Internal Friction in Metallic Materials - A Handbook
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1. M.S. Blanter I.S. Golovin H. Neuhauser H.-R. Sinning
Internal Friction in Metallic Materials A Handbook With 65 Figures
and 53 Tables 123
2. Professor Dr. Mikhail S. Blanter Moscow State University of
Instrumental Engineering and Information Science Stromynka 20,
107846, Moscow, Russia E-mail: [email protected] Professor Dr.
Hartmut Neuhauser Institut fur Physik der Kondensierten Materie
Technische Universitat Braunschweig Mendelssohnstr. 3 38106
Braunschweig, Germany E-mail: [email protected] Professor Dr.
Igor S. Golovin Physics of Metals and Materials Science Department
Tula State University E-mail: [email protected] Lenin ave.
92, 300600 Tula, Russia Professor Dr. Hans-Rainer Sinning Institut
furWerkstoffe Technische Universitat Braunschweig Langer Kamp 8
38106 Braunschweig, Germany Series Editors: Professor Robert Hull
University of Virginia Dept. of Materials Science and Engineering
Thornton Hall Charlottesville, VA 22903-2442, USA Professor R.M.
Osgood, Jr. Microelectronics Science Laboratory Department of
Electrical Engineering Columbia University SeeleyW. Mudd Building
New York, NY 10027, USA E-mail: [email protected] Professor Jrgen
Parisi Universitat Oldenburg, Fachbereich Physik Abt. Energie- und
Halbleiterforschung Carl-von-Ossietzky-Strasse 911 26129 Oldenburg,
Germany Professor HansWarlimont Institut fur
Festkorper-undWerkstofforschung, Helmholtzstrasse 20 01069 Dresden,
Germany ISSN 0933-033X ISBN-10 3-540-68757-2 Springer Berlin
Heidelberg New York ISBN-13 978-3-540-68757-3 Springer Berlin
Heidelberg New York Library of Congress Control Number: 2006938675
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3. To our families
4. Preface Internal friction and anelastic relaxation form the
core of the mechanical spec-troscopy method, widely used in
solid-state physics, physical metallurgy and materials science to
study structural defects and their mobility, transport phe-nomena
and phase transformations in solids. From the view-point of
Mechan-ical Engineering, internal friction is responsible for the
damping properties of materials, including applications of high
damping (vibration and noise reduc-tion) as well as of low damping
(vibration sensors, high-precision instruments). In many cases, the
highly sensitive and selective spectra of internal friction (as a
function of temperature, frequency, and amplitude of vibration)
contain unique microscopic information that cannot be obtained by
other methods. On the other hand, owing to the large variety of
phenomena, materials, and related microscopic models, a correct
interpretation of measured internal fric-tion spectra is often
difficult. An efficient use of mechanical spectroscopy may then
require both: (a) a systematic treatment of the different
mechanisms of internal friction and anelastic relaxation, and (b) a
comprehensive compila-tion of experimental data in order to
facilitate the assignment of mechanisms to the observed phenomena.
Whereas the first of these two approaches was developed since more
than half a century in several textbooks and monographs (e.g.,
Zener 1948, Krishtal et al. 1964, Nowick and Berry 1972, De Batist
1972, Schaller et al. 2001), the second requirement was met only by
one Russian reference book (Blanter and Piguzov 1991), with no real
equivalent in the international literature. The present book,
partly based on the Russian example, is intended to fill this gap
by providing readers with comprehensive information about published
experimental results on internal friction in metallic materials.
According to this objective, this handbook mainly consists of
tables where detailed internal friction data are combined with
specifications of relax-ation mechanisms. The key to understand
this very condensed information is provided, besides appropriate
lists of symbols and abbreviations, by the introductory Chaps. 13:
after the Introduction to Internal Friction in Chap. 1, defining
and delimiting the subject and clarifying the terminology, the
relevant
5. VIII Preface internal friction mechanisms are briefly
reviewed in Chaps. 2 (Anelastic Relaxation) and 3 (Other
Mechanisms). Although somewhat more space is obviously devoted to
the former than to the latter, this part should not be understood
as a systematic analysis of the physical sources of anelasticity
and damping; in that respect, the reader is referred e.g., to the
above-mentioned textbooks. The data collection itself, as the main
subject of the book, can be found in Chaps. 4 and 5. The tables,
generally in order of chemical composition, include the main
properties of all known relaxation peaks (like frequency, peak
height and temperature, activation parameters), the relaxation
mechanisms as sug-gested by the original authors, and additional
information about experimental conditions. Other (e.g., hysteretic)
damping phenomena, however, could not be considered within the
limited scope of this book, with very few exceptions. Chapter 4,
which represents the main body of data on crystalline metals and
alloys, is divided into subsections according to the group of the
main metal-lic element in the periodic table, with alphabetic order
within each subsec-tion. Chapter 5 contains several new types of
metallic materials with specific structures, which do not fit well
into the general scheme of Chap. 4. A short summary or specific
explanations are included at the beginning of each table. Although
the authors made all efforts to be consistent in style throughout
the book, some difficulties in evaluating individual relaxation
spectra led to slight deviations, concerning details of data
presentation, between the different chapters and subsections. Since
some of the data were evaluated from figures, the accuracy should
generally be regarded with care; in cases of doubt, the original
papers should be consulted. Over 2000 references published until
mid 2006 were included, among which many earlier ones are still
important be-cause certain alloys and effects are not covered by
the more recent literature. Latest information, if missing in this
book, might be found in three confer-ence proceedings published in
the second half of 2006 (Mizubayashi et al. 2006b, Igata and
Takeuchi 2006, Darinskii and Magalas 2006), as well as in
forthcoming continuations of these conference series. This book is
intended for students, researchers and engineers working in
solid-state physics, materials science or mechanical engineering.
From one side, due to the relatively short summary of the basics of
internal friction in Chaps. 13, it may be helpful for
nonspecialists and for beginners in the field. From the other side,
its probably most comprehensive data collection ever published on
this topic should also be attractive for top specialists and
experienced researchers in mechanical spectroscopy and anelasticity
of solids. The authors acknowledge gratefully the help of Ms.
Tatiana Sazonova with the list of references, of Ms. Brigitte Brust
with figures, and of Ms. Svetlana Golovina with tables. We are also
grateful to the Springer team, in particular Dr. Claus Ascheron,
Ms. Adelheid Duhm and Ms. Nandini Loganathan, for good cooperation.
Moscow, Tula, Braunschweig Mikhail S. Blanter, Igor S. Golovin
January 2007 Hartmut Neuhauser, Hans-Rainer Sinning
97. 82 2 Anelastic Relaxation Mechanisms of Internal Friction
Table 2.17. Parameters of grain-boundary maxima in some pure metals
(f = 1 Hz) (after Ashmarin 1991) Me peak Tm (K) Tm/Tmelt H (kJ mol
1) Al A 480610 0.50.65 117160 C 795 0.85 294 Cu A 473638 0.350.47
113168 B 703820 0.520.60 189210 C 9251025 0.680.76 202263 Ni A
670783 0.390.45 185294 B 843893 0.490.52 217246 C 9531075 0.550.62
260328 A: low temperature peak; B: medium temperature peak; C: high
temperature peak. In pure metals three GB-induced damping maxima
can exist (Ashmarin 1991): (a) a low-temperature peak with Tm
(0.350.65)Tmelt (sometimes called Ke peak), (b) a
medium-temperature peak with Tm (0.50.6)Tmelt associated with
special grain boundaries but not observed in all metals, and (c) a
high-temperature peak with Tm (0.550.85)Tmelt observed in
coarse-grained samples. Examples of these peaks are displayed in
Table 2.17. In low-concentrated substitutional solid solutions we
may distinguish the low-temperature peak (at the same temperature
as in a pure metal) from an additional peak at higher temperature
(so-called impurity grain-boundary peak), which might be connected
with the aforementioned change in the rate-controlling mechanism.
The role of grain-boundary relaxation may become dominant in
materials with extremely fine grains, where the GB regions
constitute a substantial part of the total sample volume. These
nanocrystalline materials (produced e.g., by extreme plastic
deformation), with GB structures mostly far from equilibrium and
particular mechanical properties, may require special model
descriptions of deformation and anelastic relaxation beyond those
mentioned earlier, and will be considered separately in Sect.
2.4.3. 2.4.2 Twin Boundary Relaxation A twin boundary is a very
special type of grain boundary, separating two twin crystallites
that are, with respect to their lattice, mirror images of each
other (which is possible only at a well-defined misorientation
angle). If the twin boundary is identical with the mirror plane,
usually a low-indexed, close-packed crystallographic plane, it is
called a coherent twin boundary. Since the twin crystals can be
transformed into each other by a shear transfor-mation parallel to
the mirror plane, the formation of twins (twinning), which may
occur under sufficiently high stress or during recrystallisation
(in metals and alloys of low stacking-fault energy), represents an
additional deformation mechanism. Also the perpendicular shift of a
twin boundary (growth of one twin at the expense of the other)
means a shear deformation.
98. 2.4 Interface Relaxation 83 For anelastic relaxation and
internal friction peaks to occur by stress-induced movement of twin
boundaries, these boundaries must be sufficiently mobile. As
crystallographic coherency exists across the twin interfaces, the
relaxation mechanism cannot involve interfacial sliding (Nowick and
Berry 1972). However, certain types of twin boundaries can be
shifted as the result of movement of partial dislocations (Hirth
and Lothe 1968); then, the corre-sponding dislocation mechanisms
will be involved in twin boundary relaxation. Examples for the few
existing experiments are those by Siefert and Worrell (1951) on
Mn12at%Cu, De Morton (1969) and Postnikov et al. (1968b, 1969,
1970) on InTl alloys. Twinning is most frequently accompanying
diffusionless phase transfor-mations (e.g., from cubic to
tetragonal structure), which themselves involve a shear that can be
accommodated by twinning in order to retain the external shape and
to avoid high residual stresses in the sample. The high density of
twin boundaries often produced in this case may give rise to large
effects of anelastic relaxation and internal friction. The relation
to martensitic trans-formations will be treated in Sect. 3.2.1.
2.4.3 Nanocrystalline Metals Subject of this section are
polycrystals with ultra-fine grain sizes in the nanometer range.
Such nanocrystalline materials form a special group of
nanostructured materials (or nanomaterials) which also include
other types of nanosized structures in one, two or three
dimensions. Owing to the extremely rapid development of the field,
a generally accepted terminology of nanomaterials has not yet been
fully established. From the viewpoint of materials science,
nanostructured materials may be classified into different groups
according to the shape (dimensionality) and chemical composition of
their constituent structural elements (Gleiter 1995, 2000);
however, a less precise, synonymous use of terms like
nanocrystalline, nanostructured, or nanophase is also found in the
literature. There is also no commonly agreed grain size limit to
define nanocrystalline materials. In the physical concept of highly
disordered solids, it is the fraction of atoms situated in the
cores of defects (grain boundaries, interfaces) which should be as
high as possible. Under this viewpoint the grain size should be
below 10nm (Gleiter 1989, 2000), but also limiting values of 15, 20
or 30nm have been mentioned. Engineers developing new materials, on
the other hand, are sometimes using the prefix nano for length
scales almost up to 1 m, which generally lacks a physical
justification. An application-oriented delimitation of
nanocrystalline grain sizes should rather be linked to spe-cific
properties, which are expected to be different from those of
conventional materials if dominated by the high density of grain
boundaries. In some recent reviews, an upper limit of about 100nm
is introduced (Tjong and Chen 2004, Suryanarayana 2005), which
seems to be a reasonable compromise.
99. 84 2 Anelastic Relaxation Mechanisms of Internal Friction
Many different methods and techniques have been employed to produce
nanocrystalline metallic materials (n-Me), like inert gas
condensation, mechanical alloying, severe plastic deformation,
devitrification of amorphous precursors and many others. They all
have their specific advantages or draw-backs concerning the
compositions, properties or shapes (e.g., porous or fully dense,
bulk or thin film) of materials produced; for example, the
amorphous route is well established to produce nanocrystalline soft
magnetic alloys since the successful development of FINEMET
(Yoshizawa et al. 1988). With respect to improved mechanical
properties, severe plastic deforma-tion (SPD) is one of the most
important and widely used routes, as bulk and fully dense materials
with ultra-fine grain (UFG) structure can be obtained in this way
(Valiev et al. 2000, Mulyukov and Pshenichnyuk 2003). The
surpris-ingly high temperature stability of such UFG structures has
been attributed to a high GB diffusivity and low driving force of
recrystallisation (Valiev 2002). In many cases it is not clear,
however, whether there is a significant difference in GB
diffusivity between the nanocrystalline and annealed states
(Kolobov et al. 2001) or not (Wurschum et al. 2002). The small
grain sizes of genuine n-Me lead to distinct changes in the
mechanical properties including increased yield strength and
hardness. A particular feature is the breakdown of the HallPetch
relation at grain sizes around 20nm or below. In this range, a
decreasing grain size leads to anom-alous softening, referred to as
inverse HallPetch behaviour, which is associ-ated with the
operation of diffusion-controlled mechanisms combined with GB
sliding (e.g., Schitz et al. 1998, 1999, Yamakov et al. 2002a,b,
Van Swygen-hoven 2002, Van Swygenhoven et al. 2003). The cross-over
from normal to inverse HallPetch behaviour has been treated in a
two-phase model (Kim et al. 2000, Kim and Estrin 2005), in which
the grain boundaries deform by a diffusion mechanism, and the grain
interiors by a combination of dislocation glide and
diffusion-controlled mechanisms. Anelastic grain-boundary
relaxation (Ke 1999) is considered, in a recent theory of
non-equilibrium GBs (Chuvildeev 2004), to be hardly detectable in
UFG metals below a certain temperature (0.35Tmelt), unless the
dislocation density at the GBs is decreased. Alternatively,
disclination concepts have also been discussed in connection with
relaxation processes in n-Me (Romanov 2002, 2003). The conditions
for GB relaxation are even less clear in multi-component n-Me
produced by the amorphous route, where the same factors which
favour glass formation may also lead to stabilised and more densely
packed GB structures, being less susceptible to relaxation. This
latter type of n-Me (for which, to our knowledge, no systematic
studies of GB relaxation exist) is not included in the following
considerations, but will be mentioned further below in connection
with the respective amorphous alloys (Sect. 2.6.1). Experimental
studies of anelastic properties of n-Me were undertaken by several
research groups using different mechanical spectroscopy techniques.
The main results of some systematic and therefore reliable studies,
including the materials studied and the main effects observed, are
summarised briefly in
100. 2.4 Interface Relaxation 85 Table 2.18. In most of these
cases, the total temperature-dependent internal friction Q1(T) can
be written as 1(T) = Q Q 1 b (T) + Q 1 r (T), (2.46) 1 b (T) and Q1
where the terms Q r (T) represent a background of internal friction
and a superposition of different anelastic relaxation peaks,
respectively. The first term, closely related to composition and
microstructure of the respective alloy phases as well as to the
dislocation structure, can depend on the annealing time, as well on
the amplitude and frequency of the imposed oscillatory strain. This
contribution was reported to be substantial enough to consider
nanostructured Cu (Mulyukov and Pshenichnyuk 2003) and Mg
(reinforced by different microparticles, Trojanova et al. 2004) as
high-damping materials (see Sect. 3.5) even for low vibration
amplitudes. The second term, which may contain contributions not
only from GB relaxation but also from almost all relaxation
mechanisms related to dislo-cations or point defects, is
time-independent but frequency-dependent and can often be described
by the Debye equation. Because of the lack of a com-bined study of
nanostructured metals by different mechanical spectroscopy
techniques, varying as many experimental parameters (frequency,
tempera-ture, amplitude, annealing conditions) as possible, it is
not easy to distinguish between pure anelastic relaxation
mechanisms (most important: GB relax-ation) and irreversible
mechanisms of structural relaxation, which are in most cases due to
changes in the density and distribution of dislocations.
Summarising the pertinent results published in the literature
(partly pre-sented in Table 2.18), one can draw the following
conclusions: Almost all UFG and nanostructured metals (except those
produced from the amorphous state, see above) exhibit an IF peak
(very roughly with an activation energy of about 1 eV), which is
not often found in well annealed (coarse-grained) metals. The
nature of this peak is still not entirely clear: some authors
report a thermally activated, reversible anelastic GB relaxation
consistent with the Ke approach (Ke 1999), while others attribute
the effect to irre-versible structural changes like recovery and
recrystallisation, connected with short-range GB diffusion in
non-equilibrium GB. Internal friction can generally be correlated
with superplastic proper-ties and thus can be used for determining
the optimum temperature for superplastic deformation. Structural
changes in severely deformed metals occur already around ambient
temperature, as indicated by a group of low-temperature IF peaks
observed after high-pressure torsion in Ti, Mg and several Fe-based
alloys, which is extremely sensitive to heating. In some
SPD-processed metals like Cu, a high damping capacity was reported
in a broad range of strain amplitude and temperature, however, has
not been reproduced all published works (see Sect. 3.5).
101. 86 2 Anelastic Relaxation Mechanisms of Internal Friction
Table 2.18. Mechanical spectroscopy studies of UFG metallic
materials materials mechanical spectroscopy short summary
references Pd 1 TDIF, 15 Hz. Several IF peaks. IF peak H =
62.7kJmol 1 (reordering phenomena) Weller et al. (1991) Cu: 99.997%
99.98% 2 TDIFat 10 Hz and 100 kHz; TD- and ADIF 5MHz. IF peak 420
K: reversible dynamic GB rearrangement Akhmadeev et al. (1993) Au
99.99% 3 TDIF, 300500 Hz: Bordoni peak 120 K, dislocation peak 460
K, GB peak 750 K. Okuda et al. (1994) Al, Ni 4 TDIF, 130 Hz ,
0.01200 Hz, 0.110 kHz. Relaxation IF peak 159 kJ mol 1 (Al: 475K 1
Hz, GB relaxation) Bonetti and Pasquini (1999) Cu (99.98%) Ni
(99.98%) 2 TDIF, 17 Hz and 1kHz (time-dependent IF). Irreversible
changes in the structure Gryaznov et al. (1999) Cu, Fe18Cr9Ni 5
TDIF, 2.5Hz, ADIF, 35 Hz. High damping; IF peaks at 54 and 475K
Mulyukov and Pshenichnyuk (2003) Mg 6 TDIF, 0.5, 5, 50 Hz.
Relaxation 1 due to peaks at 70K (116 kJ mol dislocations) and at
620K due to GB Trojanova et al. (2004) Mg alloys: Mg6Zn Mg9Al 2
TDIF, 10 Hz. Irreversible IF peak 530570 K, 87 kJ mol 1 enhanced GB
diffusion Chuvildeev et al. (2004a) Fe25Ni 7 TDIF, 0.55 Hz. IF
peaks due to martensitic transformation Wang et al. (2004a) Fe0.8C
5 TDIF, 12 kHz. Irreversible IF peak 550 K: recovery Ivanisenko et
al. 2004 Fe25Al 5 TDIF, 0.52 kHz. IF peaks (150300 K) due to
dislocations and self-interstitial atoms; unstable with respect to
heating. Golovin et al. (2006a,c) Ti grad2 5 TDIF, 2 kHz. IF
(Hasiguti) peak 210 K: dislocations and self-interstitial atoms;
possibly hydrogen-related effect at 410 K. Golovin et al. (2006a)
Fabrication methods: (1) evaporation, condensation, compaction; (2)
equal channel angular pressing (ECAP); (3) gas deposition; (4) mall
milling; (5) high-pressure torsion; (6) ball milling, compaction,
hot extrusion; (7) consolidation.
102. 2.5 Thermoelastic Relaxation 87 2.5 Thermoelastic
Relaxation 2.5.1 Theory Physical Principle In every solid, there
exists a fundamental thermoelastic coupling between the thermal and
mechanical states (e.g., between stress and temperature fields),
with the thermal expansion coefficient as the coupling constant.
The best known phenomenon of thermoelastic coupling is thermal
expansion, as the response of the mechanical state to an applied
change in temperature. Con-versely, fast adiabatic (i.e.,
isentropic) changes of the dilatational stress com-ponent result in
(small) temperature changes, known as the thermoelastic effect. If
such stress variations are spatially inhomogeneous either
exter-nally according to the mode of loading (e.g., bending) or
internally in a material with heterogeneous mechanical properties
temperature gradients are produced which can then relax by
irreversible heat flow (thermoelastic relaxation), causing entropy
production and dissipation of mechanical energy. The resulting
thermoelastic damping1 not to be confused with damping due to
thermoelastic martensite2 (Sect. 3.2.1) represents the most
fundamen-tal among all mechanical damping mechanisms, since it does
not require any defects but exists in all solids with non-zero
thermal expansion, even in the most perfect crystals. Assuming that
the mean free path of the phonons is small compared to the length
scale of the stress inhomogeneities, which is generally the case
except for very low temperatures and high frequencies, the heat
flow during thermoelastic relaxation can be described as a
classical diffusion process. Biot (1956) pointed out that it is the
entropy which satisfies the diffusion equation. Zeners Theory
Thermoelastic damping is known since the late 1930s, when Zener was
the first to give both a detailed theory (Zener 1937, 1938b) and a
collection of related experimental results (Zener et al. 1938,
Randall et al. 1939). The theory was developed in scalar
(one-dimensional) form mainly for the transversal vibra-tion of
homogeneous reeds and wires, but some other cases like spherical
cav-ities or polycrystals with randomly oriented crystallites were
also considered 1 As a fundamental thermodynamic phenomenon,
thermoelastic damping is some-times also referred to as
thermodynamic damping (e.g., Panteliou and Dimarogonas 1997, 2000).
Other authors have called it elastothermodynamic (Bishop and Kinra
1995, 1997; Kinra and Bishop 1996) because the cause is elastic and
the effect is both thermal and dynamic (i.e., time-dependent). 2 A
martensitic transformation is called thermoelastic if its thermal
hystere-sis and transformation energy is relatively small,
comparable in magnitude with usual elastic strain energies. This
alternative use of the term thermoelastic has nothing to do with
thermoelastic coupling considered here.
103. 88 2 Anelastic Relaxation Mechanisms of Internal Friction
by Zener. The simplest and best described case is certainly that of
alternating transverse thermal currents (Nowick and Berry 1972)
between the compressed and dilated sides of a homogeneous and
isotropic, rectangular beam, vibrating in flexure with the
frequency f. The thermoelastic damping of such a beam is in good
approximation given by 1(f, T) = T Q f f0 f2 + f2 0 (2.47) with the
relaxation strength T = 2EUT/Cp (2.48) and the peak frequency f0 =
/2h2Cp, (2.49) where is the linear thermal expansion coefficient,
EU the unrelaxed Youngs modulus, the density, Cp the specific heat
capacity at constant pressure (or stress)3, the thermal
conductivity and h the thickness of the beam (i.e., the distance
over which heat flow occurs). Equation (2.47) has the same
functional form as (1.8) and represents a Debye peak as a function
of frequency, with a single relaxation time T = 1/2f0 = h2/2Dth
(2.50) where Dth = /Cp is also called the thermal diffusivity. The
analogy between (2.50) for the thermoelastic and (2.17) for the
Gorsky relaxation, respectively, reflects the more general analogy
between thermal and atomic diffusion already pointed out by Zener
(1948). In the same way, the intercrystalline Gorsky effect
introduced in (2.18) and (2.19) is analogous to the case of
intercrystalline thermal currents (Zener 1948, Nowick and Berry
1972) with IT = R(3)2KUT/Cp (2.51) IT = d2/32Dth, (2.52) where R is
an elastic anisotropy factor (see Zener 1938b for an estimate for
cubic metals with randomly oriented crystallites), 3 denotes the
volumetric expansion coefficient, KU the unrelaxed bulk modulus and
d the dominating grain size in the polycrystal. Despite this
analogy between atomic and thermal diffusion, it should be noted
that the Arrhenius relation of thermal activation, (1.9), only
holds for 3 Here we understand Cp per unit mass as found in most
data collections; if con-sidered per unit volume as in Zeners
original equations, the density does not appear in these equations.
Instead of Cp, the symbol C (for constant stress) has also been
used in the literature. If, on the other hand, Cp or C is replaced
by C or C (at constant volume or strain), a small error in T (of
the order of T 2) is introduced (Lifshitz and Roukes 2000).
104. 2.5 Thermoelastic Relaxation 89 the former but not for the
latter having a comparatively weak temperature dependence. Thus,
unlike the Gorsky relaxation, the thermoelastic Debye peak is found
only as a function of frequency but not of temperature. Instead,
both T /T in (2.48) and f0 in (2.49) are only weakly
temperature-dependent (at least above the Debye temperature), so
that thermoelastic damping is nearly proportional to the
temperature. Another possibility of thermoelastic damping is
related to longitudinal thermal currents between the hills and
valleys of longitudinal elastic waves. In this case, treated in
some detail by Lucke (1956), the relaxation time is itself
frequency-dependent as 2 because the thermal diffusion dis-tance is
given by half the wavelength, which means that in contrast to the
normal case adiabatic conditions are expected here in the
low-frequency (!) limit. With expected peak frequencies in the GHz
range or even higher, longi-tudinal thermoelastic damping is
usually negligibly small (Nowick and Berry 1972). Advanced Theories
More extended and fundamental, three-dimensional and mathematically
more rigorous treatments can be found in many later theoretical
papers (e.g., Biot 1956, Alblas 1961, 1981, Chadwick 1962a,b, Lord
and Shulman 1967b, Kinra and Milligan 1994, Lifshitz and Roukes
2000, Norris and Photiadis 2005). However, although the general
thermoelastic equations and also some specific solutions (most
often for the transversely vibrating EulerBernoulli beam) are now
well known, it is up to the present date still difficult to
calculate the thermoelastic damping explicitly for more complex
cases beyond those already treated by Zener. An exact solution for
the thin EulerBernoulli beam was given by Lifshitz and Roukes
(2000), who also showed that Zeners approximation is valid within
2% in most of the relevant frequency range, except for the
high-frequency side of the peak far above f0 where the deviations
grow up to a 20% underestimation in the limit f . Therefore, the
still widely spread use of Zeners (2.47)(2.49) is sufficiently
accurate for many practical purposes, at least in the classical
case of transverse thermal currents during flexural vibration of
homogeneous samples. The analysis of Kinra and Milligan (1994)
formed the basis for further model calculations of thermoelastic
damping also in heterogeneous structures like fibre- or
particle-reinforced composites (Milligan and Kinra 1995, Bishop and
Kinra 1995), hollow spherical inclusions (Kinra and Bishop 1997),
lami-nated composite beams (Bishop and Kinra 1993, 1997; Srikar
2005b) or some specific cases of pores and cracks (Kinra and Bishop
1996, Panteliou and Dimarogonas 1997, 2000; Panteliou et al. 2001).
In the special case of flexural resonators made of polycrystals
(e.g., of silicon) with particularly low thermal conductivity
across the grain boundaries compared to that in the crystals,
105. 90 2 Anelastic Relaxation Mechanisms of Internal Friction
a preliminary fast equilibration of the transverse thermal currents
is possible inside the grains, which has been called
intracrystalline thermoelastic damp-ing (Srikar and Senturia 2002).
Another branch of theories is devoted to resonators with more
complex external shape, usually in form of planar structures made
of thin, flat plates vibrating predominantly either in flexure or
in torsion. Although the ther-moelastic loss should be zero in case
of pure shear, it is important to note that even the nominally
torsional vibration modes almost always contain some flexural
component which can produce significant thermoelastic damping. To
solve this problem, a flexural modal participation factor (MPF) has
been defined as the fraction of potential elastic energy stored in
flexure (Photiadis et al. 2002, Houston et al. 2002, 2004).
Assuming classical transverse ther-mal currents for this flexural
component, the thermoelastic damping of any particular vibration
mode is then obtained by multiplying the MPF with the classical
result for the flexural beam e.g., from Zeners theory. The MPF
itself can be calculated by integrating the curvature tensor of the
vibration mode over the volume of the sample, provided the
displacement field of the mode is known (Norris and Photiadis
2005). The problem then mainly consists of determining the mode
shape, e.g., with the help of finite element modelling and/or
advanced experimental techniques like laser-Doppler vibrometry (Liu
et al. 2001). 2.5.2 Properties and Applications of Thermoelastic
Damping To judge the practical importance of thermoelastic damping
in a given mate-rial, we have to consider primarily the magnitude
of the transverse relaxation strength T and the related peak
frequency f0 according to (2.48) and (2.49). A detailed compilation
of room-temperature relaxation strengths, including results of four
data collections from the literature as well as re-calculated data
using (2.48), is given in Table 2.19 for many pure metals and also
a limited number of non-metallic materials. It is typical that in
Table 2.19 the T values taken from different sources never match
exactly. This scatter may come from unspecified microstructural
influences (defects, textures) causing some variation mainly in the
possibly anisotropic quantities and E, among which deviations in
have a partic-ularly strong effect due to the quadratic dependence
in (2.48). For our own re-calculations of T , the underlying basic
data were checked for reliability by comparing different sources
wherever possible. Ideally, the data in Table 2.19 refer to random
polycrystals at least in case of metals. Exceptions are Si and Ge
where single crystal values are given, according to the
[100]-oriented wafers from which most of the respective resonators
are fabricated. The second practically important quantity is the
peak frequency f0 or, vice versa, the sample thickness h(f0) which
belongs to a pre-selected peak frequency according to (2.49). With
thermal diffusivities Dth usually in the
106. 2.5 Thermoelastic Relaxation 91 calc calculated from Table
2.19. Thermoelastic relaxation strengths T of pure metals and some
other selected materials at 300 K: T Lit taken directly from the
literature the intrinsic properties , E, and Cp,andT reference for
T Lit 103 T 103 1) 1 K 3) 1) 6 K calc material (10 E (GPa) (kgm Cp
(J kg T Lit Ag 18.9 83 10490 235 3.6 2.43.5 Zener (1948), Riehemann
(1996), Srikar (2005b) Al 23.1 70 2700 904 4.7 4.65.1 Zener (1948),
Kinra and Milligan (1994), Riehemann (1996), Srikar (2005b) Al2O3
2.73.7 Kinra and Milligan (1994), Srikar (2005b) Au 14.2 78 19300
129 1.9 1.72.2 Zener (1948), Kinra and Milligan (1994), Riehemann
(1996), Srikar (2005b) Be 11.3 287 1850 1820 3.3 4.6 Zener (1948)
Bi 13.4 32 9780 122 1.44 1.4 Zener (1948) Cd 30.8 50 8650 231 7.1
10 Zener (1948) Co 13.0 209 8900 421 2.8 3.4 Riehemann (1996) Cu
16.5 130 8920 384 3.1 3.03.7 Zener (1948), Riehemann (1996), Srikar
2005b Fe 11.8 211 7874 449 2.5 2.22.6 Zener (1948), Riehemann
(1996), Srikar 2005b Ge 6.0 100 (Srikar 2005b) 5323 321 0.63 0.45
Srikar 2005b In 32.1 11 7310 233 2.0 3.1 Riehemann (1996) Ir 6.4
528 22650 131 2.2 2.35 Riehemann (1996) Mg 25 (Weast 1973) 45 1738
1025 4.7 4.85.4 Zener (1948), Kinra and Milligan (1994), Riehemann
(1996) Mo 4.8 329 10280 251 0.88 0.86 Riehemann (1996)
107. 92 2 Anelastic Relaxation Mechanisms of Internal Friction
Table 2.19. Continued reference for T Lit 103 T 103 1) 1 K 3) 1) 6
K calc material (10 E (GPa) (kgm Cp (J kg T Lit Nb 7.3 105 8570 265
0.74 0.71 Riehemann (1996) Ni 13.4 200 8908 445 2.7 2.62.9 Zener
(1948), Riehemann (1996), Srikar (2005b) Pb 28.9 16 11 340 127 2.8
2.52.8 Zener (1948), Riehemann (1996) Pd 11.8 121 12 023 244 1.72
2.02.5 Zener (1948), Riehemann (1996) Pt 8.8 168 21 090 133 1.39
1.5 Zener (1948) Rh 8.2 275 12 450 243 1.8 0.7 Zener (1948) Sb 11.0
55 6697 207 1.44 1.51.8 Zener (1948), Riehemann (1996) Si 2.6 160
(Srikar 2005b) 2330 712 0.2 0.19 Srikar (2005b) SiC 0.350.6 Kinra
and Milligan (1994), Srikar (2005b) Si3N4 0.22 Srikar (2005b) SiO2
0.003 Srikar (2005b) Sn 22.0 50 7310 227 4.4 4.04.8 Zener (1948),
Riehemann (1996) Ta 6.3 186 16 650 140 0.95 0.3 Zener (1948) Ti 8.6
116 4507 522 1.1 0.81.2 Kinra and Milligan (1994), Riehemann
(1996),Srikar (2005b) TiC 1.4 Kinra and Milligan (1994) W 4.5 411
19 250 132 0.98 0.81.3 Zener (1948), Riehemann (1996), Srikar
(2005b) Zn 30.2 108 7140 388 10.7 5.818(!) Zener (1948), Riehemann
(1996), Srikar (2005b) Zr 5.7 68 6510 278 0.37 0.68 Riehemann
(1996) References: unless noted otherwise, the intrinsic properties
, E, and Cp were taken from WebElements [http://www.
webelements.com/].
108. 2.5 Thermoelastic Relaxation 93 range of 106 to 104 m2 s1,
samples have to be prepared mostly with thick-nesses between 0.05
and 0.5mm in order to have maximum thermoelastic damping at 1 kHz.
Metallic Materials In Table 2.19 the strongest effect is predicted
for Zn with a relaxation strength as high as 0.01 (according to T
calc) and a maximum thermoelastic damping Qm 1 = T /2 0.005,
followed by Cd, Al, Mg, Sn; but also for Ag, Be, Co, Fe, Ni and Pb
the thermoelastic loss factor at room temperature can exceed 103.
Although such values are easily observable and practically
significant, the interest in thermoelastic damping of metals has as
yet been rather limited from both the fundamental and applied
sides, and systematic experimental studies are very scarce. On the
fundamental side, mechanical spectroscopy is usually concerned with
thermally activated relaxation peaks, measured as a function of
tempera-ture to study defects and transformations in solids.
Thermoelastic damping is then noticed mainly as a linear background
to be subtracted, but very rarely studied for its own sake. This
has also experimental reasons: to trace out the full peak after
(2.47), flexural frequency and sample thickness have to be mutually
adjusted and varied accordingly, e.g., over at least two orders of
magnitude in frequency, which requires more effort than just
varying the temperature on a single sample. In addition, to observe
the pure thermoelas-tic losses, other kinds of damping have to be
suppressed effectively e.g., by suitable alloying. Only in the
early days before many other mechanisms were known thermoelastic
damping in metals was a subject of intense study as a main source
of energy dissipation. The probably still most careful measurements
of the thermoelastic relaxation peak come from that time, like the
study of Bennewitz and Rotger (1938) on German silver, and in
particular that of Berry (1955) on -brass which gave an
impressively exact confirmation of Zeners theory of transverse
thermal currents without any adjustable para-meters (see also
Nowick and Berry 1972). Based on this fundamental work, the height
and position of the thermoelastic peak were occasionally used later
to determine coefficients of thermal expansion and conductivity,
respectively, e.g., for some metallic glasses (Berry 1978, Sinning
et al. 1988) or commercial Al and Mg alloys (Goken and Riehemann
2002). As an example, Fig. 2.33 shows the annealing-induced shift
of the thermoelastic Debye peak, according to an increase in
thermal conductivity from 7 over 11 to 17WmK1, due to structural
relaxation and subsequent crystallisation of an amorphous Ni alloy
(Sinning et al. 1988). In this case, the measurement temperature
had been lowered to 170K to reduce the amount of other damping
contributions, partly still visible in Fig. 2.33 on the
low-frequency side of the peak for the as-quenched state;
therefore, the thermoelastic peaks in Fig. 2.33 are almost a factor
of two smaller than they would be at room temperature.
109. 94 2 Anelastic Relaxation Mechanisms of Internal Friction
Fig. 2.33. Frequency-dependent internal friction of a rapidly
quenched, meltspun Ni78Si8B14 ribbon (thickness h = 0.05mm) at T =
170K after different annealing treatments (the solid lines are fits
to (2.47)): (a) as-quenched amorphous state, f0 = 1200 Hz; (b)
after 2 h at 618K (structurally relaxed amorphous state), f0 = 1730
Hz; (c) crystallised, f0 = 2750 Hz (Sinning et al. 1988) Also worth
mentioning in this context is the early work of Randall et al.
(1939) on -brass with systematically varied grain sizes, which
seems still to represent the only known example of a reliable
observation of intercrystalline thermal currents. On the side of
application, the main problem is that damping due to trans-verse
thermal currents is available only in a relatively narrow frequency
range around f0, depending on the geometry of the respective
structural compo-nent. On the other hand, it might be possible in
certain cases to adjust the geometrical dimensions or the thermal
conductivity (by alloying) according to the technical requirements
of damping properties. Much more interesting from the applied
viewpoint are those thermoelas-tic damping contributions that occur
in heterogeneous metallic materials like composites or porous
metals. Three types of effects may be expected from such
heterogeneities: 1. The introduction of new internal length scales,
in addition to the sample dimensions, will distribute the
dissipation processes over a much wider frequency range. This
effect has been discussed qualitatively for metallic foams (Golovin
and Sinning 2003b, 2004). 2. Thermoelastic damping will no longer
be confined to flexural vibrations but will occur also in other
deformation modes. 3. Additional heterogeneities cause additional
temperature gradients and thus additional dissipation processes,
i.e., more damping will be produced. This is the most promising but
also least understood aspect: in fact, model calculations for
specific arrays of pores (Panteliou and Dimarogonas 1997, 2000)
have predicted a strong increase of thermoelastic damping with
porosity, up to much higher values than in the case of classical
trans-verse thermal currents; but the consequences for real
materials are not yet clear. There is a strong need for theoretical
as well as experimental
110. 2.6 Relaxation in Non-Crystalline and Complex Structures
95 research in this field, which then might open new perspectives
towards the development of heterogeneous metallic materials with
tailored properties of thermoelastic damping. Applications in
Microsystems The recently renewed interest in thermoelastic damping
is, in its main part, not related to the aforementioned
perspectives of metallic materials but has a completely different
reason: the rapid development of micro- and nano-electromechanical
systems (MEMS and NEMS) which include silicon-based micromechanical
resonators as central elements. Irrespective of the specific
application (e.g., force sensors, accelerometers, bolometers,
magnetometers, high-frequency mechanical filters or ultrafast
actuators), the performance of the micromechanical system (e.g.,
sensor sensitivity) critically depends on the quality factor Q of
the resonator which should be as high (i.e., the damping Q1 as low)
as possible. That is, contrary to the metallic case discussed
ear-lier, the aim is here not to produce damping but to avoid it.
If in the most perfect silicon resonators all defect-induced
sources of dissipation are removed, the thermoelastic damping
remains and can be influenced only by a proper geometrical design
and fabrication of the resonator. Especially with more
complex-shaped resonators like single- or double-paddle oscillators
(Kleiman et al. 1985, Liu et al. 2001, Houston et al. 2004)
attempting quality factors as high as 108, or in case of layered
structures including metallic or ceramic coatings (Srikar 2005b),
this is not a trivial task. Since most of the recent theoretical
progress on thermoelastic damping since about 2000 (see above) was
without doubt strongly motivated by the needs of MEMS and NEMS, we
have briefly sketched these important new developments here
although their basic material, silicon, is as a semicon-ductor not
included in the main data collections of this book. Finally, it
should be mentioned as well that thermoelastic damping is also an
important factor limiting the ultimate sensitivity of
interferometric gravi-tational wave detectors (Black et al. 2004).
2.6 Relaxation in Non-Crystalline and Complex Structures With the
important exception of the universal thermoelastic damping treated
in the preceding section, most mechanisms of anelastic relaxation
comprise the motion of defects interacting with an applied stress.
According to the classical understanding of defects as structural
imperfections in (periodic) crystals, such relaxation mechanisms
are traditionally defined for crystalline solids (Nowick and Berry
1972). This classical line was also followed in the Sects. 2.22.4
on point defects, dislocations and interfaces, where the
respec-tive microscopic processes of relaxation were introduced for
the crystalline
111. 96 2 Anelastic Relaxation Mechanisms of Internal Friction
case. An extension of such defect-related mechanisms to
non-crystalline struc-tures is not obvious, except for some special
cases like interstitial diffusion jumps of hydrogen atoms (if not
coupled with the motion of matrix defects, see Sect. 2.2.4). In
this context, the term non-crystalline is traditionally understood
as opposed to periodic crystals, which then includes both amorphous
solids and quasicrystals. To some extent this is still common
practice (and practically useful), although it deviates from
crystallographically correct terminology. In proper
crystallographic terms, quasicrystals are in fact crystals in the
wider sense of quasiperiodic crystals, which include both periodic
and aperiodic, long-range ordered structures (Lifshitz 2003). From
the viewpoint of anelastic relaxation of metals, on the other hand,
quasicrystals and amorphous structures have many things in common,
at least in case of icosahedral short-range order (cf. Sect.
2.2.4). There is a borderline, however, between common periodic
crystals (in most practical cases with rel-atively simple crystal
structures) on the one side, and other metallic structure types
amorphous alloys, quasicrystals and to some extent even
structurally complex periodic crystals with giant unit cells (Urban
and Feuerbacher 2004) on the other side: in the former case, most
defect-related mechanisms are quantitatively well understood and
classified within the systematic and well-founded concepts of
anelastic relaxation in crystalline solids (Nowick and Berry 1972;
at that time crystals were always understood as periodic
crys-tals), whereas in the latter case many details of the
theoretical concepts have still to be developed. In principle we
may distinguish roughly, in relation to the classical relax-ation
processes in crystalline solids, between three types of relaxation
mech-anisms in non-crystalline structures (in the above traditional
meaning including quasicrystals): (a) Mechanisms which are
independent of the structure type and exist in the same way in
crystalline as well as in non-crystalline structures, with only
numerical differences. Examples are thermoelastic damping and the
Gorsky effect (at least in the basic form of transverse thermal or
atomic diffusion currents), where relaxation strength and time may
vary according to the values of the respective parameters, but all
essential characteristics of the relaxation remain unchanged. (b)
Mechanisms which are modified by the structure type, i.e., which
are based on the same principle but with some conceptual
differences calling for a modified or extended theoretical
treatment. Examples are the Snoek-type relaxation in the
generalised form as introduced for hydrogen in Sect. 2.2.4, or a
hypothetical dislocation relaxation in an amorphous structure which
can only be treated using a more general dislocation concept
(independent of a crystal lattice). (c) Mechanisms which are
specifically found in non-crystalline but not in (simple)
crystalline structures. Examples are cooperative processes of
112. 2.6 Relaxation in Non-Crystalline and Complex Structures
97 directional structural relaxation or viscous flow (e.g., near
the glass tran-sition) in metallic glasses, or some types of
relaxation related to phasons in quasicrystals. While there is no
reason to mention again type (a), we will focus in the following on
mechanisms of types (b) and in particular (c) which can not always
be differentiated clearly from each other. The aim is to give an
intro-duction into those aspects of anelastic/viscoelastic
relaxation in amorphous (Sect. 2.6.1) and quasicrystalline (Sect.
2.6.2) structures that have not yet been considered in the previous
parts of this chapter. 2.6.1 Amorphous Alloys The most important
aspect to be considered in amorphous alloys, also called metallic
glasses, is the relation between structural and mechanical
relaxation which are closely connected. To discuss this relation,
it is first necessary to know the a-priori different definitions
and characteristics of both kinds of relaxation. Since mechanical
(anelastic or viscoelastic) relaxation has already been introduced
in Chap. 1, a brief introduction into structural relaxation will be
given here. Structural Relaxation In the literal sense, any
time-dependent equilibration of the atomic structure of condensed
matter, after any kind of external perturbation, may be called
structural relaxation (SR). This may in principle include
production, anni-hilation and rearrangement of defects in crystals
(like equilibration of thermal vacancies after changes in
temperature, or recovery and recrystallisation after plastic
deformation or irradiation), and even certain cases of phase
transfor-mations. However, it is more common to use the name
structural relaxation more specifically for continuous changes of
amorphous structures in partic-ular in glass-forming systems which
are not so easily expressed in terms of defect concentrations but
rather appear as integral modifications of the whole structure. For
instance, temperature changes generally give rise to SR due to the
temperature dependence of amorphous structures in (stable or
metastable) equilibrium. The Glass Physics Approach Understanding
SR in glass-forming systems is the key to understand glass per se,
i.e., the formation and nature of glasses and the glass transition
below which SR is largely frozen. According to many renowned
experts, this is still the most challenging unsolved problem in
condensed matter physics. The dif-ficult task of summarising the
state of knowledge in this complex field was
113. 98 2 Anelastic Relaxation Mechanisms of Internal Friction
tackled by Angell et al. (2000), by posing detailed key questions
and review-ing the best answers available as given by experts and
specialists in about 500 references. The subject was divided into
four parts, i.e., three tempera-ture domains AC with respect to the
glass transition temperature Tg, and a fourth part D dealing with
short time dynamics which can be skipped here. The main emphasis in
the review by Angell et al. (2000) is put on the high-temperature
domain A of the (supercooled) viscous liquid at T > Tg where the
system is ergodic (i.e., its properties have no history
dependence). Impor-tant items to be understood are the temperature
dependences of transport properties and relaxation times, e.g., in
form of the VogelFulcherTammann (VFT) equation and deviations from
it, as well as non-exponential relaxation functions of the form
exp[(t/ )] with 0 < < 1 (KohlrauschWilliams Watts (KWW) or
stretched exponential function, which was given a physical meaning
e.g., by Ngais coupling model of cooperative many-body molecular
dynamics (Ngai et al. 1991, Ngai 2000)). The VFT equation, e.g.,
for the viscosity , can be written as = 0 exp[D T0/(T T0)], (2.53)
with the so-called fragility parameter D and VFT temperature T0,
which are coupled with respect to the glass transition according to
Tg/T0 = 1+D / ln(g/0) 1 + D /39, (2.54) where g and 0 represent the
viscosities at T = Tg and T , respec-tively (Angell 1995). The
fragility parameter D is used to distinguish between strong liquids
or glasses with large D and almost Arrhenius-like behaviour (which
would be exact for D = implying T0 = 0), and fragile ones with
small D, a pronounced curvature in a Tg-scaled Arrhenius plot, and
a very rapid breakdown of shear resistance on heating directly
above Tg. A similar temperature dependence is also found for the
relaxation time , which in this range A is so short that the
structure can generally be considered to be in a relaxed state of
internal equilibrium. The low-temperature domain C of the truly
glassy state (T Tg), on the opposite side, can be defined as the
range where the cooperative SR of the viscous liquid (also called
main, primary or relaxation) is completely frozen. Here the
properties change essentially reversibly with temperature (as they
do in range A) but now depend strongly on history, i.e., on the
initial time-temperature path on which the system was frozen.
Relaxation in this glassy range is possible only by decoupled,
localised motion of easily mobile species (also called secondary
relaxations4). 4 These secondary relaxations are sometimes
classified further as , , , . . . relaxations, which is more
appropriate for polymers where the stepwise freez-ing of various
local degrees of freedom may be associated with specific molecular
groups, than for anorganic or metallic systems.
114. 2.6 Relaxation in Non-Crystalline and Complex Structures
99 In the intermediate temperature domain B near and not too far
below the glass transition (T Tg), primary SR must be considered
explicitly as it occurs continuously on all experimental time
scales, but without reaching equilibrium except for long annealing
times. This is the most difficult range in which structure and
properties depend on both history and actual time during the
measurement. A first approach relies on the principle of
thermorheologi-cal and structural simplicity (Angell et al. 2000)
which relates the molecular or atomic mobility to the structural
departure from equilibrium, as described by a single parameter like
the so-called fictive temperature Tf . As depicted in Fig. 2.34,
the fictive temperature can be found by projecting the actual value
of a certain property p (like volume, enthalpy, entropy, etc.) on
the equilibrium curve for the liquid extrapolated from range A,
using the slope p/T from the frozen range C. Structural relaxation
in range B can then be described as a relaxation of Tf , in the
simplest case according to T f = (T Tf )/ (2.55) with limiting
conditions Tf = T in range A and Tf = const. in range C,
respectively. The relaxation time now depends on both T and Tf, as
expressed first by Tool (1946) (T,Tf) = 0 exp[xA/kT + (1 x)A/kTf
)], (2.56) where x is a dimensionless non-linearity parameter (0
< x < 1, typically x 0.5), and A is an activation energy
(Jackle 1986, Angell et al. 2000). Fig. 2.34. Definition of the
fictive temperature Tf in different relaxing or frozen glassy
states: (1) during and (2) after rapid cooling, (3) during slow
cooling, (4) during heating after slow cooling. Indicated are also
the temperature ranges AC (Angell et al. 2000; see text). For
frozen states like in case (2), Tf may be considered identical with
Tg for a given heating or cooling rate
115. 100 2 Anelastic Relaxation Mechanisms of Internal Friction
In this simple form the fictive temperature concept has been useful
for mod-elling relaxation in the difficult temperature range B;
however, some ambiguity remains as regards which property p is
chosen, and also the non-exponentiality (KWW function), found here
as well, is not accounted for. The latter point is addressed by
more advanced concepts like that of hierarchically constrained
dynamics, considering elementary atomic relaxation events to occur
not in parallel but in series (Palmer et al. 1984). The link in
relaxation dynamics between ranges A and B is also underlined by
correlations between the para-meters , D, A and x (Angell et al.
2000). Up to this point, the synopsis of SR under the viewpoint of
glass physics applies to all kinds of glasses (polymers, metals,
oxides), necessarily neglecting more specific aspects in these
different classes of materials. In particular, for certain
characteristics of SR in metallic glasses, some different
viewpoints exist independently in the traditions of solid-state
physics and materials science rather than of glass physics. SR in
Metallic Glasses An obvious difference, as compared to non-metallic
glasses, is that in metallic glasses SR has long been noticed
mainly as a strong irreversible (irrecoverable) effect deep in the
solid range (T Tg) existing even at room temperature, rather than
as a phenomenon originating in the reversible properties of the
undercooled melt above Tg as introduced earlier. This is a
consequence of the high cooling rates used during production,
especially in case of rapidly quenched conventional metallic
glasses being in a highly unstable state far from equilibrium (high
Tf ). The undercooled melt, on the other side, is more difficult to
study and has been totally inaccessible before the development of
bulk metallic glasses which, although first prepared by Chen
(1974), became popular not before the 1990s (see Wang et al. 2004b
for a review). On this historical background, some conceptually
restricted usage of the term structural relaxation has partly
developed for metallic glasses, regard-ing SR as being absent in
the state of metastable equilibrium above Tg (e.g., Fursova and
Khonik 2000) as observed macroscopically. This would however
unnecessarily exclude from the term those fast dynamic processes in
the vis-cous liquid which are needed to maintain equilibrium (e.g.,
during tempera-ture changes), and which in glass physics just form
the core of SR, being only slowed down below Tg. To avoid this
obvious inconsistency, in this chapter we use structural relaxation
in its general physical meaning and only speak of different types
or components of SR if necessary. It was shown long ago that the
irreversible type of SR in metallic glasses, e.g., during annealing
of a rapidly quenched PdSi glass, can increase viscos-ity by five
orders of magnitude (Taub and Spaepen 1979, 1980), indicating
enhanced atomic mobility in the initial unrelaxed state. In other
words: this irreversible SR, affecting virtually all physical and
mechanical properties p
116. 2.6 Relaxation in Non-Crystalline and Complex Structures
101 (Cahn 1983), cannot be a secondary relaxation in the frozen
temperature range C but should be considered as a primary one in
range B, kinetically extended to lower temperatures. At this point
it seems surprising that at temperatures so far below Tg, there is
also a reversible (recoverable) component of SR being even faster
than the irreversible one, as observed e.g., for Youngs modulus
(Kursumovic et al. 1980, Scott and Kursumovic 1982) or enthalpy
(Scott 1981, Sommer et al. 1985, Gorlitz and Ruppersberg 1985), but
hardly for density or vol-ume (Cahn et al. 1984, Sinning et al.
1985). This (selective) low-temperature reversible SR component, to
be distinguished from reversible behaviour at the glass transition,
is difficult to understand in terms of fictive temperature or
primary/secondary relaxations, but at least roughly consistent with
an earlier hypothesis by Egami (1978) relating reversible and
irreversible SR, respec-tively, to changes in chemical and
topological (or geometrical, Egami 1983) short-range order. The
(also non-exponential) kinetics of such solid-state SR phenomena in
metallic glasses, extensively studied in both conventional and bulk
metallic glasses during the past three decades, have been widely
analysed in terms of an activation energy spectrum (AES) model,
introduced by Gibbs et al. (1983) on the basis of earlier work by
Primak (1955), and subjected to some later extensions and
modifications. This model is based on a wide non-equilibrium
distribution of Debye-type relaxation events, which during
annealing is gradually cut down from the low-energy side. While
mathemati-cally equivalent to the use of a KWW function, the
physics behind this model seems to be more consistent with the idea
of independent relaxation centres (see later), instead of the
picture of true cooperative motion associated with a KWW function.
For a microscopic understanding of SR in metallic glasses, the
oldest and maybe still most widely spread concept is that of free
volume, which was introduced by Cohen and Turnbull (1959) and
worked out later by van den Beukel and coworkers, incorporating
also Egamis distinction between topolog-ical and chemical
short-range order (e.g., van den Beukel 1993 and references
therein). Alternative concepts were added more recently, for
example based on interstitialcy theory (describing an amorphous
solid as a crystal contain-ing a few per cent of
self-interstitials; e.g., Granato 1992, 1994, 2002; Granato and
Khonik 2004), or on the theory of local topological fluctuations
(of atomic bonds and atomic-level stresses; Egami 2006). As SR is
closely related to diffu-sion, much can be learned from the recent
progress in understanding diffusion mechanisms in metallic glasses
(Faupel et al. 2003), which generally revealed highly collective
atomic processes (contrary to crystalline metals): according to
molecular dynamics simulation supported by critical experiments,
atomic migration mainly occurs in thermally activated displacement
chains or rings. Being rather local at low temperature, these
chains grow in size and concen-tration with increasing temperature
until they finally merge into flow.
117. 102 2 Anelastic Relaxation Mechanisms of Internal Friction
Relation Between Structural and Mechanical Relaxation Any
structural relaxation whatever the exact microscopic mechanism is
must involve atomic movements directed to lower the Gibbs free
energy under the acting external perturbation, generally including
anisotropic atomic-level distortions oriented in different
directions (like the above displacement chains). If the external
perturbation is isotropic, e.g., in case of a purely ther-mal
deviation from equilibrium, such local anisotropies may be averaged
out so that only a macroscopically isotropic volume change is
observed. In the presence of a mechanical stress, however, the
distribution of the local events may become asymmetric producing a
net distortion in the direction of energet-ically favoured
orientations, i.e., a mechanical relaxation due to a directional
structural relaxation (DSR). In this generality, and using the
widest meaning of SR which in principle applies to crystalline
structures as well (see above), every mechanical relax-ation
mechanism based on the motion of defects, including all cases
considered in Sect. 2.22.4, might be called a DSR: under this
viewpoint, DSR forms a very general principle of mechanical
relaxation which of course also applies to amorphous structures.
Thus, the connection between structural and mechan-ical relaxation
is generally a rather close and direct one. More specifically, the
different types and temperature ranges of SR in glass-forming
systems must be considered. In the range of the primary relaxation
around the glass transition, the same cooperative atomic motions
cause both viscous flow and SR (i.e., SR occurs by viscous flow),
so that relaxation time and viscosity can directly be converted
into each other (for which, in spite of non-exponential relaxation,
often a simple Maxwell model with = /EU is used, cf. Chap. 1).
Therefore, in the range where a mechanical (e.g., internal
friction) measurement is dominated by viscous flow, the result
directly reflects the structural relaxation. There is a
superabundant number of (mechanical and other) studies of the
relaxation over wide frequency and temperature ranges in more
stable non-metallic glass formers, whereas in metallic systems the
relaxation is accessible only under favourable conditions using the
best bulk metallic glass formers and low frequencies (see later).
The situation is less clear in metallic glasses at temperatures
further below Tg down to about 400K where the above-mentioned,
specific types of irreversible and reversible SR are found,
mechanical relaxation is at least partly anelastic (recoverable) in
nature (Berry 1978), but plastic de-formation still occurs mainly
by homogeneous flow. By assuming spatially separated structural
relaxation centres represented by two-well systems, Kosilov, Khonik
and coworkers developed a specific DSR model which applies in this
range not only to mechanical relaxation but to mechanical
prop-erties in general (e.g., Kosilov and Khonik 1993; Khonik 2000,
2003 and references therein). The relaxation centres (two-well
systems) were divided into irreversible (highly asymmetric) and
reversible (rather symmetric) ones, the former being responsible
for mainly viscoplastic low-frequency internal
118. 2.6 Relaxation in Non-Crystalline and Complex Structures
103 friction, plastic flow and even for reversible strain recovery
(Csach et al. 2001), whereas the latter cause anelastic processes
seen at higher frequencies (Khonik 1996, Eggers et al. 2006). At
still lower temperatures where plastic deformation of metallic
glasses is known to change to a highly localised shear band mode,
the primary SR is eventually frozen (range C in Fig. 2.34, in many
cases below about 400 K). If speaking of DSR in this range at all,
this can only mean secondary relaxations of special, easily mobile
species, like those of interstitially dissolved hydrogen which have
already been treated in Sect. 2.2.4. However, since such anelastic
processes in metallic glasses classified as type (b) in the
intro-duction to non-crystalline structures at the beginning of
this section have more in common with crystalline structures than
primary DSR, a true solid-state picture with a clear distinction of
the relaxing defect might be more appropriate in this
low-temperature range than the more general viewpoint of DSR.
Internal Friction Phenomena in Metallic Glasses General Aspects
Amorphous alloys have to be produced with the help of some
non-equilibrium procedure (like rapid cooling from the melt,
mechanical alloying, various kinds of deposition, etc.), during
which the formation of the thermodynami-cally stable crystalline
state is kinetically hindered. Therefore, all amorphous alloys
crystallise when heated into a temperature range with sufficient
atomic mobility, which is always connected with a maximum of
internal friction at a temperature close to the onset of
crystallisation (Fig. 2.35). In fact this crystallisation peak,
with a position usually depending on heating rate but not on
frequency (e.g., Zhang et al. 2002), is not a true relaxation peak
but a transitory effect. It basically reflects the irreversible
transition from the high and monotonically increasing IF in the
glassy amorphous phase to a much lower damping level in the
crystalline state, but can be a quite complex-shaped superposition
of many different effects in the frequent case of a multiple-step
crystallisation process. Once passed during heating, the
crystallisation peak completely disappears during subsequent
cooling or during a second heat-ing run. It has been used in some
cases to study details of the crystallisation process including
kinetics and activation energies (Sinning and Haessner 1985, Klosek
et al. 1989, Nicolaus et al. 1992). In contrast to the high damping
level at the onset of crystallisation, the internal friction in
metallic glasses is generally low at room temperature and below,
and at acoustic (vibrating-reed) frequencies often reduced to the
ther-moelastic background (see Sect. 2.5) if no special
low-temperature effects are there (see later). The temperature
dependence of IF is rather weak up to about 400500 K, where a
stronger, often exponential increase sets in which
119. 104 2 Anelastic Relaxation Mechanisms of Internal Friction
Fig. 2.35. Comparison of the low- and high-frequency IF behaviour,
at a heat-ing rate of 0.3Kmin 1, for two Ni-based glasses with
(Ni60Pd20P20) and without (Ni78Si8B14) a glass transition before
crystallisation. (1) Ni60Pd20P20, 0.08Hz; (2) Ni60Pd20P20, 450 Hz;
(3) Ni78Si8B14, 0.095 Hz; (4) Ni78Si8B14, 400 Hz. The maxima of all
curves (at 600K for Ni60Pd20P20 and 700K for Ni78Si8B14) correspond
to the onset of partial (primary) crystallisation followed by
further transformations (Sinning and Haessner 1988a) continues up
to crystallisation. At all temperatures damping is higher at 0.1 Hz
than at acoustic frequencies, indicating a broad spectrum of
additional low-frequency processes. In this context, two main
groups of metallic glasses have to be distin-guished: those which
crystallise from the solid state before reaching the glass
transition, and those which first show a glass transition and then
crys-tallise from the undercooled melt (which largely corresponds
to the distinction between conventional and bulk metallic glasses,
except for a few inter-mediate cases like CuTi showing a Tg in a
torsion pendulum at 0.30.5 Hz without being a bulk glass former
(Moorthy et al. 1994)). Glass Transition and Relaxation As shown in
Fig. 2.35 for a still moderate example, the occurrence of a glass
transition has a dramatic effect on the height of the
crystallisation peak at low frequencies which easily exceeds tan =
1, while the high-frequency IF peak remains unaffected and shows
about the same height (tan < 0.1) as without a glass transition.
The reason for this dramatic low-frequency IF increase, seen in
Fig. 2.35 as the strong upward bend of curve 1 at Tg which is
missing for the conventional metallic glass (curve 3), is the onset
of dominating viscous damping Qv 1 due to the relaxation (described
as Qv 1(T) = EU/(T) using a Maxwell model). It has been shown that
this viscous onset,
120. 2.6 Relaxation in Non-Crystalline and Complex Structures
105 shifting to higher temperature with increasing frequency, is
located just at the dynamic glass transition (assuming g = 1012
Nsm2) if the frequency is around 0.1 Hz; under certain conditions,
it could be used for determining Tg at heating rates much lower
than possible with the common DSC technique (Sinning and Haessner
1986, 1987, 1988b; Sinning 1991a, 1993a). It is important to note,
however, that such maxima in the loss factor tan (or Q1) remain
always transitory crystallisation peaks as mentioned earlier, even
in presence of a glass transition: there is no glass transition
peak or relaxation peak in tan in metallic (or more generally in
low molecular weight) glasses, contrary to occasional
misinterpretations in the lit-erature. The glass transition alone,
without the intervention of crystallisation, produces an relaxation
peak only in the loss modulus E (or G in case of shear) but not in
tan = E/E which would in this case grow infinitely as E goes to
zero in the supercooled liquid. The typical situation, producing a
peak in tan , is depicted in Fig. 2.36 for Zr65Al7.5Cu27.5 (a
moderate bulk glass former not very different from Ni60Pd20P20 in
Fig. 2.35): whereas the loss modulus E shows two separate peaks,
being identified with the relaxation and with losses during
crystallisa-tion, respectively (Rambousky et al. 1995), the single
maximum in tan does not reflect these two peaks. It is rather
dominated by the behaviour of the storage modulus E in the
denominator, which falls down in the supercooled liquid above Tg by
more than one order of magnitude, to a sharp minimum that is solely
determined by the onset of crystallisation (note the different,
logarithmic and linear scales for the moduli and tan ,
respectively). There-fore, only the rising part of the damping peak
may be associated with the relaxation. Fig. 2.36. Storage modulus E
, loss modulus E and damping tan of as-quenched amorphous
Zr65Al7.5Cu27.5, measured at 1 Hz during heating with 10Kmin 1
using a dynamic mechanical analyser (Rambousky et al. 1995). Tg
denotes the onset of the calorimetric glass transition
121. 106 2 Anelastic Relaxation Mechanisms of Internal Friction
For studying the relaxation by mechanical spectroscopy, it is
therefore more appropriate to look at E and E (or G and G)
separately, rather than just considering internal friction. To
trace out the full relaxation peak in the loss modulus as a
function of either temperature or frequency, it is important to
have a wide supercooled liquid range, i.e., to use the best bulk
metallic glasses available. Meanwhile such studies have been
performed on several more advanced Zr- and Pd-based bulk glasses
(e.g., Schroter et al. 1998, Pelletier and Van de Moort`ele 2002a,
Pelletier et al. 2002b, Lee et al. 2003a, Wen et al. 2004); an
example is shown in Fig. 2.37. The results follow the
time-temperature superposition principle, well known from
non-metallic glass formers: all curves fall on a master curve when
shifted by a temperature-dependent relaxation time which usually
obeys the VFT equation. Occasional low-temperature shoulders of the
peak in E are sometimes interpreted as a relaxation (Pelletier and
Van de Moort`ele 2002a). Contrary to the loss modulus, the loss
compliance J does not show an relaxation peak either, but
monotonically falls (like tan ) with increasing frequency or
decreasing temperature. An analysis of its frequency dependence,
with an exponent typically changing from 1 at low to 1/3 at high
fre-quencies, may be used to separate Newtonian viscous flow from
relaxation components and to discuss related models (Schroter et
al. 1998). In addition, the interest in more specific questions of
glassy dynamics in this range (which are beyond the scope of this
chapter), calling for advanced or extended experimental conditions,
has triggered some remarkable new experimen-tal developments in
mechanical spectroscopy: for instance, a non-resonant Fig. 2.37.
The relaxation of the Zr46.75Ti8.25Cu7.5Ni10Be27.5 bulk metallic
glass studied by dynamic mechanical analysis. (a) Temperature
dependence during heat-ing with 1Kmin 1 at different frequencies;
(b) isothermal frequency dependence and fit to the KWW equation
with an exponent = 0.5 (solid lines) at different temperatures (Wen
et al. 2004)
122. 2.6 Relaxation in Non-Crystalline and Complex Structures
107 vibrating-reed technique with an extremely wide frequency range
(Lippok 2000), or a special double-paddle oscillator for studying
thin films (Liu and Pohl 1998) applied to glassy alloys at high
temperatures and high frequencies (Rosner et al. 2003, 2004). The
results of such fundamental studies on bulk metallic glasses
generally confirm the main characteristics of the relaxation in the
high-temperature domain A as briefly outlined above, known from
non-metallic glass formers: in this respect, the underlying physics
appears to be the same for quite different classes of glass-forming
systems. Intermediate Temperature Range This is the classical range
of materials science in which the study of inter-nal friction in
metallic glasses began (Chen et al. 1971), and where most of our
knowledge is still based on results obtained on rapidly quenched
samples (although in this range there seems to be no big difference
to bulk alloys; Berlev et al. 2003, Eggers et al. 2006). The main
feature is here the exponen-tial increase of IF with temperature
mentioned earlier, e.g., in form of the (on the logarithmic scale)
linear rise of curves 24 in Fig. 2.35 towards the max-imum. The
following main characteristics have been reported for this rising
part of the IF spectrum: 1. It is reduced in its lower part (or
shifted to higher temperature) by irreversible structural
relaxation. For instance, if an as-quenched sam-ple is heated with
a constant rate to successively increasing temperatures (Fig.
2.38), in each heating run the IF is reduced compared to the
previous one, in close correlation to an irreversible increase of
Youngs modulus or resonance frequency (seen more clearly in
isothermal experiments; Morito and Egami 1984a, Neuhauser et al.
1990). Such an annealing behaviour is often analysed in terms of
the AES model mentioned earlier, resulting in a broad spectrum for
irreversible structural (not mechanical) relaxation ranging from
about 100 to 200 kJ mol1 in case of Fig. 2.38. 2. Using appropriate
isothermal anneals, Morito and Egami 1984b and Bothe (1985) have
been able to cycle the IF spectrum reversibly (Fig. 2.39), proving
an effect of reversible SR as well. 3. At frequencies about 1 Hz
and below, the IF of as-quenched samples depends on the heating
rate (Bobrov et al. 1996, Yoshinari et al. 1996b). 4. After a
stabilising anneal and subtraction of the thermoelastic background
Q 1 , the IF at constant frequency often shows a straight line in
an B Arrhenius plot (Fig. 2.40), i.e., it is of the empirical form
1 Q Q 1 B eA/kT . (2.57) The slope parameter A increases with
annealing (Berry 1978). 5. The IF increase shifts to higher
temperatures at higher frequencies, i.e., it is a thermally
activated relaxation effect (Fig. 2.40). The apparent activation
enthalpies H, taken from cuts at constant damping according to
123. 108 2 Anelastic Relaxation Mechanisms of Internal Friction
Fig. 2.38. Variation of (a) frequency and (b) damping of a
vibrating amor-phous Ni78Si8B14 reed during heating-cooling cycles
with 1Kmin 1. The vertical arrows indicate the onsets of structural
relaxation and crystallisation, respectively, (Neuhauser et al.
1990) ln(f2/f1) = (H/k)(T 1 1 T 1 2 ), (2.58) vary between 115 and
250 kJ mol1 for different Pd- and Fe-based glasses and annealing
treatments (with sometimes unphysically high attempt frequencies 1
0 ), and are in most cases much higher than the related slope
parameters A (22125 kJ mol1) (Soshiroda et al. 1976, Berry 1978,
Neuhauser et al. 1990). The problem of this discrepancy could be
solved by Kruger et al. (1993) by showing theoretically that the
r