formation of the glass. DSC analyses reveal that Al precipitation from glassy matrices
takes place at temperatures just above 150°C [1-4].
As for intermetallics, a contrasting behaviour is found for two groups of alloys. In
the binary Al-RE and the RE-rich ternary systems, metastable compounds with different
structures are formed on quenching from the melt and on heating the glass [5].
The possibility of nucleating various phases can represent a “confusion principle” for
the alloy helping in glass formation. The production of Al nanocrystals in spite of
a substantial driving force for nucleation of other phases, can be the outcome of homo-
geneous fluctuations in the undercooled melt or glass and be driven by the higher
mobility of Al with respect to transition and rare earth elements. For the other group of
alloys, those rich in transition metal or ternaries containing Fe, a “confusion principle”
may still operate since several intermetallic phases exist. However, the same phases are
found both during quenching, in competition with the glass, and during devitrification.
The driving force for their formation should predominate in any case as suggested by
the higher liquidus point with respect to the other group of alloys. They can then trigger
crystallization of Al as observed in the foundry practice of inoculation.
The formation of glassy or nanocrystalline phases is, therefore, the outcome of
a subtle interplay of thermodynamic [6], i.e. driving forces for transformation, and
kinetic, i.e. atomic mobility, factors. Progress in understanding these issues is reviewed
by considering transport and thermodynamic properties of the melt.
2. VISCOSITY OF UNDERCOOLED LIQUIDS.
STRONG AND FRAGILE MELTS
Non-glass forming liquids usually have = 10 3 Pa s at the equilibrium melting
point, Tm, whereas, for metallic glass-formers, is of the order of 10 1 Pa s at Tm.
During the undercooling of glass-forming liquids, viscosity shows an extraordinary
increase up to 1012 Pa s in correspondence of the glass transition, Tg. As a conse-
quence, their mobility is reduced, nucleation is retarded and the system can be vitrified
[6].
A general pattern of the viscosity of glasses and glass-forming liquids over a broad
temperature range involves three regimes. Below Tg the viscosity shows an exponential
behaviour
;exp 101
RT
E(1)
the values of Tg and (Tg) depend on the cooling rate: when it is decreased, (Tg)
increases and Tg decreases. Analogously, an exponential behaviour is found above Tm,
with different pre-exponential and activation energy parameters. Experimental data for
the viscosity of metallic melts cannot be obtained in the whole temperature range
between Tm and Tg due to the tendency of the liquid to crystallize, so the data obtained at
low and high temperature must be fitted using an appropriate function. Here, is de-
scribed by the Vogel-Fulcher-Tamman (VFT) relationship
Al-Rare Earth-Transition Metal Alloys 269
)(exp
,0TTBA (2)
where A and B are constants and T0, is the temperature where the viscosity would diverge because the liquid freezes to an amorphous solid.
In order to compare data relative to different systems, the viscosity is plotted as logvs. Tg/T: the so-called Angell plot (Fig. 1) [7]. The few cases found to date are those of SiO2 showing nearly Arrhenian behaviour (Eq. 1), and of o-terphenyl presenting VFT behaviour (Eq. 2). In the Angell classification of glass-forming liquids these represent limiting “strong” and “fragile” behaviour, respectively. From a structural point of view, strong liquids contain tetrahedral three-dimensional networks (e.g. SiO2, GeO2 and BeF2), that confer resistance to thermal disruption of the structure; the fragile systems on the contrary usually do not present strong directional bonding.
Figure 1. The viscosity of various glass-formers reported as a function of normal-ised temperature (Tg/T). The network glass-former SiO2 display “strong” behaviour (upper full line) whereas the molecular o-terphenyl glass-former (lower full line) is “fragile”. Alloys fall in between the two extremes being relatively fragile: Zr46.75Ti8.25Cu7.5Ni10Be22.5 (dashed line); Mg65Cu25Y10 (dotted line); Pd40Ni40P20(dot-dashed line), Al87Ni7Ce6 (full bold line)
The kinetic properties have a thermodynamic counterpart in the rate of entropy loss on undercooling which is faster when the fragility of the melt increases. Using the Kauzmann paradox which implies that the entropy of the liquid and crystal phases (having specific heat difference Cp) [8] becomes equal at temperature, T0, below the experimental Tg,
(3) 0ln0
TdCSSmT
T
pm
fragile systems are found to approach T0, the Kauzmann temperature, more rapidly than the strong ones. So, the fragile/strong behaviour of glass-formers can be quantified by the ratio Tg/T0: the higher the fragility the lower is Tg/T0. This assessment of the thermo-dynamic fragility gives similar results as the ratio Tg/T0, , with T0, the parameter enter-ing the VFT equation, representing the kinetic fragility. The resistance of the strong glass to structural changes is also shown by the small changes in heat capacity taking place at Tg (Cp(liq)/Cp(glass) 1.1), while fragile ones present higher jumps (60-80% of the glass specific heat) [9].
270 L. Battezzati et al.
The fragile/strong behaviour of the supercooled liquids can be also classified by means of dimensionless parameters: m and D. The former is defined as the slope of the viscosity curve reported in an Angell plot in the vicinity of Tg:
.)/()(log
gTTg TTddm (4)
Using the VFT equation, the slope of log( ) versus 1/T is given by E(Tg) = RBTg
2/(Tg T0, )2 so, the fragility at the glass transition can be quantitatively expressed as m = BTg/2.3(Tg T0, )2. D is the fragility parameter entering the modified VFT equation,
)(exp
,0
,0
TTDT
A , (4a)
where the product DT0, replaces the parameter B in the conventional expression. Strong glasses present low values of m and high values of D (e. g. m = 20 and D = 100 for SiO2), whereas the reverse occurs for fragile ones (e.g. m = 81 and D = 6.8 for o-terphenyl).
Table I. Fragility parameters m, D and Tg/T0 for different glass-formers. Metallic glasses fall in between strong (SiO2) and fragile glasses (o-terphenyl)
The fragility concept can be also applied to metallic melts, e.g. Al87Ni7Ce6. This alloy is taken as a representative example of glass-forming liquids based on Al-rare earth elements-transition metals for which viscosity data are not available. They can be estimated by deriving suitable parameters for the VFT equation.
The relevant transformation temperatures and enthalpies are: Tm = 911 K. This is the eutectic melting. The liquidus occurs at higher temperature, but it is not considered here because it refers to the primary formation of an intermetallic which does not compete with glass formation, Tx = 523 K, Tg = 513 K (both data collected at the heating rate of 40 K/min); Hm = 9.8 kJ/mol and Hx = 5.3 kJ/mol [10]. Assuming that the specific heat difference between the liquid and crystal phases can be expressed by
Cp = C/T 2 in the temperature range from Tm to Tx, the condition of vanishing entropy difference between the liquid and crystal phases, Eq. (3), is imposed, and the T0
Al-Rare Earth-Transition Metal Alloys 271
temperature is found at 444 K. Cp at Tg is found as high as 21 J/mol K indicating thermodynamic fragility.
The pre-exponential factor A is taken as NAh/Vm = 3.8 10 5 Pa s, with NA the Avogadro constant, h the Planck constant and Vm the molar volume. Using the Andrade formula for the viscosity at the melting point [11]
3/2m
maCm V
TwAT , (5)
where AC is the proportionality coefficient and wa the average atomic weight, and assigning the AC coefficient the average value 6.5 10 7 (JK 1mol1/3)1/2 for metallic glass-formers at the eutectic temperature, we obtain: (Tm) = 7.8 10 3 Pa s. Finally (Tg) is taken as 1.0 1012 Pa s. With the latter two positions, the quantities B and T0. are obtained as B = 2474 K and T0. = 447 K, remarkably close to T0. The viscosity is then expressed as = 3.8 10 5 exp [2472/(T 447)] Pa s. The fragility parameters m and Dturn out to be 127 and 5.6 respectively, indicating that the ternary Al87Ni7Ce6 is a fragile melt. Although there are differences in the glass-forming tendency when substituting other transition metals and rare earth elements for Ni and Ce or when changing the alloy composition, the present result is indicative of the behaviour of this family of alloys. The above expression (5) for viscosity provides values in agreement with the experi-mental ones obtained above the liquidus for a number of Al-Ni-RE alloys [12].
The viscosity parameters of Al87Ni7Ce6 are compared with literature findings for some glassy substances and alloys in Tab. I. The general strong/fragile scheme is followed; note that strong melts need a critical cooling rate much lower than fragile ones to be vitrified [6]. The undercooled melts scale in strength in the order Zr46.75Ti8.25Cu7.5Ni10Be22.5> Mg65Cu25Y10> Pd40Ni40P20 >> Al87Ni7Ce6.
3. THE RELATIONSHIP BETWEEN VISCOSITY AND DIFFUSIVITY
When diffusion and viscous flow occur with the same brownian mechanism, the Stokes-Einstein equation (SE) is employed:
rkTD6
, (6)
where k is the Boltzmann constant and r is an ionic radius. The SE equation is valid at high temperatures near the melting point, but the correlation may fail when the liquid approaches the glass transition [6, 13].
Illustrative data are displayed in Fig. 2 for molecular glass formers: a clear break-down of the SE relationship can be noticed in correspondence of about 1.3 Tg.The measured self-diffusion coefficients are larger than those calculated by means of the SE equation using the experimental shear viscosity values. In correspondence to the SE breakdown de-coupling takes place between the translational diffusion and viscosity on the one hand, and rotational diffusion and viscosity on the other. In fact, while at high temperatures both Drot and Dtrans are inversely proportional to the viscosity, for high undercooling this relationship is lost; near Tg the molecules translate faster (two
272 L. Battezzati et al.
orders of magnitude) than suggested by the SE equation. Figure 3 also shows data for a
metallic alloy above Tg in the limited range where they could be collected. It is to be
remarked that the points for Fe40Ni40P14B6 refer to viscosity and an apparent diffusion
coefficient derived from the measurements of the rate of crystal growth, i.e. of the size
of eutectic colonies as a function of time at various temperatures. The points appear to
show a trend analogous to that of molecular glasses.
Figure 2. A plot showing the breakdown of the Stokes-Einstein equation for various glass-
formers. The logarithm of the quantity ( D/T) is plotted vs. a normalised temperature
(T/Tg). The points referring to salol (+) and o-terphenyl ( ) (translational diffusion
coefficients) conform to SE behaviour for T/Tg > 1.3. The points referring to o-terphenyl
( ) (rotational diffusion coefficients) conform to SE immediately above Tg. The points
referring to the Fe40Ni40P14B6 alloy (diffusion coefficients derived from the rate of crystal
growth) deviate from SE behaviour up to T/Tg = 1.17
The recent discovery of bulk metallic glasses having high thermal resistance to
crystallization allows us to verify the -D relationship in a larger temperature range,
above Tg. In the case of the Zr46.7Ti8.3Cu7.5Ni10Be27.5 bulk amorphous alloy, the SE
equation appears to lose its validity, since there is a discrepancy between the ex-
perimental translational diffusion coefficient of various elements and the one computed
via the SE equation. This is shown in Fig. 3 where the experimental trend of diffusion
coefficient of various elements [14] is shown together with that obtained by
means of the SE equation. The discrepancy is apparent. The -D behaviour found for
the Zr46.7Ti8.3Cu7.5Ni10Be27.5 amorphous alloy appears to be very similar to that of many
molecular glass-formers.
The failure of the Stokes-Einstein law near 1.2-1.3 Tg can be explained by assuming
a change in the diffusion mechanism. It has been speculated that the mechanism for
mass transport passes from a co-operative diffusion process, typical of viscous flow, to
activated “hopping” transitions (jump diffusion) involving a few molecular units.
The jump diffusion occurring at high undercooling resembles the vacancy diffusion in
crystals and does not contribute to viscous flow, so the measured diffusion coefficients
are higher than those calculated using the SE equation.
Al-Rare Earth-Transition Metal Alloys 273
Figure 3. Arrhenius plot giving the
tracer diffusion coefficient of various
elements in Zr46.7Ti8.3Cu7.5Ni10Be27.5
(solid lines) and a diffusion coef-
ficient computed as if the SE equation
was valid (dashed line). There is no
apparent correlation of diffusivity of
alloying or other elements, such as Al
with viscosity in this temperature
range
Contrary to molecular substances where the mobility behaviour has been studied
and modelled by investigating the liquid state, for metallic alloys the breakdown of the
relationship between diffusion and viscosity was discovered and discussed at first for
glasses below the glass transition. This is an outcome of studies on structural relaxation
of amorphous alloys where free volume is lost on annealing via defect annihilation, the
material increases its density and the ideal glassy state is approached. Measurements of
the equilibrium viscosity and the diffusion coefficient of Fe and Au tracer atoms in
Fe40Ni40B20 and Pd40Ni40P20 relaxed glasses, respectively, showed that at constant
temperature the D product is not constant [15]. It was proposed that the viscosity and
the diffusivity do not follow the same mechanism and it was assumed that the atomic
transport during a viscous flow involves a pair of defects, whereas, in the case of
diffusion, only one defect is required; formally this leads to
(7).const2D
It was also verified that this correlation holds for a molecular glass-former above Tg.
With the new findings on bulk metallic glasses illustrated in Fig. 3 it can be suggested
that this correlation is general. The above considerations would turn useful in estimating
diffusion coefficients from viscosity in the neighbourhood of Tg if the significance
and value of the above product D2 were elucidated. At present it has been ascertained
that the product in Eq. (7) is constant at constant temperature, but it changes with
temperature and diffusion couple studied. If various diffusing species in the same alloy,
Zr46.7Ti8.3Cu7.5Ni10Be27.5, are considered, then the product scales roughly with the atomic
mass of the diffusing atom.
4. CRYSTALLIZATION AND MOBILITY
If D is derived for an alloy from Eq. (7), its activation energy would be compatible
with that found in the diffusion and growth studies. Crystallization is very frequently
studied by thermal analysis and activation energies are easily obtained by means of non
isothermal scans. These values are often difficult to rationalise unless they can be
clearly referred to nucleation and growth processes. In most experiments a limited
temperature range is spanned, so the data can be fitted with an Arrhenius equation. If
274 L. Battezzati et al.
above Tg the mobility were determined by viscous flow, the activation energy would
approximate the curvature of the VFT function in the temperature range of interest.
Therefore, it should become the higher the closer the temperature range is to Tg. As an
example, the apparent activation energy for viscous flow in Al87Ni7Ce6, calculated from
the example in paragraph 2, just above Tg is of the order of 640 kJ/mol. Such high
activation energy would certainly be rate limiting for all processes involving atomic
mobility, but values of this order have not been found in any crystallization or growth
processes for this and related alloys. At the present stage Al-transition metal-rare earth
alloys the activation energies reported so far are widely scattered. They range from
140-160 kJ/mol for some Al-Ni-Gd glasses to values in excess of 400 kJ/mol for
Al86Ni5Co2Y8 according to their composition and heat treatment. Considering homo-
geneous nucleation, the activation barrier can be estimated of the order of 40-50 RT at
deep undercooling, again lower than the above value. In most instances nucleation is
heterogeneous, so the barrier will be even lower with respect to the value for SE viscous
flow [16].
In a series of experiments with Al87Ni7Ce6 and Al87Ni7Nd6 amorphous alloys [10], it
has been recently found that Tg becomes manifest at heating rates above a certain value
typical of each alloy when the onset of crystallization is displaced to sufficiently high
temperatures. The kinetics of the transformation was then studied and the Kissinger
plots showed a change in the mechanism as a function of the heating rate, i.e. of peak
temperature as implied by a deviation from linearity in the points for the first DSC
peaks. The Kissinger plots could be referred to definite processes. At low rates, the
crystallization must precede or overlap Tg so the activation energies of 340-370 kJ/mol
for the two alloys should be related to the motion of Al in a medium characterised by
short range order of different species, and therefore by the need of cooperative
displacements of the alloy constituents. Above Tg, the viscous flow in the highly
undercooled melt apparently has no effect since lower values of activation energies,
140-190 kJ/mol, are obtained. This suggests that the need for cooperative diffusion is
removed in the molten state and that the hopping of single Al atoms suffice for
promoting crystallization. It is apparent that estimating diffusivities via the SE equation
is not appropriate for these materials. Extending the work to an Al87Ni7La6 alloy
confirms these findings [17].
Concluding, a mobility different from that expressed by the SE relationship is rele-
vant for crystallization of Al based glasses. The SE supplant (cf. Eq. (7)) may be opera-
tive or crystallization may depend on single diffusion steps not related to viscous flow.
5. FORMATION OF PRIMARY NANOCRYSTALS
In most alloys where primary nanocrystals are obtained from an amorphous matrix
the transformation is reported to be fast and then to slow down progressively. This is
reflected in highly asymmetric peaks in DSC. The glass transition is seldom detected
before the onset of precipitation. It should be remarked that the primary phase is often,
but not always Al. In fact, there have been findings in various laboratories that
metastable intermetallics form as primary phases in Al-Ni-La and Al-Ni-Ce alloys [10,
18-21]. Examples of DSC traces are shown in Fig. 4 for Al87Ni7La6 and Al87Ni7La5Ti1
Al-Rare Earth-Transition Metal Alloys 275
Figure 4. DSC traces for the alloys listed
in the insert at the heating rate of 20 K min 1
amorphous alloys. The first crystallization step of the ternary alloy is due to the pre-
cipitation of a complex cubic phase, rich in the rare earth element. Substituting some Ti
for La causes destabilization of such a phase and precipitation is then due to fcc Al.
Nevertheless, both peaks are skewed and a substantial background signal occurs on their
high temperature side indicating a similar mechanism of transformation. It has been
shown that the background signal does not involve major precipitation of new crystals
but rather the coarsening of those already existing, and homogenization of the remain-
ing matrix where compositional gradients should occur because of solute rejection by
the precipitates. It has been shown earlier that rare earth atoms can accumulate ahead of
the nanocrystalline Al interface whereas transition metals diffuse faster in the matrix so
their concentration is more readily homogenized. The new finding that a homogeniza-
tion process occurs also when a primary La-rich phase forms, implies that a gradient in
the Ni content of the matrix will retard its transformation as well. So, the two phase
materials are resistant to crystallization because compositional gradients are established.
It has been also suggested that during primary crystallization stress builds up around
precipitates because of compositional gradients. This would be analogous to the stress
effect occurring in multilayered thin film made of early and late transition metals where
the interdiffusivity depends on the length scale of modulation, being fast for short dis-
tances, comparable to precipitate size, and dropping steadily for long range motion as
implied in homogenization [22].
6. NUCLEATION THERMODYNAMICS
Precipitation from the amorphous Al87Ni7Ce6 matrix involves only Al at low rates,
i.e. below Tg, but it implies the formation of a metastable intermetallic, besides Al,
when the transformation occurs above Tg. Therefore there is a clear change in
the transformation mechanism across the glass transition. It is likely that below Tg only
the growth of Al crystals occurs on pre-existing seeds whereas, in addition to Al
growth, the two phases nucleate and grow in the undercooled melt. In Al87Ni7Nd6
the first transformation does not imply the formation of an intermetallic compound at
any rate, so the change in the mechanism can be more clearly attributed to the formation
of Al by either a growth in a glassy matrix or nucleation and a growth in a liquid matrix
[10]. In addition, the tendency of the melt to undercooling shows that the nucleation of
276 L. Battezzati et al.
intermetallic compounds which is possible at high temperature, just below the liquidus,
is sluggish.
Figure 5. The driving force for
nucleation computed by means of
the common tangent construction
for fcc-Al and Al11Nd3 in the al-
loy Al91Nd9
The nucleation scenario in these alloys can be described thermodynamically by
means of free energy curves representing the driving force for nucleation of either Al or
Al11Nd3 computed according to the parallel tangent construction as a function of tem-
perature (Fig. 5). The free energy of the liquid and crystal phases was optimized by
means of a CALPHAD software and an appropriate account was made of the excess
specific heat which is specific of glass-forming melts as well as of the occurrence of
a glass transition [23]. At the glass-forming composition of 9 at.% Nd, the driving force
for nucleation of the intermetallic builds up below the liquidus but levels off at high
undercooling because of the progressive stabilization of the liquid phase due to the
excess specific heat. Meanwhile, the driving force for nucleation of Al increases
steadily and becomes close to that of Al11Nd3 at temperatures close to the glass
transition. The two phases can actually compete for nucleation and factors such as glass
(liquid)-crystal interfacial tension can become decisive in phase selection. It is expected
that the simple fcc structure of Al will involve a lower interfacial tension with respect to
the complex Al11Nd3 compound. On the other hand, metastable compounds may display
short range order closer to that of the glass (liquid) and, therefore, nucleate preferen-
tially.
7. CONCLUSIONS
Al-transition metal-rare earth melts have been suggested to be fragile in the Al-rich
composition range where amorphisation can occur by rapid quenching. This is mostly
determined by the low value of the ideal glass transition temperature, T0, corresponding
to the point of vanishing entropy difference between liquid and crystal phases which
was estimated from thermodynamic data. Although there is limited information on
the viscosity of such alloys, referring to temperature ranges above the liquidus, the T0,
parameter entering the VFT for viscosity can be recovered. This stems from the as-
sumption that the viscosity behaviour at the eutectic and glass transition temperature
should be typical of glass forming alloys. The T0, is close to the thermodynamic T0 and
Al-Rare Earth-Transition Metal Alloys 277
the overall viscosity in the undercooling regime conforms to the fragile type in Angell
classification.
Evidences have then been reviewed of the relationship between viscosity and dif-
fusivity. At high temperature the proportionality expressed by the Stokes-Einstein
equation appears valid within the scatter of experimental data but it breaks down close
to Tg. In fact, the data obtained on the diffusion and the rate of crystal growth (including
activation energies for crystallization) show that mobility is due to either single hopping
jumps (vacancy-like mechanism) or the movement of groups of atoms (cooperative
mechanism) instead of viscous flow.
In Al-Ni-RE amorphous alloys, primary crystallization occurs frequently involving
either fcc Al or an intermetallic compound. Such two-phase materials are rather stable
kinetically in that they resist crystallization in a large temperature range. This has been
attributed to sluggishness in nucleation of stable and metastable intermetallics
depending on the existence of compositional gradients in the amorphous matrix which
is enriched in the slow diffusing elements, and on the lowering of the driving force for
nucleation due to the shape of the liquid free energy curve.
ACKNOWLEDGMENTS
This work is funded by the Research Training Network of the European Commission “Nano
Al” HPRN-CT-2000-00038.
REFERENCES
1. Inoue. A., (1998) Amorphous, nanoquasicrystalline and nanocrystalline alloys in Al-based
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