Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 1 doc

506 29 Diffusion in Metallic GlassesA second group of conventional metallic glasses consists of alloys of earlytransition metals ETM and late transition metals LTM.. Appli-cations of bulk

504 29 Diffusion in Metallic Glasses

Fig 29.2 Structure of a binary metallic glass (schematic)

They can be produced as homogeneous, metastable materials in compositionranges, where the equilibrium phase diagram requires heterogeneous phasemixtures of crystalline phases

Historical Remarks: The first liquid metal alloy vitrified by cooling from

the molten state to the glass transition was Au-Si as reported by Duwezand coworkersin 1960 [5] These authors made the discovery as a result ofdeveloping rapid quenching techniques for chilling metallic melts at very highcooling rates of 105to 106Ks−1 The work of Turnbull and of Chen [6–8]

was another crucial contribution to the field and illustrated the similaritiesbetween metallic and silicate glasses This work clearly demonstrated theexistence of a glass transition in rapidly quenched Au-Si glasses as well asother glass-forming alloys such as Pd-Si and Pd-Cu-Si, synthesised initially

by the Duwez group Already around 1950, Turnbull and Fisher had

predicted that as the ratio between the glass-transition temperature, Tg, and the liquidus temperature, Tl, of an alloy increased from Tg /T l ≈ 1/2 to 2/3,

homogeneous nucleation of crystals in the undercooled melt should become

very sluggish on laboratory time scales [6] This Turnbull criterion for the

suppression of crystallisation in undercooled melts is still today one of thebest ‘rules of thumb’ for predicting the glass-forming ability of a liquid.The field of metallic glasses gained momentum in the early 1970s whencontinuous casting processes for commercial manufacture of metal glass rib-bons such as melt spinning were developed [10] During the same periodChen [9] used simple suction casting methods to form millimeter diameterrods of ternary Pd-Cu-Si alloys at cooling rates in the range of 103 Ks−1.

If one arbitrarily defines the ‘millimeter scale’ as ‘bulk’, then the Pd-based

ternary glasses were the first examples of bulk metallic glasses Experiments

on Pd-Ni-P alloy melts, using boron oxide fluxing to dissolve heterogeneousnucleants into a glassy surface coating, showed that, when heterogeneous nu-cleation was suppressed, this ternary alloy with a reduced glass-transition

temperature of Tg /T ≈ 2/3 would form bulk glass ingots of centimeter size

at cooling rates in the range of 10 Ks−1 [11, 12] At the time, this work was

perceived by many to be a laboratory curiosity

During the late 1980s Inoue and coworkers investigated the cation of amorphous aluminium alloys In the course of this work, Inoue’steam studied ternary alloys of rare earth materials with aluminium and fer-rous metals They found exceptional glass forming ability in rare-earth-richalloys, e.g., in La-Al-Ni [13] From there, they studied similar quaternarymaterials (e.g., La-Al-Cu-Ni) and developed alloys that formed glasses atcooling rates of under 100 Ks−1with critical thicknesses ranging up to 1 cen-

fabri-timeter A similar family with the rare-earth metal partially replaced by thealkaline-earth metal Mg (Mg-Y-Cu, Mg-Y-Ni, ) [14] along with a parallelfamily of Zr-based alloys (e.g., Zr-Cu-Ni-Al) [15] were also developed Thesemulticomponent glass-forming alloys demonstrated that bulk-glass formationwas far more ubiquitous than previously thought and not confined to exoticPd-based alloys Building on the work of Inoue, Johnson and cowork-ers[16, 17] developed a family of ternary and higher order alloys of Zr, Ti,

Cu, Ni, and Be These alloys were cast in the form of fully glassy rods ofdiameters ranging up to 5 to 10 centimeters No fluxing is required to formsuch bulk metallic glasses by conventional metallurgical casting methods Theglass-forming ability and processability is comparable to that of many silicateglasses Metallic glasses can now be processed by common methods available

in a foundry [18]

Families of Metallic Glasses: The number and diversity of metallic glasses

are continually increasing We make no attempt to present a comprehensivelist because of the complexity in ternary, quaternary, and higher order alloys

We simply mention several families of alloy systems in which glass formationfrom the melt occurs readily (see also Chap 28)

Metallic glasses that require rapid cooling with rates of about 106 Ks−1 are denoted as conventional metallic glasses For conventional metallic glasses

the ‘nose’ of the nucleation curve of the TTT diagram lies in the range of 0.1

to 1 milliseconds (see Chap 28) They are usually produced by melt-spinningfor laboratory and commercial manufacture in the form of thin ribbons orsheets of about 40µm thickness

The first class of this type were alloys of late transition metals (LTM) cluding group VIIB, group VIII, and noble metals) and metalloids (M) such

(in-as Si, B, and P Metallic gl(in-asses of the type LTM-M are perhaps cally still the most important ones Many glasses based on Fe, Co, and Niand on B and P with excellent soft magnetic properties belong to this group

technogi-It was at one time believed that the glass formation range is centered around

a deep eutectic at about 20 at % metalloid Examples are Au80Si20, Pd80Si20,

Pd80P20, or Fe80B20 When further solute species are added (late transitionmetals or metalloids), the glass-forming ability may increase further Exam-ples are Fe40Ni40B20 and Pd40Ni40P20

506 29 Diffusion in Metallic Glasses

A second group of conventional metallic glasses consists of alloys of earlytransition metals (ETM) and late transition metals (LTM) The former havehigh melting temperatures and addition of a LTM generally leads to a rapiddecrease of the liquidus temperature down to an eutectic The liquidus tem-perature then remains relatively low across one or more intermetallic phases

of relatively low stability Examples of this type are Zr-Co, Zr-Cu, Zr-Ni,Zr-Fe, and Nb-Ni alloys

Most of the binary alloy systems of rare earth metals with late transitionand group IB metals have also deep eutectics They have been shown to

be readily glass forming, if the composition is centered around the eutecticcomposition Examples are La-Au, La-Ni, Gd-Fe, and Gd-Co alloys

Bulk metallic glasses exhibit TTT diagrams with a crystallisation ‘nose’

in the range between 1–100 seconds or more These alloys have an exceptionalglass-forming ability and undercooled melts, which are relatively stable [18].This permits diffusion studies even in the undercooled melt of bulk metallicglass-forming alloys By contrast, conventional metallic glasses undergo crys-tallisation before the glass-transition temperature is reached and thus can bestudied only below the glass-transition temperature High glass-forming abil-ity was recently found for bulk metallic glasses based on copper [21] Appli-cations of bulk metallic glasses benefit from their excellent elastic properties

and the good formability in the supercooled liquid state Bulk amorphous

steels is a recent development [19, 20] with potential to replace conventional

steels for some critical structural or functional applications

29.2 Structural Relaxation and Diffusion

Glasses are thermodynamically metastable in a twofold sense: (i) They can

undergo crystallisation, during which the material transforms to (a)

crys-talline phase(s) (ii) The properties of a glass may depend on its thermalhistory (see Chap 28) Upon reheating a glass to the glass-transformation

range, the glass properties may change due to a process which is called

struc-tural relaxation.

Structural relaxation of an amorphous material leads to a more ble amorphous state Structural relaxation is accompanied by a number ofchanges in physical properties Clearly, the extent of property changes for

sta-a given msta-aterista-al depends on its thermsta-al history sta-and on the method of glsta-assproduction Changes due to structural relaxation are understandable by con-

sidering the volume (or enthalpy)-versus-temperature diagram of Fig 29.3.

The volume can be altered by a heat treatment, which allows equilibration

of the structure to that pertaining to the heat treatment temperature A fastcooled glass has a higher fictive temperature, a larger volume, and a lower

density The volume difference is sometimes denoted as the excess volume If

we reheat such a sample to a temperature within the transformation range,but below the original fictive temperature, the sample will readjust to the

Fig 29.3 Schematic illustration of structural relaxation in the V-T (or H-T)

diagram of a glass-forming material

structure appropriate for the new temperature Its volume will decrease though the changes in density occurring during structural relaxation are notparticularly large (typically less than 1 %), they can be important for vis-cosity and ductility, as well as for magnetic, elastic, electric, and diffusionproperties A review of the effects of structural relaxation on various prop-erties of metallic glasses has been given by Chen [22] In this section, weconcentrate on structural relaxation of diffusion properties

Al-If structural relaxation occurs during the diffusion annealing of a sample,the diffusivity depends on time Under such conditions the thin-film solution

of Fick’s second law (Chap 3) remains valid, if the diffusivity D is replaced

by its time average given by

1

t

t

0

508 29 Diffusion in Metallic Glasses

Fig 29.4 Time-averaged diffusivitiesD of59Fe in as-cast Fe

continuous decrease of D(t) to a plateau value In the following this plateau value is denoted as DR and attributed to the relaxed amorphous state Thefeatures described above are common to many diffusion studies on metallicglasses The diffusivity decreases during diffusion annealing as a result ofstructural relaxation This effect may be described by the relationship

D(t, T ) = D R(T ) + ∆D(t, T ) (29.3)

The diffusivity enhancement, ∆D(t, T ), drops to zero upon sufficient ing and the diffusivity in the relaxed state, DR(T ), depends on temperature

anneal-only Usually, within the experimental accuracy the temperature dependence

of DR can be described by an Arrhenius relation (see below)

The diffusivity enhancement in conventional metallic glasses is correlatedwith the excess free volume present in the as-quenched material (Fig 29.3).This excess volume anneals out during structural relaxation and leads to anincrease in density Atoms can move more easily through a more open (lessdense) structure than through a more dense structure As a consequence, thediffusivity decreases during an annealing treatment at a temperature belowthe fictive temperature of the as-quenched glass Sometimes the excess volume

is also said to be due to ‘quasi-vacancies’ envisaged as localised defects beingstable over several jumps [25] In the language of quasi-vacancies the latter are

Fig 29.5 Instantaneous diffusivities D(t) of several conventional metallic glasses

as functions of annealing time according to Horvath et al [24]

mobile during structural relaxation and remove the diffusivity enhancement

In contrast to self-diffusion in crystalline metals, which occurs via vacanciespresent in thermal equilibrium, quasi-vacancies in an as-quenched amorphousalloy are present in supersaturation and anneal out when they become mobile

As a result, the diffusivities slow down until they have reached their state values

relaxed-The diffusivity enhancement depends on the material, its thermal history,and on the technique of glass production According to Fig 29.3 differentfictive temperatures lead to different amounts of structural relaxation For

a given material with low fictive temperatures the diffusivity enhancementmay be insignificant As a consequence, some conflicting results in the lit-erature about the magnitude of structural relaxation effects in diffusion arelikely due to different techniques of alloy production such as melt-spinning,splat cooling, or co-evaporation

29.3 Diffusion Properties of Metallic Glasses

Temperature Dependence: Diffusion measurements on conventional metallic glasses are usually carried out below the glass-transition temper-

ature due to the limitations imposed by incipient crystallisation of the glass

at higher temperatures The diffusion coefficients in the structurally relaxedglassy state follow an Arrhenius-type temperature dependence

510 29 Diffusion in Metallic Glasses

thus yielding pre-exponential factors D0 and activation enthalpies ∆H

Ex-amples of Arrhenius plots for both metal-metal and metal-metalloid typeamorphous alloys are shown in Fig 29.6 The temperature range in whichdiffusion measurements have been performed is often limited to 200 K or less

At high temperatures the onset of crystallisation and at low temperatures thevery small diffusivity prevents meaningful measurements

The error margins imposed on the diffusion parameters are relativelylarge, being of the order of 0.2 eV for the activation enthalpy and aboutone order of magnitude for the pre-exponential factor It was shown that theobserved Arrhenian temperature dependence within these error bars is com-patible with a narrow height distribution of jump barriers in the disorderedstructure of an amorphous alloy [26–28] Another reason for the ‘surprising’linearity of the Arrhenius plots are compensation effects between site andsaddle-point disorder [29] The most likely reason, however, is the collectivity

of the atom-transport mechanism leading to an averaging of disorder effects

in the atomic migration process (see below)

For bulk metallic glasses it is possible to carry out diffusion ments in a temperature range that covers both the undercooled melt re-

measure-Fig 29.6 Arrhenius diagram of self- and impurity diffusion in relaxed

metal-metalloid and metal-metal-type conventional metallic glasses according to Faupel

et al.[37]

gion and the glassy state We illustrate diffusion in bulk metallic glassesfor an alloy that has attracted great interest – namely the five compo-nent alloy Zr46.75Ti8.25Cu7.5Ni10Be27.5(commercially denoted as ‘Vitreloy4’).Centimeter-size rods of this alloy can be produced by casting techniques [18]and the TTT diagram in the range of the glass transition and crystallisa-tion is known from the work of Busch and Johnson [30] The temperaturedependence of diffusion for a variety of elements in Vitreloy4 is displayed

in Fig 29.7 The following diffusers have been studied; Be [31], Ni [32, 33],

Co [31, 34], Fe [31], Al [35], Hf [36] and the data have been assembled in tworeviews [37, 54] An important feature of Fig 29.7 is that the diffusivities

of several elements can be split into two different linear Arrhenius regionsbelow and above a ‘kink temperature’ The kink temperature correspond tothe transition between the glassy and supercooled liquid states The activa-tion enthalpies and pre-exponential factors in the supercooled liquid state arehigher than those below the kink temperature In addition, the kink temper-ature separating the glassy and the supercooled region is higher for elements,which diffuse faster in the amorphous state

It has been demonstrated that the diffusion times applied at low peratures were too short to reach the metastable state of the undercooledliquid at these temperatures [33, 39] A test of this interpretation of the non-linear Arrhenius behaviour is shown in Fig 29.8, in which the diffusivities

tem-Fig 29.7 Arrhenius diagram of tracer diffusion of Be, B, Fe, Co, Ni, Hf in the

bulk metallic glass Zr46.75Ti8.25Cu7.5Ni10Be27.5 (Vitreloy4) according to Faupel[37]

512 29 Diffusion in Metallic Glasses

Fig 29.8. Arrhenius diagram of tracer diffusion of B and Fe in

Zr46.75Ti8.25Cu7.5Ni10Be27.5 (Vitreloy4) according to Faupel et al [37] Open

symbols: as-cast material from [31]; filled symbols: pre-annealed material from [39]

of Fe and B in ‘as-cast’ and ‘pre-annealed’ Vitreloy4 are displayed For ciently long annealing times the material finally relaxes into the supercooledliquid state (see also Fig 29.3) Open symbols in Fig 29.8 represent diffu-sivities in the as-cast material, full symbols represent diffusivities measured

suffi-after pre-annealing between 1.17 × 106s and 2.37 × 107s at 553 K, i.e low the calorimetric glass-transition temperature The diffusivities obtainedafter extended pre-annealing below 550 K are smaller than those of the as-cast material, whereas in the high-temperature region the diffusivities of theas-cast and the pre-annealed material coincide Furthermore, the diffusivities

be-in the relaxed material can be described by one Arrhenius equation, whichalso fits the high-temperature data of the as-cast material This provides ev-idence that the kink in the temperature dependence of the diffusivity is notrelated to a change in the diffusion mechanism but depends on the thermalhistory of the material It is caused by incomplete relaxation to the state ofthe undercooled liquid

Correlation between D0 and ∆H: Reported values of the activation

enthalpy in conventional metallic glasses and supercooled glass melts, ingeneral, range from 1 to 3 eV for different diffusers (excluding hydrogen)

The pre-exponential factors D0 show a wide variation from about 10−15 to

1013m2s−1[37] This variation is much larger than the one reported for

crys-Fig 29.9 Correlation between D0and ∆H for amorphous and crystalline metals according to [37] Solid line: conventional metallic glasses; dotted line: bulk metallic glasses; dashed line: crystalline metals

talline metals and alloys (about 10−6to 102m2s−1) The experimental values

of D0 and ∆H have been found to obey the following correlation:

A and B are constants This relationship has a universal character in the

sense that it is valid not only for metallic glasses but also for self- and rity diffusion in crystalline metals and alloys involving both interstitial and

impu-substitutional diffusion (see [37] and [38] for references) The values of D0

and ∆H in both conventional metallic glasses and in the undercooled liquid

state of bulk metallic glasses do follow the same relationship as shown in

Fig 29.9 However, the fitting parameters for metallic glasses (A ≈ 10 −19

to 10−20 m2s−1 , B ≈ 0.055 eV) and crystalline metals (A ≈ 10 −7 m2s−1,

B ≈ 0.41 eV) are quite different (Fig 29.9).

The fact that the parameters A and B differ considerably for crystalline

and amorphous metals indicates that the diffusion mechanism of metallicglasses is different from the interstitial or vacancy mechanisms operating incrystals

Pressure Dependence: Studies of the pressure dependence of diffusion and

the activation volumes deduced therefrom have been key experiments for cidating diffusion mechanims of crystalline solids For vacancy-mediated dif-fusion the activation volume equals the sum of the formation and migrationvolumes of the vacancy (see Chap 8) The major contribution to the activa-

elu-514 29 Diffusion in Metallic Glasses

tion volume of self-diffusion for metallic elements comes from the formationvolume, which typically lies between 0.5 and 1 atomic volumes For intersti-tial diffusion no defect formation is involved and the activation volume equalsthe migration volume of the interstitial, which is small A small activationvolume implies a weak pressure dependence of the diffusion coefficient.Measurements of the pressure dependence of diffusion in metallic glassescan be grouped into two categories [37]:

1 Systems with almost no pressure dependence: activation volumes close tozero were reported for metallic glasses, which mainly contain late transi-tion elements and for tracers of similar size as the majority component

A typical example is displayed in Fig 29.10 Small activation volumesallow vacancy-mediated diffusion to be ruled out and have been taken asevidence for a diffusion mechanism, which does not involve the formation

of a defect

2 Systems with significant pressure dependence: activation volumes parable to those of vacancy-mediated diffusion in crystalline solids weremainly reported for diffusion in Zr-rich Co-Zr and Ni-Zr metallic glasses.They have tentatively been attributed to the formation of diffusion-mediating defects which are delocalised On the other hand, molecular-dynamics simulations for Ni-Zr glasses suggest that diffusion takes place

com-by thermally activated collective motion of chains of atoms (see Fig 29.12and Chap 6) It has been proposed that the migration volume of chain-like motion is associated with a significant activation volume [41]

Fig 29.10 Pressure dependence of Co diffusion in Co81Zr19 at 563 K according

to [37] The dashed line would corresponds to an activation volume of one atomic

volume

Isotope Effects: Isotope effect measurements proved to be useful in

deduc-ing atomic mechanisms of diffusion in crystals (Chap 9) Such studies havebeen performed on metallic glasses as well

Almost vanishing isotope effects have been reported for Co diffusion invarious relaxed, conventional metallic glasses by Faupel and cowork-ers[40, 42–45] The small isotope effects can be attributed to strong dilution

of the mass dependence of diffusion due to the participation of a large ber of atoms in a collective diffusion process Isotope effect experiments arealso reported for the deeply undercooled liquid state of bulk metallic glassesEhmler et al [46, 47] (Fig 29.11) The magnitude of the isotope effectparameter is similar to the isotope effects found for (relaxed) conventionalmetallic glasses This lends support to the view that the diffusion mechanismdoes not change at the calorimetric glass transition and demonstrates thecollective nature of diffusion processes in metallic glasses [37]

num-So far, we have mentioned isotope effects in structurally relaxed metallicglasses On the other hand, as-cast metallic glasses contain excess volumequenched-in from the liquid state Magnitudes of the isotope effect param-eter comparable to values observed for crystalline metals were reported forthe as-quenched metal-metalloid glass Co76.7Fe2Nb14.3B7[48] Such observa-tions suggest that during diffusion annealing of unrelaxed glasses quenched-inquasi-vacancies serve as diffusion vehicles until they have annealed out

Atomic Mechanisms: Experiments and computer simulations show that

diffusion mechanisms in metallic glasses contrast with diffusion in crystals

It requires other concepts, based on thermally activated highly collectiveprocesses

Fig 29.11 Isotope effect parameter as function of temperature for Co diffusion

in bulk metallic glasses according to [37]; data taken from Ehmler et al [46, 47]