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The density of potassium borate glasses related to atomic arrangements
Journal
324
of Non-Crystalline
THE DENSITY OF POTASSIUM BORATE RELATED TO ATOMIC ARRANGEMENTS
Hun P. LIM, Ajoy KARKI, and Gustav0 SUMCAD Physics
Deparrmenr,
Coe College,
Steve FELLER, Cedar
Received 28 September 1986 Revised manuscript received 22 December
Rapidr,
Solids 91 (1987) 324-332 North-Holland. Amsterdam
GLASSES
James E. KASPER
Iowa 52402,
*
USA
1986
Densities of potassium borate glasses have been measured over an extremely wide range of alkali content: from boron oxide through the orthoborate composition. As with the lithium and sodium borates a semiempirical model has been applied to the densities in order to determine the volumes of the structural units present. These calculated values, in conjunction with volumes found in the lithium and sodium cases. were used in a general discussion of the packing fractions of the units. The fractions indicate that the size increase of the structural groupings as one goes from lithium to sodium to potassium is primarily due to the alkali being used and that the same volumes are present for each of the boron oxygen configurations.
1. Introduction Recent papers from this laboratory have dealt with the relationship between density and atomic arrangements in lithium and sodium borate glasses [1,2]. It was shown that the density of the glass can directly be expressed as a composite of the microscopic atomic arrangements in a simple manner. The analyses reported in those articles yielded the various volumes of the structural groupings over extremely diverse glass compositions. This paper deals with the densities of potassium borate glasses and the relevant atomic arrangements present. A wide range of compositions were prepared spanning from R = 0 through R = 2.90, where R is the molar ratio of alkali to boron. The analysis of the density data for potassium borates yielded volumes of the structural units present. Furthermore, a comparison of the density data of the three glass systems has resulted in generalizations of the behavior of the structural arrangements. * Present
address:
Physics
Department,
University
0022-3093/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
of Iowa,
B.V.
Iowa
City,
IA, USA
H. P. Lim et al. / Density
2. Experimental
of pom.&m
borare glasses
325
method
The starting materials for the binary potassium borate glasses were potassium carbonate and boric acid. A total of 4 g using appropriate amounts of these chemicals along with less than 0.5 wt% cupric oxide were mixed in a platinum crucible and heated for 25 min at 1000” C. The function of the cupric oxide was to make the glasses more visible during the density measurements. Each melt was weighed after 20 min to ensure sample composition. It is noted that above R = 1 glass weights were more than expected due to carbon dioxide retention. The resulting melts were quickly cooled by either pouring through a roller quencher or by being squashed between two brass plates. Twenty-two glasses were prepared over the composition range R = 0.05 through R = 2.90. There is a compositional range near R = 1 where the cooling rates associated with the methods described here are too small to produce glassy samples. This is very much like the case for sodium borates [2,3,4]. However, Martin and Angel1 [5] have reported that small additions of aluminum oxide added to sodium borate melts near the metaborate composition allow glasses to be formed with relative ease. Similarly, potassium borate glasses can be made near the metaborate composition by this technique. Between 2 and 12 wt% aluminum oxide was added to the melt for R = 0.85, 0.95, and 1.10. Several samples with varying amounts of alumina were made at each value of R and the resulting densities were back-extrapolated to the binary borate glass. When compared with sodium borate glasses of the same value of R, potassium borates required more aluminum oxide in order to form glasses. The densities of glasses with R less than 0.90 were measured using a sink-float method which used acetone and diiodomethane as the light and dense fluids respectively [l]. For larger values of R a modification of the sink-float method was found to be necessary as the glasses began to react with the density fluids. A similar problem occurred for sodium borate glasses (although at a higher value of R). The modification involved bracketing the fluid densities above and below that of the glass [2]. This allowed very rapid measurements of the density to be made. Conservative estimates of the error are f0.03 g/cm3 for the low alkali glasses and t-O.04 g/cm3 for the high potassium content glasses. In order to measure carbon dioxide retention in potassium borates, a set of six samples was prepared between R = 1 and R = 5 by melting the appropriate mixtures of potassium carbonate and boric acid as described above. The samples were then sent to the Coors Spectra-Chemical Lab at Golden, Colorado for carbon analysis using the Leco Combustion technique. 3. Results The density data of the binary potassium borate glasses are depicted in fig. 1. The error limits shown are the conservative values given before. It is noted
326
H. P. Lim et al. / Density
2.40 I
of potassium
o
go000
borate glasses
0 0 0 0 0
P
1.803 0 Fig. 1. The density
of potassium
2
1 borate
glasses as a function alkali to boron,
R of R, where
3 R is the molar
ratio
of
that the measurements above R = 0.6 are on glasses not previously reported in the literature [6,7]. Figure 2 presents the density measurements derived from glasses to which aluminum oxide was added in small quantities (between 2 and 12 wtW) and from which samples the density was back-extrapolated to the binary glasses. Carbon dioxide retention is graphed in fig. 3. In this figure the reported error in carbon dioxide retention was derived from the raw carbon analyses done at Coors and corresponds to the diameters of the data points as plotted in the figure.
% CO, retained lOO-/
755 50.-
04
/' 0 lo ii P i
25-i I Fig. 2. The density of potassium borate glasses from R = 0.7 through R = 1.6 (R defined as the molar ratio of potassium to boron). The model presented in the paper is shown as‘a smooth curve. 0, binary potassium borate glasses, 0, extrapolated data from potassium borate glasses prepared with small amounts of ahuninum oxide.
Fig. 3. Percent carbon potassium borate samples (R is the molar fraction The solid curve is a the equation %CO, exp( - b(R - R,))] with 0.54.
dioxide retained in as a function of R of alkali to boron). least-squares fit to retained = lOO[l b = 0.59 and R, =
H. P. Lim et al. / Density
4. Discussion
o/potassium
borate glasses
327
of the results
4. I. Carbon dioxide retention Along with the carbon dioxide retention data given in fig. 3 is plotted a model taken from ref. [l]. This model expresses “Percent CO, retention”, P, in the sample, compared with the batch, by the following expression: P=lOO[l-exp(-b(R-R,))].
(1) In this equation R, represents the value at which CO, retention begins and 1OOb is the initial slope of the retention curve. For potassium the best values obtained from a least-squares analysis are: R, = 0.54, b = 0.59. These values are to be compared with results borates in which the parameters are [l]: Lithium:
R, = 2.30, b = 0.28.
Sodium:
R, = 2.05, b = 0.41.
from the lithium
and sodium
It is clear that the propensity to retain carbon dioxide is greater in the order: potassium, sodium, lithium. These results suggest that carbon dioxide retention is significant even at relatively low alkali content for the potassium case. 4.2. Density data The density data for the potassium system closely follow the trend for sodium borate glasses while differing from the trend of the lithium borates [1,2]. As can be seen in fig. 1, below R = 0.50 the density rises from 1.81 g/cm3 to near 2.35 g/cm3. Above R = 0.50 the glass density remains relatively flat until R = 2.90. Sodium borate glasses [2] level off above R = 0.50 with a density near 2.40 g/cm3. Densities of lithium borates [l] fall above R = 0.80 from near 2.30 g/cm3 to about 2.10 g/cm3 for R = 2.70. Between R = 0 and 0.1 the density rises very rapidly and is above that of sodium or lithium borates. Above R = 0.1 the density of potassium borates is greater than the lithium borates while below that of the sodium borates. Other workers have reported densities for potassium borate glasses with R < 0.6. A useful summary of such work is found in a paper by Shaw and Ublmann [8]. The data reported in their paper are in excellent agreement with the low alkali densities given here. 4.3. A model for the density data A model for the density of alkali borate glasses based on atomic arrangements has successfully been used with lithium and sodium borates [1,2]. This representation is based on the ideas of Krogh-Moe in which the glass is viewed
H. P. Lim et al. / Dens+
328
of poiawum
borate glawes
as a random assembly of structural groupings. These groupings are chosen from corresponding crystalline configurations. Nuclear magnetic resonance studies of potassium borate glasses of low alkali content are in close agreement with sodium and lithium borates [9]. Therefore, it will be assumed that the same structural groupings found in lithium and sodium borate glasses are also present in potassium borate glasses. These units are (fractional notations are given in parentheses): 1) boron with three bridging oxygens (f,); 2) boron with four bridging oxygens and one alkali ion (fz ); 3) boron with two bridging and one non-bridging oxygen and one alkali ion (f3); 4) boron with one bridging and two non-bridging oxygens and two alkali ions (f4); 5) boron with three non-bridging oxygens and three alkali ions (f,). Table 1 lists the fractions used in the modelling process as taken from ref. [l]. Another assumption used was that the retention of carbon dioxide which starts at R = R, = 0.54 does not affect the density. Evidence for this assumption comes from the sodium system in which it was found that the density was insensitive to whether the source of the alkali was sodium carbonate or sodium oxide [2]. This was attributed to the close similarity between the density of sodium carbonate and the sodium borate glasses. It is also true that the density of potassium borate glasses (p = 2.35 g/cm3) is extremely close to the density of potassium carbonate (p = 2.428 g/cm’) [lo] in the region of compositions where carbon dioxide retention takes place. The above assumption was found to be necessary because a source of potassium oxide (K,O) could not be found to fabricate glasses free of carbon dioxide. The following equation is used to describe the glass density [2]: f,M’ +f&P+ f,W+ . . . +f,M’+ . .. P(R) = (2) f,V’ + f,V+ f,P+ . . . +f,V’+ .. . ’ Table 1 The fractions
fl f, f f: f5
of boron
Units
0.4 I R I 0.7
0.7 5 R ~1.0
1-R R 0 0 0
1-R R/6+1/3 5R/6-l/3 0 0
1-R - R/4+5/8 5R/4-5/8 0 0
1.0 I R < 2.14
fl f2 f3 f, fs
[l]
0.0 I R < 0.4
0
-R/4+5/8 - 0.55R 0.6(R 0.2(R
+ 1.175 -1) -1)
2.14 I R I 2.5
2.5 I R I 3.0
0
0
-R/4+5/8 0 -R/2+7/4 3R/4-11/8
0 0 3-R R-2
H. P. Lim et al. / Densily Table 2 Volumes of structural sodium [2] borates
groupings
Fractional symbol
found
of potassium
in potassium
borate
borate glasses
329
glasses compared
with lithium
[l] and
Relative volume of unit compared with boron with 3 bridging oxygens K
Na
Li
1 1.52 2.15 2.85 3.85
1 1.05 1.78 2.04 2.77
1 0.84 1.37 1.68 1.95
where M’ is the mass of the ith structural grouping and V’ its volume. In eq. (2) the masses are known as well as the fractions leaving the volumes to be determined. A least-squares analysis of the density data was performed in order to find the ratios of the volumes with respect to I”. Reference (1) gives the explicit details on how this calculation was carried out. The results of the calculation are presented in table 2 along with values from lithium [l] and sodium [2] glasses. The densities calculated with the volumes given in table 2 are overlaid onto the density data as shown in fig. 4. 4.4. Comments on the density model A consistent trend can be seen in table 2 for the volumes of the structural groupings. As sodium is substituted for lithium and as potassium replaces sodium, the volumes increase. This is apparently so because of the larger sizes of the alkali ions as one goes down the first column of the periodic table.
DENSITY hd) 2.40 et----=-t-
Fig. 4. The density of potassium borate glasses as a function of R (R = mol% alkali/mol% compared with the semiempirical model (solid curve) presented in this paper.
boron)
330
H. P. Lim et al. / Density
of potassium
borare glawes
More can be said about relationships among the various volumes. Mozzi and Warren [ll], using X-ray diffraction techniques on glassy B,O,, report the boron-oxygen separation within a BO,,, triangle as 1.37 A while the oxygen-oxygen separation is 2.37 A. Taking one-half of the oxygen-oxygen separation as the radius of the oxygen ion and using the above X-ray data leads to a boron radius. The results accurate to two decimal places are: radius of 0 ion = r,, = 1.19 A, radius of B ion = rb = 0.19 A. Since the volume of the boron ion is negligible compared with oxygen, the volume of the BO,., unit is very nearly 1:5((4/3)7rri). Using the above value for the oxygen radius yields 10.6 cubic A. However, utilizing the density of boron oxide glass (1.81 g/cm3) along with the known atomic masses of oxygen and boron results in a volume for this unit, including unused space, of 32.0 cubic A. The “packing fraction” will be defined as the ratio of the volume calculated from ionic radii to the volume found from the density measurements. Thus the packing fraction of the BO,., group is 10.6/32.0 or 0.33. Similar calculations for the other structural groupings (units designated f,, f,, f,, and f,) have been done. X-ray diffraction data from lithium metaborate yields almost the same value for r, as above [12] for both bridging and non-bridging oxygens. Thus it was assumed that the oxygen radius in a trigonal unit remains at 1.19 A for both bridging and non-bridging oxygens. The radius of an oxygen ion in four-coordination with boron can also be found from X-ray measurements and is 1.28 A [13]. The ionic radii of lithium, sodium, and potassium were also needed as these ions are associated with the other structural arrangements. The radii are [14]: radius of Li ion = TLi = 0.68 A, radius of Na ion = rNa = 0.97 A, radius of K ion = rK = 1.33 A. Table 3 summarizes the results of the calculations. The constancy of the packing fractions for all of the structural groupings independent of the ion present is an important result of this study. This is consistent with the idea that changes in the size of the structural units are due to the differing sizes of the alkali ion. Furthermore, the packing fractions of planar trigonal groups are nearly the same while the tetrahedral structures of the four-coordinated borons show noticeably higher packing fractions. In addition, from R = 0 to about 0.10 the volume of the four-coordinated boron unit (V’) is similar to the sodium and lithium cases of like R. To see this, fig. 5 contrasts densities calculated with V2 = 1.52 V’ (the least squares value found from the potassium data of R = 0 up to R = 0.4) with V2 = 0.84 V’ as found earlier for lithium [l] and sodium [2] borates of extremely low R. For R I 0.1 the agreement of the potassium density data with the model
H.P. Lim et al. / Density
o/potassium
borate glusses
331
Table 3 Volumes of structural units calculated from ionic radii compared with volumes determined from density measurements (in terms of volume of boron with three-bridging oxygens). (a, Volume of unit determined from density measurements; b, Volume of unit determined from ionic radii calculations; c. Packing fraction defined as b/a.) Fraction
f2 f3 f4
fs
K
Na
Li
a
b
C
a
b
C
a
b
C
1.52 2.15 2.85 3.85
0.85 0.74 1.15 1.56
0.56 0.34 0.40 0.41
1.05 1.78 2.04 2.77
0.67 0.56 0.79 1.02
0.64 0.31 0.39 0.37
0.84 1.37 1.68 1.95
0.59 0.48 0.63 0.78
0.70 0.35 0.38 0.40
calculated using the smaller value for V2 is evidence that there is a fundamental volume in these alkalis for the four-coordinated borons of extremely low alkali content. Apparently, the ions simply fill already available voids in the structure. As more potassium is added above R = 0.1 the volume of the unit is enlarged in order to accommodate the ion. The data derived from aluminum oxide doped glasses and extrapolated back to the binary potassium borates are plotted in fig. 2. The predicted curve which results from eq. (2) when the derived values for I/‘, V3, V4, and V5 are used is also shown in this figure. This curve shows a local minimum at the metaborate composition. However, the data clearly do not show a well of such a size. While the data seem to indicate a shallow well, such a trend is not outside the experimental error.
DENSITY bc.J)
2.40 4
,,,1I"
v2=oJ34v ,’
2.20 2.00 i
1.80
Fig. 5. A close-up of alkali to boron. with greater slope with V2 = 1.52V’. oxygens
-a
,*," $,,"a...' I' 0 v~=1.!52v’ /-f 1/( 0.0
I
0!2
R
d.4
of the potassium borate data from R = 0 to R = 0.4 where R is the molar ratio Two curves derived from the model are superimposed on the data. The curve was found by using V2 = 0.84V’ while the smaller slope curve was calculated (V’ and V2 are the volumes of the following units: boron with three bridging and boron with four bridging oxygens and one alkali ion, respectively.)
H. P. Lim et al. / Densiry
332
o/potassium
borate glasses
5. Conclusions Densities of potassium borate glasses have been measured over an extremely wide range of compositions. The ideas of Krogh-Moe were used in a semiempirical model for the densities and the resulting analysis yielded the relative volumes of the structural groupings present in the glass. It was found that the arrangements are larger than in the corresponding lithium and sodium cases except for very low concentrations of potassium. These low R glasses exhibit the same fundamental volume for the four-coordinated boron as in the lithium and sodium borates. Packing fractions found by taking the ratio of the volume of the structural unit calculated from ionic radii to the volume of the structural group derived from the density data are remarkably similar for glasses prepared from any of the alkalis studied. A simple hypothesis is that the increase in volume for the structural groupings as one goes from lithium to sodium to potassium is due to the size of the alkali present. We express gratitude to Dr. Steve Martin of Iowa State University for useful discussions. Hun C. Lim and Cristina Sanchez are thanked for assistance with the manuscript. The Chemistry Research team at Coe College is credited with listening to many talks on glass and for asking questions. Coe College and the Richter Foundation are acknowledged for financial assistance. We acknowledge the financial help provided by grants of the Research Corporation and the Iowa Science Foundation. This material is based upon work supported by the National Science Foundation under Grant No. DMR-
8603218.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14]
M. Shibata, 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, to be published. R. Ota and N. Soga, Yogyo Kyokaishi 90 (9) (1982) 531. J.E. Kasper, S.A. Feller and G. Sumcad, J. Am. &ram. Sot. 67 (4) (1984) C71. S.W. Martin and C.A. Angell, J. Non-Cryst. Solids 66 (1984) 429. M. Imaoka, Yogyo Kyokaishi 69 (9) (1961) 282. H. Rawson, Inorganic Glass-Forming systems (Academic Press, New York, 1967) p. 98. R.R. Shaw and D.R. Uhlmann, J. Non-Cryst. Solids 1 (1969) 474. P.J. Bray and J.G. O’Keefe, Phys. Chem. Glasses 4 (1963) 37. R. Weast, ed., Handbook of Chemistry and Physics, Vol. 49 (CRC, Cleveland, 1968/69) B-230. R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 3 (1970) 251. W.H. Zachariasen, Acta Cryst. 17 (6) (1963) 749. W.H. Zachariasen, Acta Cryst. 16 (1963) 385. R. Weast, ed., Handbook of Chemistry and Physics, Vol. 49 (CRC, Cleveland, 1968/69) F-152.
324
of Non-Crystalline
THE DENSITY OF POTASSIUM BORATE RELATED TO ATOMIC ARRANGEMENTS
Hun P. LIM, Ajoy KARKI, and Gustav0 SUMCAD Physics
Deparrmenr,
Coe College,
Steve FELLER, Cedar
Received 28 September 1986 Revised manuscript received 22 December
Rapidr,
Solids 91 (1987) 324-332 North-Holland. Amsterdam
GLASSES
James E. KASPER
Iowa 52402,
*
USA
1986
Densities of potassium borate glasses have been measured over an extremely wide range of alkali content: from boron oxide through the orthoborate composition. As with the lithium and sodium borates a semiempirical model has been applied to the densities in order to determine the volumes of the structural units present. These calculated values, in conjunction with volumes found in the lithium and sodium cases. were used in a general discussion of the packing fractions of the units. The fractions indicate that the size increase of the structural groupings as one goes from lithium to sodium to potassium is primarily due to the alkali being used and that the same volumes are present for each of the boron oxygen configurations.
1. Introduction Recent papers from this laboratory have dealt with the relationship between density and atomic arrangements in lithium and sodium borate glasses [1,2]. It was shown that the density of the glass can directly be expressed as a composite of the microscopic atomic arrangements in a simple manner. The analyses reported in those articles yielded the various volumes of the structural groupings over extremely diverse glass compositions. This paper deals with the densities of potassium borate glasses and the relevant atomic arrangements present. A wide range of compositions were prepared spanning from R = 0 through R = 2.90, where R is the molar ratio of alkali to boron. The analysis of the density data for potassium borates yielded volumes of the structural units present. Furthermore, a comparison of the density data of the three glass systems has resulted in generalizations of the behavior of the structural arrangements. * Present
address:
Physics
Department,
University
0022-3093/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
of Iowa,
B.V.
Iowa
City,
IA, USA
H. P. Lim et al. / Density
2. Experimental
of pom.&m
borare glasses
325
method
The starting materials for the binary potassium borate glasses were potassium carbonate and boric acid. A total of 4 g using appropriate amounts of these chemicals along with less than 0.5 wt% cupric oxide were mixed in a platinum crucible and heated for 25 min at 1000” C. The function of the cupric oxide was to make the glasses more visible during the density measurements. Each melt was weighed after 20 min to ensure sample composition. It is noted that above R = 1 glass weights were more than expected due to carbon dioxide retention. The resulting melts were quickly cooled by either pouring through a roller quencher or by being squashed between two brass plates. Twenty-two glasses were prepared over the composition range R = 0.05 through R = 2.90. There is a compositional range near R = 1 where the cooling rates associated with the methods described here are too small to produce glassy samples. This is very much like the case for sodium borates [2,3,4]. However, Martin and Angel1 [5] have reported that small additions of aluminum oxide added to sodium borate melts near the metaborate composition allow glasses to be formed with relative ease. Similarly, potassium borate glasses can be made near the metaborate composition by this technique. Between 2 and 12 wt% aluminum oxide was added to the melt for R = 0.85, 0.95, and 1.10. Several samples with varying amounts of alumina were made at each value of R and the resulting densities were back-extrapolated to the binary borate glass. When compared with sodium borate glasses of the same value of R, potassium borates required more aluminum oxide in order to form glasses. The densities of glasses with R less than 0.90 were measured using a sink-float method which used acetone and diiodomethane as the light and dense fluids respectively [l]. For larger values of R a modification of the sink-float method was found to be necessary as the glasses began to react with the density fluids. A similar problem occurred for sodium borate glasses (although at a higher value of R). The modification involved bracketing the fluid densities above and below that of the glass [2]. This allowed very rapid measurements of the density to be made. Conservative estimates of the error are f0.03 g/cm3 for the low alkali glasses and t-O.04 g/cm3 for the high potassium content glasses. In order to measure carbon dioxide retention in potassium borates, a set of six samples was prepared between R = 1 and R = 5 by melting the appropriate mixtures of potassium carbonate and boric acid as described above. The samples were then sent to the Coors Spectra-Chemical Lab at Golden, Colorado for carbon analysis using the Leco Combustion technique. 3. Results The density data of the binary potassium borate glasses are depicted in fig. 1. The error limits shown are the conservative values given before. It is noted
326
H. P. Lim et al. / Density
2.40 I
of potassium
o
go000
borate glasses
0 0 0 0 0
P
1.803 0 Fig. 1. The density
of potassium
2
1 borate
glasses as a function alkali to boron,
R of R, where
3 R is the molar
ratio
of
that the measurements above R = 0.6 are on glasses not previously reported in the literature [6,7]. Figure 2 presents the density measurements derived from glasses to which aluminum oxide was added in small quantities (between 2 and 12 wtW) and from which samples the density was back-extrapolated to the binary glasses. Carbon dioxide retention is graphed in fig. 3. In this figure the reported error in carbon dioxide retention was derived from the raw carbon analyses done at Coors and corresponds to the diameters of the data points as plotted in the figure.
% CO, retained lOO-/
755 50.-
04
/' 0 lo ii P i
25-i I Fig. 2. The density of potassium borate glasses from R = 0.7 through R = 1.6 (R defined as the molar ratio of potassium to boron). The model presented in the paper is shown as‘a smooth curve. 0, binary potassium borate glasses, 0, extrapolated data from potassium borate glasses prepared with small amounts of ahuninum oxide.
Fig. 3. Percent carbon potassium borate samples (R is the molar fraction The solid curve is a the equation %CO, exp( - b(R - R,))] with 0.54.
dioxide retained in as a function of R of alkali to boron). least-squares fit to retained = lOO[l b = 0.59 and R, =
H. P. Lim et al. / Density
4. Discussion
o/potassium
borate glasses
327
of the results
4. I. Carbon dioxide retention Along with the carbon dioxide retention data given in fig. 3 is plotted a model taken from ref. [l]. This model expresses “Percent CO, retention”, P, in the sample, compared with the batch, by the following expression: P=lOO[l-exp(-b(R-R,))].
(1) In this equation R, represents the value at which CO, retention begins and 1OOb is the initial slope of the retention curve. For potassium the best values obtained from a least-squares analysis are: R, = 0.54, b = 0.59. These values are to be compared with results borates in which the parameters are [l]: Lithium:
R, = 2.30, b = 0.28.
Sodium:
R, = 2.05, b = 0.41.
from the lithium
and sodium
It is clear that the propensity to retain carbon dioxide is greater in the order: potassium, sodium, lithium. These results suggest that carbon dioxide retention is significant even at relatively low alkali content for the potassium case. 4.2. Density data The density data for the potassium system closely follow the trend for sodium borate glasses while differing from the trend of the lithium borates [1,2]. As can be seen in fig. 1, below R = 0.50 the density rises from 1.81 g/cm3 to near 2.35 g/cm3. Above R = 0.50 the glass density remains relatively flat until R = 2.90. Sodium borate glasses [2] level off above R = 0.50 with a density near 2.40 g/cm3. Densities of lithium borates [l] fall above R = 0.80 from near 2.30 g/cm3 to about 2.10 g/cm3 for R = 2.70. Between R = 0 and 0.1 the density rises very rapidly and is above that of sodium or lithium borates. Above R = 0.1 the density of potassium borates is greater than the lithium borates while below that of the sodium borates. Other workers have reported densities for potassium borate glasses with R < 0.6. A useful summary of such work is found in a paper by Shaw and Ublmann [8]. The data reported in their paper are in excellent agreement with the low alkali densities given here. 4.3. A model for the density data A model for the density of alkali borate glasses based on atomic arrangements has successfully been used with lithium and sodium borates [1,2]. This representation is based on the ideas of Krogh-Moe in which the glass is viewed
H. P. Lim et al. / Dens+
328
of poiawum
borate glawes
as a random assembly of structural groupings. These groupings are chosen from corresponding crystalline configurations. Nuclear magnetic resonance studies of potassium borate glasses of low alkali content are in close agreement with sodium and lithium borates [9]. Therefore, it will be assumed that the same structural groupings found in lithium and sodium borate glasses are also present in potassium borate glasses. These units are (fractional notations are given in parentheses): 1) boron with three bridging oxygens (f,); 2) boron with four bridging oxygens and one alkali ion (fz ); 3) boron with two bridging and one non-bridging oxygen and one alkali ion (f3); 4) boron with one bridging and two non-bridging oxygens and two alkali ions (f4); 5) boron with three non-bridging oxygens and three alkali ions (f,). Table 1 lists the fractions used in the modelling process as taken from ref. [l]. Another assumption used was that the retention of carbon dioxide which starts at R = R, = 0.54 does not affect the density. Evidence for this assumption comes from the sodium system in which it was found that the density was insensitive to whether the source of the alkali was sodium carbonate or sodium oxide [2]. This was attributed to the close similarity between the density of sodium carbonate and the sodium borate glasses. It is also true that the density of potassium borate glasses (p = 2.35 g/cm3) is extremely close to the density of potassium carbonate (p = 2.428 g/cm’) [lo] in the region of compositions where carbon dioxide retention takes place. The above assumption was found to be necessary because a source of potassium oxide (K,O) could not be found to fabricate glasses free of carbon dioxide. The following equation is used to describe the glass density [2]: f,M’ +f&P+ f,W+ . . . +f,M’+ . .. P(R) = (2) f,V’ + f,V+ f,P+ . . . +f,V’+ .. . ’ Table 1 The fractions
fl f, f f: f5
of boron
Units
0.4 I R I 0.7
0.7 5 R ~1.0
1-R R 0 0 0
1-R R/6+1/3 5R/6-l/3 0 0
1-R - R/4+5/8 5R/4-5/8 0 0
1.0 I R < 2.14
fl f2 f3 f, fs
[l]
0.0 I R < 0.4
0
-R/4+5/8 - 0.55R 0.6(R 0.2(R
+ 1.175 -1) -1)
2.14 I R I 2.5
2.5 I R I 3.0
0
0
-R/4+5/8 0 -R/2+7/4 3R/4-11/8
0 0 3-R R-2
H. P. Lim et al. / Densily Table 2 Volumes of structural sodium [2] borates
groupings
Fractional symbol
found
of potassium
in potassium
borate
borate glasses
329
glasses compared
with lithium
[l] and
Relative volume of unit compared with boron with 3 bridging oxygens K
Na
Li
1 1.52 2.15 2.85 3.85
1 1.05 1.78 2.04 2.77
1 0.84 1.37 1.68 1.95
where M’ is the mass of the ith structural grouping and V’ its volume. In eq. (2) the masses are known as well as the fractions leaving the volumes to be determined. A least-squares analysis of the density data was performed in order to find the ratios of the volumes with respect to I”. Reference (1) gives the explicit details on how this calculation was carried out. The results of the calculation are presented in table 2 along with values from lithium [l] and sodium [2] glasses. The densities calculated with the volumes given in table 2 are overlaid onto the density data as shown in fig. 4. 4.4. Comments on the density model A consistent trend can be seen in table 2 for the volumes of the structural groupings. As sodium is substituted for lithium and as potassium replaces sodium, the volumes increase. This is apparently so because of the larger sizes of the alkali ions as one goes down the first column of the periodic table.
DENSITY hd) 2.40 et----=-t-
Fig. 4. The density of potassium borate glasses as a function of R (R = mol% alkali/mol% compared with the semiempirical model (solid curve) presented in this paper.
boron)
330
H. P. Lim et al. / Density
of potassium
borare glawes
More can be said about relationships among the various volumes. Mozzi and Warren [ll], using X-ray diffraction techniques on glassy B,O,, report the boron-oxygen separation within a BO,,, triangle as 1.37 A while the oxygen-oxygen separation is 2.37 A. Taking one-half of the oxygen-oxygen separation as the radius of the oxygen ion and using the above X-ray data leads to a boron radius. The results accurate to two decimal places are: radius of 0 ion = r,, = 1.19 A, radius of B ion = rb = 0.19 A. Since the volume of the boron ion is negligible compared with oxygen, the volume of the BO,., unit is very nearly 1:5((4/3)7rri). Using the above value for the oxygen radius yields 10.6 cubic A. However, utilizing the density of boron oxide glass (1.81 g/cm3) along with the known atomic masses of oxygen and boron results in a volume for this unit, including unused space, of 32.0 cubic A. The “packing fraction” will be defined as the ratio of the volume calculated from ionic radii to the volume found from the density measurements. Thus the packing fraction of the BO,., group is 10.6/32.0 or 0.33. Similar calculations for the other structural groupings (units designated f,, f,, f,, and f,) have been done. X-ray diffraction data from lithium metaborate yields almost the same value for r, as above [12] for both bridging and non-bridging oxygens. Thus it was assumed that the oxygen radius in a trigonal unit remains at 1.19 A for both bridging and non-bridging oxygens. The radius of an oxygen ion in four-coordination with boron can also be found from X-ray measurements and is 1.28 A [13]. The ionic radii of lithium, sodium, and potassium were also needed as these ions are associated with the other structural arrangements. The radii are [14]: radius of Li ion = TLi = 0.68 A, radius of Na ion = rNa = 0.97 A, radius of K ion = rK = 1.33 A. Table 3 summarizes the results of the calculations. The constancy of the packing fractions for all of the structural groupings independent of the ion present is an important result of this study. This is consistent with the idea that changes in the size of the structural units are due to the differing sizes of the alkali ion. Furthermore, the packing fractions of planar trigonal groups are nearly the same while the tetrahedral structures of the four-coordinated borons show noticeably higher packing fractions. In addition, from R = 0 to about 0.10 the volume of the four-coordinated boron unit (V’) is similar to the sodium and lithium cases of like R. To see this, fig. 5 contrasts densities calculated with V2 = 1.52 V’ (the least squares value found from the potassium data of R = 0 up to R = 0.4) with V2 = 0.84 V’ as found earlier for lithium [l] and sodium [2] borates of extremely low R. For R I 0.1 the agreement of the potassium density data with the model
H.P. Lim et al. / Density
o/potassium
borate glusses
331
Table 3 Volumes of structural units calculated from ionic radii compared with volumes determined from density measurements (in terms of volume of boron with three-bridging oxygens). (a, Volume of unit determined from density measurements; b, Volume of unit determined from ionic radii calculations; c. Packing fraction defined as b/a.) Fraction
f2 f3 f4
fs
K
Na
Li
a
b
C
a
b
C
a
b
C
1.52 2.15 2.85 3.85
0.85 0.74 1.15 1.56
0.56 0.34 0.40 0.41
1.05 1.78 2.04 2.77
0.67 0.56 0.79 1.02
0.64 0.31 0.39 0.37
0.84 1.37 1.68 1.95
0.59 0.48 0.63 0.78
0.70 0.35 0.38 0.40
calculated using the smaller value for V2 is evidence that there is a fundamental volume in these alkalis for the four-coordinated borons of extremely low alkali content. Apparently, the ions simply fill already available voids in the structure. As more potassium is added above R = 0.1 the volume of the unit is enlarged in order to accommodate the ion. The data derived from aluminum oxide doped glasses and extrapolated back to the binary potassium borates are plotted in fig. 2. The predicted curve which results from eq. (2) when the derived values for I/‘, V3, V4, and V5 are used is also shown in this figure. This curve shows a local minimum at the metaborate composition. However, the data clearly do not show a well of such a size. While the data seem to indicate a shallow well, such a trend is not outside the experimental error.
DENSITY bc.J)
2.40 4
,,,1I"
v2=oJ34v ,’
2.20 2.00 i
1.80
Fig. 5. A close-up of alkali to boron. with greater slope with V2 = 1.52V’. oxygens
-a
,*," $,,"a...' I' 0 v~=1.!52v’ /-f 1/( 0.0
I
0!2
R
d.4
of the potassium borate data from R = 0 to R = 0.4 where R is the molar ratio Two curves derived from the model are superimposed on the data. The curve was found by using V2 = 0.84V’ while the smaller slope curve was calculated (V’ and V2 are the volumes of the following units: boron with three bridging and boron with four bridging oxygens and one alkali ion, respectively.)
H. P. Lim et al. / Densiry
332
o/potassium
borate glasses
5. Conclusions Densities of potassium borate glasses have been measured over an extremely wide range of compositions. The ideas of Krogh-Moe were used in a semiempirical model for the densities and the resulting analysis yielded the relative volumes of the structural groupings present in the glass. It was found that the arrangements are larger than in the corresponding lithium and sodium cases except for very low concentrations of potassium. These low R glasses exhibit the same fundamental volume for the four-coordinated boron as in the lithium and sodium borates. Packing fractions found by taking the ratio of the volume of the structural unit calculated from ionic radii to the volume of the structural group derived from the density data are remarkably similar for glasses prepared from any of the alkalis studied. A simple hypothesis is that the increase in volume for the structural groupings as one goes from lithium to sodium to potassium is due to the size of the alkali present. We express gratitude to Dr. Steve Martin of Iowa State University for useful discussions. Hun C. Lim and Cristina Sanchez are thanked for assistance with the manuscript. The Chemistry Research team at Coe College is credited with listening to many talks on glass and for asking questions. Coe College and the Richter Foundation are acknowledged for financial assistance. We acknowledge the financial help provided by grants of the Research Corporation and the Iowa Science Foundation. This material is based upon work supported by the National Science Foundation under Grant No. DMR-
8603218.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14]
M. Shibata, 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, to be published. R. Ota and N. Soga, Yogyo Kyokaishi 90 (9) (1982) 531. J.E. Kasper, S.A. Feller and G. Sumcad, J. Am. &ram. Sot. 67 (4) (1984) C71. S.W. Martin and C.A. Angell, J. Non-Cryst. Solids 66 (1984) 429. M. Imaoka, Yogyo Kyokaishi 69 (9) (1961) 282. H. Rawson, Inorganic Glass-Forming systems (Academic Press, New York, 1967) p. 98. R.R. Shaw and D.R. Uhlmann, J. Non-Cryst. Solids 1 (1969) 474. P.J. Bray and J.G. O’Keefe, Phys. Chem. Glasses 4 (1963) 37. R. Weast, ed., Handbook of Chemistry and Physics, Vol. 49 (CRC, Cleveland, 1968/69) B-230. R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 3 (1970) 251. W.H. Zachariasen, Acta Cryst. 17 (6) (1963) 749. W.H. Zachariasen, Acta Cryst. 16 (1963) 385. R. Weast, ed., Handbook of Chemistry and Physics, Vol. 49 (CRC, Cleveland, 1968/69) F-152.