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The density of low metal content rubidium, cesium, silver, and thallium borate glasses related to atomic arrangements
Journal
36
of Non-Crystalline
Solids 94 (1987) 36-44 North-Holland, Amsterdam
THE DENSITY OF LOW METAL CONTENT RUBIDIUM, SILVER, AND THALLIUM BORATE GLASSES RELATED TO ATOMIC ARRANGEMENTS
CESIUM,
Hun C. LIM and Steve FELLER Pltq,.sics Deparmenr,
Coe College.
Received
1987
23 January
Cedar
Rap&
Iowa
52402.
USA
Reported densities from rubidium, cesium, silver. and thallium borate glasses were used quantitative model involving atomic arrangements. The analysis yielded the volumes of the basic structural units present in all the glasses: three- and four-coordinated borons. The volume a four-coordinated boron enlarges as the radius of the metal ion increases while the volumes of three-coordinated borons for different metals are constant. Packing fractions which compare volumes calculated from ionic radii to the values found from the density model were obtained.
in a two of the the
1. Introduction Recently, a model was developed to connect density and atomic arrrangements in lithium, sodium, and potassium borate glasses [l-3]. This model was able to yield the relative volumes of the structural groupings present over very wide-ranging glass compositions. It was also possible to calculate a packing fraction [3] of each of the units, where the packing fraction was defined as the ratio of the volumes calculated from ionic radii to the volumes determined from the densities. Several unifying trends were seen including the important one that each of the glasses has the same structure at a given composition. The variation in volumes was attributed to the differing sizes of the metal ions present. It is the purpose of this paper to apply this model to existing data from other alkali and pseudoalkali borate glasses. The other systems being considered are rubidium, cesium, silver, and thallium borate glasses. In these systems, data are available to about 30 mol.% metal oxide.
2. Density data A thorough compilation of physical properties of non-silicate glasses was published by Mazurin, Streltsina and Shvaiko-Shvaikovskaya [4] in 1985. A summary of the reported density data from rubidium [5-71, cesium [6,7], silver [8-lo] and thallium [ll-131 borates can be found in fig. 1. In this figure, R is 0022-3093/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
H.C.
Lim.
S. Feller
/ Density
of low metal content
borate
glasses
37
DENSITY (g/cm’)
DENSITY (g/cm’)
SO.-.
. n n
4.0-
. .
00 2.0 I
1
I
012
0
R
0.4
Fig. 1. Density of rubidium, cesium. silver, and thallium borate glasse; as a function of R (R is defined as the molar ratio of metal to rubidium borates; 0, cesium boron). 0, borates; 0, silver borates; W, thallium borates.
0
I
0.2
R
0.4
Fig. 2. A comparison of the densities of lithium, sodium and potassium borate glasses with rubidium. cesium, silver. and thallium borates (R is defined as the molar ratio of metal to boron).
the molar ratio of metal to boron. When compared with data from lithium [l], sodium [2], and potassium [3] borates of like compositions, similar trends are noted (see fig. 2). In each system the density rises monotonically for R < 0.4.
3.
A
model
for the densities
Nuclear magnetic resonance (NMR) studies [14,15] have found that rubidium, cesium and silver borate glasses are all similar in structure to each other and to lithium, sodium and potassium borates up to R = 0.4. This similarity is best expressed by the fractions of borons three- and four-coordinated with bridging oxygens, f, and fi. These fractions follow the simple relations:
f,=l-R, f,=R,
R ~0.4, RI 0.4.
(la) (lb)
The equations in (1) are equivalent to charge conservation if the only charged borate unit present is a four-coordinated boron. Thallium is curious in that the fraction of four-coordinated borons is given by [16]: fi = +R, implying
RI 0.25,
(24
that
f,=l-+R,
R ~0.25.
(2b)
H. C. Lim.
38
S. Feller
/ Densi!,,
To explain these results, Baugher three-coordinated oxygens. Above in that thallium becomes covalent done. The model proposed earlier for
f,M’ +f*lw
lJ= f,V’ + f2V2
01 lobv meral content
borate
glasses
and Bray [16] postulated
the formation
of
R = 0.25, the situation is more complex [16] and no analysis of density in this region was density can be written [2]
R I 0.4 for Rb, Cs and Ag; R I 0.25 for Tl.
(3)
In eq. (3) M’ and M’ are the masses of the three- and four-coordinated boron units, with V’ and V’ the respective volumes. Equation (3) can be rewritten using eqs. (1) and (2) to show the explicit dependence of density on R for rubidium, cesium, and silver borates as:
P(R) =P(O)
v'/v'(O)
l+R(M'/M'-1) + R(V2/V'(0)-
V'/V'(O))
' R'
o'4'
(da)
where p(O) is the density of boron oxide glass and V’(O) the volume of the three-coordinated boron unit in that glass. Thallium is somewhat different in that M’ is a function of R since three-coordinated oxygens are thought to be formed [16]. This lowers the value of M’. For thallium borates, the density is written: p(R)
= p(o)
M'/M'(O)
+ (3/2)R(M2/M’@)
v'/V'(O)+(~/~)R(V~/V'(O)-
- M'/M'(O))
Y'/v'(O))
' R 5o'25'
(4b) where M’(O) is the mass of a three-coordinated boron unit in boron oxide glass. In eqs. (4a) and (4b) the masses of the atoms are known as are the density and associated volume at R = 0. V’ and I” remain as unkowns to be determined from the density data. M2/M’ for rubidium, cesium, and silver is:
M2/M’ =
Mass(B + 20 + Mt) Mass(B + 1.50) ’
and for thallum (see Appendix): M2,Ml(0)
= Mass(B + 2(20 + Mt)/3 + (C)$)/3) Mass(B + 1.50)
M,,M,(0)
= Ma@
+ ((3 - 29W6)/(2 Mass( B + 1.50)
(5b)
- W)O)
where B, 0, and Mt represent boron, oxygen, and the metal ion under study. The relative masses, M2/M’ and M’/M’(O), expressed by metal ion, are [17] rubidium = 3.685, cesium = 5.048, silver = 4.329, thallium = 4.659. The density of boron oxide glass was determined by averaging measurements on unannealed samples as reported in ref. [4]. This was done since the reported
H.C. Table 1 Relative volumes V’ fixed.
V’
Lim.
of borate
S. Feller
units
/ Density
o/low
metal
contem
borate
glasses
39
for Rb, Cs, Ag and Tl systems
varied
V’ and V2 varied
IOIl
v’/
Rb cs ‘% Tl
1.84 2.14 1.12 1.11
V’(0)
V’/
V’(0)
v’/
0.96 0.97 0.97 0.94
V’(0)
1.96 2.23 1.20 1.28
data from rubidium, cesium, silver and thallium are on quenched samples. The average density found was 1.823 g/cm3. For each of the metal systems a best value for the relative volumes was found by performing least-squares analyses of the data. Two methods were used. The first method followed the method reported in ref. [3]; V’ was fixed at the value found for R = 0 and only V2 was varied. The second method allowed I/’ to vary also and a two-parameter least-squares analysis was carried out. In both cases, values are reported relative to V’(0) and the results are given in table 1. The two methods give very similar results. Figure 3 shows a visual comparison of the two-parameter model with the data. The standard deviations of the density data with respect to the model are (in g/cm3) 0.03, 0.03, 0.08, and 0.08 for rubidium, cesium, silver and thallium respectively. The values for the volume ratios are contrasted with the values found for lithium [18], sodium [2], and potassium [3], see table 2.
DENSITY (y/cm3)
6
0.2
R
0.4
Fig. 3. The density model presented in this paper superimposed rubidium, cesium, silver, and thallium borate glasses (R is defined as boron). Standard deviations of the density data (compared with model) as 1. 0, rubidium borates; l , cesium borates; q I, silver borates;
on the density data from the molar ratio of metal to are shown for each system n , thallium borates.
40
H.C.
Table 2 Relative volumes
of borate
V’ fixed,
Lim,
S. Feller
units
/ Density
of low metal
for Li, Na and K systems
V2 varied
cement
borate
glasses
[2,3.18]
V’ and V2 varied
Ion
V’/
Li Na K
0.82 1.05 1.52
V’(0)
V’/
V’(0)
V2/V’(0)
0.98 0.96 0.95
0.88 1.17 1.64
A comparison among the metal systems suggests that V2/V*(0) is related to the size of the metal ion. Along these lines it was noted by Piguet and Shelby [19] that the molar volumes of silver and sodium borates are nearly identical over the composition range being considered due to the similarity of the ionic radii of sodium and silver. To clarify this comparison the concept of the packing fraction will be used. As noted before, the packing fraction is defined as the ratio of the volume of a unit calculated from ionic radii to the same volume found from the density measurements. This procedure was carried out earlier for the lithium, sodium, and potassium glasses [3]. The calculation requires values for the ionic radii and these are: radius of bridging oxygen bonded to three-coordinated boron [3,20] = 1.19 A, radius of bridging oxygen bonded to four-coordinated boron [3,21] = 1.28 A, radius of rubidium ion [22] = 1.47 A, radius of cesium ion [22] = 1.67 A, radius of silver ion [19,22] = 1.20 A, radius of thallium ion [22] = 1.47 A. In the calculation the boron was neglected since its radius (= 0.19 A [3,20]) is very much smaller than the oxygen radii as to make its volume negligible. Further details of the calculation can be found in ref. [3]. In the calculation values for Vi/V’(O) and V2/V’(0) were taken from the least-squares analysis in which both parameters were varied. The resulting packing fractions, shown
Table 3 Packing fractions
of borate
units
Borate system
Packing
lithium sodium silver potassium rubidium thallium cesium
0.67 0.57 0.65 0.52 0.49 0.63 0.52
fraction
Four-coordinated
a) Since the number of oxygens fractions was taken.
boron
Three-coordinated
boron
0.34 0.34 0.34 0.35 0.34 0.35 a’ 0.34 making
up this unit
changes
with
R, an average
of the packing
H.C.
Lim,
S. Feller
/ Density
of low metal content
borate
glasses
41
in order of increasing size of metal ion, are (along with results found earlier for lithium, sodium and potassium [3,18]) shown in table 3. The packing fractions for the alkali and silver borates reveal that there is a general decrease followed by a leveling off in efficiency in the filling of space for the four-coordinated borons as the metal ion gets larger. The packing fraction for thallium is considerably larger than would have been expected on the basis of ionic radii. However, as noted earlier, thallium is special in that the fraction of four-coordinated borons follows a steeper slope as R is varied than do the other systems. This process is thought to result in the formation of three-coordinated oxygens [16] and thus the unit is considerably different. The packing fractions for the three-coordinated borons are remarkably uniform and close to the value for pure boron oxide glass (0.33). This is believed to result from the absence of metal ions associated with this unit. The consistency of the packing fractions lends considerable support to the notion that the densities can be directly related in a quantitative way to the atomic arrangements in the glass.
4. Further discussion of the model More information about the ratio of the volumes can be found by plotting the trend of V’/V’(O) as a function of R leaving I/’ fixed at V’(O). This is shown in fig. 4. For each of the systems but thallium the volume ratio
Fig. 4. The ratio of the volumes of four-coordinated function of R (R is molar ratio of metal to boron). was fixed at the value found
borons to three-coordinated borons as a The volume of the three-coordinated borons for boron oxide glass.
H. C. Lbn.
42
S. Feller
/ Density
o/low
metal
contenr
borare
glasses
increases to a limiting value as ,R increases. As can be seen in fig. 4 the amount of change in V2/V’(0) is directly related to the size of the ion with the greater change coming for the larger ion (except thallium). This indicates that for low metal contents the ions are either filling available voids without further enlarging the boron-oxygen network or enlarging it to a smaller degree than at larger R. Such a trend was previously noted for lithium, sodium, and potassium [3] borates. Apparently, the density model which contains structural information from NMR, is sensitive enough to discern continuous changes in volumes as the composition is varied. However, it is noted that this analysis has certain assumptions built in such as that I” is fixed at the value found from boron oxide glass and that the ionic radii are indeed constants.
5. Conclusions Densities of rubidium, cesium, silver, and thallium borate glasses were used in a quantitative model involving atomic arrangements. The model, using reported data up to R = 0.4, indicates that rubidium, cesium, and silver act very much like the other alkalis with thallium showing some differences. The model uses NMR and density data in a consistent manner to get the volumes of the two basic structural units present: three- and four-coordinated borons with bridging oxygens. Packing fractions for the four-coordinated borons generally decline as the size of the metal ion increases. The three-coordinated borons have a uniform packing fraction in all systems. Further work is in progress to obtain density measurements at higher values of R in these systems. We express our gratitude to A. Karki, H.P. Lim, B.C. Liang, K.H. Chong, B. Teoh. and S.H. Choo of the glass research team at Coe College for reading the manuscript. M. Affatigato is thanked for help with the proofreading. Coe College and the Richter Foundation are acknowledged for financial assistance. Supported in part by grants of the Research Corporation and the Iowa Science Foundation. This material is based upon work supported by the National Science Foundation under Grand No. DMR-8603218.
Appendix Derivation of M’ /M’(O)
and M’ /M’(O)
for thallium borutes
It is assumed that thallium entering the glass forms four-coordinated borons in the same manner as in the other alkalis. fi can be written for this process: fr=R,
Rs0.25.
(Al)
H.C.
Lim. S. Feller
/ Densip
of low metal content
borate
glasses
43
However, eq. (2a) gives the experimental result that fi = 3/2 R for thallium borates [16]. The process of forming three-coordinated oxygens is thought to be responsible for the extra contribution to fi of R/2. It is assumed that each four-coordinated boron unit formed with three-coordinated oxygens contains one such oxygen. Two thirds of the four-coordinated borons will contain thallium and four bridging oxygens while one third will contain three two-coordinated oxygens and one three-coordinated oxygen. Thus M’/M’(O) can be written:
Mz,Ml(0)=
Mass(B+2(2O+Mt)/3
+ (0?)/3)
(5b)
Mass( B + 1 SO) Ml/M’(O) will be a function of R and is found by considering of oxygen number. The glass reaction is: R (TQ.,
PO, .5 +
$R [bcoordinated boron unit] + (1 - :R) [3-coordinated boron unit],
conservation
(AZ)
from which oxygen conservation is
0.5R-tl.5 = $R[($)2+(+)2]
+ [l-
+R]X,
where X is the number of oxygens in a three-coordinated readily found from eq. (A3) and is given by
(A31 boron unit. X can be
(A4) Hence
Ml,Ml(0)
= Ma=@+ ((3 -29R/W(2-3R))O) Mass( B + 1.50)
References [l] [2] [3] [4]
[5] [6] [7] [8] [9] [lo]
M. Sbibata. C. Sanchez, H. Patel. S. Feller, J. Stark, G. Sumcad and J. Kasper. J. Non-Cryst. Solids 85 (1986) 29. A. Karki, S. Feller, H.P. Lim, J. Stark, C. Sanchez and M. Shibata, J. Non-Cryst. Solids 92 (1987) 11. H.P. Lim, A. Karki. S. Feller, J. Kasper and G. Sumcad. J. Non-Cryst. Solids 91 (1987) 324. O.V. Mazurin. M.V. Streltsina and T.P. Shvaiko-Shvaikovskaya, Handbook of Glass Data. Part B: Single Component and Binary Non-Silicate Oxide Glasses (Elsevier. Amsterdam, 1985). B.1. Marki, Fiz. Tverd. Tela, Leningrad 3 (2) (1961) 450. S. Takeuchi, T. Yamate and M. Kunugi, J. Sot. Mater. Sci. Jpn. 14 (1965) 225. M. Kunu& A. Konishi. S. Takeuchi and T. Yamate, J. Sot. Mater. Sci. Jpn. 21 (1972) 978. B.I. Markin, Zh. Obshch. Khim. 11 (4) (1941) 285. E.N. Boulos and N.J. Kreidl, J. Am. Ceram. Sot. 54 (8) (1971) 368. K. Kamiya. S. Sakka, K. Matusita and Y. Yoshinaga. J. Non-Cryst. Solids, 38 & 39 (1980) 147.
44
H. C. Lim,
S. Feller
/ Density
of low metal
contetu
borate
glasses
[ll] B.I. Markin, Zh. Tekhn. Fiz. 22 (6) (1952) 941. [12] S. Sakka, K. Matusita and K. Karniya, Phys. Chem. Glasses 20 (2) (1979) 25. [13] K. Kamiya, S. Sakka, T. Mizuno and K. Matusita, Phys. Chem. Glasses 22 (1) (1981) 1. [14] P.J. Bray and J.G. O’Keefe, Phys. Chem. Glasses 4 (1963) 37. [15] K.S. Kim and P.J. Bray, J. Non-Metals 2 (1974) 95. [16] J.F. Baugher and P.J. Bray, Phys. Chem. Glasses 10 (1969) 77. [17] R. Weast, M. Astle, and W. Beyer, eds., Handbook of Chemistry and Physics, vol. 66 (CRC, Boca Raton, Florida, 1985-1986) B-2. [18] Value found from ref. (1) slightly adjusted by additional data. [19] J. Piguet and J. Shelby, J. Am. Ceram. Sot. 68 (8) (1985) 450. [ZO] R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 3 (1970) 251. [21] W.H. Zachariasen, Acta Cryst. 16 (1963) 385. [22] R. Weast. M. Astle, W. Beyer, eds., ibid., ref. 17, p. F-169.
36
of Non-Crystalline
Solids 94 (1987) 36-44 North-Holland, Amsterdam
THE DENSITY OF LOW METAL CONTENT RUBIDIUM, SILVER, AND THALLIUM BORATE GLASSES RELATED TO ATOMIC ARRANGEMENTS
CESIUM,
Hun C. LIM and Steve FELLER Pltq,.sics Deparmenr,
Coe College.
Received
1987
23 January
Cedar
Rap&
Iowa
52402.
USA
Reported densities from rubidium, cesium, silver. and thallium borate glasses were used quantitative model involving atomic arrangements. The analysis yielded the volumes of the basic structural units present in all the glasses: three- and four-coordinated borons. The volume a four-coordinated boron enlarges as the radius of the metal ion increases while the volumes of three-coordinated borons for different metals are constant. Packing fractions which compare volumes calculated from ionic radii to the values found from the density model were obtained.
in a two of the the
1. Introduction Recently, a model was developed to connect density and atomic arrrangements in lithium, sodium, and potassium borate glasses [l-3]. This model was able to yield the relative volumes of the structural groupings present over very wide-ranging glass compositions. It was also possible to calculate a packing fraction [3] of each of the units, where the packing fraction was defined as the ratio of the volumes calculated from ionic radii to the volumes determined from the densities. Several unifying trends were seen including the important one that each of the glasses has the same structure at a given composition. The variation in volumes was attributed to the differing sizes of the metal ions present. It is the purpose of this paper to apply this model to existing data from other alkali and pseudoalkali borate glasses. The other systems being considered are rubidium, cesium, silver, and thallium borate glasses. In these systems, data are available to about 30 mol.% metal oxide.
2. Density data A thorough compilation of physical properties of non-silicate glasses was published by Mazurin, Streltsina and Shvaiko-Shvaikovskaya [4] in 1985. A summary of the reported density data from rubidium [5-71, cesium [6,7], silver [8-lo] and thallium [ll-131 borates can be found in fig. 1. In this figure, R is 0022-3093/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
H.C.
Lim.
S. Feller
/ Density
of low metal content
borate
glasses
37
DENSITY (g/cm’)
DENSITY (g/cm’)
SO.-.
. n n
4.0-
. .
00 2.0 I
1
I
012
0
R
0.4
Fig. 1. Density of rubidium, cesium. silver, and thallium borate glasse; as a function of R (R is defined as the molar ratio of metal to rubidium borates; 0, cesium boron). 0, borates; 0, silver borates; W, thallium borates.
0
I
0.2
R
0.4
Fig. 2. A comparison of the densities of lithium, sodium and potassium borate glasses with rubidium. cesium, silver. and thallium borates (R is defined as the molar ratio of metal to boron).
the molar ratio of metal to boron. When compared with data from lithium [l], sodium [2], and potassium [3] borates of like compositions, similar trends are noted (see fig. 2). In each system the density rises monotonically for R < 0.4.
3.
A
model
for the densities
Nuclear magnetic resonance (NMR) studies [14,15] have found that rubidium, cesium and silver borate glasses are all similar in structure to each other and to lithium, sodium and potassium borates up to R = 0.4. This similarity is best expressed by the fractions of borons three- and four-coordinated with bridging oxygens, f, and fi. These fractions follow the simple relations:
f,=l-R, f,=R,
R ~0.4, RI 0.4.
(la) (lb)
The equations in (1) are equivalent to charge conservation if the only charged borate unit present is a four-coordinated boron. Thallium is curious in that the fraction of four-coordinated borons is given by [16]: fi = +R, implying
RI 0.25,
(24
that
f,=l-+R,
R ~0.25.
(2b)
H. C. Lim.
38
S. Feller
/ Densi!,,
To explain these results, Baugher three-coordinated oxygens. Above in that thallium becomes covalent done. The model proposed earlier for
f,M’ +f*lw
lJ= f,V’ + f2V2
01 lobv meral content
borate
glasses
and Bray [16] postulated
the formation
of
R = 0.25, the situation is more complex [16] and no analysis of density in this region was density can be written [2]
R I 0.4 for Rb, Cs and Ag; R I 0.25 for Tl.
(3)
In eq. (3) M’ and M’ are the masses of the three- and four-coordinated boron units, with V’ and V’ the respective volumes. Equation (3) can be rewritten using eqs. (1) and (2) to show the explicit dependence of density on R for rubidium, cesium, and silver borates as:
P(R) =P(O)
v'/v'(O)
l+R(M'/M'-1) + R(V2/V'(0)-
V'/V'(O))
' R'
o'4'
(da)
where p(O) is the density of boron oxide glass and V’(O) the volume of the three-coordinated boron unit in that glass. Thallium is somewhat different in that M’ is a function of R since three-coordinated oxygens are thought to be formed [16]. This lowers the value of M’. For thallium borates, the density is written: p(R)
= p(o)
M'/M'(O)
+ (3/2)R(M2/M’@)
v'/V'(O)+(~/~)R(V~/V'(O)-
- M'/M'(O))
Y'/v'(O))
' R 5o'25'
(4b) where M’(O) is the mass of a three-coordinated boron unit in boron oxide glass. In eqs. (4a) and (4b) the masses of the atoms are known as are the density and associated volume at R = 0. V’ and I” remain as unkowns to be determined from the density data. M2/M’ for rubidium, cesium, and silver is:
M2/M’ =
Mass(B + 20 + Mt) Mass(B + 1.50) ’
and for thallum (see Appendix): M2,Ml(0)
= Mass(B + 2(20 + Mt)/3 + (C)$)/3) Mass(B + 1.50)
M,,M,(0)
= Ma@
+ ((3 - 29W6)/(2 Mass( B + 1.50)
(5b)
- W)O)
where B, 0, and Mt represent boron, oxygen, and the metal ion under study. The relative masses, M2/M’ and M’/M’(O), expressed by metal ion, are [17] rubidium = 3.685, cesium = 5.048, silver = 4.329, thallium = 4.659. The density of boron oxide glass was determined by averaging measurements on unannealed samples as reported in ref. [4]. This was done since the reported
H.C. Table 1 Relative volumes V’ fixed.
V’
Lim.
of borate
S. Feller
units
/ Density
o/low
metal
contem
borate
glasses
39
for Rb, Cs, Ag and Tl systems
varied
V’ and V2 varied
IOIl
v’/
Rb cs ‘% Tl
1.84 2.14 1.12 1.11
V’(0)
V’/
V’(0)
v’/
0.96 0.97 0.97 0.94
V’(0)
1.96 2.23 1.20 1.28
data from rubidium, cesium, silver and thallium are on quenched samples. The average density found was 1.823 g/cm3. For each of the metal systems a best value for the relative volumes was found by performing least-squares analyses of the data. Two methods were used. The first method followed the method reported in ref. [3]; V’ was fixed at the value found for R = 0 and only V2 was varied. The second method allowed I/’ to vary also and a two-parameter least-squares analysis was carried out. In both cases, values are reported relative to V’(0) and the results are given in table 1. The two methods give very similar results. Figure 3 shows a visual comparison of the two-parameter model with the data. The standard deviations of the density data with respect to the model are (in g/cm3) 0.03, 0.03, 0.08, and 0.08 for rubidium, cesium, silver and thallium respectively. The values for the volume ratios are contrasted with the values found for lithium [18], sodium [2], and potassium [3], see table 2.
DENSITY (y/cm3)
6
0.2
R
0.4
Fig. 3. The density model presented in this paper superimposed rubidium, cesium, silver, and thallium borate glasses (R is defined as boron). Standard deviations of the density data (compared with model) as 1. 0, rubidium borates; l , cesium borates; q I, silver borates;
on the density data from the molar ratio of metal to are shown for each system n , thallium borates.
40
H.C.
Table 2 Relative volumes
of borate
V’ fixed,
Lim,
S. Feller
units
/ Density
of low metal
for Li, Na and K systems
V2 varied
cement
borate
glasses
[2,3.18]
V’ and V2 varied
Ion
V’/
Li Na K
0.82 1.05 1.52
V’(0)
V’/
V’(0)
V2/V’(0)
0.98 0.96 0.95
0.88 1.17 1.64
A comparison among the metal systems suggests that V2/V*(0) is related to the size of the metal ion. Along these lines it was noted by Piguet and Shelby [19] that the molar volumes of silver and sodium borates are nearly identical over the composition range being considered due to the similarity of the ionic radii of sodium and silver. To clarify this comparison the concept of the packing fraction will be used. As noted before, the packing fraction is defined as the ratio of the volume of a unit calculated from ionic radii to the same volume found from the density measurements. This procedure was carried out earlier for the lithium, sodium, and potassium glasses [3]. The calculation requires values for the ionic radii and these are: radius of bridging oxygen bonded to three-coordinated boron [3,20] = 1.19 A, radius of bridging oxygen bonded to four-coordinated boron [3,21] = 1.28 A, radius of rubidium ion [22] = 1.47 A, radius of cesium ion [22] = 1.67 A, radius of silver ion [19,22] = 1.20 A, radius of thallium ion [22] = 1.47 A. In the calculation the boron was neglected since its radius (= 0.19 A [3,20]) is very much smaller than the oxygen radii as to make its volume negligible. Further details of the calculation can be found in ref. [3]. In the calculation values for Vi/V’(O) and V2/V’(0) were taken from the least-squares analysis in which both parameters were varied. The resulting packing fractions, shown
Table 3 Packing fractions
of borate
units
Borate system
Packing
lithium sodium silver potassium rubidium thallium cesium
0.67 0.57 0.65 0.52 0.49 0.63 0.52
fraction
Four-coordinated
a) Since the number of oxygens fractions was taken.
boron
Three-coordinated
boron
0.34 0.34 0.34 0.35 0.34 0.35 a’ 0.34 making
up this unit
changes
with
R, an average
of the packing
H.C.
Lim,
S. Feller
/ Density
of low metal content
borate
glasses
41
in order of increasing size of metal ion, are (along with results found earlier for lithium, sodium and potassium [3,18]) shown in table 3. The packing fractions for the alkali and silver borates reveal that there is a general decrease followed by a leveling off in efficiency in the filling of space for the four-coordinated borons as the metal ion gets larger. The packing fraction for thallium is considerably larger than would have been expected on the basis of ionic radii. However, as noted earlier, thallium is special in that the fraction of four-coordinated borons follows a steeper slope as R is varied than do the other systems. This process is thought to result in the formation of three-coordinated oxygens [16] and thus the unit is considerably different. The packing fractions for the three-coordinated borons are remarkably uniform and close to the value for pure boron oxide glass (0.33). This is believed to result from the absence of metal ions associated with this unit. The consistency of the packing fractions lends considerable support to the notion that the densities can be directly related in a quantitative way to the atomic arrangements in the glass.
4. Further discussion of the model More information about the ratio of the volumes can be found by plotting the trend of V’/V’(O) as a function of R leaving I/’ fixed at V’(O). This is shown in fig. 4. For each of the systems but thallium the volume ratio
Fig. 4. The ratio of the volumes of four-coordinated function of R (R is molar ratio of metal to boron). was fixed at the value found
borons to three-coordinated borons as a The volume of the three-coordinated borons for boron oxide glass.
H. C. Lbn.
42
S. Feller
/ Density
o/low
metal
contenr
borare
glasses
increases to a limiting value as ,R increases. As can be seen in fig. 4 the amount of change in V2/V’(0) is directly related to the size of the ion with the greater change coming for the larger ion (except thallium). This indicates that for low metal contents the ions are either filling available voids without further enlarging the boron-oxygen network or enlarging it to a smaller degree than at larger R. Such a trend was previously noted for lithium, sodium, and potassium [3] borates. Apparently, the density model which contains structural information from NMR, is sensitive enough to discern continuous changes in volumes as the composition is varied. However, it is noted that this analysis has certain assumptions built in such as that I” is fixed at the value found from boron oxide glass and that the ionic radii are indeed constants.
5. Conclusions Densities of rubidium, cesium, silver, and thallium borate glasses were used in a quantitative model involving atomic arrangements. The model, using reported data up to R = 0.4, indicates that rubidium, cesium, and silver act very much like the other alkalis with thallium showing some differences. The model uses NMR and density data in a consistent manner to get the volumes of the two basic structural units present: three- and four-coordinated borons with bridging oxygens. Packing fractions for the four-coordinated borons generally decline as the size of the metal ion increases. The three-coordinated borons have a uniform packing fraction in all systems. Further work is in progress to obtain density measurements at higher values of R in these systems. We express our gratitude to A. Karki, H.P. Lim, B.C. Liang, K.H. Chong, B. Teoh. and S.H. Choo of the glass research team at Coe College for reading the manuscript. M. Affatigato is thanked for help with the proofreading. Coe College and the Richter Foundation are acknowledged for financial assistance. Supported in part by grants of the Research Corporation and the Iowa Science Foundation. This material is based upon work supported by the National Science Foundation under Grand No. DMR-8603218.
Appendix Derivation of M’ /M’(O)
and M’ /M’(O)
for thallium borutes
It is assumed that thallium entering the glass forms four-coordinated borons in the same manner as in the other alkalis. fi can be written for this process: fr=R,
Rs0.25.
(Al)
H.C.
Lim. S. Feller
/ Densip
of low metal content
borate
glasses
43
However, eq. (2a) gives the experimental result that fi = 3/2 R for thallium borates [16]. The process of forming three-coordinated oxygens is thought to be responsible for the extra contribution to fi of R/2. It is assumed that each four-coordinated boron unit formed with three-coordinated oxygens contains one such oxygen. Two thirds of the four-coordinated borons will contain thallium and four bridging oxygens while one third will contain three two-coordinated oxygens and one three-coordinated oxygen. Thus M’/M’(O) can be written:
Mz,Ml(0)=
Mass(B+2(2O+Mt)/3
+ (0?)/3)
(5b)
Mass( B + 1 SO) Ml/M’(O) will be a function of R and is found by considering of oxygen number. The glass reaction is: R (TQ.,
PO, .5 +
$R [bcoordinated boron unit] + (1 - :R) [3-coordinated boron unit],
conservation
(AZ)
from which oxygen conservation is
0.5R-tl.5 = $R[($)2+(+)2]
+ [l-
+R]X,
where X is the number of oxygens in a three-coordinated readily found from eq. (A3) and is given by
(A31 boron unit. X can be
(A4) Hence
Ml,Ml(0)
= Ma=@+ ((3 -29R/W(2-3R))O) Mass( B + 1.50)
References [l] [2] [3] [4]
[5] [6] [7] [8] [9] [lo]
M. Sbibata. C. Sanchez, H. Patel. S. Feller, J. Stark, G. Sumcad and J. Kasper. J. Non-Cryst. Solids 85 (1986) 29. A. Karki, S. Feller, H.P. Lim, J. Stark, C. Sanchez and M. Shibata, J. Non-Cryst. Solids 92 (1987) 11. H.P. Lim, A. Karki. S. Feller, J. Kasper and G. Sumcad. J. Non-Cryst. Solids 91 (1987) 324. O.V. Mazurin. M.V. Streltsina and T.P. Shvaiko-Shvaikovskaya, Handbook of Glass Data. Part B: Single Component and Binary Non-Silicate Oxide Glasses (Elsevier. Amsterdam, 1985). B.1. Marki, Fiz. Tverd. Tela, Leningrad 3 (2) (1961) 450. S. Takeuchi, T. Yamate and M. Kunugi, J. Sot. Mater. Sci. Jpn. 14 (1965) 225. M. Kunu& A. Konishi. S. Takeuchi and T. Yamate, J. Sot. Mater. Sci. Jpn. 21 (1972) 978. B.I. Markin, Zh. Obshch. Khim. 11 (4) (1941) 285. E.N. Boulos and N.J. Kreidl, J. Am. Ceram. Sot. 54 (8) (1971) 368. K. Kamiya. S. Sakka, K. Matusita and Y. Yoshinaga. J. Non-Cryst. Solids, 38 & 39 (1980) 147.
44
H. C. Lim,
S. Feller
/ Density
of low metal
contetu
borate
glasses
[ll] B.I. Markin, Zh. Tekhn. Fiz. 22 (6) (1952) 941. [12] S. Sakka, K. Matusita and K. Karniya, Phys. Chem. Glasses 20 (2) (1979) 25. [13] K. Kamiya, S. Sakka, T. Mizuno and K. Matusita, Phys. Chem. Glasses 22 (1) (1981) 1. [14] P.J. Bray and J.G. O’Keefe, Phys. Chem. Glasses 4 (1963) 37. [15] K.S. Kim and P.J. Bray, J. Non-Metals 2 (1974) 95. [16] J.F. Baugher and P.J. Bray, Phys. Chem. Glasses 10 (1969) 77. [17] R. Weast, M. Astle, and W. Beyer, eds., Handbook of Chemistry and Physics, vol. 66 (CRC, Boca Raton, Florida, 1985-1986) B-2. [18] Value found from ref. (1) slightly adjusted by additional data. [19] J. Piguet and J. Shelby, J. Am. Ceram. Sot. 68 (8) (1985) 450. [ZO] R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 3 (1970) 251. [21] W.H. Zachariasen, Acta Cryst. 16 (1963) 385. [22] R. Weast. M. Astle, W. Beyer, eds., ibid., ref. 17, p. F-169.