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The contribution of viscous flow to creep deformation of silicon nitride
Materials Science and Engineering A271 (1999) 257 – 265 www.elsevier.com/locate/msea
The contribution of viscous flow to creep deformation of silicon nitride Qiang Jin 1, D.S. Wilkinson *, G.C. Weatherly Department of Materials Science and Engineering, McMaster Uni6ersity, 1280 Main Street W, Hamilton, Ont. L8S 4L7, Canada Received 21 December 1998; received in revised form 30 April 1999
Abstract For silicon nitride ceramics, in which a less refractory grain boundary amorphous film bonds more deformation-resistant Si3N4 grains, one of the possible creep mechanisms is the redistribution of this amorphous film. Although this mechanism has been confirmed by measuring the film thickness distribution before and after creep, it is not clear whether viscous flow occurs only in the initial stage of creep or throughout the creep process. In this paper, compression creep tests were conducted and the film thickness distributions at different strain values were examined using high-resolution electron microscopy. A bimodal distribution of the film widths was observed after creep for all the strains examined, with the peak positions essentially unchanged after creep ranging from 7to 200 h at 1400°C. This demonstrates that viscous flow contributes principally to the initial stage of creep. In general, creep continues at a much lower rate after this cavity-free viscous flow process, entering the second stage. However, once cavities have nucleated, a local film thickness change may occur in the neighbouring areas by viscous flow. Therefore, viscous flow due to cavitation and its contribution to creep deformation are also discussed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Viscous flow; Creep deformation; Silicon nitride
1. Introduction The macroscopic creep behaviour of silicon nitride ceramics has been extensively studied [1 – 8]. Although the overall shape of the creep curve varies considerably with different materials and testing conditions (e.g. stress level and stress state), there are some common features in most materials. An initial period of steadystate creep is usually observed at low strain (0.1%), followed by a rapid decrease in strain rate by about one order of magnitude [5,7,9 – 11]. This behaviour has been termed ‘exhaustion creep’ [7]. Microstructurally, these materials contain intergranular amorphous films. Extensive electron microscopy investigations [9,11–17] indicate that the grain boundary films are of uniform thickness, about 1 nm in most Si3N4 ceramics. Viscous flow of the grain boundary amorphous film has been * Corresponding author. Tel.: +1-905-525-9140; fax: + 1-905-5268404. E-mail address: [email protected] (D.S. Wilkinson) 1 Present address: Exxon Research and Engineering Company, Clinton Township, Route 22 East, Annandale, NJ 08801, USA.
proposed as a possible mechanism for creep deformation [4,5,7,18–21]. The creep response predicted by the viscous flow models is consistent with the observed creep behaviour [5,7,9,11,21]. ‘Exhaustion creep’ can be explained as due to flow of the amorphous phase from grain boundaries in compression to grain boundaries in tension. The sharp decrease in strain rate reflects the situation where the most amorphous phase at compressive grain boundaries has been squeezed out and a new, stress-dependent film thickness is established. The low strains for ‘exhaustion creep’ ( 0.1%) are also consistent with the concept of constrained deformation brought about by facet-contact percolation, assuming an average grain size of Si3N4 is about 1 mm. In recent contributions [9,11,21], we have provided direct evidence for viscous flow of the grain boundary phase during creep of silicon nitride. Using both the Fresnel fringe imaging and lattice fringe imaging techniques, we measured the grain boundary film thickness distributions before and after creep for several different materials in both compression and tension tests. In all cases examined to date, the film widths displayed a bimodal distribution after creep, whereas they were
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confined to a narrow range before creep. We concluded that, within the experimental error associated with the techniques employed, the grain boundary films are uniformly distributed along grain boundaries in the as-sintered materials. After creep, however, some grain boundaries become thinner and others become thicker, in response to the local normal stress. Although this cannot be proven unequivocally, it suggests that the boundaries where the films are thinner correspond to those under local compression while the boundaries where the films are thicker correspond to those under local tension. A bimodal distribution of the film widths after creep indicates that the final equilibrium thickness of a grain boundary subject to a compressive stress may not be zero, as assumed in the viscous flow models [4,18], but has a finite, stress-dependent value. The effect of this on the model has been considered in a recent review [22]. The ‘exhaustion creep’ concept is based on the assumption that viscous flow governs the initial creep stage. In general, creep continues at a much lower rate during the second stage of creep [5,7,9 – 11]. Either a dissolution-reprecipitation or a cavitation mechanism has been proposed to explain this stage [7,10,23–25]. Although the redistribution of the grain boundary phase after creep confirms the viscous flow mechanism [9,11,21], it is not clear whether this process only occurs in the initial stage or extends to the second stage, since the film thickness distribution was always measured after substantial second stage creep. Additionally, if cavitation is involved in the second stage, this may also contribute to the overall film thickness redistribution. In this paper, measurements of the film thickness distribution at several different creep times (or strain values) are reported. These correspond to creep deformation at the end of the initial stage and at two points in the second stage. The results obtained at these different times were compared, and the possible contribution from local cavitation to the film thickness change was also considered. This leads to a better understanding of the viscous flow process during creep of silicon nitride.
2. Experimental procedure Samples of silicon nitride, composed of both equiaxed and elongated b-Si3N4 grains, were prepared by hot-isostatically pressing high-purity Si3N4 powder with 800 wt ppm Ba, at 1925°C for 1 h. The addition of Ba enhances densification without introducing secondary crystalline phases, thereby providing a dense (96% of the theoretical density) but simple microstructure for this investigation. Creep testing was conducted at 1400°C under different compressive stresses (50, 100 and 200 MPa). For the
stress level of 100 MPa, the tests were interrupted at three different creep times (7, 40 and 200 h). A detailed description of the creep testing procedure can be found elsewhere [9]. The samples were cooled under load to avoid any viscoelastic recovery. Transmission electron microscopy (TEM) thin foils of both as-sintered and crept samples were prepared by the standard procedure of grinding, dimpling, argon-ion-beam milling followed by carbon coating to prevent charging during observation. The TEM samples were cut parallel to the stress axis from the crept samples. Grain boundary film widths were measured by using the lattice fringe imaging technique in a high-resolution electron microscope (JEOL 2010F FEG-STEM), with a point resolution of 0.24 nm. The film thickness distribution was evaluated by a statistical analysis, which has been described elsewhere [9].
3. Results and discussion
3.1. Amorphous film measurements The creep behaviour of this material exhibits a continuous decreasing strain rate with time. Presenting the data in terms of strain rate as a function of time (or strain) on a log–log plot, two plateaus of strain rate can be seen. A typical creep response is shown in Fig. 1, which was obtained under 100 MPa for different creep times. In this figure, the solid curve is for creep of 40 h, while the dashed line is for creep of 200 h. The results are very reproducible. The difference of about one order of magnitude in strain rate between the two plateaus demonstrates that there are two distinct creep stages, defined, respectively, as the initial and second stage of creep. A similar creep behaviour has also been observed in several other grades of silicon nitride [5,7,9,21]. The three different creep times chosen for the interrupted tests (7, 40 and 200 h) correspond to three different strain values; one at the end of the first stage (o=0.0049) and the other two in the second stage (o=0.0104 and 0.0307). Tests at different stress levels (50, 100 and 200 MPa) indicate that the stress exponent of strain rate in the second stage of creep is about 2. The film widths in the as-sintered state of this material were fit to a Gaussian distribution, with a mean value of 1.2 90.1 nm, in agreement with previous observations [9,11,21] (Fig. 2). The standard deviation of 0.1 nm is thought to be equal to the experimental error of the lattice fringe imaging technique [16,26], indicating a uniform film thickness along any two-grain boundary. A representative grain boundary film (1.2 nm thick) is shown in Fig. 3. In previous work [9], the film thickness in a silicon nitride doped with the same amount of Ba (800 ppm) was measured by the Fresnel fringe imaging technique, a value of 1.4 nm being
Q. Jin et al. / Materials Science and Engineering A271 (1999) 257–265
observed. However, it should be noted that the processing parameters were different in these two cases (0.5 h, 1925°C for the previous work and 1 h, 1925°C in the present work). Although the two grades of Si3N4 have the same nominal composition of the starting powder, the real composition of the residual intergranular amorphous phase may be different after sintering, thereby influencing the film thickness. The film thickness distributions were measured at different creep strains (see Fig. 1) and the results are shown in Fig. 4. A bimodal thickness distribution was found at all the strains examined. More importantly, there was no apparent difference in the positions of the two peaks in the distribution after creep from 7 to 200 h. In all cases, the first peak is around 0.85 nm, while the second one is about 1.4 nm. This strongly suggests that viscous flow mainly occurs in the initial stage of creep. Once a new equilibrium film thickness distribution is established under an applied stress, it should be stable during the second stage of creep unless other processes occur which could cause a redistribution of the boundary phase. A thinner (compressively loaded) boundary and a thicker (tensilely loaded) boundary are shown in Fig. 5 after creep.
259
Fig. 2. Histogram of the grain-boundary film thickness distribution in the as-sintered Ba doped material. The mean thickness value and the S.D. are shown.
A description of the initial stage of creep by viscous flow has been presented elsewhere [9,11]. Direct evidence for the model comes from the thickness measurement of grain boundary films before and after creep. In addition, the viscous flow models [4,18] predict an ‘exhaustion creep’ behaviour, consistent with the initial stage of creep observed experimentally (Fig. 1). In this paper, we have shown that viscous flow contributes predominantly to the first stage of creep. The increase in film thickness at tensile boundaries (0.2 nm) is about half of the decrease in film thickness at compressive boundaries (0.35 nm). This is consistent with the models [4,18], which predict that in compression, two-thirds of the grain boundaries receive the amorphous phase from the remaining third.
3.2. Effects of ca6itation
Fig. 1. Creep response for the Ba doped material at 1400°C, 1000 MPa for different time periods (7, 40 and 200 h): (a) strain rate vs. time curves; (b) strain rate vs. strain curves. The solid curve is for creep of 40 h while the dash curve is for creep of 200 h.
After creep of 200 h, some very thick grain boundaries (3–4 nm) were occasionally observed (Fig. 4c). This is associated with cavitation at multigrain junctions during the second stage of creep, which can be ascribed to grain boundary sliding accommodated by cavitation. The experimental evidence for these mechanisms includes the direct observation of cavities at triple junctions, a stress exponent of about two and a large number of strain whorls at grain boundaries due to grain-to-grain contact [27]. This is consistent with
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our previous investigation on a similar Ba-doped material [9]. Assuming that the intergranular phase is incompressible, the excess viscous phase displaced by cavity formation must be squeezed into the adjacent grain boundaries, thus causing a local increase in the film thickness at these boundaries. It is now clear that viscous flow contributes to creep deformation in two different ways. The main contribution occurs in the initial stage. During this process, the grain boundary phase redistributes throughout the sample, while the volume remains constant. In contrast, cavity-induced viscous flow occurs stochastically in nature and is accompanied by dilation of the sample. It is not so clear how cavitation influences the film thickness distribution. To understand the cavity-related viscous flow process, a simplified model has been developed. This is presented in Appendix A, from which we derive an expression for the rate of thickening of the grain boundary film when cavitation occurs in triple junctions. The microstructure of the Si3N4 is assumed to consist of hexagonal grains with incompressible grain boundary films of uniform thickness. Based on the experimental observations that small cavities are almost exclusively observed at triple grain junctions with only a small proportion along two-grain interfaces [29], the model assumes that the growth of cavities can only occur at triple junctions by viscous flow of the amor-
phous phase into the adjacent two-grain boundaries. Only grain boundaries in tension are predicted to have a significant increase in their film thickness, while boundaries in compression (which are at equilibrium) essentially are unchanged. The film thickness change at the adjacent grain boundaries is shown in Fig. 6 as a function of cavity size predicted by the model. As can be seen, a cavity, 40 nm in diameter, could increase the boundary film under tension (boundary A) from 1 to 3 nm. Fig. 7a is a lattice fringe image of a thick grain boundary (3 nm thick) observed in the sample crept for 200 h. At lower magnification (Fig. 7b) this boundary is seen to interconnect with a triple junction cavity (50 nm in diameter). This reflects how cavity formation influences the grain boundary film thickness. Since cavitation only occurs at multigrain junctions which are highly stressed and its influence on the grain boundary film thickness is highly localized, most grain boundaries will not be influenced by cavitation provided the density of cavities is very low. The measured film thickness distributions at different strain values found in the present investigation can thus be reasonably explained. Although the model itself is an idealized one and does not reflect the stochastic nature of the cavitation process, it does shed light on the explanation of other cavity-related phenomena as well as the current experimental results. Experimental observations have shown that cavitation during tensile creep of silicon nitride produces strain primarily in the axial direction of the specimen, rather than a uniform outward expansion [30]. This can be easily explained by the current model since grain boundaries in tension are expected to accommodate almost all the glass which flows out from the cavitated multigrain junctions. 4. Conclusions
Fig. 3. High-resolution lattice image of a Si3N4 grain boundary in the as-sintered Ba doped material, showing an amorphous intergranular film thickness of 1.2 nm.
The thickness distributions of the grain boundary films at different creep strains have been measured in a Ba-doped Si3N4 using the high-resolution lattice fringe imaging technique. In contrast to the uniform thickness of grain boundary films in the as-sintered material, the film widths after creep displayed a bimodal distribution at all the strains examined. In particular, no significant difference was found in the film thickness distribution at different strains corresponding to creep immediately after the initial stage and well into the second stage. This observation demonstrates that viscous flow contributes only to the first stage of creep. However, viscous flow is also involved in creep deformation associated with cavitation. Once cavitation occurs at multigrain junctions during the second stage of creep, the excess amorphous phase can flow into the adjacent tensile grain boundaries and increase the local film thickness at these boundaries. This cavity-related viscous flow is highly localized and stochastic in nature.
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Fig. 4. Film thickness distribution in the Ba doped material after creep for different time; (a) 7 h (o = 0.0049); (b) 40 h (o=0.0104); (c) 200 h (o =0.0307).
Appendix A. A model for cavity growth by viscous flow and its effect on grain boundary film thickness The microstructure of silicon nitride is assumed to consist of rigid hexagonal grains (facet length, 2L) and
a continuous intergranular amorphous phase (Fig. A1) [28]. The amorphous phase is present both at two-grain boundaries as thin films of uniform thickness (2d0) and at triple junctions as glass pockets of radius R (L\ \ R). To facilitate the calculations, we use a two-dimen-
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Fig. 5. High-resolution lattice fringe images of grain boundary films in the Ba doped material crept under 100 MPa for 40 h, showing (a) a film thickness of 0.8 nm at one grain boundary, and (b) a film thickness of 1.7 nm at another grain boundary.
sional model in which triple junctions are simplified as cylinders. Experiments have shown that small cavities are found almost exclusively at triple junctions with a small proportion along two-grain interfaces in glass-containing ceramics [29]. Cavity growth along two-grain boundaries by viscous flow has been modelled by other investigators [31,32]. Observations [3,23] suggest that the triple junctions in typical liquid phase sintered materials are generally of sufficient size (7 – 100 nm in diameter) to accommodate a critical cavity nucleus having a spherical morphology. We now wish to model the growth of cavities which are nucleated in the triple junctions and analyze its influence on the film thickness change. It is assumed that the growth of cavities only occurs at triple junctions and all triple junctions cavitate simultaneously by viscous flow of the amorphous phase into the adjacent two-grain boundaries. The assumptions of incompressible primary grains and a linear viscous fluid are also used. Suppose that the material is loaded by a far-field stress s , and the radius of the triple junction cavities is r (as shown in Fig. A1). The boundary conditions for the fluid pressure in the amorphous film at x = 0 and x =L are given by
s x = L : g/r
and
ds =0 dx x = 0
(A1)
Fig. 6. Effect of cavity growth on the film thickness change, showing the boundary in tension (dA) takes almost all the glass which flows out from the pockets while boundaries in compression (dB or dC) essentially do not change their thickness. In calculations, s= 100 MPa, d0 =0.5 nm, L =0.5 mm, g =0.2 J.m − 2 [29] are used. In addition, the size of cavity nucleus r0 =5 nm is arbitrarily assumed whole a fluid viscosity at 1400°C (h =105 Pa.s) is used based on the result of Mosher et al. [33].
Q. Jin et al. / Materials Science and Engineering A271 (1999) 257–265
d2s 3h dd + =0 dx 2 d 3 dt
263
(A2)
where h is the fluid viscosity and 2d is the boundary thickness at time t. Using the boundary conditions, the stress distribution along a boundary is g 3hd: s(x)= + 3 (L 2 − x 2) r 2d
(A3)
When a uniaxial tensile stress s is applied in the direction perpendicular to boundary A, the average stresses at the three boundaries A, B and C can be obtained following the analysis in Ref. [18]: 3 sA = s 2
and sB = sC = 0
(A4)
For boundary A, mechanical equilibrium requires sAL=
& L
0
n
g 3hd: A + (L 2 − x 2) dx r 2d 3A
(A5)
Combining this expression with Eq. (A2), we have 3 g 3 s − dA 2 r :dA = hL 2
(A6)
In the same way, we obtain g − d 3B r d: B = d: C = hL 2
(A7)
Conservation of mass requires that p(r 2 − r 20)= (2dA − 2d0)L+2(2dB − 2d0)L
(A8)
where r0 is the size of cavity nucleus. Here, it is assumed that the cavity nucleation does not influence the initial film thickness.
Fig. 7. (a) High-resolution lattice image of a thick grain boundary film in the Ba doped material crept at 1400°C, 100 MPa for 200 h, (b) lower magnification of the grain boundary, showing it is connected to a triple junction cavity.
where g is the surface energy of the viscous phase, and s is the stress at location x. The condition at x = 0 follows from symmetry considerations; the condition at x =L ensures continuity of the fluid pressure between the two-grain channel and the triple junction. The governing differential equation for viscous flow of a fluid layer at a grain boundary is given by [31]
Fig. A1. Schematic diagram for stress analysis during viscous flow process.
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264 Table 1 Values of assumed parameters Parameter
Value
Reference
d0 L g (h/2d0) at 1100–1200°C
0.5×10−9 m 0.5×10−6 m 0.2 J m−2 1016 to 1017 Pa s/m
[12] [29] [33]
Combining Eqs. (A6), (A7) and (A8), we find that
− 1/2 − 1/2 Á 1 − 3s t  + 2dB − 3d0 à à 2 2 d hL ddB − g d 3B = 2à B + r 20à dt hL à p Ã Ä É 2L (A9)
This differential equation expresses the thickness change of boundary B (2dB) as a function of time. Although an analytical solution is difficult to obtain, the equation can be solved numerically by using reasonable values of the physical parameters (Table 1). In the same way, the thickness change at boundary A and the radius of a cavity can also be obtained as a function of time. Assume that the initial cavity size r0 =5 nm and the viscosity of the boundary phase h= 105 Pa s, the relationship between the cavity size and corresponding film thickness change at adjacent grain boundaries can be obtained (see Fig. 7). The value of 105 Pa s used for the viscosity corresponds to the temperature of 1400°C, based on the results of Mosher et al. [33].
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The contribution of viscous flow to creep deformation of silicon nitride Qiang Jin 1, D.S. Wilkinson *, G.C. Weatherly Department of Materials Science and Engineering, McMaster Uni6ersity, 1280 Main Street W, Hamilton, Ont. L8S 4L7, Canada Received 21 December 1998; received in revised form 30 April 1999
Abstract For silicon nitride ceramics, in which a less refractory grain boundary amorphous film bonds more deformation-resistant Si3N4 grains, one of the possible creep mechanisms is the redistribution of this amorphous film. Although this mechanism has been confirmed by measuring the film thickness distribution before and after creep, it is not clear whether viscous flow occurs only in the initial stage of creep or throughout the creep process. In this paper, compression creep tests were conducted and the film thickness distributions at different strain values were examined using high-resolution electron microscopy. A bimodal distribution of the film widths was observed after creep for all the strains examined, with the peak positions essentially unchanged after creep ranging from 7to 200 h at 1400°C. This demonstrates that viscous flow contributes principally to the initial stage of creep. In general, creep continues at a much lower rate after this cavity-free viscous flow process, entering the second stage. However, once cavities have nucleated, a local film thickness change may occur in the neighbouring areas by viscous flow. Therefore, viscous flow due to cavitation and its contribution to creep deformation are also discussed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Viscous flow; Creep deformation; Silicon nitride
1. Introduction The macroscopic creep behaviour of silicon nitride ceramics has been extensively studied [1 – 8]. Although the overall shape of the creep curve varies considerably with different materials and testing conditions (e.g. stress level and stress state), there are some common features in most materials. An initial period of steadystate creep is usually observed at low strain (0.1%), followed by a rapid decrease in strain rate by about one order of magnitude [5,7,9 – 11]. This behaviour has been termed ‘exhaustion creep’ [7]. Microstructurally, these materials contain intergranular amorphous films. Extensive electron microscopy investigations [9,11–17] indicate that the grain boundary films are of uniform thickness, about 1 nm in most Si3N4 ceramics. Viscous flow of the grain boundary amorphous film has been * Corresponding author. Tel.: +1-905-525-9140; fax: + 1-905-5268404. E-mail address: [email protected] (D.S. Wilkinson) 1 Present address: Exxon Research and Engineering Company, Clinton Township, Route 22 East, Annandale, NJ 08801, USA.
proposed as a possible mechanism for creep deformation [4,5,7,18–21]. The creep response predicted by the viscous flow models is consistent with the observed creep behaviour [5,7,9,11,21]. ‘Exhaustion creep’ can be explained as due to flow of the amorphous phase from grain boundaries in compression to grain boundaries in tension. The sharp decrease in strain rate reflects the situation where the most amorphous phase at compressive grain boundaries has been squeezed out and a new, stress-dependent film thickness is established. The low strains for ‘exhaustion creep’ ( 0.1%) are also consistent with the concept of constrained deformation brought about by facet-contact percolation, assuming an average grain size of Si3N4 is about 1 mm. In recent contributions [9,11,21], we have provided direct evidence for viscous flow of the grain boundary phase during creep of silicon nitride. Using both the Fresnel fringe imaging and lattice fringe imaging techniques, we measured the grain boundary film thickness distributions before and after creep for several different materials in both compression and tension tests. In all cases examined to date, the film widths displayed a bimodal distribution after creep, whereas they were
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confined to a narrow range before creep. We concluded that, within the experimental error associated with the techniques employed, the grain boundary films are uniformly distributed along grain boundaries in the as-sintered materials. After creep, however, some grain boundaries become thinner and others become thicker, in response to the local normal stress. Although this cannot be proven unequivocally, it suggests that the boundaries where the films are thinner correspond to those under local compression while the boundaries where the films are thicker correspond to those under local tension. A bimodal distribution of the film widths after creep indicates that the final equilibrium thickness of a grain boundary subject to a compressive stress may not be zero, as assumed in the viscous flow models [4,18], but has a finite, stress-dependent value. The effect of this on the model has been considered in a recent review [22]. The ‘exhaustion creep’ concept is based on the assumption that viscous flow governs the initial creep stage. In general, creep continues at a much lower rate during the second stage of creep [5,7,9 – 11]. Either a dissolution-reprecipitation or a cavitation mechanism has been proposed to explain this stage [7,10,23–25]. Although the redistribution of the grain boundary phase after creep confirms the viscous flow mechanism [9,11,21], it is not clear whether this process only occurs in the initial stage or extends to the second stage, since the film thickness distribution was always measured after substantial second stage creep. Additionally, if cavitation is involved in the second stage, this may also contribute to the overall film thickness redistribution. In this paper, measurements of the film thickness distribution at several different creep times (or strain values) are reported. These correspond to creep deformation at the end of the initial stage and at two points in the second stage. The results obtained at these different times were compared, and the possible contribution from local cavitation to the film thickness change was also considered. This leads to a better understanding of the viscous flow process during creep of silicon nitride.
2. Experimental procedure Samples of silicon nitride, composed of both equiaxed and elongated b-Si3N4 grains, were prepared by hot-isostatically pressing high-purity Si3N4 powder with 800 wt ppm Ba, at 1925°C for 1 h. The addition of Ba enhances densification without introducing secondary crystalline phases, thereby providing a dense (96% of the theoretical density) but simple microstructure for this investigation. Creep testing was conducted at 1400°C under different compressive stresses (50, 100 and 200 MPa). For the
stress level of 100 MPa, the tests were interrupted at three different creep times (7, 40 and 200 h). A detailed description of the creep testing procedure can be found elsewhere [9]. The samples were cooled under load to avoid any viscoelastic recovery. Transmission electron microscopy (TEM) thin foils of both as-sintered and crept samples were prepared by the standard procedure of grinding, dimpling, argon-ion-beam milling followed by carbon coating to prevent charging during observation. The TEM samples were cut parallel to the stress axis from the crept samples. Grain boundary film widths were measured by using the lattice fringe imaging technique in a high-resolution electron microscope (JEOL 2010F FEG-STEM), with a point resolution of 0.24 nm. The film thickness distribution was evaluated by a statistical analysis, which has been described elsewhere [9].
3. Results and discussion
3.1. Amorphous film measurements The creep behaviour of this material exhibits a continuous decreasing strain rate with time. Presenting the data in terms of strain rate as a function of time (or strain) on a log–log plot, two plateaus of strain rate can be seen. A typical creep response is shown in Fig. 1, which was obtained under 100 MPa for different creep times. In this figure, the solid curve is for creep of 40 h, while the dashed line is for creep of 200 h. The results are very reproducible. The difference of about one order of magnitude in strain rate between the two plateaus demonstrates that there are two distinct creep stages, defined, respectively, as the initial and second stage of creep. A similar creep behaviour has also been observed in several other grades of silicon nitride [5,7,9,21]. The three different creep times chosen for the interrupted tests (7, 40 and 200 h) correspond to three different strain values; one at the end of the first stage (o=0.0049) and the other two in the second stage (o=0.0104 and 0.0307). Tests at different stress levels (50, 100 and 200 MPa) indicate that the stress exponent of strain rate in the second stage of creep is about 2. The film widths in the as-sintered state of this material were fit to a Gaussian distribution, with a mean value of 1.2 90.1 nm, in agreement with previous observations [9,11,21] (Fig. 2). The standard deviation of 0.1 nm is thought to be equal to the experimental error of the lattice fringe imaging technique [16,26], indicating a uniform film thickness along any two-grain boundary. A representative grain boundary film (1.2 nm thick) is shown in Fig. 3. In previous work [9], the film thickness in a silicon nitride doped with the same amount of Ba (800 ppm) was measured by the Fresnel fringe imaging technique, a value of 1.4 nm being
Q. Jin et al. / Materials Science and Engineering A271 (1999) 257–265
observed. However, it should be noted that the processing parameters were different in these two cases (0.5 h, 1925°C for the previous work and 1 h, 1925°C in the present work). Although the two grades of Si3N4 have the same nominal composition of the starting powder, the real composition of the residual intergranular amorphous phase may be different after sintering, thereby influencing the film thickness. The film thickness distributions were measured at different creep strains (see Fig. 1) and the results are shown in Fig. 4. A bimodal thickness distribution was found at all the strains examined. More importantly, there was no apparent difference in the positions of the two peaks in the distribution after creep from 7 to 200 h. In all cases, the first peak is around 0.85 nm, while the second one is about 1.4 nm. This strongly suggests that viscous flow mainly occurs in the initial stage of creep. Once a new equilibrium film thickness distribution is established under an applied stress, it should be stable during the second stage of creep unless other processes occur which could cause a redistribution of the boundary phase. A thinner (compressively loaded) boundary and a thicker (tensilely loaded) boundary are shown in Fig. 5 after creep.
259
Fig. 2. Histogram of the grain-boundary film thickness distribution in the as-sintered Ba doped material. The mean thickness value and the S.D. are shown.
A description of the initial stage of creep by viscous flow has been presented elsewhere [9,11]. Direct evidence for the model comes from the thickness measurement of grain boundary films before and after creep. In addition, the viscous flow models [4,18] predict an ‘exhaustion creep’ behaviour, consistent with the initial stage of creep observed experimentally (Fig. 1). In this paper, we have shown that viscous flow contributes predominantly to the first stage of creep. The increase in film thickness at tensile boundaries (0.2 nm) is about half of the decrease in film thickness at compressive boundaries (0.35 nm). This is consistent with the models [4,18], which predict that in compression, two-thirds of the grain boundaries receive the amorphous phase from the remaining third.
3.2. Effects of ca6itation
Fig. 1. Creep response for the Ba doped material at 1400°C, 1000 MPa for different time periods (7, 40 and 200 h): (a) strain rate vs. time curves; (b) strain rate vs. strain curves. The solid curve is for creep of 40 h while the dash curve is for creep of 200 h.
After creep of 200 h, some very thick grain boundaries (3–4 nm) were occasionally observed (Fig. 4c). This is associated with cavitation at multigrain junctions during the second stage of creep, which can be ascribed to grain boundary sliding accommodated by cavitation. The experimental evidence for these mechanisms includes the direct observation of cavities at triple junctions, a stress exponent of about two and a large number of strain whorls at grain boundaries due to grain-to-grain contact [27]. This is consistent with
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our previous investigation on a similar Ba-doped material [9]. Assuming that the intergranular phase is incompressible, the excess viscous phase displaced by cavity formation must be squeezed into the adjacent grain boundaries, thus causing a local increase in the film thickness at these boundaries. It is now clear that viscous flow contributes to creep deformation in two different ways. The main contribution occurs in the initial stage. During this process, the grain boundary phase redistributes throughout the sample, while the volume remains constant. In contrast, cavity-induced viscous flow occurs stochastically in nature and is accompanied by dilation of the sample. It is not so clear how cavitation influences the film thickness distribution. To understand the cavity-related viscous flow process, a simplified model has been developed. This is presented in Appendix A, from which we derive an expression for the rate of thickening of the grain boundary film when cavitation occurs in triple junctions. The microstructure of the Si3N4 is assumed to consist of hexagonal grains with incompressible grain boundary films of uniform thickness. Based on the experimental observations that small cavities are almost exclusively observed at triple grain junctions with only a small proportion along two-grain interfaces [29], the model assumes that the growth of cavities can only occur at triple junctions by viscous flow of the amor-
phous phase into the adjacent two-grain boundaries. Only grain boundaries in tension are predicted to have a significant increase in their film thickness, while boundaries in compression (which are at equilibrium) essentially are unchanged. The film thickness change at the adjacent grain boundaries is shown in Fig. 6 as a function of cavity size predicted by the model. As can be seen, a cavity, 40 nm in diameter, could increase the boundary film under tension (boundary A) from 1 to 3 nm. Fig. 7a is a lattice fringe image of a thick grain boundary (3 nm thick) observed in the sample crept for 200 h. At lower magnification (Fig. 7b) this boundary is seen to interconnect with a triple junction cavity (50 nm in diameter). This reflects how cavity formation influences the grain boundary film thickness. Since cavitation only occurs at multigrain junctions which are highly stressed and its influence on the grain boundary film thickness is highly localized, most grain boundaries will not be influenced by cavitation provided the density of cavities is very low. The measured film thickness distributions at different strain values found in the present investigation can thus be reasonably explained. Although the model itself is an idealized one and does not reflect the stochastic nature of the cavitation process, it does shed light on the explanation of other cavity-related phenomena as well as the current experimental results. Experimental observations have shown that cavitation during tensile creep of silicon nitride produces strain primarily in the axial direction of the specimen, rather than a uniform outward expansion [30]. This can be easily explained by the current model since grain boundaries in tension are expected to accommodate almost all the glass which flows out from the cavitated multigrain junctions. 4. Conclusions
Fig. 3. High-resolution lattice image of a Si3N4 grain boundary in the as-sintered Ba doped material, showing an amorphous intergranular film thickness of 1.2 nm.
The thickness distributions of the grain boundary films at different creep strains have been measured in a Ba-doped Si3N4 using the high-resolution lattice fringe imaging technique. In contrast to the uniform thickness of grain boundary films in the as-sintered material, the film widths after creep displayed a bimodal distribution at all the strains examined. In particular, no significant difference was found in the film thickness distribution at different strains corresponding to creep immediately after the initial stage and well into the second stage. This observation demonstrates that viscous flow contributes only to the first stage of creep. However, viscous flow is also involved in creep deformation associated with cavitation. Once cavitation occurs at multigrain junctions during the second stage of creep, the excess amorphous phase can flow into the adjacent tensile grain boundaries and increase the local film thickness at these boundaries. This cavity-related viscous flow is highly localized and stochastic in nature.
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261
Fig. 4. Film thickness distribution in the Ba doped material after creep for different time; (a) 7 h (o = 0.0049); (b) 40 h (o=0.0104); (c) 200 h (o =0.0307).
Appendix A. A model for cavity growth by viscous flow and its effect on grain boundary film thickness The microstructure of silicon nitride is assumed to consist of rigid hexagonal grains (facet length, 2L) and
a continuous intergranular amorphous phase (Fig. A1) [28]. The amorphous phase is present both at two-grain boundaries as thin films of uniform thickness (2d0) and at triple junctions as glass pockets of radius R (L\ \ R). To facilitate the calculations, we use a two-dimen-
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Fig. 5. High-resolution lattice fringe images of grain boundary films in the Ba doped material crept under 100 MPa for 40 h, showing (a) a film thickness of 0.8 nm at one grain boundary, and (b) a film thickness of 1.7 nm at another grain boundary.
sional model in which triple junctions are simplified as cylinders. Experiments have shown that small cavities are found almost exclusively at triple junctions with a small proportion along two-grain interfaces in glass-containing ceramics [29]. Cavity growth along two-grain boundaries by viscous flow has been modelled by other investigators [31,32]. Observations [3,23] suggest that the triple junctions in typical liquid phase sintered materials are generally of sufficient size (7 – 100 nm in diameter) to accommodate a critical cavity nucleus having a spherical morphology. We now wish to model the growth of cavities which are nucleated in the triple junctions and analyze its influence on the film thickness change. It is assumed that the growth of cavities only occurs at triple junctions and all triple junctions cavitate simultaneously by viscous flow of the amorphous phase into the adjacent two-grain boundaries. The assumptions of incompressible primary grains and a linear viscous fluid are also used. Suppose that the material is loaded by a far-field stress s , and the radius of the triple junction cavities is r (as shown in Fig. A1). The boundary conditions for the fluid pressure in the amorphous film at x = 0 and x =L are given by
s x = L : g/r
and
ds =0 dx x = 0
(A1)
Fig. 6. Effect of cavity growth on the film thickness change, showing the boundary in tension (dA) takes almost all the glass which flows out from the pockets while boundaries in compression (dB or dC) essentially do not change their thickness. In calculations, s= 100 MPa, d0 =0.5 nm, L =0.5 mm, g =0.2 J.m − 2 [29] are used. In addition, the size of cavity nucleus r0 =5 nm is arbitrarily assumed whole a fluid viscosity at 1400°C (h =105 Pa.s) is used based on the result of Mosher et al. [33].
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d2s 3h dd + =0 dx 2 d 3 dt
263
(A2)
where h is the fluid viscosity and 2d is the boundary thickness at time t. Using the boundary conditions, the stress distribution along a boundary is g 3hd: s(x)= + 3 (L 2 − x 2) r 2d
(A3)
When a uniaxial tensile stress s is applied in the direction perpendicular to boundary A, the average stresses at the three boundaries A, B and C can be obtained following the analysis in Ref. [18]: 3 sA = s 2
and sB = sC = 0
(A4)
For boundary A, mechanical equilibrium requires sAL=
& L
0
n
g 3hd: A + (L 2 − x 2) dx r 2d 3A
(A5)
Combining this expression with Eq. (A2), we have 3 g 3 s − dA 2 r :dA = hL 2
(A6)
In the same way, we obtain g − d 3B r d: B = d: C = hL 2
(A7)
Conservation of mass requires that p(r 2 − r 20)= (2dA − 2d0)L+2(2dB − 2d0)L
(A8)
where r0 is the size of cavity nucleus. Here, it is assumed that the cavity nucleation does not influence the initial film thickness.
Fig. 7. (a) High-resolution lattice image of a thick grain boundary film in the Ba doped material crept at 1400°C, 100 MPa for 200 h, (b) lower magnification of the grain boundary, showing it is connected to a triple junction cavity.
where g is the surface energy of the viscous phase, and s is the stress at location x. The condition at x = 0 follows from symmetry considerations; the condition at x =L ensures continuity of the fluid pressure between the two-grain channel and the triple junction. The governing differential equation for viscous flow of a fluid layer at a grain boundary is given by [31]
Fig. A1. Schematic diagram for stress analysis during viscous flow process.
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264 Table 1 Values of assumed parameters Parameter
Value
Reference
d0 L g (h/2d0) at 1100–1200°C
0.5×10−9 m 0.5×10−6 m 0.2 J m−2 1016 to 1017 Pa s/m
[12] [29] [33]
Combining Eqs. (A6), (A7) and (A8), we find that
− 1/2 − 1/2 Á 1 − 3s t  + 2dB − 3d0 à à 2 2 d hL ddB − g d 3B = 2à B + r 20à dt hL à p Ã Ä É 2L (A9)
This differential equation expresses the thickness change of boundary B (2dB) as a function of time. Although an analytical solution is difficult to obtain, the equation can be solved numerically by using reasonable values of the physical parameters (Table 1). In the same way, the thickness change at boundary A and the radius of a cavity can also be obtained as a function of time. Assume that the initial cavity size r0 =5 nm and the viscosity of the boundary phase h= 105 Pa s, the relationship between the cavity size and corresponding film thickness change at adjacent grain boundaries can be obtained (see Fig. 7). The value of 105 Pa s used for the viscosity corresponds to the temperature of 1400°C, based on the results of Mosher et al. [33].
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