ScriptaMetahrgica et Materialia, Vol. 32, No. 2, pp. 271276,1995 Copyright CI 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0956716X/‘% $9.50 t .OO
Pergamon
NONISOTHERMAL ALLOY
VISCOUS
CONSIDERED
FLOW
BEHAVIOUR
AS A FREE
*, B.J.Zappel
K.Russew * Institut
for Metals
MaxPlanckInstitut
VOLUME
Seestr.
1574 Sofia, Bulgaria Institut
92, D70174
GLASSY
PHENOMENON
and F.Sommer
Science,
fur Metallforschung,
OF Pd4eNi4sPzs
RELATED
fiir Werkstoffwissenschaft,
Stuttgart,
FRG
(Received May 12, 1994) (Revised August 31, 1994) 1. Introduction The viscous
flow behaviour
years as it determines, a great
extent
fact that connected
time. with
structure. alloys.
the formation
metallic
for a given
This Under
with annealing
of amorphous
on one hand,
of their properties
subtle
is possible terms
during
changes
of the atomic
them
relaxation
structure
structural
relaxation
is reflected
isothermal
annealing,
most
time
reached
[l], PdNiPSi the isothermal
to produce
glasses are thermodynamically metastable, Before the crystallization occurs structural
[1,2]. It should
prior
the kinetics
of the free volume
theory
of viscosity
sensitively
the metastable a linear
the viscosity
et al. approach
equilibrium
increase
71 of glassy of viscosity value
of isothermal annealing. equilibrium viscosity has
studies
of AuGeSi shown
[3], PdCuSi that
studying
alloy near its quasiequilibrium,
towards
[9,10]. The basic equations
to
Due to the
will reach a constant
[6] h ave recently glassy
[l5].
by the viscosity
exhibit
were the viscosity
P.Duine
for many
they crystallize upon annealing relaxation takes place which is
at the temperature glasses in which
of PddsNi4ePse
research
and, on the other hand, annealings
towards
alloys that
equilibrium of metallic
to crystallization
(5,6,8]. [4], and PdNiP viscous flow behaviour
to analyze
most
glassy
be expected
when the glass has reached metastable However, the only isothermal studies been clearly
alloys has been an area of intense
the possibility
its quasiequilibrium
of the kinetic
analysis
it
value in of Duine
et
al. [S] are as follows. The most generalized
temperature
dependence
of the viscosity
n can be represented
as [5,11]
Here Q,, is the activation energy for the viscous flow, 70 is a preexponential factor whose physical meaning is discussed in detail in [ll], and cf is the concentration of the socalled “flowdefects” responsible for viscous flow taking place under applied shear stress r. According to the free volume theory the amount of free volume in the quasiequilibrium state is given by
271
vrscous
212
Vf,eq = xeq(T)yv’
To are two model parameters
where
B and
(VFT)
constant
respectively,
necessary
“~/TV*. The relation
overlap
which
factor between
eqs.
to the VogelFulcherTamman of the VFT
empirical
0.5 and 1, and u* the socalled
The reduced
1
free volume
equation, “critical
3: is defined
as z =
(> 
(I), (2) and (3), th e socalled viscosity qeq is obtained
quasiequilibrium
(2)
cf and z is given by [5] cf =exp
Combining
correspond
temperature
to form a flow defect.
between
(T B To)yTJ*
=
and to the ideal glass transition
y is the geometrical
free volume”,
Vol.32,No. 2
FLOW
.
2
(3)
“hybride”
temperature
dependence
of the
(4 In the ca.ses when the glassy with increasing the most
annealing
suitable
time following
differential
time (due to defect qeq by sudden
alloy has not reached
changes
a bimolecular
equation,
annihilation
the quasiequilibrium
describing
kinetics
of the annealing
dcf
dt=
kinetics
which properly
temperature,

bCj(Cj
Cf,e)
where
i(t)
is the rate
of annealing Text c(t) =
annealing.
for the viscous
eqs.
(l),
v, the jump
(2), (3) and
frequency,
concentration (5), together
Qr
at the with the
flow
of elongation time
Combining
of
(5) flow defect
of isothermal
with
the variation
. with
function
is the rate constant of relaxation $$ ( > energy of relaxation, and cf,e the equilibrium
of Newton
[6] that
is
Here k, = v,exp
equation
along
of cf change
reflects
the activation temperature
cf changes
[1,5]. It has been shown
the bimolecular
or defect production),
state
of the sample,
one obtains
for the elongation
c(t) as a
[6]
(0, > kT
cf,et
rloT
+
$, F
1 
Cflo  Cf,e exp( k,cf,et) Cf,O

LZns b
Cf,O
(7)
where cf,o is the concentration of flow defects at the beginning of an isothermal annealing. Duine et al. [6] have calculated the time dependence of the q approach towards equilibrium at two different temperatures
of isothermal
annealing
near equilibrium
from fits to the measured
elonga
namely 70, Q,,, v,, Qr, B,Z’o, tion data according to eq.(7). Eq.(7) includes 7 fitting parameters, and cf,o. These fitting parameters have been successfully estimated by Duine et al. [6] and Koebrugge et al. [7] by using a nonlinear, multiparameter regression analysis. Taking into
Vol. 32, No. 2
VISCOUS FLOW
account
eq.(3)
studied,
as a function
it becomes
of free volume
possible
yields
release
tion [5], A. van den Beukel specific heat
cp measured
been experimentally
heating
and production
and JSietsma
free volume
rates.
of free volume
have proposed
[5,8] to be almost
a simple
quantitatively
gives an additional
as annealing
requires linear
energy
relation
alloy out
consump
between
the
(dx/dT)q, which has
change
fulfilled.
confirmation
3: of the glassy
Furthermore,
with DSC, and the rate of free volume
proved
[6] obtained
The successful
the reduced
at different
of heat,
rate used in DSC. This result parameters
to calculate
of temperature
213
q is the constant
for the reliability
heating
of the fitting
with the aid of eq.(7).
application
of the approach
of P.Duine
et al. [6] is limited
to glassy alloys with
extremely high thermal stability. Most amorphous alloys with practical significance crystallize in the close proximity of their glass transition temperatures, thus making any study of their viscous
flow behaviour
under
isothermal
conditions
practically
impossible.
The experimental
difficulties imposed by crystallization can be overcome to a great extent ing q under continuous heating rather than under isothermal conditions. earlier
works
described
above,
we have undertaken
this
study
aiming
[1214) by measurStimulated by the
to carry
out viscosity
measurements on PddsNitsPzs glassy alloy under constant heating rate conditions (heating 10 Km+’ ). Special attention was given to the possibility for theoretical interpretation the experimental and Spaepen
results.
Thus,
we have further
[l], A. van den Beukel
nonisothermal viscosity measurements alloys including those with relatively
Under
constant with
heating
time
under
rate
conditions
isothermal
the change
of cf with temperature.
differential
equation
approach
et al.
for practically
of Tsao
[6] for the case of
all well known glassy
Considerations
conditions,
eq.(5),
the flow defect concentration d escribing has to be transformed in order to describe
Taking
account
into
that
(2)
= (g)
q, the following
is obtained g
with P(T)
the theoretical
[5], and P.Duine
making it applicable low thermal stability.
2. Theoretical
change
developed
and J.Sietsma
rate of
= :exp
+ P(T)C~ = and Q(T)
$1 = 
J;
Q(+e
[email protected](T)
= Fexp
. Its analytical
solution
is
J:P(z)dz& $, J’(z)dz
1 J; P(z)dz + cf,oe
with To used as the starting temperature of heating. By combining eq.(9) with es.(l) it is possible to represent the measured temperature dependence of 77at constant q in the temperature range around T,. The parameters could be obtained in a way quite similar to the nonlinear multiparameter least square regression analysis of Duine et al. [6]. However, in order to be able to prove the validity of eq.(9) we have simply performed nonisothermal viscous flow measurements
under
a constant
heating
rate
using
the Pd40Ni40Pze
glassy
alloy, which
has been
VISCOUS
214
well studied
under
isothermal
basis of the parameter
conditons
values
FLOW
Vol. 32, No. 2
[6], and have calculated
given by Duine
the q(T)
et al. [6], and Koebrugge
dependence
on the
et al. [7] using eqs.(9)
and (1).
3. Experimental The glassy PddsNidePzs of a narrow
ribbon
were carried described
out
in detail
Approach
alloy used in this study
with crosssection at q =
dimensions
lOKmin’
elsewhere
using
[12].
The
was kindly of 1.45
a Heraeus
x
supplied
0.049 mm’.
TMA
temperature
Viscosity
500 silica
accuracy
glass
was f2K.
elongation measurements was better than flpm. The apparent as a function of temperature using a modified Newtonian equation viscosity measurements as described elsewhere [13]
TJm=
by J.Sietsma
in the form
measurements dilatometer
The
as
accuracy
of
viscosity 71,~~ was calculated for the case of nonisothermal
&
00)
where Ar is the shear stress difference of two successive elongation measurements performed under different loads, and Ai is the strain rate difference of the same creep experiments as a function
of temperature
(time).
More details
4. Experimental Figure cording tively.
1 shows to es.(l)
the calculated and
eq.(9)
In the calculations
the values
the two viscosity
, as obtained values
ity data
at temperatures between
higher
rates
as obtained
of the glassy
alloy ac
100 Kmin‘,
Qq = 1931cJmolC’,
respecv, = 3.4 .
by Duine
by Duine
than
dependencies
of 0.01, 0.1, 1, 10, and
by Koebrugge
determined
the calculated
and Discussion
of 70 = 3.26 . 1023Pas,
representative points of our nonisothermal a good agreement between the calculated difference
Results
viscositytemperature
for heating
1025s11, B = 6600K, Te = 355K, c~,~ = 2.135 . lol1
are given in [12141.
et al. [6], and Q+ = 1581cJmoZ‘, and purposes et al. [7], were used. For comparison
et al.
[6] at 553 and 563 K, as well as several
viscosity measurements qen values and the data
Tg exists.
At temperatures
and experimentally
obtained
are given. It is obvious that of Duine [6] and our viscosaround
Tg there is a slight
nonequilibrium
viscosity
data.
At temperatures
lower than 550K the experimental values are by a factor of approximately in Figure 2 that the viscosities 2 higher than the calculated ones. It is clearly demonstrated measured do not depend on the applied stress, thus confirming the fact that the stress level used is within the Newtonian viscous regime. It should be noted as well that small changes of the fitting parameters cause relatively great changes in the viscositytemperature dependence. The proper way to obtain better agreement between q,,lC and qezp is to determine the whole set of fitting parameters from the set of experimental viscosity range Tg  40K to To,, the onset temperature of crystallization.
values
within
the temperature
Vol. 32, No. 2
vK!ous
215
FLOW
‘I’, K 600
550
500 1014
0  Duine et al. 30
0  this work
10’2 F2
v) 2
25 10’0 I=
20 106 1.7
1.8
1.9
Tl,
Fig.1.
Calculated
viscositytemperature
6  qe9 calculated
according
563 K are also given,
together
to eq.(4).
1O3 K’
dependencies
rates OE 1  O.OlKmin‘, 2  O.lKmin‘,
2.0
of PdJsNi4sPzs
3  lKmin_‘,
The q,, data
with nonisothermal
glassy
ahoy
at heating
4  lOKmin_‘, 5  lOOKmin_’
of Duine
qezr, data
et al. (61, measured obtained
and
at 553 and
at q = lOKmin_‘.
T, K 600
550
500
1’
1o12
F
c 0 1.66 YPa
Duine et al.

109 20
1
I
,
I
1.7
1.8
1.9
Tl,
Fig.2. Calculated viscositytemperature Duine et al. [6] and Koebrugge et al. stresses are also shown.
2.0
1O3 Kr
dependencies
using
[7]. Experimental
the original values
fitting
measured
parameters
at different
of
shear
Vol. 32. No. 2
VISCOUS FLOW
276
5. Conclusions
Nonisothermal tions
provide
approach
viscosity
an attractive
of Tsao
measurements possibility
and Spaepen
alloys
viscosity
under
constant
the theoretical
[l], A. van den Beukel
[6] for the case of nonisothermal exhibiting
of glassy to extend
measurements
heating
and isothermal
and J.Sietsma
rate
condi
experimental
[5], and P.Duine
on practically
alI known
et al.
glassy alloys
glass transition.
Acknowledgment The authors
are indebted
One of us (K. Russew)
to Dr.
gratefully
J. Sietsma
acknowledges
for providing a 3 month
the amorphous
EU PECO
research
ribbon grant
studied. NR.:1069.
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2.
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