- Email: [email protected]
A study of a new phenomenological compacting equation
Powder Technology 114 Ž2001. 255–261 www.elsevier.comrlocaterpowtec
Letter
A study of a new phenomenological compacting equation Renato Panelli a , Francisco Ambrozio Filho b,) b
a Mahle Metal LeÕe Miba Sinterizados, Rua Brasılio ´ Luz 535, CEP 04746-901 Sao ˜ Paulo, Brazil Comissao e Nucleares, DiÕisao ˜ Nacional de Energia Nuclear, Instituto de Pesquisas Energeticas ´ ˜ MMM, C.P. 11049 Cidade UniÕersitaria, ´ CEP 05422-970 Sao ˜ Paulo, Brazil
Received 20 June 1999; received in revised form 13 December 1999; accepted 13 December 1999
Abstract A new phenomenological compacting equation has been proposed. The equation is lnŽ1rŽ1 y D .. s A'P q B where P is the applied pressure, D is the relative density of the compact, A is a parameter related to densification of the compact by plastic deformation, and B is a parameter related to powder density at the start of compaction. Linear regression analysis has been used to compare the new equation with the four compacting equations often used and proposed by Balshin, Heckel, Kawakita and Ge. The results show that the new equation gives linear correlation coefficients very close to unity. This equation, together with parameters A and B, permits improved evaluation of the compacting characteristics, compared to that performed by other equations. q 2001 Elsevier Science S.A. All rights reserved. Keywords: Compaction; Mathematical modeling
1. Introduction The powder compaction process plays an important role in the manufacture of a variety of products that include ceramics, metallic parts, fertilizers and pharmaceuticals w1x. In the case of ceramics and metallic parts, the microstructure of sintered powder compacts depend strongly on the quality of the green compact, which in turn depends on the behavior of the powder during compaction, that is, by the density–pressure relationship of the powder w2x. According to Bockstiegel w3x, the interest originally in density–pressure relationships stemmed from the practical problem of being able to predict the pressures required for achieving a certain density, later on, the interest seemed to shift more towards the analytical problem of finding an adequate but simple mathematical description for experimentally obtained density–pressure curves. Considerable effort has been made to characterize powders and their compaction behavior using a compacting equation w2,4–10x. In this paper the four often used compaction equations are compared, in a manner similar to that carried out by Ge
w8x in 1995. A new phenomenological compacting equation has also been included in this paper w9x.
2. Compacting equations Balshin attempted to correlate the relative density of compact powders with externally applied pressures and proposed the following equation: ED EP
Corresponding author. Fax: q55-11-816-9370. E-mail address: [email protected] ŽF. Ambrozio Filho..
D2
Ž 1.
P
ln P s y
A1 D
q B1
Ž 2.
where P is the applied pressure, D is the relative density of the compact and A1 , B1 are constants. Balshin called A1 the pressing modulus and considered it to be analogous to the Young’s modulus w2x. Balshin’s equation can be rewritten as: 1 D
)
sk
s A 2 ln P q B2
Ž 3.
Heckel, in 1961 w5x, considered the compaction of powders to be analogous to a first-order chemical reaction. The pores are the reactant and the densification of the bulk
0032-5910r01r$ - see front matter q 2001 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 0 . 0 0 2 0 7 - 2
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
256
is the product. The AkineticsB of the process may be described as a proportionality between the change in density with pressure and the pore fraction by: ED EP
sK Ž1 yD.
Ž 4.
This simplification done by Ge in his equation is not theoretically solved. In 1998, Panelli and Ambrozio Filho w9x, based upon observations of typical compaction curves, proposed the following differential equation to represent such curves: Eq
where Ž1 y D . is the pore fraction and K is a constant. The final form of Heckel’s equation can be written as follows: ln
ž
1 1yD
/
s A 3 P q B3
Ž 5.
where A 3 and B3 are constants. Heckel postulated that the constant A 3 gave a measure of the ability of the compact to densify by plastic deformation and constant B3 represented the degree of packing achieved at low pressures as a result of rearrangement before appreciable amounts of inter-particle bonding occurred. w6x derived a compaction equation Kawakita and Ludde ¨ from the observed relationship between applied pressure and volume or relative density. Kawakita obtained the following equation: D s D y D0
A4 P
q B4
Ž 6.
where A 4 and B4 are constants and D 0 the relative powder apparent density without application of pressure. Attention must be paid to the experimental determination of D 0 , and deviations from Kawakita’s equations are sometimes due to fluctuations in the measured value of D 0 , which is usually inaccurate. Ge w1x proposed a new differential equation for the pressing of powders: ED EP
sK
Ž1 y D. Dn Pm
Ž 7.
Kq s
EP
Ž 10 .
Pm
where q is relative porosity, P the applied pressure and K, m are constants. By integrating Eq. Ž10. from q Žporosity at P . to q0 Žporosity at P s 0. at applied pressures from 0 to P, a general equation is obtained: ln
q0 q
s A7
Pym q1 ym q 1
q B7 .
Ž 11 .
This equation has an integration constant B7 , which must be zero to satisfy the condition q0 s q when P s 0. Fitting experimental data to it by regression analysis tested Eq. Ž11.. Linear correlation coefficients, R, give values better than 0.99 when m s 0.5. So the value 0.5 was adopted for m in Eq. Ž10. and rewritten as: ln
q0 q
s A g'P
Ž 12 .
where q0 s Ž1 y D 0 . , q s Ž1 y D . and D 0 and D are the relative densities of the powder at zero pressure and of the compact at pressure P. Upon substitution, Eq. Ž12. becomes: ln
Ž 1 y D0 . s A g'P . Ž1 yD.
Ž 13 .
Eq. Ž13. has limited applicability owing to the necessity to determine the parameter D 0 . It is therefore important to
where n and m are constants. From Eq. Ž7., Ge obtained the following equation:
Ž 1 y D0 . log ln s A 5 log P q B5 Ž1 yD.
Ž 8.
Ge, in a second paper published in 1995 w8x, proposed a further simplified form of Eq. Ž8.. He observed that his equation was subject to the same level of imprecision as Kawakita’s equation, due to fluctuations in the measured value of D 0 . Ge considered that D 0 approached zero when total pressure reached zero, and his equation could be rewritten as: log ln
1
Ž1 yD.
s A 6 log P q B6
where A 6 and B6 are constants.
Ž 9.
Table 1 The compacting equations that have been compared. P sapplied pressure, Ds relative density of the compact, D 0 s relative apparent density of powder in absence of applied pressure, and A n and Bn are constant parameters; ns 2, 3, 4, 6 and 10 Author
Equation
Equation number
Reference
Balshin Heckel
1r Ds A 2 ln P q B2 lnŽ1rŽ1y D .. s A 3 P q B3 DrŽ Dy D 0 . s Ž A 4 r P .q B4 logwlnŽ1rŽ1y D .x s A 6 log P q B6 lnŽ1rŽ1y D ..
Ž3. Ž5.
w4x w5x
Ž6.
w6x
Ž9.
w8x
Ž15.
w9x
Kawakita Ge Panelli and Ambrozio Filho
s A10 'P q B10
Table 2 Experimental data on the compaction of various powders. P sapplied pressure ŽMPa. and Ds relative density of compact Powder Material no.
Reference Experimental data on compaction of powders
Silver bromide, 8–10 mesh, 958C
w4x
2
Ammonium nitrate
w11x
3
Silver bromide, 10–14 mesh, 258C
w4x
4
Silver bromide, 8–10 mesh, 258C
w4x
5
Silver bromide, 14–20 mesh, 258C
w4x
6
Atomized tin
w1x
7
Lead Žmetallic.
w11x
8
Atomized lead
w1x
9
Ammonium Chloride
w11x
10
Potassium chloride
w11x
11
Potassium nitrate
w11x
12
Sodium chloride
w11x
13
Barium nitrate
w11x
14
Electrolytic copper
w1x
15
Stainless steel
w1x
16
AISI M2
w9x
17
Calcium carbonate
w11x
18
Spray-dried alumina
w12x
19
M2q10 vol% NbC, mechanical alloyed 5h w10x
20
Tungsten carbide
w1x
21
Titanium carbide
w1x
22
Alumina
w9x
P 21 D 0.800 P 1.0 D 58.8 P 21 D 0.740 P 21 D 0.743 P 21 D 0.743 P 0 D 0.501 P 2.1 D 0.625 P 0 D 0.439 P 0.7 D 0.455 P 1.2 D 0.435 P 0.6 D 0.526 P 4.3 D 0.526 P 0.7 D 0.556 P 0 D 0.300 P 0 D 0.374 P 260.4 D 0.620 P 1.9 D 0.435 P 14 D 0.511 P 260 D 0.603 P 0 D 0.201 P 0 D 0.370 P 365.6 D 0.502
31 0.841 1.9 0.625 31 0.789 31 0.791 31 0.781 14.7 0.739 5.1 0.667 32.34 0.937 0.9 0.476 1.9 0.455 2.1 0.556 7.0 0.556 1.5 0.588 59.0 0.607 154.8 0.648 365.6 0.677 3.1 0.455 28 0.557 365 0.631 49.0 0.471 18.0 0.518 474.3 0.513
47 0.908 3.0 0.667 47 0.861 47 0.858 47 0.852 19.6 0.799 8.9 0.714 64.68 0.974 1.6 0.500 2.6 0.476 4.4 0.588 10.2 0.588 2.7 0.625 108 0.664 308.7 0.740 474.3 0.716 4.8 0.476 49 0.580 474 0.650 68.6 0.489 36.0 0.547 567.3 0.521
61 0.949 4.7 0.714 61 0.909 61 0.905 61 0.900 29.4 0.853 13.4 0.769 96.04 0.986 2.2 0.526 3.6 0.500 7.4 0.625 14.8 0.625 4.7 0.667 114 0.674 463.5 0.800 567.3 0.755 7.4 0.500 69 0.609 567 0.673 98.0 0.512 72.0 0.587 655.7 0.528
78 0.971 7.2 0.769 78 0.934 78 0.934 78 0.927 49.0 0.920 20.5 0.833 130.3 0.991 2.9 0.556 4.9 0.526 11.6 0.667 21.4 0.667 9.2 0.714 187 0.753 617.4 0.831 655.7 0.778 12.3 0.526 96 0.625 655 0.690 147 0.538 144 0.614
94 0.985 11.6 0.833 94 0.954 94 0.954 94 0.954 68.6 0.951 32.5 0.909 160.7 0.993 4.0 0.588 6.6 0.556 17.8 0.714 31.2 0.714 22.3 0.769 294 0.839 772.2 0.870
110 0.988 19.6 0.909 110 0.967 110 0.969 110 0.965 98.0 0.975
126 0.977 126 0.977 126 0.976 117.6 0.984
142 0.985 142 0.980 142 0.980 147.0 196.0 245.0 294.0 0.990 0.993 0.996 0.997
196.0 0.995 5.5 8.0 12.1 0.625 0.667 0.714 8.9 11.7 15.9 0.588 0.625 0.667 26.8 41.2 76.4 0.769 0.833 0.909 45.0 65.9 101.9 0.769 0.833 0.909 68.6 0.833 372 1000 0.874 0.964 926.1 0.890
18.7 0.769 21.3 0.714
32.0 0.833 29.1 0.769
63.7 0.909 42.1 0.833
76.9 0.909
21.9 41.2 0.556 0.588 140 206 0.650 0.673
86.0 178.2 0.625 0.667 278 343 441 447 481 503 0.694 0.695 0.713 0.714 0.716 0.722
196 245 0.559 0.576 288 576 0.650 0.699
294 392 490 588 0.586 0.609 0.627 0.641 864 0.719
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
1
257
258
Powder no.
Eq. Ž3. A2
B2
R
Eq. Ž5. A 3 Ž=10y2 .
B3
R
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
y0.1520 y0.2080 y0.1780 y0.1751 y0.1787 y0.1077 y0.1868 y0.0339 y02546 y0.3098 y0.1766 y0.2637 y0.1350 y0.2302 y0.2350 y0.3550 y0.1769 y0.1510 y0.2216 y0.2278 y0.1393 y0.1673
1.7017 1.7179 1.8672 1.8556 1.8752 1.5589 1.7751 1.1765 2.0759 2.3783 1.9003 2.3100 1.7343 2.5584 2.7125 3.5815 2.3772 2.3096 2.8919 3.0013 2.3225 2.9797
y0.983 y0.998 y0.990 y0.982 y0.985 y0.924 y0.988 y0.950 y0.993 y0.997 y0.983 y0.999 y0.992 y0.977 y0.995 y0.998 y0.994 y0.992 y0.996 y0.999 y0.998 y0.999
0.0338 0.0802 0.0233 0.0222 0.0221 1.6113 0.0467 1.4920 0.0283 0.0246 0.0218 0.0164 0.0130 0.2521 0.1493 0.1360 0.0027 0.0959 0.0616 0.0699 0.0598 0.0184
0.8635 0.8484 0.8810 0.9383 0.9017 1.6838 0.8549 2.5854 0.7560 0.6438 0.8148 0.7265 0.9799 0.9122 0.8641 0.6208 0.6730 0.8404 0.7667 0.6512 0.8818 0.6308
0.995 0.998 0.999 0.996 0.997 0.963 0.999 0.971 0.971 0.988 0.993 0.999 0.916 0.985 0.995 0.998 0.906 0.942 0.998 0.974 0.950 0.998
Eq. Ž6. A4
16.398
B4
R
1.9063
0.991
1.7681
0.996
34.031 117.86
1.4547 1.6157
0.974 0.997
12.259 25.406
1.4645 2.2157
0.970 0.959
3.6059
Eq. Ž9. A6
B6
R
Eq. Ž15. A10
B10
R
0.6520 0.3317 0.6015 0.5858 0.5866 0.4998 0.3186 0.3612 0.3018 0.3448 0.2386 0.3921 0.1778 0.4700 0.4179 0.4779 0.1447 0.1599 0.2509 0.1924 0.1434 0.1262
y0.6855 y0.0947 y0.6923 y0.6670 y0.6760 y0.4424 y0.1601 y0.0962 y0.2114 y0.3393 y0.1615 y0.4570 y0.0758 y0.8999 y0.9041 y1.1771 y0.2858 y0.3251 y0.6426 y0.5260 y0.3225 y0.4805
0.992 0.978 0.996 0.996 0.994 0.997 0.952 0.997 0.989 0.979 0.940 0.983 0.998 0.993 0.997 0.998 0.999 0.998 0.992 0.999 0.997 0.999
0.5102 0.4381 0.3864 0.3698 0.3674 0.3474 0.3321 0.3027 0.2500 0.2373 0.2086 0.1984 0.1305 0.1032 0.0647 0.0568 0.0435 0.0276 0.0256 0.0224 0.0212 0.0083
y0.9275 0.3591 y0.5871 y0.4758 y0.4985 0.1267 0.3584 1.1676 0.3886 0.2256 0.4586 0.2361 0.7753 0.0516 0.2189 0.0431 0.5638 0.6816 0.0506 0.4934 0.6768 0.5378
0.992 0.991 0.994 0.996 0.994 0.995 0.975 0.993 0.999 0.994 0.988 0.986 0.982 0.998 0.998 0.998 0.977 0.984 0.995 0.997 0.991 1.000
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
Table 3 Linear regression analysis of the five-compaction equations. Rs linear regression coefficient. A n and Bn are the equations coefficients, where ns 2, 3, 4, 6 and 10
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
simplify Eq. Ž13. by eliminating D 0 . This can be done as follows: ln ln
ž ž
1 1yD 1 1yD
/ /
s A 9'P q ln
ž
1 1 y D0
s A10 'P q B10
/
Ž 14 . Ž 15 .
where A10 and B10 are constants characteristics of the powder w9x. Constants A10 and B10 can be used as parameters to characterize the powder behavior during pressing. It can be stated that parameter A10 Žinclination of the compressibility curve. provides the plastic deformation capacity of the powder in compaction. Thus, as A10 increases, the powder undergoes increasing plastic deformation during compaction. It is also postulated that parameter B10 Žinterception of the curve resulting from Eq. Ž15. at zero pressure. expresses the density in the absence of pressure. Table 1 presents the compacting equations that have been compared.
3. Linear regression analysis The compaction equations proposed by Balshin, Eq. Ž3., by Heckel, Eq. Ž5., by Kawakita, Eq. Ž6., by Ge, Eq. Ž9., and by Panelli and Ambrozio Filho, Eq. Ž15., can be rewritten in a general form as: f 1 Ž D . s A n f 2 Ž P . q Bn
Ž 16 .
where f 1Ž D ., f 2 Ž P ., A n and Bn are dependent on the equations, so that f 1Ž D . s 1rD, f 2 Ž P . s ln P, A 2 and B2 for Balshin’s equation, f 1Ž D . s lnw1rŽ1 y D .x, f 2 Ž P . s P, A 3 and B3 for Heckel’s equation, f 1Ž D . s DrŽ D y D 0 ., f 2 Ž P . s 1rP, A 4 and B4 for Kawakita’s equation, f 1Ž D . s log lnw1rŽ1 y D .x , f 2 Ž P . s log P, A 6 and B6 for Ge’s equation, f 1Ž D . s lnw1rŽ1 y D .x , f 2 Ž P . s 'P , A10 and B10 for Panelli and Ambrozio Filho’s equation. If the values of f 1Ž D . are plotted against the values of f 2 Ž P ., a straight-line relationship will be attained for each of the compaction equations. Thus, a quantitative comparison between the equations can be made by linear regression analysis. Different sets of experimental data on compaction are shown in Table 2. Eqs. Ž3., Ž5., Ž6., Ž9. and Ž15. tested each set of data and the results are shown in Table 3. Both tables are constructed ordering A10 values of Eq. Ž15. in a decreasing way from top to bottom of the tables .
4. Discussion Due to the non-availability of D 0 only a fraction of the data sets presented in Table 2 could be tested with Eq. Ž6. as shown in Table 3. The application of Eq. Ž6. is quite limited, because the equation can be used only when D 0 ,
259
the relative density of the loose powder at zero pressure, is known. The other four equations need only the parameters D, relative density of the compact and P, the applied pressure. The correlation coefficients, R, Žas shown in Table 3. for the five equations analyzed have a mean value and range of variation as follow: Eq. Ž3. mean: 0.986, range: 0.924 to 0.999; Eq. Ž5. mean: 0.979, range: 0.906 to 0.999; Eq. Ž6. mean: 0.981, range: 0.970 to 0.997; Eq. Ž9. mean: 0.989, range: 0.940 to 0.999; and Eq. Ž15. mean: 0.993, range: 0.975 to 0.999. From these values it can be concluded that Eq. Ž15. shows the highest mean correlation coefficient, R, and narrowest range of variation. Furthermore, the lowest value of R found by Eq. Ž15. is 0.975, which is higher than the lowest values of R when using ŽEqs. Ž3., Ž5., Ž6. and Ž9.. Nevertheless, all the equations provide reasonable fits and could be used. The equations can be evaluated taking into consideration that two processes are usually involved in the compaction: particle rearrangement followed by plastic deformation andror fragmentation. Plastic deformation is the main mechanism in pressing of ductile powders and fragmentation is the main mechanism in pressing of brittle powders. Generally, rearrangement occurs at pressures less than 1 MPa and densification due to rearrangement is dependent on powder characteristics, but corresponding to the first 5% to 10% decrease in porosity. At higher pressures, plastic deformation or fragmentation is the major form of densification for powders. A plot of f 1Ž D . versus f 2 Ž P . from the general Eq. Ž16. for the five compacting equations would give two straight lines, one for smaller pressures corresponding to particle rearrangement and the other for higher pressures corresponding to plastic deformation or fragmentation of the particles. Fig. 1a shows a qualitative compression curve by plotting the relative density D of the powder against compacting pressure P and Fig. 1b shows the two straight lines by plotting f 1Ž D . against f 2 Ž P .. As more than one mechanism can operate in the compaction, no single equation can perfectly represent the phenomenon over the complete range of pressures. On the other hand the rearrangement represents a small amount of the total densification, causing only a small error in the fit of the equation. The significance of the two constants that appear in all the equations can also be considered. As shown in the introduction, parameter A10 from Eq. Ž15. represents the powder’s ability to densify by plastic deformation. Plastic materials, especially soft metals, with the highest A values are at the top of Table 2, and the brittle materials Žceramics. with the lowest A10 values are at the bottom of the table. This aspect can also be seen in Fig. 2, where some plots of 'P versus lnŽ1rŽ1 y D.. using Eq. Ž15. are presented. Eq. Ž5. has a format similar to that of Eq. Ž15., showing that the parameter A 3 of that equation would also show the
260
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
Fig. 1. Ža. Qualitative compression curve of the relative density D of the powder against compacting pressure P and Žb. schematic representation of f 1Ž D . against f 2 Ž P ., when plastic deformation or fragmentation follows particle rearrangement.
powder’s ability to densify by plastic deformation. However, there is no similar order in the A 3 parameter, as shown by A10 in Eq. Ž15.. The other equations also do not present a parameter that can be used to characterize the powder in the same way as A10 from Eq. Ž15.. Parameter B has no significance in ŽEqs. Ž3., Ž6. and Ž9., because the equations are not defined at zero pressure. In Eqs. Ž5. and Ž15., parameters B3 and B10 can be used to calculate the relative density of the powder at zero pressure D 0 . Nevertheless, as pointed out earlier, at the beginning of compaction rearrangement must be considered ŽFig. 1b.. Therefore, D 0 calculated by using B3 and B10 , would be different from the relative density of loose powder Ž Dap ., determined by ASTM B212 or similar. In Heckel’s equation w5x, as he obtained B3 always somewhat higher than lnŽ1rŽ1 y Dap .., it was considered
that D 0 would represent the degree of packing achieved at low pressures due to the rearrangement processes. In the case of B10 of Eq. Ž15., their values are higher or lower than lnŽ1rŽ1 y Dap .. giving D 0 values higher or lower than Dap , or even negative D 0 . The obtained values depend on the rearrangement and on inaccuracies in the regression analysis due to the use of high pressures in the determination of the curves Žas discussed in a previous publication w9x.. Considering that rearrangement represents only a small fraction of total densification in compaction w2x, D 0 values calculated from Eq. Ž15. could be used as an approximation of the loose powder density Ž Dap .. Actually, for the real determination of Dap it would be necessary to obtain the rearrangement equation. The possibility to characterize the powders during compaction with the constants of Eq. Ž15. together with its
Fig. 2. Some compressibility curves.
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
good fit with experimental results are the main reasons to use this equation in modeling the compaction process.
5. Conclusions A new phenomenological compaction equation is compared with the four often used compaction equations. From this comparison it is possible to conclude that: Eq. Ž15., i.e., lnŽ1rŽ1 y D .. s A10 'P q B10 , is the one which best represents the density–pressure relationship for powders. Its linear correlation coefficient is almost close to the unity. Only Eq. Ž15. has constants representing powder characteristics during compaction: parameter A10 represents the ability of the powder to densify by plastic deformation and parameter B10 represents the density of the powder in the beginning of the compaction. However, due to the rearrangement mechanism at the beginning of compaction, B determined by the equation can be inaccurate.
6. List of symbols An Bn D D0
Constants for n varying from 1 to 10 Constants for n varying from 1 to 10 Relative density of compact at pressure P Relative powder density in absence of applied pressure
Dap K, k m P q q0 R
261
Relative apparent density Constants Constant Applied pressure wMPax Relative porosity of compact at pressure P Relative porosity of compact in absence of applied pressure Coefficient of linear regression
References w1x R. Ge, Int. J. Powder Metall. 27 Ž3. Ž1991. 216–221. w2x D.E. Niesz, Kona 14 Ž1996. 44–51. w3x G. Bockstiegel, Modern developments in powder metallurgy, Proc. Int. Powder Metall. Conf. 1 Ž1966. 155–187, Plenum, New York. w4x I. Shapiro, I.M. Kolthoff, J. Phys. Colloid. Chem. 51 Ž1947. 483– 493. w5x R.W. Heckel, Trans. Metall. Soc. AIME 221 Ž1961. 671–675. w6x K. Kawakita, K. Ludde, Powder Technol. 4 Ž1971. 61–68. ¨ w7x L. Parilak, ´ E. Dudrova, ´ PM ’94 1 Ž1994. 737–740. w8x R. Ge, Powder Metall. Sci. Technol. 6 Ž3. Ž1995. 20–24. w9x R. Panelli, F. Ambrozio Filho, Powder Metall. 41 Ž2. Ž1998. 131– 133. w10x R. Panelli, F. Ambrozio Filho, Mater. Sci. Forum 299r300 Ž1999. 463–469. w11x E.E. Walker, Trans. Faraday Soc. 19 Ž83. Ž1923. 73–82. w12x R.P. Mort, R. Sabia, D.E. Niesz, R.E. Riman, Powder Technol. 79 Ž1994. 111–119.
Letter
A study of a new phenomenological compacting equation Renato Panelli a , Francisco Ambrozio Filho b,) b
a Mahle Metal LeÕe Miba Sinterizados, Rua Brasılio ´ Luz 535, CEP 04746-901 Sao ˜ Paulo, Brazil Comissao e Nucleares, DiÕisao ˜ Nacional de Energia Nuclear, Instituto de Pesquisas Energeticas ´ ˜ MMM, C.P. 11049 Cidade UniÕersitaria, ´ CEP 05422-970 Sao ˜ Paulo, Brazil
Received 20 June 1999; received in revised form 13 December 1999; accepted 13 December 1999
Abstract A new phenomenological compacting equation has been proposed. The equation is lnŽ1rŽ1 y D .. s A'P q B where P is the applied pressure, D is the relative density of the compact, A is a parameter related to densification of the compact by plastic deformation, and B is a parameter related to powder density at the start of compaction. Linear regression analysis has been used to compare the new equation with the four compacting equations often used and proposed by Balshin, Heckel, Kawakita and Ge. The results show that the new equation gives linear correlation coefficients very close to unity. This equation, together with parameters A and B, permits improved evaluation of the compacting characteristics, compared to that performed by other equations. q 2001 Elsevier Science S.A. All rights reserved. Keywords: Compaction; Mathematical modeling
1. Introduction The powder compaction process plays an important role in the manufacture of a variety of products that include ceramics, metallic parts, fertilizers and pharmaceuticals w1x. In the case of ceramics and metallic parts, the microstructure of sintered powder compacts depend strongly on the quality of the green compact, which in turn depends on the behavior of the powder during compaction, that is, by the density–pressure relationship of the powder w2x. According to Bockstiegel w3x, the interest originally in density–pressure relationships stemmed from the practical problem of being able to predict the pressures required for achieving a certain density, later on, the interest seemed to shift more towards the analytical problem of finding an adequate but simple mathematical description for experimentally obtained density–pressure curves. Considerable effort has been made to characterize powders and their compaction behavior using a compacting equation w2,4–10x. In this paper the four often used compaction equations are compared, in a manner similar to that carried out by Ge
w8x in 1995. A new phenomenological compacting equation has also been included in this paper w9x.
2. Compacting equations Balshin attempted to correlate the relative density of compact powders with externally applied pressures and proposed the following equation: ED EP
Corresponding author. Fax: q55-11-816-9370. E-mail address: [email protected] ŽF. Ambrozio Filho..
D2
Ž 1.
P
ln P s y
A1 D
q B1
Ž 2.
where P is the applied pressure, D is the relative density of the compact and A1 , B1 are constants. Balshin called A1 the pressing modulus and considered it to be analogous to the Young’s modulus w2x. Balshin’s equation can be rewritten as: 1 D
)
sk
s A 2 ln P q B2
Ž 3.
Heckel, in 1961 w5x, considered the compaction of powders to be analogous to a first-order chemical reaction. The pores are the reactant and the densification of the bulk
0032-5910r01r$ - see front matter q 2001 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 0 . 0 0 2 0 7 - 2
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
256
is the product. The AkineticsB of the process may be described as a proportionality between the change in density with pressure and the pore fraction by: ED EP
sK Ž1 yD.
Ž 4.
This simplification done by Ge in his equation is not theoretically solved. In 1998, Panelli and Ambrozio Filho w9x, based upon observations of typical compaction curves, proposed the following differential equation to represent such curves: Eq
where Ž1 y D . is the pore fraction and K is a constant. The final form of Heckel’s equation can be written as follows: ln
ž
1 1yD
/
s A 3 P q B3
Ž 5.
where A 3 and B3 are constants. Heckel postulated that the constant A 3 gave a measure of the ability of the compact to densify by plastic deformation and constant B3 represented the degree of packing achieved at low pressures as a result of rearrangement before appreciable amounts of inter-particle bonding occurred. w6x derived a compaction equation Kawakita and Ludde ¨ from the observed relationship between applied pressure and volume or relative density. Kawakita obtained the following equation: D s D y D0
A4 P
q B4
Ž 6.
where A 4 and B4 are constants and D 0 the relative powder apparent density without application of pressure. Attention must be paid to the experimental determination of D 0 , and deviations from Kawakita’s equations are sometimes due to fluctuations in the measured value of D 0 , which is usually inaccurate. Ge w1x proposed a new differential equation for the pressing of powders: ED EP
sK
Ž1 y D. Dn Pm
Ž 7.
Kq s
EP
Ž 10 .
Pm
where q is relative porosity, P the applied pressure and K, m are constants. By integrating Eq. Ž10. from q Žporosity at P . to q0 Žporosity at P s 0. at applied pressures from 0 to P, a general equation is obtained: ln
q0 q
s A7
Pym q1 ym q 1
q B7 .
Ž 11 .
This equation has an integration constant B7 , which must be zero to satisfy the condition q0 s q when P s 0. Fitting experimental data to it by regression analysis tested Eq. Ž11.. Linear correlation coefficients, R, give values better than 0.99 when m s 0.5. So the value 0.5 was adopted for m in Eq. Ž10. and rewritten as: ln
q0 q
s A g'P
Ž 12 .
where q0 s Ž1 y D 0 . , q s Ž1 y D . and D 0 and D are the relative densities of the powder at zero pressure and of the compact at pressure P. Upon substitution, Eq. Ž12. becomes: ln
Ž 1 y D0 . s A g'P . Ž1 yD.
Ž 13 .
Eq. Ž13. has limited applicability owing to the necessity to determine the parameter D 0 . It is therefore important to
where n and m are constants. From Eq. Ž7., Ge obtained the following equation:
Ž 1 y D0 . log ln s A 5 log P q B5 Ž1 yD.
Ž 8.
Ge, in a second paper published in 1995 w8x, proposed a further simplified form of Eq. Ž8.. He observed that his equation was subject to the same level of imprecision as Kawakita’s equation, due to fluctuations in the measured value of D 0 . Ge considered that D 0 approached zero when total pressure reached zero, and his equation could be rewritten as: log ln
1
Ž1 yD.
s A 6 log P q B6
where A 6 and B6 are constants.
Ž 9.
Table 1 The compacting equations that have been compared. P sapplied pressure, Ds relative density of the compact, D 0 s relative apparent density of powder in absence of applied pressure, and A n and Bn are constant parameters; ns 2, 3, 4, 6 and 10 Author
Equation
Equation number
Reference
Balshin Heckel
1r Ds A 2 ln P q B2 lnŽ1rŽ1y D .. s A 3 P q B3 DrŽ Dy D 0 . s Ž A 4 r P .q B4 logwlnŽ1rŽ1y D .x s A 6 log P q B6 lnŽ1rŽ1y D ..
Ž3. Ž5.
w4x w5x
Ž6.
w6x
Ž9.
w8x
Ž15.
w9x
Kawakita Ge Panelli and Ambrozio Filho
s A10 'P q B10
Table 2 Experimental data on the compaction of various powders. P sapplied pressure ŽMPa. and Ds relative density of compact Powder Material no.
Reference Experimental data on compaction of powders
Silver bromide, 8–10 mesh, 958C
w4x
2
Ammonium nitrate
w11x
3
Silver bromide, 10–14 mesh, 258C
w4x
4
Silver bromide, 8–10 mesh, 258C
w4x
5
Silver bromide, 14–20 mesh, 258C
w4x
6
Atomized tin
w1x
7
Lead Žmetallic.
w11x
8
Atomized lead
w1x
9
Ammonium Chloride
w11x
10
Potassium chloride
w11x
11
Potassium nitrate
w11x
12
Sodium chloride
w11x
13
Barium nitrate
w11x
14
Electrolytic copper
w1x
15
Stainless steel
w1x
16
AISI M2
w9x
17
Calcium carbonate
w11x
18
Spray-dried alumina
w12x
19
M2q10 vol% NbC, mechanical alloyed 5h w10x
20
Tungsten carbide
w1x
21
Titanium carbide
w1x
22
Alumina
w9x
P 21 D 0.800 P 1.0 D 58.8 P 21 D 0.740 P 21 D 0.743 P 21 D 0.743 P 0 D 0.501 P 2.1 D 0.625 P 0 D 0.439 P 0.7 D 0.455 P 1.2 D 0.435 P 0.6 D 0.526 P 4.3 D 0.526 P 0.7 D 0.556 P 0 D 0.300 P 0 D 0.374 P 260.4 D 0.620 P 1.9 D 0.435 P 14 D 0.511 P 260 D 0.603 P 0 D 0.201 P 0 D 0.370 P 365.6 D 0.502
31 0.841 1.9 0.625 31 0.789 31 0.791 31 0.781 14.7 0.739 5.1 0.667 32.34 0.937 0.9 0.476 1.9 0.455 2.1 0.556 7.0 0.556 1.5 0.588 59.0 0.607 154.8 0.648 365.6 0.677 3.1 0.455 28 0.557 365 0.631 49.0 0.471 18.0 0.518 474.3 0.513
47 0.908 3.0 0.667 47 0.861 47 0.858 47 0.852 19.6 0.799 8.9 0.714 64.68 0.974 1.6 0.500 2.6 0.476 4.4 0.588 10.2 0.588 2.7 0.625 108 0.664 308.7 0.740 474.3 0.716 4.8 0.476 49 0.580 474 0.650 68.6 0.489 36.0 0.547 567.3 0.521
61 0.949 4.7 0.714 61 0.909 61 0.905 61 0.900 29.4 0.853 13.4 0.769 96.04 0.986 2.2 0.526 3.6 0.500 7.4 0.625 14.8 0.625 4.7 0.667 114 0.674 463.5 0.800 567.3 0.755 7.4 0.500 69 0.609 567 0.673 98.0 0.512 72.0 0.587 655.7 0.528
78 0.971 7.2 0.769 78 0.934 78 0.934 78 0.927 49.0 0.920 20.5 0.833 130.3 0.991 2.9 0.556 4.9 0.526 11.6 0.667 21.4 0.667 9.2 0.714 187 0.753 617.4 0.831 655.7 0.778 12.3 0.526 96 0.625 655 0.690 147 0.538 144 0.614
94 0.985 11.6 0.833 94 0.954 94 0.954 94 0.954 68.6 0.951 32.5 0.909 160.7 0.993 4.0 0.588 6.6 0.556 17.8 0.714 31.2 0.714 22.3 0.769 294 0.839 772.2 0.870
110 0.988 19.6 0.909 110 0.967 110 0.969 110 0.965 98.0 0.975
126 0.977 126 0.977 126 0.976 117.6 0.984
142 0.985 142 0.980 142 0.980 147.0 196.0 245.0 294.0 0.990 0.993 0.996 0.997
196.0 0.995 5.5 8.0 12.1 0.625 0.667 0.714 8.9 11.7 15.9 0.588 0.625 0.667 26.8 41.2 76.4 0.769 0.833 0.909 45.0 65.9 101.9 0.769 0.833 0.909 68.6 0.833 372 1000 0.874 0.964 926.1 0.890
18.7 0.769 21.3 0.714
32.0 0.833 29.1 0.769
63.7 0.909 42.1 0.833
76.9 0.909
21.9 41.2 0.556 0.588 140 206 0.650 0.673
86.0 178.2 0.625 0.667 278 343 441 447 481 503 0.694 0.695 0.713 0.714 0.716 0.722
196 245 0.559 0.576 288 576 0.650 0.699
294 392 490 588 0.586 0.609 0.627 0.641 864 0.719
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
1
257
258
Powder no.
Eq. Ž3. A2
B2
R
Eq. Ž5. A 3 Ž=10y2 .
B3
R
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
y0.1520 y0.2080 y0.1780 y0.1751 y0.1787 y0.1077 y0.1868 y0.0339 y02546 y0.3098 y0.1766 y0.2637 y0.1350 y0.2302 y0.2350 y0.3550 y0.1769 y0.1510 y0.2216 y0.2278 y0.1393 y0.1673
1.7017 1.7179 1.8672 1.8556 1.8752 1.5589 1.7751 1.1765 2.0759 2.3783 1.9003 2.3100 1.7343 2.5584 2.7125 3.5815 2.3772 2.3096 2.8919 3.0013 2.3225 2.9797
y0.983 y0.998 y0.990 y0.982 y0.985 y0.924 y0.988 y0.950 y0.993 y0.997 y0.983 y0.999 y0.992 y0.977 y0.995 y0.998 y0.994 y0.992 y0.996 y0.999 y0.998 y0.999
0.0338 0.0802 0.0233 0.0222 0.0221 1.6113 0.0467 1.4920 0.0283 0.0246 0.0218 0.0164 0.0130 0.2521 0.1493 0.1360 0.0027 0.0959 0.0616 0.0699 0.0598 0.0184
0.8635 0.8484 0.8810 0.9383 0.9017 1.6838 0.8549 2.5854 0.7560 0.6438 0.8148 0.7265 0.9799 0.9122 0.8641 0.6208 0.6730 0.8404 0.7667 0.6512 0.8818 0.6308
0.995 0.998 0.999 0.996 0.997 0.963 0.999 0.971 0.971 0.988 0.993 0.999 0.916 0.985 0.995 0.998 0.906 0.942 0.998 0.974 0.950 0.998
Eq. Ž6. A4
16.398
B4
R
1.9063
0.991
1.7681
0.996
34.031 117.86
1.4547 1.6157
0.974 0.997
12.259 25.406
1.4645 2.2157
0.970 0.959
3.6059
Eq. Ž9. A6
B6
R
Eq. Ž15. A10
B10
R
0.6520 0.3317 0.6015 0.5858 0.5866 0.4998 0.3186 0.3612 0.3018 0.3448 0.2386 0.3921 0.1778 0.4700 0.4179 0.4779 0.1447 0.1599 0.2509 0.1924 0.1434 0.1262
y0.6855 y0.0947 y0.6923 y0.6670 y0.6760 y0.4424 y0.1601 y0.0962 y0.2114 y0.3393 y0.1615 y0.4570 y0.0758 y0.8999 y0.9041 y1.1771 y0.2858 y0.3251 y0.6426 y0.5260 y0.3225 y0.4805
0.992 0.978 0.996 0.996 0.994 0.997 0.952 0.997 0.989 0.979 0.940 0.983 0.998 0.993 0.997 0.998 0.999 0.998 0.992 0.999 0.997 0.999
0.5102 0.4381 0.3864 0.3698 0.3674 0.3474 0.3321 0.3027 0.2500 0.2373 0.2086 0.1984 0.1305 0.1032 0.0647 0.0568 0.0435 0.0276 0.0256 0.0224 0.0212 0.0083
y0.9275 0.3591 y0.5871 y0.4758 y0.4985 0.1267 0.3584 1.1676 0.3886 0.2256 0.4586 0.2361 0.7753 0.0516 0.2189 0.0431 0.5638 0.6816 0.0506 0.4934 0.6768 0.5378
0.992 0.991 0.994 0.996 0.994 0.995 0.975 0.993 0.999 0.994 0.988 0.986 0.982 0.998 0.998 0.998 0.977 0.984 0.995 0.997 0.991 1.000
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
Table 3 Linear regression analysis of the five-compaction equations. Rs linear regression coefficient. A n and Bn are the equations coefficients, where ns 2, 3, 4, 6 and 10
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
simplify Eq. Ž13. by eliminating D 0 . This can be done as follows: ln ln
ž ž
1 1yD 1 1yD
/ /
s A 9'P q ln
ž
1 1 y D0
s A10 'P q B10
/
Ž 14 . Ž 15 .
where A10 and B10 are constants characteristics of the powder w9x. Constants A10 and B10 can be used as parameters to characterize the powder behavior during pressing. It can be stated that parameter A10 Žinclination of the compressibility curve. provides the plastic deformation capacity of the powder in compaction. Thus, as A10 increases, the powder undergoes increasing plastic deformation during compaction. It is also postulated that parameter B10 Žinterception of the curve resulting from Eq. Ž15. at zero pressure. expresses the density in the absence of pressure. Table 1 presents the compacting equations that have been compared.
3. Linear regression analysis The compaction equations proposed by Balshin, Eq. Ž3., by Heckel, Eq. Ž5., by Kawakita, Eq. Ž6., by Ge, Eq. Ž9., and by Panelli and Ambrozio Filho, Eq. Ž15., can be rewritten in a general form as: f 1 Ž D . s A n f 2 Ž P . q Bn
Ž 16 .
where f 1Ž D ., f 2 Ž P ., A n and Bn are dependent on the equations, so that f 1Ž D . s 1rD, f 2 Ž P . s ln P, A 2 and B2 for Balshin’s equation, f 1Ž D . s lnw1rŽ1 y D .x, f 2 Ž P . s P, A 3 and B3 for Heckel’s equation, f 1Ž D . s DrŽ D y D 0 ., f 2 Ž P . s 1rP, A 4 and B4 for Kawakita’s equation, f 1Ž D . s log lnw1rŽ1 y D .x , f 2 Ž P . s log P, A 6 and B6 for Ge’s equation, f 1Ž D . s lnw1rŽ1 y D .x , f 2 Ž P . s 'P , A10 and B10 for Panelli and Ambrozio Filho’s equation. If the values of f 1Ž D . are plotted against the values of f 2 Ž P ., a straight-line relationship will be attained for each of the compaction equations. Thus, a quantitative comparison between the equations can be made by linear regression analysis. Different sets of experimental data on compaction are shown in Table 2. Eqs. Ž3., Ž5., Ž6., Ž9. and Ž15. tested each set of data and the results are shown in Table 3. Both tables are constructed ordering A10 values of Eq. Ž15. in a decreasing way from top to bottom of the tables .
4. Discussion Due to the non-availability of D 0 only a fraction of the data sets presented in Table 2 could be tested with Eq. Ž6. as shown in Table 3. The application of Eq. Ž6. is quite limited, because the equation can be used only when D 0 ,
259
the relative density of the loose powder at zero pressure, is known. The other four equations need only the parameters D, relative density of the compact and P, the applied pressure. The correlation coefficients, R, Žas shown in Table 3. for the five equations analyzed have a mean value and range of variation as follow: Eq. Ž3. mean: 0.986, range: 0.924 to 0.999; Eq. Ž5. mean: 0.979, range: 0.906 to 0.999; Eq. Ž6. mean: 0.981, range: 0.970 to 0.997; Eq. Ž9. mean: 0.989, range: 0.940 to 0.999; and Eq. Ž15. mean: 0.993, range: 0.975 to 0.999. From these values it can be concluded that Eq. Ž15. shows the highest mean correlation coefficient, R, and narrowest range of variation. Furthermore, the lowest value of R found by Eq. Ž15. is 0.975, which is higher than the lowest values of R when using ŽEqs. Ž3., Ž5., Ž6. and Ž9.. Nevertheless, all the equations provide reasonable fits and could be used. The equations can be evaluated taking into consideration that two processes are usually involved in the compaction: particle rearrangement followed by plastic deformation andror fragmentation. Plastic deformation is the main mechanism in pressing of ductile powders and fragmentation is the main mechanism in pressing of brittle powders. Generally, rearrangement occurs at pressures less than 1 MPa and densification due to rearrangement is dependent on powder characteristics, but corresponding to the first 5% to 10% decrease in porosity. At higher pressures, plastic deformation or fragmentation is the major form of densification for powders. A plot of f 1Ž D . versus f 2 Ž P . from the general Eq. Ž16. for the five compacting equations would give two straight lines, one for smaller pressures corresponding to particle rearrangement and the other for higher pressures corresponding to plastic deformation or fragmentation of the particles. Fig. 1a shows a qualitative compression curve by plotting the relative density D of the powder against compacting pressure P and Fig. 1b shows the two straight lines by plotting f 1Ž D . against f 2 Ž P .. As more than one mechanism can operate in the compaction, no single equation can perfectly represent the phenomenon over the complete range of pressures. On the other hand the rearrangement represents a small amount of the total densification, causing only a small error in the fit of the equation. The significance of the two constants that appear in all the equations can also be considered. As shown in the introduction, parameter A10 from Eq. Ž15. represents the powder’s ability to densify by plastic deformation. Plastic materials, especially soft metals, with the highest A values are at the top of Table 2, and the brittle materials Žceramics. with the lowest A10 values are at the bottom of the table. This aspect can also be seen in Fig. 2, where some plots of 'P versus lnŽ1rŽ1 y D.. using Eq. Ž15. are presented. Eq. Ž5. has a format similar to that of Eq. Ž15., showing that the parameter A 3 of that equation would also show the
260
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
Fig. 1. Ža. Qualitative compression curve of the relative density D of the powder against compacting pressure P and Žb. schematic representation of f 1Ž D . against f 2 Ž P ., when plastic deformation or fragmentation follows particle rearrangement.
powder’s ability to densify by plastic deformation. However, there is no similar order in the A 3 parameter, as shown by A10 in Eq. Ž15.. The other equations also do not present a parameter that can be used to characterize the powder in the same way as A10 from Eq. Ž15.. Parameter B has no significance in ŽEqs. Ž3., Ž6. and Ž9., because the equations are not defined at zero pressure. In Eqs. Ž5. and Ž15., parameters B3 and B10 can be used to calculate the relative density of the powder at zero pressure D 0 . Nevertheless, as pointed out earlier, at the beginning of compaction rearrangement must be considered ŽFig. 1b.. Therefore, D 0 calculated by using B3 and B10 , would be different from the relative density of loose powder Ž Dap ., determined by ASTM B212 or similar. In Heckel’s equation w5x, as he obtained B3 always somewhat higher than lnŽ1rŽ1 y Dap .., it was considered
that D 0 would represent the degree of packing achieved at low pressures due to the rearrangement processes. In the case of B10 of Eq. Ž15., their values are higher or lower than lnŽ1rŽ1 y Dap .. giving D 0 values higher or lower than Dap , or even negative D 0 . The obtained values depend on the rearrangement and on inaccuracies in the regression analysis due to the use of high pressures in the determination of the curves Žas discussed in a previous publication w9x.. Considering that rearrangement represents only a small fraction of total densification in compaction w2x, D 0 values calculated from Eq. Ž15. could be used as an approximation of the loose powder density Ž Dap .. Actually, for the real determination of Dap it would be necessary to obtain the rearrangement equation. The possibility to characterize the powders during compaction with the constants of Eq. Ž15. together with its
Fig. 2. Some compressibility curves.
R. Panelli, F. Ambrozio Filhor Powder Technology 114 (2001) 255–261
good fit with experimental results are the main reasons to use this equation in modeling the compaction process.
5. Conclusions A new phenomenological compaction equation is compared with the four often used compaction equations. From this comparison it is possible to conclude that: Eq. Ž15., i.e., lnŽ1rŽ1 y D .. s A10 'P q B10 , is the one which best represents the density–pressure relationship for powders. Its linear correlation coefficient is almost close to the unity. Only Eq. Ž15. has constants representing powder characteristics during compaction: parameter A10 represents the ability of the powder to densify by plastic deformation and parameter B10 represents the density of the powder in the beginning of the compaction. However, due to the rearrangement mechanism at the beginning of compaction, B determined by the equation can be inaccurate.
6. List of symbols An Bn D D0
Constants for n varying from 1 to 10 Constants for n varying from 1 to 10 Relative density of compact at pressure P Relative powder density in absence of applied pressure
Dap K, k m P q q0 R
261
Relative apparent density Constants Constant Applied pressure wMPax Relative porosity of compact at pressure P Relative porosity of compact in absence of applied pressure Coefficient of linear regression
References w1x R. Ge, Int. J. Powder Metall. 27 Ž3. Ž1991. 216–221. w2x D.E. Niesz, Kona 14 Ž1996. 44–51. w3x G. Bockstiegel, Modern developments in powder metallurgy, Proc. Int. Powder Metall. Conf. 1 Ž1966. 155–187, Plenum, New York. w4x I. Shapiro, I.M. Kolthoff, J. Phys. Colloid. Chem. 51 Ž1947. 483– 493. w5x R.W. Heckel, Trans. Metall. Soc. AIME 221 Ž1961. 671–675. w6x K. Kawakita, K. Ludde, Powder Technol. 4 Ž1971. 61–68. ¨ w7x L. Parilak, ´ E. Dudrova, ´ PM ’94 1 Ž1994. 737–740. w8x R. Ge, Powder Metall. Sci. Technol. 6 Ž3. Ž1995. 20–24. w9x R. Panelli, F. Ambrozio Filho, Powder Metall. 41 Ž2. Ž1998. 131– 133. w10x R. Panelli, F. Ambrozio Filho, Mater. Sci. Forum 299r300 Ž1999. 463–469. w11x E.E. Walker, Trans. Faraday Soc. 19 Ž83. Ž1923. 73–82. w12x R.P. Mort, R. Sabia, D.E. Niesz, R.E. Riman, Powder Technol. 79 Ž1994. 111–119.