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Non-exponential structural relaxation, anomalous light scattering and nanoscale inhomogeneities in glass-forming liquids
Journal of Non-Crystalline Solids 160 (1993) 52-59 North-Holland
iou~N,~ L
or
NON-CRYS LL ESOLIDS
Non-exponential structural relaxation, anomalous light scattering and nanoscale inhomogeneities in glass-forming liquids C.T. M o y n i h a n a n d J. S c h r o e d e r Departments of Materials Engineering and Physics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
Received 13 November 1992 Revised manuscript received 1 March 1993
Light scattering from glass-forming liquids exhibits an anomalous time dependence in the glass transition region, e4 maxima in the scattering intensity versus temperature curves during heating. It is shown that this behavior is consistent with the presence of nanoscale inhomogeneities (density fluctuations) which relax at different rates. It is suggested that this could be the source of non-exponential structural relaxation kinetics. An expression relating the size of these regions to structural relaxation kinetic parameters has been developed and predicts sizes in excellent agreement with those determined by other methods.
1. Introduction Structural relaxation is the kinetically impeded rearrangement of the structure of a liquid in response to changes in external variables such as t e m p e r a t u r e or pressure. It is commonly associated with the glass transition and manifests itself as a time dependence of properties such as specific volume or enthalpy following a t e m p e r a t u r e change. The solid lines in figs. 1 and 2 show schematically the variation in the macroscopic specific volume, v, as a liquid is cooled and then reheated through the glass transition region [1,2]. On cooling the rapid increase in the structural relaxation time with decreasing t e m p e r a t u r e initially prevents the structure from fully equilibrating (at the high t e m p e r a t u r e end of the transition region) and eventually results in freezing of a non-equilibrium structure (at the low temperature end of the transition region). The result is the smooth v versus T cooling curve of fig. 1, whose slope monotonically changes from liquidCorrespondence to: Professor C.T. Moynihan, Departments of
Materials Engineering and Physics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. Tel: + 1-518 276 6125.
like to glass-like values with decreasing temperature. On reheating through the glass transition region the v versus T curve does not retrace the cooling curve. R a t h e r hysteresis is observed, and the heating curve has the characteristic sigmoidal shape shown in fig. 2. The kinetics of structural relaxation are nonexponential, and their description requires a relaxation function q~(t) = E g g e x p ( - - t / r i ) i
(1)
involving a weighted distribution of relaxation times r i, where t is time and the gi are weighting coefficients [1,2]. Equivalently, the kinetics may be described by an intrinsically non-exponential relaxation function, such as the K W W expression [1,2]: q~(t) = e x p [ - ( t / r ) e ] ,
(2)
where 0
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
C.T. Moynihan, J, Schroeder / Light scattering from glass-forming liquids
or the cooperatively rearranging region model of Donth [5]. Other models propose an inherently nonexponential mechanism, e.g., the defect diffusion model of Glarum [6,7] or the coupling model of Ngai and co-workers [8]. Some models are intermediate in character, e.g., ref. [9]. We discuss here some recent light scattering results in the glass transition region and their relevance to this question.
53 V+
/ V AV - - - " - ~ l / //ViI
//
ii
/ / ~ / ~ , , ~ AV
/1 ' /
//
>4:
>-
1! t/
J / 1/
GLASS
2. Anomalous scattering in the glass transition region ol
>
Shown in fig. 3 are plots reported by Bokov and Andreev [10,11] of light scattering intensity versus temperature for B20 3 glass as it is cooled and subsequently reheated through the glass transition region. Figure 4 shows a plot originally reported from our laboratory at a conference in 1989 [12] of the Landau-Placzek ratio RLp, (= IR/2IB, where I R and I B are respectively the Rayleigh and Brillouin scattering intensities [13]) versus temperature of a 53ZrF4-20BaF2-4LaF 33A1F3-20NaF (ZBLAN) glass during heating through the glass transition region. Tg in both
V+
/ ////
+~
V
/V-
//// "~"" LIQUID
>
/'/
GLASS >
1 Fig. 1. Schematic plot during cooling through the glass transition region of the temperature dependence of the average specific volume, v, the specific volumes, v+ and v_, of regions whose specific volumes deviate from the average, and the square of the deviation, (ac) 2.
T Fig. 2. Schematic plot during reheating through the glass transition region of the temperature dependence of v, v+, t , and ( A u ) 2.
figures is the glass transition temperature measured by differential scanning calorimetry. The cooling curve of fig. 3 exhibits an anomalous sigmoidal shape compared to the monotonic v versus T cooling curve of fig. 1, while the heating curves of figs. 3 and 4 exhibit anomalous maxima compared to the sigmoidal u versus T heating curve of fig. 2. (Note that RLp in fig. 4 has to be considered relative to the Rue curve of the glass (the dashed line), which decreases markedly with increasing temperature because of the temperature dependence of the Brillouin peak intensity, I B, due to scattering from phonons [12,13].) It should be noted that the data in figs. 3 and 4 were highly reproducible. The maxima in the heating curves were replicated five separate times for B203 [10] and three separate times for ZBLAN [12]. Similar light scattering maxima (or relative maxima) on heating through Tg have also been observed for 1,3,5-tri-oL-naphthyl benzene [14], for potassium borate and sodium germanate glasses [15], and for other heavy metal fluoride glasses [16]. Undoubtedly related anomalous behavior has also been observed by Golubkov [17] for small angle X-ray scattering (SAS) from BeO 3
54
C.T.
Moynihan, J. Schroeder / Light scattering from glass-forming liquids heating
B2 03
.~_
ro 100
200
300
400
T(°C) Fig. 3. Intensity (arbitrary units) of light scattered at 90° from vitreous B203 during cooling and subsequent reheating through the glass transition region. Redrawn from data in ref. [10]. Cooling and reheating curves are displaced for clarity.
glass at very sma!l angles (40') during isothermal annealing in the transition region - minima in the scattering intensity versus time curves following a quench and maxima following a step function increase in temperature. In a subsequent publication [18] a maximum in the SAS intensity at 40' was reported for an Sb203-B203 glass during heating through Tg. It appears that anomalous scattering in the transition region may be a universal feature of glass-forming liquids. Intrinsic light scattering from liquids and glasses is due to density and concentration fluctuations [13]. Concentration fluctuations freeze at temperatures well above T,, so that time-dependent scattering in the transition region is proportional to the time-dependent mean square fluctuations, (A2p) or (A2v), in the density, p, or specific volume, v, due to local liquid-like structural variations [13]. These are given by [4,19] ( A2p ) / p 2 = ( A2U ) /U 2 = k T A K / V ,
the scattering intensity during both cooling and heating in the transition region. Consider, 3 la Robertson [4], two regions in the liquid or glass which due to liquid-like structural variations have specific volumes v+ and v_ respectively one standard deviation greater than and smaller than the mean specific volume, u, and which are large enough that, to a first approximation, they may relax independently of neighboring regions. Given the general correlation between volume and liquid transport properties [20], the region with the larger specific volume, v+, should have a structural relaxation time shorter than average and depart from the equilibrium liquid curve on cooling through the transition region at a lower temperature than the region with the smaller specific volume, v_, which should have a relaxation time longer than average. This behavior is shown by the dashed lines in fig. 1. Given the additional expectation [1,2] that the u+ and u_ cooling curves have a monotonic curvature, so will that for the average macroscopic specific volume u ( = ( v + + v _ ) / 2 ) . However, the square (Av) 2 ( = (A2u)) of the deviations of u+ and v from the average specific volume during cooling will exhibit a sigmoidal
6O
o ._J or" 4(
(3)
where k is the Boltzmann constant, Ax is the difference between the liquid and glass isothermal compressibilities, and V is the sampling or reference or correlation volume. Mazurin and Porai-Koshits [11] recently discussed the light scattering results during heating of B203 glass [10] and suggested that the anomalous maximum was expected from the fact that small regions of low density could relax faster than regions of high density. What follows here is a more extensive treatment in this vein of the expected variation in
ZBLAN 2c 280
,,
I
380
480 T(K)
580
Fig. 4. Landau-Placzek ratio, RLo , versus temperature during heating of a ZBLAN glass through the glass transition region. Dashed line is the extended RLp versus T curve for the glass. Solid line is a guide to the eye. Data from ref. [12]. The accuracy of the RLp values is determined by the accuracy of measurement of the Brillouin peak intensity, IB, and is approximately +5%.
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids
shape when plotted against temperature, as shown in the lower part of fig. 1. Since the light scattering intensity is proportional to (A z v), one thus anticipates that the scattering intensity cooling curves will also display this sigmoidal shape, as is borne out experimentally in fig. 3. A similar line of reasoning can be applied to the v+ and v reheating curves, as illustrated in fig. 2. Given that the v+ and v_ heating curves exhibit the expected sigmoidal shape, so will the heating curve for the average specific volume. This will cause the (Av) 2 heating curve to pass through a maximum, leading one in turn to expect the maxima in the experimental scattering intensity heating curves shown in figs. 3 and 4. Arguments similar to these can likewise explain the minima and maxima observed in the small angle X-ray scattering curves during isothermal annealing of B203 glass [17]. These arguments are somewhat simplistic, since they ignore, for example, the fact that the regions may thermally fluctuate in specific volume on a timescale comparable to that on which they relax internally and that relaxation processes occurring in one region may be communicated to other regions bordering it, leading to some degree of 'coupling' in the sense of Ngai and co-workers [8,9]. Nonetheless, we feel that the arguments are of sufficient validity to account qualitatively for the anomalous light scattering behavior.
specific volume fluctuation, (A2v), and the dependence of the relaxation time on free volume. Subsequently Donth [5], in a much simpler treatment, did this via a connection between the mean square temperature fluctuation, (A2T), and the temperature dependence of the relaxation time (parameterized as an Arrhenius activation energy, A H * ) . In what follows we go through a similar exercise, but in a fashion that will allow a broader perspective with regard to the temperature dependence of the relaxation time distribution and can take into account the non-linear character of structural relaxation in a glass well removed from equilibrium. Of the various models for the temperature dependence of the mean structural relaxation time, r, of liquids and glasses, the A d a m - G i b b s theory [21] seems to be the most successful [2,2224]. The theory gives In r = In r 0 + s * A ~ / m k T s c ,
The interpretation of anomalous light scattering given here supports the contentions of Robertson [4] and of Donth [5] that the apparent distribution of structural relaxation times in liquids and glasses is not due to an inherently nonexponential relaxation mechanism, but rather corresponds to a thermally induced spatial distribution of small regions which vary in their properties. One may relate the two distributions via an expression for the dependence of the relaxation time on the liquid and glass properties. This was first done by Robertson [4] in an elaborate model appropriate to chain polymers in which he postulated a connection between the mean square
(4)
where r 0 is a constant, Atz an activation energy, s* the minimum local configurational entropy needed for a structural rearrangement, s c the mean specific configurational entropy (i.e., per unit mass) of the liquid, and m the mass of a molecule or structural unit. The cross-correlation between fluctuations in the configurational or relaxational parts of the entropy and the specific volume is given by the expression [19] (AscAtg) = k T v Z A a / V ,
3. Fluctuations and relaxation time distributions
55
(5)
where Aa is the relaxational part of the thermal expansion coefficient. Hence fluctuations in specific relaxational volume and specific configurational entropy are positively correlated if Aa is positive. Since this condition is satisfied for nearly all liquids for which Aa has been determined near T~, then a region whose relaxational specific volume deviates from the mean will exhibit a corresponding deviation of the same sign in its specific configurational entropy. In line with eq. (4) and the physical picture developed above, let us assume that a region with local specific configurational entropy, Sci, has a local relaxation time, ri, given by In r i = In r o + s * A l ~ / m k T s c i
(6)
56
C T. Moynihan, J. Schroeder / Light scatteringfrom glass-forming liquids
so that
parameters which appear in the Narayanaswamy equation [1,2,22]:
In ri - In r = ( s * A l z / m k T ) ( 1 / s c i - 1/sc) =
-
(s*A#z/mkTs~)(sc~
(7)
- s~).
(A 2 In ~') = [(1 - x ) A H * / R T ] 2 ( k u / A C p V A G ) . (is)
The mean square fluctuation in the specific configurational entropy is [19] (AZsc) = kvAcp/V,
(8)
where Act, is the difference between the equilibrium liquid and glass specific heats. From this and eq. (7) it follows that the mean square deviation of In r; from its average is (A 2 In r ) = (s*Atz/mkTs2c)2(kvAcp/VaG),
(9)
where we have now designated the correlation volume as VAG to indicate that eq. (9) was obtained using the Adam-Gibbs theory. The specific configurational entropy is given by
sc = f ~ ( a c , / V )
dr,
(10)
where T2 is a temperature where the configurational entropy of the equilibrium liquid would vanish. If one assumes that Act, is approximately inversely proportional to temperature ( A c p = a / T ) [22,23], combination of eqs. (9) and (10) gives (A 2 In r) = [ B T 2 / ( T _ r
2 2
)l(kvl
c.v
o
),
where B =S*Al~T21mka. Equation (4) then hecomes the familiar Fuleher equation [23]:
T2).
(12)
If one approximates the temperature dependence of r over a small temperature range by an Arrhenius expression, d In r d(1/T)
AH* -
-
R
BT 2 -
( T - T2) 2'
(13)
and makes the substitution suggested by Hodge [221,
x = 1 - T2/T,
In r = In r o + 7 v * / ( v -
(14)
then eq. (11) may also be written in terms of the
Vo),
(16)
and for the local relaxation time in a region with local specific volume vi, In r i = In r o + ~lV*//( v i - Vo).
(17)
Here v 0 is the random close-packed specific volume of the liquid, v* is the minimum local specific free volume needed for structural relaxation, and y a factor that corrects for free volume overlap between neighboring relaxing structural elements. If we combine eqs. (16) and (17) in the same way that eqs. (4) and (6) were combined, utilize eq. (3) and assume that the average free volume is given by v - Vo =
(11)
In r = l n r o + B / ( T -
Here A H * is an Arrhenius activation energy and x (0 < x < 1) a non-linearity parameter which partitions the relative dependences of r on temperature and structure. (Equation (15) is equivalent to the expression derived by Donth [5], except for his omission of the non-linearity factor, (1 - x ) . ) An expression for (A 2 In r ) may also be developed using a free volume theory expression [20] for the mean structural relaxation time,
T2),
(18)
we obtain ( k 2 In r ) = [ B I ( T -
T2)2]2[kTAK/( AOL)2VFv] . (19)
Here B ( = y v * / v A a ) and T2 are the same as in eqs. (11) and (12), Tz is now interpreted as the temperature at which the free volume of the equilibrium liquid would vanish, and we have designated the correlation volume, VFV, to signify that eq. (19) was obtained using the free volume theory. Using eq. (13), eq. (19) may also be written in an approximate form valid over a small temperature range:
( A2 In r ) = ( A H * / R T 2 ) 2 [ k T A K / ( A a ) 2 V w ] . (20)
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids
Note that one of the differences between eq. (15) from the A d a m - G i b b s theory and eq. (20) from the free volume theory is the absence of the (1 - x ) non-linearity factor in eq. (20). This underscores one of the many criticisms leveled against the free volume theory [2,24], namely, that it fails to account correctly for the non-linear character of structural relaxation and for the temperature dependence of the isostructural viscosity and relaxation time. (A 2 In r ) is a measure of the width of the spectrum or distribution of relaxation times and can be directly related to the KWW/3 parameter of eq. (2) (/3 ~ 1 as ( A 2 In r ) ~ 0, /3 + 0 as (A 2 In r)--+oo). A table of (A 2 In r ) ( = Var ln(r/ro)) as a function of /3 is given in ref. [25]. A number of things are noteworthy with regard to eqs. (11), (15), (19) and (20). First, it is predicted that there should be a correlation between (A 2 In r ) (or the KWW/3 parameter) and the parameters (B and Tz in eqs. (11) and (19) or x and A H * in eqs. (15) and (20)) describing the temperature and structure dependence of the mean relaxation time, a fact extensively documented by Hodge [22]. Second, a decrease in (42 In z ) (increase in/3) with increasing temperature is predicted, a fact again abundantly documented in the literature (e.g., refs. [3,9,26-30]). Note via eqs. (11) and (19), which are the expressions for (A 2 In z) applicable over an extended temperature range, that this is true even if the correlation volumes, VAG or VFV, vary as ( T T2) -2, as suggested by Donth and co-workers [5,31]. Finally, for liquids whose relaxation time temperature dependence is described by eq. (12), the width of the distribution of relaxation times is expected via eqs. (11) and (19) to diverge ( ( A 2 In r ) - - ) % / 3 - - + 0 ) as T ~ T 2 , behavior recently noted for o-terphenyl and salol by Nagel and Dixon [28-30]. In table 1 are listed values of the linear dimensions, VA1/3 a n d VF~3, of the reference or correlation volumes calculated respectively from eqs. (11) or (15) and (19) or (20) in the glass transition region for three glass-forming liquids - B203, glycerol and poly(vinyl acetate) (PVAc). The thermodynamic and structural relaxation kinetic parameters appearing in these equations were taken
57
Table 1 T h e r m o d y n a m i c and kinetic parameters associated with structural relaxation in the glass transition region for three glasses
T (K): 103v (m3/kg): 10 2Ac o ( J / k g K): 104ha ( K - 1): 1011AK ( m 2 / N ) : H: /3: (A 2 In r ) : 10 3B (K): T 2 (K): I O - 3 A H * / R (K): x: vxG/3 (nm): V I ~ 3 (nm): Vlo/3 (nm):
B203
Glycerol
PVAc
550 0.558 6.3 3.5 2.8 4.7 0.62 2.65
200 0.758 9.5 3.8 8.1 3.5 0.65 2.25 2.50 128 2.7 5.4 0.49
304 0.843 5.0 4.32 20.9 2.2 0.51 4.67 71.3 0.41 4.6 8.4 0.49
45.3 0.39 2.3 5.3 0.32
from the literature [25,32-37] and are also listed in table 1. The values of V2/3 are comparable to those obtained by Donth (Va1/3 in his notation) [5]. The ~71/3 • FV values calculated from eqs. (19) or (20), which are based on the free volume theory, are similar in magnitude to but about a factor of two larger than the VA~/3 values calculated from eqs. (11) or (15), which are based on the A d a m Gibbs theory. The source of this discrepancy is evident if one takes the ratio of v1/3 1/3 FV to VAG calculated from the respective equations •
VFV1/3/ I/" 1/3
/ " AG = ( T 2 / T ) - 2/3 H 1/3 = (1 --X)-2/317f 1/3,
(21)
where /7 is the Prigogine-Defay ratio [36,38]:
II = AcpAK//Tu( Aot) 2.
(22)
As shown in table 1, near Tg H typically has values in the range 2-5, while T 2 / T and (1 - x ) are both roughly 0.6. Hence one expects VF1/3/~11/3 V / r A G to lie in the range 1.8-2.4. In view of the difficulties with the free volume theory mentioned above, we shall henceforth consider the V2/3 values to be physically more realistic. In the context of our model, the VAG 1/3 values are the sizes of the independently relaxing regions. Also listed in table 1 are the sizes 1/1/3 • tool of the basic molecular or structural units (the B 0 3 / 2 group, the glycerol molecule and" the PVAc repeat unit) calculated from the macroscopic spe-
C.T. Moynihan, Z Schroeder / Light scattering from glass-forming liquids
58
cific volumes Vml/3 ol __ ( -
M v / N A ) 1/3
(23)
-
where M is the formula weight of the molecule or structural unit and NA is Avogadro's number. The relative sizes of the relaxing regions 1/1/3/ "AG / Vml/3 o~ are all about the same - five to ten structural units wide for each liquid. This is a sufficient size that a given region should be able to relax relatively independently with regard to neighboring regions. An obvious question is whether there is any independent evidence for nanoscale heterogeneity of this sort in glasses and glassforming liquids, and indeed there does seem to be such evidence. Robertson [4] estimated from structural considerations a minimum size of 2-2.5 nm for a region capable of independent rearrangement in polystyrene (a polymer very similar to PVAc). Golubkov [17] calculated that 'regions of non-uniformity' responsible for the anomalous X-ray scattering at very small angles during annealing of B20 3 must be roughly 3 nm in diameter. Malinovsky and Sokolov [39] and Sokolov et al. [40] have estimated sizes in the range 1.5-3.0 nm for structural fluctuations responsible for the low frequency boson peak in the Raman spectra and the width of the first sharp X-ray diffraction peak of polymers and various oxide glasses, including B20 3. These authors identified the structural fluctuations with the socalled medium-range order in liquids and glasses. Duval and co-workers [41-43] have interpreted inelastic neutron scattering and Raman scattering boson peak results for inorganic glasses and polymers as indicating discontinuities in the glass structure corresponding to regions (which they call 'blobs') 2-5 nm in size. Significantly, both of these last two groups of researchers find that the sizes of the structural fluctuations or 'blobs' for high polymers are roughly twice the sizes of those for inorganic glasses, in agreement with the relative values in table 1 of V~ 1/3 for the high polyAC mer PVAc and for B 2 0 3 . 4. Conclusions
In summary, the anomalous time dependences of light scattering in the glass transition region
discussed here appear to be the first results to offer evidence as to the physical origin of the non-exponential character of structural relaxation kinetics. Indications are that the apparent distribution of relaxation times does indeed have a physical correspondence to a distribution of thermally induced regions of varying density and entropy. As noted by Donth [5] and evident from eqs. (11), (15), (19) and (20), the actual magnitude of the distribution of relaxation times, ( A 2 In r), depends strongly on the size of the regions. The experimental values of (A 2 In ~-) demand regions which are typically a few nm across, and the question remains as to why this size and not, perhaps, larger or somewhat smaller. It seems likely that the answer to this question is intimately connected to the scale of the mediumrange order which seems to be endemic to the vitreous state. This research was supported by Grant No. DMR-8801004 from the National Science Foundation. The authors are grateful to O.V. Mazurin and M.E. Lines for calling their attention respectively to the B20 3 scattering data and the boson peak data.
References [1] C.T. Moynihan et al., Ann. NY Acad. Sci. 279 (1976) 15. [2] G.W. Scherer, J. Non-Cryst. Solids 123 (1990) 75. [3] J.H. Simmons and P.B. Macedo, J. Res. Natl. Bur. Stand. 75A (1971) 175. [4] R.E. Robertson, J. Polym. Sci.: Polym. Sym. 68 (1978) 173; J. Polym. Sci.: Polym. Phys. 17 (1979) 597. [5] E. Donth, J. Non-Cryst. Solids 53 (1982) 325. [6] S.H. Glarum, J. Chem. Phys. 33 (1960) 1371. [7] M.F. Shlesinger and E.W. Montroll, Proc. Natl. Acad. Sci. USA 81 (1984) 1280. [8] K.L. Ngai, R.W. Rendell, A.K. Rajagopal and S. Teitler, Ann. NY Acad. Sci. 484 (1986) 150. [9] K.L. Ngai, R.W. Rendell and D.J. Plazek, J. Chem. Phys. 94 (1991) 3018. [10] N.A. Bokov and N.S. Andreev, Sov. J. Glass Phys. Chem. 15 (1989) 243. [11] O.V. Mazurin and E.A. Porai-Koshits, in: The Physics of Non-Crystalline Solids. ed. L.D. Pye, W.C. LaCourse and H.J. Stevens (Taylor and Francis, Washington, DC, 1992) p. 22. [12] J. Schroeder et al., Mater. Sci. Forum 67&68 (1991) 471.
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids [13] J. Schroeder, in: Treatise on Materials Science and Technology, Vol. 12, ed. M. Tomozawa and R.H. Doremus (Academic Press, New York, 1977) p. 157. [14] R.-J. Ma, T.-J. He and C.H. Wang, J. Chem. Phys. 88 (1988) 1497. [15] N.A. Bokov, Sov. J. Glass Phys. Chem. 17 (1991) 498. [16] J. Schroeder, M. Whitmore, M.R. Silvestri and C.T. Moynihan, J. Non-Cryst. Solids, in press. [17] V.V. Golubkov, Sov. J. Glass Phys. Chem. 15 (1989) 280. [18] V.V. Golubkov and M.M. Pivavarov, Sov. J. Glass Phys. Chem. 17 (1991) 135. [19] L.D. Landau and E.M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, MA, 1969) pp. 343-354. [20] M.H. Cohen and D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [21] G. Adam and J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [22] I.M. Hodge, Macromolecules 20 (1987) 2897; J. NonCryst. Solids 131-133 (1991) 435. [23] G.W. Scherer, J. Am. Ceram. Soc. 75 (1992) 1060. [24] P.K. Gupta, Rev. Solid State Sci. 3 (1989) 221. [25] C.T. Moynihan, LP. Boesch and N.L Laberge, Phys. Chem. Glasses 14 (1973) 122. [26] J. Tauke, T.A. Litovitz and P.B. Macedo, J. Am. Ceram. Soc. 51 (1968) 158. [27] R. Weiler, R. Bose and P.B. Macedo, J. Chem. Phys. 53 (1970) 1258. [28] P.K. Dixon and S.R. Nagel, Phys. Rev. Lett. 61 (1988) 341. [29] S.R. Nagel and P.K. Dixon, J. Chem. Phys. 90 (1989) 3885.
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[30] P.K. Dixon, Phys. Rev. B42 (1990) 8179. [31] E.W. Fischer, E. Donth and W. Steffen, Phys. Rev. Lett. 68 (1992) 2344. [32] C.T. Moynihan, S.N. Crichton and S.M. Opalka, J. NonCryst. Solids 131-133 (1991) 420. [33] A.K. Schulz, J. Chim. Phys. 51 (1954) 324. [34] R. Piccirelli and T.A. Litovitz, J. Acoust. Soc. Am. 29 (1957) 1009. [35] N.O. Birge, Phys. Rev. B34 (1986) 1631. [36] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (1976) 4136. [37] H. Sasabe and C.T. Moynihan, J. Polym. Sci. Polym. Phys. 16 (1978) 1447. [38] C.T. Moynihan and A.V. Lesikar, Ann. NY Acad. Sci. 371 (1981) 151. [39] V.K. Malinovsky and A.P. Sokolov, Solid State Commun. 57 (1986) 757. [40] A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Quitmann, Phys. Rev. Lett. 69 (1992) 1540. [41] E. Duval, A. Boukenter and T. Achibat, J. Phys.: Condens. Matter 2 (1990) 10227. [42] T. Achibat, A. Boukenter, E. Duval, G. Lorentz and S. Etienne, J. Chem. Phys. 95 (1991) 2949. [43] E. Duval, A. Boukenter, T. Achibat and B. Champagnon, in: The Physics of Non-Crystalline Solids, ed. L.D. Pye, W.C. LaCourse and H.J. Stevens (Taylor and Francis, Washington, DC, 1992) p. 82.
iou~N,~ L
or
NON-CRYS LL ESOLIDS
Non-exponential structural relaxation, anomalous light scattering and nanoscale inhomogeneities in glass-forming liquids C.T. M o y n i h a n a n d J. S c h r o e d e r Departments of Materials Engineering and Physics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
Received 13 November 1992 Revised manuscript received 1 March 1993
Light scattering from glass-forming liquids exhibits an anomalous time dependence in the glass transition region, e4 maxima in the scattering intensity versus temperature curves during heating. It is shown that this behavior is consistent with the presence of nanoscale inhomogeneities (density fluctuations) which relax at different rates. It is suggested that this could be the source of non-exponential structural relaxation kinetics. An expression relating the size of these regions to structural relaxation kinetic parameters has been developed and predicts sizes in excellent agreement with those determined by other methods.
1. Introduction Structural relaxation is the kinetically impeded rearrangement of the structure of a liquid in response to changes in external variables such as t e m p e r a t u r e or pressure. It is commonly associated with the glass transition and manifests itself as a time dependence of properties such as specific volume or enthalpy following a t e m p e r a t u r e change. The solid lines in figs. 1 and 2 show schematically the variation in the macroscopic specific volume, v, as a liquid is cooled and then reheated through the glass transition region [1,2]. On cooling the rapid increase in the structural relaxation time with decreasing t e m p e r a t u r e initially prevents the structure from fully equilibrating (at the high t e m p e r a t u r e end of the transition region) and eventually results in freezing of a non-equilibrium structure (at the low temperature end of the transition region). The result is the smooth v versus T cooling curve of fig. 1, whose slope monotonically changes from liquidCorrespondence to: Professor C.T. Moynihan, Departments of
Materials Engineering and Physics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. Tel: + 1-518 276 6125.
like to glass-like values with decreasing temperature. On reheating through the glass transition region the v versus T curve does not retrace the cooling curve. R a t h e r hysteresis is observed, and the heating curve has the characteristic sigmoidal shape shown in fig. 2. The kinetics of structural relaxation are nonexponential, and their description requires a relaxation function q~(t) = E g g e x p ( - - t / r i ) i
(1)
involving a weighted distribution of relaxation times r i, where t is time and the gi are weighting coefficients [1,2]. Equivalently, the kinetics may be described by an intrinsically non-exponential relaxation function, such as the K W W expression [1,2]: q~(t) = e x p [ - ( t / r ) e ] ,
(2)
where 0
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
C.T. Moynihan, J, Schroeder / Light scattering from glass-forming liquids
or the cooperatively rearranging region model of Donth [5]. Other models propose an inherently nonexponential mechanism, e.g., the defect diffusion model of Glarum [6,7] or the coupling model of Ngai and co-workers [8]. Some models are intermediate in character, e.g., ref. [9]. We discuss here some recent light scattering results in the glass transition region and their relevance to this question.
53 V+
/ V AV - - - " - ~ l / //ViI
//
ii
/ / ~ / ~ , , ~ AV
/1 ' /
//
>4:
>-
1! t/
J / 1/
GLASS
2. Anomalous scattering in the glass transition region ol
>
Shown in fig. 3 are plots reported by Bokov and Andreev [10,11] of light scattering intensity versus temperature for B20 3 glass as it is cooled and subsequently reheated through the glass transition region. Figure 4 shows a plot originally reported from our laboratory at a conference in 1989 [12] of the Landau-Placzek ratio RLp, (= IR/2IB, where I R and I B are respectively the Rayleigh and Brillouin scattering intensities [13]) versus temperature of a 53ZrF4-20BaF2-4LaF 33A1F3-20NaF (ZBLAN) glass during heating through the glass transition region. Tg in both
V+
/ ////
+~
V
/V-
//// "~"" LIQUID
>
/'/
GLASS >
1 Fig. 1. Schematic plot during cooling through the glass transition region of the temperature dependence of the average specific volume, v, the specific volumes, v+ and v_, of regions whose specific volumes deviate from the average, and the square of the deviation, (ac) 2.
T Fig. 2. Schematic plot during reheating through the glass transition region of the temperature dependence of v, v+, t , and ( A u ) 2.
figures is the glass transition temperature measured by differential scanning calorimetry. The cooling curve of fig. 3 exhibits an anomalous sigmoidal shape compared to the monotonic v versus T cooling curve of fig. 1, while the heating curves of figs. 3 and 4 exhibit anomalous maxima compared to the sigmoidal u versus T heating curve of fig. 2. (Note that RLp in fig. 4 has to be considered relative to the Rue curve of the glass (the dashed line), which decreases markedly with increasing temperature because of the temperature dependence of the Brillouin peak intensity, I B, due to scattering from phonons [12,13].) It should be noted that the data in figs. 3 and 4 were highly reproducible. The maxima in the heating curves were replicated five separate times for B203 [10] and three separate times for ZBLAN [12]. Similar light scattering maxima (or relative maxima) on heating through Tg have also been observed for 1,3,5-tri-oL-naphthyl benzene [14], for potassium borate and sodium germanate glasses [15], and for other heavy metal fluoride glasses [16]. Undoubtedly related anomalous behavior has also been observed by Golubkov [17] for small angle X-ray scattering (SAS) from BeO 3
54
C.T.
Moynihan, J. Schroeder / Light scattering from glass-forming liquids heating
B2 03
.~_
ro 100
200
300
400
T(°C) Fig. 3. Intensity (arbitrary units) of light scattered at 90° from vitreous B203 during cooling and subsequent reheating through the glass transition region. Redrawn from data in ref. [10]. Cooling and reheating curves are displaced for clarity.
glass at very sma!l angles (40') during isothermal annealing in the transition region - minima in the scattering intensity versus time curves following a quench and maxima following a step function increase in temperature. In a subsequent publication [18] a maximum in the SAS intensity at 40' was reported for an Sb203-B203 glass during heating through Tg. It appears that anomalous scattering in the transition region may be a universal feature of glass-forming liquids. Intrinsic light scattering from liquids and glasses is due to density and concentration fluctuations [13]. Concentration fluctuations freeze at temperatures well above T,, so that time-dependent scattering in the transition region is proportional to the time-dependent mean square fluctuations, (A2p) or (A2v), in the density, p, or specific volume, v, due to local liquid-like structural variations [13]. These are given by [4,19] ( A2p ) / p 2 = ( A2U ) /U 2 = k T A K / V ,
the scattering intensity during both cooling and heating in the transition region. Consider, 3 la Robertson [4], two regions in the liquid or glass which due to liquid-like structural variations have specific volumes v+ and v_ respectively one standard deviation greater than and smaller than the mean specific volume, u, and which are large enough that, to a first approximation, they may relax independently of neighboring regions. Given the general correlation between volume and liquid transport properties [20], the region with the larger specific volume, v+, should have a structural relaxation time shorter than average and depart from the equilibrium liquid curve on cooling through the transition region at a lower temperature than the region with the smaller specific volume, v_, which should have a relaxation time longer than average. This behavior is shown by the dashed lines in fig. 1. Given the additional expectation [1,2] that the u+ and u_ cooling curves have a monotonic curvature, so will that for the average macroscopic specific volume u ( = ( v + + v _ ) / 2 ) . However, the square (Av) 2 ( = (A2u)) of the deviations of u+ and v from the average specific volume during cooling will exhibit a sigmoidal
6O
o ._J or" 4(
(3)
where k is the Boltzmann constant, Ax is the difference between the liquid and glass isothermal compressibilities, and V is the sampling or reference or correlation volume. Mazurin and Porai-Koshits [11] recently discussed the light scattering results during heating of B203 glass [10] and suggested that the anomalous maximum was expected from the fact that small regions of low density could relax faster than regions of high density. What follows here is a more extensive treatment in this vein of the expected variation in
ZBLAN 2c 280
,,
I
380
480 T(K)
580
Fig. 4. Landau-Placzek ratio, RLo , versus temperature during heating of a ZBLAN glass through the glass transition region. Dashed line is the extended RLp versus T curve for the glass. Solid line is a guide to the eye. Data from ref. [12]. The accuracy of the RLp values is determined by the accuracy of measurement of the Brillouin peak intensity, IB, and is approximately +5%.
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids
shape when plotted against temperature, as shown in the lower part of fig. 1. Since the light scattering intensity is proportional to (A z v), one thus anticipates that the scattering intensity cooling curves will also display this sigmoidal shape, as is borne out experimentally in fig. 3. A similar line of reasoning can be applied to the v+ and v reheating curves, as illustrated in fig. 2. Given that the v+ and v_ heating curves exhibit the expected sigmoidal shape, so will the heating curve for the average specific volume. This will cause the (Av) 2 heating curve to pass through a maximum, leading one in turn to expect the maxima in the experimental scattering intensity heating curves shown in figs. 3 and 4. Arguments similar to these can likewise explain the minima and maxima observed in the small angle X-ray scattering curves during isothermal annealing of B203 glass [17]. These arguments are somewhat simplistic, since they ignore, for example, the fact that the regions may thermally fluctuate in specific volume on a timescale comparable to that on which they relax internally and that relaxation processes occurring in one region may be communicated to other regions bordering it, leading to some degree of 'coupling' in the sense of Ngai and co-workers [8,9]. Nonetheless, we feel that the arguments are of sufficient validity to account qualitatively for the anomalous light scattering behavior.
specific volume fluctuation, (A2v), and the dependence of the relaxation time on free volume. Subsequently Donth [5], in a much simpler treatment, did this via a connection between the mean square temperature fluctuation, (A2T), and the temperature dependence of the relaxation time (parameterized as an Arrhenius activation energy, A H * ) . In what follows we go through a similar exercise, but in a fashion that will allow a broader perspective with regard to the temperature dependence of the relaxation time distribution and can take into account the non-linear character of structural relaxation in a glass well removed from equilibrium. Of the various models for the temperature dependence of the mean structural relaxation time, r, of liquids and glasses, the A d a m - G i b b s theory [21] seems to be the most successful [2,2224]. The theory gives In r = In r 0 + s * A ~ / m k T s c ,
The interpretation of anomalous light scattering given here supports the contentions of Robertson [4] and of Donth [5] that the apparent distribution of structural relaxation times in liquids and glasses is not due to an inherently nonexponential relaxation mechanism, but rather corresponds to a thermally induced spatial distribution of small regions which vary in their properties. One may relate the two distributions via an expression for the dependence of the relaxation time on the liquid and glass properties. This was first done by Robertson [4] in an elaborate model appropriate to chain polymers in which he postulated a connection between the mean square
(4)
where r 0 is a constant, Atz an activation energy, s* the minimum local configurational entropy needed for a structural rearrangement, s c the mean specific configurational entropy (i.e., per unit mass) of the liquid, and m the mass of a molecule or structural unit. The cross-correlation between fluctuations in the configurational or relaxational parts of the entropy and the specific volume is given by the expression [19] (AscAtg) = k T v Z A a / V ,
3. Fluctuations and relaxation time distributions
55
(5)
where Aa is the relaxational part of the thermal expansion coefficient. Hence fluctuations in specific relaxational volume and specific configurational entropy are positively correlated if Aa is positive. Since this condition is satisfied for nearly all liquids for which Aa has been determined near T~, then a region whose relaxational specific volume deviates from the mean will exhibit a corresponding deviation of the same sign in its specific configurational entropy. In line with eq. (4) and the physical picture developed above, let us assume that a region with local specific configurational entropy, Sci, has a local relaxation time, ri, given by In r i = In r o + s * A l ~ / m k T s c i
(6)
56
C T. Moynihan, J. Schroeder / Light scatteringfrom glass-forming liquids
so that
parameters which appear in the Narayanaswamy equation [1,2,22]:
In ri - In r = ( s * A l z / m k T ) ( 1 / s c i - 1/sc) =
-
(s*A#z/mkTs~)(sc~
(7)
- s~).
(A 2 In ~') = [(1 - x ) A H * / R T ] 2 ( k u / A C p V A G ) . (is)
The mean square fluctuation in the specific configurational entropy is [19] (AZsc) = kvAcp/V,
(8)
where Act, is the difference between the equilibrium liquid and glass specific heats. From this and eq. (7) it follows that the mean square deviation of In r; from its average is (A 2 In r ) = (s*Atz/mkTs2c)2(kvAcp/VaG),
(9)
where we have now designated the correlation volume as VAG to indicate that eq. (9) was obtained using the Adam-Gibbs theory. The specific configurational entropy is given by
sc = f ~ ( a c , / V )
dr,
(10)
where T2 is a temperature where the configurational entropy of the equilibrium liquid would vanish. If one assumes that Act, is approximately inversely proportional to temperature ( A c p = a / T ) [22,23], combination of eqs. (9) and (10) gives (A 2 In r) = [ B T 2 / ( T _ r
2 2
)l(kvl
c.v
o
),
where B =S*Al~T21mka. Equation (4) then hecomes the familiar Fuleher equation [23]:
T2).
(12)
If one approximates the temperature dependence of r over a small temperature range by an Arrhenius expression, d In r d(1/T)
AH* -
-
R
BT 2 -
( T - T2) 2'
(13)
and makes the substitution suggested by Hodge [221,
x = 1 - T2/T,
In r = In r o + 7 v * / ( v -
(14)
then eq. (11) may also be written in terms of the
Vo),
(16)
and for the local relaxation time in a region with local specific volume vi, In r i = In r o + ~lV*//( v i - Vo).
(17)
Here v 0 is the random close-packed specific volume of the liquid, v* is the minimum local specific free volume needed for structural relaxation, and y a factor that corrects for free volume overlap between neighboring relaxing structural elements. If we combine eqs. (16) and (17) in the same way that eqs. (4) and (6) were combined, utilize eq. (3) and assume that the average free volume is given by v - Vo =
(11)
In r = l n r o + B / ( T -
Here A H * is an Arrhenius activation energy and x (0 < x < 1) a non-linearity parameter which partitions the relative dependences of r on temperature and structure. (Equation (15) is equivalent to the expression derived by Donth [5], except for his omission of the non-linearity factor, (1 - x ) . ) An expression for (A 2 In r ) may also be developed using a free volume theory expression [20] for the mean structural relaxation time,
T2),
(18)
we obtain ( k 2 In r ) = [ B I ( T -
T2)2]2[kTAK/( AOL)2VFv] . (19)
Here B ( = y v * / v A a ) and T2 are the same as in eqs. (11) and (12), Tz is now interpreted as the temperature at which the free volume of the equilibrium liquid would vanish, and we have designated the correlation volume, VFV, to signify that eq. (19) was obtained using the free volume theory. Using eq. (13), eq. (19) may also be written in an approximate form valid over a small temperature range:
( A2 In r ) = ( A H * / R T 2 ) 2 [ k T A K / ( A a ) 2 V w ] . (20)
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids
Note that one of the differences between eq. (15) from the A d a m - G i b b s theory and eq. (20) from the free volume theory is the absence of the (1 - x ) non-linearity factor in eq. (20). This underscores one of the many criticisms leveled against the free volume theory [2,24], namely, that it fails to account correctly for the non-linear character of structural relaxation and for the temperature dependence of the isostructural viscosity and relaxation time. (A 2 In r ) is a measure of the width of the spectrum or distribution of relaxation times and can be directly related to the KWW/3 parameter of eq. (2) (/3 ~ 1 as ( A 2 In r ) ~ 0, /3 + 0 as (A 2 In r)--+oo). A table of (A 2 In r ) ( = Var ln(r/ro)) as a function of /3 is given in ref. [25]. A number of things are noteworthy with regard to eqs. (11), (15), (19) and (20). First, it is predicted that there should be a correlation between (A 2 In r ) (or the KWW/3 parameter) and the parameters (B and Tz in eqs. (11) and (19) or x and A H * in eqs. (15) and (20)) describing the temperature and structure dependence of the mean relaxation time, a fact extensively documented by Hodge [22]. Second, a decrease in (42 In z ) (increase in/3) with increasing temperature is predicted, a fact again abundantly documented in the literature (e.g., refs. [3,9,26-30]). Note via eqs. (11) and (19), which are the expressions for (A 2 In z) applicable over an extended temperature range, that this is true even if the correlation volumes, VAG or VFV, vary as ( T T2) -2, as suggested by Donth and co-workers [5,31]. Finally, for liquids whose relaxation time temperature dependence is described by eq. (12), the width of the distribution of relaxation times is expected via eqs. (11) and (19) to diverge ( ( A 2 In r ) - - ) % / 3 - - + 0 ) as T ~ T 2 , behavior recently noted for o-terphenyl and salol by Nagel and Dixon [28-30]. In table 1 are listed values of the linear dimensions, VA1/3 a n d VF~3, of the reference or correlation volumes calculated respectively from eqs. (11) or (15) and (19) or (20) in the glass transition region for three glass-forming liquids - B203, glycerol and poly(vinyl acetate) (PVAc). The thermodynamic and structural relaxation kinetic parameters appearing in these equations were taken
57
Table 1 T h e r m o d y n a m i c and kinetic parameters associated with structural relaxation in the glass transition region for three glasses
T (K): 103v (m3/kg): 10 2Ac o ( J / k g K): 104ha ( K - 1): 1011AK ( m 2 / N ) : H: /3: (A 2 In r ) : 10 3B (K): T 2 (K): I O - 3 A H * / R (K): x: vxG/3 (nm): V I ~ 3 (nm): Vlo/3 (nm):
B203
Glycerol
PVAc
550 0.558 6.3 3.5 2.8 4.7 0.62 2.65
200 0.758 9.5 3.8 8.1 3.5 0.65 2.25 2.50 128 2.7 5.4 0.49
304 0.843 5.0 4.32 20.9 2.2 0.51 4.67 71.3 0.41 4.6 8.4 0.49
45.3 0.39 2.3 5.3 0.32
from the literature [25,32-37] and are also listed in table 1. The values of V2/3 are comparable to those obtained by Donth (Va1/3 in his notation) [5]. The ~71/3 • FV values calculated from eqs. (19) or (20), which are based on the free volume theory, are similar in magnitude to but about a factor of two larger than the VA~/3 values calculated from eqs. (11) or (15), which are based on the A d a m Gibbs theory. The source of this discrepancy is evident if one takes the ratio of v1/3 1/3 FV to VAG calculated from the respective equations •
VFV1/3/ I/" 1/3
/ " AG = ( T 2 / T ) - 2/3 H 1/3 = (1 --X)-2/317f 1/3,
(21)
where /7 is the Prigogine-Defay ratio [36,38]:
II = AcpAK//Tu( Aot) 2.
(22)
As shown in table 1, near Tg H typically has values in the range 2-5, while T 2 / T and (1 - x ) are both roughly 0.6. Hence one expects VF1/3/~11/3 V / r A G to lie in the range 1.8-2.4. In view of the difficulties with the free volume theory mentioned above, we shall henceforth consider the V2/3 values to be physically more realistic. In the context of our model, the VAG 1/3 values are the sizes of the independently relaxing regions. Also listed in table 1 are the sizes 1/1/3 • tool of the basic molecular or structural units (the B 0 3 / 2 group, the glycerol molecule and" the PVAc repeat unit) calculated from the macroscopic spe-
C.T. Moynihan, Z Schroeder / Light scattering from glass-forming liquids
58
cific volumes Vml/3 ol __ ( -
M v / N A ) 1/3
(23)
-
where M is the formula weight of the molecule or structural unit and NA is Avogadro's number. The relative sizes of the relaxing regions 1/1/3/ "AG / Vml/3 o~ are all about the same - five to ten structural units wide for each liquid. This is a sufficient size that a given region should be able to relax relatively independently with regard to neighboring regions. An obvious question is whether there is any independent evidence for nanoscale heterogeneity of this sort in glasses and glassforming liquids, and indeed there does seem to be such evidence. Robertson [4] estimated from structural considerations a minimum size of 2-2.5 nm for a region capable of independent rearrangement in polystyrene (a polymer very similar to PVAc). Golubkov [17] calculated that 'regions of non-uniformity' responsible for the anomalous X-ray scattering at very small angles during annealing of B20 3 must be roughly 3 nm in diameter. Malinovsky and Sokolov [39] and Sokolov et al. [40] have estimated sizes in the range 1.5-3.0 nm for structural fluctuations responsible for the low frequency boson peak in the Raman spectra and the width of the first sharp X-ray diffraction peak of polymers and various oxide glasses, including B20 3. These authors identified the structural fluctuations with the socalled medium-range order in liquids and glasses. Duval and co-workers [41-43] have interpreted inelastic neutron scattering and Raman scattering boson peak results for inorganic glasses and polymers as indicating discontinuities in the glass structure corresponding to regions (which they call 'blobs') 2-5 nm in size. Significantly, both of these last two groups of researchers find that the sizes of the structural fluctuations or 'blobs' for high polymers are roughly twice the sizes of those for inorganic glasses, in agreement with the relative values in table 1 of V~ 1/3 for the high polyAC mer PVAc and for B 2 0 3 . 4. Conclusions
In summary, the anomalous time dependences of light scattering in the glass transition region
discussed here appear to be the first results to offer evidence as to the physical origin of the non-exponential character of structural relaxation kinetics. Indications are that the apparent distribution of relaxation times does indeed have a physical correspondence to a distribution of thermally induced regions of varying density and entropy. As noted by Donth [5] and evident from eqs. (11), (15), (19) and (20), the actual magnitude of the distribution of relaxation times, ( A 2 In r), depends strongly on the size of the regions. The experimental values of (A 2 In ~-) demand regions which are typically a few nm across, and the question remains as to why this size and not, perhaps, larger or somewhat smaller. It seems likely that the answer to this question is intimately connected to the scale of the mediumrange order which seems to be endemic to the vitreous state. This research was supported by Grant No. DMR-8801004 from the National Science Foundation. The authors are grateful to O.V. Mazurin and M.E. Lines for calling their attention respectively to the B20 3 scattering data and the boson peak data.
References [1] C.T. Moynihan et al., Ann. NY Acad. Sci. 279 (1976) 15. [2] G.W. Scherer, J. Non-Cryst. Solids 123 (1990) 75. [3] J.H. Simmons and P.B. Macedo, J. Res. Natl. Bur. Stand. 75A (1971) 175. [4] R.E. Robertson, J. Polym. Sci.: Polym. Sym. 68 (1978) 173; J. Polym. Sci.: Polym. Phys. 17 (1979) 597. [5] E. Donth, J. Non-Cryst. Solids 53 (1982) 325. [6] S.H. Glarum, J. Chem. Phys. 33 (1960) 1371. [7] M.F. Shlesinger and E.W. Montroll, Proc. Natl. Acad. Sci. USA 81 (1984) 1280. [8] K.L. Ngai, R.W. Rendell, A.K. Rajagopal and S. Teitler, Ann. NY Acad. Sci. 484 (1986) 150. [9] K.L. Ngai, R.W. Rendell and D.J. Plazek, J. Chem. Phys. 94 (1991) 3018. [10] N.A. Bokov and N.S. Andreev, Sov. J. Glass Phys. Chem. 15 (1989) 243. [11] O.V. Mazurin and E.A. Porai-Koshits, in: The Physics of Non-Crystalline Solids. ed. L.D. Pye, W.C. LaCourse and H.J. Stevens (Taylor and Francis, Washington, DC, 1992) p. 22. [12] J. Schroeder et al., Mater. Sci. Forum 67&68 (1991) 471.
C.T. Moynihan, J. Schroeder / Light scattering from glass-forming liquids [13] J. Schroeder, in: Treatise on Materials Science and Technology, Vol. 12, ed. M. Tomozawa and R.H. Doremus (Academic Press, New York, 1977) p. 157. [14] R.-J. Ma, T.-J. He and C.H. Wang, J. Chem. Phys. 88 (1988) 1497. [15] N.A. Bokov, Sov. J. Glass Phys. Chem. 17 (1991) 498. [16] J. Schroeder, M. Whitmore, M.R. Silvestri and C.T. Moynihan, J. Non-Cryst. Solids, in press. [17] V.V. Golubkov, Sov. J. Glass Phys. Chem. 15 (1989) 280. [18] V.V. Golubkov and M.M. Pivavarov, Sov. J. Glass Phys. Chem. 17 (1991) 135. [19] L.D. Landau and E.M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, MA, 1969) pp. 343-354. [20] M.H. Cohen and D. Turnbull, J. Chem. Phys. 31 (1959) 1164. [21] G. Adam and J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [22] I.M. Hodge, Macromolecules 20 (1987) 2897; J. NonCryst. Solids 131-133 (1991) 435. [23] G.W. Scherer, J. Am. Ceram. Soc. 75 (1992) 1060. [24] P.K. Gupta, Rev. Solid State Sci. 3 (1989) 221. [25] C.T. Moynihan, LP. Boesch and N.L Laberge, Phys. Chem. Glasses 14 (1973) 122. [26] J. Tauke, T.A. Litovitz and P.B. Macedo, J. Am. Ceram. Soc. 51 (1968) 158. [27] R. Weiler, R. Bose and P.B. Macedo, J. Chem. Phys. 53 (1970) 1258. [28] P.K. Dixon and S.R. Nagel, Phys. Rev. Lett. 61 (1988) 341. [29] S.R. Nagel and P.K. Dixon, J. Chem. Phys. 90 (1989) 3885.
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[30] P.K. Dixon, Phys. Rev. B42 (1990) 8179. [31] E.W. Fischer, E. Donth and W. Steffen, Phys. Rev. Lett. 68 (1992) 2344. [32] C.T. Moynihan, S.N. Crichton and S.M. Opalka, J. NonCryst. Solids 131-133 (1991) 420. [33] A.K. Schulz, J. Chim. Phys. 51 (1954) 324. [34] R. Piccirelli and T.A. Litovitz, J. Acoust. Soc. Am. 29 (1957) 1009. [35] N.O. Birge, Phys. Rev. B34 (1986) 1631. [36] P.K. Gupta and C.T. Moynihan, J. Chem. Phys. 65 (1976) 4136. [37] H. Sasabe and C.T. Moynihan, J. Polym. Sci. Polym. Phys. 16 (1978) 1447. [38] C.T. Moynihan and A.V. Lesikar, Ann. NY Acad. Sci. 371 (1981) 151. [39] V.K. Malinovsky and A.P. Sokolov, Solid State Commun. 57 (1986) 757. [40] A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Quitmann, Phys. Rev. Lett. 69 (1992) 1540. [41] E. Duval, A. Boukenter and T. Achibat, J. Phys.: Condens. Matter 2 (1990) 10227. [42] T. Achibat, A. Boukenter, E. Duval, G. Lorentz and S. Etienne, J. Chem. Phys. 95 (1991) 2949. [43] E. Duval, A. Boukenter, T. Achibat and B. Champagnon, in: The Physics of Non-Crystalline Solids, ed. L.D. Pye, W.C. LaCourse and H.J. Stevens (Taylor and Francis, Washington, DC, 1992) p. 82.