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Glass transition temperature and viscosity of supercooled melts
890
Journal of Non-Crystalline Solids 117/118 (1990) 890-893 North-Holland
GLASS TRANSITION TEMPERATURE AND VISCOSITY OF SUPERCOOLED MELTS Jfirgen WACHTER and Ferdinand SOMMER Max-Planck-Institut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestr. 75, D-7000 Stuttgart 1, F.R.G. Based on the hole model of liquids and few generally available thermodynamic properties to fix the model parameters the iso-entropic and glass transition temperatures and viscosities of supercooled liquid metals and alloys are determined. Comparison with given experimental results shows the good feasibility of this simple approach.
1. INTRODUCTION
degrees of undercooling and results in the following
The solidification behaviour of rapidly solidified alloys
expression of the entropy difference 2
can only be described quantitatively if the thermodynamic and atomic transport properties of hypercooled
AS(T) = AS m
-
melts are known. A simple extrapolation of the data -
nRexp(-(1
- 1))[(1 + 7 8 ) e x p ( - 7 8 )
"/To 7To (1 + -y-)exp(--T-)]
T < T m (2)
above the melting temperature for the undercooled regime is only a poor approximation. For example, specific
Eh 8 = TT---&~n, = "~ Eh = formation enthalpy with 3' = ~-~o' vh
heat increases strongly with decrease in temperature be-
of one mole holes, va = mean hard core volume of atoms,
low melting temperature and viscosity changes by several
Vh = hole volume.
order of magnitude in a non-Arrhenius-type behaviour
The model parameters V5 and n can be fixed if two expe-
till the glass transition temperature if crystallization is
rimental values of A c p ( T ) at T < T m are known . The
bypassed. The purpose of this paper is to use simple mo-
Acp-value at T'* can not be used for A c p m < 1 . For
dels to determine the glass transition temperatures and
high melting metals only A c p m is generally known and
viscosities of supercooled metallic liquids.
for these elements a mean value for 78 = 0.18 is used, which has been obtained from the evaluation of low mel-
2. DETERMINATION OF THE IDEAL GLASS TEM-
T < T m (e.g. Table 1). The hole model of Dubey et al.2
PERATURE The iso-entropic temperature where the entropy difference AS between undercooled liquid and the corresponding crystalline phase(s) becomes zero is often called the ideal glass temperature To. To can be determined from
/
can also directly applied for liquid alloys at concentrations of congruent melting intermetallic phases, of miuima and maxima in the liquidus and eutectic concentrations because only at these compositions and temperatures the integral Gibbs free energies of liquid and solid alloys are
Tm
S/To) =
ting metals the Acp of which are experimentally given at
=0
equal. A H m for alloys of eutectic composition is the (1)
*/
To
melting enthalpy of the corresponding equilibrium phases, T r" is the eutectic temperature and cp" is the heat
with A S " = melting entropy, cp t = heat capacity of the
capacity of the solid mixture. The experimental data
undercooled liquid, cp* = heat capacity of the crystalline phase, A c p ( T ) = cpl(T) - cp'(T).
used for evaluation of exemplary binary and ternary alloys at eutectic concentration are given in Table 1. The
Because only few experimental results of cp t of undercoo-
calculated ideal glass temperatures are shown in Table
led melts exist so several approximations for A c p have
1. The ratio ~
been developed and tested 1. The hole theory of liquids
pared to glass forming liquid alloys. The reason for this is that A c p of undercooled liquid metals shows a smaller
gives the best approach describing A c p even at large 0022-3093/90/$03.50 (~) Elsevier SciencePublishers B.V. (North-Holland)
of pure metals is distincly smaller com-
J. Wachter, F. Sommer/ Glass transition temperature and viscosity of supercooled melts
AH ~ [Jmo1-1] Pb Sn Ga Ni Cu Mgs5.sCu14.5 P dso S i 2o PdTsCusSil8
4800 7040 5540 17497.4 13156.6 7840 8170 7250
T~ [ T1 ] T2 601 505 303 1726 1356 758 1071 1015
[K] 601 400 303 1726 1356 758 1071 1015
6[nj
Acpm [ Acp(T1) Acp(T2) I [Jmol-'K-L]
250 200 120
378 658 642
1.21 -1.04 2.89 6.91 0.30 7.67 7.63 6.80
1.21 3.01 2.89 6.91 0.30 7.67 7.63 6.80
5.57 10.85 12.00
11.04 13.67 13.40
0.16 0.10 0.28 0.18 0.18 1.02 0.62 0.41
16.7 64.0 14.8 40.2 13.5 5.6 10.9 19.0
891
To I [Jmo1-1] 809.6 436.9 707.7 2583.0 2030.1 6440.4 5549.4 3453.5
I [K] 160 139 74 441 344 370 378 647 658 621 642
137 117 61 380 298 271 557 538
Table 1: Experimental data, model parameter and calculated data 7 increase as a function of undercooling compared to glass forming alloys. The approximation ~
= 0.33,
lue corresponds well with experimental values at Tg
(85.10 -6) 5
which has been often used 3, is therefore not specific. I 0 25O In
3. FREE VOLUME
200
The hole theory enables to connect thermodynamic quantities via the hole volume with transport properties of
. - -
vf/v a
150
liquids. The number of holes Nh in thermal equilibrium 100
according to the hole theory is given by Tm
1)
Nh = nN~[exp(75--f-)exp(1 - n
50
1]-1
-
(3) 0
N~ is the number of atoms and the pressure dependent
0 Temperature
,K
term in eqn.(3) is neglected. The free volume introduced by Cohen and TurnbuU 4, which is free to redistribute, should be zero at To. The ratio between the average free
Figure I : Calculated hole volume and free volume of PdTsCuoSia6
volume v! per atom and v~ can be obtained from eqn.(3) by
4. VISCOSITY
v! Va
(Nh(T)
Nh(To))Vh Nav~
(4)
-
[exp(7~
+ ( l - - n )1 ) --1] - 1 -
[exp(7 + (1
-
1))
_
For the representation of the viscosity 1/of glass forming undercooled liquids the Vogel-Fulcher-Tamman relationship is frequently used
11_1
y = yoexp(- T----~-o)
(6)
The obtained temperature dependence of N ~ from eqn.(3)
It is often not possible to assess the parameter Yo, B,
with V ~ Nava + Nhvh and YJVa from eqn.(4) of liquid and supercooled PdTsCu6Si16 is given in Fig. 1. From
med value of 1012 [Pas] at the experimentally observable
To in eqn.(6) unambiguously because beside the assu-
temperature dependence of the hole volume the following
glass transition temperature Tg the y-values are known
expression of the volume expansion coefficient av results
only above T m, in the range of 10-1 [Pas] which show
oN ~h~ Eh Eh 1 ,~v(T) = (-:-~T ')p = -k-~exp(--R--f" - (1 - -~))
Arrhenius-type behaviour. Spaepen has derived using
(5)
The obtained mean value of ~v of PdrsCu6Si16 in the temperature range Tg + 30 K is 72 • 10-e K -1 . This va-
the free volume theory an expression for the viscosity in the homogeneous flow regime at low stresses s
892
J. Wachter, F. Sommer / Glass transition temperature and viscosity of supercooled melts RT
= --
exp(
vo
AG,,
)exp(-sy-
)
(7)
with VD = Debeyefrequency, fZ = avaxage volume per mole, AGm = activation energy of an atomic jump.
The viscosity of undercooled liquid metals can be obtained from the hole model and eqn.(7) even at Tg which is not acessible by rapid solidification (Fig. 4).
AGm of liquid metals is approximately 10% of the total activation energy of viscous flow and is approximated by Eh in the calculations because Eh is also 10% of the obtained total activation energy 0 0 (®o(T) = ~v l R T + Eh) (see eqn.(7) and Fig. 2).
0 exp. -- cllcullted
12 10
i6
g
300
i
4
s 250 2 0
C;~" 200
-2
lSO
-4
'6oo
9o0
12oo 1~o ~
100
21oo
Temperature
,K
50
Viscosity of nickel
Figure 4 : ....... , ......... , ......... ,. . . . . . . . . , ......... , ......... i,...,.,i..,. 700
600
900
Tl~
ltO0
1200
TemperaLure
1300
,K
The diffusivity D of undercooled liquid metals can be calTotal activation energy of viscous flow of
Figure 2 :
culated with the Stokes-Einstein equation from eqn.(7)
PdrsCu6Si16
2 2 , Va, , Eh, D(T) = -~aoVDexp(--~l)exp(--~)
(8)
The Debeyefrequency of the alloys is obtained additi-
with ao = mean atomic diameter.
vely from the values of the components. The comparison
The diffusivity of Mgss.sCula.s calculated with eqn.(8)
between calculated and measured r/-values shows a good
show the derivation from Arrhenius-type behaviour at
correspondence (see Fig. 3) in high and low temperature
large undercooling (see Fig.5).
region. -8 12
O
exp.
__
cmlculated
i
( ~ -10
10 -12 ...1
6
4
-18
2 0
-20
-2
-22 -24
-4
i ....
700
Figure 3 :
600
900
T la 1100 1200 1300 Temperature ,K
Viscosity of PdrsCu6Sia6
400
~ ....
500
i ....
600
, ....
~
, ....
i.llll
900
....
Figure 5 :
,
1000 1100
Temperatur
Diffusivity of Mgss.sCu14.s
,K
J. Wachter, F. Sommer/ Glass transition temperature and viscosity of supercooled melts
893
The nucleation frequency and growth velocity of crystal-
The calculated r(T)-value and experimental time con-
lization processes at large undercooling are strongly in-
stants from two different cooling rates (ql = 0.33 K s -1,
fluenced by the values of the viscosity or diffusivity of undercooled metals and the Gibbs free energy difference
q2 = 106 K s -1) are shown in Fig. 7. The obtained glass transition temperatures are compaired with Tg-values ob-
AG between the undercooled liquid and the correspon-
tained by differential-scanning- calorimetric measurements
ding crystalline phase(s). Fig. 6 shows the velocity u of the crystal-melt interface calculated with
(see Table 1).
AG(T)
u(T) --- 6D(T)(1 - e x p ( - - - R - T - - )
(9)
ao
The u(T)-values obtained by extrapolating the experi-
T
mental diffusivities above T m to lower temperatures and using Acp = Acp "~ for the calculation of A G ( T ) ( . . . .
................... _qJ.................
,-3
),
_~~..
differ considerable from the values obtained from eqn. (8) and AG(T)-values from the hole model (
__q_2................
-5
) at
large undercooling. -10
-15 700
800
900
~
1100 1200 1300 .K
Temperature
7
5
Figure 7 : Calculated time constants of PdTsCu6Si16 0
REFERENCES -10
350 400 450 500 550 600 650 700
i
Temperatur ,K
Figure 6 : Calculated growth velocities of
1.
L. Battezzati and E. Garrone, Z. Metallkde. 75 (1984) 305.
2.
K.S. Dubey and P. Rama~chandrarao, Acta metall. 32 (1984) 91.
Mgss.sCu14.5
5. GLASS TEMPERATURE At sufficient undercooling, if crystallization is bypassed, the liquid is frozen at the glass transition temperature Tg where the time constant of the liquid r ( T ) is equal to the experimental time constant t. r ( T ) is the average time between two successive atomic jumps in the undercooled liquid. The diffusion coefficient in a three dimensional random walk process is given by D ( T ) = la2or(T)-I
(10)
Combining eqns. (8) and (10) gives an expression for the time constant of undercooled metals 4 ~Va~ , Zh r ( T ) = ~vDCXp[-~l)exp(-R-~)
(n)
3.
J. Agren, Phys. Chem. Liq. 18 (1988) 123.
4.
M.H. Cohen and D. Turnbull, J. Chem. Phys. 31
5. 6.
H.S. Chen, J. appl. Phys. 46 (1978) 3289. F. Spaepen, Acta metall. 25 (1977) 407.
(1959) 1164.
7. J. Wachter, PhD Thesis University of Stuttgart
(1989)
Journal of Non-Crystalline Solids 117/118 (1990) 890-893 North-Holland
GLASS TRANSITION TEMPERATURE AND VISCOSITY OF SUPERCOOLED MELTS Jfirgen WACHTER and Ferdinand SOMMER Max-Planck-Institut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestr. 75, D-7000 Stuttgart 1, F.R.G. Based on the hole model of liquids and few generally available thermodynamic properties to fix the model parameters the iso-entropic and glass transition temperatures and viscosities of supercooled liquid metals and alloys are determined. Comparison with given experimental results shows the good feasibility of this simple approach.
1. INTRODUCTION
degrees of undercooling and results in the following
The solidification behaviour of rapidly solidified alloys
expression of the entropy difference 2
can only be described quantitatively if the thermodynamic and atomic transport properties of hypercooled
AS(T) = AS m
-
melts are known. A simple extrapolation of the data -
nRexp(-(1
- 1))[(1 + 7 8 ) e x p ( - 7 8 )
"/To 7To (1 + -y-)exp(--T-)]
T < T m (2)
above the melting temperature for the undercooled regime is only a poor approximation. For example, specific
Eh 8 = TT---&~n, = "~ Eh = formation enthalpy with 3' = ~-~o' vh
heat increases strongly with decrease in temperature be-
of one mole holes, va = mean hard core volume of atoms,
low melting temperature and viscosity changes by several
Vh = hole volume.
order of magnitude in a non-Arrhenius-type behaviour
The model parameters V5 and n can be fixed if two expe-
till the glass transition temperature if crystallization is
rimental values of A c p ( T ) at T < T m are known . The
bypassed. The purpose of this paper is to use simple mo-
Acp-value at T'* can not be used for A c p m < 1 . For
dels to determine the glass transition temperatures and
high melting metals only A c p m is generally known and
viscosities of supercooled metallic liquids.
for these elements a mean value for 78 = 0.18 is used, which has been obtained from the evaluation of low mel-
2. DETERMINATION OF THE IDEAL GLASS TEM-
T < T m (e.g. Table 1). The hole model of Dubey et al.2
PERATURE The iso-entropic temperature where the entropy difference AS between undercooled liquid and the corresponding crystalline phase(s) becomes zero is often called the ideal glass temperature To. To can be determined from
/
can also directly applied for liquid alloys at concentrations of congruent melting intermetallic phases, of miuima and maxima in the liquidus and eutectic concentrations because only at these compositions and temperatures the integral Gibbs free energies of liquid and solid alloys are
Tm
S/To) =
ting metals the Acp of which are experimentally given at
=0
equal. A H m for alloys of eutectic composition is the (1)
*/
To
melting enthalpy of the corresponding equilibrium phases, T r" is the eutectic temperature and cp" is the heat
with A S " = melting entropy, cp t = heat capacity of the
capacity of the solid mixture. The experimental data
undercooled liquid, cp* = heat capacity of the crystalline phase, A c p ( T ) = cpl(T) - cp'(T).
used for evaluation of exemplary binary and ternary alloys at eutectic concentration are given in Table 1. The
Because only few experimental results of cp t of undercoo-
calculated ideal glass temperatures are shown in Table
led melts exist so several approximations for A c p have
1. The ratio ~
been developed and tested 1. The hole theory of liquids
pared to glass forming liquid alloys. The reason for this is that A c p of undercooled liquid metals shows a smaller
gives the best approach describing A c p even at large 0022-3093/90/$03.50 (~) Elsevier SciencePublishers B.V. (North-Holland)
of pure metals is distincly smaller com-
J. Wachter, F. Sommer/ Glass transition temperature and viscosity of supercooled melts
AH ~ [Jmo1-1] Pb Sn Ga Ni Cu Mgs5.sCu14.5 P dso S i 2o PdTsCusSil8
4800 7040 5540 17497.4 13156.6 7840 8170 7250
T~ [ T1 ] T2 601 505 303 1726 1356 758 1071 1015
[K] 601 400 303 1726 1356 758 1071 1015
6[nj
Acpm [ Acp(T1) Acp(T2) I [Jmol-'K-L]
250 200 120
378 658 642
1.21 -1.04 2.89 6.91 0.30 7.67 7.63 6.80
1.21 3.01 2.89 6.91 0.30 7.67 7.63 6.80
5.57 10.85 12.00
11.04 13.67 13.40
0.16 0.10 0.28 0.18 0.18 1.02 0.62 0.41
16.7 64.0 14.8 40.2 13.5 5.6 10.9 19.0
891
To I [Jmo1-1] 809.6 436.9 707.7 2583.0 2030.1 6440.4 5549.4 3453.5
I [K] 160 139 74 441 344 370 378 647 658 621 642
137 117 61 380 298 271 557 538
Table 1: Experimental data, model parameter and calculated data 7 increase as a function of undercooling compared to glass forming alloys. The approximation ~
= 0.33,
lue corresponds well with experimental values at Tg
(85.10 -6) 5
which has been often used 3, is therefore not specific. I 0 25O In
3. FREE VOLUME
200
The hole theory enables to connect thermodynamic quantities via the hole volume with transport properties of
. - -
vf/v a
150
liquids. The number of holes Nh in thermal equilibrium 100
according to the hole theory is given by Tm
1)
Nh = nN~[exp(75--f-)exp(1 - n
50
1]-1
-
(3) 0
N~ is the number of atoms and the pressure dependent
0 Temperature
,K
term in eqn.(3) is neglected. The free volume introduced by Cohen and TurnbuU 4, which is free to redistribute, should be zero at To. The ratio between the average free
Figure I : Calculated hole volume and free volume of PdTsCuoSia6
volume v! per atom and v~ can be obtained from eqn.(3) by
4. VISCOSITY
v! Va
(Nh(T)
Nh(To))Vh Nav~
(4)
-
[exp(7~
+ ( l - - n )1 ) --1] - 1 -
[exp(7 + (1
-
1))
_
For the representation of the viscosity 1/of glass forming undercooled liquids the Vogel-Fulcher-Tamman relationship is frequently used
11_1
y = yoexp(- T----~-o)
(6)
The obtained temperature dependence of N ~ from eqn.(3)
It is often not possible to assess the parameter Yo, B,
with V ~ Nava + Nhvh and YJVa from eqn.(4) of liquid and supercooled PdTsCu6Si16 is given in Fig. 1. From
med value of 1012 [Pas] at the experimentally observable
To in eqn.(6) unambiguously because beside the assu-
temperature dependence of the hole volume the following
glass transition temperature Tg the y-values are known
expression of the volume expansion coefficient av results
only above T m, in the range of 10-1 [Pas] which show
oN ~h~ Eh Eh 1 ,~v(T) = (-:-~T ')p = -k-~exp(--R--f" - (1 - -~))
Arrhenius-type behaviour. Spaepen has derived using
(5)
The obtained mean value of ~v of PdrsCu6Si16 in the temperature range Tg + 30 K is 72 • 10-e K -1 . This va-
the free volume theory an expression for the viscosity in the homogeneous flow regime at low stresses s
892
J. Wachter, F. Sommer / Glass transition temperature and viscosity of supercooled melts RT
= --
exp(
vo
AG,,
)exp(-sy-
)
(7)
with VD = Debeyefrequency, fZ = avaxage volume per mole, AGm = activation energy of an atomic jump.
The viscosity of undercooled liquid metals can be obtained from the hole model and eqn.(7) even at Tg which is not acessible by rapid solidification (Fig. 4).
AGm of liquid metals is approximately 10% of the total activation energy of viscous flow and is approximated by Eh in the calculations because Eh is also 10% of the obtained total activation energy 0 0 (®o(T) = ~v l R T + Eh) (see eqn.(7) and Fig. 2).
0 exp. -- cllcullted
12 10
i6
g
300
i
4
s 250 2 0
C;~" 200
-2
lSO
-4
'6oo
9o0
12oo 1~o ~
100
21oo
Temperature
,K
50
Viscosity of nickel
Figure 4 : ....... , ......... , ......... ,. . . . . . . . . , ......... , ......... i,...,.,i..,. 700
600
900
Tl~
ltO0
1200
TemperaLure
1300
,K
The diffusivity D of undercooled liquid metals can be calTotal activation energy of viscous flow of
Figure 2 :
culated with the Stokes-Einstein equation from eqn.(7)
PdrsCu6Si16
2 2 , Va, , Eh, D(T) = -~aoVDexp(--~l)exp(--~)
(8)
The Debeyefrequency of the alloys is obtained additi-
with ao = mean atomic diameter.
vely from the values of the components. The comparison
The diffusivity of Mgss.sCula.s calculated with eqn.(8)
between calculated and measured r/-values shows a good
show the derivation from Arrhenius-type behaviour at
correspondence (see Fig. 3) in high and low temperature
large undercooling (see Fig.5).
region. -8 12
O
exp.
__
cmlculated
i
( ~ -10
10 -12 ...1
6
4
-18
2 0
-20
-2
-22 -24
-4
i ....
700
Figure 3 :
600
900
T la 1100 1200 1300 Temperature ,K
Viscosity of PdrsCu6Sia6
400
~ ....
500
i ....
600
, ....
~
, ....
i.llll
900
....
Figure 5 :
,
1000 1100
Temperatur
Diffusivity of Mgss.sCu14.s
,K
J. Wachter, F. Sommer/ Glass transition temperature and viscosity of supercooled melts
893
The nucleation frequency and growth velocity of crystal-
The calculated r(T)-value and experimental time con-
lization processes at large undercooling are strongly in-
stants from two different cooling rates (ql = 0.33 K s -1,
fluenced by the values of the viscosity or diffusivity of undercooled metals and the Gibbs free energy difference
q2 = 106 K s -1) are shown in Fig. 7. The obtained glass transition temperatures are compaired with Tg-values ob-
AG between the undercooled liquid and the correspon-
tained by differential-scanning- calorimetric measurements
ding crystalline phase(s). Fig. 6 shows the velocity u of the crystal-melt interface calculated with
(see Table 1).
AG(T)
u(T) --- 6D(T)(1 - e x p ( - - - R - T - - )
(9)
ao
The u(T)-values obtained by extrapolating the experi-
T
mental diffusivities above T m to lower temperatures and using Acp = Acp "~ for the calculation of A G ( T ) ( . . . .
................... _qJ.................
,-3
),
_~~..
differ considerable from the values obtained from eqn. (8) and AG(T)-values from the hole model (
__q_2................
-5
) at
large undercooling. -10
-15 700
800
900
~
1100 1200 1300 .K
Temperature
7
5
Figure 7 : Calculated time constants of PdTsCu6Si16 0
REFERENCES -10
350 400 450 500 550 600 650 700
i
Temperatur ,K
Figure 6 : Calculated growth velocities of
1.
L. Battezzati and E. Garrone, Z. Metallkde. 75 (1984) 305.
2.
K.S. Dubey and P. Rama~chandrarao, Acta metall. 32 (1984) 91.
Mgss.sCu14.5
5. GLASS TEMPERATURE At sufficient undercooling, if crystallization is bypassed, the liquid is frozen at the glass transition temperature Tg where the time constant of the liquid r ( T ) is equal to the experimental time constant t. r ( T ) is the average time between two successive atomic jumps in the undercooled liquid. The diffusion coefficient in a three dimensional random walk process is given by D ( T ) = la2or(T)-I
(10)
Combining eqns. (8) and (10) gives an expression for the time constant of undercooled metals 4 ~Va~ , Zh r ( T ) = ~vDCXp[-~l)exp(-R-~)
(n)
3.
J. Agren, Phys. Chem. Liq. 18 (1988) 123.
4.
M.H. Cohen and D. Turnbull, J. Chem. Phys. 31
5. 6.
H.S. Chen, J. appl. Phys. 46 (1978) 3289. F. Spaepen, Acta metall. 25 (1977) 407.
(1959) 1164.
7. J. Wachter, PhD Thesis University of Stuttgart
(1989)