HEAT CAPACITY AND MEDIUM RANGE ORDER IN OXIDE GLASSES

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HEAT CAPACITY AND MEDIUM RANGE ORDER IN OXIDE GLASSES

N. Soga

To cite this version:

N. Soga. HEAT CAPACITY AND MEDIUM RANGE ORDER IN OXIDE GLASSES. Journal de

Physique Colloques, 1982, 43 (C9), pp.C9-557-C9-568. �10.1051/jphyscol:19829112�. �jpa-00222417�

JOURNAL DE PHYSIQUE

Colloque C9, supplement au n°12, Tome 43, deoembre 1982 page C9-557

HEAT CAPACITY AND MEDIUM RANGE ORDER IN OXIDE GLASSES

N. Soga

Department of Industrial Chemistry,Faoulty of Engineering, Kyoto University, Japan

Résumé.- Pour analyser les données déjà existantes et nouvelles de la capacité calorifique aux basses températures des silicates,des borates et des germanates aux états vitreux et cristallin, on a appliqué la théorie des trois-bandes qui utilise trois températures caractéristiques, 0E, ^ et ©3. Les valeurs de eE pour différents cations modificateurs dans les verres ont été semblables à celles trouvées dans les cristaux. Le changement de l'état de coordination ou de l'ordre à courte distance des cations formateurs de réseau a été observable dans 61. Les valeurs de ©3 pour certains verres ont été beaucoup plus petites que celles observées dans les cristaux comparables. Ces données ont été utilisées pour discuter la répar- tition statistique ou l'ordre à moyenne distance dans la structure du verre. En outre, on a présenté l'applicabilité des résultats pour interpréter le comportement élastique du verre.

Abstract.- The three-band theory using three characteristic tem- peratures 9E, B± and 93 has been applied to analyze the existing and new data of low. temperature heat capacity of various silicates, borates and germanates in both glassy and crystalline states. The values of B„ for various modifying cations in glasses were similar to those in crystals. The change in coordination state or short range order of network forming cations was observable in &]_. The values of 93 for some glasses were considerably smaller than those of comparable crystals. These data were used to discuss the ran- domness or medium range order in glass structure. Furthermore, the applicability of the results to interpret the elastic behavior of glass was presented.

1. Introduction.- The short range ordering of atoms or ions in glass is often considered to be similar to that in the crystal having the same chemical composition, and many physical properties of glass has been analyzed as though glass were an isotropic single crystal. From this view point, the knowledge of similarity or dissimilarity in physical properties between the crystalline and glassy states should be useful to discuss glass structure, although this approach is indirect compared with those using the small angle X-ray or neutron diffraction technique and infrared spectroscopy.

Among various physical properties, heat capacity provides useful infor- mations, because it is directly related to atomic vibrations associated with the interatomic forces between atoms or ions in solids. In order to analyze the temperature dependence of heat capacity of solids, a usual way is to assume that each atom is a point mass connected each other by a spring having an appropriate attractive force and oscillates its eigen frequencies. This simple theory of heat capacity was intro-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19829112

C9-558 JOURNAL DE PHYSIQUE

duced first by Einstein/l/ and then by Debye/2/ and it was successfully applied to interpret the temperature dependence of heat capacityofvar- ious solids such as alkali halides and metals. However, it has been found that the Debye theory does not hold for high polymers with chain or layer structures. The real macromolecular structure of these poly- mers does not correspond to a three-dimensional continuum as postulated by the Debye theory. So it must be modified to include the vibrations nf a one-dimensional continuum suitable forthese polymers. Several such modifications have been attempted in the past. For example Tarasov/3/

developed a theoretical model for substances with chain or layer struc- ture by considering both of the continuum distributions.

The previous X-ray diffraction (XRD) studies on silicate glasses have indicated that most of them possess a modified three-dimensional Si02 network structure. The glass forming compounds usually take some kinds of network structure, and thus the Debye theory does not hold for most of inorganic glasses. In fact, Westrum/4/ and WhiteandBirch/5/ pointed out that the heat capacity curve of vitreous silica has acharacteristic form different from a normal heat capacity curve of a simple ionic com- pound and closely resembles that of a substance with polymer-like chain structure. In order to apply Tarasov's two-band theory tothesubstances containing network formers and modifiers, suchas alkali silicates, some modification is necessary because additional vibrational modes are expected to appear due to the existence of modifying ions. As described in the previous paper/6/, an Einstein function was introduced to the two-band theory as another term representing the vibrational modes of modifying ions, and a theoretical expression for this three-band theory was formulated.

In thepast years, the present author andhisco-workers have been trying to accumulate systematic data of heat capacity for different types of oxide glasses with a hope that the three-band theory collld be used to interpret these data and to obtain an appropriate infornidtion about the interatomic forces between atoms or ions which is needed to perform molecular dynamic calculations for oxide glasses. In the mean time, it was realized that the compositional dependence of these characteristic temperatures deduced by the three-band theory for various oxide glasses could provide valuable informations about glass structure, particularly those of short range order as well as medium range order when compared with the results of crystalline compounds. The present paper describes

some of the results obtained so far.

2. Three-Band Theory and its Meaning.- tihen the frequency distribution function for vibrations of atoms constituting a solid is known, the temperature dependence of vibrational properties such as heat capacity can be calculated. According to the Debye theory/2/, the number of vibrational modes per frequency interval is proportional to the square of frequency of normal modes up to a certain maximum frequency. This cut-off frequency is related to the interatomic force, mass and lattice spacing of constituent atoms, and thus it is characteristic of a solid.

Although this Debye theory describes the temperature dependence of heat capacity for a number of solids, itis no more applicable whentheatomic structure of a solid becomes complicated.

For example, let us assume a network structure shown schematically in Fig. l(a). In the Debye model, a solid is usually assumed to be repre- sented by a monoatomic lattice composed of atoms where the volume per cell is the average atomic volume. Thus, inthis model the above network structure is modified by the one like Fig. l(b)

,

where all atoms are distributed homogeneously with even spacing. Clearly, its nearest interatomic distance becomes larger than the original one. Since the interatomic force is a function of interatomic distance, differentkinds

Fig.1. Schematic mode

2

of network s t r u c t u r e .

of interatomic potential have to be used for (a) and (b) to give the same heat capacity. This is not reasonable.

To solve this discrepancy, the frequency distribution function is con- sidered to consist of two different types of vibrational modes: one due to the strong interaction force between atoms or ions and the other due to the weak interaction force between the repeated units of atomic assemblies, as shown in Fig. 1(C). By following Tarasov's approach,the former is assumed to take the one-dimensional continuum distribution having atotal of 3N1 high frequency modes from

vmax

to vl and thelatter the three-dimensional continuum distribution having a total of 3N2 low- frequency modes from vl to 0 . Then, the total vibrational energy U1,3 of the system can be given by the following equation.

V

U1.3 = 3N1( vmax-vl) max h

v

[exp (h~/kT) -11 -ldv V1

+ 9~2v;~j;(l h

v3

[exp (hv/kt)

-11

-lei,

....

⁽¹⁾

where h and

k

are the Planck and Boltzmann constants, respectively. The heat capacity C1,3 of the system is obtained by setting x = hv/kT, 81=

hvmax/k and 83 = hvl/k as follows.

If the above model is applicable, the characteristic temperatures cal- culated from the heat capacity data should conform to the crystalla- graphic structure of solids. So, an attempt was made to analyze the heat capacity data of crystalline compounds with different crystallo- graphic structures. The simplest case is the one where each repeated unit contains only one atom with a close packed structure. In this case, the number of the repeated units is the same as the number of atoms and the distance between the repeated units is the same as the interatomic distance. Thus, 83 should be equal to 81 and these two characteristic temperatures should converge into one value, or the Debye temperature OD, even if the heat capacity data were analyzed by eq. (2). On the other hand, the heat capacity data of a solid with

a

linear chain structure should be described by 81 without

e3.

If a crystal takes a layer structure, eq. (2) should be modified to include the vibrational modes of the two-dimensional continuumdistributionwith the parameter 82 in place of those of the three dimensional continuum distribution with 83.

The analyses of heat capacity data were made for NaC1, MgO, Se, As203, Bi203, graphite, As and Sb, and the results are shown in Table I. AS expected, the values of and 93 became equal for NaCl and MgO, whlle only

el

appeared for Se, As203 and Bi203. The data of graphite, AS and Sb metals fit for a model of

the

two-dimensional continuum and the values of and G2 became equal. These results agree with the above consideration.

JOURNAL DE PHYSIQUE

Table I. Characteristic Temperatures for Some Crystals with Simple Structures.

Material Structure Characteristic Temperature Data

81 83 92 from

NaCl NaCl type 281 K 281 K - /7/

MgO NaCl type 758 758 - /8/

se Herical chain 375 0 - /9/

As203 Chain of Valentinite 1003 0 - /lo/

Bi203 Chain of Valentinite 661 0 - /11/

Graphite Layer 1370 - 1370 /12/

As metal Layer 331 - 331 /lo/

Sb metal Layer 223 - 223 /lo/

To extend the preceding treatment to various compounds, the additional vibrational energies of other ions (network modifiers) than the network formers must be considered. The bond strengths of these ions are much weaker than that of network formers and it is assumed that the vibra- tional modes of these modifying cations are local modes andcontribute independently to the heat capacity in the form of an Einstein function with the characteristic temperature

eE.

Consequently, 8~ depends on the kind of modifying cations but not on their concentration, nor on the kind of network formers. Thus, the representative expression of heat capacity by the present three-band theory with

y

= QE/T is

Once the heat capacity of a solid is measured at various temperatures, el,

e3

and BE can be determined numerically by a computer to fit the experimental data. Although the real structure of oxide glass is unknown, a model structure consisting of chains of network forming metal polyhedra, similar to those of crystalline states, maybeassumed.

If so, the glass structure changes depending upon bond length, bond angle, number of polyhedra in the repeated units, and type of inter- connection of polyhedra. Such differences should cause a change in the lattice vibrational frequency spectra and thus the value of 81 and 83. Therefore, it is possible to discuss the structure of the glass network by comparing 81 and 83 for various glasses and crystals.

3. Characteristic Temperatures for Various Glasses.- The heat capacity data of various glasses at low temperatures were determined in the same manner as described in the previous papers/6,13Q15/. The temperature range was from

77R

to 300K for most cases, but in some cases the meas- urements were made at lower temperatures than 77K. All the data were analyzed by the three-band theory using eq.(3). Fig.2 shows an example

Fig. 2. Temperature dependence o f heat capacity for NagO.

Si02 glass.

0 1 ^{I I I I}

J

60 100 140 180 220 260 300 Temperature K

of the fitness of the three-band theory with the experimental data for Na2O-2Si02 glass. The values of the characteristic temperatures for various glasses thus obtained were summarized in Table

11. For

the sake of comparison, theheat capacity data forvarious silicate, alumino- silicate, germanate and borate compounds were compiled from literature and were analyzed also by means of eq.(3). The results are given in Table

111.

T a b l e . 11. C h a r a c t e r i s t i c T e m p e r a t u r e s f o r V a r i o u s G l a s s e s

-

G l a s s C o m p o s i t i o n 0 1 €I3 BE G l a s s C o m p o s i t i o n Q1

e 3

QE

MgO

-

CaO - SiOZ

0 50 5 0

1 2 . 5 37.5 50

25 25 50

33.5 1 6 . 6 50 3 7 . 5 1 2 . 5 5 0

T a b l e . 111. c h a r a c t e r i s t i c T e m p e r a t u r e s f o r V a r i o u s C r y s t a l s

-

l a t t i c e d i m e n s i o n D a t a

C r y s t a l

s y s t e m a b c 01 ' 3 'E from

S i 0 2 ( q u a r t z ) h e x . 5 . 0 0 - 5 . 4 6 1450 2 4 1 /16/

( t r i d y m i t e ) h e x . 5 . 0 5 - 8 . 2 6 1440 230 ( c r y s t o b a l i t e ) c u b . 7.18 -

-

1480 2 2 5

Na2Si205 o r t h . 6.44 1 5 . 0 4 4.96 1360 340 253 /17/

Na2Si03 h e x . 6 . 0 8 - 4 . 8 3 1320 396 256 /17/

K2Si205 1 3 1 0 315 1 5 3 /18/

KAlSi 0

2 6 t e t . 13.12 - 1 2 - 7 9 1270 283 1 5 0 /13/

CsA1S12o6 c u b . 1 3 . 7 - 1 1 0 5 2 3 1 5 0 /13/

MqSi03 o r t h . 8 . 8 3 1 8 . 2 5.19 1250 352 702 /20/

CaSi03 t r i . 6 . 9 0 1 1 . 7 8 1 9 . 6 5 1240 3 0 3 570 /21/

Ge02 t e t . 4 . 9 9 - 7 . 0 7 1250 500 /22/

Na2Ge03 h e x . 6.26 - 4.92 1030 425 232 /23/

K2Ge03 890 427 1 3 5 /23/

MgGeO3 h e x . 4.94 - 1 3 . 7 6 1 1 3 0 5 3 1 7 6 1 /23/

CaGe03 t r i . 8 . 2 3 7 . 5 8 7.34 1125 421 525 /23/

B2°3 cub. 1 0 . 0 6 -

-

1860 240 /13/

NaBO? - 1 6 0 0 1 4 0 2 1 0 /24/

C9-562 JOURNAL DE PHYSIQUE

4. C o m p o s i t i o n a l a n d S t r u c t u r a l Dependence ofcharacteristic temperatures.

4.1. Network Modifiers and 9 ~ . - The network modifiers, such as aLkali ions, are assumed here to occupy the interstices of network structures.

If the network structure remains unchanged when network modifiers are substituted from one kind to another, the change in internal energy arises mainly from the difference in bond energy between two kinds of network modifiers. The interatomic bonding nature of network modifiers is generally considered ionic rather than covalent, so that their bond strength may be approximately represented by the Born potential/25/.

where Z1 and 22 are the effective charges of cations and anions, A the Madelung constant and B an empirical constant. If we use thispotential, the change in bulk modulus due to such substitution can be expressed in terms of volume V as follows/l9/.

where C is the constant related to the packing conditionofatoms orions as well as to the repulsive force. The validity of eq.(5) was shown for alkaline-earth silicate glasses in the previous paper/26/.

In the three-band theory, such change should appear in CJE. The assump- tion of the three-band theory is that the modifying cations vibrate independently and thus BE depends on the kind of alkali ions but not on their concentrations. In Fig. 3, the values of 0~ for various alkali ions listed in Tables I1 and 111 are plotted as a function of alkali content. The resonant frequency of an independent oscillator with the mass of m and the spring constant of g is proportional to

m.

^As

described above, the main interaction of alkali-oxy~en bonds is repre- sentable by the Born potential. Since g is given by d 2 ~ / d r ~ , g becomes inversely proportional to the cube of interatomic distance or to the mean atomic volume. ~ h u s BE is proportional to &GS. The values of Jl/mr3 for Li-0, Na-0, K-0 and Cs-0 bonds were calculated from the atomic weight of alkali ions and the ionic radii of alkali and oxygen ions. The relation between BE and is shown in Fig. 4. Clearly, a linear dependence between these two quantities can be seen.

Fig. 3. Dependence o f BE on a l k a l i content for various a Z k a l i s i l i c a t e glasses and c r y s t a l s .

Fig. 4 . Relationship between B8 and resonant frequency of aZkaZz i o n s .

The comparison i n QE between t h e g l a s s y and t h e c r y s t a l l i n e s t a t e s s h o w n i n T a b l e s I1 a n d 1 1 1 o r i n F i g . 3 i n d i c a t e s t h a t

eE

f o r a g l a s s i s s l i g h t l y lower t h a n t h a t f o r t h e c r y s t a l h a v i n g t h e samecomposition. T h i s seems t o mean t h a t t h e a v e r a g e bond l e n g t h of a l k a l i - o x y g e n bonds i n g l a s s s t r u c t u r e i s s l i g h t l y l o n g e r t h a n t h a t i n c r y s t a l s t r u c t u r e .

4.2. Network Formers and 81.- S i n c e 8 1 i s a s s o c i a t e d w i t h t h e bond s t r e n g t h of network f o r m e r s , t h e r e l a t i v e v a l u e s of

el

f o r B-0, Si-0 and Ge-0 bonds may b e e s t i m a t e d i n a s i m i l a r way a s i n t h e c a s e of network m o d i f i e r s . However, it i s n o t e a s y t o do s o b e c a u s e of uncer- t a i n t y i n p o t e n t i a l which governs t h e bond s t r e n g t h of t h e s e bonds. By assumln t h a t t h e Born p o t e n t i a l a l s o h o l d s f o r t h e s e o x i d e s , t h e v a l u e s of

+

Z 1 Z 2 / m r 3 f o r t h e s e bonds were c a l c u l a t e d from t h e i n t e r a t o m i c d i s - t a n c e based on t h e i r i o n i c r a d i i , t h e mean a v e r a g e mass of t h e s e i o n s and t h e c h a r g e s o f i o n s . The r e s u l t s a r e p l o t t e d i n F i g . 5 a g a i n s t 8 1 f o r B2O3, S i 0 2 and Ge02. A l i n e a r r e l a t i o n s h i p i s o b s e r v e d .

The v a l u e of

el

d e c r e a s e d c o n s i d e r a b l y w i t h t h e a d d i t i o n o f network m o d i f i e r s f o r s i l i c a t e g l a s s e s a s w e l l a s f o r c r y s t a l s , a s shown i n T a b l e s I1 and 111. When t h e network m o d i f i e r s a r e added t o a c o n t i n u o u s network s t r u c t u r e o f S i 0 2 , s o - c a l l e d n o n - b r i d g i n g oxygens a p p e a r , c a u s i n g t h e changes i n e f f e c t i v e c h a r g e o f oxygen i o n s and i n t e r a t o m i c d i s t a n c e among o t h e r e f f e c t s . These changes c e r t a i n l y a f f e c t t h e bond s t r e n g t h o f S i - 0 bonds and t h u s t h e v a l u e of

el.

According t o e q . ( 4 ) , t h e p o t e n t i a l depends a l s o on t h e Madelung c o n s t a n t A , which changes w i t h a t o m i c a r r a n g e m e n t s . T h e r e f o r e , i f t h e c o o r d i - n a t i o n s t a t e of t h e network forming c a t i o n s changes, t h e s p r i n g c o n s t a n t g should change a c c o r d i n g l y , r e s u l t i n g i n t h e change i n 81. I n F i g . 6 , t h e v a l u e s of 01 f o r t h e g l a s s e s i n t h e systems Na2O-B203, Na20-Si02

20 40

1600 1500

Y

1400

?

CD

1300 2000

Y

1500

m 7

1000

Ca-0

~~~~~~

0 1 2 3 4 x 1 0 ~ ' 1200

JZ1Z2/mr3

Fig. 5 . Re Zationship between 1100

2 0 40

01 and resonant frequency o f Na20 mol%

cation-oxygen i o n bonds.

Fig. 6 . Change i n 01 for various oxide gZasses containing Na2O.

-

/ I B - ~ -

/*y~e:i-O -

/*

Mg-0

C9-564 JOURNAL DE PHYSIQUE

and Na2O-Ge02 are shown as a function of Na20 content. For silicate glasses, 81 decreases monotonously with Na20 content. On the other hand, for borate and germanate glasses, increases first with Na20 content and then decreases. As described above, the addition of weak Na-0 bonds to glass network causes a decrease in 61 as seen in silicate glasses. Thus, such an increase in 81 seems to be associated with the well-accepted change in coordination number of boron/27/ or germanium ions/28,29/. The amount of Na20 giving the maximum value of 81 corre- sponds reasonably well with that for other physical properties. A close look of Fig.6 shows a tendency that 81 of B-0 bonds for .the glasses containing more than 20 mol% Na20 decreases only slightly, while

el

of Ge-0 bonds for the glasses containing more than 20 mol% Na20 decreases considerably and becomes much lower than that of Ge02 glass. This dif- ference seems to indicate that most of boron ions retain the four coor- dinated state even if Ba20 content becomes more than that required to make the coordination number of boronfrom t h r e e t o f o u r , w h i l e g e r m a n i u m ions tend to change its coordination number from six to four after reaching the maximum amount. This result is in accordance with the experimental results of NMR forborateglasses/27/ and X-ray diffraction study for germanate glasses/29/.

The intermediate oxides such as A1203 act sometimes as a network former and sometimes as a network modifier. It is generally considered that aluminum ions are incorporated into network and no breaking of network structure takes place,when A1203 content is equivalent to Na20 content in silicate glasses.

If

so, both 81 and BE for alkali aluminosilicate glasses having Na20-A1203-nSiO2 compositions should not change so much even though n is varied. This can be seen in the results of lithium- and sodium-aluminosilicate glasses shown in Table 11. The slightly lower values of for aluminosilicate glasses than those of silicate glasses containing equivalent amounts of Na2O are attributable to the introduction of weak A1-0 bonds in Si02 glass network.

4.3. Medium Range Order and €I3.- According to the theory of lattice dynamics, themaximum vibrational frequency of adiscrete lattice depends upon the size of vibrational units : the larger the size, the lower the frequency/30/. Since the characteristic temperature 63 is proportional to the maximum vibrational frequency ofthethree-dimensional continuum, a low value of 83 means that the average size of the vibrational units is large. Thus, the comparison of the values of 83 for various glasses and crystals gives an information about the size of repeated assemblies

of

atoms or ions. As shown in Table 11, 63 for Na2O-SiO2 glass is close to that for Na2Si03 crystal. Thus, it is considered that the average size of the repeated units is similar for both states. In other words, the medium range structure of Na20-Si02 glass is similar to that of Na2Si03 crystal. On the otherhand,thevalue of 83 for fused silica is much smaller than that ofany crystalline Si02, indicating that fused silica has a medium range structure differentfromSiO2 crystal. Quali- tatively speaking, these results suggests that fused silica has a more disordered or random structure than Na20.Si02 glass.

In order to discuss this medium range structure more quantitati~ely~the estimation of the size ofrepeated units is necessary. This may be done if the relationship between 81 and 63 becomes known. At this moment, an explicit relationship is difficult to derive, and so some reasonable assumptions have to be incorporated. In thepresent study, the following simple assumption was applied; the minimum wavelength of vibrational modes responsible for

e3

is comparable totheaverage sizeoftherepeated units and it is multiple of the minimum wave length for el, which is related with the interatomic distance. This assumption may beeasily visualized from the model shown in Fig. l(c). The choice of the inter- atomic distance affects the size. As described in the previous section

The i n t e r a c t i o n o f a l k a l i i o n s t o f u s e d s i l i c a d e c r e a s e s t h e s i z e t o a b o u t 5A up t o t h e amount of 20 mol% Na20, beyond which it remains a l m o s t c o n - s t a n t . T h i s s i z e i s comparable t o t h e s i z e found from 61/83 f o r N a 2 S i 0 3 o r Na2Si205 c r y s t a l . I t i s a l s o c l o s e t o t h e minimum c r y s t a l l o g r a p h i c c e l l dimension of s i l i c a t e s a s shown i n T a b l e 111, as w e l l a s t h e mean f r e e p a t h o f p h o n o n s f o r s i l i c a t e g l a s s e s / 3 2 / . The same c o n c l u s i o n c a n b e o b t a i n e d w h e n t h e d a t a o f a l k a l i n e - e a r t h s i l i c a t e i n b o t h s t a t e s i n T a b l e I1 and I11 a r e compared. Such dependence o f t h e s i z e of r e p e a t e d u n i t s on t h e amount of network m o d i f i e r s i n d i c a t e s t h a t t h e b r i d g i n g a n g l e of Si-0-Si bonds v a r i e s widely f o r f u s e d s i l i c a b u t becomes narrow w i t h a d d i t i o n of network m o d i f i e r s and approaches t o t h a t o f t h e c o r r e s p o n d i n g c r y s t a l s . T h i s tendency i s i n a c c o r d a n c e w i t h t h e f a c t t h a t t h e d i f - f e r e n c e i n d e n s i t y between t h e g l a s s y and c r y s t a l l i n e s t a t e s of Si02 is q u i t e l a r g e b u t it becomes s m a l l w i t h a d d i t i o n o f network m o d i f i e r s . 8 1 d e c r e a s e s w i t h i n c r e a s i n g t h e a m o u n t

of network f o r m e r s , r e s u l t i n g prob-

a b l y from t h e changes i n e f f e c t i v e l o c h a r g e of oxygen i o n s and i n t e r a t o m i c

d i s t a n c e . I f t h e former e f f e c t i s 8 dominant, t h e S i - 0 bond l e n g t h remains

a l m o s t c o n s t a n t and t h e bond l e n g t h f o r f u s e d s i l i c a may b e used t o e s t i - mate t h e s i z e o f r e p e a t e d u n i t s . On m t h e o t h e r hand, i f t h e l a t t e r e f f e c t 4 . i s dominant, t h e c h a n g e i n bond l e n g t h &

h a s t o b e e s t i m a t e d . T h i s may b e done by t a k i n g t h e l i n e a r r e l a t i o n s h i p 2 between €11 and J ~ 1 ~ 2 / r m T , which g i v e s

a b o u t 10% change i n bond l e n g t h when

The r a t i o o f 81/63 f o r GeO2 g l a s s i s l a r g e r t h a n t h a t of Si02 g l a s s , and t h a t f o r B2O3 g l a s s i s s t i l l l a r g e r . The a d d i t i o n of Na20 t o t h e s e g l a s s e s d e c r e a s e s t h e s i z e b u t it i s s t i l l much l a r g e r t h a n t h o s e of t h e c o r r e s p o n d i n g c r y s t a l s . Such a d i f f e r e n c e between s i l i c a t e g l a s s e s and b o r a t e o r germanate g l a s s e s may be a s s o c i a t e d w i t h t h e f a c t t h a t b o t h boron and germanium i o n s t a k e two d i f f e r e n t c o o r d i n a t e d s t a t e s , which might g i v e more d i s o r d e r e d g l a s s s t r u c t u r e t h a n t h a t o f s i l i c a t e g l a s s e s . T h i s r e s u l t s u g g e s t s t h a t a much l a r g e r number of atoms have t o be used a s t h e system s i z e f o r t h e m o l e c u l a r dynamic s i m u l a t i o n s f o r b o r a t e o r germanate g l a s s e s t h a n f o r s i l i c a t e g l a s s e s .

5. C h a r a c t e r i s t i c Temperatures and Anomalous E l a s t i c Behavior o f G l a s s . Heat c a p a c i t y and e l a s t i c c o n s t a n t s a r e r e l a t e d e a c h o t h e r b e c a u s e b o t h p r o p e r t i e s depend upon t h e i n t e r a t o m i c f o r c e s . The Debye t e m p e r a t u r e i s u s u a l l y used t o r e l a t e them. T h i s problem f o r g l a s s h a s been d i s - c u s s e d by Anderson/33/. S i n c e g l a s s i s e l a s t i c a l l y i s o t r o p i c , one k i n d of continuum i s enough t o d e s c r i b e i t s e l a s t i c b e h a v i o r . Thus, u s u a l l y t h e e l a s t i c p r o p e r t i e s o f g l a s s e s a r e d i s c u s s e d based on two modes of v i b r a t i o n s , l o n g i t u d i n a l and s h e a r modes, o f a t h r e e - d i m e n s i o n a l con-

4 1

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\ Glassy

- '\

^states

.

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\ ⁰ Crystalline

6 ' h \ states

*-L

\.\.\.

- ', .--

^0..

^{.- -}

.

^\h*

---.---

^1-0

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'

-15

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v,

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_-1

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8 1 v a r i e s from 1550K t o 1300X. 2 0 40

Na20 mol%

I n F i g . 7 , t h e e s t i m a t e d s i z e o f r e - Fig. 7. The e s t i m a t e d l a t t i c e s i z e p e a t e d u n i t s f o r g l a s s e s and c r y s t a l s o f Na20-Si02 compounds.

i n Na20-Si02 system i s shown a s a

f u n c t i o n of Na20 c o n t e n t . A c o n s t a n t bond l e n g t h o f 1.6A, which i s t h e a v e r a g e of S i - 0 bond l e n g t h f o r v a r i o u s s i l i c a t e compounds, was used.

The s i z e may i n c r e a s e a b o u t 1 0 % when t h e change i n b o n d l e n g t h i s a s s u m e d t o t a k e p l a c e . The e s t i m a t e d s i z e o f r e p e a t e d u n i t s of a b o u t 1 7 A f o r f u s e d s i l i c a i s a b o u t t w i c e of t h a t of S i 0 2 c r y s t a l s and i s comparable t o t h e s i z e of c h a r a c t e r i s t i c o r d e r o f 15%30A i n d i c a t e d by P h i l l i p s / 3 1 / f o r c o - v a l e n t n o n - c r y s t a l l i n e s o l i d s .

C9-566 JOURNAL DE PHYSIQUE

tinuum. However, some g l a s s e s show anomalous e l a s t i c b e h a v i o r s , which c a n n o t be e x p e c t e d from such a s i m p l e model. To e x p l a i n such anomalous b e h a v i o r s , a more c o m p l i c a t e d model h a s t o be u s e d . The p r e s e n t t h r e e - band t h e o r y may b e used t o e s t a b l i s h such a model. The f o l l o w i n g s a r e two examples.

5.1. Mixed A l k a l i E f f e c t . - The mixed a l k a l i e f f e c t i n s i l i c a t e g l a s s h a s been found t o a p p e a r on v a r i o u s p h y s i c a l p r o p e r t i e s . E l a s t i c c o n s t a n t s a r e no e x c e p t i o n , a l t h o u g h it i s s m a l l compared w i t h t h e p r o p e r r i e s i n v o l v i n g a l k a l i d i f f u s i o n / 3 4 / . To c l a r i f y t h i s e f f e c t , it i s i m p o r t a n t t o know t h e p l a c e i n g l a s s s t r u c t u r e where t h e mixing o f a l k a l i i o n s g i v e s t h e most pronounced e f f e c t . Such i n f o r m a t i o n may be o b t a i n e d by a n a l y z i n g h e a t c a p a c i t y d a t a w i t h t h e ' t h r e e - b a n d t h e o r y . The e f f e c t s o f mixed a l k a l i i o n s on c h a r a c t e r i s t i c t e m p e r a t u r e s a r e shown i n F i g . 8. The v a l u e s of f o r mixed i o n s f a l l on a l i n e drawn between t h e v a l u e s o f two end members, i n d i c a t i n g t h a t t h e b o n d s t r e n g t h of mixed R-0 bonds i s t h e a v e r a g e o f two k i n d s o f R-0 bonds. I n o t h e r words, R-0 bonds of one k i n d c a n n o t be s t r e n g t h e n e d n o r weakened by t h e e x i s t e n c e of a n o t h e r k i n d of R-0 bonds. On t h e o t h e r hand, b o t h 81 and 8-3 show a p o s i t i v e d e v i a t i o n from t h e l i n e a r a d d i t i v i t y o f two end members, i n d i c a t i n g t h a t t h e bond s t r e n g t h of S i - 0 bonds becomes s t r o n g e r by mixing two k i n d s of a l k a l i i o n s . The maximum d e v i a t i o n of a b o u t 7 % i s observed f o r Na-Cs g l a s s e s . I n o r d e r t o e s t i m a t e t h e i n c r e a s e i n bond s t r e n g t h , t h e v a l u e s o f 8 1 f o r S i - 0 , B-0 and Ge-0 a r e p l o t t e d i n F i g . 9 a g a i n s t t h e s i n g l e bond s t r e n g t h r e p o r t e d by Sun/35/.

Also i n c l u d e d a r e t h e v a l u e s of 8 1 (=83) f o r MgO and CaO c r y s t a l s . When a l i n e a r dependence i s a s s u m e d t o e x i s t , t h e above i n ~ r e a s e i n 8 ~ i n d i - c a t e s t h a t S i - 0 bond becomes a b o u t 7-8 % ( o r a b o u t 10 kcal/mol) s t r o n g e r by mixing Na and C s i o n s and a b o u t 3-4 % ( o r a b o u t 5 k c a l / m o l ) s t r o n g e r by mixing Na-K o r K - C s i o n s . Such i n c r e a s e i n bond s t r e n g t h i s i n

0

0.5 1

A 1 k a l i mol r a t i o

\Fig. 8. Characteristic temperahres for mixed a l k a l i - s i l i c a t e glasses o f R20.

2Si02 composition.

Fig. 9. Relationship between B I and single bond strength for various oxides.

Open c i r c l e f o r Ge-0 i s for four coordi- nated s t a t e , and t h a t for B-0 for four coordinated s t a t e .

accordance with the increase in bulk modulus observed for R20e5Si02 glasses (R = Na, K , Cs)/36/. The value of remains about the same even though R is changed from one kind to two kinds, indicating that the medium range structure remains unchanged. Consequently, it may be said that the tightening of network structure may be responsible for the mixed alkali effects. In comparison, alkaline-earth silicate glasses show no such deviation from the linearity for both 81 ande3 as given in Table 11. Thus little mixed alkaline-earth effects should be observed, which is in accordance with the results of elastic proeprties for mixed alkaline-earth silicate glasses/26/.

5.2 Pressure and Temperature dependences of Elastic Constants.- The most pronounced elastic anomaly appears on fused silica and silica-rich glasses as the negative pressure derivative and positive temperature derivative of bulk modulus. However, such anomaly disappears when network modifiers are introduced into glass. Several attempts have been made to explain this anomaly from structural viewpoint, such as on the basis of the change in bridging angle of Si-0-Si/37/. However, it is difficult to do so only from the data of elastic constants, which are the average properties associatedwithvibrational modes of long wave lengths.

According to Jorgensen/38/, the compression of a-quartz results from a cooperative rotation or tilting of the Si04 tetrahedra around their shared oxygen corners, with individual tetrahedra remaining relatively rigid. This means that the pressure or temperature derivativeofelastic moduli depends on how the network structure changes with pressure or temperature. As described in the previous section, the glasses containing a large amount of network modifiers have almostthe same values of 81/e3 or the medium range order as those of crystals, and furthermore they have the similar values of 81. Thus, if the crys- talline state of these compounds shows normal behaviors, then these glasses are expected to behave normally. In fact, the pressure and temperature derivatives of alkali silicate glasses or alkaline-earth silicate glasses with more than 25 mol% network modifiers have been found to be close to those of oxide or silicate crystals/39,40/. On the other hand, the values of for fused silica and silica-rich glasses are much larger thanthose of crystalline Si02, although the values of 81 are about the same for these two states. Thus, theelastic anomaly may be discussed from the difference in 83 or the medium range order. More quantitative discussion may be made by means of an appro- priate averaging scheme for elastic moduli of heterogeneous materials once the volume fractions of the three types of bonds become known, or by means of the finite element analysis once a structural model satis- fying the data of €I1,

e3

and

eE

is given. These approaches are now being undertaken.

6. Conclusion.- The three-band theory described in this paper is based on a model that heat capacity of oxide glass arises from three types of vibrational modes related to glass structure: the independent Einstein- type interatomic vibrations of modifying cations, the interatomic one- dimensional vibrations of network forming ions and the intermolecular three-dimensional vibrations of glass network. The characteristic temperatures of interatomic vibrations for network modifiers andnetwork formers havebeen found to be 50%600K and 1000%2000K, respectively,which reflect the difference in single bond strength of these cations. The characteristic temperature of intermolecular vibrations were 50~400K, which reflect the size of vibrational units. These values show that the precise measurements in a wide temperature range fromquitelow tem- peratures to room temperature are required to deduce these characteris- tic temperatures from heat capacity data.

C9-568 JOURNAL DE PHYSIQUE

Once t h e s e c h a r a c t e r i s t i c t e m p e r a t u r e s a r e o b t a i n e d , t h e i r c o m p o s i t i o n a l v a r i a t i o n s m a y b e u s e d t o d i s c u s s t h e change i n s h o r t r a n g e b o n d i n g n a t u r e of b o t h network f o r m e r s and m o d i f i e r s . The d i f f e r e n c e i n c h a r a c t e r i s t i c t e m p e r a t u r e of i n t e r m o l e c u l a r v i b r a t i o n s between t h e g l a s s y and c r y s - t a l l i n e s t a t e s r e s u l t s from t h e change i n c o n n e c t i n g ways of network forming metal-oxygen p o l y h e d r a . Thus, t h e comparison among v a r i o u s g l a s s e s and c r y s t a l s a r e u s e f u l f o r d i s c u s s i n g t h e change inmedium r a n g e o r d e r o f g l a s s s t r u c t u r e . By s e t t i n g a s i m p l e m o d e 1 , t h e s i z e o f r e p e a t e d u n i t s f o r o x i d e g l a s s e s h a s been e s t i m a t e d t o be s e v e r a l t o s e v e r a l t e n s angstroms.

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