Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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272 Franz Faupel and Klaus R¨atzke

T (K)

1.4 1.6 1.8 2.0

1000 / T (K )

-1

Fig. 6.7. Arrhenius

plot summarizing the

diﬀusivities of various

tracers in amorphous

7.5

27.5

Thedataweretaken

from the following ref-

erences: Be [143], B,

Co, Fe, Al [92], Co [90].

The sequence of the

curves is as in the leg-

end. Data for Ni [91],

being close to those of

Co, are omitted for sake

of clarity.

deeply supercooled liquid state of metallic bulk-glass-forming alloys. Results

for Vitrelloy 4 and a well known ‘Inoue glass’ are displayed in Fig. 6.8. For

experimental details the reader is referred to [90, 136].

One notes that the isotope eﬀect is very small in the whole temperature

range and exhibits no temperature dependence. The magnitude of the isotope

eﬀect is similar to the isotope eﬀects found in the glassy state of conventional

metallic glasses [103–105,116]. This support the view that the diﬀusion mech-

anism does not change at the caloric glass transition temperature and that the

highly collective hopping processes occurring in the glassy state still deter-

mine long-range diﬀusion in the deeply supercooled melt. Collective hopping

of many atoms has recently also been observed in deeply supercooled melts

in MD simulations [137]. The order of magnitude of 10 atoms participating in

the hopping processes is in agreement with the measured E values [90,136].

Recently, we measured the isotope eﬀect in a Pd

alloy at

one and the same temperature in the glassy state and after long-time relax-

ation in the supercooled liquid state [60]. Although the diﬀusivity dropped

by about an order of magnitude, the isotope eﬀect did not change and was

close to zero in both states (Fig. 6.9). This provides further evidence of the

absence of a change in the diﬀusion mechanism at the caloric glass transition

temperature.

Since this macroscopic glass transition is a transition from a metastable

equilibrium state above T

to a non-equilibrium state below T

there is an ob-

vious explanation of the kinks in Arrhenius plots. In the glassy state diﬀusion

6 Diﬀusion in Metallic Glasses and Supercooled Melts 273

Fig. 6.8. Isotope eﬀect as function of temperature for Co diﬀusion in bulk metal-

lic glasses. The data are taken from [90]. The diﬀusivities in the Be-containing

“Johnson glass” are displayed as squares.

takes place in a frozen-in structure which does not change with temperature

(except for irreversible structural relaxation [59, 131, 138]). Above T

,how-

ever, any increase in temperature gives rise to structural changes, e.g., an

increase in free volume which is certainly expected to enhance diﬀusion. In

these terms the increase in slope in the Arrhenius plots near the calorimet-

ric glass transition temperature (see Fig. 6.7) is simply due to temperature

dependent structural changes. Therefore, the slope −k

∂ ln D/∂T

−1

above

can no longer be regarded as an activation energy and should be termed

apparent or eﬀective activation energy.

UsinganovelPd

alloy, which exhibits exceptional stability

against crystallization, we were recently able to study for the ﬁrst time long-

range diﬀusion at and above T

in a metallic system [139]. The results are

shown in Fig. 6.9. Below T

one notes Arrhenius behavior in conjunction with

a very low isotope eﬀect as observed before (see Figs. 6.7 and 6.8, and the

data for Pd

in Fig. 6.9). This was attributed to thermally acti-

vated highly collective hopping in the preceding chapter. Above T

, however,

our data provide clear evidence of a change in the atomic dynamics. The ef-

fective activation energy −k

∂ ln D/∂T

−1

(slope in Fig. 6.9) drops gradually

as expected from the onset of viscous ﬂow in the mode coupling scenario. Due

to the increasing inﬂuence of liquid-like motion the nearest neighbor barriers

should gradually decay as the temperature is increased above T

. Our diﬀu-

sion data are even in good quantitative agreement with a ﬁt of the idealized

274 Franz Faupel and Klaus R¨atzke

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

1/T

 1000 [1/K]

Fig. 6.9. Temperature dependence of (top) Co diﬀusivity and (bottom) isotope

eﬀect in Pd

and Pd

. The diﬀusivity is plotted on a

semi-logarithmic scale vs. 1/T . Open squares show data for Ni and Cu diﬀusion

from quasielastic neutron scattering (QENS) in Pd

. The dashed line

is a ﬁt of the idealized mode coupling theory to the QENS data (see text). The

melting temperature T

(more precisely, the quasi-eutectic melting temperature)

and T

, determined from the ﬁt to the QENS, data are displayed. Below T

the tracer

diﬀusivity data were ﬁtted by an Arrhenius law. Open circles represent samples in

the glassy state at diﬀerent stages of structural relaxation.

6 Diﬀusion in Metallic Glasses and Supercooled Melts 275

...

..... ...

Fig. 6.10. Comparison of measured tracer diﬀusivities with diﬀusivities calculated

from the Stokes-Einstein relation [141, 144].

mode coupling theory to quasielastic neutron scattering data [140] above T

The idealised MCT neglects hopping processes and predicts a [(T −T

)/T

]

dependence, as shown in Fig. 6.9 (dashed line). Thus, the present results

strongly support the mode coupling approach for metallic glass formers.

The transition from solid-like hopping below T

to liquid-like atomic mo-

tion above T

is also reﬂected in the behavior of the viscosity. Above T

dif-

fusivities calculated from the Stokes-Einstein relation (see (6.5)) agree well

with the tracer diﬀusion data (Fig. 6.10). Below T

, however, Co diﬀusion

and viscous ﬂow are clearly decoupled.

We point out that the excellent agreement between our long-range dif-

fusion data and the diﬀusivities from quasielastic neutron scattering, which

measures short-range diﬀusion, allows convection eﬀects to be ruled out. The

absence of convection eﬀects, which are a severe problem in diﬀusion mea-

surements in ordinary liquids and melts under gravity conditions, is in accord

with the about two orders of magnitude higher viscosity of the present melt,

compared to ordinary melts, at the liquidus temperature [141].

The high viscosity of the Pd

melt is reﬂected in a remark-

ably low isotope eﬀect close to zero even in the equilibrium melt. This shows

that the present Pd-Cu-Ni-P melt is still far away from the hydrodynamic

regime of uncorrelated binary collisions as much as 200 K above T

.Forthe

276 Franz Faupel and Klaus R¨atzke

hydrodynamic regime kinetic theories predict [7, 16] predict a m

−1/2

mass

dependence and hence an isotope eﬀect close to unity (see (6.2)

The behavior of the excellent glass-forming alloy contrasts sharply with

that of simple melts such as liquid Sn, which exhibits an isotope eﬀect of the

expected order of unity [13]. Molecular dynamics simulations [142] indicate a

strong correlation of isotope eﬀect and density of the liquid and suggest the

present very low E value to be due to the small density diﬀerence between

glassy and liquid state of only about 3%. The strongly coordinated atomic

motion up to the equilibrium liquid, evidenced by the very low isotope eﬀect,

appears to be the key to exceptional glass forming ability of the present alloy.

6.7 Conclusion

Amorphous metallic alloys, also termed metallic glasses, are the paradigm

of dense random packing. Conventional metallic glasses are very prone to

crystallization and do not lend themselves for diﬀusion studies above the

glass transition temperature. With the discovery of novel bulk-glass-forming

alloys diﬀusion in metallic systems can now be investigated from the glassy

state up to the equilibrium melt. This is of considerable interest not only

from the technological point of view but also in terms of fundamental science,

particularly in connection with the glass transition. According to the mode

coupling theory the glass transition is a kinetic phenomenon characterized

by the arrest of viscous ﬂow at a critical temperature T

well above the

calorimetric glass transition temperature T

.BelowT

an extension of the

original theory predicts cooperative hopping processes.

In this chapter we focus on results from radiotracer measurements of dif-

fusion and frequently refer to molecular dynamics simulations, which have

provided much insight into the underlying diﬀusion mechanisms. The tracer

diﬀusivity is not aﬀected by thermodynamic eﬀects and is hence directly

related to the atomic dynamics. In particular, we present results from iso-

tope eﬀect measurements which indeed conﬁrm the highly collective nature

of diﬀusion in structurally relaxed metallic glasses. The highly cooperative

hopping processes seem to be closely related to the universal low-frequency

excitations.

We also address the role of defects based on measurements of the isother-

mal pressure dependence which can be interpreted in terms of the activation

volume of diﬀusion. The corresponding results do not reﬂect a uniform pic-

ture. Depending on the structure of the glass, essentially vanishing activation

volumes and those of the order of an atomic volume have been observed. The

vanishing activation volumes found in several conventional metallic glasses

clearly allow diﬀusion via thermal defects to be ruled out. Large activation

volumes in conjunction with a very small isotope eﬀect have mainly been

interpreted as being indicative of cooperative hopping that takes place by

assistance of thermally generated defects. This interpretation is not unique,

6 Diﬀusion in Metallic Glasses and Supercooled Melts 277

judging from recent molecular dynamics simulations which reveal a signiﬁcant

migration volume for a direct collective hopping process. In line with the pos-

sibility of defect-mediated diﬀusion we provide evidence of the existence of an

opposite Kirkendall eﬀect in interdiﬀusion between certain amorphous alloys

that combine slow diﬀusion via thermal defects and fast direct diﬀusion.

We also address the role of quenched-in excess volume in metallic glasses.

On the one hand, isotope eﬀect measurements during structural relaxation in

melt-spun ribbons support the notion of quenched-in vacancy-like defect that

enhance diﬀusion and anneal out during irreversible structural relaxation by

migration to the outer surface. On the other hand, similar measurements in

thin metal-metal glasses clearly point to highly cooperative diﬀusion even in

the as-quenched glassy state and to internal annihilation of excess volume.

In accord with the mode coupling scenario cooperative hopping processes

are also observed in the supercooled liquid state up to T

. Kinks in Arrhenius

plots are shown to be related to structural changes above T

but not to a

change in the diﬀusion mechanism. Above T

the eﬀective activation energy

drops gradually as expected from the decay of barriers, induced by the onset

of liquid-like atomic motion. Isotope eﬀect measurements in Pd-based alloys

show atomic transport up to the equilibrium melt to be far away from the

hydrodynamic regime of uncorrelated binary collisions. This appears to be a

prerequisite of excellent glass forming abilities.

Notation

a atomic radius (in Stokes-Einstein eq.)

D diﬀusion coeﬃcient

prefactor of Arrhenius law

diﬀusion coeﬃcient at zero frequency

E isotope eﬀect parameter

f correlation factor

g(r) pair correlation function

H activation enthalpy of diﬀusion

activation enthalpy distribution

Boltzmann’s constant

mass of isotope i

m + M eﬀective mass

p (hydrostatic) pressure

S(q) structure factor

T absolute temperature

critical temperature of mode coupling theory

caloric glass transition temperature

quasi-eutectic melting temperature

Vogel-Fulcher temperature

reference temperature for free volume theory

278 Franz Faupel and Klaus R¨atzke

act

activation volume

¯v mean velocity

mean free volume per molecule

β parameter of Kohlrausch law

η viscosity

λ mean free path

τ relaxation time

attempt frequency

Φ(t) relaxation function

Ω atomic volume

ω frequency

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