Defect Microdynamics in Minerals and Solid&State ...geophysics.wustl.edu/seminar/1990_Karato_RG.pdfplications for the mechanisms of seismic wave attenuation (section 3). The importance

DEFECT MICRODYNAMICS IN MINERALS AND SOLID-

STATE MECHANISMS OF SEISMIC WAVE ATTENUATION

AND VELOCITY DISPERSION IN THE MANTLE

S. Karato • and H. A. Spetzler Cooperative Institute for Research in Environmental Sciences and Department of Geological Sciences University of Colorado, Boulder

Abstract. The propagation of seismic waves in the Earth's mantle can be significantly affected by relaxation processes, causing attenuation and velocity dispersion (reduction). This paper reviews the solid-state mechanisms of relaxation processes based on the theory of defect microdynamics in solids together with some experimental observations on defects in minerals (particularly in olivine). For a given mechanism to have a significant effect on seismic wave propagation, both the density and the mobility of the defects must be in an appropriate range. The examination of the densities (and geometry) and mobilities of defects in olivine shows that dislocation and/ or grain boundary mechanisms can have a significant effect on seismic wave propagation, although wide distributions of geometrical factors (such as spacing of pinning points) and of mobilities are required to explain all available data. Point defect mechanisms, however, are unlikely to be important because their densities are too small and/or their mobilities are too large. Since the dislocation density and/or grain size are determined in most cases by the long-term tectonic stress, seismic wave attenuation and velocity dispersion (reduction) involving these defects are

likely to depend on the magnitude of the tectonic stress as well as the temperature. Theoretical considerations suggest a wide range of dependence of seismic wave attentua- tion (and velocity dispersion) on the long-term tectonic stress. This is particularly the case for dislocation mechanisms and warrants careful experimental investigation. Dislocations and/or grain boundaries cause anelastic behavior (relaxation peaks) when they are pinned or blocked at some points. Pinning or blocking becomes ineffective at high temperatures and/or low frequencies, causing a transition to viscoelastic behavior. Both laboratory and seismological observations of internal friction are dominated by the "high-temperature background" where internal friction increases monotonically with temperature, which can be interpreted as a gradual transition to viscoelastic behavior or to a wide distribution of relaxation times. However, in most experimental studies to date, the dislocation densities or the grain sizes were not well controlled, making it difficult to identify the attenuation mechanisms and preventing any quantitative applications to Earth. The need for better characterization of defect microstructures in experimental specimens is emphasized.

1. INTRODUCTION

Although the solid Earth behaves almost like an elastic body at seismic frequencies, there is ample evidence for deviation from ideal elasticity. This includes the attenuation of seismic waves [e.g., Anderson and Hart, 1978] and velocity dispersion, i.e., the frequency dependence of seismic wave velocities [Kanamori and Anderson, 1977]. The attenuation of seismic waves and the associated

velocity dispersion in rocks are likely to be due to

1Now at Department of Geology and Geophysics, University of Minnesota, Minneapolis.

relaxation process(es). In many cases these processes involve the motion of defects (including melts) in rocks, and therefore Earth's internal friction and velocity dispersion depend on the concentration, geometry, and mobility of relevant defects [e.g., Jackson and Anderson, 1970]. These defect-related properties are known to be sensitive to the thermochemical environment (temperature, pressure, fugacity of oxygen and water, etc.) and the differential stress, both of which are also closely related to tectonic activity. Therefore the observed seismic wave attenuation (and velocity anomalies), if properly interpreted, can provide important clues to Earth's thermochemical and stress states. The principal aim of

Copyright 1990 by the American Geophysical Union.

8755-1209/90/90RG-01675 $05.00 399 ß

Reviews of Geophysics, 28, 4 / November 1990 pages 399-421

Paper number 90RG01675

400 ß Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS 28, 4 / REVIEWS OF GEOPHYSICS

material science studies in this field is to establish the rela-

tionship between the internal friction (and velocity dispersion) and the thermochemical and stress state.

Until recently, there were very few relevant experimental studies of internal friction and velocity dispersion in rocks, and proper interpretation of seismic wave attenuation and velocity anomalies has been difficult. Some pioneering attempts have been made to correlate seismic wave attenuation and velocity anomalies with tectonic activity or thermal structures [e.g., Utsu, 1967; Solomon and Toks6z, 1970; Karato, 1980; Sato et al., 1988a, b]. Ex- perimental studies to measure the internal friction and velocity dispersion in rocks and minerals have recently been initiated by a few groups [Gueguen et al., 1981; Berckhemer et al., 1982a; Jackson et al., 1984; Sato et al., 1989; Getting et al., 1987, 1989; Jackson and Paterson, 1987, 1990]. Also, the nature of defects in minerals (particularly in olivine) has been studied in detail (for a review, see Karato [1989]). It appears timely at this stage to explore the possible implications of defect-related properties for internal friction and velocity dispersion. This will provide a guideline for the experimental studies and their interpretation.

In this paper we first discuss some of the fundamental concepts of internal friction (section 2), followed by a brief summary of the seismological observations and their implications for the mechanisms of seismic wave attenuation (section 3). The importance of solid-state mechanisms is stressed. In section 4 the nature of deformation associated

with seismic wave propagation is examined and compared with longer-term deformation such as postglacial rebound and mantle convection. In section 5 the models of solid-

state internal friction are reviewed, and their relevance to seismic wave attenuation is examined in light of recent studies on defects in minerals. Finally, some of the experimental results on upper mantle minerals and rocks are reviewed in section 6. It is concluded that the mechanisms

of internal friction are not well constrained in these studies

and that these results cannot be directly applied to Earth, although they clearly demonstrate the importance of some solid-state mechanisms. The need for careful examination

of defect microstinctures in experimental specimens together with reliable measurements is emphasized.

2. SOME FUNDAMENTALS OF INTERNAL

FRICTION IN SOLIDS

The elastic deformation of a solid is characterized by the instantaneous and unique response of strain to stress. Deviation from ideal elasticity may occur through time- dependent response or nonunique response (or both). At high temperatures and low frequencies, as in the case of seismic wave attenuation in the mantle, time-dependent response is important. (Assumption of linearity is often made in the formal theory of nonelastic behavior [Nowick and Berry, 1972, chapter 1]. Mechanical response of a

rock for elastic waves is in most cases linear, but there are two important exceptions to this. One is the internal friction due to frictional sliding at cracks [e.g., Johnson and ToksOz, 1980], and the other is the internal friction due to unpinning of dislocations (see section 5.2.3). In both cases the internal friction increases with strain magnitude. The effects of cracks are not likely to be important in the mantle, but the effects of dislocation unpinning may be quite important in the manfie (section 5.2.3).)

Two types of time-dependent behavior can be distinguished. One is the case where the deformation is not instantaneous but has a unique equilibrium state (standard linear solid). In this case a transition from one (unrelaxed) state to another (relaxed) state will occur via some kinetic process(es). This behavior is referred to as anelasticity [Nowick and Berry, 1972, chapter 1]. Anelastic behavior is characterized by two parameters: the relaxation strength A (= (anelastic strain)/(elastic strain)) and the relaxation time x. The internal friction Q-• has a maximum at the frequency •00 (= l/x), and the modulus M changes with frequency, resulting in the dispersion of elastic wave velocities (Figure la). A mechanical model for anelasticity consists of two springs and one viscous element (Figure la).

The other type of time-dependent behavior is viscoelasticity in which, at constant stress, the strain becomes infinite at infinitely long time (Maxwell body [Nowick and Berry, 1972, chapter 1]). In this case the internal friction increases with the decrease of frequency, and at zero frequency all the energy is dissipated and the modulus goes to zero (Figure lb). A mechanical model for viscoelasticity is a spring and a viscous element in series (Figure lb). This can be viewed as a specific case of an anelastic body with a vanishingly small spring constant M 1 ß

In a more realistic case, both anelastic and viscoelastic behavior occur in a solid depending on the frequency (or the temperature). Such a solid is called a Burgers body, which has two characteristic times (or two viscous elements) associated with anelastic and viscoelastic behavior (Figure lc).

Physical mechanisms of anelasticity include thermoelasticity and some defect-related mechanisms [e.g., Jackson and Anderson, 1970; Nowick and Berry, 1972]. The intergranular thermoelasticity arises from stress relaxation due to intergranular thermal currents caused by the anistropy or heterogeneity of compressibility of the grains [Zener, 1948]. Therefore this mechanism dominates in compressional waves. Most of the defect-related mechanisms, in contrast, dominate in shear waves because the defects, particularly dislocations or grain boundaries, move under shear stress but not under hydrostatic stress. These defect- related mechanisms also cause viscoelasticity.

The way in which a given solid deviates from ideal elasticity depends on the microscopic mechanisms. Anelastic behavior results when the motion of the defects

under stress stops at sufficiently long time, yielding a

28, 4/REVIEWS OF GEOPHYSICS Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS ß 401

Stondord Lineor Solid (onelostic)

(A)

(B)

(c)

velocity

Co o

log Frequency

Moxwell Body (viscoelostic)

• velocity

)

COl log Frequency

Burgers Body

velocity

.• • Q-I

I I coP-_ col

log Frequency

M I

M •7

Figure 1. Mechanical models for typical nonelastic behaviors of solids. (a) A standard linear solid in which relaxation occurs

with a characteristic time x = o•0 -1 = 'q/[M•(M• + M•.)] m is shown. Internal friction Q-I is maximum at the frequency o• 0. The elastic modulus (and elastic wave velocities) has a high (unrelaxed) value M•. at high frequencies (o• >> o•0). (b) A Maxwell body in which a solid behaves like an elastic body at high frequency (o• >> %, o• -1 = •I/M) and like a viscous body at low frequency (o• << %). (c) A Burgers body which contains the characteristics of both a standard linear solid and a Maxwell body is also illustrated. At low frequency (o• << o•., o•2 -1 ~ '1•. (1/M• + 1/M•.)) the solid behaves like a viscous solid, and at higher frequencies it behaves like a standard linear solid with the characteristic frequency {00 -1 ~ TI1/[Ml(M 1 + M2)] m. Arrows indicate the direction of the shift of velocity or internal friction due to an increase in temperature.

well-defined relaxed state. Thermoelasticity, for example, results in anelastic behavior, since at sufficiently long time the temperature is homogenized and thermal currents cease. Viscoelastic behavior results when the defect mo-

tion continues indefinitely. All solids show viscoelasticity at sufficiently high temperatures and/or low frequencies.

In either case, solid-state internal friction due to defects is sensitive to the temperature T and pressure P through the temperature and pressure dependence of the characteristic time (•: = •:0 exp [E(P)/R7], where E is the activation enthalpy, and R is the gas constant). In addition, internal friction is also sensitive to the long-term stress, because it changes the density or geometry of defects.

The frequency dependence of internal friction shown in Figure 1 is for ideal cases where a single characteristic time is defined for anelasticity and the viscoelasticity is characterized by a well-defined viscosity. In the more realistic case, there is a distribution of characteristic times, and there is a gradual transition from anelastic to viscoelastic behavior. In either case the frequency dependence of internal friction is weaker than in the ideal case. Takeuchi

[1972], Liu et al. [1976], and Minster and Anderson [1981] proposed that distributions of characteristic times in an anelastic material yield the weak frequency dependence of internal friction observed in the Earth.

It is important to note that internal friction is necessarily accompanied by a velocity dispersion (see Figure 1). The amount of velocity dispersion in the Earth associated with internal friction of Q = 100-500 is approximately 0.5-1.5% for the frequency range 1-10 -3 Hz [Kanarnori and Anderson, 1977]. A similar amount of velocity variation (through relaxation processes) should occur if the relaxation time varies by a factor of 103. This is a significant amount since the magnitude of the velocity anomalies observed by seismic tomography is approximately +0.5% for the lower mantle [Dziewonski, 1984] and approximately +2% for the upper mantle (except for the topmost layer) [Woodhouse and Dziewonski, 1984]. As will be shown later, the velocity changes due to defect-related relaxation processes are sensitive to the long-term stress as well as the temperature. This is in contrast to the velocity change due to anharmonicity which is insensitive to the stress and is not associated with

seismic wave attenuation. Thus the study of internal friction or relaxation processes has important implications for the interpretation of seismic tomography as well as of seismic wave attenuation.

3. CONSTRAINTS ON MECHANISMS OF SEISMIC ATTENUATION IMPOSED BY GEOPHYSICAL OBSERVATIONS

Several geophysical observations can be used to estimate the magnitude of internal friction and its dependence on depth, tectonic setting, and frequencies. They include the amplitude decay of body and surface waves, the broadening of free oscillation peaks, and the damping of the Chandler wobble [e.g., Anderson and Hart, 1978; Canas and Mitchell, 1978; Sipkin and Jordan, 1979; Jordan, 1981; Smith and Dahlen, 1981; Anderson and Given, 1982; Ulug and Berckhemer, 1984; Choy and Cormier, 1986]. Also, the dependence of seismic wave


velocities on frequencies and the resultant change in the periods of free oscillations and of the Chandler wobble provide important information on Earth's nonelastic properties [Liu et al., 1976; Kanamori and Anderson, 1977; Anderson and Minster, 1979; Smith and Dahlen, 1981].

Figure 2 shows an Earth model based mainly on seismological observations (body waves, surface waves, and free oscillations) in which the following simplifying assumptions are made: (1) no lateral variation and (2) no frequency dependence of internal friction nor of seismic

. I I I I

'T 0

-2

-4

•ooo 2000 3000

Depth (km )

Figure 2. A gross Earth slxucture model based on seismological observations, showing both (top) the depth variation of seismic velocities and (bottom) the depth variation of internal friction Q-1 [after Jordan, 1981]. Note the two high internal friction (and relatively low velocity) zones near the top and bottom of the mantle. The internal fr•CtlOn for compression, Q•c, •s s•g- nificantly smaller than that for shear, Q•I. Note also that the internal friction in the lower mantle is smaller than that in the

upper mantle. The model represents a regionally averaged structure for a given depth and assumes frequency-independent internal friction.

velocities. This figure shows the depth variation of seismic wave velocities and of internal friction. Internal

friction, Q-•, is shown for shear deformation (Q•) and for dilatational deformation (Q[•) which are related to the internal friction for shear waves Q;• and for compressional waves Q7, • through

Qj1 = Q•i (1)

and

Q,7,1 = L Q;1 + (1 - L)Q•c 1 (2) with L - (4/3)(V .)2 [Anderson and Hart, 1978] In this - •/V/. . gross structural m6del the following points are relevant with respect to seismic wave attenuation. (1) Q-• is much larger than Q•i in the mantle. (2) Q-1 is larger in the upper mantle than in the lower mantle (except at the bottom of the lower mantle). (3) There are two anomalous layers; one near the surface (around 100-200 km depth) and the other near the bottom of the mantle, where the attenuation is large and the velocities are anomalously low.

There are also large lateral variations in the topmost upper mantle (approximately 0-200 kin) and near the bottom of the mantle [Woodhouse and Dziewonski, 1984; Dziewonski, 1984]. In the upper mantle where detailed studies have been made, a correlation of high attenuation (low Q) and low velocity has been observed with respect to lateral variation [Utsu, 1967; Solomon and Toks6z, 1970; Canas and Mitchell, 1978; Regan and Anderson, 1984]. We also note that recent surface wave studies have shown

that the depth of the lithosphere/asthenosphere boundary in the oceanic upper manfie is very shallow (-50 km at 100-m.y.-old mantle [Regan and Anderson, 1984; Kawasaki, 1986]), where the temperature is 900-1000 K, far below the solidus.

The frequency dependence of internal friction is difficult to estimate because of the trade-off between the

frequency dependence and the depth variation of internal friction. In the estimation shown in Figure 3, the Earth's mantle is divided into two parts (the upper and lower mantle), and the averaged internal friction in each part is analyzed (the depth variation in each layer is not taken into accoun0. The data used include body waves (~10 -•- 102 S-1), free oscillations (-10 -4-10 -3 $-1), the solid Earth tide (-10 4 s-i), and the Chandler wobble (-10 -8 s-i). Q-• depends weakly on frequency in both the upper and the lower mantle. When the frequency dependence is written as a power law form, we find Q;1 ~ (l).•x ((1) is th(5 fr(5 o quency) with •x = 0.2-0.3. Anderson and Given [1982] proposed an absorption band model with more •complex frequency dependence, including a broad peak of attenuation in the upper mantle around t0 ~ 10 -2 S -I, but the observational evidence to support this is rather weak.

From the seismological observations summarized in Figures 2 and 3, we make the following deductions:

1. Since significant seismic wave attenuation occurs in areas where the temperature is significantly below the solidus, much of the seismic wave attenuation (and associated velocity dispersion) must be attributed to solid-state mechanisms, rather than to partial melting (for discussions of partial melting, see Shankland et al. [1981] and Karato [1990]). (The effects of vapor phases might be important in the crest but are not likely to be important in the mantle.)

2. Among the solid-state mechanisms, thermoelasficity cannot be a major contributor because it is much more

28, 4 / REVIEWS OF GEOPHYSICS

important for compressional waves than for shear waves, which is contrary to observation, Q-1 • >> Q•l. Solid-state mechanisms involving defects are likely to be responsible for the observed seismic wave attenuation.

3. Since large attenuation is associated with low velocities (Figure 2), either the seismic frequencies must be higher than the characteristic frequencies of anelastic peaks (Figure la) or the relaxation mechanism must be viscoelastic (Figure lb). However, the frequency dependence in the mantle is not Q•i ~0)-1 as the simple models imply (Figure 1). A distribution of peak frequencies (in anelastic behavior) and/or a gradual transition from them to viscoelastic behavior may be involved.

We also note that the seismic wave attenuation in the

lower mantle at the lowest frequencies (i.e., the Chandler wobble) is close to that expected for a Maxwell body with a viscosity q of 1021-22 Pa s. These viscosities are similar to those estimated from postglacial rebound [Peltier et al., 1981; Nakada and Lainbeck, 1989], suggesting that the damping of the Chandler wobble and the viscous relaxation in the lower mantle associated with postglacial rebound may involve similar microscopic processes.

-I

I • I X I I I I I I

• o

'":i:i:i:i:•:•:i:i:i:i:i:!:i:!:i

% ":'-".:i:i:i::-'.-': ::.':::

rebound

-12 -I0 -8 -6 -4

log to (s -I )

I ' "-'-'-'t :;:x ...... I

-2 0 2

Figure 3. The frequency dependence of intemal friction Q-1 of shear velocity for the upper and the lower mantle [after Anderson and Given (AG), 1982; Anderson and Minster (AM), 1979; Sipkin and Jordan (SJ), 1979; Smith and Dahlen (SD), 1981]. The data used include the body waves (10-1-10 e s-l), surface waves (10 -1 - 10 -2 s -1), free oscillations (10 -3-10 -4 s -1), and the Chandler wobble (10- 8 s-i). Also shown are the theoretical values of internal friction corresponding to a Maxwell body with viscosities of 10 • Pa s and 1022 Pa s for M = 100 GPa. Note that the intemal friction for the Chandler wobble is close to that for a

Maxwell body (viscoelastic body) with a viscosity of 10 el Pa s.

Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS ß 403

4. TIME SCALES AND STRAIN MAGNITUDES

Before going into details of mechanisms of seismic wave attenuation, it is instructive to examine briefly the nature of the deformation associated with seismic wave

attenuation. The main concerns here are the time scale, the strain (and stress) amplitude, and the amount of defect motion. These aspects are relevant because many of the experimental observations of defect motions are made under conditions that are vastly different from seismic wave attenuation and also because we are interested in

extracting some information on long-term deformation from seismic wave attenuation.

Table 1 summarizes the time scales and the strain (and stress) magnitudes associated with seismic wave attenuation, postglacial rebound, and mantle convection. Shown together in this table are the mean distances of dislocation motion and of groin boundary sliding assuming that the inelastic strain is due to either of these processes alone. In this estimation, use has been made of the relation e ~ pbl•l (e is the strain, p is the dislocation density, b is the length of the Burgers vector, and l• 1 is the mean distance of dislocation motion) and e ~ lg•/d (/g• is the mean distance of grain boundary sliding, and d is the groin size). The following values are assumed on the basis of the

TABLE 1. Time Scales, the Magnitude of Defect Motion and of Shear Stress Associated With Three Geophysical Processes

Internal Friction Postglacial Mantle for Seismic Waves Rebound Convection

Time scale, s 10 -2 -10 4 ~10 TM ~10 is Strain l0 s -10 -6 ~10 -5 ~1 /dial ' m 10-10-10 -6 10-5-10 -3 1-102 ldlsl/<ldlsi > 10 -5 - 10 -2 10 -1 - 1 10 s - 1 O' Ibs ' m 10-12-10 -10 ~10- 7 ~10- 2 •gb/d 10-10-10- s -1• 5 •hear strew, Mpa 10 -3-10 -1 1-10 1-10

For internal friction, Q = 100 is assumed. Variables are def'med as follows: ld• d, the mean distance of

dislocation motion; </aa•>, the mean dislocation spacing (10-4 -10 -5 m); l., the mean distance of grain boundary sliding; and

ß

d, the gram s•ze (10 -2 m).

microstructures of upper mantle xenoliths: p = 108-10 lø m -2, b = 5 x 10 -lø m, and d = 10 -2 m. (The estimation of dislocation densities and groin size in the Earth's mantle is not straightforward. The upper mantle xenoliths provide direct information, but their microstinctures reflect a complicated deformation history with a wide range of dislocation densities (p = 108-10 m m -2) and of groin sizes (d = 10-4-10 -2 m) [e.g., Nicolas, 1978; Avd Lallemant et al., 1980; Karato, 1984]. The combined petrological and microstructural studies suggest that the high dislocation densities and small groin sizes within these variations are


the result of short-term deformation events such as occur in

mantle diapirs [Goetze, 1975; Avd Lallemant et al., 1980]. Small dislocation densities (say, 10s-10 •ø m-:) and large grain sizes (d- 10-: m) are believed to be representative of the long-term deformation such as the one related to plate tectonic motion.)

The strain magnitude and therefore the amount of defect motion associated with seismic wave attenuation are much

smaller than those involved in longer-term deformation. For example, the dislocations move only a small fraction of their mean distances when a seismic wave passes through a rock. Therefore the dislocation structures are not likely to be significantly altered by the passage of seismic waves (or free oscillations). In contrast, the dislocations move much larger distances than their average spacings in the case of the deformation involved in mantle convection

and in laboratory creep and other large-strain experiments. Dislocation structures are therefore totally reorganized by this deformation, resulting in dislocation densities determined by the shear stresses associated with the long-term deformation. Similarly, the amount of grain boundary sliding associated with seismic waves is much smaller than the average grain size, but that associated with convective motion can be quite large and may lead to recrystallization, fabric formation, and reorganization of grain boundary structure.

For the deformation associated with postglacial rebound, the situation is marginal. The dislocations move roughly the same distance as their average spacing. Therefore it is questionable whether steady state dislocation densities can be attained during the deformation associated with postglacial rebound. Transient creep is a distinct possibility.

To conclude, the deformation associated with seismic wave attenuation involves very small strain and therefore occurs without significantly changing the defect microstructures (the dislocation density, the grain size, etc.), which are largely determined by the long-term tectonic stresses.

Another consequence of the small strains associated with seismic wave attenuation is that the defect mobility involved in seismic waves might be entirely different from that involved in longer-term (larger strain) deformations. This question is related to the detailed microphysics of defect motion and will be discussed in the next section.

properties in minerals (particularly in olivine), and (3) geophysical constraints.

Generally, the internal friction (and velocity dispersion) due to solid-state mechanisms involving defects depends on (1) the concentration, (2) the mobility, and (3) the geometry of defects. These factors determine the characteristic time and relaxation strength (in the case of artelasticity) or the relevant "viscosity" (in the case of viscoelasticity). We discuss these relations for point defect mechanisms, dislocation mechanisms, and grain boundary mechanisms in the following sections.

5.1. Point Defect Mechanisms

Point defects (including interstitial atoms or substitutional atoms) or a group of point defects may interact with the applied stress when they are associated with an appropriate strain field. Nowick and Berry [1972, chapter 8] give a detailed account of the criteria for the occm'mnc½ of anelasticity due to point defects. According to their results, an½lastic relaxation can be produced only by a defect whose symmetry is lower than that of the host crystal.

An example is an interstitial atom in a cubic crystal, which has a tetragonal symmetry (Figure 4a). The relaxation due to this type of defect is referred to as the Snoek relaxation. In this case the interstitial atoms can

assume two different sites with different energies depending on the orientation with respect to the stress field. The transition between the two states occurs via diffusion.

Similarly, a pair of substitutional atoms has a defect structure with lower symmetry than that of a host crystal (Figure 4b). The anelastic relaxation due to this type of defect is referred to as Zener relaxation. Similarly, Southgate [1966] discussed the anelastic relaxation in

(a) (b) (c)

5. SOLID-STATE MECHANISMS OF INTERNAL

FRICTION

In this section we review some of the solid-state

mechanisms of internal friction and examine their

geophysical importance. The arguments here are based on (1) theoretical models of internal friction which have been developed from experimental studies of materials with industrial interest (metals, alloys, ionic or covalent crystals), (2) experimental results on defect-related

Figure 4. A point defect or a group of point defects are often associated with strain fields. When the strain field has a lower

symmetry than the host crystal, the defect has an interaction with the applied stress which causes a change in configuration of the defect through diffusion, resulting in artelastic relaxation. Shown are (a) an interstitial atom (shown by a small circle) in the bcc lattice causing a Snoek peak, (b) a pair of substitutional defects in the fcc lattice causing a Zener peak, and (c) a pair of vacancy and aliovalent impurity atoms (impurity atoms with a charge different from the host, for example, Fe z+ in olivine) in ionic solids causing a relaxation similar to a Zener peak [Southgate, 1966].

28, 4 ! REVIEWS OF GEOPHYSICS

MgO due to defect complexes (vacancy-impurity pairs; Figure 4c). In either case the relaxation strength A can be written as [Nowick and Berry, 1972]

(anelastic strain) (elastic strain) ~ •(Co - C)/(c•/M)

~ Co (MI2/RT)(8œ) 2 (3)

where

concentration of defects with the relaxed

configuration; total defect concentration; elastic modulus; molar volume of defect; strain associated with defect.

Use has been made of the relation C ~ C O exp (- 5•

If the defect is formed by a reaction with the surround- ing chemical environment, C O is related to the fugacity (or the activity) of that element (or of the species containing that element) and depends on temperature as well (A increases with increasing T). Important examples of this type of defect are hydrogen-related defects in olivine and quartz [Karato, 1989; Paterson, 1989] and presumably in other minerals. When the defect is a pair of substitutional atoms, C O increases with their mole fraction X.

The long-term tectonic stress c• T does not strongly affect the point defect concentration and hence A. However, •T might cause anisotropy in A by changing the population of defects (at the ground state) at different sites.

When the interaction between defects is large, a different temperature dependence of A appears through an order-disorder transition [Nowick and Berry, 1972, chapter 16]. In this case, (3) is replaced with A ~ Co(M•/RIT- T•l)(5•) 2, where T• is the transition temperature which depends on the interaction energy. Note that the anelasticity can be very large near the transition temperature. Such a large anelasticity was in fact observed in an A1-Zn alloy [Nowick and Berry, 1972, chapter 16]. We discuss the potential importance of this phenomenon in minerals later.

The characteristic time x for point defect mechanisms is given by [Nowick and Berry, 1972]

x ~ a2/D (4)

where a is the distance a defect must move to change from an unrelaxed to a relaxed state and D is the relevant diffusion coefficient. When the defect is a interstitial atom

as shown in Figure 4a, D is the diffusion coefficient of the interstitial atom. When a defect complex such as a pair of substitutional atoms (Figure 4b or Figure 4c) is involved, it is difficult to relate D to the self diffusion coefficient in the

material because the local atomic configuration near the


defect deviates from that of the perfect crystal (see, for example, Brailsford [1986]). In any case it is seen from (4) that the effect of the thermochemical environment and of the long-term differential stress •r comes from their effect on D. Therefore x will decrease with increasing T (and presumably with the increase of the fugacity of water or of hydrogen). The long-term tectonic stress •r will not have an important effect on x.

Now let us examine the possible point defect mechanisms in minerals and in the upper mantle. The following conditions must be met for a point defect to produce a significant anelastic effect on seismic waves: (1) the concentration of the defect (C o) must be significant, (2) the symmetry of the defect must be lower than that of the host crystal, (3) the strain (5•) associated with the defect must be significantly large, and (4) the characteristic x must be within or close to the seismic frequency band.

The relaxation strength A and the characteristic time x for possible point defects in olivine are estimated, and the results are shown in Table 2. In this estimation the

diffusion coefficients relevant to hydrogen-related defects are assumed to be the self diffusion coefficient of the atom

for that site. If the self diffusion coefficients of hydrogen or of vacancies are assumed, the relaxation times will be much smaller [e.g., Karato and Sato, 1982; Mackwell and Kohlstedt, 1990] and thus much less important at seismic frequencies.

TABLE 2. The Relaxation Strength and Relaxation Time for Point Defect Mechanisms of Anelasticity

Relaxation Relaxation

Strength Time, s

M site defects

F%-F% pair* -10 -3 10 -5-10 -4 Hydrogen-related defects -10-4

Si site defects

Hydrogen-related defects -10-4 1-10 The strain associated with a defect is arbitrarily assumed to be

0.1. The diffusion coefficients are for olivine at T/T,• = 0.7 (i.e., -1500 K) (data from Karato [1989]).

* All Fe ions are assumed to form pairs, giving the upper bound for the relaxation strength.

It is concluded from Table 2 that the possible point defect(s) (or defect complex(es)) that might cause significant anelasticity at seismic frequencies are those related to the Si site in olivine, particularly those related to hydrogen. However, even with the most optimistic estimate (C O ~ 10-4, • ~ 10-•), the relaxation strength A is ~10 -4 (corresponding to Q values in excess of 104), which is very small.

In pryoxenes the situation is different. First, the cation (Mg, Ca, Fe) diffusion in pyroxenes is much slower than that in olivine [Freer et al., 1982]. This brings the relaxation time into the range of periods of body waves.


Second, the distribution of cations in pyroxenes is known to cause some lattice distortion, order-disorder transitions although their transition temperatures appear rather low (T/T• < 0.4; T• is the melting temperature) [Ghose, 1965; Morimoto et al., 1960]. Therefore the redistribution of cations in pyroxenes is a possible mechanism of seismic wave attenuation in the upper mantle, particularly in the low-temperature portions of the upper mantle.

The point defect mechanisms discussed so far cause anelasfic relaxation since the defects can assume a

well-defined relaxed configuration(s) at sufficiently long times. However, there are some processes in which point defects are generated and annihilated indefinitely within a solid. An example of these processes is diffusion creep. In such a case, there is no longer a well-defined relaxed state, and therefore the solid behaves like a viscoelastic material.

These processes are discussed in section 5.3 in relation to grain boundary mechanisms.

5.2. Dislocation Mechanisms

5.2.1. General Considerations Dislocations in

crystals move viscously under shear stress and cause both macroscopic strain and anelasficity [e.g., Nabarro, 1967; Hirth and Lothe, 1968]. Dislocations have the following characteristics that are important with respect to internal friction: (1) unlike point defects, all dislocations interact with shear stress, and (2) the dislocation density is mainly determined by the applied long-term shear stress but not by the temperature (or hydrostatic pressure).

Under an applied shear stress o a dislocation will move with velocity v which is determined by o and other thermochemical variables (T, P, fugacity of water, etc.). When dislocation motion occurs on a macroscopic scale (i.e., a scale comparable to grain size), a macroscopic deformation will result:

(5)

(• is the strain rate; p is the dislocation density; b is the magnitude of the Burgers vector). At low stress and high temperature the dislocation velocity depends linearly on the stress •:

v=B, (6)

(B is the mobility of dislocation). Hence

(7)

The mobility B of a dislocation is determined by the resistances to its motion. They include phonon scattering, impurity drag, resistance due to lattice periodicity (the Peierls potential), jog drag, and the interaction with other dislocations [e.g., Hirth and Lothe, 1968]. In the case of macroscopic dislocation motion the velocity relevant to macroscopic strain (i.e., (5)) is an averaged velocity of a

dislocation that moves a long distance (grain scale motion), overcoming various obstacles (resistances).

When the applied stress is low and/or the time scale is short, as in the case of the interaction with seismic waves, a dislocation may not be able to overcome some of the obstacles at that time scale. These obstacles that are

difficult to overcome include impurity atoms, jogs, and nodes, most of which have a discrete nature (as opposed to continuous obstacles like phonon scattering and the Peierls potential). "Pinning" may occur at these points, limiting the dislocation motion to a microscopic scale determined by the spacing of pinning points and the magnitude of the applied stress. This pinning therefore can be responsible for anelasfic behavior.

In contrast, when the applied stress is high and/or the time scale is long (or the temperature is high), pinning will not be effective, and continuous dislocation motion will result, causing viscoelastic behavior.

It is important to realize that the anelasticity due to dislocation motion occurs only under restricted conditions. Since the seismic wave attenuation in the Earth's mantle

occurs at relatively high temperatures (T/T,, = 0.6-0.8), and low frequencies (to < 10 z s-•), viscoelasticity, rather than, or in addition to, anelasticity, is a distinct possibility. We shall discuss this point later (section 5.2.3) in the light of experimental results.

By whatever particular mechanisms the energy loss due to the interaction with dislocations occurs, it is evident that the magnitude of the nonelastic effects is proportional to the dislocation density and that the characteristic time 'c is inversely proportional to the dislocation mobility. However, to understand the dependence of these parameters on the thermochemical and stress state, we need to examine the mechanisms in more detail. Readers who are not interested in the details of dislocation

microdynamics may wish to skip to section 5.2.4. 5.2.2. Artelasticity Due to Dislocations First

consider the case where pinning is effective, i.e., at relatively low temperatures and/or high frequencies and/or low stress. Anelasficity will result since the dislocation stops for a sufficiently long time at a pinning point to yield a well-defined relaxed state. The nature of anelasticity is determined by the dislocation mobility and by geometrical factors such as the spacings between pinning points. Also, the magnitude of the Peierls energy will affect the nature of anelasficity, since it determines the kinetics and geometry of the relaxation process.

We can distinguish two cases with respect to the geometry of dislocation (Figures 5 and 6): Peierls potential control and line tension control. When the Peierls

potential (.•<•t,b 2, where % is the Peierls stress and b is the length of Burgers vector) is large and not negligible compared to the self energy of a dislocation (~ gb2; I.t is the shear modulus), a dislocation assumes a shape largely controlled by the Peierls potential and will be straight and parallel to the Peierls potential valley. On the other hand, when the Peierls potential is negligible compared to the

28, 4 / REVIEWS OF GEOPHYSICS

(A)


[001]

..L [1001 Figure •. Dislocations in olivine: (a) straight edge dislocations parallel to [001] (Balsam Gap dunRe) and (b) straight screw dislocations parallel to [ 100] (dunira nodule from Mount Erebus, Antarctica). Not• the cusps on dislocations (see arrows) suggesting pinning. Dislocations near the pinning points have

20' :•m

mixed character and are curved. The pinning at right (large arrows) is due to the interaction with a (100) tilt boundary, and pinning at other points (small arrows) is probably due to jogs formed by interactions with other dislocations.

self energy, a dislocation assumes a shape that is controlled by its self energy (fine tension) and the applied stress. Table 3 summarizes the magnitudes of the Peierls potential (or stress) relative to the self energy in a variety of solids. It is seen that the Peierls potential is large in those crystals where bonding is covalent and/or the unit cell dimensions are large. Silicates generally have large Peierls potentials because of largely covalent S i-O bonds and large unit cell dimensions. Therefore straight dislocations are common in silicates, but curved dislocations are also seen, particularly when motion perpendicular to the glide planes (e.g., dislocation climb) is possible, i.e., at high temperatures (Figure 5).

5.2.2.1. tine Tension Control (String Model) This includes solids with low Peierls potentials like fcc metals and alkali halides. It also includes solids with high Peierls potentials (e.g., silicates), when dislocations oblique to the Peierls potential valley are present. In the latter case, kinks are already present for a geometrical reason (geometrical kinks), and therefore the dislocation motion does not

TABLE 3. The Peierls Stress o r to Shear Modulus I.t Ratios in Various Crystals

Solids fcc metals (Ag, AI, Cu) <10 -5 bcc metals (ot-Fe, W, Mo) -5 x 10 -• IV group (Ge, Si) ~1 III-V group (GaAs, GaP) ~0.1 II-VI group (CdTe, ZnO) -5 x 10 -a Alkali ha!ides (NaC!, LiF) -(1-5) x 10 -4 O!ivine ~0.1

Quartz -0.2 MgA!:On --0.1 Corrundum (AI:O•, Cr:Oa, Fe:O•) ~5 x 10 -2 Ice ~0.1

Data are after Evans and Goetze [1979], Frost and Ashby [1982], and Takeuchi [1985]. The larger the values of the ratio, the greater is the influence of the Peiefis potential (over the self energy) in determining the shape of the dislocations. Large values tend to restrict the dislocations to Peierls potential valleys resulting in straight dislocations.


require the kink nucleation. In these cases the Peierls potential does not largely control the dislocation shape, and a dislocation behaves like a string with a line tension -gb 2. A segment of a dislocation between pinning points bows out until the force due to the applied stress balances the force due to line tension (Figure 6a). The magnitude of relaxation strength is proportional to the area that a

(a) Line tension controi

Low Peierls potential

segment of dislocation sweeps and can be calculated as [e.g., Nowick and Berry, 1972, chapter 12]

A - (anelastic strain)/(elastic strain) - pbl'/(c•/M) - p/2 (8)

where l' is the distance of dislocation motion in the

relaxation process, I the distance between pinning points,, the stress, b the magnitude of Burgers vector, and M the elastic modulus. Here use has been made of the relations c•

= Mb/l and l' ~ I. The exact formula for relaxation strength depends on the geometry of the slip systems relative to the

High Peierls potential ( a applied shear stress and, in the case of geometrical kinks, dislocation oblique to the On the angle of a dislocation with respect to the Peierls Peierls potential valley) potential valley [Fantozziet al., 1982].

The relaxation time is determined by the ratio of restoring force to the viscous force (see Figure 1). The restoring force in this case is the line tension of the dislocation which depends on the spacing between the

--- pinning points (c• ~ Mb/l). The viscous force is given by c• -v/B. Balancing these forces yields:'v - MbB/l, from which the relaxation time 'c(- l/v) is given by

(b) Peieris potential control 'c- 12/gbB (9)

Figure 6. Schematic illustrations showing the dislocation mechanisms of anelasticity. The dislocations move viscously under applied stress and thus absorb energy. When the dislocations are pinned at some points as shown in these figures, their motion stops at certain points, giving well-defined relaxed states and hence anelastic behavior. (When pinning is not effective, long-range motion of dislocations is possible at low frequencies, yielding a viscoelastic behavior.) The solid lines show the dislocation line. The dashed lines show the Peierls potential valleys. The dislocations are assumed to be pinned (pinning points are shown by solid circles). Two cases are distinguished: (a) where the dislocation shape is controlled by the line tension and (b) where the dislocation shape is controlled by the Peiefis potential. The case in Figure 6a includes a solid with low Peierls potential (e.g., fcc metals, alkali halides) and also a solid with high Peierls potential (e.g., silicates) with a dislocation oblique to the Peierls potential valley. In the former case (Figure 6a) the distance that a dislocation moves under stress increases with the

distance I between pinning points, and the characteristic time therefore depends on I. In the case in Figure 6b, on the other hand, the distance is determined by the lattice spacing and is independent of l, 'and the characteristic time is determined by the nucleation rate of kinks and therefore does not depend on I.

where B is the mobility of the dislocation defined by (6). A slightly different formula will apply to the motion of a dislocation oblique to the Peiefis potential in a high Peierls potential solid, in which case B is proportional to the mobility of kinks [Fantozzi et al., 1982]. It is seen from (9) that the relaxation times have a distribution through the distributions of the spacings between the pinning points l and of the mobilities B. When the dislocation structure is

well organized (i.e., regular arrays or networks) and only a single slip system operates (i.e., the case of single crystals oriented for a single slip), the characteristic time will have a narrow distribution. On the other hand, when the dislocation structure is not well organized and multiple slip occurs (i.e., the case of polycrystals), the relaxation times will have a wide distribution.

Let us first discuss the constraint on I and B imposed by the seismological observations. For a mechanism to be efficient for seismic wave attenuation, A must be significant (say A > 10 -3) and 'c must be in or near the seismic frequency band. For a given dislocation density the constraint with respect to A gives a lower bound for I. Likewise the characteristic time gives a constraint on 12/B. From the range of seismic wave frequencies (including those of free oscillations and of the Chandler wobble), we have 'c = 10 -2- 10 s s. We can also assume that the upper bound for I is the grain size (~! cm). Thus the spacing between pinning points I and the dislocation mobility B must have the values in the range shown in Figure 7 in order for this mechanism to be effective for seismic wave

attenuation. Also shown in this figure is a range of long-term dislocation mobility B r estimated from a manfie viscosity of .-.1019-22 Pa s [Peltier et al., 1981; Nakada and


Lambeck, 1989], from the relation/• = (•/•1 = pbv. It is noted that in order to have significant internal friction at body wave frequencies (x = 10-2-102 s), the dislocation mobility must be significantly higher than the long-term mobility. One obvious possibility to explain this observation is that the motion of geometrical kinks is responsible for the seismic wave attenuation, while other resistances, including kink nucleation, control the longer-term dislocation motion.

We now discuss the dependence of A and x on the thermochemical and stress states. From (8) and (9) it is seen that the variation of A and x comes mainly through the variation of p (dislocation density), l (spacing between pinning points), and B (dislocation mobility).

The dislocation density p depends mainly on long-term stress (•r, but not on the thermochemical environment [Nabarro, 1967; Hirth and Lothe, 1968], i.e.,

p ~ b-20jT/g) 2 (10)

(D

c-

O

for long term deformation

8•T (s)

g ro in / )0.,. Oz,.0o,

(f o-

I I I I I

Dislocation Mobility (m/s)/(Pa)

Figure 7. The range of dislocation mobility and distance between pinning points (for a string model) compatible with seismological observations. The dislocation density of p = 108 m -2 and the elastic modulus of M = 10 n Pa axe assumed. Constraints axe obtained from the relaxation strength A and relaxation time x. The long-term dislocation mobility inferred from postglacial rebound [e.g., Peltier et al., 1981; Nakada and Lambeck, 1989] is shown for compaxison. The dislocation mobility must be significantly higher than the long-term mobility for this dislocation mechanism to be effective for seismic wave attenuation.

The dislocation velocity is sensitive to temperature (and pressure) and other thermochemical environmental factors

including the fugacity of water frigo)-

a: aft, (11)

Generally,

•B fOT > 0 •B fO f }i2 o > 0 •B fOP < 0

It is not well known how the spacing between pinning points changes with these variables. But we can distinguish two cases: (1) pinning due to impurities and (2) pinning due to jogs (or nodes) formed by dislocation interactions.

In the first case, at geological time scales the characteristic time of dislocation motion (~ l/v, where I is the mean spacing of dislocation and v is the dislocation velocity) is much larger than the time scale of diffusion of impurities (-12/D, where D is the diffusion coefficient). One can thus assume that the concentration of impurities bound at dislocations, C, is determined by the thermodynamic equilibrium as C = C0K(T) where C O is the number of impurities per unit volume and K(T) is the equilibrium constant for the binding of impurities at dislocations (which depends on T). Therefore

p/l ~ CoK(T) (12)

hence

l ~ ((•'r)2/CoK(• (13)

Note that for shorter time scales like those in laboratory experiments, thermodynamic equilibrium may not be attained. In these cases I will be independent of the dislocation and therefore of stress.

In the second case the mean distance between pinning points is proportional to the mean distance between dislocations, i.e.,

1~ p-1/e _ ((•T)-I (14)

One of the most important characteristics of dislocation mechanisms of anelasticity is its dependence on the long-term tectonic stress c• r. In the case where the string model applies, this comes from the dependence of the dislocation density p and from the stress dependence of the spacing between the pinning points.

5.2.2.2. Peieds Potential Control (Kink Model) When the Peierls potential is the major barrier to dislocation motion, the nucleation (and migration) of kinds over the Peierls potential hill controls the dislocation motion. In this case the dislocation mobility is related to the intrinsic properties of dislocations (the Peierls potential). This is in contrast to the case of a low Peierls potential where the dislocation mobility is likely to be controlled by


impurities. Anelastic relaxation due to the intrinsic properties of dislocations is referred to as Bordoni relaxation (including the anelasticity due to geometrical kinds discussed in the previous section). Detailed accounts of Bordoni relaxation can be found in the work of Nowick

and Berry [1972, chapter 13] and Fantozzi et al. [1982]. Imagine a dislocation lying in a deep Peierls potential

valley. Unlike the case for a low Peierls potential, the dislocation will not bow out under stress since the barrier

to the motion perpendicular to the Peierls potential is too large. Instead, the dislocation motion will occur at a more microscopic scale, through the nucleation of a pair of kinks (double-kink nucleation) and their migration along the Peierls potential (Figure 6b). The nucleation of kinks involves motion over the Peierls potential hill and is therefore in general more difficult than motion along the Peierls potential. As a result, kinks migrate a long distance before the next nucleation occurs. A large number of obstacles along a dislocation may inhibit the migration of kinks, yielding a well-defined relaxed state. Thus inelastic behavior will result.

Since the nucleation of a pair of kinks is the most difficult process in the motion of dislocations along Peierls potentials, the relaxation time x of the entire process is equal to the characteristic time for the nucleation of kinks. It is given by [e.g., Nowick and Berry, 1972, chapter 13; Fantozzi et al., 1982]

There are several points in which the mechanisms in a kink model differ functionally from those in a string model. First, in the kink model the relaxation time x is independent of the spacings of the pinning points, and second, the relaxation strength A depends on the spacing of the pinning points I as A ~ p/, not as A ~ pl 2 as in the string model (equation (8)).

This results in a difference in the stress dependence of relaxation strength. For the pinning due to jogs, for example, the string model predicts no dependence of relaxation strength on stress, while the kink model predicts A ~ a (see (8), (10), (14) and (16)). Also, the stress dependence of relaxation time is different. In the kink model the relaxation time does not depend on the pinning distance (equation (15)) and therefore does not change with stress. In contrast, the strong model predicts that the relaxation time depends on the pinning point distance and therefore changes with stress (equation (9)).

Another complexity is the possible dependence of activation energies on stress which' change the relaxation strength and/or relaxation time with stress [Pard, 1961]. Experimental evidence for this effect is summarized by Fantozzi et al. [1982].

Let us now discuss the constraints of I and the activation

energy of kink nucleation. First, using (16) and a lower limit for the relaxation strength of 10 -3 , we get the lower bound for l:

x = (l/v) exp (E¾/RT) (15)

where v is the frequency of dislocation vibration (-1011 s -1 [Kocks et al., 1975]) and E• is the energy of formation of a pair of kinks. Note that the relaxation time is independent of the distance between pinning points.

The relaxation strength A is proportional to an anelastic strain, which is proportional to the area swept by a dislocation (b/) and to the population of dislocations in the relaxed configuration relative to the unrelaxed configuration. This ratio is determined by the thermodynamic equilibrium, and therefore the relaxation strength is given by [e.g., Nabarro, 1967, chapter 7]

A ~ (b4Bpl/RT) exp (-H/RT) (16)

where H is the energy difference between the relaxed and the unrelaxed configuration, i.e., the energy of two kinks and the work done by the stress. (Note that in a string model, all the dislocations are assumed to have a relaxed configuration at a sufficiently long time, while in the kink model, some dislocations remain in an unrelaxed state. The difference comes from the scale of dislocation motion.

In a string model, dislocations move a long distance, and therefore the configumtional entropy does not determine their position: the mechanical equilibrium determines the dislocation configuration. In contrast, in a kink model the unit step of dislocation motion occurs at a small scale, and the configurational entropy becomes significant.)

l > 10 -5-10 -• m

To obtain this lower bound, we furthermore assumed H(o) = 0 (i.e., assuming the maximum A for a given /), p = 108-101ø m_2, T = 2000 K, g = 101• Pa, and b = 5 x 10-•ø m. By using this model, large values of l were obtained because of the assumptions that dislocations move only by one lattice spacing.

Second, from the characteristic time (equation (15)) we can get a constraint on the formation energy of a kink pair. The preexponential term is 1/v ~ 10 -11 s [e.g., Kocks et al., 1975]; hence

x, s E¾/RT 10 -2 21 10 2 30 lO t 35 108 43

Assuming T/T• = 0.75 (T• is the melting temperature), we get

10 -2 16 10 2 22 104 26 108 32


The values of E•JRT_ in the body wave and surface wave frequency range•x ='•0 -2-10 z s) are significantly smaller than those for creep (Ec•/RT • = 24-31 [Karato, 1989]), but E•JRT• in the range for free oscillations and the ChandTer wobble (x = t04-108 s) appear to be comparable to those for creep. Note also that the activation energy for hardness which is presumably controlled by kink pair nucleation is E/RT• ~ 3 t [Evans and Goetze, 1979].

In any case, in order to have a wide range of x in this mechanism, a wide distribution of the activation energies (i.e., dislocation mobility) is required. It appears difficult to account for all of the observed seismic attenuation by a single mechanism involving kink pair nucleation, even if the effect of internal stress is considered. One possibility is that the low-frequency seismic wave attenuation is due to kink pair nucleation but that the high-frequency attenuation involves faster kinetics such as the migration of geometrical kinks.

5.2.3. Dislocation Mechanisms Involving #Unpin- ning": 'rransition to Viscoelastic Behavior When the temperature and/or the stress amplitude is high (or the frequency is low), pinning will not be effective. The dislocations may break away from the pinning agents (impurities, jogs, or nodes), or the pinning agents them- selves may move along with the dislocation. We call these two processes "unpinning." Under such circumstances the energy loss mechanisms have the following characteristics:

t. Now that the pinning is not effective, the unrelaxed state is no longer well defined. Continuous dislocation motion will occur over the distances longer than that in the pinned conditions. Thus the mechanical behavior will change into viscoelastic behavior, and hence the energy loss will increase monotonously with the increase of temperature (or with the decrease of frequency).

2. Generally, the unpinning process is more significant at higher stresses. This will cause nonlinear (amplitude dependent) energy loss.

Unpinning is a complex phenomenon which depends on the nature of the dislocation-pinning agent (impurities, etc.) interaction, the mobility of pinning agent, and the distance between the pinning points [Teutonico et al., 1964; Southgate and Mendelson, 1965; Liicke et al., 1968; Blair et al., 1971]. To illustrate the fundamental physics and get some idea about under what conditions unpinning might occur, we shall consider a simple model described by Nowick and Berry [1972, chapter 12]. In this model a dislocation is assumed to interact with impurities whose spacing is l, and the binding energy is E•. Then the frequency c0• at which unpinning occurs is given by [Nowick and Berry, 1972, chapter 12]

•0,= v exp [-(E b -ob21)/RT] (17)

This simple model contains some of the important physics of unpinning. First at 0 K, unpinning occurs only when the stress exceexls a critical value o c = E•/bel. Second, at f'mite temperatures, unpinning will be more effective at higher temperatures, higher stresses, lower frequencies, and larger pinning distances. Figure 8 shows the unpinning conditions due to thermal breakaway based on this simple model. Although the parameters in this model are poorly known, it is seen from Figure 8 that unpinning is quite likely to occur during seismic wave propagation and many of the high-temperature and low-frequency experiments.

Figure 8 indicates that strain amplitudes in ex•rimental studies must be sufficiently small, • << e c = E•/b IM (M is the elastic modulus), in order to observe linear anelasticity.

T: 1500 K

•• 12_..00•.•• 2-

I- Eb: 2--0 0

-2__

T: 1500 K•

-4 Eb= 400 kd/mol

i•// I I I !• [ I !

-z -• -s -4

Io• 0 I{m} {for I I I I ! I

-S -4 -3 -E -I 0

IO•l 0 • { MP•} (for I = I0 -4 m I I I I I I

-•o -5 -8 -z

lotto •{= •/M}(for I = IO -4m)

Figure 8. The frequency of unpinning events, •0, , based on simple model (equation (17)). When •0, becomes comparable or exceeds the frequency of elastic waves, pinning becomes unimportant. This diagram shows that pinning is ineffective at relatively high temperature, high stress o (or high strain •), and large pinning point spacing I. Unpinning will occur at T = 0 K when the stress (or pinning point distance) exceeds a critical value øc (or It). Et, is the binding energy of the pinning agent with a dislocation. Two cases are illustrated: a small binding

where v is the characteristic vibration frequency of energy (E• = 200 kJ mol -• as in pinning due to impurities and a dislocation (~t0 • s -• [Kocks et al., 1975]). When this large binding energy (E• = 400 kJ mol -•) as in pinning due to frequency exceeds or becomes comparable to the fre- jogs or nodes. Unpinning is likely under typical experimental quency of the seismic wave, unpinning becomes important. conditions and also in the propagation of seismic waves.

412 ß Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS 28, 4/REVIEWS OF GEOPHYSICS

Also, it is seen from this diagram that for the pinning agents with low binding energies (i.e., impurities), pinning is not effective already at body wave frequencies. Therefore among the possible mechanisms of pinning discussed in section 5.2.2, pinning due to dislocation interaction (jogs or nodes) appears to be the most likely mechanism to cause anelastic behavior for seismic wave

propagation. When unpinning is effective, there will be no well-

defined relaxed state, and the material behavior will be viscoelastic. In the viscoelastic regime, Q-• is related to the dislocation mobility B or the viscosity •1 as

•-• = MpbB/o• = M/ozrl (18)

where the mobility B is smaller (and the viscosity •1 is larger) than that in the anelastic regime. This transition from the anelastic to the viscoelastic regime is gradual since the unpinning occurs first at the weakest pinning point on a dislocation (i.e., large l) and will successively go into stronger pinning points as the temperature increases (or the frequency decreases). Furthermore, a distribution of pinning point spacings on a dislocation as well as the existence of different kinds of pinning points will cause a gradual transition from anelastic to viscoelastic behavior (see (17)). The gradual transition will lead to a weak frequency dependence of Q since the mobility will decrease with the decrease of frequency as stronger pinning points get involved.

Such a gradual decrease in dislocation mobility with the decrease of frequency (or the increase of time) is equiva- lent to a transient creep behavior. Jeffreys [1976] showed that the frequency dependence of Q as Q -(tox) a cor- responds to a creep function •P(t) of the form

•(0- (1 + t/x) a- 1 (19)

and therefore

t

/•(0 "' (1 + •)-0-'*) (20a) ~ (t/x) (a-0 t >> x (20b)

The transient creep of •:(t) ~ F I• type behavior (or œ(0 ~ t a) has been observed in olivine and olivine-rich rocks [Post, 1977; Berckhemer et al., 1979; Toriurni et al., 1984]. However, in most of these experiments the strain (or the stress) amplitudes were much higher than those involved in seismic wave attenuation.

At lower frequencies (or higher strain amplitude or higher temperatures) when the dislocation mobility is independent of frequency, a transition to steady state creep occurs. Thermal unpinning thus provides a physical explanation for the Burgers body behavior. At present, transient creep in mantle materials has not been studied in enough detail to delineate the conditions for the transition from transient to steady state creep [Post, 1977; Toriumi et al., 1984; Smith and Carpenter, 1987].

We note that the gradual transition to the viscoelastic regime due to progressive unpinning is a possible mechanism for the "high-temperature background" observed in many of the high-temperature experiments on internal friction [Chang, 1961; Southgate and Mendelson, 1965; Woirgard et al., 1981; Gueguen et al., 1981; Kampfrnann and Berckhemer, 1985]. Since the unpinning processes are generally sensitive to the stress amplitude (see Figure 8), special attention must be paid to the possible nonlinear effects in interpreting experimental results and seismological data.

5.2.4. Summary of Dislocation Mechanisms Dislocations can cause a wide range of nonelastic (anelastic or viscoelastic) behavior. Factors which may have an important influence on the behavior of dislocations and therefore on the nonelastic properties of solids include the Peierls potential, the interaction of dislocations with impurities, and the interaction between dislocations (the latter two determine the characteristics of pinning). Among the various mechanisms a common characteristic in dislocation mechanisms is their dependence on the long-term tectonic stress (•r. The higher the tectonic stress is, the more significant the nonelastic effects will be. However, the details of the stress dependence is poorly constrained by theoretical arguments.

One can deduce some constraints on dislocation

mobility and geometry (e.g., the spacing of pinning points) assuming that dislocation mechanisms are responsible for seismic wave attenuation. The results show that we can

get reasonable constraints on the spacing of pinning points but that the dislocation mobility at relatively high frequencies (e.g., body and surface waves) must be significantly higher than that for long-term deformation. Therefore microscopic mechanisms of dislocation motion must be different between long-term deformation (i.e., postglacial rebound, mantle convection) and short-term deformation (body and surface wave attenuation). One possible explanation is that dislocation motion parallel to the Peierls potential (motion of geometrical kinks) is responsible for short-term deformation and that motion perpendicular to the Peierls potential is responsible for long-term deformation. This hypothesis can be tested experimentally.

In addition, a simple calculation shows that the pinning necessary for anelastic behavior is not effective at high temperature and/or low frequencies. Thus a transition to viscoelastic behavior should be a natural consequence of unpinning. As we will see, most experimental data and seismological observations (Figures 3 and 10) are consistent with this suggestion.

5.3. Grain Boundary Mechanisms In a polycrystalline solid, internal friction may occur

through the motion of grain boundaries. Two contrasting modes of grain boundary motion can be distinguished: one is the grain boundary sliding (motion parallel to grain boundaries) and the other is grain boundary migration (motion perpendicular to grain boundaries).


5.3.1. Grain Boundary Sliding Grain boundary sliding can relax the shear stress and will cause internal friction. In examining the nature of internal friction due to grain boundary sliding, it is important to recognize that a grain boundary cannot generally be a perfectly flat plane. There must be some topographical irregularities which obstruct grain boundary sliding. Thus grain boundary sliding can be considered to be composed of two succes- sive processes, i.e., the sliding of flat portions of the grain boundaries and the accommodation of the deformation at

irregularities [Raj and Ashby, 1971]. If viscous accommodation at irregularities is difficult for a given time scale, the accommodation will be elastic. Then grain boundary sliding will stop when the applied force balances the force due to the elastic strain at the irregularities. This causes anelastic behavior. If, on the other hand, viscous accommodation is easy, then continuous grain boundary sliding will occur, resulting in viscoelastic behavior. Thus the mechanical behavior associated with grain boundary sliding is related to the nature of the accommodation of deformation at irregularities. We shall discuss this point later.

First we discuss the anelastic situation, i.e., the case for a relatively short time scale (or low temperature) or a large irregularity. When the accommodation of the deformation at the irregularities is elastic, the amount of sliding U is given by [Raj and Ashby, 1971]

anelasticity due to grain boundary sliding. This point was clearly demonstrated by the classical work by K• [1947].

The nature of the sliding mobility B s is not well known [e.g., Sun and K•, 1981], except that it is sensitive to the temperature through thermally activated processes. The sliding mobility is also sensitive to impurities and/or seconda• phases [e.g., Turnbaugh and Norton, 1968; Mosher and Raj, 1974]. Careful examination of impurities and, particularly, of secondary phases at grain boundaries is necessary in order to assess experimental results.

When the temperature is high and/or the frequency is low, the accommodation of deformation at irregularities will be viscous, involving diffusive mass transport or dislocation motion. In the case where grain shape is the major irregularity, the time scale of viscous accommodation is given by diffusion creep. Therefore in this case the criterion for viscoelastic behavior is given by

Doff• > d2(RT/Mfl) (24)

where Dof f is the effective diffusion coefficient for diffusion creep and fl is the volume of the diffusing species. Assuming that Dofr ~ 10 -•5 m2/s at T/T• = 0.7 [Karato et al., 1986], the transition times can be estimated

Transition Time Size of Irregularity •, s d, m

v - (Xlh2)(.laO (2

where )• is the wavelength of the boundary shape, h is its amplitude, and M is the elastic modulus. The most important irregularity is that due to neighboring grains, and in this case, X ~ h ~ d (grain size), giving

U/d ~ o/M (22)

This relation means that the amount of anelastic strain

(-U/d) is proportional to the elastic strain (-o/M) and is independent of the grain size. This leads to the important conclusion that the relaxation strength A due to groin boundary sliding is independent of the grain size [Raj and Ashby, 1971; Nowick and Berry, 1972, chapter 15]. A more detailed analysis [e.g., Raj and Ashby, 1971; Nowick and Berry, 1972, chapter 15] shows the magnitude of relaxation strength due to this mechanism to be quite large A~ 10 -!.

The kinetics of groin boundary sliding are not well understood, but assuming that the rate of sliding is proportional to the shear stress • = B sa (B s is the sliding mobility), the characteristic time x becomes (see, for example, Nowick and Berry [1972])

'r = d/B sM (23)

That is, the characteristic time is inversely proportional to the sliding mobility B s. Thus it is the dependence of the characteristic time on the grain size that is diagnostic for

10 +•ø 10-2 10 +8 10-3 10 +• 10-n 10 +4 10-s 10 +2 10 •

It is seen that viscoelastic behavior will appear only at very low frequencies if d is identified with typical groin sizes (d = 10 -3-10 -2 m). However, the grain boundaries in rocks deformed by dislocation creep have smaller-scale irregularities [Masuda and Fujimura, 1981; Karato et al., 1986], which are caused by grain boundary migration (Figure 9). It is possible therefore that the transition to viscoelastic behavior occurs at higher frequencies.

Now let us examine if anelasticity due to grain boundary sliding is consistent with seismological observations. First, the relaxation strength of this mechanism is very large (A ~ 10 -•) and can be consistent with seismological observations if the frequencies where maximum attenuation occurs are slightly beyond the seismic frequency band. Second, regarding the characteristic time (equation (23)), we can estimate sliding mobilities corresponding to characteristic times as

'•, s B s, rn s -• Pa -• 10-2 10-12 102 10-16 104 10-18 108 10-22


where the grain size is assumed to be 10 4 m and M = 10 -11 Pa. It is difficult to assess the above results since no

experimental studies are available on grain boundary sliding in minerals. When the sliding rate is controlled by diffusion accommodation at irregularities, the mobility B s is given as [Raj and Ashby, 1971]

B s - g•/RT )•/h 2 D •rr (25)

The above results require small topographic heights, h/)• = 10-4-10 -I, which can be consistent with observations of grain boundary shape (see Figure 9).

A "•i':•i; ) ...... " ....-.:.: .... ::!i •.•.;•.•,•::;:•:•::•'•,•::..• :•11'"-5 '"""" . .

-'." .---.•- :::"::•:"-"" '•'?-.-..-'??.•:•i'i•i•.: ..... :•,:•:.•j•i•': .. -•4:' '• .... ' ........... .: . .......... "::' ........... -:•. -. ...•:•: .... "•::•-.: '•'::? ......... :......:. .... .. .. •.•. •:'•?•;:.:•.:•:•. ...... :.: ß ..::.::.-' . ... • .... -•:;:.•:.;• ..... .

:../.% .... . ......-

.•.

.• . ...•.- ........ ...... :: :.'"' ..: .-: ......... " .t :•....-::•" ..'

• ..•- : .,..--.. ...... -4•::•::• ß - .:. . •. .:...• ..... ::::•3::•..:.•.•:: ....:?....•. :::.:.. ..... :.?:..?..:.•.: .;•:::':. ....... :.... . ... ;:.. ::•. .... .......... .:::::"--.?: .f:% :'... .:' ........

.... ':' .... ..':'•'%.•;;:..;..5:;:'::-:":" ; .......... -. ."-•-.:-.•..•:'; .... ;•:•:•:::•;:.•;:..: ................. ' ..... "-":---'"•:*::-'"• ':•:---;..:.' .....

....... ß •.;-. :.•.. ........ .

..•.:•:.:-'• . ........... ).•?•::; .....? :.....::.: ....... .•..•.:...-

:•3• ....... •:.•t•5;•& '?".'•'•.. •":•L:• .... '""• .. ..... •.= ::/'"'" "7•'"•'•:':.'"'""77.•:"•"%•"'"•'"• • ..:'f"•

......... ..... , ............. •'5•'•::•"•'• .................. ':•::'•':':'•':•'•'::::" ':":•:/;t:•'. •'"":•' ' ...... •';-'/ ....... •':.'"--....3'•3 .' 7 •. :: ....... :v:: •'""•......

'"" ....... f':":"'-'•' .'" ':..: 4 • ..... ':' •. ':•' • '•' • ': ..... •..

. : --• ..:.•v:.•,:.:.:: 3 ....... . .... '27'• ..• • ::' ..-"': ' '; •"•::•":"':• .•...;:;•., ' •.•.• .... m-. " .

•:.; -: ..

.. ...... .. •: . . ....... . •. •

- .:• '• •... .. ....... . . • , •..;:;:•: ..

•...< .......... .

-. 5... :<-::..'. :?•. -•-•. "'-...... ...•. '•-..:; ..

..... :'- '7'47:-: .... .-: .... .--"'.•'.... ....... . • • ;':- ..: .. ..:? •;:•:.-: ..•

........ :• '..-.%;.:::...4.::.'•:•.:-.. :..• ,. •..

50 •m

The sliding mobility B s will increase with temperature. But the effect of shear stress on B s is unknown, although it can be significant since the shear stress can change the grain boundary topography [Masuda and Fujimura, 1981; Karato et al., 1986].

5.3.2. Grain Boundary Migration In contrast to grain boundary sliding, grain boundary migration does not generally cause strain (see, however, Means and Jessell [1986]) and therefore by itself does not result in stress relaxation. However, there are two ways in which grain boundary migration may contribute to relaxation. One is that the sliding mobility B s might be controlled by grain boundary migration, because grain boundary migration can eliminate the boundary irregularities that obstruct grain boundary sliding. The other is that in a material where elastic anisotropy is large, the heterogeneity in strain energy will cause grain boundary migration that may result in anelasticity. However, the magnitude of this effect should be very small at the stress levels of seismic waves ((•- 103 Pa; see Table 1). Since the surface energy O, d2/c•sw. cific surface energy = ,•! to elastic strain energy ratio e_d 3 is very large (-10), the amount of grain boundary migration should be very small.

6. ANALYSIS OF THE EXPERIMENTAL RESULTS ON

MANTLE MINERALS AND ROCKS

From the foregoing discussion it is clear that one of the most important aspects of the experimental studies of internal friction is to investigate correlations between seismic wave attenuation and defect microstinctures. The

microstructures which need to be studied include those

dislocation structures (dislocation density, types of dislocations, dislocation shape), grain boundary structures (grain size, grain boundary shape, secondary phases), and point defects that significantly affect the mobilities of dislocations and/or grain boundaries (i.e., water-related defects). Unfortunately, such studies are not available for mantle minerals or rocks. But some preliminary pioneering work has been done which we shall discuss below.

Figure 9. Grain boundary structures of olivine aggregates: (a) an isostatically hot-pressed specimen, showing nearly equilibrium grain boundary morphology, and (b) a specimen deformed at T = 1573 K, P = 300 MPa, and/• = 10 -5 s -1, showing irregular grain boundary due to strain-induced grain boundary migration.

Regarding the dependence on the thermochemical and stress states, we conclude that (1) the relaxation strength A is independent of the thermochemical and stress states and (2) the characteristic time depends on the thermochemical and stress states through d (grain size) and B s (sliding mobility). When the grain size is determined by dynamic recrystallization [e.g., Karato et al., 1980], the grain size is related to the shear stress (•r as d- ((•r)-m (m = 1-1.5).

6.1. Gueguen et a!.'s Studies Gueguen and his group have made pioneering measure-

ments of internal friction at seismic frequencies and at high temperatures. A torsion pendulum type apparatus was used in the frequency range of 10 -4-10 Hz and from room temperature to 1400øC (at atmospheric pressure) [Woir- gard and Gueguen, 1978; Gueguen et al., 1981, 1989].

The samples studied include synthetic forsterite single crystals and aggregates, a natural enstatite, and a natural peridotite. The major results are summarized as follows: (1) the deformed single crystal of fosterite shows much larger internal friction than an undeformed counterpart [Gueguen et al., 1989]; (2) the internal friction in a single crystal of deformed fosterite is dominated by the "high- temperature background" superposed with a minor

28, 4 / REVIEWS OF GEOPHYSICS Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS ß 415

absorption peak around 10 -• Hz [Gueguen et al., 1989] (see Figure 10); and (3) after annealing of a natural peridotite, a significant decrease in internal friction was observed, although the dislocation densities were not much reduced [Woirgard and Gueguen, 1978].

IO ̧

10-3

SSMF (1989) Peridotire

(d = 5 to 800p. m)

BKA (1982)

X, \ •,• • Aheim Dunire

%•._ • GWD (1981)

%••'--• Deformed Peridotire • • • BKA (1982)

• • • •heim Dunite • 'N J' (d<.6mm)

G DMW (1989) •,.,,..,..',.I• 989) • A BKAS(1982) (si' "•;::'•tal) ••, Synthetic Forsterite .

•' "' ß .... / ,•.,.'"• (polycrystalline) Deformed //'""/' - (d-'.4 to 1.6mm) Undeformed '

Q-• normalized to 1200 øC

I I I ! I I I • I -2 0 2 4 6

log Frequency (Hz)

Figure 10. Internal friction in olivine and olivine-rich rocks. All data are reduced to 1200øC using the reported activation energies. Note that most data show a monotonic increase of internal friction with a decrease of frequency, a characteristic of "high-temperature background." Data sources are abbreviated as follows: BKA, Berckhemer et al. [1982b]; BKAS, Berckhemer et al. [1982a]; GWD, Gueguen et al. [1981]; GDMW, Gueguen et al. [1989]; SSMF, Sato et al. [1989] and JP, Jackson and Paterson [ 1990].

On the basis of the last observation (observation 3), Gueguen and his coworkers suggested that the absorption peak is due to dislocation climb in subboundaries [Woir- gard and Gueguen, 1978; Gueguen et al., 1981]. However, the interpretation of the experimental results on polycrystals under atmospheric pressure is not straightforward since heat treatment of polycrystals will produce open cracks or result in crack healing which may have significant effects on internal friction (see, for example, Jackson [1986]). In addition, annealing may have reduced the concentration of water-related point defects which generally enhance the defect mobilities.

The results of Gueguen et al. [1989] on single crystals clearly demonstrate the importance of dislocations. The internal friction is dominated by the "high-temperature background," i.e., the frequency and temperature depend-

dence can be given by Q-Z ~ 0)-• exp (-aE/RT), with ot ~ 0.2 and E ~ 440 kJ mol -z.

Gueguen et al. [1989] considered that their results are in contradiction with seismological observations because of (1) the frequency dependence Q-Z ~ 0) -0'2 and (2) values that are too large for internal friction. However, as we have seen, the observed frequency dependence is in fact in reasonable agreement with the seismological observations (Figure 3). The second point, which Gueguen et al. [1989] attributed to a nonlinear effect, can be interpreted as indicating that the dislocation density (or long-term stress) in the Earth's mantle is much smaller than that in this

particular experimental condition.

6.2. Berckhemer et al.'s Studies

Berckhemer and his group have made extensive studies on high-temperature and low-frequency internal friction in mantle rocks [Berckhemer et al., 1979, 1982a, b; Kampfmann and Berckhemer, 1985]. Their studies were on polycrystalline rocks (either synthetic aggregates or natural rocks) at atmospheric pressure in argon. The results are subject to reservation because of possibly significant effects of open cracks (see, for example, Jackson [1986]) and oxidation. However, their results contain important aspects which are examined below. The following points are noted (see Figure 10): (1) The internal friction is composed of two components: "high- temperature background" and a peak around 700ø-900øC (depending on rocks and frequencies). (2) The internal friction does not show a drastic change when the temperature crosses the solidus, although the modulus shows a drastic decrease [Berckhemer et al., 1982a]. (3) At the same temperature the internal friction in ultrabasic rocks is significantly smaller than that in basic rocks. (4) The activation energies of high-temperature background E] (Q-Z ~ 0)-• exp (-(zE]/RT)) and the activation energies of the absorption peak E 2 (0)peak - exp (-E2/RT)) are both of the order of 500-700 ld mol -•, which are values similar to those for creep. (5) The magnitudes of internal friction are considerably larger than those expected from creep. (6) The internal friction is independent of strain amplitude. (7) Generally, fine-grained rocks show higher internal friction than coarse-grained rocks.

It is clear from item 2 that the main mechanisms of

internal friction under their experimental conditions must be solid-state mechanisms. Berckhemer and his coworkers

suggested dislocation mechanisms. Although their results (items 1-5) are largely consistent with dislocation mechanisms (see section 5.2), this explanation must be considered hypothetical since no experiments were made to test the effect of dislocation densities. The observed

correlation between grain size and internal friction (see item 7) suggests that grain boundary mechanisms may also be important. However, since naturally deformed rocks have often undergone dynamic recrystallization and fine-groin size is correlated with high dislocation density [e.g., Nicolas, 1978], it is difficult to distinguish these two

416 ß Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS 28, 4/REVIEWS OF GEOPHYSICS

factors when naturally deformed rocks are used. It is necessary to control the grain size and dislocation density in- dependently to deduce the mechanisms of internal friction unambiguously. A particularly interesting aspect of this group's results is that item 1 above is consistent with the results of recent surface wave studies [Regan and Ander- son, 1984; Kawasaki, 1986] which show a thin oceanic plate (transition to low-velocity zone occurs, in these models, at-50 km depth in the 50- to 100-m.y.-old oceanic mantle).

6.3. $ato et al.'s Study Sato et al. [1989] measured internal friction in a

hot-pressed peridotite at high temperatures (up to 1280øC), high pressures (0.2-0.73 GPa), and high frequencies (co = 50-800 kHz), using a pulse transmission and spectral ratio technique. Their results are summarized as follows: (1) the internal friction increases monotonously with temperature, (2) no drastic changes occur when the temperature Crosses the solidus, (3) there is a relatively large pressure effect (the internal friction correlates with TiCI' m where T• is the solidus temperature, rather than the melting point of the constituent minerals), and (4) the frequency dependence is weak (Q-• ~ c0"% o• < 0.2).

Mainly on the basis of the observed large pressure effects, Sato et al. [1989] suggested a grain boundary mechanism. However, the experimental evidence to support their suggestion is not convincing since they did not test the effect of grain size. A large pressure effect could well be attributed to a dislocation mechanism [see Green and Borch, 1987].

One aspect of Sato et al.'s observations that is difficult to interpret is the very small frequency dependence with a significant temperature (and pressure) dependence. Assuming that Q-• ~ co• exp (-o.E*/RT), their results of o• < 0.2 and cdr*/R = 13.3T• imply that E* > 780 kJ mol -•, which significantly exceeds the known activation energies in olivine [e.g., Karato, 1989].

Since their measurements were made at much higher frequencies than the seismic frequencies, careful determination of the frequency dependence and the microstructural studies on its origin are essential when the results are to be applied to Earth. Unfortunately, these points were not well explored in Sato et al.'s [1988a, b, 1989] study.

6.4. lackson and Paterson's Study Jackson and Paterson [1990] have succeeded in

building an apparatus which allows them to measure internal friction on rock samples at simultaneous high temperature and pressure. This development is the result of a long and intensive effort to achieve measurement conditions which approximate conditions in the deep crest and the mantle. Being able to measure internal friction under high hydrostatic pressure circumvems the many possible contributions to internal friction coming from the presence of open cracks. In their preliminary study they chose Aheim dunite, a rock from the same area as the

Aheim dunite used by Berckhemer et al. (BKA) [1982b]. This allows a direct comparison of their respective results.

The measurements of Jackson and Paterson (JP) were made at 1000øC and at 300 MPa over a frequency range from 0.03 to 1 Hz. Their frequency dependence changes from a value of o[ = 0.19 to 0.15 as the sample is held at these experimental conditions for 1-5 hours. This compares with a value for {x of 0.25 for the data of Berckhemer et al. [1982b]. In Figure 10 we used the expression Q-• = co--a exp (-u.E*/RT) to extrapolate the JP results to 1200øC, using E* = 440 kJ mol -•. Note the good agreement with the data from BKA [1982b]. The good agreement may be fortuitous since we used 440 kJ mol -•, the value obtained by Gueguen et al. [1989] for single- crystal forsterite, instead of 700 kJ mol -•, which was obtained by BKA [1982b] for their sample.

The preliminary results from Jackson and Paterson [1990] point the way toward a plethora of data to come which Will be directly applicable to Earth. Careful sample characterization in terms of chemical composition, buffering, and microdefect structures will surely accom- pany new data from this group.

6.5. Comparison of the Experimental Results Figure 10 compares the results of the above four sets of

studies. The comparison is made at a common temperature (T = 1200øC). When necessary, interpolation or extrapola- tion of the results is made assuming Q-• ~ co--a exp (-aE*/RT). A common feature is the predominance of the "high-temperature background," namely, the general in- c• of internal friction with a decrease of frequency. This point is not clear in Sato et al.'s [1989] results, but the observed temperature effect strongly suggests this behavior. The measurements of both Berckhemer's and Sato's groups showed no drastic change in Q-• upon partial melting, which strongly suggests a subsolidus mechanism(s).

Another interesting observation common to studies of Berckhemer's and Gueguen's groups is that the activation energy of internal friction E (Q-• ~ co--a exp (-uE/RT)) is approximately equal to that of dislocation creep. A similar observation was made in metals [Lloyd and McElroy, 1976]. This may appear somewhat puzzling because it is often considered that the activation energy for Q is smaller than that for creep because of the faster kinetics of dislocation motion involved in Q than in creep [Gueguen and Mercier, 1973]. One notes, however, that when the characteristic times of relaxation have a wide distribution, the activation energy for Q is largely determined by those of the characteristic times at the long-period side. For example, for an absorption band model by Anderson and his coworkers [Anderson and Minster, 1979; Anderson and

Given, 1982], Q ~ [(0re2) •- (cox,) a] ~ (0re2) • (•:• is the short-period cutoff time; •2 is the long-period cutoff time). Thus it is possible that the activation energy for Q is significantly larger than that for a typical relaxation time at a given frequency (• - l/t0).

Alternatively, the observed internal friction in the


experiments may involve unpinning processes (thermal breakaway from or diffusive drag of pinning points) as discussed in section 5.2.3. In this case the activation

energy for Q is similar to those for unpinning processes [e.g., Southgate and Mendelson, 1965]. We have seen that pinning (or unpinning) is most likely due to dislocation interaction, in which case unpinning involves dislocation climb which has a relatively large activation energy similar to that for creep.

Regardless of the details of the microscopic basis for it, the near agreement of activation energies for internal friction and creep implies that the seismologically observed internal friction may provide useful information about long-term deformation properties of Earth's interior.

Some differences can also be seen. First, $ato et al.'s [1989] results show significantly higher internal friction than the other studies. This is true even when a possible effect of grain size is taken into account. We offer two possible explanations. One is that Sato et al.'s samples may contain a significantly larger amount of water than the other samples. They made hot-pressed samples from a natural peridotite without drying it at sufficiently high temperatures. It is well known that natural peridotites contain some hydrous silicates which, upon dehydration, significantly alter the defect mobilities [Chopra and Paterson, 1981, 1984; Karato et al., 1986]. Drying at high temperatures (T > 1000øC) is necessary to remove hydrous components. The other alternative is that Sato et al.'s samples may have much larger dislocation densities than the other samples. During hot pressing a significant number of dislocations (p ~ 10 • m -e) can be generated because of plastic deformation [Karato et al., 1986]. Careful examination of water content (by infrared spectroscopy) and of dislocation densities is necessary to assess these points.

Second, the results on single crystals of forsterites by Gueguen et al. [1989] agree reasonably well with the results on hot-pressed forsterite aggregates and a natural dunite by Kampfmann and Berckhemer [1985] and the results on the dunite by Jackson and Paterson [1990]. However, when the comparison is made at higher temperatures, Gueguen et al.'s results show significantly smaller internal friction than those of Kampfmann and Berckhemer. This is because the measured temperature dependences are significantly different between the two studies. The reason for this discrepancy is not understood.

From the foregoing discussion it is quite clear that in any experimental study on internal friction, careful examination of defect microstructures is absolutely necessary to identify the mechanisms. Unless one understands the mechanisms, experimental results should not be applied to Earth.

7. CONCLUDING REMARKS

The internal friction in Earth occurs over a wide range of conditions (co = 10 -8-10 +2 Hz, e ~ 10 -7 or less, T/T,,, ~ 0.5-0.8 in the mantle below approximately 100 km).

Current experimental studies show that significant internal friction occurs at subsolidus temperatures. Also, the frequency dependence found in laboratories is such that Q-• = to -• (t• = 0.1-0.3) which is quite consistent with seismological observations. However, the physical mechanisms of internal friction are not well understood, and this prevents an unambiguous application of laboratory data to Earth.

We have reviewed solid-state models of internal friction

with emphasis on those applicable at high TITs. It has been shown that either dislocation mechanisms or grain boundary mechanisms (or both) are likely to be responsible for seismic wave attenuation. However, a wide range of behavior is predicted, particularly with respect to the effects of long-term tectonic stress on dislocation mechanisms. This possible stress dependence of internal friction and velocity dispersion has important bearings on the interpretation of seismological observations. The conventional view in which velocity anomalies and/or Q distribution are solely attributed to the temperature variation may need revision. It is necessary to can'y out well-controlled experimental studies before a physically reasonable interpretation of seismological observations can be made on the basis of laboratory studies. Such studies are now under way in Canberra on polycrystalline rocks [Jackson et al., 1984; Jackson and Paterson, 1987, 1990] and in Boulder on single crystals [Getting et al., 1987, 1990].

An important aspect of this paper is that viscoelasticity in addition to anelasticity may be very important for seismic wave attenuation and velocity dispersion. The observed "high-temperature background" (Q-t ~ to -• exp (-txE/RT)) that has often been attributed to a wide distribution of anelasticity peaks [e.g., Minster and Anderson, 1981] may be due to a gradual transition from anelastic to viscoelastic behavior. If so, the relation between Q and (transient) creep is closer than in the case where Q is due to anelasticity alone. To test this idea, very low strain creep experiments will be useful in addition to Q measurements.

GLOSSARY

In this section we give a brief guide to a terminology in nonelastic behavior in solids.

nonelastic deformation: All deformation which is not

elastic and includes anelastic and viscoelastic deformation.

anelasticity: Mechanical behavior of a material in which a material can assume two different equilibrium states: one is the instantaneous response (unrelaxed state) and the other is a long-term equilibrium (relaxed) state. The nature of anelasticity is characterized by the relaxation time between the two states which determines the characteristic

frequency at which this relaxation occurs and by the difference in strain (anelastic strain) between the two states which gives the magnitude of the anelastic effect. A


standard linear solid (Figure la) is a typical model for anelastic behavior.

viscoelasticity: Mechanical behavior of a material in which material behavior changes from elastic at high frequencies to viscous at low frequencies. No equilibrium strain is defined at lower frequencies. Viscoelasticity is sometimes used to describe more general mechanical behavior including anelasticity [Nowick and Berry, 1972, chapter 1]. However, realizing the importance of distin- guishing the microscopic processes with well-defined relaxed states from those without them, we refer to viscoelasticity as mechanical behavior in which no equilibrium state exists at low frequencies. A Maxwell body (Figure lb) is a typical model for viscoelastic behavior.

Bordoni relaxation (peak): Anelastic relaxation in solids first studied by Bordoni. The magnitude of relaxation is very sensitive to the deformation state (amount of strain, annealing treatment, etc.) and impurities, but the peak frequency is relatively insensitive to these variables. This mechanism is therefore considered to be due to disloca-

tions. More specifically, the elementary process of relaxation in this process is considered to be related to the intrinsic properties of dislocations since the peak frequencies are insensitive to impurities.

called Peiefis potential, and the stress needed to move a dislocation over the Peierls potential without thermal agitation (i.e., at T= 0 K) is called the Peierls stress. Peierls potential (stress) is sensitive to the nature of bonding and crystal structure. Silicates generally have a large Peierls potential (stress) because of largely covalent Si-O bonds and large unit cells.

pinning: The resistance to dislocation motion is not al- ways homogeneous. Sometimes discrete obstacles like im- purifies and jogs exert significant resistance forces to dislocation motion. Then a dislocation is pinned at these points. In this case the motion of a dislocation is limited, and a well-defined relaxed state appears at sufficiently long time. Thus pinning is responsible for anelastic behavior. Pinning is no longer effective at high temperatures and/or high stresses, causing a transition to viscoelastic behavior.

point defect: Defects in crystal with zero dimension. They include impurity atoms, vacancies (vacant lattice sites), interstitials, and their combinations. Typical impurity atoms in minerals include substitutional ions (Fe in M site in olivine and orthopyroxene) and OH-related defects which presumably form some defect complexes with vacancies.

Burgers vector: A slip vector associated with a dislocation in a solid which is one of the unit vectors that has a

periodicity of lattice.

dislocation: A line defect in a solid which defines a

boundary of an area over which slip has taken place. A dislocation is characterized by its slip (glide) plane and slip vector (Burgers vector). A dislocation parallel to the slip vector (Burgers vector) is called screw dislocation, and the one perpendicular to the slip vector is called edge dislocation. A dislocation intermediate between the two is

referred to as a mixed dislocation.

high-temperature background' Internal friction often observed in relatively high temperature (T/T• > 0.5, where T• is the melting temperature) experiments characterized by a monotonous increase in internal friction with an increase in temperature or with a decrease of frequency.

jog: A step on a dislocation normal to its glide plane.

kink: A step on a dislocation in its glide plane.

Peierls potential (stress): A dislocation in a crystal has the lowest energy when it lies in one of the low-index crystallographic orientations. When a dislocation moves out of its low-energy stable configuration, the energy of the crystal changes, and it exerts a resistance force on the dislocation motion. The potential energy change due to the change in position of a straight dislocation in a crystal is

NOTATION

Symbols are given in order of appearance in the text (or figures).

A relaxation strength. relaxation time.

internal friction.

frequency. exponent describing frequency dependence of Q (Q-1 ~ c0•).

• viscosity. M modulus.

E activation energy. T temperature. P pressure. R gas constant.

V/, compressional wave velocity. V s shear wave velocity. L = (4/3)(Vsfgv) •. d grain size. e strain.

P b

l

dislocation density. Burgers vector. distance of pinning points on a dislocation, mean distance of dislocations. strain associated with defect. defect concentration. molar volume of defect.

28, 4 / REVIEWS OF GEOPHYSICS Karato and Spetzler: DEFECT MICRODYNAMICS IN MINERALS ß 419

D

v

u

h

stress.

distance defect moves from unrelaxed to relaxed state.

diffusion coefficient.

velocity of dislocation motion. mobility of dislocation. shear modulus.

equilibrium constant for binding of impurities at dislocations.

frequency of vibration of dislocations. energy of kink pair formation. energy difference between relaxed and unrelaxed state.

amount of sliding on a grain boundary. wavelength of grain boundary shape. amplitude of grain boundary shape. specific surface energy.

ACKNOWII:DGMI:NTS. The content of this paper was developed when S.K. was visiting CIRES, which was supported by CIRES and NSF. The editors H. ]'. Melosh and M. Neugebauer provided helpful comments to improve the manuscript. I. Jackson and M. Paterson made a reprint of their latest paper available to us. Thank you all.

H. ]'. Melosh was the editor in charge of this paper. He thanks Y. Gueguen and I. Jackson for their assistance in evaluating the technical content and D. Sandwell for serving as a cross- disciplinary referee.

REFERENCES

Anderson, D. L., and J. W. Given, Absorption band Q model for the Earth, J. Geophys. Res., 87, 3893-3904, 1982.

Anderson, D. L., and R. S. Hart, Q of the Earth, J. Geophys. Res., 83, 5869-5882, 1978.

Anderson, D. L., and J. B. Minster, The frequency dependence of Q in the Earth and implications for mantle rheology and Chandler wobble, Geophys. JJt. Astron. Soc., 58, 431-440, 1979.

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Defect Microdynamics in Minerals and Solid&State ...geophysics.wustl.edu/seminar/1990_Karato_RG.pdfplications for the mechanisms of seismic wave attenuation (section 3). The importance