- Email: [email protected]
Silicate melts: The “anomalous” pressure dependence of the viscosity
Geochimicaet CosmochimicaActa, Vol. 59, No. 13, pp. 2725-2731, 1995 Copyright © 1995ElsevierScienceLtd Printedin the USA.All rightsreserved 0016-7037/95 $9.50 + .00
Pergamon
~16-7~7(~)~1~-9
Silicate melts: The "anomalous" pressure dependence of the viscosity Y. BOTTINGAand P. RICHET Institut de Physique du Globe de Pads, D6partementdes G-6omat6riaux,75252 PARIS Cedex 05, France (Received November 3, 1994; accepted in revisedform April 5, 1995)
Abstract--The decrease of the specific volume, when the extent of polymerization diminishes, is a cause of the pressure sensitivity of the viscosity of silicate melts. This effect can be explained by means of the Adam and Gibbs (1965) theory, taking into account the pressure dependence of the degree of polymerization of the melt and its influence on the configurational entropy. At temperatures close to their glass transitions, liquid silica and SiO2-Na20 melts have configurational entropies that are probably due to the mixing of their bridging and nonbridging oxygen atoms. 1. INTRODUCTION
1975), and S i - O bond weakening due to pressure-induced distortion of the liquid structure (Woodcock et al., 1976; Sharma et al., 1979). The Gupta (1987) suggestion can not be experimentally verified because there is no way to measure the coefficient of thermal expansion of a silicate glass at temperatures in excess of the glass transition, where the negative P effect on silicate melt viscosity has been observed. Available data for the coefficient of thermal expansion of glasses and of compositionally equivalent liquids (Knoche et al., 1992, 1994) are not in favor of this suggestion. The relation between the pressure dependence of the viscosity and the coefficients of thermal expansion of liquid and glass, is evident by inspection of Eqn. 8, see section III. Sharma et al. (1979) have shown that the observed pressure induced viscosity decrease in liquid GeO2 was not due to a change in the coordination of Ge. This does not rule out this mechanism to be a cause of a pressure effect on the viscosity, but it means that it is not an explanation for the observations by Kushiro (1976) and Kushiro et al. (1976). However, Molecular Dynamics (MD) simulation of SiO2 showed a pressure enhancement of the diffusion coefficients of Si and O (Woodcock et al., 1976), implying a decrease of the viscosity. The influence of pressure on the mobility of Si and O was believed to be due to observed changes of bond angles and lengths and ephemeral coordination changes. Similar MD results were published by Angell et al. ( 1982, 1983). These simulations showed an adjustment of silicate melt structure at T = 6000 K and 80 GPa pressure, but no quantitative relation was obtained to express the extent of structural adjustment per unit viscosity change. Therefore, it is not simple to extrapolate to 1700 K and 0.5 GPa, the P and T values of the Kushiro (1976) and Kushiro et al. (1976) experiments. Hence, it is not sure that the observed effects of pressure on the silicate melt viscosity compiled by Scarfe et al. (1987) and the influence of melt composition can be explained by means of the Angell et al. (1982, 1983) MD simulations.
Naturally occurring silicate liquids cover an enormous range of compositions that has given rise to a bewildering collection of esoteric names. However, the physical and chemical properties of only relatively few of these liquids have been studied experimentally. Therefore, one resorts to interpolation and more often to extrapolation to estimate how silicate melts in nature will react to changes of P, T, and composition. Unfortunately, such extrapolations in P-T-composition space are not straightforward because of possible changes in the internal structure of the liquid and in particular, the extent of polymerization. Qualitatively, it is fairly well known how compositional changes affect melt polymerization, but the effect of pressure has been less studied. If cooled at a high enough rate, all liquids can be transformed into glasses; this transformation does not depend on bonding type. On the contrary, the polymerization shown by silicate liquids depends very much on the bonding. At not too high pressures, i.e., at magmatic temperatures and P < 1 GPa, the great majority of the silicon atoms in silicate melts are surrounded by four nearest neighbor oxygen atoms. Polymerization of silicate liquids is the consequence of the formation of "'bridging oxygens", these oxygen atoms bonded to two silicon atoms join two SiO4 tetrahedra. As in crystalline silicates, in melts, two tetrahedra may share a vertex, but edge or plane sharing does not occur. This arrangement of threedimensionally interconnected SiO4 tetrahedra, gives rise to entities with a very open structure. These entities may form eventually a continuous network such as found in fully polymerized melts like SiOz. The physical-chemical nature of bonding in silicates with its directional characteristics is responsible for these structural features. The SiO4 network in silica has been described in detail by Mozzi and Warren (1969). Theoretical aspects of S i - O bonding and polymerization have been discussed by Harrison (1980) and De Jong and Brown (1980a,b). Kushiro (1976) and Kushiro et al. (1976) discovered that the viscosity of certain silicate melt compositions becomes smaller when pressure is applied. Different explanations have been suggested, namely the possibility that the coefficient of thermal expansion of the glass can be larger than that of the liquid at the same pressure and temperature (Gupta, 1987), the effect of pressure-induced coordination changes (Waff,
2. POLYMERIZATION AND MOLAR VOLUME In silicate melts, one distinguishes two different types of cations, the network formers and the network modifiers. At P = O.1 MPa and magmatic temperatures the structure of liquid Si02 does not depend much on temperature, but the influence of network modifiers on the 2725
2726
Y. Bottinga and P. Richet i
.
.
.
.
i
.
.
.
.
i
.
.
.
.
i
.
.
.
.
SiOz-Na 20, 1400"C
27.8
,t•~ •
m
~ 27.6 o
im
"6 27.4 E
27.2 (al i
i
i
i
I
10
20
30
40
50
Mol 96 NazO 27.4 .
.
.
.
i
.
.
.
.
i
SiOz-BaO 1700" C
A
'
A
t
~uE27.2
•i
e o
> 27.0 o
a network-modifying oxide is dissolved in liquid SiO2. Apparently, the transformation of bridging oxygen ( B O ) into nonbridging oxygen ( N B O ) gives rise to contraction. Measurements on liquid solutions belonging to the binary systems SiO2-N20 ( N = Li, Na, K) at 1400°C (Bockris et al., 1956) and S i O 2 - M O ( M = Mg, Ca, Sr, Ba) at 1700°C (Tomlinson et al., 1958) show perfect linear variations of the mean molar melt volumes with composition, indicating a constant SiO2 partial molar volume of about 26.6 cm 3, for 0.75 ~ X(SiO2) -> 0.50; see Fig. 2. This value is in essential agreement with the partial molar volumes of SiO2 at 1700 -- T (°C) -> 1400, reported by Bottinga et al. (1983) and Lange et al. (1987), for Fe-free melts with X(SiO2) --< 0.8. Hence, the data indicate that the partial molar volume of SiO2 decreases from 27.3 cm 3 at X(SiO2) = 1.0 to about 26.6 cm 3 at X (SiO2) --- 0.8. Therefore, the decrease of the mean molar volumes shown in Fig. 1 cannot be interpreted as being caused exclusively by a possible variation of the partial molar volumes of Na20 or BaO at X(SiO2) -> 0.8. The observations shown in Fig. 2 confirm the interpretation offered in the previous paragraph. Admittedly, these observed volume variations (Figs. 1, 2) are quite small; from a practical point of view, they do not affect the partial molar values for the oxide components of natural silicate melts deduced by Bottinga et al. ( 1982, 1983) or Lange and Carmichael (1987). Notwithstanding the smallness of these variations and a possible lack of accuracy, these data are valuable because of their precision and coherence. Huang and Cormack (1990, 1991 ) reported MD data for alkalisilicate glasses. These authors used the Vessal et al. (1989) three
E
~ 26.8
36
=E
. . . .
i
. . . .
i
. . . .
i
. . . .
F
, ~ 1 1 r
. . . .
(b) 26.6
l
0
l
l
l
I
15
,
i
i
i
I
i
h
30
b
i
i . . . . 45
32
60
Mol 96 BaO
FIG. 1. (a) SiO2-Na20, Mean molar volume at P = 0.l MPa, T = 14000C. Data from Bockris et al. (1956) and Brtickner (1964). (b) SiO2-BaO, Mean molar volume at P = 0.1 MPa, T = 1700°C. Data from Tomlinson et al. (1958) and Brtickner (1964).
silica structure has been well documented, particularly in recent work (Mysen and Frantz, 1995; Frantz and Mysen, 1994; Huang and Cormak, 1990, 1991; Farnan et al., 1992; Xue and Stebbins, 1993). A comprehensive review of earlier work can be found in Mysen (1988). The polymerization of silicate liquids causes an increase of the partial molar volume of dissolved SiO2, or in other words the partial molar volume of a bridging oxygen is larger than that of a nonbridging oxygen. This feature is discernible in the systematic variations of the mean molar volumes of liquids in the binary systems SiO2-Na20 and SiO2-BaO; see Fig. 1. For both systems one notices a decrease of the mean molar volume of the melt when the silica mole fraction (X (SiOz)) changes from 1 to 0.8. Unfortunately, mean molar volume measurements are very rare in this X (SiO:) range. The measurements on the SiO2-BaO melt system are noisy, but it is the only system for which there are so many observations for X(SiO2) ~ 0.8, and they confirm the trend shown in the SiO2-Na20 system. At 1400°C, the mean molar volumes of SiOz-NazO solutions decrease from 27.3 cm 3 at X(SiO2) = 1 (Briickner, 1964), to about 27.1 cm 3 at X(SiOz) = 0.8 (Bockris et al., 1956). The partial molar volume of Na20 in Fe-free silicate melts is about 29 cm 3 at X(SiO2) = 0.8 and 1400°C (Bottinga et al., 1983; Lange and Carmichael, 1987). The SiO2-BaO system at 1700°C (Tomlinson et al., 1958) shows for the same X(SiOz) range a diminution from 27.3 to 26.6 cm3/mol, while the partial molar volume of BaO equals 27.6 cm 3 at 1700°C (Bottinga et al., 1983). The molar volume of pure liqui d silica at 1700°C is the same as at 1400°C (Briiekner, 1964). Figure 1 shows that observed mean molar volume data have a tendency of becoming smaller when a network-modifying oxide is dissolved in pure SiO2, in spite of the fact that the network-modifier oxide has a partial molar volume larger than the molar volume of SiO2. The simplest interpretation of these observations is that the partial molar volume of SiO2 diminishes when
o
•
NazO
28
o
E ~ 24
LizO
SiOz-NzO
I[ 20 20
"
25
30
~
~
35
~
.
~
40
45
50
Mol % MzO 28
E
i
.
.
.
.
i
.
.
.
.
r
.
,
BaO
u
26
E
24
~
22
0
E e-
g
20
"/8
30
SiOz'MO 1 700"C . . . .
MgO (b) 3t5 . . . .
t . . . . 40
Mol % MO
I . . . . 45
50
FIG. 2. Mean molar volumes of binary silicate melts as a function of the network modifier concentration. (a) SiO2-N20, N = K, Na, Li at P = 0.1 MPa, T = 1400°C. Intercepts of linear fits at mol% N20 = 0; K20:26.61 cm 3, Na20:26.84 cm 3, Li20:26.84 cm 3. Mean molar volume measurements from Bockris et al. (1956). (b) SiO2-MO, M = Ba, Sr, Ca, Mg at P = 0.1 MPa, T = 1700°C. Intercepts of linear fits at mol% M O = 0; BaO: 25.76 cm 3, SrO: 25.90 cm 3, CaO: 26.77 cm 3, MgO: 26.65 cm 3. Mean molar volume values from Tomlinson et al. (1958).
Pressure dependence of viscosity of silicate melts body interatomic potential, i.e., the potential energy function does not depend exclusively on distances between atoms but also on directions. The typical aspects of bonding in silicates are better reproduced with a three-body potential than with the more commonly used two-body potentials (Kubicki and Lasaga, 1988). The fictive temperatares of the Huang and Cormak (1990, 1991) simulated glass samples are not known. These samples were prepared at T = 6000 K and subsequentlycooled to 3000, 1500, and 293 K. To achieve structural relaxation, the samples stayed for 100,000 timesteps of 2 fs at each of these temperatures. Huang and Cormak ( 1990, 1991) noted that at all temperatures the Si-NBO bond lengths were shorter than the Si-BO ones. The conclusion that the partial molar volume of NBO is smaller than that of BO, is in harmony with X-ray work on crystalline Na2SiO3, ot-NazSizOs,/~-Na~Si2Os,and Li2Si205, which indicates a Si-BO bond length of about 0.167 nm and a Si-NBO bond length of 0.156 nm (McDonald and Cruickshank, 1967; Plant and Cruickshank, 1968; Plant, 1968; Liebau, 1961). Similar bond lengths have been determined for pyroxenes; for crystalline diopside Cameron et al. (1973) found Si-BO and Si-NBO bond lengths of 0.169 and 0.159 nm, respectively, which were nearly independent of temperature at 25 < T < 1000°C. 3. POLYMERIZATION AND VISCOSITY In the following paragraphs it will be shown that a minor extension of the Adam-Gibbs configurational entropy theory (Adam and Gibbs, 1965) suffices to explain at least qualitatively most of the available observations of the effect of pressure on viscosity. Lack of thermodynamic data for silicate melts makes a quantitative verifcation of the proposed model not yet possible. One deduces easily from the Adam and Gibbs (1965) theory that the viscosity of a liquid is given by In (rl) = Ae + Be/(TSco,f),
(1)
where r/ is the viscosity, Ae and Be are constants, T is the absolute temperature, and Sconfis the configurational entropy at temperature T. This equation has been discussed in the earth sciences literature many times since 1972 (Bottinga and Weill, 1972), and more recently in Richet (1984), Richet and Neuville (1992), and Bottinga et al. ( 1995 ). Scoffcan be evaluated calorimetrically, whereas A~ and Be are obtained by fitring Eqn. 1 to viscosity observations. Available evidence for liquid silicates indicates that the temperature dependence of Ae or Be are negligible in comparison with that of the denominator in Eqn. 1 (Richet, 1984). For reasons of simplicity, the discussion in this section will be limited to SiO2 and the binary alkali silicates. Using Eqn. l, one obtains (Richet, 1984; Gupta, 1987) (0 In (rl)/OP)r = [-Be/(TS2on~)]/(cOSco,f/cOP)r,
(2)
assuming that both A~ and Be do not depend on P. In our discussion we consider only the P-T region T < 2000 K and 0 < P -< 2 GPa; we assume that in this region Be is constant. Evidently, Sconfdepends on P, T, and on the extent of polymerization ((), hence dSco,f = ( cOSco,f/ O P )r.¢d P + ( oqSconf/oqT )p,~dT
+ (OS~o,f/O~)e,r[(C~/OP)rdP + (O~/OT)edT].
(3)
2727
polymerization is equal to the fraction of bridging oxygen atoms. Hence, changes in the extent of polymerization cause volume changes, therefore the pressure dependence of should be taken care of. Because of the large S i - O bond energy of about 450 kJ/mol (Balta and Balta, 1976), polymerization of binary silicate liquids may be considered to be temperature independent for T < 2000 K. The temperature independence of the heat capacity of liquid SiO2 (Richet and Bottinga, 1986) and of the partial molar heat capacity of SiO2 in binary silicate melts suggest that the distribution of SiO4 tetrahedra in these melts does not change radically in this temperature range. We suppose that the configurational entropy associated with polymerization is given by the entropy of mixing of NBOs and BOs. The entropy effect of mixing is additive, so we can write Sconf(P, T, () = Sc*onf(P, T) + S*n*f(((P, T ) ) ,
where Sc*onf(P,T) does not depend on ~; it is the part of the configurational entropy which is usually only considered, and one has
S * ~ ( ~ ( P , T ) ) = -RNo[~ In (()
+(1-~)In(1-~)],
(5)
R is the ideal gas constant and S*o~ the mean oxide molar configurational entropy of polymerization. No stands for the number of oxygen atoms per gram formula weight (gfw) of melt divided by the number of oxide moles per gfw of melt. Hence, for a SiO2 melt No equals 2, and for Na2Si2Os, No is equal to 5/3. Obviously, one can not apply Eqn. 5 to binary alkaline earth silicates because the two NBOs, due to the presence of one alkaline earth metal atom, are not distributed randomly. This effect causes the configurational entropies for liquid wollastonite, enstatite, and diopside (see Richet and Neuville, 1992) per oxide mole to be significantly less than the values listed in Table 1 for the system SiO2-Na20. Differentiating Eqn. 5, one obtains
(cOS*~/c~)e.r = -RNo In (~/(1 - ~)), where0<(<
(6)
1.
TABLE 1. Comparison between calculations of the configurational entropy (see text) in liquid XSiO2-(1 - X)Na20 at T = Tg. mol fract.
Tg
Calc. conf. ex%tro£~ (c) Diff.
X
(K)
eqn. (i) (a) ecru.(5) (b)
.80 .75 .70 .67 .65 .60 .55
745 734 730 730 707 694 686
9.0 8.3 8.9 7.9 10.1 8.9 8.9
7.9 8.7 9.2 9.3 9.4 9.2 8.8
% -12.2 + 4.8 + 3.4 +17.7 - 6.9 + 3.4 - i.i
(a) V i s c o s i t y m e a s u r e m e n t s by ~OC~ et al. (1994), BOCKRIS et al. (1956), N~X/VILLE (1993) a n d MEILING a n d UHLMANN (1977) w e r e u s e d a s
ir~ut data in eqn. (1).
Generally, the extent of polymerization is used only in a qualitative sense. For the simple silicate liquids considered in this paper, we define ~ = BO/(BO + NBO), i.e., the extent of
(4)
(b) q h e BO, ~ 0 abundances of ~ (1990) were u s e d in eqn. (5)
(c) Mean oxide molar e n ~ in Joules/K.
and COR~CK
values (see text)
2728
Y. Bottinga and P. Richet TAaLE 2. Observed changes of the viscosity of silicate melts due to increases of pressure, compiled by Scarfe et al. (1987). Ccm~ositicn
Ge~ Na20. Si02 Na20.2SIO2 Na20.3SIO2
cet~i2~ K21¢~Si5012 NaCaAISi207 NaAISi206 NaA/Si308
01. r'a~-,el. ALk. o1 .basalt O1. tholeiite
thol. An~eaite Obsid/an
~ (c)
1.0 0.33 0.60 0.71 0.33 0.67 0.71 1.0 1.0 0.71 0.79 0.81 0.84 0.93 0.97
Viscosity c h a n ~ ( d P a s)
6270; 1160 (a) 2; 15 151; 90 485; 170 3; ii 2890;1000 36; 26 64700;5090 113000;18400 i0; 6 i0; 8 35; 17 161; 80 1820; 810 -80%(b)
Pressure interval (GPa)
0 0 0 0 0 5 0 0 0 0 0 0 0 0 0
-
1 1 2 2 1.5 2 1.5 2.4 2 2 1.5 2 1.2 2 2
Temp. (°C)
1425 1300 1200 1175 1640 1300 1450 1350 1400 1450 1400 1400 1300 1350 1400
(a) ist v a l u e m e a s u r e d a t l o w pressure, 2 n d v a l u e at h i g h pressure. (b) the o b s e r v e d d e c r e a s e of the viscosity is g i v ~ in % of initial value at P = 0.i MPa; no other details available. (c) ~ = B O / T o t a l oxygen, s e e text; e s t i m a t e d from the stoichiQmetry.
Using a Maxwell relation gives
( OS*onflOe)r = - ( OVco.flOT)v,
(7)
where Vco.f = VL -- V~, V ~ is the configurational molar volume, and VL and V~ are the liquid and glass molar volumes, respectively. Hence,
( OSeonf/OP)T = - [ ( OVL/OT)p -- ( OVJOT)e] -
RNoln (~/(1 - 4))
(8)
and (0 In (rl)lOP)r = [Bd(TS~o., ) ] [ ( OVLIOT)v,~
- (OVJOT)v,¢ + RNo In (4/(1 - 4))(O~/OP)T],
(9)
where Sconeis the configurational entropy at T, P, and 4. At T = Ts and P = 0.1 MPa, Sco.f (Eqn. 4) is known, it is equal to the residual entropy at 0 K, or can be obtained from viscosity observations, see Richet (1984), and S*o.~ can be calculated with Eqn. 5. At T > Tg, the configurational entropy can be calculated by integrating
S~o,f( P, T, 4) = Sconf(P, rg, 4(P, rg)) + f (C~one(T)lT)dT
(10)
from Tg to T. In Eqn. 10 Ce~o.f = c ~ - cm, i.e., the difference between the heat capacities for the liquid and vitreous phase at constant pressure. At T >-- Tg, CeL has been measured for many liquid silicates or can be evaluated with simple models (Richet and NeuviUe, 1992; Richet and Bottinga, 1986), and cm is given by the Dulong and Petit limiting value, cm = 3 N . R (Haggerty et al., 1968; Richet, 1984), where N equals the number of gatoms per gfw. For the simple binary
silicate liquids discussed here, the extent of polymerization is independent of temperature, but the configurational entropy increases with temperature because ceco,a > 0 (Richet and Bottinga, 1984b). The measured coefficients of thermal expansion ( a ) of silicate liquids and their corresponding glasses are positive for all compositions showing a negative variation of the viscosity with pressure. Knoche et al. ( 1992, 1994) reported aL > aG > 0 for their observations on melts in the systems NaA1Si3Os-CaAl2Si2Os-CaMgSi206 and SiO2-Na20 outside a narrow temperature region associated with the glass transition. Hence, the difference between the two partial derivatives of the molar volumes of the liquid and the glass (Eqn. 9) is positive. The partial molar volume of bridging oxygen atoms is larger than that of nonbridging ones, see section II, consequently, (04/OP)r < 0 according to Le Ch~telier's principle. When 4 > 0.5, an increase in P causes a diminution of 4 and an augmentation of the configurational entropy (Eqn. 6), resuiting in a lowering of the viscosity (Eqn. 9). But if 4 < 0.5, a decrease of the extent of polymerization due to an increase in pressure gives a decrease of the configurational entropy and hence an increase of the viscosity, conform to the behavior of ordinary liquids. These conclusions are in qualitative agreement with the data compiled by Scarfe et al. (1987) on the pressure dependence of the viscosity of fourteen silicate melts with 4 larger as well as smaller than 0.5 (Table 2). Certainly, the dependence of the configurational entropy on the extent of polymerization in Eqn. 9 is not sufficiently general for treating all the multicomponent silicate liquids listed in the Scarfe et al. (1987) compilation. At high temperature and for 4 > 0.5 the anomalous pressure effect tends to disappear, while for ~ < 0.5 the normal viscosity increase with pressure
Pressure dependence of viscosity of silicate melts should become less important (Eqn. 9). The reason is that high T means large Scone,and large values of T and Scoa cause a diminution of the absolute value of ( 0 In (77)/0 P)r. Pressure or addition of network modifiers to the melt will decrease and cause eventually the melt viscosity to change to a normal P dependence. 4. CONFIGURATIONAL ENTROPY The configurational entropy is a measure of the number of configurational states to which a system has access, or in other words the number of configurations the system can adopt without affecting its average internal energy. In numerous papers we have already discussed this subject and how one can evaluate Sco,f at different temperatures (Richet, 1984; Richet and Bottinga, 1986; Richet and Neuville, 1992). When upon cooling a liquid passes the glass transition its configurational entropy is frozen in, i.e., this entropy does not change when the temperature is further decreased. Hence by measuring the residual entropy of an amorphous phase at T = 0 K one measures the configurational entropy at T = Tg. The residual entropy of silica glass at 0 K is 5.1 J/(mol. K) (Richer et al., 1982). The source of this entropy is not known. Bell and Dean (1968) have built a spokes and ball random network model (Mozzi and Warren, 1969); according to them, the number of ways one can add a single SiO4 tetrahedron at a typical site of the model surface is very small, because the restricted range of permissible S i - O - S i angles and because the infinite extent of the model. This point of view was shared by Thathachari and Tiller (1985), who found it very difficult to construct a space filling random network of SiO4 tetrahedra without NBOs for amorphous SiO2 because of the stringent limitations imposed by the minimum tolerated O - O distance in such a network. Therefore, we have simply assumed that the residual configurational entropy is due to the freezing in of dangling bonds at the glass transition, i.e., 1480 K (Richet et al., 1982). The configurational entropy of SiO2, due to the random distribution of NBO over the oxygen positions of the SiO4 tetrahedra can be calculated with Eqn. 5; with ( = 0.91 one obtains S~onf(Tg) = 5.0 J/(mol K). Bell and Dean (1968) calculated a value of 5.78 J/(mol. K) using their spoke and ball model for SiO2 and some rather obvious approximations. The presence of NBOs in amorphous SiO2 is controversial; Mikkelsen and Galeener (1980) argue that the 606 cm-I Raman line in the amorphous pure SiO2 spectrum is due to NBOs. They noticed that the 606 cm-1 peak intensity for different vitreous SiO2 samples is proportional to the reciprocal fictive temperatures of the samples. Solution in liquid silica of network modifier oxides, i.e., alkali metal oxides and alkali earth metal oxides, causes the formation of NBOs, and thus alleviates the steric difficulties mentioned in Thathachari and Tiller (1985). The observations by Farnan and Stebbins (1990) and Farnan et al. (1992) on vitreous K2Si409 and KMg0.sSi409 indicate that the distribution of NBOs and BOs is frozen in when the cooling silicate liquid is transformed into a glass. The distribution of these two types of oxygens is a source of residual entropy in silicate glasses. Equation 5 and the Huang and Cormack (1990) NBO and BO abundances were used to calculate the configurational entropy of binary alkali silicates. These NBO and BO abun-
2729
dances are compatible with those one can deduce from the Q species concentrations published by Mysen and Frantz (1994) inferred from Raman spectra of melts in the system SiO2Na20 at magmatic temperatures. The NMR observations by Brandriss and Stebbins (1988) for vitreous Na2Si205 are compatible with those by Mysen and Frantz (1994). Available evidence indicates that all oxygen atoms are bonded to one silicon atom at least in binary silicate melts (Huang and Cormack, 1990; Misawa et al., 1980). To a first approximation, MD simulations (Huang and Cormak, 1990, 1991 ) show that globally the NBO concentration is given by the stoichiometry. The configurational entropies for xSiO2 - ( 1 - x)Na20, with 0.81 > x > 0.54, calculated with Eqn. 5 are compared (Table 1 ) with those obtained by fitting Eqn. 1 to the observed viscosities of these melts as outlined in Richet (1984). The average of the absolute values (Table 1 ) of the differences between these two sets of entropy data is about 7%, which in the present context should be considered as an excellent agreement. Hence, the configurational entropies of these melts could be due to the distribution of the two structural types of oxygen atoms, i.e., NBO and BO. Recent measurements by Gaskell et al. (1991) and earlier observations reviewed in Bottinga et al. ( 1981 ), indicate that alkali and alkaline earth metal cations are not randomly distributed in simple binary silicate melts. This implies the clustering of NBOs (Greaves and Ngai, 1994), which has been observed by Farnan et al. (1992). These observations are in harmony with the results of Huang and Cormak ( 1991 ), who noticed also the clustering tendencies of alkali ions and NBOs. Hence, Eqn. 5 is at best only a good approximation. As far as we are aware, evidence in favor of ideal mixing in silicate liquids is indirect. Unfortunately, there are no simple ways to evaluate the entropy of mixing other than making use of the ideal solution model. However, application of this model gives often physically acceptable results, which may lack the desired accuracy but are qualitatively correct. While the N B O - B O distribution is frozen in when a silica or silicate liquid transforms into a glass the self diffusion of network modifier atoms is only slightly effected by the glass transition, see Dingwell (1990) and Dingwell and Webb (1990). The alkali and alkali earth ions are confined to reside in the neighborhood of NBOs (Farnan et al., 1992; Greaves and Ngai, 1994); apparently, they do not contribute to the generation of configurational states playing a role in the process of viscous deformation. This conclusion is not the same as, but similar to, Ryerson's (1985) statement that the entropic effect of the mixing of network-modifying cations does not influence the activity of SiO2, but that this latter quantity is mainly affected by the enthalpic effects associated with variations in bond energies due to the presence of the modifiers. The observed effect of cation mixing on the viscosity of silicate melts (Richet, 1984; Neuville and Richet, 1991 ) may be interpreted as being the result of the fact that these cations reflect the distribution of the NBOs. 5. EPILOGUE We have shown, using some reasonable assumptions, that the Kushiro discovery can be explained with the Adam and Gibbs (1965) theory, when the pressure sensitivity of the
2730
Y. Bottinga and P. Richet
polymerization shown by the data of Bockris et al. (1956) and Tomlinson et al. (1958) is taken into account and the appropriate configurational entropy change is introduced into Eqn. 5. Hence, at the time of the discovery of the anomalous P-viscosity effect, theory was sufficiently advanced to explain the observations by Kushiro (1976) and Kushiro et al. (1976). However, the association of BOs and NBOs with configurational entropy became possible only after the publication of the N M R observations by Stebbins and his colleagues (Stebbins et al., 1992) and the realization of the importance of residual entropy of silicate glasses (Richet, 1984). Acknowledgments--We would like to thank Mark Ghiorso for his scientific and professional editorial handling of our manuscript. The work published in this contribution benefited from financial support by the CNRS-DBT program. Editorial handling: M. S. Ghiorso
REFERENCES Adam G. and Gibbs J. H. (1965) On the temperature dependence of cooperative relaxation properties in glass-forming liquids. Z Chem. Phys. 43, 139-146. Angell C. A., Cheeseman P. A., and Tamaddon S. (1982) Pressure enhancement of ion mobilities in liquid silicates from computer simulation studies to 800 kilobars. Science 218, 885-887. Angell C. A., Cheeseman P. A., and Tamaddon S. (1983) Water-like transport property anomalies in liquid silicates investigated at high T and P by computer techniques. Ball MindraL 106, 87-98. Balta P. and Balta E. (1976) Introduction to the Physical Chemistry of the Vitreous State. Abacus Press. Bell R. J. and Dean P. (1968) The configurational entropy in the random network theory. Phys. Chem. Glasses 9, 125-127. Bockris J. O'M., Tomlinson J. W., and White J. L. (1956) The structure of liquid silicates: partial molar volumes and expansivities. Trans. Faraday Soc. 53, 299-310. Bottinga Y. and Weill D. F. (1972) The viscosity of magmatic silicate liquids: a model for calculation. Amer. Z Sci. 272, 438-475. Bottinga Y., Weill D. F., and Ricbet P. ( 1981 ) Thermodynamic modeling in silicate melts. In Thermodynamics of Minerals and Melts (ed. R. C. Newton et al.); Adv. Phys. Geochem. 1, pp. 207-245. Springer Verlag. Bottinga Y., Weill D. F., and Richet P. (1982) Density calculations for silicate liquids: I. Revised method for aluminosilicate compositions. Geochint Cosmochim. Acta 46, 909-920. Bottinga Y., Riehet P., and Weill D. F. (1983) Calculation of the density and thermal expansion coefficient of silicate liquids. Bull. MindraL 106, 129-138. Bottinga Y., Richet P., and Sipp A. (1995) Viscosity regimes of homogeneous silicate melts. Amer. Mineral 80, 305-318. Brandriss M. E. and Stebbins J. F. (1988) Effects of temperature on the structures of silicate liquids: 29SiNMR results. Geochim. Cosmochim. Acta $2, 2659-2670. Briickner R. (1964) Cliarakteristische physikalisehe Eigenschaften der oxydisehen Hauptglasbildner und ihre Beziehungen zur Strukmr der Gltisem H: Mechanische und optische Eigenschaften als Funktion der thermische Vorgeschichte. Glastechn. Ber. 37, 459475. De Jong B. H. W. S. and Brown G. E. (1980a) Polymerization of silicate and aluminate tetrabedra in glasses, melts, and aqueous solutions--I. Electronic structure of I-I6Si207, I-I6AISiO~-, and I-I6A120~-. Geochim. Cosmochim. Acta 44, 491-511. De Jong B. H. W, S. and Brown G. E. (1980b) Polymerization of silicate and aluminate tetralaedra in glasses, melts and aqueous solutions--H. The network modifying effects of Mg 2+, K + , Li + , H +, OH-, F - , H20, CO2, and H30 + on silicate polymers. Geochim. Cosmochim. Acta 44~ 1627-1642.
Dingwell D. B. (1990) Effects of structural relaxation on cationic tracer diffusion in silicate melts. Chem. GeoL 82, 209-216. Dingwell D. B. and Webb S. L. (1990) Relaxation in silicate melts. Eur. J. Mineral 2, 427-449. Faman I. and Stebbins J. F. (1990) A high temperature 29Si NMR investigation of solid and molten silicates. J. Amer. Chem. Soc. 112, 32-39. Farnan I. et al. (1992) Quantification of the disorder in networkmodified silicate glasses. Nature 358, 31-35. Frantz J. D. and Mysen B. O. (1995) Raman spectra of BaO-SiO2, SrO-SiO2, and CaO-SiO2 melts to 1600"(2. Chem. Geol. (in press) Gaskell P. H., Eckersley M. C., Barnes A. C., and Chieux P. ( 1991 ) Medium-range order in the cation distribution of a calcium silicate glass. Nature 350, 675-677. Greaves G. N. and Ngal K. L. (1994) Ionic transport properties in oxide glasses derived from atomic structure. J. Non-Cryst. Solids 172-174, 1378-1388. Gupta P. K. (1987) Negative pressure dependence of the viscosity. J. Amer. Ceram. Soc. 70, C152-C153. Gupta P. K. (1989) Models of the glass transition. Rev. Solid State Sci. 3, 221-257. World Scientific Publishing Co. Haggerty J. S., Cooper A. R., and Heasley J. H. (1968) Heat capacity of three inorganic glasses and supercooled liquids. Phys. Chem. Glasses 5, 130-136. Harrison W. A. (1980) Electronic Structure and the Properties of Solids. W. H. Freeman and Company. Huang C. and Cormack A. N. (1990) The structure of sodium silicate glass. J. Chem. Phys. 93, 8180-8186. Huang C. and Cormack A. N. (1991) Structural differences and phase separation in alkali silicate glasses. J. Chem. Phys. 95, 3634-3642. Knoche R., Dingwell D. B., and Webb S. L. (1992) Non-linear temperature dependence of liquid volumes in the system albite-anorthite-diopside. Contrib. Mineral. Petrol. 111, 61-73. Knoche R., Dingwell D. B., Seifert S. A., and Webb S. L. (1994) Non-linear properties of supercooled liquids in the system Na20SiO2. Chem. Geol. 116, 1-16. Kubicki J. D. and Lasaga A. C. (1988) Molecular dynamics simulations of SiO2 melt and glass: Ionic and covalent models. Amer. Mineral 73, 941-955. Kushiro I. (1976) Changes in viscosity and structure of melt of NaA1Si206 composition at high pressures. J. Geophys. Res. 81, 6347-6350. Kushiro I., Yoder H. S., and Mysen B. O. (1976) Viscosities of basalt and andesite melts at high pressures. J. Geophys. Res. 81, 63516356. Lange R. A. and Carmichael I. S. E. (1987) Densities of K20-Na20CaO-MgO-FeO-Fe203-A1203-TiO2-SiO2 liquids: new measurements and derived partial molar properties. Geochim. Cosmochim. Acta 51, 2931-2946. Lieban F. (1961) Untersuchungen an Schichtsilikaten des Formeltyps Am(Si2Os)n. I. Die Kristallstruktur der Zimmertemperaturform des Li2Si2Os. Acta Cryst. 14, 389-395. McDonald W. S. and Cruickshank D. W. J. (1967) A reinvestigation of the structure of sodium metasilicate, Na2SiO3. Acta Cryst. 22, 37 -43. Meiling G. S. and Uhlmann D. R. (1977) Crystallisation and melting kinetics of sodium disilicate. Phys. Chem. Glasses 8, 62-68. Mikkelsen J. C. and Galeener F. L. (1980) Thermal equilibration of Raman active defects in vitreous silica. J. Non-Cryst. Solids 37, 71-84. Misawa M., Price D. L., and Suzuki K. (1980) The short range structure of alkali disilicate glasses by pulsed neutron total scattering. J. Non-Cryst. Solids 37, 85-97. Mozzi R. L. and Warren B. E. (1969) The structure of vitreous silica. J. Appl. Cryst. 2, 164-172. Mysen B. O. (1988) Structure and Properties of Silicate Melts. Elsevier. Mysen B. O. and Frantz J. D. (1994) Silicate melts at magmatic temperatures: in-situ structure determination to 1651"(2 and effect of temperature and bulk composition on the mixing behavior of structural units. Contrib. Mineral. Petrol. 117, 1-14.
Pressure dependence of viscosity of silicate melts Neuville D. R. (1992) Etude des Proprittts Thermodynamiques et Rhtologiques des Silicates Fondus. Thtse de Doctorat, Univ. Paris 7. Neuville D. R. and Richet P. ( 1991 ) Viscosity and mixing in molten (Ca, Mg) pyroxenes and garnets. Geochim. Cosmochim. Acta 55, 1011-1019. Plant A. K. (1968) A reconsideration of the crystal structure of/3Na2SizOs. Acta. Cryst. B24, 1077-1083. Plant A. K. and Cruickshank D. W. J. (1968) The crystal structure of a-Na2Si2Os. Acta Cryst. B24, 13-19. Richet P. (1984) Viscosity and configurational entropy of silicate melts. Geochim. Cosmochim. Acta 48, 471-483. Richet P. (1990) GeO2 vs. SiO2: Glass transistions and thermodynamic properties of polymorphs. Phys. Chem. Minerals 17, 7 9 88. Richet P. and Bottinga Y. (1984a) Glass transition and thermodynamic properties of amorphous SiO~, NaA1Si,O2,+2 and KAISi3Os. Geochim. Cosmochim. Acta 48, 453-470. Richet P. and Bottinga Y. (1984b) Heat capacity of aluminum-free silicates. Geochim. Cosmochim. Acta 49, 471-486. Richet P. and Bottinga Y. (1984c) Anorthite, andesine, wollastonite, diopside, cordierite and pyrope: thermodynamics of melting, glass transitions, and properties of the amorphous phases. Earth Planet. Sci. Lett. 67, 415-432. Richet P. and Bottinga Y. (1986) Thermochemical properties of silicate glasses and liquids: a review. Rev. Geophys. 24, 1-26. Richet P. and Neuville D. R. (1992) Thermodynamics of silicate melts: configurational properties. In Thermodynamic Data Systematics and Estimation (ed. S. Saxena), pp. 132 - 160. Springer-Verlag. Richet P., Bottinga Y., Denielou L., Petitet J. P., and Tequi C. (1982) Thermodynamic properties of quartz, cristobalite and amorphous SiO2: drop calorimetry measurements between 1000 and 1800 K and a review from 0 to 2000 K. Geochim. Cosmochim. Acta 46, 2639-2658.
2731
Ryerson F. J. (1985) Oxide solution mechanisms in silicate melts: Systematic variations in the activity coefficient of SiO2. Geochim. Cosmochim. Acta 49, 637-649. Scarfe C. M., Mysen B. O., and Virgo D. (1987) Pressure dependence of the viscosity of silicate melts. In Magmatic Processes: Physicochemical Principles (ed. B. O. Mysen); Spec. Publ. No. 1, pp. 59-67. Geochem. Soc. Scherer G. (1990) Theories of relaxation. J. Non-Cryst. Solids 123, 75-89. Sharma S. K., Virgo D., and Kushiro I. (1979) Relationship between density, viscosity and structure of GeO2 melts at low and high pressure. J. Non-Cryst. Solids 33, 235-248. Stebbins J. F., Farnan I., and Xue X. (1992) The structure and dynamics of silicate liquids: A view from NMR spectroscopy. Chem. Geol. 96, 371-386. Thathachari T. T. and Tiller W. A. (1985) Importance of oxygenoxygen interactions in silica structures. J. Appl. Phys. 57, 18051811. Tomlinson J. W., Heines M. S. R., and Bockris J. O'M. (1958) The structure of liquid silicates, pt. 2. Trans. Faraday Soc. 54, 18221833. Urbain G., Bottinga Y., and Richet P. (1982) Viscosity of liquid silica, silicates and alumino-silicates. Geochim. Cosmochim. Acta 46, 1061-1072. Vessal B., Leslie M., and Catlow C, R. A. (1989) Molecular dynamics simulation of silica glass. Molec. Simul. 3, 123-136. Waft H. S. (1975) Pressure-induced coordination changes in magmatic liquids. Geophys. Res. Lett. 2, 193-196. Woodcock L. V., Angell C. A., and Cheeseman P. (1976) Molecular dynamics of the vitreous state: Simple ionic systems and silica. J. Chem. Phys. 65, 1565-1577. Xue X. and Stebbins J. F. (1993) 23Na NMR chemical shifts and local Na coordination environments in silicate crystals, melts and glasses. Phys. Chem. Minerals 20, 297-307.
Pergamon
~16-7~7(~)~1~-9
Silicate melts: The "anomalous" pressure dependence of the viscosity Y. BOTTINGAand P. RICHET Institut de Physique du Globe de Pads, D6partementdes G-6omat6riaux,75252 PARIS Cedex 05, France (Received November 3, 1994; accepted in revisedform April 5, 1995)
Abstract--The decrease of the specific volume, when the extent of polymerization diminishes, is a cause of the pressure sensitivity of the viscosity of silicate melts. This effect can be explained by means of the Adam and Gibbs (1965) theory, taking into account the pressure dependence of the degree of polymerization of the melt and its influence on the configurational entropy. At temperatures close to their glass transitions, liquid silica and SiO2-Na20 melts have configurational entropies that are probably due to the mixing of their bridging and nonbridging oxygen atoms. 1. INTRODUCTION
1975), and S i - O bond weakening due to pressure-induced distortion of the liquid structure (Woodcock et al., 1976; Sharma et al., 1979). The Gupta (1987) suggestion can not be experimentally verified because there is no way to measure the coefficient of thermal expansion of a silicate glass at temperatures in excess of the glass transition, where the negative P effect on silicate melt viscosity has been observed. Available data for the coefficient of thermal expansion of glasses and of compositionally equivalent liquids (Knoche et al., 1992, 1994) are not in favor of this suggestion. The relation between the pressure dependence of the viscosity and the coefficients of thermal expansion of liquid and glass, is evident by inspection of Eqn. 8, see section III. Sharma et al. (1979) have shown that the observed pressure induced viscosity decrease in liquid GeO2 was not due to a change in the coordination of Ge. This does not rule out this mechanism to be a cause of a pressure effect on the viscosity, but it means that it is not an explanation for the observations by Kushiro (1976) and Kushiro et al. (1976). However, Molecular Dynamics (MD) simulation of SiO2 showed a pressure enhancement of the diffusion coefficients of Si and O (Woodcock et al., 1976), implying a decrease of the viscosity. The influence of pressure on the mobility of Si and O was believed to be due to observed changes of bond angles and lengths and ephemeral coordination changes. Similar MD results were published by Angell et al. ( 1982, 1983). These simulations showed an adjustment of silicate melt structure at T = 6000 K and 80 GPa pressure, but no quantitative relation was obtained to express the extent of structural adjustment per unit viscosity change. Therefore, it is not simple to extrapolate to 1700 K and 0.5 GPa, the P and T values of the Kushiro (1976) and Kushiro et al. (1976) experiments. Hence, it is not sure that the observed effects of pressure on the silicate melt viscosity compiled by Scarfe et al. (1987) and the influence of melt composition can be explained by means of the Angell et al. (1982, 1983) MD simulations.
Naturally occurring silicate liquids cover an enormous range of compositions that has given rise to a bewildering collection of esoteric names. However, the physical and chemical properties of only relatively few of these liquids have been studied experimentally. Therefore, one resorts to interpolation and more often to extrapolation to estimate how silicate melts in nature will react to changes of P, T, and composition. Unfortunately, such extrapolations in P-T-composition space are not straightforward because of possible changes in the internal structure of the liquid and in particular, the extent of polymerization. Qualitatively, it is fairly well known how compositional changes affect melt polymerization, but the effect of pressure has been less studied. If cooled at a high enough rate, all liquids can be transformed into glasses; this transformation does not depend on bonding type. On the contrary, the polymerization shown by silicate liquids depends very much on the bonding. At not too high pressures, i.e., at magmatic temperatures and P < 1 GPa, the great majority of the silicon atoms in silicate melts are surrounded by four nearest neighbor oxygen atoms. Polymerization of silicate liquids is the consequence of the formation of "'bridging oxygens", these oxygen atoms bonded to two silicon atoms join two SiO4 tetrahedra. As in crystalline silicates, in melts, two tetrahedra may share a vertex, but edge or plane sharing does not occur. This arrangement of threedimensionally interconnected SiO4 tetrahedra, gives rise to entities with a very open structure. These entities may form eventually a continuous network such as found in fully polymerized melts like SiOz. The physical-chemical nature of bonding in silicates with its directional characteristics is responsible for these structural features. The SiO4 network in silica has been described in detail by Mozzi and Warren (1969). Theoretical aspects of S i - O bonding and polymerization have been discussed by Harrison (1980) and De Jong and Brown (1980a,b). Kushiro (1976) and Kushiro et al. (1976) discovered that the viscosity of certain silicate melt compositions becomes smaller when pressure is applied. Different explanations have been suggested, namely the possibility that the coefficient of thermal expansion of the glass can be larger than that of the liquid at the same pressure and temperature (Gupta, 1987), the effect of pressure-induced coordination changes (Waff,
2. POLYMERIZATION AND MOLAR VOLUME In silicate melts, one distinguishes two different types of cations, the network formers and the network modifiers. At P = O.1 MPa and magmatic temperatures the structure of liquid Si02 does not depend much on temperature, but the influence of network modifiers on the 2725
2726
Y. Bottinga and P. Richet i
.
.
.
.
i
.
.
.
.
i
.
.
.
.
i
.
.
.
.
SiOz-Na 20, 1400"C
27.8
,t•~ •
m
~ 27.6 o
im
"6 27.4 E
27.2 (al i
i
i
i
I
10
20
30
40
50
Mol 96 NazO 27.4 .
.
.
.
i
.
.
.
.
i
SiOz-BaO 1700" C
A
'
A
t
~uE27.2
•i
e o
> 27.0 o
a network-modifying oxide is dissolved in liquid SiO2. Apparently, the transformation of bridging oxygen ( B O ) into nonbridging oxygen ( N B O ) gives rise to contraction. Measurements on liquid solutions belonging to the binary systems SiO2-N20 ( N = Li, Na, K) at 1400°C (Bockris et al., 1956) and S i O 2 - M O ( M = Mg, Ca, Sr, Ba) at 1700°C (Tomlinson et al., 1958) show perfect linear variations of the mean molar melt volumes with composition, indicating a constant SiO2 partial molar volume of about 26.6 cm 3, for 0.75 ~ X(SiO2) -> 0.50; see Fig. 2. This value is in essential agreement with the partial molar volumes of SiO2 at 1700 -- T (°C) -> 1400, reported by Bottinga et al. (1983) and Lange et al. (1987), for Fe-free melts with X(SiO2) --< 0.8. Hence, the data indicate that the partial molar volume of SiO2 decreases from 27.3 cm 3 at X(SiO2) = 1.0 to about 26.6 cm 3 at X (SiO2) --- 0.8. Therefore, the decrease of the mean molar volumes shown in Fig. 1 cannot be interpreted as being caused exclusively by a possible variation of the partial molar volumes of Na20 or BaO at X(SiO2) -> 0.8. The observations shown in Fig. 2 confirm the interpretation offered in the previous paragraph. Admittedly, these observed volume variations (Figs. 1, 2) are quite small; from a practical point of view, they do not affect the partial molar values for the oxide components of natural silicate melts deduced by Bottinga et al. ( 1982, 1983) or Lange and Carmichael (1987). Notwithstanding the smallness of these variations and a possible lack of accuracy, these data are valuable because of their precision and coherence. Huang and Cormack (1990, 1991 ) reported MD data for alkalisilicate glasses. These authors used the Vessal et al. (1989) three
E
~ 26.8
36
=E
. . . .
i
. . . .
i
. . . .
i
. . . .
F
, ~ 1 1 r
. . . .
(b) 26.6
l
0
l
l
l
I
15
,
i
i
i
I
i
h
30
b
i
i . . . . 45
32
60
Mol 96 BaO
FIG. 1. (a) SiO2-Na20, Mean molar volume at P = 0.l MPa, T = 14000C. Data from Bockris et al. (1956) and Brtickner (1964). (b) SiO2-BaO, Mean molar volume at P = 0.1 MPa, T = 1700°C. Data from Tomlinson et al. (1958) and Brtickner (1964).
silica structure has been well documented, particularly in recent work (Mysen and Frantz, 1995; Frantz and Mysen, 1994; Huang and Cormak, 1990, 1991; Farnan et al., 1992; Xue and Stebbins, 1993). A comprehensive review of earlier work can be found in Mysen (1988). The polymerization of silicate liquids causes an increase of the partial molar volume of dissolved SiO2, or in other words the partial molar volume of a bridging oxygen is larger than that of a nonbridging oxygen. This feature is discernible in the systematic variations of the mean molar volumes of liquids in the binary systems SiO2-Na20 and SiO2-BaO; see Fig. 1. For both systems one notices a decrease of the mean molar volume of the melt when the silica mole fraction (X (SiOz)) changes from 1 to 0.8. Unfortunately, mean molar volume measurements are very rare in this X (SiO:) range. The measurements on the SiO2-BaO melt system are noisy, but it is the only system for which there are so many observations for X(SiO2) ~ 0.8, and they confirm the trend shown in the SiO2-Na20 system. At 1400°C, the mean molar volumes of SiOz-NazO solutions decrease from 27.3 cm 3 at X(SiO2) = 1 (Briickner, 1964), to about 27.1 cm 3 at X(SiOz) = 0.8 (Bockris et al., 1956). The partial molar volume of Na20 in Fe-free silicate melts is about 29 cm 3 at X(SiO2) = 0.8 and 1400°C (Bottinga et al., 1983; Lange and Carmichael, 1987). The SiO2-BaO system at 1700°C (Tomlinson et al., 1958) shows for the same X(SiOz) range a diminution from 27.3 to 26.6 cm3/mol, while the partial molar volume of BaO equals 27.6 cm 3 at 1700°C (Bottinga et al., 1983). The molar volume of pure liqui d silica at 1700°C is the same as at 1400°C (Briiekner, 1964). Figure 1 shows that observed mean molar volume data have a tendency of becoming smaller when a network-modifying oxide is dissolved in pure SiO2, in spite of the fact that the network-modifier oxide has a partial molar volume larger than the molar volume of SiO2. The simplest interpretation of these observations is that the partial molar volume of SiO2 diminishes when
o
•
NazO
28
o
E ~ 24
LizO
SiOz-NzO
I[ 20 20
"
25
30
~
~
35
~
.
~
40
45
50
Mol % MzO 28
E
i
.
.
.
.
i
.
.
.
.
r
.
,
BaO
u
26
E
24
~
22
0
E e-
g
20
"/8
30
SiOz'MO 1 700"C . . . .
MgO (b) 3t5 . . . .
t . . . . 40
Mol % MO
I . . . . 45
50
FIG. 2. Mean molar volumes of binary silicate melts as a function of the network modifier concentration. (a) SiO2-N20, N = K, Na, Li at P = 0.1 MPa, T = 1400°C. Intercepts of linear fits at mol% N20 = 0; K20:26.61 cm 3, Na20:26.84 cm 3, Li20:26.84 cm 3. Mean molar volume measurements from Bockris et al. (1956). (b) SiO2-MO, M = Ba, Sr, Ca, Mg at P = 0.1 MPa, T = 1700°C. Intercepts of linear fits at mol% M O = 0; BaO: 25.76 cm 3, SrO: 25.90 cm 3, CaO: 26.77 cm 3, MgO: 26.65 cm 3. Mean molar volume values from Tomlinson et al. (1958).
Pressure dependence of viscosity of silicate melts body interatomic potential, i.e., the potential energy function does not depend exclusively on distances between atoms but also on directions. The typical aspects of bonding in silicates are better reproduced with a three-body potential than with the more commonly used two-body potentials (Kubicki and Lasaga, 1988). The fictive temperatares of the Huang and Cormak (1990, 1991) simulated glass samples are not known. These samples were prepared at T = 6000 K and subsequentlycooled to 3000, 1500, and 293 K. To achieve structural relaxation, the samples stayed for 100,000 timesteps of 2 fs at each of these temperatures. Huang and Cormak ( 1990, 1991) noted that at all temperatures the Si-NBO bond lengths were shorter than the Si-BO ones. The conclusion that the partial molar volume of NBO is smaller than that of BO, is in harmony with X-ray work on crystalline Na2SiO3, ot-NazSizOs,/~-Na~Si2Os,and Li2Si205, which indicates a Si-BO bond length of about 0.167 nm and a Si-NBO bond length of 0.156 nm (McDonald and Cruickshank, 1967; Plant and Cruickshank, 1968; Plant, 1968; Liebau, 1961). Similar bond lengths have been determined for pyroxenes; for crystalline diopside Cameron et al. (1973) found Si-BO and Si-NBO bond lengths of 0.169 and 0.159 nm, respectively, which were nearly independent of temperature at 25 < T < 1000°C. 3. POLYMERIZATION AND VISCOSITY In the following paragraphs it will be shown that a minor extension of the Adam-Gibbs configurational entropy theory (Adam and Gibbs, 1965) suffices to explain at least qualitatively most of the available observations of the effect of pressure on viscosity. Lack of thermodynamic data for silicate melts makes a quantitative verifcation of the proposed model not yet possible. One deduces easily from the Adam and Gibbs (1965) theory that the viscosity of a liquid is given by In (rl) = Ae + Be/(TSco,f),
(1)
where r/ is the viscosity, Ae and Be are constants, T is the absolute temperature, and Sconfis the configurational entropy at temperature T. This equation has been discussed in the earth sciences literature many times since 1972 (Bottinga and Weill, 1972), and more recently in Richet (1984), Richet and Neuville (1992), and Bottinga et al. ( 1995 ). Scoffcan be evaluated calorimetrically, whereas A~ and Be are obtained by fitring Eqn. 1 to viscosity observations. Available evidence for liquid silicates indicates that the temperature dependence of Ae or Be are negligible in comparison with that of the denominator in Eqn. 1 (Richet, 1984). For reasons of simplicity, the discussion in this section will be limited to SiO2 and the binary alkali silicates. Using Eqn. l, one obtains (Richet, 1984; Gupta, 1987) (0 In (rl)/OP)r = [-Be/(TS2on~)]/(cOSco,f/cOP)r,
(2)
assuming that both A~ and Be do not depend on P. In our discussion we consider only the P-T region T < 2000 K and 0 < P -< 2 GPa; we assume that in this region Be is constant. Evidently, Sconfdepends on P, T, and on the extent of polymerization ((), hence dSco,f = ( cOSco,f/ O P )r.¢d P + ( oqSconf/oqT )p,~dT
+ (OS~o,f/O~)e,r[(C~/OP)rdP + (O~/OT)edT].
(3)
2727
polymerization is equal to the fraction of bridging oxygen atoms. Hence, changes in the extent of polymerization cause volume changes, therefore the pressure dependence of should be taken care of. Because of the large S i - O bond energy of about 450 kJ/mol (Balta and Balta, 1976), polymerization of binary silicate liquids may be considered to be temperature independent for T < 2000 K. The temperature independence of the heat capacity of liquid SiO2 (Richet and Bottinga, 1986) and of the partial molar heat capacity of SiO2 in binary silicate melts suggest that the distribution of SiO4 tetrahedra in these melts does not change radically in this temperature range. We suppose that the configurational entropy associated with polymerization is given by the entropy of mixing of NBOs and BOs. The entropy effect of mixing is additive, so we can write Sconf(P, T, () = Sc*onf(P, T) + S*n*f(((P, T ) ) ,
where Sc*onf(P,T) does not depend on ~; it is the part of the configurational entropy which is usually only considered, and one has
S * ~ ( ~ ( P , T ) ) = -RNo[~ In (()
+(1-~)In(1-~)],
(5)
R is the ideal gas constant and S*o~ the mean oxide molar configurational entropy of polymerization. No stands for the number of oxygen atoms per gram formula weight (gfw) of melt divided by the number of oxide moles per gfw of melt. Hence, for a SiO2 melt No equals 2, and for Na2Si2Os, No is equal to 5/3. Obviously, one can not apply Eqn. 5 to binary alkaline earth silicates because the two NBOs, due to the presence of one alkaline earth metal atom, are not distributed randomly. This effect causes the configurational entropies for liquid wollastonite, enstatite, and diopside (see Richet and Neuville, 1992) per oxide mole to be significantly less than the values listed in Table 1 for the system SiO2-Na20. Differentiating Eqn. 5, one obtains
(cOS*~/c~)e.r = -RNo In (~/(1 - ~)), where0<(<
(6)
1.
TABLE 1. Comparison between calculations of the configurational entropy (see text) in liquid XSiO2-(1 - X)Na20 at T = Tg. mol fract.
Tg
Calc. conf. ex%tro£~ (c) Diff.
X
(K)
eqn. (i) (a) ecru.(5) (b)
.80 .75 .70 .67 .65 .60 .55
745 734 730 730 707 694 686
9.0 8.3 8.9 7.9 10.1 8.9 8.9
7.9 8.7 9.2 9.3 9.4 9.2 8.8
% -12.2 + 4.8 + 3.4 +17.7 - 6.9 + 3.4 - i.i
(a) V i s c o s i t y m e a s u r e m e n t s by ~OC~ et al. (1994), BOCKRIS et al. (1956), N~X/VILLE (1993) a n d MEILING a n d UHLMANN (1977) w e r e u s e d a s
ir~ut data in eqn. (1).
Generally, the extent of polymerization is used only in a qualitative sense. For the simple silicate liquids considered in this paper, we define ~ = BO/(BO + NBO), i.e., the extent of
(4)
(b) q h e BO, ~ 0 abundances of ~ (1990) were u s e d in eqn. (5)
(c) Mean oxide molar e n ~ in Joules/K.
and COR~CK
values (see text)
2728
Y. Bottinga and P. Richet TAaLE 2. Observed changes of the viscosity of silicate melts due to increases of pressure, compiled by Scarfe et al. (1987). Ccm~ositicn
Ge~ Na20. Si02 Na20.2SIO2 Na20.3SIO2
cet~i2~ K21¢~Si5012 NaCaAISi207 NaAISi206 NaA/Si308
01. r'a~-,el. ALk. o1 .basalt O1. tholeiite
thol. An~eaite Obsid/an
~ (c)
1.0 0.33 0.60 0.71 0.33 0.67 0.71 1.0 1.0 0.71 0.79 0.81 0.84 0.93 0.97
Viscosity c h a n ~ ( d P a s)
6270; 1160 (a) 2; 15 151; 90 485; 170 3; ii 2890;1000 36; 26 64700;5090 113000;18400 i0; 6 i0; 8 35; 17 161; 80 1820; 810 -80%(b)
Pressure interval (GPa)
0 0 0 0 0 5 0 0 0 0 0 0 0 0 0
-
1 1 2 2 1.5 2 1.5 2.4 2 2 1.5 2 1.2 2 2
Temp. (°C)
1425 1300 1200 1175 1640 1300 1450 1350 1400 1450 1400 1400 1300 1350 1400
(a) ist v a l u e m e a s u r e d a t l o w pressure, 2 n d v a l u e at h i g h pressure. (b) the o b s e r v e d d e c r e a s e of the viscosity is g i v ~ in % of initial value at P = 0.i MPa; no other details available. (c) ~ = B O / T o t a l oxygen, s e e text; e s t i m a t e d from the stoichiQmetry.
Using a Maxwell relation gives
( OS*onflOe)r = - ( OVco.flOT)v,
(7)
where Vco.f = VL -- V~, V ~ is the configurational molar volume, and VL and V~ are the liquid and glass molar volumes, respectively. Hence,
( OSeonf/OP)T = - [ ( OVL/OT)p -- ( OVJOT)e] -
RNoln (~/(1 - 4))
(8)
and (0 In (rl)lOP)r = [Bd(TS~o., ) ] [ ( OVLIOT)v,~
- (OVJOT)v,¢ + RNo In (4/(1 - 4))(O~/OP)T],
(9)
where Sconeis the configurational entropy at T, P, and 4. At T = Ts and P = 0.1 MPa, Sco.f (Eqn. 4) is known, it is equal to the residual entropy at 0 K, or can be obtained from viscosity observations, see Richet (1984), and S*o.~ can be calculated with Eqn. 5. At T > Tg, the configurational entropy can be calculated by integrating
S~o,f( P, T, 4) = Sconf(P, rg, 4(P, rg)) + f (C~one(T)lT)dT
(10)
from Tg to T. In Eqn. 10 Ce~o.f = c ~ - cm, i.e., the difference between the heat capacities for the liquid and vitreous phase at constant pressure. At T >-- Tg, CeL has been measured for many liquid silicates or can be evaluated with simple models (Richet and NeuviUe, 1992; Richet and Bottinga, 1986), and cm is given by the Dulong and Petit limiting value, cm = 3 N . R (Haggerty et al., 1968; Richet, 1984), where N equals the number of gatoms per gfw. For the simple binary
silicate liquids discussed here, the extent of polymerization is independent of temperature, but the configurational entropy increases with temperature because ceco,a > 0 (Richet and Bottinga, 1984b). The measured coefficients of thermal expansion ( a ) of silicate liquids and their corresponding glasses are positive for all compositions showing a negative variation of the viscosity with pressure. Knoche et al. ( 1992, 1994) reported aL > aG > 0 for their observations on melts in the systems NaA1Si3Os-CaAl2Si2Os-CaMgSi206 and SiO2-Na20 outside a narrow temperature region associated with the glass transition. Hence, the difference between the two partial derivatives of the molar volumes of the liquid and the glass (Eqn. 9) is positive. The partial molar volume of bridging oxygen atoms is larger than that of nonbridging ones, see section II, consequently, (04/OP)r < 0 according to Le Ch~telier's principle. When 4 > 0.5, an increase in P causes a diminution of 4 and an augmentation of the configurational entropy (Eqn. 6), resuiting in a lowering of the viscosity (Eqn. 9). But if 4 < 0.5, a decrease of the extent of polymerization due to an increase in pressure gives a decrease of the configurational entropy and hence an increase of the viscosity, conform to the behavior of ordinary liquids. These conclusions are in qualitative agreement with the data compiled by Scarfe et al. (1987) on the pressure dependence of the viscosity of fourteen silicate melts with 4 larger as well as smaller than 0.5 (Table 2). Certainly, the dependence of the configurational entropy on the extent of polymerization in Eqn. 9 is not sufficiently general for treating all the multicomponent silicate liquids listed in the Scarfe et al. (1987) compilation. At high temperature and for 4 > 0.5 the anomalous pressure effect tends to disappear, while for ~ < 0.5 the normal viscosity increase with pressure
Pressure dependence of viscosity of silicate melts should become less important (Eqn. 9). The reason is that high T means large Scone,and large values of T and Scoa cause a diminution of the absolute value of ( 0 In (77)/0 P)r. Pressure or addition of network modifiers to the melt will decrease and cause eventually the melt viscosity to change to a normal P dependence. 4. CONFIGURATIONAL ENTROPY The configurational entropy is a measure of the number of configurational states to which a system has access, or in other words the number of configurations the system can adopt without affecting its average internal energy. In numerous papers we have already discussed this subject and how one can evaluate Sco,f at different temperatures (Richet, 1984; Richet and Bottinga, 1986; Richet and Neuville, 1992). When upon cooling a liquid passes the glass transition its configurational entropy is frozen in, i.e., this entropy does not change when the temperature is further decreased. Hence by measuring the residual entropy of an amorphous phase at T = 0 K one measures the configurational entropy at T = Tg. The residual entropy of silica glass at 0 K is 5.1 J/(mol. K) (Richer et al., 1982). The source of this entropy is not known. Bell and Dean (1968) have built a spokes and ball random network model (Mozzi and Warren, 1969); according to them, the number of ways one can add a single SiO4 tetrahedron at a typical site of the model surface is very small, because the restricted range of permissible S i - O - S i angles and because the infinite extent of the model. This point of view was shared by Thathachari and Tiller (1985), who found it very difficult to construct a space filling random network of SiO4 tetrahedra without NBOs for amorphous SiO2 because of the stringent limitations imposed by the minimum tolerated O - O distance in such a network. Therefore, we have simply assumed that the residual configurational entropy is due to the freezing in of dangling bonds at the glass transition, i.e., 1480 K (Richet et al., 1982). The configurational entropy of SiO2, due to the random distribution of NBO over the oxygen positions of the SiO4 tetrahedra can be calculated with Eqn. 5; with ( = 0.91 one obtains S~onf(Tg) = 5.0 J/(mol K). Bell and Dean (1968) calculated a value of 5.78 J/(mol. K) using their spoke and ball model for SiO2 and some rather obvious approximations. The presence of NBOs in amorphous SiO2 is controversial; Mikkelsen and Galeener (1980) argue that the 606 cm-I Raman line in the amorphous pure SiO2 spectrum is due to NBOs. They noticed that the 606 cm-1 peak intensity for different vitreous SiO2 samples is proportional to the reciprocal fictive temperatures of the samples. Solution in liquid silica of network modifier oxides, i.e., alkali metal oxides and alkali earth metal oxides, causes the formation of NBOs, and thus alleviates the steric difficulties mentioned in Thathachari and Tiller (1985). The observations by Farnan and Stebbins (1990) and Farnan et al. (1992) on vitreous K2Si409 and KMg0.sSi409 indicate that the distribution of NBOs and BOs is frozen in when the cooling silicate liquid is transformed into a glass. The distribution of these two types of oxygens is a source of residual entropy in silicate glasses. Equation 5 and the Huang and Cormack (1990) NBO and BO abundances were used to calculate the configurational entropy of binary alkali silicates. These NBO and BO abun-
2729
dances are compatible with those one can deduce from the Q species concentrations published by Mysen and Frantz (1994) inferred from Raman spectra of melts in the system SiO2Na20 at magmatic temperatures. The NMR observations by Brandriss and Stebbins (1988) for vitreous Na2Si205 are compatible with those by Mysen and Frantz (1994). Available evidence indicates that all oxygen atoms are bonded to one silicon atom at least in binary silicate melts (Huang and Cormack, 1990; Misawa et al., 1980). To a first approximation, MD simulations (Huang and Cormak, 1990, 1991 ) show that globally the NBO concentration is given by the stoichiometry. The configurational entropies for xSiO2 - ( 1 - x)Na20, with 0.81 > x > 0.54, calculated with Eqn. 5 are compared (Table 1 ) with those obtained by fitting Eqn. 1 to the observed viscosities of these melts as outlined in Richet (1984). The average of the absolute values (Table 1 ) of the differences between these two sets of entropy data is about 7%, which in the present context should be considered as an excellent agreement. Hence, the configurational entropies of these melts could be due to the distribution of the two structural types of oxygen atoms, i.e., NBO and BO. Recent measurements by Gaskell et al. (1991) and earlier observations reviewed in Bottinga et al. ( 1981 ), indicate that alkali and alkaline earth metal cations are not randomly distributed in simple binary silicate melts. This implies the clustering of NBOs (Greaves and Ngai, 1994), which has been observed by Farnan et al. (1992). These observations are in harmony with the results of Huang and Cormak ( 1991 ), who noticed also the clustering tendencies of alkali ions and NBOs. Hence, Eqn. 5 is at best only a good approximation. As far as we are aware, evidence in favor of ideal mixing in silicate liquids is indirect. Unfortunately, there are no simple ways to evaluate the entropy of mixing other than making use of the ideal solution model. However, application of this model gives often physically acceptable results, which may lack the desired accuracy but are qualitatively correct. While the N B O - B O distribution is frozen in when a silica or silicate liquid transforms into a glass the self diffusion of network modifier atoms is only slightly effected by the glass transition, see Dingwell (1990) and Dingwell and Webb (1990). The alkali and alkali earth ions are confined to reside in the neighborhood of NBOs (Farnan et al., 1992; Greaves and Ngai, 1994); apparently, they do not contribute to the generation of configurational states playing a role in the process of viscous deformation. This conclusion is not the same as, but similar to, Ryerson's (1985) statement that the entropic effect of the mixing of network-modifying cations does not influence the activity of SiO2, but that this latter quantity is mainly affected by the enthalpic effects associated with variations in bond energies due to the presence of the modifiers. The observed effect of cation mixing on the viscosity of silicate melts (Richet, 1984; Neuville and Richet, 1991 ) may be interpreted as being the result of the fact that these cations reflect the distribution of the NBOs. 5. EPILOGUE We have shown, using some reasonable assumptions, that the Kushiro discovery can be explained with the Adam and Gibbs (1965) theory, when the pressure sensitivity of the
2730
Y. Bottinga and P. Richet
polymerization shown by the data of Bockris et al. (1956) and Tomlinson et al. (1958) is taken into account and the appropriate configurational entropy change is introduced into Eqn. 5. Hence, at the time of the discovery of the anomalous P-viscosity effect, theory was sufficiently advanced to explain the observations by Kushiro (1976) and Kushiro et al. (1976). However, the association of BOs and NBOs with configurational entropy became possible only after the publication of the N M R observations by Stebbins and his colleagues (Stebbins et al., 1992) and the realization of the importance of residual entropy of silicate glasses (Richet, 1984). Acknowledgments--We would like to thank Mark Ghiorso for his scientific and professional editorial handling of our manuscript. The work published in this contribution benefited from financial support by the CNRS-DBT program. Editorial handling: M. S. Ghiorso
REFERENCES Adam G. and Gibbs J. H. (1965) On the temperature dependence of cooperative relaxation properties in glass-forming liquids. Z Chem. Phys. 43, 139-146. Angell C. A., Cheeseman P. A., and Tamaddon S. (1982) Pressure enhancement of ion mobilities in liquid silicates from computer simulation studies to 800 kilobars. Science 218, 885-887. Angell C. A., Cheeseman P. A., and Tamaddon S. (1983) Water-like transport property anomalies in liquid silicates investigated at high T and P by computer techniques. Ball MindraL 106, 87-98. Balta P. and Balta E. (1976) Introduction to the Physical Chemistry of the Vitreous State. Abacus Press. Bell R. J. and Dean P. (1968) The configurational entropy in the random network theory. Phys. Chem. Glasses 9, 125-127. Bockris J. O'M., Tomlinson J. W., and White J. L. (1956) The structure of liquid silicates: partial molar volumes and expansivities. Trans. Faraday Soc. 53, 299-310. Bottinga Y. and Weill D. F. (1972) The viscosity of magmatic silicate liquids: a model for calculation. Amer. Z Sci. 272, 438-475. Bottinga Y., Weill D. F., and Ricbet P. ( 1981 ) Thermodynamic modeling in silicate melts. In Thermodynamics of Minerals and Melts (ed. R. C. Newton et al.); Adv. Phys. Geochem. 1, pp. 207-245. Springer Verlag. Bottinga Y., Weill D. F., and Richet P. (1982) Density calculations for silicate liquids: I. Revised method for aluminosilicate compositions. Geochint Cosmochim. Acta 46, 909-920. Bottinga Y., Riehet P., and Weill D. F. (1983) Calculation of the density and thermal expansion coefficient of silicate liquids. Bull. MindraL 106, 129-138. Bottinga Y., Richet P., and Sipp A. (1995) Viscosity regimes of homogeneous silicate melts. Amer. Mineral 80, 305-318. Brandriss M. E. and Stebbins J. F. (1988) Effects of temperature on the structures of silicate liquids: 29SiNMR results. Geochim. Cosmochim. Acta $2, 2659-2670. Briickner R. (1964) Cliarakteristische physikalisehe Eigenschaften der oxydisehen Hauptglasbildner und ihre Beziehungen zur Strukmr der Gltisem H: Mechanische und optische Eigenschaften als Funktion der thermische Vorgeschichte. Glastechn. Ber. 37, 459475. De Jong B. H. W. S. and Brown G. E. (1980a) Polymerization of silicate and aluminate tetrabedra in glasses, melts, and aqueous solutions--I. Electronic structure of I-I6Si207, I-I6AISiO~-, and I-I6A120~-. Geochim. Cosmochim. Acta 44, 491-511. De Jong B. H. W, S. and Brown G. E. (1980b) Polymerization of silicate and aluminate tetralaedra in glasses, melts and aqueous solutions--H. The network modifying effects of Mg 2+, K + , Li + , H +, OH-, F - , H20, CO2, and H30 + on silicate polymers. Geochim. Cosmochim. Acta 44~ 1627-1642.
Dingwell D. B. (1990) Effects of structural relaxation on cationic tracer diffusion in silicate melts. Chem. GeoL 82, 209-216. Dingwell D. B. and Webb S. L. (1990) Relaxation in silicate melts. Eur. J. Mineral 2, 427-449. Faman I. and Stebbins J. F. (1990) A high temperature 29Si NMR investigation of solid and molten silicates. J. Amer. Chem. Soc. 112, 32-39. Farnan I. et al. (1992) Quantification of the disorder in networkmodified silicate glasses. Nature 358, 31-35. Frantz J. D. and Mysen B. O. (1995) Raman spectra of BaO-SiO2, SrO-SiO2, and CaO-SiO2 melts to 1600"(2. Chem. Geol. (in press) Gaskell P. H., Eckersley M. C., Barnes A. C., and Chieux P. ( 1991 ) Medium-range order in the cation distribution of a calcium silicate glass. Nature 350, 675-677. Greaves G. N. and Ngal K. L. (1994) Ionic transport properties in oxide glasses derived from atomic structure. J. Non-Cryst. Solids 172-174, 1378-1388. Gupta P. K. (1987) Negative pressure dependence of the viscosity. J. Amer. Ceram. Soc. 70, C152-C153. Gupta P. K. (1989) Models of the glass transition. Rev. Solid State Sci. 3, 221-257. World Scientific Publishing Co. Haggerty J. S., Cooper A. R., and Heasley J. H. (1968) Heat capacity of three inorganic glasses and supercooled liquids. Phys. Chem. Glasses 5, 130-136. Harrison W. A. (1980) Electronic Structure and the Properties of Solids. W. H. Freeman and Company. Huang C. and Cormack A. N. (1990) The structure of sodium silicate glass. J. Chem. Phys. 93, 8180-8186. Huang C. and Cormack A. N. (1991) Structural differences and phase separation in alkali silicate glasses. J. Chem. Phys. 95, 3634-3642. Knoche R., Dingwell D. B., and Webb S. L. (1992) Non-linear temperature dependence of liquid volumes in the system albite-anorthite-diopside. Contrib. Mineral. Petrol. 111, 61-73. Knoche R., Dingwell D. B., Seifert S. A., and Webb S. L. (1994) Non-linear properties of supercooled liquids in the system Na20SiO2. Chem. Geol. 116, 1-16. Kubicki J. D. and Lasaga A. C. (1988) Molecular dynamics simulations of SiO2 melt and glass: Ionic and covalent models. Amer. Mineral 73, 941-955. Kushiro I. (1976) Changes in viscosity and structure of melt of NaA1Si206 composition at high pressures. J. Geophys. Res. 81, 6347-6350. Kushiro I., Yoder H. S., and Mysen B. O. (1976) Viscosities of basalt and andesite melts at high pressures. J. Geophys. Res. 81, 63516356. Lange R. A. and Carmichael I. S. E. (1987) Densities of K20-Na20CaO-MgO-FeO-Fe203-A1203-TiO2-SiO2 liquids: new measurements and derived partial molar properties. Geochim. Cosmochim. Acta 51, 2931-2946. Lieban F. (1961) Untersuchungen an Schichtsilikaten des Formeltyps Am(Si2Os)n. I. Die Kristallstruktur der Zimmertemperaturform des Li2Si2Os. Acta Cryst. 14, 389-395. McDonald W. S. and Cruickshank D. W. J. (1967) A reinvestigation of the structure of sodium metasilicate, Na2SiO3. Acta Cryst. 22, 37 -43. Meiling G. S. and Uhlmann D. R. (1977) Crystallisation and melting kinetics of sodium disilicate. Phys. Chem. Glasses 8, 62-68. Mikkelsen J. C. and Galeener F. L. (1980) Thermal equilibration of Raman active defects in vitreous silica. J. Non-Cryst. Solids 37, 71-84. Misawa M., Price D. L., and Suzuki K. (1980) The short range structure of alkali disilicate glasses by pulsed neutron total scattering. J. Non-Cryst. Solids 37, 85-97. Mozzi R. L. and Warren B. E. (1969) The structure of vitreous silica. J. Appl. Cryst. 2, 164-172. Mysen B. O. (1988) Structure and Properties of Silicate Melts. Elsevier. Mysen B. O. and Frantz J. D. (1994) Silicate melts at magmatic temperatures: in-situ structure determination to 1651"(2 and effect of temperature and bulk composition on the mixing behavior of structural units. Contrib. Mineral. Petrol. 117, 1-14.
Pressure dependence of viscosity of silicate melts Neuville D. R. (1992) Etude des Proprittts Thermodynamiques et Rhtologiques des Silicates Fondus. Thtse de Doctorat, Univ. Paris 7. Neuville D. R. and Richet P. ( 1991 ) Viscosity and mixing in molten (Ca, Mg) pyroxenes and garnets. Geochim. Cosmochim. Acta 55, 1011-1019. Plant A. K. (1968) A reconsideration of the crystal structure of/3Na2SizOs. Acta. Cryst. B24, 1077-1083. Plant A. K. and Cruickshank D. W. J. (1968) The crystal structure of a-Na2Si2Os. Acta Cryst. B24, 13-19. Richet P. (1984) Viscosity and configurational entropy of silicate melts. Geochim. Cosmochim. Acta 48, 471-483. Richet P. (1990) GeO2 vs. SiO2: Glass transistions and thermodynamic properties of polymorphs. Phys. Chem. Minerals 17, 7 9 88. Richet P. and Bottinga Y. (1984a) Glass transition and thermodynamic properties of amorphous SiO~, NaA1Si,O2,+2 and KAISi3Os. Geochim. Cosmochim. Acta 48, 453-470. Richet P. and Bottinga Y. (1984b) Heat capacity of aluminum-free silicates. Geochim. Cosmochim. Acta 49, 471-486. Richet P. and Bottinga Y. (1984c) Anorthite, andesine, wollastonite, diopside, cordierite and pyrope: thermodynamics of melting, glass transitions, and properties of the amorphous phases. Earth Planet. Sci. Lett. 67, 415-432. Richet P. and Bottinga Y. (1986) Thermochemical properties of silicate glasses and liquids: a review. Rev. Geophys. 24, 1-26. Richet P. and Neuville D. R. (1992) Thermodynamics of silicate melts: configurational properties. In Thermodynamic Data Systematics and Estimation (ed. S. Saxena), pp. 132 - 160. Springer-Verlag. Richet P., Bottinga Y., Denielou L., Petitet J. P., and Tequi C. (1982) Thermodynamic properties of quartz, cristobalite and amorphous SiO2: drop calorimetry measurements between 1000 and 1800 K and a review from 0 to 2000 K. Geochim. Cosmochim. Acta 46, 2639-2658.
2731
Ryerson F. J. (1985) Oxide solution mechanisms in silicate melts: Systematic variations in the activity coefficient of SiO2. Geochim. Cosmochim. Acta 49, 637-649. Scarfe C. M., Mysen B. O., and Virgo D. (1987) Pressure dependence of the viscosity of silicate melts. In Magmatic Processes: Physicochemical Principles (ed. B. O. Mysen); Spec. Publ. No. 1, pp. 59-67. Geochem. Soc. Scherer G. (1990) Theories of relaxation. J. Non-Cryst. Solids 123, 75-89. Sharma S. K., Virgo D., and Kushiro I. (1979) Relationship between density, viscosity and structure of GeO2 melts at low and high pressure. J. Non-Cryst. Solids 33, 235-248. Stebbins J. F., Farnan I., and Xue X. (1992) The structure and dynamics of silicate liquids: A view from NMR spectroscopy. Chem. Geol. 96, 371-386. Thathachari T. T. and Tiller W. A. (1985) Importance of oxygenoxygen interactions in silica structures. J. Appl. Phys. 57, 18051811. Tomlinson J. W., Heines M. S. R., and Bockris J. O'M. (1958) The structure of liquid silicates, pt. 2. Trans. Faraday Soc. 54, 18221833. Urbain G., Bottinga Y., and Richet P. (1982) Viscosity of liquid silica, silicates and alumino-silicates. Geochim. Cosmochim. Acta 46, 1061-1072. Vessal B., Leslie M., and Catlow C, R. A. (1989) Molecular dynamics simulation of silica glass. Molec. Simul. 3, 123-136. Waft H. S. (1975) Pressure-induced coordination changes in magmatic liquids. Geophys. Res. Lett. 2, 193-196. Woodcock L. V., Angell C. A., and Cheeseman P. (1976) Molecular dynamics of the vitreous state: Simple ionic systems and silica. J. Chem. Phys. 65, 1565-1577. Xue X. and Stebbins J. F. (1993) 23Na NMR chemical shifts and local Na coordination environments in silicate crystals, melts and glasses. Phys. Chem. Minerals 20, 297-307.