history dependence of the field-cooling/zero-field-cooling (FC/ZFC) curves.16 (b)Simulation result for the FC/ZFC curves. Here, T0denotes the equilibrium transition temperature of themodel in the absence of disorder (defined in thesection on phenomenological modeling andcomputer simulations); it is used as a referencetemperature. In this figure, Tg (transition temperature)is located at the maximum of the ZFC curve. (c)FC/ZFC curves of a relaxor ferroelectric, PLZT (leadlanthanum zirconate titanate). (d) FC/ZFC curves ofa cluster spin glass (Reference 16 and therein). T,temperature; E, applied electric field; H, appliedmagnetic field.

0.0055 4

3

ZFC

FC

low anisotropy

2

Str

ain

(%)

Strain glass Ti48.5Ni51.5

1

0 0.5T/T0

1 1.5

0.0054

0.0053

Str

ain

0.0052

0.0051

0.0050

frozen

frozen

120 140 180T (K)Tg

200 220 240 260 280

ZFC

FC

unfrozen

Stress = 40 MPa

a b

160

Relaxor FerroelectricsPLZT 8/65/35

Cluster Spin GlassLa0.7Ca0.3Mn0.7Co0.3O3

0.3

0.2

Pol

ariz

atio

n (C

/m2 )

ZFC

FC

E = 3 kV/cm0.1

0.0–100 0 100

Temperature (°C)200 0

0 50 100 150Temperature (K)

H = 0.03 T

FC

ZFC

200 250 300

1

2

3

4

Mag

netiz

atio

n (A

m2 /k

g)

5

6

c d



describes a defect-free ferroelastic transi-tion and the resultant domain morphology.(2) Random point defects (or inhomo-geneities) are assumed to cause a spatialdistribution of Ms temperatures over thesystem. Some details of the model aregiven later.

Given that the inhomogeneous strainfluctuations of interest here are confinedinto certain (two-dimensional) planes,25

we shall illustrate the problem for a 2Dsquare-to-rectangle transition that mimicsa cubic-tetragonal transition (where therectangular cross-section of the tetragonalphase is used in 2D). Suppose that e1, e2,and e3 are the symmetry-adapted strainsrepresenting, respectively, hydrostatic,deviatoric, and shear distortions of thecubic lattice. The change of symmetry atthe transition is described by e2, which willbe the (primary) order parameter e2. Softdirections will be along {11}⟨11̄⟩ displace-ments corresponding to a small elasticconstant C', defined as

C' = (C11 − C12)/2, (1)

where C11 and C12 are, respectively, thecomponents Cxxxx and Cxxyy of the second-order elastic constant tensor. The elasticanisotropy is then measured by the ratio A = C44/C' (where C44 is the componentCxyxy of the elastic constant tensor associ-ated with shear distortions). A generalLandau free-energy density for this transi-tion will be of the form

F = Fh + Fgrad, (2)

ture (x < xc, T > Ms) and the nanostructureof unfrozen strain glass (x > xc, T > Tg),look very similar; both exhibit quasistaticnano domains with a ferroelastic structure.The only difference is that the unfrozenstrain glass nanostructure corresponds toa higher defect concentration; this leads toa smaller nanodomain size. Recent studieswith FC/ZFC and mechanical susceptibil-ity experiment showed that both types ofnanostructures have similar ergodicbehavior, but a small history dependencestill exists.18 This means that both nano -structures correspond to essentially a“strain liquid” state (i.e., ergodic), but theliquid has a certain “stickiness” (i.e., aquasistatic component exists). The nano -structure of the unfrozen strain glass (T > Tg) corresponds to a stickier state than the premartensitic nano structure.18

Quantitative analysis of the dynamics ofthe strain glass has shown that extremelylong-time relaxation (>103 seconds) existsin the unfrozen glass state (T > Tg).37 Thislong-lived strain fluctuation or quasistaticcomponent in the more viscous strain liq-uid corresponds to the “static” nanostruc-ture observed by TEM. This picture is alsotrue for the premartensitic nanostructure,but it is less viscous. Therefore, there is nofundamental difference in physical naturebetween the premartensitic nanostructure(x < xc) and the nanostructure of theunfrozen strain glass (x > xc); both corre-spond to a more viscous strain liquidstate, but a higher defect concentrationmakes an even more viscous strain liquid.This is very similar to an everyday phe-

nomenon: adding more gelatin into waterwill make a more viscous liquid, and athigh gelatin doping, the viscous liquiddoes not freeze into the crystal ice, ratherit is frozen into a structural glass—jelly.The difference in viscosity results in differ-ent con sequences: the less-viscous strainliquid (x < xc) undergoes a normal marten-sitic transition, but the viscous liquid (x > xc) undergoes a freezing transitioninto a strain glass. Point defects areresponsible for such a crossover.18

Phenomenological Modeling and Computer Simulations ofPremartensitic Nanostructures and Strain Glass Nanostructures

Different phenomenological modelshave been proposed to account fornanoscopic textures in ferroelastic andmartensitic materials. All of these modelsincorporate the requirements that the system must sensitively respond to localcoupling to some kind of disorder.21–25

The model used here19,20 is a generalizationof the model proposed by Kartha et al.21 Itsuccessfully reproduces the experi -mentally observed premartensitic nano -structures and strain glass structures.In particular, it explains why some systemsexhibit a tweed nanostructure, whereasothers exhibit a mottled or spotty nanos-tructure.

The model has the following key fea-tures: (1) The free energy of the systemis described by a Landau-Ginzburg en -ergy function extended to consider long-range elastic interactions. This model

Figure 5. (a) Experimental results on the shape-memory effect (plastic deformation at T < Tg and the shape recovery at T > Tg) and superelasticity(deformation at T > Tg) of strain glass Ti48.5Ni51.5. (b) Simulation results for plastic deformation at low temperature. Insets show snapshots ofconfigurations at a given value of strain. The image size is scaled down by a factor of five as compared to Figure 1c and 1d. T, temperature.

350

300

250

200

150

100

50

0

200

150

100

50

00 1 2 3 4 0 1 2 3 45 6 7 8

Strain (%)Strain (%)

263 K (>>Tg)

188 K (>Tg) 173 K (>Tg)

138 K (<Tg)

Heating to above Tg

Str

ess

(MP

a)

Str

ess

(MP

a)

a b



where the first term is the homogeneouscontribution adequate for a first-ordertransition, which includes both the nonlin-ear contribution associated with the pri-mary order parameter (e2) and theharmonic elastic energy associated withnonsymmetry breaking strains, nonorderparameter strain components, (e1, e3). Thesecond term is a gradient (or Ginzburg)term accounting for the energy cost ofinterfaces, such as twin boundaries. Whenthis free energy is minimized, imposingthe requirement of smoothly fitting unitcells without creating (structural) defectsby means of the elastic compatibility(Saint-Vénant) condition,29 the depend-ence on the nonsymmetry breakingstrains gives rise to a nonlocal, long-rangeanisotropic interaction. Here, it is denotedby the function U(r–r') that correlates anytwo unit cells at positions r and r', respec-tively.28 The following free energy densityis obtained:

F = C'(T,η) e22 − B e2

4 + D e26 + K |∇e2|2

+ dr' e2(r) U(r− r') e2(r'). (3)

T is the temperature, η the disorder field,and B, D, and K are phenomenologicalparameters.

In this model, point defects are takeninto account by means of a quenched-in,spatially fluctuating field included in theharmonic coefficient C', which is assumedto be of the form

C'(T,η(r)) = a[T – Tc – η(r)], (4)

where Tc is the lower stability limit of thehigh-temperature phase in the clean(absence of disorder) limit, and a isdefined by Equation 4 and is the tempera-ture derivative of C'. Usually, the disorderfield η is assumed to be spatially corre-lated and Gaussian distributed with zeromean. Thus, disorder has the effect of pro-ducing a distribution of transition temper-atures T(r) = T0 + η(r), where T0 is theequilibrium transition temperature of themodel in the clean limit. Therefore,regions with different degrees of metasta-bility separated by finite free energy barri-ers exist in the system. In the previousexpression, the long-range kernel U(r–r')goes as Acos(4θ)/|r–r'|2. This term isessential to providing a global responsedue to a local coupling of strain to disor-der, which is necessary for textures tooccur. Note that this long-range term isminimized for a cross-hatched (tweed)pattern due to the cos(4θ) leading term.

This means that order parameter corre-lations will be favored along the {11} diag-onal directions (θ = ± π/4). However, sincethe strength of this long-range interaction

is controlled by the anisotropy factor A,these correlations along the diagonals willbe strongly screened for low enough val-ues of the elastic anisotropy. It was previ-ously demonstrated that in the limit ofinfinite elastic anisotropy, this model canbe exactly mapped to a random-bond ver-sion of the antiferromagnetic spin-glassSherrington-Kirkpatrick model.21

In the following, we summarize a num-ber of key results reported in References19 and 20 that arise from the numericalsimulations of the present model andcompare them with the experimental datawhen available. We shall find that the sim-ulation results are in good agreement withexperiments.(1) Static premartensitic nanostructuresare absent in a defect-free (i.e., η(r) = 0) fer-roelastic system.(2) Static premartensitic nanostructuresexist only in a defect-containing system(i.e., η(r) ≠ 0) prior to the martensitic transition. The value of elastic anisotropy A = C44/C' determines the morphology ofthe premartensitic nanostructures. Highanisotropy leads to cross-hatched pre-martensitic tweed (Figure 1c) before themartensitic transition; low anisotropyresults in a mottled/spotty nanostructure(Figure 1d) prior to the transition. Thisis in good agreement with the experimen-tal results shown in Figure 1a (highanisotropy) and Figure 1b (low anisotropy).(3) There exists a crossover at a criticaldefect concentration xc, from a normalmartensitic transition into a freezing transition of the nano-sized ferroelasticdomains—the strain glass transition.Figure 2c and 2d show the simulationresults for an intermediate anisotropy system (such as NiTi(Fe) with low Fe con-tent) with low and high defect con -centration, respectively. At low defect concentration (x < xc), the system under-goes a normal martensitic transition froma premartensitic nanostructure (spottydomains) to a normal martensite withlarge twins (Figure 2c). At high defect con-centration (x > xc), the normal martensitictransition vanishes, and instead the systemexhibits a freezing transition during whichthe system retains the nanodomains downto 0 K (Figure 2d). The simulated results inFigure 2c and 2d reproduce the crossoverfrom the normal martensitic transition intothe nanoscale transition of strain glass.(4) The proof that the nanoscale transitionshown in Figure 2d is a strain glass transition can be found in Figure 4b. Thecalculated FC/ZFC curves show a charac-teristic history dependence below Tg,which should correspond approximatelyto the temperature of the maximum of the ZFC curve in Figure 4b. The deviation

between both FC and ZFC curves is indica-tive that the nanoscale transition shown inFigure 2d is indeed a strain glass transition.This simulation result is in good agreementwith the experimental one for Ti58.5Ni51.5(Figure 4a). No signatures of glassy behav-ior have been observed in the premarten-sitic nanostructure (Figure 2c), whichtransforms into a long-range twinnedmartensitic state. This is also in agreementwith experimental observation.18

(5) External stresses can change a strainglass with nanodomains into normalmartensite with large domains or even asingle domain, as shown in Figure 5b. Thepresence of a stress favors a particularstrain domain and at the same time over-comes the energy barrier for domainswitching. As a result, the nano-sizedmulti-domains of the frozen strain glassare switched into a favorable singledomain of a normal martensite. With heat-ing to a temperature above Tg, the systemreverts to the nanodomained state withaverage cubic structure (such as Figure 2d,the state at T > Tg). This causes a recoveryof the original sample shape and leads tothe shape-memory effect. Depending ontemperature, superelastic or pseudoelastic(plastic) behavior is found. Such a result isin good agreement with the experimentalone shown in Figure 5a.(6) Elastic anisotropy can affect the criticaldefect concentration for the crossover froma normal martensitic transition to a strainglass transition, as shown in Figure 6. Highanisotropy (A) requires a high defect con-centration (accounted for in the model by ahigh enough amount of disorder) to renderthe system into a strain glass, and lower

0.7

0.6

0.5

glassy

twinning (non-glassy)

0.4

0.3

Am

ount

of D

isor

der

0.2

0.1

00 1 2 3

A4 5

Figure 6. The influence of elasticanisotropy A on the critical amount ofdisorder (point defects) to induce acrossover from a normal martensite(twinning) to a strain glass state.19,20

The values denoted by circles havebeen computed from the FC/ZFCcurves, whereas the values denoted bycrosses have been obtained from thefree-energy metastability analysis. Formore details, see Reference 20.



anisotropy requires a lower critical defectconcentration (amount of disorder) to makea strain glass. The reason for this effect isthat high anisotropy increases the strengthof the long-range elastic interaction andhence favors a long-range strain ordering(i.e., the normal martensitic transition).

Generic Phase Diagram for Point-Defect-Containing FerroicSystems—Crossover from FerroicTransition to Ferroic GlassTransition

A ferroelastic (or martensitic) transition,a ferroelectric transition, and a ferromag-netic transition involves long-range order-ing of lattice distortion (or strain), electricdipole, and magnetic moment, respectively.The long-range interaction arises from elastic compatibility (i.e., elastic dipoles),electric dipoles, and magnetic dipoles,respectively. Except for a difference in theorder parameter, the behavior of theseordering transitions and the associatedmicrostructures, as well as physical proper-ties, are very similar or parallel. Thus theyare called ferroic transitions as a whole.38

An appealing question arises as towhether point defects (or inhomo-geneities) in these different systems cangenerate similar consequences as those forferroelastic systems (shown in the phasediagram of Figure 2a). The answer is yes,and it seems that point defects can pro-duce strikingly similar consequences forthese ferroic systems (Figure 2a),39,40 butsuch similarity has not been generallyacknowledged in the past.

For ferroelastic systems, we haveknown from the previous discussion thatpoint defects produce two prominentconsequences: (1) Below a critical defectconcentration (x < xc), the normal ferro-elastic or martensitic transition remains,but the defects cause the formation of apremartensitic nanostructure above thetransition temperature. (2) Above the crit-ical defect concentration (x > xc), long-range strain ordering (ferroelastictransition) is suppressed; instead, ananoscale freezing transition, the strainglass transition, takes place.

For ferroelectric systems, low-level dop-ing does not suppress the ferroelectrictransition but causes a static short-rangeordering of electric dipoles above the ferro-electric transition temperature Tc. Suchshort-range ordering corresponds to thenano-sized ferroelectric domains abovethe transition temperature and can extendto a rather high temperature.41 On theother hand, a high concentration of suit-able dopants (as point defect) can create aglassy phase called a relaxor, where nano-sized dipolar domains are frozen below a

freezing temperature Tg.39 Therefore, thereexists a crossover from a normal ferroelec-tric transition into a relaxor (glass) transi-tion at a critical defect concentration,41

being similar to its ferroelastic counterpart.For ferromagnetic systems, a similar sit-

uation exists. It has been known for a longtime that doping with a sufficiently highconcentration of nonmagnetic elements(as point defects) can destroy the long-range magnetic order of a ferromagneticsystem, resulting in a locally ordered statecalled cluster spin glass.40 But little isknown about whether there is a “precur-sor nanostructure” prior to the normal fer-romagnetic transition for insufficientdoping. Fortunately, a magnetic precursornanostructure (called “magnetic tweed”)recently has been discovered,6,42 and a the-oretical model has been established.6Again, we find a very similar situation asthe other two classes of ferroic materials.

From the previous discussion, we cansee a beautiful, common picture emergingabout the effect of point defects on thethree kinds of ferroic materials, as shownin Figure 7. Below a critical defect concen-tration xc, the normal ferroic transition ismaintained, but in the high-temperaturephase, there is short-range ordering (man-ifesting itself as quasistatic nanostructureor tweed). Above the critical defect con-centration xc, the system crosses over into aferroic glass transition, where the local fer-roic order is frozen below the glass transi-tion temperature Tg; the resultant ferroicglass is a nanostructure of ferroic domains.

Finally, similar nanostructures causedby point defects seem to exist in an even

wider range of transforming materials,including many strongly correlated sys-tems such as high-Tc superconductorsand CMR manganites;7–9 the correspon-ding glass phases are also expected.Indeed, tweed has been reported inyttrium- barium-copper-oxide (YBCO)superconductors, and there exists a critical doping level above which the system shows a tweed-only state withthe absence of a normal tetragonal-orthorhombic transition.7,8 It is highlylikely that this tweed-only state is a strainglass. Therefore, the nanostructures andrelated glass phenomena due to pointdefects are expected to be a common fea-ture for a wide range of transforming sys-tems, including CMR9 and possiblymultiferroic materials. Exploration of thenew physics and novel propertiesinvolved is expected to be a very promis-ing area of materials research.

ConclusionsThe physics of undoped or defect-free

ferroelastic or martensitic materials is, in general, transparent, and all themicrostructural features as well as thephysical properties have been almostfully understood. However, when suchsystems contain point defects ordopants, many puzzling phenomenaappear, such as the well-observed pre-martensitic nanostructures and therecently discovered strain glass—a kindof nanoscale transition. Understandingsuch phenomena is of fundamentalinterest and may lead to novel applica-tions. We have reviewed the recentprogress in this emerging field anddemonstrated that there is a beautifulcommon picture about the effect of pointdefects, which create premartensiticnanostructures below a critical level andstrain glass above the critical level. Thepicture may be common among all fer-roic systems containing point defects,such as relaxor ferroelectrics and ferro-magnets, and similar phenomena mayalso be found in an even wider range oftransforming materials, such as high-Tcsuperconductors and colossal magne-toresistance manganites.

AcknowledgmentsWe thank T. Lookman, D. Sherrington,

T. Suzuki, D. Wang, Y.Z. Wang, Z. Zhang,and Y.M. Zhou for discussions andS. Sarkar for providing some of the micrographs of Figure 2a. X. Ren andY. Wang acknowledge the support fromJSPS of Japan. A. Planes, T. Castán, andP. Lloveras acknowledge the financialsupport from CICyT (Spain), Project No. MAT2007-61200, Marie-Curie RTN

Figure 7. Generic phase diagram of adefect-containing ferroic system showingthe crossover from a ferroic transition toa ferroic glass transition. Tc is the tran -sition temperature of the normal ferroictransition; Tg is the glass transition tem -perature; and xc is the crossover defectconcentration.18 It is noted that this phasediagram can be applied to ferroelastic,ferroelectric, and ferromagneticsystems. x is the defect concentration.

Ferroic glass

Normal parent phase(dynamic disorder)

T

0x (defect concentration)

xc

Unfrozen ferroic glass/Precursory tweed

(quasi-dynamic local order)

Ferroic phase(frozen local order)(long-range order)

Tg

Tc


MULTIMAT (EU), Contract No. MRTN-CT-2004-505226, and DURSI (Catalonia),Project No. 2005SGR00969. A. Saxenaacknowledges the JSPS InvitationalFellowship at Tsukuba, Japan. This workwas supported, in part, by the U.S.Department of Energy.

References1. L.E. Tanner, D. Schryvers, S.M. Shapiro,Mater. Sci. Eng. A 127, 205 (1990).2. Y. Murakami, H. Shibuya, D. Shindo,J. Microsc. 203, 22 (2001).3. T. Castán, E. Vives, L. Mañosa, A. Planes, A. Saxena, in Magnetism and Structure in FunctionalMaterials, A. Planes, L. Mañosa, A. Saxena, Eds.,(Springer Verlag, Berlin, 2005), pp. 27–48.4. X.H. Dai, Z. Xu, D. Viehland, Philos. Mag. A 70, 33 (1994).5. X.H. Dai, Z. Xu, J.F. Li, D. Viehland, Philos.Mag. A 74, 395 (1996).6. A. Saxena, T. Castán, A. Planes, M. Porta, Y. Kishi, T.A. Lograsso, D. Viehland, M. Wuttig,M. De Graef, Phys. Rev. Lett. 92, 197203 (2004).7. W.W. Schmahl, A. Putnis, E. Salje, P.Freeman, A. Graeme-Barber, R. Jones, K.K.Singh, J. Blunt, P.P. Edwards, J. Loram, K. Mirza, Philos. Mag. Lett. 60, 241 (1989).8. Y.W. Xu, M. Suenaga, J. Tafto, R.L. Sabatini,A.R. Moodenbaugh, P. Zolliker, Phys. Rev. B 39,6667 (1989).9. F. Millange, V. Caignaert, B. Domengès, B. Raveau, E. Suard, Chem. Mater. 10, 1974(1998).10. N. Mathur, P. Littlewood, Nat. Mater. 3, 207(2004).

11. K.H. Ahn, T. Lookman, A.R. Bishop, Nature428, 401 (2004).12. E. Dagotto, Science 309, 257 (2005).13. A.R. Bishop, T. Lookman, A. Saxena, S.R. Shenoy, Europhys. Lett. 63, 289 (2003).14. S. Sarkar, X. Ren, K. Otsuka, Phys. Rev. Lett.95, 205702 (2005).15. Y. Wang, X. Ren, K. Otsuka, Phys. Rev. Lett.97, 225703 (2006).16. Y. Wang, X. Ren, K. Otsuka, A. Saxena,Phys. Rev. B 76, 132201 (2007).17. Y. Wang, X. Ren, K. Otsuka, A. Saxena, ActaMater. 56, 2885 (2008).18. X.B. Ren, Y. Wang, Y. Zhou, Z. Zhang, D. Wang, G. Fan, K. Otsuka, T. Suzuki, Y. Ji,J. Zhang, Y. Tian, S. Hou, X. Ding, Philos. Mag.(2009), in press.19. P. Lloveras, T. Castán, M. Porta, A. Planes,A. Saxena, Phys. Rev. Lett. 100, 165707 (2008).20. P. Lloveras, T. Castán, M. Porta, A. Planes,A. Saxena, Phys. Rev. B 80, 054107 (2009).21. S. Kartha, T. Castán, J.A. Krumhansl, J.P. Sethna, Phys. Rev. Lett. 67, 3630 (1991).22. S. Kartha , J.A. Krumhansl, J.P. Sethna, L.K.Wickham, Phys. Rev. B 52, 803 (1995).23. S. Semenovskaya, A.G. Khachaturyan, ActaMater. 45, 4367 (1997).24. S. Semenovskaya, A.G. Khachaturyan,J. Appl. Phys. 83, 5125 (1998).25. E.K.H. Salje, Phase Transitions in Ferroelasticand Co-Elastic Crystals (Cambridge UniversityPress, Cambridge, 1990).26. K. Otsuka, C.M. Wayman, Eds., ShapeMemory Materials (Cambridge University Press,Cambridge, 1998).27. A. Planes, L. Mañosa, Solid State Phys. 55,159 (2001).

28. A. Saxena, T. Lookman, in Handbook ofMaterials Modeling, S. Yip, Ed. (Springer-Verlag,2005), pp. 2143–2154.29. S.R. Shenoy, T. Lookman, A. Saxena, A.R.Bishop, Phys. Rev. B 60, R12537 (1999); K.O.Rasmussen, T. Lookman, A. Saxena, A.R.Bishop, R.C. Albers, S.R. Shenoy, Phys. Rev. Lett.87, 055704 (2001).30. S.M. Shapiro, B.X. Yang, Y. Noda, L.E.Tanner, D. Schryvers, Phys. Rev. B 44, 9301(1991).31. W. Cai, Y. Murakami, K. Otsuka, Mater. Sci.Eng. A 275, 186 (1999).32. T. Kakeshita, T. Fukuda, H. Tetsukawa, T. Saburi, K. Kindo, T. Takeuchi, M. Honda, S. Endo, T. Taniguchi, Y. Miyako, Jpn. J. Appl.Phys. 37, 2535 (1998).33. K. Otsuka, X. Ren, Progr. Mater. Sci. 50, 511(2005).34. M.S. Choi, T. Fukuda, T. Kakeshita, H. Mori, Philos. Mag. 86, 67 (2006).35. W. Petry, J. Phys. IV 5, C2–15 (1995).36. J.S. Zhang, PhD thesis, University ofTsukuba, 2000.37. Y. Wang, PhD thesis, Xi’an JiaotongUniversity, 2008.38. V.K. Wadhawan, Introduction to FerroicMaterials (Gordon and Breach, Armsterdam, 2000).39. D. Viehland, S.J. Jang, L.E. Cross, M.Wuttig, Phys. Rev. B 46, 8003 (1992).40. J.A. Mydosh, Spin Glasses (Taylor & Francis,London, 1993).41. G. Burns, F.H. Dacol, Phys. Rev. B 28, 2527(1983).42. Y. Murakami, D. Shindo, K. Oikawa, R. Kainuma, K. Ishida, Acta Mater. 50, 2173 (2002). ■■

ferroelastic nanostructures and nanoscale transitions ... xiaobing/papers... · symmetry breaking...

Documents