Lattice instability in supersaturated solid solutions [Elektronische Ressource] / vorgelegt von Fanni Jurányi
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LATTICE INSTABILITYIN SUPERSATURATED SOLID SOLUTIONSvon der Fakult at fur Naturwissenschaften der Technischen Universit at Chemnitzgenehmigte Dissertation zur Erlangung des akademischen Gradesdoktor rerum naturalium(Dr. rer. nat.)vorgelegt von Dipl.-Phys. Fanni Jur anyigeboren am 26.08.1974 in Baja (Ungarn)eingereicht am 11.07.2003Gutachter: Prof. Dr. Jens-Boie SuckProf. Dr. Walter HoyerProf. Dr. Ulrich HerrTag der Vereteidigung: 10.12.2003http://archiv.tu-chemnitz.de/pub/2Bibliogra sc he BeschreibungThemaFanni Jur anyi: Lattice instability in supersaturated solid solutionsTechnische Universit at Chemnitz, Fakult at fur NaturwissenschaftenDissertation 2003; 78 Seiten; 57 Abbildungen; 5 Tabellen; 96 Literaturquellen.KurzfassungDer Ein u von Unordnung ist ein wichtiges Forschungsgebiet der Materialwissenschaft.Bei ub ers attigten festen L osungen handelt es sich um einen metastabilen Zustand, bei demdas Wirtsgitter gezwungen ist, mehr Fremdatome aufzunehmen, als termodynamisch sta-bil ist. Die Fremdatome im Wirtsgitter sind Punktdefekte, deren Menge mit der Konzen-tration variiert werden kann.Die hier untersuchten bin aren Legierungen wurden mit einer Kugelmuhle hergestellt.Im Gegensatz zu anderen Pr aparationsmethoden wie z.B. erm oglicht es die Methodedes Kugelmahlens, eine fur inelastische Neutronenstreuexperimente ausreichende Proben-menge herzustellen, bei der die Proben makroskopisch homogen und isotrop sind.
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LATTICE INSTABILITY IN SUPERSATURATED SOLID SOLUTIONS
vonderFakultatfurNaturwissenschaftenderTechnischenUniversitatChemnitz genehmigte Dissertation zur Erlangung des akademischen Grades
doktor rerum naturalium
(Dr. rer. nat.)
vorgelegtvonDipl.-Phys.FanniJuranyi
geboren am 26.08.1974 in Baja (Ungarn)
eingereicht am 11.07.2003
Gutachter: Prof. Dr. Jens-Boie Suck Prof. Dr. Walter Hoyer Prof. Dr. Ulrich Herr
Tag der Vereteidigung: 10.12.2003 http://archiv.tu-chemnitz.de/pub/
2
Bibliograsc he Beschreibung
Thema FanniJuranyi:Latticeinstabilityinsupersaturatedsolidsolutions TechnischeUniversitatChemnitz,FakultatfurNaturwissenschaften Dissertation 2003; 78 Seiten; 57 Abbildungen; 5 Tabellen; 96 Literaturquellen.
Kurzfassung Der Ein
u von Unordnung ist ein wichtiges Forschungsgebiet der Materialwissenschaft. Bei ubersattigten festen Losungen handelt es sich um einen metastabilen Zustand, bei dem das Wirtsgitter gezwungen ist, mehr Fremdatome aufzunehmen, als termodynamisch sta-bil ist. Die Fremdatome im Wirtsgitter sind Punktdefekte, deren Menge mit der Konzen-tration variiert werden kann. DiehieruntersuchtenbinarenLegierungenwurdenmiteinerKugelmuhlehergestellt. Im Gegensatz zu anderen Praparationsmethoden wie z.B. ermoglicht es die Methode des Kugelmahlens, eine fur inelastische Neutronenstreuexperimente ausreichende Proben-menge herzustellen, bei der die Proben makroskopisch homogen und isotrop sind. Die Zunahme substitutionaler Defekte bewirkt eine Gitterinstabilitat gegenuber Scher-kraftenundanschliessendmoglicherweiseeinenOrdnungs-Unordnungs-Ubergang. In dieser Arbeit wurden vor allem dynamische Aspekte betrachtet: die Gitterdynamik wurde auf atomarer Ebene mittels inelastischer Neutronenstreuung untersucht. Die da-raus erhaltene verallgemeinerte Schwingungszustandsdichte beschreibt die Einzelteilchen-Dynamik, der dynamische Strukturfaktor erhalt daruberhinaus Informationen uber die kollektive Bewegung. Desweiteren wurde die Dynamik ubersattigter Losungen mit der eines Kristalls im termodynamisch stabilen Zustand und dem eines Glases, das keine langreichweitige Ordnung besitzt, verglichen. Zwei Systeme wurden untersucht: Zr100 xAlx(als stabiler Kristall, als ubersattigte festeLosungundalsGlas)sowieCu100 xFexattebsreftsgiet(u-edeihcsngsuoeLertvmien nen Konzentrationen an Fe-Atomen). Bei beiden Systemen konnte ein Weichwerden (softening) von Phononen beobachtet werden, insbesondere von transversalen Moden, dieeinendirektenNachweisderobenerwahntenGitterinstabilitatergeben.Dievon derverallgemeinertenZustandsdichteausberechnetenthermodynamischenGroen,wie SpezischeWarmeundDebye-TemperaturzeigenqualitativeUbereinstimmungmitden Werten,diemanausmakroskopischenMessungenerhalt.DieerhohteMengeanDe-fekten verursacht weiterhin eine lineare Skalierung der Schwingungszustandsdichte. Die dadurchentstehendemikroskopischeInhomogenitatzeigtsichalsVerbreiterungderSpek-tren. Allgemeine Merkmale von ungeordnete Materie wie der sogenannte Boson-Peak oder dietennkonnbeobichtteewcatha,cudrneeR-noitaxalichnfhtdeurliutuhcsrebttaetgi Kristalle.
Schlagworte Gitterinstabilitat,ubersattigtefestenLosungen,Ordnungs-UnordnungsUbergang,We-ichwerden (softening) der Phononen, unelastische Neutronenstreuung, Neutronen Flugzeit-Spektrometrie, binare Legierungen
Contents 1 Introduction 5 2 Theoretical background 7 2.1 Lattice instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Thermodynamical considerations . . . . . . . . . . . . . . . . . . . 7 2.1.2 Mechanical considerations . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Vibrations in disordered systems . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Small amount of substitutional defects . . . . . . . . . . . . . . . . 10 2.2.2 Amorphous solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Experimental techniques 16 3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 X-Ray di raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Di eren tial Scanning Calorimetry . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Vibrating sample magnetometer . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5Inelastic neutron scattering 19. . . . . . . . . . . . . . .. . . . . . . . . 3.6 Neutron di raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Sample characterization 25 4.1 Zr100 xAlx. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 25 4.2 Cu100 xFex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Results 39 5.1 The order-disorder transition (Zr100 xAlx) .. . . . . . . . . . . . . . . . . 39 5.1.1 Generalized vibrational density of states . . . . . . . . . . . . . . . 39 5.1.2 Dynamic structure factor . . . . . . . . . . . . . . . . . . . . . . . . 44 5.1.3 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 The supersaturated state (Cu100 xFex) . . . . . . . . . . . . . . . . . . . . 49 5.2.1 Generalized vibrational density of states . . . . . . . . . . . . . . . 49 5.2.2 Dynamic structure factor . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.3 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . 56 6 Discussion 57 6.1 Small amount of substitutional defects . . . . . . . . . . . . . . . . . . . . 57 6.2 Amorphous solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Predicted lattice instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3
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CONTENTS
7 Summary 60 A Details of a developed software 62 A.1 Data transformation from , EtoQ, Espace . . . . . . . . . . . . . . . . 62 A.2 E ect of smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.3 Dispersion: Maxima of S(Q,E) . . . . . . . . . . . . . . . . . . . . . . . . . 66 B Q dependent intensity of transversal and longitudinal modes 67 C Transmission of the Zr100 xAlxsamples 68 Bibliography 68 Danksagung 74 EidesstattlicheErklarung75 Lebenslauf 76 Vero entlichungenundVortrage77
Chapter 1
Introduction
Most of the microscopic and the macroscopic properties of ideal crystals are nowadays well understood and even predictable. A certain degree of disorder leads to di eren t physical properties. Explanation and improvement of these unusual properties are the maintasksofmaterialsresearch.Whilesmallamountofdi erentkindofdefectscanbe well described by perturbation theory, there is no model of disordered condensed matter which is valid in general [61]. In this work a material class is investigated, in which the degree of disorder can be tuned. Under certain conditions, the solubility of guest atoms in a host lattice can be ex-tended beyond the thermodynamically stable limit, resulting in a metastable state, called supersaturated solid solution. They can be prepared by di eren t methods like irradiation, rapid quenching or mechanical attrition [37]. The increasing number of substitutional defectsisaccompaniedbyalatticeinstability,andnallymaycauseanorder-disorder transition (ODT). Theaimofthisworkistoinvestigatethein
uenceofthisdestabilizationofthelattice on the lattice vibrations and to try to get this way some insight into the microscopic mechanism of this order-disorder transition by means of neutron scattering. Furthermore, the dynamics of an unstable crystal is compared to that of a crystal with small disorder and to that of a glass as representative for a completely disordered material. The changes inthedynamicsthroughtheorder-disordertransitionhavebeenfollowedforthersttime on an atomic level. The single particle and the collective atomic dynamics is described by the vibrational density of states and the dynamic structure factor, respectively. Both are obtained from inelastic neutron scattering experiments. The change in the dynamics reveals features in addition to those expected from introducing a small amount of point defects.Accordingtotheorytheresistivityofthelatticeagainstshear
uctuationsis remarkably reduced approaching the critical concentration, where the transition occurs. This is conrmed by the observation of softening of long wavelength transverse surface modes [63] and of the transverse phonons [9]. The characteristic properties of the dynamics of glasses are not observed in the crystalline samples. Experimental evidence for this kind of order-disorder transition was provided by Meng et al.[53]. They partly amorphysed Zr3 mi- TEMAl sample by hydrogen absorption. crographs show clear evidence of homogeneous amorphization of whole grains in contrast to the heterogeneous nucleation of the amorphous phase at higher production temper-ature. Linkeret al. Theystudied the amorphization of Nb by B implantation [49]. observedalinearincreaseofthestrainversusBconcentration,andnallydrasticstrain
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CHAPTER 1. INTRODUCTION
release by amorphization. A reversible crystal-to-glass transition could be achieved also by applying pressure [25, 76]. Ma and Atzmon has measured the enthalpy-composition curves for supersaturated and amorphous Zr100 xAlxsystems. The maxima at about the critical composition, where the amorphous phase forms, together with the continuous de-crease of the lattice parameters supported, that the phase formation by ball milling can be determined by polymorphous constrains, rather than by nucleation and growth under metastable equilibrium [52]. Egami and Waseda proposed a model based on the atomic level stresses in a solid solution, employing an elastic continuum approximation, which givesastabilitylimitingoodagreementwithanempiricalsizedi erencecriteria[16]. The ODT was successfully simulated by molecular dynamics calculations [40, 91]. It was conrmed that amorphization occurs only when the atomic size di erence is sucien tly large. The stability limit of a supersaturated solid solution furthermore depends on the structural similarities of the host lattice and the intermetallic compound [27]. The ODT isaccompaniedandmostlikelycausedbyareducedresistivityagainstshear
uctuations in the sample [3, 40]. This theory agrees with results of indirect, macroscopic experiments. The sound ve-locity - which is related to the Debye temperature, and therefore also to the shear mod-ulus - in irradiated Zr3Al measured by Brillouin scattering drops to about 50% at the transition[63].EttlandSamwer[20]haveobtainedfromlowtemperaturespecicheat measurements a distinct decrease of the Debye temperature at the critical concentration as predicted by theory. A. Caroet al.report the simulation of the density of states of Al99Si1solid solution [7]. They observe a softening of especially the transverse acoustic peak in comparison to pure Al. The presence of a Si atom destroys the symmetrical neighborhood of Al atoms, and therefore causes asymmetric changes in the restoring forces. Vacancies in Al with the same concentration show a similar, but more pronounced behavior. Such a behavior of the density of states was observed also experimentally in the same system, studied by inelastic neutron scattering [10, 75]. This solid solution is demixing too fast to allow to approach the OTD at about 21 at. % Si. Here Zr100 xAlxalloys are investigated, which is one of the few systems, in which the metastable amorphous state can be reached and persists at room temperature. This system was well suited for the study of the transition, but not for the detailed investi-gation of the supersaturated region before the transition, because ideal samples, which di er just in the concentration, could not be produced. Therefore also the Cu100 xFex system was investigated. The atomic size di erence between the Cu and Fe atoms is quite small, therefore a supersaturated solid solution is formed over a wide concentration range. According to theory, the softening of phonons (their shift toward lower energy) is very small except close to the critical concentration. Therefore in case of the CuFe system was necessary to produce samples with similar properties, like crystal size, defects (except thosecausedbythesupersaturation)andstresses,whichcanalsoin
uencethedynamics atlowenergies,wherethemaine ectsareobserved[30].
Chapter 2
Theoretical background
2.1 Lattice instability One can distinguish two types of melting [91]: Upon heating athermodynamicalof a crystal takes place at temperatures,melting where the free energy of the crystal and that of the liquid are equal. The melt nucleatesheterogeneouslyat surfaces or defects, and the liquid-solid interface propagatesintothecrystalwithanitevelocity. Intheabsenceofnucleationsitesorsucientatomicmobilitythecrystalwillbe superheated up to the temperature, where it reaches a critical volume by thermal expansion, which causes it to meltmechanically Itdue to an elastic instability. occurshomogeneouslyin the whole crystal on the time scale corresponding to the propagation of lattice vibrations, which is considerable shorter than the propagation time of the solid-liquid interface in thermodynamical melting. There exist also two kinds of amorphization, analog to the two types of melting. Here the disordered phase after the ODT is not a liquid, but a solid state, which is reached under certain constraints. In the following only the amorphization caused by supersaturation of binary solutions will be discussed. In this case instead of the volume the concentration of the foreign atoms (supersaturation) can be used as controlling parameter. This can be measured by X-ray di raction through the lattice parameter.
2.1.1 Thermodynamical considerations Supersaturated solid solutions are metastable crystalline phases, where atoms of di eren t elementscanoccupylatticepositionsoveraniteconcentrationregion,outsidethesolu-bility region in thermodynamical equilibrium. Consequently the system is characterized by the polymorphous phase diagram, which describes the metastable equilibriums of a crystalline solid solution, and the corresponding liquid and glass at a certain temperature and pressure [22]. The equilibrium condition at low temperatures, i.e. considering only the crystalline phase (X) and the glass (G) is dG= SdT+V dp+GdnG+XdnX= 0.(2.1)
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CHAPTER 2. THEORETICAL BACKGROUND
G: Gibbs free energy; S: entropy; T: temperature; p: pressure;=n∂G∂: chemical potential; n: amount of a phase. At constant temperature and pressure (dT=0;dp=0) Eq. 2.2 has to be solved taking into account, thatdnX= dnG.
∂G ∂∂GXdnX+n∂GGdnG= 0. nX
(2.2)
HereGG( pT ,) andGX(T , p) are the molar Gibbs free energies for the glass and the crystalline phase, respectively. This approach leads to the tangent method, which is sketched in Fig. 1 [23]. It shows the free energy diagram of a hypothetical system with two crystalline phases and one amorphous phase in between. At an intermediate composition both the amorphous and the crystalline phase are present. The total free energy will be minimized by the ratio of these phases, and will follow the straight line between the concentrationsc0Bandc00Bwith increasing concentrationcB. Considering partitionless melting (without changing the local compositions) of a binary system the following non-equilibrium condition has to be fullled.
GG(T , p, cG=cB) =GX( p, cT ,X=cB),
(2.3)
wherecBis the critical concentration where the ODT takes place. This applies to the order-disorder transi-tion in solid solutions, where the nucleation of a second crystalline phase is kinetically prohibited (no phase separation is allowed). According to the polymorphous condition, the system will undergo an amorphization at the critical concentration,cB. Before reaching this transition the system forms a metastable supersaturated solid solution. Its free energy follows the curve between c0BandcB. After the amorphization its free energy changes betweencBandc0B0 along the curve, which belongs to the amor-phous phase. The equilibrium between the di erentphasesisshownintheconstant pressure phase diagram in Fig. 2. as a func-tion of temperature.
Fig. 1: Free energy diagram of a hypothetic system [23]
2.1. LATTICE INSTABILITY
Fig. 2: Polymorphous phase diagram of a Fig. 3: Phase diagram of a vacancy enriched binary system. [22] system. [24]
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In the extreme case, where the foreign ”atoms” are vacancies, a similar diagram was obtained[24]whichisshowninFig.3.Introductionofvacancies,asinnitelysmall atoms leads to a volume expansion, and at a critical volume, i.e. a temperature dependent concentration to a glass transition.
2.1.2 Mechanical considerations
Mechanical instability of a crystal was proposed rst by Lindemann [48]: a crystal becomes unstable if the amplitude of the thermally induced atomic displacements reach a critical fraction of the typical distance be-tween the atoms. Krillet al.showed, that in case of the crystal-to-amorphous transi-tion the static displacement should be con-sidered instead of the total one [46]. Born suggested, that at the critical point one of the independent shear moduli of the crystal vanishes [3]. More generalized arguments in this direction by Tallon [80] agree with ex-perience. Fig. 4 shows, that the shear mod-Fig. 4: Dependence of the shear modulus onulus is extrapolated to zero at the volume the relative volume [60]hT..taelmelto0fKtahtVi0ohsshtswlereheftyscr-sihtmuoeveloationshipbe-tween melting and amorphization, and be-tween superheating and supersaturation of a crystal. An overview can be found in [38, 66]. The order-disorder transition of a binary solution was successfully simulated [40]. Fig. 5 shows the obtained structure depending on the concentration and atomic size ratio. Amorphization occurs only if the size ratio of the atoms is sucien tly large. The critical size di erence depends on the chemical properties of the atoms, which can be described by the electronegativity and electron-to-atom (e/a The solution is the more) ratio [4, 69].
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