Residual and configurational entropy: Quantitative checks through applications of Adam–Gibbs theory to the viscosity of silicate melts

Journal of Non-Crystalline Solids 355 (2009) 628–635

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Residual and configurational entropy: Quantitative checks through applications of Adam–Gibbs theory to the viscosity of silicate melts Pascal Richet Physique des Minéraux et des Magmas, Institut de Physique du Globe, 4, Place Jussieu, 75252 Paris cedex 05, France

a r t i c l e

i n f o

Article history: Available online 9 April 2009 PACS: 61.43 Fs 65.60.+a 66.20.Cy Keywords: Mixed-alkali effect Alkali silicates Aluminosilicates Calorimetry Thermodynamics Structural relaxation Viscosity

a b s t r a c t In this paper the traditional view that glasses possess residual entropy, which can be determined by calorimetric means, is quantitatively supported by applications of Adam and Gibbs configurational entropy theory to the temperature, composition and pressure dependences of the viscosity of silicate melts. This theory is also in harmony with the mechanisms of viscous flow, as understood from NMR experiments, according to which viscosity is controlled by the rate of bond rearrangements between network-forming cations and oxygens. As a matter of fact, Adam and Gibbs basic expression relating structural relaxation times to the reciprocal of the product of temperature and configurational entropy can be derived from a phenomenological analysis of the temperature dependence of the activation energy for viscous flow. Adam–Gibbs theory thus works well for silicate melts because network-modifying cations also play a role in bond rearrangements such that, as a bulk property, configurational entropy is actually relevant to structural relaxation and flow. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Rekindling an old debate, Gupta and Mauro [1,2] have recently denied the existence of frozen in configurational entropy in glasses and disordered solids on the basis of the non-equilibrium nature of these phases [e.g. [1,2]]. As discussed by Goldstein [3], however, the traditional view that glass does possess residual entropy at 0 K is supported by classical thermodynamics reasoning along with a number of basic thermochemical results. Without repeating Goldstein’s arguments, the purpose of this paper is to show how properties of silicate melts can contribute to the current debate. Phenomenologically, a variety of calorimetric data and applications of Adam–Gibbs theory [4] to the viscosity of these liquids will be shown to be inconsistent with Gupta and Mauro’s thesis whereas they fit instead within the traditional picture. In particular, this paper will show that (i) calorimetrically residual entropies of silicate glasses lend themselves to structural inferences that are confirmed by structural studies; and (ii) that these entropies are quantitatively consistent with values derived from completely independent experimental data, namely, viscosity measurements made either on relaxed melts or on glasses.

E-mail address: [email protected] 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.01.027

Interest in molten silicates stems from the fact that they are the most important glass-forming liquids from either a natural or an industrial standpoint. In both contexts a difficult challenge is to arrive at a quantitative understanding of viscosity, the property that controls heat and mass transfer, as a function of temperature and composition. It is in this respect that the configurational entropy theory proposed by Adam and Gibbs [4] on the basis of a statistical mechanical model of polymers has been most successful since no other theory has proven to enable so many different features to be accounted for quantitatively [5,6]. The fundamental tenet of Adam–Gibbs theory is that relaxation in viscous liquids becomes increasingly sluggish below the glass transition range as a result of a concomitant dearth of atomic configurations. Specifically, structural relaxation is described in terms of cooperative rearrangements of the melt in mutually independent regions whose minimum size (z) decreases with increasing temperature. The probability w of these rearrangements at temperature T was calculated by Adam and Gibbs to be

wðTÞ ¼ A expðz Dl=kTÞ;

ð1Þ

where A is a pre-exponential factor, k Boltzmann constant and Dl the Gibbs free-energy barriers hindering the cooperative rearrangements referred to an appropriately chosen unit of matter. Associating z with a critical value of the configurational entropy (Sconf)

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of the liquid, Adam and Gibbs finally derived an equation of the form

wðTÞ ¼ A expðBe =TSconf Þ;

ð2Þ

where Be represents the molar Gibbs free-energy barriers opposing structural rearrangements. Because structural relaxation times (s) are inversely proportional to the average probability of these rearrangements, one obtains

s ¼ A0 expðBe =TSconf Þ;

ð3Þ 0

where the pre-exponential factor A , representing relaxation times at infinite temperature, is of the order of 1013 s, the period of atomic vibrations which would remain the only relevant characteristic times in the high temperature limit. The versatility of Adam–Gibbs theory results in part from the fact that its key parameter, configurational entropy, is not a fitting parameter, but is amenable to calorimetric measurement and simple modeling. However, practical applications of Adam–Gibbs theory have long been hampered by the difficulties affecting these experimental determinations. This is why we first describe in this paper how such calorimetric determinations can be performed and the kind of checkable structural inferences that can be drawn from these results. We then present examples of these applications to the temperature, composition and pressure dependences of the viscosity of molten silicates. We finally make use of the known microscopic mechanisms of viscous flow to rederive Adam–Gibbs basic relation in a purely phenomenological manner. In this way one can summarize the reasons why this theory works so well for liquid silicates in contrast to the more ambiguous results obtained for other systems, and why the conventional concept of configurational entropy of glasses at 0 K remains of basic importance.

melting of the crystal (DHf) at the melting point; (iii) heat capacity measurements on the supercooled liquid between Tf and the glass transition temperature Tg; (iv) heat capacity measurements on the glass from Tg to 0 K. Assuming that the crystal is perfectly ordered at 0 K, such that its entropy can be taken as 0 at 0 K, the residual entropy of the glass at 0 K is:

Sg ð0Þ ¼

Z

Tf

C pc =T dT þ DHf =T f þ

0

Z

Tg

C pl =TdT þ Tf

Z

0

C pg =T dT;

ð4Þ

Tg

where the subscripts c, l and g designate the crystal, liquid and glass phases, respectively. The residual entropy is purely configurational in origin. Since the assumption that the melt structure becomes frozen in at the glass transition is fully justified by available structural data [7], this entropy is usually considered constant between 0 K and Tg. Residual entropy then represents the configurational entropy of the melt at Tg

Sconf ðT g Þ ¼ Sg ð0Þ:

ð5Þ

If the configurational heat capacity of the melt, C conf , is known, one P calculates the configurational entropy at any temperature above Tg with

Sconf ðTÞ ¼ Sconf ðT g Þ þ

Z

T Tg

C conf =T dT: P

ð6Þ

For molten silicates, C conf is given to a very good approximation by P the difference between the heat capacities of the melt at the temperature considered and that of the glass at the glass transition [8]. Of particular interest in this respect is the fact that the glass transition of silicates takes place when the heat capacity of the glass becomes close to the Dulong-and-Petit limit of 3 R/g atom K [9,10]. Hence, the configurational Cp can be written as

2. Adam–Gibbs theory

C conf ¼ C pl  C pg ðT g Þ ¼ C pl  3 nR; P

2.1. Calorimetric configurational entropy

where R is the gas constant and n the number of atoms in the used gram formula weight. Considering the limited magnitude of anharmonic effects in such strongly bonded materials, this fact leaves litin the liquid phase [8]. tle room for further variations of C conf P The data plotted in Fig. 1 illustrate the fact that the actual configurational entropy of amorphous CaMgSi2O6 differs by about 50% from the entropy difference between the liquid (or glass) and crystal phases. Similar discrepancies are found for the 10 silicate compositions for which calorimetric data are available. As emphasized for a number of organic substances [11], they originate in large vibrational contributions to the entropy differences between the amorphous and crystal phases. Hence these entropy differences represent very poor approximations for Sconf which should not be used. In passing, we note that the procedure summarized with Eq. (4) might seem inconsistent because calculation of an entropy change requires the existence of a reversible pathway between the initial and final thermodynamic states considered. As far as heat transfer is concerned, such pathways exist within both the glass and supercooled liquid domains, but not across the glass transition which is an intrinsically irreversible phenomenon. Although heat capacities measured on heating and cooling across the glass transition differ (e.g. [12]), the entropy created as a result of this irreversibility appears small enough compared to residual and configurational entropies that it can safely be neglected in calculations of thermodynamic functions [3,13,14]. Besides, the extensive calorimetric measurements now available for silicate glasses rule out any significant configurational contributions to the relative entropy of silicate glasses below the glass transition range [15].

Configurational entropy of glasses and liquids can be determined by calorimetric means when there exists a congruently melting compound of the same composition. As illustrated in the calorimetric cycle depicted in Fig. 1 for the CaMgSi2O6 composition, the residual entropy of the glass at 0 K is first determined from: (i) heat capacity (Cp) measurements for the crystal from 0 K to the congruent melting temperature Tf; (ii) the enthalpy of

700

CaMgSi 2O6

600

Liquid

Tg

500

²S

f

Crystal 400 300

Glass

200

TK

100

S l,g - Sc Sconf

0 0

500

1000

1500

2000

T (K) Fig. 1. Calorimetric determination of the residual entropy of CaMgSi2O6 glass and of the configurational entropy of the liquid. For comparison, the entropy difference between either the liquid or the glass and the crystal phases is shown as a dotted curve. The arrows indicate the calorimetric glass transition (Tg) and the temperature of the so-called Kauzmann paradox (TK) at which configurational entropy would vanish. Data sources: crystal [16–17], glass and liquid [6,18].

ð7Þ

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g ¼ G1 s;

2.2. Residual entropy and glass structure The extreme case of NaAlSiO4 glass gives a first completely independent check of the validity of residual entropy data. From the calorimetric methods summarized in Fig. 1, a residual entropy of 9.7 ± 2.0 J/mol K, or 1.4 J/g atom K, has been measured [19]. Now, the interesting feature is that the latter figure is even lower than that determined for pure SiO2, namely 1.4 J/g atom K [20]. It was thus concluded that Si, Al mixing cannot not contribute much to the residual entropy of NaAlSiO4 glass and that Si and Al have an essentially ordered distribution in this material [19]. Interestingly, this conclusion drawn from calorimetric data only has been fully confirmed by 17O Nuclear Magnetic Resonance (NMR) experiments [21,22]. On the other hand, the contribution of cation mixing on configurational and residual entropies is clearly shown by the (Ca, Mg)SiO3 metasilicate glass series. Calorimetric measurements indicate that the residual entropies at 0 K of CaSiO3, (Ca, Mg)0.5 SiO3 and MgSiO3 glasses are 8.5 ± 3, 12.2 ± 1.5, and 8.7 ± 5 J/mol K, respectively [23,8,15]. Now, all three glasses have similar anionic frameworks [7] and differ mainly by the existence of alkaline earth mixing in the intermediate composition. If this mixing is assumed to be random, then its contribution to configurational entropy is 5.8 J/mol K as given by

Sconf mix ¼ R

X

xi ln xi ;

ð8Þ

ð9Þ

where G1 is the shear modulus at infinite frequency. By combining Eqs. (3) and (9), one finally writes [5]

log g ¼ Ae þ Be =TSconf ;

ð10Þ

conf

where S is given by Eqs. (6) and (7) and Ae and Be are now considered as fit parameters. The validity of Eq. (10) is readily checked through plots of log g against 1/TSconf. The data shown in Fig. 2 for CaAl2Si2O8 melt illustrate the quality of the linear fits made in this way, even over 15 orders of magnitude for a liquid whose viscosity is strongly non-Arrhenian. With two instead of three fit parameters, agreement with experimental data is at least as good as with empirical TVF equations, viz

log g ¼ A þ B=ðT  T 1 Þ:

ð11Þ

Actually, the deviations of fits to experimental viscosities made with Eq. (10) for compositionally simple and complex melts are within the accuracy of the available measurements. The agreement is in fact so good that, alternatively, one can use Eq. (10) as a threeparameter equation and determine not only Ae and Be, but also Sconf at a given temperature (Tg, for instance) as a third fit parameter. A plot of the values derived in this markedly different way against calorimetric entropies does show a 1:1 correlation (Fig. 3). A note-

where xi is the Ca/(Ca + Mg) mol fraction of the material. This result is not only consistent with the aforementioned residual entropies, but also with the viscosity variations described below for the same system. Unless extremely large CP anomalies are assumed to exist in the vicinity of 0 K, these features provides additional support for the operational determination of residual entropy of glasses. As a matter of fact, there is not any experimental evidence for such anomalies. On the contrary, the large Cp differences between glasses with different thermal histories pictured by Gupta and Mauro’s Fig. 1 [2] are not borne out by calorimetric data for silicates [8] as found previously for selenium and organic substances [11]. And, anyway, it would remain to show how such anomalies could account for the various aspects of the composition dependence of configurational entropy that are described in the rest of this paper.

60

Mg3 Al2Si 3O12

Scal (J/mol K)

50

CaAl2Si 2O8

40

NaAlSi3O8

30

CaMgSi2O6

CaSiO3

MgSiO3

10

0

0

10

20

30

40

50

60

Svis (J/mol.K)

2.3. Configurational entropy and viscosity Application of Adam–Gibbs theory to viscosity (g) is straightforward if configurational entropy is known. With the Maxwell relationship, one first relates viscosity to relaxation time

Fig. 3. Comparison between configurational entropies at the glass transition determined from calorimetry and viscosity measurements. Data sources: CaSiO3 [23]; MgSiO3 [15]; CaMgSi2O6 [8]; KAlSi3O8, NaAlSi3O8 and CaAl2Si2O8 [5], Mg3Al2Si3O12 [28].

16

16

b

a 12

log η (Pa.s)

12

log η (Pa.s)

KAlSi 3O8

20

8

4

CaAl 2Si 2O 8

0 4

5

6

7

104/T (K-1)

8

9

8

4

0 10

0

5

10

15

20

25

30

1/TSconf (J/mol)

Fig. 2. Viscosity of liquid CaAl2Si2O8 plotted either as a function of reciprocal temperature (a) and of 1/TSconf (b), where Sconf is the calorimetrically determined configurational entropy. Data sources: solid squares [24]; open circles [25]; open squares [26]; solid circles [27].

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P. Richet / Journal of Non-Crystalline Solids 355 (2009) 628–635

small difference between two large numbers. Hence, a unique usefulness of Adam–Gibbs theory as applied to the viscosity of silicate melts is the possibility of determining accurately a property otherwise as elusive as configurational entropy.

20

Window glass

log η (Pa.s)

16

12

2.4. Temperature and composition effects Viscosity is generally measured with experimental timescales of the order of 102 s. Because G1 is about 1010 Pa and varies by less than a factor of ten with either temperature or composition [29], the Maxwell relationship implies that relaxation times become longer than experimental timescales when the viscosity becomes higher than 1012 Pa s. Time-dependent viscosities are then observed until the melt structure has adjusted to the new temperature conditions and the constant viscosity of a melt in internal equilibrium is measured. Not only such measurements define a single trend with the data at temperatures above the glass transition range (Fig. 4), provided they actually refer to internal thermodynamic equilibrium, but these values are in excellent agreement with extrapolations up to 1014 Pa s of Adam–Gibbs equations obtained from measurements made above the glass transition range [30]. The picture is different if measurements are performed rapidly at temperatures low enough that structural relaxation does not take place [31]. Arrhenian variations are observed in this manner

8

4

0 4

6

8

10

12

14

16

-1

4

10 /T (K ) Fig. 4. Transition from equilibrium to isostructural viscosity when relaxation times become longer than the experimental timescale [26]. Data for window glass. Open squares: equilibrium viscosities [30,31,33], even at the lowest temperatures; solid squares [31]: viscosity of samples whose configuration has been frozen in at Tg = 777 K. Solid line: values given by Eq. (10), with the constant Sconf(Tg) value derived from fits made to the equilibrium viscosities.

worthy advantage of the former entropies is that their error bars are smaller than those of the latter which suffer from representing a

16

14

a

NTS2

b Viscosity (log Pa.s)

(J/g atom K) conf

Cp

KTS2 12

10 N3TS 4 NTS4 KTS4

6

4

2 600

NTS2 N 3TS4

12

14

8

KTS4

10 8 KTS2

6

NTS4

4 2

NS2 KS2 800

NS2

0

1000

1200

1400

1600

6

1800

8

10

12

14

104/T (K-1)

T (K) 30

c NTS

N TS

2

3

4

25 KTS

KTS

4

20 NTS

4

15

NS

2

S

conf

(J/mol K)

2

KS

2

10

5 800

1000

1200

1400

1600

1800

T (K) Fig. 5. Viscosity and configurational entropy of alkali titanosilicate melts [35]. Data for sodium and potassium disilicates are included for comparison. Abbreviations: N (Na2O), K (K2O), T (TiO2), S (SiO2). (a) Configurational heat capacities as derived with Eq. (7) from Cp determinations. (b) Viscosity as measured at high temperatures and near the glass transition. (c) Configurational entropies derived from fits made to the measured viscosities with Eq. (10).

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for a material whose configurational entropy has become constant data and equilibrium viscosities, (Fig. 4). From experimental C conf P Sconf(Tg) and the parameters Ae and Be of Eq. (10) can be determined [30]. Using these parameters, along with a constant Sconf value, one predicts with Eq. (10) the viscosity of a sample with a constant fictive temperature of 777 K (solid line in Fig. 4). Excellent agreement with the experimental isostructural data [31] is thus achieved without any adjustable parameters [32]. This is another stringent check of the fact that the configurational entropy of the melt is actually frozen in at the glass transition and remains constant below Tg. Referring to previous papers for more extensive accounts [4,5], we will illustrate by two more striking examples the close connection between configurational entropy and the temperature and composition dependences of viscosity. Alkali titanosilicate melts will be dealt with first. They exhibit anomalously high heat capacity changes at the glass transition and then, in the liquid range, marked Cp decreases with increasing temperature such that their heat capacities and configurational heat capacities become similar near 1800 K to those of their Ti-free counterparts (Fig. 5(a)). It follows that, on cooling from high temperatures, molten alkali titanosilicates lose entropy at a unusually high rate compared to Ti-free melts [34,35]. According to Eq. (10), the deviations of viscosity from Arrhenius laws should be stronger than for Ti-free melts. As shown by Fig. 5(b), these features are quantitatively borne out by experimental viscosity data. The question is then to know whether alkali titanosilicates have anomalously high configurational entropies at high temperatures, or anomalously low entropies near the glass transition. The entropies derived from fits of Eq. (10) to the data indicate that the former assumption holds true (Fig. 5(c)). In view of the large energy involved in the Cp effects, the alkali titanosilicate anomaly necessarily originates in short-order interactions such as temperature-dependent mixing of silicate and titanate entities [7,35]. Other effects of cation mixing on viscosity are well documented for simple systems for which calculation of the resulting excess entropy is straightforward [5]. Consider for example mixing in

14

12 1025 K 10

1050 K

Sconf ðTÞ ¼

X

xi Sconf ðTÞ þ Sconf i mix ;

ð12Þ Sconf i

where xi is the Ca/(Ca + Mg) mol fraction, the configurational entropy of endmember i, whose temperature dependence is given by Eq. (6), and Sconf mix is the configurational entropy of mixing as given (T) terms of the endby Eq. (8). The salient feature is that the Sconf i members increase considerably with temperature whereas Sconf mix remains constant. The relative contribution of this excess term to the total configurational entropy of the intermediate compositions thus is much higher at low than at high temperature. As a result, the depressing effect on viscosity caused by this term is most effective at lower temperatures, i.e., at high viscosities, and becomes vanishingly small at high temperatures [5,36]. 2.5. Pressure effects Although it does not shed direct light on the entropy debate, we finally turn to the pressure dependence of viscosity which depends sensitively on composition, being either positive or negative [37]. As a first step, one can differentiate Eq. (9) with the assumption that both Ae and Be do not depend on pressure [5,38] to obtain

ð@ ln g=@PÞT ¼ ðBe =TSconf 2 Þð@Sconf =@PÞT ¼ ðBe =TSconf 2 Þð@V conf =@TÞP :

ð13Þ

conf

With the assumption that V , the configurational volume, is the difference between the liquid and glass phases, Eq. (13) yields only viscosity increases with pressure because thermal expansion coefficients are greater for liquids than for glasses. Hence, the observed viscosity decreases are not accounted for by Eq. (13) unless (oBe/oP)T is considered as a fit parameter. Alternatively, however, a more fundamental approach can be followed once recognition is made that the positive and negative pressure dependences of viscosity correlate with the degree of polymerization of the melt as defined by the ratio x = BO/(BO + NBO) of bridging oxygens (BO), which connect two SiO4 tetrahedra, and non-bridging oxygens (NBO), see Fig. 7. To consider the relationship between this ratio and configurational entropy at high pressure, the starting point is the fact that the molar volume of bridging oxygens is about 0.7 cm3/mol higher than that of nonbridging oxygens [39]. Le Châtelier’s principle thus implies that x

1075 K

7 6

6

1/η (dη/dP) (GPa-1)

log η (Pa.s)

8

(Ca, Mg)SiO3 melts where a linear variation of viscosity at high temperatures as a function of the Ca/(Ca + Mg) ratio contrasts with the deep minimum of about two orders of magnitude observed near the glass transition (Fig. 6). Now, the configurational entropy of an intermediate composition is expressed as

1225 K

4

T

4

5

2

1500 K

0

3 2 1 0

1750 K -1 0.3

-2 0

20

40

60

80

100

mol % CaSiO3 Fig. 6. Mixed-alkaline earth effect on viscosity for (Ca, Mg)SiO3 melts near the glass transition range and at superliquidus temperatures [36].

0.4

0.5

0.6

0.7

0.8

0.9

1

ξ Fig. 7. Pressure dependence of viscosity against degree of polymerization as deduced from the compilation of Scarfe et al. [37]. Data for n = 1 refer to GeO2 and NaAlSi3O8.

P. Richet / Journal of Non-Crystalline Solids 355 (2009) 628–635

decreases with increasing pressure. To account for this decrease, the configurational entropy can be split into two parts:

Sconf ðP; T; xÞ ¼ Sconf ðP; TÞ þ Sconf ðnÞ;

ð14Þ

conf

where S (P, T) is that part that does not depend on x. As a first approximation, Sconf(x) may be assumed to represent the entropy of mixing of bridging and non-bridging oxygens:

Sconf ðnÞ ¼ RNO ½n ln n þ ð1  nÞ lnð1  nÞ;

dSconf ¼ ð@Sconf =@PÞT;n dP þ ð@Sconf =@TÞP;n dT ð16Þ

3. Adam–Gibbs theory and the microscopic mechanisms of viscous flow For silicate melts attention has long focused on superliquidus viscosities, which generally conform to Arrhenius equations

g ¼ Aa expðEa =RTÞ:

Because the degree of polymerization varies little with temperature, in view of the high strength of Si–O bonds, one obtains eventually by neglecting the (on/oT)P term:

ð@ ln g=@PÞT ¼ ðBe =TSconf 2 Þfð@V conf =@TÞP;n þ RNO ln½n=ð1  nÞ  ð@n=@PÞT g;

ready noted, (on/oP)T < 0. From Eq. (17) one concludes that Sconf increases or decreases with pressure depending on whether n is higher or lower than 0.5, whereas viscosity shows the opposite variation [39]. Data are lacking to make quantitative calculations of these effects. Qualitatively, however, the predicted trends account for the positive or negative pressure dependences of the measured viscosities (Fig. 7).

ð15Þ

where NO is the number of oxygen atoms per mole of oxide components. By differentiating Eq. (14), one writes:

þ ð@Sconf =@nÞP;T ½ð@n=@PÞT dP þ ð@n=@TÞP dT:

633

ð17Þ

where Sconf is the configurational entropy at given T, P and n, which is derived at room pressure from viscosity measurements. As al-

ð18Þ

It was often assumed that variations in activation energies Ea reflect temperature-dependent changes in the energy barriers involved in viscous flow [e.g. [25]]. But the decreases of Ea by a factor of 5 or more that can be observed when viscosity is considered from the glass transition to superliquidus temperatures are quite inconsistent with such interpretations [8]. In contrast, one deduces from Eq. (10) that the Be terms representing the Gibbs free-energy opposing structural rearrangements are constant and that the apparent

Fig. 8. Bond exchange and viscous flow of a sodium silicate melt subjected to a shear stress as pictured from NMR experiments [40]. In (a) an entity with a non-bridging oxygen (dark) and its charge-compensating sodium gets close to the bridging oxygen (clear) of another entity. After the sodium has moved away (b), the two entities form an activated complex in which a silicon atom is five-fold coordinated by oxygens. In (c) the activated complex has decomposed into two new entities in which the two oxygens considered have exchanged their non-bridging and bridging characters, the sodium atom being bonded to the newly non-bridging oxygen. Because all atoms participate in the rearrangement, the energetics of all bonds, and not only of Si–O bonds, play a role in the process, which is similar when other network-forming cations such as Al are involved.

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P. Richet / Journal of Non-Crystalline Solids 355 (2009) 628–635

changes in activation energies result instead from increases of configurational entropy. Likewise, it has been believed that the strong lowering of the viscosity of silicate melts associated with decreasing SiO2 contents results from depolymerization of the anionic framework and, thus, from a decrease of the average size of the flowing units. This picture is also incorrect because it is static, not dynamic, and implicitly puts the emphasis on the existence of long-lived microscopic entities while neglecting the very high rate of bond rearrangements typical of melts. In this regard, a major advance has been the demonstration made by NMR experiments that viscosity scales as the rate of bond rearrangement between oxygen and the network-forming cations Si and Al [40–42]. Hence, viscous flow is not governed by the relative ease with which entities of varying shape and size move with respect to one another, but by the manner in which these entities exchange bonds through mutual interpenetration (Fig. 8). On this basis, one can readily derive Eq. (10) from simple phenomenological considerations. If the Arrhenius equation is used as a starting point, one has to conclude that the activation energy decreases with increasing temperature. Now, activation energy depends on the average strength of atomic bonds within the melt. But it also depend on a steric factor that accounts for the greater or lesser ease with which entities can get into contact to form the activated complex through which bond rearrangement actually takes place. One can thus express Ea as a function of two terms

Ea =R ¼ Be =X:

ð19Þ

The first term, Be, represents the energy barriers opposing bond rearrangement; it is constant since bond strengths do not vary significantly with temperature. Hence, the second term, X, has to increase with increasing temperatures to ensure the observed net decrease of Ea. As a steric factor, it may be assumed to be related to configurational entropy, which is the thermodynamic measure of atomic disorder. Because Sconf increases continuously with temperature, a simple proportionality relationship with X may even be assumed. And because X has the same dimension as energy, the simplest proportionality factor is temperature. By writing in this way

Ea =R ¼ Be =X ¼ Be =TSconf ;

ð20Þ

14

BaS2

Sr 4 S6

KS9

12

CaS

log η (Pa s)

10

8

MgS

6

NaS5 NaS2 4

2

LiS 2 0

-2 4

6

8

10

12

14

one arrives directly at Eq. (10) and then, with Eq. (9), to Eq. (3). Within this framework, the decreasing size of the regions of the melt in which cooperative rearrangements take place is simply due to the fact that, with increasing temperature, increasingly larger entities can engage in bond exchange, without requiring any change in the overall degree of polymerization. As a first approximation bonding in silicates can be assumed to be ionic and, thus, to depend primarily on ionic radius and charge [7], with the consequence that bonding with oxygen is much weaker for alkali than for alkaline earth cations. That Be is a measure of average bond strength is borne out by experimental data (Fig. 9) which show systematically higher viscosities for alkaline earth than for alkali systems at lower temperatures, where configurational entropy is small in all melts. Because stronger bonding gives rise to higher configurational heat capacity and steeper variations of configurational entropy, the viscosity of alkaline earth melts exhibits stronger non-Arrhenian character so that they eventually become similar to that of alkali melts in the high temperature limit. Contrary to Arrhenius laws, which assume hopping of a single atom over a constant energy barrier as the fundamental step of fluid flow, Adam and Gibbs theory relies on the availability of a great many different configurational states to account for the cooperative nature of rearrangements in which all atoms in the melt are involved. All atoms actually take part in viscous flow lies because of the cohesive strength of the silicate network which is made up of a variety of anionic entities built on SiO4 (or AlO4) tetrahedra to which other cations are bonded. The structural fluctuations through which shear relaxation takes place are thus the same as those involved in volume and enthalpy relaxation, with the result that these bulk properties show the same relaxation kinetics as viscosity [43]. In this respect, viscosity markedly contrasts with electrical conductivity, which is determined only by the mobility of the fastest moving cations, i.e., the network-modifiers [44]. As a result, electrical conductivity does not scale as viscosity and a bulk property such as configurational entropy is irrelevant to account for its variations. The fact that the glass transition takes place in the vicinity of the Dulong-and-Petit limit is also consistent with the overall strength of the silicate network and has the important practical consequence to allow the configurational entropy, or at least its temperature dependence, to be evaluated readily. As noted previously [5], in all these respects application of Adam–Gibbs theory to silicate melts is not beset by the problems found when considering other kinds of glass-forming liquids in which there is strong decoupling between local and bulk dynamics at all temperatures. As an example, in a liquid such as Ca0.4 K0.6(NO3)1.4 the onset of rotational or librational relaxation of ions is observed at much lower temperatures than that of NO2 3 shear relaxation, with the result that structural, volume, shear and NO3 rotational relaxation times differ considerably at low temperatures [45]. In the same way, rotational or librational degrees of freedom are active in organic chain polymers below the glass transition range and their contribution to configurational heat capacity and entropy thus is irrelevant to shear relaxation. Only that part of the total configurational entropy which is related to shear relaxation should be taken into account in Adam–Gibbs modeling. This would make calorimetric determinations of configurational entropy difficult because Eq. (7) would be inappropriate. And, even when there exists an isochemical crystalline phase, the entropy difference between amorphous and crystalline phase would approximate configurational entropy too badly to remedy to this situation. Acknowledgments

-1

104/T (K ) Fig. 9. Viscosity of alkali and alkaline earth silicate melts. See [5], chapter 6, for data sources.

This paper is dedicated to Y. Bottinga who first arose the author’s interest in the rheology of silicate melts. Fruitful interactions on this topic with M.A. Bouhifd, B.O. Mysen, D.R. Neuville, J.

P. Richet / Journal of Non-Crystalline Solids 355 (2009) 628–635

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