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Comment on the role of intragranular dislocations in superplastic yttria-stabilized zirconia
Scripta Materialia 48 (2003) 599–604 www.actamat-journals.com
Comment on the role of intragranular dislocations in superplastic yttria-stabilized zirconia N. Balasubramanian, T.G. Langdon
*
Departments of Aerospace and Mechanical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089-1453, USA Received 20 August 2002; accepted 24 September 2002
Abstract It is shown by analysis, including through the incorporation of dislocation pile-ups, that the intragranular movement of dislocations plays little or no significant role in the deformation of high purity yttria-stabilized zirconia when testing at 1673 K at stresses below 100 MPa. It is proposed instead that deformation occurs through interface-controlled Coble diffusion creep. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Ceramics; Creep; High-temperature mechanical properties; Superplasticity; Yttria-stabilized zirconia
1. Introduction Although the occurrence of superplastic flow was first reported in ceramic materials several decades later than in metals [1,2], nevertheless the development of superplastic ceramics has made rapid advances in recent years. For example, a recent report describes the occurrence of high strain rate superplasticity in a composite of zirconia, alumina and spinel with a tensile elongation of 390% at 1923 K when using a strain rate of 1 s1 and a remarkably high elongation of 1050% in the same composite at the slightly lower strain rate of 0.4 s1 [3].
Despite these experimental advances, there has been little progress in identifying the precise flow mechanisms occurring in superplastic ceramics. Furthermore, this uncertainty relates even to materials subjected to very extensive investigations such as yttria-stabilized tetragonal polycrystalline zirconia (henceforth designated Y-TZP, where the numeral preceding Y denotes the mole percentage of yttria). Accordingly, the present report addresses this problem with an emphasis on examining recent reports discussing the role of lattice dislocations during creep of superplastic 3Y-TZP [4–8].
2. Flow behavior in superplastic Y-TZP *
Corresponding author. Tel.: +1-213-740-0491; fax: +1-213740-8071. E-mail address: [email protected] (T.G. Langdon).
As in metals, the flow mechanism in superplastic Y-TZP is generally characterized by a relationship containing an Arrehenius term and two
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exponents characterizing the dependence of strain rate on stress and grain size. The steady-state strain rate, e_, at an absolute temperature, T , is expressed as a function of the applied stress, r, and the grain size, d, as: p ADGb b r n ð1Þ e_ ¼ kT d G where D is the appropriate diffusion coefficient ð¼ D0 expðQ=RT Þ, where D0 is the frequency factor, Q is the activation energy and R is the gas constantÞ, G is the shear modulus, b is the Burgers vector, k is BoltzmannÕs constant, A is a dimensionless constant and n and p are the exponents for the stress and the inverse grain size, respectively. Although Eq. (1) appears relatively simple, it is often difficult to determine the magnitudes of the various experimental parameters for superplastic Y-TZP. For example, it may be difficult to establish a true steady-state condition, problems may arise from the occurrence of necking in tension or inhomogeneous deformation due to barreling and/ or end constraints in compression, internal cavities or cracks may develop and the flow rate may be markedly influenced by the occurrence of concurrent grain growth. As a consequence of these problems, it becomes difficult to use the apparent experimental values of Q, n and p for an unam-
biguous determination of the rate-controlling flow mechanisms. Four distinct flow mechanisms have been proposed to explain the behavior observed in superplastic Y-TZP at 1673 K at stresses below 3 102 MPa [8–11] and these various mechanisms, and their implications, are summarized in Table 1. In practice, it is known that grain boundary sliding cannot occur in a polycrystalline matrix without accommodation through the movement of intragranular dislocations [12] and evidence for this concurrent slip is well-documented in superplastic metallic alloys [13,14]. Inspection of the four proposed mechanisms in Table 1 shows that only the second model, based on interface-controlled Coble diffusion creep, leads to polycrystalline flow without any concomitant intragranular dislocation activity. The fourth model in Table 1 is based specifically on recent experimental observations by Morita et al. [4–6,8] and the relevant creep data forming the background to this model are shown as the experimental points in Fig. 1 for samples having a spatial grain size of 0.35 lm tested at a temperature of 1673 K. Morita and Hiraga [8] interpreted these data in terms of an intragranular dislocation recovery mechanism at high stresses (16–80 MPa) with n 2:7, a threshold stress at intermediate stresses (9–16 MPa) where n 5:0
Table 1 Proposed flow mechanisms in superplastic 3Y-TZP at 1673 K up to 3 102 MPa Proposed mechanism
Implication
Intragranular dislocation activity?
Reference
Grain boundary sliding at high stresses: interface-reaction controlled process at low stresses
n increases from 2 to 3 and p decreases from 3 to 1 with decreasing stress; constant Q over entire range of stress n increases from 1 to 3 and p decreases from 3 to 1 with decreasing stress; constant Q over entire range of stress n increases with decreasing stress so that n ! 1; Q increases in the threshold stress region
Yes, in order to accommodate grain boundary sliding
Owen and Chokshi [9]
No
Berbon and Langdon [10]
Yes, in order to accommodate grain boundary sliding
Jimenez-Melendo and DomınguezRodrıguez [11] Morita and Hiraga [8]
Interface-controlled Coble diffusion creep at all stresses
Grain boundary sliding at high stresses; a threshold stress at low stresses Dislocation recovery mechanism at high stresses with a threshold stress; Nabarro-Herring diffusion creep at low stresses
Increasing n and Q with decreasing stress at high stresses; a transition to n ¼ 1 at low stresses
Yes but only at higher stresses (>9 MPa) above the region of diffusion creep
N. Balasubramanian, T.G. Langdon / Scripta Materialia 48 (2003) 599–604
601
observations appear to generally support the proposed mechanism given in the last row of Table 1, Mu~ noz et al. [7] have questioned whether an intragranular dislocation flow mechanism is viable in 3Y-TZP at these low stresses at a temperature of 1673 K. Accordingly, this question is now examined.
3. The potential for intragranular dislocation activity in 3Y-TZP at low stresses The shear stress, s, required to nucleate a dislocation is given by the relationship [17] s¼
GbflnðLp =bÞ 1:67g 2pLp ð1 mÞ
ð2Þ
where Lp is the distance between pinning points which may be approximated by d=3 and m is PoissonÕs ratio. For 3Y-TZP, the temperature dependence of the shear modulus was determined recently as [18] Fig. 1. Strain rate versus stress for 3Y-TZP at 1673 K with a grain size of 0.35 lm: the experimental points are from Morita and Hiraga [8] and the solid line shows the prediction using Eq. (6) for interface-controlled Coble diffusion creep [10].
and a transition to Nabarro–Herring diffusion creep at low stresses ( 6 9 MPa) where n 1:3. The grain size exponent, p, was reported as 1.8 at low stresses and 2.2–3.0 at high stresses and the same activation energy of 580 kJ mol1 was reported for all three regions. To substantiate this interpretation, Morita and Hiraga [6] used transmission electron microscopy to examine samples of 3Y-TZP deformed to selected strains at 1673 K and then cooled rapidly under load at an initial rate of 5 K s1 . As in earlier reports on 3Y-TZP [15] and 2.5Y-TZP [16], there was no evidence in the deformed or undeformed samples for the presence of an amorphous phase either along the grain boundaries or at triple-grain junctions. However, there was evidence for intragranular dislocation activity after deformation at 15 MPa and there was an increasing density of intragranular dislocations as the stress was increased to 30 and 50 MPa. Although these
G ¼ 84 103 13:3T
ð3Þ
giving G ¼ 62 GPa at 1673 K which is close to the value of G ¼ 65 GPa reported earlier [19]. The Burgers vector for an ða=2Þh1 1 0i-type lattice dislocation in 3Y-TZP is 0.36 nm [20,21] and, using the Tresca criterion to yield the minimum possible stress, the tensile stress, r, is twice the shear stress, s. Thus, the value calculated for r from Eq. (2) is at least 130 MPa which suggests a factor at least 9 higher than the lowest stress of 15 MPa where dislocation activity was reported by Morita and Hiraga [6]. On the other hand, the present calculation is consistent with experimental results showing the stress needed to activate dislocation movement is 300 MPa in 4Y-TZP single crystals at 1673 K when using a strain rate of 1:5 105 s1 [7] and the yield stress in 6.8% yttria-zirconia pseudo-cubic crystals is 150 MPa at 1673 K with a strain rate of 6 105 s1 [22]. The experimental stress of 15 MPa reported for intragranular dislocation activity by Morita and Hiraga [6] is also exceptionally low based on two other considerations. First, by extrapolating yield stress data for yttria-stabilized cubic ZrO2 single crystals, it was estimated earlier that the yield
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stress for 3Y-TZP at 1673 K is 90 MPa when using a strain rate of 105 s1 [10]. Second, noting the crystal structures suggest the Peierls potential for tetragonal ZrO2 is only slightly higher than for cubic ZrO2 , a procedure was developed earlier for estimating the minimum value of the Peierls stress [10] and this procedure, when used with a shear modulus for 3Y-TZP of 62 GPa [18] and data from cubic ZrO2 , leads to a Peierls stress at 1673 K of not less than 180 MPa. Thus, these calculated values and the experimental values [7,22] are reasonably consistent but they are at least a factor of 10 higher than the stresses where dislocation activity was reported by Morita and Hiraga [6]. Recognizing a potential difficulty because of the low stresses, Morita and Hiraga [6] proposed that stress concentrations may be present at multiplegrain junctions to increase the applied stress by factors of 14–25 and they provided experimental evidence for these stress concentrations by documenting the presence of dislocation pile-ups. To check the possibility of a significant stress concentration, it is first noted that the shear stress at the head of a pile-up is given by [23] spile-up ¼
2Ls2 Gb
ð4Þ
where L is the pile-up length and b is the appropriate Burgers vector. As in the earlier analysis [10], it may be assumed initially that the dislocations pile-up along the grain boundaries so that the value of b is equal to bgb for grain boundary dislocations. Taking bgb ¼ 0:18 nm representing one-half of the value for lattice dislocations, and using the reported value of 0.2 lm for the length of the pileup at 15 MPa [6], spile-up is estimated as 4.5 MPa so that there is no pile-up and no stress concentration. Morita and Hiraga [6] suggested the Burgers vector of grain boundary dislocations may be ‘‘much smaller’’ than b=2 but even if bgb is taken as one-tenth of the lattice value there remains no significant stress concentration. Similar discrepancies are apparent also when the calculation is repeated for the other experimental stresses examined by Morita and Hiraga [6] or when b is taken as the lattice Burgers vector. Thus, these calculations provide a very clear demonstration of the difficulties encountered in attempting to use
conventional pile-up theory for materials with submicrometer grain sizes. There is also a problem with the numbers of dislocations, N , reported in each intragranular pile-up [6]. The conventional theory for N gives a relationship of the form [24] N¼
Lspð1 mÞ Gb
ð5Þ
Using the experimental values reported for L for the intragranular pile-ups [6] and taking m ¼ 0:33, N is estimated as 0.13, 0.24 and 0.28 for stresses of 15, 30 and 50 MPa, respectively. This calculation demonstrates that dislocation pile-ups are not formed under these conditions.
4. Implications for the flow mechanisms in 3Y-TZP The calculations in the preceding section, when considered together with the results from several experimental investigations, provide strong evidence that intragranular dislocation activity is unlikely in superplastic 3Y-TZP at a temperature of 1673 K when using stresses below 100 MPa. By contrast, there are well-documented reports of high values of n, indicative of an intragranular dislocation mechanism, in experiments conducted at even higher stress levels: for example, n 10 at 800 MPa for 2Y-TZP at 1300 K [25] and n 7 with p ¼ 0 above 300 MPa for 3Y-TZP at 1673 K [26]. It is important to note also that an intragranular dislocation recovery mechanism such as climb, as proposed by Morita and Hiraga [8] for stresses above 16 MPa, will take place within the grains so that there is no dependence on grain size and p ¼ 0. This latter requirement is clearly inconsistent with the numerous investigations reporting p 2–3 at these stress levels [9,11,26] including in the experiments of Morita and Hiraga [8] where p 2:2–3:0 in the high stress region. In addition, Morita and Hiraga [8] proposed a threshold stress at intermediate stresses in 3Y-TZP but this would lead, as indicated in Table 1, to an increase in both n and Q at intermediate stresses [27,28] whereas their experiments, and other results [9], show a constant value of Q over the entire stress range.
N. Balasubramanian, T.G. Langdon / Scripta Materialia 48 (2003) 599–604
An alternative interpretation, as given in the second row of Table 1, is to consider flow occurring through an interface-controlled model for Coble diffusion creep [29] where this model has been shown to work well for some other results on 3Y-TZP [10]. In this model, the strain rate is expressed as ! 3 2 Ngb 33:4DZrðgbÞ Gb dZr b r e_ ¼ 2 d G kT b Ngb þ 0:5 ð6Þ where DZrðgbÞ is the grain boundary diffusion coefficient, dZr is the effective grain boundary width for the Zr4þ ions and Ngb is the number of grain boundary dislocations in a single grain boundary wall which is given by [10] Ngb ¼
rd 2Gbgb
ð7Þ
Experimental values for dZr DZrðgbÞ are not directly available but the activation energy for lattice diffusion of the Zr4þ cations in 2.8Y-TZP was reported recently as 6.3 eV and 6.5 eV from tracer diffusion and internal friction measurements, respectively [30]. Data on tetragonal CeO2 – ZrO2 –HfO2 [31] and cubic yttria-stabilized ZrO2 [32] suggest an activation energy for boundary diffusion equal to 0.8 of the value for lattice diffusion. Therefore, the appropriate activation energy for cation grain boundary diffusion in 3YTZP is estimated as 5.1 eV corresponding to 500 kJ mol1 . This value is in excellent agreement with the value of 506 kJ mol1 used in earlier calculations [10] and taken from the grain boundary diffusion coefficient for Zr4þ ions in tetragonal CeO2 –ZrO2 –HfO2 solid solutions [31]. Accordingly, using this value for dZr DZrðgbÞ [31], the solid line in Fig. 1 shows the prediction of Eq. (6) where the upper broken line with a slope of n ¼ 1 is for unmodified Coble diffusion creep and the higher slope of the solid line at lower stresses is due to the interface control. It is apparent from Fig. 1 that Eq. (6) gives good agreement with the experimental datum points, to within one order of magnitude, over the entire range of stresses used in these experiments.
603
5. Summary and conclusions Calculations suggest there is little or no movement of intragranular dislocations in fine-grained 3Y-TZP at stresses below 100 MPa when testing at 1673 K. A flow model based on interface-controlled Coble diffusion creep gives good agreement with published experimental data.
Acknowledgement This work was supported by the United States Department of Energy under grant no. DE-FG0392ER45472.
References [1] Wakai F, Sakaguchi S, Matsuno Y. Adv Ceram Mater 1986;1:259. [2] Wakai F, Kato H. Adv Ceram Mater 1988;3:71. [3] Kim B-N, Hiraga K, Morita K, Sakka Y. Nature 2001;413:288. [4] Morita K, Hiraga K, Sakka Y. Mater Res Soc Symp Proc 2000;601:93. [5] Morita K, Hiraga K. Scripta Mater 2000;42:183. [6] Morita K, Hiraga K. Phil Mag Lett 2001;81:311. [7] Mu~ noz A, Wakai F, Domınguez-Rodrıguez A. Scripta Mater 2001;44:2551. [8] Morita K, Hiraga K. Acta Mater 2002;50:1075. [9] Owen DM, Chokshi AH. Acta Mater 1998;46:667. [10] Berbon MZ, Langdon TG. Acta Mater 1999;47:2485. [11] Jimenez-Melendo M, Domınguez-Rodrıguez A. Acta Mater 2000;48:3201. [12] Langdon TG. Acta Metall Mater 1994;42:2437. [13] Falk LKL, Howell PR, Dunlop GL, Langdon TG. Acta Metall 1986;34:1203. [14] Valiev RZ, Langdon TG. Acta Metall Mater 1993;41:949. [15] Primdahl S, Th€ olen A, Langdon TG. Acta Metall Mater 1995;43:1211. [16] Ikuhara Y, Thavorniti P, Sakuma T. Acta Mater 1997;45:5285. [17] Hirth JP, Lothe J. Theory of dislocations. New York: McGraw-Hill; 1968. [18] Hayakawa M, Miyauchi H, Ikegami A, Nishida M. Mater Trans JIM 1998;39:268. [19] Fukuhara M, Yamauchi I. J Mater Sci 1993;28:4681. [20] G omez-Garcıa D, Jimenez-Melendo M, DomınguezRodrıguez A, Heuer AH. J Am Ceram Soc 1990;73:2452. [21] Parthasarathy TA, Hay RS. Acta Metall Mater 1996;44: 4663.
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[22] Baither D, Baufeld B, Messerschmidt U. J Am Ceram Soc 1995;78:1375. [23] Friedel J. Dislocations. Oxford: Pergamon Press; 1964. [24] Eshelby JD, Frank FC, Nabarro FRN. Phil Mag 1951;42:351. [25] Hwang C-MJ, Chen I-W. J Am Ceram Soc 1990;73:1626. [26] Charit I, Chokshi AH. Acta Mater 2001;49:2239. [27] Li Y, Langdon TG. Acta Mater 1998;46:3937. [28] Li Y, Langdon TG. Acta Mater 1999;47:3403.
[29] Arzt E, Ashby MF, Verrall RA. Acta Metall 1983;31:1977. [30] Kilo M, Weller M, Borchardt G, Damson B, Weber S, Scherrer S. Defect and Diffusion Forum 2001;194– 199:1039. [31] Sakka Y, Oishi Y, Ando K, Morita S. J Am Ceram Soc 1991;74:2610. [32] Oishi Y, Ando K, Sakka Y. In: Yan MF, Heuer AH, editors. Advances in ceramics (Additivies and interfaces in electronic ceramics), vol. 7. Columbus, OH: The American Ceramic Society; 1983. p. 208.
Comment on the role of intragranular dislocations in superplastic yttria-stabilized zirconia N. Balasubramanian, T.G. Langdon
*
Departments of Aerospace and Mechanical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089-1453, USA Received 20 August 2002; accepted 24 September 2002
Abstract It is shown by analysis, including through the incorporation of dislocation pile-ups, that the intragranular movement of dislocations plays little or no significant role in the deformation of high purity yttria-stabilized zirconia when testing at 1673 K at stresses below 100 MPa. It is proposed instead that deformation occurs through interface-controlled Coble diffusion creep. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Ceramics; Creep; High-temperature mechanical properties; Superplasticity; Yttria-stabilized zirconia
1. Introduction Although the occurrence of superplastic flow was first reported in ceramic materials several decades later than in metals [1,2], nevertheless the development of superplastic ceramics has made rapid advances in recent years. For example, a recent report describes the occurrence of high strain rate superplasticity in a composite of zirconia, alumina and spinel with a tensile elongation of 390% at 1923 K when using a strain rate of 1 s1 and a remarkably high elongation of 1050% in the same composite at the slightly lower strain rate of 0.4 s1 [3].
Despite these experimental advances, there has been little progress in identifying the precise flow mechanisms occurring in superplastic ceramics. Furthermore, this uncertainty relates even to materials subjected to very extensive investigations such as yttria-stabilized tetragonal polycrystalline zirconia (henceforth designated Y-TZP, where the numeral preceding Y denotes the mole percentage of yttria). Accordingly, the present report addresses this problem with an emphasis on examining recent reports discussing the role of lattice dislocations during creep of superplastic 3Y-TZP [4–8].
2. Flow behavior in superplastic Y-TZP *
Corresponding author. Tel.: +1-213-740-0491; fax: +1-213740-8071. E-mail address: [email protected] (T.G. Langdon).
As in metals, the flow mechanism in superplastic Y-TZP is generally characterized by a relationship containing an Arrehenius term and two
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exponents characterizing the dependence of strain rate on stress and grain size. The steady-state strain rate, e_, at an absolute temperature, T , is expressed as a function of the applied stress, r, and the grain size, d, as: p ADGb b r n ð1Þ e_ ¼ kT d G where D is the appropriate diffusion coefficient ð¼ D0 expðQ=RT Þ, where D0 is the frequency factor, Q is the activation energy and R is the gas constantÞ, G is the shear modulus, b is the Burgers vector, k is BoltzmannÕs constant, A is a dimensionless constant and n and p are the exponents for the stress and the inverse grain size, respectively. Although Eq. (1) appears relatively simple, it is often difficult to determine the magnitudes of the various experimental parameters for superplastic Y-TZP. For example, it may be difficult to establish a true steady-state condition, problems may arise from the occurrence of necking in tension or inhomogeneous deformation due to barreling and/ or end constraints in compression, internal cavities or cracks may develop and the flow rate may be markedly influenced by the occurrence of concurrent grain growth. As a consequence of these problems, it becomes difficult to use the apparent experimental values of Q, n and p for an unam-
biguous determination of the rate-controlling flow mechanisms. Four distinct flow mechanisms have been proposed to explain the behavior observed in superplastic Y-TZP at 1673 K at stresses below 3 102 MPa [8–11] and these various mechanisms, and their implications, are summarized in Table 1. In practice, it is known that grain boundary sliding cannot occur in a polycrystalline matrix without accommodation through the movement of intragranular dislocations [12] and evidence for this concurrent slip is well-documented in superplastic metallic alloys [13,14]. Inspection of the four proposed mechanisms in Table 1 shows that only the second model, based on interface-controlled Coble diffusion creep, leads to polycrystalline flow without any concomitant intragranular dislocation activity. The fourth model in Table 1 is based specifically on recent experimental observations by Morita et al. [4–6,8] and the relevant creep data forming the background to this model are shown as the experimental points in Fig. 1 for samples having a spatial grain size of 0.35 lm tested at a temperature of 1673 K. Morita and Hiraga [8] interpreted these data in terms of an intragranular dislocation recovery mechanism at high stresses (16–80 MPa) with n 2:7, a threshold stress at intermediate stresses (9–16 MPa) where n 5:0
Table 1 Proposed flow mechanisms in superplastic 3Y-TZP at 1673 K up to 3 102 MPa Proposed mechanism
Implication
Intragranular dislocation activity?
Reference
Grain boundary sliding at high stresses: interface-reaction controlled process at low stresses
n increases from 2 to 3 and p decreases from 3 to 1 with decreasing stress; constant Q over entire range of stress n increases from 1 to 3 and p decreases from 3 to 1 with decreasing stress; constant Q over entire range of stress n increases with decreasing stress so that n ! 1; Q increases in the threshold stress region
Yes, in order to accommodate grain boundary sliding
Owen and Chokshi [9]
No
Berbon and Langdon [10]
Yes, in order to accommodate grain boundary sliding
Jimenez-Melendo and DomınguezRodrıguez [11] Morita and Hiraga [8]
Interface-controlled Coble diffusion creep at all stresses
Grain boundary sliding at high stresses; a threshold stress at low stresses Dislocation recovery mechanism at high stresses with a threshold stress; Nabarro-Herring diffusion creep at low stresses
Increasing n and Q with decreasing stress at high stresses; a transition to n ¼ 1 at low stresses
Yes but only at higher stresses (>9 MPa) above the region of diffusion creep
N. Balasubramanian, T.G. Langdon / Scripta Materialia 48 (2003) 599–604
601
observations appear to generally support the proposed mechanism given in the last row of Table 1, Mu~ noz et al. [7] have questioned whether an intragranular dislocation flow mechanism is viable in 3Y-TZP at these low stresses at a temperature of 1673 K. Accordingly, this question is now examined.
3. The potential for intragranular dislocation activity in 3Y-TZP at low stresses The shear stress, s, required to nucleate a dislocation is given by the relationship [17] s¼
GbflnðLp =bÞ 1:67g 2pLp ð1 mÞ
ð2Þ
where Lp is the distance between pinning points which may be approximated by d=3 and m is PoissonÕs ratio. For 3Y-TZP, the temperature dependence of the shear modulus was determined recently as [18] Fig. 1. Strain rate versus stress for 3Y-TZP at 1673 K with a grain size of 0.35 lm: the experimental points are from Morita and Hiraga [8] and the solid line shows the prediction using Eq. (6) for interface-controlled Coble diffusion creep [10].
and a transition to Nabarro–Herring diffusion creep at low stresses ( 6 9 MPa) where n 1:3. The grain size exponent, p, was reported as 1.8 at low stresses and 2.2–3.0 at high stresses and the same activation energy of 580 kJ mol1 was reported for all three regions. To substantiate this interpretation, Morita and Hiraga [6] used transmission electron microscopy to examine samples of 3Y-TZP deformed to selected strains at 1673 K and then cooled rapidly under load at an initial rate of 5 K s1 . As in earlier reports on 3Y-TZP [15] and 2.5Y-TZP [16], there was no evidence in the deformed or undeformed samples for the presence of an amorphous phase either along the grain boundaries or at triple-grain junctions. However, there was evidence for intragranular dislocation activity after deformation at 15 MPa and there was an increasing density of intragranular dislocations as the stress was increased to 30 and 50 MPa. Although these
G ¼ 84 103 13:3T
ð3Þ
giving G ¼ 62 GPa at 1673 K which is close to the value of G ¼ 65 GPa reported earlier [19]. The Burgers vector for an ða=2Þh1 1 0i-type lattice dislocation in 3Y-TZP is 0.36 nm [20,21] and, using the Tresca criterion to yield the minimum possible stress, the tensile stress, r, is twice the shear stress, s. Thus, the value calculated for r from Eq. (2) is at least 130 MPa which suggests a factor at least 9 higher than the lowest stress of 15 MPa where dislocation activity was reported by Morita and Hiraga [6]. On the other hand, the present calculation is consistent with experimental results showing the stress needed to activate dislocation movement is 300 MPa in 4Y-TZP single crystals at 1673 K when using a strain rate of 1:5 105 s1 [7] and the yield stress in 6.8% yttria-zirconia pseudo-cubic crystals is 150 MPa at 1673 K with a strain rate of 6 105 s1 [22]. The experimental stress of 15 MPa reported for intragranular dislocation activity by Morita and Hiraga [6] is also exceptionally low based on two other considerations. First, by extrapolating yield stress data for yttria-stabilized cubic ZrO2 single crystals, it was estimated earlier that the yield
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stress for 3Y-TZP at 1673 K is 90 MPa when using a strain rate of 105 s1 [10]. Second, noting the crystal structures suggest the Peierls potential for tetragonal ZrO2 is only slightly higher than for cubic ZrO2 , a procedure was developed earlier for estimating the minimum value of the Peierls stress [10] and this procedure, when used with a shear modulus for 3Y-TZP of 62 GPa [18] and data from cubic ZrO2 , leads to a Peierls stress at 1673 K of not less than 180 MPa. Thus, these calculated values and the experimental values [7,22] are reasonably consistent but they are at least a factor of 10 higher than the stresses where dislocation activity was reported by Morita and Hiraga [6]. Recognizing a potential difficulty because of the low stresses, Morita and Hiraga [6] proposed that stress concentrations may be present at multiplegrain junctions to increase the applied stress by factors of 14–25 and they provided experimental evidence for these stress concentrations by documenting the presence of dislocation pile-ups. To check the possibility of a significant stress concentration, it is first noted that the shear stress at the head of a pile-up is given by [23] spile-up ¼
2Ls2 Gb
ð4Þ
where L is the pile-up length and b is the appropriate Burgers vector. As in the earlier analysis [10], it may be assumed initially that the dislocations pile-up along the grain boundaries so that the value of b is equal to bgb for grain boundary dislocations. Taking bgb ¼ 0:18 nm representing one-half of the value for lattice dislocations, and using the reported value of 0.2 lm for the length of the pileup at 15 MPa [6], spile-up is estimated as 4.5 MPa so that there is no pile-up and no stress concentration. Morita and Hiraga [6] suggested the Burgers vector of grain boundary dislocations may be ‘‘much smaller’’ than b=2 but even if bgb is taken as one-tenth of the lattice value there remains no significant stress concentration. Similar discrepancies are apparent also when the calculation is repeated for the other experimental stresses examined by Morita and Hiraga [6] or when b is taken as the lattice Burgers vector. Thus, these calculations provide a very clear demonstration of the difficulties encountered in attempting to use
conventional pile-up theory for materials with submicrometer grain sizes. There is also a problem with the numbers of dislocations, N , reported in each intragranular pile-up [6]. The conventional theory for N gives a relationship of the form [24] N¼
Lspð1 mÞ Gb
ð5Þ
Using the experimental values reported for L for the intragranular pile-ups [6] and taking m ¼ 0:33, N is estimated as 0.13, 0.24 and 0.28 for stresses of 15, 30 and 50 MPa, respectively. This calculation demonstrates that dislocation pile-ups are not formed under these conditions.
4. Implications for the flow mechanisms in 3Y-TZP The calculations in the preceding section, when considered together with the results from several experimental investigations, provide strong evidence that intragranular dislocation activity is unlikely in superplastic 3Y-TZP at a temperature of 1673 K when using stresses below 100 MPa. By contrast, there are well-documented reports of high values of n, indicative of an intragranular dislocation mechanism, in experiments conducted at even higher stress levels: for example, n 10 at 800 MPa for 2Y-TZP at 1300 K [25] and n 7 with p ¼ 0 above 300 MPa for 3Y-TZP at 1673 K [26]. It is important to note also that an intragranular dislocation recovery mechanism such as climb, as proposed by Morita and Hiraga [8] for stresses above 16 MPa, will take place within the grains so that there is no dependence on grain size and p ¼ 0. This latter requirement is clearly inconsistent with the numerous investigations reporting p 2–3 at these stress levels [9,11,26] including in the experiments of Morita and Hiraga [8] where p 2:2–3:0 in the high stress region. In addition, Morita and Hiraga [8] proposed a threshold stress at intermediate stresses in 3Y-TZP but this would lead, as indicated in Table 1, to an increase in both n and Q at intermediate stresses [27,28] whereas their experiments, and other results [9], show a constant value of Q over the entire stress range.
N. Balasubramanian, T.G. Langdon / Scripta Materialia 48 (2003) 599–604
An alternative interpretation, as given in the second row of Table 1, is to consider flow occurring through an interface-controlled model for Coble diffusion creep [29] where this model has been shown to work well for some other results on 3Y-TZP [10]. In this model, the strain rate is expressed as ! 3 2 Ngb 33:4DZrðgbÞ Gb dZr b r e_ ¼ 2 d G kT b Ngb þ 0:5 ð6Þ where DZrðgbÞ is the grain boundary diffusion coefficient, dZr is the effective grain boundary width for the Zr4þ ions and Ngb is the number of grain boundary dislocations in a single grain boundary wall which is given by [10] Ngb ¼
rd 2Gbgb
ð7Þ
Experimental values for dZr DZrðgbÞ are not directly available but the activation energy for lattice diffusion of the Zr4þ cations in 2.8Y-TZP was reported recently as 6.3 eV and 6.5 eV from tracer diffusion and internal friction measurements, respectively [30]. Data on tetragonal CeO2 – ZrO2 –HfO2 [31] and cubic yttria-stabilized ZrO2 [32] suggest an activation energy for boundary diffusion equal to 0.8 of the value for lattice diffusion. Therefore, the appropriate activation energy for cation grain boundary diffusion in 3YTZP is estimated as 5.1 eV corresponding to 500 kJ mol1 . This value is in excellent agreement with the value of 506 kJ mol1 used in earlier calculations [10] and taken from the grain boundary diffusion coefficient for Zr4þ ions in tetragonal CeO2 –ZrO2 –HfO2 solid solutions [31]. Accordingly, using this value for dZr DZrðgbÞ [31], the solid line in Fig. 1 shows the prediction of Eq. (6) where the upper broken line with a slope of n ¼ 1 is for unmodified Coble diffusion creep and the higher slope of the solid line at lower stresses is due to the interface control. It is apparent from Fig. 1 that Eq. (6) gives good agreement with the experimental datum points, to within one order of magnitude, over the entire range of stresses used in these experiments.
603
5. Summary and conclusions Calculations suggest there is little or no movement of intragranular dislocations in fine-grained 3Y-TZP at stresses below 100 MPa when testing at 1673 K. A flow model based on interface-controlled Coble diffusion creep gives good agreement with published experimental data.
Acknowledgement This work was supported by the United States Department of Energy under grant no. DE-FG0392ER45472.
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