Be diffusivity data in the bulk metallic glass forming alloys Zr_(41.2)Ti_(13.8)Cu_(12.5)Ni_(10)Be_(22.5) and Zr_(46.7)Ti_(8.3)Cu_(7.5)Ni_(10)Be_(27.5) are reported for temperatures between 530K and 710K, extending up to 80K into the supercooled liquid states of the alloys. At the glass transition temperature, T_g, a change in temperature dependence of the data is observed in both alloys, and above T_g the diffusivity increases faster with temperature than below. The data in the supercooled liquid can be described by a modified Arrhenius expression containing the communal entropy of the supercooled liquid and based on a diffusion mechanism suggested earlier. The comparison with viscosity data in the supercooled liquid state of Zr_(46.7)Ti_(8.3)Cu_(7.5)Ni_(10)Be_(27.5) reveals a breakdown of the Stokes- Einstein relation, whereas D(T) and η(T) follow a relation close to van den Beukel's. The breakdown of the Stokes- Einstein relation indicates a cooperative diffusion mechanism in the supercooled liquid state of the ZrTiCuNiBe alloys

Geyer, U.

Johnson, W. L.

Macht, M.-P.

Qiu, Y.

Schneider, S.

Tombrello, T. A.

Caltech Authors - Main

SMALL ATOM DIFFUSION AND BREAKDOWN ON STOKES-EINSTEIN 
RELATION IN THE SUPERCOOLED LIQUID STATE OF Zr-Ti-Cu-Ni-Be ALLOYS 
U. GEYER*, S. SCHNEIDER* Y. QIU**, M.-P. MACHT***, 
T.A. TOMBRELLO**, W.L. JOHNSON** 
*I. Phys. Institut and SFB 345, Universitat Gi:ittingen, 37073 Gi:ittingen, Germany 
**California Institute of Technology, Pasadena, CA 91125, USA 
***Hahn-Meitner-Institut Berlin, Abt. Strukturforschung, 14109 Berlin, Germany 
ABSTRACT 
Be diffusivity data in the bulk metallic glass forming alloys Zr4i.2Ti13.sCu125Ni10Be22.s 
and Zr46.7Tis.3Cu75Ni10Bez7.s are reported for temperatures between 530K and 710K, extending 
up to SOK into the supercooled liquid states of the alloys. At the glass transition temperature, 
T g' a change in temperature dependence of the data is observed in both alloys, and above T g the 
diffusivity increases faster with temperature than below. The data in the supercooled liquid can 
be described by a modified Arrhenius expression containing the communal entropy of the su-
percooled liquid and based on a diffusion mechanism suggested earlier. The comparison with 
viscosity data in the supercooled liquid state of Zr46.7Ti8.3Cu75Ni10Be275 reveals a breakdown 
of the Stokes- Einstein relation, whereas D(T) and ll(T) follow a relation close to van den Beu-
kel's. The breakdown of the Stokes- Einstein relation indicates a cooperative diffusion mecha-
nism in the supercooled liquid state of the ZrTiCuNiBe alloys. 
INTRODUCTION 
Recent investigations of multicomponent deep eutectic metallic systems have led to the 
development of bulk metallic glasses [1-5] with superior glass forming abilities and an excellent 
stability with respect to crystallization. This has opened new opportunities for fundamental 
study of the supercooled liquid state above the glass transition temperature, Tg, as well as of 
the glass transition in metallic systems, which both were experimentally almost inaccessible 
before. 
For the ZrTiCuNiBe alloy system [4] atomic diffusion [6-8] and viscosity [9], the time-
temperature-transformation (TTT) diagram for nucleation and growth of crystals [10] and other 
thermophysical properties [11] of the supercooled melt have already been investigated. Studies 
of crystallization in the Zr4i.2Ti13.8Cu125Ni10Be22.5 alloy around Tg reveal that crystallization is 
preceded by phase separation [12, 13]. Further, the availability of bulk specimens has made 
possible serious mechanical testing which demonstrates that the new alloys are very promising 
for engineering applications, even more as they can easily be processed and formed in the su-
percooled liquid state. 
This paper reports investigations of atomic diffusion in the glassy and supercooled liquid 
states of the Zr4i.2Ti13.8Cu12.sNi10Be22.5 (Vl) [6, 7] and Zr46.7Tis.3Cu75Ni10Be27.s (V4) [8] alloys, 
using Be as the diffusing species. Consequences of phase separation for these and other investi-
gations of the supercooled liquid state of Vl are briefly discussed, and diffusivity, entropy and 
viscosity data are compared with regard to the diffusion mechanism in the supercooled liquid 
state of these metallic alloys. 
301 
Mat. Res. Soc. Symp. Proc. Vol. 455"1997 Materials Research Society 
EXPERIMENT 
Experimental details of sample preparation and Be diffusion profile measurements by 
high energy backscattering spectrometry have been given before [6, 14]. In the course of our 
investigations the onset of a time dependence of diffusivity values after increased annealing 
times became obvious. This observation initiated more detailed small angle neutron scattering 
studies of the thermal stability of Vl during isothermal annealing near the glass transition tem-
perature [12, 13], revealing phase separation followed by nanocrystallization. Isothermal an-
nealing of the samples thus must be restricted to the early stages of phase separation where the 
composition changes grow very slowly and the system stays almost homogeneous. In case of 
fast Be diffusion we were able to apply annealing times short enough to avoid influence of de-
composition and long enough to produce measurable diffusion profiles (Fig. 1). However, in-
vestigations on diffusion of the slower diffusors Co and Al in Vl show a decrease of diffusivity 
with time [15]. In general, one must consider possible influence of compositional changes due 
to phase separation in ZrTiCuNiBe and other multicomponent glass forming alloys during all 
isothermal experiments around the glass transition temperature. 
JQ-16 
• 533K 6 530K 
• 573K 0 570K 
JQ-17 •• • 623K D 622K 
~ 643K v 640K 
<> <> 
• 663K <> 660K ,......_ 
JQ-18 ~~ · · · · · · · · onset of SANS signal VJ N-
s D "· D [[] 
'-' 
" ·-~ oo 0 .,:i J0-19 I- • 0 
10-20 t- . ~ !!. ·!:!. .. 
• •• 
J0-21 
JOI J02 J03 104 105 
duration of heat treatment (s) 
RES UL TS AND DISCUSSION 
J06 
Fig.1: 
Self diffusivity of beryllium in 
Vl (solid symbols) and V4 
(open symbols) as a function 
of annealing time. The dotted 
line indicates onset of 
decomposition in Vl, as re-
vealed by small angle neutron 
scattering (SANS). 
In rapidly quenched conventional metallic glasses diffusion is known to be sensitive to 
the relaxation state of the samples. In a nonrelaxed sample diffusivity drops as a function of 
time due to structural relaxation and annealing of excess free volume. It approaches a constant 
value when the sample is fully relaxed. It is thus essential to check for relaxation effects in 
evaluating diffusion data. As shown in Fig. 1 no dependence of Be diffusivity on the duration 
of heat treatment could be found within the experimental time-temperature window in both 
alloys. This ensures that no significant relaxation effects occur. At 623K annealing times ex-
302 
ceeding the early stages of decomposition have been applied to Vl, but no significant influence 
of decomposition on Be diffusivity is visible. 
Figure 2 shows an Arrhenius plot of Be diffusivity in Vl and V4. For both alloys, the re-
spective data can be divided into two subsets. The subsets for temperatures below about 625K 
fit an Arrhenius law Dse(T)=D0xexp(-Q/k8T) with D0=1.8xlff11m 2/s and Q=l.OSeV (Vl) and 
D0=8xlff
10
m
2/s and Q=l.leV (V4), respectively. Above 625K an enhanced temperature depend-
ence of Dse is observed in both alloys. If interpreted in terms of Arrhenius behavior, this leads 
to higher activation energies of 4.47eV (Vl) and l.9eV (V4), and to much higher D0 values of 
l.lxl017m2/s (Vl) and l.7xlff'm2/s (V4). 
The change in temperature dependence is associated with the glass transition which in 
both alloys occurs at about 625K for isothermal annealing experiments. The present data extend 
40K into the supercooled liquid state of Vl and SOK into that of V4. This again demonstrates 
that V 4 is more stable against phase separation and crystallization than Vl. 
T (K) 
700 650 600 550 500 
10-'6 
• Zr41 .2Tin8Cu125Ni10Be225 Fig.2: 
• Zr46.7Ti8.3Cu7.5Ni10Be275 10-17 Arrhenius plot of Be diffusiv-
T; ity in Vl and V4. Tgi is the 
:g glass transition temperature 
~ 
[Jl 10·'" for isothermal annealing. The NS solid lines represent best Ar-
'--" rhenius fits to the data. The 
" a:i 10-'9 fitting parameters are given 0 in the text. 
I 0-'" 
10·21 
1.4 1.5 1.6 1.7 1.8 1.9 2.0 
10oorr (K1) 
The small D0 and Q values for Be diffusion in the solid states of both alloys suggest single 
atomic jumps of the small Be atoms in the amorphous matrix. The about one order of magni-
tude higher diffusivity in the solid state of V4 indicates a higher fraction of free volume in this 
alloy, in accordance with the higher average atomic size in V4 [8]. In the supercooled liquid 
states (SLS) of both alloys the Arrhenius parameters show increased values. Particularly for Vl 
the respective numbers seem unphysically high. This indicates that interpretation of the data in 
the SLS in terms of thermally activated single atomic jumps is not appropriate and a different 
diffusion mechanism must be involved. We have suggested that diffusion of Be (and probably 
of other small atoms) in the SLS can be explained by single atomic jumps of the diffusing atoms 
in a slowly changing configuration of neighboring atoms [6]. The continuous atomic rear-
rangements in the SLS support the atomic jumps by providing a higher frequency of critical free 
volume fluctuations. The respective contribution to Be diffusivity is related to the configura-
303 
tional entropy of the supercooled liquid. By taking into account the temperature dependent en-
tropy change D.SSLS(T) due to the glass transition, the data can be fitted by the following modi-
fied Arrhenius expression: 
where D0SS and LlttSS are the preexponential factor and migration enthalpy for diffusion in the 
solid state, N is a typical number of neighbor atoms supporting the single jumps by viscous re-
arrangements, and D.SSLS(T) is the temperature dependent configurational entropy per mole of 
the supercooled liquid. Only the fraction NINA of the total configurational entropy which is 
related to the rearrangements of N nearest and next-nearest neighbor atoms is considered in 
this expression. Near the glass transition temperature, Tg, the entropy can be approximated by 
a linear expression, D.SSLS(T)~Llcp(Tg) x(T-Tg) /Tg, with Llcp(Tg) the specific heat capacity differ-
ence at T g of the supercooled liquid and crystalline states. Figure 3 shows that this expression 
fits the Be diffusion data of both alloys if one takes N=22 for Vl and N=13 for V4, respectively. 
For Vl the fit is based on the fully experimentally determined entropy function [11], in the case 
of V4 it is based on the linear approximation. Beryllium diffusivity in the SLS of the ZrTiCu-
NiBe alloys scales with the slope of the entropy function above T,, and the large activation en-
ergies above Tg are mimicked by the increase of configurational entropy with temperature. 
Since Llcp(Tg) is by 2.9 smaller in V4 than in Vl [8], the slope of the entropy vs. temperature 
function of V4 is smaller and the change in temperature dependence of D8e(T) is less. The 
smaller number of neighbor atoms involved in Be diffusion in V4 again points to a higher frac-
tion of free volume in this alloy. The change in temperature dependence at Tg with decreasing 
temperature is related to the rapidly increasing viscosity of the system: below Tg mainly single 
atomic jumps contribute to diffusion, above T g cooperative atomic rearrangements dominate 
the scenario. This change of diffusion mechanism at Tg should only occur for atoms that are 
able to perform single atomic jumps. Ni diffusivity in Zr55Al10Ni10Cu25 also shows a distinct 
increase in temperature dependence above T g [16], and data in [17] indicate a small increase in 
temperature dependence at T, of Ni diffusivity in Vl. No effect is observed for Co diffusion in 
Vl and Co and Al diffusion in V4 [15], the Arrhenius parameters being comparable to those of 
Be in the SLS. 
The present diffusivity data and the equilibrium viscosity data of V4 in the SLS [9] over-
lap in the temperature regime 650-710K. Thus, we can check both, the Stokes-Einstein relation 
[18] and van den Beukel's relation [19], for atomic transport in the SLS of V4 at these tempera-
tures. Figure 4 shows our measured data in the SLS of V4 and the diffusivity as predicted by the 
Stokes-Einstein relation for translational diffusion, D(T)=k8T /6nxri(T)xR, by choosing the 
atomic radius of Be, l.lA, for the molecular radius R and using the calculated TJ(T) from the 
Vogel-Fulcher fit to the experimentally determined viscosity. According to Fig. 4, the Be diffu-
sion is much faster in comparison to the prediction of the Stokes-Einstein relation. We believe 
that the observed breakdown of the Stokes-Einstein relation is another indication for the pro-
posed cooperative diffusion mechanism [6] in the SLS of ZrTiCuNiBe bulk glass formers [8]. 
304 
,---.. 
"' N....__ 
a 
'-' 
"' o:> 
Cl 
,---.. 
"' 
....__ 
N 
a 
'-' 
"' o:> 
Cl 
T (K) 
1000 750 500 
10-•.---.----.,,.------.---------. 
lff9 
lffl() 
!ff" 
lffl2 
!ff" 
lff14 
lff 15 
lffl6 
10-17 
10-18 
I ff1• 
' 
' 
' 
' 
' 
' 
• Zr41 _2Ti 13.KCu 125Ni 10Be225data 
----- Zr412Ti 1311Cu 125NimBe225 fit 
• Zr467Ti1uCu75Ni 10Be275 data 
--Zr46.7Ti83Cu7.5Ni 10Be275 fit 
T' 
: g 
\ SLS SS 
' 
(b)N=13 ~'• I \.. 
700 
• ---~ 
-·- -
680 
... . 
:•. 
T (K) 
660 640 
- - - - - · Tl(T)xD '''(T)=const 
. -
-- ... 
--
• 
• 
-~~--Stokes-Einstein 
prediction from viscosity 
data (R= l.lA) 
2.0 
I 0-21 '--_.__..._ _ _,_ _ __,__~---'--~-----' 
1.40 1.45 1.50 
1ooorr (K1) 
1.55 1.60 
305 
Fig. 3: 
Dae data and fit curves to the 
modified Arrhenius law for 
data in the supercooled liq-
uid. The fit for Vl is based on 
the fully experimentally de-
termined entropy function 
[ll], the fit for V4 is based on 
a linear approximation of the 
entropy function . 
Fig.4: 
Stokes-Einstein prediction for 
diffusivity (R=l.lA), as calcu-
lated from the Vogel-Fulcher fit 
for V4 viscosity data [9] (solid 
line) and result of a fitting pro-
cedure of T)(D/m2s-1)"=const to 
the data (dashed line, n=l.85, 
const=3.8xI0-13poise). 
Based on D(T) and TJ(T) measurements on conventional metallic glasses, van den Beukel 
suggested the relation rixD2=const for the correlation of diffusivity and viscosity data [19]. In 
terms of the free volume model it can be derived by assuming two different defect types with 
different concentrations responsible for diffusion and viscosity. The present data on V4 follow a 
relation rixDl.85=const which is close to van den Beukel's suggestion (Fig. 4). The interpretation 
of this experimentally found relation needs to be clarified. 
ACKNOWLEDGMENTS 
This work was supported by the U.S. Army Research Office [Grant DAAH04-95-1-0233], 
by the U.S. Dept. of Energy under Grant No. DE-FG03-86ER45242, by the DFG via SFB 345, and 
by the National Science Foundation [Grant DMR93-18931]. 
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Small atom diffusion and breakdown on Stokes-Einstein relation in the supercooled liquid state of Zr-Ti-Cu-Ni-Be alloys

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Small atom diffusion and breakdown on Stokes-Einstein relation in the supercooled liquid state of Zr-Ti-Cu-Ni-Be alloys

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