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Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica
]OURNA L OF
Journal of Non-Crystalline Solids 143 (1992) 65-74 North-Holland
NON-CRYSTALLINE SOLIDS
Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica Q. W a n g , G . A . S a u n d e r s , H . B . S e n i n a n d E.F. L a m b s o n School of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK Received 4 January 1991 Revised manuscript received 6 D e c e m b e r 1991
M e a s u r e m e n t s of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components (TOEC) of vitreous silica between 77 and 293 K. In the absence of such data previous practice has been to assume that these T O E C are independent of temperature; however a significant temperature dependence has been observed. With the exception of the smallest C456, each of the T O E C is anomalously positive: a feature consistent with the well established aberrant negative values for the hydrostatic pressure derivatives (OCij/8P)e-o of the second-order elastic stiffness tensor components (which reveal softening of the long-wavelength acoustic modes u n d e r pressure). The T O E C increase as the temperature is reduced showing that the pressure-induced acoustic mode softening becomes enhanced at lower temperatures. T h e largest T O E C is Cll I which is 4 6 . 7 x 10 l° Pa at 293 K and increases to 99.1 × 101° Pa at 77 K. The hydrostatic pressure derivatives of the second-order stiffnesses have been calculated from the T O E C ; (~Ca~/OP)P-O is m u c h larger than (i~C44/OP)P-O over the whole temperature range: the longitudinal acoustic mode softens more with pressure than the shear mode. This effect is emphasised by the finding that the acoustic mode Griineisen parameters are negative with [YLI > l y s l . As the temperature is reduced the longitudinal acoustic mode Griineisen parameter, YL, increases in magnitude considerably, reaching a value of - 5 . 5 at 77 K. T h e m e a n acoustic mode Griineisen parameter, T el, increases to a larger negative value ( - 3 . 1 at 77 K) than previously suspected on the basis of the assumption of t e m p e r a t u r e - i n d e p e n d e n t third-order elastic constants: the discrepancy between the low-temperature limits of the thermal, YL, th and m e a n acoustic mode, YL, el Griineisen parameters is not quite so large as had been thought. The vibrational anharmonicity of the acoustic modes plays an important part in causing the thermal expansion of vitreous silica to be negative at low temperatures.
1. Introduction In general experimental determinations of the t e m p e r a t u r e dependences of the third-order elastic stiffness tensor components ( T O E C ) are sparse because it is demanding to make m e a s u r e m e n t s of the pressure dependences of ultrasonic wave velocities as a function of temperature. The normal procedure used to obtain a complete set of T O E C is to measure the effects of hydrostatic and uniaxial pressure on the velocity of ultrasonic waves p r o p a g a t e d in the solid of interest. E m p h a sis is usually placed on making as many measurements under hydrostatic pressure as possible because the effects induced are larger, and conse-
quently are easier to measure with accuracy, than those obtained from application of uniaxial pressure. However, at low temperatures the liquids used as pressure-transmitting media freeze making it difficult to measure the effects of pressure on ultrasonic wave velocity or even to ensure that the applied pressure is truly hydrostatic. Although it is possible to use gaseous systems to apply hydrostatic pressures, for safety reasons this is best avoided. However we have noticed that it is possible to obtain all three independent components of the third-order elastic stiffness tensor for an isotropic solid by making ultrasonic wave velocity m e a s u r e m e n t s solely under uniaxial pressures, a possibility which does not seem to
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
66
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity o f vitreous silica
have been explored before. Hence our major objective has been to construct an equipment to measure the effects of uniaxial pressure on ultrasonic wave velocities below room temperature in glass specimens, and to assess experimentally whether this approach can be used to obtain the complete sets of T O E C needed to quantify the nonlinear acoustic properties of glasses at lower temperatures. Vitreous silica has been chosen as the candidate material on which to make such experiments because it is of great technological importance, it is a pure, single-phase glass and furthermore has remarkable acoustic mode vibrational anharmonicity effects not yet studied at low temperature. We have succeeded in determining the effects of uniaxial stress on the velocities of enough ultrasonic mode configurations to derive complete sets of T O E C as a function of temperature between 77 and 293 K. The non-linear acoustic properties of vitreous silica SiO 2 are singularly anomalous. The temperature dependences of the elastic stiffness constants are negative up to about 70 K and then become positive [1-5]. The hydrostatic pressure dependences are negative: when subjected to pressure, vitreous silica becomes easier to squeeze [6-8]: This extraordinary behaviour also occurs in other tetrahedrally coordinated glasses G e O 2 and BeF2, but not for triangular or 'modified' tetrahedral glasses [5]. It is shown also by vitreous samarium phosphate [9]. While the elastic moduli of fused quartz decrease under moderate applied pressure, normal behaviour is assumed at higher pressures. In his early compression studies Bridgman [6] first observed that the bulk modulus decreases with increasing pressure until a reversal of slope at about 3 GPa. This reversal was confirmed by ultrasonic measurements as a function of high pressure which showed non-linear decreases of the elastic stiffnesses Cll , C44 and the bulk modulus to a minimum value at about (2.3 _+ 0.2) GPa, and thereafter an increase with increasing pressure; elastic behaviour was observed up to pressures of 3 GPa [10]. Bogardus [7] first obtained a set of mixed adiabatic-isothermal T O E C for fused silica at room temperature from measurements using the ultrasonic pulse superposition technique as a function of both hydro-
static and uniaxial pressure. He found that each TOEC, except the smallest C456, has a positive value consistent with the anomalous negative sign of the hydrostatic pressure derivatives (OCjJ 0P)p= 0 of the second-order elastic stiffnesses (SOEC). Subsequently a combination of results obtained from ultrasonic second harmonic generation [11] with those of ultrasonic beam mixing [12] was used to obtain a complete set of adiabatic T O E C for fused silica [11]. Cantrell and Breazeale [13] have determined the temperature dependence of the adiabatic T O E C ClSll alone between 3 and 300 K from measurements of the second harmonic generation from the fundamental of longitudinal ultrasonic waves. Recently the pressure dependence of the longitudinal sound wave velocity in vitreous silica has been measured using high-resolution Brillouin spectroscopy between 50 K and room temperature [14]; the reduction of velocity induced by application of pressure was found to occur throughout this temperature range. Negative hydrostatic pressure derivatives (OCiJOP)e=o and positive temperature dependences of the second-order elastic stiffness tensor components and positive T O E C reveal anomalous effects in the vibrational anharmonicity: the frequencies of the long-wavelength acoustic modes decrease under pressure so that the mode Grfineisen parameters
y(p, q) = - ( ~ In w(p, q)/a In V ) r , e = 0 ( a In o~(p, q) ) =
-
V
OP
(0~) T,P=O
=B ~
T,P=O
(1) OP
T,P=O
are negative. Here o~(p, q) is the vibrational frequency of a mode p of wavevector q, P is the applied pressure, V is the volume of the solid and B T is the isothermal bulk modulus which is defined as
In addition to being responsibl e for non-linear acoustic properties, vibrational anharmonicity is
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous sifica
the source of thermal expansion. Those glasses which show the anomalous elastic behaviour under pressure or with temperature may also exhibit a negative thermal expansion at low temperature [5] where contributions from the long-wavelength, low-frequency acoustic modes play a dominant role. Therefore, knowledge of the acoustic mode Griineisen parameters is a prerequisite for understanding thermal expansion at low temperatures. Although the acoustic mode Griineisen parameters have been determined for vitrious silica at room temperature [5,10,15], their temperature dependences are not known. One goal of the present work has been to fill this gap. Acoustic mode Griineisen parameters can be determined from measurements of either the hydrostatic pressure derivatives of the second-order elastic constants or the TOEC. The strain energy per unit mass U(rl)(= U(S, 7q)- U(S, 0)) is defined in terms of Lagrangian strains ~q as p0g(~)
= 1
S 1
1
S
s
+ 5zC~t~E;no~7~¢~7~7~C~Tno.
(3)
Hence the T O E C
CSt~c=P°
( 03U(S' rt) ) 0~7~/30r/~~0~7~ s;n=0
(4)
constitute the cubic coefficients of this strain Hamiltonian and quantify the lowest-order anharmonic term of the potential with respect to change in the particle separation. Therefore T O E C data, and the acoustic mode Griineisen parameters, determine the vibrational anharmonicity of the long-wavelength acoustic modes to cubic order in strain.
2. Experimental techniques A rectangular parallelepiped of fused silica of dimensions of about 1 cm 3 with three pairs of orthogonal faces polished flat and parallel to optical precision was prepared. Parallelism of the faces was examined by using an optical interference method and found to be within one wave-
67
length of sodium light. Quartz transducers, driven at their fundamental frequency of 10 MHz (X-cut for longitudinal, Y-cut for shear waves) were bonded to the specimen. An indium bond was made between the specimen and transducer by first coating them with thin films of nichrome, gold and indium in that order and then cold welding under vacuum. Uniaxial stresses were applied in a screw press [16]. The internal stress patterns inside the specimen under uniaxial stress in the press were examined using two pieces of polaroid. No colour patterns should be observable when the uniaxial pressure is uniform. When stress patterns could be seen, the position of the specimen between settling shims was adjusted until they were removed. Ultrasonic experiments under uniaxial stress were carried out in a liquidnitrogen cryostat in which it was possible to stabilise the temperature to within +0.1 K over a time period of several minutes during which the measurements were made. Changes induced in ultrasonic wave velocity by application of a uniaxial stress were measured to the required sensitivity of 1 part in 10 7 using an automated gated pulse superposition technique [17]. To bypass the requirement for determination of changes induced in sample dimensions, the 'natural velocity W' technique [18] was employed. The effects of the uniaxial stress applied (up to about 1 × 10 7 Pa) on ultrasonic wave velocity are small; although the systematic errors in absolute magnitude of the T O E C are quite large, the error in the relative changes with temperature is much smaller.
3. Experimental configurations for determination of the TOEC of an isotropic solid An isotropic solid has the following third-order elastic stiffness tensor components: C l l 1 = C222 = C333;
C144 = C255 = C366;
C l l 2 = C233 = C133 = C l l 3 = C122 = C233; C155 = C244 = C344 = C166 = C266 = C355; C123;
C456 .
(5)
68
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
However only three of these are independent and can be taken as C123 =/)1;
C144 = P2;
(6)
C456 =/"3.
Then the remaining non-zero tensor components are given by linear combinations of these three parameters: Cl12 =/)1 q- 2/)2,
(7)
C155 =/)2 q- 2/)3,
(8)
C111 = /)1 q- 6/)2
+ 8/)3"
(9)
The T O E C , which are sixth rank tensor ponents Cuvprqs , c a n be obtained from the derivatives (p0W2)~,=0 evaluated at zero using [19] t -- ( P o W 2 ) p = o = ( N " M ) 2 + 2 w F u c + G u c ,
comstress stress (10)
with
w=(poW2)o=(pV2)o
s = C;.rsN,
NqUrV
(11)
,
F u c = SfbrsM,,MbUrU ~ ,
(12)
G u c = SabuvCuvprqsMaMbNpNqUrV r s.
(13)
H e r e W is the natural velocity, V is the measured velocity, N and U are unit vectors along the wave propagation and polarisations directions respectively; M is a unit vector along the applied stress direction; C s and S f are isentropic stiffness and isothermal compliance tensor components respectively. This procedure [18] leads to the solutions for the effect of uniaxial pressure on the velocity of ultrasonic modes propagated in an isotropic solid given in table 1. To determine the three independent T O E C of vitreous silica, measurements were made of the effects of uniaxial pressure on the velocities of these three ultrasonic m o d e configurations.
4. Experimental results for the second- and third-order elastic stiffness tensor components of vitreous silica as a function of temperature
T h e s e c o n d - o r d e r elastic constants, C lJ' s Young's modulus, E s, bulk modulus, B s, and Poisson's ratio, or, calculated from the measured ultrasonic wave velocities are given in table 2. The velocities of both longitudinal and shear wave propagated in vitreous silica (table 2) increase approximately linearly with increasing temperature in the t e m p e r a t u r e range studied here. This is a well-known p h e n o m e n o n associated with the relaxation processes which produce the broad p e a k characteristic of the ultrasonic attenuation in silicate glasses [2-4,20,21]. It contrasts markedly with the behaviour of a pure dielectric crystal for which the velocity of sound decreases with increasing t e m p e r a t u r e due to the effects of vibrational anharmonicity. The three independent T O E C Vl, /)2 and u 3, defined in eq. (6), have been determined from the elastic constants (table 2) and the uniaxial pressure derivatives { ( 1 / W ) d W / d P } e = o of the measured ultrasonic wave natural velocity (table 3) using the expressions given in table 1. The T O E C , determined using eqs. (7-9), are compared in table 4 with those of obtained at room temperature by previous workers. The third-order Cauchy relationships C123 = C 1 4 4 = C456 are not obeyed, implying a large non-central force contribution to the vibrational anharmonicity. The central aim of this work to determine the t e m p e r a t u r e dependence of the T O E C of vitreous silica has been achieved down to 77 K; the results are given in table 5. The hydrostatic pressure derivatives (OCi]/OP)p= 0 of the elastic stiffness tensor com-
Table 1 The experimental configurations and the relationships used for calculation of the three TOEC (vl, /'2, /'3) defined in eq. (6) of an isotropic medium under uniaxial compression. E is Young's modulus and ~r is Poisson's ratio. The stress is applied in a direction M perpendicular to the ultrasonic mode propagation direction N Mode polarisation Mode 1: L Mode 2: S Mode 3: S
Displacement direction, U parallel to N parallel to m perpendicular to M
w = (poVa)e_o
{d(poW2)/dP}p=o
C1~
[~r(2w + 8v3)+ vi(2o- - 1)+ Uz(8~r-2)]/E [ -- 2W + v2(2o" - 1) + 2 v 3 ( c r - 1)]/E [o-(2w +4v3)+ v2(2o-- 1)]/E
C44
C44
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
69
Table 2 Longitudinal (V L) and shear (Vs) wave velocities (in units of ms 1), elastic st±finesses and bulk (B s) and Young's (E s) moduli (in units of 101° Pa) and Poisson's ratio of vitreous SiO 2 as a function of temperature. The errors are: AVL,AVs = _+2 ms-Z; Cll,C12,C44 = _+0.003 × 10 l° Pa; ES,B s = _+0.01 × 10 l° Pa; ~r = _+0.001 T (K)
VL
Vs
CII
C12
C44
Young's modulus, E
Bulk modulus
Poisson's ratio,
293 273 253 233 213 193 173 143 123 113 103 93 77
5974 5971 5957 5947 5940 5938 5927 5912 5909 5902 5911 5899 5893
3733 3723 3688 3681 3675 3669 3665 3651 3645 3641 3632 3623 3620
7.862 7.853 7.817 7.791 7.774 7.768 7.740 7.700 7.692 7.674 7.669 7.665 7.652
1.723 1.735 1.826 1.821 1.824 1.837 1.822 1.826 1.838 1.832 1.855 1.883 1.878
3.069 3.059 2.996 2.985 2.975 2.965 2.959 2.937 2.927 2.921 2.907 2.891 2.887
7.24 7.23 7.13 7.10 7.08 7.07 7.05 7.00 6.98 6.97 6.95 6.92 6.91
3.77 3.77 3.82 3.81 3.81 3.81 3.80 3.78 3.79 3.78 3.79 3.81 3.80
0.180 0.181 0.189 0.189 0.190 0.191 0.191 0.192 0.193 0.193 0.195 0.198 0.197
ponents, calculated from the TOEC data, are given in table 6. Measurement of the change in ultrasonic wave velocity induced by application of a constant stress leads to mixed adiabatic (S) isothermal (T) moduli ( c S f ) (this has been written without superscripts S and T throughout this paper) making it necessary to known the corrections required to determine adiabatic third-order elastic constants cSk. Following the technique detailed by Shull
Table 3 Experimental data for the pressure gradient { ( 1 / W ) d W / dP/e=0 of natural velocity (in units of 10 -11 Pa -1) of ultrasonic modes propagated in vitreous silica under uniaxial pressure. The modes are labelled as in table 1 T (K)
Mode 1
Mode 2
Mode 3
293 273 253 233 213 193 173 143 123 113 103 93 77
-0.693__+0.001 - 0.652 _+0.001 - 0.826 ± 0.001 - 1.073 _+0.001 -1.132_+0.001 -1.388_+0.001 -1.653__+0.001 -2.283-+0.001 -2.254_+0.001 - 2.343 ± 0.001 - 2.412±0.001 -2.607__+0.001 -2.726_+0.001
-2.161_+0.003 - 2.129 ± 0.006 - 1.881 ± 0.003 - 1.958 ± 0.005 -2.111+0.004 -2.247+0.003 - 2.265 ± 0.003 -2.335-+0.005 -2.365±0.001 -2.347+0.001 -2.463_+0.001 -2.463±0.001 -2.546_+0.001
-0.944__+0.005 - 1.026 _+0.001 - 1.031 _+0.001 - 1.131 _+0.001 -1.194__+0.001 -1.236__+0.001 -1.341-+0.001 -1.573-+0.001 -1.773±0.001 -1.903_+0.001 -1.991_+0.001 - 2.093 ± 0.001 -2.236_+0.001
[22], it can be shown that the relationship giving the difference between the measured ClS/~" and the adiabatic quantity cS11 is
( c S l - c s l r l )= - - ~ T - 1
(c111+2c112)]3 Here BS/B r= (1 + 3 a y t h T ) = 1.0013. At room temperature (8Cll/8T) P has been obtained as 10.7 × 106 Pa K -1 from a polynomial fit to the data for the temperature dependence of Cll. The coefficient a of linear thermal expansion is 5 × 10 -7 K -1 and ~th is 2.918. These results lead to a value for the difference (CSll - C l iST1) of only - 0 . 8 9 × 101° Pa. The magnitudes of this correction ( c S k - cS:~) are the same for the other T O E C and are much less than the experimental error. Therefore for computational purposes the data for the T O E C given in table 5 can be taken as being the purely adiabatic constants, cSk. Second harmonic generation from the fundamental of longitudinal ultrasonic waves propagated in four different types of fused silica has been used [13] to determine the non-linearity parameter/3 = - ( 3 C l l + Cm)/3Cll between 4.2
70
Q. Wang et aL / Elastic constants and acoustic vibrational anharmonicity o f vitreous silica
Table 4 Comparison between the third-order elastic stiffness components for vitreous silica at room temperature obtained by previous workers. Units: 109 Pa Source
Sample type
6111
Cll 2
CI23
C144
ClSs
C456
Present results [7] ~) [22] a) [11] b) [11] b) [23] c) [13] b) [13] b) [13] b) [13] u)
Fused silica Fused silica Vitreous silica G.E. 151 Optical quality Fused silica Suprasil W l Suprasil W2 Suprasil 1 Suprasil 2
470 +_10 526 _+40 620 648+- 5 487+_ 4 550 +_ 1 670+_ 12.4 685 +_ 5.8 731_+ 7.3 675 +_ 6.7
234_+20 239+_30 261 537 +- 26
81± 5 54+_13 72 428 +- 32
76+_6 93+_8 95 54 +- 3
59+_10 72_+ 2 90 28 +_ 5
-8.7+_0.6 - 1 1 _+3 - 2.5 - 13.2 +_0.8
a) By ultrasonic pulse superposition method. b) By ultrasonic harmonic generation method. c) By shock-wave compression.
and 300 K. In combination with longitudinal wave velocity, VL, measurements which give C n from C l l -- P I/'.2 L,
TOEC (with the exception of silica (table 5).
C456) of
vitreous
(15)
the adiabatic third-order elastic constant, cS11, could be determined as a function of temperature. It was found that cSlt showed a small decrease with decreasing temperature for Suprasil W l , W2 and 2 but an increase for Suprasil 1. The present results show that C n l increases with decreasing temperature (table 5), as do the other
5. Discussion: the vibrational anharmonicity of vitreous silica as a function of temperature
One of the strikingly anomalous features of the vibrational properties of fused silica is that the hydrostatic pressure derivatives of the elastic stiffness constants are negative [6,7]. With decreasing temperature each pressure derivative
Table 5 The third-order elastic constants of vitreous silica between 77 and 293 K. The units are 101° Pa. The errors are C m = 2 × 10 m Pa, Cn2 = ± 2 × 1 0 1 ° Pa, C123 = _+0.8x 101° Pa, C144 = ± 0 . 8 x 1 0 I° Pa, C155 = +0.1×101° Pa, C456 = _+0.6×10 m Pa T (K)
Cm
C~12
C123
C144
C155
C456
293 273 253 233 213 193 173 143 123 113 103 93 77
47.0 45.4 41.4 48.3 54.0 62.9 68.0 81.7 83.2 84.4 87.9 94.2 99.1
23.4 22.5 24.7 29.6 31.7 37.4 41.9 53.5 53.6 54.9 56.9 61.4 64.0
8.1 7.1 10.4 14.2 15.2 19.9 23.5 32.9 31.1 31.4 32.5 35.9 36.9
7.6 7.7 7.1 7.7 8.3 8.8 9.2 10.3 11.2 11.7 12.2 12.8 13.5
5.9 5.7 4.2 4.7 5.6 6.4 6.5 7.1 7.4 7.4 7.8 8.2 8.8
- 0.9 - 1.0 - 1.5 - 1.5 - 1.4 - 1.2 - 1.3 - 1.6 - 1.9 -2.2 - 2.2 - 2.3 - 2.4
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica Table
6
Hydrostatic components from
the
The
errors
pressure and
bulk
experimental are
derivatives modulus data
of the elastic of vitreous
for the
TOEC
stiffness
tensor
silica determined given
in table
5.
_+0.01
T
(i~Cll//
(i~C12/
(~C44/
(~B/
(K)
OP)p= o
OP)P=O
OP)P=O
i~P)P=O
239
- 9.98
- 3.99
- 2.99
- 5.99
273
-9.69
-3.76
-2.96
-5.74
253
-9.59
-4.37
-2.61
-6.11
233
- 11.08
- 5.57
- 2.75
- 7.41
213
- 11.96
- 6.04
- 2.96
- 8.01
193
- 13.71
- 7.43
- 3.14
- 9.53
173
- 15.02
- 8.59
- 3.22
- 10.74
143
- 18.29
- 11.48
- 3.41
- 13.75
123
- 18.42
- 11.32
- 3.55
- 13.69
113
- 18.81
- 11.61
- 3.59
- 14.01
103
- 19.39
- 12.02
-3.69
- 14.48
93
- 20.65
- 13.04
- 3.8i
- 15.58
77
- 21.58
- 13.62
- 3.98
- 16.27
(OCll/OP)p=o, (OC12/OP)p=o, (OC44/OP)p=o and (OB/OP)p=o gets larger (table 6): the extraordinary behaviour becomes enhanced at lower temp e r a t u r e s . T h e e x p e r i m e n t a l facts that (OC11/aP)p=o is much greater than (OC44/OP)p=o and that Cll I is by far the largest TOEC, and is positive, show that the acoustic mode softening induced by pressure is substantially greater for the longitudinal than for the shear acoustic modes. The temperature dependence of the ultrasonic attenuation in vitreous silica is dominated by a broad maximum with Tmax at about 50 K [2] whose origin lies in classical, thermally activated, relaxation processes in double-welled asymmetric potentials [24-27]. Although a wide range of low-temperature thermodynamical and transport properties of glasses can be understood within the framework of this phenomenological model, the effects of pressure on the two-level systems have yet to be investigated theoretically. The problem has recently been addressed experimentally [14,28]. Measurements of the effect of pressure on the ultrasonic attenuation in the quartz glass Suprasil Wl have shown that the broad absorption peak shifts from about 50 K at atmospheric pressure to 90 K under a pressure of 0.42
71
GPa [28]. The anomalous positive temperature dependence of the second-order elastic moduli of vitreous silica is probably also a consequence of the double-well relaxation processes which produce this attenuation peak. Furthermore since this positive temperature dependence of the elastic moduli extends right up to room temperature (table 2) the relaxation processes would seem to remain operative at that temperature. The same processes are probably responsible for the anomalous non-linear acoustic behaviour under pressure, namely the positive signs of the TOEC (table 5) and the negative values found for the hydrostatic pressure d e r i v a t i v e s (OCij/OP)p= 0 (table 6). The observation that (OC11/OP)p=o has a much larger negative value than (OC44/OP)p_o indicates that the double-well relaxation processes are more effective in softening the longitudinal than the shear acoustic waves. TOEC data have as yet been obtained only down to 77 K (table 6), still somewhat above the temperature Tmax at which the maximum relaxation occurs where the effects of the relaxation processes on the vibrational anharmonicity could be much enhanced. Knowledge of the TOEC below room temperature helps us to clarify further the origins of the anomalous behaviour of the thermal expansion of vitreous silica because the thermal expansion and non-linear acoustic properties both relate to the vibrational anharmonicity. The thermal Griineisen parameter, , ) / t h , which can be obtained from data for the linear coefficient, a, of thermal expansion, the compressibility, K, and the specific heat, C, using yth = 3aV/KSCp = 3aV/KrCv,
(16)
results from the sum of excited modes:
yth = E yC,(T)I E Ci(T). i
(17)
i
The second- and third-order elastic stiffnesses provide a measure, which can be expressed by acoustic mode Griineisen parameters, of the anharmonicity of the long-wavelength acoustic modes alone. For an isotropic solid there are two components: "/i~ and Ys, which refer to the Iongi-
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
72 -1.0
•
"6 -2.0
II
i
|
I.
¢
i
i
g
•
.~ -3.(1
"
•
¢ i
i
:7
q-5 X 10 - 7 K -1) at room temperature, decreases as the temperature is reduced and changes sign at about 180 K, reaches a minimum at about 40-50 K and then tends to zero as the temperature tends to 0 K [29-32]. The thermal expansion accrues from the summation (eq. (17)) over the excited vibrational states including the long-wavelength acoustic modes. The infrared spectrum of vitreous silica is dominated by three peaks at about 100, 800 and 460 cm-1. Each of the vibrational modes responsible for these peaks involves both stretching and bending of the silicon-oxygen bond, and in general the higher the frequency, the larger the contribution made by stretching [33]. The mode Griineisen gammas for these modes have been determined from the pressure dependences of the peak positions [34,35]. While the sign of the Yi for the 800 cm -1 mode is positive, those for 1100 and 470 cm -1 are negative. A physical description of how these signs arise in terms of the stretching and bending of the silicon-oxygen bond has been given by Chow, Phillips and Bienenstock [35] who showed that the density-of-states calculation [33] can be used to obtain a specific heat and thermal expansion which are in reasonable agreement with experiment, indicating that the small value for the thermal expansion of vitreous silica comes about as a result of summation over the effects of modes with different Griineisen parameter signs. As the temperature is lowered, the long-wavelength acoustic phonons with negative Griineisen parameters can be expected to make an increasingly important contribution to the thermal expansion. At low temperature when the excited phonon population includes a high proportion of longwavelength acoustic modes, the values of the elastic, YL, el and thermal, YL, th Griineisen parameters would be expected to become equal, like the elastic and thermal values of ( C / T 3 ) r _ , O K [3638]. For crystalline dielectric solids, in which the Debye approximation is reasonably valid at low temperatures, ~/}h and 3,[~ do tend to become equal. In general amorphous materials show anomalous thermal properties at low temperatures [39,40] which have been accounted for by the model of low-energy, two-level systems with a broad spectrum of their energy splitting [24-27]. (~
-a.0
.o_ ,A
A
-6.0
so
2;0 Temperoture
2;0
300
(K}
Fig. 1. The temperature dependences of the Griineisen parameters YL (triangles) and Ys (circles) of the long-wavelength longitudinal and shear acoustic modes respectively and the mean y el (squares) for vitreous silica.
tudinal and shear elastic waves, given by [11] "YL = -- ( 1 / 6 C l l ) [ 3 B
+ 2Cll + Clll -{- 2Cl12],
(18) and Ts = - (1/6C44)[3B
+ 2C44 + (Cll I - C123)/2 ] .
(19) These acoustic mode Griineisen parameters, obtained from the experimental second- and thirdorder elastic constant results, are plotted as a function of temperature in fig. 1 for vitreous silica. Also shown is the mean acoustic mode Griineisen parameter, ye~, obtained using
on the basis of the Debye continuum model. The results that YL, YS and Tel are negative and become larger when the temperature is reduced, are consistent with the behaviour of the thermal expansion with temperature. The thermal expansion coefficient of vitreous silica is over an order of magnitude smaller than that of crystalline quartz which in turn is comparable to that of many other solids. The dependence of the thermal expansion coefficient upon temperature can be summarised as follows: it is small and positive
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
Thus the Debye model is not valid: the specific heat is dominated by the high density of these low-energy states, which also contribute to the thermal expansion [41] and to the ultrasonic wave velocities and attenuation [42]. For vitreous silica, Brugger and Fritz [15] noted a complete lack of agreement between the estimated low-temperath ture limits of the elastic, TL, el and thermal, TL, Grfineisen parameters. The rapid increase of the negative thermal expansion at low temperatures [30,43] is evidence for a large increase in T th a s the t e m p e r a t u r e is decreased [11,5]. The magnitude of the discrepancy between Y~ and T~ has been examined by Phillips [41]. Using White's [43] thermal expansion data, and the specific heat measured on the same sample, he finds at 1.5 K that Y~ is - 1 7 while T [ 1 is only - 2 . 2 . In the estimation of TL, el these previous workers [15,41] assumed that the T O E C are independent of temperature for vitreous silica. The present measurements of the T O E C (table 5) show that this assumption is not valid. The m e a n long-wavelength acoustic m o d e Griineisen parameter, y el, calculated from the present measurements of the T O E C decreases somewhat as the t e m p e r a t u r e is reduced (fig. 1). So the discrepancy between Y~ and y~! may not be as large as previously envisaged. A p r o g r a m m e to measure the third-order elastic constants of vitreous silica at liquid-helium temperatures is now in progress to quantify the problem through a determination of the low-temperature limit of the mean long-wavelength acoustic m o d e Griineisen parameter, y}.~.
6. Conclusions
(1) The T O E C of vitreous silica measured between 77 and 293 K are found to be markedly dependent upon temperature. (2) The T O E C (except the smallest C456) are anomalously positive. This is consistent with the negative vlaues found for the hydrostatic pressure derivatives (OCii/OP)p= o. (3) The largest T O E C is C m . (OC11/OP)p= o is much larger than (OC44/OP)p=oover the whole t e m p e r a t u r e range: the longitudinal acoustic
73
mode softens much more with pressure than the shear mode. (4) The acoustic m o d e Griineisen parameters are negative throughout the whole t e m p e r a t u r e range, ]Yi~l for the longitudinal m o d e being numerically substantially larger than l ysl for the shear mode. As the t e m p e r a t u r e is reduced [7L [ increases considerably, reaching - 5 . 5 at 77 K. (5) The mean acoustic mode Griineisen parameter, yel, increases to a larger negative value ( - 3 . 1 at 77 K) than previously suspected on the basis of the assumption of t e m p e r a t u r e independent T E O C : the discrepancy between the lowt e m p e r a t u r e limits of the thermal, T th, and mean, 7 e~, Grfineisen parameters is not quite so large as had been thought. We are most grateful to Dr M. Cutroni (Universit?~ di Messina) for informing us that an indium bond between a quartz transducer to a vitreous silica specimen facilitates low-temperature ultrasonic experiments and Dr B.J. James ( G E C Hirst Research Centre) for describing the cold-welding technique for making such a bond.
References
[1] M.E. Fine, H. Van Duyne and N.T. Kenney, J. Appl. Phys. 25 (1954) 402. [2] O.L. Anderson and H.E. B6mmel, J. Am. Ceram. Soc. 38 (1955) 125. [3] J.T. Krause and C.R. Kurkjian, J. Am. Ceram. Soc. 51 (1968) 226. [4] J.T. Krause, J. Appl. Phys. 42 (1971) 3035. [5] C.R. Kurkjian, J.T. Krause, H.J. McSkimin, P. Andreatch and T.B. Bateman, in: Amorphous Materials, ed. R.M. Douglas and B. Ellis (Wiley, New York, 1972) p. 463. [6] P.W. Bridgman, Proc. Am. Acad. Arts Sci. 76 (1948) 55; 76 (1948) 71. [7] E.H. Bogardus, J. Appl. Phys. 36 (1965) 2504. [8] L. Peselnick, R. Meister and W.H. Wilson, J. Phys. Chem. Solids 28 (1967) 635. [9] A. Mierzeiewski, G.A. Saunders, H.A.A. Sidek and B. Bridge, J. Non-Cryst. Solids 104 (1988) 333. [10] K. Kondo, S. Iio and A. Sawaoka, J. Appl. Phys. 52 (1981) 2826. [11] W.T. Yost and M.A. Breazeale, J. Appl. Phys. 44 (1973) 1909. [12] R.W. Dunham and H.B. Huntington, Phys. Rev. B2 (1970) 1098.
74
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
[13] J.H. Cantrell and M.A. Breazeale, Phys. Rev. B17 (1978) 4864. [14] D. Tielbiirger, R. Merz and S. Hunklinger, in: Proc. 3rd Int. Conf. on Phonon Physics, Vol. 1 (World Scientific, Singapore, 1990) p. 447. [15] K. Brugger and T.C. Fritz, Phys. Rev. 157 (1967) 524. [16] Y.K. Yogurtcu, A.J. Miller and G.A. Saunders, J. Phys. C13 (1980) 6585. [17] Y.K. Yogurtcu, E.F. Lambson, A.J. Miller and G.A. Saunders, Ultrasonics 18 (1980) 155. [18] R.N. Thurston and K. Brugger, Phys. Rev. A133 (1964) 1604. [19] K. Brugger, J. Appl. Phys. 36 (1965) 768. [20] R.E. Strakna, Phys. Rev. 123 (1961) 2020. [21] C.A. Maynell, G.A. Saunders and S. Scholes, J. NonCryst. Solids 12 (1973) 271. [22] H.E. Shull, PhD thesis, Pennsylvania State University, Dissertation Abstr. Int. B31 (1970) 2228. [23] R.A. Graham, J. Acoust. Soc. Am. 51 (1972) 1576. [24] W.A. Phillips, J. Low-Temp. Phys. 7 (1972) 351. [25] P.W. Anderson, B.I. Halperin and C.M. Varma, Philos. Mag. 25 (1972) 1. [26] K.S. Gilroy and W.A. Phillips, Philos. Mag. B43 (1981) 735. [27] W.A. Phillips, Proc. 3rd Int. Conf. on Phonon Physics, Vol. 1 (World Scientific, Singapore, 1990) p. 367. [28] U. Bartell and S. Hunklinger, J. Phys. (Paris) 43 (1988) C9-489.
[29] G.K. White and J.F. Birch, Phys. Chem. Glasses 6 (1965) 85. [30] G.K. White, Cryogenics 4 (1964) 2. [31] G.K. White, Cryogenics 16 (1976) 487. [32] D.F. Gibbons, J. Phys. Chem Solids 11 (1959) 246. [33] R.J. Bell, N.F. Bird and P. Dean, J. Phys. C1 (1968) 299. [34] J.R. Ferraro and M.H. Manghnani, J. Phys. Chem. Glasses 13 (1972) 6. [35] K. Chow, W.A. Phillips and A.I. Bienestock, in: Phase Transitions, ed. H.K. Henisch and R. Roy (Pergamon, New York, 1973) p. 333. [36] P. Flubacher, A.J. Leadbetter, J.A. Morrison and B.P. Stoicheff, J. Phys. Chem. Solids 12 (1959) 53. [37] O.L. Anderson and G.J. Dienes, in: Non-Crystalline Solids, ed. V.D. Frechette (Wiley, New York 1960) p. 449. [38] A.A. Antoniou and J.A. Morrison, J. Appl. Phys. 36 (1965) 1873. [39] R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. [40] R.B. Stephens, Phys. Rev. B8 (1973) 2896. [41] W.A. Phillips, in: Amorphous Solids, ed. W.A. Phillips, (Springer, Berlin, 1981) p. 53. [42] L. Pieh4, R. Maynard, S. Hunklinger and J. J~ickle, Phys. Rev. Lett. 32 (1974) 1426. [43] G.K. White, Phys. Rev. Lett. 34 (1975) 204.
Journal of Non-Crystalline Solids 143 (1992) 65-74 North-Holland
NON-CRYSTALLINE SOLIDS
Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica Q. W a n g , G . A . S a u n d e r s , H . B . S e n i n a n d E.F. L a m b s o n School of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK Received 4 January 1991 Revised manuscript received 6 D e c e m b e r 1991
M e a s u r e m e n t s of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components (TOEC) of vitreous silica between 77 and 293 K. In the absence of such data previous practice has been to assume that these T O E C are independent of temperature; however a significant temperature dependence has been observed. With the exception of the smallest C456, each of the T O E C is anomalously positive: a feature consistent with the well established aberrant negative values for the hydrostatic pressure derivatives (OCij/8P)e-o of the second-order elastic stiffness tensor components (which reveal softening of the long-wavelength acoustic modes u n d e r pressure). The T O E C increase as the temperature is reduced showing that the pressure-induced acoustic mode softening becomes enhanced at lower temperatures. T h e largest T O E C is Cll I which is 4 6 . 7 x 10 l° Pa at 293 K and increases to 99.1 × 101° Pa at 77 K. The hydrostatic pressure derivatives of the second-order stiffnesses have been calculated from the T O E C ; (~Ca~/OP)P-O is m u c h larger than (i~C44/OP)P-O over the whole temperature range: the longitudinal acoustic mode softens more with pressure than the shear mode. This effect is emphasised by the finding that the acoustic mode Griineisen parameters are negative with [YLI > l y s l . As the temperature is reduced the longitudinal acoustic mode Griineisen parameter, YL, increases in magnitude considerably, reaching a value of - 5 . 5 at 77 K. T h e m e a n acoustic mode Griineisen parameter, T el, increases to a larger negative value ( - 3 . 1 at 77 K) than previously suspected on the basis of the assumption of t e m p e r a t u r e - i n d e p e n d e n t third-order elastic constants: the discrepancy between the low-temperature limits of the thermal, YL, th and m e a n acoustic mode, YL, el Griineisen parameters is not quite so large as had been thought. The vibrational anharmonicity of the acoustic modes plays an important part in causing the thermal expansion of vitreous silica to be negative at low temperatures.
1. Introduction In general experimental determinations of the t e m p e r a t u r e dependences of the third-order elastic stiffness tensor components ( T O E C ) are sparse because it is demanding to make m e a s u r e m e n t s of the pressure dependences of ultrasonic wave velocities as a function of temperature. The normal procedure used to obtain a complete set of T O E C is to measure the effects of hydrostatic and uniaxial pressure on the velocity of ultrasonic waves p r o p a g a t e d in the solid of interest. E m p h a sis is usually placed on making as many measurements under hydrostatic pressure as possible because the effects induced are larger, and conse-
quently are easier to measure with accuracy, than those obtained from application of uniaxial pressure. However, at low temperatures the liquids used as pressure-transmitting media freeze making it difficult to measure the effects of pressure on ultrasonic wave velocity or even to ensure that the applied pressure is truly hydrostatic. Although it is possible to use gaseous systems to apply hydrostatic pressures, for safety reasons this is best avoided. However we have noticed that it is possible to obtain all three independent components of the third-order elastic stiffness tensor for an isotropic solid by making ultrasonic wave velocity m e a s u r e m e n t s solely under uniaxial pressures, a possibility which does not seem to
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
66
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity o f vitreous silica
have been explored before. Hence our major objective has been to construct an equipment to measure the effects of uniaxial pressure on ultrasonic wave velocities below room temperature in glass specimens, and to assess experimentally whether this approach can be used to obtain the complete sets of T O E C needed to quantify the nonlinear acoustic properties of glasses at lower temperatures. Vitreous silica has been chosen as the candidate material on which to make such experiments because it is of great technological importance, it is a pure, single-phase glass and furthermore has remarkable acoustic mode vibrational anharmonicity effects not yet studied at low temperature. We have succeeded in determining the effects of uniaxial stress on the velocities of enough ultrasonic mode configurations to derive complete sets of T O E C as a function of temperature between 77 and 293 K. The non-linear acoustic properties of vitreous silica SiO 2 are singularly anomalous. The temperature dependences of the elastic stiffness constants are negative up to about 70 K and then become positive [1-5]. The hydrostatic pressure dependences are negative: when subjected to pressure, vitreous silica becomes easier to squeeze [6-8]: This extraordinary behaviour also occurs in other tetrahedrally coordinated glasses G e O 2 and BeF2, but not for triangular or 'modified' tetrahedral glasses [5]. It is shown also by vitreous samarium phosphate [9]. While the elastic moduli of fused quartz decrease under moderate applied pressure, normal behaviour is assumed at higher pressures. In his early compression studies Bridgman [6] first observed that the bulk modulus decreases with increasing pressure until a reversal of slope at about 3 GPa. This reversal was confirmed by ultrasonic measurements as a function of high pressure which showed non-linear decreases of the elastic stiffnesses Cll , C44 and the bulk modulus to a minimum value at about (2.3 _+ 0.2) GPa, and thereafter an increase with increasing pressure; elastic behaviour was observed up to pressures of 3 GPa [10]. Bogardus [7] first obtained a set of mixed adiabatic-isothermal T O E C for fused silica at room temperature from measurements using the ultrasonic pulse superposition technique as a function of both hydro-
static and uniaxial pressure. He found that each TOEC, except the smallest C456, has a positive value consistent with the anomalous negative sign of the hydrostatic pressure derivatives (OCjJ 0P)p= 0 of the second-order elastic stiffnesses (SOEC). Subsequently a combination of results obtained from ultrasonic second harmonic generation [11] with those of ultrasonic beam mixing [12] was used to obtain a complete set of adiabatic T O E C for fused silica [11]. Cantrell and Breazeale [13] have determined the temperature dependence of the adiabatic T O E C ClSll alone between 3 and 300 K from measurements of the second harmonic generation from the fundamental of longitudinal ultrasonic waves. Recently the pressure dependence of the longitudinal sound wave velocity in vitreous silica has been measured using high-resolution Brillouin spectroscopy between 50 K and room temperature [14]; the reduction of velocity induced by application of pressure was found to occur throughout this temperature range. Negative hydrostatic pressure derivatives (OCiJOP)e=o and positive temperature dependences of the second-order elastic stiffness tensor components and positive T O E C reveal anomalous effects in the vibrational anharmonicity: the frequencies of the long-wavelength acoustic modes decrease under pressure so that the mode Grfineisen parameters
y(p, q) = - ( ~ In w(p, q)/a In V ) r , e = 0 ( a In o~(p, q) ) =
-
V
OP
(0~) T,P=O
=B ~
T,P=O
(1) OP
T,P=O
are negative. Here o~(p, q) is the vibrational frequency of a mode p of wavevector q, P is the applied pressure, V is the volume of the solid and B T is the isothermal bulk modulus which is defined as
In addition to being responsibl e for non-linear acoustic properties, vibrational anharmonicity is
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous sifica
the source of thermal expansion. Those glasses which show the anomalous elastic behaviour under pressure or with temperature may also exhibit a negative thermal expansion at low temperature [5] where contributions from the long-wavelength, low-frequency acoustic modes play a dominant role. Therefore, knowledge of the acoustic mode Griineisen parameters is a prerequisite for understanding thermal expansion at low temperatures. Although the acoustic mode Griineisen parameters have been determined for vitrious silica at room temperature [5,10,15], their temperature dependences are not known. One goal of the present work has been to fill this gap. Acoustic mode Griineisen parameters can be determined from measurements of either the hydrostatic pressure derivatives of the second-order elastic constants or the TOEC. The strain energy per unit mass U(rl)(= U(S, 7q)- U(S, 0)) is defined in terms of Lagrangian strains ~q as p0g(~)
= 1
S 1
1
S
s
+ 5zC~t~E;no~7~¢~7~7~C~Tno.
(3)
Hence the T O E C
CSt~c=P°
( 03U(S' rt) ) 0~7~/30r/~~0~7~ s;n=0
(4)
constitute the cubic coefficients of this strain Hamiltonian and quantify the lowest-order anharmonic term of the potential with respect to change in the particle separation. Therefore T O E C data, and the acoustic mode Griineisen parameters, determine the vibrational anharmonicity of the long-wavelength acoustic modes to cubic order in strain.
2. Experimental techniques A rectangular parallelepiped of fused silica of dimensions of about 1 cm 3 with three pairs of orthogonal faces polished flat and parallel to optical precision was prepared. Parallelism of the faces was examined by using an optical interference method and found to be within one wave-
67
length of sodium light. Quartz transducers, driven at their fundamental frequency of 10 MHz (X-cut for longitudinal, Y-cut for shear waves) were bonded to the specimen. An indium bond was made between the specimen and transducer by first coating them with thin films of nichrome, gold and indium in that order and then cold welding under vacuum. Uniaxial stresses were applied in a screw press [16]. The internal stress patterns inside the specimen under uniaxial stress in the press were examined using two pieces of polaroid. No colour patterns should be observable when the uniaxial pressure is uniform. When stress patterns could be seen, the position of the specimen between settling shims was adjusted until they were removed. Ultrasonic experiments under uniaxial stress were carried out in a liquidnitrogen cryostat in which it was possible to stabilise the temperature to within +0.1 K over a time period of several minutes during which the measurements were made. Changes induced in ultrasonic wave velocity by application of a uniaxial stress were measured to the required sensitivity of 1 part in 10 7 using an automated gated pulse superposition technique [17]. To bypass the requirement for determination of changes induced in sample dimensions, the 'natural velocity W' technique [18] was employed. The effects of the uniaxial stress applied (up to about 1 × 10 7 Pa) on ultrasonic wave velocity are small; although the systematic errors in absolute magnitude of the T O E C are quite large, the error in the relative changes with temperature is much smaller.
3. Experimental configurations for determination of the TOEC of an isotropic solid An isotropic solid has the following third-order elastic stiffness tensor components: C l l 1 = C222 = C333;
C144 = C255 = C366;
C l l 2 = C233 = C133 = C l l 3 = C122 = C233; C155 = C244 = C344 = C166 = C266 = C355; C123;
C456 .
(5)
68
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
However only three of these are independent and can be taken as C123 =/)1;
C144 = P2;
(6)
C456 =/"3.
Then the remaining non-zero tensor components are given by linear combinations of these three parameters: Cl12 =/)1 q- 2/)2,
(7)
C155 =/)2 q- 2/)3,
(8)
C111 = /)1 q- 6/)2
+ 8/)3"
(9)
The T O E C , which are sixth rank tensor ponents Cuvprqs , c a n be obtained from the derivatives (p0W2)~,=0 evaluated at zero using [19] t -- ( P o W 2 ) p = o = ( N " M ) 2 + 2 w F u c + G u c ,
comstress stress (10)
with
w=(poW2)o=(pV2)o
s = C;.rsN,
NqUrV
(11)
,
F u c = SfbrsM,,MbUrU ~ ,
(12)
G u c = SabuvCuvprqsMaMbNpNqUrV r s.
(13)
H e r e W is the natural velocity, V is the measured velocity, N and U are unit vectors along the wave propagation and polarisations directions respectively; M is a unit vector along the applied stress direction; C s and S f are isentropic stiffness and isothermal compliance tensor components respectively. This procedure [18] leads to the solutions for the effect of uniaxial pressure on the velocity of ultrasonic modes propagated in an isotropic solid given in table 1. To determine the three independent T O E C of vitreous silica, measurements were made of the effects of uniaxial pressure on the velocities of these three ultrasonic m o d e configurations.
4. Experimental results for the second- and third-order elastic stiffness tensor components of vitreous silica as a function of temperature
T h e s e c o n d - o r d e r elastic constants, C lJ' s Young's modulus, E s, bulk modulus, B s, and Poisson's ratio, or, calculated from the measured ultrasonic wave velocities are given in table 2. The velocities of both longitudinal and shear wave propagated in vitreous silica (table 2) increase approximately linearly with increasing temperature in the t e m p e r a t u r e range studied here. This is a well-known p h e n o m e n o n associated with the relaxation processes which produce the broad p e a k characteristic of the ultrasonic attenuation in silicate glasses [2-4,20,21]. It contrasts markedly with the behaviour of a pure dielectric crystal for which the velocity of sound decreases with increasing t e m p e r a t u r e due to the effects of vibrational anharmonicity. The three independent T O E C Vl, /)2 and u 3, defined in eq. (6), have been determined from the elastic constants (table 2) and the uniaxial pressure derivatives { ( 1 / W ) d W / d P } e = o of the measured ultrasonic wave natural velocity (table 3) using the expressions given in table 1. The T O E C , determined using eqs. (7-9), are compared in table 4 with those of obtained at room temperature by previous workers. The third-order Cauchy relationships C123 = C 1 4 4 = C456 are not obeyed, implying a large non-central force contribution to the vibrational anharmonicity. The central aim of this work to determine the t e m p e r a t u r e dependence of the T O E C of vitreous silica has been achieved down to 77 K; the results are given in table 5. The hydrostatic pressure derivatives (OCi]/OP)p= 0 of the elastic stiffness tensor com-
Table 1 The experimental configurations and the relationships used for calculation of the three TOEC (vl, /'2, /'3) defined in eq. (6) of an isotropic medium under uniaxial compression. E is Young's modulus and ~r is Poisson's ratio. The stress is applied in a direction M perpendicular to the ultrasonic mode propagation direction N Mode polarisation Mode 1: L Mode 2: S Mode 3: S
Displacement direction, U parallel to N parallel to m perpendicular to M
w = (poVa)e_o
{d(poW2)/dP}p=o
C1~
[~r(2w + 8v3)+ vi(2o- - 1)+ Uz(8~r-2)]/E [ -- 2W + v2(2o" - 1) + 2 v 3 ( c r - 1)]/E [o-(2w +4v3)+ v2(2o-- 1)]/E
C44
C44
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
69
Table 2 Longitudinal (V L) and shear (Vs) wave velocities (in units of ms 1), elastic st±finesses and bulk (B s) and Young's (E s) moduli (in units of 101° Pa) and Poisson's ratio of vitreous SiO 2 as a function of temperature. The errors are: AVL,AVs = _+2 ms-Z; Cll,C12,C44 = _+0.003 × 10 l° Pa; ES,B s = _+0.01 × 10 l° Pa; ~r = _+0.001 T (K)
VL
Vs
CII
C12
C44
Young's modulus, E
Bulk modulus
Poisson's ratio,
293 273 253 233 213 193 173 143 123 113 103 93 77
5974 5971 5957 5947 5940 5938 5927 5912 5909 5902 5911 5899 5893
3733 3723 3688 3681 3675 3669 3665 3651 3645 3641 3632 3623 3620
7.862 7.853 7.817 7.791 7.774 7.768 7.740 7.700 7.692 7.674 7.669 7.665 7.652
1.723 1.735 1.826 1.821 1.824 1.837 1.822 1.826 1.838 1.832 1.855 1.883 1.878
3.069 3.059 2.996 2.985 2.975 2.965 2.959 2.937 2.927 2.921 2.907 2.891 2.887
7.24 7.23 7.13 7.10 7.08 7.07 7.05 7.00 6.98 6.97 6.95 6.92 6.91
3.77 3.77 3.82 3.81 3.81 3.81 3.80 3.78 3.79 3.78 3.79 3.81 3.80
0.180 0.181 0.189 0.189 0.190 0.191 0.191 0.192 0.193 0.193 0.195 0.198 0.197
ponents, calculated from the TOEC data, are given in table 6. Measurement of the change in ultrasonic wave velocity induced by application of a constant stress leads to mixed adiabatic (S) isothermal (T) moduli ( c S f ) (this has been written without superscripts S and T throughout this paper) making it necessary to known the corrections required to determine adiabatic third-order elastic constants cSk. Following the technique detailed by Shull
Table 3 Experimental data for the pressure gradient { ( 1 / W ) d W / dP/e=0 of natural velocity (in units of 10 -11 Pa -1) of ultrasonic modes propagated in vitreous silica under uniaxial pressure. The modes are labelled as in table 1 T (K)
Mode 1
Mode 2
Mode 3
293 273 253 233 213 193 173 143 123 113 103 93 77
-0.693__+0.001 - 0.652 _+0.001 - 0.826 ± 0.001 - 1.073 _+0.001 -1.132_+0.001 -1.388_+0.001 -1.653__+0.001 -2.283-+0.001 -2.254_+0.001 - 2.343 ± 0.001 - 2.412±0.001 -2.607__+0.001 -2.726_+0.001
-2.161_+0.003 - 2.129 ± 0.006 - 1.881 ± 0.003 - 1.958 ± 0.005 -2.111+0.004 -2.247+0.003 - 2.265 ± 0.003 -2.335-+0.005 -2.365±0.001 -2.347+0.001 -2.463_+0.001 -2.463±0.001 -2.546_+0.001
-0.944__+0.005 - 1.026 _+0.001 - 1.031 _+0.001 - 1.131 _+0.001 -1.194__+0.001 -1.236__+0.001 -1.341-+0.001 -1.573-+0.001 -1.773±0.001 -1.903_+0.001 -1.991_+0.001 - 2.093 ± 0.001 -2.236_+0.001
[22], it can be shown that the relationship giving the difference between the measured ClS/~" and the adiabatic quantity cS11 is
( c S l - c s l r l )= - - ~ T - 1
(c111+2c112)]3 Here BS/B r= (1 + 3 a y t h T ) = 1.0013. At room temperature (8Cll/8T) P has been obtained as 10.7 × 106 Pa K -1 from a polynomial fit to the data for the temperature dependence of Cll. The coefficient a of linear thermal expansion is 5 × 10 -7 K -1 and ~th is 2.918. These results lead to a value for the difference (CSll - C l iST1) of only - 0 . 8 9 × 101° Pa. The magnitudes of this correction ( c S k - cS:~) are the same for the other T O E C and are much less than the experimental error. Therefore for computational purposes the data for the T O E C given in table 5 can be taken as being the purely adiabatic constants, cSk. Second harmonic generation from the fundamental of longitudinal ultrasonic waves propagated in four different types of fused silica has been used [13] to determine the non-linearity parameter/3 = - ( 3 C l l + Cm)/3Cll between 4.2
70
Q. Wang et aL / Elastic constants and acoustic vibrational anharmonicity o f vitreous silica
Table 4 Comparison between the third-order elastic stiffness components for vitreous silica at room temperature obtained by previous workers. Units: 109 Pa Source
Sample type
6111
Cll 2
CI23
C144
ClSs
C456
Present results [7] ~) [22] a) [11] b) [11] b) [23] c) [13] b) [13] b) [13] b) [13] u)
Fused silica Fused silica Vitreous silica G.E. 151 Optical quality Fused silica Suprasil W l Suprasil W2 Suprasil 1 Suprasil 2
470 +_10 526 _+40 620 648+- 5 487+_ 4 550 +_ 1 670+_ 12.4 685 +_ 5.8 731_+ 7.3 675 +_ 6.7
234_+20 239+_30 261 537 +- 26
81± 5 54+_13 72 428 +- 32
76+_6 93+_8 95 54 +- 3
59+_10 72_+ 2 90 28 +_ 5
-8.7+_0.6 - 1 1 _+3 - 2.5 - 13.2 +_0.8
a) By ultrasonic pulse superposition method. b) By ultrasonic harmonic generation method. c) By shock-wave compression.
and 300 K. In combination with longitudinal wave velocity, VL, measurements which give C n from C l l -- P I/'.2 L,
TOEC (with the exception of silica (table 5).
C456) of
vitreous
(15)
the adiabatic third-order elastic constant, cS11, could be determined as a function of temperature. It was found that cSlt showed a small decrease with decreasing temperature for Suprasil W l , W2 and 2 but an increase for Suprasil 1. The present results show that C n l increases with decreasing temperature (table 5), as do the other
5. Discussion: the vibrational anharmonicity of vitreous silica as a function of temperature
One of the strikingly anomalous features of the vibrational properties of fused silica is that the hydrostatic pressure derivatives of the elastic stiffness constants are negative [6,7]. With decreasing temperature each pressure derivative
Table 5 The third-order elastic constants of vitreous silica between 77 and 293 K. The units are 101° Pa. The errors are C m = 2 × 10 m Pa, Cn2 = ± 2 × 1 0 1 ° Pa, C123 = _+0.8x 101° Pa, C144 = ± 0 . 8 x 1 0 I° Pa, C155 = +0.1×101° Pa, C456 = _+0.6×10 m Pa T (K)
Cm
C~12
C123
C144
C155
C456
293 273 253 233 213 193 173 143 123 113 103 93 77
47.0 45.4 41.4 48.3 54.0 62.9 68.0 81.7 83.2 84.4 87.9 94.2 99.1
23.4 22.5 24.7 29.6 31.7 37.4 41.9 53.5 53.6 54.9 56.9 61.4 64.0
8.1 7.1 10.4 14.2 15.2 19.9 23.5 32.9 31.1 31.4 32.5 35.9 36.9
7.6 7.7 7.1 7.7 8.3 8.8 9.2 10.3 11.2 11.7 12.2 12.8 13.5
5.9 5.7 4.2 4.7 5.6 6.4 6.5 7.1 7.4 7.4 7.8 8.2 8.8
- 0.9 - 1.0 - 1.5 - 1.5 - 1.4 - 1.2 - 1.3 - 1.6 - 1.9 -2.2 - 2.2 - 2.3 - 2.4
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica Table
6
Hydrostatic components from
the
The
errors
pressure and
bulk
experimental are
derivatives modulus data
of the elastic of vitreous
for the
TOEC
stiffness
tensor
silica determined given
in table
5.
_+0.01
T
(i~Cll//
(i~C12/
(~C44/
(~B/
(K)
OP)p= o
OP)P=O
OP)P=O
i~P)P=O
239
- 9.98
- 3.99
- 2.99
- 5.99
273
-9.69
-3.76
-2.96
-5.74
253
-9.59
-4.37
-2.61
-6.11
233
- 11.08
- 5.57
- 2.75
- 7.41
213
- 11.96
- 6.04
- 2.96
- 8.01
193
- 13.71
- 7.43
- 3.14
- 9.53
173
- 15.02
- 8.59
- 3.22
- 10.74
143
- 18.29
- 11.48
- 3.41
- 13.75
123
- 18.42
- 11.32
- 3.55
- 13.69
113
- 18.81
- 11.61
- 3.59
- 14.01
103
- 19.39
- 12.02
-3.69
- 14.48
93
- 20.65
- 13.04
- 3.8i
- 15.58
77
- 21.58
- 13.62
- 3.98
- 16.27
(OCll/OP)p=o, (OC12/OP)p=o, (OC44/OP)p=o and (OB/OP)p=o gets larger (table 6): the extraordinary behaviour becomes enhanced at lower temp e r a t u r e s . T h e e x p e r i m e n t a l facts that (OC11/aP)p=o is much greater than (OC44/OP)p=o and that Cll I is by far the largest TOEC, and is positive, show that the acoustic mode softening induced by pressure is substantially greater for the longitudinal than for the shear acoustic modes. The temperature dependence of the ultrasonic attenuation in vitreous silica is dominated by a broad maximum with Tmax at about 50 K [2] whose origin lies in classical, thermally activated, relaxation processes in double-welled asymmetric potentials [24-27]. Although a wide range of low-temperature thermodynamical and transport properties of glasses can be understood within the framework of this phenomenological model, the effects of pressure on the two-level systems have yet to be investigated theoretically. The problem has recently been addressed experimentally [14,28]. Measurements of the effect of pressure on the ultrasonic attenuation in the quartz glass Suprasil Wl have shown that the broad absorption peak shifts from about 50 K at atmospheric pressure to 90 K under a pressure of 0.42
71
GPa [28]. The anomalous positive temperature dependence of the second-order elastic moduli of vitreous silica is probably also a consequence of the double-well relaxation processes which produce this attenuation peak. Furthermore since this positive temperature dependence of the elastic moduli extends right up to room temperature (table 2) the relaxation processes would seem to remain operative at that temperature. The same processes are probably responsible for the anomalous non-linear acoustic behaviour under pressure, namely the positive signs of the TOEC (table 5) and the negative values found for the hydrostatic pressure d e r i v a t i v e s (OCij/OP)p= 0 (table 6). The observation that (OC11/OP)p=o has a much larger negative value than (OC44/OP)p_o indicates that the double-well relaxation processes are more effective in softening the longitudinal than the shear acoustic waves. TOEC data have as yet been obtained only down to 77 K (table 6), still somewhat above the temperature Tmax at which the maximum relaxation occurs where the effects of the relaxation processes on the vibrational anharmonicity could be much enhanced. Knowledge of the TOEC below room temperature helps us to clarify further the origins of the anomalous behaviour of the thermal expansion of vitreous silica because the thermal expansion and non-linear acoustic properties both relate to the vibrational anharmonicity. The thermal Griineisen parameter, , ) / t h , which can be obtained from data for the linear coefficient, a, of thermal expansion, the compressibility, K, and the specific heat, C, using yth = 3aV/KSCp = 3aV/KrCv,
(16)
results from the sum of excited modes:
yth = E yC,(T)I E Ci(T). i
(17)
i
The second- and third-order elastic stiffnesses provide a measure, which can be expressed by acoustic mode Griineisen parameters, of the anharmonicity of the long-wavelength acoustic modes alone. For an isotropic solid there are two components: "/i~ and Ys, which refer to the Iongi-
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
72 -1.0
•
"6 -2.0
II
i
|
I.
¢
i
i
g
•
.~ -3.(1
"
•
¢ i
i
:7
q-5 X 10 - 7 K -1) at room temperature, decreases as the temperature is reduced and changes sign at about 180 K, reaches a minimum at about 40-50 K and then tends to zero as the temperature tends to 0 K [29-32]. The thermal expansion accrues from the summation (eq. (17)) over the excited vibrational states including the long-wavelength acoustic modes. The infrared spectrum of vitreous silica is dominated by three peaks at about 100, 800 and 460 cm-1. Each of the vibrational modes responsible for these peaks involves both stretching and bending of the silicon-oxygen bond, and in general the higher the frequency, the larger the contribution made by stretching [33]. The mode Griineisen gammas for these modes have been determined from the pressure dependences of the peak positions [34,35]. While the sign of the Yi for the 800 cm -1 mode is positive, those for 1100 and 470 cm -1 are negative. A physical description of how these signs arise in terms of the stretching and bending of the silicon-oxygen bond has been given by Chow, Phillips and Bienenstock [35] who showed that the density-of-states calculation [33] can be used to obtain a specific heat and thermal expansion which are in reasonable agreement with experiment, indicating that the small value for the thermal expansion of vitreous silica comes about as a result of summation over the effects of modes with different Griineisen parameter signs. As the temperature is lowered, the long-wavelength acoustic phonons with negative Griineisen parameters can be expected to make an increasingly important contribution to the thermal expansion. At low temperature when the excited phonon population includes a high proportion of longwavelength acoustic modes, the values of the elastic, YL, el and thermal, YL, th Griineisen parameters would be expected to become equal, like the elastic and thermal values of ( C / T 3 ) r _ , O K [3638]. For crystalline dielectric solids, in which the Debye approximation is reasonably valid at low temperatures, ~/}h and 3,[~ do tend to become equal. In general amorphous materials show anomalous thermal properties at low temperatures [39,40] which have been accounted for by the model of low-energy, two-level systems with a broad spectrum of their energy splitting [24-27]. (~
-a.0
.o_ ,A
A
-6.0
so
2;0 Temperoture
2;0
300
(K}
Fig. 1. The temperature dependences of the Griineisen parameters YL (triangles) and Ys (circles) of the long-wavelength longitudinal and shear acoustic modes respectively and the mean y el (squares) for vitreous silica.
tudinal and shear elastic waves, given by [11] "YL = -- ( 1 / 6 C l l ) [ 3 B
+ 2Cll + Clll -{- 2Cl12],
(18) and Ts = - (1/6C44)[3B
+ 2C44 + (Cll I - C123)/2 ] .
(19) These acoustic mode Griineisen parameters, obtained from the experimental second- and thirdorder elastic constant results, are plotted as a function of temperature in fig. 1 for vitreous silica. Also shown is the mean acoustic mode Griineisen parameter, ye~, obtained using
on the basis of the Debye continuum model. The results that YL, YS and Tel are negative and become larger when the temperature is reduced, are consistent with the behaviour of the thermal expansion with temperature. The thermal expansion coefficient of vitreous silica is over an order of magnitude smaller than that of crystalline quartz which in turn is comparable to that of many other solids. The dependence of the thermal expansion coefficient upon temperature can be summarised as follows: it is small and positive
Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
Thus the Debye model is not valid: the specific heat is dominated by the high density of these low-energy states, which also contribute to the thermal expansion [41] and to the ultrasonic wave velocities and attenuation [42]. For vitreous silica, Brugger and Fritz [15] noted a complete lack of agreement between the estimated low-temperath ture limits of the elastic, TL, el and thermal, TL, Grfineisen parameters. The rapid increase of the negative thermal expansion at low temperatures [30,43] is evidence for a large increase in T th a s the t e m p e r a t u r e is decreased [11,5]. The magnitude of the discrepancy between Y~ and T~ has been examined by Phillips [41]. Using White's [43] thermal expansion data, and the specific heat measured on the same sample, he finds at 1.5 K that Y~ is - 1 7 while T [ 1 is only - 2 . 2 . In the estimation of TL, el these previous workers [15,41] assumed that the T O E C are independent of temperature for vitreous silica. The present measurements of the T O E C (table 5) show that this assumption is not valid. The m e a n long-wavelength acoustic m o d e Griineisen parameter, y el, calculated from the present measurements of the T O E C decreases somewhat as the t e m p e r a t u r e is reduced (fig. 1). So the discrepancy between Y~ and y~! may not be as large as previously envisaged. A p r o g r a m m e to measure the third-order elastic constants of vitreous silica at liquid-helium temperatures is now in progress to quantify the problem through a determination of the low-temperature limit of the mean long-wavelength acoustic m o d e Griineisen parameter, y}.~.
6. Conclusions
(1) The T O E C of vitreous silica measured between 77 and 293 K are found to be markedly dependent upon temperature. (2) The T O E C (except the smallest C456) are anomalously positive. This is consistent with the negative vlaues found for the hydrostatic pressure derivatives (OCii/OP)p= o. (3) The largest T O E C is C m . (OC11/OP)p= o is much larger than (OC44/OP)p=oover the whole t e m p e r a t u r e range: the longitudinal acoustic
73
mode softens much more with pressure than the shear mode. (4) The acoustic m o d e Griineisen parameters are negative throughout the whole t e m p e r a t u r e range, ]Yi~l for the longitudinal m o d e being numerically substantially larger than l ysl for the shear mode. As the t e m p e r a t u r e is reduced [7L [ increases considerably, reaching - 5 . 5 at 77 K. (5) The mean acoustic mode Griineisen parameter, yel, increases to a larger negative value ( - 3 . 1 at 77 K) than previously suspected on the basis of the assumption of t e m p e r a t u r e independent T E O C : the discrepancy between the lowt e m p e r a t u r e limits of the thermal, T th, and mean, 7 e~, Grfineisen parameters is not quite so large as had been thought. We are most grateful to Dr M. Cutroni (Universit?~ di Messina) for informing us that an indium bond between a quartz transducer to a vitreous silica specimen facilitates low-temperature ultrasonic experiments and Dr B.J. James ( G E C Hirst Research Centre) for describing the cold-welding technique for making such a bond.
References
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Q. Wang et al. / Elastic constants and acoustic vibrational anharmonicity of vitreous silica
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