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On maximum supercooling in liquids
Solid State Communications, Vol 67, No 7, pp 731-733, 1988 Printed in Great Britain
0038-1098/88 $3 00 + 00 Pergamon Press plc
ON M A X I M U M S U P E R C O O L I N G IN LIQUIDS P K Dixlt, B A Vald and K C Sharma Department of Physics, H P University, Shlmla-171 005, India
(Received 16 February 1988, m revised form 3 March 1988 by A R Verma) A reahstlc approximation has been utlhzed within the framework of homogeneous nucleation theory, to calculate the change in volume free energy A Q on sohdlfication, and the maximum supercoohng temperature To for a few metallic and semIconducting systems The approximation Includes the maximization of the nucleation rate and the hyperbohc form for the speofic heat change i e ACp = C/T It yields unique values of To~T,. which are found to be consistent with those between 1/3 and 1/2 as often quoted In the hterature
IN R E C E N T years there has been a lot of interest in supercooled liquids [1-3] and liquid-glass transition [4, 5] because of the applied nature of the problem, this interest has covered the static and dynamic points of view at the experimental and theoretical level The theoretical maximum supercooling is closely related to the stability limit which can be achieved through the consideration of thermodynamics and kinetic based arguments In general, the change In volume free energy and ItS extremum behavlour at the maximum supercooling imply a continuous transition from the metastable to the stable state of the system Gibbs and Dimarzlo [6, 7] gave the arguments that liquid to glass transition, although influenced by kinetic factors such as cooling rate, is a reflection of genuine (secondorder) thermodynamic transition at which the configuratlonal entropy of the liquid would vanish in an experiment o f infinite time scale The amorphous phase below the transition is characterized by extremely long relaxation times relative to the observation time and therefore stability against crystalhzatlon However, the stability (structural) hmlt [8] in simple llqmds is believed to be due to density fluctuations (of a particular size) which are reflected through the static structure factor [9, 10] S(q) Further, Wendt and Abraham [11] gave a criterion for the onset of liquid to glass transition with pressure and/or temperature, that is R(T, P) = (g~./gm~x) = 0 14 where gm,. and gmaxare values of pair distribution function at its first mlmmum and first maximum respectively The stability limit has also been attributed to the soft relaxatlonal mode [12] which appears to Indicate a discontinuous process [13] For pure, single component, metastable systems it is well-known that the supercooled liquid transforms into a stable solid phase via homogeneous nucleation
and one must have appreciable rates of nucleation and crystal growth However, in hquld metals, the crystal growth time is supposed to be much smaller compared to the tlme required for nucleation Turnbull [14] was the first to observe that the maximum supercooling (at a given pressure) was around 0 18 Tm where Tm is the melting temperature Later the maximum supercoolmg was noticed [2], in many cases to be as large as twice that given by Turnbull and there have been a few more attempts [15-17] to understand such a phenomena From the thermodynamic point of view the major difficulty in the calculation o f the energy change, AG~, and the maximum supercooling temperature has been the lack o f knowledge of heat capacity change ACp for supercooled liquids Recently, Battezzattl and Garrone [18] pointed out that, for simple fluids, the form ACpOtl/T yields accurate results for the heat capacity Concentrating on the nucleation aspect and using ACp = C/T, we will try to have a unique value fo the maximum supercooling temperature To for different liquids The steady state rate for homogeneous nucleation is given by [3]
I = Io e - a G c / R r
(1)
where 10 is essentially temperature independent within the temperature range of interest AGe is the critical free energy for the formation of nucleus, gwen as AGe =
167to-3 3AG2
(2)
Here a corresponds to the liquid-solid mterfaclal energy, AG, IS the change m volume free energy on solidification, which may be expressed at constant pressure, as
731
732
ON M A X I M U M S U P E R C O O L I N G IN LIQUIDS
Vol 67, No 7
Table 1 Thermodynamw data used for the calculation of maxlmum supercoohng AT~¢(= T . - To) at To Properties
Systems
T. (K) AS=(Cal/mol) ACp = B . (Cal/mol K) x = To/T. ATN = (1 - x ) T .
Hg
Pb
Ga
BI
Sn
In
Se
Te
Ausl 4SIls 6
234 1 23 037
600 4 19 053
302 8 4 41 069
544 1 4 83 1 39
505 1 3 33 11
429 7 1 81 044
493 3 04 27
725 58 20
636 3 69 1 85
0 38 145
0 42 351
0 38 188
0 41 318
0 43 288
0 40 174
0 55 222
0 43 412
0 47 337
89 2
249 2
115 1
226 1
217 1
255 7
271 2
313 2
299
(K) T0(K)
AG~ = A H -
TAS,
(3)
with the enthalpy change T
AH =
(4)
Tm~S. + f d T ACp, r,.
and the entropy change AS =
AS. +
i d T DC, T '
(51
rm
for supercooled liquids at temperature T Here AS. is the entropy of fusion at the equlhbnum freezing temperature T. ACp is the &fference in specific heat of the supercooled liquid and the solid phase We take the realistic form AC v = C / T for simple fluids, where C is some constant It may be noted from equations (4) and (5) that the thermodynamm quantities m the supercooled phase are to be interpreted in terms of quantitms at the melting temperature T. In vmw of this, the constant C may be chosen at the melting point, T., as C = Tm Bm where B . is the specific heat at T. It yields
with x = To/Tm, which can be solved graphically for umque value of x The calculated values of AG~ for simple supercooled hqmds usmg the hyperbolic form of ACp, are found qmte close to the experimental data The corresponding values of x for them, Hg, Pb, Ca, BI, Sn, In, Se and Te, along with the other constants used in this work are given in Table 1 We have also included a supercooled alloy Aus~ 4 Sl~s6 for the sake of completeness The range of values of x obtained for these systems is found to be consistent with that quoted by I
I
I
I
,ooI
o_
o o
o
A r ,xS. + Bm r m l n ~
+ ar
(6)
The maximum supercooling temperature To below which the system can not exist in the liquid phase, may be obtained by max~mlzmg the nucleation rate I with respect to temperature T It yields, using equation (1) and (2),
This equation may be simplified using equation (6) to 3x-
[
1 + ~
3 - 3x + ln x
)]
=
0
(8)
*
300f X
o!
° Xl
2006 o z
t
X
o
AGo =
X
I
100
Hg 200
I
',,o~'
,
j
300
400
Go
'L
Se n f
500 TM oK
'
B,
I ,
81-&18'6,,,
PbAu i
600
SJ
i T¢
700
Fig 1 The maximum supercooling TN vS the equilibrium freezing temperature Tm Open circles represent the present calculation, sohd circles are the data of Turnbull [14], crosses are the results of Perepezko [2] Stars represents the experimental results after correction of results obtained from crosses for a faster cooling rate
Vol 67, No 7
ON MAXIMUM SUPERCOOLING IN LIQUIDS
Dubey and Ramachandrarao and others These values, m the form ATN =Tm - To, are gwen along with, the experimental data (for all but Se) m Fig 1 as a function of Tm and he in the proper range between 1/3 and 1/2 given in earlier works It IS qmte slgmficant to note that the supercoohng gwen by the parameter x ( = To~TIn) is solely determined by the characterlsUc thermodynamic quantmes at the melting point
7 8
9 10 11
Acknowledgements - - One of us (KCS) thanks Professor P A Egelstaff for helpful discussions on the subject
12 13
REFERENCES 1 2 3 4 5 6
D H Rassmussen, J H Perepezko & C R Lopar Jr, Raptdly Quenched Metals, Vol 1, p 51, MIT press (1967) J H Perepezko, Raptd Sohdtficatton Processmg, Prmctple and Technologws (Edited by M Cohen, B K Kear and R Mehrablan) (1984) A C Zettlemoyer (Ed), Nucleatwn, Marcell Dedder, N Y (1969) M H Cohen & G Grest, Phys Rev B20, 1077 (1979) V Bengtzehus, W Gotze & A Sjolander, J Phys C17, 5914 (1984) J H Gibbs & E A Dlmarzlo, J Chem Phys 28, 373 (1958)
14 15 16 17 18 19 20 21
733
E A Dlmarzlo & J H Gibbs, J Chem Phys 28, 807 (1958) J K Klrkwood, Collected works Vol 1, Quantum Stattsttcs and Cooperative Phenomena, (Edited by E H Stdhnger, Jr ) Gordon and Breach, N Y (1965) J M Jordon, Phys Rev AI$, 1272 (1978) U Bengtzehus, Phys Rev A33, 3433 (1986) and references cited thereto H R Wendt & F F Abraham, Phys Rev Lett 41, 1244 (1976) J B Suck, J H Perepezko, I E Anderson & C A Angell, Phys Rev Lett 47, 424 (1981) and references cited thereto H B Smgh & A Holz, Sohd State Commun 45, 985 (1983) D Turnbull & R E Cech, J Appl Phys 21, 804 (1950) J D Haffman, J Chem Phys 29, 1192 (1958) D R H Jones & G A Chandwlch, Phd Mag 24, 995 (1971) C V Thomson & F Spaepen, Acta Met 27, 1855 (1979) L Battezzatt~ & E Garrone, Z Metahkde 75, 305 (1984) K S Dubey & P Ramachandrarao, Acta Met 32, 91 (1984) P Ramachandrarao & K S Dubey, Int J Raptd Sohdteatton 1, 1 (1984) Y Mlyazawa & G M Pound, J Cryst Growth 23, 45 (1974)
0038-1098/88 $3 00 + 00 Pergamon Press plc
ON M A X I M U M S U P E R C O O L I N G IN LIQUIDS P K Dixlt, B A Vald and K C Sharma Department of Physics, H P University, Shlmla-171 005, India
(Received 16 February 1988, m revised form 3 March 1988 by A R Verma) A reahstlc approximation has been utlhzed within the framework of homogeneous nucleation theory, to calculate the change in volume free energy A Q on sohdlfication, and the maximum supercoohng temperature To for a few metallic and semIconducting systems The approximation Includes the maximization of the nucleation rate and the hyperbohc form for the speofic heat change i e ACp = C/T It yields unique values of To~T,. which are found to be consistent with those between 1/3 and 1/2 as often quoted In the hterature
IN R E C E N T years there has been a lot of interest in supercooled liquids [1-3] and liquid-glass transition [4, 5] because of the applied nature of the problem, this interest has covered the static and dynamic points of view at the experimental and theoretical level The theoretical maximum supercooling is closely related to the stability limit which can be achieved through the consideration of thermodynamics and kinetic based arguments In general, the change In volume free energy and ItS extremum behavlour at the maximum supercooling imply a continuous transition from the metastable to the stable state of the system Gibbs and Dimarzlo [6, 7] gave the arguments that liquid to glass transition, although influenced by kinetic factors such as cooling rate, is a reflection of genuine (secondorder) thermodynamic transition at which the configuratlonal entropy of the liquid would vanish in an experiment o f infinite time scale The amorphous phase below the transition is characterized by extremely long relaxation times relative to the observation time and therefore stability against crystalhzatlon However, the stability (structural) hmlt [8] in simple llqmds is believed to be due to density fluctuations (of a particular size) which are reflected through the static structure factor [9, 10] S(q) Further, Wendt and Abraham [11] gave a criterion for the onset of liquid to glass transition with pressure and/or temperature, that is R(T, P) = (g~./gm~x) = 0 14 where gm,. and gmaxare values of pair distribution function at its first mlmmum and first maximum respectively The stability limit has also been attributed to the soft relaxatlonal mode [12] which appears to Indicate a discontinuous process [13] For pure, single component, metastable systems it is well-known that the supercooled liquid transforms into a stable solid phase via homogeneous nucleation
and one must have appreciable rates of nucleation and crystal growth However, in hquld metals, the crystal growth time is supposed to be much smaller compared to the tlme required for nucleation Turnbull [14] was the first to observe that the maximum supercooling (at a given pressure) was around 0 18 Tm where Tm is the melting temperature Later the maximum supercoolmg was noticed [2], in many cases to be as large as twice that given by Turnbull and there have been a few more attempts [15-17] to understand such a phenomena From the thermodynamic point of view the major difficulty in the calculation o f the energy change, AG~, and the maximum supercooling temperature has been the lack o f knowledge of heat capacity change ACp for supercooled liquids Recently, Battezzattl and Garrone [18] pointed out that, for simple fluids, the form ACpOtl/T yields accurate results for the heat capacity Concentrating on the nucleation aspect and using ACp = C/T, we will try to have a unique value fo the maximum supercooling temperature To for different liquids The steady state rate for homogeneous nucleation is given by [3]
I = Io e - a G c / R r
(1)
where 10 is essentially temperature independent within the temperature range of interest AGe is the critical free energy for the formation of nucleus, gwen as AGe =
167to-3 3AG2
(2)
Here a corresponds to the liquid-solid mterfaclal energy, AG, IS the change m volume free energy on solidification, which may be expressed at constant pressure, as
731
732
ON M A X I M U M S U P E R C O O L I N G IN LIQUIDS
Vol 67, No 7
Table 1 Thermodynamw data used for the calculation of maxlmum supercoohng AT~¢(= T . - To) at To Properties
Systems
T. (K) AS=(Cal/mol) ACp = B . (Cal/mol K) x = To/T. ATN = (1 - x ) T .
Hg
Pb
Ga
BI
Sn
In
Se
Te
Ausl 4SIls 6
234 1 23 037
600 4 19 053
302 8 4 41 069
544 1 4 83 1 39
505 1 3 33 11
429 7 1 81 044
493 3 04 27
725 58 20
636 3 69 1 85
0 38 145
0 42 351
0 38 188
0 41 318
0 43 288
0 40 174
0 55 222
0 43 412
0 47 337
89 2
249 2
115 1
226 1
217 1
255 7
271 2
313 2
299
(K) T0(K)
AG~ = A H -
TAS,
(3)
with the enthalpy change T
AH =
(4)
Tm~S. + f d T ACp, r,.
and the entropy change AS =
AS. +
i d T DC, T '
(51
rm
for supercooled liquids at temperature T Here AS. is the entropy of fusion at the equlhbnum freezing temperature T. ACp is the &fference in specific heat of the supercooled liquid and the solid phase We take the realistic form AC v = C / T for simple fluids, where C is some constant It may be noted from equations (4) and (5) that the thermodynamm quantities m the supercooled phase are to be interpreted in terms of quantitms at the melting temperature T. In vmw of this, the constant C may be chosen at the melting point, T., as C = Tm Bm where B . is the specific heat at T. It yields
with x = To/Tm, which can be solved graphically for umque value of x The calculated values of AG~ for simple supercooled hqmds usmg the hyperbolic form of ACp, are found qmte close to the experimental data The corresponding values of x for them, Hg, Pb, Ca, BI, Sn, In, Se and Te, along with the other constants used in this work are given in Table 1 We have also included a supercooled alloy Aus~ 4 Sl~s6 for the sake of completeness The range of values of x obtained for these systems is found to be consistent with that quoted by I
I
I
I
,ooI
o_
o o
o
A r ,xS. + Bm r m l n ~
+ ar
(6)
The maximum supercooling temperature To below which the system can not exist in the liquid phase, may be obtained by max~mlzmg the nucleation rate I with respect to temperature T It yields, using equation (1) and (2),
This equation may be simplified using equation (6) to 3x-
[
1 + ~
3 - 3x + ln x
)]
=
0
(8)
*
300f X
o!
° Xl
2006 o z
t
X
o
AGo =
X
I
100
Hg 200
I
',,o~'
,
j
300
400
Go
'L
Se n f
500 TM oK
'
B,
I ,
81-&18'6,,,
PbAu i
600
SJ
i T¢
700
Fig 1 The maximum supercooling TN vS the equilibrium freezing temperature Tm Open circles represent the present calculation, sohd circles are the data of Turnbull [14], crosses are the results of Perepezko [2] Stars represents the experimental results after correction of results obtained from crosses for a faster cooling rate
Vol 67, No 7
ON MAXIMUM SUPERCOOLING IN LIQUIDS
Dubey and Ramachandrarao and others These values, m the form ATN =Tm - To, are gwen along with, the experimental data (for all but Se) m Fig 1 as a function of Tm and he in the proper range between 1/3 and 1/2 given in earlier works It IS qmte slgmficant to note that the supercoohng gwen by the parameter x ( = To~TIn) is solely determined by the characterlsUc thermodynamic quantmes at the melting point
7 8
9 10 11
Acknowledgements - - One of us (KCS) thanks Professor P A Egelstaff for helpful discussions on the subject
12 13
REFERENCES 1 2 3 4 5 6
D H Rassmussen, J H Perepezko & C R Lopar Jr, Raptdly Quenched Metals, Vol 1, p 51, MIT press (1967) J H Perepezko, Raptd Sohdtficatton Processmg, Prmctple and Technologws (Edited by M Cohen, B K Kear and R Mehrablan) (1984) A C Zettlemoyer (Ed), Nucleatwn, Marcell Dedder, N Y (1969) M H Cohen & G Grest, Phys Rev B20, 1077 (1979) V Bengtzehus, W Gotze & A Sjolander, J Phys C17, 5914 (1984) J H Gibbs & E A Dlmarzlo, J Chem Phys 28, 373 (1958)
14 15 16 17 18 19 20 21
733
E A Dlmarzlo & J H Gibbs, J Chem Phys 28, 807 (1958) J K Klrkwood, Collected works Vol 1, Quantum Stattsttcs and Cooperative Phenomena, (Edited by E H Stdhnger, Jr ) Gordon and Breach, N Y (1965) J M Jordon, Phys Rev AI$, 1272 (1978) U Bengtzehus, Phys Rev A33, 3433 (1986) and references cited thereto H R Wendt & F F Abraham, Phys Rev Lett 41, 1244 (1976) J B Suck, J H Perepezko, I E Anderson & C A Angell, Phys Rev Lett 47, 424 (1981) and references cited thereto H B Smgh & A Holz, Sohd State Commun 45, 985 (1983) D Turnbull & R E Cech, J Appl Phys 21, 804 (1950) J D Haffman, J Chem Phys 29, 1192 (1958) D R H Jones & G A Chandwlch, Phd Mag 24, 995 (1971) C V Thomson & F Spaepen, Acta Met 27, 1855 (1979) L Battezzatt~ & E Garrone, Z Metahkde 75, 305 (1984) K S Dubey & P Ramachandrarao, Acta Met 32, 91 (1984) P Ramachandrarao & K S Dubey, Int J Raptd Sohdteatton 1, 1 (1984) Y Mlyazawa & G M Pound, J Cryst Growth 23, 45 (1974)