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Generic grain size distribution for liquid phase sintering
Scripta METALLURGICA et MATERIALIA
Vol. 26, pp. 95-98, 1992 Printed in the U.S.A.
Pergamon Press plc All rights reserved
GENERIC GRAIN SIZE DISTRIBUTION FOR LIQUID PHASE SINTERING
Sung-Chul yangl and R. M. German 2 1Research and Development Center, Mando Co., 95 Deokso, Kynng-Gi, Korea 2Brash Chair in Materials, Engineering Science and Mechanics Deparunent, Pennsylvania State University, University Park, PA 16802 (Received September 24, 1991) (Revised October 22, 1991)
The LSW (Lifshitz-Slyozov and Wagner) [1,2] and various theories [3-7] exist for modelling the microstructural coarsening and asymptotic grain size dislributions for ideal systems. Many models have been proposed for more realistic conditions using various assumptions. The models are useful up to about 50 vol% solid. Beyond this point, too many of the assumptions fail. In the present work, an empirical model is used to represent the generic grain size distribution for liquid phase sintering. Experimental data and theoretical models were analysed using the plot of standard normal versus normalized grain size. From the experimental (3-D) grain size plot a representative line was deduced. These results best fit the LSEM (Lifshitz-Slyozov Encounter Modified) model [5] except for the small size ranges at the lower 15% of the distribution. This may indicate preferential growth by coalescence for the smaller grains. Analysis of previous Work It is important to predict a grain size distribution for practiead liquid phase sintering systems. As shown in Table I, many assumptions used in the theaxetical models fail in practical systems in which the volume of solid exceeds more than 80%. In practical liquid phase sinlering we deal with microslxuctures where a typical system might be 90W-7Ni-3Fe, at 1500°C, which gives -20 vol% liquid, dihedral angle of 25 °, contiguity of 0.25 or higher, and over 4 contacts per grain [8-I I]. We cannot see the true grain size using metallography, so a (2-D) slice is used and the intercept size distribution is measured using random test lines. This is a tedious and inaccurate basis for testing theories which arises because the theories give a sphere diameter distribution while the measurement is a random intercept length on non-spherical grains. As an alternative, a mathematical distribution function is considered an effective way to represent ~alistic size distributions. In this work, we have used a standard normal versus log (L/Ls0) plot to analyze the grain size distribution from previous work. Here L is the grain size, and 1.50 is the distribution median. The cumulative fractions of distributions were converted to standard normal using the values of the standard normal distribution function [ 12]. As a demonstration, the distribution of grain intercept length with various compositions, sintering times, and sintering temperatures in W- Ni-Fe systems [8,13-16] arc plotted in Figure 1. In this plot, the y-axis is the standard normal (approximately 2% of the grains are smaller than the size at -2 standard normal, 16% smaller than the size at -1 standard normal, and so on with 50% at 0, 84% at +1, and 98% at +2). The x-axis is the grain size divided by the size at the distribution median. The distributions at the volume fraction of 0.8 from both the LSEM model [5] and the BWEM (BrallsfordWynblatt Encounter Modified) model [4,7] were converted into the standard normal versus log (L/Ls0) plot for comparison with the experimental data. All of the grain sizes in Figure 1 are random linear intercept lengths; therefore, the fit of the data with the BWEM model is not valid. Figure 2 shows important data of 3-D grain size in liquid phase sIntered ZnO-BaO [16], and 30%Fe-70%Cu [13] systems. In the ZnO-BaO system, the grains are separated by dissolving the matrix phase, and in the Fe-Cu system, size was measured by circumscribing grains with circles of known diameter which were drawn sequentially in size on a Iransparency (2-D) and were corrected to 3-dimensional size (3-D) using the Swartz-Saltikov method [13,17]. The result shows reasonable agreement with the LSEM model. Data for the size distribution of AI3Li precipitates in the AI-Li system [18] also show good agreement with the LSEM model. We can approximate it as 3-D size because it is gathered from a transmitted image and the shape is almost perfectly spherical. From the standard normal versus log (L/Ls0) plots of the 3-D experimental data, a representative line was deduced which can best fit to the data as shown in Figure 2. This empirical line provides a generic grain size distribution for liquid phase sintering.
95 0036-9748/92 $5.00 + .00 Copyright (c) 1991 Pergamon Press plc
96
LIQUID PHASE SINTERING
Vol. 26, No. 1
Reviewing all the comparisons above, it can be concluded that the LSEM model represents best the real 3-D grain size distribution except for the small size ranges at the lower 15% of the distribution. The apparent lack of agreement for the smaller grains is systematic among several studies and may reflect a gravity-induced coalescence effect [11]. This analysis is quite empirical; however, it provides important information to predict the spatial arrangement of grains in liquid phase sintering together with other information responsible for this arrangement, such as volume fraction of the solid, surface tension, and gravitational forces. Further development of the model distribution function and geometry for liquid phase sintering will be performed by a computer simulation technique using the parameters described above [19].
To develop an empirical model to represent a generic grain size distribution for liquid phase sintering, experimental data and theoreucal models were analyzed using the plot of standard normal versus normalized size. From the experimental 3-D grain size plot, the most representative line was deduced. This line fits best to the LSEM (Lifshitz-Slyozov Encounter Modified) model except for the small size rallges of the lower 15% of the disuibution. Recent research shows that grain coalescence nught be a cause for this departure fn3m theory. Aelmowled~nent The authors wish to thank NASA for supporting this work under grant NAG 3-1095. References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
I.A, Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids, 19, 35 (1961). C. Wagner, Z. Elektrochem, 65, 581 (1961). AJ. Aredll, Acta Metall., 20, 61 0972). A.D. Brailsford and P. Wynblatt, Acta Metall., 27, 489 (1979). C.K.L. Davies, P. Nash, and R.N. Stevens, Acta Metall., 28, 179 (1980). P.W. Voorhees and M.E. Glicksman, Metall. Trans., 15A, 1081 (1984). C.S. Jayanth and P. Nash, J. Mater. Sci., 24, 3041 (1989). Sung-Chul Yang, S.S. Mani, and R.M. German, Adv. in Powder MeL, published by MPIF, Princeton, New Jersey, vol. 1,469 (1990). R.M. German, Metall. Trans., 16A, 1247 (1985). R.M. German, Metall. Trans., 18A, 909 (1987). C.M. Kipphut, A. Bose, S. Farooq, and R.M. German, Mccall. Trans., 19A, 1905 (1988). B.W. Lindgren, "Statistical Theory," chap. 3, The Macmillan Company, New York (1968). Sung-Chul Yang, Ph.D. thesis, Illinois Institute of Technology (1989). Eugene G. Zukas and Haskell Sheinberg, Powder Technology, 13, 85 (1976). T.K. Kang and D.N. Yoon, MetalL Trans., 9A, 433 (1978). Sung-Chul Yang and R.M. German, accepted for publication in J. Am. Ceram. Soc. (1991). E.E. Underwood, Quantitative Microscopy, Ed. by Dehoff and F.N. Rhines, p162-200, New York, McGraw-Hill (1968). K. Mahalingam, B.P. Gu, G.L. Liedl, and T.H. Sanders, Jr., Acta Met., 35, 483-98 (1987). R.M. German, M. Zivkovic, and S.C. Yang, "Computer Simulation of Solid Volume Fraction Limit in Liquid Phase Sintering," to be published. TABLE 1 Difficulties between Model and Experiment assumptions in model
prac~cal
no contact no contiguity zero~ an~e no settling isotropic surface energy spherical 8rains
over 4 contacts per grains 20 to 80~ contiguity usually 30° settling due to gravity anisotropic surface energy shape accommodated
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Vol. 26, pp. 95-98, 1992 Printed in the U.S.A.
Pergamon Press plc All rights reserved
GENERIC GRAIN SIZE DISTRIBUTION FOR LIQUID PHASE SINTERING
Sung-Chul yangl and R. M. German 2 1Research and Development Center, Mando Co., 95 Deokso, Kynng-Gi, Korea 2Brash Chair in Materials, Engineering Science and Mechanics Deparunent, Pennsylvania State University, University Park, PA 16802 (Received September 24, 1991) (Revised October 22, 1991)
The LSW (Lifshitz-Slyozov and Wagner) [1,2] and various theories [3-7] exist for modelling the microstructural coarsening and asymptotic grain size dislributions for ideal systems. Many models have been proposed for more realistic conditions using various assumptions. The models are useful up to about 50 vol% solid. Beyond this point, too many of the assumptions fail. In the present work, an empirical model is used to represent the generic grain size distribution for liquid phase sintering. Experimental data and theoretical models were analysed using the plot of standard normal versus normalized grain size. From the experimental (3-D) grain size plot a representative line was deduced. These results best fit the LSEM (Lifshitz-Slyozov Encounter Modified) model [5] except for the small size ranges at the lower 15% of the distribution. This may indicate preferential growth by coalescence for the smaller grains. Analysis of previous Work It is important to predict a grain size distribution for practiead liquid phase sintering systems. As shown in Table I, many assumptions used in the theaxetical models fail in practical systems in which the volume of solid exceeds more than 80%. In practical liquid phase sinlering we deal with microslxuctures where a typical system might be 90W-7Ni-3Fe, at 1500°C, which gives -20 vol% liquid, dihedral angle of 25 °, contiguity of 0.25 or higher, and over 4 contacts per grain [8-I I]. We cannot see the true grain size using metallography, so a (2-D) slice is used and the intercept size distribution is measured using random test lines. This is a tedious and inaccurate basis for testing theories which arises because the theories give a sphere diameter distribution while the measurement is a random intercept length on non-spherical grains. As an alternative, a mathematical distribution function is considered an effective way to represent ~alistic size distributions. In this work, we have used a standard normal versus log (L/Ls0) plot to analyze the grain size distribution from previous work. Here L is the grain size, and 1.50 is the distribution median. The cumulative fractions of distributions were converted to standard normal using the values of the standard normal distribution function [ 12]. As a demonstration, the distribution of grain intercept length with various compositions, sintering times, and sintering temperatures in W- Ni-Fe systems [8,13-16] arc plotted in Figure 1. In this plot, the y-axis is the standard normal (approximately 2% of the grains are smaller than the size at -2 standard normal, 16% smaller than the size at -1 standard normal, and so on with 50% at 0, 84% at +1, and 98% at +2). The x-axis is the grain size divided by the size at the distribution median. The distributions at the volume fraction of 0.8 from both the LSEM model [5] and the BWEM (BrallsfordWynblatt Encounter Modified) model [4,7] were converted into the standard normal versus log (L/Ls0) plot for comparison with the experimental data. All of the grain sizes in Figure 1 are random linear intercept lengths; therefore, the fit of the data with the BWEM model is not valid. Figure 2 shows important data of 3-D grain size in liquid phase sIntered ZnO-BaO [16], and 30%Fe-70%Cu [13] systems. In the ZnO-BaO system, the grains are separated by dissolving the matrix phase, and in the Fe-Cu system, size was measured by circumscribing grains with circles of known diameter which were drawn sequentially in size on a Iransparency (2-D) and were corrected to 3-dimensional size (3-D) using the Swartz-Saltikov method [13,17]. The result shows reasonable agreement with the LSEM model. Data for the size distribution of AI3Li precipitates in the AI-Li system [18] also show good agreement with the LSEM model. We can approximate it as 3-D size because it is gathered from a transmitted image and the shape is almost perfectly spherical. From the standard normal versus log (L/Ls0) plots of the 3-D experimental data, a representative line was deduced which can best fit to the data as shown in Figure 2. This empirical line provides a generic grain size distribution for liquid phase sintering.
95 0036-9748/92 $5.00 + .00 Copyright (c) 1991 Pergamon Press plc
96
LIQUID PHASE SINTERING
Vol. 26, No. 1
Reviewing all the comparisons above, it can be concluded that the LSEM model represents best the real 3-D grain size distribution except for the small size ranges at the lower 15% of the distribution. The apparent lack of agreement for the smaller grains is systematic among several studies and may reflect a gravity-induced coalescence effect [11]. This analysis is quite empirical; however, it provides important information to predict the spatial arrangement of grains in liquid phase sintering together with other information responsible for this arrangement, such as volume fraction of the solid, surface tension, and gravitational forces. Further development of the model distribution function and geometry for liquid phase sintering will be performed by a computer simulation technique using the parameters described above [19].
To develop an empirical model to represent a generic grain size distribution for liquid phase sintering, experimental data and theoreucal models were analyzed using the plot of standard normal versus normalized size. From the experimental 3-D grain size plot, the most representative line was deduced. This line fits best to the LSEM (Lifshitz-Slyozov Encounter Modified) model except for the small size rallges of the lower 15% of the disuibution. Recent research shows that grain coalescence nught be a cause for this departure fn3m theory. Aelmowled~nent The authors wish to thank NASA for supporting this work under grant NAG 3-1095. References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
I.A, Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids, 19, 35 (1961). C. Wagner, Z. Elektrochem, 65, 581 (1961). AJ. Aredll, Acta Metall., 20, 61 0972). A.D. Brailsford and P. Wynblatt, Acta Metall., 27, 489 (1979). C.K.L. Davies, P. Nash, and R.N. Stevens, Acta Metall., 28, 179 (1980). P.W. Voorhees and M.E. Glicksman, Metall. Trans., 15A, 1081 (1984). C.S. Jayanth and P. Nash, J. Mater. Sci., 24, 3041 (1989). Sung-Chul Yang, S.S. Mani, and R.M. German, Adv. in Powder MeL, published by MPIF, Princeton, New Jersey, vol. 1,469 (1990). R.M. German, Metall. Trans., 16A, 1247 (1985). R.M. German, Metall. Trans., 18A, 909 (1987). C.M. Kipphut, A. Bose, S. Farooq, and R.M. German, Mccall. Trans., 19A, 1905 (1988). B.W. Lindgren, "Statistical Theory," chap. 3, The Macmillan Company, New York (1968). Sung-Chul Yang, Ph.D. thesis, Illinois Institute of Technology (1989). Eugene G. Zukas and Haskell Sheinberg, Powder Technology, 13, 85 (1976). T.K. Kang and D.N. Yoon, MetalL Trans., 9A, 433 (1978). Sung-Chul Yang and R.M. German, accepted for publication in J. Am. Ceram. Soc. (1991). E.E. Underwood, Quantitative Microscopy, Ed. by Dehoff and F.N. Rhines, p162-200, New York, McGraw-Hill (1968). K. Mahalingam, B.P. Gu, G.L. Liedl, and T.H. Sanders, Jr., Acta Met., 35, 483-98 (1987). R.M. German, M. Zivkovic, and S.C. Yang, "Computer Simulation of Solid Volume Fraction Limit in Liquid Phase Sintering," to be published. TABLE 1 Difficulties between Model and Experiment assumptions in model
prac~cal
no contact no contiguity zero~ an~e no settling isotropic surface energy spherical 8rains
over 4 contacts per grains 20 to 80~ contiguity usually 30° settling due to gravity anisotropic surface energy shape accommodated
Vol.
26, No.
1
LIQUID
PHASE SINTERING
97
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26, No.
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