Excerpt
Contents
1 Abstract
2 Objectives
3 Fundamentals
3.1 Structure of metallic glass
3.2 Glass formation and crystallization
3.2.1 Glass transition
3.2.2 Classical nucleation theory
3.2.3 Crystal growth in undercooled liquids
3.2.4 Isothermal and isochronal devitrification of metallic glasses
3.2.5 Fragility concept
3.2.6 Glass-forming ability
3.3 Mechanical properties
3.3.1 Deformation of bulk metallic glass
3.3.2 Measures to affect the plasticity of bulk metallic glass
3.4 Cu-Zr-Al-based alloys
3.5 Flash-annealing
4 Experimental
4.1 Sample preparation
4.2 X-ray diffraction
4.3 Calorimetry
4.4 Flash-annealing
4.5 Electro-static levitation
4.6 Microscopy
4.7 Mechanical testing
4.8 Optical profilometry
5 Results and Discussion
5.1 Development of the flash-annealing device
5.1.1 Description of the temperature-time heating curve
5.1.2 Inductive heating
5.1.3 Estimation of the cooling rate
5.1.4 Surface oxidation of Cu-Zr-Al-based BMGs during flash-annealing
5.2 Characterization of the as-cast bulk metallic glasses
5.2.1 Development of the Cu44Zr44Al8Hf2Co2 alloy
5.2.2 Amorphicity of the Cu46Zr46Al8 and Cu44Zr44Al8Hf2Co2 BMGs
5.2.3 Mechanical behaviour of as-cast Cu44Zr44Al8Hf2Co2 BMGs . .
5.3 Structural changes during flash-annealing below Tx
5.3.1 Calorimetric investigation of structural changes
5.3.2 Mechanical properties of flash-annealed bulk metallic glass
5.4 Crystallization of BMGs during flash-annealing
5.4.1 Phase formation of Cu-Zr-Al-based BMGs
5.4.2 Crystallization kinetics of Cu-Zr-Al-based BMGs
5.4.3 Mechanical behaviour of Cu44Zr44Al8Hf2Co2 BMG composites
6 Summary
7 Outlook
8 Appendix
8.1 Chemical analysis
8.2 Device development
8.2.1 Temperature control
8.2.2 Inductive heating - derivation of the skin depth
8.2.3 Estimation of the cooling rate
8.3 Calorimetric analysis
8.3.1 Isochronal transformation kinetics (Kissinger)
8.3.2 Isothermal transformation kinetics (Johnson-Mehl-Avrami-Kolmogorov)
8.3.3 Fragility
8.4 Uniformity of partially crystallized BMG
8.5 Crystal growth rate
8.6 Stereology
Bibliography
1 Abstract
(Bulk) metallic glasses ((B)MGs) are known to exhibit the highest yield strength of any metallic material (up to 5 GPa), and show an elastic strain at ambient conditions, which is about ten times larger than that of crystalline materials. Despite these intriguing mechanical properties, BMGs are not used as structural materials in service, so far. The major obstacle is their inherent brittleness, which results from severe strain localization in so-called shear bands. MGs fail due to formation and propagation of shear bands. A very effective way to attenuate the brittle behaviour is to incorporate crystals into the glass. The resulting BMG composites exhibit high strength as well as plasticity. Cu-Zr-Al-based BMG composites are special to that effect, since they combine high strength, plasticity and work-hardening. They are comprised of the glass and shape-memory B2CuZr crystals, which can undergo a deformation-induced martensitic transformation. The work-hardening originates from the martensitic transformation and overcompensates the work-softening of the glass. The extent of the plasticity of BMG composites depends on the volume fraction, size and particularly on the distribution of the B2 CuZr crystals. Nowadays, it is very difficult, if not impossible to prepare BMG composites with uniformly distributed crystals in a reproducible manner by melt-quenching, which is the standard preparation method. Flash-annealing of BMGs represents a new approach to overcome this deficiency in the preparation of BMG composites and is the topic of the current thesis.
Cu46Zr46Al8 and Cu44Zr44Al8Hf2Co2 BMGs were flash-annealed and afterwards invest- igated in terms of phase formation, crystallization kinetics and mechanical properties. Flash-annealing is a process, which is characterized by the rapid heating of BMGs to predefined temperatures followed by instantaneous quenching. A temperature-controlled device was succesfully developed and built. The Cu-Zr-Al-based BMGs can be heated at rates ranging between 16 K/s and about 200 K/s to temperatues above their melting point. Rapid heating is followed by immediate quenching where cooling rates of the order of 10[3] K/s are achieved.
As a BMG is flash-annealed, it passes the glass-transition temperature, Tg, and transforms to a supercooled liquid. Further heating leads to its crystallization and the respective temperature, the crystallization temperature, Tx, divides the flash-annealing of BMGs into two regimes:
(1) sub-Tx-annealing and (2) crystallization.
The structure of the glass exhibits free volume enhanced regions (FERs) and quenched-in nuclei. Flash-annealing affects both heterogeneities and hence the structural state of the glass. FERs appear to be small nanoscale regions and they can serve as initiation sites for shear bands. Flash-annealing of Cu-Zr-Al-based BMGs to temperatures below Tg leads to structural relaxation, the annihilation of FERs and the BMG embrittles. In contrast, the BMG rejuvenates, when flash-annealed to temperatures of the supercooled liquid region (SLR). Rejuvenation is associated with the creation of FERs. Compared to the as-cast state, rejuvenated BMGs show an improved plasticity, due to a proliferation of shear bands, which are the carrier of plasticity in MGs. Flash-annealing enables to probe the influence of the free volume in bulk samples on their mechanical properties, which could not be studied, yet.
In addition, B2CuZr nanocrystals precipitate during the deformation of flash-annealed Cu44Zr44Al8Hf2Co2 BMGs. Deformation-induced nanocrystallization does not occur for the present as-cast BMGs. Flash-annealing appears to stimulate the growth of quenched-in nuclei, which are subcritical in size and can also dissolve, once the BMG is heated to temperatures in the SLR. Rejuvenation represents a disordering process, whereas the growth of quenched-in nuclei is associated with ordering. There is a competition between both processes during flash-annealing. The ordering seems to lead to a “B2-like” clustering of the medium range of Cu44Zr44Al8Hf2Co2 BMGs with increasing heating duration. So far, there does not exist another method to manipulate the MRO of BMGs.
If Cu44Zr44Al8Hf2Co2 BMGs are flash-annealed to temperatures near Tx, most likely compressive resiudal stresses develop near the surface, which is cooled faster than the interior of the BMG specimen. They hinder the propagation of shear bands and increase the plasticity of flash-annealed BMGs in addition to rejuvenation and deformation-induced nanocrystallization.
If BMGs are heated to temperatures above Tx, they start to crystallize. Depending on the exact temperature to which the BMG is flash-annealed and subsequently quenched, one can induce controlled partial crystallization. Consequently, BMG composites can be prepared. Both Cu-Zr-Al-based BMGs are flash-annealed at various heating rates to study the phase formation as a function of the heating rate. In addition, Tg and Tx are identified for each heating rate, so that a continuous heating transformation diagram is constructed for both glass-forming compositions. An increasing heating rate kinetically constrains the crystallization process, which changes from eutectic (Cu10Zr7 and CuZr2) to polymorphic (B2CuZr). If the Cu-Zr-Al-based BMGs are heated above a critical heating rate, exclusively B2 CuZr crystals precipitate, which are metastable at these temperatures. Thus, flash-annealing of Cu46Zr46Al8 and Cu44Zr44Al8Hf2Co2 BMGs followed by quenching enables the preparation of B2 CuZr BMG composites. The B2 precipitates are small, high in number and uniformly distributed when compared to conventional BMG composites prepared by melt-quenching.
Such composite microstructures allow the direct observation of crystal sizes and numbers, so that crystallization kinetics of deeply supercooled liquids can be studied as they are flash-annealed. The nucleation kinetics of devitrified metallic glass significantly diverge from the steady-state and at high heating rates above 90K/s transient nucleation effects become evident. This transient nucleation phenomenon is studied experimentally for the first time in the current thesis.
Once supercritical nuclei are present, they begin to grow. The crystallization temperature, which depends on the heating rate, determines the crystal growth rate. At a later stage of crystallization a thermal front traverses the BMG specimen. In levitation experiments, this thermal front is taken as the solid-liquid interface and its velocity as the steady-state crystal growth rate. However, the thermal front observed during flash-annealing, propagates through the specimen about a magnitude faster than is known from solidification experiments of levitated supercooled liquids. As microstructural investigations show, crystals are present in the whole specimen, that means far ahead of the thermal front. Therefore, it does not represent the solid-liquid interface and results from the collective growth of crystals in confined volumes. This phenomenon originates from the high density of crystals and becomes evident during the heating of metallic glass. It could be only observed for the first time in the current thesis due to the high temporal resolution of the high-speed camera used.
The heating rate and temperature to which the BMG is flash-annealed determine the nucleation rate and the time for growth, respectively. The size and number of B2 CuZr crystals can be deliberately varied. Thus mechanical properties of B2 CuZr BMG composites can be studied as a function of the volume fraction and average distance of B2 particles. Cu44Zr44Al8Hf2Co2 BMG specimens were flash-annealed at a lower and higher heating rate (35 K/s and 180 K/s) to different temperatures above Tx and subsequently subjected to uniaxial compression. BMG composites prepared at higher temperatures show a lower yield strength and larger plastic strain due to the higher crystalline volume fraction. They not only exhibit plasticity in uniaxial compression, but also ductility in tension as a preliminary experiment demonstrates. Furthermore, nanocrystals precipitate in the amorphous matrix of BMG composites during deformation. They grow deformation-induced from quenched-in nuclei, which are stimulated during flash-annealing.
In essence, flash-annealing of BMGs is capable of giving insight into most fundamental scientific questions. It provides a deeper understanding of how annealing affects the structural state of metallic glasses. The number and size of structural heterogeneities can be adjusted to prepare BMGs with improved plasticity. Furthermore, crystallization kinetics of liquids can be studied as they are rapidly heated. Transient nucleation effects arise during rapid heating of BMGs and they cannot be described using the steady-state nucleation rate. Therefore, an effective nucleation rate was introduced. Besides, the flash-annealing process rises the application potential of BMGs. The microstructure of BMG composites comprised of uniformly distributed crystals and the glass, can be reliably tailored. Thus, flash-annealing constitutes a novel method to design the mechanical properties of BMG composites in a reproducible manner for the first time. BMG composites, which exhibit high strength, large plasticitiy and as in the case of B2 CuZr BMG composites as well work-hardening behaviour, can be prepared, so that the intrinsic brittleness of monolithic BMGs is effectively overcome.
2 Objectives
(Bulk) metallic glasses ((B)MGs) offer appealing mechanical properties due to their unique structure. In the absence of long-range ordered structures, for instance grain boundaries and dislocations being typical of crystalline alloys, metallic glasses exhibit room-temperature strength much closer to the theoretical limit of the material than their crystalline counter- parts [1, 2]. This outstanding strength combined with high hardness, large elastic strain and high fracture toughness are attractive characteristics for structural materials [2, 3, 4, 5, 6]. Mainly due to their brittle behaviour at room-temperature and limited sample dimensions, they have not replaced traditional materials in service so far. The specific deformation mechanism, which is based on shear band formation and propagation, is responsible for the brittle behaviour of metallic glasses [7, 8, 9]. Many efforts have been made to overcome this “vulnerability” in order to control the evolution and propagation of dominant shear bands, which quickly lead to failure.
Over the past years, scientists have proposed several concepts with the aim to increase plasticity or ultimately ductility in tension and toughness while maintaining the high strength of BMGs [9]. These manifold concepts range from changing the sample shape and size [10, 11, 12, 13] over mechanical [14, 15, 16, 17, 18, 19, 20, 21] and/ or surface [22, 23, 24, 25] pre-treatment and alloy modification [26, 27, 28, 29, 30] to the preparation of composites [31, 32, 33, 34, 35]. Particularly, the preparation of BMG composites, which consist of one or several crystalline phases embedded in the glass, is a promising approach. Cu50Zr50-based BMG composites are of high interest in this respect [33, 34, 36, 37, 38, 39, 40]: They contain B2 CuZr shape-memory crystals, which not only exhibit a pronounced plastic strain but also work-hardening due to a deformation-induced martensitic transformation [37, 39, 41]. Ductile crystals impede the propagating shear bands and ensure their proliferation. In order to optimize the mechanical properties of BMG composites, small and uniformly distributed precipitates are necessary [40, 39]. During deformation, they lead to a higher population of shear bands being equivalent to a less localized deformation and greater plasticity is achieved [42].
So far, it is very challenging to prepare uniform BMG composites. Casting technologies using copper-moulds, which enable high cooling rates, represent the standard method to produce BMG composites, nowadays. During quenching of the melt, crystals nucleate and grow from the supercooled liquid and the residual liquid vitrifies when the glass-transition temperature is passed. Adjusting a uniform distribution of crystals in the glass is extremely difficult, since the heat is extracted from the outside of the liquid contacting the mould to its interior. The resulting non-uniform temperature field within the supercooled liquid upon quenching makes it difficult to obtain uniform BMG composites. Furthermore, the melt does not wet the whole mould at once, but flows gradually into it. In addition to that, high crystal growth rates and low nucleation rates are accessed during cooling from the melt. Thus, often only a few and relatively big crystals with a size of several 100 µm in diameter crystallize within the supercooled liquid, before it vitrifies. However, a uniform distribution of many small crystals is vital to exploit the full potential of BMG composites.
Heating metallic glasses above their crystallization temperature leads to their devitrifica- tion involving nucleation and crystal growth. Compared to solidification, both processes occur at an extreme undercooling leading to reduced growth rates and higher nucleation rates [43, 44]. Thus partial devitrification followed by quenching in order to freeze-in the obtained microstructure, represents a new and promising strategy to prepare uniform BMG composites with small crystals.
The first aim of the present work is to design, built and test a device being capable of rapidly heating and subsequently immediately quenching BMG specimens in a uniform and reproducible manner. The heating and ejection process of the BMG specimen must be temperature-controlled, so that the heating rate and temperature to which the BMG is heated to, can be varied.
BMG composites consisting of B2 CuZr particles embedded in the glass shall be prepared.
The occurrence of the shape-memory B2 CuZr phase and a good glass-forming ability limit the selection of the alloy system to Cu-Zr-Al. The high-temperature B2 CuZr phase is metastable at room temperature [45], yet rapid cooling of Cu-Zr-Al-based alloys enables to cast B2 CuZr BMG composites [37, 40, 41, 46, 36, 39]. One could speculate that the B2 CuZr phase forms as well during the devitrification of Cu-Zr-Al-based metallic glass at high heating rates. Indeed, a recent work yields hope [47] rendering the preparation of B2 CuZr BMG composites by devitrification possible. Therefore, a rapid heating or flash-annealing device as it is termed here, shall be developed first. Cu-Zr-Al-based BMGs are flash- annealed at different heating rates to temperatures above the crystallization temperature and the influence of the heating rate on the phase formation shall be unravelled. The exact ejection temperature of the BMG specimen determines the volume fraction of the crystalline phase(s) and the crystal size distribution of the so-obtained composite. The immediate subsequent quenching at a high cooling rate is crucial here, since the partially crystallized supercooled liquid must be frozen in. The mechanical properties of the BMG composites and deformation mechanisms involved are studied. Moreover, the microstructure of these BMG composites allows to gain knowledge about crystallization kinetics of highly supercooled liquids during rapid heating.
Theoretical studies have suggested that the devitrification of metallic glass at high heating rates leads to transient effects which reduce the nucleation rate [48]. The flash-annealing setup should enable to investigate whether non-steady state nucleation rates are present and whether high heating rates also affect the crystal growth rates of the devitrifying metallic glass.
The last aspect on crystallization kinetics exemplifies quite well why metallic glasses have been in the focus of scientists for almost six decades: Besides their application potential already mentioned, metallic glasses are also capable of giving insight into understanding most fundamental scientific questions. Only recently, the structure of metallic glasses is in strong focus of scientific research. Many works indicate the presence of less dense regions on a nanoscale [49, 50, 51]. These regions are enhanced in free volume and their presence seems to improve the plasticity of BMGs [52]. Cu-Zr-Al-based MGs are special to that effect, since in addition quenched-in nuclei appear to be present [34, 53]. Free volume enhanced regions (FERs) and quenched-in nuclei constitute structural heterogeneities of MGs.
Another aim of the present work is to investigate whether and how the structural state of Cu-Zr-Al-based MGs changes during flash-annealing and subsequent compression testing. As MGs are heated, they embrittle first, owing to structural relaxation being equivalent to the annihilation of FERs. If temperatures of the supercooled liquid region are accessed, the supercooled liquid is present and rejuvenation should occur. It shall be investigated whether the applied quenching rate is sufficient to prepare rejuvenated bulk samples. They should exhibit larger plasticity [52]. Rejuvenation is defined as the creation of free volume and hence increases the disorder of MGs. In contrast, quenched-in nuclei are ordered regions. Do these nuclei, which are subcritical in size, grow during flash-annealing or do they dissolve? This is one question, which shall be addressed in the course of this thesis.
Both competing processes, the growth of quenched-in nuclei (ordering) and rejuvenation (disordering) represent antagonists and it will be investigated whether one process is favoured with increasing heating duration. Furthermore, the mechanical properties of BMGs having different structural states shall be analyzed. Not only the presence of FERs, but also quenched-in nuclei, which could be stimulated during flash-annealing, should affect the plasticity of BMGs.
3 Fundamentals
Metallic glass can be prepared by many different processing routes like amorphization via irradiation, vapour deposition, solid state amorphization or, most commonly applied, rapid quenching of the liquid [54, 55]. Heat is extracted rapidly from the melt to inhibit crystallization. Due to the liquid-like structure which is termed “amorphous”, this state of matter exhibits some unique mechanical, corrosive and electrical properties for instance [1, 2, 4].
Metallic glasses first attracted significant attention after Klement and Duwez [56] obtained an amorphous Au-Si alloy by rapid quenching of the melt in 1960. After that, the systematic search for melt-quenched metallic glass compositions began and more and more researchers became interested in investigating them. In 1988, Inoue and co-workers prepared for the first time bulk metallic glasses, which are defined as metallic glasses with a critical casting thickness of more than 1mm in the smallest dimension [57]. Their development provided specimens suitable for mechanical testing and revealed their potential as structural material in service. Over the past six decades, significant scientific progress has been made regarding casting technology and alloy development, so that more and more metallic glasses and especially bulk metallic glasses with a variety of compositions have been discovered and studied to date [2]. Beside the search of novel glass-forming liquids, intense efforts have been made to investigate the disordered structure [4, 58, 59, 60, 61].
The first section focuses on the structure of metallic glasses (MGs) including recent models, which explain the presence of structural heterogeneities [52, 62]. MGs can be prepared by melt-quenching. Prior to the vitrification of supercooled liquids as they are cooled, heterogeneous dynamics, which most likely cause heterogeneities, become evident [63, 64, 65]. The transformation at which the supercooled liquid vitrifies, is termed glass transition. Crystallization constitutes the other prominent transformation in metallic glasses. It is a first-order transformation and can be described by classical nucleation and growth theory [66]. Both phase transformations are addressed at the next section 3.2. Furthermore, the fragility of supercooled liquids and their glass-forming ability (GFA) are briefly introduced.
After that, the mechanical deformation mechanism of BMGs and BMG composites are explained, as this sets the context for measures to improve the plasticity of BMGs. As we will show, Cu50Zr50-based BMGs are special, owing to their extraordinary GFA and deformation mechanism [34, 67]. After a short introduction of the Cu-Zr-Al system, finally
3.1. STRUCTURE OF METALLIC GLASS
fast annealing as a rapid heat treatment method, which can be applied to various material classes, will be introduced in detail. Thereby, the focus is on the flash-annealing of MGs.
The lack of long-range order is the defining structural characteristic of glass [68]. For this reason, no sharp reflections, which could be attributed to lattice planes, are present in the X-ray diffraction (XRD) pattern. Instead, one can observe two characteristic broad “maxima” (Fig.3.1.1, marked with black arrows). Metallic glass shows no discernable microstructure like for instance grains, precipitates, interfaces, twins or dislocations as they are typical characteristics of polycrystalline alloys. Instead, high-resolution transmission electron micrographs display mazelike patterns (Fig. 3.1.1b).
illustration not visible in this excerpt
Figure 3.1.1: Characteristics of the structure of metallic glass. a) XRD pattern of an ascast Cu46Zr46Al8 metallic glass ribbon. Except of two broad “maxima”, no sharp reflections typical for crystalline alloys are present. b) High-resolution transmission electron micrograph of an as-cast Cu44Zr44Al8Hf2Co2 BMG. Mazelike patterns are visible.
Bernal and Scott have been the first to propose a model to describe disordered structures [60, 61]. They approximated the atoms in metals as identical hard spheres and arranged them as densely as possible in a random manner. The three-dimensional space, however, cannot be filled solely with these densely randomly packed clusters without introducing long-range translational symmetry. This problem is addressed in literature as “packing frustration” [59, 62, 69]. Therefore Bernal suggested that holes smaller than a hard sphere have to be incorporated into the dense random packing. Bernal termed them “canonical holes” [61]. Altogether, according to this model five different types of holes with edges of equal length are present as can be seen from Fig. 3.1.2. This pioneering concept known as “the dense random packing of hard spheres” (DRPHS) model [69], has been analyzed by
CHAPTER 3. FUNDAMENTALS
means of molecular dynamics (MD) simulation to determine the fraction of each hole. The “empty” space necessary to avoid packing frustration is also termed as “free volume” [59] and is redistributed among all hole types whereas the tetrahedra are found to be dominant [70].
illustration not visible in this excerpt
Figure 3.1.2: Schematic of the five types of holes, which are necessary to fill the three- dimensional space according to the DRPHS-model. a) tetrahedron, b) octahedron, c) tetragonal dodecahedron, d) trigonal prism capped with three half octahedra and e) Archimedean antiprism capped with two half octahedra. Each panel depicts the corres- ponding hard-sphere packing surrounding the hole (left) and the hole as purple sphere in the center (right). Taken from [69].
Nevertheless, this first model is not valid for multicomponent systems with atoms of significantly different radii. Furthermore, metal atoms are not spheres, and packing is clearly not as random as following model considers.
The high density and non-directional bonding of metallic glasses already imply a structural order [1, 59, 71, 72]. The first compelling atomic structural model for metallic glasses has been proposed by Miracle [59]. His efficient packing (ECP) topological model, which represents nowadays the most prevalent model, considers the differences of atomic radii while not taking into account chemical effects. The structure of metallic glass on a short-range order (SRO < 0.5 nm [1]) is described by densely packed atomic clusters which consist of a solute atom (α) as the centre and a first coordination shell of nearest neighbours, the solvent atoms (Ω) (Fig. 3.1.5a) [59, 72]. These solute-centred clusters arise from the strong tendency to form as many bonds as possible between unlike species due to the large negative heat of mixing, which is distinctive for chemical elements of good glass-forming alloys [58]. The size and polyhedra type of these efficient packed clusters depend on the ratio of the atomic radii between the solute and solvent atoms and are determined by the coordination number (CN) of the solvent atoms (Fig.3.1.3). Thereby, the CN ranges from 8 to 20 [59, 73] and depens on the composition of the MG.
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Figure 3.1.3: A series of circles depicts the influence of relative size on local packing efficiency in two dimensions. The radius of the solvent atom (blue) increases from left to right, whereas the radius of the solute atoms (red) is constant. Redrawn after [72].
State-of-the-art experiments [74, 75, 76, 77, 78] and especially computer simulations [74, 79, 80] have contributed to a fundamental understanding of the SRO. Sheng et al. [74] have analyzed the packing of solute-centered clusters of a number of model MGs using ab-initio molecular dynamics simulations. They have validated their results by comparing the calculated CN with measured ones and ascertained that the preference for a particular type is controlled by the effective atomic size ratio between the solute and slovent atoms, in accordance with the topological ECP-model. Despite the large number of polyhedra types, certain polyhedra appear more often. For Zr-Pt [74, 77] and Cu-Zr [79, 81] MGs, for instance, icosahedra (CN = 12) are favoured (Fig.3.1.4) and Sheng et al. have termed these MGs “icosahedral ordered” [74].
illustration not visible in this excerpt
Figure 3.1.4: Schematic of an icosahedral cluster - the structural motif of Cu-Zr-based metallic glasses. The blue and red spheres represent Zr- and Cu-atoms. The Cu-centre is purple highlighted. Redrawn after [81].
Cheng et al. have conducted MD simulations on Cu-Zr- and Cu-Zr-Al-based MGs in order to investigate the corresponding SRO and the influence of Al [79]. They have concluded that aside Cu-centered icosahedra, which are the characteristic solute-centered clusters or “structural motif” as they term it, also a high population of distorted or incomplete icosahedra are present in Cu-Zr-based MGs. Alloying with Al promotes icosahedral ordering and thus greatly enhances the number of “complete” Cu-centered coordination polyhedra. Nowadays, experimental techniques like for instance quantitative nanodiffraction, confirm simulated structural motifs which characterize the SRO. Indeed, they show that the icosahedral cluster is the structural motif of Cu-Zr-based MGs [82].
Coming back to the ECP-Model; Miracle suggested that solute-centered clusters penetrate and interconnect with each other by sharing faces, vertices, and edges, so that an extended structure, termed the medium-range order (MRO ∼1 nm [1]), results [59]. In order to efficiently fill the three-dimensional space, the solute-centred clusters are themselves arranged into simple cubic (sc), hexagonal closed-packed (hcp) or face-centred cubic (fcc) arrays [72]. The fcc cluster packing is most common for clusters with at least 12 nearest neighbours [72]. According to Miracle, two additional topological distinct solutes are necessary for ordering
- the secondary (β) and tertiary (γ) solute, which occupy octahedral and tetrahedral interstices of the clusters, respectively (Fig. 3.1.5b). The structural order is presumed to extend over a few clusters which have no orientational order [59, 71].
illustration not visible in this excerpt
Figure 3.1.5: The fcc-cluster-packing of a Zr-(Al, Ti)-(Cu,Ni)-Be alloy according to the ECP-model. a) Two- and b) three- dimensional illustration of a dense cluster-packing. The α-, β- and γ-sites are occupied by the blue, purple and orange spheres, respectively. Zr represents the solvent atoms which form icosahedra around each α-solute. Taken from [59].
In the case of CuZr-based MGs, icosahedral clusters interconnect with each other by face-sharing as well as interpenetration [78] and thus serve as the backbone of the MG structure [79]. The ECP-model is able to predict measured structural properties. Partial coordination numbers, which are obtained from synchrotron radiation experiments, are in good agreement with the ones predicted by the ECP-model [59, 72, 73]. Thus, experiments give a quantitative verification of the predicted short-range order [73, 72]. The ECP-model has also deficiencies, since it estimates the extent of the MRO, which is about 1 nm [72], but is not able to exactly characterize the structure of the MRO [69]. For multicomponent BMGs with at least four different elements, it is very challenging, if not impossible, to differentiate between solute and solvent [79]. Cheng and Ma [79] have therefore drawn the conclusion that the ECP-model has to be refined and consequently, Ma has just recently suggested a different model describing the SRO and MRO of MGs [62].
Similar to Miracle [59], he perceives clusters as fundamental motifs comprising the structure of MGs [62]. According to Ma’s results derived from MD simulations [62, 81], all fundamental motifs consist of tetrahedral clusters representing the basic packing unit.
Allowing each atom to participate in several clusters, enables to achieve a high packing density. In other words, tetrahedral clusters are fragments of, for instance, icosahedral clusters, which are the most stable and thus populous SRO type of Cu-Zr-based glasses [80]. MGs tend to maximize the number of favoured fundamental motifs, which are not ideal in shape and distorted, so that they can be accomodated in the overall structure [79, 81]. However, since MGs are densely packed and favoured motifs cannot fill the three-dimensional space, in addition geometrically unfavoured motifs (GUMs) are necessary. The CNs of GUMs deviate significantly from the favoured motif and are therefore either extremely under- or over-coordinated. They are needed for filling the three-dimensional space [62] and connect the backbone structure comprised of favoured motifs [79, 80, 81]. The number and kind of atoms determine which cluster represents a favoured motif [69]. Metallic atoms are not hard spheres and interact via manybody, “soft” potentials. They prefer a chemical local order to select favourable neighbours to lower their energy [62].
illustration not visible in this excerpt
Figure 3.1.6: Schematic of a tetrahedral cluster.
According to Ma, GUMs which are less ordered regions retained during the fast cooling of the liquid, are unstable and prone to reconfiguration evoked by heating or stress [62]. They contain on average more free volume and behave more “liquid-like” under external stimuli [62]. Dmowski et al. [83] as well have hypothesized the presence of liquid-like soft regions within the structure of MGs. By means of high-energy X-ray diffraction, they have investigated in-situ the structure of metallic glasses during elastic deformation and show that only[3] of the volume fraction of MGs deform elastically. The residual volume fraction behaves anelastically indicating the presence of liquid-like regions. Liu et al. [49] conducted amplitude-modulation atomic force microscopy (AM-AFM) on a thin Zr-Cu-Ni-Al metallic glass film. A vibrating cantilever-tip ensemble scans across the sample surface and enables to obtain the dissipated energy. The resulting energy dissipation map (Fig. 3.1.7a) shows randomly distributed viscoelastic heterogeneities with a domain size of about 1-2 nm [49]. These structural heterogeneous domains, which are termed just heterogeneites in the following, show significant anelastic deformation as is consistent with the results of Dmowski et al. [83]. Heterogeneities correspond to more defective regions with relatively low viscosity and elastic modulus and consequently behave liquid-like [49]. Similar findings of Wagner et al. [50] confirm the presence of structural heterogeneities representing “soft spots”. They detected a nanometre-scale inhomogeneous distribution of the contact resonance frequency of Pd-Cu-Si-based MGs by means of atomic force acoustic microscopy and concluded that soft heterogeneities on a nm-scale are present. MD simulations of Ding et al. [81] show that these soft heterogeneities observed in experiments correlate to regions with a high content of GUMs. The simulation box of a Cu-Zr-based MG model during deformation shows that the atoms with the strongest participation in soft vibrational modes are mostly in GUMs as Fig. 3.1.7b depicts. The colours indicate the different degree of participation in soft vibrational modes. GUMs are non-uniformly distributed and form soft spots on a nm-scale, as is consistent with Ma’s model and aforementioned experiments [62, 81]. Furthermore, Ding et al. have superposed the locations of local motifs that have clearly experienced plastic deformation (white circles in Fig. 3.1.7b) with the locations of soft spots and they overlap remarkably. This finding is consistent with the GUMs-model, since GUMs are unstable and prone to stress-induced reconfiguration.
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Figure 3.1.7: Heterogeneities as an inherent characteristic of the structure of metallic glass.
a) Energy dissipation map of a Zr-Cu-Ni-Al metallic glass film. The map was obtained from amplitude-modulation atomic force microscopy (AM-AFM) and local randomly distributed soft domains (red regions) with large energy dissipation are clearly visible. Taken from [49].
b) MD simulation enables to localize regions with a high content of soft GUMs. Contour map depicts the heterogeneous spatial distribution of Cu and Zr atoms that participate the most in soft modes. The colours indicate the different degree of participation in soft vibrational modes (see scale bar). The white circles mark the locations that have clearly experienced plastic deformation. Taken from [81].
3.1. STRUCTURE OF METALLIC GLASS
Experiments and MD simulation definitely prove the existence of structural heterogeneities which can be explained by Ma’s GUMs-model [62]. GUMs may be retained during the fast cooling of the liquid [62], defintiely necessary to bypass crystallization. Heterogeneous dynamics of metallic liquids rapidly cooled towards the glass-transition, are supposed to lead to the formation of heterogeneities as will be shown in the next section on the glass formation and crystallization.
3.2 Glass formation and crystallization
A glass can be formed by cooling a liquid fast enough without intervening crystallization [2, 58, 66, 84, 85, 86]. At cooling from the liquid state, the viscosity of the liquid increases rapidly and, at some point, the liquid transforms into a glass. The first subsection 3.2.1 elaborates on this change termed “the glass transition” and introduces the free volume concept. The crystallization, which is the other prominent phase change in metallic glasses is in the focus of the following subsections. It proceeds in two consecutive stages, namely nucleation and crystal growth. After introducing both stages and their basic kinetics more in detail, the isothermal and isochronal devitrification kinetics of metallic glasses are considered. At last the fragility concept (subsection 3.2.5) and glass-forming ability (subsection 3.2.6) of supercooled liquids will be considered. The temperature dependence of the viscosity for a supercooled liquid is captured by the so-called fragility being an important parameter to classify glass-forming liquids.
3.2.1 Glass transition
The glass transition can be detected by changes of properties such as volume, enthalpy or for instance entropy as a function of temperature and cooling rate [84, 85, 87]. Fig. 3.2.1 depicts the volume depending on the temperature.
illustration not visible in this excerpt
Figure 3.2.1: Change of volume, entropy and enthalpy as a function of temperature and cooling rate. The higher the cooling rate, the faster the glass vitrifies and the higher the glass-transition temperature, Tg. Adapted from Ref. [88].
As a liquid is cooled, its volume decreases continuously until the melting point, Tm, is reached. Depending on the cooling rate and its composition, the liquid can either crystallize or vitrify. Crystallization usually occurs at a defined undercooling and configurational rearrangements are necessary to form viable nuclei. Crystallization represents a first-order transformation associated with a discrete change of properties like for instance the volume as can be seen from Fig. 3.2.1. The volume of the liquid (red curve) drops abruptly when crystallization takes place (black curve) and the corresponding crystallization kinetics are addressed in subsections 3.2.2 and 3.2.3.
Assuming fast cooling and the appropriate composition of the liquid, crystallization can be circumvented. As the supercooled liquid is cooled to lower temperatures, it densifies via atomic rearrangements and its viscosity increases. The volume continues to decrease with augmenting undercooling (purple curve) and the atoms that comprise the supercooled liquid move more and more slowly. At some temperature the slope of the volume-temperature curve (Fig. 3.2.1) decreases gradually. The experimentally observed volume begins to deviate from the “equilibrium” value at this point, because the atomic rearrangements necessary for the liquid to find the “equilibrium” volume [63] are retarded and finally cease to exist. In other words, the atoms move so slowly that they are not able to rearrange during cooling [63, 89, 90] and the supercooled liquid transforms into a glass (Fig. 3.2.1, blue curve). This change is called glass transition and is not a phase formation but rather a kinetic phenomenon [63].
The glass transition occurs continuously within a temperature range in contrast to crystallization with its discrete transition temperature, as Fig. 3.2.1 depicts. Although the glass transition takes place within a temperature range, one tries to assign a specific temeperature, the glass-transition temperature, Tg (Fig. 3.2.1). Several ways are discussed in literature how to determine Tg, which does not exactly facilitate the comparison of Tg values [85]. The difficulties in determining the glass-transition temperature can be avoided by defining that Tg is associated with a fixed viscosity, η, of 10[12] Pa·s [66, 85].
From a thermodynamic point of view, the glass-transition is absolutely necessary. Oth- erwise at the so-called Kauzmann temperature, TK, the entropy of the amorphous state would become lower than that of the crystalline state, which is impossible (Fig. 3.2.1).
This contradiction is known as the Kauzmann’s paradox [91].
The viscous slowdown occuring at the glass transition, can be explained by relaxation processes, which are addressed first, and the free-volume model. Johari and Goldstein [92] have introduced that the glass formation can be described by relaxation processes [90, 92]. They have been the first to recognize that the vitrification of the supercooled liquid can be expressed by the α-relaxations which disappear after the glass is formed. They have also realized that there is a second relaxation process, the β-relaxation, which is still present below the glass transition and is believed to be the principal source of dynamics in the glassy state [90, 92]. Recent findings have shown that both relaxation processes are related to each other [93] and that the β-relaxation occurs in the glass at shorter time scales and lower temperatures [90]. Furthermore, it is believed that β-relaxations are governed by ramified, string-like clusters, while α-relaxations take place through activated events involving compact regions [94].
Another approach to explain the viscous slowdown at the glass transition is the free- volume concept. The basic idea behind it is that atoms need “free volume” to rearrange [95]. Turnbull and Cohen [96] defined the free volume as that part of the volume, which can be redistributed without energy cost, so that voids with a critical value can form. These voids enable then atoms to “jump” to another site [96, 97]. However, as the liquid is cooled, relaxation processes annihilate the free volume. As a consequence less and less diffusional atom jumps occur and the viscosity of the supercooled liquid increases as is in compliance with the Doolitle equation (Eq. 3.2.1) [98]:
illustration not visible in this excerpt
where η0 is the viscosity at infinite temperature and represents a pre-exponential constant, C is a constant, and vf is the free volume.
Many liquids require very fast cooling rates, but any liquid is theoretically able to form a glass, if cooled rapidly enough [99]. Thereby, the glass transition is not only continuous, but also depends on the cooling-rate [55]. The faster the cooling, the “earlier” the supercooled liquid transforms to glass and consequently the higher Tg (Fig. 3.2.1). A supercooled liquid exhibits a lower viscosity and higher volume at higher temperatures. When the liquid is cooled faster, the glass transition takes place at higher temperatures and more free volume is trapped in the glass.
Although the free volume theory is widely accepted, it is not free of controversy. It is not possible to define the free volume rigorously on an atomic scale [85] and recent experiments have shown that it does not provide a fundamentally sound explanation of the low-temperature dynamics in fragile glass-forming liquids (subsection 3.2.5) [63, 100]. Furthermore, the free-volume concept does not describe the heterogeneous distribution of the free volume, as recent experiments [49, 50, 83] suggest. The MRO of metallic glasses shows a clustering of the free volume. In other words, the structure of metallic glasses seems to show free volume enhanced regions (FERs), which correspond to GUM-enriched regions, according to Ma’s structural model [62].
The free volume or just FERs are annihilitated when metallic glasses are annealed. This process is called relaxation and can be detected by isochronal calorimetric measurements as indicated by the green curve in Fig. 3.2.2. Tg is passed at further heating and the glass transforms into the supercooled liquid, which crystallizes at higher temperatures. In the following, the relaxation process and glass transition as the glass is heated are explained more in detail by means of the concept proposed by van den Beukel and Sietsma [101]. The straight black line represents an approximation of the free volume being in ther-
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Figure 3.2.2: Free volume and heat flow as a function of temperature as a glass is heated. T g,S and T g,E indicate the start and end of the glass transition, respectively. v f correponds to the free volume content trapped in the glass and v eq is the equilibrium concentration of the free volume. Adapted from Ref. [101].
modynamic equilibrium [101]. At high temperatures above the glass transition the free volume is annihilated fast enough, so that a thermodynamic equilibrium is maintained during cooling (Fig. 3.2.2, blue curve A-B). On further cooling, the atomic mobility is reduced and an excess amount of free volume is trapped as the supercooled liquid vitrifies (blue curve deviates from black curve). When the glass is subsequently heated (Fig. 3.2.2, red curve) at a constant heating rate, the excess free volume is annihilated (Fig. 3.2.2, green curve). The free volume gradually annihilates (CD) under release of heat as the exothermic event of the DSC curve indicates. This process is called structural relaxation of the glass. When MGs are prepared at higher cooling rates, more free volume, which is equivalent to a higher number of FERs, is trapped during vitrification. They show then a stronger structural relaxation on thermal activation. At D, the free volume curve (red) crosses the equlibrium curve. Free volume cannot be produced as easily in the glass as in the supercooled liquid, due to the frozen-in configurational rearrangements. Therefore, the free volume in the relaxed glass lags behind the “equilibrium” concentration of the free volume, v eq. As the glass is further heated, the glass-transition starts (Fig. 3.2.2, green curve, T g,S), the viscosity drops and new free volume is generated (Fig. 3.2.2, red curve D-E). The glass transition is an endothermic event as indicated by the DSC curve (green). At E the glass transition is completed (T g,E on the green DSC curve), the supercooled liquid is present and the kinetics of the free volume generation are not retarded any more. Thus, the “equilibrium” concentration of the free volume is attained, until finally the supercooled liquid crystallizes (not shown in Fig. 3.2.2).
The annihilation of free volume reduces the number of FERs and it is called relaxation. The number of FERs strongly affects the mechanical behaviour, particularly the plasticity, of BMGs, as subsection 3.3.2 will show. Relaxation leads to the embrittlement of BMGs [102], whereas the increase of the number of FERs, which is termed rejuvenation, represents a measure to enhance the plasticity of BMGs.
Several apporaches to rejuvenate MGs are reported, and they are based on mechanical processing or thermal treatments. Concustell et al. [103] have conducted shot-peening on relaxed Pd-based metallic glass. At first, they heated the glass to a temperature below Tg and held it there for a certain duration to induce structural relaxation. Afterwards, they shot-peened it and detected an increase of ΔHrelax, which is equivlaent to rejuvenation. Chen cold-rolled as well a Pd-based BMG and observed an increase of ΔHrelax compared to the as-cast state upon reheating in a DSC. He has suggested that plastic deformation in metallic glasses is accompanied by atomic regrouping being analogous to the viscous flow of liquids [104]. Another example is high-pressure torsion, which as well induces an increase of ΔHrelax [105]. Magagnosc et al. achieved rejuvenation of metallic glass by ion irradiation, which causes an increase of structural disorder [106]. Only recently, Ketov et al. [52] have demonstrated rejuvenation caused by cryogenic cycling of La-, Zr- and Cu-Zr-Al-based metallic glasses. They suspect the existence of FERs which they term just heterogeneities. Since the glass is non-uniform on a nanoscale, the thermal expansion coefficient varies locally, as illustrated in Fig. 3.2.3.
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Figure 3.2.3: Heterogeneous thermal expansion coefficients in the as-cast state. The thermal expansion coefficient (represented by ellipsoids) of a glass varies locally due to the heterogeneous nature of the structure. When the temperature is changed, these local variations in thermal expansion cause internal stresses (red and blue regions indicate compression and tension, respectively) to develop in the glass. Taken from [107].
Changing the temperature then induces internal stresses, which cause irreversible local atomic rearrangements and hence rejuvenation (Fig. 3.2.4a) [52, 107]. MGs are continuously rejuvenated, the more cryogenic cycles are executed. However, there is a critical number of cycles, after which the glass begins to relax [52]. This number depends on the compositon and particularly dimension of the BMG specimen [52]. The bigger the specimens are, the less effective the cryogenic cycling is (Fig. 3.2.4b) [52].
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Figure 3.2.4: Rejuvenation due to cryogenic cycling. a) DSC traces depict the specific heat Cp as a function of the temperature. The heat of relaxation, ΔHrelax, is indicated by the shaded areas and is increased after cryogenic thermal cycles. b) Influence of the dimension of La-based BMG specimens and number of cryogenic cycles on ΔHrelax. A higher relaxation enthalpy is attained for ribbons than for rods, and relaxation sets in at a lower number of cycles for the ribbons. The data points are for individual measurements and the changes lie outside the error range of ± 30 J/mol. Taken from [52].
Aside the above mentioned relaxation of BMGs during annealing to temperatures below Tg (sub-Tg annealing), MGs can additionally form nanocrystals when heated to temperatures below Tg. Wei et al. [108] have isothermally annealed Cu48Zr48Al4 BMG specimens at a temperature of 15 K below Tg for different durations ranging from 2 to 12 hours. The metastable B2 CuZr phase forms first and decomposes into the stable low-temperature Cu10Zr7 and CuZr2 phases as can be seen from the corresponding XRD patterns (Fig. 3.2.5a). The B2 CuZr nanocrystals are randomly distributed in the annealed glass and their size ranges from 20 to 40 µm in diameter (Fig. 3.2.5b) [108]. A sub-Tg anneal was also carried out for Cu44Zr44Al8Hf2 BMG specimens which have been held at 25, 50, 75, and 100 K below Tg for a duration of 5 min and have been subsequently water-quenched [53]. Due to structural relaxation, all specimens show a lower ΔHrelax than the as-cast sample. For the sample annealed at 50 K below Tg, annealing-induced B2 CuZr nanocrystals with a size ranging between 2 and 20 nm in diameter, could be detected by means of HRTEM (Fig. 3.2.6). It is interesting to note that the B2 CuZr phase forms first, though it is metastable at this temperature. This finding corroborates well with the work of Pauly et al. on deformation-induced nanocrystallization of B2 CuZr [34], as will be shown in subsection 3.3.2. B2 clusters are frozen-in during the preparation of the BMG and their growth appears to be kinetically favoured during subsequent isothermal annealing
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Figure 3.2.5: Nanocrystallization of Cu48Zr48Al4 BMGs due to sub-Tg annealing. a) XRD pattern of specimens isothermally annealed at a temperature of 15K below Tg. At first B2 CuZr forms and decomposes into Cu10Zr7 and CuZr2 with increasing annealing duration. b) Bright-field TEM-image depicts nanocrystals (dark entities). According to the corresponding SAED pattern, they can be identified as B2 CuZr. Taken from [108].
(subsection 3.5).
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Figure 3.2.6: Annealing-induced precipitation of B2 CuZr nanocrystals. The HRTEM image shows that nanocrystals with a size ranging from 2 to 5 nm precipitated during sub-Tg annealing in the Cu44Zr44Al8Hf2 BMG specimen. The inset depicts the corresponding SAED pattern which confirms the ordered nature of the nanocrystals. The B2 CuZr phase has been identified from the lattice spacing of 0.326 nm. Taken from [53].
During the annealing of Cu-Zr-Al-based BMGs to temperatures below Tg, two interacting processes seem to be effective: (1) the structural relaxation, and (2) the precipitation of B2 CuZr nanocrystals. Atoms in FERs rearrange faster into ordered nanocrystalline structures and relaxation annihilates this free volume.
One focus of the current thesis is to investigate the structural changes occuring in Cu- Zr-Al-based BMG specimens as they are flash-annealed to temperatures below Tx. With increasing temperature, at first FERs are annihilated at temperatures below Tg. At higher annealing temperatures the supercooled liquid is present and the subsequent quenching rate determines the free volume content or rephrased number of FERs. If the quenching rate is higher than the cooling rate during the preparation of the BMG specimens, then rejuvenation (disordering) will arise. Since quenched-in nuclei seem to grow already due to sub-Tg annealing, it is probable that they also survive and even grow at temperatures within the supercooled liquid region. These clusters are subcritical in size and of course some of them should dissolve.
In summary, FERs and queched-in nuclei, which constitute structural heterogeneities, are essential elements of the structure of metallic glasses. It shall be investigated how these heterogeneties behave when BMGs are flash-annealed. Two competing processes seem to be present as BMGs are flash-annealed between Tg and Tx: (1) disordering as the supercooled liquid rejuvenates, and (2) ordering as nanocrystals grow from already existing quenched-in nuclei.
3.2.2 Classical nucleation theory
The glass transition was treated more in detail, yet, we did not refer to the crystallization, which is the other prominent transformation in metallic glasses. Therefore, the present and the following subsections elaborate on the nucleation, crystal growth and devitrification kinetics of metallic glass, respectively.
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Fig. 3.2.1 illustrates the change of the volume as a function of temperature. Coming from the melt, a discontinuous change in volume can be observed during crystallization (Fig. 3.2.1 black curve). The corresponding slope or first derivative with respect to the temperature, dV dT, exhibits a discontinuity at the melting point, Tm. Hence, crystallization is defined as a first-order phase transition according to the Ehrenfest classification [109, 110, 111].
Crystallization proceeds by nucleation at which a solid-liquid interface is formed and the advancement of the interface termed crystal growth. This phase transformation can be described by classical nucleation theory and crystal growth theory which are the topic of the present and next subsections, respectively [31, 44, 66, 112, 113].
According to the classical crystallization theory, nucleation results from statistical thermal fluctuations of the liquid leading to the transient appearance and disappearance of small regions (clusters) of a new phase within a parent phase [114]. Clusters appear and decay continuously and spontaneously. The duration of a fluctuation mainly depends on its size and the fluctuation only evolves into a macroscopic region of the new phase when exceeding a critical size. The clusters are then termed critical nuclei.
As the liquid is undercooled below Tm, the formation of critical nuclei and particularly their growth are thermodynamically favoured over decay [110, 111, 112, 115, 116, 117, 118, 119, 120]. Fig. 3.2.7 exemplifies the driving force for crystallization per volume, which is defined as the difference between the free energy of the liquid and crystalline state, ΔGv [31, 44, 55, 109, 110, 111, 112, 116]. Below the melting temperature, Tm, the melt is instable and strives to transform to the crystalline state, since its Gibbs energy, G, is lower. ΔGv is defined as the difference in Gibbs enthalpy of the crystalline and liquid state and is a function of the undercooling ΔT [110, 111, 112, 115, 116, 117, 118, 119, 120].
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Figure 3.2.7: Gibbs enthalpy per unit volume, ΔGv , of the liquid and the crystalline phase as a function of the temperature. Every system tends to minimize its Gibbs free energy and a pure metallic liquid is crystallizing when sufficiently supercooled. ΔGv depicts the driving force for crystallization. Redrawn after [120].
On the one hand, at T < Tm, the crystalline state is associated with lower free energy (Fig. 3.2.7) and thus the formation and growth of critical nuclei implies a release of energy.
On the other hand, a solid-liquid interface is created when a cluster forms, which “costs” energy. Both contributions, which depend differently on the cluster size, determine the change in free energy. Nucleation is only thermodynamically favoured when ΔG < 0. For simplicity, spherical clusters are assumed and the energy balance during the formation of a cluster can be then expressed as a function of its radius, r, as follows:
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where σ > 0 and ΔGv (T < Tm) < 0 is the previously mentioned driving force for crystallization (both parameters are taken as independent of the cluster size). The first and second summands of Eq. 3.2.2 are termed volume and interface term, respectively, in the following. The volume term corresponds to the reduction of the free enthalpy and has a third-power dependence on the radius, while the interface term, which describes the formation of the solid-liquid interface, has only a second-power dependence. As r increases (Fig. 3.2.8), ΔG is initially positive and goes through a maximum, at a critical radius r∗. When r > r∗, then applies d(ΔG) dr < 0 and the growth of a nucleus is favoured, since it will lower the energy of the system (Fig. 3.2.8) [120]. In the following r∗ and ΔG∗ are deduced from Eq. 3.2.2.
Figure 3.2.8: Free enthalpy of a crystalline cluster as a function of its radius at temperatures below Tm. The reader is referred to this paragraph for more information. Adapted from [119].
We obtain r∗ by differentiation of Eq. 3.2.2 with respect to r and solving for r∗after equating to zero gives:
Substitution of Eq. 3.2.3 into Eq. 3.2.2 yields the activation energy of crystallization:
ΔG∗ is also considered as the work necessary for the formation of a critical nucleus and represents the height of the nucleation barrier [114, 119]. Clusters with a radius smaller than r∗ are called undercritical nuclei or embryos. They are unstable and decay spontaneously, since their interface-to-volume ratio is large and hence the interface contribution dominates. In contrast, clusters with a radius larger than r∗ are termed supercritical nuclei and are able to lower the free energy by growing. However, though being of supercritical size, clusters can be still unstable unless their size is larger than r0 = 1.5·r∗ [119]. ΔG∗ and r∗ decrease drastically with increasing undercooling ΔT, so that more and more nuclei result. This leads after complete crystallization to a fine-grained microstructure.
So far, we have introduced the spontaneous production of supercritical clusters [116], which is referred to as homogeneous nucleation. The same thermodynamic considerations apply to heterogeneous nucleation which is the case when the melt contains solid particles like, for instance, impurities, or is in contact with a crucible or oxide layer and nucleation is facilitated. Instead of creating a nucleus, the melt only has to wet the heterogeneous nucleation site, which requires a lower activation energy [120]. For more information the reader is referred to [120]. The results of the present work do not show any hints for heterogeneous nucleation. Therefore, in the following only the homogeneous nucleation is considered.
After introducing the thermodynamics, we now focus on the crystallization kinetics. At first, we are going to introduce the steady-state homogeneous nucleation rate and complete this subsection with transient nucleation effects. Such effects are theoretically predicted when MGs are devitrified at high heating rates [48, 121].
The first kinetic model of nucleation was derived by Volmer and Weber [122], who assumed that clusters of n atoms grow or shrink by the addition or detachment of a single atom [114, 116, 117, 118, 122, 123, 124]. The rates of atom-addition and -detachment are termed forward and backward rates, respectively. Both are known as conversion rates [114, 119]. The description of nucleation by means of conversion rates is consistent with thermodynamics: The forward and backward rates are equal at clusters with a critical size, r∗. Subcritical clusters are dissolving since the backward rate is larger than the forward rate and supercritical clusters continue to grow due to the higher forward rate [114]. The cluster size and time stronlgy affect the nucleation rate. Volmer and Weber [122] have only considered the cluster size and not its time-dependence. Moreover, they have perceived that the cluster population diverges at cluster sizes beyond r∗ and simply did not consider them [114, 119, 122]. All atoms of supercritical clusters are replaced by an equivalent number of atoms, which are continously added to the system, so that a dynamic equilibrium prevails.
As already mentioned, supercritical nuclei can be still unstable unless their size is larger than r0 = 1.5·r∗.
The model of Becker and Döring [123, 114, 119] considers that supercritical nuclei can be unstable and the time-independet nucleation rate, the steady-state rate, ISS , in a condensed system is as follows [66, 119]:
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η·e(−kB·T ),
A being constant, η is the viscosity, kB is the Boltzmann constant, and T is the temperature. Two terms govern the steady-state nucleation rate, ISS : At high temperatures in the vicinity of the melting temperature, Tm, the exponential term known as the “thermodynamical contribution” [114, 119] dominates ISS. This term contains the activation energy for the formation of critical nuclei, which strongly depends on the undercooling (see Eq. 3.2.4). In contrast, the viscosity, η, which represents the “kinetic contribution” determines ISS at large undercoolings, particularly in the vicinity of the glass-transition temperature, Tg. This contribution reflects the atomic diffusion necessary for the transition of an atom from the liquid state to the solid nucleus [110, 111, 112, 119, 114, 116, 125, 126, 127]. Fig. 3.2.9 schematically depicts I SS as a function of temperature. I SS rises sharply with increasing undercooling, passes through a maximum and decreases again with further undercooling towards T g due to the decreasing atomic mobility. The viscosity of a supercooled liquid is a macroscopic measure to characterize the atomic diffusion, since the diffusion coefficient, D, can be described as a function of the inverse of the viscosity, η − [1] [128]. The equation is known as the Stokes-Einstein relation [129]:
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Figure 3.2.9: Schematic representation of the homogeneous steady-state nucleation rate in an undercooled melt or glass. The melting temperature, T m, and the glass-transition temperature, T g, are indicated. The blue and red regions mark the temperature regions for devitrification of amorphous alloys and conventional solidification, respectively. Redrawn
after [43].
During conventional solidification, nucleation kinetics can only be analyzed at limited undercoolings as can be seen from the red highlighted area in Fig. 3.2.9. In contrast, owing to the sluggish crystal growth in glasses on annealing, nucleation can be studied by crystallization statistics of partially devitrified glasses near T g [43, 130]. In these cases, crystallization can be interrupted at any fraction transformed by lowering the temperature [112]. The variation of the number density of nuclei per volume, N V, with time, t, can be determined directly by counting crystals and analysis of the crystal size distribution [43, 130]. Köster [130] for instance has shown that in Co-Zr, N V increases with time, t, as is consistent with homogeneous nucleation:
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In the early stages of nucleation, NV can increase linearly with t (steady-state nucleation) due to the dynamic equilibrium [119], as Fig. 3.2.10 displays. In the case of steady-state nucleation, the plot of NV as a function of time yields a straight line with the slope being equal to the steady-state value, ISS [75, 117, 118, 131] (Fig. 3.2.10b).
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Figure 3.2.10: Number of nuclei as a function of time for the following types of nucleation:
(a) quenched-in active nuclei, (b) steady-state homogeneous, and (c) transient homogeneous. The steady-state nucleation rate ISS and the induction time, θ, are indicated. Adapted from [112].
Transient effects can also occur at the early stages of nucleation, in which dNV /dt increases with t (Fig. 3.2.10 (b)), as is well known from crystallization studies on silicate glasses [132]. The nucleation rate initially is low and increases with time to ISS. At long durations, NV (t) can be approximated by [118, 131]:
NV (t) = ISS·(t − θ), (3.2.8)
where θ is an effective time-lag and represents the induction time, also known as incubation time. θ strongly depends on the annealing temperature, since it scales with the atomic mobility [131] and it can be obtained by extrapolating the number of nuclei as a function of time in the steady-state regime to the time axis [131]. Such an incipient transient nucleation behaviour has been experimentally conformed by Shen et al. [75]. They have counted the number of nuclei as a function of time for various isothermal anneals. θ becomes shorter and NV augments with increasing annealing temperature for a Zr59Ti3Cu20Al10 BMG. Kelton has calculated the temperature-dependance of θ and NV for LiSi2-glasses and confirmed Shen et al.’s experiments [131].
According to the work of Greer [44] on the crystallization kinetics of Fe80B20 glass, NV can be also constant (Fig. 3.2.10 (a)) due to the growth of quenched-in nuclei in the absence of additional formation of new nuclei. There is evidence that homogeneous nucleation can occur in the supercooled liquid during rapid quenching to the glassy state as is the case of Fe80B20 [44, 112]. Quenched-in nuclei will start growing when the annealing temperature is reached and the resulting particle size distribution (PSD) is very narrow [130]. Greer [44] has investigated the effect of quenched-in nuclei, on the nucleation rate during devitrification of glassy Fe80B20 ribbons. He analyzed the number of crystals as a function of the cooling rate at which the ribbons were produced [44] and has prepared metallic glass ribbons with different thicknesses [44]. The cooling rate was estimated from the corresponding ribbon thickness. Furthermore, he assumes that nucleation and the subsequent growth of nuclei do not invole solute partitioning. Otherwise, a compositional gradient and accompanying diffusion field would develop around the growing nuclei. Then, extensions to the classical nucleation theory would be required to address such a problem. Particularly, one must analyze whether the difference in composition is manifested mainly as a change in the work of cluster formation or even requires a new kinetic nucleation model [133]. Greer [44] carried out isochronal measurements at a heating rate of 10 K/min of Fe80B20 ribbons prepared at different cooling rates. According to Greer’s analysis, the number of quenched-in nuclei per unit volume, NV, is of the order of 10[18]m-[3]and it scales with the cooling rate, dT dt,as follows [44]:
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where c ranges between 2 to 4. Consequently, he has confirmed that the nuclei have to be produced during quenching, since their number depends on the cooling rate. If steady-state nucleation applied at every temperature during quenching, then c would be 1. He concluded from the large population of quenched-in nuclei and its strong dependence on the quenching rate that transient homogeneous nucleation occurs as the supercooled liquid is quenched [44].
Assuming no quenched-in nuclei, Shneidmann et al. have analyzed analytically the influence of the heating rate on the homogeneous nucleation rate. The starting point of their study is a thermodynamic consideration of the critical radius, r∗. As the supercooled melt is cooled from the melting temperature, Tm, r∗ decreases with temperature as Fig.
3.2.11 and following Eq. shows [134]:
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Figure 3.2.11: Critical radius, r ∗, as a function of temperature, T. Due to a higher undercooling (T m -T), r ∗ deacreases as the supercooled liquid is cooled. If a glass is heated and crystallization sets in, r ∗ will increase.
Of course, Eq. 3.2.10 also applies to the supercooled liquid, which is heated towards T m. Then, r ∗ increases with augmenting temperature. When a glass is heated to temperatures above the glass-transition temperature, T g, it transforms to a deeply supercooled liquid and further heating eventually leads to crystallization. At such large undercoolings, r ∗ is relatively small and consequently many crystals should nucleate (Fig. 3.2.11). If a glass is heated faster, T g and T x even more, will shift to higher temperatures suggesting a larger r ∗. By heating a highly supercooled liquid at high rates, Shneidmann has shown by means of analytical solutions based on the Becker-Döring equation that nucleation can be effectively suppressed [48, 121] and Fig. 3.2.12a depicts his results. The nucleation rate as a function of temperature is given for various heating rates up to 10 K/s.
This transient nucleation effect, which can only be encountered during heating, is termed “choking” of nucleation [48]. As the highly supercooled liquid is heated, the temperature increases with time, so does r ∗ and one obtaines the rate of increase of r ∗, dr ∗ dt.Onlywhen the nucleus grows faster than dr ∗ dt, it will remain supercritical (r > r ∗) and continues to grow [48,121]. In other words, the crystal growth rate has to exceed dr ∗ dt.Byapplyinghigh heating rates to a glass, T x becomes relatively high, dr ∗ dt isthenlargeandnucleationcanbe effectively suppressed [48, 121]. Above a critical heating rate, r ∗ increases faster with time than the nuclei can grow. Shneidman et al. [48] have described this transient nucleation effect by defining a cut-off temperature, T 0(ϕ), below which nucleation is suppressed. T 0(ϕ) increases the faster the glass is heated [121] (Fig. 3.2.12b):
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Figure 3.2.12: Transient nucleation during the devitrivication of metallic glass. a) Calcu- lated time-dependent nucleation rate multiplied by 10[17] for different heating rates. The inset magnifies the nucleation rate resulting from a heating rate of 10 K/s. Adapted from [48]. b) The cut-off temperature, T0, as a function of the heating rate, ϕ. Redrawn after [121].
NV = [1]
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where ϕ is the heating rate and T1 is the temperature to which the supercooled liquid is heated.
3.2.3 Crystal growth in undercooled liquids
Nucleation precedes crystal growth being the focus of this subsection, which is limited to the morphology and stability of the solid-liquid interface, the crystal growth rate, and methods to determine it.
Once critical nuclei are formed, they will continue to grow accompanied by the attachment of atoms. Thus, the solid-liquid interface propagates and latent heat of fusion, ΔHf , is released at the interface [135]. The latent heat can be either extracted through the melt or the crystal, and hence one distinguishes two cases [120]: In directional solidification, the heat is transferred from the solid-liquid interface through the crystal to the crucible walls and finally to the environment. In the other case, non-directional solidification, the undercooled liquid acts as a sink for the latent heat [119, 135]. We are interested in non-directional solidfication of metallic melts, and therefore limit what follows to it.
The solid-liquid interface of metallic melts is rough on an atomic scale and exposes a lot of favourable sites for the attachment of atoms from the liquid [119, 120]. The propagation of such a solid-liquid interface, which we name in the following “interface”, gives rise to non-faceted crystals, and the corresponding crystal growth type is termed continuous [119, 120]. For more information the reader is referred to [119, 120, 135]. For simplicity, we only consider the precipitation and morphology of just one phase in the following. The morphology of the growing crystal depends on the velocity of the interface and its interface becomes morphologically unstable for non-directional solidification [120, 119]. The negative temperature and concentration gradients (in the case of non-polymorphic crystallization) ahead of the interface destabilize the interface since the undercooled liquid serves as a heat sink and mass redistribution [120, 135]. As a consequence, protrusions form on the initially planar interface (Fig. 3.2.13) and cause an even steeper temperature gradient.
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Figure 3.2.13: Initial evolution of an unstable solid-liquid interface as a function of time. Protursions are amplified until a marked difference in growth of the tips and depressions of the perturbed interface has occurred. Taken from [120].
The developing tip of the protursion rejects more heat and the local growth rate increases [120]. The temperature and concentration gradients destabilize the interface and the capillary effect stabilizes it [119, 120, 135, 136]. The advancement of an unstable interface leads to unstable growth resulting in various crystalline structures like, cellular, dendritic, banded and fractal morphologies [119, 120, 137]. The morphology depends on the velocity of the interface as depicted in Fig. 3.2.14. At small velocities, an interface remains planar up to a velocity equal to the critical velocity, uc [138]. Above uc, the smooth interface becomes instable, protursions form and develop into cells. The interface of cells becomes instable and dendrites with trunks and side-branches develop as the velocity increases further. At high interface velocities, dendritic patterns degenerate back to cells and with even further increasing velocity a planar interface front results again. uA, the critical velocity for absolute stability of the planar interface marks the velocity at which the interface is stable against small protrusions. [120, 136, 138]. Kurz et al. [120] and Herlach et al. [119]
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Figure 3.2.14: Morphological interface diagram for solidification of binary systems. A microstructural transition occurs from the planar front to cellular structure to dendrites and again back to cellular structure and finally to planar front with increasing solidification velocity, u. uc is the velocity according to the criterion for constitutional undercooling and uA is the velocity corresponding to the velocity for absolute morphological stability of the interface. Adapted from [119].
Various experimental methods for the determination of growth velocities in undercooled melts have been developed. They aim at measuring growth rates as a function of the undercooling and direct contact between the supercooled liquid and crucible walls has to be avoided [119, 135]. Otherwise, heterogeneous nucleation on containerwalls, which initiates crystallization, avoids to achieve large undercoolings of the supercooled liquid and can give rise to directional solidification. Nowadays, containerless levitation techniques as for instance electrostatic levitation (ESL) [139] are established as standard methods for measuring the steady-state crystal growth rate, which is addressed in the following paragraph.
The ESL method is used in the present work and will be briefly described. Fig. 3.2.15 depicts a schematic of the setup of a ESL system. The specimen is placed between two electrodes and molten. Coulomb forces act on the charged droplet within the electrostatic field and counteract the gravitational force, so that the specimen is able to levitate [119]. Stable positioning of the specimen is accomplished by a feedback control system. In our case, the position of the sample is controlled by a CCD (charge coupled device) camera picking up an image of the object inside the chamber with a frame rate of 120 fps. The position and velocity of the sample are analyzed by a microcomputer, which controls the position and damping of the sample through electrostatic forces (Fig. 3.2.15) [119]. A pyrometer, which measures the surface temperature of the sample and a high-speed-camera are essential parts of the ESL system (not shown in Fig. 3.2.15).
The propagation of the solidification front can be tracked by its thermal field using photodiodes, pyrometers and/ or high-speed cameras. Particularly, high-speed cameras
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Figure 3.2.15: Electro-static levitation system. a) Schematic shows the levitated sample (red) between both electrodes and the positioning system. The pyrometer, which measures the surface temperature of the sample, and the high-speed-camera recording the propagating interface are not shown. Adapted from [119]. b) Photograph taken from the inside of an electrostatic levitator. A liquid drop of molten alloy levitates within the electrostatic field. Taken from [140].
are capable of recording the advancement of the solid/ liquid interface at a rate of up to several thousands frames per seconds (fps) [139, 141, 142]. They perceive the difference of emissivity of solid (bright) and undercooled liquid phase (dark) and hence the thermal front can be allocated to the interface [143]. Figs. 3.2.16a-c show a series of images during the solidification of a Cu50Zr50 droplet. The propagation of the thermal front as a function of time is clearly visible.
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Figure 3.2.16: Solidification process of the Cu50Zr50 alloy triggered at an undercooling of 180 K. The images a-c were taken by a high-speed-camera at a rate of 3000 fps. The bright regions display the crystallizing volume of the levitated droplet at which latent heat is released. The propagation of the interface is clearly visible from the difference in emissivity of the crystallizing volume and the undercooled melt. The subfigures d-f depict the corresponding simulations. The solidification path, s, is indicated. Taken from [144].
After Herlach et al., surface-induced heterogeneous nucleation is the predominant process initiating solidification in levitated liquid specimens [119]. Generally, one can distinguish between triggered crystallization, where a needle touches the surface at a given undercooling, and spontaneous crystallization being caused by contaminations accumulating at the surface of the levitated undercooled specimens [119]. Once the crystallization starts from a single nucleus at the surface, dendrites propagate isotropically through the volume of the sample [119]. One obtains the measured growth velocity, u, from the solidification pathway, Δs, being the sample diameter divided by the time interval, Δt, necessary for the solidification front to propagate throughout the whole sample. Simulations are used to evaluate the images recorded by the high-speed-camera and to determine the solidification path, as Figs.
3.2.16d-f display. Δt is given by the number of recorded images and the frame rate of the
camera. The recorded thermal front is approximated by the envelope of the dendrite tips [119, 143] and hence the solidification front velocity is taken as the crystal growth rate. The work of Herlach et al. is recommended for more information to the ESL technique.
They give a comprehensive overview [142] to containerless undercooling methods.
An undercooling, ΔT , is present at the interface and the crystal growth rate depends on that undercooling, which can be expressed by four contributions within the framework of the sharp interface model [120, 145, 146, 147] (Eq. 3.2.12):
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ΔTk is the kinetic undercooling, ΔTt is the thermal undercooling, ΔTr is the curvature undercooling, and ΔTc is the solutal undercooling. All investigations of the crystal growth rate in the present thesis, are carried out on Cu50Zr50-based alloys. All contributions are briefly explained and, in the following, it will be discussed how and if they affect the overall undercooling at the interface of Cu50Zr50 -based droplets. Concentration gradients can be present in alloys as solute diffusion occurs ahead of the interface [134]. Solute can be rejected at the interface into the liquid under equilibrium conditions and a concentration gradient arises. If the liquidus temperature decreases with increasing solute concentration, then one speaks of solutal undercooling, ΔTc [119]. The temperature at the interface is lowered due to the higher solute concentration and protrusions are more likely to occur. The supercooled Cu50Zr50 liquid solidifies polymorphically into B2 CuZr, regardless of the undercooling (Fig. 5.4.23) [148]. Due to the polymorphic crystallization, there is no solutal undercooling ΔTc = 0 [141]. In order to determine the interface temperature during its advancement, Ivantsov [149] has conducted calculations under the assumption of equilibrium conditions at the parabolic interface of a needle-shaped pure metal crystal, whose interface is isothermal and moves at a constant velocity. His analysis yields stationary solutions of the transport equations that relate the thermal Péclet number to the undercooling, which he termed thermal undercooling, ΔTt [119, 135]:
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where Ti is the temperature at the interface, T∞m is the temperature of the undercooled liquid far away of the interface, Clpisthespecificheatoftheliquidatconstantpressure,I [28] is the Ivantsov-function and Pt = u·r/2·aL is the Péclet number. u and r are the velocity and radius of the parabolic interface, respectively and aL is the thermal diffusivity of the liquid.
Attachment kinetics of atoms from the liquid to the crystalline phase affect the crystal growth. According to the classical model of Wilson and Frenklel for continuous growth in a one-component system [124, 150, 151], these attachment processes are governed by atomic diffusion at the interface. Atoms have to overcome an activation barrier, ΔGa, to attach to the crystalline phase, so that they lower their Gibbs energy. The growth velocity, u, of the interface can be obtained from rate theory [125, 126] and under the assumption of polymorphic or eutectic growth, u is given by [152]
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where u0 is the maximum growth velocity within the diffusion-limited model of Wilson and Frenkel [150, 151]. At small undercoolings, ΔG/(R·Ti) ≪ 1, Eq. 3.2.14 can be approximated as linear in ΔG as follows [119]:
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Thus, the kinetic undercooling ΔTk = Tm − Ti for a planar interface with a temperature Ti, which advances at a constant velocity, u, is defined. µ, the proportionality constant, is also called the kinetic coefficient and is a measure for the mobility of the phase boundary [119].
At large undercoolings, the driving force for crystallization, ΔG, is very high and the exponential term in Eq. 3.2.14 can be neglected. Then u ≈ ukin, so that diffusion is the ratelimiting process [141]. Wilson [150] already assumed in 1900 that ukin would be inversely proportional to the viscosity of the liquid, η, and indeed this is verified experimentally [153] and can be expressed as follows [152, 136]:
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where a0 is the atomic diameter. At very large undercoolings as the liquid is cooled towards Tg, this simple inverse proportionality can break down. ukin is then higher as predicted by
η. Ediger et al. [128] have suggested that this decoupling is possibly related to a breakdown of the relationship between the diffusivity and viscosity. He describes this decoupling by a fractional Stokes-Einstein relation, where ukin ∝ η−ξ . The exponent ξ ≤ 1 and the value 1 represent full coupling of crystal growth and viscous flow [128].
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[...]
- Quote paper
- Konrad Kosiba (Author), 2017, Flash-Annealing of Copper-Zirconium-Aluminium based Bulk Metallic Glasses, Munich, GRIN Verlag, https://www.grin.com/document/369387
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