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A discussion about the structural model of “nuclear magnetic resonance studies of glasses in the system Na2OB2O3SiO2”
Journal of Non-Crystalline Solids 45 (1981) 29-38 North-Holland Publishing Company
29
A DISCUSSION ABOUT THE STRUCTURAL MODEL OF "NUCLEAR MAGNETIC RESONANCE STUDIES OF GLASSES IN THE SYSTEM Na20-B203-SiO2" S.Z. Xiao Glass Lahoratoo,, Be(ring Glass Research Institute, Hong-Qiao, Bei]ing 100062, People 2 Republic ~?f China Received 25 December 1979 Revised manuscript received 3 April 1981
The structural model for the system Na20-B203-SiO 2 suggested by Yun, Feller and Bray is discussed. A different structural model is suggested in this paper the results of which are in better agreement with experiment.
1. Introduction Yun and Bray [1] have proposed a structural model for the glasses in the system x N a z O - y B 2 0 3 - z S i O 2 to explain their N M R results, which were expressed as the fraction of boron atoms in boron oxygen tetrahedra (N4) and the fraction of boron atoms in asymmetrical boron oxygen triangles containing one or two non-bridging oxygen atoms (N3A). Recently, Yun et al. presented another paper [2] on the same subject, where they corrected the model proposed in ref. 1. In the present paper a different structural model for R ~ R0, K ~< 8 (R = x / y , R o is a value of R at which all of the diborate units are exhausted, and K = z'/y) is presented, which is in better agreement with the experimental data presented in ref. 1.
2. Relevant reactions and equations in refs. 1 and 2 The model in refs. 1 and 2 consists of a few reactions and a set of equation. The relevant reactions and equations in refs. 1 and 2 are summarized below.
+[1.5(BO 2)'
+ l . 5 N a 1+],
(1)
where (B407) z-, (B306) 3 and (BO2) 1- denote a diborate unit, ring-type metaborate unit and loose tetrahedron unit, respectively. 0022-3093/81/0000- 0000/$02.50 © 1981 North-Holland
30
S.Z. Xiao / The structural model of NMR studies
According to the suggestion proposed in refs. 1 and 2, the starting and ending point of reaction (1) correspond to Rma~ and R0, respectively. In the region
: Rma×~R~4(K+l.Z5Rma×) and K <~ 8, additional Na 1+ ions are proportionally shared between the borate and silicate networks. In the region R 0 ~< R ~<4 ( K + 1.25 Rmax) and K ~ 8 , (BO3) 3- and (BO2) 1- units are destroyed to form pyroborate (B205)4- units and orthoborate (BO 3)3- units. From the above assumptions, the relevant equations derived in refs. 1 and 2 are: for Rmax <~R~Ro, K<~8 N4(c)-----N4max - [ 0 . 2 5 / ( 1 + K ) ] ( R - Rmax) ,
(2a)
N3A(c, =[1.25/(1 + K ) ] ( R - Rmax);
(2b)
and for R o ~
(3a)
N3A(C) =[1.25/(1 + K ) ] ( R 0 - Rmax) + [0.05/(1 + K ) ] ( R - R0);
(3b)
and 1 Rma x = -i--~g -~"0.5, R 0 : Rrnax + (1 -[- g ) ( 1 -
(4) Rmax).
(5)
3. An improved treatment of the experimental data in ref. I The model in ref. 2 is generalized from the Li20 - B203 binary system. In this paper an attempt is made to extract directly information about the structural changes in the region R ~> R0, K<~ 8 from the experimental data in ref. 1. Hence, it is necessary to recalculate the empirical equation of the experimental data in ref. 1 using a least-squares fit; each family of data having the same value of Kis divided in two groups: Rmax ~ R <~ R o and R >~ R o. The superscripts (l) and (2) are employed to correspond to the regions Rmax ~ R R o and R 1> R 0, respectively. In all of the 36 experimental data points in ref. l, only data points nos. 20, 30, 35 and 36 belong to the case of R/> R o. these four points, unfortunately, are distributed among three groups of K (K = 1, 2, 3). Therefore, the empirical equations for N~(I~)or N(~]E) for R m~1~< R~< R o are first calculated, then, their values at R o (i.e. N4(~,R and N~,]E)R ) can be found by extrapolation. The empirical equations for'N°4({~)or N(2~E)' f(~r R i> R o are then calculated, using these values N(~E),Ro or N(~),R~ and the relevant
31
S.Z. Xiao / The structural model o f N M R studies
Table 1 E m p i r i c a l s t r a i g h t line e q u a t i o n s f o r N~2~), NJ2A~E) a n d t h e i r slopes as c a l c u l a t e d b y the a u t h o r
R~R
o ~-
K = 1 K= 2 K=3
-- 0 . 4 4 R + 1.08 - 0 . 1 5 R + 0.77 - 0 . 1 2 R + 0.81
- 0.44 - 0.15 -0.12
' "3A(E) --
~3A(E)
0 . 8 1 R - 0.79 0.24R - 0.070 0.18R - 0.049
0.8 l 0.24 0.18
data points in the region R/> R 0. For the data group K = 3, when the equation for N3C~E)for Rmax ~< R ~< R o is calculated, the theoretical initial point (R = Rmax = 0.69, N3A = 0) is used because there are only two points nos. 33 and 34. These equation for R t> R o and their slopes as calculated by the author are listed in table 1.
4. An analysis for the loose boron oxygen symmetric triangles Since N4 +N3A +N3s = 1, thenN3s = 1 - ( N 4 +N3A ).
(6)
Inserting eqs. (3a) and (3b) into eq. (6), we derive the equation for N3~s2~c)as follows: N(2)
3s(c) -
0.2
1 + 1~ ( R -
R0).
(7)
On the other hand, inserting the relevant empirical equations listed in table 1 into eql (6), we can derive the empirical equation for NCSZ~E).The equations for N3~E) and their slopes are shown in table 2. From eq. (7) it can be seen that N~sZ~c)increases for R t> R 0, according to the model of ref. 2. Obviously, this is consistent with the presumption about eqs. (3a) and (3b), which involves the formation of orthoborate (BO3) 3- units. A (BO3) 3- unit consists of one symmetric triangle with three non-bridging oxygen atoms [3]. In contrast to eq. (7), table 2 shows that N3~E) decreases. Hence, the author suggests that a new approach is required to reflect the case
Table 2 E q u a t i o n s f o r NJS2~E) a n d their slopes
R>~ R o
K= 1 K= 2 K= 3
N3(2) S(E) =
S3(~E)
- 0.37R + 0.71 - 0 . 0 9 0 R + 0.30 -- 0 . 0 6 0 R + 0.24
-0.37 - 0.090 -- 0 . 0 6 0
32
S.Z. Xiao / The structural model of NMR studies
of N3(~0E)decreasing, and that for R >~R 0 the number of (BO3) 3- units can be neglected in the region where N3(s2~)i>0. Then N(2}E) can be assigned to the fraction of loose boron oxygen symmetric triangles (BO,.5) °.
5. Two new reactions for R ~>R o, K~< 8
It is suggested in this paper that the following two reactions may take place simultaneously: forRo <~R,K<~8
2(BO,.5) ° + 2Na20~[(B205) 4 + 4 Y a ' + ] , 2[(BO2)'
+ N a '+] +
Na20---~[(B205)4- +4 Nal+].
(8) (9)
Now, it is still assumed that additional Na I + ions are proportionally shared between the silicate and borate networks as claimed in refs. 1 and 2. However, the sharing correlation of Na20 being introduced between reactions (8) and (9) ~"' by trial and error. Thus, the equation for the shares of can be selected as ~.~ Na20 obtained by reactions (8) or (9) is
(Na20)(8) -- (Na20)(9)= ½[l/(I + K)](Na20).
(I0)
Hence, for reaction (8) we have S(2) (8)0 4(C)
and "2)
(8)
s A(c) : 0.25/(1 + K).
(11)
For reaction (9) we have 4(2 (9)--0.5/(1 (C) ---~
+K)
and $3A, C)(2) (_--9)0.5/(1 + K ) .
(12)
According to reaction (8) the following expression holds: _ q(2) (8) V(2) °33(C) -- ~3A(C).
Taking eqs. (11) into account, we have $3(~{c)(8-)- 0.25/(1 + K).
(13)
Table 3 shows the values calculated from eq. (13) and their comparison with the corresponding empirical values in table 2.
33
S.Z. Xiao / The structural model of NMR studies Table 3 Values calculated from eq. (13) and their comparison with the corresponding values in table 2 (2) S3S(C) K= 1 K= 2 K ~- 3 Average
,~,(2) / ~'(21 o3S(C)/°3S~E)
--0.13 35.1% - 0.083 92.2% - 0.063 105% (Only for groups of K = 2 and 3) 98.6%
(Note: The author suggests that the earlier crystallization phenomenon may take place in glass no. 20 (see appendix for details), and for R ~> R0 the group of K = 1 has only one data point from glass no. 20, hence, the empirical slope of this group cannot be compared with the others.)
From table 3 it can be seen that eq. (13) may expect the rate of change of the loose symmetric triangles in the region R ~> R o, K ~< 8. Since reactions (8) and (9) proceed simultaneously, then the total rate of change of tetrahedra and of asymmetric triangles should be calculated from these two reactions. Thus, the two sides of eq. (11) should be added to the corresponding sides of eq. (12), then S~i(2!,, (t.) • - 0 . 5 / ( 1 + K),
(14a)
S(-1 3A(C) = 0 . 7 5 / ( 1 + K ) .
(1481
Now we will consider the region in which reaction (9) occurs. Let the end point of reaction (9) correspond to Rt. Rt is a value of R at which all of the (BO2) I- units are exhausted. The equation for R t is derived below. Solving eqs. (6) in ref. 1 we have m = l y --~6z,
(15)
where m denotes the number of (B407) 2- units corresponding to the start of reaction (1). According to reactions (9) and (1) it can be determined respectively that (Na20)/[(BO2)'-] = 1/2 and [(BO2)'-l/[(B407) z-] = 1.5/1. Taking eq. (10) into account, then it is easy to write that 1
1
(Xt --AT°) 1 + K 2 - m
1.5g,1
where Xt and Xo are the NazO contents corresponding to R t and R o, respectively. The two sides are divided by y, yielding R , - - R o = 1.5(1 +K)m/y, where R, = X , / y ,
Ro = Xo/y.
(161
S.Z. Xiao / The structural model of NMR studies
34
Inserting eq. (4) into eq. (5), we have R o = 1 +½K-~K
(17)
2.
Inserting eqs. (15) and (17) into eq. (16), we can derive that :
+~K
(18)
¢~K 2.
Let R't denote a value of R at which Nj(2~E)= 0. [At this moment, all of the (BO1.5)° units in the glasses are exhausted.] From the relevant equations for N3~S2~E)in table 2, let N3~s2~E~= 0, for K = 2 and 3, R', can be found to be 3.33 and 4.00, respectively. On the other hand, from eq. (18), for K = 2 and 3, the values of R t equal 3.44 and 3.81, respectively. The corresponding values of R', and R, are approximately equal. Therefore, it seems to be reasonable to expect that in the region R o <~ R <~ R t, (BO2) 1- and (BOLs) ° units may take part in the structural.reactions simultaneously. This favors the assumption that reactions (8) and (9) occur simultaneously in the region R o <~ R <~ R t.
6. Comparison of the two different approaches with the experiment data [1] Table 4 shows the comparison of the values of theoretical slopes calculated from eqs. (3a) and (3b) and eqs. (14a) and (14b) with the slopes of the relevant empirical equations in table 1. In table 4, columns nos. 1 and 3 and columns nos. 2 and 4 correspond to the model in ref. 2 and the one in the present paper, respectively. Finally, we can consider how to establish equations for N4¢~/ and N3¢~c) for R >~ Ro, K < 8. From eqs. (2a) and (2b), it can be shown that at R o N4(c).Ro+ N3A(C),ao= 1; therefore, from eq.(6) N3s(c),Ro=0. But, from the relevant empirical equations for N3(~E)in table 2, it can be calculated that for K = 2 and
Table 4 Comparison of the values of theoretical slopes calculated from eqs. (3a) and (3b) and (14a) and (14b) with the slopes of the relevant empirical equations in table 1
sg~,
s~,
s~A,c,'2'
.2, °3A(C) °3A(E)
K= 2
-0.083 - 55% -0.15
-0.17 - 113% -0.15
0.017 - 7.1% 0.24
0.25 - 104% 0.24
K=3
-0.063 - 53% -0.12
-0.13 - 108% -0.12
0.013 - 7.2% 0.18
0.19 - 106% 0.18
Average Note a
54% (1)
111% (2)
7.2% (1)
105% (2)
a Note: The values of S are calculated or selected respectively from: (1) the relevant eqs. (3a) and (3b) and table l; (2) the relevant eqs. (14a) and (14b) and table 1.
S.Z. Xiao / The structural model of NMR studies
35
3, NaStE).Ro equal 0.14 and 0.12, respectively. Therefore, we now face two choices. (1) If we take into account the contribution of the loose symmetric triangles in the region R ~> R 0, K~< 8, as described in reaction (8), then we must modify reaction (1). For example, on the right side of reaction (1), we change a proper quantity of (B306) 3- over to a corresponding quantity of (BOi s) ° and (B205) 4- , according to the quantitative relation
1(B306)3- _>3 (BO1.5)0 + 3 (B205 )4-. (2) If we use reaction (1) for Rmax ~< R ~< R 0, K~< 8, then we must theoretically exclude the existence of the loose symmetric triangles. The second case is selected in the present paper. Consequently, we have to select the slope described in eq. (12), which is the single contribution of reaction (9), to establish the equations for N4~{~)and N3~2~c).Then, we have: forR 0 <~R<~Rt,K<~8 N4((2c ) ) = N4~81-[0.5/(1 + K ) ] ( R -
N3(2~c)--N3sA)C)+ [ 0 . 5 / ( 1
R0),
(19a)
+ K ) ] ( R - Ro),
(19b)
where N4~8) and N3~]c) denote the values at R 0 of N4(c) and N3A(C) calculated from eqs. (2a) and (2b), respectively. A comparison of these equations with the experimental data of ref. 1 is
~44¢) 0.6
A
•
•
0
o.#
0.2
°'.~.
°!,t
o.~
ll/~ c~.> A/3a~;
Fig. 1. Plot of experimental values of N4 and N3A of ref. 1 against corresponding values calculated from eqs, (3a) and (3b) and (19a) and (19b), respectively. Open circles and triangles correspond to the values calculated from eqs. (3a) and (3b), respectively. Full circles and triangles correspond to the values calculated from eqs. (19a) and (19b), respectively.
36
S.Z, Xiao / The structural model of NMR studies
shown in fig. 1. The straight line in fig. 1 represents N4(E) = N4(c) o r
N3A(E ) ~-
N3A(C)-
From fig. 1 it can be seen that the full circles and triangles corresponding to this model are closer to straight lines than those from the model of ref. 2. 7. Discussion
When the experimental conditions (e.g. the thermohistory of the glass) varying, the shares of Na20 obtained by reactions (8) and (9) might not be equal, then the function Rtlk) may also be modified, whereas it would still be of 2nd order. For eq. (18), let d R J d K = O, the maximum condition of R t with variation of K can be resolved as K=3.7 R t ] m a x "~
3.89.
The glass composition corresponding to R t] max is within the glass-forming area [4] in this ternary system. For R>~Rt, K<~8, the probable reactions occurring with Na20 being introduced in the glasses are 2[(B306)3- + 3 Nal+] + 3 Na20 ---~3[(B205)4- + 4 Nal+], [(B306)3- +3 Na'+] + 3 Na20 ----~3[(BO3)3- + 3 Nal+], and [(B205) 4- + 4 Na '+] + Na20--* 2[(BO3) 3- + 3 Na'+]. It is impossible to derive the sharing correlation of additional Na20 among the three reactions and other details, because of the absence of experimental data in this region. However, it can still be presumed that during this course of reaclion, (BSi4010) 1 units (which should be rewritten as [(BO2) 1(4 SiO2PNa20)] because 4 SiO2 have already combined with a certain number of Na20 units at this time) remain unchanging, then N 4 ~ constant < 1. It should be pointed out that this expression for R >~ Rt, K~< 8 is strictly theoretical. It would be rather interesting if this could be verified experimentally. However, if the changes proceeding in the real glasses are wholly or sufficiently described by reaction (1), then no loose symmetric triangles remain; consequently this model is probably not applicable, and the model proposed in ref. 2 is perhaps applicable. It appears that the experiments dealing with both N 4 and N3A systematically as reported in ref. 1 are to be desired. This model, of course, is simplified, and is merely one model of several possible. The variation in the real glasses is more complicated as mentioned in refs. 1 and 5. The author agrees with the other points of view in refs. 1 and 2.
S.Z, Xiao / The structural model of NMR studies
37
8. Conclusion The structural model and corresponding equations suggested in this paper enable the transformation among the boron oxygen coordinate polyhedra for R o < ~ R < ~ R t, K < ~ 8 to be better explained than that proposed in ref. 2. The distinct difference between the two models is that when R passes R 0 the present model claims that both N4~c) and N3A(c) break to form two straight lines which intersect at R 0, while the model in ref. 2 claims only N3A~CI shows this behaviour; in addition, the slope of N3a(c) for R 0 ~< R ~< R, for this model is 10 times greater than that of ref. 2. Correspondingly, according to this model, after the exhaustion of the diborate units, the loose tetrahedron units and metaborate units may take part in reactions one after another rather than together as suggested in ref. 2, and the point at which the orthoborate units start to be created may be at higher values of R than suggested in ref. 2. The authors expresses his deep appreciation to Mr. S.H. Liu and Mr. S.X. Wang for encouraging him to write this article, and also to Miss X. Xu for assistance in preparing the manuscript.
Appendix Fig. 1 in ref. 1 shows that, of all the 36 experimental glasses, glass no. 20 is the nearest to the boundary of the glass-forming area. Although it has been shown by means of X-ray diffraction techniques that there is no sign of devitrification in all of the glasses [1], this check might not be absolutely reliable, if the crystallization in question were stopped at the initial stage by quenching. For example, it has been reported that earlier crystallization could be detected by electron diffraction techniques [8] and electron microscopy [9], but by the X-ray diffraction techniques in the same glasses [8,9]. Since the composition point of glass no. 20 is located within the first crystallization area of sodium metaborate in the phase diagram [6] of this ternary system, it seems that this crystalline phase would deposit if the said crystallization could take place. This crystal wholly consists of ring-type metaborate units [7]. It may be presumed that in the course of the crystallization, reaction (8) would change into 2(BO~.5)° + Na20 ~ [ 2 (B306)3- +2 Na' + 1 [Now S3s is twice that calculated from eq. (13)], and some preferential sharing of introduced Na ~+ ions for borate network and for this reaction would occur as well. Thereby, it allows N3s~z) to decrease sharply. Then, this can explain the particular fact that for K = 1, the absolute value of S~21c~calculated from eq. (13) is far less than that of S3~IE), the mutual ratio is only 35.1%, as shown in table 3.
38
S,Z. ~Xiao / The structural model of NMR studies
Because the composition point of glass no. 20 is located far from the area of phase separation recognized in the general literature [10,11], the probability of phase separation occurring in glass no. 20 is suggested to be negligible by the present author. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Y.H. Yun and P.J. Bray, J. Non-Crystalline Solids 27 (1978) 363. Y.H. Yun, S.A. Feller and P.J. Bray, J. Non-Crystalline Solids 33 (1979) 273. M.J. Park and P.J. Bray, Phys. Chem. Glasses 13 (1972) 50. M. Imaoka, J. Ceram. Ass. Japan 69 (1961) 282. W.L. Konijnendijk, Philips Res. Rep. Suppl. No. 1 (1975). E.M. Leivin et al., Phase diagram for ceramcists, 184, Second ed. (Am. Ceram. Soc., 1969) fig. 515. M. Marrezio et al., Acta Cryst. 16 (1963) 594. S.M. Brekhovskikh et al., Fiz. Khim. Stekla 3 (1977) 172. M.E. Milberg et al., Phys. Chem. Glasses 13 (1972) 79. W. Haller et al., J. Am. Ceram. Soc. 53 (1970) 34. Z.D. Alekseeva et al., Fiz. Khim. Stekla 3 (1977) 114.
29
A DISCUSSION ABOUT THE STRUCTURAL MODEL OF "NUCLEAR MAGNETIC RESONANCE STUDIES OF GLASSES IN THE SYSTEM Na20-B203-SiO2" S.Z. Xiao Glass Lahoratoo,, Be(ring Glass Research Institute, Hong-Qiao, Bei]ing 100062, People 2 Republic ~?f China Received 25 December 1979 Revised manuscript received 3 April 1981
The structural model for the system Na20-B203-SiO 2 suggested by Yun, Feller and Bray is discussed. A different structural model is suggested in this paper the results of which are in better agreement with experiment.
1. Introduction Yun and Bray [1] have proposed a structural model for the glasses in the system x N a z O - y B 2 0 3 - z S i O 2 to explain their N M R results, which were expressed as the fraction of boron atoms in boron oxygen tetrahedra (N4) and the fraction of boron atoms in asymmetrical boron oxygen triangles containing one or two non-bridging oxygen atoms (N3A). Recently, Yun et al. presented another paper [2] on the same subject, where they corrected the model proposed in ref. 1. In the present paper a different structural model for R ~ R0, K ~< 8 (R = x / y , R o is a value of R at which all of the diborate units are exhausted, and K = z'/y) is presented, which is in better agreement with the experimental data presented in ref. 1.
2. Relevant reactions and equations in refs. 1 and 2 The model in refs. 1 and 2 consists of a few reactions and a set of equation. The relevant reactions and equations in refs. 1 and 2 are summarized below.
+[1.5(BO 2)'
+ l . 5 N a 1+],
(1)
where (B407) z-, (B306) 3 and (BO2) 1- denote a diborate unit, ring-type metaborate unit and loose tetrahedron unit, respectively. 0022-3093/81/0000- 0000/$02.50 © 1981 North-Holland
30
S.Z. Xiao / The structural model of NMR studies
According to the suggestion proposed in refs. 1 and 2, the starting and ending point of reaction (1) correspond to Rma~ and R0, respectively. In the region
: Rma×~R~4(K+l.Z5Rma×) and K <~ 8, additional Na 1+ ions are proportionally shared between the borate and silicate networks. In the region R 0 ~< R ~<4 ( K + 1.25 Rmax) and K ~ 8 , (BO3) 3- and (BO2) 1- units are destroyed to form pyroborate (B205)4- units and orthoborate (BO 3)3- units. From the above assumptions, the relevant equations derived in refs. 1 and 2 are: for Rmax <~R~Ro, K<~8 N4(c)-----N4max - [ 0 . 2 5 / ( 1 + K ) ] ( R - Rmax) ,
(2a)
N3A(c, =[1.25/(1 + K ) ] ( R - Rmax);
(2b)
and for R o ~
(3a)
N3A(C) =[1.25/(1 + K ) ] ( R 0 - Rmax) + [0.05/(1 + K ) ] ( R - R0);
(3b)
and 1 Rma x = -i--~g -~"0.5, R 0 : Rrnax + (1 -[- g ) ( 1 -
(4) Rmax).
(5)
3. An improved treatment of the experimental data in ref. I The model in ref. 2 is generalized from the Li20 - B203 binary system. In this paper an attempt is made to extract directly information about the structural changes in the region R ~> R0, K<~ 8 from the experimental data in ref. 1. Hence, it is necessary to recalculate the empirical equation of the experimental data in ref. 1 using a least-squares fit; each family of data having the same value of Kis divided in two groups: Rmax ~ R <~ R o and R >~ R o. The superscripts (l) and (2) are employed to correspond to the regions Rmax ~ R R o and R 1> R 0, respectively. In all of the 36 experimental data points in ref. l, only data points nos. 20, 30, 35 and 36 belong to the case of R/> R o. these four points, unfortunately, are distributed among three groups of K (K = 1, 2, 3). Therefore, the empirical equations for N~(I~)or N(~]E) for R m~1~< R~< R o are first calculated, then, their values at R o (i.e. N4(~,R and N~,]E)R ) can be found by extrapolation. The empirical equations for'N°4({~)or N(2~E)' f(~r R i> R o are then calculated, using these values N(~E),Ro or N(~),R~ and the relevant
31
S.Z. Xiao / The structural model o f N M R studies
Table 1 E m p i r i c a l s t r a i g h t line e q u a t i o n s f o r N~2~), NJ2A~E) a n d t h e i r slopes as c a l c u l a t e d b y the a u t h o r
R~R
o ~-
K = 1 K= 2 K=3
-- 0 . 4 4 R + 1.08 - 0 . 1 5 R + 0.77 - 0 . 1 2 R + 0.81
- 0.44 - 0.15 -0.12
' "3A(E) --
~3A(E)
0 . 8 1 R - 0.79 0.24R - 0.070 0.18R - 0.049
0.8 l 0.24 0.18
data points in the region R/> R 0. For the data group K = 3, when the equation for N3C~E)for Rmax ~< R ~< R o is calculated, the theoretical initial point (R = Rmax = 0.69, N3A = 0) is used because there are only two points nos. 33 and 34. These equation for R t> R o and their slopes as calculated by the author are listed in table 1.
4. An analysis for the loose boron oxygen symmetric triangles Since N4 +N3A +N3s = 1, thenN3s = 1 - ( N 4 +N3A ).
(6)
Inserting eqs. (3a) and (3b) into eq. (6), we derive the equation for N3~s2~c)as follows: N(2)
3s(c) -
0.2
1 + 1~ ( R -
R0).
(7)
On the other hand, inserting the relevant empirical equations listed in table 1 into eql (6), we can derive the empirical equation for NCSZ~E).The equations for N3~E) and their slopes are shown in table 2. From eq. (7) it can be seen that N~sZ~c)increases for R t> R 0, according to the model of ref. 2. Obviously, this is consistent with the presumption about eqs. (3a) and (3b), which involves the formation of orthoborate (BO3) 3- units. A (BO3) 3- unit consists of one symmetric triangle with three non-bridging oxygen atoms [3]. In contrast to eq. (7), table 2 shows that N3~E) decreases. Hence, the author suggests that a new approach is required to reflect the case
Table 2 E q u a t i o n s f o r NJS2~E) a n d their slopes
R>~ R o
K= 1 K= 2 K= 3
N3(2) S(E) =
S3(~E)
- 0.37R + 0.71 - 0 . 0 9 0 R + 0.30 -- 0 . 0 6 0 R + 0.24
-0.37 - 0.090 -- 0 . 0 6 0
32
S.Z. Xiao / The structural model of NMR studies
of N3(~0E)decreasing, and that for R >~R 0 the number of (BO3) 3- units can be neglected in the region where N3(s2~)i>0. Then N(2}E) can be assigned to the fraction of loose boron oxygen symmetric triangles (BO,.5) °.
5. Two new reactions for R ~>R o, K~< 8
It is suggested in this paper that the following two reactions may take place simultaneously: forRo <~R,K<~8
2(BO,.5) ° + 2Na20~[(B205) 4 + 4 Y a ' + ] , 2[(BO2)'
+ N a '+] +
Na20---~[(B205)4- +4 Nal+].
(8) (9)
Now, it is still assumed that additional Na I + ions are proportionally shared between the silicate and borate networks as claimed in refs. 1 and 2. However, the sharing correlation of Na20 being introduced between reactions (8) and (9) ~"' by trial and error. Thus, the equation for the shares of can be selected as ~.~ Na20 obtained by reactions (8) or (9) is
(Na20)(8) -- (Na20)(9)= ½[l/(I + K)](Na20).
(I0)
Hence, for reaction (8) we have S(2) (8)0 4(C)
and "2)
(8)
s A(c) : 0.25/(1 + K).
(11)
For reaction (9) we have 4(2 (9)--0.5/(1 (C) ---~
+K)
and $3A, C)(2) (_--9)0.5/(1 + K ) .
(12)
According to reaction (8) the following expression holds: _ q(2) (8) V(2) °33(C) -- ~3A(C).
Taking eqs. (11) into account, we have $3(~{c)(8-)- 0.25/(1 + K).
(13)
Table 3 shows the values calculated from eq. (13) and their comparison with the corresponding empirical values in table 2.
33
S.Z. Xiao / The structural model of NMR studies Table 3 Values calculated from eq. (13) and their comparison with the corresponding values in table 2 (2) S3S(C) K= 1 K= 2 K ~- 3 Average
,~,(2) / ~'(21 o3S(C)/°3S~E)
--0.13 35.1% - 0.083 92.2% - 0.063 105% (Only for groups of K = 2 and 3) 98.6%
(Note: The author suggests that the earlier crystallization phenomenon may take place in glass no. 20 (see appendix for details), and for R ~> R0 the group of K = 1 has only one data point from glass no. 20, hence, the empirical slope of this group cannot be compared with the others.)
From table 3 it can be seen that eq. (13) may expect the rate of change of the loose symmetric triangles in the region R ~> R o, K ~< 8. Since reactions (8) and (9) proceed simultaneously, then the total rate of change of tetrahedra and of asymmetric triangles should be calculated from these two reactions. Thus, the two sides of eq. (11) should be added to the corresponding sides of eq. (12), then S~i(2!,, (t.) • - 0 . 5 / ( 1 + K),
(14a)
S(-1 3A(C) = 0 . 7 5 / ( 1 + K ) .
(1481
Now we will consider the region in which reaction (9) occurs. Let the end point of reaction (9) correspond to Rt. Rt is a value of R at which all of the (BO2) I- units are exhausted. The equation for R t is derived below. Solving eqs. (6) in ref. 1 we have m = l y --~6z,
(15)
where m denotes the number of (B407) 2- units corresponding to the start of reaction (1). According to reactions (9) and (1) it can be determined respectively that (Na20)/[(BO2)'-] = 1/2 and [(BO2)'-l/[(B407) z-] = 1.5/1. Taking eq. (10) into account, then it is easy to write that 1
1
(Xt --AT°) 1 + K 2 - m
1.5g,1
where Xt and Xo are the NazO contents corresponding to R t and R o, respectively. The two sides are divided by y, yielding R , - - R o = 1.5(1 +K)m/y, where R, = X , / y ,
Ro = Xo/y.
(161
S.Z. Xiao / The structural model of NMR studies
34
Inserting eq. (4) into eq. (5), we have R o = 1 +½K-~K
(17)
2.
Inserting eqs. (15) and (17) into eq. (16), we can derive that :
+~K
(18)
¢~K 2.
Let R't denote a value of R at which Nj(2~E)= 0. [At this moment, all of the (BO1.5)° units in the glasses are exhausted.] From the relevant equations for N3~S2~E)in table 2, let N3~s2~E~= 0, for K = 2 and 3, R', can be found to be 3.33 and 4.00, respectively. On the other hand, from eq. (18), for K = 2 and 3, the values of R t equal 3.44 and 3.81, respectively. The corresponding values of R', and R, are approximately equal. Therefore, it seems to be reasonable to expect that in the region R o <~ R <~ R t, (BO2) 1- and (BOLs) ° units may take part in the structural.reactions simultaneously. This favors the assumption that reactions (8) and (9) occur simultaneously in the region R o <~ R <~ R t.
6. Comparison of the two different approaches with the experiment data [1] Table 4 shows the comparison of the values of theoretical slopes calculated from eqs. (3a) and (3b) and eqs. (14a) and (14b) with the slopes of the relevant empirical equations in table 1. In table 4, columns nos. 1 and 3 and columns nos. 2 and 4 correspond to the model in ref. 2 and the one in the present paper, respectively. Finally, we can consider how to establish equations for N4¢~/ and N3¢~c) for R >~ Ro, K < 8. From eqs. (2a) and (2b), it can be shown that at R o N4(c).Ro+ N3A(C),ao= 1; therefore, from eq.(6) N3s(c),Ro=0. But, from the relevant empirical equations for N3(~E)in table 2, it can be calculated that for K = 2 and
Table 4 Comparison of the values of theoretical slopes calculated from eqs. (3a) and (3b) and (14a) and (14b) with the slopes of the relevant empirical equations in table 1
sg~,
s~,
s~A,c,'2'
.2, °3A(C) °3A(E)
K= 2
-0.083 - 55% -0.15
-0.17 - 113% -0.15
0.017 - 7.1% 0.24
0.25 - 104% 0.24
K=3
-0.063 - 53% -0.12
-0.13 - 108% -0.12
0.013 - 7.2% 0.18
0.19 - 106% 0.18
Average Note a
54% (1)
111% (2)
7.2% (1)
105% (2)
a Note: The values of S are calculated or selected respectively from: (1) the relevant eqs. (3a) and (3b) and table l; (2) the relevant eqs. (14a) and (14b) and table 1.
S.Z. Xiao / The structural model of NMR studies
35
3, NaStE).Ro equal 0.14 and 0.12, respectively. Therefore, we now face two choices. (1) If we take into account the contribution of the loose symmetric triangles in the region R ~> R 0, K~< 8, as described in reaction (8), then we must modify reaction (1). For example, on the right side of reaction (1), we change a proper quantity of (B306) 3- over to a corresponding quantity of (BOi s) ° and (B205) 4- , according to the quantitative relation
1(B306)3- _>3 (BO1.5)0 + 3 (B205 )4-. (2) If we use reaction (1) for Rmax ~< R ~< R 0, K~< 8, then we must theoretically exclude the existence of the loose symmetric triangles. The second case is selected in the present paper. Consequently, we have to select the slope described in eq. (12), which is the single contribution of reaction (9), to establish the equations for N4~{~)and N3~2~c).Then, we have: forR 0 <~R<~Rt,K<~8 N4((2c ) ) = N4~81-[0.5/(1 + K ) ] ( R -
N3(2~c)--N3sA)C)+ [ 0 . 5 / ( 1
R0),
(19a)
+ K ) ] ( R - Ro),
(19b)
where N4~8) and N3~]c) denote the values at R 0 of N4(c) and N3A(C) calculated from eqs. (2a) and (2b), respectively. A comparison of these equations with the experimental data of ref. 1 is
~44¢) 0.6
A
•
•
0
o.#
0.2
°'.~.
°!,t
o.~
ll/~ c~.> A/3a~;
Fig. 1. Plot of experimental values of N4 and N3A of ref. 1 against corresponding values calculated from eqs, (3a) and (3b) and (19a) and (19b), respectively. Open circles and triangles correspond to the values calculated from eqs. (3a) and (3b), respectively. Full circles and triangles correspond to the values calculated from eqs. (19a) and (19b), respectively.
36
S.Z, Xiao / The structural model of NMR studies
shown in fig. 1. The straight line in fig. 1 represents N4(E) = N4(c) o r
N3A(E ) ~-
N3A(C)-
From fig. 1 it can be seen that the full circles and triangles corresponding to this model are closer to straight lines than those from the model of ref. 2. 7. Discussion
When the experimental conditions (e.g. the thermohistory of the glass) varying, the shares of Na20 obtained by reactions (8) and (9) might not be equal, then the function Rtlk) may also be modified, whereas it would still be of 2nd order. For eq. (18), let d R J d K = O, the maximum condition of R t with variation of K can be resolved as K=3.7 R t ] m a x "~
3.89.
The glass composition corresponding to R t] max is within the glass-forming area [4] in this ternary system. For R>~Rt, K<~8, the probable reactions occurring with Na20 being introduced in the glasses are 2[(B306)3- + 3 Nal+] + 3 Na20 ---~3[(B205)4- + 4 Nal+], [(B306)3- +3 Na'+] + 3 Na20 ----~3[(BO3)3- + 3 Nal+], and [(B205) 4- + 4 Na '+] + Na20--* 2[(BO3) 3- + 3 Na'+]. It is impossible to derive the sharing correlation of additional Na20 among the three reactions and other details, because of the absence of experimental data in this region. However, it can still be presumed that during this course of reaclion, (BSi4010) 1 units (which should be rewritten as [(BO2) 1(4 SiO2PNa20)] because 4 SiO2 have already combined with a certain number of Na20 units at this time) remain unchanging, then N 4 ~ constant < 1. It should be pointed out that this expression for R >~ Rt, K~< 8 is strictly theoretical. It would be rather interesting if this could be verified experimentally. However, if the changes proceeding in the real glasses are wholly or sufficiently described by reaction (1), then no loose symmetric triangles remain; consequently this model is probably not applicable, and the model proposed in ref. 2 is perhaps applicable. It appears that the experiments dealing with both N 4 and N3A systematically as reported in ref. 1 are to be desired. This model, of course, is simplified, and is merely one model of several possible. The variation in the real glasses is more complicated as mentioned in refs. 1 and 5. The author agrees with the other points of view in refs. 1 and 2.
S.Z, Xiao / The structural model of NMR studies
37
8. Conclusion The structural model and corresponding equations suggested in this paper enable the transformation among the boron oxygen coordinate polyhedra for R o < ~ R < ~ R t, K < ~ 8 to be better explained than that proposed in ref. 2. The distinct difference between the two models is that when R passes R 0 the present model claims that both N4~c) and N3A(c) break to form two straight lines which intersect at R 0, while the model in ref. 2 claims only N3A~CI shows this behaviour; in addition, the slope of N3a(c) for R 0 ~< R ~< R, for this model is 10 times greater than that of ref. 2. Correspondingly, according to this model, after the exhaustion of the diborate units, the loose tetrahedron units and metaborate units may take part in reactions one after another rather than together as suggested in ref. 2, and the point at which the orthoborate units start to be created may be at higher values of R than suggested in ref. 2. The authors expresses his deep appreciation to Mr. S.H. Liu and Mr. S.X. Wang for encouraging him to write this article, and also to Miss X. Xu for assistance in preparing the manuscript.
Appendix Fig. 1 in ref. 1 shows that, of all the 36 experimental glasses, glass no. 20 is the nearest to the boundary of the glass-forming area. Although it has been shown by means of X-ray diffraction techniques that there is no sign of devitrification in all of the glasses [1], this check might not be absolutely reliable, if the crystallization in question were stopped at the initial stage by quenching. For example, it has been reported that earlier crystallization could be detected by electron diffraction techniques [8] and electron microscopy [9], but by the X-ray diffraction techniques in the same glasses [8,9]. Since the composition point of glass no. 20 is located within the first crystallization area of sodium metaborate in the phase diagram [6] of this ternary system, it seems that this crystalline phase would deposit if the said crystallization could take place. This crystal wholly consists of ring-type metaborate units [7]. It may be presumed that in the course of the crystallization, reaction (8) would change into 2(BO~.5)° + Na20 ~ [ 2 (B306)3- +2 Na' + 1 [Now S3s is twice that calculated from eq. (13)], and some preferential sharing of introduced Na ~+ ions for borate network and for this reaction would occur as well. Thereby, it allows N3s~z) to decrease sharply. Then, this can explain the particular fact that for K = 1, the absolute value of S~21c~calculated from eq. (13) is far less than that of S3~IE), the mutual ratio is only 35.1%, as shown in table 3.
38
S,Z. ~Xiao / The structural model of NMR studies
Because the composition point of glass no. 20 is located far from the area of phase separation recognized in the general literature [10,11], the probability of phase separation occurring in glass no. 20 is suggested to be negligible by the present author. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Y.H. Yun and P.J. Bray, J. Non-Crystalline Solids 27 (1978) 363. Y.H. Yun, S.A. Feller and P.J. Bray, J. Non-Crystalline Solids 33 (1979) 273. M.J. Park and P.J. Bray, Phys. Chem. Glasses 13 (1972) 50. M. Imaoka, J. Ceram. Ass. Japan 69 (1961) 282. W.L. Konijnendijk, Philips Res. Rep. Suppl. No. 1 (1975). E.M. Leivin et al., Phase diagram for ceramcists, 184, Second ed. (Am. Ceram. Soc., 1969) fig. 515. M. Marrezio et al., Acta Cryst. 16 (1963) 594. S.M. Brekhovskikh et al., Fiz. Khim. Stekla 3 (1977) 172. M.E. Milberg et al., Phys. Chem. Glasses 13 (1972) 79. W. Haller et al., J. Am. Ceram. Soc. 53 (1970) 34. Z.D. Alekseeva et al., Fiz. Khim. Stekla 3 (1977) 114.