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Viscous flow and crystallization of bulk metallic glass forming liquids
Citation
Masuhr, Andreas
(1999)
Viscous flow and crystallization of bulk metallic glass forming liquids.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/S22B-5C59.
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
An experimental setup was designed and implemented to measure the flow behavior of liquids in the viscosity range from [...] Pa s to [...] Pa s. The viscosity of the [...] (V1) bulk metallic glass forming alloy was measured over a temperature range from 927 K to 1173 K. At the liquidus temperature, the viscosity is 2.3 Pa s, which is about three orders of magnitude larger than the viscosity of a pure metallic liquid. The free volume theory as formulated by Cohen and Grest describes the temperature dependence of the viscosity of V1 over 14 orders in magnitude.
The high viscosity of V1 above the liquidus temperature stabilizes the liquid against convective flow due to temperature gradients and allows for diffusion experiments in the equilibrium liquid. The temperature dependence of the diffusivity of large atoms like A1 or Au scales with the viscosity. The time scales obtained from the viscosity measurements suggest that above the calorimetric glass transition region the diffusion of small and medium sized atoms is governed by thermally activated jumps.
Liquid V1 could be successfully supercooled inside high purity graphite crucibles without changing the stability of the supercooled liquid with respect to crystallization compared to levitated samples. The sluggish kinetics that are reflected in the high viscosity in the supercooled liquid state contribute significantly to the good glass forming ability of the alloy. The critical cooling rate is about 1 K/s.
The onset of crystallization under isothermal conditions as well as upon heating from the amorphous state was studied in detail. The critical heating rate to bypass crystallization was measured to be 200 K/s and the difference between the critical cooling and critical heating rate can be qualitatively understood in the framework of nucleation and growth.
However, the observed deviations from classical steady state nucleation behavior indicate a more complex crystallization mechanism. Rheological and crystallization studies at constant shear rate suggest that changes in the morphology of the supercooled liquid of V1 occur as a precursor of crystallization.
Item Type:
Thesis (Dissertation (Ph.D.))
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Johnson, William Lewis
Thesis Committee:
Unknown, Unknown
Defense Date:
8 December 1998
Record Number:
CaltechETD:etd-06242005-094416
Persistent URL:
DOI:
10.7907/S22B-5C59
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2716
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VISCOUS FLOW AND CRYSTALLIZATION
.OF BULK METALLIC GLASS FORMING LIQUIDS
Thesis by
Andreas Masuhr
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1999
(submitted December 8, 1998)
ii
Andreas Masuhr
iti
To Susanne
ve
ACKNOWLEDGMENTS
I am grateful to Professor Bill Johnson for giving me the opportunity to pursue
thy Ph.D. at Caltech. It has been a privilege to work with him and experience his
unparalleled intuition for science. I thank him for giving me the freedom to explore
the subject and for his faith in my abilities and decisions.
A special thanks goes to Ralf Busch for his continuing support and for making the
last four years enjoyable. Parts of the experimental work were made possible through
fruitful collaborations with Eric Bakke, Achim Rehmet, Andy Waniuk, Jan Schroers,
and the invaluable help of Mike Gerfen. My memories are with Y.J. Kim who helped
to get my research project started in 1994,
I profited from many stimulating discussions with Ulrich Geyer, Susanne
Schneider, Atakan Peker, Mo Li, Konrad Samwer, Helmut Mehrer, Kathrin Knorr, and
many others. Also, I would like to thank all members of the Materials Science Group
for making my stay at Caltech a wonderful experience. |
The financial support from the Department of Energy under grant No. DE-FG-03-
86ER45242 and from the ALCOA Technical Center is greatly appreciated.
Finally, I would like to acknowledge the support and encouragement of my
parents ‘and my wife Susanne. None of this work would have been possible without
them.
ABSTRACT
An experimental setup was designed and implemented to measure the flow
behavior of liquids in the viscosity range from 10’ Pas to 10*Pas. The viscosity of
the Za sTir3 sCU0 0Ni 12 sBen s (V1) bulk metallic glass forming alloy was measured
over a temperature range from 927K to 1173 K. At the liquidus temperature, the
viscosity is 2.3 Pa s, which is about three orders of magnitude larger than the viscosity
of a pure metallic liquid. The free volume theory as formulated by Cohen and Grest
describes the temperature dependence of the viscosity of V1 over 14 orders in
magnitude.
The high viscosity of V1 above the liquidus temperature stabilizes the liquid
against convective flow due to temperature gradients and allows for diffusion
experiments in the equilibrium liquid. The temperature dependence of the diffusivity
of large atoms like Al or Au scales with the viscosity. The time scales obtained from
the viscosity measurements suggest that above the calorimetric glass transition region
the diffusion of small and medium sized atoms is governed by thermally activated
jumps.
Liquid V1. could be successfully supercooled inside high purity graphite crucibles
without changing the stability of the supercooled liquid with respect to crystallization
compared to levitated samples. The sluggish kinetics that are reflected in the high
viscosity in the supercooled liquid state contribute significantly to the good glass
forming ability of the alloy. The critical cooling rate is about 1 K/s.
| ; The onset of crystallization under isothermal conditions as well as upon heating
from the amorphous state was studied in detail. The critical heating rate to bypass
crystallization was measured to be 200 K/s and the difference between the critical
cooling and critical heating rate can be qualitatively understood in the framework of
nucleation and growth.
However, the observed deviations from classical steady state nucleation behavior
indicate a more complex crystallization mechanism. Rheological and crystallization
studies at constant shear rate suggest that changes in the morphology of the
— supercooled liquid of V1 occur as a precursor of crystallization.
_ Vii
CONTENTS
. Introduction vesesecenesneescesecsscesecnsecnsessacesecasecnscssecnsessecssenscencaseesecnecssscenssaneensansessnsntees 1
« Experimental... seeseseseesesenerensenenes eeeesessesecaeesessesecseesecseensessseeseeseeaessensesesenenees 6
2.1. Experimental Setup ...... ccc csscsecssssesssccesesesscsscssasseessseecssssesssssseseeesseseseessnese 7
2.2. Concentric Cylinder Rheometry......... cc ecesesesseecseeeeeseneessesesesesesseseessesneseaees 15
2.3, Graphite as a Crucible Material... cc ccesescsseccssesseesssccsssesssseceeseseeeesseeeensees 21
2.4. Capillary Flow Rheomety ........ eee eeceeceeeseeseeeseseeseseesseesessecessesseseessseeeeeses 23
. Viscous Flow of Zr-based Glass Forming ALLOYS ..........cecceeeeseeseeseseeseeeeseeeeeeees 26
3.1. Viscosity: Experimental Results 00.0... cessssseseessesescesseeesessscsssseesecsssereeesenes 27
3.2. High Temperature Limit of the VisCOSILY 00.0... cece seeeseeesseeseeseeeeeeseseseeeeees 30
3.3. Entropy Model for Viscous FlOW....... 0. cecesccsseeeeeseeesssesseessseseneseseeeeseseeseesene 36
3.4. Free Volume Model for Viscous FlOW..........ccccesscesssecsseeessseseeesseeeeseesseeeeeees 43
. Time Scales in the Liquid and the Supercooled Liquid ...... ee eee eeeseereeeeeeees 53
4.1. Diffusion in the Liquid State... ees ceseceseecsseeeeseecesesensesesesereeeernsseseeeees 53
4.2. Time Scales for Viscous Flow and Atomic Transpoft 0.0.0.0... cceeesseeeseeeeeeees 56
. Crystallization of the supercooled liquid... eles eeseeeesseteeseseesessesseeesenseseneeens 62
5.1. Crystallization: Tsothermal ............ccsccesccssecsrecsceeseesescessessscceseeseesseeseseaeseeeessees 7\
5.2. Crystallization: Linear Cooling and Heating... cece eeccesesseeeeeeseeseeeeeeees 82
5.3. Crystallization at constant shear rate.........cccecsesseeseeseeeessesseseeeseeeeeseeeeneeesees 93
SUMMALY....... cc ceccescccseseseessesesseesssecesscsesssevsrccssessnseseseeecseessesesssesseeseesseeesepeneseseeseeenees 98
Fig.
Fig.
Fig.
Fig.
Fig.
_ Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
10
11
12
13
14
15
16
17.
18
19
20
21
22
, vill
* FIGURES
The crystalline, gaseous, and liquid state... sssessssesseseseesseesteeatserstenee 2
Experimental setup of the high temperature viscometer............cccccccsccseeseeeees 8
Photograph of the high temperature viscometer ............cccsccssssessescessessseeseeseees 9
Calibration of the RF power SUPPLY ............:cccccssecsssesssessecsscessessseeseesssssseesneses 11
Temperature control and data acquisition SYSteM............cceseceseeeeeeteeeeeenees 13
Torque meter Calibration CULVE .0......:cseseeseesceseeeseeeessseaeeeasessseeeseeseeeaseeeeseesees 18
Viscosity measurements Of glycerol ....... eee eeescesesseesseeseseesseesseseessseseseesee LO
Photograph and schematic of the liquid—graphite interface... 22
Capillary flow rheometry seceeeensessseaseseevesssenssesssasessanssnecesseessecsseeesseseesaseeeeeeeaee 24
Isothermal torque reading at constant shear rate... ec eeeeeseceseneeeneeseneeersees 28
Viscosity measurements upon linear COOLING............cccesseceeeeseseeseeesseeseeessens 29
Capillary flow rheometry: experimental results... ceccecsetseseeetesseeeeeneee 31
Viscosities of liquid metals at high temperatures 0.0.0.0... ceeeeesseseeeteeeeeeeeeees 32
Arrhenius plot of the viscosity of V1 and metallic elements ..............:.:00000 33
Normalized Arrhenius plot of the viscosity of glass forming systems .......... 37
Specific heat Capacity... cseeeeeeseeesssseessseesesareneenseneeneseenesetsneeennen 40
Viscosity of V1 and least-squares fits of entropy and free volume model .... 42
Configurational entropy Of V1 ou. eee eesceenceeseeteceeseeseceeaseeescseeseeeeseraeeeenees 44
Free volume of V1 as determined by the viscosity sesuusssssssssssuevesessessesssevecees 47
Interatomic spacing and thermal expansion in liquids and crystals............... 49
Apparent activation energies for VISCOUS flOW..........cceseeesceeteeseeeseneeseeeeeeees 52
Concentration profile Of AU if V1 ool eessesceeeecnseeeeseeeeseeeseeecesseseeeaeesees 55
Fig.
Fig.
Fig.
Fig.
Fig
Fig.
_ Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
23
24
26
27
28
.29
30
31
32
33
34
35
36
37
38
Time’ scales in the liquid and supercooled liquid Of V1... eeeseeeeeeeseeneeees 58
Isothermal crystallization studies of the supercooled liquid state........0........ 63
Differential thermoanalysis upon linear heating ............eeeeeceeeseeeneeeeteeeesenis 66
Cooling curves from the equilibrium liquid state .........0.. cee eeeeseeseeeeeeereeeees 68
Driving force for crystallization... eee eeeeseeereesceeesceseereeeecenserseeeeseanesses 70
Time-temperature profile for isothermal anneals of supercooled V1............ 72
Time-temperature-transformation (TTT) diagram... cece eeeeeeseeeeeeeees 73
Nucleation rates and growth velocities in supercooled V1 ou... eee eeeeeeeees 79
Crystallized volume fraction and rate of transformation ......... eee 81
Continuous-cooling-transformation (CCT) diagram... cee eeseceeesereeees 84
Time-temperature profile of a fast heating and cooling experiment.............. 86
Thermoanalysis of V1 upon fast heating «0.0.0.0... esse eeeeeeseeeeseeeeeeestateeeeeeees 88
Continuous-heating-transformation (CHT) diagram......... eee eeeeeeseeeeeeeees 89
Crystallized volume fraction for continuous cooling and heating ................. 91
Linear cooling curves at constant SHOAL ate oe eee eeeeeeeeseeteeeeeeeeeeeeetereeeaeees 94
Shear rate dependent solidification eeseseeeenesees sessessaseseenuannesetsnnsstunsessnenenees 96
1. INTRODUCTION -
In thermodynamic equilibrium, a one component substance may be present as a
ctystal or a fluid. As shown in the pressure-temperature, p-T, phase diagram in
Fig 1 (a), a distinction between the liquid and gaseous state is strictly justifiable only
along the vapor pressure curve, where the two states of matter can be found in co-
existence. At ambient pressure, indicated by the dotted line in Fig 1 (a), however, a
distinction between the three states, crystalline, liquid, and gaseous, is appropriate and
will be used in the following.
Quantum mechanics and statistical mechanics give a mathematical framework to
relate the macroscopic properties of a substance to its microscopic structure and
dynamics. The periodicity of a perfect crystal at T=O on the one hand, and the
stochastic behavior of atoms in an ideal gas for T—> © on the other, yield simple
analytical expressions for the equations of state for these two states of matter. For the
liquid state, the notion of an ‘ideal liquid’ based on a simple and mathematically
tractable model has not yet been established [1]. Models based on hard spheres or
other pair potentials may serve as a substitute, particularly in computer simulations,
but have not reached the degree of universality as the theories of crystals and gases.
Experimentally, the transformation from the liquid to the crystalline state may be
suppressed for extended periods of time when the liquid is cooled below its melting
point (‘supercooling’) [2]. The physical properties of the liquid are continuous across
the melting point, and the system can be found in metastable equilibrium for extended
temperature ranges. Fig 1(b) schematically shows the specific volume
‘(a
© (a)
. critical point
Sf. crystal liquid vapour
| |
T. T, temperature
(b)
o)
E liquid
ro) supercooled |
> liquid
glass
crystal
Th, T, temperature
Fig. 1 The crystalline, gaseous, and liquid state
(a) Crystalline and fluid phase regions as a function of pressure and temperature.
The crystalline, liquid, and gaseous phases at ambient pressure (---) are indicated. (b)
Temperature dependent specific volume of the crystalline and liquid phase (—) as
well as of the supercooled liquid and the glass (---). The melting temperature, Tm, and
the boiling temperature, Ty, at ambient pressure are marked.
. versus temperature curve for a one-component substance. Below the glass transition
region, the volume shows a weaker temperature dependence than expected from an
extrapolation of the high temperature values. It is important to note that. the
temperature rangé in which the glass transition can be observed experimentally
depends on the cooling rate that is applied to the liquid.
With the liquid state an average internal relaxation time is associated that
describes the time scale on which the system samples its available phase space in
(metastable) thermodynamic equilibrium. While for equilibrium liquids this time is
_ often found to be less than 10° s, it increases dramatically in the supercooled liquid.
A continuous increase of more than 14 orders of magnitude in the relaxation time with
decreasing temperature can be observed using, e.g., dielectric measurements. The
viscosity increases — to a good approximation — proportional to this relaxation time,
making it a macroscopic probe for rearrangements on a microscopic scale [3].
It is widely accepted that any liquid can - in principal - be quenched to the
amorphous state by applying an appropriate ‘critical’ cooling rate to prevent
significant crystallization. This rate may be as small as 10° K/s for good glass
formers like commercial soda-lime-silica glasses. In order to form a metallic glass
from a monatomic metallic liquid, the (estimated) critical cooling rate is as large as
10” Kis [4]. Duwez [5] found that Au-Si alloys with near-eutectic composition can be
quenched into the amorphous state with cooling rates of ca. 10° K/s. These high rates
allow only for the formation of foils with a maximum thickness of some 10 Um.
Recently, multicomponent metallic alloys were found to form metallic glass with rates
as small as 10° K/s resulting in bulk amorphous glass (BMG) with smallest
“dimensions on’the order of 1cm. One of the important concepts in understanding the
glass forming ability includes Turnbull’s [4] discussion of the reduced glass transition
temperature, T,/Tm, and its influence on the nucleation kinetics.
In chapter 2, the experimental apparatus and procedure used to study the flow
behavior and crystallization of Zr-based metallic alloys are described. Calibration
proceduires are presented and the importance of a compatible crucible material for the
rheometric measurements is discussed in sub-chapters. In chapter 3.1 results from the
high temperature viscometer on liquid Zr41.2Ti13.gCuio0.0Ni12,sBe22.5 (V1) are presented.
A model of the high temperature limit of the viscosity, discussed in chapter 3.2, yields
a ‘fix-point’ in the discussion of the rheological properties of a wide range of liquids.
It allows for a phenomenological classification of the present results in the framework
of ‘strong’ and ‘fragile’ liquids. In the subsequent chapters, 3.3 and 3.4, the
temperature dependence of the viscosity is analyzed according to entropy and free
volume models for viscous flow. Results from Cohen and Grest’s theory of the liquid
state are used in chapter 4 to compare the time scales of viscous flow with relaxation
times from calorimetric experiments and atomic transport data in the liquid and
supercooled liquid.
Results from crystallization studies of V1 are presented in chapter 5. First,
isothermal measurements of the onset of crystallization are analyzed in chapter 5.1
according to classical nucleation and growth theory. The difference in crystallization
behavior upon cooling from the equilibrium melt on the one. side and upon heating
amorphous samples on the other, is the subject of chapter 5.2. A modification of the
experimental apparatus, allowing for high heating rates and precise temperature
“measurements, allows for an unprecedented temperature range of the supercooled
liquid in a metallic system upon heating. Finally, results from crystallization studies at
constant shear rate are presented and discussed in chapter 5.3.
Ae
2... EXPERIMENTAL —
In most. viscometers either the rotating (or oscillating) cylinder method or the
capillary method is employed [6]. The resolution of capillary viscometers is, in
general, superior to that of the rotating cylinder viscometers. While the first are
primarily used for measuring low viscosity liquids with viscosities, 7, of less than
1 Pas, the latter are used for liquids with more sluggish flow behavior. There are no
commercially available viscometers for liquid metals with melting points on the order
of 1000K. For the present work, an experimental setup was designed to
simultaneously measure the flow behavior as well as heat releases upon crystallization
and melting of metallic alloys with melting temperatures up to 1400 K.
Liquid metals are known to be susceptible to impurities like oxygen if processed
in air or in an incompatible crucible. The effects of contaminations induced through
the liquid surface or the container interface will decrease with increasing sample sizes
and — in industrial applications - the purity of the bulk liquid may only be remotely
dependent on the processing environment. On a laboratory scale with sample sizes of
a few cm’, however, the liquid has to be processed either in vacuum or under an inert
atmosphere. The choice of a compatible crucible is equally important.
In the present work, the apparatus designed to process the liquid samples encloses
a volume of approximately 3x10 m? or about 0.1 mole of an inert atmosphere at
ambient pressure. For comparison, the mass of the graphite containers (shear cell) of
ca. 60 g corresponds to 5 mole of carbon atoms and a typical sample of 30g of V1
contains a total of 0.5 mole of atoms. The size of the container in conjunction with the
high atomic mobility of impurities in both the graphite and the sample at temperatures
of more than 1000 K imply that the purity of the graphite is decisive for the purity of
the metallic liquid. An impurity concentration of, e.g., 10° (1 ppm) in the solid or
liquid state corresponds to a partial pressure of impurities in the atmosphere of 10" Pa.
2.1. | Experimental Setup
Viscosity measurements are based on either pressure-driven (Poiseuille) flow or
wall-driven (Couette) flow of the fluid under inspection [7, 8]. Experiments
employing steady state Poiseuille flow through quartz capillaries will be briefly
discussed in chapter 2.4 while in the following the setup for viscosity measurements
involving wall-driven flow between concentric cylinders is desribed.
At the heart of the high temperature Couette viscometer that is shown in Fig 2 and
Fig 3 is the shear cell, consisting of a cylindrical crucible (outer cylinder) and a
concentric rotating bob (inner cylinder). The liquid metal fills the space between the
two cylinders. Under continuous rotation of the inner cylinder, a steady-state flow
field is established within the liquid, resulting in a static torque on the fixed outer
cylinder. For a Newtonian fluid, this torque is proportional to the shear viscosity
coefficient of the liquid as will be shown in chapter 2.2.
As all graphite parts used in this work, both cylinders are machined from graphite,
grade DFP-1, supplied by POCO Graphite, Inc. The graphite is attached to titanium
(99.9%) adapters. Besides giving ample mechanical support up to temperatures of
1400 K, the titanium parts serve as a getter for residual oxygen. A stainless steel shaft
connects the titanium adapter of the inner cylinder to a ferrofluid feedthrough, type
microstepping
motor
RF heating
coil a
RF b
TC
Ar ,
vacuum
pump system
PC AIDTa
torque
temperature
9..10 VDC
§..100 kHz TTL
- Fig. 2 Experimental setup of the high temperature viscometer
Two concentric cylinders are mounted inside the vacuum chamber and are inductively
heated by the RF heating coil. A static torque sensor is positioned at the bottom of the
chamber. A microstepping motor is mounted on a vertical drive (not shown) and is
connected to the rotating, inner graphite cylinder through a ferrofluid feedthrough.
Fig. 3 Photograph of the high temperature viscometer
Experimental setup used for the high temperature rheometry and for the
crystallization studies of V1 and V4. The photograph shows the shear cell’s outer
cylinder heated by the RF coil with the inner cylinder inserted into the liquid. An o-
ring sustained ball bearing mount is attached to the rotating shaft above the shear cell
to ensure concentric alignment of the two cylinders.
ho 10
os 080-50C103912. ‘A microstepping motor, type P315-M233, is attached to the air side
of the feedthrough via an 82:50 gear and is driven by a P315 microstep drive, supplied
by American Precision Industries, Inc. The resolution of the drive of 50800 steps per
revolution allows smooth rotary motion of the inner cylinder with angular frequencies
from ca. 5X107 s” to 2x10’s".
The titanium adapter of the outer cylinder is connected to a static torque sensor at
the bottom of the apparatus. Depending on the scope of the experiment, either an
Eaton 2121-100 or an Eaton 2127-10 torque sensor is used for maximum torque
_ readings of 100 Lbs-In. (1.15 Nm) and 10 Oz-In. (7.2 x 10° Nm). The torque sensor
is mounted inside the vacuum chamber to avoid the influence of a feedthrough on the
torque reading. Both sensors use four strain gauges each, mounted at 45° to the torque
axis and wired in the form of a Wheatstone bridge. An OMEGA DP41 signal
conditioner that also processes the torque sensor signal supplies the excitation voltage
of 24 V.
The graphite shear cell is inductively heated using a Lepel MR-7.5 radio
frequency (RF) power supply and a seven turn heating coil made out of 6.35 mm
diameter copper tubing. The RF power supply has a maximum output of 7.5 kW at ca.
~ 200 kHz. The RF coil is wound around the quartz tube enclosing the shear cell. In
this way, RF vacuum feedthroughs are avoided and the coil’s position can easiliy be
adjusted to yield a homogeneous temperature distribution in the shear cell. The
measured temperature as a function of the RF power supply setting is shown in Fig 4
for two different graphite crucibles. A type K thermocouple, OMEGA SCAIN-040U-
12-SHX, is inserted into the outer cylinder’s crucible wall, measuring the temperature
' l t | ' 1 ' co oi a | Vv efe
0.05 + a ° at
® 04. < 30 5 (b) or
8 0.03 4 J S20- ors oe
ae) te)? bs 7 ° q 9 . 3 te q
= # 0.02 - 7 o q nh _
&S Soo1+ J §105° 3ae 7
0.00 + + wm | iiss. :
T | qT | T ] T oc 0) TT + | T | v
1000. 2000 3000 {000 2000 3000
time (s) time (s)
100 3 a ee a T TTT z
= 24 (©) E
& 805 3
ioe 4
> 4
om E
© A
= 3
roy E
a :
LL q
a 4
| v J v T 1 t J qT J | J v T T ] qT J qT qT | i qT qT T T
0.00 0.01 0.02 0.03 0.04 0.05
thermocouple voltage (V)
Fig. 4 Calibration of the RF power supply
(a) Thermocouple voltage measured as a function of time for a 31.8mm diameter
graphite crucible heated by a 62 mm diameter, 7-turn RF coil. The temperature is
~ raised in steps of 50 K from 600 K to 1200 K. (b) Corresponding power setting of the
RF power supply. (c) Power setting as a function of thermocouple voltage (0). For
comparison, power Settings for a 4mm diameter graphite crucible heated bya3lmm
diameter, 5 turn RF coil are also shown (a). Solid lines in (c) are least-squares fits of
third order polynomials to the data points.
a 12.
* of the shear cell (9]. Particular care has been given to a ‘loose’ placement of the lead
wires of the thermiocouple inside the vacuum chamber in order to minimize the
resulting offset in the torque reading below the detection limit of the torque sensors of
3X 10° Nm. Temperature gradients of +4 K of the nominal temperature have been
observed on the surface of the shear cell using a Leeds & Northrup 8622 pyrometer.
The temperature reading of the thermocouple has been found to be in accordance with
the melting points of Sn, Al, and Ag within +0.5 K, +0.5 K, and +0.8 K, respectively.
The vacuum chamber was constructed using high vacuum components, supplied
by MDC Vacuum Products. Flexible bellows support the metal-glass connectors
attached to the ends of the quartz tube that surrounds the shear cell. A Balzers
TPG 064 turbo pump with a TCP 0/5 drive unit, connected to the vacuum system,
yields a vacuum of 8x10 Pa. The turbo pump is backed up by a Leybold-Heraeus
Trivac D2A rotary pump equipped with a forline trap to prevent oil-backstreaming.
High purity Ar (99.9999%) is used as an inert atmosphere in experiments, which
require fast cooling rates. The Ar passes a heated titanium-sponge getter, supplied by
API Controls Division, Inc., before reaching the vacuum chamber. He (99.9999%)
flow through an overpressure outlet valve (2.2 x 10° Pa) may further increase the
- cooling rate of the graphite crucible up to 80 K/s.
| The data acquisition system, schematically shown in Fig 5, is built around a
National Instruments AT-MIO-16E-50 multipurpose data acquisition board, installed
in a PC, running Labview 4.0. Via a National Instruments SCB-68 connector box, that
contains a cold junction temperature sensor for the thermocouple readout, two
13
(type K)
time-temperature-frequency
profile
xs) .| TCC) | Qs").
30 25 0
240 900 0.4
120° 850 0.4
120 800 0.4 5
“30 15 0 power setting =F +P, Unc. +B Cina. +B, ney
PID temperature control,
TTL pulse generator
(custom Labview VI
analog-digital converter
AT-MIO-16E-50 RF power supply
- I
| > Lepel MR-7.5
connector box ]] 0..10 V P
SCB68 0..20 kHz microstepping
0.50 mV j 0.5 vt controller / motor
thermocouples pressure gauges
PX 213, PKR 250
- Fig. 5 Temperature control and data acquisition system
The custom written PID Labview VI compares the measured thermocouple voltages
with the set points of the time-temperature-frequency profile. According to Eq. (1), a
voltage (0..10 V) is applied to the remote input of the RF power supply. The ‘power
fit’ eliminates the need to adjust the integrational parameter, I, in Eq. (1).
po 14.
thermocouples’ two pressure sensors, and the torque sensor are connected for data
acquisition at 16 bit with a maximum sample rate of 20 kHz.
: The first pressure gauge is an OMEGA PX213 with an OMEGA DP4] signal
conditioner for total pressures of up to 3 x 10° Pa and for rough vacuum. For accurate
high vacuum readings, the second gauge, a Balzers PKR 250 Full Range Gauge with
readout unit Balzers TPG 251, is used.
The nominal temperature-time profile along with the desired rotation frequencies
of the inner cylinder are programmed prior to the start of the experiment. The times,
_ temperatures, and frequencies are contained in a spreadsheet that can easily be
modified using, e.g., ORIGIN 4.0. The spreadsheet is read by a Labview virtual
instrument (VI) that employs a modified proportional-integrational-differential (PID)
control algorithm. Depending on the actual temperature measurement (thermocouple
voltage, Uy,) and the set-point temperature, U,p, the PID algorithm computes a voltage
(power setting) in the range from 0 V to 10 V, according to Eq. (1), that is applied to
the remote control input of the Lepel RF power supply.
d(U,, —U,,)
power setting = PU, — Uy) + 1f(Us - U,, )dt +D it
(1)
; To reduce the number of control parameters in the PID algorithm from three (P, J, D)
to. two (P, D), a so-called ‘power-fit’ is performed after the crucible or shear cell has
been mounted. For this power-fit, the temperature is raised in steps of 50 K and the
RF power output required to maintain the temperature is recorded. Examples are
shown in Fig 4 (a) and Fig 4 (b). Typical results for the equilibrium power setting as a
function of the thermocouple voltage, i.e., the temperature, are plotted in Fig 4 (c) for
: . 15.
“two different chucibles. The smaller the difference between the radius of the heating
| coil and the outside diameter of the crucible, the more effectively is the RF power
absorbed by the graphite. Third order polynomials are used to fit the power settings as
a function of the thermocouple voltage in Fig 4 (c). The four fit Parameters (Po, P1,
P>, P3) of the polynomial are fed into the PID algorithm, eliminating the need to adjust
the integrational constant, /, in the PID loop in Eq. (1) by setting J =0.
2.2. Concentric Cylinder Rheometry
The viscosity of glycerol, C3H;(OH)3, has been measured previously and its
temperature dependence around room temperature is well-established [2]. At 23°C the
viscosity is 7 = 2.8 Pas. As glycerol wets graphite, it may serve as a test fluid for the
present concentric cylinder Couette viscometer. The following analysis of the flow
field inside the shear cell is the basis for the viscosity measurements of V1.
The liquid fills the gap between the inner and the outer cylinders of the shear cell,
with radii, R, and R2, respectively. The inner cylinder rotates with angular frequency,
Q, while the outer cylinder is fixed, Assuming the no-slip boundary conditions
uy (R,) = 2 R, (2)
Ug(R,)=O (3)
the velocity in 6-direction, u,, within the liquid can be integrated from the equations
of motion in radial and tangential direction [10]:
ro 16.
QR (R
u, = —-r}|. 4
goals ”
Using cylindrical coordinates (r, 8, z), the velocity in z- and r-direction is zero for the
case of infinitely long cylinders:
u, =u, =0. (5)
At the flow velocities used in the present work, one can neglect the compressibility of
the liquid. From Eq. (4) the radius dependent shear rate, y , can be calculated as
_ 0 (Uy 2R>R, 1
=y,=r—|—|=- =a. 6
YE Tso re(*) R}-R r (0)
For small spacings between the two cylinders, i.e., for (Ro-R1) « Rj, the variation of
the shear rate with the radius, r, is small, and an average shear rate,
(R+R
Y ave “(43% 5 | ; (7)
may be used. For, e.g., a shear cell with R, =1111x10~° m, R, =12.70x10~ m used
in the viscosity measurements of V1, the average shear rate in Eq.(7) is
V we = O.177Q.
For a Newtonian fluid, the shear stress, s, is proportional to the shear strain rate
; where the proportionality constant is called the shear viscosity coefficient, 7:
s=Yn. (8)
At the wall (r = R,) of the outer cylinder the shear stress (shear force per area) is
2R;
R ~ R
s(R,) = (Ry) = - Qn (9)
and the static torque, M, on the outer cylinder is calculated from the shear stress at
r = R, times the area, 27 R, L, and the radius, R,, of the outer cylinder:
4n LR? R?
M =——— 2. On. 10
RR n (10)
Equation (10) relates the measured torque, M, to the viscosity, 7, through the angular
frequency, Q, and the dimensions of the shear cell. It was derived by Couette [11],
who also performed the first viscosity measurements using the concentric cylinder
geometry. The ratio of the length, LZ, of the shear cell to the spacing between the
cylinders, L/(R2-R}), is typically on the order of 30 for the present shear cells.
For calibration, the torque meter is mounted horizontally outside the vacuum
chamber and a horizontal bar of 12 cm length, attached to the torque meter, is loaded
with standard weights. The corresponding torque sensor signals are plotted in Fig 6 as
a function of the applied torque, M. A linear relation
M =0.01446 torque sensor signal
Nm (11)
is found to describe the data accurately over four orders of magnitude in torque.
The- torque transferred through the glycerol to the outer cylinder
(R, =12.70 x10 m) is plotted in Fig 7 as a function of the angular frequency of the
inner cylinder. Two different inner cylinders were used with radii 6.35 mm and
12.7 mm, respectively. The resolution of the torque sensor allows for measurements
down to 10°Nm. Plotted in the same figure is the torque calculated from Eq. (10)
E T T VS | T T Ue | T J | T T Us
-— 10°4 -
0 , 1
Cc
Roy 7 7
~ 1074 =
® 7 4
= 4 J
® 5 4
2 10° z
2 7 J
10° T qT a | qT t CVUery T qT VPrrery i TOT Pere
10° 10° 10° 10° 10"
torque (Nm)
Fig.6 Torque meter calibration curve
~ Static torques are applied by loading a 12 cm beam with standard weights. The
amplified torque signal (0) using an OMEGA DP41 signal conditioner is well
described by the linear relation, Eq. (11).
T Perry T | T Vor Terry T TUTE
10°.
» ©
posal
torque (Nm)
TT a |
it pial
risiul 1 dep iil 1 toil 4 et
10° 10" 10° 10! 10°
angular frequency (s’)
Fig. 7 Viscosity measurements of glycerol
. Static torque reading on the outer cylinder ( R, = 12.70 x 10 m) as a function of the
rotation frequency of the inner cylinder at room temperature. Two different inner
cylinders with radius R,, =11.11X10™ m (0) and R,,, = 6.35 x 107 m (a) are used.
The solid lines are calculated torque curves according to Eq. (10) and the literature
value Of Neyer = 2-8 Pa s.
ro 20.
| “using the literature value for the viscosity of glycerol and the given geometry of the
shear cell. The solid lines in Fig 7 do not represent fits to the experimental data but
merely the expected torque based on the above calculation of the flow field. Good
agreement between the torque data and the calculation confirms the validity of the no-
slip boundary conditions and justifies the assumption of laminar flow for the present
experiments, Effects due to the finite length of the cylinders have also been neglected.
‘The energy per unit time, P, which is dissipated inside the liquid, is equal to the
rate of work that is performed by the inner, rotating cylinder:
An LR; R; 2?
R 7 R
P =-20 R, Lu,(R,)s(R,)= (12)
Typical dimensions of the shear cell used for the BMG forming alloys are
R,=1111x10°m, R, =1270x10°m, and L=35x10°m. With a maximum
angular frequency of 10 s’, the total power dissipated in the fluid with a viscosity of
10 Pas is less than 0.12Js". If thermal conduction from the shear cell mount and the
inert atmosphere is neglected, electromagnetic radiation from the graphite is the only
means of establishing energy balance in the shear cell. With the power,
radiation
=0,, € AT", (13)
- emitted from the cell, an energy increase of 0.1 J per second yields a temperature rise
of the apparatus of about 0.2 K with the above parameters. This can be neglected
compared to thermal gradients inside the shear cell of about +4K. In Eq. (13),
O,, =5.670x10°Js'm~K* is the Stefan-Boltzmann constant, € the total
hemispherical emissivity of the graphite, and A, the total outer area of the shear cell.
2.3. Graphite as a Crucible Material
The crucible material used inthe construction of the concentric cylinder shear cell
has to fulfill several requirements in order to be used in conjunction with Zr-based
metallic liquids. The no-slip boundary conditions described in chapter 2.2 require an
intimate contact between the liquid and the crucible walls. In most applications
involving liquid metals it is not desirable that the liquid wets the container. However,
wetting between the shear cell and the liquid is a prerequisite for the present viscosity
measurements. At elevated temperatures reactions between the metallic liquid and the
| crucible material are therefore of concern.
The wetting of V1 on the graphite is shown in the photograph and the schematic
in Fig 8. The sample, processed in the liquid at 1300 K for 12 min, shows a wetting
angle of ca. 3° on the crucible interface. A high vacuum anneal of the graphite at a
temperature of 1300 K was employed prior to the experiment. In Fig 8, a ca. 0.5 mm
deep zone within the porous graphite, where the metallic liquid has penetrated the
crucible wall, is visible. The schematic demonstrates that the penetration of the liquid
into the graphite serves as a precursor for macroscopic wetting on the porous surface.
In this process, the microscopic contact angle between the liquid and the non-porous
substrate can be significantly larger than zero.
Scanning electron microscopy analysis has shown that a Zr-C reaction layer with
a width of ca.0.5\um forms between the graphite and V1. Therefore, upon
penetrating the porous structure, the liquid’s composition will drift to a composition
with less favorable glass forming ability. The changed composition of the alloy is
BUA
b ALBAAL
(b) ABUABE
BAAR
AAA EZ
VM VA WA
Ma) A
VA WA WA
Vm) A
Ve, VA
VM A A
——_
Fig.8 Photograph and schematic of the liquid-graphite interface
(a) Photograph of the cross section of a cylindrical graphite crucible containing
480 mg of V1. The liquid has infiltrated the porous graphite up to a depth of
ca. 0.5mm. (b) The schematic illustrates (in two dimensions) the wetting of a liquid
on a porous substance.
en 23.
likely to correspond to a higher liquidus temperature and the flow of the liquid inside
the porous graphite will eventually be limited by crystallization at the penetration front
for a given processing temperature.
2.4... Capillary Flow Rheometry
Most of the viscosity measurements for a variety of liquids at or near room
temperature “have been performed with capillary tube viscometers. In these
experiments, the liquid flows under the influence of a pressure gradient from one
reservoir through a capillary into a second reservoir. For laminar flow of a Newtonian
liquid and no-slip boundary conditions at the capillary wall, the volume flow rate,
AV/At, through the capillary is inversely proportional to the liquid’s viscosity (Hagen-
Poiseuille law [12]):
AV _ Rey Ap
At 8n Al
(14)
The pressure gradient, Ap, is measured along the length, Al, of the capillary having an
inner radius Reap.
In Fig 9 a schematic of the setup used for the capillary flow studies of metallic
liquids in this work is shown. An ingot of ca. 40 g of the alloy is melted inside an
inductively heated cylindrical graphite crucible. The capillary is inserted vertically
into the melt and subsequently the pressure of the Ar atmosphere is increased. The
liquid flows with a rate, AV/At, into the upper reservoir. The upper part of the graphite
crucible is equipped with a 1.5 mm wide slit that allows for the observation of the
height, h, of the liquid inside the upper reservoir. A motor driven single lens reflex
RF coil
| O | Py O
© Pp O
graphite _o O
art ©
quartz 4 Lo. |
capillary “o™~ yf height, A
° O length, /
liquid —O 6
O O
O O
Fig. 9 Capillary flow rheometry
Schematic setup to measure the flow rate of a liquid through a quartz capillary at
elevated temperatures. The liquid is contained in an inductively heated graphite
crucible. Along the length, I, of the capillary, a pressure gradient, Ap = pi-p2, is
established through an Ar atmosphere. The increase in height within the upper
reservoir with time is monitored with a motor-driven SLR camera.
a 25.
(SLR) camera photographs the momentary height at intervals of 2.52 s. The volume
flow rate,
AV = im Ah ; (15)
At ‘At
can be determined from the increase in height, Ah/At, of the liquid column inside the
upper cylindrical reservoir with diameter, Rres. Consequently, the viscosity can be
determined from a measurement of h(t) by:
~ 8R2, Al
R* -1
= “2( =) . (16)
At
A type K thermocouple inserted into the crucible wall measures the temperature
of the graphite crucible. For the case of a vertical capillary, corrections to Eq. (14) can
be applied to account for the finite size of the reservoirs, i.e., for a time dependent
pressure gradient per length, Ap/Al [13].
3. VISCOUS FLOW OF ZR-BASED GLASS FORMING ALLOYS
Recently, viscosity measurements [14, 15, 16, 17, 18] in the deeply supercooled
liquid of multicomponent bulk metallic glass (BMG) forming alloys have found
considerable attention. Beam bending and parallel plate rheometry are useful
techniques for studying the flow of deeply supercooled samples in the viscinity of the
glass transition regime. For those measurements, the material is first quenched into
the amorphous state and then heated to temperatures where the equilibrium viscosity
of the supercooled liquid can be studied within typical timescales of some 10 seconds
to several hours [16]. Even for the most stable amorphous metallic alloys, the onset of
crystallization limits the experimentally accessible temperature range of the
supercooled liquid to about 100 K [14, 19]. Relatively few data are available from
high temperature rheometry in the equilibrium liquid or slightly supercooled liquid
state of metallic systems [18, 20, 21]. |
In chapter 3.1 experimental results from the concentric cylinder viscometer and
from capillary flow will be presented. In the following analysis, the high temperature
limit of the viscosity of the liquid will play an important role. For this reason, in
chapter 3.2 the rate theory of flow will be revisited for the limiting case of high
temperatures. The concept of classifying liquids according to their fragility follows
from this high temperature limit of the viscosity. The viscosity of V1 is analyzed
according to the entropy and free volume model of flow in chapters 3.3 and 3.4,
respectively. Published heat capacity data and specific volume measurements
an 27
| ‘complement the discussion and allow for a detailed comparison of the two
phenomenological models.
3.1. Viscosity: Experimental Results
Typical torque readings from a concentric cylinder shear cell containing liquid V1
are shown in Fig 10. The inner cylinder is alternately rotated in clockwise and
counterclockwise direction for 60s each. In this way, determination of the torque
baseline is straightforward and the resolution of the viscosity measurement is doubled
compared to simple continuous motion. The torque meter signal (mV) is converted
via the calibration curve of Fig 6 and Eq. (11) into a torque reading (Nm). Analysis of
the flow field between the two cylinders from chapter 2.2, which resulted in Eq. (10)
yields the viscosity, 7, as a function of the temperature. The decrease in the torque
signal with increasing temperature in Fig 10 arises from the decrease in viscosity of
the liquid alloy. The resolution of the torque sensor in conjunction with the increasing
electrical noise from the RF heating coil does not allow for viscosity measurements in
excess of 1200 K for the present system.
The viscosity of the supercooled liquid state of V1 was also measured upon
cooling into the supercooled liquid state. The measured viscosity as a function of
temperature for various shear rates is plotted in Fig11. No dependence of the
viscosity on shear rate is observed indicating Newtonian flow behavior for the present
temperature range and the shear rates indicated in Fig 11. The sharp increase in the
viscosity signal in Fig 11, representing the onset of solidification, will be discussed in
chapter 5.2.
0.16 |
1033 K :
O : hace, SIE
0.14 2, RE Stig OE, iy SSO en )
S porto cccrccn o-
@ :
© 0.12-
” 1053 K
s + Sitter thadhh Fe
g Ogata
cD en
E 0.10-
a)
(oF
, a
[e)
0.06 3 :
60 120 180 240
Fig. 10 Isothermal torque reading at constant shear rate
Clockwise and counterclockwise rotation of the inner cylinder for 60's results in an
alternating torque signal. The temperature is raised in steps of 50 K and the system is
held at each temperature for 180s prior to each experiment to allow for thermal
equilibrium within the shear cell. The baselines (---) of the torque meter signals are
shifted by arbitrary offsets.
- temperature (K)
1000 950 900
{ 0° ~ l | n tl l 1 ‘| l Ll Ll t | 5
— | —2— 0.546 s" :
| —°— 0.274 s" g 1
—»— 0.110 s" i | J
>» | —v—0.055s"
@ 10 7 | é i F
g ] ; | :
a 7 @ 7
4 Oo 4
oO
QD
> 10'> 7
1 0° LJ | UJ LI q T LJ qT T J | qT
1.00 1.05 1.10
1000 /T (1/K)
Fig. 11. Viscosity measurements upon linear cooling
Viscosities during linear cooling with a cooling rate of 0.66 K/s (see chapter 5.3). The
average shear rates according to Eq. (7) are indicated.
? ; 30.
7 . Resulis from a typical capillary flow experiment of the equilibrium liquid of V1
are shown in Fig 12. The height of the liquid in the upper reservoir (see Fig 9) is
determined from the image analysis of 35 mm slides taken by a motor driven SLR
camera. The time interval between two frames is 2.52s. Consequently, Eq. (16)
yields the viscosity of the liquid.
| In Fig 13 the present high temperature viscosity measurements are plotted in
conjunction with published results for several metallic elements [22] and for metallic
glass forming systems [18, 21, 23]. The viscosity of V1 at its liquidus temperature is
about 3 orders of magnitude larger compared to metallic elements at their melting
point. Extrapolating the viscosity of V1 to higher temperatures in Fig 14 according to
the free volume model (see chapter 3.4), one finds a ‘normal’ melt viscosity for a
metallic system at about the average melting point of V1’s constituents. An
extrapolation of the viscosity curve of Ni is also plotted to guide the eye. The
comparison to a so-called fragile liquid (see chapter 3.2) like Ni shows that it is the
proximity of V1’s composition to a deep eutectic in conjunction with strong liquid
behavior that lead to sluggish kinetics in the equilibrium melt.
3.2. High Temperature Limit of the Viscosity
In this chapter, Eyring’s concept of rate theory [24] is applied towards the high
temperature limit of the viscosity of the liquid state. For that purpose, the liquid is
envisaged to be stable in the limit T > ~, i.e., no phase transformation to the gaseous
state occurs. The temperature dependence of the density or of a typical inter-atomic
length, a, is neglected to keep the liquid in the condensed state.
heigth (m)
Fig. 12
Height, h, of the liquid inside the upper reservoir as obtained by image analysis of
photographs taken at 2.52 intervals. A setup according to Fig. 9 with
Rep = 1.0X 107 m, Rog = 15X10” m, Ap = 6800 Pa, and Al =1.73x 10° m was
0.012
1 1 1 1 i if 1 1 i l Ll l i 1 I l l 1 i | 1 L i 1
Capillary flow rheometry: experimental results
used. The viscosity, according to Eq. (16), is n = 4.4 Pas.
temperature (K)
2000 1600 1400 1200 1000 900
Losiuel pool Lait
viscosity (Pa s)
bol io TT
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
1000 /T (1/K)
Fig. 13 Viscosities of liquid metals at high temperatures
Results from viscosity measurements of V1 obtained by the rotating cup viscometer (@)
and the capillary method (A) as well as a Cohen-Grest fit (—). Data for Pd7gCugSirs
(-------) [18], PdgoSiig (---) [21], Tiz4sZr1,Cug Nig (---) [23], and the viscosities of several
metallic elements at their melting points (+), see Fig. 14, are plotted for comparison.
The viscosity of Ni is extrapolated into its supercooled liquid state (-----).
temperature (K)
4000 2000 | 1000 500
teres a a ae 1 | it 1
viscosity (Pa s)
0.5 1.0 1.5 2.0
1000 / T (1/K)
Fig. 14. Arrhenius plot of the viscosity of V1 and metallic elements
Viscosity of V1 as measured by beam bending (1) and concentric cylinder rheometry
(0) as well as a Cohen-Grest fit (—), Eq. (32), Eq. (36). Melt viscosities (+) of Sn, Bi,
Tl, Cd, Pb, Zn, Te, Sb, Mg, Ag, Ac, Au, Cu, Mn, Be, Ni, Co, Fe, Sc, Pd, V, Ti, Pt, Zr,
Cr, Rh, B, Ru, Ir, Mo, Os, Re, and W near their melting point.(in order of increasing
melting point) [22] and an extrapolation of the viscosity of Ni (---) into its supercooled
liquid state.
- 34
A shear strain rate, 7, is then applied to the material resulting in a macroscopic shear
stress s=7ny. It is assumed that at sufficiently high temperature viscous flow is
controlled by single particle motion, i.e., the shear strain rate is homogeneous down to
distances on the order of an interatomic length a,
where the velocity difference, Au, per length a is measured perpendicular to the
direction of flow. The rates, ks and ky, of atoms moving forward and backward in the
direction of flow under the presence of the shear stress,
k, =k, exp| —- 7 and k, = ky, exp] — , (18)
are written according to Eyring. In Eq. (18), the effect of the shear stress is to create a
potential energy gradient, sa*, per inter-atomic length, a, that favors forward
compared to backward motion. Consequently, the velocity gradient can be written as
the difference between the two rates, multiplied by the average displacement, a:
: sa’
Au =(k, —k,)a = 2k, asin ar | (19)
In the limit of high temperatures, kT >s a’, the first term of a Taylor series yields:
lim Au = koS@ : (20)
Tee kT
while solving for the viscosity coefficient in Eq. (17) results in:
> (21)
Eyring found a universal expression for the rate constant, ko, from the frequency, ¥/a
(average velocity per length), times the one-dimensional. translational partition
function, fi:
=f v_ 2a m'kT 5
«= (22)
The average velocity,
p= |X (23)
27m
results from the thermal energy per degree of freedom and leads to:
kT
k,=—. 24
oF Fy (24)
Finally, Eq. (24) inserted in Eq. (21), with v,, =a’ as the average atomic volume,
results in a universal expression for the high temperature limit of the viscosity of the
liquid state:
My =. (25)
_ For the present system, V1, the atomic volume, Van = M/(N, p), can be derived
from specific volume measurements [25] that yielded a density, p, of about 6gcm”
for the liquid. The average molar mass, M, amounts to 60gg-atom™ and Ng is
Avogadro’s number. With this, Eq. (25) leads to a lower limit of the viscosity of Vitl
of 7) =4x10" Pas.
po 36
The classification of liquids according to various degrees of ‘fragility’, made
popular by Angell [26], relies on the applicability of Eq. (25) for a wide range of
liquids. The specific volume is known to vary by only about one order of magnitude
between different liquids. Therefore, the high temperature limit of the viscosity is
always on the order of 10° to 10% Pas.
Scaling the viscosity-temperature curves to an (arbitrary) viscosity of 10°? Pas
results in the normalized Arrhenius plot as shown in Fig 15. All curves meet — by the
above definition of T, — at 10’ Pas (or 10'° Poise). So-called ‘strong’ liquids, like
S102, exhibit an Arrhenius-like temperature dependence. The viscosity of ‘fragile’
liquids has a stronger temperature dependence in the vicinity of T, and a weaker
temperature dependence at high temperatures.
The viscosities of V1 and Zr46.75Tig 25Cu7.5NiioBeo7.5 (V4) [14] indicate that these
liquids are less fragile than other metallic melts [27]. The difference in the specific
heat capacity between the amorphous and the supercooled liquid state in the glass
transition regime is another measure of the fragility of a system. Again, the bulk
metallic glass forming alloys V1 and v4 are strong liquids compared to normal
metallic melts.
~ 3.3. Entropy Model for Viscous Flow
The kinetic slowdown reflected in the increase in viscosity with decreasing
temperature can be connected to the thermodynamic functions of the supercooled
liquid through the so-called entropy model for viscous flow. For a variety of systems,
it has been found [26] that with an increase in viscosity the isobaric entropy decreases
12
, mes!
10." | .
VA
10
io)
HN
—_
oO
‘i
NJ
IN
—_
io)
re)
LN
viscosity (Pa s)
°,
—_
Nh
oO
fl
—_)
oO
0.0 0.2 0.4 0.6 0.8 1.0
T, /T
Fig. 15 Normalized Arrhenius plot of the viscosity of glass forming systems
The inverse temperature, 1/T, is normalized to the glass transition temperature, T,,
which is defined for this plot as the temperature where the viscosity reaches a value of
10’? Pas. The viscosity of V1(e) is plotted in comparison with a strong liquid,
SiO? (a), and a fragile liquid, o-terphenyl (A) [2]. -
i 38
7 relatively sharply in the deeply supercooled melt. Adam and Gibbs 18] developed a
theory on cooperative relaxation in liquids and supercooled liquids that quantifies the
dependence of transport properties like the viscosity on the entropy of the system. The
increase in relaxation times and viscosity with increasing supercooling is directly
linked to a decrease in configurational entropy, S,, of the system. Within this model,
the viscosity,is expressed as
Cc
= — |, 26
N=No cxf TS. (26)
where the constant C represents an effective free enthalpy barrier for cooperative
rearrangements. The pre-factor, 7,, in Eq. (26) can be calculated according to
Eq. (25) to be 4-10° Pas. It is further assumed that the vibration of the atoms in a
(supercooled) liquid is modified only by a slight anharmonicity compared to vibrations
in the crystalline state. Neglecting this difference, the configurational entropy can
therefore be approximated by the difference, AS , between the total entropies of the
liquid and the crystal,
S, = Sig, —8 (27)
oryst. ?
assuming that their vibrational parts of the entropy are equal. One should note,
~ however, that in a multicomponent system like V1, the ‘crystalline state’ is composed
of a variety of crystalline phases each with a certain homogeneity range with respect
to its composition. Except for one-component systems or simple alloys where the
structure of the equilibrium phases and their respective compositions are known, the
entropy, Scryst., is in practice treated as a phenomenological quantity. In this spirit, AS
: 39.
results from an ‘integration of the measured heat capacity difference Ac, = c, i, — Cy xu
between the liquid and the crystal [29]:
Trig Ac, ~
S.=S.(Tiq)- J iat. (28)
Inserting Eq. (28) in Eq. (26), the configurational entropy at the liquidus, S.(7j,,), 1s
treated as a fitting parameter. This takes into account that in a multicomponent system
the crystalline state can have a considerable amount of entropy of mixing. In fact, a
disordered solid solution can have a larger entropy of mixing than the supercooled
liquid of the same composition with strong tendency to topological and chemical short
range order [30].
If the specific heat capacity difference is proportional to the inverse temperature,
ie., if Ac, ~1/T , Eq. (26) in conjunction with Eq. (28) is formally equivalent to a
Vogel-Fulcher-Tammann (VFT) relation:
N=MNo ex = =| . (29)
For the present alloy, however, a 1/7 dependence clearly underestimates the
temperature dependence of Ac, as shown in Fig 16. Consequently, a quantitative
analysis of the flow behavior of Vit by a VFT relation cannot be justified on the basis
of the Adam-Gibbs entropy model. In Fig 16, the low temperature part of the heat
capacity (dashed curve) was calculated according to the Debye model with an
estimated Tpebye = 305 K in analogy to V4.
heat capacity (J / g-atom K)
10-— ; Tebye T cauzmann 7
0 é Lj | Li | UJ | T | t | U
0 200 400 600 800 1000 1200
temperature (K)
Fig. 16 Specific heat capacity
Specific heat capacities of V1 in the liquid (0) and crystalline (A) state. Equation (30)
- and Eq. ¢ 31) are shown as solid curves (—). The measured heat capacity during the
calorimetric glass transition is shown for various heating rates (x, V, +) [29]. The
low temperature part of the heat capacity (---) was calculated according to the Debye
model with an estimated Tpebye = 305 K. A Acy ~ I/T least squares fit to the specific
heat capacity of the supercooled liquid is indicated (- - -).
ro . 41.
* In order to'comply with the high temperature limit of the viscosity according to
chapter 3.2, the specific heat capacity difference in Eq. (28) has to vanish for To.
a To avoid the condition Acp = 0 for finite T, ic., to avoid a minimum in the viscosity,
the following analytical expressions are used to fit the experimental specific heat
measurements:
Cox =3R+B,T + B,T’, (30)
Cotiq = Cpa + B3T + BT. (31)
From a least-squares fit of Eq. (30) and Eq. (31) to the experimental data, the
parameters are (in SI units) R=8314, B,=-8.021-10°, B, =2.076-10°,
B, =2940-10°, and B, =3934-10’. They describe the experimental data very well
(Fig 16) and yield a finite high temperature limit 7, for the viscosity. Kubaschewski’s
formula [29] describes the heat capacity data equally well, but fails to comply with
lim Ac, =0 and therefore with lim N(T) = No-
Teo
A fit of Eq. (26) to the viscosity data of V1 yields the two fit parameters
C=203kJ g-atom’ (or 21leVatom’) and S!=158JK'g-atom’. The
corresponding viscosity curve is shown in Fig 17, in comparison to results from the
~ free volume model, which is discussed in chapter 3.3. Results obtained from the high
temperature Couette viscometer as well as from beam bending rheometry [16] of V1
are included in Fig 17. Qualitative agreement between the measured viscosity and the
entropy model is found, but, e.g., the tendency of the viscosity values to show
Arrhenius behavior at temperatures below ca. 700 K is not entirely accounted for in
Eq. (29).
temperature (K)
14001200 1000 800 600
10
15
10
13
10
“415
10
oO
viscosity (Pa s)
o 8
0.8 1.0 1.2 1.4 1.6 1.8
1000 / T (1/K)
Fig. 17 Viscosity of V1 and least-squares fits of entropy and free volume model
Viscosity of V1.as measured by beam bending (]) and concentric cylinder rheometry
(0). The data are fitted to the entropy model (---), Eq. (26), and the free volume model.
The dashed curve (---) represents a VFT fit and the solid curve (—) a Cohen-Grest fit
(see chapter 3.4).
ho 43.
; Using the fit parameters C and S™ and Eq. (26) to solve for the configurational
entropy, we ‘obtain Fig 18. The data points in Fig 18 result from the individual
viscosity measurements converted via Eq. (26) into configurational entropies and the
solid line isthe configurational entropy according to the best fit of the viscosity data.
Although the temperature dependence of the viscosity in Fig 17 is in qualitative
agreement with Adam’s and Gibb’s theory, it is clear from Fig 18 that the decrease in
entropy with supercooling cannot quantitatively account for the exact form of (7).
3.4. Free Volume Model for Viscous Flow
Within the free volume model, viscous flow in a liquid occurs as a consequence
of density fluctuations or rearrangements of so-called free volume. It is essential to
the theory that the atomic rearrangements take place spontaneously, i.e., without
appreciable activation energy compared to kT. In this model, fluid motion in the
liquid and supercooled liquid can occur, because in localized regions of configuration
space, the free energy landscape of the atoms is flat compared to the thermal energy,
kT. The amount of free volume in the supercooled liquid decreases with decreasing
temperature and can eventually lead to structural arrest. The viscosity can be
_ expressed as
N= No caf ) (32)
Ve
with v, as the average free volume per atom and bv,, as the critical volume for flow
(31). The atomic volume v,, of V1 was measured by Ohsaka et al. [25] in levitation
experiments to be 167x10”m?* near the liquidus. The pre-factor 1) =h/v,, in
25 TT
20 - 90°C
, 1 oo 1
5 14
© 1
5S
= 10
> 107
oO 4
2 4
5-
0 yt
400 600 g00 1000 1200
temperature (K)
Fig. 18 Configurational entropy of V1
Entropy difference between the liquid and the crystalline state of V1 (.--) and
configurational entropy, Se (—) of the liquid as obtained from a fit of Eq. (26) to the
viscosity data in Fig. 17. The viscosity data are converted via Eq.(26) into
configurational entropies, S, (©), (0).
mo . 45.
“Bg. (32) is set to 4x 10°Pas as discussed in chapter 3.2. Under the assumption of a
linear relation, |
| Vp =V,0¢(T - Ty), (33)
between the free volume and the temperature, Eq. (32) takes the form of a VFT
equation:
1=No Sr Eaeeal (34)
Here, a, characterizes the increase in free volume with temperature. Often, the
- dimensionless parameter
D, = b
Os Ty
(35)
is introduced in Eq. (34), to classify a liquid according to its fragility [26, 32]. At the
temperature 7,, the free volume vanishes and viscous flow is no longer possible,
according to this model. The so-called VFT temperature, To, is usually treated as a fit
parameter. For the V1 data, shown in Fig 14, one finds D, =165 and 7) = 426.3 K.
The present values of D« and To differ slightly from previously published results [24]
due to the recent high temperature measurements of 7 in the equilibrium liquid and
. supercooled liquid. Compared to other metallic systems, the relatively high value of
Ds reflects ‘strong’ liquid (see chapter 3.2). It has been demonstrated [27] that the
glass forming ability of a metallic system represented by the critical cooling rate
correlates with its fragility parameter D+. The stronger the liquid (i.e., the higher Ds),
the lower the critical cooling rate to bypass crystallization.
ho 46
In an extended version of the free volume model proposed by Cohen and Grest
[3 lL, the free volume decreases to zero, only for T = 0, and the viscosity is expressed
through a temperature-dependent free volume [31]
k 4y
VY, = £{r-n+ fer-ny Ber. (36)
A least-squares fit of the logarithm of the viscosity data to Eq. (32) in conjunction
with Eq. (36) describes the data very well as shown in Fig 14. The three fit parameters
are found to be bv,,¢,/k =4933 K, T, =672 K, and 4v,¢,/k =162 K. Cohen and
_Grest’s concept of a volume dependent free energy for liquid- and solid-like cells,
which led to Egq.(36) models the viscosity of V1 better than a
Vogel-Fulcher-Tammann (VFT) relation as the comparison in Fig 14 shows.
In the limit of low temperatures, T — 0, the free volume in Eq. (36) becomes
proportional to the inverse temperature, v, ~ T-'. This corresponds to an Arrhenius-
like temperature dependence of the viscosity in Eq. (32). In the limit of high
temperatures, T — oo, Eq. (36) yields v, ~ (T - Ty) , corresponding to a VFT relation
in Eq. (32). The transition region from v, ~ (T- T,) to v; ~ T"', ie., from VET to
Arrhenius behavior, is found to be around 7, = 672 K.
From the viscosity measurements, the normalized free volume ratio, v, / (bv,,),
in Eq. (32) can be calculated in conjunction with the above fit. Data corresponding to
the present viscosity data of V1 are shown in Fig 19 (left axes), The high temperature
viscosity data yield
4 1.2
4 1.0
4 0.8
= 0.6
v./bv. (%)
v./ Vv (%)
- 0.4
+ 0.2
400 600 800 1000 1200
temperature (K)
Fig. 19 Free volume of V1 as determined by the viscosity
Viscosity data from the concentric cylinder viscometer (0) and from beam bending (0)
converted into relative free volume according to Eq. (32).
LM iggy jot KO, (G7)
b bv, dT
with @,as the average expansion coefficient for the free volume in the temperature
range from 800 K to 1200 K. This increase in free volume with temperature can be
approximated by the difference between the thermal expansion coefficients of the
liquid and the glass [33],
Os = Aiig — O& gtass> (38)
with a, =532x10° K” and a,,.. =339x10° K" [25]. From Eg. (37) and
liq glass
Eq. (38) the value of b =0.105 can be calculated and the corresponding average free
volume per atom, v,/v,,, is shown in Fig 19 (right axes). At the liquidus
m?
temperature, one finds v, = 9.6 x 10° V,,» 1.€., only about 1% of the sample’s volume is
‘free’. This small amount of free volume is consistent with the picture of a dense
metallic liquid with slow kinetics.
The average interatomic spacing in liquid metals is — to a first approximation —
determined by the size of the individual atoms. The latter may be calculated from the
nearest neighbor distance in the crystalline state. In Fig 20 the specific volume of the
liquid state, Vmjig, to the 1/3 power is plotted as a function of the nearest neighbor
' distance, ayn.xu, in the crystalline state for a number of metallic elements [34]. The
relation
(Vang )3 = 0.925 Ayn . (39)
describes the literature data of Vmjiq and @nnxu to within +4% as shown in Fig 20.
Taking the weighted average of the nearest neighbor distances (atom diameters) of the
axio' | , , :
44 i CkQ, Mg
=~ 1x10° 4 si Plo 5 & Ro "Brg dg J
x : pe Pog, =
—Sixtot4 aay Cup Aso 5 © hod
ea ; ) ite 20 AUg Bato :
exo" 7 Weige Ovt q
0 T T T T T T T I T a"
oO
2x10° 4x10° 6x10” 8x10° 1x10”
-1
OW (K )
Fig. 20 Interatomic spacing and thermal expansion in liquids and crystals
(a) Average interatomic spacing as a function of the nearest neighbor distance in the
liquid and crystalline state of V1 (¢) and of the metallic elements Be, Fe, Ni, Cr, Co,
Cu V, Zn, Rh, Ru, Ir, Mo, Os, W, Pd, Re, Pt, Nb, Ta, Al, Au, Ag, Ti, Cd, Hg, Li, Hf,
Mg, Zr, Tl, Pb, Th, Ce, Nd, Pr, Na, La, Ca, Sr, Ba, K, Rb, and Cs (0) [34].
(b) Thermal expansion coefficient in the liquid [25] and crystalline state of V1 (¢) and
several metallic elements (e) [22, 35].
oe 50
“elements, Tr, Ti, Cu, Ni, and Be, in the ratio of the atomic composition of V1, the
| specific volume of V1 in the liquid state is only slightly (2%) smaller than predicted
by Eq. (39), In contrast, the thermal expansion coefficient, jg, of liquid V1 is
considerably smaller than for most transition metals [35] as the comparison in Fig 20
(b) shows. The thermal expansion coefficient of the liquid enters through Eq. (38)
into the fragility parameter, D+, and a small Ojig is consistent with a large value of D+,
1.€., with strong liquid behavior.
As discussed in chapter 3.1 an extrapolating of the viscosity of V1 to higher
- temperatures in Fig 14 and Fig 17 according to the Cohen-Grest model, Eq. (32) and
Eq. (36), yields a normal melt viscosity for a metallic system on the order of 10°Pas
at about the average melting point of V1’s constituents [22]. For viscous flow one
defines an apparent activation energy as
_ o(in 7)
~ O(L/ KT) (40)
Q, (7)
which can be determined from the slope of the logarithm of the viscosity plotted
versus inverse temperature. The apparent activation energies of V1 in the liquid and
supercooled liquid state are plotted in Fig 21 in units of electron-volts and of the
_ thermal energy, kT. The Cohen-Grest fit from Fig 17 was taken as a basis for
calculating Q, as defined in Eq. (40). At temperatures between 600 K and 650 K
where the glass transition of V1 can be observed in conventional calorimetric
experiments the apparent activation energy for viscous flow of the supercooled liquid
is on the order of 4eV. The finite low temperature limit of the apparent activation
. ‘eriergy In Fig?1 (a) represents the transition to an Arrhenius-like temperature
dependence of the viscosity of V1 within the Cohen-Grest model.
| The temperature dependence of the viscosity of the metallic elements in the
experimentally accessible temperature range above the melting point is characterized
by apparent activation energies, Q, of less than 0.5 eV. Similar to the viscosity itself
(see Fig. 14), the apparent activation energy for viscous flow of V1 approaches normal
values for a metallic system at about the average melting point of the constituents.
10 TTT res anc ha nto nde n-ne nln n-
Set
— * one Cu id
oO 0.1 eegdezn 98
CyRKP Ne Hie ge
0.04 L} qT tT 1 | UJ
200 400 600 8001000 2000
7 2. T - T T T T
100 _ (b) o
o ae
x -
~ 10 mee
= me.
'S)
e e
4 PAO Nco Lin g APPZ” ona Conky Ae
LJ : v q U | LJ
200 400 600 8001000 2000
temperature (K)
Fig. 21 Apparent activation energies for viscous flow
Apparent activation energies as calculated by Eq. (40) for V1 in units of eV and of kT.
The solid parts of the curves (—) represent the experimentally accessible temperature
; range while the. dotted parts (.--) are extrapolations based on the Cohen-Grest model.
For comparison, the measured apparent activation energies of several metallic
elements are included (+) [22].
4... TIMESCALES IN THE LIQUID AND THE SUPERCOOLED LIQUID
4.1. Diffusion in the Liquid State
Tn order to quantify the time scale for atomic transport in the equilibrium melt, Au
interdiffusion in liquid V1 was studied at 1050 K. Usually, the diffusion studies in
liquid metals. are complicated due to the influence of convection in the melt. Thermal
gradients upon heating the sample to the diffusion temperature or unwanted thermal
gradients during the isothermal anneal imply density gradients in the liquid that may
trigger convective flow. In order to estimate the stability of the liquid against density
driven convective flow, a spherical sub-volume with radius a of the liquid is
considered. A buoyancy force,
4 3
is exerted on the sub-volume if a temperature difference, AT, compared to the
surrounding liquid exists. In Eq. (41), p is the density of the surrounding liquid, Otig
its thermal expansion coefficient, and g the gravitational constant. When the sub-
volume has a velocity, U, relative to its surrounding, a frictional force,
F,=6rnau, . (42)
according to Stoke’s law will hinder its motion. The larger the viscosity, 7, the larger
the frictional forces that damp the convective flow.. Under steady-state conditions, the
time, (a/U), for the sub-volume to move a distance, a, will be on the order of the
thermal diffusion time, (C, pa’ /x). If the frictional force is larger than the
: 54
. buoyancy force, F, > F,,, the system will be stable against density driven convective
flow. Rearranging Eq. (41) and Eq. (42) leads to the stability criterion:
3 42
a’ p’ gC, a, AT eo.
NK 2
(43)
The radius of the diffusion crucible of 3x10 m serves as an upper limit for a. The
density, p, of V1 at 1050 K is 5965 kg m™ [25], the heat capacity, Cp, is 771] kg? K”
and the thermal expansion coefficient is ,,, = 532 x 10° K. With k=10WmK"
and a viscosity of 7 =15 Pas at the diffusion temperature, Eq. (43) implies that the
temperature differences, AT, inside the liquid should be smaller than 175K. For
comparison, within a system having a viscosity of 10° Pas, typical for most metals,
the temperature distribution has to be uniform within ca. 0.1 K in order to suppress
convective flow.
With a modification of the present setup for the capillary flow experiments, a Au
foil was inserted vertically into the liquid V1 [36]. After 400s the sample with the Au
foil was cooled to room temperature. The resulting spatial Au concentration, Cau, was
analyzed using a Jeol JXA-733 microprobe and is shown in Fig 22. The maximum Au
concentration within the liquid was less than 1.5 at-% in this experiment. An error
function profile,
C,, (x) = c, erfc] ———— .}, 44
au(X) = Cy ; | (44)
with
In (c,.)
0.0 2.0x10" 4.0x10™ 6.0x10™
penetration length (m)
Fig. 22 Concentration profile of Au in VI
- Atomic fraction, cau, of Au in V1 after diffusion at 1050 K and 400 s as measured by
electron microprobe (0). The solid line (—) represents a complementary error
function fit to the data, Eq. (44).
. _;__2_f 2
erfe(z) = 1 [, exp (-u?) du (45)
was used to obtain the diffusion coefficient, D,,=61x10"'ms*. The
corresponding fit is also plotted in Fig 22.
Quasielastic neutron scattering experiments on V4 revealed that above the
liquidus temperature the differences in the diffusion coefficients of the various
elements in the liquid alloy are less than one order of magnitude [37]. This finding is
in accordance with the merging tracer diffusion coefficients in Fig 22 and is well
known for foreign diffusion in the equilibrium melts of Ag, Hg, Sn, Pb, and Bi [38].
4.2. Time Scales for Viscous Flow and Atomic Transport
In the following, time scales for viscous flow are compared to typical diffusion
times in the liquid and supercooled liquid state of V1. From the temperature
dependence of the viscosity data presented in chapter 3 and from published tracer
diffusion data, the notion of solid-like and liquid-like atomic transport in the
supercooled liquid will be discussed.
At temperatures below 650 K, the isothermal relaxation of the viscosity of V1 to
its equilibrium values has been measured [16]. Similar to the viscosity relaxation in
' V4 [39], the internal equilibration times, tT, , associated with this relaxation into the
supercooled liquid state are found to be proportional to the viscosity:
T=
. 46
" G, (0)
The experimental values of t, and 7 yield G, =55x 10°Pa where the temperature
fT
"dependence of Gy ‘is neglected compared to 7 and tT). Also, for simplicity, the
following discussion will be based on a single equilibration time constant, t, at each
temperature, thereby neglecting non-exponential relaxation behavior. A more detailed
analysis would include a spectrum of relaxation times [40].For long annealing and
observation times, t >> T, the system is said to be in the supercooled liquid state,
while for short times, t < T, the sample reveals properties of a solid-like amorphous
State.
The viscosity data from Fig 17 including the Cohen-Grest fit have been converted
via Eq. (46) into equilibration times and the times are plotted in the form of an
Arrhenius plot in Fig 23. The times T,, in Fig. 23 are in good agreement with the total
times for the calorimetric glass transition upon linear heating with rate R. From
differential scanning calorimetry the latter are defined by (T,"" - Tee )/ R [29].
Note that T, is proportional but not equal to the relaxation time as defined within
Maxwell’s model of viscoelasticity [41, 42]. From the shear modulus of V1 of
G =33x10'°Pa [43] one expects the Maxwell time (n/G) for shear stress relaxation
after small deformations to be about 2 orders of magnitude faster than the internal
- equilibration time %. .
Recently, studies of Be interdiffusion and Ni, Co, and Al radiotracer diffusion in
deeply supercooled V1 were performed. Al diffusion in V1 should be only slightly
faster than Zr due to the similarity of their atomic radii. Tracer diffusion studies of Al
and Zr in V4 support this assumption [44]. Regardless of the exact diffusion
mechanism of the various elements, a time t, may be defined for the successful
temperature (K)
1200 1000 800 600
| 0° [1 4 i | i al ! |
time (s)
10 Prey rr be p rrr yp rr bb por borer yp Poe bp rs
0.8 1.0 1.2 1.4 1.6 1.8
1000 /T (1/K)
~ Fig.23 Time scales in the liquid and supercooled liquid of V1
Thermal equilibration times, %,, (+) according to Eq. (46) and the Cohen-Grest fit (—)
as well as times for the glass transition (¢) [29] and relaxation times (a) to reach the
equilibrium viscosity [16]. Diffusion times from interdiffusion of Be (x) and Au (A)
and tracer diffusion of Ni (©), Co (V), and Al (9), (---), see Eq. (47) and Eq. (48).
a | 59.
a —_
displacement of a tracer atom by a distance | = V1 * . For the i-th tracer, the Einstein
equation for random walk reads: _
Tpi = . (47)
The times, T,;, correspond to the mean displacement of an average atomic diameter,
I. Note that this definition of 7,, does not necessarily imply a jump mechanism.
From the atomic volume, v,,, / is estimated to be 3.2 x 107°m and the corresponding
Tp; are added in Fig 23. Around 600 K the diffusivities of Al and Ni differ by about
3 orders of magnitude, while they show a tendency to merge at higher temperatures.
The Al diffusion shows temperature dependence similar to the internal relaxation time:
Toa = 714X107 T,. (48)
The proportionality factor implies that the mean displacement of an Al (Zr) atom in
the supercooled liquid during a typical relaxation time is (7.14 x 107) ? J or roughly 4
interatomic diameters.
Ni and Co show significantly smaller absolute values of Tt, and activation
energies of about 2.0eV to 2.2eV. Clearly, the time scales for viscous flow and
- diffusion of medium-sized atoms have different temperature dependencies in the
deeply supercooled liquid state of V1. A collective hopping mechanism is likely to
control the diffusion of Ni and Co in this temperature range as this could be confirmed
for Co in V4 by isotope effect measurements [45]. Similarly, one expects the
migration of the smallest atom in the alloy, Be, to be controlled by thermally activated
jumps on a potential energy surface that fluctuates on a time scale given by T,.
ce 60.
: - Guided by'the differences among the diffusion coefficients for various elements at
low temperatures on the one hand and the scaling between viscosity and diffusion in
the equilibrium liquid, a hybrid equation for the diffusion times, T, , of medium sized
atoms in V1. is considered. Within this model, the time for a successful displacement,
l, of, e.g., Ni can either be limited by viscous flow or by a solid-like jump, where
Eq. (4) takes into account that both processes may occur parallel:
| = $m + I exe{ 2). (49)
Ton = T_T io kT
_At high temperatures Ty, is limited by the first term in Eq. (4), i.e., by the time scale
for changes in the surrounding matrix, t,. The value of g,, =14 (see above) may
serve as a lower limit and a first estimate for g,; as one does not expect Ni diffusion
to be faster than Au near the liquidus (see Fig 22). The corresponding times, Tpx;,
are plotted in Fig 23 with Q,, =21eV. At sufficiently low temperatures, below about
800 K, thermally activated jumps of Ni with an average activation energy Q,, can
occur as the potential energy barriers for Ni diffusion become fixed for the time
between two successive jumps,. | Both processes in Eq. (49) act in parallel in a
relatively wide intermediate temperature range between about 700 K to 800 K, which
can be regarded as the glass transition regime for Ni diffusion. This temperature range
for the change in the atomic transport mechanism is independent of the time scale of
the experiment but differs among the various components in the alloy. In fact, for the
fast diffusing species (Ni, Cu, Be) in the alloy one expects a cascade of transitions.
The jumps of a medium sized atom like Ni in the dense packed supercooled liquid will
7 61
| probably also affect the positions of the fastest diffusor, Be. Description of the Be
diffusion is therefore particularly difficult as its motion is affected by the collective
movement of all other elements in the cascade of transitions.
Ae
5. CRYSTALLIZATION OF THE SUPERCOOLED LIQUID
The ability to form metallic glass by cooling from an equilibrium liquid is
equivalent to suppressing crystallization within (certain regions of) the supercooled
liquid state of the system. Consequently, experimental crystallization studies are a
powerful tool for understanding the glass forming ability of metallic systems.
Monatomic or binary metals may be supercooled - in some cases up to several
100 K - below their melting points by containerless processing or fluxing techniques
but critical cooling rates to form a glass are typically on the order of 10’ to 10K".
To characterize the stability with respect to crystallization of a conventional
monatomic or binary metallic systems in its supercooled liquid state, oftentimes only
the minimum crystallization temperature, T,, upon linear undercooling below the
liquidus temperature is cited [46, 47].
Recently, crystallization studies [48, 49, 50] in the deeply supercooled liquid of
multicomponent BMG forming alloys have found considerable attention. The
experiments are made possible by the alloys’ increased stability with respect to
crystallization.
Most of the experiments have been performed by heating amorphous samples
with a rate on the order of 1 K/s through the glass transition region into the
supercooled liquid state [48, 51, 52]. This method is denoted by ‘path A’ in Fig 24.
Commercially available calorimeters are well suited for these studies and results are
found to be independent of the crucible material used. Only few data are available for
thermophysical properties in the equilibrium or slightly supercooled liquid of BMG
-T,
Iq
5 0.8 supercooled liquid
© 5
QO } 3 4fL___iii ie Ape ----- +--+ -------- +--+ +--+ - === --
5 0.6 + glass transition region
o4b glass
0.2
time
Fig. 24 . Isothermal crystallization studies of the supercooled liquid state
Path A: Heating of amorphous samples, prepared, e.g., by melt quenching through the
glass transition region. Path B: supercooling of the liquid below the liquidus
- temperature.
"forming alloys. The high affinity of the transition metal alloys for impurities like
oxygen poses severe challenges for experimental studies at high temperatures.
However, the temperature range between the liquidus temperature, 7jjg, and the
temperature. with the minimum time to crystallization, T;, is decisive for the glass
forming ability of the system [53, 54] as will be discussed below.
In this work, isothermal anneals of the supercooled liquid along path B in Fig 24
are performed in which the equilibrium liquid serves as the ‘as-prepared’ state
compared to the amorphous solid in path A. Experiments in the supercooled liquid are
_ oftentimes performed under non-isothermal conditions, i.e., either cooling from the
equilibrium liquid or heating from the amorphous solid state. Subsequent isothermal
anneals of the supercooled liquid, schematically shown in Fig 24, are possible only for
relatively good glass formers, that allow for an experimental time window of at least
some 10 seconds before the onset of crystallization. The good glass forming ability of
V1, represented by a critical cooling rate of 1 K/s [55], allows for these isothermal
experiments.
The critical cooling rate for glass formation, as used in the present work, is
defined as the minimum rate at which an equilibrium liquid has to be cooled to keep
. the crystalline volume fraction to a small but finite value, e.g., 10°. In this sense, an
amorphous solid may contain a small amount of crystals. Whether a finite volume of
an amorphous sample is completely crystalline-free or whether it contains a small
volume fraction of crystals depends on the details of the crystallization process, e.g.,
the nucleation and growth rates. In most metals the presence of a single crystalline
nucleus may trigger the crystallization of the entire sample because of the large growth
HF 65:
. “velocities of the crystals in the supercooled metallic liquid. In good glass formers like
commercial soda-lime-silica glasses, on the other hand, unwanted crystalline
imputities have a marginal effect on the glass forming ability because the growth
velocity of the crystalline phase(s) is small. The growth velocity will play a central
role in the interpretation of the experimental results presented in this chapter.
Of particular concern in the study of supercooled liquids is the liquidus
temperature of the alloy defining the boundary between the equilibrium and the
supercooled liquid. While in principal well defined within the Zr-Ti-Cu-Ni-Be phase
diagram, this transition is difficult to detect experimentally. Differential
thermoanalysis (DTA) scans with heating rates of 20 K/min on as-quenched V1 are
shown in Fig 25. For this experiment, a Setaram DSC 2000 was equipped with
graphite crucibles machined from POCO DFP-I graphite. The He (99.9999%) purge
gas was purified using a heated titanium getter type Matheson 830] and the instrument
was calibrated using the melting points of pure Al (99.9999%) and Ag (99.99%).
The first run (solid line in Fig 25) depicts the glass transition region from ca.
625 K to 655 K and at least four exothermic events from 716 K to 810 K. In the past,
much attention has been given to the onset of crystallization from the amorphous state
. of V1 in the temperature range from ca. 600 K to 750 K [29, 48, 51, 56]. At these
(low) temperatures, metal or ceramic containers may be used to anneal the material in
conventional furnaces. In contrast, experimental studies in the vicinity of the liquidus
temperature require either levitation techniques or the use of compatible crucible
materials. A series of endothermic peaks from 935 K to 1035 K in Fig 25 indicates
the melting of crystalline phases.
150
100
pe)
© 50
ay
z 0
oC
oO .
= -50
is
(@)
Oo
S -100 +
-150 1 | l | i | i | 1 | L i
500 600 700 800 900 1000 1100
temperature (K)
Fig. 25 Differential thermoanalysis upon linear heating
_ Heating of as-quenched, amorphous V1 (—) and slowly (0.1 K/s) cooled, crystallized
V1 (-- ) in a Setaram DSC 2000 differential thermal analyzer with heating rates of
0.33 K/s. Both scans have been corrected by the baseline obtained from empty
crucibles.
fT
- Subsequenily, the sample is slowly cooled to room temperature (ca. 0.1 K/s) and
then the second DTA scan (dotted curve) is recorded upon reheating. The absence of
the glass transition and the exothermic crystallization events shows that the sample
completely crystallized upon cooling from the liquid state after the first run. The
endothermic melting signal in the second DTA trace has a different shape compared to
the first run. In particular, it is somewhat sharper and the highest peak (1035 K) can
no longer be resolved. The slow cooling after the first run apparently resulted in a
different crystalline microstructure compared to the crystals that formed upon heating
_from the amorphous state. As only the relative heights of peaks up to 1026 K shift, the
liquidus temperature was taken to be 7, =1026K. The good stability of the DTA
baselines for T > 1050 K, obtained by using high purity graphite instead of alumina
crucibles, leads to a slightly higher liquidus temperature than previously reported [29].
Cooling from the equilibrium liquid of V1 to room temperature with rates as low
as 0.9 K/s to 1.2 K/s is sufficient to form metallic glass [57]. In Fig 26 time-
temperature profiles with a slightly higher cooling rate of about 2 K/s are shown. For
these experiments, the shear cell in the apparatus that was described in chapter 2.1 was
exchanged with graphite crucibles of 4mm diameter, containing ca. 140 mg of V1.
- After annealing the samples in vacuum (107 Pa) at the indicated temperatures for
360 s, the RF power supply was switched off and the cooling by radiation and
conduction via the crucible mount is monitored. The recalescence upon crystallization
can be best seen in the derivative of the time-temperature signal, shown for two
cooling curves in the inset of Fig 26. The endothermic heat signal reduces the cooling
rate by about 0.2 K/s for ca. 30s. The shape of the peak in the cooling rate curve
‘ . . UJ qv U v | v v v T if LU T
. .
1200 Fs. a ff 7
. ny
= . . =
ae Q
*. . . *. {as}
‘ . . ‘ =
M00R RNB 4
‘. ‘5's ‘ ‘ ‘ 8 2 +
temperature (K)
| °
oO
oO
ice)
(o>)
800
Fig.26 Cooling curves from the equilibrium liquid state
Cooling of V1 contained in graphite crucibles from various temperatures.
Recalescence events ( . ) observed in the temperature-time profiles are indicated. The
inset shows the (negative) derivatives of the temperature with respect to time, i.e., the
cooling rate. The curves 1-10 correspond to starting temperatures of 1300 K, 1250 K,
1200 K, 1150 K, 1100 K, 1075 K, 1050 K, 1000 K, 950 K, and 900 K. Experiments
1-6 (---) resulted in amorphous and 7-10 (—) in crystalline material.
. “depends. on the*size and geometry of the crucible used and the amount of V1 used. In
addition, the wetting of the (liquid) sample on the graphite is crucial for a precise time
and therefore temperature measurement of the onset of crystalliztion upon cooling. As
this wetting is not achieved with, e.g., Zn, Al, Ag, or Au, a reproducible calibration of
the peak area to the amount of the heat released by the sample was not possible.
In Fig 26 crystallization is not observed upon cooling from initial temperatures
exceeding 1050 K. This temperature is only slightly higher than the liquidus
temperature measured upon heating in the DTA. To form metallic glass by cooling
from the liquid, the sample has been superheated above its liquidus temperature by
only 30 K.
While the increased stability of V1 compared to simple metallic systems makes a
detailed experimental analysis of the temperature and rate dependence of
crystallization from the supercooled liquid possible, characterization of the primary
crystalline phase and the thermodynamic driving force is complicated in this
multicomponent system. The basic crystallographic structure of the primary phases
that form in the Zr-Ti-Cu-Ni-Be system is known [58], but a detailed thermophysical
analysis is not yet available. From, e.g., heat capacity measurements, so far only the
. specific Gibbs free energy difference, AG, between the liquid and a crystalline phase
mixture is known (see Fig 27). The interpretation of AG as the driving force for the
crystallization is valid only in the case of a polymorphic transformation. Similarly,
one usually has to use a crystalline phase mixture of V1 as a reference state for
specific volume measurements [25].
3x1 0° qT 1 | qT T T | T T T qT if T v T 1 | qT v T T ] 7 qT
2x10°- 4
2 -_ -_
O / ]
a :
1x10 7 7
600 700 800 900 1000
temperature (K)
Fig. 27 Driving force for crystallization
_ Specific Gibbs free energy difference between the supercooled liquid and the
crystalline state (—) as obtained by integrating Eq. (50) with the experimental results
for the specific heat capacity difference, Ac, given by Eq. (30) and Eg. (31). Linear
approximation (---), Eq. (51), used in the classical nucleation and growth model.
7 The. specific Gibbs. free energy difference, AG, between the liquid and the
crystalline phase results from differential scanning calorimetry (DSC) [29], yielding a
lower limit for the driving force for primary crystallization. In calculating AG from
the measured specific heat capacities (Eq. 30 and Eq. 31) via
_ Tig sons 7 Tig Ac,(T ) ad
AG = AH, — AS;Ty, — i, Ac,(T) aT +T{ df, (50)
it will be sufficient to use a linear approximation for the temperature dependence,
Ti —T
AG = 7.992 Tig=7)__J (51)
Kg-atom’
for the following phenomenological discussion of chapters 5.1 and 5.2.
5.1. Crystallization: Isothermal
With the computer controlled induction furnace designed for the viscosity
measurements described in chapter 2, it is also possible to study the temperature
dependence of the onset of crystallization in the supercooled liquid state of V1. For
this purpose 4 mm diameter graphite crucibles containing ca. 0.05 em? of V1 replace
the concentric cylinder shear cell. The small crucibles allow for cooling rates of up to
40 Ks" to be applied to the liquid prior to the isothermal anneals. The recalescence
~ upon crystallization can be observed in the thermocouple signal. An example of a
measured time-temperature profile is shown in Fig 28 (a). The onset of crystallization
is best observed in the time derivative of the temperature reading, i.e., in the heating
rate curve shown in Fig 28 (b).
The logarithm of the onset times, t,, of the first recalescence event versus the
inverse annealing temperature are summarized in Fig 29. As the comparison to results
(a)
temperature (K)
oO
l [ lL
| | | Litt l Ltt | l
rreerprerrrprrrryurrrygrrryrprrrryprrarryerey
200 300 400 500 600 700 800 900 1000
TUeTTyT erry rrrryprrrryprrrrypreenrprrerrprreris
0.004 (py 7
— foo bp crystallization 4
S$ fre oe 1
° 8 7
0.00218 .2% 68 J
_ I> oS oe 7
Ke 7 0022 © g, abe 2 1
~ FT ao 0 0, 280 ° 8 . SP od
4 “Po 90 ° ° ° Po ) me & So ° g
k- am ee @ ro Bo & Fo 0 COP @ RURAL
% 0.000 2 Base ae ‘ ota , “Eg? aoe a
4 4° % QB go BERR, 6 %~ “s ost
: ° ° ep 6 5 o' "
-0.0024 - g _
rrierprrveryprvrrprrrryprerryprvrryprvvry ere
200 300 400 500 600 700 800 900 1000
time (s)
Fig. 28 Time-temperature profile for isothermal anneals of supercooled V1
_ (a) Temperature, T, of a 4mm diameter graphite crucible containing 86 mg V1 as a
function of the annealing time, t. (b) Derivative, dT/dt, showing the temperature rise
due to the heat release upon crystallization.
| temperature (K)
11001000 900 800 700 600
10° 2 T T vt T T T T i T T T F
; / J
10° q z
10° q 3
_ 7
J glass transition |
10° 4 (1K/s) 4
fig :
J T. A
10° T y T T T T T
1.0 1.2 1.4 1.6
1000 /T (1/K)
Fig. 29 Time-temperature-transformation (TTT) diagram
Primary crystallization from the supercooled liquid state of V1 contained in graphite
crucibles (0) and processed by ESL (0). Least-squares fits of Eq. (58) with
x = 0.001 (—). The contribution from the viscous (---) and from the Arrhenius
term (---) in Eq. (58) are indicated.
i 74
from electrostatic levitation experiments [55] shows, the graphite containers do not
appreciably alter the crystallization behavior in comparison to containerless processing
[59]. In spite of the presence of the crystalline crucible walls, the transformation
kinetics of the supercooled liquid are unchanged within the resolution of the
experiment. Obviously, the crystallization of the bulk sample is not triggered by
surface crystallization at the container interface.
The experimental data in Fig 29 show a minimum at T, = 895 K. This behavior
is well known for a large variety of materials [2] and arises from the competition
between increasing thermodynamic driving force for crystallization and decreasing
atomic mobility upon supercooling. The shortest times to crystallization are as large
as 60 s at this temperature, reflecting a critical cooling rate for glass formation of only
1 K/s. This rate is at least 6 orders of magnitude larger than values estimated for
simple metallic systems. At the temperature 7, the viscosity of V1 is about 4 orders of
magnitude larger than the viscosity of, e.g., Ni at a similar degree of supercooling (see
Fig 10). Apparently, the sluggish kinetics at elevated temperatures discussed above
acccount for a significant part of V1’s exceptional glass forming ability.
Classical nucleation theory is often used to quantify the time and temperature
- dependence of crystallization from a supercooled liquid [50]. In its simplest form, the
steady-state nucleation rate,
I, =ADwy eal - a ) (52)
is written as a product of an effective diffusivity, D.,., and a thermodynamic
Boltzmann factor to overcome a nucleation barrier for crystal formation. For
75:
| homogeneous nucleation of Spherical particles [2], the latter includes the Gibbs free
energy,
._ l6z0°
AG’ =———,
3AG
(53)
to form a critical nucleus. The exact form of the specific Gibbs free energy difference,
AG(T), between the liquid and the crystal and the values for o in Eq. (53) are of
particular importance for the shape of the TTT diagram at the high temperature end,
T > Ty. The bottleneck for nucleation at low temperatures, T < Ty, is the kinetic factor,
Derr, in Eq. (52).
The interfacial energy, o, is treated as a fit parameter in Eq. (53). A temperature
dependent interfacial energy can be included [60], but for simplicity it is omitted in the
following discussion.
The growth velocity, u, of the crystalline phase can, to a first approximation, be
expressed as a product of a kinetic and a thermodynamic factor:
_ SDs). (- Vin =|
— f exp iw . (54)
An effective diffusivity, D,,, as above is used and the fraction of sites at the liquid-
crystal interface where atoms are preferentially added or removed is taken to be f=1.
The crystallized volume fraction, x, will depend on both the nucleation rate and
the growth velocity. In general, the nucleation rate and the growth velocity can be
both time- and temperature-dependent. Considering three-dimensional growth and the
steady-state nucleation rate from Eq. (52), the time-dependent ‘extended’ volume
pO T6
fraction, x., of crystallized material is obtained by integrating over all nucleation
events:
f t 3
== j 7.2) f WT’) dt" dt. (55)
The double time integral sums over all nucleations events, occurring at time, T, and
takes into account the crystal growth from time, T, to time, ¢. In the derivation of the
quantity x. in Eq. (55), it is assumed that all growth centers continue to grow
unimpeded for all times, t, giving rise to the term ‘extended’ crystallized volume
| fraction, xe. For isothermal anneals and steady state nucleation, Eq. (52) and Eq. (54)
inserted in Bq. (55) yield with a simple integration [53]:
x, = sh “tt. (56)
Consequently, the time, t,, to crystallize a small volume fraction, x, can be written as:
t= (24:,) . (57)
Except for the effective diffusivity, D
e!
» one is left with A and o as the only two fit
parameters in Eq. (57).
In order to simulate the minimum in the nucleation time at 895 K in Fig 29 with
Eq. (57) and the commonly used relation, D,, ~ 7', the parameter o = 0.040 Jm”
has to be chosen. The relatively high temperature of the ‘nose’ in the TTT diagram
implies a rather small interfacial energy compared to values of 0.06 Jm” to 0.3 Jm”
reported for monatomic liquid metals [46]. The smaller the interfacial energy, o, the
smaller the nucleation barrier in Eq. (53). This suggests that the good glass forming
Po - 77 -
| ability of V1 is not based on-the interfacial properties between the supercooled liquid
and the nucleating crystals.
_ In Fig 29, for temperature T < 800 K, the logarithm of the onset of crystallization
is approximately proportional to the inverse temperature: log(t, ) ~1/T. In this
temperature range, the thermodynamic driving force, AG, has virtually no influence on
the slope of the log(t, ) versus 1/T plot, because the kinetic factor, Derr, dominates in
Eq. (52) and Eq. (54). Therefore, Eq. (57) yields t, ~ Deg, Which implies an
Arrhenius-like effective diffusivity: D,, ~ exp(—Q.,/kT) for the low temperature
_ part of the TTT diagram.
The effective activation energy, Qe, in Fig29 is on the order of leV,
substantially smaller than the apparent activation energies for viscous flow of more
than 3 eV shown in Fig 21. All viscosity and crystallization data were obtained on
samples in their supercooled liquid state and the failure of the viscosity to describe the
onset times of crystallization can therefore not be connected with the calorimetric
glass transition [61]. |
The temperature dependence of the tracer diffusion coefficients of medium-sized
_ atoms was discussed in chapter 4.2. A hybrid equation, Eq. (49), for the diffusivity
was proposed that combined the contributions from viscous flow at high temperature
with thermally activated hopping at low temperatures within the supercooled liquid. It
is tempting to use an expression similar to Eq. (49) for the effective diffusivity, Deer.
However, the effective diffusivities and activation energies used to describe kinetics at
liquid-solid interfaces during crystallization in a multicomponent system are not
i 78
equivalent to (bulk) tracer diffusion coefficients. The collective transport mechanisms
that were proposed to control atomic transport in the bulk of the Zr-based glass
forming alloys [45] are likely to be somewhat altered near a crystalline cluster.
From Eq. (56) one finds x, ~ Di, and with the proposed hybrid equation, the
temperature and time dependence of x, takes the form:
0.) =P T| +B exp -2)]
x(T,t) =F | +P, exe iT ) ves
v,, AG l6zo°
-|1-en- UT ) cxf - 3AG? t*. (58)
As in Eq. (49), the kinetic term (first line) in Eq. (58) is an additive superposition
of a viscous and an Arrhenius term. For simplicity, a VFT relation may be used in
Eq. (58) for the temperature dependence of the viscosity instead of the Cohen-Grest
formula.
In Fig 29 the crystallization times were fitted with the aid of Eq. (58) solved for
the time, ¢ (solid curve). The contributions from the viscous term (dashed curve) and
the Arrhenius term (dotted curve) in Ea. (58) are indicated in Fig 29. The large
apparent activation energies for viscous flow at low temperatures overestimate the
- temperature dependence of the crystallization times. The assumption that D,, ~ 7”
is therefore not valid in the vicinity of the glass transition temperature.
The fit of the hybrid equation, Eq. (58), to the data in Fig 29 yields the nucleation
rate, I(T), and the crystal growth velocity, u(T). Both curves are shown in Fig 30.
Again the contributions from the viscous and the Arrhenius term are indicated. One
notes that the maximum of the nucleation rate is located close to the temperature, T,,,
0 Le 5
~~ 7 ——_
= 4 Nn
” ° °
E | o£
— —
2 | g
© ©
o 7% ¢
> 4 2
=) - ©
ae o
7 Oo
© , 5
2 =
cs) , 8
-20 re f T 7 Prerrryert Tryp ry rrr yt -15
500 600 700 800 900 1000
temperature (K)
Fig. 30 Nucleation rates and growth velocities in supercooled V1
_ Left axis: Growth velocity, u, calculated according to Eq. ( 54) with a maximum at
985 K. Right axis: nucleation rate as calculated according to Eq. (52).
ro | 80.
“with the mininium time to crystallization. In contrast, the growth velocity has its
maximum at a significantly higher temperature of 985 K. The growth velocity will
». playa central role in the discussion of the critical cooling and heating rates to bypass
crystallization, presented in the following chapter.
It was assumed in the derivation of Eq. (57) and Eq. (58) that the volume fraction,
Xe, is much smaller than unity, i.e., that Eq. (57) is only applicable to the early stages
of crystallization. With increasing volume fraction of the crystalline phase, individual
crystals will impinge each other and the rate of crystalline growth will decrease as x
approaches unity. Frequently, the later stages of crystallization are analyzed according
to the formalism proposed by Johnson, Mehl, Avrami, and Kolmogorov (JMAK) [62].
The transformed volume fraction, x, is written as
x =1-exp[-x,()], (59)
where x is given by Eq. (56) in the case of isothermal anneals or, more generally, by
Eq. (55). The crystallized volume fraction, x, according to the JMAK formalism of
Eg. (59), is plotted in Fig 31 (a) as a function of temperature and the logarithm of the
time elapsed. Although the results in Fig31 are based on the assumption of a
continuous transformation from the liquid to the crystalline state (x, ~ t*), most of the
~ volume of the supercooled liquid crystallizes in a fairly narrow time interval for a
given temperature.
The rate, dx/dt, of the volume fraction crystallized per unit time, plotted in
Fig 31 (b), is approximately equal to the heat release observed in an isothermal
calorimetric crystallization experiment. One notes that at each temperature, T < Tig, a
well defined crystallization peak arises after an apparent incubation time, even under
@.
900
1000
(b)
Fig. 31 — Crystallized volume fraction and rate of transformation
‘(a) Crystallized volume fraction, x, of V1 according to Eq. (59) versus temperature
and the logarithm of the time elapsed. (b) Crystallization rate, dx/dt, as a function of
the temperature and the logarithm of the time.
p82
| ‘the assumption ‘of steady state nucleation and growth under isothermal conditions, i.e.,
Xe - t* in Eq. (58), The height of the crystallization peak, dx/dt, in Fig 31 shows a
maximum in the vicinity of the temperature T,. The latent heat of crystallization is
released in a-‘minimum amount of time near T,. Thermoanalysis studies at significantly
lower or higher temperatures than 7, result in crystallization peaks that are more
spread out in time and therefore smaller in magnitude (see, e.g., Fig 28).
On the basis of the present experimental data. it is yet unclear to what extent the
assumption of a steady state nucleation rate has to be modified for the present system.
‘Results from microstructure analysis indicate that the nucleation rate used above
serves as.a lower limit and that transient nucleation events are partly responsible for
the long nucleation times, i.e., for the good glass forming ability of the alloy [58].
5.2. Crystallization: Linear Cooling and Heating
One of the central quantities in theoretical and experimental studies on glass
formation is the critical cooling rate, Ror, to bypass crystallization upon cooling from
the equilibrium liquid [2]. It is commonly used to quantify the glass forming ability of
a liquid. |
The practical importance of the critical cooling rate stems from the fact that it is
linked to the maximum thickness, dmax, of a quenched amorphous ingot. The size of
the smallest dimension of the ingot is limited by the requirement that heat must be
conducted from the center of the sample to its surface. In the case of perfect heat
extraction from the surface, dmax is (up to a factor of order unity) given by [63]
IR
K,;, M,. fi
d wax ir liq mol ~ liq ; (60)
‘c,cr Co stig p liq
In Eq. (60), Kijiq is the thermal conductivity (W mK’), Ptiq the density (kg m”°), and
Cp.iq the specific heat capacity (W g-atom! K') of the liquid. Mmor is the alloys molar
mass (kg g-atom’'). With typical values for the Zr-based metallic glass forming
mo}
systems af K,,=10Wm'K", M,,,=0.06kgg-atom", 7,, =1000K,
Cotiq = 40 J g- atom" K", and p,, = 6000 kgm”, Eq. (60) yields:
fn 0.05 R, . (61)
Ks"
With a critical cooling rate of ca. 1K/s this results in a maximum thickness of
ca. 50 mm for V1 and with R, = 9 K/s of ca. 18 mm for V4. These dimensions agree
well with the maximum size ingots prepared by quenching the liquid in copper molds.
In Fig 32 the experimentally observed critical cooling rate and the onset
temperatures of crystallization for smaller cooling rates are shown in the form of a
semi-logarithmic time-temperature plot. As for the isothermal crystallization studies,
the liquid metal was contained in graphite crucibles heated inside the computer-
controlled induction furnace described in chapter 2.
The. continuous-cooling-transformation (CCT) diagram, Fig 32, shows that the
decisive temperature range for glass formation is between ca. 850 K and the liquidus
temperature. Once the liquid is successfully supercooled below ca. 850K with a
constant cooling rate, the crystallization kinetics have practically become slow enough
for the sample to reach the amorphous state.
1100 T TTT U Perengy t UL) | UJ PL | q Urn La LL
=<
temperature (K)
try Lay pp Lia tp pt
500 UL TTT TTT TTT TT arti rm
10" 10° 10' 10° 10° 10° 10°
log (time/s )
Fig. 32 Continuous-cooling-transformation (CCT) diagram
_ Linear cooling experiments from the equilibrium liquid of VI. Experimentally
observed onsets of crystallization (0) with varying cooling rate. Time temperature
profile for the critical cooling rate is indicated (—). Continuous cooling
transformation (CCT) diagram as calculated from Eq. (55) for x = 0.001 (---). The
average structural equilibration times marking the glass transition region are also
indicated (---).
; 85.
: ; The crystallized volume fraction during linear cooling with cooling rate, a, can be
calculated with the aid of Eq. (55). For this purpose the time-temperature profile,
- T(t) =T;, —&t, (62)
is inserted into Eq. (55) together with the nucleation rate, J;, and the growth velocity,
u, from Fig 30. The integrals in Eq. (55) can be solved numerically, lowering the
temperature from the liquidus temperature in steps of 1 K/s of duration 1 K/a. For
each temperature step the crystallized volume fraction, x, is calculated according to
Eq. (55).
The times, t,, where the numerical solution of the integrals yields x = 0.001 have
been calculated for various cooling rates, a, and are indicated in Fig 32 (dotted curve).
The agreement between this calculated CCT diagram and the experimental data in
Fig 32 is satisfactory - the critical cooling rate, e.g., predicted by the calculations is
only a factor of two larger than the experimentally observed rate.
With the present setup, it is possible to study the crystallization behavior of the
supercooled liquid upon heating with rates up to 200 K/s. An example for heating an
8 mm diameter graphite crucible in conjunction with a 5-turn, 42 diameter RF coil at
70% power setting of the LEPEL RF power supply is shown in Fig 33 (a). To achieve
' the highest accuracy in the temperature reading a Labview VI without a PID control
algorithm was used and the thermocouple voltages were written to the hard drive only
after the experiment had finished. With this configuration, it is possible to sample the
thermocouple voltage with a sampling rate of 6000 Hz. Upon completion of the
temperature scan, digital data smoothing was performed by averaging 100 adjacent
temperature readings, yielding an effective scan rate of 60 Hz.
Lititytrp tit.
0 10 20 30 40
time (s)
am 1 20 . | TT et | Trt 1 Ld ee | | J ao 35 a Xap. if TT gt | Tri? | tt
‘wh 4 @ , (b) J ‘hn 7 a (c) 7
< 4 Pee, “ystallization 4 < 304 Re 4
[o) 1 00 7 \ ee o) en 7
© . 4 © 95 -J "Ren 4
+ ‘| glass ae “ ; ee 4
2 80 4 transition 4 ™® 2 20 glass J
o J melting 9 “J transition a
< , oO J J
60 | Le | a | Tr Lo eee | } qv 1 5 Lm 1 rree | rere | Ll
600 800 1000 1200 800 700 600
temperature (K) temperature (K)
Fig. 33 Time-temperature profile of a fast heating and cooling experiment
Time-temperature profile (a) as measured by a thermocouple inserted in the graphite
crucible wall.. Heating rate (b) and cooling rate (c) as obtained by differentiation
showing the glass transition, crystallization and melting of the sample.
co ; 87.
. The differentiated time-temperature curve upon heating, Fig 33 ©), shows the
glass transition, crystallization, and melting upon heating. At the indicated heating
_ Tate of ca. 60 Kis, the temperature difference, AT,, between the midpoint of the glass
transition and the onset of crystallization is as large as 205 K. This is roughly double
the temperature range achieved with conventional heating rates on the order of
0.1 K/s.
In Fig 34 heating rate curves with rates up to ca. 300 K/s for both amorphous and
crystallized samples are shown. At the highest heating rate, the onset of crystallization
of the supercooled liquid and subsequent melting is no longer detectable. For
comparison, the heating rate curves of crystallized V1 samples are also shown in
Fig 34. Here, the melting of the crystalline phases can clearly be identified in the
differentiated temperature signal for all heating rates. The times and temperatures of
the onset of crystallization are summarized in Fig 35 together with previously
published data from crystallization studies using a Perkin Elmer DSC 7 [29]. The
supercooled liquid is slightly more stable against crystallization when processed in the
high purity graphite crucibles. At heating rates on the order of 0.1 K/s, the
temperature interval, AT;, is larger by ca. 30 K compared to the published results [29].
- The question whether this effect is due to the incorporation of a small amount of
carbon into the alloy [64] or a consequence of the higher purity of the processing
environment in the present setup is the subject of further investigations [65].
The stability of the supercooled liquid upon heating above the glass transition
region is oftentimes characterized by the size of the temperature interval,
'* ©
co)
100 - ~
heating rate (K s')
0 v | UJ ] qv | qT
600 800 1000 1200 1400
temperature (K)
Fig. 34 Thermoanalysis of V1 upon fast heating
Heating rates, dT/dt, as determined from measured time-temperature profiles. The
solid curves (—) represent experiments starting with amorphous V1 and the dotted
curves (---) are heating rate curves starting with completely crystallized samples.
< ev 4
ev a
2 A y 7
> de
© O eA Vv i
® ° Mm yy -
2. 0 i
= fe) J
® “Hh - {@) 7
_ a
~-H._ =
a ae
T TTT] TV TTiiny T ory ay
10" 10° 10" 10° 10° 10° 10°
log (time/s )
Fig. 35 Continuous-heating-transformation (CHT) diagram
_ Onset of crystallization (¢) as measured with the present setup and published results
(°,A,V) for the three main exothermic events in DSC scans as well as the time-
temperature profile (—) for the critical heating rate of 200 K/s. Average times for the
glass transition region are also plotted (m).
a 90
AT, (R,) = T,(R,)~T,(R,); (63)
between the onset of the glass transition and onset of the first crystallization event.
The notion of the temperatures, Tz, T;, and AT; in Eq. (63), is only meaningful in
conjunction with a heating rate, Ry. |
Usually, the temperature range that is decisive for the glass forming ability of the
material is not accessible by heating amorphous metallic samples through the glass
transition region... The width of the supercooled liquid region upon heating, AT,, is
therefore only a crude and indirect measure of the glass forming ability of the system.
The present experimental results for the onset times of crystallization in Fig 32
and Fig 35 show a difference between the critical cooling and the critical heating rate
of about two orders of magnitude. The time dependence of the crystallized volume
fraction according to Eq. (55) can qualitatively explain this difference between the
critical cooling and the critical heating rate as described in the following.
Starting with x=0 at T=Tjjq, the extended crystallized volume fraction for
successive temperature steps of one Kelvin of duration 1 K/a are computed via
Eq. (55). The volume fraction, x, is easily computed via Eq. (59) and the results are
plotted in Fig 36 (curve a). Upon cooling from the liquidus temperature, x increases
continuously and becomes approximately constant at 600K. Note the logarithmic
scale of the volume fraction from 10° to 1. The simulation is terminated at 450 K.
This ‘freezing’ of the crystallization kinetics can be.attributed to the strong decrease in
Dee in Eq. (52) and Eq. (54), hindering further nucleation and growth of crystals. The
total crystallized volume fraction of 0.035% for the simulated cooling curve in Fig 36
will not be observed in the thermocouple signal of the experimental setup used. Also,
10".5 7
¢ 10°7 4
© 10°7 7
@M *
€ 10° 4
S 10° a
Oo 4
4g 6
= 10 -
a 1 i
S107 4
© i a”
10" 7 ;
10° To ttt ttt
500 600 700 800 900 1000
temperature (K)
Fig. 36 Crystallized volume fraction for continuous cooling and heating
Cooling from the equilibrium melt with 1 K/s (a) resulting in a volume fraction of
"crystalline material of 3.52 x 107% and re-heating with 1 K/s (b). For comparison,
heating from a crystalline- free solid with I K/s (c). All curves are calculated with the
aid of Eq. (55) and Eq. (59).
. ‘such a small atiount of crystals will 50 undetected in a conventional x-ray diffraction
experiment. .
; The simulation is continued, now starting at 450 K with the crystal distribution
obtained during cooling. As can be seen in Fig 36 (curve b), the crystallized volume
fraction increases during heating as new nuclei are formed and the existing crystals
continue to grow. The temperature of 820 K at which x = 0.001 is indicated.
For comparison, a simulation starting with a perfectly amorphous sample, x = 0,
at 450 K is also shown in Fig 36 (curve c). The sample reaches x = 0.001 at 880 K.. In
this simulation, during heating from the purely amorphous state, the exact same
number of nuclei form as during cooling. This stems from the assumption of steady-
state nucleation. However, the nuclei formed are exposed to different growth upon
cooling and heating, respectively.
As shown in Fig 30, the growth rate calculated according to Eq. (54) has a
maximum at a relatively high temperature of 985 K within the supercooled liquid
range. Upon heating, nuclei forming at, e.g., the minimum in TTT diagram at 7,, will
be exposed to these high growth rates before reaching the liquidus temperature. In
contrast, the nuclei that form upon cooling at 7, will experience less favorable growth
. rates. Starting with a crystal-free amorphous solid, the volume fraction crystallized
upon heating to the liquidus temperature with a constant rate will be higher than the
volume fraction crystallized upon cooling with the same (negative) rate.
Mathematically, this difference results from the difference in integrating over the
growth rate, u, in Eq. (55) for heating and cooling respectively. In Fig 36 the sample
cooled with 1 K/s from Tig to 450 K contains 0.035% crystals, while the one heated
a 93.
: : from 450 K to Tig is 90% crystallized. if the effect of quenched-in crystals is taken
into account (see Fig 36), the difference is even more severe. The critical cooling rate
required to keep the average crystallized volume fraction to a certain (small) value is
therefore smaller than the corresponding critical heating rate. For the present system,
the calculations yield Re,cooling = 1 K/s and Repeating = 9 K/s, which are shown in Fig 32
and Fig 35, respectively.
5.3. Crystallization at constant shear rate
Shear induced changes in the microstructure have been experimentally observed
jin a variety of polymers as well as silica-based liquids [66, 67, 68] and have been the
subject of theoretical calculations [69, 70, 71]. Both shear-induced mixing as well as
shear-induced decomposition has been observed for various systems. It has been
suggested that in metallic systems the presence of a (high) shear rate should suppress
the formation of crystals in the supercooled liquid.
With the setup described in chapter 2, a constant shear rate may be applied to the
supercooled liquid while, simultaneously, heat releases during solidification are
recorded from the thermocouple signal. The viscosity of V1 as measured with the
concentric cylinder viscometer during linear cooling is shown in Fig 11. The onset of
solidification can be seen in the deviation from Newtonian behavior.
The time-temperature profiles recorded during viscosity measurements shown in
Fig 11 are plotted in Fig 37 (a). Initially, the cooling rate, shown in Fig 37 (b), is
0.66 K/s. With the deviation from Newtonian behavior, marked by open circles in
Fig 37 (a), a small decrease in the cooling rate, i.e., a small exothermic event, can be
1050
_ 1000 -
® 950-
© 900 -
QO. *
- € g50-
Qo.
800 Prreprerrryprrrereprrrryperrryprrrrprrrrprereryerry
600 650 700 750 800 850 900 950 1000 1050
4040 Fa we
0.8 -
0.6 ater
0.4 -
0.24
0.0 -
-0.2 -
cooling rate (K/s)
TrTeperrryprrerperenryprrerprrrryprereprerrpryer
600 650 700 750 800 850 900 950 1000 1050
time (s)
Fig. 37 Linear cooling curves at constant shear rate
Temperature (a) and cooling rate (b) during cooling from the equilibrium melt of V1
under constant shear rate. The curves 1-4 correspond to shear rates shown in Fig. 11.
The onset of deviations from the equilibrium viscosity ( 0 ) and the onset (¢) and
endpoint (") of the main crystallization peak are indicated. The rotation of the inner
cylinder was stopped manually (2). The cooling rate data in (b) were smoothed (—)
to guide the eye.
ot 95.
: “detected in Fig’37 (b). The sharp increase in viscosity in Fig 11 by over one order of
magnitude occurs prior to the onset of the main recalescence, as shown in Fig 37.
A similar behavior of V1 was already observed at temperatures in the vicinity of
the glass transition region [16]. It was suggested that the supercooled liquid phase
separates into two liquids prior to the onset of crystallization. The strong increase in
viscosity is likely to arise from the continuous phase having a higher viscosity than the
initial supercooled liquid. Long relaxation times of the viscosity to its new
equilibrium value after the phase separation support this view.
Whether the precursor to crystallization at high temperatures, shown in Fig 11 and
Fig 37, is also connected with phase separation in the supercooled liquid is yet
unknown. In experiments in the vicinity of the glass transition temperature, it is
possible to quench the supercooled liquid during the onset of crystallization in to room
temperature without significant changes in the microstructure during the quench.
Subsequently, the microstructure can be investigated with standard techniques. In
contrast, cooling the liquid from temperatures Tz 850K during the onset of
crystallization with cooling rates of less than 100 K/s resulted in completely
crystallized material. The mechanism for primary crystallization at elevated
. temperatures is not yet understood in full detail.
The shear rate dependence of the onset of deviations from the equilibrium
viscosity and the onset and endpoints of the main recalescence are summarized in
Fig 38. A larger shear rate leads to deviations from Newtonian flow at higher
temperatures. The position of the main crystallization peak remains unchanged within
the resolution of the experiments. One should note that the stepping motor was
oO = =
© |
& te—* —s ad
2 900 - 4
7~"—a = a
880 —- 7
0.0 0.1 0.2 0.3 0.4 0.5 0.6
shear rate (s’)
Fig. 38 Shear rate dependent solidification
Temperatures of onset of deviations from the equilibrium viscosity (©) as a function of
the applied strain rate for a cooling rate of 0.66 K/s. Corresponding onset (¢) and
endpoint (=) temperatures of the main recalescence event are also indicated.
‘switched off, ie, the shear flow was terminated (open squares in Fig 37 (a)), shortly
after the increase in viscosity in Fig 11 to protect the shear cell from damage. The
strain rates applied in the present experiments did not exceed 1 s' and are therefore
much smaller than the strain rates proposed to influence crystallization in, e.g., melt-
spinning.
The viscosity of the V1 bulk metallic glass forming alloy was measured over a
temperature range from 927 K to 1173 K in the equilibrium and supercooled liquid.
The viscosity at the liquidus temperature of 2.3 Pas is about three orders of magnitude
larger than the viscosity of any pure metallic liquid. The free volume theory as
formulated by Cohen and Grest describes the temperature dependence of the viscosity
of V1 over 14 orders in magnitude.
The temperature dependence of the viscosity is found to scale with the time scales
for diffusion of large atoms like Au or Al. The relaxation times obtained from the
viscosity measurements suggest that in the deeply supercooled liquid the diffusion of
small and medium sized atoms is governed by thermally activated jumps. A hybrid
equation is proposed that takes into account the displacement of atoms through
spontaneous fluctuations in the surrounding matrix as well as through thermally
activated processes.
Liquid V1 could be successfully supercooled inside high purity graphite crucibles.
The container walls are found to have no influence on the times to crystallization
under isothermal conditions. The critical cooling rate of V1 contained in graphite is
less than 2 K/s. The sluggish kinetics that are reflected in the high viscosity in the
supercooled liquid state contribute significantly to the good glass forming ability of
the alloy. |
The onset of crystallization under isothermal conditions as well as upon heating
from the amorphous state was studied in detail. The critical heating rate to bypass
"99
. crystallization ‘was measured to be 200 K/s and the difference between the critical
cooling and the critical heating rate can be qualitatively understood in the framework
of nucleation and growth. Deviations from classical nucleation and growth theory
found in rheological and crystallization studies at constant shear rate suggest that
changes in the morphology of the supercooled liquid of V1 occur as a precursor of
crystallization.
[1]
(2)
[3]
[4]
[5]
[6]
(71
[8]
(9]
[10]
(11)
[12]
[13]
[14]
[15]
[16]
‘100
Ae
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