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Constitutive modeling of spark-plasma sintering of conductive materials
Scripta Materialia 55 (2006) 1175–1178 www.actamat-journals.com
Constitutive modeling of spark-plasma sintering of conductive materials Eugene Olevskya,* and Ludo Froyenb a
San Diego State University, Powder Technology Laboratory, 5500 Campanile Dr., San Diego, CA 92182-1323, USA b Katholieke Universiteit Leuven, Department of Metallurgy and Materials Engineering, Kasteelpark Arenberg 44, B-3001 Leuven, Belgium Received 17 May 2006; revised 5 July 2006; accepted 7 July 2006 Available online 18 September 2006
A constitutive model for spark-plasma sintering (SPS) taking into consideration various mechanisms of material transport is developed. The contributions of sintering stress (surface tension), external load, and electromigration to sintering shrinkage are jointly analyzed. It is shown that electromigration-related material flux can be a significant component of the electric-current-accelerated diffusion. A spark-plasma sintering mechanism transport map is designed for aluminum powder. The results of modeling agree satisfactorily with the experimental data on Al powder’s SPS in terms of shrinkage kinetics. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Electromigration; Diffusion; Powder consolidation
Spark-plasma sintering (SPS) [1], also known as electric-discharge sintering [2,3] or Field-Assisted Sintering [4] is a promising technique of powder consolidation. This manufacturing approach involves rapid heating of powder by electric current with simultaneous application of external pressure. Numerous experimental investigations1 point to the ability of SPS to render highly-dense powder products with the potential of grain size retention. The latter ability is of significance for the consolidation of nano-powder materials [1], where the grain growth is one of the major problems. Due to the complex nature of various phenomena involved in SPS, only few modeling attempts have been undertaken until presently. This includes contributions of Munir, Anselmi-Tamburini, Groza, Zavaliangos with co-workers [5–7] and others’. The conducted theoretical studies are mostly reduced to the modeling of temperature and electric current density distributions [3]. In practically all of the publications the role of electrical field is narrowed down to the generation of Joule heat, which thereby reduces the theoretical framework, required for the description of shrinkage and grain growth, to the existing constitutive models of powder consolidation [8,9]. However, as indicated by a number of experimental studies [1], there exists a direct contribu* Corresponding author. E-mail: [email protected] 1 An ISI Web of KnowledgeSM survey indicates about 1000 archival journal publications related to SPS.
tion of electric current to the diffusion mass transport leading to densification. This can substantially amend the materials’ behavior during SPS. The present work describes the results of a preliminary study in this direction. The developed model pursues the purpose of outlining a concept of a combined account of various material transport mechanisms in electric-current-assisted sintering, omitting a number of factors (such as spatial nonuniformities, different sources of grain growth, role of surface diffusion, phase transformations, possible (still debatable) plasma formation, presence of surface oxides, ‘‘electron wind’’, etc). We consider two major components of densification-contributing mass transfer during SPS: (i) grainboundary diffusion and (ii) power-law creep. The driving sources for these material transport mechanisms are: (i) externally applied load, (ii) sintering stress (surface tension), and (iii) steady-state electromigration (electric field contribution to diffusion). Similarly to sintering, following Hummel [10], electromigration is a result of diffusion, but in this case it is driven by an electric field. The commonness of physical mechanisms allows the combined description of both phenomena and their analysis in the same framework. The main driving forces of the diffusion are high levels of grain–pore and grain–grain interface areas in a material (electron wind that causes a drift of atoms by momentum transfer [11] for the purpose of this work is considered to be negligible).
1359-6462/$ - see front matter Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.07.009
1176
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178
Methodologically, the cases of both free sintering and sintering under load are rather important for distinguishing the unique contributions of the three above mentioned factors influencing material behavior under SPS. It is therefore natural to use the earlier developed framework for modeling the grain-boundary diffusion in sintering under applied pressure (see Olevsky and co-workers [12]) as the theoretical background for the consideration of SPS. Below we derive a simplified micro-mechanical model for a powder compact of simple-packed, rectangular grains with semi-axes a and c and elliptical pores located at the grain quadra-junctions whose unit cell is shown in Figure 1. The maximum and minimum curvature radii ra and rc of the elliptical pore contour are defined as: ra ¼
c2p ; ap
rc ¼
a2p cp
ð1Þ
where ap and cp are the pores’ semi-axes. Based on the results of Olevsky and co-workers [12], one can obtain the following relationship describing stresses in the x-direction: 3a 1 1 / 3 sin þ ðc þ c Þ y2 r þ rx ¼ x p 2c2 rc c 2 2c3 3 / 1 1 3 x ðc þ cp Þ sin þa ð2Þ r 2c 2 2 rc 2c where a is the surface tension, / is the dihedral angle, a x -effective (far-field) external is the grain semi-axis; r x is negative). stress in the x-direction (compressive r cþc x c p is a local stress on the grain boundary; Parameter r cþcp is the stress concentration factor. c The flux of matter ~ J caused by the grain boundary diffusion is determined by Nernst–Einstein equation [13] including the chemical potential gradient along the grain boundaries due to the aforementioned normal stresses and the electromigration: ~ ~ ð3Þ E þ C r rr J ¼ CE~ Here ~ E is the component of the electric field in the ~ is the gradient tangent plane of the grain boundary, rr d D of stresses normal to the grain boundary, C r ¼ gbkT gb , where Dgb is the coefficient of the grain boundary diffuapplied load y
cp c
electric current O
σx(y)
x
grains σy(x) a
sion, dgb is the grain boundary thickness, k is the Boltzman’s constant, T is the absolute temperature. Parameter CE is determined by Blech’s formula [14]: dgb Dgb Z eq ð4Þ CE ¼ XkT where X is the atomic volume, Z* is the valence of a migrating ion, and eq is the electron charge (the product Z*eq is called ‘‘the effective charge’’). In the case of the structure shown in Figure 1: dgb Dgb 1 U orx Z þ ¼ e J gb ð5Þ q y X l kT oy Here U and l are the electric potential and the characteristic length along the electric field. Substituting Eq. (2) in Eq. (5), and taking into consideration the relationship between the flux and the strain rate e_ gbx (_egbx should be negative in case of shrinkage in x-direction) in the orthogonal direction [12]: e_ gbx ¼
Despite the latter relationship accounts for the structure anisotropy, for simplicity, let us assume the equi-axiality of the pore–grain structure (reserving the analysis of the impact of the pore–grain texture for future investigations). In this case: dgb Dgb X e_ gbx ¼ kT ðG þ rp Þ2 Z eq U 3a 1 1 G þ rp x þ ð8Þ r G rp 2G X l G2 Here G = a = c is the grain size, rp = ap = cp is the pore radius, and the dihedral angle is assumed / = 60°. Eq. (8) incorporates the contributions of the three factors to the overall densification under SPS. This includes _ stgbx , e_ dl the shrinkage rate components e_ em gbx , e gbx due to electromigration, surface tension, and due to the contribux to the diffusion, respectively: tion of the external load r
Figure 1. Representative unit cell including rectangular grain and ellipsoidal pores in quadra-junctions. The pore–grain structure is subjected to a simultaneous action of external load, surface tension, and electromigration. The electric field is macroscopically unidirectional, and can branch locally.
dgb Dgb Z eq U ; kT ðG þ rp Þ2 l 3dgb Dgb X a 1 1 ¼ ; 2 kT ðG þ rp Þ G rp 2G
e_ em gbx ¼
ap
pores
ð6Þ
We obtain the following expression for the shrinkage kinetics (for SPS processing conditions – pressing in a rigid die, the axial component of the shrinkage rate equals to the overall volume shrinkage rate): dgb Dgb X e_ gbx ¼ kT ða þ ap Þðc þ cp Þ Z eq U 3a 1 1 / c þ cp x þ sin r ð7Þ c rc c 2 X l c2
e_ stgbx electric field
J gb X y ðcÞ 2ða þ ap Þ c
e_ dl gbx ¼
x dgb Dgb X r kT ðG þ rp Þ G2
ð9Þ
The derived expressions for the components of the axial strain rate are valid for SPS pressing schematics due to the proximity to zero of the equivalent Poisson’s ratio’s viscous analogy (see [12]).
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178
SPS is a process involving hot deformation of a powder under pressure. Under these conditions, in accord with Ashby deformation maps [15], power-law (dislocation) creep also should be one of the important mechanisms of the material flow. Based on the continuum theory of sintering [9]: 1 m1 x ¼ AW u_ecrx þ w u ð_ecrx þ e_ cry Þ þ P L ð10Þ r 3 where u and w are normalized shear and bulk viscosity moduli depending on porosity h; e_ crx and e_ cry are components of the shrinkage rate corresponding to the mechanism of power-law creep; PL is the effective sintering stress depending on porosity h; A and m are power-law creep frequency factor and power-law creep exponent, respectively; and W is the equivalent effective strain rate, which in the considered case is given by expression: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 uj_ecrx e_ cry j þ wð_ecrx e_ cry Þ ð11Þ W ¼ pffiffiffiffiffiffiffiffiffiffiffi 3 1h For SPS pressing mode (pressing in a rigid die), the deformation occurs only in the x-direction, while e_ cry ¼ 0. Therefore (also taking into account the negative signs of the shrinkage rate and the compressive axial stress from (10) and (11) and employing the following relationships for u, w, and PL [9]): 2
u ¼ ð1 hÞ ; w ¼
2 ð1 hÞ3 3a 2 ð1 hÞ ; PL ¼ 3 2G h
ð12Þ
we obtain: ( e_ crx ¼
3h 2
32
3a 2 x ð1 hÞ r 2G
Að1 hÞ
5 2
)m1 ð13Þ
Following Ashby [15], the power-law frequency factor can be written as: Qcr ð14Þ A ¼ A0 exp RT In our model framework, the total shrinkage rate during SPS is equal to the superposition of the shrinkage rates corresponding to the grain-boundary diffusion and power-law creep mechanisms: dgb Dgb X e_ x ¼ e_ gbx þ e_ crx ¼ kT ðG þ rp Þ2 Z eq U 3a 1 1 G þ rp x þ r G rp 2G X l G2 ( 3 )m1 5 3h 2 3a 2 x ð1 hÞ r ð15Þ Að1 hÞ2 2 2G In Eq. (15) the power-law creep component is related to the macroscopic powder volume subjected to SPS (this component, in particular, depends on porosity). The macroscopic homogenization of the component corresponding to the grain-boundary diffusion in a simplified way can be achieved by introducing the following formula [16] for the dependence of the pore radius on the grain size and porosity: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ rp ¼ G 3 h=½6ð1 hÞ
1177
Fortunately, the theory of electromigration is quite well-developed (however, never applied in models of sintering, as mentioned before), therefore the necessary model parameters and stress–strain assessments are readily available for a number of materials. The material parameters related to electromigration, grain-boundary diffusion, and power-law creep in aluminum powder are given in Table 1. Eq. (15) allows the direct comparison of the contribution of different mechanisms to the SPS shrinkage kinetics. Our analysis indicates that the external load’s contribution to the grain-boundary diffusion (corresponding to e_ dl gbx Þ is much smaller than the contribution of other factors. The calculations are based on the materials data V , from Table 1; in addition it is assumed that Ul ¼ 417 m x ¼ 283 MPa. The calculations have been T = 673 K, r conducted for different grain sizes: (a) G = 40 lm, (b) G = 1 lm, and (c) G = 100 nm. Our analysis indicates how the magnitude of the contributions of three different factors change with changing the grain size. For conventional, micron-size powders (G = 40 lm), power-law is the dominant mechanism of material transport. For smaller porosities and ultrafine powders, (G < 1 lm), the electromigration plays the dominant role, while for smaller porosity in the nano-powder range sintering stress dominates. The above-mentioned interplay (see the densification map in Fig. 2) is possible when porosity is lower than 30% (for larger void volume fractions, power-law creep is always the main mechanism of the material flow). For porosities lower than 30%, both surface tension (sintering stress) and electromigration can become the main contributors to shrinkage depending on the average grain size. Here we should note a possible limitation of this preliminary model assessment. Since both locally and macroscopically the generation of Joule heat (and hence, temperature distribution) cannot be decoupled from the electric current (electric field intensity) [1], parameter Ul should evolve with densification. Therefore, for very small porosities in the electromigration-dominating zone, the ultimate collapse of voids may require externally applied load as the primary factor. Future, more advanced SPS models should include a heat-balance equation along with the developed constitutive relationship for densification. Based on the continuity equation [9] (for the considered pressing schematics, e_ y ¼ 0): Table 1. Material properties for aluminum Effective charge Atomic volume Surface tension Grain-boundary diffusion frequency factor Activation energy for grain-boundary diffusion Activation energy for power-law creep Power-law creep frequency factor Power-law creep exponent a
Z*qe X a ½dgb Dgb 0 a
8E 019 C [13] 1.66E 29 m3 [13,17] 1.12 J/m2 [17] 3.00E 14 m3/s [17]
Qgb a
60 KJ/mol [17]
Qcr
1.20E + 02 KJ/mol [15]
A0
566 MPa/sm [15]
m
2.27E 01 [15]
[dgbDgb] = [dgbDgb]0exp (Qgb/RT) [17].
1178
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178 0.35
898
0.3
798
0.35 0.30 Temperature Porosity-Model
Temperature, K
0.2
external load
0.15 0.1
surface tension
1.00E-07
0.20 598 0.15 498 0.10
1.00E-06
1.00E-05
0.00 0
Figure 2. Densification map for aluminum powder, x ¼ 283 MPa. T = 673 K, r
200
400
600
800
1000
Time, s
1.00E-04
Grain Size, m
h_ ¼ ð1 hÞð_ex þ e_ y Þ ¼ ð1 hÞð_ex Þ
0.05
298
electromigration
0.25
Porosity-Experiment
398
0.05 0 1.00E-08
698
Porosity
Porosity
0.25
U l
V ¼ 417 m ,
ð17Þ
and taking into account Eqs. (14)–(16), the following relationship for the densification kinetics can be obtained:
8 < X h_ ¼ ð1 hÞ dgb Dgb p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : kT
3 G þ G h=½6ð1 hÞ ( " # Z eq U 3a 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ X l G G 3 h=½6ð1 hÞ 2G pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) G þ G 3 h=½6ð1 hÞ rx G2 9 ( 3 , )m1 = 5 3h 2 3a Qcr 2 x ð1 hÞ2 ð1 hÞ r A0 exp ; RT 2 2G
ð18Þ The latter differential equation has been solved numerically by the Runge–Kutta method of the fourth order. The results of the calculations are shown in Figure 3. The temperature and pressure time cyclograms used in the calculations replicate the conditions of the experiment on SPS of an aluminum powder [18,19] with the average particle size of 55 lm. The applied field is V (Joule heat generation balance accepted to be of 500 m – based estimation), the pressure is constant and equal to 23.5 MPa. Material parameters correspond to Table 1. The experimental data on the shrinkage kinetics are also shown in Figure 3 and indicate satisfactory agreement with the modeling results. A constitutive model of spark-plasma sintering taking into consideration the direct contribution of electric current into grain-boundary diffusion is developed. The calculated densification map for an alumina powder indicates three possible porosity – grain size domains of dominancy of one of the three considered driving factors of material transport: externally applied load, surface tension, and electromigration. The power-law creep induced by an external stress always dominates for higher porosities. For lower porosity values, the electromigration can become the dominant mechanism, and for smaller particle sizes and low porosity, surface tension is the main driving factor for densification. For very small porosities in the electromigration-dominating zone, the ultimate collapse of voids may require externally applied load as the primary factor.
Figure 3. Porosity kinetics during SPS of aluminum powder. Comparison of the developed model with experimental data of Xie et al. [18,19].
The developed model has been utilized to calculate the shrinkage kinetics of an aluminum powder. The results of the calculations agree satisfactorily with the available experimental data. The support of the Research Council of the Catholic University Leuven is gratefully appreciated. The support of the National Science Foundation, Division for Materials Research (grant DMR-0315290) is gratefully appreciated. [1] Z.A. Munir, U. Anselmi-Tamburini, M. Ohyanagi, J. Mater. Sci. 41 (2006) 763–777. [2] I.M. Fedorchenko, G.L. Burenkov, A.I. Raichenko, A.F. Khirienko, V.M. Kriachek, Doklady Akademii Nauk SSSR 236 (3) (1977) 585–588. [3] A. Raichenko, E. Chernikova, E. Olevsky, J. Phys. IV C7 (v.3) (1993) 1235–1239. [4] L.A. Stanciu, V.Y. Kodash, J.R. Groza, Metall. Mater. Trans. A 32 (10) (2001) 2633–2638. [5] U. Anselmi-Tamburini, S. Gennari, J.E. Garay, Z.A. Munir, Mater. Sci. Eng. A 394 (1–2) (2005) 139–148. [6] J.R. Groza, A. Zavaliangos, Mater. Sci. Eng. A 287 (2) (2000) 171–177. [7] B. McWilliams, A. Zavaliangos, K.C. Cho, R.J. Dowding, JOM 58 (4) (2006) 67–71. [8] J. Besson, M. Abouaf, J. Am. Ceram. Soc. 75 (8) (1992) 2165–2172. [9] E. Olevsky, Mater. Sci. Eng. R. 23 (1998) 41–100. [10] R.E. Hummel, Int. Mater. Rev. 39 (3) (1994) 97–111. [11] W.W. Mullins, Metall. Mater. Trans. 26A (1995) 1917– 1929. [12] E.A. Olevsky, B. Kushnarev, A. Maximenko, V. Tikare, M. Braginsky, Philos. Mag. 85 (2005) 2123–2146. [13] M. Scherge, C.L. Bauer, W.W. Mullins, Acta Metall. Mater. 43 (9) (1995) 3525–3538. [14] I.A. Blech, C. Herring, Appl. Phys. Lett. 29 (3) (1976) 131–133. [15] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps, Pergamon Press, NY, 1982. [16] F.B. Swinkels, M.F. Ashby, Acta Metall. 29 (2) (1981) 259–281. [17] R.M. German, Sintering Theory and Practice, John Wiley & Sons, NY, 1996. [18] G. Xie, O. Ohashi, T. Yoshioka, M. Song, K. Mitsuishi, H. Yasuda, K. Furuya, T. Noda, Mater. Trans. 42 (9) (2001) 1846–1849. [19] G. Xie, O. Ohashi, N. Yamaguchi, Trans. Mater. Res. Soc. Jpn 27 (4) (2002) 743–746.
Constitutive modeling of spark-plasma sintering of conductive materials Eugene Olevskya,* and Ludo Froyenb a
San Diego State University, Powder Technology Laboratory, 5500 Campanile Dr., San Diego, CA 92182-1323, USA b Katholieke Universiteit Leuven, Department of Metallurgy and Materials Engineering, Kasteelpark Arenberg 44, B-3001 Leuven, Belgium Received 17 May 2006; revised 5 July 2006; accepted 7 July 2006 Available online 18 September 2006
A constitutive model for spark-plasma sintering (SPS) taking into consideration various mechanisms of material transport is developed. The contributions of sintering stress (surface tension), external load, and electromigration to sintering shrinkage are jointly analyzed. It is shown that electromigration-related material flux can be a significant component of the electric-current-accelerated diffusion. A spark-plasma sintering mechanism transport map is designed for aluminum powder. The results of modeling agree satisfactorily with the experimental data on Al powder’s SPS in terms of shrinkage kinetics. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Electromigration; Diffusion; Powder consolidation
Spark-plasma sintering (SPS) [1], also known as electric-discharge sintering [2,3] or Field-Assisted Sintering [4] is a promising technique of powder consolidation. This manufacturing approach involves rapid heating of powder by electric current with simultaneous application of external pressure. Numerous experimental investigations1 point to the ability of SPS to render highly-dense powder products with the potential of grain size retention. The latter ability is of significance for the consolidation of nano-powder materials [1], where the grain growth is one of the major problems. Due to the complex nature of various phenomena involved in SPS, only few modeling attempts have been undertaken until presently. This includes contributions of Munir, Anselmi-Tamburini, Groza, Zavaliangos with co-workers [5–7] and others’. The conducted theoretical studies are mostly reduced to the modeling of temperature and electric current density distributions [3]. In practically all of the publications the role of electrical field is narrowed down to the generation of Joule heat, which thereby reduces the theoretical framework, required for the description of shrinkage and grain growth, to the existing constitutive models of powder consolidation [8,9]. However, as indicated by a number of experimental studies [1], there exists a direct contribu* Corresponding author. E-mail: [email protected] 1 An ISI Web of KnowledgeSM survey indicates about 1000 archival journal publications related to SPS.
tion of electric current to the diffusion mass transport leading to densification. This can substantially amend the materials’ behavior during SPS. The present work describes the results of a preliminary study in this direction. The developed model pursues the purpose of outlining a concept of a combined account of various material transport mechanisms in electric-current-assisted sintering, omitting a number of factors (such as spatial nonuniformities, different sources of grain growth, role of surface diffusion, phase transformations, possible (still debatable) plasma formation, presence of surface oxides, ‘‘electron wind’’, etc). We consider two major components of densification-contributing mass transfer during SPS: (i) grainboundary diffusion and (ii) power-law creep. The driving sources for these material transport mechanisms are: (i) externally applied load, (ii) sintering stress (surface tension), and (iii) steady-state electromigration (electric field contribution to diffusion). Similarly to sintering, following Hummel [10], electromigration is a result of diffusion, but in this case it is driven by an electric field. The commonness of physical mechanisms allows the combined description of both phenomena and their analysis in the same framework. The main driving forces of the diffusion are high levels of grain–pore and grain–grain interface areas in a material (electron wind that causes a drift of atoms by momentum transfer [11] for the purpose of this work is considered to be negligible).
1359-6462/$ - see front matter Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.07.009
1176
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178
Methodologically, the cases of both free sintering and sintering under load are rather important for distinguishing the unique contributions of the three above mentioned factors influencing material behavior under SPS. It is therefore natural to use the earlier developed framework for modeling the grain-boundary diffusion in sintering under applied pressure (see Olevsky and co-workers [12]) as the theoretical background for the consideration of SPS. Below we derive a simplified micro-mechanical model for a powder compact of simple-packed, rectangular grains with semi-axes a and c and elliptical pores located at the grain quadra-junctions whose unit cell is shown in Figure 1. The maximum and minimum curvature radii ra and rc of the elliptical pore contour are defined as: ra ¼
c2p ; ap
rc ¼
a2p cp
ð1Þ
where ap and cp are the pores’ semi-axes. Based on the results of Olevsky and co-workers [12], one can obtain the following relationship describing stresses in the x-direction: 3a 1 1 / 3 sin þ ðc þ c Þ y2 r þ rx ¼ x p 2c2 rc c 2 2c3 3 / 1 1 3 x ðc þ cp Þ sin þa ð2Þ r 2c 2 2 rc 2c where a is the surface tension, / is the dihedral angle, a x -effective (far-field) external is the grain semi-axis; r x is negative). stress in the x-direction (compressive r cþc x c p is a local stress on the grain boundary; Parameter r cþcp is the stress concentration factor. c The flux of matter ~ J caused by the grain boundary diffusion is determined by Nernst–Einstein equation [13] including the chemical potential gradient along the grain boundaries due to the aforementioned normal stresses and the electromigration: ~ ~ ð3Þ E þ C r rr J ¼ CE~ Here ~ E is the component of the electric field in the ~ is the gradient tangent plane of the grain boundary, rr d D of stresses normal to the grain boundary, C r ¼ gbkT gb , where Dgb is the coefficient of the grain boundary diffuapplied load y
cp c
electric current O
σx(y)
x
grains σy(x) a
sion, dgb is the grain boundary thickness, k is the Boltzman’s constant, T is the absolute temperature. Parameter CE is determined by Blech’s formula [14]: dgb Dgb Z eq ð4Þ CE ¼ XkT where X is the atomic volume, Z* is the valence of a migrating ion, and eq is the electron charge (the product Z*eq is called ‘‘the effective charge’’). In the case of the structure shown in Figure 1: dgb Dgb 1 U orx Z þ ¼ e J gb ð5Þ q y X l kT oy Here U and l are the electric potential and the characteristic length along the electric field. Substituting Eq. (2) in Eq. (5), and taking into consideration the relationship between the flux and the strain rate e_ gbx (_egbx should be negative in case of shrinkage in x-direction) in the orthogonal direction [12]: e_ gbx ¼
Despite the latter relationship accounts for the structure anisotropy, for simplicity, let us assume the equi-axiality of the pore–grain structure (reserving the analysis of the impact of the pore–grain texture for future investigations). In this case: dgb Dgb X e_ gbx ¼ kT ðG þ rp Þ2 Z eq U 3a 1 1 G þ rp x þ ð8Þ r G rp 2G X l G2 Here G = a = c is the grain size, rp = ap = cp is the pore radius, and the dihedral angle is assumed / = 60°. Eq. (8) incorporates the contributions of the three factors to the overall densification under SPS. This includes _ stgbx , e_ dl the shrinkage rate components e_ em gbx , e gbx due to electromigration, surface tension, and due to the contribux to the diffusion, respectively: tion of the external load r
Figure 1. Representative unit cell including rectangular grain and ellipsoidal pores in quadra-junctions. The pore–grain structure is subjected to a simultaneous action of external load, surface tension, and electromigration. The electric field is macroscopically unidirectional, and can branch locally.
dgb Dgb Z eq U ; kT ðG þ rp Þ2 l 3dgb Dgb X a 1 1 ¼ ; 2 kT ðG þ rp Þ G rp 2G
e_ em gbx ¼
ap
pores
ð6Þ
We obtain the following expression for the shrinkage kinetics (for SPS processing conditions – pressing in a rigid die, the axial component of the shrinkage rate equals to the overall volume shrinkage rate): dgb Dgb X e_ gbx ¼ kT ða þ ap Þðc þ cp Þ Z eq U 3a 1 1 / c þ cp x þ sin r ð7Þ c rc c 2 X l c2
e_ stgbx electric field
J gb X y ðcÞ 2ða þ ap Þ c
e_ dl gbx ¼
x dgb Dgb X r kT ðG þ rp Þ G2
ð9Þ
The derived expressions for the components of the axial strain rate are valid for SPS pressing schematics due to the proximity to zero of the equivalent Poisson’s ratio’s viscous analogy (see [12]).
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178
SPS is a process involving hot deformation of a powder under pressure. Under these conditions, in accord with Ashby deformation maps [15], power-law (dislocation) creep also should be one of the important mechanisms of the material flow. Based on the continuum theory of sintering [9]: 1 m1 x ¼ AW u_ecrx þ w u ð_ecrx þ e_ cry Þ þ P L ð10Þ r 3 where u and w are normalized shear and bulk viscosity moduli depending on porosity h; e_ crx and e_ cry are components of the shrinkage rate corresponding to the mechanism of power-law creep; PL is the effective sintering stress depending on porosity h; A and m are power-law creep frequency factor and power-law creep exponent, respectively; and W is the equivalent effective strain rate, which in the considered case is given by expression: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 uj_ecrx e_ cry j þ wð_ecrx e_ cry Þ ð11Þ W ¼ pffiffiffiffiffiffiffiffiffiffiffi 3 1h For SPS pressing mode (pressing in a rigid die), the deformation occurs only in the x-direction, while e_ cry ¼ 0. Therefore (also taking into account the negative signs of the shrinkage rate and the compressive axial stress from (10) and (11) and employing the following relationships for u, w, and PL [9]): 2
u ¼ ð1 hÞ ; w ¼
2 ð1 hÞ3 3a 2 ð1 hÞ ; PL ¼ 3 2G h
ð12Þ
we obtain: ( e_ crx ¼
3h 2
32
3a 2 x ð1 hÞ r 2G
Að1 hÞ
5 2
)m1 ð13Þ
Following Ashby [15], the power-law frequency factor can be written as: Qcr ð14Þ A ¼ A0 exp RT In our model framework, the total shrinkage rate during SPS is equal to the superposition of the shrinkage rates corresponding to the grain-boundary diffusion and power-law creep mechanisms: dgb Dgb X e_ x ¼ e_ gbx þ e_ crx ¼ kT ðG þ rp Þ2 Z eq U 3a 1 1 G þ rp x þ r G rp 2G X l G2 ( 3 )m1 5 3h 2 3a 2 x ð1 hÞ r ð15Þ Að1 hÞ2 2 2G In Eq. (15) the power-law creep component is related to the macroscopic powder volume subjected to SPS (this component, in particular, depends on porosity). The macroscopic homogenization of the component corresponding to the grain-boundary diffusion in a simplified way can be achieved by introducing the following formula [16] for the dependence of the pore radius on the grain size and porosity: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ rp ¼ G 3 h=½6ð1 hÞ
1177
Fortunately, the theory of electromigration is quite well-developed (however, never applied in models of sintering, as mentioned before), therefore the necessary model parameters and stress–strain assessments are readily available for a number of materials. The material parameters related to electromigration, grain-boundary diffusion, and power-law creep in aluminum powder are given in Table 1. Eq. (15) allows the direct comparison of the contribution of different mechanisms to the SPS shrinkage kinetics. Our analysis indicates that the external load’s contribution to the grain-boundary diffusion (corresponding to e_ dl gbx Þ is much smaller than the contribution of other factors. The calculations are based on the materials data V , from Table 1; in addition it is assumed that Ul ¼ 417 m x ¼ 283 MPa. The calculations have been T = 673 K, r conducted for different grain sizes: (a) G = 40 lm, (b) G = 1 lm, and (c) G = 100 nm. Our analysis indicates how the magnitude of the contributions of three different factors change with changing the grain size. For conventional, micron-size powders (G = 40 lm), power-law is the dominant mechanism of material transport. For smaller porosities and ultrafine powders, (G < 1 lm), the electromigration plays the dominant role, while for smaller porosity in the nano-powder range sintering stress dominates. The above-mentioned interplay (see the densification map in Fig. 2) is possible when porosity is lower than 30% (for larger void volume fractions, power-law creep is always the main mechanism of the material flow). For porosities lower than 30%, both surface tension (sintering stress) and electromigration can become the main contributors to shrinkage depending on the average grain size. Here we should note a possible limitation of this preliminary model assessment. Since both locally and macroscopically the generation of Joule heat (and hence, temperature distribution) cannot be decoupled from the electric current (electric field intensity) [1], parameter Ul should evolve with densification. Therefore, for very small porosities in the electromigration-dominating zone, the ultimate collapse of voids may require externally applied load as the primary factor. Future, more advanced SPS models should include a heat-balance equation along with the developed constitutive relationship for densification. Based on the continuity equation [9] (for the considered pressing schematics, e_ y ¼ 0): Table 1. Material properties for aluminum Effective charge Atomic volume Surface tension Grain-boundary diffusion frequency factor Activation energy for grain-boundary diffusion Activation energy for power-law creep Power-law creep frequency factor Power-law creep exponent a
Z*qe X a ½dgb Dgb 0 a
8E 019 C [13] 1.66E 29 m3 [13,17] 1.12 J/m2 [17] 3.00E 14 m3/s [17]
Qgb a
60 KJ/mol [17]
Qcr
1.20E + 02 KJ/mol [15]
A0
566 MPa/sm [15]
m
2.27E 01 [15]
[dgbDgb] = [dgbDgb]0exp (Qgb/RT) [17].
1178
E. Olevsky, L. Froyen / Scripta Materialia 55 (2006) 1175–1178 0.35
898
0.3
798
0.35 0.30 Temperature Porosity-Model
Temperature, K
0.2
external load
0.15 0.1
surface tension
1.00E-07
0.20 598 0.15 498 0.10
1.00E-06
1.00E-05
0.00 0
Figure 2. Densification map for aluminum powder, x ¼ 283 MPa. T = 673 K, r
200
400
600
800
1000
Time, s
1.00E-04
Grain Size, m
h_ ¼ ð1 hÞð_ex þ e_ y Þ ¼ ð1 hÞð_ex Þ
0.05
298
electromigration
0.25
Porosity-Experiment
398
0.05 0 1.00E-08
698
Porosity
Porosity
0.25
U l
V ¼ 417 m ,
ð17Þ
and taking into account Eqs. (14)–(16), the following relationship for the densification kinetics can be obtained:
8 < X h_ ¼ ð1 hÞ dgb Dgb p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : kT
3 G þ G h=½6ð1 hÞ ( " # Z eq U 3a 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ X l G G 3 h=½6ð1 hÞ 2G pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) G þ G 3 h=½6ð1 hÞ rx G2 9 ( 3 , )m1 = 5 3h 2 3a Qcr 2 x ð1 hÞ2 ð1 hÞ r A0 exp ; RT 2 2G
ð18Þ The latter differential equation has been solved numerically by the Runge–Kutta method of the fourth order. The results of the calculations are shown in Figure 3. The temperature and pressure time cyclograms used in the calculations replicate the conditions of the experiment on SPS of an aluminum powder [18,19] with the average particle size of 55 lm. The applied field is V (Joule heat generation balance accepted to be of 500 m – based estimation), the pressure is constant and equal to 23.5 MPa. Material parameters correspond to Table 1. The experimental data on the shrinkage kinetics are also shown in Figure 3 and indicate satisfactory agreement with the modeling results. A constitutive model of spark-plasma sintering taking into consideration the direct contribution of electric current into grain-boundary diffusion is developed. The calculated densification map for an alumina powder indicates three possible porosity – grain size domains of dominancy of one of the three considered driving factors of material transport: externally applied load, surface tension, and electromigration. The power-law creep induced by an external stress always dominates for higher porosities. For lower porosity values, the electromigration can become the dominant mechanism, and for smaller particle sizes and low porosity, surface tension is the main driving factor for densification. For very small porosities in the electromigration-dominating zone, the ultimate collapse of voids may require externally applied load as the primary factor.
Figure 3. Porosity kinetics during SPS of aluminum powder. Comparison of the developed model with experimental data of Xie et al. [18,19].
The developed model has been utilized to calculate the shrinkage kinetics of an aluminum powder. The results of the calculations agree satisfactorily with the available experimental data. The support of the Research Council of the Catholic University Leuven is gratefully appreciated. The support of the National Science Foundation, Division for Materials Research (grant DMR-0315290) is gratefully appreciated. [1] Z.A. Munir, U. Anselmi-Tamburini, M. Ohyanagi, J. Mater. Sci. 41 (2006) 763–777. [2] I.M. Fedorchenko, G.L. Burenkov, A.I. Raichenko, A.F. Khirienko, V.M. Kriachek, Doklady Akademii Nauk SSSR 236 (3) (1977) 585–588. [3] A. Raichenko, E. Chernikova, E. Olevsky, J. Phys. IV C7 (v.3) (1993) 1235–1239. [4] L.A. Stanciu, V.Y. Kodash, J.R. Groza, Metall. Mater. Trans. A 32 (10) (2001) 2633–2638. [5] U. Anselmi-Tamburini, S. Gennari, J.E. Garay, Z.A. Munir, Mater. Sci. Eng. A 394 (1–2) (2005) 139–148. [6] J.R. Groza, A. Zavaliangos, Mater. Sci. Eng. A 287 (2) (2000) 171–177. [7] B. McWilliams, A. Zavaliangos, K.C. Cho, R.J. Dowding, JOM 58 (4) (2006) 67–71. [8] J. Besson, M. Abouaf, J. Am. Ceram. Soc. 75 (8) (1992) 2165–2172. [9] E. Olevsky, Mater. Sci. Eng. R. 23 (1998) 41–100. [10] R.E. Hummel, Int. Mater. Rev. 39 (3) (1994) 97–111. [11] W.W. Mullins, Metall. Mater. Trans. 26A (1995) 1917– 1929. [12] E.A. Olevsky, B. Kushnarev, A. Maximenko, V. Tikare, M. Braginsky, Philos. Mag. 85 (2005) 2123–2146. [13] M. Scherge, C.L. Bauer, W.W. Mullins, Acta Metall. Mater. 43 (9) (1995) 3525–3538. [14] I.A. Blech, C. Herring, Appl. Phys. Lett. 29 (3) (1976) 131–133. [15] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps, Pergamon Press, NY, 1982. [16] F.B. Swinkels, M.F. Ashby, Acta Metall. 29 (2) (1981) 259–281. [17] R.M. German, Sintering Theory and Practice, John Wiley & Sons, NY, 1996. [18] G. Xie, O. Ohashi, T. Yoshioka, M. Song, K. Mitsuishi, H. Yasuda, K. Furuya, T. Noda, Mater. Trans. 42 (9) (2001) 1846–1849. [19] G. Xie, O. Ohashi, N. Yamaguchi, Trans. Mater. Res. Soc. Jpn 27 (4) (2002) 743–746.