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A kinetic analysis of the glass-liquid transition
Volume 96A, number 6
PHYSICS LETTERS
11 July 1983
A KINETIC ANALYSIS OF THE G L A S S - L I Q U I D TRANSITION P. DESRE, M. SHIMOJI 1, A. PASTUREL and P. HICTER Laboratoire de Thermodynamique et Physico-Chimie Mdtallurgiques, L TPCM associ~ au CNRS {LA 29), ENSEEG Domaine Universitaire bp 75, 38402 Saint-Martin-d'Hkres, France Received 4 January 1983
We show that the glass-liquid transition may be analyzed from a first-order kinetic law and we calculate in a simple way the variation of the apparent heat capacity with the temperature.
The thermal properties o f amorphous materials are strongly controlled by kinetic conditions imposed in the process of preparation. The glass transition temperature is known to be not a constant of the material but a function of experimental conditions: Tg is higher for faster heating or cooling rates. This value depends also on thermal history of the sample. In fig. 1 we have schematized the change in specific heat Cp, at constant pressure as function of the temperature as presented from experimental measurements. Most of the measurements are done at constant heating rate and show a pronounced peak in the neighbourhood of Tg. For a few materials, which do not crystallize easily (glassy polymers [1] ) Cp is a monotonic de-
i Present address: Department of Chemistry, Faculty of Science, HokkMdo University, Sappora 060, Japan. Cp
T Fig. 1. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
creasing function of Tg when measured at constant cooling rate. A microscopic calculation of thermodynamic functions of amorphous materials (metallic and non-metallic glasses) were made by Cohen and Grest [2] using a percolation theory within the framework of the "free volume" model. The percolation approach was also employed in discussing the glass transition of metals qualitatively by Cyrot [3] and Chen [4]. Following an equivalent approach, we call f the fraction of "liquid" produced during the glass-liquid transition observed at a constant heating rate/~ = dT/ dt. The fraction of "liquid" in this representation has to be understood as the fraction of the volume of the system whose atomic configurations are defrozen. This is a simple and global way of representing all effects near Tg. Owing to the fact that the observed behaviour of Cp is irreversible near the glass transition and the fact that the amorphous state is a non-equilibrium state compared to the relaxated structure and undercooled liquid, we consider that the transformation is spontaneous on heating. Kinetic effects are included in the measurements of specific heat at constant heating rate. In this case the apparent (and observed) change in heat capacity is the consequence o f the spontaneous defrustration of atomic configurations frozen at T < Tg. This analysis is according to the curve of the fig. 1. In this case, the total differential of the enthalpy H is written at constant pressure: 299
Volume 96A, number 6
PHYSICS LETTERS
dH = (8H/OT)L pdT + (all~Of)T, p dr,
11 July 1983
&C~
J Kh~ole "~
A.IO0 Q=50o0 J
or , ,,T. 0.1 a.u,
60
dtt (~f)T df/dt dT - Cp + dT-~ ,p
~ /",t. 1 a.u.
"
Cp = dH/dT, the apparent heat capacity of the system,
40
~
'~]I............. T: 5 a ,U '
/t
is written, using the specific heat at constant pressure of amorphous and liquid phases, i.e. C(A) and Cp(L):
i \ \ //
",,
%
20
C ; = c ( A ) + f (c(L ) -- C(pA)) + q T - l df[ at ,
: f ."/ ]," /'
where q = (aH]af)T,p is a positive quantity of heat, taken as constant in first approximation. The more simple law for df]dt is a first-order kinetic law:
dfldt = r ( 1
-f).
It is well known that in the neighbourhood of Tg, the viscosity obeys a Vogel-Fulcher law and not an Arrhenius law. It is certainly due to the presence of the relaxation spectrum pointed out by many authors. Thereby we choose to write:
df/dt = e x p [ - A / ( T - TO) ] (I - f ) . In this relation, T O is the so-called "ideal" transition temperature. In view of simplicity and to study the general behaviour of the phenomena we have chosen a preexponential term equal to one for the V o g e l Fulcher law. But for the kinetic behaviour there is another preexponential term which has the inverse of time for unit - on this point we do not have any knowledge. So, with a choice of a total preexponential term equal to one, we are constrained to use an arbitrary time unit (a.u.) With the following conditions:
T=To,
t=O,
f(To,O)=O
and T = T o + f ' t ,
/ . i ........ 2,s
e x p [ - A / ( T - TO) ]
AC~ J K:=mole ' F\ I! ~Q.75oo J /
60
4o
+ (A/T) Ei(-A/(T - TO))}, 20
x e- t
Ei(-x) = f -7- dt.
i /
AT
i ., soooJ ',,~
// i! I •f
//
A.100 1" .1 a.u.
\
\
,,,\
/ " " " ,,. ',,\ , \
! ?: /i /
... ,,.N
/ Q=2sooJ
/,'i
.... j i:~ 29
49, Fig. 3.
300
190
We now want to study the effects o f q and T on the apparent specific heat at constant pressure C~. For this we use the ACp=C(L)-C(p A) ~ 24 J I ( - 1 s o l - 1 value generally observed. The constant A o f the V o g e l Fulcher law is taken to be equal to 100, which is a reasonable value for the chosen writing (reminding that we have taken the preexponential term in the V o g e l Fulcher law equal to one). In fig. 2, we give the curves AC/~(AT).= C~ - C ( h ) for q = 5 kJ and for differents values of T (from 0.1 a.u. to 5 a.u.). The Cp peak moves towards higher temperatures as T mcreases. In fig. 3, we study the effect of variation of q for a heating rate T = 1 a.u.. Obviously, the characteristic peak of C~ increases with q. After a thermal treatment at T < Tg, the metallic glass undergoes a structural relaxation and there is
,
with
75 Fig. 2.
f can be written as: f= 1 - exp(-[(T-T0)/i~]
,s9
69.
AT
Volume 96A, number 6
PHYSICS LETTERS
a great reduction of "free volume". This new state can be considered as energetically more far from the liquid state than the state obtained before annealing. A greater energy is necessary to have the transition A ~ L (for example 5 kJ rather than 1 k J) and the peak is more important. This phenomenon is well observed by Chen [4] on annealed samples at 540 K for 1 h or 50 h (Tg = 583 K). More detailed studies will be necessary to determine theoretically q, taken in our work as a parameter, including all the effects. If we consider that q is a function o f f , T and T the correct variation o f ACt~ with T should be represented by a combination o f curves represented on figs. 2 and 3. To conclude, we can say that the first-order kinetic analysis gives a satisfactory description o f the observed
11 July 1983
phenomena and shows that the glass transition cannot be considered as a thermodynamic "diffuse" transition o f second order. Mr. Shimoji is grateful to the French authority for financial assistance for his stay in the Grenoble University.
References [1] H.S. Chen and T.T. Wang, J. Appl. Phys. 52 (1981) 5898. [2] M.H. Cohen, G.S. Grest, Phys. Rev. B20 (1979) 1077; B21 (1980) 4113. [3] M. Cyrot, J. Phys. (Paris) 41 (1980) C8-107. [4] H.S. Chen, Proc. 4th Int. Conf. on Rapidly quenched metals (Sendal), eds. T. Matsumoto and K. Suzuki, Japan. Inst. Metals (1982) 495.
301
PHYSICS LETTERS
11 July 1983
A KINETIC ANALYSIS OF THE G L A S S - L I Q U I D TRANSITION P. DESRE, M. SHIMOJI 1, A. PASTUREL and P. HICTER Laboratoire de Thermodynamique et Physico-Chimie Mdtallurgiques, L TPCM associ~ au CNRS {LA 29), ENSEEG Domaine Universitaire bp 75, 38402 Saint-Martin-d'Hkres, France Received 4 January 1983
We show that the glass-liquid transition may be analyzed from a first-order kinetic law and we calculate in a simple way the variation of the apparent heat capacity with the temperature.
The thermal properties o f amorphous materials are strongly controlled by kinetic conditions imposed in the process of preparation. The glass transition temperature is known to be not a constant of the material but a function of experimental conditions: Tg is higher for faster heating or cooling rates. This value depends also on thermal history of the sample. In fig. 1 we have schematized the change in specific heat Cp, at constant pressure as function of the temperature as presented from experimental measurements. Most of the measurements are done at constant heating rate and show a pronounced peak in the neighbourhood of Tg. For a few materials, which do not crystallize easily (glassy polymers [1] ) Cp is a monotonic de-
i Present address: Department of Chemistry, Faculty of Science, HokkMdo University, Sappora 060, Japan. Cp
T Fig. 1. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
creasing function of Tg when measured at constant cooling rate. A microscopic calculation of thermodynamic functions of amorphous materials (metallic and non-metallic glasses) were made by Cohen and Grest [2] using a percolation theory within the framework of the "free volume" model. The percolation approach was also employed in discussing the glass transition of metals qualitatively by Cyrot [3] and Chen [4]. Following an equivalent approach, we call f the fraction of "liquid" produced during the glass-liquid transition observed at a constant heating rate/~ = dT/ dt. The fraction of "liquid" in this representation has to be understood as the fraction of the volume of the system whose atomic configurations are defrozen. This is a simple and global way of representing all effects near Tg. Owing to the fact that the observed behaviour of Cp is irreversible near the glass transition and the fact that the amorphous state is a non-equilibrium state compared to the relaxated structure and undercooled liquid, we consider that the transformation is spontaneous on heating. Kinetic effects are included in the measurements of specific heat at constant heating rate. In this case the apparent (and observed) change in heat capacity is the consequence o f the spontaneous defrustration of atomic configurations frozen at T < Tg. This analysis is according to the curve of the fig. 1. In this case, the total differential of the enthalpy H is written at constant pressure: 299
Volume 96A, number 6
PHYSICS LETTERS
dH = (8H/OT)L pdT + (all~Of)T, p dr,
11 July 1983
&C~
J Kh~ole "~
A.IO0 Q=50o0 J
or , ,,T. 0.1 a.u,
60
dtt (~f)T df/dt dT - Cp + dT-~ ,p
~ /",t. 1 a.u.
"
Cp = dH/dT, the apparent heat capacity of the system,
40
~
'~]I............. T: 5 a ,U '
/t
is written, using the specific heat at constant pressure of amorphous and liquid phases, i.e. C(A) and Cp(L):
i \ \ //
",,
%
20
C ; = c ( A ) + f (c(L ) -- C(pA)) + q T - l df[ at ,
: f ."/ ]," /'
where q = (aH]af)T,p is a positive quantity of heat, taken as constant in first approximation. The more simple law for df]dt is a first-order kinetic law:
dfldt = r ( 1
-f).
It is well known that in the neighbourhood of Tg, the viscosity obeys a Vogel-Fulcher law and not an Arrhenius law. It is certainly due to the presence of the relaxation spectrum pointed out by many authors. Thereby we choose to write:
df/dt = e x p [ - A / ( T - TO) ] (I - f ) . In this relation, T O is the so-called "ideal" transition temperature. In view of simplicity and to study the general behaviour of the phenomena we have chosen a preexponential term equal to one for the V o g e l Fulcher law. But for the kinetic behaviour there is another preexponential term which has the inverse of time for unit - on this point we do not have any knowledge. So, with a choice of a total preexponential term equal to one, we are constrained to use an arbitrary time unit (a.u.) With the following conditions:
T=To,
t=O,
f(To,O)=O
and T = T o + f ' t ,
/ . i ........ 2,s
e x p [ - A / ( T - TO) ]
AC~ J K:=mole ' F\ I! ~Q.75oo J /
60
4o
+ (A/T) Ei(-A/(T - TO))}, 20
x e- t
Ei(-x) = f -7- dt.
i /
AT
i ., soooJ ',,~
// i! I •f
//
A.100 1" .1 a.u.
\
\
,,,\
/ " " " ,,. ',,\ , \
! ?: /i /
... ,,.N
/ Q=2sooJ
/,'i
.... j i:~ 29
49, Fig. 3.
300
190
We now want to study the effects o f q and T on the apparent specific heat at constant pressure C~. For this we use the ACp=C(L)-C(p A) ~ 24 J I ( - 1 s o l - 1 value generally observed. The constant A o f the V o g e l Fulcher law is taken to be equal to 100, which is a reasonable value for the chosen writing (reminding that we have taken the preexponential term in the V o g e l Fulcher law equal to one). In fig. 2, we give the curves AC/~(AT).= C~ - C ( h ) for q = 5 kJ and for differents values of T (from 0.1 a.u. to 5 a.u.). The Cp peak moves towards higher temperatures as T mcreases. In fig. 3, we study the effect of variation of q for a heating rate T = 1 a.u.. Obviously, the characteristic peak of C~ increases with q. After a thermal treatment at T < Tg, the metallic glass undergoes a structural relaxation and there is
,
with
75 Fig. 2.
f can be written as: f= 1 - exp(-[(T-T0)/i~]
,s9
69.
AT
Volume 96A, number 6
PHYSICS LETTERS
a great reduction of "free volume". This new state can be considered as energetically more far from the liquid state than the state obtained before annealing. A greater energy is necessary to have the transition A ~ L (for example 5 kJ rather than 1 k J) and the peak is more important. This phenomenon is well observed by Chen [4] on annealed samples at 540 K for 1 h or 50 h (Tg = 583 K). More detailed studies will be necessary to determine theoretically q, taken in our work as a parameter, including all the effects. If we consider that q is a function o f f , T and T the correct variation o f ACt~ with T should be represented by a combination o f curves represented on figs. 2 and 3. To conclude, we can say that the first-order kinetic analysis gives a satisfactory description o f the observed
11 July 1983
phenomena and shows that the glass transition cannot be considered as a thermodynamic "diffuse" transition o f second order. Mr. Shimoji is grateful to the French authority for financial assistance for his stay in the Grenoble University.
References [1] H.S. Chen and T.T. Wang, J. Appl. Phys. 52 (1981) 5898. [2] M.H. Cohen, G.S. Grest, Phys. Rev. B20 (1979) 1077; B21 (1980) 4113. [3] M. Cyrot, J. Phys. (Paris) 41 (1980) C8-107. [4] H.S. Chen, Proc. 4th Int. Conf. on Rapidly quenched metals (Sendal), eds. T. Matsumoto and K. Suzuki, Japan. Inst. Metals (1982) 495.
301