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Ultrasonic velocity in potassium borate glasses
Journal of Non-Crystalline Solids 127 (1991) 65-74 North-Holland
65
Ultrasonic velocity in potassium borate glasses Masao Kodama D e p a r t m e n t of Industrtal Chemtstry, K u m a m o t o Institute o f Teehnolog)', Ikeda, K u m a m o t o 860, J a p a n
Received 23 February 1990 Revised 11 September 1990
Ultrasomc velocity m potassium borate glasses denoted by the composition formula x2K20-xlB203 (x I and x 2 are mole fractions of B203 and K20, respectively, and x 1 + x 2 = 1) is measured as a function of x 2 at a frequency of 10 MHz and at a temperature of 298 K by the ultrasomc pulse echo overlap method. The elastic property of these glasses is analyzed m terms of the three structural units defined as Bf03 ~ a, K + B O 2 0 - ~- b, and K + BO 4 ~ c, where O represents a bridging oxygen and O - a non-bridging oxygen, on the assumption that these structural umts have their respective elastic constants. The elastic constants of these three structural units are defined on the basis of the elastic internal energy due to deformation. It is shown numerically that the structural unit c increases the rigidity of the glass whereas the structural unit b decreases it. By use of the values of the three elastic constants, the fraction, N 4, of boron atoms in tetrahedral coordination is calculated as a function of x2/(1-x2). It is shown especially that non-bridging oxygen begins to form at x 2 = 0.065 and also the rate at which N4 increases w~th increasing x 2 changes suddenly at x 2 = 0.20.
1. Introduction
We shall denote the composition of alkali borate glasses by the formula x 2 R 2 0 " X l B 2 0 3, where R denotes one of the alkali metals (Li, Na, K, Rb, and Cs), x 2 denotes the mole fraction of R20, and x I = 1 - x 2 denotes the mole fraction of B203. One property affected by changes in structure is the velocity of sound in the resulting glass. Studies on the velocity of sound in alkali borate glasses are not numerous at present. Lort~sch et al. [1] carried out Brillouin scattering experiments on glasses of the R20-B203 systems (R = Li, Na, K, Rb, and Cs) and reported the hypersonic velocity in glasses of each system as a function of composition. In the case of potassium borate glasses, however, the number of compositions studied by them is insufficient for a discussion of elastic properties of these glasses as a function of coffaposition. The purpose of the present paper is to measure the ultrasonic velocity in potassium borate glasses as a function of composition and to study structural changes of these glasses.
It has been pointed out that the structure of alkali borate glasses is strongly dependent upon the species of alkali ion [1-3]. An early study by Bray and O'Keefe [4] on the nuclear magnetic resonance ( N M R ) of alkali borate glasses led to the conclusion that the fraction. N 4, of boron atoms in tetrahedral coordination followed the relation of N 4 = x2/(1 --X2) up tO x 2 ~ 0.30 with respect to all alkali ion species and then deviated from this relation as x 2 increased further. However, Zhong and Bray [2] have recently refined their N M R techniques and have shown that the measured N 4 value is always less than the value of x2/(1 -x2) except for a region of low x 2 where no data are reported (x 2 = 0.17, 0.25, 0.33, 0.40, and 0.45) and also that the difference, x2/(1 - x 2) - N 4, becomes larger as the size of alkali ion increases. This result indicates that a small amount of non-bridging oxygen is already formed even at x 2 = 0.17 for all alkali ion species and also that the amount of non-bridging oxygen increases as the size of alkali ion increases. Also Chryssikos et al. [3] have interpreted their R a m a n spectra at
0022-3093/91/$03.50 © 1991 - Elsevier Science Pubhshers B.V. (North-Holland)
M. Kodama / Ultrasomc velocity m potasstum borate glasses
66 \
\ 0\ /
/
0\ B--O-
0
/
/
a
\0 B--O-
0
K"
/0
b
0/ \B// \
K÷
potassium borate glasses are discussed in relation to the behavior of N4 as a function of x2/(1 - x2).
O\
c
2. Theoretical background
Fig. 1. Three s t r u c t u r a l units m p o t a s s m m b o r a t e glasses. The
structural units a, b, and c have the formulae B O 3, K + B O 2 0 , and K+BO~, respectively, where ~ represents a bridging oxygen and O- a non-bridgingoxygen. Both the units b and c can be written also as K+BO2-. It should be noted that potassmm ion is also included in the structural units b and c.
x 2 = 0.17 to conclude that N4 decreases and the amount of non-bridging oxygen increases as R is changed from Li to Cs. In alkali borate glasses, there exist several borate groups such as boroxol ring, pentaborate group, triborate group, and diborate group as explained by Krogh-Moe [5] and by Konijnendijk [6]. Raman scattering studies by Lor/~sch et al. [1] and by Chryssikos et al. [3] on these glasses have indicated that, when R is changed from Li to Cs, different kinds of borate groups are formed even if x 2 is the same. Thus, we cannot discuss the structure and properties of alkali borate glasses in the same way without taking notice of the difference in the alkali ion species. With regard to potassium borate glasses studied in this paper, an approximate relation of N 4 as a function of x 2 can be obtained from the study by Zhong and Bray [2], while on the other hand the kinds of borate groups being present in this system are not fully studied as a function of x 2 at present. Thus, this paper assumes the three structural units [7-9] shown in fig. 1 as the primary entities present in potassium borate glasses. These structural units are the most abundant ones in the composition range studied in this paper (0 < x 2 < 0.34) [3,9]. We shall assume that these structural units exhibit their inherent elastic properties and hence have their respective elastic constants. The elastic constants of these three structural units are defined on the basis of a theory of elastic internal energy described in a previous paper [10]. By use of these elastic constants, structural changes in
It is shown in the previous paper [10] that three different expressions for elastic internal energies can be obtained by taking the internal energy relative to unit mass, unit volume, and unit amount of substance. The expression for the elastic internal energy relative to unit amount of substance is extended in the following in order to obtain the elastic constants of the structural units. The amount of a substance is expressed always in the unit of mole in this paper. In considering a deformed body at some temperature, we choose the undeformed state to be the state of the body in the absence of external forces and at the temperature. Let a,, i = 1, 2, 3, denote the Cartesian coordinates of a material particle before the deformation and let x, (al, a2, a3) denote the coordinates of the same particle after the deformation. The Lagrangean strain components ,/,j that describe finite elastic strains [11,12] are defined as 0xk Oxk
77,/=½ 3a, 3a/
) 8,, ,
i, j = 1 , 2 , 3 .
(1)
Here the summation is taken over the repeated subscript k. The symbol 8,/denotes the Kronecker delta. Instead of the tensor notation, we use in the following the abbreviated Voigt notation denoted by the subscript a which runs from 1 to 6 : 1 1 - 1, 22-2, 33-3, 23-4, 31-5, 12-6. The Lagrangean strain in the abbreviated notation is defined by Brugger [12] as
~/,, = 2 ~ , , / ( 1 + 8u).
(2)
Since the sound wave propagates under adiabatic conditions, the internal energy can be chosen as the most convenient thermodynamic potential. Let the molar mass, the molar internal energy, and the molar entropy be denoted by M, Um, and Sm, respectively. Then, the fundamental thermody-
67
M. Kodama / Ultrasomc velocttv m potassium borate glasses
namic equation for the molar internal energy of a deformed body can be expressed as d U m = TdS~ + ( M / p ) G d ~ .
(3)
Here, T is the temperature, p is the density of the undeformed state, t~ = (o/M)(OUm/3~l~)Sm is the quantity called the thermodynamic tension, and the summation is taken over the subscript a from 1 to6.
In order to be able to apply the general formula of the molar internal energy to any particular deformation, we must know the molar internal energy of the body as a function of the strain. This expression can be obtained by using the fact that the deformation is small and hence by expanding the molar internal energy in powers of ~ . Since the glass is elastically isotropic, we shall consider only isotropic bodies in the following. Since t~ = ( p / M ) ( ~ U m / 3 1 1 a ) s m is zero when no external stresses are present, it follows that there is no first order term in the expansion of the molar internal energy in powers of %. The second derivative of the molar internal energy with respect to the Lagrangean strain, keeping the entropy constant, is given by
( ~2Um) 3~
=MV
2.
(4)
1 2(1 - x 2 )
=
(
1
{ x 2 K 2 0 + (1 - x2)B203 ) x2 ) B O 3 + x2 K + B O 2 , 1 --x z 1 -- x 2
Here V is the velocity of a sound wave having the strain component ~/~ ( a = 1, 2, 3 refers to normal strains or stresses for the longitudinal wave and a = 4, 5, 6 refers to the shear components for the transverse wave). If the strain occurs under adiabatic condition, the molar internal energy of the body increases. Expanding the molar internal energy in powers of the Lagrangean strain ~/~ at constant entropy about the state of zero strain and neglecting all terms above the second order, we obtain the increment, bU m, of the molar internal energy due to the elastic deformation, or the elastic internal energy relative to unit amount of the substance:
(1/2)MV2 1 . (5)
(6)
where BO 3 represents the structural unit a and K+BOz- represents both of the structural units b and c. By use of N4, the second term on the right of eq. (6) can be expressed further in terms of the structural units b and c:
x2 K+BO~=( ~
1-x 2
sm
AUm = Um(S m, 71,,) - U m ( S m, 0) =
The constant MV z expresses the elastic constant relative to unit amount of the substance since this constant determines the increment of the molar internal energy caused by a given strain. Equation (5) holds for both the longitudinal velocity, V~, and the transverse velocity, V~. We shall now consider how to express the elastic constants of the three structural units in potassium borate glasses. For this purpose, it is necessary to derive a chemical formula representing the structural units from the composition formula. It may be reasonable to use such a formula in which the amount of boron is always kept constant even if x 2 is varied. We shall first convert the composition formula into the formula in which 1 mol of boron is always contained:
12
1-x 2
)
N4 K + B O 2 0
+ N4K+BO4 .
(7)
For the sake of simplicity, we shall write the three structural units as
BO 3 ~ a,
(8)
K + B O 2 0 ---- b,
(9)
K + B O 4 --- c.
(10)
Now the composition formula can be expressed in terms of the three structural units as 1 2(1 - x2) ( x 2 K 2 0 + (1 - x2)B20 ,
=
( x2)(x2 ) a+ 12~c2 N4 1
1---x,
b+N4c. In)
68
M. Kodama / Ultrasomc velocity in potasstum borate glasses
We shall notice that the molar mass, M c, of the composition formula x2K20- (1 - x2)B203 is defined as
M¢ = x2MK: o + (1 -- x2) MB2O,,
(12)
where MK2 o and MB:o3 denote the molar masses of the component K 2 0 and the component B203, respectively. This definition of Mc contains actually 2x 2 mol of potassium, 2(1 - x2) mol of boron, and (3 - 2x2) mol of oxygen. We shall define the molar mass, M s, of the formula concerning the structural units on the right of eq. (11) as M~=2(1M~
__X2)
-(1
x2 ) l__X2 M~ X2
+(1
N4) Mb + Na M ~
X2
(13) where M a, M b, and M c are the molar masses of the structural units a, b, and c, respectively. This definition of M s contains always 1 mol of the structural units a, b, and c. Let Urn.s be the molar internal energy of the mixture of the structural units on the right of eq. (11). Also let Um,a, Urn,b, and Urn.¢ be the molar internal energies of the structural units a, b, and c, respectively. The molar internal energy of the formula on the right of eq. (11) can then be expressed
For the sake of brevity, we shall omit the subscript Sin.~ tO the derivatives in the following. By noticing eq. (4), the second-order derivative on the left of eq. (15) can be expressed as 02Um's
The quantity MsV 2, which can be measured directly, represents the elastic constant relative to 1 mol of the mixture of the structural units on the right of eq. (11). We see, from eqs. (4) and (5), that the second-order derivative of the molar internal energy of a substance with respect to the Lagrangean strain is equal to the elastic constant relative to 1 mol of the substance. Thus we can define the elastic constants E a, E b, and E¢ of the structural units a, b, and c respectively as
02Um,~ 0g/2 -- E a , 0rt2
Um, s =
(
1
t
(
1 - x2
t
3,/2"
(Omat + N . / ~
E~.
(19)
Substituting eqs. (16)-(19) into eq. (15), we can finally relate Ms V2 to the elastic constants of the three structural units:
(
\1
X2
1 - x2_
_
X2
N4)E b (20)
(14)
Let Sin,~ be the molar entropy of the formula on the right of eq. (11). Differentiating eq. (14) twice with respect to r/~ at constant Sin,~, we have
"+- 1 -- X 2
(18)
+ N4E c .
-1- N4Um, c .
)
E b,
0 2Um,c
Msv 2
N4 Umb "
(17)
02Um,b
as
x2 Um,a+ 1 - x2
(16)
M,V:.
0n2~
N4
0,rl 2
.
3. Experimental
3.1. Glass preparation
t
Sm.s
(15)
A series of potassium borate glasses denoted by the composition formula x2K20" x|B203 was prepared at such intervals that the increment of x 2 equals 0.02, where glasses are formed below x 2 0.35 [13]. In order to obtain good transmission of ultrasonic waves, it is necessary to prepare glasses
M. Kodama / UItrasomc velocttv m potasstum borate glasse~
with high homogeneity and without strains or bubbles. With the aim of preparing glasses with high homogeneity, the starting materials were initially made to react in an aqueous solution. A potassium hydroxide solution of about 1.0 m o l / d m 3 was prepared by dissolving electronic grade K O H (about 99% purity) in distilled water and this solution was used as the starting material for K20. After the concentration of this solution had been determined with a potentiometric titration method, the required volume was taken from a burette. As the starting material for BzO3, electronic grade H3BO 3 (more than 99.7% purity) was used. Amounts of the starting materials calculated to give 30 g in the melts were dissolved by adding distilled water in a beaker. Then the solution was transferred to a dry box and after complete evaporation of the water a chemically reacted powder was obtained. By making use of a SiC resistance electric furnace, the powder was fused in a 20 cm 3 platinum crucible which was placed in an alumina crucible. The fusion was carried out at temperatures from 900 to 1300°C for about 4 h with occasional stirring using a platinum wire; these temperatures were decreased as x 2 increased. Then the liquid was poured into a cylindrical graphite mold 15 m m in diameter and 23 m m in depth which had been preheated at the glass transition temperature in an electric muffle furnace. This glass transition temperature was determined from Shelby's experimental points [13] through which a smooth curve was drawn. Subsequently, the cast glass in the mold was held at the glass transition temperature for 2 h, then cooled at a rate of 1 K / m i n to room temperature while passing dry nitrogen through the muffle furnace. The residual liquid was poured onto an aluminium plate and later used for chemical analysis. Sometimes small bubbles were observed in the cast glasses, but these glasses were not used in the present experiment. The glass of each composition was stored in a hermetic vial. The compositions of all the glasses were analyzed with respect to both x 1 and x 2 with a neutralization titration described in the previous paper [10]. Each composition was determined by five analyses; the probable error was less than
69
2 × 10 - 4 mole fraction for all the compositions studied. Although the density, p, of each glass is not the requisite quantity for the present theory, it is necessary in order to calculate ordinary elastic constants such as Young's modulus, For this reason, the density of the cast glass was also measured at 298 K by a hydrostatic weighing method described in the previous paper [10]. The error in the measurements of density was within 8 × 10 4 g cm -3.
3.2. Ultrasonic velocity In order to measure the ultrasonic velocity, each glass specimen must have a pair of end faces that are flat and mutually parallel. Each glass was first ground on a glass plate using SiC abrasives by setting it in a holder to maintain the two faces parallel and subsequently polished by hand with fine alumina abrasive and machine oil on a flat glass plate. During polishing, the length of the specimen was occasionally measured at the center and the four corners with a micrometer reading to 1 ~m. The polishing was continued until the lengths of five portions coincided to within 2 ~tm; the final length at the center was taken as the length of the specimen. Dimensions of polished glasses were 15 m m in diameter and normally 12 m m long. Inspection with a strain viewer showed that all the specimens were transparent and almost free from stress. Ultrasonic travel time was measured at a frequency of 10 M H z and at a temperature of 298 K by use of the pulse echo overlap method [14]. The apparatus used was constructed in the present author's laboratory and the circuitry has been described in a previous paper [15]. X-cut and Y-cut quartz transducers resonating at a fundamental frequency of 10 M H z were used for the generation and detection of the longitudinal wave and of the transverse wave, respectively. A pair of electrodes of the transducer was deposited, by evaporating at first chromium and then gold, onto each quartz disk with a diameter of 10 m m so as to have a circular active area with a diameter of 8 m m and so as to have two electrode terminals on one surface of the disk. The transducer was bonded
M. Kodama / Ultrasonic velooty m potasstum borate glasses
70
Table 1 Analyzed composition, molar mass of composition formula, density, and ultrasonic velocities of longitudinal and transverse waves of potassium borate glasses at 298 K
x2 0 0.0192 0.0405 0.0614 0.0823 0.1016 0.1227 0.1415 0.1609 0.1780 0.1984 0.2206 0.2391 0.2599 0.2800 0.3011 0.3209 0.3402
M+
p
v~
v,
(g mol - t )
(g cm -3)
(km s - t )
(km s - t )
69.62 70.09 70.62 71.13 71.64 72.12 72.64 73.10 73.57 73.99 74.50 75.04 75.50 76.01 76.50 77.02 77.51 77.98
1.838 1.894 1.948 1.992 2.028 2.050 2.072 2.088 2.103 2.118 2.138 2.168 2.197 2.229 2.257 2.282 2.297 2.306
3.469 3.730 3.964 4.132 4.264 4.332 4.398 4.439 4.472 4.512 4.560 4.656 4.749 4.856 4.920 4.962 4.958 4.914
1.901 2.050 2.186 2.280 2.351 2.375 2.400 2.412 2.423 2.444 2.480 2.546 2.605 2.669 2.705 2.719 2.701 2.653
4. Results
Table 1 compiles the basic data obtained with the present series of experiments. Figure 2 shows the results of the velocity of sound measured by Lor~Ssch et al. [1] at a frequency in the order of 10 G H z and by the present author at a frequency of 10 MHz. These two sets of experimental data are in close agreement especially in the case of longitudinal velocity, from which we see that the frequency dispersion effect on the velocity of sound is small for potassium borate glasses. In order to elucidate the elastic properties of the structural units in potassium borate glasses, we shall plot MsV 2 against x 2 / ( 1 - x 2 ) . Figure 3 shows the plot of the longitudinal elastic constant, MsVl2, against x 2 / ( 1 - xz). Figure 4 shows the plot of the shear elastic constant, M y t 2, against X2/(1X2). The difference between msVl 2 and MsVt2 stems from the volume effect that a change
I
I
i
I
I
I
5.0
0
0 0
0
0
0 0 0 Q
to the specimen on one of the two parallel faces with phenyl benzoate [16]. The specimen with the transducer attached was set in a hermetic specimen holder and controlled at 25 ° C by placing the holder on a water bath. McSkimin [17] proposed a criterion for determining the correct cyclic overlap between echoes with the purpose of measuring the ultrasonic travel time with high accuracy. The McSkimin criterion was extended in the previous paper [10] to become applicable to ordinary glass specimens and this was used in the present measurements. Once a pair of echoes are properly overlapped according to the McSkimin criterion, it is possible to measure the ultrasonic travel time within an error of 0.02% for round trips greater than 5 ~s [17]. Since the velocity of sound is equal to the propagation distance divided by the travel time and the error in the measurements of sample length is within 0.02%, the error in the present measurements of ultrasonic velocity is within 0.04%.
0 0
0
0
0 0 U~
E
0
o=
~4.0
3.0
0
0
0
I 0
El 0
El
I• 13 0
0
0
0
U
0
3.0~-
0
T
2O
I
I
O.I
I
I
I
I
02 03 g2 Fig. 2. Velocities of sound in potassium borate glasses as a function of x 2. V~ ((D) and Vt (El) measured by this author. VI (e) and Vt (m) measured by Lor/Ssch et al. [1]. 0
71
M. Kodama / Ultrasomc veloctty m potassmm borate glasses I
I
I
I
1.5
/o
I
5. Discussion o
o
o
/ o/
o
E 1.0 ~E
/o.O .o~°~°/
05
00
?
I
0,l
I
02
I
0.3
1
0.4
[
05
X21(l ~X 2 )
Bray and O'Keefe [4] proposed a model of the structural change common to all alkali ion species that, on addition of R 2 0 to B203, the structural unit a is converted only into the structural unit c up to x 2 - - 0 . 3 0 and, on further addition of R20, the structural unit c is converted into the structural unit b. On the other hand, the recent N M R study by Zhong and Bray [2] has shown, first, that N 4 < x2/(1 - x 2) always for the composition range above x2/(1 - x 2) = 0.20 and hence the structural unit a can be converted into both the structural units b and c for this composition range and, second, that N 4 is a function not only of x J ( 1 x2) but also of the size of alkali ion species. However, we cannot ascertain from the study by Zhong and Bray [2] whether the structural unit a is converted into only the structural unit c or into both the structural units b and c when a slight
Fig. 3. The plot of the longitudinal elastic constant MsVl 2 against x2/(1 - x2). The hne (a) indicates the first term on the
right of eq. (20), which represents the contribution of the structural unit a to MsVi2. The three straight lines through datapoints show only the trend of the points.
0.5
I
I
I
I
I
O 0.4
in volume due to compressions and expansions is involved in longitudinal strains while no change in volume is involved in shear strains. We see however that both plots have almost the same characteristic. Neither MsVI2 nor MsVt2 change in a monotonic way with an increase in x 2 / ( 1 - x2) but exhibit sudden changes in slope at definite compositions. Each plot can be divided into the four segments having different slopes; the first three segments can be approximated by straight lines except for a few points nearest their boundaries while the last segment has a curvature. These composition ranges are 0 < x2/(1 - x2) < 0.07 (or 0 < x 2 < 0.065), 0.07 < x 2 / ( 1 - x 2 ) < 0.24 (or 0.065 < x 2 < 0.20), 0.24 < x2/(1 - x2) < 0.38 (or 0 . 2 0 < x 2<0.28), and 0 . 3 8 < x 2 / ( 1 - x 2 ) < 0.52 (or 0.28 < x 2 < 0.34).
O
o
/;
o.3 ~
/
~0.2 Ig
/
of°F
o
? 0.11
00
~
~ 1
0.1
.
(
a
) I
I
012 0.3 0.4 x2/(1 - X 2 )
I
0 5
Fig. 4. The plot of the shear elastic c o n s t a n t MsVt2 against x2). The line (a) indicates the first term on the right of eq. (20), which represents the contribution of the structural unit a to MsVt2. The three straight hnes through datapoints show only the trend of the points.
x2/(1 -
72
M. Kodama / Ultrasomc t,eloctty m potasstum borate glasses
amount of R : O is added to B203 at a composition below x2/(1 - x 2) = 0.20. We shall first assume that the relation of N0 = X 2 / ( 1 -- X2) holds in the range 0 < x 2 < 0.30 and thus in the range 0 < x2/(1 - x2) < 0.43 as the early N M R study by Bray and O'Keefe has shown [4]. Substitution of N 4 = x2/(1 - x 2 ) into eq. (20) gives
Table 2 The values of Ea, Eb, and Ec
My 2= 1
calculated by fitting the least squares parabola to nearby points. The value of E b c a n now be calculated by use of eq. (20) at the point of x2/(1 - x 2) = 0.50. Table 2 shows the values of E~, E b , and E c calculated in this way for each of the longitudinal and the transverse waves. It seems unusual that E b takes a negative value. However, the negative value is permissible since the structural unit b does not arise alone but coexists always with the structural unit c. Since E b is negative, we conclude that the structural unit b decreases the rigidity of the glass, which can be attributed to the fact that the structural unit b acts to break down a covalent bond of the network. The value of E c is ten times as great as the value of E a for each of the longitudinal and the transverse waves. This difference indicates that the structural unit c increases the rigidity of the glass, which can be attributed to the fact that the structural unit c forms a three-dimensional covalent bond in the network. Rewriting eq. (20) into the form
1 -x2x 2 Ea + T-Z~x2Ec,
(21)
so that the plot of M~V 2 against X 2 / ( 1 - - X 2 ) should be a single straight line in the composition range 0 < x2/(1 - x2) < 0.43 if this assumption is valid. Since each plot of MsVi 2 and MsVt2 against X 2 / ( 1 -- X2) is by no means a single straight line in the composition range 0 < x2/(1 - x2) < 0.43, it is evident that not only the structural unit c but also the structural unit b forms in this composition range. Discussions about the viscosity of alkali borate glasses as a function of composition [8] and on the acidity and basicity of the structural units [9] point out that, when a slight amount of K 20 is added t o B203, the structural unit a is converted only into the structural unit c. Thus, in the first segment of the composition range 0 < x2/(1 - x 2) < 0.07, the structural unit a is converted only into the structural unit c and therefore this segment can be described by eq. (21). If the structural unit b forms, then the slope of the plot of MsV 2 against X 2 / ( 1 - - X 2 ) should decrease since the structural unit b destroys the network to decrease the rigidity of the glass. The slopes of the second and the third segments are smaller than that of the first segment. This decrease indicates that the structural unit b now forms above the composition of x 2 / ( 1 - x 2 ) = 0.07. Then the relation of M y 2 as a function of x2/(1 - x2) should be described by eq. (20). We shall evaluate E~, E b, and E c. By use of eq. (21), the values of E~ and E c can be calculated from the first segment of the plot of M y 2 against X 2 / ( 1 -- X2). In order to calculate the value of E b, it is necessary to use one known value of N4 at a given composition. For this purpose, we shall use the value of N0 = 0.34 at the composition of x 2 / ( 1 -x2) = 0.50 determined by Zhong and Bray [2]. The value of Ms V2 at this composition can be
Ea (MJmo1-1) Longitudinal wave 0.42 Transverse wave 0.13
N4 -
I ( MsV2_ (1 Ec - E b
Eb (MJmo1-1)
E~ (MJmo1-1)
- 1.4 - 0.6
4.2 1.3
x ~2 1 -- X 2 ) Ea
we can calculate N 4 as a function of x2/(1 - x 2 ) by use of the values of E a, E b, and E c given in table 2. Figures 5 and 6 show the plots of N4 as a function of x z / ( 1 - x2) calculated by use of the values for the longitudinal wave and by use of the values for the transverse wave, respectively. We see that the difference between fig. 5 and 6 is small. Both plots of N0 against x 2 / ( 1 - x2) can again be approximated by the four segments; the compositions at which the plot of N0 against X 2 / ( 1 - - X 2 ) changes its slope are consistent with
M. Kodama / Ultrasomc oeloctty In potassium borate glasses I
I
I
i
0.5 N4=x2/(1-x2)~
03 02 0.1 I
[
01
1
0.2
I
I
03 04 x2/(1-x 2)
05
Fig. 5. N4 plotted against x 2 / ( 1 - x2). Q, N 4 calculated from eq, (22) by use of the values for the longitudinal wave; E3, N4 determined from N M R spectroscopy by Zhong and Bray [2], the bar being the error m the determination of N4, When the structural u m t a is converted only into the structural umt c, the relation of N4 as a function of x 2 / ( 1 - Xz) can be represented by the straight line N4 = x 2 / ( l - x2) and this is also shown The two straight lines through the calculated points of N 4 m the composition range 0,07 < x 2 / ( 1 - x2) < 0 38 show only the trend of these points.
i
f
I
i
I
0.5
04
N4=x2~'-~ / o
0.3
z~
0
0
the compositions at which the plot of MsV 2 against - - X 2 ) changes its slope. Three points of N4 determined from the area under the N M R absorption curves by Zhong and Bray [2] are also shown in figs. 5 and 6, where the point a t x 2 / ( ] - x 2 ) = 0.50 is used for the calculation of E w We see that the value of N4 at the composition of x 2 / ( ] - - X 2 ) = 0.20 determined from the N M R experiment is in close agreement with the value of N4 calculated from the ultrasonic velocity. At the composition of X 2 / ( 1 -- x 2 ) = 0.334, however, there is a considerable discrepancy between the value of N4 measured by N M R spectroscopy and the value calculated from the ultrasonic velocity. For this discrepancy, the following two reasons can be considered. The first reason is that the present analysis is made on the basis of three structural untts without taking notice of the fact that the borate network is composed of several borate groups. It appears likely that each borate group also has its characteristic elastic property. Admitting that the elastic property of the present glasses can be analyzed in terms of borate groups, we will arrive at essentially the same conclusion at least for the relation of N4 as a function of X 2 / ( 1 - - X 2 ) since borate groups are made up of structural units as explained by Konijnendijk [6]. The second reason for the discrepancy is that the N M R technique of Zhong and Bray [2] still has an error of _+0.025 in the determination of N4. We notice from eq. (11) that the quantity, X2/(1-x2)-N 4, indicates the fraction of the structural unit b, or the fraction of boron atoms with one non-bridging oxygen. The calculated behavior of N4 as a function of x2/(1 - x 2) can be summarized in terms of non-bridging oxygen as follows. (1) For the first composition range 0 < x ~ / ( l - x : ) < 0.07, no non-bridging oxygen forms. (2) At x2/(1 - x 2) = 0.07, non-bridging oxygen begins to form. (3) For the second composition range 0.07 < x 2 / ( 1 - x 2) <0.24, the amount of non-bridging oxygen increases at a constant rate with increasing X2/(]
0.4
0102 /
73
~
~
/
I
01
Fig. 6. N4 plotted against
°
1 0.2
/
-
I
I
03 04 X2/(I - X2)
x2/(1
-
I
05
x2). G, N4 calculated from
eq. (22) by use of the values for the transverse wave; [], N 4 determined from N M R spectroscopy by Zhong and Bray [2], the bar being the error m the determination of N4. When the structural unit a ~s converted only mto the structural umt c, the relation of N4 as a function of x 2 / ( l - x2) can be represented by the straight line N4 = x 2 / ( 1 - x 2 ) and this is also shown. The two straight hnes through the calculated points of N4 in the composition range 0.07 < x 2 / ( l - x 2 ) < 0,38 show only the trend of these points.
x J ( 1 - x2 ). (4) At x2/(1 - x 2) = 0.24, the rate at which the amount of non-bridging oxygen increases with increasing x2/(1 - x 2) becomes different.
74
M. Kodama / U l t r a s o n t c veloctt)' m potasstum borate glasses
(5) For the third composition range 0.24 <
x2/(1 - x2) < 0.38, the amount of non-bridging oxygen increases also at a constant rate with increasing x2/(1 - x 2 ) but the rate of this increase in the third region is less than that in the second region. (6) Above x 2 / ( 1 - x 2 ) = 0.38, the amount of non-bridging oxygen increases to a high degree with increasing x2/(1 - x2). Figure 2 shows that the rate at which the velocity of sound increases with increasing x 2 decreases in the second composition range 0.065 < x 2 < 0.20, which can now be attributed to the fact that non-bridging oxygen easily forms in this composition range. Shelby [13] has shown that the thermal expansion coefficient of potassium borate glasses exhibits a minimum at the composition of x 2 = 0.20 or x J ( 1 - x 2 ) = 0.24. The present analysis indicates that the rate at which non-bridging oxygen increases with an increase in xz/(1 - x 2) decreases suddenly at this composition or, in other words, the rate at which N 4 increases with an increase in xz/(1 - x 2 ) increases suddenly at this composition. However the cause of this minimum cannot be made clear only from the present analysis.
6. Conclusions The elastic constants of the three structural units in potassium borate glasses have been defined on the basis of the elastic internal energy due to deformation. The velocity of sound in potassium borate glasses has been measured as a function of composition from which the elastic
constants of the three structural units have been evaluated to conclude that the structural unit c increases the rigidity of the glass whereas the structural unit b decreases it. By use of the elastic constants of the three structural units, N4 has been calculated as a function of x2/(1 - x 2 ) , the relation of which exhibits sudden changes in slope at definite compositions. It has been shown especially that non-bridging oxygen forms above the composition of x 2 = 0.065 and also the rate at which N 4 increases with increasing X 2 / ( 1 - X2) changes suddenly at x 2 = 0.20.
References [1] J. LorOsch, M. Couzl, J Pelous, R. Vacher and A. Levasseur, J. Non-Cryst. Solids 69 (1984) 1. [2] J. Zhong and P.J, Bray, J. Non-Cryst. Solids 111 (1989) 67. [3] G.D. Chryssikos, E.I. Kamitsos and M.A. Karakassides, Phys. Chem. Glasses 31 (1990) 109. [4] P.J. Bray and J.G, O'Keefe, Phys. Chem. Glasses 4 (1963) 37. [5] J. Krogh-Moe, Phys. Chem. Glasses 6 (1965) 46. [6] W.L. Konijnendijk, Phihps Res. Rep. Suppl. 1 (1975) 1. [7] P. Beekenkamp, Phllips Res. Rep. Suppl, 4 (1966) 1. [8] T.J.M. Visser and J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 376. [9] A. Paul and R.W. Douglas, Phys. Chem. Glasses 8 (1967) 151. [10] M. Kodama, Phys. Chem. Glasses 26 (1985) 105. [11] R.N Thurston, m: Physical Acoustics, Vol. 1, Pt A, ed. W.P. Mason (Academic Press, New York, 1964) p. 1. [12] K. Brugger, Phys. Rev. 133 (1964) A1611. [13] J.E. Shelby, J. Am. Ceram. Soc. 66 (1983) 225. [14] E.P. PapadaMs, in: Physical Acoustics, Vol. 12, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1976) p. 277. [15] M. Kodama and S. Hayashi, Bull. Kumamoto Inst. Tech. 3 (1978) 121. [16] M. Kodama, Jpn. J. Appl, Phys. 21 (1982) 1247. [17] H.J. McSkimin, J. Acoust. Soc. Am. 33 (1961) 12.
65
Ultrasonic velocity in potassium borate glasses Masao Kodama D e p a r t m e n t of Industrtal Chemtstry, K u m a m o t o Institute o f Teehnolog)', Ikeda, K u m a m o t o 860, J a p a n
Received 23 February 1990 Revised 11 September 1990
Ultrasomc velocity m potassium borate glasses denoted by the composition formula x2K20-xlB203 (x I and x 2 are mole fractions of B203 and K20, respectively, and x 1 + x 2 = 1) is measured as a function of x 2 at a frequency of 10 MHz and at a temperature of 298 K by the ultrasomc pulse echo overlap method. The elastic property of these glasses is analyzed m terms of the three structural units defined as Bf03 ~ a, K + B O 2 0 - ~- b, and K + BO 4 ~ c, where O represents a bridging oxygen and O - a non-bridging oxygen, on the assumption that these structural umts have their respective elastic constants. The elastic constants of these three structural units are defined on the basis of the elastic internal energy due to deformation. It is shown numerically that the structural unit c increases the rigidity of the glass whereas the structural unit b decreases it. By use of the values of the three elastic constants, the fraction, N 4, of boron atoms in tetrahedral coordination is calculated as a function of x2/(1-x2). It is shown especially that non-bridging oxygen begins to form at x 2 = 0.065 and also the rate at which N4 increases w~th increasing x 2 changes suddenly at x 2 = 0.20.
1. Introduction
We shall denote the composition of alkali borate glasses by the formula x 2 R 2 0 " X l B 2 0 3, where R denotes one of the alkali metals (Li, Na, K, Rb, and Cs), x 2 denotes the mole fraction of R20, and x I = 1 - x 2 denotes the mole fraction of B203. One property affected by changes in structure is the velocity of sound in the resulting glass. Studies on the velocity of sound in alkali borate glasses are not numerous at present. Lort~sch et al. [1] carried out Brillouin scattering experiments on glasses of the R20-B203 systems (R = Li, Na, K, Rb, and Cs) and reported the hypersonic velocity in glasses of each system as a function of composition. In the case of potassium borate glasses, however, the number of compositions studied by them is insufficient for a discussion of elastic properties of these glasses as a function of coffaposition. The purpose of the present paper is to measure the ultrasonic velocity in potassium borate glasses as a function of composition and to study structural changes of these glasses.
It has been pointed out that the structure of alkali borate glasses is strongly dependent upon the species of alkali ion [1-3]. An early study by Bray and O'Keefe [4] on the nuclear magnetic resonance ( N M R ) of alkali borate glasses led to the conclusion that the fraction. N 4, of boron atoms in tetrahedral coordination followed the relation of N 4 = x2/(1 --X2) up tO x 2 ~ 0.30 with respect to all alkali ion species and then deviated from this relation as x 2 increased further. However, Zhong and Bray [2] have recently refined their N M R techniques and have shown that the measured N 4 value is always less than the value of x2/(1 -x2) except for a region of low x 2 where no data are reported (x 2 = 0.17, 0.25, 0.33, 0.40, and 0.45) and also that the difference, x2/(1 - x 2) - N 4, becomes larger as the size of alkali ion increases. This result indicates that a small amount of non-bridging oxygen is already formed even at x 2 = 0.17 for all alkali ion species and also that the amount of non-bridging oxygen increases as the size of alkali ion increases. Also Chryssikos et al. [3] have interpreted their R a m a n spectra at
0022-3093/91/$03.50 © 1991 - Elsevier Science Pubhshers B.V. (North-Holland)
M. Kodama / Ultrasomc velocity m potasstum borate glasses
66 \
\ 0\ /
/
0\ B--O-
0
/
/
a
\0 B--O-
0
K"
/0
b
0/ \B// \
K÷
potassium borate glasses are discussed in relation to the behavior of N4 as a function of x2/(1 - x2).
O\
c
2. Theoretical background
Fig. 1. Three s t r u c t u r a l units m p o t a s s m m b o r a t e glasses. The
structural units a, b, and c have the formulae B O 3, K + B O 2 0 , and K+BO~, respectively, where ~ represents a bridging oxygen and O- a non-bridgingoxygen. Both the units b and c can be written also as K+BO2-. It should be noted that potassmm ion is also included in the structural units b and c.
x 2 = 0.17 to conclude that N4 decreases and the amount of non-bridging oxygen increases as R is changed from Li to Cs. In alkali borate glasses, there exist several borate groups such as boroxol ring, pentaborate group, triborate group, and diborate group as explained by Krogh-Moe [5] and by Konijnendijk [6]. Raman scattering studies by Lor/~sch et al. [1] and by Chryssikos et al. [3] on these glasses have indicated that, when R is changed from Li to Cs, different kinds of borate groups are formed even if x 2 is the same. Thus, we cannot discuss the structure and properties of alkali borate glasses in the same way without taking notice of the difference in the alkali ion species. With regard to potassium borate glasses studied in this paper, an approximate relation of N 4 as a function of x 2 can be obtained from the study by Zhong and Bray [2], while on the other hand the kinds of borate groups being present in this system are not fully studied as a function of x 2 at present. Thus, this paper assumes the three structural units [7-9] shown in fig. 1 as the primary entities present in potassium borate glasses. These structural units are the most abundant ones in the composition range studied in this paper (0 < x 2 < 0.34) [3,9]. We shall assume that these structural units exhibit their inherent elastic properties and hence have their respective elastic constants. The elastic constants of these three structural units are defined on the basis of a theory of elastic internal energy described in a previous paper [10]. By use of these elastic constants, structural changes in
It is shown in the previous paper [10] that three different expressions for elastic internal energies can be obtained by taking the internal energy relative to unit mass, unit volume, and unit amount of substance. The expression for the elastic internal energy relative to unit amount of substance is extended in the following in order to obtain the elastic constants of the structural units. The amount of a substance is expressed always in the unit of mole in this paper. In considering a deformed body at some temperature, we choose the undeformed state to be the state of the body in the absence of external forces and at the temperature. Let a,, i = 1, 2, 3, denote the Cartesian coordinates of a material particle before the deformation and let x, (al, a2, a3) denote the coordinates of the same particle after the deformation. The Lagrangean strain components ,/,j that describe finite elastic strains [11,12] are defined as 0xk Oxk
77,/=½ 3a, 3a/
) 8,, ,
i, j = 1 , 2 , 3 .
(1)
Here the summation is taken over the repeated subscript k. The symbol 8,/denotes the Kronecker delta. Instead of the tensor notation, we use in the following the abbreviated Voigt notation denoted by the subscript a which runs from 1 to 6 : 1 1 - 1, 22-2, 33-3, 23-4, 31-5, 12-6. The Lagrangean strain in the abbreviated notation is defined by Brugger [12] as
~/,, = 2 ~ , , / ( 1 + 8u).
(2)
Since the sound wave propagates under adiabatic conditions, the internal energy can be chosen as the most convenient thermodynamic potential. Let the molar mass, the molar internal energy, and the molar entropy be denoted by M, Um, and Sm, respectively. Then, the fundamental thermody-
67
M. Kodama / Ultrasomc velocttv m potassium borate glasses
namic equation for the molar internal energy of a deformed body can be expressed as d U m = TdS~ + ( M / p ) G d ~ .
(3)
Here, T is the temperature, p is the density of the undeformed state, t~ = (o/M)(OUm/3~l~)Sm is the quantity called the thermodynamic tension, and the summation is taken over the subscript a from 1 to6.
In order to be able to apply the general formula of the molar internal energy to any particular deformation, we must know the molar internal energy of the body as a function of the strain. This expression can be obtained by using the fact that the deformation is small and hence by expanding the molar internal energy in powers of ~ . Since the glass is elastically isotropic, we shall consider only isotropic bodies in the following. Since t~ = ( p / M ) ( ~ U m / 3 1 1 a ) s m is zero when no external stresses are present, it follows that there is no first order term in the expansion of the molar internal energy in powers of %. The second derivative of the molar internal energy with respect to the Lagrangean strain, keeping the entropy constant, is given by
( ~2Um) 3~
=MV
2.
(4)
1 2(1 - x 2 )
=
(
1
{ x 2 K 2 0 + (1 - x2)B203 ) x2 ) B O 3 + x2 K + B O 2 , 1 --x z 1 -- x 2
Here V is the velocity of a sound wave having the strain component ~/~ ( a = 1, 2, 3 refers to normal strains or stresses for the longitudinal wave and a = 4, 5, 6 refers to the shear components for the transverse wave). If the strain occurs under adiabatic condition, the molar internal energy of the body increases. Expanding the molar internal energy in powers of the Lagrangean strain ~/~ at constant entropy about the state of zero strain and neglecting all terms above the second order, we obtain the increment, bU m, of the molar internal energy due to the elastic deformation, or the elastic internal energy relative to unit amount of the substance:
(1/2)MV2 1 . (5)
(6)
where BO 3 represents the structural unit a and K+BOz- represents both of the structural units b and c. By use of N4, the second term on the right of eq. (6) can be expressed further in terms of the structural units b and c:
x2 K+BO~=( ~
1-x 2
sm
AUm = Um(S m, 71,,) - U m ( S m, 0) =
The constant MV z expresses the elastic constant relative to unit amount of the substance since this constant determines the increment of the molar internal energy caused by a given strain. Equation (5) holds for both the longitudinal velocity, V~, and the transverse velocity, V~. We shall now consider how to express the elastic constants of the three structural units in potassium borate glasses. For this purpose, it is necessary to derive a chemical formula representing the structural units from the composition formula. It may be reasonable to use such a formula in which the amount of boron is always kept constant even if x 2 is varied. We shall first convert the composition formula into the formula in which 1 mol of boron is always contained:
12
1-x 2
)
N4 K + B O 2 0
+ N4K+BO4 .
(7)
For the sake of simplicity, we shall write the three structural units as
BO 3 ~ a,
(8)
K + B O 2 0 ---- b,
(9)
K + B O 4 --- c.
(10)
Now the composition formula can be expressed in terms of the three structural units as 1 2(1 - x2) ( x 2 K 2 0 + (1 - x2)B20 ,
=
( x2)(x2 ) a+ 12~c2 N4 1
1---x,
b+N4c. In)
68
M. Kodama / Ultrasomc velocity in potasstum borate glasses
We shall notice that the molar mass, M c, of the composition formula x2K20- (1 - x2)B203 is defined as
M¢ = x2MK: o + (1 -- x2) MB2O,,
(12)
where MK2 o and MB:o3 denote the molar masses of the component K 2 0 and the component B203, respectively. This definition of Mc contains actually 2x 2 mol of potassium, 2(1 - x2) mol of boron, and (3 - 2x2) mol of oxygen. We shall define the molar mass, M s, of the formula concerning the structural units on the right of eq. (11) as M~=2(1M~
__X2)
-(1
x2 ) l__X2 M~ X2
+(1
N4) Mb + Na M ~
X2
(13) where M a, M b, and M c are the molar masses of the structural units a, b, and c, respectively. This definition of M s contains always 1 mol of the structural units a, b, and c. Let Urn.s be the molar internal energy of the mixture of the structural units on the right of eq. (11). Also let Um,a, Urn,b, and Urn.¢ be the molar internal energies of the structural units a, b, and c, respectively. The molar internal energy of the formula on the right of eq. (11) can then be expressed
For the sake of brevity, we shall omit the subscript Sin.~ tO the derivatives in the following. By noticing eq. (4), the second-order derivative on the left of eq. (15) can be expressed as 02Um's
The quantity MsV 2, which can be measured directly, represents the elastic constant relative to 1 mol of the mixture of the structural units on the right of eq. (11). We see, from eqs. (4) and (5), that the second-order derivative of the molar internal energy of a substance with respect to the Lagrangean strain is equal to the elastic constant relative to 1 mol of the substance. Thus we can define the elastic constants E a, E b, and E¢ of the structural units a, b, and c respectively as
02Um,~ 0g/2 -- E a , 0rt2
Um, s =
(
1
t
(
1 - x2
t
3,/2"
(Omat + N . / ~
E~.
(19)
Substituting eqs. (16)-(19) into eq. (15), we can finally relate Ms V2 to the elastic constants of the three structural units:
(
\1
X2
1 - x2_
_
X2
N4)E b (20)
(14)
Let Sin,~ be the molar entropy of the formula on the right of eq. (11). Differentiating eq. (14) twice with respect to r/~ at constant Sin,~, we have
"+- 1 -- X 2
(18)
+ N4E c .
-1- N4Um, c .
)
E b,
0 2Um,c
Msv 2
N4 Umb "
(17)
02Um,b
as
x2 Um,a+ 1 - x2
(16)
M,V:.
0n2~
N4
0,rl 2
.
3. Experimental
3.1. Glass preparation
t
Sm.s
(15)
A series of potassium borate glasses denoted by the composition formula x2K20" x|B203 was prepared at such intervals that the increment of x 2 equals 0.02, where glasses are formed below x 2 0.35 [13]. In order to obtain good transmission of ultrasonic waves, it is necessary to prepare glasses
M. Kodama / UItrasomc velocttv m potasstum borate glasse~
with high homogeneity and without strains or bubbles. With the aim of preparing glasses with high homogeneity, the starting materials were initially made to react in an aqueous solution. A potassium hydroxide solution of about 1.0 m o l / d m 3 was prepared by dissolving electronic grade K O H (about 99% purity) in distilled water and this solution was used as the starting material for K20. After the concentration of this solution had been determined with a potentiometric titration method, the required volume was taken from a burette. As the starting material for BzO3, electronic grade H3BO 3 (more than 99.7% purity) was used. Amounts of the starting materials calculated to give 30 g in the melts were dissolved by adding distilled water in a beaker. Then the solution was transferred to a dry box and after complete evaporation of the water a chemically reacted powder was obtained. By making use of a SiC resistance electric furnace, the powder was fused in a 20 cm 3 platinum crucible which was placed in an alumina crucible. The fusion was carried out at temperatures from 900 to 1300°C for about 4 h with occasional stirring using a platinum wire; these temperatures were decreased as x 2 increased. Then the liquid was poured into a cylindrical graphite mold 15 m m in diameter and 23 m m in depth which had been preheated at the glass transition temperature in an electric muffle furnace. This glass transition temperature was determined from Shelby's experimental points [13] through which a smooth curve was drawn. Subsequently, the cast glass in the mold was held at the glass transition temperature for 2 h, then cooled at a rate of 1 K / m i n to room temperature while passing dry nitrogen through the muffle furnace. The residual liquid was poured onto an aluminium plate and later used for chemical analysis. Sometimes small bubbles were observed in the cast glasses, but these glasses were not used in the present experiment. The glass of each composition was stored in a hermetic vial. The compositions of all the glasses were analyzed with respect to both x 1 and x 2 with a neutralization titration described in the previous paper [10]. Each composition was determined by five analyses; the probable error was less than
69
2 × 10 - 4 mole fraction for all the compositions studied. Although the density, p, of each glass is not the requisite quantity for the present theory, it is necessary in order to calculate ordinary elastic constants such as Young's modulus, For this reason, the density of the cast glass was also measured at 298 K by a hydrostatic weighing method described in the previous paper [10]. The error in the measurements of density was within 8 × 10 4 g cm -3.
3.2. Ultrasonic velocity In order to measure the ultrasonic velocity, each glass specimen must have a pair of end faces that are flat and mutually parallel. Each glass was first ground on a glass plate using SiC abrasives by setting it in a holder to maintain the two faces parallel and subsequently polished by hand with fine alumina abrasive and machine oil on a flat glass plate. During polishing, the length of the specimen was occasionally measured at the center and the four corners with a micrometer reading to 1 ~m. The polishing was continued until the lengths of five portions coincided to within 2 ~tm; the final length at the center was taken as the length of the specimen. Dimensions of polished glasses were 15 m m in diameter and normally 12 m m long. Inspection with a strain viewer showed that all the specimens were transparent and almost free from stress. Ultrasonic travel time was measured at a frequency of 10 M H z and at a temperature of 298 K by use of the pulse echo overlap method [14]. The apparatus used was constructed in the present author's laboratory and the circuitry has been described in a previous paper [15]. X-cut and Y-cut quartz transducers resonating at a fundamental frequency of 10 M H z were used for the generation and detection of the longitudinal wave and of the transverse wave, respectively. A pair of electrodes of the transducer was deposited, by evaporating at first chromium and then gold, onto each quartz disk with a diameter of 10 m m so as to have a circular active area with a diameter of 8 m m and so as to have two electrode terminals on one surface of the disk. The transducer was bonded
M. Kodama / Ultrasonic velooty m potasstum borate glasses
70
Table 1 Analyzed composition, molar mass of composition formula, density, and ultrasonic velocities of longitudinal and transverse waves of potassium borate glasses at 298 K
x2 0 0.0192 0.0405 0.0614 0.0823 0.1016 0.1227 0.1415 0.1609 0.1780 0.1984 0.2206 0.2391 0.2599 0.2800 0.3011 0.3209 0.3402
M+
p
v~
v,
(g mol - t )
(g cm -3)
(km s - t )
(km s - t )
69.62 70.09 70.62 71.13 71.64 72.12 72.64 73.10 73.57 73.99 74.50 75.04 75.50 76.01 76.50 77.02 77.51 77.98
1.838 1.894 1.948 1.992 2.028 2.050 2.072 2.088 2.103 2.118 2.138 2.168 2.197 2.229 2.257 2.282 2.297 2.306
3.469 3.730 3.964 4.132 4.264 4.332 4.398 4.439 4.472 4.512 4.560 4.656 4.749 4.856 4.920 4.962 4.958 4.914
1.901 2.050 2.186 2.280 2.351 2.375 2.400 2.412 2.423 2.444 2.480 2.546 2.605 2.669 2.705 2.719 2.701 2.653
4. Results
Table 1 compiles the basic data obtained with the present series of experiments. Figure 2 shows the results of the velocity of sound measured by Lor~Ssch et al. [1] at a frequency in the order of 10 G H z and by the present author at a frequency of 10 MHz. These two sets of experimental data are in close agreement especially in the case of longitudinal velocity, from which we see that the frequency dispersion effect on the velocity of sound is small for potassium borate glasses. In order to elucidate the elastic properties of the structural units in potassium borate glasses, we shall plot MsV 2 against x 2 / ( 1 - x 2 ) . Figure 3 shows the plot of the longitudinal elastic constant, MsVl2, against x 2 / ( 1 - xz). Figure 4 shows the plot of the shear elastic constant, M y t 2, against X2/(1X2). The difference between msVl 2 and MsVt2 stems from the volume effect that a change
I
I
i
I
I
I
5.0
0
0 0
0
0
0 0 0 Q
to the specimen on one of the two parallel faces with phenyl benzoate [16]. The specimen with the transducer attached was set in a hermetic specimen holder and controlled at 25 ° C by placing the holder on a water bath. McSkimin [17] proposed a criterion for determining the correct cyclic overlap between echoes with the purpose of measuring the ultrasonic travel time with high accuracy. The McSkimin criterion was extended in the previous paper [10] to become applicable to ordinary glass specimens and this was used in the present measurements. Once a pair of echoes are properly overlapped according to the McSkimin criterion, it is possible to measure the ultrasonic travel time within an error of 0.02% for round trips greater than 5 ~s [17]. Since the velocity of sound is equal to the propagation distance divided by the travel time and the error in the measurements of sample length is within 0.02%, the error in the present measurements of ultrasonic velocity is within 0.04%.
0 0
0
0
0 0 U~
E
0
o=
~4.0
3.0
0
0
0
I 0
El 0
El
I• 13 0
0
0
0
U
0
3.0~-
0
T
2O
I
I
O.I
I
I
I
I
02 03 g2 Fig. 2. Velocities of sound in potassium borate glasses as a function of x 2. V~ ((D) and Vt (El) measured by this author. VI (e) and Vt (m) measured by Lor/Ssch et al. [1]. 0
71
M. Kodama / Ultrasomc veloctty m potassmm borate glasses I
I
I
I
1.5
/o
I
5. Discussion o
o
o
/ o/
o
E 1.0 ~E
/o.O .o~°~°/
05
00
?
I
0,l
I
02
I
0.3
1
0.4
[
05
X21(l ~X 2 )
Bray and O'Keefe [4] proposed a model of the structural change common to all alkali ion species that, on addition of R 2 0 to B203, the structural unit a is converted only into the structural unit c up to x 2 - - 0 . 3 0 and, on further addition of R20, the structural unit c is converted into the structural unit b. On the other hand, the recent N M R study by Zhong and Bray [2] has shown, first, that N 4 < x2/(1 - x 2) always for the composition range above x2/(1 - x 2) = 0.20 and hence the structural unit a can be converted into both the structural units b and c for this composition range and, second, that N 4 is a function not only of x J ( 1 x2) but also of the size of alkali ion species. However, we cannot ascertain from the study by Zhong and Bray [2] whether the structural unit a is converted into only the structural unit c or into both the structural units b and c when a slight
Fig. 3. The plot of the longitudinal elastic constant MsVl 2 against x2/(1 - x2). The hne (a) indicates the first term on the
right of eq. (20), which represents the contribution of the structural unit a to MsVi2. The three straight lines through datapoints show only the trend of the points.
0.5
I
I
I
I
I
O 0.4
in volume due to compressions and expansions is involved in longitudinal strains while no change in volume is involved in shear strains. We see however that both plots have almost the same characteristic. Neither MsVI2 nor MsVt2 change in a monotonic way with an increase in x 2 / ( 1 - x2) but exhibit sudden changes in slope at definite compositions. Each plot can be divided into the four segments having different slopes; the first three segments can be approximated by straight lines except for a few points nearest their boundaries while the last segment has a curvature. These composition ranges are 0 < x2/(1 - x2) < 0.07 (or 0 < x 2 < 0.065), 0.07 < x 2 / ( 1 - x 2 ) < 0.24 (or 0.065 < x 2 < 0.20), 0.24 < x2/(1 - x2) < 0.38 (or 0 . 2 0 < x 2<0.28), and 0 . 3 8 < x 2 / ( 1 - x 2 ) < 0.52 (or 0.28 < x 2 < 0.34).
O
o
/;
o.3 ~
/
~0.2 Ig
/
of°F
o
? 0.11
00
~
~ 1
0.1
.
(
a
) I
I
012 0.3 0.4 x2/(1 - X 2 )
I
0 5
Fig. 4. The plot of the shear elastic c o n s t a n t MsVt2 against x2). The line (a) indicates the first term on the right of eq. (20), which represents the contribution of the structural unit a to MsVt2. The three straight hnes through datapoints show only the trend of the points.
x2/(1 -
72
M. Kodama / Ultrasomc t,eloctty m potasstum borate glasses
amount of R : O is added to B203 at a composition below x2/(1 - x 2) = 0.20. We shall first assume that the relation of N0 = X 2 / ( 1 -- X2) holds in the range 0 < x 2 < 0.30 and thus in the range 0 < x2/(1 - x2) < 0.43 as the early N M R study by Bray and O'Keefe has shown [4]. Substitution of N 4 = x2/(1 - x 2 ) into eq. (20) gives
Table 2 The values of Ea, Eb, and Ec
My 2= 1
calculated by fitting the least squares parabola to nearby points. The value of E b c a n now be calculated by use of eq. (20) at the point of x2/(1 - x 2) = 0.50. Table 2 shows the values of E~, E b , and E c calculated in this way for each of the longitudinal and the transverse waves. It seems unusual that E b takes a negative value. However, the negative value is permissible since the structural unit b does not arise alone but coexists always with the structural unit c. Since E b is negative, we conclude that the structural unit b decreases the rigidity of the glass, which can be attributed to the fact that the structural unit b acts to break down a covalent bond of the network. The value of E c is ten times as great as the value of E a for each of the longitudinal and the transverse waves. This difference indicates that the structural unit c increases the rigidity of the glass, which can be attributed to the fact that the structural unit c forms a three-dimensional covalent bond in the network. Rewriting eq. (20) into the form
1 -x2x 2 Ea + T-Z~x2Ec,
(21)
so that the plot of M~V 2 against X 2 / ( 1 - - X 2 ) should be a single straight line in the composition range 0 < x2/(1 - x2) < 0.43 if this assumption is valid. Since each plot of MsVi 2 and MsVt2 against X 2 / ( 1 -- X2) is by no means a single straight line in the composition range 0 < x2/(1 - x2) < 0.43, it is evident that not only the structural unit c but also the structural unit b forms in this composition range. Discussions about the viscosity of alkali borate glasses as a function of composition [8] and on the acidity and basicity of the structural units [9] point out that, when a slight amount of K 20 is added t o B203, the structural unit a is converted only into the structural unit c. Thus, in the first segment of the composition range 0 < x2/(1 - x 2) < 0.07, the structural unit a is converted only into the structural unit c and therefore this segment can be described by eq. (21). If the structural unit b forms, then the slope of the plot of MsV 2 against X 2 / ( 1 - - X 2 ) should decrease since the structural unit b destroys the network to decrease the rigidity of the glass. The slopes of the second and the third segments are smaller than that of the first segment. This decrease indicates that the structural unit b now forms above the composition of x 2 / ( 1 - x 2 ) = 0.07. Then the relation of M y 2 as a function of x2/(1 - x2) should be described by eq. (20). We shall evaluate E~, E b, and E c. By use of eq. (21), the values of E~ and E c can be calculated from the first segment of the plot of M y 2 against X 2 / ( 1 -- X2). In order to calculate the value of E b, it is necessary to use one known value of N4 at a given composition. For this purpose, we shall use the value of N0 = 0.34 at the composition of x 2 / ( 1 -x2) = 0.50 determined by Zhong and Bray [2]. The value of Ms V2 at this composition can be
Ea (MJmo1-1) Longitudinal wave 0.42 Transverse wave 0.13
N4 -
I ( MsV2_ (1 Ec - E b
Eb (MJmo1-1)
E~ (MJmo1-1)
- 1.4 - 0.6
4.2 1.3
x ~2 1 -- X 2 ) Ea
we can calculate N 4 as a function of x2/(1 - x 2 ) by use of the values of E a, E b, and E c given in table 2. Figures 5 and 6 show the plots of N4 as a function of x z / ( 1 - x2) calculated by use of the values for the longitudinal wave and by use of the values for the transverse wave, respectively. We see that the difference between fig. 5 and 6 is small. Both plots of N0 against x 2 / ( 1 - x2) can again be approximated by the four segments; the compositions at which the plot of N0 against X 2 / ( 1 - - X 2 ) changes its slope are consistent with
M. Kodama / Ultrasomc oeloctty In potassium borate glasses I
I
I
i
0.5 N4=x2/(1-x2)~
03 02 0.1 I
[
01
1
0.2
I
I
03 04 x2/(1-x 2)
05
Fig. 5. N4 plotted against x 2 / ( 1 - x2). Q, N 4 calculated from eq, (22) by use of the values for the longitudinal wave; E3, N4 determined from N M R spectroscopy by Zhong and Bray [2], the bar being the error m the determination of N4, When the structural u m t a is converted only into the structural umt c, the relation of N4 as a function of x 2 / ( 1 - Xz) can be represented by the straight line N4 = x 2 / ( l - x2) and this is also shown The two straight lines through the calculated points of N 4 m the composition range 0,07 < x 2 / ( 1 - x2) < 0 38 show only the trend of these points.
i
f
I
i
I
0.5
04
N4=x2~'-~ / o
0.3
z~
0
0
the compositions at which the plot of MsV 2 against - - X 2 ) changes its slope. Three points of N4 determined from the area under the N M R absorption curves by Zhong and Bray [2] are also shown in figs. 5 and 6, where the point a t x 2 / ( ] - x 2 ) = 0.50 is used for the calculation of E w We see that the value of N4 at the composition of x 2 / ( ] - - X 2 ) = 0.20 determined from the N M R experiment is in close agreement with the value of N4 calculated from the ultrasonic velocity. At the composition of X 2 / ( 1 -- x 2 ) = 0.334, however, there is a considerable discrepancy between the value of N4 measured by N M R spectroscopy and the value calculated from the ultrasonic velocity. For this discrepancy, the following two reasons can be considered. The first reason is that the present analysis is made on the basis of three structural untts without taking notice of the fact that the borate network is composed of several borate groups. It appears likely that each borate group also has its characteristic elastic property. Admitting that the elastic property of the present glasses can be analyzed in terms of borate groups, we will arrive at essentially the same conclusion at least for the relation of N4 as a function of X 2 / ( 1 - - X 2 ) since borate groups are made up of structural units as explained by Konijnendijk [6]. The second reason for the discrepancy is that the N M R technique of Zhong and Bray [2] still has an error of _+0.025 in the determination of N4. We notice from eq. (11) that the quantity, X2/(1-x2)-N 4, indicates the fraction of the structural unit b, or the fraction of boron atoms with one non-bridging oxygen. The calculated behavior of N4 as a function of x2/(1 - x 2) can be summarized in terms of non-bridging oxygen as follows. (1) For the first composition range 0 < x ~ / ( l - x : ) < 0.07, no non-bridging oxygen forms. (2) At x2/(1 - x 2) = 0.07, non-bridging oxygen begins to form. (3) For the second composition range 0.07 < x 2 / ( 1 - x 2) <0.24, the amount of non-bridging oxygen increases at a constant rate with increasing X2/(]
0.4
0102 /
73
~
~
/
I
01
Fig. 6. N4 plotted against
°
1 0.2
/
-
I
I
03 04 X2/(I - X2)
x2/(1
-
I
05
x2). G, N4 calculated from
eq. (22) by use of the values for the transverse wave; [], N 4 determined from N M R spectroscopy by Zhong and Bray [2], the bar being the error m the determination of N4. When the structural unit a ~s converted only mto the structural umt c, the relation of N4 as a function of x 2 / ( l - x2) can be represented by the straight line N4 = x 2 / ( 1 - x 2 ) and this is also shown. The two straight hnes through the calculated points of N4 in the composition range 0.07 < x 2 / ( l - x 2 ) < 0,38 show only the trend of these points.
x J ( 1 - x2 ). (4) At x2/(1 - x 2) = 0.24, the rate at which the amount of non-bridging oxygen increases with increasing x2/(1 - x 2) becomes different.
74
M. Kodama / U l t r a s o n t c veloctt)' m potasstum borate glasses
(5) For the third composition range 0.24 <
x2/(1 - x2) < 0.38, the amount of non-bridging oxygen increases also at a constant rate with increasing x2/(1 - x 2 ) but the rate of this increase in the third region is less than that in the second region. (6) Above x 2 / ( 1 - x 2 ) = 0.38, the amount of non-bridging oxygen increases to a high degree with increasing x2/(1 - x2). Figure 2 shows that the rate at which the velocity of sound increases with increasing x 2 decreases in the second composition range 0.065 < x 2 < 0.20, which can now be attributed to the fact that non-bridging oxygen easily forms in this composition range. Shelby [13] has shown that the thermal expansion coefficient of potassium borate glasses exhibits a minimum at the composition of x 2 = 0.20 or x J ( 1 - x 2 ) = 0.24. The present analysis indicates that the rate at which non-bridging oxygen increases with an increase in xz/(1 - x 2) decreases suddenly at this composition or, in other words, the rate at which N 4 increases with an increase in xz/(1 - x 2 ) increases suddenly at this composition. However the cause of this minimum cannot be made clear only from the present analysis.
6. Conclusions The elastic constants of the three structural units in potassium borate glasses have been defined on the basis of the elastic internal energy due to deformation. The velocity of sound in potassium borate glasses has been measured as a function of composition from which the elastic
constants of the three structural units have been evaluated to conclude that the structural unit c increases the rigidity of the glass whereas the structural unit b decreases it. By use of the elastic constants of the three structural units, N4 has been calculated as a function of x2/(1 - x 2 ) , the relation of which exhibits sudden changes in slope at definite compositions. It has been shown especially that non-bridging oxygen forms above the composition of x 2 = 0.065 and also the rate at which N 4 increases with increasing X 2 / ( 1 - X2) changes suddenly at x 2 = 0.20.
References [1] J. LorOsch, M. Couzl, J Pelous, R. Vacher and A. Levasseur, J. Non-Cryst. Solids 69 (1984) 1. [2] J. Zhong and P.J, Bray, J. Non-Cryst. Solids 111 (1989) 67. [3] G.D. Chryssikos, E.I. Kamitsos and M.A. Karakassides, Phys. Chem. Glasses 31 (1990) 109. [4] P.J. Bray and J.G, O'Keefe, Phys. Chem. Glasses 4 (1963) 37. [5] J. Krogh-Moe, Phys. Chem. Glasses 6 (1965) 46. [6] W.L. Konijnendijk, Phihps Res. Rep. Suppl. 1 (1975) 1. [7] P. Beekenkamp, Phllips Res. Rep. Suppl, 4 (1966) 1. [8] T.J.M. Visser and J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 376. [9] A. Paul and R.W. Douglas, Phys. Chem. Glasses 8 (1967) 151. [10] M. Kodama, Phys. Chem. Glasses 26 (1985) 105. [11] R.N Thurston, m: Physical Acoustics, Vol. 1, Pt A, ed. W.P. Mason (Academic Press, New York, 1964) p. 1. [12] K. Brugger, Phys. Rev. 133 (1964) A1611. [13] J.E. Shelby, J. Am. Ceram. Soc. 66 (1983) 225. [14] E.P. PapadaMs, in: Physical Acoustics, Vol. 12, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1976) p. 277. [15] M. Kodama and S. Hayashi, Bull. Kumamoto Inst. Tech. 3 (1978) 121. [16] M. Kodama, Jpn. J. Appl, Phys. 21 (1982) 1247. [17] H.J. McSkimin, J. Acoust. Soc. Am. 33 (1961) 12.