CHINESE JOURNAL OF PHYSICS
VOL. 23. NO. 4
WINTER 1985
Anomalous Temperature Dependence of Ultrasonic Velocity
in Potassium Silicate Glasses
M. J. Lin ( $$t;s8% )
Department of Physics, National Taiwan Normal University, Taipei, Taiwan I I7
(Received 12 June 1986)
ABSTRACT
The results of ultrasonic experiment on a series of K, 0 xSi0, (x = 4 to 9.5) glass samples are presented. The glass density
is conspicuously proportional to the contained molar percentage of
potassium oxide. At temperatures above 210K, the ultrasonic
velocity shows anomalous linear increase with temperature for
some of the glasses, with low K, 0 concentrations. For those with
K, 0 molar percentages exceeding about 18%, the ultrasonic velocities decrease with temperature, showing a tendency of approaching
normal elastic properties of a dielectric crystal. The temperature
coefficient of ultrasonic velocity change at high temperatures has
an approximate linear dependence on the K, 0 mole %, and there is
a strong linear relationship between the ultrasonic velocity and the
K, 0 concentration. Possible explanations in terms of thermal
fluctuations in the glass network structure are discussed.
I. INTRODUCTION
I
The study of amorphous or glassy solids has become an intriguing field in the recent
decade. Measurements of thermal, acoustic, and dielectric properties of amorphous solids at
low temperatures below 20K show some characteristic features which have no counterpart
in corresponding measurements in crystals.’ These low temperature behaviours are suscessfully explained by the two-level system model, although the microscopic origin of the so
called two-level system remains unknown. Pure fused quartz is very informative in investigating the general amorphous properties. Its ultrasonic measurements at low temperatures give
the two-level system model strong support.2 Moreover, its high temperature data show
another unusual temperature dependence of sound velocity, which has not been clearly
explained.3 Actually, no quantitative ultrasonic analysis covering the whole temperature
range has been performed so far. To approach this problem, it might be very useful investigating the elastic properties of K, 0-SiO, glass system, in which the K, 0 is used as a
glass modifier.
236
ANOMALOUS TEMPERATURE DEPENDENCE OF ULTRASONIC VELOCITY
IN POTASSIUM SILICATE GLASSES
The paper presents the experimental results of ultrasonic study on a series of potassiiim
silicate glasses with different concentrations of potassium oxide. At high temperatures, the
sound velocity increases linearly with temperature in most of the glass samples, similar to
that of pure fused quartz. But the temperature coefficient decreases as the concentration of
potassium oxide increases. Based on the frozen-in fluctuation model proposed by Kulbitskaya et a1.,4 the addition of potassium oxide causes a new fluctation in the glass structure,
which is explained to be linearly proportional to the K, 0 mole % contained in the glass, and
the temperature dependence of this fluctuation is positive, opposite to the negative value of
the fluctuations caused by the whole glass network composed of SiO, tetrahedral units.
c1
II. SAMPLE PREPARATION
Glasses of molar compositions K, 0-xSi0, (x = 4,4.5, 5, 5.5, 6, 7, 8, 9, and 9.5) were
prepared from appropriate fractions of extra pure SiO 2 and reagent grade K, CO, powders.
The two components were melted together in a crucible at about 1500°C for four to five
hours. The melt was quickly poured into a mold and annealed at about 550°C for a few
hours. The glass was then furnace-cooled overnight to room temperature. Samples for the
ultrasonic study, approximately lcm x OScm x 0.5cm in size, were cut from the central
portion of each material. The two opposing faces of each sample were polished to be exactly
parallel and mirror-like.
Quantitative chemical analyses were done on all glass samples. The impurities, mainly
in the form of R, 0,) occupy less than 1% by weight. Table I lists the chemical compositions, molar p$rcentages, and densities of the investigated glass samples.
’
TABLE I.
me
P-
fused quartz was boqht
fran
valpey-Fisher
Colporation.
Chemical compositions, molar percentages, and densities of the investigated
glass samples.
*
M. J. LIN
III.
231
EXPERIMENTAL TECHNIQUES
Density measurements were made by using the method of Archimedes, in which the
sample was weighed on an electrobalance, first in air and then suspended in glycerol of
known density.
Measurements of ultrasonic velocity variation were carried out by using a phasecomparison technique.’ A block diagram of the system is shown in Fig. 1. A HewlettPackard model 8656A signal generator provides a very stable continuous rf signal. This
stable signal is then split into two channels, one of which is gated by a diode switch to form
pulses which are then amplified, injected into the sample, and the echo train received. At
this point, the ethos and the other channel of the cw reference signal are heterodyned to 30
MHz in separated mixers by the same local oscillator. This mixing step leaves the phase
relationship of the two signals unchanged. Both channels are then amplified and mixed in a
Hewlett-Packard 1015A double balanced mixer which acts as a phase detector. Its output,
0
sample
I
I
3oMHz
Amp.
output
Phase
Detector
*
FIG. 1
I
Flow chart of the measurement of ultrasonic velocity change.
238
ANOMALOUS TEMPERATURE DEPENDENCE OF ULTRASONIC VELOCITY
IN POTASSIUM SILICATE GLASSES
after being amplified, is the interference echo pattern, which is then displayed on an
oscilloscope. Its phase can be precisely monitored by PAR model 162 boxcar averager.
Velocity changes are measured by bringing a single echo’s phase to null by adjusting
frequency of the signal source. After a velocity change the selected echo’s phase is again
brought to null by adjusting the frequency. The change in velocity is given by Av/v = Af/f +
A!?/!?, where f is the frequency and P the sample’s length. At low temperatures, the thermal
expansion AQ/~? is typically 10’ times smaller than the velocity change Av/v in most solids,
and can usually be neglected. The resolution of the sound velocity measurement system
depends on the stability of the signal source. Velocity changes of one part in lo6 can be
achieved in our present system.
0
IV. RESULTS AND DISCUSSION
.
A. Density results
Figure 2 shows the relationship of the glass density versus the concentration of the
glass modifier. The conspicuous linear dependence of the density upon the molar percentage
of potassium oxide can be explained in terms of random-network model of glass structure,
proposed by Warren.6 ,7. In the case of fused quartz, each silicon is tetrahedrally surrounded
by 4 oxygens in three dimensions and each oxygen bonded between two silicons. In other
words, each tetrahedral unit is linked to 4 neighboring similar units at its four corners, and
thus a random network is formed. The addition of potassium oxides to SiOZ increases the
ratio of oxygen to silicon to a value greater than 2 and modifies the three-dimensional
network with formation of singly bonded oxygens. The potassium ions are held rather
loosely in the various holes in the irregular silicon-oxygen network. For reasons of local
I
I
2.42c
Q
8
6
lI
0
4
I
I
8
I
I
12
I
I
16
I
I
20
FIG. 2 Relationship between density and molar percentage of potassium oxide for potassium
silicate glasses.
--._.-_.
M.J.LIN
239
charge neutrality, the modifying potassium ions are located in the vicinity of the singly
bonded oxygens. If the molar percentage of potassium oxide is not too great, most of the
affected tetrahedral unit will have only one single bonded oxygen. Conceivably, they are
still strongly connected to other units to form a network. Starting from the structure of
fused quartz, as potassium oxide is added, the potassium ions take the best places they can
find in the various holes in the silicon-oxygen network. Provided that the Kz 0 molar
percentage is less than a certain value, the network structure remains essentially unchanged
as K2 0 is added, except a few of oxygen ions are unbridged. Thus, on the average, the
number of particles in a unit volume found in the potassium glass will be greater than that in
the fused quartz. The excess particles should be the added potassium oxide. Therefore, the
density of the potassium glass increases linearly with the molar percentage of the contained
potassium oxide.
B. Ultrasonic velocity results
The ultrasonic velocities for both shear and longitudinal waves in an isotropic solid can
be calculated by the following relation,
MV )I/2
v = ($)1/Z = (_
(1)
m
’
where M is the shear or longitudinal modulus, P the density, m the mass, and V the volume
of the solid. The relative ultrasonic velocity change with temperature can thus be calculated
as
- 1 dv(2)
v dT
where (Y is the thermal expansion coefficient of the solid. Since, in general, M = M(V, T),
aM
aM
dM
-=aV-+=
(3)
dT
Hence,
1 dv
-_
v dT
(4)
Take vL representing the longitudinal velocity, L the longitudinal modulus, G the shear
modulus, and K the adiabatic bulk modulus, then,
v aL K aL =
4 G aL
--=_-- ( 1 -x) ap’
(5)
L aP
L av
and
(6)
where P represents pressure.
From the data of Gamberg et a1.,8 one can estimate the magnitude of the bracketed
term on the right-hand side of Eq. (6). Typical values of (Y, G, L, and aL/aP at room
temperature for potassium silicate glasses are 1 x 10” K-i, 266 kb, 665 kb, and -8.8,
respectively. Hence,
240
ANOMALOUS TEMPERATURE DEPENDENCE OF ULTRASONIC VELOCITY
IN POTASSIUM SILICATE GLASSES
clrl-(l-~)~, =
5 x lO”K-‘.
A typical value of vi1 dvL/dT is in the order of 104KK1. Hence, we can conclude that
the longitudinal ultrasonic velocity change with temperature is mainly due to the tempera_
ture dependence of longitudinal modulus.
A glass is microscopically inhomogeneous solid, consisting of micro-regions having
different degrees of order and, accordingly, different densities and local elastic moudli. In
such a medium the relationship between the mechanical stresses and strains differs in
different regions. Ultrasonic velocity data of this kind of medium actually measure the
average of the local elastic moduli. Kulbitskaya et. a1.4 have shown that an average macroscopic modulus for an elastically inhomogeneous medium can be represented in the form
<M>=Mo-<Al>,
.
8
(8)
where M, is the elastic modulus of an equivalent homogeneous system, and <A2> the mean
square amplitude of the spatial inhomogeneities of elasticity. The M, could also be thought
as the elastic modulus of its corresponding crystalline counterpart.
The inhomogeneities in glass may be conceived to be originated from the result of
frozen-in fluctuations of the order parameter that are present in the liquid state of glass. The
mean square amplitude of these fluctuations is proportional to the absolute temperature T,
and their relaxational time is proportional to the viscosity of the glass9 When glass is
quenched from its liquid state, the viscosity becomes very large as the crystallizing temperature is approached, and hence prevents the vitrifying liquid from acquiring an equilibrium, homogeneous structure. Its fluctuations of the order parameter, having an amplitude
cc T’12 , turn out in effect to be frozen at about the temperature Tg, the glass transition
temperature, and to be preserved in the solid state of the glass. Due to the above postulated
mechanism of formation of the fluctuations, as the glass is heated, the glassy state has a
tendency of relaxing toward an equilibrium state, thus relieving the fluctuations. Therefore,
a <A’>/dT is negative. The sign of aM,/aT is also negative which is typical behavior of a
pure dielectric crystal’ ’ : the ultrasonic velocity decreases monotonically with temperature.
If the absolute value of a <A’>/aT is larger than that of aM,/aT, a <M>/aT is positive, or
the ultrasonic velocity increases with temperature. On the other hand, as the glass is cooled
down, the fluctuation will continually grow until a certain temperature is reached, at which
the viscosity is large enough to prevent atoms from being rearranged. Thus, at low temperatures, aM,/aT dominates, and the elastic property of glass behave toward that of the
crystalline counterpart, i.e., the ultrasonic velocity decreases with temperature. The changing role of the fluctuations with temperature causes a minimum in ultrasonic velocity.
With addition of the glass modifier, K20, part of the oxygen ions in the glass network
turn out to be singly bonded as described previously. These tetrahedral SiO, units with
singly bonded oxygen will have more degrees of freedom than those of normally bonded
SiO, units. Hence, the addition of K2 0 causes more fluctuations in the elastic inhomogeneity in the glass. If this newly added fluctuation is represented by A’, Eq. (8) can be rewritten
as
0
e
l
-
M. J. LIN
241
< M(K,O * xSiO,)> = M, -<A2> -<A”>.
(9)
Here we have neglected the effect of potassium ions upon ultrasonic velocity. Since the
bond strength of K-O is about 10 times weaker than that of Si-0,’ ’ and the atomic percentage of potassium ion is less than 14% for the investigated glass samples. It is reasonable to
neglect the K-O bond’s contribution to the ultrasonic velocity of the glasses.
It is worthy of note to distinguish <A”> from <A’>. The latter is caused by the
thermal fluctuation of whole glass network, while the former by individual tetrahedral SiO,
units with singly-bonded oxygens. Therefore, <A”> should be linearly proportional to the
number of such SiO, units, which, in turn, is proportional to the molar percentage of K, 0
contained in the glass structure. Furthermore, the temperature behavior of <A”> is
different from that of <A*>. As temperature increases, the less bound SiO, units acquire
more thermal energy and cause more fluctuations. Hence, the sign of 8 <A12>/aT is positive, opposite to that of 8 <A*>/aT, and its magnitude proportional to the molar percentage of the contained Kz 0.
By the previous arguments about glass density, we note that the network structure of
fused quartz is essentially the same as those of potassium glasses of different concentrations
of K2 0. Therefore, statistically, the <A*> may not be changed significantly in the considered glasses. Equation (9) may be approximated as follows:
< M(K* 0 - xSi0, ) > = < M(fuse quartz) > - <A” >.
(10)
Equation (10) indicates that the average macroscopic elastic modulus, or, equivalently,
the ultrasonic velocity in a potassium glass is less than that of fused quartz, and decreases
linearly with the molar percentage of K, 0. This conclusion seems to be verified in Fig. 3,
which shows a strong linear relationship between the longitudinal ultrasonic velocity and the
contained KZ 0 mole %.
T
0
1
1
1
1
1
8
1
1
12
1
1
16
1
I
20
I
K,O mole%
FIG. 3 Relationship between longitudinal ultrasonic velocity (at 300K) and the contained
K, 0 mole % in potassium silicate glasses.
ANOMALOUS TEMPERATURE DEPENDENCE OF ULTRASONIC VELOCITY
IN POTASSIUM SlLICATE GLASSES
242
By the discussions of the previous paragraphs, at high temperatures, the temperature
derivatives of < M (fused quartz) > and <A” > are positive. Thus, by Eqs. (6) and (lo), the
temperature coefficient of ultrasonic velocity change in potassium glass is less than that of
fused quartz and decreases linearly with its K, 0 concentration.
Figure 4 shows the temperature dependences of longitudinal ultrasonic velocity
changes in the investigated glasses and the pure Z-cut quartz crystal. For reasons of being
easier to compare the temperature coefficients of ultrasonic velocity, the point of 60K is
chosen as the common reference point. As shown in the figure, the quartz single crystal
shows a typical temperature behavior of ultrasonic velocity in a pure dielectric crystal, while
the glasses behave quite differently. At temperatures above 2 1 OK, the ultrasonic velocities in
these glasses linearly depend upon temperature. The temperature coefficients of the glasses,
taken as the averages of v-l (Av/AT) between 210K and 300K, decrease linearly with their
molar percentages of Kz 0 as shown in Fig. 5. As more K, 0 is added, this temperature
coefficient becomes less positive, and it even turns out to be negative as the K2 0 mole %
exceeds 18%, approaching toward normal elastic hehavior of a dielectric crystal. This experimental result agrees with the predictions made by Saga,’ ’ using a different approach
from that used in this paper.
60
.
.
20
.
t
.
.
.
F”SCd
Oua,,r
.
.
.
l
.
l
l
’
.
.
1’.
::.
0. l
. .
.*
-8O-
.
.
.
.
1
*.*
l
.
.
.
l
.
.
l
,
110 ,
,
. **.
‘.*.._
0.
00..
180 ,
,
T(K)
FIG. 4
.
I
1289%
.
**..
100 ,
.
. ll.b3%
.-
I
oum* Crystal
.
. . . . . : . . . . . . . . .
1206+
.
9.66%
. lO.OL%
:
.
.
.
.
.
.
.
.
.
a:*.
loo-
.
.
,
220
,
*
l
:.
*
*..
l
l
l
. . ,B.,R%
,
-1- 1 300 : xuB%
260
Temperature dependence of 30 MHz longitudinal ultrasonic velocity change in potassium silicate glasses. The number indicated on each curve represents the molar
percentage of the potassium oxide contained in the glasses.
The elastic anomalies of glasses have puzzled glass scientists for a few decades.
243
M. J. LIN
:
Although the elastic properties at very low temperatures have been recently explained by the
two-level system model, those at high temperatures are still not clear. Several attempts to
explain these high temperature phenomena are reported in literature, but none of them has
gained satisfactory success. With alkaline oxides as glass modifiers, the alkaline silicate
glasses are good glass samples to approach this problem. Further ultrasonic study of
potassium silicate glasses with high concentrations of K2 0 should yield more valuable informations about the structures of glasses and their changes with temperature.
0.60
-z:[, , , , , , , ,A
0
L
8
12
16
20
K,O mole~/o
FIG. 5
Relationship between the temperature coefficient of longitudinal ultrasonic velocity
change, taken between 2 1 OK and 300K, and the contained K2 0 mole % in potassium
silicate glasses.
V. ACKNOWLEDGEMENTS
The author is grateful to Mr. W. S. Hsu and Mr. C. S. Yang of Allied Industrial Institute
for their kind assistance in fabricating the glass samples and performing quantitative
chemical analyses. He also thanks Miss M. L. Shaw for her help in part of the ultrasonic data
measurements and Dr. W. M. Wang for helpful discussions. This work is supported by
National Science Council of Republic of China under the contract number NSC74-0208M003-07.
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-_
_
244
ANOMALOUS TEMPERATURE DEPENDENCE OF ULTRASONIC VELOCITY
IN POTASSWM SILICATE GLASSES
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a