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Fictive pressure effects in structural relaxation
Journal of Non-Crystalhne Solids 102 (1988) 231-239 North-Holland, Amsterdam
231
FICTIVE P R E S S U R E E F F E C T S I N S T R U C T U R A L R E L A X A T I O N P.K. G U P T A Department of Ceramic Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210, USA
Structural relaxation is treated in terms of two independent internal parameters, namely fictive temperature, Tf, and fictive pressure, Pf. Rate equations for Tf and Pf are derived. It is shown that, in general, the fictive pressure changes during isobaric cooling experiments and the key parameter controlling this change is the configurational expansion coefficient. Furthermore it is pointed out that by examining the shape of the relaxation paths in the A V - / ~ S space the relaxation coefficients can be evaluated readily.
1. Introduction
Since the classic papers of Davies and Jones [1], putting the thermodynamic and kinetic properties of glasses on a firm phenomenological basis, much understanding has been gained and summarized [2-8]. Phenomenologically a glassy state is described by two sets of parameters; the external parameters (which are necessary and sufficient to describe an equilibrium state) such as temperature (T), pressure ( P ) and composition ( C ) and the internal (also called the order or structural) parameters, Z - ( Z l . . . . . Z N). In an equilibrium state (i.e., the supercooled liquid state at some T and P ) Z assumes its equilibrium value, Z~e)(T, P, C ) , for which the Gibbs' free energy of the system is minimized. That the description of glasses using only a single internal parameter ( N = 1) is adequate for m a n y technological applications has been amply demonstrated by the use of the concept of fictive temperature (Te), which was first introduced by Tool a little more than fifty years ago [9]. Davies and Jones pointed out that one could define fictive pressure pf as an equivalent and alternative way to describe a single internal parameter system. However there is growing evidence that most (and possibly all) glass forming systems require more than one internal parameter ( N > 1) for a valid characterization of their behavior. This evidence appears in two forms: 0022-3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(i) the measured values of the Prigogine-Defay Ratio are larger than one [10,11], and (ii) the crossover experiments in which two glass samples with different histories but prepared in a way that at the same T and P they have the same value of one property (for example volume V), show differences in their subsequent relaxation behaviors [12-14]. While it is clear that one internal parameter is insufficient for description of a glassy state, no other information is available at present about the value of N. There is one case however where one expects N = 2. This case is that of a glass produced by quenching a pure melt at a rate fast enough to freeze all the configurational degrees of freedom but slow enough for the vibrational degrees of freedom to remain in equilibrium. The two internal parameters are clearly the temperature ( T0) and the pressure (P0) at the start of the quenching. Motivated partly by this example and partly by the enormous utility of the fictive temperature concept, a treatment of the thermodynamics and kinetics of the glassy state with two internal parameters is presented in this paper. This treatment casts the two internal parameters in the form of fictive temperature (Tf) and fictive pressure (Pf). While more general multi-parameter ( N > 2) approaches have been reported by others [6,8], we, by limiting our treatment to two parameters, hope to capture much of the intrinsic simplicity of single parameter models while at the same
232
P.K. Gupta / Fictive pressure effects in structural relaxation
time generalize to account for the important memory effects as are shown in cross-over type of experiments.
2. Thermodynamics of two internal parameter systems
For convenience but without loss of generality we present our treatment for the case of a homogeneous one component system. The external parameters for this system are T and P. Let the internal parameters be Z a and Z 2. All properties of a non-equilibrium state including the Gibbs free energy per mole (G) are functions of T, P, Z 1 and Z2:
G = G(T, P, Z,, Z2).
(1)
2.2. Glassy state A glassy state is characterized by specifying the values of Z 1 and Z 2. Therefore the free energy G (g) of a glass depends on both the external and the internal parameters:
G (g)= G(g)(T, P, Zl, Z2).
2.3. Near-equilibrium description of glasses At a given T and P, one can expand the free energy of a glass about the equilibrium state as follows: G(g)(T, P, Z 1, Z2)
2 +1 E Bible)(r, P)[Z~-z(ie)(T, p)]2 i=1
Equilibrium state, for a specified set of external parameters, is defined by the minimum in G and is characterized by two equations: (i) the extremum condition: (e) = 0
?)
=
2.1. Equilibrium state
aG
(6)
( i = 1, 2).
(2)
+ ....
(7)
If the expansion is considered only up to the second order term, one obtains a near-equilibrium description of a glass.
OZi T , P
2.4. Definitions of Tf and P/
(ii) The stability condition: the symmetric second derivative matrix [B] where
For a glass (specified by values of Z 1 and Z2) the fictive temperature Tf(Z1, Z2) and the fictive pressure Pf(Za, Z2) are defined as the temperature and pressure where the equilibrium values Z~e) and Z2(e) are equal to the specified values Z 1 and Z 2. In other words
~2G
(e)
B~;)(T' P)=- aZ, OZk T, P
(i, k = 1 2)
is positive definite. Without any loss of generality, one can choose Z1 and Z 2 such that [B] is diagonal. The stability condition can then be written as
B(iie)(T, P ) > 0
(i = 1, 2).
z~e)(T, P )
(i = 1, 2).
(4)
Substitution of Z~e) in eq. (1) gives the equilibrium free energy G (e) which depends only on the external parameters:
G '°~ = G(°)(T, e).
for
i
i = 1, 2.
(8)
(3)
Eq. (2) can be solved to give the equilibrium values of the internal parameters: Z (e) =
z(e)(Tf(Zl, Z2), Pf(Zl, Z 2 ) ) = Z
(5)
2.5. Expression of free energy in terms of (T/, Pf) Eq. (7) can be expressed in terms of ( Tf, Pf) easily if one postulates a linear dependence on T and P of Z~e). In other words (for i = 1, 2):
z?)(T, P) =Z~e)(T0, P o ) + ( T - r o ) O i + ( P - P o ) f l , ,
(9)
P.K. Gupta / Fictivepressure effects in structural relaxation where
Eq. (18) is the desired expression for the thermodynamic Gibb's free energy of a glassy state. Eqs. (15), (16) and (17) also show that AC and AK are positive definite while Zia could assume either sign. It is important to emphasize that all coefficients in eq. (18) are experimentally obtainable quantities.
Oi= ~Z~) 3T Vo,po fli - 3z(e) OP To,Po are assumed constants. Substitution of eq. (9) in eq. (8) gives
Z , - z}e)(T, P) = (Tf - To)O, + (Pf - Po)fl,(10) Substitution of eq. (10) into eq. (7) gives:
G(g)(T, P, Tf, Pf)
2.6. Expressions for other thermodynamic functions of glass By using definitions of entropy S ( = - 3G/3T) and of volume V(= 3G/3P), it is straight forward to derive their expressions by using eq. (18): AC
Sg(T, P, Tf, Pf)=s(e)(T, P) + - - T - ( T f - T)
: G,e)(T, P ) 2
+½ ~-~ B . [ ( T f - T ) O i + ( P r - P ) f l i ] 2.
= V(e)(T, P ) - v ( e ) A K ( P - P )
AC = Cp(~)- C~(g),
(12)
AK---- K (e) - U ( g ) ,
(13)
,~O/ ~ Ot(pe) -- Ot(g) ,
(14)
where Cp, K r and ap are the isobaric heat capacity, isothermal compressibility and isobaric expansion coefficient, respectively, then, by taking the appropriate temperature and pressure derivatives of eq. (11), it can be shown that 2 T
(19)
v g ( r , P, Tf, Pt)
If one defines
=
-- v(e)Aa(Pf - P),
(11)
i=1
AC
233
~_, B, Oi2 > 0,
+ v(e)Aa(Tf - T).
(20)
Expression for the enthalpy H (g) c a n be obtained similarly by using its definition, H = G + TS.
2.7. Expressions for the affinities The expression for the affinities corresponding to Tf and Pf can also be obtained from eq. (18). These are:
(15)
i=1
T,P,Pf
V
2
a K = a / V (¢) Y'~ BiiBi2 > O,
(21)
(16)
\
i=1
3 o i = V ( e ) a K ( p f - P) _ v ( e , a , ~ ( r f _ r ) .
2
Aa = - 1 / V (~) E B,O,B,.
(17)
]
(22)
i=1
Substituting eqs. (15), (16) and (17) in eq. (11), we get
(3G)_~ = - ( S ( e ) - s ( g ) )
G(g)(T, P, Tf, Pf) = G
'
(23)
Similarly by comparing eq. (22) and (20), we obtain
V(¢)AK + - AC.T. ~( f - T)2 + -----~ ~ ( p f _ p): - V(~)Aa(Tr -- T ) ( P r - P ) .
By comparing eqs. (21) and (19), we see that
( ~G ) = v ( e ) (18)
(24)
234
P.K. Gupta / Fictivepressure effects in structuralrelaxation
(25)
(ii) thermodynamic coupling due to nonvanishing Aa. This causes the driving force for say Tf to depend on Pf - P (see eq. (21)). In general both Lrp and Aa are nonzero and could be of either sign. Eqs. (27) and (28) are linear if Lij's depend only on T and P. However, as mentioned earlier, they can also depend on Tf and Pf in which case eqs. (27) and (28) become nonlinear.
(26)
3.2. Properties of the relaxation coefficient matrix
3. Kinetics of structural relaxation
3.1. Linear laws Using the principles of the Non-Equilibrium Thermodynamics [15], linear laws of relaxation can be written for time variations of Tr and Pf.
aTf
aG + Lre_~f
at
aPe
at
LTTvtf
[
aG
'
aG}
Le:r~ff + L e e ~ f f
[A]
'
where the kinetic coefficient are such that Lre = LeT and the [L] matrix is positive definite. This implies that L r r > 0, Lee > 0 and LrrLee - L2p > 0. The positive definiteness of [L] guarantees amonotonic decrease of free energy during the relaxation process. These kinetic coefficients depend on T and P; approximately as the reciprocal of the viscosity. They may also depend on Tf and Pf. Expressions for Lij can be postulated following the approaches of Narayanaswamy [16] and of Scherer [17]. Substituting eqs. (21) and (22) into eqs. (25) and (26), the linear kinetic laws for structural relaxation follow:
0Tf _An(Tf - T ) + A 1 2 ( P f - P ) ,
(27)
apf at
(28)
at
A21(Tf -- T) + A22(P f - P ) ,
where
AC
All = Lrr T
LreV(e)Aa'
A12 = - LTTV(e)AOt + LrpV(e)AK, A21 = LTe AC T
(29)
LeeV(~)Aa,
Eq. (29) shows that the matrix of relaxation coefficients is a product of two matrices:
[A]=[LI[aG"],
(3O)
where [L] is the real symmetric, positive definite matrix of kinetic coefficients and [AG"] is a real symmetric, positive definite matrix of thermodynamic coefficients such as
[ 02[G(g)- G(e)] ] a<',
=
Tf,e,
and
,,
[ 02[G(g)- G(e)] ]
By virtue of the stability condition (eq. (3)), AG" is a positive definite matrix. Although [A] is not necessarily symmetric, it follows, from the properties of [L] and [AG"] matrices, that [A] will always have real positive eigenvalues [8]. Positive definiteness of [A] implies that:
All --kA22 ~>0, ALIA22 - AI2A21 > 0.
(31)
If )`1 and )`2 are the two eigenvalues of [A] and
A22 = - LTeV(e)Aa + Leev(e)AK.
)`1 -~ )`2 then )`11 is the largest relaxation time and
Eqs. (27) and (28) are the desired relaxation equations. They represent coupled first order differential equations and can be solved for given Tf, Pf initial conditions. The coupling between the relaxations of Tf and Pf originates from two sources: (i) kinetic coupling due to nonvanishing kinetic coefficient LTe, and
)`21 is the smaller relaxation time. The solutions of eqs. (27) and (28) will in general be represented as the sum of two exponentials. Relaxation behavior at short times will be dominated by )`71 and at large times the relaxation will be controlled by )`~-1. Only in the cases when either )`1 =)'2, or when the initial state lies on an eigen-direction of A will the relaxation be single exponential.
P.K. Gupta / Fictivepressure effects in structural relaxation
235
3.3. Relaxation equation for an experimentally measurable property (qO
3.4. Some additional constraints on the kinetic coefficients
For any observable property (q,) such as volume or enthalpy, it can be shown that the deviation from the equilibrium value is given by:
Intrinsic constraints on [L] and [AG"] matrices have already been mentioned. According to these:
,~ -
#~)(T, P)
: [A(~bT)](Tr - T) + [A(ffp)](pf-- p )
(32)
(38)
(LTp) 2 < LrrLpp and AC>O, AK>0,
where
(39)
(Aa) 2 < A C A K / T V (e).
and
(
t 'e, - (
can be treated as constants with respect to Tf and Pf. The isobaric and isothermal relaxation equation follows from eq. (32):
aep T,P-- [A(~T)] aTf 0t
aPf at'
(33)
where the expressions for aTe/at and aPf/at are given by eqs. (27) and (28). More specifically, the relaxation equations for volume and entropy (two properties which can be measured experimentally) are: aV at-(A12~--~-CT-A22V(e)Aa)(Tf- T )
+ (A~v(°)a/~- ~v(~>a~)(t,, - p), (34)
as
LTT > O, Lpp > 0,
ac)(7. ~
at - ( A21V(e)A°t - A l l - T --
- T)
+ (AllV(e)zaa - A z , V ( e ) A K ) ( p f - p ) .
(35) These equations can be rewritten by using eqs. (19) and (20) as follows: aS -- A l l ( S - S(e)) .'J-A21(V- V(e)), at 0V at - A I 2 ( S - S(e)) - A 2 2 ( V - v(e))"
(36) (37)
Eqs. (36) and (37) show that the entropy and volume relaxations are also coupled.
These restrictions permit undershoots (or overshoots) in properties in single temperature (or pressure) jump experiments. To see this clearly it is useful to rewrite eq. (25) and (26) by using eq. (23) and (24) as follows:
OTf at
L r T ( S - S(~)) + LTp(V-- V~e))'
(40)
OPf L T p ( S - S (e)) + L p p ( V - V (e)) (41) at Consider a system for which Aa > 0 and LTp = O. These values are permitted by eqs. (38) and (39) and imply Alz < 0 and A21 <0. When such a system undergoes relaxation following an isobaric single-step downward temperature jump, eq. (28) shows that aPf/atlt= o is positive. Thus Pf initially increases and then goes through a maximum before relaxing back to the initial value (which is also the equilibrium value). Clearly in the final approach to equilibrium aPt~at < 0. Eq. (41) shows (because LTp = 0) that during this phase (when aPf/at < 0). v is less than V (~). This implies that the volume must undershoot before approaching equilibrium. While there are no fundamental reasons not to expect such an undershoot, experimentally this phenomenon has never been observed [18]. Thus it appears that additional restrictions must be placed on LTp (since Aa is known to be of either sign) to ensure monotonic relaxation of properties during simple pressure or temperature jump experiments. It is easy to show from eqs. (40) and (41) that:
LTp > 0
(if Aa > 0)
P.K. Gupta / Fictive pressure effects in structural relaxation
236
and
LTp < 0
(Pf-P) (if Aa < 0)
(42)
AV=O
eP~o
:':
I
(Tf -T~
(43a)
Similar reasoning using isothermal pressure experiment leads to the following inequality (from eq. (34)): AotA12 < A22A K .
f.
/, // / 7" /
are necessary (but not sufficient) conditions for monotonic relaxation of volume. Eqs. (34) and (35) imply some constraints on A12 and A21. Recall from eq. (29) that the signs of A12 and A21 are still unconstrained inspite of the knowledge, for example, that LTe > 0 if Aa > 0. In an isobaric downward temperature jump experiment (OS/3t)t= o must be negative. Therefore it follows from eq. (35) that AogTVA21 < A11AC.
AS=O
(43b)
4. Relaxation paths in the fictive plane It is useful to examine relaxation of a system by following its path in the fictive plane which is defined by the axes corresponding to (Tf - T) and (Pf - P). A point in this plane specifies the deviation of a glass from its equilibrium state. Clearly the origin is the only point which corresponds to the equilibrium state. All other points represent nonequilibrium states. During structural relaxation the state of the system traces a path in the fictive plane from some initial state towards the origin. The curve is termed the relaxation path. Some properties of relaxation paths are: (i) All relaxation paths end at the origin. (ii) A relaxation path is a straight line if the line joining the origin to the initial state vector is an eigenvector of matrix A. This implies that all relaxation paths are straight fines if the eigenvalues of A are equal. (iii) The tangent to the relaxation path at t = 0 is along the major eigenvector of A matrix. This is the direction of fast relaxation. (iv) In general relaxation paths will be C-shaped (and not S-shaped). Fig. 1 shows some features of the fictive plane. Point " I " represents the initial nonequilibrium
~Tf=n Ot v Fig. 1. Relaxation path in the fictive plane defined by (Tf - T) and (Pf - P ) coordinates. The path shown is for a system (with Aa > 0 and A21 < 0) undergoing relaxation following an isobaric downward jump in temperature. The relaxation path begins at the initial state marked I and ends at the equilibrium state at the origin. It shows that the fictive pressure increases initially and subsequently returns to the original value which is also the equilibrium value. The straight lines labelled A x = 0 are drawn assuming A a > 0 and A21 < 0. The arrow on a straight line indicates that for all points in the direction of the arrow A x > 0.
state of a system which has undergone an isobaric downward jump in temperature by an amount corresponding to ( T f - T ) I . The curved path with arrow shows the relaxation path of a system (for which A21 < 0). On the straight line labelled AV= 0, V (g) = V (e). The slope of this straight line is (Aa/AK). For all states to the right of this line V (g) > V (~). This follows from eq. (20) which can be rewritten as (Pf-
P ) --
Tf-- T)-
[
"g----~( A - - ~ -
The straight line AS = 0 is the line corresponding to S (g) = S (~). Again all points to the right of this line corresponds to states S (g) > S (e) (provided Aa > 0). The equation of this line follows from eq. (19) which can be rewritten as ( P f - P ) = ( vza~x~](TfAC 1 ] _ T) - [ S(g) ] v (-e -S(~) T~ A -
1
P.K. Gupta / Fictive pressure effects in structural relaxation
Also shown in fig. 1 is a line passing through the origin which corresponds to states where OPf/~t = 0. Its equation can be obtained by rewriting eq. (28):
A2,(Tf_T) -
(pf_p)_
I
AV
{OPel
=[A,2AC/r= A2VA.)
= |
+
(V-V~e))
ielaxatioPath" n
77j"
]
( T f - T)
(ov/ot) [A22AKV- Aa2V/ia] "
All points above this line correspond to (~V/~t) > 0. It can be shown using inequalities (43) that the slope of this line is less than the slope of a line for AV = 0. This is because (~V/~t) < 0 is a more restrictive condition than V (g) > V (e).
5. Relaxation path in the AV-AS plane It is clear from eqs. (27) and (28) that the relaxation behaviour is completely determined by knowledge of the elements of the [A] matrix. In this section we show that considerable information can be obtained about these coefficients by studying simultaneously the relaxation of two independent properties such as volume and entropy and examining the relaxation path of the system in the space defined by A V ( - V (g)- V (e)) and A S ( - S (g) - S(e)). Fig. 2 shows the AV-AS plane. All points (except the origin) represent nonequilibrium states. All equilibrium states are represented by the origin. In order to appreciate the usefulness of the AV-AS space, consider again relaxation of a system following isobaric down-
/ P':P I /'-
This line coincides with the line / i V = 0 when L r e = 0, thus causing undershooting of the volume (when Aa > 0) during isobaric temperature jump relaxation curves. All points to the right of this line correspond to states where (aPf/St) > 0, provided A 22 > 0. The line which corresponds to states (3Tf/Ot) = 0 is also shown in fig. 1. All points below this line correspond to (~Tf/8t)< 0 if A12 < 0. Using eq. (34), a line can be drawn corresponding to 8V/Ot = 0. The equation for this line will be
(Pf-P)
---
237
/
AS -~ (S- S~e~<
jJ
/
Fig. 2. Relaxation path in the A V - A S plane for a system (Aa > 0 and A21 < 0) undergoing relaxation following an isobaric downward temperature jump. The origin corresponds to the equilibrium state. The slope of Pf = P line is given by T V Aa/AC. The shape of the relaxation path gives direct information about the elements of the relaxation matrix [A] or [A ~]. The dashed arrow labelled (1) represents the major eigen-direction and the one labelled (2) corresponds to the minor eigen-direction of [A 1].
ward temperature jump. Furthermore Aa will be considered positive for the purpose of illustration. The initial state (I) of such a system lies on the line corresponding to Pf = constant. The equation of this line is obtained from eqs. (19) and (20):
Av=[TV(e)Aa] ASAC - (~r - I)
T(V(e)/ia)2(ef
--
P),
AC where ~r is thc Prigogine-Defay Ratio, i.e. ,ff
ACA K TV(e)( Aa)2 "
The line corresponding to Pf = P goes through the origin and all points to the right of the line correspond to states for which Pf > P. The relaxation path will be a straight line (joining I to the origin) only if Pf remained constant. This is true
238
P.K. Gupta / Fictive pressure effects in structural relaxation
only if A a = 0 and LTe=O in which case the Pf = P line will coincide with the horizontal axis (AV= 0). In general the relaxation path will be curved. When A21 < 0, the relaxation will deviate as shown in fig. 2 towards the fight side of Pf = P line. The relaxation path, of course, will end at the origin. It is also clear that if there is no overshoot (or undershoot) in AV or AS during relaxation, the relaxation path must be contained entirely within the rectangle formed by the vertical and horizontal lines passing through the origin and the initial state. Furthermore cross-over type of experiments can be readily interpreted in AV-AS space. This will be demonstrated by different relaxation paths of two samples (prepared with different histories) but in the same initial state. Eqs. (36) and (37) describe the shape of the relaxation path. The relaxation is described by the matrix [A 1] defined as:
(AH -A2 / [ A 1] =
_A12
A22 1"
It has the same eigenvalues as [A] matrix and the eigenvectors of [A 1] are simply related to the eigenvectors of [A l matrix. In other words knowledge of [A 1] is equivalent to the knowledge of [A]. The tangents to the relaxation paths at the initial state and the origin represent the major and minor eigenvectors of [ A1] matrix. Furthermore the maximum deviation of the relaxation path from the straightline path gives additional information about [A ~] matrix. It can be shown that three of the four elements of [A 1] or [A] matrix can be obtained from the shape of the relaxation path. Since the relaxation path can be readily plotted in the AV-AS plane from experimental data, it provides a convenient and quick method for determining the elements of the [A] matrix to within a scale factor (i.e., three of the four elements).
6. Summary and conclusions A two-internal parameter (namely the fictive temperature and fictive pressure) treatment is formulated using the principles of Nonequilibrium Thermodynamics. The relaxation equations ob-
tained for Tf and Pf are coupled because of thermodynamic (Aa ~ 0) and kinetic (Lrp -4:O) reasons. The coupled equations imply that in an isobaric temperature jump experiment the fictive pressure initially deviates from its equilibrium value and reapproaches the final value in the limit of long times. Similarly Tf deviates in an isothermal pressure jump experiment. By examining the influence of the coupled relaxation equations on volume and entropy changes, several new restrictions are derived for the kinetic coefficients. These are given by eqs. (42) and (43). Furthermore it is shown that these constraints are necessary but not sufficient to ensure a monotonic variation of volume during an isobaric temperature jump experiment. The concept of relaxation paths in the fictive plane ( T f - T, P e - P) is introduced and properties of relaxation paths are examined. It is shown that the elements of the relaxation matrix can be readily determined (to within a scale factor) by the shape of the relaxation path in the AV-AS space. This two parameter treatment retains much of the simplicity contained in one parameter models while it is more general and accounts for complex memory effects as observed in crossover type of experiments. It is proposed that the use of this model over a single parameter model (which is inadequate to explain memory effects) and over multiparameter models (where analytical complexity hinders clarity) should enhance our understanding of the technological aspects of the glassy state associated with the structural relaxation phenomenon.
References [1] R.O. Davies and G.O. Jones, (a) Adv. Phys. 2 (1953) 370; (b) Proc. Roy. Soc. (London) A217 (1953) 26. [2] G.W. Scherer, Relaxation in Glasses and Composites (John Wiley, New York, 1986). [3] S. Brawer, Am. Ceram. Soc. (1985). [4] C.T. Moynihan et al., Ann. NY Acad. Sci. 279 (1976) 15. [5] G.P. Johari, Les Houches Lectures, (Editions de Physique, Paris, 1982) p. 109. [6] A.J. Kovacs, Ann. NY Acad. Sci. 371 (1981) 38. [7] S.M. Rekhson, Glass Science and Technology, Vol. 3, eds. D.R. Uhlmann and N.J. Kreidl (Academic Press, New York, 1986) p. 1.
P.K. Gupta / Fictive pressure effects in structural relaxation [8] C.T. Moynihan and P.K. Gupta, J. Non-Cryst. Solids 29 (1978) 143. [9] A.Q. Tool and C.G. Eichlin, J. Am. Ceram. Soc. 14 (1931) 276. [10] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (10) (1976) 4136. [11] C.T. Moynihan and A.V. Lesikar, Ann. NY Acad. Sci. 371 (1981) 151. [12] H.N. Ritland, J. Am. Cer. Soc. 39 (12) (1956) 403. [13] P.B. Macedo and A. Napolitano, J. Res. NBS 71A (3) (1967) 231.
239
[14] J.A. Leake et al., J. Non-Cryst. Solids 61 & 62 (1984) 787. [15] S.R. DeGroot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1963). [16] O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 910) (1971) 491. [17] G.W. Scherer, J. Am. Ceram. Soc. 69 (5) (1986) 374. [18] (a) C.T. Moynihan, private communication (1987); (b) S.M. Rekhson, private communication (1987).
231
FICTIVE P R E S S U R E E F F E C T S I N S T R U C T U R A L R E L A X A T I O N P.K. G U P T A Department of Ceramic Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210, USA
Structural relaxation is treated in terms of two independent internal parameters, namely fictive temperature, Tf, and fictive pressure, Pf. Rate equations for Tf and Pf are derived. It is shown that, in general, the fictive pressure changes during isobaric cooling experiments and the key parameter controlling this change is the configurational expansion coefficient. Furthermore it is pointed out that by examining the shape of the relaxation paths in the A V - / ~ S space the relaxation coefficients can be evaluated readily.
1. Introduction
Since the classic papers of Davies and Jones [1], putting the thermodynamic and kinetic properties of glasses on a firm phenomenological basis, much understanding has been gained and summarized [2-8]. Phenomenologically a glassy state is described by two sets of parameters; the external parameters (which are necessary and sufficient to describe an equilibrium state) such as temperature (T), pressure ( P ) and composition ( C ) and the internal (also called the order or structural) parameters, Z - ( Z l . . . . . Z N). In an equilibrium state (i.e., the supercooled liquid state at some T and P ) Z assumes its equilibrium value, Z~e)(T, P, C ) , for which the Gibbs' free energy of the system is minimized. That the description of glasses using only a single internal parameter ( N = 1) is adequate for m a n y technological applications has been amply demonstrated by the use of the concept of fictive temperature (Te), which was first introduced by Tool a little more than fifty years ago [9]. Davies and Jones pointed out that one could define fictive pressure pf as an equivalent and alternative way to describe a single internal parameter system. However there is growing evidence that most (and possibly all) glass forming systems require more than one internal parameter ( N > 1) for a valid characterization of their behavior. This evidence appears in two forms: 0022-3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(i) the measured values of the Prigogine-Defay Ratio are larger than one [10,11], and (ii) the crossover experiments in which two glass samples with different histories but prepared in a way that at the same T and P they have the same value of one property (for example volume V), show differences in their subsequent relaxation behaviors [12-14]. While it is clear that one internal parameter is insufficient for description of a glassy state, no other information is available at present about the value of N. There is one case however where one expects N = 2. This case is that of a glass produced by quenching a pure melt at a rate fast enough to freeze all the configurational degrees of freedom but slow enough for the vibrational degrees of freedom to remain in equilibrium. The two internal parameters are clearly the temperature ( T0) and the pressure (P0) at the start of the quenching. Motivated partly by this example and partly by the enormous utility of the fictive temperature concept, a treatment of the thermodynamics and kinetics of the glassy state with two internal parameters is presented in this paper. This treatment casts the two internal parameters in the form of fictive temperature (Tf) and fictive pressure (Pf). While more general multi-parameter ( N > 2) approaches have been reported by others [6,8], we, by limiting our treatment to two parameters, hope to capture much of the intrinsic simplicity of single parameter models while at the same
232
P.K. Gupta / Fictive pressure effects in structural relaxation
time generalize to account for the important memory effects as are shown in cross-over type of experiments.
2. Thermodynamics of two internal parameter systems
For convenience but without loss of generality we present our treatment for the case of a homogeneous one component system. The external parameters for this system are T and P. Let the internal parameters be Z a and Z 2. All properties of a non-equilibrium state including the Gibbs free energy per mole (G) are functions of T, P, Z 1 and Z2:
G = G(T, P, Z,, Z2).
(1)
2.2. Glassy state A glassy state is characterized by specifying the values of Z 1 and Z 2. Therefore the free energy G (g) of a glass depends on both the external and the internal parameters:
G (g)= G(g)(T, P, Zl, Z2).
2.3. Near-equilibrium description of glasses At a given T and P, one can expand the free energy of a glass about the equilibrium state as follows: G(g)(T, P, Z 1, Z2)
2 +1 E Bible)(r, P)[Z~-z(ie)(T, p)]2 i=1
Equilibrium state, for a specified set of external parameters, is defined by the minimum in G and is characterized by two equations: (i) the extremum condition: (e) = 0
?)
=
2.1. Equilibrium state
aG
(6)
( i = 1, 2).
(2)
+ ....
(7)
If the expansion is considered only up to the second order term, one obtains a near-equilibrium description of a glass.
OZi T , P
2.4. Definitions of Tf and P/
(ii) The stability condition: the symmetric second derivative matrix [B] where
For a glass (specified by values of Z 1 and Z2) the fictive temperature Tf(Z1, Z2) and the fictive pressure Pf(Za, Z2) are defined as the temperature and pressure where the equilibrium values Z~e) and Z2(e) are equal to the specified values Z 1 and Z 2. In other words
~2G
(e)
B~;)(T' P)=- aZ, OZk T, P
(i, k = 1 2)
is positive definite. Without any loss of generality, one can choose Z1 and Z 2 such that [B] is diagonal. The stability condition can then be written as
B(iie)(T, P ) > 0
(i = 1, 2).
z~e)(T, P )
(i = 1, 2).
(4)
Substitution of Z~e) in eq. (1) gives the equilibrium free energy G (e) which depends only on the external parameters:
G '°~ = G(°)(T, e).
for
i
i = 1, 2.
(8)
(3)
Eq. (2) can be solved to give the equilibrium values of the internal parameters: Z (e) =
z(e)(Tf(Zl, Z2), Pf(Zl, Z 2 ) ) = Z
(5)
2.5. Expression of free energy in terms of (T/, Pf) Eq. (7) can be expressed in terms of ( Tf, Pf) easily if one postulates a linear dependence on T and P of Z~e). In other words (for i = 1, 2):
z?)(T, P) =Z~e)(T0, P o ) + ( T - r o ) O i + ( P - P o ) f l , ,
(9)
P.K. Gupta / Fictivepressure effects in structural relaxation where
Eq. (18) is the desired expression for the thermodynamic Gibb's free energy of a glassy state. Eqs. (15), (16) and (17) also show that AC and AK are positive definite while Zia could assume either sign. It is important to emphasize that all coefficients in eq. (18) are experimentally obtainable quantities.
Oi= ~Z~) 3T Vo,po fli - 3z(e) OP To,Po are assumed constants. Substitution of eq. (9) in eq. (8) gives
Z , - z}e)(T, P) = (Tf - To)O, + (Pf - Po)fl,(10) Substitution of eq. (10) into eq. (7) gives:
G(g)(T, P, Tf, Pf)
2.6. Expressions for other thermodynamic functions of glass By using definitions of entropy S ( = - 3G/3T) and of volume V(= 3G/3P), it is straight forward to derive their expressions by using eq. (18): AC
Sg(T, P, Tf, Pf)=s(e)(T, P) + - - T - ( T f - T)
: G,e)(T, P ) 2
+½ ~-~ B . [ ( T f - T ) O i + ( P r - P ) f l i ] 2.
= V(e)(T, P ) - v ( e ) A K ( P - P )
AC = Cp(~)- C~(g),
(12)
AK---- K (e) - U ( g ) ,
(13)
,~O/ ~ Ot(pe) -- Ot(g) ,
(14)
where Cp, K r and ap are the isobaric heat capacity, isothermal compressibility and isobaric expansion coefficient, respectively, then, by taking the appropriate temperature and pressure derivatives of eq. (11), it can be shown that 2 T
(19)
v g ( r , P, Tf, Pt)
If one defines
=
-- v(e)Aa(Pf - P),
(11)
i=1
AC
233
~_, B, Oi2 > 0,
+ v(e)Aa(Tf - T).
(20)
Expression for the enthalpy H (g) c a n be obtained similarly by using its definition, H = G + TS.
2.7. Expressions for the affinities The expression for the affinities corresponding to Tf and Pf can also be obtained from eq. (18). These are:
(15)
i=1
T,P,Pf
V
2
a K = a / V (¢) Y'~ BiiBi2 > O,
(21)
(16)
\
i=1
3 o i = V ( e ) a K ( p f - P) _ v ( e , a , ~ ( r f _ r ) .
2
Aa = - 1 / V (~) E B,O,B,.
(17)
]
(22)
i=1
Substituting eqs. (15), (16) and (17) in eq. (11), we get
(3G)_~ = - ( S ( e ) - s ( g ) )
G(g)(T, P, Tf, Pf) = G
'
(23)
Similarly by comparing eq. (22) and (20), we obtain
V(¢)AK + - AC.T. ~( f - T)2 + -----~ ~ ( p f _ p): - V(~)Aa(Tr -- T ) ( P r - P ) .
By comparing eqs. (21) and (19), we see that
( ~G ) = v ( e ) (18)
(24)
234
P.K. Gupta / Fictivepressure effects in structuralrelaxation
(25)
(ii) thermodynamic coupling due to nonvanishing Aa. This causes the driving force for say Tf to depend on Pf - P (see eq. (21)). In general both Lrp and Aa are nonzero and could be of either sign. Eqs. (27) and (28) are linear if Lij's depend only on T and P. However, as mentioned earlier, they can also depend on Tf and Pf in which case eqs. (27) and (28) become nonlinear.
(26)
3.2. Properties of the relaxation coefficient matrix
3. Kinetics of structural relaxation
3.1. Linear laws Using the principles of the Non-Equilibrium Thermodynamics [15], linear laws of relaxation can be written for time variations of Tr and Pf.
aTf
aG + Lre_~f
at
aPe
at
LTTvtf
[
aG
'
aG}
Le:r~ff + L e e ~ f f
[A]
'
where the kinetic coefficient are such that Lre = LeT and the [L] matrix is positive definite. This implies that L r r > 0, Lee > 0 and LrrLee - L2p > 0. The positive definiteness of [L] guarantees amonotonic decrease of free energy during the relaxation process. These kinetic coefficients depend on T and P; approximately as the reciprocal of the viscosity. They may also depend on Tf and Pf. Expressions for Lij can be postulated following the approaches of Narayanaswamy [16] and of Scherer [17]. Substituting eqs. (21) and (22) into eqs. (25) and (26), the linear kinetic laws for structural relaxation follow:
0Tf _An(Tf - T ) + A 1 2 ( P f - P ) ,
(27)
apf at
(28)
at
A21(Tf -- T) + A22(P f - P ) ,
where
AC
All = Lrr T
LreV(e)Aa'
A12 = - LTTV(e)AOt + LrpV(e)AK, A21 = LTe AC T
(29)
LeeV(~)Aa,
Eq. (29) shows that the matrix of relaxation coefficients is a product of two matrices:
[A]=[LI[aG"],
(3O)
where [L] is the real symmetric, positive definite matrix of kinetic coefficients and [AG"] is a real symmetric, positive definite matrix of thermodynamic coefficients such as
[ 02[G(g)- G(e)] ] a<',
=
Tf,e,
and
,,
[ 02[G(g)- G(e)] ]
By virtue of the stability condition (eq. (3)), AG" is a positive definite matrix. Although [A] is not necessarily symmetric, it follows, from the properties of [L] and [AG"] matrices, that [A] will always have real positive eigenvalues [8]. Positive definiteness of [A] implies that:
All --kA22 ~>0, ALIA22 - AI2A21 > 0.
(31)
If )`1 and )`2 are the two eigenvalues of [A] and
A22 = - LTeV(e)Aa + Leev(e)AK.
)`1 -~ )`2 then )`11 is the largest relaxation time and
Eqs. (27) and (28) are the desired relaxation equations. They represent coupled first order differential equations and can be solved for given Tf, Pf initial conditions. The coupling between the relaxations of Tf and Pf originates from two sources: (i) kinetic coupling due to nonvanishing kinetic coefficient LTe, and
)`21 is the smaller relaxation time. The solutions of eqs. (27) and (28) will in general be represented as the sum of two exponentials. Relaxation behavior at short times will be dominated by )`71 and at large times the relaxation will be controlled by )`~-1. Only in the cases when either )`1 =)'2, or when the initial state lies on an eigen-direction of A will the relaxation be single exponential.
P.K. Gupta / Fictivepressure effects in structural relaxation
235
3.3. Relaxation equation for an experimentally measurable property (qO
3.4. Some additional constraints on the kinetic coefficients
For any observable property (q,) such as volume or enthalpy, it can be shown that the deviation from the equilibrium value is given by:
Intrinsic constraints on [L] and [AG"] matrices have already been mentioned. According to these:
,~ -
#~)(T, P)
: [A(~bT)](Tr - T) + [A(ffp)](pf-- p )
(32)
(38)
(LTp) 2 < LrrLpp and AC>O, AK>0,
where
(39)
(Aa) 2 < A C A K / T V (e).
and
(
t 'e, - (
can be treated as constants with respect to Tf and Pf. The isobaric and isothermal relaxation equation follows from eq. (32):
aep T,P-- [A(~T)] aTf 0t
aPf at'
(33)
where the expressions for aTe/at and aPf/at are given by eqs. (27) and (28). More specifically, the relaxation equations for volume and entropy (two properties which can be measured experimentally) are: aV at-(A12~--~-CT-A22V(e)Aa)(Tf- T )
+ (A~v(°)a/~- ~v(~>a~)(t,, - p), (34)
as
LTT > O, Lpp > 0,
ac)(7. ~
at - ( A21V(e)A°t - A l l - T --
- T)
+ (AllV(e)zaa - A z , V ( e ) A K ) ( p f - p ) .
(35) These equations can be rewritten by using eqs. (19) and (20) as follows: aS -- A l l ( S - S(e)) .'J-A21(V- V(e)), at 0V at - A I 2 ( S - S(e)) - A 2 2 ( V - v(e))"
(36) (37)
Eqs. (36) and (37) show that the entropy and volume relaxations are also coupled.
These restrictions permit undershoots (or overshoots) in properties in single temperature (or pressure) jump experiments. To see this clearly it is useful to rewrite eq. (25) and (26) by using eq. (23) and (24) as follows:
OTf at
L r T ( S - S(~)) + LTp(V-- V~e))'
(40)
OPf L T p ( S - S (e)) + L p p ( V - V (e)) (41) at Consider a system for which Aa > 0 and LTp = O. These values are permitted by eqs. (38) and (39) and imply Alz < 0 and A21 <0. When such a system undergoes relaxation following an isobaric single-step downward temperature jump, eq. (28) shows that aPf/atlt= o is positive. Thus Pf initially increases and then goes through a maximum before relaxing back to the initial value (which is also the equilibrium value). Clearly in the final approach to equilibrium aPt~at < 0. Eq. (41) shows (because LTp = 0) that during this phase (when aPf/at < 0). v is less than V (~). This implies that the volume must undershoot before approaching equilibrium. While there are no fundamental reasons not to expect such an undershoot, experimentally this phenomenon has never been observed [18]. Thus it appears that additional restrictions must be placed on LTp (since Aa is known to be of either sign) to ensure monotonic relaxation of properties during simple pressure or temperature jump experiments. It is easy to show from eqs. (40) and (41) that:
LTp > 0
(if Aa > 0)
P.K. Gupta / Fictive pressure effects in structural relaxation
236
and
LTp < 0
(Pf-P) (if Aa < 0)
(42)
AV=O
eP~o
:':
I
(Tf -T~
(43a)
Similar reasoning using isothermal pressure experiment leads to the following inequality (from eq. (34)): AotA12 < A22A K .
f.
/, // / 7" /
are necessary (but not sufficient) conditions for monotonic relaxation of volume. Eqs. (34) and (35) imply some constraints on A12 and A21. Recall from eq. (29) that the signs of A12 and A21 are still unconstrained inspite of the knowledge, for example, that LTe > 0 if Aa > 0. In an isobaric downward temperature jump experiment (OS/3t)t= o must be negative. Therefore it follows from eq. (35) that AogTVA21 < A11AC.
AS=O
(43b)
4. Relaxation paths in the fictive plane It is useful to examine relaxation of a system by following its path in the fictive plane which is defined by the axes corresponding to (Tf - T) and (Pf - P). A point in this plane specifies the deviation of a glass from its equilibrium state. Clearly the origin is the only point which corresponds to the equilibrium state. All other points represent nonequilibrium states. During structural relaxation the state of the system traces a path in the fictive plane from some initial state towards the origin. The curve is termed the relaxation path. Some properties of relaxation paths are: (i) All relaxation paths end at the origin. (ii) A relaxation path is a straight line if the line joining the origin to the initial state vector is an eigenvector of matrix A. This implies that all relaxation paths are straight fines if the eigenvalues of A are equal. (iii) The tangent to the relaxation path at t = 0 is along the major eigenvector of A matrix. This is the direction of fast relaxation. (iv) In general relaxation paths will be C-shaped (and not S-shaped). Fig. 1 shows some features of the fictive plane. Point " I " represents the initial nonequilibrium
~Tf=n Ot v Fig. 1. Relaxation path in the fictive plane defined by (Tf - T) and (Pf - P ) coordinates. The path shown is for a system (with Aa > 0 and A21 < 0) undergoing relaxation following an isobaric downward jump in temperature. The relaxation path begins at the initial state marked I and ends at the equilibrium state at the origin. It shows that the fictive pressure increases initially and subsequently returns to the original value which is also the equilibrium value. The straight lines labelled A x = 0 are drawn assuming A a > 0 and A21 < 0. The arrow on a straight line indicates that for all points in the direction of the arrow A x > 0.
state of a system which has undergone an isobaric downward jump in temperature by an amount corresponding to ( T f - T ) I . The curved path with arrow shows the relaxation path of a system (for which A21 < 0). On the straight line labelled AV= 0, V (g) = V (e). The slope of this straight line is (Aa/AK). For all states to the right of this line V (g) > V (~). This follows from eq. (20) which can be rewritten as (Pf-
P ) --
Tf-- T)-
[
"g----~( A - - ~ -
The straight line AS = 0 is the line corresponding to S (g) = S (~). Again all points to the right of this line corresponds to states S (g) > S (e) (provided Aa > 0). The equation of this line follows from eq. (19) which can be rewritten as ( P f - P ) = ( vza~x~](TfAC 1 ] _ T) - [ S(g) ] v (-e -S(~) T~ A -
1
P.K. Gupta / Fictive pressure effects in structural relaxation
Also shown in fig. 1 is a line passing through the origin which corresponds to states where OPf/~t = 0. Its equation can be obtained by rewriting eq. (28):
A2,(Tf_T) -
(pf_p)_
I
AV
{OPel
=[A,2AC/r= A2VA.)
= |
+
(V-V~e))
ielaxatioPath" n
77j"
]
( T f - T)
(ov/ot) [A22AKV- Aa2V/ia] "
All points above this line correspond to (~V/~t) > 0. It can be shown using inequalities (43) that the slope of this line is less than the slope of a line for AV = 0. This is because (~V/~t) < 0 is a more restrictive condition than V (g) > V (e).
5. Relaxation path in the AV-AS plane It is clear from eqs. (27) and (28) that the relaxation behaviour is completely determined by knowledge of the elements of the [A] matrix. In this section we show that considerable information can be obtained about these coefficients by studying simultaneously the relaxation of two independent properties such as volume and entropy and examining the relaxation path of the system in the space defined by A V ( - V (g)- V (e)) and A S ( - S (g) - S(e)). Fig. 2 shows the AV-AS plane. All points (except the origin) represent nonequilibrium states. All equilibrium states are represented by the origin. In order to appreciate the usefulness of the AV-AS space, consider again relaxation of a system following isobaric down-
/ P':P I /'-
This line coincides with the line / i V = 0 when L r e = 0, thus causing undershooting of the volume (when Aa > 0) during isobaric temperature jump relaxation curves. All points to the right of this line correspond to states where (aPf/St) > 0, provided A 22 > 0. The line which corresponds to states (3Tf/Ot) = 0 is also shown in fig. 1. All points below this line correspond to (~Tf/8t)< 0 if A12 < 0. Using eq. (34), a line can be drawn corresponding to 8V/Ot = 0. The equation for this line will be
(Pf-P)
---
237
/
AS -~ (S- S~e~<
jJ
/
Fig. 2. Relaxation path in the A V - A S plane for a system (Aa > 0 and A21 < 0) undergoing relaxation following an isobaric downward temperature jump. The origin corresponds to the equilibrium state. The slope of Pf = P line is given by T V Aa/AC. The shape of the relaxation path gives direct information about the elements of the relaxation matrix [A] or [A ~]. The dashed arrow labelled (1) represents the major eigen-direction and the one labelled (2) corresponds to the minor eigen-direction of [A 1].
ward temperature jump. Furthermore Aa will be considered positive for the purpose of illustration. The initial state (I) of such a system lies on the line corresponding to Pf = constant. The equation of this line is obtained from eqs. (19) and (20):
Av=[TV(e)Aa] ASAC - (~r - I)
T(V(e)/ia)2(ef
--
P),
AC where ~r is thc Prigogine-Defay Ratio, i.e. ,ff
ACA K TV(e)( Aa)2 "
The line corresponding to Pf = P goes through the origin and all points to the right of the line correspond to states for which Pf > P. The relaxation path will be a straight line (joining I to the origin) only if Pf remained constant. This is true
238
P.K. Gupta / Fictive pressure effects in structural relaxation
only if A a = 0 and LTe=O in which case the Pf = P line will coincide with the horizontal axis (AV= 0). In general the relaxation path will be curved. When A21 < 0, the relaxation will deviate as shown in fig. 2 towards the fight side of Pf = P line. The relaxation path, of course, will end at the origin. It is also clear that if there is no overshoot (or undershoot) in AV or AS during relaxation, the relaxation path must be contained entirely within the rectangle formed by the vertical and horizontal lines passing through the origin and the initial state. Furthermore cross-over type of experiments can be readily interpreted in AV-AS space. This will be demonstrated by different relaxation paths of two samples (prepared with different histories) but in the same initial state. Eqs. (36) and (37) describe the shape of the relaxation path. The relaxation is described by the matrix [A 1] defined as:
(AH -A2 / [ A 1] =
_A12
A22 1"
It has the same eigenvalues as [A] matrix and the eigenvectors of [A 1] are simply related to the eigenvectors of [A l matrix. In other words knowledge of [A 1] is equivalent to the knowledge of [A]. The tangents to the relaxation paths at the initial state and the origin represent the major and minor eigenvectors of [ A1] matrix. Furthermore the maximum deviation of the relaxation path from the straightline path gives additional information about [A ~] matrix. It can be shown that three of the four elements of [A 1] or [A] matrix can be obtained from the shape of the relaxation path. Since the relaxation path can be readily plotted in the AV-AS plane from experimental data, it provides a convenient and quick method for determining the elements of the [A] matrix to within a scale factor (i.e., three of the four elements).
6. Summary and conclusions A two-internal parameter (namely the fictive temperature and fictive pressure) treatment is formulated using the principles of Nonequilibrium Thermodynamics. The relaxation equations ob-
tained for Tf and Pf are coupled because of thermodynamic (Aa ~ 0) and kinetic (Lrp -4:O) reasons. The coupled equations imply that in an isobaric temperature jump experiment the fictive pressure initially deviates from its equilibrium value and reapproaches the final value in the limit of long times. Similarly Tf deviates in an isothermal pressure jump experiment. By examining the influence of the coupled relaxation equations on volume and entropy changes, several new restrictions are derived for the kinetic coefficients. These are given by eqs. (42) and (43). Furthermore it is shown that these constraints are necessary but not sufficient to ensure a monotonic variation of volume during an isobaric temperature jump experiment. The concept of relaxation paths in the fictive plane ( T f - T, P e - P) is introduced and properties of relaxation paths are examined. It is shown that the elements of the relaxation matrix can be readily determined (to within a scale factor) by the shape of the relaxation path in the AV-AS space. This two parameter treatment retains much of the simplicity contained in one parameter models while it is more general and accounts for complex memory effects as observed in crossover type of experiments. It is proposed that the use of this model over a single parameter model (which is inadequate to explain memory effects) and over multiparameter models (where analytical complexity hinders clarity) should enhance our understanding of the technological aspects of the glassy state associated with the structural relaxation phenomenon.
References [1] R.O. Davies and G.O. Jones, (a) Adv. Phys. 2 (1953) 370; (b) Proc. Roy. Soc. (London) A217 (1953) 26. [2] G.W. Scherer, Relaxation in Glasses and Composites (John Wiley, New York, 1986). [3] S. Brawer, Am. Ceram. Soc. (1985). [4] C.T. Moynihan et al., Ann. NY Acad. Sci. 279 (1976) 15. [5] G.P. Johari, Les Houches Lectures, (Editions de Physique, Paris, 1982) p. 109. [6] A.J. Kovacs, Ann. NY Acad. Sci. 371 (1981) 38. [7] S.M. Rekhson, Glass Science and Technology, Vol. 3, eds. D.R. Uhlmann and N.J. Kreidl (Academic Press, New York, 1986) p. 1.
P.K. Gupta / Fictive pressure effects in structural relaxation [8] C.T. Moynihan and P.K. Gupta, J. Non-Cryst. Solids 29 (1978) 143. [9] A.Q. Tool and C.G. Eichlin, J. Am. Ceram. Soc. 14 (1931) 276. [10] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (10) (1976) 4136. [11] C.T. Moynihan and A.V. Lesikar, Ann. NY Acad. Sci. 371 (1981) 151. [12] H.N. Ritland, J. Am. Cer. Soc. 39 (12) (1956) 403. [13] P.B. Macedo and A. Napolitano, J. Res. NBS 71A (3) (1967) 231.
239
[14] J.A. Leake et al., J. Non-Cryst. Solids 61 & 62 (1984) 787. [15] S.R. DeGroot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1963). [16] O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 910) (1971) 491. [17] G.W. Scherer, J. Am. Ceram. Soc. 69 (5) (1986) 374. [18] (a) C.T. Moynihan, private communication (1987); (b) S.M. Rekhson, private communication (1987).