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Quantitative analysis of ultrasonic attenuation in TiO2-doped CaO–Al2O3–B2O3 glasses A. Abd El-Moneim ∗ Physics Department, Faculty of Science, Zagazig University, Zagazig, Egypt Received 11 February 2005; received in revised form 23 August 2005; accepted 6 September 2005

Abstract A comprehensive quantitative analysis of the composition dependence of ultrasonic attenuation in TiO2 -doped 30CaO–30Al2 O3 –40B2 O3 glasses was achieved according to the central force theory and ring deformation model. Empirical formulas relating ultrasonic attenuation at room temperature and 4 MHz frequency with a number of physical and structural parameters, such as the bulk modulus, oxygen density, average first order stretching force constant and atomic ring size of the network, number of loss centers, i.e. double-well systems (oxygen atoms vibrating in asymmetric double-well potentials) and the barrier height of the asymmetric double-well potential are reported. The good agreement between the observed and theoretically calculated values of bulk modulus evidences the validity of the derived empirical equations for the studied glasses. © 2005 Elsevier B.V. All rights reserved. Keywords: Ultrasonic attenuation; Loss centers; Double-well systems; Bulk modulus; Structural parameters; Polycomponent borate glasses

1. Introduction Ultrasonic pulse-echo technique is one of the most useful experimental tools for structural studies of glasses. The ultrasonic attenuation and elastic moduli as a function of composition and temperature are directly related to the glass structure, thus, their measurements allow evaluating the influence of modifiers on glass networks [1–12]. The term ultrasonic attenuation (UA) is used throughout to mean the quantified energy losses after the ultrasonic waves have been propagated through the glasses. Attempts [1–9] have been made to study the effect of composition, temperature and frequency on the UA in glasses. It was shown that the UA depends linearly on the applied frequency (w) according to the relation, α = G(x, T)w + αr , where G is constant that depends on the temperature and composition of the glass and αr is the residual attenuation [1–4]. On the other hand, the attenuation versus temperature curves revealed the presence of broad absorption peaks in the temperature range between 20 and 300 K [2–8]. The peak positions were shifted to higher temperatures and large magnitudes of attenuation with increasing frequency.

∗

Present address: Teacher’s College in Dammam, P.O. Box 2375, Dammam 31451, Saudi Arabia. Tel.: +966 3818 1998; fax: +966 3826 9936. E-mail address: [email protected] 0254-0584/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2005.09.016

Bridge and Patel [8,9] explained the behaviors of UA on the basis of the existence of loss centers (double-well systems) in the network of the glass, which are oxygen atoms moving onto asymmetric and atomic dimensional double-well potentials, as shown in Fig. 1. Each double-well system has a potential barrier height (activation energy), V, and asymmetry, , which is the free-energy difference between corresponding particle states in the two wells, i.e. the separation of the two minima and exists for both longitudinal and transverse motion of oxygen atoms for all kind of bonds. The presence of asymmetry is reasonable because the local environment to the left of a double-well system of atomic dimensions generally differ from the environment to the right. At the absorption peak, the oxygen atom can surmount the potential barrier between the two wells in a thermally activated relaxation process. The relaxation time depends on the sample temperature T and may be written as τ = τo exp−V/KB T , where τo−1 is the attempt frequency (classical vibration frequency of the atom related to how often an attempt is made to exceed the barrier) and KB is Boltzmann’s constant. The physical and ultrasonic properties of the 30CaO–30Al2 O3 –40B2 O3 glasses doped with 5, 10, 15 and 20 mol% of TiO2 were reported recently [13–15]. The ultrasonic velocities and elastic moduli results showed to be composition dependent, which suggested that Al3+ and Ti4+ ions act as network formers by formation of AlO4 tetrahedra and TiO6 octahedra. This is a result of the conversion of

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semi-theoretical relationship, V = χ1 Fm

(1)

where χ1 is a constant and m is a high positive power. The bulk modulus K for a three-dimensional network could be obtained according to the ring deformation model [12] as, K = χ2 F−4

(2)

where χ2 is another constant. Bridge and Patel [9] combined Eqs. (1) and (2) and derived the following relationship between K, V, F and Fig. 1. (a) Schematic two-dimensional representation of the structure of noncrystalline material with atomic ring size . Full circles represent cations and open circles represent anions; a and b are the transverse and longitudinal vibrations of oxygen atoms. (b) Double-well system with barrier height V and asymmetry .

B–O–B linkages into Al–O–B–O–Ti linkages, which leads to an increase in the three-dimensional linkage (connectivity) of the glass, and consequently, stiffens the network of these glasses with increasing TiO2 concentration. The present work is an attempt to correlate the measured UA at room temperature with the physical and structural properties of these glasses. The total number of loss centers (double-well systems) per unit volume, oxygen density, fractional number of loss centers per oxygen atom, barrier height of the asymmetric double-well potential, average atomic ring size and cation–anion stretching force constant have been calculated and discussed as a function of the TiO2 content. Moreover, the agreement between observed and theoretically calculated values of bulk modulus has been investigated for these glasses. 2. Experimental The investigated titanium calcium aluminoborate (TCAB) glasses were prepared by the melt quenching method using analytical reagent (AR) grade materials having purity of more than 99.9% as described earlier in Refs. [13,14]. The two opposite faces of each glass sample were made parallel using a lapping machine with 600-grade SiC powder and the surfaces were polished using a precision polishing machine. The ultrasonic measurements were carried out at room temperature and at 4 MHz frequency using the pulse-echo technique. A single transducer was used both as the source of the initial pulse and the receiver for all the successive echoes. The transducer was coupled to one face of the specimen with a good acoustic coupling material (hydrate silicon fluid) to avoid any air gap between the transmitted pulse and the specimen. The UA was calculated from the relation, α = 2t1 ln hh1 where t is the thickness of the sample, while h1 and h2 are the 2 heights of two consecutive echoes.

3. Theory The correlation between the physical properties of the glasses at room temperature and the activation energy, V, of ultrasonic relaxation at low temperature has been well studied [9–11]. Bridge and Patel [9] proposed quantitatively that the average value of V increases with the first order stretching force constant, F, and ring size, , of the network and reported the following

KV = χ3 F(m−4)

(3)

where χ3 = χ1 χ2 is a constant. Their model was tested on the following glass formers: SiO2 , P2 O5 , As2 O3 , B2 O3 , GeO2 and binary Mo–P–O glass system in order to calculate the value of the exponent m, resulting in the following equation: KV = χ3 F 2 −0.5

(4)

For glasses with approximately the same values of atomic ring sizes, KV ∼ χ4 F 2

(5)

where χ4 is a constant. Recently, Abd El-Moneim and Abd ElLatif [10] have extended the model proposed by Bridge and Patel [9] to vitreous TeO2 and the tri-component TeO2 –V2 O5 –Sm2 O3 glasses and reported the following relationships between V and K: V = 113F2.6422 KV = 1.2 × 10

−27

(6) 2 −1.36

F

(7)

The total number of loss centers (asymmetric double-well systems) per unit volume of the glass, N, was found to be proportional to both the oxygen density, [O], and according to the following equation: N = χ5 [O]m

(8)

where χ5 is a constant and [O] is given by [10]: ρ (9) [O] = Na NO M where Na is Avogadro’s number, No the number of oxygen atoms per glass formula unit, ρ the density of the glass and M is its molecular weight. Recently [13], we have reported the following empirical equation, which correlate the UA at room temperature with both N and F, N (10) F where χ6 is a frequency dependent constant. Eliminating N from Eqs. (8) and (10), we obtain, α = χ6

m (11) F where χ7 = χ5 χ6 is a constant. The fractional number of doublewell systems per oxygen atom, N , is related to the stretching α = χ7 [O]

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force constant and bulk modulus according to a schematic relationship reported by Bridge and Patel [8] in the form, m/n F N = χ8 (12) K where n is another positive power and χ8 is a constant. Since the number of double-well systems per oxygen atom is a fraction of the total number of loss centers per unit volume N, Eq. (10) can be rewritten as, N 1 F m/n α = χ9 (13) = χ9 F F K where χ9 is a constant. The above considerations suggest that the UA dependent on the bulk modulus, total number of loss centers, barrier height of the asymmetric double-well systems, oxygen density, average atomic ring size and stretching force constant of the glasses. 4. Analysis and discussion Fig. 2 shows the composition dependence of the room temperature UA at a frequency of 4 MHz in the investigated TCAB glasses. As the TiO2 content increase up to 20 mol%, there is a decrease in the UA of about 32%. The line in the figure can be represented by the relation: α = 11.5 exp(−0.02x)

(dB cm−1 )

(14)

with correlation ratio of 98.5%, where x is the mol% of TiO2 . The change of UA through the studied composition range can be explained quantitatively in terms of the change in the structure and physical properties of the glass network as follows. According to the ring deformation model [12], the investigated TCAB glasses are regarded as three-dimensional networks of B O, Al O and Ti O covalent bonds and atomic rings. In these glasses, B O B, B O Al, B O Ti and Ti O Al bond angles have a spread of values around the fixed values obtained in the crystalline titanium calcium aluminoborates. The variability of B O B, B O Al, BO Ti and Ti O Al bond angles means that there is a spread values for B B, B Ti, B Al and Ti Al

Fig. 2. Composition dependence of UA in the investigated TCAB glasses [13,14]. The solid line represents the fitting of the data.

spacing and atomic ring sizes in these glasses. Bridge et al. [12] suggested that any change in the average stretching force constant, F, and bulk modulus, K, will consequently change the average atomic ring size of the network according to the following relation: 0.0106F 0.26 = (15) K where F is given by: (nf xf )i F = i i (nf x)i

(16)

where nf is the number of network bonds per formula unit (3 for B2 O3 , 4 for Al2 O3 and 6 for both CaO and TiO2 [12,13]), whereas x and f are the mole fraction and average first order stretching force constant of the oxide component i, respectively. Eq. (15) suggests that glasses with small values of F and large bulk modulus will have a smaller atomic ring sizes . It has been found that the values of f are 197, 660, 240 and 282 N m−1 for

Table 1 Physical, ultrasonic and structural properties of the investigated TCAB and pure B2 O3 glasses TiO2 (mol%) (g cm−3 )

ρ [13,14] α (dB cm−1 ) [13,14] K (GPa) [13] F (N m−1 ) [13] [O] (1028 m−3 ) [13–15] (nm) [13] N (%) (F/K)1.555 [O]6.15 (1026 ) αcal (dB cm−1 ) (Eq. (19)) αcal (dB cm−1 ) (Eq. (20)) V (eV) D (eV) Kcal (GPa)

5

10

2.747 10.35 53.1 365.9 5.22 0.506 1.79 20.1 7.9 10.38 10.39 0.071 0.255 62.1

2.764 8.74 59.8 355.4 5.20 0.487 1.64 16.0 6.2 8.49 8.43 0.062 0.233 70.6

15 2.810 8.00 60.3 346 5.24 0.483 1.61 15.1 6.0 8.27 8.29 0.059 0.225 71.1

20

Pure B2 O3 glass

2.857 7.07 66.5 337.8 5.28 0.468 1.50 12.5 5.0 7.00 7.05 0.053 0.210 78.8

1.83 [12] 13.2 [12] 660 [12] 0.876 [12] 5.61

0.545 0.999 12.5

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TiO2 , B2 O3 , Al2 O3 and CaO oxides, respectively [12,13]. Thus, the average force constant of the newly formed Al–O–B–O–Ti linkages are lower than the already existing Al–O–B linkages. Table 1 shows that the calculated F values decrease from 365.9 to 337.8 N m−1 as the TiO2 content increases from 5 to 20 mol%. Consequently, the average atomic ring size of these glasses decreases from 0.506 to 0.468 nm through the studied composition range. This is attributed to the gradual increase in the three-dimensional character of the network due to the conversion of Al–O–B linkages into Al–O–B–O–Ti linkages. The decrease of ultrasonic attenuation with increasing TiO2 concentration confirms the strengthening nature of these glasses as suggested from the composition dependence ultrasonic velocities and elastic moduli [13,14]. Rajendran et al. [16] expressed the attenuation coefficient in a solid material as: α = a1 αa + a2 αb + αc + αd + αe

(17)

where αa is the true absorption, αb the scatter loss, αc the coupling loss, αd the diffraction loss and αe the losses due to non-parallelism and surface roughness. Due to the amorphous character of the investigated samples, parallelism of the opposite faces and minute thickness of the bonding material between the sample and the transducer, the coefficients αb , αd , and αe are negligible. Consequently, Eq. (17) can be rewritten for the present TCAB glasses as α = a1 αa + αc

(18)

We shall now consider whether the presented model in Section 3 can explain the composition dependence of UA in the present TCAB glasses. Fig. 3 shows that ␣F increases as the ratio between the stretching force constant and bulk modulus F/K increases. Fitting these experimental data using Eq. (13) and performing least-square linear regression, with ln(αF) and ln(F/K), one obtains χ9 = 1.889 and m/n = 1.555, and correlation ratio of 98.2%. Consequently, Eq. (13) can be rewritten for the

Fig. 3. Variation of αF with the ratio F/K in the investigated TCAB glasses. The solid line represents the fitting of the data.

Fig. 4. Variation of the quantity Fα/[O] with the average atomic ring size in TCAB glasses. The solid line represents the fitting of the data.

present glasses as 1 F 1.555 α = 188.9 F K

(dB cm−1 )

(19)

Fig. 4 shows that the ratio αF/[O] increase systematically with the average atomic ring size, , of the network, which is in good agreement with Eq. (11). The least-square linear regression performed on ln(αF/[O]) and ln() yields the following relation: α = 4.8035 × 10−24

[O] 6.15 F

(dB cm−1 )

(20)

with correlation coefficient of 98.2%. Consequently, a close fit of Eqs. (19) and (20) to the data of Figs. 3 and 4 is given by m = 6.15 and n = 3.95. Substituting the obtained values of m and n in Eqs. (8) and (12) yields the following equations for N and N , N = χ5 [O]6.15 1.555 F N = χ8 K

(21) (22)

According to Table 1, the calculated value of F 1.555 [O]6.15 and K decreases with increasing the TiO2 content in the glass. Consequently, both the total number of loss centers per unit volume and the fractional number of double-well systems per oxygen atoms will decrease with increasing TeO2 concentration. This behavior is inconsistent with the composition dependence of UA in the same glasses (Fig. 2) and fit Eqs. (11) and (13). The values of ultrasonic attenuation, αcal , which have been calculated theoretically for the investigated TCAB glasses using Eqs. (19) and (20), are tabulated in Table 1. It can be seen from the table that the correlation ratio between the theoretically calculated and experimentally observed values of UA ranges between 96 and 99%. Thus, if we take into account the uncertainty that is inherent to experimental measurements and the approximations underlying Eqs. (8)–(20), we can consider that the presented model predict correctly the experimental UA in these glasses. The values of V that have been obtained from Eq. (6) for both the quaternary TCAB glasses are tabulated in Table 1, which

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can be represented by α = 338.31V 1.3182

(dB cm−1 )

(24)

shows that V decreases from 0.071 to 0.053 eV with increasing TiO2 content in TCAB glasses from 5 to 20 mol%. Fig. 5 illustrates how the value of V varies with experimental bulk modulus, K, in the present TiO2 -doped 30CaO–30Al2 O3 –40B2 O3 glasses. The data for P2 O5 –MoO3 , GeO2 , B2 O3 and As2 O3 glasses were taken from our Refs. [8–10] and included in the same figure for comparison. It is clear from this figure that the data of these glasses lays on a single hyperbolic curve. The fitted curve can be represented by the relation:

with correlation ratio of 99.6%. This behavior is reasonable and is in accordance with the above discussion. According to the central force theory [8], the motion of each loss center in an asymmetric double-well potential can be described as an oscillation around either of the two double-well minima. The ultrasonic waves disturb the equilibrium and produce a relative energy shift between the minima of the double wells by an amount δU = DE, where D = 1.5V2/3 is the deformation potential and E is the magnitude of the strain field [8,17]. One can note from Table 1 that D decreases as the TiO2 concentration increases, which corroborates with UA results for this glass system. In Ref. [14], we have used Makishima and Mackenzie model [18,19] and investigated the agreement between the experimental and theoretical values of elastic moduli and Poisson’s ratio of the present TiO2 -doped 30CaO–30Al2 O3 –40B2 O3 glasses. It was found that the correlation coefficient between the theoretical and experimental values of bulk modulus ranges between 62 and 72%. It is worth evaluating the use of Eqs. (6) and (7) for TCAB glasses. Table 1 shows the values of bulk modulus, Kcal , which have been calculated theoretically from these equations. The correlation coefficient between the experimental and theoretical values of bulk modulus, K, becomes greater than 84%. These results proved that the schematic relationships (6) and (7) are more suitable for predicting and correlating the bulk modulus of these glasses when compared to Makishima and Mackenzie theory [18,19].

K = 1.9477V −1.216

5. Conclusions

Fig. 5. Variation of bulk modulus with the barrier height of double-well potential in pure, binary and quaternary glasses. The solid line represents the fitting of the data.

(GPa)

(23)

with correlation ratio of 98.1%. This result confirms the validity of Eq. (6) for these glasses. Fig. 6 shows that the UA increases with increasing the barrier height V. The fitted curve in this figure

Quantitative analysis of the room temperature UA has been carried out in TCAB glasses according to the central force theory [8] and ring deformation model [12]. The structure changes of these glasses are deduced by relating the UA to the bulk modulus and critical structure parameters. The total number of loss centers vibrating in asymmetric double-well systems, barrier height of the asymmetric double-well systems, oxygen density, average atomic ring size and stretching force constant of the network may be regarded as indicators to the value of ultrasonic attenuation in glasses. References

Fig. 6. Variation of UA with the barrier height of the double-well potential in the investigated TCAB glasses. The solid line represents the fitting of the data.

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