ARTICLE IN PRESS
Physica B 362 (2005) 123–132 www.elsevier.com/locate/physb
Density of mixed alkali borate glasses: A structural analysis H. Doweidar, G.M. El-Damrawi, Y.M. Moustafa, R.M. Ramadan Glass Research Group, Physics Department, Faculty of Science, Mansoura University, P.O. Box 83, Mansoura 35516, Egypt Received 29 May 2004; received in revised form 13 December 2004; accepted 2 February 2005
Abstract Density of mixed alkali borate glasses has been correlated with the glass structure. It is assumed that in such glasses each alkali oxide associates with a proportional quantity of B2O3. The number of BO3 and BO4 units related to each type of alkali oxide depends on the total concentration of alkali oxide. It is concluded that in mixed alkali borate glasses the volumes of structural units related to an alkali ion are the same as in the corresponding binary alkali borate glass. This reveals that each type of alkali oxide forms its own borate matrix and behaves as if not affected with the presence of the other alkali oxide. Similar conclusions are valid for borate glasses with three types of alkali oxide. r 2005 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 66.30.Hs Keywords: Borate glasses; Mixed alkali glasses; Density
1. Introduction Density of solids is mostly the simplest physical property that can be measured. However, it would be a highly informative property if the structure of the material could be well defined. Density data were used to calculate the volumes of structural units present in various types of glass. It has been shown that any of the structural units in alkali[1,2] and alkaline earth [3,4] silicate glasses as well as in alkali alumino-silicate glasses [5] has a
constant volume over the entire region where it exists. Furthermore, the volume of any unit in a binary silicate glass remains the same in silicate glasses [1,3,5] containing two modifier oxides or more. In contrast, in alkali borate glasses the volume of BO3 unit and also that of BO4 unit change markedly with the glass composition. The present work is a trial to correlate the density with the structure of mixed alkali borate glasses.
2. Experimental Corresponding author. Tel.: +20 50 2319702;
fax: +20 50 2246781. E-mail address:
[email protected] (H. Doweidar).
Binary and mixed alkali borate glasses (Tables 1–3) were prepared from reagent grade
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.02.001
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132
124
Table 1 Composition, density and molar volume of R2O–B2O3 glasses R2O (mol%)
Li2O
20 24 30 35 40 45 50
Na2O
K2O
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
2.12 2.17 2.24 2.27 2.29 2.29 2.29
29.12 27.66 25.81 24.54 23.48 22.57 21.69
2.21 2.20 2.35 2.41 2.44 2.46 2.47
30.86 30.16 28.62 27.82 27.30 26.87 26.61
2.14 2.20 2.27 2.32 2.32 2.35 2.38
34.76 34.38 33.86 33.72 34.22 34.40 34.49
Table 2 Composition, density and molar volume of 0.24[(R2O)a+(R2O)b]0.76B2O3 glasses X
0 0.167 0.333 0.417 0.5 0.667 0.833 1
Li/(Li+Na)
Li/(Li+K)
Na/(Na+K)
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
2.25 2.24 2.22 2.22 2.21 2.20 2.18 2.17
30.16 29.74 29.33 29.11 28.92 28.52 28.14 27.66
2.20 2.21 2.18 2.19 2.18 2.17 2.17 2.17
34.38 33.02 32.26 31.59 31.15 30.02 28.81 27.66
2.20 2.21 2.22 2.22 2.22 2.23 2.25 2.25
34.37 33.64 32.93 32.62 32.27 31.51 30.77 30.17
Table 3 Composition, density and molar volume of 0.4[(R2O)a+(R2O)b]0.6B2O3 glasses X
0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
Li/(Li+Na)
Li/(Li+K)
Na/(Na+K)
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
D (g/cm3)
Vm (cm3/mol)
2.44 2.42 2.41 2.39 2.37 2.35 2.33 2.31 2.29
27.30 26.88 26.34 25.84 25.42 24.92 24.43 23.96 23.48
2.32 2.32 2.31 2.31 2.30 2.30 2.30 2.29 2.29
34.22 32.91 31.57 30.23 28.90 27.55 26.20 24.83 23.48
2.32 2.34 2.35 2.37 2.38 2.39 2.41 2.43 2.44
34.22 33.31 32.47 31.51 30.71 29.87 29.01 28.06 27.30
chemicals. Li2CO3, Na2CO3, K2CO3, and H3BO3 were used as sources for the oxides in the glasses. The glasses were melted in porcelain crucibles in an electric furnace. Selected compositions of
Li2O–B2O3 and K2O–B2O3 glasses were prepared in silica crucibles to compare with those molten in porcelain crucibles. Melting was carried out at temperatures ranged between 800 and 920 for
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132
3. Results Fig. 1 shows the dependence of density D on the modifier oxide content for Li2O–B2O3, Na2O– B2O3 and K2O–B2O3 glasses. In all cases the density increases with a decreasing rate when increasing the content of alkali oxide (R2O). The
2.6 2.5
D (g/cm3)
Na2O-B2O3 2.4
K2O-B2O3
36 D Dc Vm Vm(c)
2.35 2.30 2.25
D
34 32
2.20 30
2.15 2.10
Vm
Vm (cm3/mol)
2.40
D (g/cm3)
binary glasses and 800–980 1C for mixed alkali glasses. The crucible and its content were kept in the furnace for about 30 min. The melt was swirled frequently before being poured in preheated steel molds. The glasses were allowed to cool normally to room temperature. Glasses with alkali oxide content greater than 35 mol% were obtained by pressing the melt between two steel blocks. The obtained samples were transparent and apparently homogeneous. The samples were kept in a desiccator until required. The glass samples were sealed with silica jell in plastic sacks and kept in a desiccator until required. The density (D) of glasses was determined, directly after preparation, at room temperature using the Archimedes method with xylene as an immersion fluid. At least, three samples of each glass were used to determine the density. Density values are precise to 70.02 g/cm3.
125
28
2.05 2.00 0.0
0.2
0.4
0.6
0.8
26 1.0
Li/(Li+Na) Fig. 2. Density and molar volume of xLi2O (0.24x)Na2O 0.76B2O3 glasses as a function of the ratio Li2O/ (Li2O+Na2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (2) and (3) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 24 mol% R2O.
results of density agree well with those of previous studies ([6] and references therein). In Fig. 1 is shown that the density tends to be constant for Li2OX35 mol%. In the case of Na2O–B2O3 and K2O–B2O3 glasses there is a marked decrease in the rate of change of density in the same region (R2OX35 mol%). Fig. 1 also shows that the densities of the glasses prepared in silica crucibles agree, within the experimental error limits, with those of glasses prepared in porcelain crucibles. Fig. 2 shows, as an example, the dependence on composition of the density and molar volume (Vm) of xLi2O (0.24x)Na2O 0.76B2O3 glasses. There is a linear change in both of D and Vm when changing the ratio Li2O/(Li2O+Na2O). Similar behavior is observed for all the investigated glasses.
2.3 Li2O-B2O3 2.2
4. Discussion
2.1
4.1. Alkali borate glasses 2.0 20
25
30
35
40
45
50
R2O (mol%) Fig. 1. Density of Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses as a function of the alkali oxide (R2O) content. Empty symbols are the results of the present study and the filled symbols are data taken from Ref. [6]. The symbols (*) and (n) denote, respectively, density data for Li2O–B2O3 and K2O–B2O3 glasses prepared in silica crucibles.
In the present work, it is aimed to investigate borate glasses having two modifier oxides. Li2O, Na2O and K2O are candidates for such purpose. Hence Li2O–Na2O–B2O3, Li2O–K2O–B2O3 and Na2O–K2O–B2O3 glasses (Tables 2 and 3) are selected as a material for this work. It is preferred to start the study by investigating the corresponding
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132
binary glasses, i.e. Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses (Table 1). In borate glasses alkali oxide is assumed to convert BO3 units into BO4 units up to about 33 mol% R2O [7] with a rate of two BO4 units for each R2O molecule. The fraction N4 of the BO4 units increases with increasing the alkali oxide content and reaches its maximum around 40 mol% R2O. It is accepted [7] that up to 33 mol% R2O the formation of BO4 units follows the relation N4 ¼ x=ð1 xÞ; where x is the molar fraction of the alkali oxide in glass. The trend observed for N4 up to about 40 mol% R2O resembles that of D (Fig. 1). Fig. 3 shows a linear increase in D with increasing N4 up to about 38 mol% R2O in Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses. The BO4 unit is the denser one among the borate units [8]. It is then to correlate the increase in density, up to 33 mol% R2O, with the increase in N4. It is to notice that in this region the fraction of BO4 units increases at the expense of the BO3 units that have a lower density. In Fig. 4 is shown that the change with composition of the molar volume Vm can be approximated to two linear regions with a change in the slope around 33 mol% R2O. Above 33 mol% R2O it is assumed [7] that the added alkali oxide causes the formation of nonbridging oxygen ions (NBOs) in the borate matrix. This type of ions represents broken bonds in the network. Units containing NBOs might have
2.7
2.5 Modifier ion Li Na K
D (g/cm3)
2.3
2.6 2.5
2.2
2.4
2.1
2.3
2.0
2.2
1.9
2.1
D (g/cm3)
2.4
2.0
1.8 0.0
0.1
0.2
0.3
0.4
0.5
N4 Fig. 3. Correlation between density and the fraction (N4) of the BO4 units in Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses, up to 38 mol% R2O. N4 values are taken from Ref. [7].
38
34
Vm (cm3/mol)
126
30
26 Modifier ion Li Na K
22
18 20
24
28
32
36
40
44
48
52
R2O (mol%) Fig. 4. Dependence of the molar volume on the alkali oxide content in Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses. Empty symbols are the results of the present study and the filled symbols are data taken from Ref. [6].
greater volume than those without NBOs. Formation of such units appears as a change in the rate with which Vm varies with composition. It is shown in Fig. 4 that the rate of change in Vm is reduced above 33 mol% R2O for Li2O–B2O3 and Na2O–B2O3 glasses. In contrast, there is an increase in Vm of K2O–B2O3 glasses in the same region. These effects might be due to the volume of the units containing NBO, which in turn is related to the modifier ion size. It is worth mentioning that NBOs do form at the expense of BO4 units. The latter are denser and have smaller volume than units having NBOs [8]. Thus one may expect an increase in Vm when increasing the R2O content. However, this is not the sole factor that affects Vm since the number of borate units do also contribute to it. The molar volume of a glass is given [9] as X Vm ¼ N uV u, (1) where Nu is the number of the unit (u) per mole of glass and Vu is its volume. The number of borate units per mole of glass decreases linearly with the increase in the R2O content. It can be said that the formation of NBOs in the borate network for R2O433 mol% is the reason for the lower rate of the density increase (Fig. 1) as well as the change in the rate with which Vm varies with composition (Fig. 4).
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132 2.40
D ¼ ½N 4a M 4a þ N 4b M 4b þ N 3 M 3 =½N 4a V 4a þ N 4b V 4b þ N 3a V 3a þ N 3b V 3b . ð2Þ Here xa and xb are, respectively, the mole fractions of the alkali oxides (R2O)a and (R2O)b. N4a and N4b are, respectively, the numbers, per mole of glass, of the BO4 units that form from (R2O)a and 2.40
36 D Dc Vm Vm(c)
Vm
D (g/cm3)
2.30
34
2.25
32
2.20
D
2.15
30
2.10
Vm (cm3/mol)
2.35
28
2.05 2.00
26 0.0
0.2
0.4
0.6
0.8
1.0
D (g/cm3)
2.35
The above bases can be employed to calculate the density of mixed alkali glasses with R2O content up to 33 mol%. Figs. 2,5 and 6 show the density and molar volume of xLi2O (0.24x)Na2O 0.76B2O3, xLi2O (0.24x)K2O 0.76B2O3 and xNa2O (0.24x)K2O 0.76B2O3 glasses, respectively. The total alkali oxide content in these glasses is 24 mol%, i.e. it converts BO3 into BO4 units without formation of nonbridging oxygen ions in the borate matrix. In Figs. 2,5 and 6 there is a linear change in both D and Vm when increasing the content of one of the modifier oxides at the expense of the other. This behavior suggests a systematic difference in these properties when substituting one molecule of a modifier oxide with a molecule of the other oxide. The trends observed in Figs. 2,5 and 6 are consistent with the results of other studies [10]. The density of a glass having the molar formula xa(R2O)a xb(R2O)b (1xaxb)B2O3, 0o(xa+ xb)p0.333, can be given as
35 D Dc Vm Vm(c)
Vm
2.30
34 33 32
2.25 31
D 2.20
30
2.15 2.10 0.0
Vm (cm3/mol)
4.2. Mixed alkali borate glasses
127
29
0.2
0.4
0.6
0.8
28 1.0
Na/(Na+K) Fig. 6. Density and molar volume of xNa2O (0.24x)K2O 0.76B2O3 glasses as a function of the ratio Na2O/ (Na2O+K2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (2) and (3) and the volumes obtained from corresponding binary R2OB2O3 glasses having 24 mol% R2O.
(R2O)b. M4a and M4b are the masses and V4a and V4b are the volumes of those units, respectively. M3 is the mass of BO3 unit. M4 is taken as the mass of (B+2O+R) and M3 as (B+1.5O). The volume of a structural unit is the volume of the constituting ions and its allied space in the matrix. In mixed alkali borate glasses B2O3 associates with both (R2O)a and (R2O)b. Then N3a represents the number of BO3 units, per mole of glass, in the (R2O)a–B2O3 portion and N3b is that in the (R2O)b–B2O3 one. In such glasses the part of B2O3 that associates with (R2O)a and (R2O)b depends on the total alkali oxide content. For example the glass 0.08Li2O 0.16Na2O 0.76B2O3 can be looked as composed of two matrices (Li2O–B2O3 and Na2O–B2O3) in a random distribution. Both matrices contain 24 mol% of alkali oxide. The same can be said for all glasses in the series xLi2O (0.24x)Na2O 0.76B2O3. It may be adequate to take the glass 0.08Li2O 0.16Na2O 0.76B2O3 as an example for calculating the parameters in Eq. (2). These are N 4Li ¼ 2 0:08N A ,
(2a)
N 4Na ¼ 2 0:16N A ,
(2b)
N 3 ¼ 2ð0:76 0:24ÞN A ,
(2c)
M 4Li ¼ 8:2613 1023 g,
(2d)
Li/(Li+K) Fig. 5. Density and molar volume of xLi2O (0.24x)K2O 0.76B2O3 glasses as a function of the ratio Li2O/ (Li2O+K2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (2) and (3) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 24 mol% R2O.
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132
128
M 4Na ¼ 1:0927 1022 g,
(2e)
M 3 ¼ 5:7804 1023 g,
(2f)
N 3Li ¼ 2½xLi ð0:76=0:24Þ xLi N A , ¼ 0:3467N A ,
change of both D and Vm with composition is due only to a change in the number of units. Let ðxa þ xb Þ ¼ xR and xB is the molar fraction of B2O3 in glass, i.e. xB ¼ ð1 xa xb Þ ¼ ð1 xR Þ: Eq. (2) can then be reformed to
ð2gÞ
V m ¼ 2N A fxa ½ðV 4a V 4b Þ þ ðxB =xR 1ÞðV 3a V 3b Þ þ ½xR V 4b þ ðxB xR ÞV 3b g. ð4Þ
ð2hÞ
This relation gives Vm as a function of xa. It represents a straight-line equation. The slope is 2NA[V4aV4b+(xB/xR1)(V3aV3b)] and its intercept is 2NA[xRV4b+(xBxR)V3b]. This relation is valid for 0oxRp0.333. It implies a linear change in Vm when changing xa at the expense of xb. It is to notice that, for certain types of alkali oxides, both the slope and intercept depend on the value of xR (and consequently xB). On the other hand, for a specific value of xR, the slope and intercept are determined by V4a, V4b, V3a and V3b, i.e. they would depend on the types of alkali ions. The volumes given for BO3 units are nearly equal whereas there are marked differences between the volumes of BO4 units. This indicates that (V4aV4b) would have a great effect on the slope of Eq. (4). The magnitude of slope would be greater for greater values of (V4aV4b). This is the case for Li2O–K2O–B2O3 glasses. In addition, the intercept is determined by the type and concentration of (R2O)b. This is plausible since the intercept represents the value of Vm at ðR2 OÞa ¼ 0: Fig. 7 shows the dependence of Vm on the concentration (R2O)a, in mol%, for glasses having xR ¼ 0:24: (R2O)a is the content of the alkali oxide that increases in concentration. The slopes and intercepts of the lines in Fig. 7 agree well with those predicted from Eq. (4) and the volumes given above for the structural units. This reveals that the model presented is able to describe the variation of D and Vm with composition of such glasses. Some investigators [10,12] reported slight deviations from linearity in the density of mixed alkali glasses. In the light of the present analysis it is assumed that these deviations arise due to tolerance in the experimental results. The linear change in density and molar volume has been proven also for mixed modifier silicate glasses [1,3,4].
N 3Na ¼ 2½xNa ð0:76=0:24Þ xNa N A , ¼ 0:6933N A .
It is not possible to solve Eq. (2) for the volumes V4a, V4b, V3a and V3b. An alternative way is to get these volumes from the corresponding binary borate glasses. For 24 mol% R2O it was found that V4Li ¼ 30.64 1024 cm3, V3Li ¼ 29.90 1024 cm3, V4Na ¼ 39.09 1024 cm3, V3Na ¼ 29.86 1024 cm3, V4K ¼ 53.34 1024 cm3 and V3K ¼ 30.20 1024 cm3 [6,11]. By using these values in Eq. (1) densities of the glasses presented in Figs. 2,5 and 6 can be calculated. Within the experimental error limit, the obtained densities agree with the experimental data. A better agreement could be achieved by making minor changes (p0.5%) in the volumes of the structural units. The volumes used to get the calculated densities are V4Li ¼ 30.7 1024 cm3, V3Li ¼ 30.1 1024 cm3, V4Na ¼ 39.3 1024 cm3, V3Na ¼ 29.9 1024 cm3, V4K ¼ 53.5 1024 cm3 and V3K ¼ 30.2 1024 cm3. The agreement between the experimental and calculated densities reveals that the structural units in mixed alkali borate glasses do maintain their volumes as in the corresponding binary borate glasses. This means that the interaction between the alkali ion and the nearestneighbor ions in a structural unit is the predominant factor that affects the volume of a structural unit. On the basis of Eq. (1) the molar volume of xa(R2O)a xb(R2O)b (1xaxb)B2O3 glasses, 0o (xa+xb)p0.333, can be given as V m ¼ N 4a V 4a þ N 4b V 4b þ N 3a V 3a þ N 3b V 3b . (3) Calculated Vm values are presented in Figs. 2,5 and 6. They are obtained from Eq. (3) and the given volumes. Because of the constant volumes of the units in these glasses it is concluded that the
ARTICLE IN PRESS H. Doweidar et al. / Physica B 362 (2005) 123–132 36
2.6 D Dc Vm Vm(c)
2.5
32
D (g/cm3)
Vm (cm3/mol)
34
Na2O– K2O– B2O3 Li2O– K2O– B2O3
30 Li2O– Na2O– B2O3
D
2.4
34 32 30
2.3 28 2.2
26
Vm
2.1
Vm (cm3/mol)
36
129
24
28 2.0
22 0.0
0.2
0.4
0.6
0.8
1.0
Li/(Li+Na)
26 5
10
15
20
25
(R2O)a (mol%) Fig. 7. Molar volume of xa(R2O)a xb(R2O)b (1xaxb)B2O3 glasses, ðxa þ xb Þ ¼ 0:24; in dependence of the concentration (in mol%) of (R2O)a. Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eq. (3) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 24 mol% R2O.
Fig. 8. Density and molar volume of xLi2O (0.4x) Na2O 0.6B2O3 glasses as a function of the ratio Li2O/ (Li2O+Na2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (6) and (7) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 40 mol% R2O.
36
2.6 2.5
D (g/cm3)
Above 33 mol% R2O there is another distribution of the structural units, where NBOs would form. In this case, Eqs. (2) and (3) must be modified to include the terms related to NBOs. Figs. 8–10 show a linear change in both D and Vm when changing the ratio of alkali ions in borate glasses having 40 mol% R2O. This behavior suggests that also in these glasses the volumes of the units related to each type of alkali ions remain constant. It means that Vu does not change when varying the content of (R2O)a at the expense of that of (R2O)b. As the case for the glasses containing 24 mol% R2O it may be assumed that Vu in borate glasses having 40 mol% R2O would be the same as would be in the corresponding binary alkali borate glasses. Unfortunately the volumes of units containing NBOs in alkali borate glasses are not yet known. It is therefore not possible, at the present time, to go forward with analyzing the density data of the mixed alkali glasses with R2O433 mol% in such a way like that used with Eqs. (2) and (3). We can however try another way to solve this problem. In xLi2O (0.4x)Na2O 0.6B2O3 glasses, for example, B2O3 is divided proportionally between Li2O and Na2O. This means that each alkali oxide incorporates itself in an amount of B2O3 depending on its amount in glass. In all cases the relative concentration of Li2O or Na2O in their borate
D Dc Vm Vm(c)
Vm
34 32
2.4 30 2.3
D
28
2.2
26
2.1
Vm (cm3/mol)
0
24 22
2.0 0.0
0.2
0.4
0.6
0.8
1.0
Li/(Li+K) Fig. 9. Density and molar volume of xLi2O (0.4x) K2O 0.6B2O3 glasses as a function of the ratio Li2O/ (Li2O+K2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (6) and (7) and the volumes obtained from corresponding binary R2OB2O3 glasses having 40 mol% R2O.
matrices would be 40 mol%. We can then try to use the average volume per alkali ion (VR40) in the corresponding binary alkali borate. This can be obtained as V R40 ¼ V m =ð2 0:4N A Þ.
(5)
Here Vm refers to the molar volume of the concerned 0.4R2O 0.6B2O3 glass. It might be clear that VR40 represents the average volume of all the structural units present in glass at 40 mol% alkali oxide. These are symmetric BO3 (without NBOs), BO4 and asymmetric BO3 (with NBOs)
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130
34 32
2.4
30
D
2.3
28 2.2 D Dc Vm Vm(c)
2.1
26
M Li40 ¼ ½ð0:1=0:4Þ 0:6 69:62 þ 0:1 29:88 Vm (cm3/mol)
Vm
2.5
=ð2 0:1 6:022 1023 Þ ðgÞ and M Na40 ¼ ½ð0:3=0:4Þ 0:6 69:62 þ 0:3 61:98 =ð2 0:3 6:022 1023 Þ ðgÞ.
24 22
0.0
0.2
0.4
0.6
0.8
1.0
Na/(Na+K) Fig. 10. Density and molar volume of xNa2O (0.4x) K2O 0.6B2O3 glasses as a function of the ratio Na2O/ (Na2O+K2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (6) and (7) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 40 mol% R2O.
units. If also in these glasses the volumes in binary borate glasses would remain the same in mixed alkali glasses, then we can calculate Vm of the latter by using VR40. It is given as X N R40 V R40 , (6) Vm ¼ where NR40 is the number of alkali ion per mole of glass (2 0.4NA) at 40 mol% alkali oxide. From the obtained results (Table 1) it is concluded that V Li40 ¼ 48:729 1024 cm3 ; 24 V Na40 ¼ 56:666 10 cm3 and V K40 ¼ 71:022 1024 cm3 : By means of these values and Eq. (6) Vm can be calculated for xLi2O (0.4x)Na2O 0.6B2O3, xLi2O (0.4x)K2O 0.6B2O3 and xNa2O (0.4x)K2O 0.6B2O3 glasses. In Figs. 8–10 it is shown that the calculated molar volumes (Vm(c)) agree well with the experimental values. This result indicates that also in mixed alkali borate glasses with NBOs the volumes of the structural units are the same as the binary glasses of the same alkali content. Similar procedure can be followed to calculate the density of these glasses. The density can be given as .X X D¼ N R40 M R40 N R40 V R40 , (7) where MR40 is the average mass per alkali ion in the corresponding borate matrix at 40 mol% alkali oxide. As an example, in the glass 0.1Li2O 0.3Na2O 0.6B2O3, MR40 can be
Here 69.62, 29.88 and 61.98 g are, respectively, the molecular weights of B2O3, Li2O and Na2O. It is easy to show that M Li40 ¼ 1:115 1022 g; M Na40 ¼ 1:382 1022 g and M K40 ¼ 1:649 1022 g: These values are constant for R2 O ¼ 40 mol; regardless of the individual content of each alkali oxide. This is because in any case the relative concentration of each oxide in its borate matrix would be 40 mol%. The calculated densities presented in Figs. 8–10 are obtained from these values and Eq. (7). The agreement is quite good with the experimental densities. To test the validity of the above conclusions a glass series containing three alkali oxides has been investigated. Fig. 11 shows a linear change, with Li/(Li+K), in both D and Vm of xLi2O (0.2x)K2O 0.2Na2O 0.6B2O3 glasses. The total alkali oxide content in these glasses is 40 mol%. We can then use the volume per alkali ion obtained above for Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses at 40 mol% R2O. With these volumes and 36
2.6
34
2.5
32
D
2.4
30 2.3 28
Vm
2.2
Vm (cm3/mol)
2.0
D (g/cm3)
D (g/cm3)
calculated as
36
2.6
26
2.1
24
2.0
22 0.0
0.2
0.4
0.6
0.8
1.0
Li/(Li+K) Fig. 11. Dependence of D and Vm of xLi2O (0.2x) K2O 0.2Na2O 0.6B2O3 glasses on the ratio Li2O/(Li2O+ K2O). Empty symbols represent experimental results and the filled ones are calculated. The latter are obtained by means of Eqs. (6) and (7) and the volumes obtained from corresponding binary R2O–B2O3 glasses having 40 mol% R2O.
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Eq. (6) one can calculate the molar volume of the glasses. Similarly, the density of xLi2O (0.2x)K2O 0.2Na2O 0.6B2O3 glasses can be calculated from Eq. (7) and the volumes and the masses per alkali ion obtained for R2 O ¼ 40 mol: There is a good agreement between the calculated and the experimental density and molar volume (Fig. 11). This reveals that also in multiple modifier borate glasses the volumes of structural units related to an alkali ion are the same as in the corresponding binary alkali borate glasses. The ideas treated in the present work can then be used to calculate both Vm and D of mixed and multiple alkali borate glasses with any content of R2O. The present study indicates that the volumes of units in mixed and multiple alkali borate glasses are the same as in the binary glasses with the same concentration of R2O. This means that the volumes of structural units in borate glasses depend only on the ratio R2O/B2O3, regardless of the types of the other modifiers present in glass. In other words, it seems as if each alkali oxide associates with B2O3 to form a matrix that is independent of those related to the other alkali oxides. Mixed alkali effect is also observed in the change with composition of the glass transition temperature Tg of mixed alkali glasses. It appears either as a negative deviation from additivity [13] or a minimum in Tg at certain value of X for various types of borate glasses [14–16]. Such changes of Tg represent an example where the mixed alkali effect appears for a non-transport property. Tg is certainly correlated to the structure of glass. It increases with increasing the bond strength and both the cross-link and the packing densities of the glass [17]. Investigations on Tg of Li-alkali borate glasses [18] indicated that at low contents of alkali oxides the transition temperature varies for glasses having alkalis in the succession Li2Na4Li2K4Li2Rb4Li2Cs: A reversed trend is observed for glasses having high alkali contents. The change of Tg of mixed alkali glasses has been correlated with the mass of modifier ions [14,19]. Heavier ions cause a greater reduction in Tg than the lighter ones. The factors that affect the transition temperature might be the same affecting the density of glass. However, there is a main difference that has to be considered.
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Density, which is a bulk property, is often determined at room temperature whereas Tg is obtained as a result of a thermally activated process. The latter includes, at first, dissociation of bonds of the weakest bound species. These are the bonds between modifier cations and oxygen ions forming the network. It can be said that while density is a property that results from averaging the packing of all structural units, Tg depends mainly on the bond strength of the weakest bond in the matrix. Such bonds would vary in mixed alkali glasses from those in single alkali ones. The fraction N4 of four-coordinated boron atoms also shows deviation from additivity for mixed alkali glasses. Zhong and Bray [20] indicated that N4 of mixed alkali glasses is less than the value that can be expected by considering additivity of the values of the alkalis in glass. In some cases, N4 of mixed alkali glasses was found less than the value of any of the corresponding binary alkali borate glasses. Kamitsos et al. [13] came to similar conclusions from analyzing the infrared spectra of xNa2O (1x)Cs2O 2B2O3 glasses. A negative departure of N4 from additivity is assumed to be due to a conversion of BO4 units into asymmetric BO3 units. These are BO3 triangles having a non-bridging oxygen ion per unit. A decrease in the fraction of the rigid BO4 units in favor of the more flexible asymmetric BO3 units is rather inconsistent with the chemical durability of such glasses. It is well known that mixed alkali glasses have superior durability with respect to the binary glasses of the same alkali oxide content. MAS-NMR investigations on 0.3[xNa2O (1x)Li2O] 0.7B2O3 and 0.3[xNa2O (1x)K2O] 0.7B2O3 glasses [21] revealed a linear change in the isotropic chemical shift of 23Na upon increasing the Na2O content. It increases for 0.3[xNa2O (1x)Li2O] 0.7B2O3 glasses and decreases for 0.3[xNa2O (1x)K2O] 0.7B2O3 glasses. Retai et al. [21] deduced that each type of alkali ions occupies its own unique site, regardless of the glass composition. In addition the oxygen environments of Na+ sites change when the sites in the vicinity are occupied by Li+ or K+ ions. Similar effects were reported for silicate, aluminosilicate, borate, germanate and
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tellurite glasses [22]. It is suggested that when a smaller cation substitutes for a larger one the mean size of the sites for the larger one increases and vice versa. The effect is related to the cation field strength and is due to a competition of cations for the negative sites of oxygen ions. However, it is accepted that the presence of dissimilar alkali ions affects both the local and intermediate range orders of the glass network [23]. The above discussion shows that both N4 and the site size in mixed alkali glasses are not the same as in the relevant binary glasses. The results of the present study and previous studies on various mixed alkali silicate glasses [1,3,5] revealed that both the density and molar volume can be calculated by using the number and volume of the individual structural units in the corresponding single alkali glasses. It appears therefore that there is inconsistency between the presented model and the results of NMR for mixed alkali glasses. The origin of discrepancy may lie, in part, in the concept of site size and the volume of a structural unit. It seems that the change in the size of one type of sites would be at the expense of the size of the other type, but the total size of both sites would remain constant. This is based on the fact that the molar volume of mixed alkali glasses can be additively calculated from the parameters of the binary glasses [1,3,5]. This conclusion leads to another one, that N4 of mixed alkali glasses would be obtained additively. If this were the case, then a question would arise about the possibility that different types of BO4 units preclude the signals of each other. It appears that the mixed alkali effect still needs more investigations to be clarified.
5. Conclusion The matrix of mixed alkali borate glasses can be looked as being composed of alkali borate networks incorporated in each other. Each type of alkali oxide is assumed to associate itself with a proportional quantity of B2O3. This means that the ratio of alkali oxide/boron oxide in any case
would be that of the total alkali oxide/total boron oxide. In spite of the great change with composition of the volumes of structural units in alkali borate glasses, the volumes of the borate structural units in mixed alkali glasses are the same found in the corresponding binary alkali borate glasses. This conclusion would be useful for calculating the density and molar volume of multiple alkali borate glasses.
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