This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 131.215.225.9

This content was downloaded on 20/08/2014 at 10:09

Please note that terms and conditions apply.

The relationship between the porosity and elastic moduli of the Bi-Pb-2212 high-Tc

superconductor

View the table of contents for this issue, or go to the journal homepage for more

1995 Supercond. Sci. Technol. 8 101

(http://iopscience.iop.org/0953-2048/8/2/006)

Home Search Collections Journals About Contact us My IOPscience

Supercond. Sd.Technol. 8 (1995) 101-107. Printed in the UK

The relationship between the porosity 1 and elastic moduli of the Bi-Pb-2212

high- Tc superconductor

R Ravinder Reddy, M Muralidhar, V Hari Babu and P Venugopal Reddy

Department of Physics, Osmania University, Hyderabad 500 007, India

Received 13 October 1994

Abstract. A series of Bi-Pb-2212 high-T, superconductors, having different porosities has been prepared by the solid-state reaction method. After the usual characterization of the samples by XRD, AC susceptibility, SEM, bulk-density and porosity studies, longitudinal (6) and shear (Vs) wave velocity measurements have also been undertaken by the pulse-transmission technique. Using the experimental values of V; and V,, the Young's moduli (E) of all the samples have been calculated and are later corrected to zero porosity by employing a number of theoretical equations. The corrected moduli are compared with the literature value. which itself has been arrived at by averaging the available single-crystalline stiffness constants. It is concluded from the results that among all the formulae, Mackenzie's equation and that proposed by the authors of the present investigation are in closest agreement with the literature value.

1. Introduction

Ceramic materials frequently contain porosity, either by design or due to processing limitations. The strength of a porous material, in turn, depends on pore size and distribution and pore morphology, with high porosity leading to low strength. In contrast to this, Almond er al [l] has the view that the porosity of a ceramic oxide material in general and an oxide superconductor in particular is beneficial because the connectivity among the pores provides an opportunity for the free movement of oxygen into the bulk of the sample. This in turn is very useful, especially in the case of samples such as high-T, superconductors (HTscs), for the diffusion of 0 by slow solid-state diffusion during the annealing process. As a result, 0 is distributed throughout the volume of the sample, resulting in a uniform 0 stoichiometry in such porous samples.

Apart from this, a knowledge of the effect of porosity on physical properties of structural ceramics is essential in order to use them as gas sensors, heat exchangers, filters etc; moreover, to improve the mechanical properties of HTSCs superconductors, it is essential to understand the influence of porosity on their mechanical behaviour. Therefore, the investigation of the effect of porosity on mechanical properties of solids in general, and superconductors in particular. is of broad scientific as well as technological interest.

There are several theoretical methods for the evaluation of the void-free elastic moduli of ceramics. They include

0953-2048/95/020101+07519.50 0 1995 IOP Publishing Ltd

theoretical, linear, non-linear, exponential and power-law types of modeVequations. Methods employing the linear, non-linear and power-law equations for the evaluation of zero-porosity elastic module of Y-123 superconductors have already been discussed in an earlier publication published from this laboratory [2]. Theoretical methods are those purely based on experimental data. The advantage of these methods if that one can evaluate the non-porous elastic-moduli values even for a single sample, provided its bulk and x- ray^ density values are available, whereas, in the case of the other three types of equation, the experimental data of at least seven or eight samples are required in order to evaluate these parameters with reasonable accuracy by the least-squares-fitting method. In the present investigation, details of methods involving various types of theoretical equation for the evaluation of void-free elastic moduli are discussed.

In general, it is known that the preparation of a pure Bi- 2212- or Bi-2223-based HTSC is not an easy task, because the two phases occur simultaneously. It has also been observed that the introduction of Pb is expected to stabilize the high-T, 2223 phase by suppressing the other, i.e. the low-T, 2212 [3,4]. Similarly, it has also been reported that Pb plays an important role not only in stabilizing the low-T, 2212 phase by suppressing the high-T, 2223 .- one,-but .. also enhance the values of both Tc as well as density without affecting the structural properties of Bi-2212. In fact, it has also been reported that the addition of Pb up to a concentration of 0.35 mol% slightly improves the T, and density values without affecting

101

R Ravinder Reddy et a/

the 2212 structure crystallographically [5]. As such, a systematic investigation of either a pure Bi-2212 or 2223 sample is interesting and essential from the viewpoint of technological and scientific importance. Although a number of researchers have investigated the Bi-2223Rb system, no systematic study of the Bid'bO.3Sr2CaCu208 ceramic superconducting material is available. As mentioned earlier, a study of the relationship between the elastic moduli and porosity in general is not only important but also essential in understanding the mechanical behaviour of the material. Results of such an investigation are presented here.

2. Experimental details

The samples with nominal composition Bi1.7Pbo.3Srz CaCuzOs were prepared by the high-temperature solid- state reaction method using highly pure (99%) BizO3, PbO, SrcOs, CaO and CuO. Stoichiometric mixtures were calcined at 800 "C for 24 h and 820 "C for 50 h with two intermediate grindings. The resulting powder was pressed into three pellets of 1 cm diameter each by applying a varying pressure. The pellets were then sintered at 840 "C for 100 h followed by a quenching to room temperature. As a result three samples with varying porosities, hereafter designated as samples A, B and C were obtained.

The phase formation was tested by XRD, using a CPS 120 INEL diffractometer having Cu Kor radiation. Superconducting transition temperatures were recorded in the range 80 5 T 5 150 K by an AC susceptometer using the standard mutual-inductance technique. The ultrasonic longitudinal shear wave velocity measurements were carried out by the pulse-transmission technique on samples having dimensions of 4 mm in thickness and 10 mm in diameter. X- and Y-cut PZT transducers, with a fundamental frequency of 1 MHz, were used for transmitting and receiving the longitudinal (K) and shear (V,) waves respectively. Both the surfaces of the samples were polished to form parallel faces before making the ultrasonic measurements. The transit time of ultrasound has been measured up to an accuracy of 0.1 ps using a 100 MHz digital storage oscilloscope (Tektronix model No 2221) and the overall accuracy of these measurements is &lo m s-'. The bulk densities of all three samples have also been determined by the well known immersion method, the details of which are given elsewhere [6]. The x-ray densities of the samples have been computed from the known values of lattice parameters using the Bell formula 171-

3. Results and discussion

3.1. X-ray diffraction

As the XRD patterns of all three samples are similar, the diffractogram of just one of the samples, viz. sample A, is shown in figure 1. Using the d-spacing values, the lattice constants of the sample were obtained using the standard least-squares method and assuming that the sample's structure is orthorhombic. The values obtained are a = 5.41 A, b = 5.42 A and c = 30.68 A.

102

It can be seen that from the XRD patterns of the samples that, even after the addition of the 0.3 mol% Pb to the Bi- 2212 samples, there are no additional peaks. From XRD one can also note that no trace of either Bi-2223 phase or an unreacted mass can be recognized. This observation clearly indicates that the addition of Pb is useful in suppressing the high-T, phase, which in turn results in stabilizing the Bi- 2212 phase structurally, and the result is in conformity with the findings of the literature. As such, it may be logically concluded that for all practical purposes the samples of the present investigation can be treated as Bi-2212 samples. As such, all the properties, including the elastic properties of these samples, could be compared with those of Bi-2212.

32. AC Susceptibility

With a view to understanding the nature and purity of the samples and also to determining the values of T,, AC susceptibility studies have been undertaken. The real ( x ' ) and imaginary (x") parts of the susceptibility are shown in figure 2. One can see from the figure that there is a clear step in both x' and x" versus T plots. Further, it can also be seen from the figure that the sharpness of the susceptibility peak, and hence the area under the loss peak, is found to decrease continuously with increasing porosity. Since the area under the loss peak is a direct measure of the defect nature of the sample, it may be concluded that the quality of the last sample is inferior when compared with that of the first one. Thus the susceptibi1ity)measurements indirectly confirm the porosity results. The peak heights and the widths of the peaks at half maximum are given in table 1.

3.3. Microstructure

The scanning electron micrograph studies of all the samples under investigation have been undertaken mainly to study the morphology of the grain structure on a fractured surface and are shown in figure 3. It can be seen from the micrograph of sample A that it has reasonably well grown grains with a small number of pores when compared with the other two micrographs. This observation is in conformity with the relatively high value of experimental bulk density (4650 kg N3). It can also be seen from the figure that the grain growth of the other two samples (B and C), with very low intergrain contact and a porosity of the order of 30%, is consistent with the measured bulk density values.

3.4. Room-temperature elastic moduli

The experimental values of longitudinal (V,) and shear (V,) wave velocities of all three samples along with the computed values of Young's ( E ) and rigidity (C) moduli and Poisson's ration ( U ) are presented in table 2. It can be seen from the table that both elastic moduli are found to decrease with increasing porosity values; this result is in conformity with well known theories. It can also be seen from the table that both the elastic moduli of the first sample are almost in agreement with those reported by Fanggao et al [SI. However, as the measurements are made on a highly

Porosity and elastic moduli of Bi-Pb-2212

0

2 6 (Degree) - Figure 1. The XRD pattern of sample A.

Table 1. AC susceptibility results for Bil.,Pbo,3Sr2CaCu20y samples. Width of

T,(onset) Peak temperature Peak peak at half from x‘ against T from x ‘ against T height maximum

Sample (K) (K) (cm) (K)

A 81 B 80

74 74

7.6 2 5.7 2.2

C 81 75 4.9 3

Table 2. Experimental elastic data.

Sample (m s-‘) (m s-’) (IO3 kg m3) (lo3 kg m3) (GPa) (GPa) U (‘3’3)

A 2990 1850 4.65 6.41 15.91 41.57 0.18 27.50 B 2800 1780 4.50 6.41 13.78 35.28 0.17 29.85 C 2750 1725 4.46 6.41 13.27 33.72 0.17 30.55

VI V S P Pm G E C

porous sample, it is thought that the comparison may not be relevant unless the moduli are corrected to zero porosity.

3.5. Elastic data corrected to the void-free state

As the porosity of the samples of present investigation is about 30%. the measured elastic moduli do not have much significance unless they are corrected to a void-free state. As such, they are corrected to zero porosity using different theoretical models and the details are given below. As there are eight different theoretical equations reported by different authors, they are designated as TEI, TEz etc for the sake of convenience.

3.5.1. Mackenzie’s formula (TE~). Mackenzie [9] calculated the effective bulk and shear moduli of a solid by a self-consistent method. The method is based on the

assumption that the real material contains isolated spherical pores distributed randomly throughout the volume of the material. It has also been assumed that the sintered porous material is homogeneous and isotropic in nature and that the volume of all the holes present in a sample is small when compared to its total volume. Fnrther, each hole is supposed to be surrounded by a spherical shell of a real material and the reaction of the rest of the material is estimated by replacing it with an equivalent homogeneous material. The method thus depends on the fact that any microscopic volume containing a large number of holes can be replaced by equivalent homogeneous material without essentially changing the elastic behaviour of the solid. The equations proposed by Mackenzie are

Bo - B/B = P(3Bo +4Go)/4Go(l - P) Go - G/Go = 5P(3Bo +4Go)/(gBo + 8Go)

(1)

(2)

103

R Ravinder Reddy et a/

Figure 2. The real (x ' ) and imaginary (x") parts of the AC susceptibility against temperature for samples A, 6 and C.

Figure 3. Scanning electron micrograph studies of samples (a) A, (bj B and (cj C.

104

Porosity and elastic moduli ot Bi-Pb-2212

where Bo and Go are the bulk and rigidity moduli of a non-porous matrix respectively. while B and G are, the experimental values of bulk and rigidity modulus respectively. P is the porosity of the material. The corrected values obtained using these two equations are given in table 3.

3.5.2. Hasselman and Fulrath’s formula ( T E ~ ) Hasselman and Fulrath [lo], while studying the influence of porosity on elastic moduli of certain ceramic materials such as glass, formulated the following two formulae for the evaluation of void-free elastic moduli. These authors assumed that mainly spherical types of pore are likely to occur in the material at the time of preparation of the samples by a ceramic method.

E(P) = Eo(1 - L Y E P ) (3)

G ( P ) = Go(l -01cP) (4)

where Eo and Go are the Young’s and rigidity moduli values of a non-porous matrix respectively and P is the porosity of the sample. It has been assumed that the value of both OIE and (YG ranges around two. Using this value, the zero- porosity Young’s moduli of all the samples of the present investigation have been evaluated and are given in table 3.

3.5.3. Uemura and Takayangi’s equation (TE~) There have been many theoretical as well a.s experimental investigations in the literature on the elastic properties of a solid containing soft inclusions or voids. According to Uemura and Takayanagi [ l I], since voids are theoretically regarded as soft inclusions, the formula meant for soft inclusions can be used even to account for the presence of a large number of spherical pores. The expressions given by these authors are

G ( P ) = [(7-5Uo)(l-P)]/[15(1 - Q ) P f ( 7 - 5Uo)(1 - P)] (5)

(6) The corrected values of both the moduli using these

K ( P ) = Ko[4Go(l - P)]/(4Co f 3KoP).

equations are also included in table 3.

3.5.4. Ledbetter and Datta’s formula (TE~) In this method, the authors [I21 consider the ceramic material as a composite material containing spherical voids. If we assume that the void density is pp, the x-ray mass density is pmUs and the volume pore fraction if C, the bulk density of the ceramic material ( p ) is given by the expression

P = Pm(1 - C ) + pp. (7)

Assuming that the void density of an ideal material is

(8 )

zero, the macroscopic density expression reduces to

P = Pm(1 - C).

The authors assumed that the randomly distributed spherical voids possess zero resistance to both dilation and shear. The equations given by these authors are

where

A I = (8/3)(1 - C )

Az = (3 - 2C)B - (8/3 f4C)G

A3 = -3(1+ C)BG

(10) B, = 4GmB/4(1 - C)Gm - 3CB

E, = 9B,G, (3% + Gm)

and B, G and E represent the experimental values of bulk, rigidity and Young’s modulus, respectively, while G,, B, and E, are the respective elastic moduli of a non-porous material.

3.5.5. Alford’s equation ( T E ~ ) , Alford et al [I31 also suggested another equation for the evaluation of non-porous elastic moduli, and it is given by

E(P) = Eo(1 - PP.

According to these authors, the non-porous elastic moduli of Y-123 superconducting samples can be obtained by substituting m = 3 in a general power-law equation of the type mentioned above. As Bi-based superconductors are also ceramic in nature with a high value of Tc, it is thought that the same power law may be used to evaluate the zero- porosity elastic moduli of even the Bi-2212 samples of the present investigation. However, when this equation is applied, it has been surprisingly observed that the corrected values are very high.

3.5.6. The equation of the present investigation (m) Another available power-law equation for the porosity dependence of the elastic moduli of superconductors has been arrived at by the authors of the present investigation, based on the theory of elasticity of continuum:

,

E(P) = Eo(1 - P)”.

According to Wagh et al [14], who have used the power law to study the porosity dependence of elastic moduli of number of a ceramic materials, the value of the exponent in the case of ceramics fabricated without any sintering aids is approximately two. On the other hand, Phani and Mukerjee [15] showed that the value of m is 1.85 in the case of porous epoxy resin. Wagh et a1 also observed that the value of the exponent depends on the grain size and pore size of the material under investigation and that the value could v“y from two to 4.6. From a close observation of all the above results, one may conclude that the value of the exponent (m) could vary from a low value of two to a very high value of approximately five. Keeping this and the experimental data of the present investigation in view, a value of 2.25 for the,exponent has been arrived at for the samples of the present investigation.

105

R Ravinder Reddy et a/

Table 3. Void-free elastic data. A 6 C

TEI 80.67 37.51 77.26 37.82 77.11 TEp 106.39 35.35 98.41 34.10 96.43 T& 65.97 34.37 57.13 31.23 54.12 TEq 66.50 28.02 61.03 26.61 58.39

... TEg 109.08 - 101.98 - 100.44 TEs 85.70 - 78.21 - 76.45 TE7 77.03 - 63.69 - 61 55 TEg 118.15 - 109.53 - 107.50

38.26 34.02

25.03 28.86

- - - - -

3.5.7. Ishani and Cohen’s equation (T%) Ishani and Cohen [I61 also proposed a representative model by assuming the shape of the pore as a cube and derived a relation

E ( P ) = Eo(l - P2’3).

Using the above power-law formula zero-porosity values have been estimated and are given in table 3.

However, it was observed that this type of approach, with equations derived by taking regular shapes for the pores, has limited application [17]. The difficulties encountered in these approaches reveal a need for a new model that will explain the porosity dependence of mechanical and fracture properties of ceramics, and predict microstructure-property relationships by retaining the random structure of ceramics, i.e. the regular shapes and sizes of the pores and their random distribution.

3.5.8. Ramakrishnan and Arnnachalam’s equation (TEg) Ramalaishnan and Arunachalam [18] have derived an equation for the effective elastic moduli and Poisson’s ratio of porous solids using the principle of statistical continuum mechanics, and it is given by a general equation

where M ( P ) is the effective modulus, MO is the zero- porosity modulus and P is the porosity of the sample. The constant term b,,, depends on the modulus concerned and is a function of only the dense-material Poisson’s ratio. The calculated value of both the moduli are given in table 3.

3.6. A comparison of the Young’s modulus with the literature

The values corrected to zero porosity obtained using different formulae are relevant and meaningful only when they are compared with the litenturdstandard values. The comparison also helps us to judge which one of the theoretical formulae is suitable for predicting the correct void-free elastic moduli of HTSCs in general and the Bi- 2212 sample of the present investigation in particular. As mentioned earlier, since on the introduction of Pb no major structural changes of the Bi-2212 phase are found, it is thought that elastic properties of the samples of the present

106

investigation could be compared with those of Bi-2212 samples available in the literature. Unfortunately, although a number of researchers have reported the Young’s moduli values of polycrystalline Bi-2212 samples, the values are found to vary from 30 to 75 GPa [19-231. As such, it is very difficult to arrive as a definite conclusion about the validity of the values predicted by a particular formula. In view of this, it is thought that it would be more appropriate to compare the values of the present investigation with those of the single-crystalline ones. Unfortunately, there are fewer elastic stiffness data on single crystals, because it is very difficult to grow a sufficiently large and defect- free single-crystal of Bi-2212 for accurate ultrasonic wave velocity measurements. The short pulse transit times associated with small dimensions hinder the precision of measurement of elastic stiffness tensor components.

However, Fanggao et a1 [24] and Saunders er ol 1251 estimated the ultrasonic longitudinal velocity of a dense and highly textured polycrystalline Bi-2212 sample: this is in reasonably good agreement with the single-crystalline data available in the literature. They also observed that the ultrasonic velocity, in general, is higher in the a-b plane than along the c axis. According to these authors, as the longitudinal elastic stiffness within the layers is substantially larger than that normal to the layers, the velocity of a longitudinal elastic wave transmitted within the layers could be much greater than that of a wave transmitted along the direction of weak binding, namely the C axis. In fact, Saunders et a1 obtained a Young’s modulus of 101 GPa along the a-b plane and a value of 44 GPa along the C axis.

Now in order to calculate the value of the average Young’s modulus, a procedure based on three following assumptions has been adopted by Fanggao et al [24].

(i) The elastic properties are identical for all the crystalline directions subtending a particular angle 0 with the C axis.

(ii) In a perfectly random polycrystalline sample, the probability of a crystalline having a plane normally aligned within dB of the direction of propagation of the incident ultrasonic wave is given by sin 0 dB.

(iii) An ellipsoidal velocity surface with major and minor axes of V, and V, provides an estimation of sound velocity along an arbitrary direction in a crystallite. The velocity,of sound propagation for such an arbitrary direction

Porosity and elastic moduli of Bi-Pb-2212

Table 4. The average Young's modulus and percentage mean deviation.

Average value Standard value Mean deviation Equation' (GPa) (GPa) (%)

TEt 78.34 4.12 E 2 100.41 22.8 T h 59.07 27.70 TE4 61.97 81.71 TES 103.83 ~. TE6 80.12 TE, 65.79

24.15 27.07

1.9 19.48

TEE 111.72 36.70

v# may be resolved into components Ve cos e and V, sin e parallel to the c axis and U axis, which are related by the equation of the ellipsoidal velocity surface

v,2 cos' e v,2 sin' 6 + = 1 v,2 v," from which v# = va/[l + [(Va/vc)* - 1]cos2 ep2 .

direction in a polycrystalline sample is then given by The average sound velocity (V,) along the longitudinal

V, = UV, [ In [(l + J1 + u 2 ) ] / u )

where u2 = V,'/(V,' - V:),

Bi-2212 sample, having a bulk density at 6208 kg Fanggao ef ul [24] found the following values for a

V, = 4370 m s-'

and

V, = 2670m s-' .

Using the values of V,, V, and p , the value of Young's modulus has been obtained and is found to be 81.71 GPa. The average Young's modulus value (EO) obtained for the samples of the present investigation using different types of theoretical equation are given in table 3.

The average Young's modulus obtained by each method along with the values of percentage deviation from the standard value are given in table 4.' From this, it can be concluded that the power-law equation of the present investigation followed by Mackenzie's formula are in best agreement with the standard elastic data, and hence they are more appropriate than the remaining ones, as regards the prediction of the void-free elastic moduli of the Bi-2212 samples of the present investigation.

Acknowledgmen t s

The authors are grateful to the Department of Science and Technology, Government of India, for providing funds to carry out the research project. One of the authors (RRR)

thanks the Council of Scientific and Indushial Research (CSJR) of the government of India for awarding a senior research fellowship.

R e f e r e n c e s

Ill Almond D P, Wang Q, Freestone F, Lambson E F, Champman B and Saunders G A 1989 J. Phys.: Condens. Mutter 1 6853

[21 Ravinder Reddy R, Prakash Om and Venugopal Reddy P 1994 AppL Supercond. at press

[3] Nobumasa H, Simizu K, Kitano Y and Kawai T 1988 Jupun. J. Appl. Phys. 27 1609

[41 Endo U, Koyama S and Kawai T 1988 Japan J. Appl. Phys. 27 L1476

151 Muralidhar M, Narasimha Reddy K and Hari Babu V 1991 Phys. Status Solidi a 126 I

[61 Shekar S and Venugopal Reddy P 1993 Mod. Phys. Len. B 7 935

[7] Bell A M T 1990 Supercond Sci. TechnoL 3 55 [SI Fanggao C, C a n k u " M, Saunders G A, AI-Kheffaji A,

Almond D P and Ford P J 1991 Supercond Sci. Technol. 4 13

[91 Mackenzie J K 1950 Proc. Phys. Soc. B 63 2 [IO] Hasselman D P H and Fulrath R M 1964 J. A m Cerum.

[I 11 Uemura S and Takayanagi M 1966 J. AppL Polym. Sci. 10

1121 Ledbetter H and Datta S 1986 J. ACOUI. Soc. Am, 79 239 [I31 Alford N McN, Birchall J D, Clegg W J, Harmer M A,

Kendall K and Jones D H 1988 J. Muter. Sci 23 761 [141 Wagh A S, Singh I P and Poeppel R B 1993 J. Mater. Sci.

28 3589 [151 Phani K K and Mukerjee R N 1987 J. Muter. Sci 22 3453 [161 Ishani 0 and Cohen L J 1967 Int. J. Mech. Sci. 9 539 [I71 Bert C W 1985 J. Muter. Sci. 20 2220 [I81 Ramakrishnan N and Arunachalam V S 1990 J. Muter. Sci.

[I91 Dominec I 1993 Supercond Sci. Technol. 6 153 [201 He Y, Xiang J, Jin S, We A and Zhang J 1990 Physicu B

1211 Ledbetter H. Kim S and Toeano K 1991 Phvsica C

Soc. 47 52

113

25 3930

165 & 166 1283 - . _ 185-189 935

1221 Dominec J. Vasek P, Svoboda P. Plechacek V and . . Laermans C 1992 Mod. Phys.'Lett. B 6 1049

[231 Fanggao C, Cankurtaran M. Saunders G A, Almond D P, Ford P I and AI-Kheffaji A 1990 Supercond. Sci Technol. 3 3

D P, Champman B, Cankurtaran M, Poeppel R B and Goretta K C 1993 Suoercond. Sci. TechnoL 6 484

[?AI Fanggao C. Ford P I, Saunders G A, Li Jiaqiang, Almond

1251 Saunders G A, Fanggao C, Jiaqiang Li, Wang 0, Cankurtaran M, Lambson EF. Ford P J aid kmond D P 1994 Phys. Rev. B 49 9862

1 07